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Godfrey Lowell CABOT SCIENCE LIBRARY
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FRAGILE
and circulates only with permission.
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SMITHSONIAN INSTITUTION.
Washington City, November^ 18Q4.
This work (No. 854), Smithsonian Geographical Ta-
bles, forms the second part of Volume XXXV, Smithsonian
Miscellaneous Collections. The "Smithsonian Meteoro-
logical Tables," issued in 1893, formed the first part of this
volume, and the third or concluding part, "Smithsonian
Physical Tables," is in preparation.
LIBRARY CATALOGUE SLIPS.
Smithsonian Institution.
Smithsonian Geographical tables. Prepared by
R. S. Woodward. City of Washington, published
by the Smithsonian Institution, 1894. 8*^. cv -j-
182 pp.
From Smithsonian Miscellaneous Collections, vol. 35.
(Number 854.)
Woodward, R. S.
Smithsonian Geographical tables. Prepared by
R. S. Woodward. City of Washington, published
by the Smithsonian Institution, 1894. 8°. cv -|-
182 pp.
From Smithsonian Miscellaneous Collections, vol. 35.
(Number S54.)
Smithsonian Geographical tables. Prepared by
R. S. Woodward. City of Washington, published
by the Smithsonian Institution, 1894. 8°. cv +
182 pp.
From Smithsonian Miscellaneous Collections, vol. 35.
(Number 854.)
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cmc-ift^
854
SMITHSONIAN
GEOGRAPHICAL TABLES
PREPARED BY
R. S. WOODWARD
CITY OF WASHINGTON
PUBLISHED BY THE SMITHSONIAN INSTITUTION
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n6 // cj^, ^^^S
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TJU Riversiie Prtss^ Cambridge ^ Mau.^ U.S.A.
Electrotyped and Printed by H. O. Houghton & Ca
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ADVERTISEMENT.
In connection with the system of meteorological observations established by
the Smithsonian Institution about 1850, a series of meteorological tables was
compiled by Dr. Arnold Guyot, at the request of Secretary Henry, and was pub-
lished in 1852 as a volume of the Miscellaneous Collections.
A second edition was published in 1857, and a third edition, with further
amendments, in 1859.
Though primarily designed for meteorological observers reporting to the
Smithsonian Institution, the tables were so widely used by meteorologists and
physicists that, after twenty-five years of valuable service, the work was again re-
vised, and a fourth edition was published in 1884.
In a few years the demand for the tables exhausted the edition, and it appeared
to me desirable to recast the work entirely, rather than to undertake its revision
again. After careful consideration I decided to publish the new work in three
parts : Meteorological Tables, Geographical Tables, and Physical Tables, each
representative of the latest knowledge in its field, and independent of the others ;
but the three forming a homogeneous series.
Although thus historically related to Doctor Guyot's Tables, the present work
is so entirely changed with respect to material, arrangement, and presentation,
that it is not a fifth edition of the older tables, but essentially a new publication.
The first volume of the new series of Smithsonian Tables (the Meteorological
Tables) appeared in 1893. The present volume, forming the second of the
series, the Geographical Tables, has been prepared by Professor R. S. Woodward,
formerly of the United States Coast and Geodetic Survey, but now of Columbia
College, New York, who has brought to the work a very wide experience both in
field work and in the reduction of extensive geodetic observations.
S. P. Langley, Secretary,
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PREFACE.
In the preparation of the following work two difHculties of quite different
kinds presented themselves. The first of these was to make a judicious selec-
tion of matter suited to the needs of the average geographer, and at the same
time to keep the volume within prescribed limits. Of the vast amount of
material available, much must be omitted from any work of limited dimen-
sions, and it was essential to adopt some rule of discrimination. The rule
adopted and adhered to, so far as practicable, was to incorporate little material
already accessible in good form elsewhere. Accordingly, while numerous ref-
erences are made in the volume to such accessible material, an attempt has
been made wherever feasible to introduce new matter, or matter not hitherto
generally available.
The second difficulty arose from the present uncertainty in the relation of the
British and metric units of length, or rather from the absence of any generally
adopted ratio of the British yard to the metre. The dimensions of the earth
adopted for the tables are those of General Clarke, published in 1866, and now
most commonly used in geodesy. These dimensions are expressed in English
feet, and in order to convert them into metres it is necessary to adopt a ratio of
the foot to the metre. The ratio used by (General Clarke, and hitherto gener-
ally used, is now known to be erroneous by about one one hundred thousandth
part. The ratio used in this volume is that adopted provisionally by the Office
of Standard Weights and Measures of the United States and legalized by Act
of Congress in 1866. But inasmuch as a precise determination of this ratio is
now in progress under the auspices of the International Bureau of Weights and
Measures, and inasmuch as the value for the ratio found by this Bureau will
doubtless be generally adopted, it has been thought best in the present edition
to restrict quantities expressed in metric measures to limits which will require
no change from the uncertainty in question. In conformity with this decision
the dimensions of the earth are given in feet only, and, with a few unimportant
exceptions, to which attention is called in the proper places, tables giving quan-
tities in metres are limited to such a number of figures as are definitely known.
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VI PREFACE.
It is a matter of regret that, owing to the cause just stated, less prominence
has been given in the tables to metric than to British units of length. On the
other hand, it seems probable that the more general use of British units will
meet the approval of the majority of those for whose use the volume is designed.
The introductory part of the volume is divided into seven sections under the
heads. Useful Formulas, Mensuration, Units, Geodesy, Astronomy, Theory of
Errors, and Explanation of Source and Use of Tables, respectively. In pre-
senting the subjects embraced under the first six of these headings an attempt
was made to give only those features leading directly to practical applications
of the principles involved. It is hoped, however, that enough has been given of
each subject to render the work of value in a broader sense to those who may
desire to go beyond mere applications.
The most of the calculations required in the preparation of the tables were
made by Mr. Charles H. Kummell and Mr. B. C. Washington, Jr. Their work
was done with skill and fidelity, and it is believed that the systematic checks
applied by them have rendered the tables they computed entirely trustworthy.
Mention of the particular tables computed by each of them is made in the
Explanation of Source and Use of Tables, where full credit is given also for
data not specially prepared for the volume.
The Appendix to the present volume is that prepared by Mr. George E. Cur-
tis for the Meteorological Tables. Its usefulness to the geographer is no less
obvious and general than to the meteorologist.
The proofs hav^ been read independently by Mr. Charles H. Kummell and
the editor. The plate proofs, also, have been read by the editor ; and while it
is difficult to avoid errors in a first edition of a work containing many formulas
and figures, it is believed that few, if any, important errata remain in this volume.
R. S. Woodward.
Columbia College, New York, N. Y., June 15, 1894
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CONTENTS.
USEFUL FORMULAS.
PAGB
1. Algebraic Formulas xiii
a. Arithmetic and geometric means xiii
b. Arithmetic progression xiii
c. Geometric progression xiii
d. Sums of special series xiii
e. The binomial series and applications xiv
f. Exponential and logarithmic series xiv
g. Relations of natural logarithms to other logarithms .... xv
2. Trigonometric Formulas xv
a. Signs of trigonometric functions xv
b. Values of functions for special angles xv
c. Fundamental formulas xv
d. Formulas involving two angles xvi
e. Formulas involving multiple angles xvi
f. Exponential values. Moivre's formula xvi
g. Values of functions in series xvii
h. Conversion of arcs into angles and angles into arcs .... xvii
3. Formulas for Solution of Plane Triangles xviii
4. Formulas for Solution of Spherical Triangles xx
a. Right angled spherical triangles xx
b. Oblique angled triangles xx
5. Elementary Differential Formulas xxi
a. Algebraic xxi
b. Trigonometric and inverse trigonometric xxi
6. Taylor's and Maclaurin's Series xxii
a. Taylor's series xxii
b. Maclaurin's series xxii
c. Example of Taylor's series xxii
d. Example of Maclaurin's series xxiii
7. Elementary Formulas for Integration xxiii
a. Indefinite integrals xxiii
b. Definite integration xxvi
MENSURATION.
I. Lines xxviii
a. In a circle xxviii
b. In regular polygon xxviii
c. In ellipse xxix
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Vlll CONTENTS.
2. Areas ••••• xxix
a. Area of plane triangle xxix
b. Area of trapezoid xxix
c. Area of regular polygon xxx
d. Area of circle, circular annul us, etc xxx
e. Area of ellipse xxx
f. Surface of sphere, etc xxxi
g. Surface of right cylinder xxxi
h. Surface of right cone xxxi
i. Surface of spheroid xxxi
3. Volumes xxxii
a. Volume of prism xxxii
b. Volume of pyramid xxxii
c. Volume of right circular cylinder xxxii
d. Volume of right cone with circular base xxxii
e. Volume of sphere and spherical segments xxxii
f. Volume of ellipsoid xxxiii
UNITS.
1. Standards of Length and Mass xxxiv
2. British Measures and Weights xxxvii
a. Linear measures xxxvii
b. Surface or square measures xxxviii
c. Measures of capacity xxxviii
d. Measures of weight xxxix
3. Metric Measures and Weights xl
4. The C. G. S. System of Units xlii
GEODESY.
1. Form of the Earth. The Earth's Spheroid. The Geoid . . xliii
2. Adopted Dimensions of Earth's Spheroid xliii
3. Auxiliary Quantities xliii
4. Equations to Generating Ellipse of Spheroid xliv
5. Latitudes used in Geodesy xliv
6. Radii of Curvature xlv
7. Lengths of Arcs of Meridians and Parallels of Latitude . xlvi
a. Arcs of meridian xlvi
b. Arcs of parallel xlix
8. Radius-Vector of Earth's Spheroid 1
9. Areas of Zones and Quadrilaterals of the Earth's Surface 1
10. Spheres of Equal Volume and Equal Surface with Earth's
Spheroid Hi
11. Co-ordinates for the Polyconic Projection of Maps . . . liii
12. Lines on a Spheroid Ivi
a. Characteristic property of curves of vertical section .... Ivi
b. Characteristic property of geodesic line Ivii
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CONTENTS. IX
13. Solution of Spheroidal Triangles Ivii
a. Spherical or spheroidal excess Iviii
14. Geodetic Differences of Latitude, Longitude, and Azimuth Iviii
a. Primary triangulation Iviii
b. Secondary triangulation Ix
15. Trigonometric Leveling Ixi
a. Computation of heights from observed zenith distances ... Ixi
b. Coefficients of refraction Ixiii
c. Dip and distance of sea horizon Ixiii
16. Miscellaneous Formulas Ixiii
a. Correction to observed angle for eccentric position of instrument Ixiii
b. Reduction of measured base to sea level Ixiv
c. The three-point problem Ixiv
17. Salient Facts of Physical Geodesy Ixv
a. Area of earth's surface, areas of continents, area of oceans . . Ixv
b. Average heights of continents and depths of oceans .... Ixv
c. Volume, surface density, mean density, and mass of earth . . Ixv
d. Principal moments of inertia and energy of rotation of earth . Ixvi
ASTRONOMY.
1. The Celestial Sphere. Planes and Circles of Reference . Ixvii
2. Spherical Co-ordinates Ixvii
a. Notation Ixvii
b. Altitude and azimuth in terms of declination and hour angle . Ixviii
c. Declination and hour angle in terms of altitude and azimuth . Ixix
d. Hour angle and azimuth in terms of zenith distance .... Ixix
e. Formulas for parallactic angle Ixix
f. Hour angle, azimuth, and zenith distance of a star at elongation Ixx
g. Hour angle, zenith distance, and parallactic angle for transit of
a star across prime vertical Ixx
h. Hour angle and azimuth of a star when in the horizon, or at the
time of rising or setting Ixxi
i. Differential formulas Ixxii
3. Relations of Different Kinds of Time used in Astronomy . Ixxii
a. The sidereal and solar days Ixxii
b. Relation of apparent and mean time Ixxiii
c. Relation of sidereal and mean solar intervals of time .... Ixxiii
d. Interconversion of sidereal and mean solar time Ixxiii
e. Relation of sidereal time to the right ascension and hour angle
of a star Ixxiv
4. Determination of Time Ixxiv
a. By meridian transits Ixxiv
b. By a single observed altitude of a star Ixxvi
c. By equal altitudes of a star Ixxvii
5. Determination of Latitude Ixxvii
a. By meridian altitudes Ixxvii
b. By the measured altitude of a star at a known time .... Ixxviii
c. By the zenith telescope by.izedbvG-Oogf^
X CONTENTS.
6. Determination of Azimuth Ixxix
a. By observation of a star at a known time bcxix
b. By an observed altitude of a star Izxad
c. By equal altitudes of a star lyxxi
THEORY OF ERRORS.
1. Laws of Error Ixzxiii
a. Probable, mean, and average errors Ixxxiv
b. Probable, mean, average, and maximum actual errors of inter-
polated logarithms, trigonometric functions, etc Ixxxv
2. The Method of Least Squares Ixxxvi
a. General statement of method Ixxxvi
b. Relation of probable, mean, and average errors , . , . . Ixxxviii
c. Case of a single unknown quantity Ixxxix
d. Case of observed function of several unknown quantities . . xc
e. Case of functions of several observed quantities xciii
f. Computation of mean and probable errors of functions of ob-
served quantities xcv
EXPLANATION OF SOURCE AND USE OF TABLES.
Tables i and 2 xcix
Table 3 . xcix
Table 4 xcix
Tables 5 and 6 xcix
Tables 7 and 8 c
Table 9 c
Tables 10 and II c
Table 12 c
Tables 13 and 14 c
Tables 15 and 16 ci
Table 17 ci
Table 18 cii
Tables 19-24 cii
Tables 25-29 ciii
Table 30 ciii
Table 31 civ
Tables 32 and 33 civ
Tables 34 and 35 civ
Tables 36 and 37 civ
Table 38 civ
Table 39 civ
Table 40 civ
Table 41 cv
Table 42 cv
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CONTENTS.
TABLES.
TABLB PACB
1. For converting U. S. Weights and Measures — Customary to
Metric 2
2. For converting U. S. Weights and Measures — Metric to Cus-
tomary 3
3. Values of reciprocals, squares, cubes, square roots, cube roots,
and common logarithms of natural numbers 4-22
4. Circumference and area of circle in terms of diameter d . , , 23
5. Logarithms of numbers, 4-place 24-25
6. Antilogarithms, 4-place 26-27
7. Natural sines and cosines 28-29
8. Natural tangents and cotangents 30-31
9. Traverse table (differences of latitude and departure) .... 32-47
10. Logarithms of meridian radius of curvature in English feet . . 48-56
1 1 . Logarithms of radius of curvature of normal section in English feet 5 7-65
12. Logarithms of radius of curvature (in metres) of sections oblique
to meridian 66-67
13. Logarithms of factors for computing spheroidal excess of triangles
(unit = English foot) 68
14. Logarithms of factors for computing spheroidal excess of triangles
(unit ^ the metre) 69
15. Logarithms of factors for computing differences of latitude, longi-
tude, and azimuth in secondary triangulation (unit = English
foot) 70-73
16. Logarithms of factors for computing differences of latitude, longi-
tude, and azimuth in secondary triangulation (unit ^ the metre) 74-77
17. Lengths of terrestrial arcs of meridian (in English feet) .... 78-80
18. Lengths of terrestrial arcs of parallel (in English feet) .... 81-83
19. Co-ordinates for projection of maps, scale = 1/250000 .... 84-91
20. Co-ordinates for projection of maps, scale = 1/125 000 .... 92-101
21. Co-ordinates for projection of maps, scale = 1/126 720 .... 102-109
22. Co-ordinates for projection of maps, scale = 1/63 360 .... 110-121
23. Co-ordinates for projection of maps, scale = 1/200000 .... 122-131
24. Co-ordinates for projection of maps, scale = 1/80 000 .... 132-141
25. Areas of quadrilaterals of the earth's surface of 10^ extent in lati-
tude and longitude 142
26. Areas of quadrilaterals of the earth's surface of 1° extent in lati-
tude and longitude 144-145
27. Areas of quadrilaterals of the earth's surface of 30' extent in lati-
tude and longitude 146-148
28. Areas of quadrilaterals of the earth's surface of 15' extent in lati-
tude and longitude 150-154
29. Areas of quadrilaterals of the earth's surface of 10' extent in lati-
tude and longitude 156-159
30. Determination of heights by the barometer (formula of Babinet) . 160
31. Mean astronomical refraction 161
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XU CONTENTS.
32. Conversion of arc into time 162
33. Conversion of time into arc 163
34. Conversion of mean time into sidereal time 164
35. Conversion of sidereal time into mean time 165
36. Length of 1° of the meridian at different latitudes (in metres,
statute miles, and geographic miles) 166
37. Length of i^ of the parallel at different latitudes (in metres, stat-
ute miles, and geographic miles) 167
38. Interconversion of nautical and statute miles 168
39. Continental measures of length, with their metric and English
equivalents 168
40. Acceleration (f) of gravity on surface of earth and derived func-
tions 169
41. Linear expansions of principal metals 170
42. Fractional change in a number corresponding to a change in its
logarithm 170
APPENDIX.
Numerical Constants 171
Goedetical Constants *. . . . 171
Astronomical Constants 172
Physical Constants 172
Synoptic conversion of English and Metric Units —
English to Metric 173
Metric to English 174
Dimensions of physical quantities 175
INDEX 177
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USEFUL FORMULAS.
I. Algebraic.
a. Arithmetic and geometric means. The arithmetic mean of n quanti-
ties ay d, Cy ,,. is
-^ (« + * + .+ ...);
their geometric mean is
(a d c. .)K
A case of special interest is
^ = H. + *){.-(2f|)'}'-
b. Arithmetic progression. If a is the first term, and a-^-dy a -{- 2 dy
a-^- $ dy . ,. 2ire the successive terms, the ;ith or last term z is
g=:a-^ (n — 1) d
The sum s of the n terms of this series is
= {z-i(n-i)d}n
c. Geometric progression. If a is the first term, and ary ar^y
successive terms, the ;ith or last term z is
The sum of the n terms is
a (r*— i ) r z — a __ z (r*— i)
If
r— I r— I
r < 1 and « ^ 00,
are the
I — r
d. Sums of special series.
1 + 2+3 + 4 + .. . + « =i«(«+i)
2+4+6+8+
X+3+S+7+
i«+2»+3»+4'+
^•+2«+3*+4'+
+ 2 « = « (» + i)
+ (2»— l) = ««
+ «« =J«(«+l)(2«+l)
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Xiv USEFUL FORMULAS.
e. The binomial series and applications.
For a> b,
' 1.2
«(«-l)(>,-2)
1.2.3 <^-|-....
For « < I,
(i ± «)• = I ±« « H — \-^ x' ± -^ J^^ '^ A^ + . . . .
(i -! xf = » + ** + 3«'+4«»+S**+...
(i + *)* = 1 + **- 4 ^ + tV** - tI» **+ . . .
(^^p^ = I - i*+ I **- w *•+ ^ ^- . . .
f. Exponential and logarithmic series.
For — 00 < jc < 00,
""" I "' 1.2"' 1.2.3"' I.2.3.4"''**"
The number e is the base of the natural or " Napierian " system of logarithms.
For :r = -|- I, the above series gives
e=. 2.718281828459 ....
In the natural system the following series hold with the limitations indicated :
' 1 ' 1.2 ^^ 1.2.3 •
— 00 < j: < op;
log (I 4-*) = *---!-^-^+^-...
o < :r < 00;
y<(2x+:yy.
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USEFUL FORMULAS.
XV
g. Relations of natural logarithms to other logarithms.
B = base of any system,
JV= any number,
L = log N\.o base B = log^-A^
/ = log i\r to base e = loge^
Then
Z = /l0g,K?=//l0ge^,
log^= i/logeB = fly say, which is called the modulus of the system whose base
is B. In the common, or Briggean system,
M = logio^ = 0.43429448
log /A = 9.6377843 — 10.
2. Trigonometric Formulas.
a. Signs of trigonometric functions.
Function.
1st Quadrant.
2d Quadrant.
3d Quadrant.
4th Quadrant.
sine
cosine ....
tangent . . .
cotangent . . .
I
+
+
+
+
+
+
1 1 + 1
b. Values of functions for special angles.
0°
90°
180^
270°
3^°
30°
45°
60°
sine ....
+ 1
— I
i
iV2
iVs
cosine . . .
+ 1
— I
+ t
iV3
\>fi
i
tangent . . .
00
00
JV3
I
V3
cotangent . .
00
00
00
V3
I
4V3
c. Fundamental formulas.
sin* a -|- cos* a = i,
cos a sec a = i,
sin a
tan a:
cos a
tan a cot a ^ I,
sin a cosec a = i,
cos a
cot a= -rr — >
sm a
1 + tan* a = T— 5- =sec*a,
* cos* a ^
I + C0t*a=: .0 = cosec* a,
' Sin* a ^
versed sin a = i — cos <
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XVI USEFUL FORMULAS.
d. Formulas involving two angles.
sin (a±i 0) = sin a cos fi ± cos a sin fi,
cos (a ± j8) = cos a COS ^ =f sin a sin /S.
tan (a ± j8) = (tan a ± tan /3)/(i :f tan a tan /?),
cot (a ± J8) = (cot a cot iS T l)/(cot a ± COt /3).
sin a -|- sin /? = 2 sin ^(a -|- /5) cos J(a — ^),
sin a — sin /5 = 2 cos J(a -|- i^) sin J(a — )3).
cos a -|- cos P=2 COS J(a -|- fi) COS ^(a — fi),
cos a — COS /8 = — 2 sin ^(a + j8) sin J(a — /?).
^ sin (a ± B)
tan a ± tan /? = ^^^^ ;r, A>
'^ cos a COS p
sin 09 ± a)
cot a ± cot j8 = ,:^^,- p -
'^ sin a sm p
2 sin a sin ^ = cos (a — /5) — cos (a -|- p\
2 cos a COS /? = COS (a — j8) + COS (a + jS),
2 sin a COS j8 = sin (a — /5) + sin (a -|- /8).
Slf±|il| = ,..K. + «co.K.-«,
e. Formulas involving multiple angles.
sin 2 a = 2 sin a cos a,
sin 3 a = 3 sin a cos* a — sin* a,
cos 2 a = cos' a — sin* a = i — 2 sin* a = 2 cos* a— i,
cos 3 a = cos* a — 3 sin* a cos cu
sin a I — cos a / l — COS a \*
tan J a = J _|.cosa~ sin a ~ \i + cos a) '
2 tan a ^ cot* a — I
tan 2 a = zzzrz* cot 2 a =
I — tan* a 2 cot a
2 tan i a I — tan* j a
^^^^ = r+-SEn^ ^^'"=i+tan*ia
2 sin* a = I — cos 2 a, 2 COS* a = I + COS 2 a,
4 sin* a = 3 sin a — sin 3 a, 4 cos* a = 3 cos a + cos 3 eu
f. Exponential values. Moivre's formula.
e = base of natural logarithms,
/= V^ ^'^= - i» ^'*= - '» ^■*= ^ etc.
cos ;c = i (^ + ^-*'), sin j. = iU^ - ^"**),
cos m; = i (<f-* + <?*), sin m: = ^V (^"* — <?*)•
(cos X ±i sin ^)'" = cos mx ±$ sin »w.
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USEFUL FORMULAS. XVll
g. Values of functions in series.
For X in arc the following series hold within the limits indicated.
^ JC* x^
sin* = *-g- + — - — + ...,
cos*=i-- + ---^+...,
— 00 < a: < + 00.
tanjr = * + J^ + A^ + 3¥y*^ + ....
sec Jc=i-fi:jf*-|-Aa:* + VW *•+•••»
— Jir<jc<+iir.
cot ^ = - (i - J :!:* - A •** - viir -^^ - • • • )i
COSec :«: = ^ (l + J :J^ + 3 Jiy X* + y/iViy ^* + - . . ).
— IT < a: < + IT.
arc sin ;r = ;c + i j^ + A -^ + T^y -^^^ + • • • »
arctan^ = ^--+--y+--...,
- I <:r < + i.
or =: sin ^ + J sin* * + tfe sin* x + xf y sin^ or + • • • i
--i«-<^<4-iir.
ic = tan X '~'\ tan* jp + J tan* j: — | tan' :i: + • • • >
-Jir<Jt:< + iir.
log sin a: = log or - fi (J :c* + y^^ a:* + 7^7 :«•+...),
X positive and < ir,
fi = modulus of common logarithms. See p. xv.
log tan :«: = log ^ + ^ (i x« + /^ ^* + ^f f 5 ^* + . . . X
:c positive and < ^ ir.
h. Conversion of arcs into angles and angles into arcs.
Denote by x^^ a:', and x" respectively the angle (in degrees, minutes, or sec-
onds) corresponding to the arc x. Then by equality of ratios
360° _ 360 X 60^ _ 360 X 60 X 60'' _ 2jr
x^
whence
^
— «"
a:° =
180°
9! =
180 X 60'
«" =
180 X 60 X 60"
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Then
XVlll USEFUL FORMULAS.
Put l8o^ ^ t. rj . t. ^.
= p = number of degrees m the radius,
=zp' = number of mmutes in the radius,
= p = number of seconds in the radius.
*° = a: p^ x'zzzx p\ x" = X p".
p" = 57-°2957795» log p"" = 1.75812263,
p' = 3437-'74677, log p' = 353627388,
p" = 2o6264."8o6, log p" = 5.3 14425 13-
3. Formulas for Solution of Plane Triangles.
a, dyC = sides of triangle,
a, /8, y = angles opposite to <?, 3, c, respectively,
A = area of triangle,
r = radius of inscribed circle,
J? = radius of circumscribed circle,
a b^ c_^
sin a sin /8 sin y * *
a:=^b cos y + ^ cos /8, ^ = ^ cos a -|- <* cos y, ^ = « cos /5 + ^ cos o.
r = 4 ^ sm ^ a sm ^ /3 sin ^ y = ^ » •»
(« + ^) cos i (a + /8) = r cos i (a - )S),
(a — ^) sin i (a -j- /?) = ^- sin J (a — /?).
g + ^ _ tan j^ (fl +i^) ^ tan ^y
a — ^ ~ tan i (a—fi) ~ tan i (a — j8)'
tf* = ^ -|- ^ — 2 ^ ^ cos o = (^ + f)* — 4 ^ r cos* ^ a.
V s (s — a) s — a
, , . df* sin )8 sin y _, . . /> .
^ = ^ a ^ sin Y = 7- '- -=12 Ic sin a sin a sm y
■ ' 2 sm a '^ '
= r» cot i a cot i /8 cot J y = ^ j (j — ^z) (j — ^) (j — ^)
=.rs:=\ahc IR.
Digitized by VjOOQIC
sm
In right angled triangles let
Then
USEFUL FORMULAS.
a = altitude,
b = base,
c = hypothenuse,
a = ^ sin a = ^ cos p=z b tan a = ^ cot /5,
b =zcsm p-=c cos a=z a tan p = acotcu
^ = Jtf^ = Jfl«cota = J^tana = J^sin2a.
Table for solution of oblique triangles.
zix
Given.
Sought.
Formula.
a, ^, ^
a
A
sini«-v/('-'^j^'-'\ .-i(a + 3 + .),
tf, ^, a
/8
y
sin )8 = ^ sin a/a.
When tf > ^, /5 < 90** and but one value results. When b> a,
P has two values.
y=i8o'>-(a + /S).
r = tr sin y/sin a,
^= J tr ^ sin y.
a.^^
b
y
c
A
3 =1 a sin )3/sin a.
y=i8o»-(« + ^).
r = tf sin y/sin a = tf sin (a + /3)/sin a.
-^ = i df ^ sin y = J a* sin /5 sin y/sin a.
a,b,y
a
a sin y
i(« + i8) = 9o°-iy,
<• = (a* + ^* — 2 « * cos 7)',
= {(aJfSf-Aabcos*\yY,
= (a — *)/cos ^ where tan ^ = 2 V<' ^ sin ^ y/(a — b),
= a sin y/sin a.
/I = ^ a ^ sin y.
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ZX USEFUL FORMULAS.
4. Formulas for Solution of Spherical Triangles.
a. Right angled spherical triangles.
aybyC'=z sides of triangle, c being the hypotenuse,
a, /5, y = angles opposite to tf, b^ c, respectively,
7 = 90^
sin a=zsin c sin o^ sin ^ = sin c sin )3,
tan a = tan c cos /3, tan b = tan c cos a,
= sin b tan a, = sin « tan /3 ;
cos a = cos a sin ft, cos /3 = cos 3 sin a ;
cos c = cos a cos ^ = cot a cot /9.
b. Oblique angled triangles.
a, b, c = sides of triangle,
a, ft 7 = angles opposite to a, by r, respectively,
€ = a + /5-|-7— 180° = spherical excess,
S = surface of triangle on sphere of radius r.
sin a sin b sin c
sin a sin )3 sin y
cos a = cos ^ cos c-^ sin b sin ^ cos a,
..,^_ZZ.COS_a-C0S^^j-^ . COS(<T-P)cos(<r-^ y)
sin * a — : a — = > cos * d — : a — : f
' * sin /3 Sin y ^ ^ ^ ^^^^ ^ ^^^ ^
J — cos o- cos (o- — g)
tan *^ — cos(cr-i8)cos(cr-yy
. « sin (j — b) sin (^ — ^) « , sin s sin (j — a)
^^'^'^^= sin ^ sin. ^' ^^^'*"= sin ^ sin. '
fo«« r « — s^n {s - ^) sin (j - c)
tan J a— gin j sin (j -«) '
cot 1 tf cot i^ ^ 4- cos y
cot i € = -r-^ ' -^
^ sm y
tan* J €== tan i J tan J (j — a) tan i ( J — ^) tan i (^ — .).
Napier's analogies.
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USEFUL FORMULAS. XXi
Gausses formulas,
cos i (a -[- fl) COS J r = COS \{a-\' S) sin \ y,
sin J (o 4- i^) cos i ^ = cos \{a ^V) cos ^ y,
cos J (o — )8) sin J ^ = sin J (dP + ^) sin i y,
sin J (a — ^) sin i r = sin J (« — ^) cos \ y.
5. Elementary Differential Formulas.
a. Algebraic.
u^ VyW,»,.=> variables subject to diiferentiation,
a, 3, r, . . . = constants.
d(a + «) = du^ d(a u)-=,a dUj
d(u v)z=LU dv-^-v du,
J/ \ ( du , dv y dwx_ \
<«z^a/...)=^— + — + — + .•• j»i^w...,
' du - -udv du udv
(a '\-bu \ _ bh - ag
da'' = a" log a dv, d^:=^ dv
(e = base of natural logarithms),
dlogv=z dv/v.
b. Trigonometric and inverse trigonometric.
//sin ;r = cos * dx, dcos jc = — sin ^ /£r,
^an X = sec* x dx, dcot x = — cosec* x dx,
dsec X = sec" x sin x dx, //cosec :t = — cosec* :tr cos x dx.
dlog sin j; = cot x dx, dlog cos ^ = — tan :c dx.
dx ^ dx
, « » //arc cos :tr = ^ / »
. . . ^^ . dx
//arc sin a* = ± . . //arc cos :tr = =F
//arc tan :« = ^ . ^ daxc cot :r = —
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XXll USEFUL FORMULAS.
6. Taylor's and Maclaurin's Series.
a. Taylor's series.
If «=y(j;-|-^), any finite and continuous function oi x -{^ A, A being an
arbitrary increment to x; and if du/dxy if^u/dx^^ ... are finite and deter-
minate,
u =/(x + A) =/(x) +/' (x) h +/" ix) f +/'" {X) ^ + . . . .
where f(x\/' (x\ /" (jr), ... are the values of /(x + ^), du/dx, dhi/dx^^ . . .
when ^ = o. This is Taylor's series or theorem. The remainder after the first
n terms in ^ is expressed by the definite integral
//
o
b. Maclaurin's series.
If in Taylor's series we make jc = o, and ^ = ^, the result is
«=/(*)=/(o)+/'(o)*+y"(o)^^+/'"(o)j^+...,
where/(o),/'(o), /"(o), ... are the values oi/(x), du/dx^ dhijdx^ . . . when
a: = o. This is Maclaurin's series or theorem. The remainder after the first n
terms in x is expressed by the definite integral
o
c. Example of Taylor's series.
u=f{x + h) = \og(x-\-h).
du I
1% ~x-\-h'
f{x) = log X,
Hence for common logarithms, ^ being the modulus,
log (jc + >i) = log Ji: + ft (jiri^-i:r-«>i« + i;^«/5« -...),
and the sum of the remaining terms is
h
o
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USEFUL FORMULAS. Xziii
Since jp is the least value of(x'\'A^z) within the limits of this integral, the
sum of the remaining terms is negative, and numerically
^4)'-
If, for example, (^/x) = i/ioo, the remainder in question is less than
i X 0.434 X io~^, or about one unit in the ninth place of decimals.
d. Example of Maclaurin's series.
u =z/(x) = sin X,
/(o)=o,
^ =cosx, /(o) = + i,
^ = - sm X, /'(o) = o,
^-j- = - cos X, f'ip) = - I,
Hence
f(x) = sin :!C = :!C — 1
-^ ^ ^ 1.2.3 ' 1.2.3.4.5
and the sum of the remaining terms is
X
r i sin (x -^ z) sfi dz,
o
If g is the greatest value of sin (x — z) within the limits of this integral the
remainder in question is negative and numerically
^ 6 '^ 5 ! *^-
If, for example, x = ir/S (the arc of 30°), g=i, and the remainder is numeri-
cally less than 0.0000143.
7. Elementary Formulas for Integration.
a. Indefinite integrals.
I adx =:aidx = aX'^C.
f/(x) dx+f<l> (x)dx = f{/(x) + ^ (x)} dx.
If ar = ^ (y), and dx=zil/ (ji) dy^
jfix) dx = j>{* {y)} *' O) dy.
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XXiv USEFUL FORMULAS.
Since d{uv) = udv + vdu^
j udv z=,uv —x vdu ; and
if i^ = J{x) and z/ = <^ {x)^
fdxf/(x, y) dy =py^/{x, y) dx.
^^p{x) dx = *p(«) </* -Cxf(x) dx.
/(^ + ^-)"^-^^^'+^-
/<^ . ^ r — d'* , ^
Y37P = arc tan x-\- C, I ^\^ = arc cot ;c + C
— , , A = (ab)"^ arc tan (3/tf)* jc + ^» ^or a and 3 both positive,
= (cUf)"^ arc cot (^/^)* jc + C, for j; and ^ both negative,
= i (- «^)""* log (Z^^)t^^^ + ^, for ^^^ negative.
=H^'-^r*iog gig:^:^:j:g +c,for^'~^.>o.
r(a + :r*)* /& = i ;r (a + ^* + i df log {:c + (^ + ^»} + C
r(aa - ;c«)» ,/^ = i ^ («« -. ^* + i tf a arc sin ^ + C
r(tf + bxy dx = l(a^ bx)\lb + C
♦ This is the fonnula for integration by parts.
t Natural logarithms are used in this and the following integrals. For relation of natural to
common logarithms see section i, g. i ^ ^^ rri ^
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USEFUL FORMULAS. XXV
+ J (fl^r - ^lc^{a + 2 ^o: + a^" » /£r + C.
C{a + ^^)-» i& = 2 (tf + bx)^lb + C.
fCa + ^ar) (tf + ^jr)- * //:c = S (3 a^ - 2 a/5 + j8 bx) (a + ^o:) V^ + (7.
r(d!» — jc*)-» /i^ = ± arc sin ^ + C,
= T arc cos ~ + Q
= aarctan(^)' + C.
J(« + **)-» ^* = log {* + (« + ;^'} + C,
r(a + 2 *;t + c«^-» <& = T== log {3 + «;+(«:+&*+<»**)»}+C;for<:>o,
= - 7^ "c sin (^lS)t + ^' ^°^ '<°-
C<edx = a' /log a-\-C, Ci'dx =ze*-\-C.
1 1<^ X dx = x log X — x-\- C.
JOog xy :r'dx = ^^ (log *)•+» + C.
Isxax dx^=z ^ cos ^ + C, I cos ^ //a: = sin :c + ^•
1 sin" Jf //a:=: Jx — J sin 2 ;r+ (7, j cos*a:^ = 4^ + i sin 2;r+C
I tan X dx=> — log cos ^ + C, I cot jc ^ = log sin X'\'C.
/'^ • r ^ tf sin ^x — ^ COS ^jc ^ , _
^s\xibxdx= ^2 ^^2 ^+(7.
/'^, . , ^ COS bx-^b sin ^:tr ^, , ^
^ cos bx dx = ^a ! ^a ^ + (7.
I arc sin ;r ^:c = ^ arc sin a; ± (i — c^^ + C.
I arc cos :« /£r = ;ic arc cos ^ T (i — ^* + C
I arc tan xdx'=.x arc tan x —\ log (i + ^ + ^•
I arc cot X dx-=LX arc cot :t + J log (i + ^c*) + ^«
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XXVI USEFUL FORMULAS.
b. Definite Integration.
n b c n
J4, (x) dx —J4, (x) dx + J"^ {x)dx'\'.. . r^ ix) dx.
a a b m
b a
r^ (^) dx = -Cfl> (x) dx.
a b
a a
r<^ ix) dx = f<l> (a — x) dx.
o o
If ^ (;r) = ^ (— x), an " even function " of x,
a o a
JV (*) dx—^^ (x) dx = \C4>(x) dx.
o — tf — fl
If ^ (jc) = — ^ (— x\ an " odd function " of x,
o a -{-a
I ^ (a:) ^ = I ^ (:c) dx, and 1 4^ (x) dx ^=1 o.
—a o — tf
If A be the greatest and B the least value of ^ (x) within the limits a and ^^
b
Aib-'d) > JV {x)dx> £{b — a\
a
Si formula useful in determining approximate values of integrals. See, e. g.,
section 6, d.
b
If u = Cff, (x) dx,
du . du
00
o
I 00
C dx _ C__dx__.
o
00
I
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USEFUL FORMULAS. ZXVU
00
o
00
o
jP?-—'*x »•/!:«:= I. 3. 5... (2 «— l)tf— (2 a) -(" + *> Vir.
o
00
JV— * ^-1 ^jp _ V(ir/a).
o
«• IT
I sin m:c sin /UP ii[r = J cos fnx cos at ^ = o,
o o
when m and » are unequal integers.
w
I sin mx cos nx dx =, —^——^ for m and « integers and m^n odd,
o
= o, for m and « integers and m— n even.
W IT
I sin' mx dx= i cos* x^ur ^ = ^ ir, for m an integer,
o o
^ir ^TT I
fsin- X dx = fcos" xdx— C(i — ar^ »<— « dx.
000
00 00
/sin X J /*cos jf . /^ , V
o o
00 00
j sin :^ dx '=.\co^s? dx'=.\ V(7r/2).
o o
00
r?— *• cos 23* i& = J ^-(&/«)« V(ir/a).
o
00
j ^" «• *• sin 23* i£i;= o.
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MENSURATION.
I. Lines.
a. In a circle.
r = radius of circle,
€ = length of any chord,
s = arc subtended by r,
a = angle corresponding to s,
h = height of arc s above r, or perpendicular distance from middle point of
arc to chord.
Circumference = 2 «• r,
IT = 3.14 159 265, log IT = 0.49 7149^7*
2 IT = 6.28318 531, log 2 IT =0.79 817 987.
r=2rsin^a, jz=ra.
Length of perpendicular from center on chord
= r cos \ a
Hr-r(^)--»(^)'-*(^)--...}.
A = r (l — COS i a)
= 2 r sin* J a
=*.j(9+A0)'+*(^)+...}.
b. In regular polygon.
r = radius of inscribed circle,
• i?= radius of circumscribed circle,
« = number of sides,
s = length of any side,
fi = angle subtended by s,
p = perimeter of polygon.
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MENSURATION.
^ = 36o7«,
s=:2 rtan\ fiz=z2 J^ sin ^ Pf
p = ns = 2 nr ta,n i P = 2 nl^ sin \ p.
See table under c, below.
c. In ellipse.
a = semi-axis major,
d = semi-axis minor,
^ = eccentricity = (i — ^/a*)*,
jPz=: perimeter of ellipse,
■1+8 ^^'i- +■
i + ^i-^"
64
Distance from centre to focus =.ae.
Distance from focus to extremity of major axis = (i — i).
Distance from focus to extremity of minor axis = a.
= IT (<z -|- ^) ^, say, where q stands for the series in n. The values of q cor-
responding to a few values of n are : —
n
^
n
9
1. 0000
0-5
1.063s
0.1
1.0025
0.6
1.0922
0.2
I.OIOO
: 0-7
1. 1267
0.3
1.0226
0.8
1.1677
0.4
1.0404
1 0.9
I-2ISS
I.O '
1.2732
2. Areas.
a. Area of plane triangle.
(See table on p. xix.)
b. Area of Trapezoid.
bi = upper base of trapezoid,
b% = lower base of trapezoid,
a = altitude of trapezoid, or perpendicular distance between bases.
Area = J (^^ + b^ a.
Digitized by VjOOQIC
XXX
MENSURATION.
c. Area of regular polygon.
A = area,
ryE=. radii of inscribed and circumscribed circles,
s = length of any side,
n = number of sides,
P = angle subtended by x = $60° /n,
A = nr^ tsia i p = in JP sin p=zins* cot \ p.
Table of Values.
9t
^
A
jP
s
3
120°
0.4330 J*
1.2990 it'
0.5774 s
1.7321^
4
90
1. 0000
2.0000
0.7071
I.4I42
S
72
1-7205
2.3776
0.8507
1.1756
6
60
2.5981
2.5981
I.OOOO
I.OOOO
7
Sif
36339
2-7364
1.1524
0.8678
8
4S
5-8284
2.8284
1.3066
0.7654
9
40
6.i8i8
2.8925
I.46I9
0.6840
10
36
7.6942
2.9389
I.6I80
0.6180
II
32A
9-3656
2-9735
»-7747
0.5635
12
30
1 1. 1962
3.0000
1-9319
0.5176
13
28ft
13-1858
3.0207
2.0893
0.4786
14
25*
IS-334S
3-0372
2.2470
0.4450
IS
24
17.6424
3-0505
2.4049
0.4158
16
22^
20.1094
30615
2.5629
1
0.3902
d. Area of circle, circular annulus, etc.
r = radius of circle,
d = diameter,
a = angle of any sector,
ri, r, = smaller and greater radii of an annulus.
Area of circle = «• r^ = J ir ^,
W = 3.14 159 265, log XT = 0.49 714 987-
Area of sector =zar^f for a in arc,
= IT r^ W360), for a in degrees.
Area of annulus = ir (rj* — rj^.
e. Area of ellipse.
ay d = semi axes respectively
e = eccentricity = (a^ — d^^a
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MENSURATION. XXXi
Area of ellipse = w a d,
= IT a* COS ^, if ^ = sin ^
f. Surface of sphere, etc.
r = radius of sphere,
^, ^ = latitudes of parallels bounding a zone,
c = spherical excess of a spherical triangle
= sum of spherical angles less i8o°,
Total surface = 4 ir r*.
Surface of zone = 2 ir r* (sin <^ — sin <^),
= 4 IT f^ cos J (^ + <^ sin i (<^ — 4^).
Surface of spherical triangle = r* c, for c in arc,
= r* c/p", for c in seconds,
p" = 206 264.8", log p" = 5.3 1 442 s 13.
g. Surface of right cylinder.
r = radius of bases of cylinder,
h = altitude of cylinder.
Area cylindrical surface = 2 ir r ^.
Total surface = 2 ir r (r -|- ^).
h. Surface of right cone.
r =: radius of base,
h = altitude,
s = slant height.
Conical surface •=.Trrsz=nrr{h^-\' r*)\
Total surface = ir r (j + r).
i. Surface of spheroid.
a^ d=z semi axes,
e = eccentricity = {(a + ^) (« — ^)}V^*
Surface of oblate spheroid = 2 ir «• -j i -| log (--3-) \
= 4ira«(i~i^-TV^-A^'-...).
Surface of prolate spheroid = 2 vad •] (i — ^*-| >•
= 4^« ^ (i - i^ - A ^-Ti3f^* --..).
* The logarithm in this fonnula refers to the natural or *' Napierian " system. For areas of
zones and qnadrilaterals of an oblate spheroid, see pp. 1-liL
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ZXXll MENSURATION.
3. Volumes.
a. Volume of prism.
A = area of base, h = altitude, F= volume.
• V= A A.
For an oblique triangular prism whose edges a, b^ c are inclined at an angle a
to the base,
r=i (a + ^ + ^)^sino.
b. Volume of pyramid.
A = area of base, h = altitude, F= volume.
V= \Ah.
For a truncated pyramid whose parallel upper and lower bases have areas Ax
and A^ respectively and whose distance apart is ^,
The volume of a wedge and obelisk may be expressed by means of the volumes
of pyramids and prisms.
c. Volume of right circular cylinder.
r = radius of base, h = altitude, r= volume.
IT = 3.X4I59 265, log IT = 0.49 714987'
For an obliquely truncated cylinder (having a circular base) whose shortest and
longest elements are h^ and h^ respectively,
For a hollow cylinder the radii of whose inner and outer surfaces are rx and r,
respectively, and whose altitude is hy
V—irh{r\ — r\)
d. Volume of right cone with circular base,
r = radius of base, h = altitude, F= volume.
For a right truncated cone the radii of whose upper and lower parallel bases
are r^ and r, respectively, and whose altitude is h,
e. Volume of sphere and spherical segments.
r = radius of sphere, h = altitude of segment, V-=. volume.
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MENSURATION. XXXlll
For the entire sphere
y=i ^ IT r*z=: 4.1888 r* approximately.
(For V and log v see c above.)
For a spherical segment of height A
For a zone, or difference in volume of two segments whose altitudes are Ai and
Ai respectively
y=irr(Al — AD — i^(Ai — /i^
where ri and r, are the radii of the bases of the zone and ^ A = Af^ Ai.
f. Volume of ellipsoid.
a, df c = semi axes, ^= volume.
For an ellipsoid of revolution about
the a-axis, y= ^ fr a i^^
the ^axis, ^= ^v a* d.
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UNITS.
I. Standards of Length and Mass.
The only systems of units used extensively at the present day are the British
and metric. The fundamental units in these systems are those of time, length,
and mass. From these all other units are derived. The unit of time, the mean
solar second, is common to both systems.
The standard unit of length in the British system is the Imperial Yard, which
is defined to be the distance between two marks on a metallic bar, kept in the
Tower of London, when the temperature of the bar is 60° F.
The standard unit of mass in the British system is the Imperial Pound Avoirdu-
pois. It is a cylindrical mass of platinum marked " P. S. 1844, i lb.," preserved
in the office of the Exchequer at Westminster.
In the metric system the standard unit of length is the Metre, now represented
by numerous platinum iridium Prototypes prepared by the International Bureau
of Weights and Measures.
The standard of mass in the metric system is the Kilogramme, now represented
by numerous platinum iridium Prototypes prepared by the International Bureau
of Weights and Measures.
Both systems of units have been legalized by the United States. Virtually, how-
ever, the material standards of length and mass of the United States are cer-
tain Prototype Metres and certain Prototype Kilogrammes. The present status
of the two systems of units so far as it relates to the United States is set forth
in the following statement from the Superintendent of Standard Weights and
Measures, bearing the date April 5, 1893.
Fundamental Standards of Length and Mass.*
" While the Constitution of the United States authorizes Congress to * fix the
standard of weights and measures,' this power has never been definitely exer-
cised, and but little legislation has been enacted upon the subject. Washington
regarded the matter of sufficient importance to justify a special reference to it in
his first annual message to Congress (January, 1790), and Jefferson, while Secre-
tary of State, prepared a report at the request of the House of Representatives, in
which he proposed Quly, 1790) *to reduce every branch to the decimal ratio
already established for coins, and thus bring the calculation of the principal
affairs of life within the arithmetic of every man who can multiply and divide.'
The consideration of the subject being again urged by Washington, a committee
• Bulletin 26, U. S. Coast and Geodetic Survey. Washington : Government Printing Office,
1893. Published here by permission of Dr. T. C. Mendenhall, Superintendent Coast and Geo-
detic Survey.
Digitized by V^OOQ IC
UNITS. XXXV
of Congress reported in favor of Jefferson's plan, but no legislation followed.
In the mean time the executive branch of the Government found it necessary to
procure standards for use in the collection of revenue and other operations in
which weights and measures were required, and the Troughton 82-inch brass
scale was obtained for the Coast and Geodetic Survey in 18 14, a platinum kilo-
gramme and metre, by Gallatin, in 182 1, and a Troy pound from London in 1827,
also by Gallatin. In 1828 the latter was, by act of Congress, made the standard
of mass for the Mint of the United States, and although totally unfit for such pur-
pose it has since remained the standard for coinage purposes.
" In 1830 the Secretary of the Treasury was directed to cause a comparison to
be made of the standards of weight and measure used at the principal custom-
houses, as a result of which large discrepancies were disclosed in the weights and
measures in use. The Treasury Department, being obliged to execute the consti-
tutional provision that all duties, imposts, and excises shall be uniform throughout
the United States, adopted the Troughton scale as the standard of length ; the
avoirdupois pound to be derived from the Troy pound of the Mint as the unit of
mass. At the same time the Department adopted the wine gallon of 231 cubic
inches for liquid measure and the Winchester bushel of 2 150*42 cubic inches for
dry measure. In 1836 the Secretary of the Treasury was authorized to cause a
complete set of all weights and measures, adopted as standards by the Depart-
ment for the use of custom-houses and for other purposes, to be delivered to the
Governor of each State in the Union for the use of the States respectively, the
object being to encourage uniformity of weights and measures throughout the
Union. At this time several States had adopted standards differing from those
used in the Treasury Department, but after a time these were rejected, and finally
nearly all the States formally adopted by act of legislature the standards which
had been put in their hands by the National Government. Thus a good degree
of uniformity was secured, although Congress had not adopted a standard of
mass or of length other than for coinage purposes as already described.
" The next and in many respects the most important legislation upon the subject
was the Act of July 28, 1866, making the use of the metric system lawful through-
out the United States, and defining the weights and measures in common use in
terms of the units of this system. This was the first general legislation upon the
subject, and the metric system was thus the first, and thus far the only system
made generally legal throughout the country.
" In 187 s an International Metric Convention was agreed upon by seventeen
governments, including the United States, at which it was undertaken to establish
and maintain at common expense a permanent International Bureau of Weights
and Measures, the first object of which should be the preparation of a new inter-
national standard metre and a new international standard kilogramme, copies of
which should be made for distribution among the contributing governments.
Since the organization of the Bureau, the United States has regularly contributed
to its support, and in 1889 the copies of the new international prototypes were
ready for distribution. This was effected by lot, and the United States received
metres Nos. 21 and 27, and kilogrammes Nos. 4 and 20. The metres and kilo-
grammes are made from the same material, which is an alloy of platinum with ten
per cent of iridium.
". -^ . Digitized by Google
XXXVl UNITS.
*'0n January 2, 1890, the seals which had been placed on metre No. 27 and
kilogramme No. 20, at the International Bureau of Weights and Measures near
Paris, were broken in the Cabinet room of the Executive Mansion by the Presi-
dent of the United States, in the presence of the Secretary of State and the
Secretary of the Treasury, together with a number of invited guests. They were
thus adopted as the National Protot3rpe Metre and Kilogramme.
" The Troughton scale, which in the early part of the century had been tenta-
tively adopted as a standard of length, has long been recognized as quite un-
suitable for such use, owing to its faulty construction and the inferiority of its
graduation. For many years, in standardizing length measures, recourse to copies
of the imperial yard of Great Britain had been necessary, and to the copies of
the metre of the archives in the Office of Weights and Measures. The standard
of mass originally selected was likewise unfit for use for similar reasons, and
had been practically ignored.
"The recent receipt of the very accurate copies of the International Metric
Standards, which are constructed in accord with the most advanced conceptions
of modern metrology, enables comparisons to be made directly with those stand-
ards, as the equations of the National Prototypes are accurately known. It has
seemed, therefore, that greater stability in weights and measures, as well as much
higher accuracy in their comparison, can be secured by accepting the international
prototypes as the fundamental standards of length and mass. It was doubtless
the intention of Congress that this should be done when the International Metric
Convention was entered into in 1875 ; otherwise there would be nothing gained
from the annual contributions to its support which the Government has con-
stantly made. Such action will also have the great advantage of putting us in
direct relation in our weights and measures with all civilized nations, most of
which have adopted the metric system for exclusive use. The practical effect
upon our customary weights and measures is, of course, nothing. The most care-
ful study of the relation of the yard and the metre has failed thus far to show
that the relation as defined by Congress in the Act of 1866 is in error. The
pound as there defined, in its relation to the kilogramme, differs from the impe-
rial pound of Great Britain by not more than one part in one hundred thousand,
an error, if it be so called, which utterly vanishes in comparison with the allow-
ances in all ordinary transactions. Only the most refined scientific research will
demand a closer approximation, and in scientific work the kilogramme itself is
now universally used, both in this country and in England.*
* Note. — Reference to the Act of 1866 results in the establishment of the following : —
Equations,
I yard = ^— - metre.
' 3937
I pound avoirdupois = > kilo.
A more precise value of the English pound avoirdupois is ^ ^ kilo., differing from the above
by about one part in one hundred thousand, but the equation established by law is sufficiently
accurate for aU ordinary conversions.
As already stated, in work of high precision the kilogramme is now all but universally used,
and no conversion is required.
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UNITS. XXXVU
•* In view of these facts, and the absence of any material normal standards of
customary weights and measures, the Office of Weights and Measures, with the
approval of the Secretary of the Treasury, will in the future regard the Interna-
tionai Prototype Metre and Kilogramme as fundamental standards, and the cus-
tomary units, the yard and the pound, will be derived therefrom in accordance
with the Act of July 28, 1866. Indeed, this course has been practically forced
upon this office for several years, but it is considered desirable to make this for-
mal announcement for the information of all interested in the science of metrology
or in measurements of precision.
T. C. Mendenhall,
Superintendent of Standard Weights and Measures.
" Approved :
J. G. Carlisle,
Secretary of the Treasury.
April 5, 1893."
No ratios of the yard to the metre and of the pound to the kilogramme have as
yet been adopted by international agreement ; but precise values of these ratios
vrill doubtless be determined and adopted within a few years by the International
Bureau of Weights and Measures. In the mean time, it will suffice for most pur-
poses to use the values of the ratios adopted provisionally by the Office of Stand-
ard Weights and Measures of the United States. These values are —
I yard ^ f |g^ metres, or i metre = f |g J yards,
I pound ^ \%%%% kilogrammes, or i kilogramme = \%%^i pounds.
These ratios were legalized by Act of Congress in i866. Expressed decimally
these values are * —
I yard ^ 0.914402 metres, i metre = 1.093 611 yards,
I pound = 0.45 359 kilogrammes, i kilogramme = 2.20462 pounds.
The above values of the relations of the standards of the British and Metric
systems of units are adopted in this work. Tables i and 2 give the equivalents
of multiples of the standard units and also equivalents of multiples of the derived
units of surface and volume. These tables are published by the Office of Stand-
ard Weights and Measures of the United States, and are here republished by per-
mission of the Superintendent of that Office.
2. British Measures and Weights.
a. Linear measures.
The unit of linear measure is the yard. Its principal sub-multiples and multi-
ples are the inch ; the foot ; the rod, perch, or pole ; the furlong ; and the mile.
The following table exhibits the relations among these measures : —
* The actual error of the relation of the yard to the metre may be as great as T/200 0C0th part,
and the actual error of the relation of the pound to the kilogramme as great as i/ioo 000th part.
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XXXVUl
UNITS.
Inches.
Feet.
Yards.
Rods.
Furlongs.
MUes.
I
0.083
0.028
0.00505
0.00012626
0.0000157828
12
I.
0.333
0.06060
0.00151515
0.00018939
36
3-
I.
0.1818
0.00454s
0.00056818
•
198
16.S
5-5
I.
0.025
0.003125
7920
660.
220.
40.
I.
0.125
63360
5280.
1760.
320.
8.
I.
Other measures are the —
Surveyor's or Gunter's chain = 4 rods = 66 feet = 100 links of 7.92 inches
each.
Fathom = 6 feet ; Cable length =120 fathoms.
Hand = 4 inches ; Palm = 3 inches ; Span = 9 inches.
b. Surface or square measures.
The unit of square measure is the square yard. Its relations to the principal
derived units in use are shown in the following table : —
Sq. feet.
Sq. yards.
Sq. rods.
Roods.
Acres.
Sq. miles.
I.
O.IIII
0.00367309
0.000091827
0.000022957
9-
I.
0.0330579
0.000826448
0.000206612
272.25
30.25
I.
0.025
0.00625
10890.
I2ia
40.
I.
0.25
43560.
4840.
i6a
4.
I.
27878400
3097600.
102400.
2560.
640.
I.
c. Measures of capacity.
The unit of capacity for dry measure is the bushel (2150.4 cubic inches about).
The units of capacity for liquid measure are the British gallon (of 277.3 cubic
inches about) and the wine gallon (of 231 cubic inches, nominally). The latter
gallon is most commonly used in the United States. The following table shows
the relations of the sub-multiples and multiples of the bushel and gallon : —
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UNITS.
XZXIX
Dry Measures.
liquids.
Pint
= A bushel.
Gill
= ^gall.
Quart
= 2 pints = ^ "
Pint = 4 gills
= i "
Peck
= 8 quarts = J "
Quart = 2 pints
= i "
Bushel
= 4 pecks =1 "
Gallon = 4 quarts
= I "
Barrel = 31^ gallons
= 31* "
1
Hhd. = 2 barrels
= 63 "
Besides the above measures of capacity the following volumetric units are
used : —
Cubic foot ^ 1728 cubic inches.
Cubic yard = 27 cubic feet = 46656 cubic inches.
Board-measure foot = i square foot X i inch thickness = 144 cubic inches.
Perch (of masonry) = i perch (16.5 feet) length X i foot height X i-S feet
thickness^ 24.75 cubic feet ; 25 cubic feet are commonly called a perch for con-
venience.
Cord (of wood) = 8 feet length X 4 feet breadth X 4 ^eet height.
= 128 cubic feet.
d. Measures of weight.
The unit of weight is the avoirdupois pound. One 7000th part of this is called
a grain, and 5760 such grains make the troy pound. The sub-multiples and mul-
tiples of these two pounds are exhibited in the following table : —
Avoirdupois.
Troy.
Dram
=
iriir lb.
Grain
= i^M lb.
Ounce
= 16 drs.
=
A "
Pennyweight =20 grs.
= ^iiy
Pound
= 16 ozs.
=
I "
Ounce =24 dwt.
= A "
Quarter
= 28 lbs.
=
28 "
Pound =12 ozs.
= I "
Hundred-wt
.= 4qrs.
=
112 "
Long ton
= 20 cwt.
=
2240 "
Short ton
=
=
2000 "
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XI UNITS.
3. Metric Measures and Weights.
As explained in section i above, the standards of length and mass in the
metric system are the metre and the kilogramme. Two material representatives
of each of these standards are possessed by the United States and preserved at
the Office of Standard Weights and Measures at Washington, D. C.
The standards of length are Prototype Metres Nos. 21 and 27. These are
platinum iridium bars of X cross section, and their lengths are defined by lines
ruled on their neutral surfaces. Their lengths at any temperature / Centigrade
are given by the following equations : —
Prototype No. 21 == i*" + 2.^5 + S,i^66$ t -|- o.'^oo 100 /*,
Prototype No. 27 = i"* — i.'*6 -|- 8.'*657 / + o-'*oo 100 /*,
where the symbol fi stands for one micron, or one millionth of a metre.- The
probable errors of these Prototypes may be taken as not exceeding ± o.'*2, or
1/5 000 oooth of a metre for temperatures between o"^ and 30° C.
The standards of mass are Protot3rpe Kilogrammes Nos. 4 and 20. They are
cylindrical masses of platinum iridium. Their masses and volumes are given by
the following equations : —
Mass. Volume.
Prototype Kilogramme No. 4=1*^ — o."^o75, 46."'4i8,
Prototype Kilogramme No. 20 = i*^ — o«"^o39» 46."'402,
where the —
Symbol kg stands for one kilogramme,
Symbol mg stands for one milligramme ^ o.*^oooooi,
Symbol tnl stands for one millilitre = one cubic centimetre.
The definitive probable error assigned to the Prototype Kilogrammes by the
International Bureau is ± o."'oo2, or 1/500 000 oooth of a kilogramme.
The act of Congress approved July 28, 1866, authorizing the use of the metric
system in the United States, provides that the tables in a schedule annexed shall
be recognized " as establishing, in terms of the weights and measures now in use
in the United States, the equivalents of the weights and measures expressed
therein in terms of the metric system ; and said tables may be lawfully used for
computing, determining, and expressing, in customary weights and measures, the
weights and measures of the metric system." The following copy of that sched-
ule gives the denominations of the multiples and sub-multiples of the measures
of length, surface, capacity, and weight in the metric system as well as their
legalized equivalents in British units.
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UNITS.
Xli
Schedule annexed to Act of July 28, 1866.
Measures of Length.
Equivalents in Denominations in Use.
Metric Denominations.
M^ametre
Kilometre .
Hectometre
Decametre .
Metre . .
Decimetre .
Centimetre .
MUlunetre .
Values in Metres.
lOOOO.
1000.
100.
10.
0.1
0.01
0.001
6.3137 miles.
0.62137 °ule, or 3280 feet and lo inches.
328 feet and i inch.
393.7 inches.
39.37 inches.
3.937 inches.
0.3937 inch.
0.0394 inch.
Measures of Surface.
Metric Denominations.
Hectare
Are
Centare
Values in
Square Metres.
loooo
100
Equivalents in Denominations in Use.
a.471 acres.
1 19.6 square yards.
1550 square inches.
Measures of Capacity.
Metric Denominations and Values.
Equivalents in Denominations in Use.
Names.
Kilofitre or stere
Hectoiitz«
Decalitre.
Litre . .
DedKtre.
Centilitre
MiUiUtre.
No. of
Litres.
1000.
100.
0.01
0.001
Cubic Measure.
I cubic metre . .
o. I cubic metre .
10 cubic decimetres
I cubic decimetre .
o. I cubic decimetre
10 cubic centimetres
I cubic centimetre
Dry Measure.
I. ')o8 cubic yards
2 bus. and 3.35 pks.
9.08 quarts ...
0.908 quart . . .
6. 1022 cubic inches
0.6102 cubic inch
0.061 cubic inch .
Liqiiid or Wine
Measure.
264.17 gallons.
26.417 i^allons.
2.6417 gallons.
1.0567 quarts.
0.845 Kill.
0.338 fluid-ounce.
0.27 fluid-drachm.
Measures of Weight.
Metric Denominations and Values.
Names.
Millier or tonneau
<>iintal ....
Myriagramme . .
Kiiogrsnune, or kilo
Hectogranune . .
Decagramme . .
Gramme ....
Coitigramme . .
Millignunme . .
Number of
Grammes.
loooo.
1000.
100.
10.
0.1
0.01
0.001
Weight of what Quantity of Water
at Maximum Density.
I cubic metre . . .
I hectolitre . . .
10 litres ....
I litre
I decilitre ....
10 cubic centimetres
I cubic centimetre .
o. I cubic centimetre
10 cubic millimetres
I cubic millimetre .
Equivalents in Denominations
in Use.
Avoirdupois Weight.
2204.6 pounds.
220.46 pounds.
22.046 pounds.
2.2046 pounds.
3.5274 ounces.
0.3527 ounce.
1 5* 43s grains.
1.5432 grains.
0.1543 grain.
00154 S^Ain.
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xlii UNITS.
4. The C G. S. System of Units.
The C. G. S. system of units is a metric system in which the fundamental
units are the centimetre, the gramme, and the mean solar second. It is the sys-
tem now generally used for the expression of physical quantities.
The most important of the derived units in the C. G. S. system, their equiva-
lents in terms of ordinary units, and their dimensions in terms of the fundamen-
tal units of length, mass, and time, are given in the Appendix to this volume.
For an elaborate consideration of the subject of units and their interrelations
the reader may be referred to "Units and Physical Constants," by J. D. Everett,
London, Macmillan & Co., i2mo, 4th ed., 1891.
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GEODESY.
I. Form of the Earth. The Earth's Spheroid. The Geoid.
The shape of the earth is defined essentially by the sea surface, which embraces
about three fourths of the entire surface. The sea surface is an equipotential
surface due to the attraction of the earth's mass and to the centrifugal force of its
rotation. We may imagine this surface to extend through the continents, and
thus to be continuous. Its position at any continental point is the height at
which water would stand if a canal connected the point with the ocean.
Geodetic measurements show that this surface is represented very closely by
an oblate spheroid, whose shorter axis coincides with the rotation axis of the
earth. This is called the earth's spheroid. The actual sea surface, on the other
hand, is called the geoid. With respect to the spheroid the geoid is a wavy sur-
face lying partly above and partly below ; but the extent of the divergence of the
two surfaces is probably confined to a few hundred feet.
2. Adopted Dimensions of Earth's Spheroid.
The dimensions of the earth's spheroid here adopted are those of General A.
R. Clarke, published in 1866, to wit: —
Semi major axis, a = 20 926 062 English feet.
Semi minor axis, ^ = 20 855 121 " "
3. Auxiliary Quantities.
The following quantities are of frequent use in geodetic formulas : —
^ = 1/ 5 — , the eccentricity of generating ellipse,
a — d
f = > the flattening, ellipticity, or compression,
I — «
^=2/-/«.
/=^-v^^"^=^ + -f + ^ + I^ +
2 n
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xliv GEODESY.
/
2
±7 = (J/) + a/)' + a/)* + a/)* + .
'^= (F^^ = 4 (« - 2 «' + 3 «• - 4 »* + • • •)•
"^ =r=r7 — T *+■ T "^ "8" "^ 16" + • • • •
I + y^l — ^ 4 ' 8 * 64 ' 128
The numerical values of the most useful of these quantities and their logarithms
are —
log
tf = 20 926 062 feet, 7'32o6875,
3 = 20 85s 121 feet, 7.3192127,
^= 0.00676866, 7.8305030 — 10,
m = 0.00339583, 7.5309454 - 10,
n =z 0.00169792, 7.2299162 — 10.
4. Equations to Generating Ellipse of Spheroid.
With the origin at the centre of the ellipse, and with its axes as coordinate
axes, the equation in Cartesian co-ordinates is
^ + =^ = 1. 0)
a and 3 being the major and minor axes respectively, and x and y being parallel
to those axes respectively.
For many purposes it is useful to replace equation (i) by the two following : —
a: = tf cos Oj
which give (i) by the elimination of 0. This angle is called the reduced latitude.
See section 5.
5. Latitudes used in Geodesy.
Three different latitudes are used in geodesy, namely: (i) Astronomical or
geographical latitude ; (2) geocentric latitude ; (3) reduced latitude. The astro-
nomical latitude of a place is the angle between the normal (or plumb line) at that
place and the plane of the earth's equator ; or when the plumb line at the place
coincides with the normal to the generating ellipse, it is the angle between that
normal and the major axis of the ellipse. The geocentric latitude of a place is
the angle between the equator and a line drawn from the place to the earth's cen-
tre ; or it is the angle between the radius-vector of the place and the equator.
The reduced latitude is defined by equations (2) in section 4 above. The geo-
metrical relations of these different latitudes are shown in Fig. i by the notation
given below.
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GEODESY. • Xlv
In order to express the analytical relations between the different latitudes let
B <l> = the astronomical latitude,
if/ = the geocentric latitude,
$ = the reduced latitude.
Then, referring to equations (i) and (2) under
section 4 above, and to Fig. i, it appears that
Figl.
tan <^:
tan ^ :
dx _.a^y
•^v
tantf = f.
dx
Hence
tan ^ = — g- tan <^ = (i — ^ tan <^,
tan tf = (i — <f^» tan <^ = (i — r^"* tan ^.
^ — ^ = »f sin 2 <^ — /«^ sin 4 <^ + . . . ,
<^ — tf = «sin2^ — 4^«'sin4<^ + ....
For the adopted spheroid
and
log (i - ^ = 9.9970504,
^ — ^ (in seconds) = 7oo."44 sin 2 <^ — i."i9 sin 4 4h
if^ -^ tf (in seconds) = 35o."22 sin 2 ^ — ©."30 sin 4 ^
6. Radii of Curvature.
p^ = radius of curvature of meridian section of spheroid at any point whose
latitude is <f>=^PO, Fig. i,
p. = radius of curvature of normal section perpendicular to the meridian at
the same point = PQ, Fig. i,
p. = radius of curvature of normal section making angle a with the meridian
at same point.
p« = /z (i - ^ (i - <f» sin« <^)-f,
p« = d5(i — i?^sinV)-*,
£^ COS* a , sin* a
Pa~ Pm "^ Pn
I ^
^^ i (' + i"3? ^^^^ ^ ^^^* a) (i — ^ sin* <^)*.
log (i - ^ sin* <^)-* = + log (i + n)
— fin cos 2<l>
+ i A* »* cos 4<^
— ^ fin* cos 6<^
fi = modulus of common logarithms and n is same as in section 3. For the
adopted spheroid —
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Xlvi GEODESY.
Radius of curvature of meridian section p^ in feet,
log P-. = + 7-3199482
— [4.34482] cos 2<^
+ [1.274] cos 4<^
~~ • . • •
Radius of curvature of normal section pn in feet.
log p» = + 7-32^243
— [3.86^70] COS 2^
+ [0.797] COS 4<^
The numbers in brackets in these formulas are logarithms to be added to the
logarithms of cos 2<^ and cos 4<^. The numbers corresponding to the sums of
these logarithms will be in units of the seventh decimal place of the first constant
Thus, for <^ = o,
log p,= 7.3214243
- 7373-9
_+ 6^
= 7.3206875 = log a.
7. Length of Arcs of Meridians and Parallels of Latitude.
a. Arcs of Meridian.
For the computation of short meridional arcs lying between given parallels of
latitude the following simple formulas suffice :
* = K^ + *i), (i)
In these, ^ and <^ are the latitudes of the ends of the arc, A^is the required
length, and p^ is the meridian radius of curvature for the latitude ^ of the middle
point of the arc. The formula for ^M implies that A^ is expressed in parts of
the radius. If A<^ is expressed in seconds, minutes, or degrees of arc, the for-
mula becomes —
Meridional distance h.M in feet.
A »^ — Pm A<^ (in seconds)
^^— 206264.8 '
p^ A<^ (in minutes)
~ 3437-747 '
_ p^ A<^ (in degrees) .
57-29578 ' (2)
log (1/206264.8) = 4.6855749 — io»
log (1/3437-747) = 6.4637261 - 10,
log 0/57-29578) = 8.2418774 — 10.
^i, ^, = end latitudes of arc, A^ = ^, — ^„
pb = meridian radius of curvature for ^ = ^(^ + ^) * ^^r log p^ see Table xo.
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GEODESY. Xlvii
The relations (2) wfll answer most practical purposes when A^ does not exceed
5°. A comparison with the precise formula (3) below shows in fact that the error
of (2) is very nearly
i ^ Aif>* cos 2if> . /^lM,
which vanishes for ^ = 45°, and which for A^ = 5° is at most yirAinr ^-^» or
about II feet.
Numerical example. Suppose —
^ = 37°29'48/'i7,
^ = 3S%8'29."89.
Then
* = K^ + <^) = 36" 39' 09."o3,
A«^= 4^— <^= i°4i'. i8."28,
= 6o78."28.
From the first of (2)
cons't. log 4.6855749 — 10
Table 10, log p« 7.3193112
log A«^ 3.7837807
AJIf = 614705 feet, log AJfeT 5.7886668
The values of AJl/'for intervals of 10", 20" . . . 60", and for 10', 20' . . . 60' are
given in Table 17 for each degree of latitude from 0° to 90°.
For precise computation of long meridional arcs the following formula is ade-
quate : —
^M=z Aq A^ — Ai cos 2^ sin A^
+ A% cos 4<^ sin 2A^
— A^ cos 6<^ sin 3A^ (3)
+ A^ cos 8^ sin 4A^
In this, AJf, ^, and A^ have the same meanings as above, and ^0, ^j, . . . are
functions of a and ^ or of ^ and n.
Thus, in terms of a and n^
^o = « (I +«r + i «" + A «* + . . . X
^i = 3a(i + «)-»(«-i«»-...),
^,= V a (l + «)-!(««- i «^- . . . ),
Introducing the adopted values of a and », these constants become —
log.
^0 = 20 890 606 feet, 7.3 1995 10,
Ai= 106 411 feet, 5.0269880,
^, = 113 feet, 2.0528,
A^ = 0.15 feet, 9.174 — 10.
Digitized by
GooqIc
xlviii GEODESY.
It appears, therefore, that the first three terms of (3) will give AJl/'with an
accuracy considerably surpassing that of the constant A^ In the use of (3) it will
generally be most convenient to express A<^ in degrees, and in this case Aq must
be divided by the number of degrees in the radius, viz. : 57.2957795 [1.7581226].
Applying this value and writing the logarithms of Aq, A^^ etc., in rectan^lar
brackets in place of Aq, Ai, etc., (3) becomes
Meridional distance AM in feet
^Jlf=: [5.5618284] A<^ (in degrees)
— [5.0269880] cos 2<^ sin A^ (4)
-}- [2.0528] cos 4<^ sin 2A^
2^ = ♦« + ^i. A^ == ^, — ^^ ^j, ^, = end latitudes of arc.
Formula (4) will suffice for the calculation of any portion or the whole of a
quadrant. The length of a quadrant is the value of the first term of (4) when
<^ = 45° and A<^ = 90^, since all of the remaining terms vanish.
Numerical examples, — 1°. Suppose
«^i = o'' and <^ = 45°.
Then 2<^ = 4S^
log.
cons't 5.5618284
45 1.6532125
ist term + 16 407 443 feet ist term 7.2150409
cos 2^ 9.8494850 — 10
sin A<^ 9.8494850 — 10
cons't 5.0269880
2d term — 53 205.7 ^^^^ ^^ ^^"^"^ 47259580
The third term of the series vanishes by reason of the factor cos 4 <^ = cos 90**
= o. The sum of the first two terms, or length of a meridional arc from the
equator to the parallel of 45®, is 16 354 237 feet.
2°. Suppose ^i = 45° and ^ = 90°.
Then 2^ = 135°,
A^= 45".
The numerical values of the terms will be the same as in the previous example,
but the sign of the second term will be plus. Hence the length of the meridional
arc between the parallel of 45° and the adjacent pole is 16 460 649 feet. The
sum of these two computed distances, or the length of a quadrant, is 32 814886
feet.
Digitized by VjOOQIC
GEODESY. Xlix
This agrees as it should with the length given by (4) when 2^ ^ 90^ and A^
= 90**.*
b. Arcs of parallel.
The radius of any parallel of latitude is equal to the product of the radius of
curvature of the normal section for the same latitude by the cosine of that lati-
tude. That is, see Fig. i, r being the radius of the parallel —
r = p^cos«^
and the entire length of the parallel is —
2 IT r = 2 IT p, cos ^.
Designate the portion of a parallel lying between meridians whose longitudes
are \\ and A^ by A/', and call the difference of longitude A^ — • Ai, AA.
Then —
Arc of parallel tJ* in feet
. « 2 IT pL, cos ^ ^ , ,. , V
^-P= X296000 ^("'^^°'^»)'
2 IT p^ cos ^^^/. .^v /v
= aTeoG ^^ (''' minutes), (i)
2 «r p, COS ^ ^ . /. J X
= V60 ^^ ^"^ degrees).
log (2 ir/i 296000) = 4.6855749 — 10,
log (2 ir/21600) = 6.4637261 — 10,
log (2 ir/360) = 8.2418774 — 10.
A,, A^ = end longitudes of arc, AA. ^ X, — A,,
ph = radius of curvature of normal section for latitude of parallel ; for log p^k see Table 11.
NumericcU Example, — Suppose ^ = 35°, and AA = 72^ Then from the third
of (9)
log.
cons't 8.2418774 — 10
Table 11, p, 7.321 17 16
cos«^ 9.9133645 — 10
AA 1.8573325
A/'= 21 564 827 feet, tkP 7.3337460
* The best formula for computing the entire length of a meridian curve is this :
»(« + *) (i + l«« + A«* + ...),
m which a, b, and n are the same as defined in section 2. For the values here adopted— >
log.
(i 4- i »' + • • •) 0.0000003
(a + b) 7.6209807
» 0.4971499
length 8.1181309
The length of the perimeter of the generating ellipse, or the meridian circumference of the
earth, is, therefore —
131 259 550 feet = 24 859.76 miles.
Digitized by VjOOQIC
1
GEODESY.
The values of A/'for intervals of lo", 20" . . . 60", and for 10', 20'
are given in Table 18 for each degree of latitude from 0° to 90°.
8. Radius- Vector of Earth's Spheroid.
p = radius-vector
= a (i — 2^ sin« ft> + ^ sin« </>)* (i — ^ sin* <^)-».
60'
logp:
^^S I'Vv^l'Tir?- +/*(«-«) cos 2^
~" i A* (^* "■ ^*) cos 4^
+ J /* («• — «•) cos 6^
For the adopted spheroid
log (p in feet) = 7.3199520 + [3.86769] cos 2«^
— [1-2737] cos 4<^,
the logarithms for the terms in ^ corresponding to units of the seventh decimal
place. Thus, for ^ = o,
^og p = 7-3199520
+ 7373-8
— 18.8
= 7.320687s = log a.
9. Areas of Zones and Quadrilaterals of the Earth's
Surface.
An expression for the area of a zone of the earth's surface or of a quadrilateral
bounded by meridians and parallels may be found in the following manner : —
The area of an elementary zone dZ, whose middle latitude is ^ and whose
width is p„ dif>f is (see Fig. i),
dZ = 2 w r p^dtfi
= 2 7rp„,p^cosif> dff>.
By means of the relations in section 6 this becomes
J7 a/ ^ cos d4>
„ I — ^ // (<f sin A)
(x)
2 V a
e (i — ^ sin'-* ^f '
The integral of this between limits corresponding to <^ and <^ or the area of a
zone bounded by parallels whose latitudes are if>i and ^ respectively, is
Z=ira^
I - ^
f sin <^
^ sin 01
I — ^ sin' ^ I — ^ sin* ^
+ * Nap. log ?^+'f ^M'l''!"?^
I » f ^ (i — ^ sm </>,) (i + ^ sm 0i) .
>■ (2)
Digitized byLjOOQlC
GEODESY.
To get the area of the entire surface of the spheroid, make <^i = — • J ir and <^
= -|- i TT in (2). The result is
Surface of spheroid = 2 tt a' I i + — ~r ^^P- ^^S 1 j _^ ) \ (3)
For numerical applications it is most advantageous to express (3) in a series of
powers of e. Thus, by Maclaurin's theorem,
Surface
of spheroid = 4 'r a* ^i -——————... j. (4)
For the calculation of areas of zones and quadrilaterals it is also most advan-
tageous to expand (2) in a series of powers of e sin <;^] and e sin ^ and express
the result in terms of multiples of the half sum and half difference of <^ and <^.
Thus, (2) readily assumes the form
Z •=.2v a^ {\ — ^ I (sin ^ — sin <i^i) + ~ ^ (sin* <^ — sin* ^) + . • . I.
From this, by substitution and reduction, there results
wherein
_ 1 Ci cos <^ sin J A</> — Ca cos 3«^ sin I A0 )
^ — 2 ^ ) _|_ Cs cos 50 sin # A^ - . i ^S;
A«^ = 08 — <^i ,
a=.^(|- + ^ + o+...), (6)
^=.».(i^ + i + ...).
If Q be the area of a quadrilateral bounded by the parallels whose latitudes are
01 and 0s and by meridians whose difference of longitude is AA,
AX
^ 27r
Hence, using the English mile as unit of length, (5) and (6) give for the
adopted spheroid —
Area of quadrflateral in square miles.
C = AA (in degrees) j 1'°^ ^ ^'1* ^^''J "^^ ^* ''^^ * ^* 1 ,
^ / + ^8 cos 50 sm } A0 — . . . ) '
log ^ * = 5.7375398, (7)
log rj= 2.79173,
log ^3 = 9.976 — 10.
♦ = i (^, + ^1 ), A^ = ^, — ^1.
^1, ^2 = latitudes of bounding parallels,
AA. ^ difference of longitude of bounding meridians.
* ^if ^1. ^» ^^ obtained from Cj, C^ C, respectively by dividing the latter by the number of
degrees in the radius, viz : 57-29578.
Digitized by V^OOQ IC
lii GEODESY.
Numerical examples, — i°. Suppose <^ = o, ^ = 90° and AX = 360°. Then
(7) should give the area of a hemispheroid. The calculation runs thus :
log. log. log.
^1 5-7375398 c, 2.79173 c^ 9.976 - 10
cos 9.8494850 — 10 COS 3 ^ 9.84948, — 10 cos s ^ 9.849, — 10
sin \ A<^ 9.8494850 — 10 sin I A^ 9.84949 — 10 sin f A^ 9-848, — 10
360 2.5563025 360 2.55630 360 2.556
Sum 7.9928123 5.04700, 2.229
Hence —
ist term = -j- 98358591
2dterm = -- 111429
3d term = + 169
Q = sum = 98470189
Twice this is the area of the spheroidal surface of the earth ; 1. ^., 196 940 378
square miles.
2°. The last result may be checked by (4). Thus,
(y- + 7j + . • . ) = 0.00225928
1.9990177
log a* = 7.1961072
log 4 ^r = 1.0992099
log (196940407) = 8.2943348
This number agrees with the number derived above as closely as 7-place
logarithms will permit, the discrepancy between the two values being about
vjnsijnru P^^ ^^ ^^^ siea.. Hence, with a precision somewhat greater than the
precision of the elements of the adopted spheroid warrants,
Area earth's surface = 196 940 400 square miles.
The areas of quadrilaterals of the earth's surface bounded by meridians and
parallels of 1°, 30', 15', and 10' extent respectively, in latitude and longitude, are
given in Tables 25 to 29.
10. Spheres of Equal Volume and Equal Surface with
Earth's Spheroid.
rj = radius of sphere having same volume as the earth's spheroid,
rs = radius of sphere having same surface as that spheroid.
Digitized by
GooqIc
1^
GEODESY.
<j — ri = J<i^(i+A^4"---) = 0-00II3 «> about
r2 — rj = ^ tf^* -]-... = o.oooooi <z, about.
liii
II. Coordinates for the Polyconic Projection of Maps.
In the polyconic system of map projection every parallel of latitude appears on
the map as the developed circumference of the
base of a right cone tangent to the spheroid along
that parallel. Thus the parallel EF (Fig. 2)
will appear in projection as the arc of a circle
EOF (Fig. 3) whose radius 0G-=>1 is equal
to the slant height of the tangent cone EFG
(Fig. 2). Evidently one meridian and only one
will appear as a straight line. This meridian is
generally made the central meridian of the area
to be projected. The distances along this cen-
tral meridian between consecutive parallels are
made equal (on the scale of the map) to the real Aj-
distances along the surface of the spheroid. The
circles in which the parallels are developed are
not concentric, but their centres all lie on the
central meridian. The meridians . are concave
toward the central meridian, and, except near the corners of maps showing large
areas, they cross the paral-
lels at angles differing little
from right angles.
In the practical work of
map making, the meridians
and parallels are most ad-
vantageously defined by the
co-ordinates of their points
of intersection. These co-
ordinates may be expressed
in the following manner :
For any parallel, as EOF
(Fig. 3), take the origin O
at the intersection with the
central meridian, and let the rectangular axes oi V (OG) and X (OQ) be re-
spectively coincident with and perpendicular to this meridian. Call the interval
in longitude between the central meridian and the next adjacent one AA, and
denote the angle at the centre G subtended by the developed arc OF by a.
Digitized by
Google
liv • GEODESY.
Then from Fig. 3 it appears that
x= / sin a,
y= 2 / sin* Jo.
But from Figs. 2 and 3,
/=p^COt«^,
/a = r AX = p^ AA. cos <^,
whence
a = AA. sin ^
Hence, in terms of known quantities there result
x = p^ cot <f> sin (AX sin </>), ^ x
^ = 2 p, cot ^ sin* ^ (AX sin <;^).
Numerical example. — Suppose <^ = 40° and AX = 25° = 90000".
Then
log 90000" = 4.9542425,
log sin 40° = 9.8080675 — 10,
log 578So-"88 = 4.7623100 ;
AX sin.</» = 16° 04' io."88,
J (AX sin <^) = 8°o2'o5."44.
log. log.
sin (AX sin <^) 9.4421760 — 10 sin \ (AX sin <^) 9.1454305 — 10
cot ^ 0.0761865 sin i (AX sin ^) 9.1454305 — 10
pi,. Table 11 7.3212956 cot ^ 0.0761865
p^ Table 11 7.3212956
2 0.3010300
X 6.8396581 y 5-9^93731
jT = 6 912 865 feet y = 975 828 feet
The equations (i) are exact expressions for the co-ordinates. But when
AX is small, one may use the first terms in the expansions of sin (AX sin <^) and
sin* K^^ ^i'^ ^) ^"^ reach results of a much simpler form.
Thus,
sin (AX sin ^) = AX sin ^ — J(AX sin <^)* + . . . ,
sin* i(^0 sin <^) = i(AX sin <^)* - A(^ sin <^)* + . . . ;
whence, to terms of the second order,
^ = p„ AX cos <^ [i — i(AX sin <^)*],
y = iPn (AX)* sin 2<l> [i - T^KAX sin<^)*].
(2)
If the terms of the second order in these equations be neglected, the value of
X will be too great by an amount somewhat less than i(AX sin </>)* . x, and the
value of y will be too great by an amount somewhat less than -^(AX sin <^)* . y.
An idea of the magnitudes of these fractions of x and y may be gained from the
following table, which gives the values of i(AX sin <^)* for a few values of the
arguments AX and (^.
Digitized by VjOOQIC
GEODSSY.
Valiics of i(AA sin <^)«.
Iv
*
AX
20°
40°
60°
o
I
1/1680OO
1/47700
1
1/26260
2
1/42000
1/119OO
1/6560
3
1/1870O
1/5300
1/2920
It appears from this table that the first terms of (2) will suffice in computing
the coordinates for projection of all maps on ordinary scales, and of less extent
in longitude than 2° from the middle meridian. For example, the value of x for
AX = 2°, and <j> = 40°, and for a scale of two miles to one inch (i/i 26720), is
53.063 inches less 1/11900 part, or about 0.004 inch, which may properly be
regarded as a vanishing quantity in map construction. For the computation of
the co-ordinates given in the tables 19 to 24, where AX does not exceed 1°, it
is amply sufficient, therefore, to use
^ = p„ AX cos <^,
y = ipn (^y sin 2i>.
(3)
In these formulas and in (2), if AX is expressed in seconds, minutes, or degrees,
it must be divided by the number of seconds, minutes, or degrees in the radius.
The logarithms of the reciprocals of these numbers are given on p. xlvi. In the
construction of tables like 19 to 24, it is most convenient, when English units are
used, to express AX in minutes and x and y in inches. For this purpose, sup-
posing log pM to be taken from Table 11, if s be the scale of the map, or scale
factor, equations (3) become —
Co-ordinates x and^ in inches for scale s.
X =
3437-747
3
p, s AX cos <t>.
AX in minutes ;
log (12/3437.747) = 7.54291 -- 10,
log (3/(3437747)0 = 3-4046 - 10.
(4)
Tables 19 to 24 give the values of x andy for various scales and for the zone of
the earth's surface lying between 0° and 80°.
Numerical example, — Suppose <^ = 40° and AX = 15' ; and let the scale of
the map be one mile to the inch, or ^ = 1/63360. Then the calculation by (4)1
runs thus : Digitized by LjOOgic
Ivi GEODESY.
log.
log.
cons't 7.54291 — 10
cons't 3.4046 — 10
Pn 7-32130
Pn 7-3213
s 5.19818 — 10
s 5.1982 — 10
IS 1. 17609
(15)* 2.3522
COS ^ 9.88425 — 10
sin 2<^ 9.9934 — 10
X 1.12273
y 8.2697 — 10
In.
In.
X = 13.266
y = 0.01861.
These values of x and ^, it will be observed, agree with those corresponding to
the same arguments in Table 22.
When many values for the same scale are to be computed, log s should, of
course, be combined with the constant logarithms of (4). Moreover, since in (4)
X varies as AA and y as (AX)*, when several pairs of co-ordinates are to be com-
puted for the same latitude, it will be most advantageous to compute the pair cor-
responding to the greatest common divisor of the several values of AX and derive
the other pairs by direct multiplication.
12. Lines on a Spheroid.
The most important lines on a spheroid used in geodesy are (a) the curve of a
vertical section ; (S) the geodesic line ; and (c) the alignment curve. Imagine two
points in the surface of a spheroid, and denote them by /\ and P2 respectively.
The vertical plane at I*i containing J\ and the vertical plane at ^2 containing
1*1 give vertical section curves or lines. The curves cut out by these two planes
coincide only when /\ and J\ are in a meridian plane. The geodesic line is
the shortest line joining JPi and /g, and lying in the surface of the spheroid.
The alignment curve on a spheroid is a curve whose vertical tangent plane at
every point of its length contains the terminal points I*i and J\. The curve
(a) lies wholly in one plane, while (f) and (c) are curves of double curvature.
In the case of a triangle formed by joining three points on a spheroid by lines
lying in its surface, the curves of class (a) give two distinct sets of triangle
sides, while the curves of classes (d) and (c) give but one set of sides each.
For all intervisible points on the surface of the earth, these different lines differ
immaterially in length ; the only appreciable differences they present are in their
a2imuths (see formula under b below). Of the three classes of curves the first
two only are of special importance.
a. Characteristic property of curves of vertical section.
Let ttij = azimuth of vertical section at J\ through J\,
a^i = azimuth of vertical section at I2 through JF^i,
$1, $2 = reduced latitudes of /\ and 1*2 respectively,
81, 82 = angles of depression at /\ and J\ respectively of the chord joining
these points.
Then the characteristic property of the vertical section curve joining /\ and /i is
sin aui cos 61 cos Si = sin (oj.! — 180°) cos O2 cos ^ j
Digitized byLjOOQlC
GEODESY. Ivii
The azimuths a^ and 03.1, it will be observed, are the astronomical azimuths,
or the azimuths which would be determined astronomically by means of an alti-
tude and azimuth instrument.
b. Characteristic property of geodesic line.
Let aYs = azimuth of geodesic line at -Pi,
a'2.1 = azimuth of geodesic line at P^
0^ $2 = reduced latitudes of Pi and P^ respectively.
Then the characteristic property of the geodesic line is
sin ttj^ cos ^1 = sin (i 80°— 02.1) cos 62 = cos 60,
where ^0 is the reduced latitude of the point where the geodesic through Pi and
P^ is at right angles to a meridian plane.
The difference between the astronomical azimuth a^^s and the geodesic azimuth
a'l^ is expressed by the following formula :
ai^ — a'lj (in seconds) =1^ p" ^ / £ | cos^ <^ sin 201^
where s = length of geodesic line Pi P^,
a = major semi-axis of spheroid,
ez=. eccentricity of spheroid,
p" = 2o6264."8,
if> =. astronomical latitude of /\,
ttij = azimuth (astronomical or geodesic) of Pi P^
log tV p"\-\ = 7-4244 — 20, for a in feet.
Thus, for ^ = o and aj^ = 45°, for which cos* ^ sin 20^, = i, the above for-
mula gives
aij — a'ug = o."o74, for s = 100 miles,
;= 0.296, for s = 200 miles.
so that for most geodetic work this difference is of little if any importance.
13. Solution of Spheroidal Triangles.
The data for solution of a spheroidal triangle ordinarily presented are the
measured angles and the length of one side. This latter may be either a geodesic
line or a vertical section curve, since their lengths are in general sensibly equal.
Such triangles are most conveniently solved in accordance with the rule afforded
by Legendre's theorem, which asserts that the sides of a spheroidal triangle (of
any measurable size on the earth) are sensibly equal to the sides of a plane
triangle having a base of the same length and angles equal respectively to the
spheroidal angles diminished each by one third of the excess of the spheroidal
triangle. In other words, the computation of spheroidal triangles is thus made to
depend on the computation of plane triangles. Digitized byLjOOQlC
Ivili GEODESY.
a. Spherical or spheroidal excess.
The excess of a spheroidal triangle of ordinary extent on the earth is given by
€ (in seconds) = p" >
Pm Pn
where S is the area of the spheroidal or corresponding plane triangle ; p« /», are
the principal radii of curvature for the mean latitude of the vertices of the tri-
angle ; and p" = 206 26^"%. For a sphere, p^=z p^=: radius of the sphere.
Denote the angles of the spheroidal triangle by A, B, C, respectively ; the cor-
responding angles of the plane triangle by a^ P,y (as on p. xviii) ; and the sides
common to the two triangles by a, bj c. Then
5 = i <j^ sin y = i ^r sin o = J ^tf sin p.
a = A — i€, P = B — \€, y=C— Jc.
Tables 13 and 14 give the values of log Q/'/ipi^ for intervals of 1° of astro-
nomical or geographical latitude.*
14. Geodetic Differences of Latitude, Longitude, and
Azimuth.
a. Primary triangulation.
Denote two points on the surface of the earth's spheroid by Bi and P^ respec-
tively. Let
s = length of geodesic line joining /\ and /V,
^, 0, = astronomical latitudes of Bi and If,
Ai, X2 = longitudes of J\ and B3,
oij = azimuth of jPi P^ (s) at -Pi,
oj.! = azimuth of P^ Pi (s) at P^
e = eccentricity of spheroid,
p^^ p^ = principal (meridian and normal) radii of curvature at the point -Pi.
Then for the longest sides of measurable triangles on the earth the following
formulas will give <^ A,, and ojj in terms of <^, Ai, ai.s, and s. The azimuths are
astronomical, and are reckoned from the south by way of the west through 360°.
a' = 180° — ttij, and oai = 180° + **"> ^^^ a^ <i8o°
a! = ai4 — 180°, and Oj.! = 180° — a", for a^ > 180°
(«
{ = 1 j-:^^ cos^ <h sin 2a' (3)
• For the solution of very large triangles and for a full treatment of the theory thereof, consult
Die Afathematischtn und PhysiktUischen TheorUcn der Hbheren Geoddsie^ von Dr. F. R. Helmert.
Leipzig. i88o, 1884. ^.g.,.^^^ ^y V^OOg le
GEODESY. lix
(4)
'^"^'^^ sin |(a" + „' + {) {^ + A 'y' COS» Ka" - a% (5)
To express 17, ^, and 0s — <^ in seconds of arc we must multiply the right hand
sides of (2), (3), and (5) by p" = 206 264."8. For logarithmic compution of rf"
and C\ or 17 and { in seconds, we may write with an accuracy generally sufficient
log v" = log (p"s/p.) + i j^ (^J' COS« <k COS* a', (6) .
log r = log J (7^1^ + log {(VO* COS* <h sin 2 a'}, (7)
where /t in (6) is the modulus of common logarithms. For units of the 7th deci-
mal place of log V' we have for the adopted spheroid
M*'
'°si^:r?= 3-69309.
Also
Similarly, for the computation of the logarithm of the last factor in (5) we have
log {i + tV ^*
Putting for brevity
log {I + A '^^ cos« K»" - ''O} = log {1 + ^, (v'T cos* K-" - -O}.
the logarithm of the desired logarithm is given to terms of the second order
inclusive in ^ by
log log (i + i^) = log ^ ^ - i /x ^.
For the adopted spheroid
'°sn^«= 4-92975 -10
for units of the seventh decimal place.
For a line 200 miles (about 320 kilometres) long, the maximum value of the
second term in (6) is but 12.6 units in the 7th place of log ?/". For the same
length of line, the maximum value of £" is o/'Sg^, and the maximum value of the
logarithm of the last factor in (5), or log (i + ^)> is less than 922 units in the
seventh place of decimals.
For computing differences of latitude, longitude, and azimuth in primary
triangulation whose sides are 1° (about 70 miles, or 100 kilometres) or less
in length, the most convenient means are formulas giving ^ — <^i, A, — Xj, and
Digitized byLjOOQlC
Ix GKODBSY.
ojj — (i8o® — ajj), in series proceeding according to powers of the d i stance x.
Formulas of this kind with convenient tables for facilitating the computatioos
are given in the Reports of the U. S. Coast and Geodetic Survey.*
b. Secondary triangulation.
For secondary triangulation, wherein the sides are 12 miles (20 kilometres) or
less in length, and wherein differences of latitude and longitude are needed to tlxe
nearest hundredth of a second only, the following formulas may suffice. Usin^
the same notation as in the preceding section, the formulas are : —
(4j = 180° + o,j -|- Aa,
A^ = — aiS cos ajj — tfj -^ sin* au^
AX = + ^i sec ^1 s sin a^ — ^, j* sin 0^ cos au^ (2)
Aa = — ^1 tan if>i s sin a^ -|- r, J* sin a, j cos a^^
The constants entering the latter equations are defined by the following
expressions, wherein p,» and p. are the principal radii of curvature of the spheroid
at the point whose latitude is <^ and p" = 206 264."8 :
Pm Pm
p" tan ifn . p" sec <^i tan <^i p'^ ( i -|- 2 tan* ^)
(h y ^2 —s > *8 2 •
2 P- A, Pn 2 Pi,*
The logarithms of the factors <Zi, ^1, Ci, Of, df, Cf, are given in Table 15 for the
English foot as unit, and in Table 16 for the metre as unit, the argument being
the initial latitude 4>i for all of them.
When all of the differences given by (2) are computed, they may be checked
by the formula
sinK^+^,) = ^- (3)
For convenience of reference in numerical applications of the above formulas,
(2) may be written thus :
A<^ = ^1 + ji^
AX = ^, + B^
Aa = Ci + Ci,
in which, for example, Ai and A^ are the first and second terms respectively of
A^, due regard being paid to the signs of the functions of a^j.
Numerical example. The following example will serve to illustrate the use of
formulas (i) to (3). The value of log s is for s in English feet, s being in this
case about 12.3 miles.
^ 38°54'o8."38
A0 —07' 5o."2i
^ 38° 46' i8/'i7
i(*,+ *i) 38°50'i3/'27
• See Appendix 7, Report of 1884, for latest edition of .these tablp. t
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^1
88» 03' 24."is
«ij 43° 01' 46-"29
AA.
+ 09'20."23
Aa —05' 51. "32
A,
88° 12' 44."3T
<4., 223° St; S4."97
GEODESY. Ixi
log log log log
s 4.81308 s 4.81308 s sin aij 4.647 s sin a^^ 4.647
cos au 9.86392 sin ajj 9.83402 s sin a^^ 4-647 ^ cos a^ 4.677
ai 7.99495 sec ^1 0.10890 <Z2 0.279 ^8 0.688
^1 7-99316 ^s 0.733
Ai 2.6719s Bi 2.74916 Ai 9.573 Bi 0.012
sin ^ 9-79795 ^« °-°57
Ci 2.54711
log
^1 - 469."84 -^1 + 56i."25 C, - 352-"46 Aa 2.54570
^, - o."37 ^2 - i."o3 C, + i."i4 AX 2.74836
A^ — 47o."2i A\ + s6o."22 Aa — 35i."32 sin i(<^ + *i) 9-79734
15. Trigonometric Leveling.
a. Computation of heights from observed zenith distances.
Let s = sea Iqvel distance between two points /i and /i,
Z?i, Hi = heights above sea level of Fi and JPf,
Zi = observed zenith distance of P2 from -Pi,
Zi = observed zenith distance of Pi from /\,
p = radius of curvature of vertical section at /\ through P^ or at P^
through Pi^ the curvature being sensibly the same for both for this
purpose,
C = angle at centre of curvature subtended by j,
«fi, nii = coefficients of refraction at Pi and P^
Aj?i, A«i = angles of refraction at Pi and P^.
Then, the fundamental relations are
C=-, £iiZi=.miC, t^ZiZ=.miC, ^ ^x
*i + ^ + Ajj + Ajgj = 180° 4- (7,
H^^Hi = stzxi K^ + A.. - ^x - A^O (i + t^yi^^ + -i^+. . .). (2)
When the zenith distances Zi and z^ are simultaneous, or when A«i and Aiij are
assumed to be equal, (2) becomes
i9i-^,= .tan K*, - ^0 (i + ^^y^' + I¥7 + - • •)• <3)
For the case of a single observed zenith distance 2^1, say, and a known or
assumed value of »i = »ii = »?8, the following formula may be applied :
^,-^i = :rcot^,+ '~^^ x^+^-:=^^«cot«^i. (4)
The coefficient of refraction m varies very greatly under different atmospheric
conditions. Its average value for land lines is about 0.07. The following table
gives the values of log \{i — 2 w) and log (i — m) for values of m ranging from
0.05 to 0.10. It is taken from Appendix 18, Report of U. S. Coast and Geodetic
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Ixii
GEODESY.
Survey for 1876. Table 12 taken from the same source gives values of log p
needed for use in (3) and (4).
For less precise work one may use equation (4) in the form
^ — i?i = X cot «i + ^ -^t
Tai/e of values of log Ki
— 2 «) and log (i — m).
m
logi(i — 2»»).
log(i-«).
m
log i(i — 2 m).
log (!-•). 1
0.050
9-65321
9.978
0.075
9.62839
9.966
SI
65225
77
76
62737
66
52
65128
77
77
62634
65
53
65031
76
78
62531
65
54
64933
76
79
62428
64
0.0SS
9.64836
9-975
0.080
9-62325
9-964
56
64738
75
81
62221
63
57
64640
75
82
621 18
63
58
64542
74
83
62014
62
59
64444
74
84
61910
62
0.060
964345
9-973
0.085
9.61805
9.961
61
64246
73
86
61700
61
62
64147
72
87
61595
60
63
64048
72
88
61490
60
64
63949
7»
89
61384
60
0.065
9.63849
9.971
0.090
9.61278
9-959
66
63749
70
9'
61172
59
67
63649
70
92
61066
58
68
63543
69
93
60959
58
69
63448
69
94
60853
57
0.070
9-63347
9.968
0.095
9.60746
9-957
71
63246
68
96
60638
56
7a .
63144
68
97
60531
56
73
63043
67
98
60423
55
74
62941
67
99
60315
55
t
0.1 00
9.60206
9-954
(s)
wherein, if we make m = 0.07 and use for p its average value, or SPmpZ for
latitude 45^,
log c = 2.313 — 10 for s in feet,
= 2.829 — 10 for J in metres.
Thus, for a distance (s) of 10 miles the value of the term cs^'m (5) is 57.3 feet.
If altitudes a^, say, are observed in the place of zenith distances ^i, it is most
convenient to write (5) thus : —
^ — ZTi = ± J tan tti -|- r J*,
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(6)
GEODESY. Ixiii
where the upper sign is used when oi is an angle of elevation and the lower sign
when Oi is an angle of depression.
b. Coefficients of refraction.
When Zi and z^ are both observed for a given line, a coefficient of refraction may
be computed from the assumption of equality of coefficients at the two ends of
the line. Thus, equations (i) give
AiTi + A^, = i8o° + C - (^1 + «,),
or
(^1 + ^«) ^ = ^So° + ^ - (*i + ^)'
whence
»ll + ^2 = I — - (^l + *8 "~ lS0°)'
Assuming wi = jw, = «, and supposing *i + ^a "" ^^^^ expressed in seconds
of arc,
^ = j|i -^(5, + ^,- i8o°)j.
p"= 2o6264."8, log p" = 5.3144251.
c. Dip and distance of sea horizon.
Let
Then
^ = height of eye above sea level,
8 = dip or angle of depression of horizon,
s = distance of horizon from observer.
8 (in seconds) = 58.82 ^A in feet,
= 106.54 V^ in metres.
s (in miles) = 1.3 17 V>4 in feet,
s (in kilometres) = 3.839 ^A in metres.
The above formulas take account of curvature and refraction. They depend
on the value 0.0784 for the coefficient of refraction, and are quite as accurate as
the uncertainties in such data justify. For convenience of memory, and for an
accuracy amply sufficient in most cases, the coefficients of the radicals in the last
two formulas may be written | and V respectively.
16. Miscellaneous Formulas.
a. Correction to observed angle for eccentric position of instrument
Let C be the eccentric position of the instrument, and Co the observed value of
the angle at that point between two other points A and B. Let C denote the
central point as well as the angle ACB desired. Call the distance CC r and
denote the angle ACC by 6. Denote the lines BC and AC, which are as-
sumed to be sensibly the same as BC and AC, by a and d respectively. Then
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Iziv GBODESY.
n n r a^ p"r sin (6 ^ Q) p"r sin
C— Q (m seconds) = ^^ ^ — - — r 1
p" = 206 264."8, log p" = 5.3144251.
Attention must be paid to the signs of sin (0 — Co) and sin B, and to tbe fact
that angles are counted from A towards B through 360°. A diagram drawn in
accordance with the above specifications will elucidate any special case.
b. Reduction of measured base to sea level.
Let / be the length of the bar, tape or other unit used in measuring the base.
Let 4 be the corresponding length reduced to sea level for a height A, this latter
being the observed height of /. Then if p denote the radius of curvature of the
earth's surface in the direction of tlie base,
*=,-^.=(.-^+...)'
with sufficient accuracy. Hence, for the whole length of the base,
2^ = 2/- -UA.
P
If Z denote the total measured length, Zq the corresponding total sea level
length, and If the mean value of the heights A, the above equation gives
Zo = Z-Z ^.
P
c. The three-point problem.
In this problem the positions of three points A, By C, and hence the elements
of the triangle they form, are given 'together with the two angles y^-PC and BPC
at a point B whose position is required. Denote the angles and the sides of the
known triangle by A^ B^ C, and a, b^ c, respectively. Also put
APC=P, BPC=a,
PAC = x, BBC=zy.
Then the sum of the angles in the quadrilateral PACB is
a + )3 + x+>+C=36o^
whence
K* + >) = 180° - i(a + iS + C). (i)
Compute an auxiliary angle z from the equation
a sin /3
Then
tan z = , ^. ; (2)
sin a ^ ^
tan ^x - J') = tan {z - 45°) tan y^x + y\ (3)
These three equations give all the data essential to a complete determination
of the position of P, Any special case should be elucidated by a diagram drawn
in accordance with the specifications given above.
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GEODESY.
IXF
When the position? of thQ points A, B, C are given on a map, the position of
J^ on the same map may be found graphically by drawing lines making angles
with each other equal to the given angles a and fi from a point on a piece of
tracing paper, and then placing this tracing on the map so as to meet the required
conditions. This ready method of solving the problem is often sufficient.
17. Salient Facts of Physical Geodesy.
a. Area of earth's surface, areas of continents, area of oceans.*
Square miles.
Total area of earth's surface 196 940 000
Area continent of Europe 3 820 000
" " Asia 17230000
" " Africa 11 480 000
" " Australia 3406000
" " America 15950000
Total area of continents 51886000
Total area of oceans 145054000
b. Average heights of continents and depths of oceans.t
Feet. Metres.
Average height of continent of Europe .
** " Asia . .
" Africa .
" " Australia
" " America .
Average height of all
980
1640
1640
820
1340
1440
300
500
500
250
410
440
Feet. Metres.
Average depth of Atlantic Ocean 12 100 3680
" " Pacific Ocean 12 700 3890
" " Indian Ocean 11 000 3340
Average depth of all 11 300 3440
c. Volume, surface density, mean density, and mass of earth.
Volume of earth = 259 880 000 000 cubic miles.
= I 083 200 000 000 cubic kilometres.
= 260 X !©• cubic miles (about).
= 108 X 10" cubic kilometres (about).
Surface density of earth = 2.56 ± 0.16 t
Mean density of earth = 5.576 ± 0.016.
• Derived from relative areas given in Helmert's Geoddsie^ Band II. p. 313.
t Helmert's Geoddsiey Band II. p. 313.
X These densities are given by Professor Wm. Harkness in his memoir on The Solar Parallax
and Related Constants, The surface density applies to that portion of the earth's crust which lies
above and within a shell ten miles thick, the lower surface of this shell being ten miles below sea
level.
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Ixvi GEODESY.
Assuming the mass of a cubic foot of water to be 62.28 pounds (at 62^ ¥.),
Mass of earth* = 13 284 X 10** pounds.
= 6642 X 10" tons (of 2000 lbs.).
= 60 258 X 10" kilogrammes.
d. Principal moments of inertia and energy of rotation of earth.
M= mass of earth,
A = moment of inertia of earth about an axis in its equator,
C = moment of inertia about axis of rotation,
a = equatorial axis of earth,
w = angular velocity of earth,
= (2 ir/86164) for mean solar second as unit of time.
Thent
^ = 0.325 Ma^f
0=0.326 Afa\
Energy of rotation of earth = i (JC.
= 0.163 w'J/a*.
= 504 X lo** foot-poundals.
= 217 X lo** kilogramme-metres.
= 212 X 10" ergs.
The most exhaustive treatise on the theory of geodesy is found in " Die Mathe*
matischen und Physikalischen Theorieen der Hoheren Geodasie," von Dr. F. R.
Helmert. Leipzig : B. G. Teubner ; 8vo, 1880 (vol. i.), 1884 (vol. ii.). An excel-
lent work on the practical as well as theoretical features of the subject is *' Die
geodatischen Hauptpunkte und ihre Co-ordinaten," von G. Zachariae ; autorisirte
deutsche Ausgabe, von £. Lamp. Berlin : Robert Oppenheim, 8vo, 1878. Of
works in English the most comprehensive is " Geodesy," by A, R. Clarke. Ox-
ford : The Clarendon Press, 8vo, 1880-
* The mass of the earth's atmosphere is about one-millionth part of the entire mass, or about
66 X lo^* tons.
t The values of A and C are those given by Harkness, /oc. ci/.t but they are here abridged to
three places of decimals.
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ASTRONOMY.
I. The Celestial Sphere. Planes and Circles of Reference.
The celestial sphere is a sphere to which it is convenient to refer stars and
other celestial objects. Its centre is assumed to be coincident with the eye of
the observer, and the objects referred to it are supposed to lie in its surface.
The orientation of this sphere is defined by its equator, which is assumed to be
parallel to the earth's equator. The equator is thus the principal plane of refer-
ence. Other planes of reference are the plane of the horizon, which is perpen-
dicular to the plumb line at the place ; the meridian, which is a plane through
the place and the earth's axis of rotation ; the prime-vertical, which is a vertical
plane at the place at right angles to the meridian ; and the ecliptic, which is a
plane parallel to the plane of the earth's orbit. These planes cut the surface of
the sphere in great circles called the equator, the horizon, the meridian, etc. The
points on the sphere defined by the intersection of the meridians, or the points
where the axis of the equator pierces the sphere, are called the poles. Similarly,
the prolongation of the plumb line upwards pierces the sphere in the zenith, and
its prolongation downwards pierces the sphere in the nadir. Great circles pass-
ing through the zenith are called vertical circles.
2. Spherical Co-ordinates.
a. Notation.
The position of a celestial body may be defined by several systems of co-ordi-
nates. The most important of these in practical astronomy are the azimuth
and altitude system and the hour angle and declination system. In the first of
these the azimuth of a star or other body is the angle between the meridian
plane of the place and a vertical plane through the star. It is measured, in gen-
eral, from the south around by the west through 360°. The altitude of a star is
its angular distance above the horizon, and its zenith distance is the complement
of the altitude. In the second system the hour angle of a star is the angle
between the meridian plane of the place and a meridian plane through the star.
It is measured towards the west through 360°. The declination of a star is its
angular distance above or below the equator ; the complement of the declination
is called the polar distance.
The angular distance of the pole above the horizon is equal to the zenith dis-
tance of the equator, or to the latitude of the place. Likewise, the altitude of
the equator and the zenith distance of the pole are each equal to the comple-
ment of the latitude at any place.
Digitized by VjOOQIC
Ixviii ASTRONOMY.
These quantities are usually designated by the following notation : —
A = the azimuth of a star or object,
h=^\Xs altitude,
z = its zenith distance = 90° — ^,
/ = its hour angle,
3 = its declination,
/ = its polar distance = 90° — 8,
^ = the parallactic angle, or angle at the star between the pole and the
zenith,
^ = the latitude of the place of observation.
b. Altitude and azimuth in terms of declination and hour angle.
The fundamental relations for this problem are —
sin ^ = sin ^ sin 8 -|- cos ^ cos 8 cos /,
cos h cos ^ = — cos ^ sin 8 -|- sin ^ cos 8 cos /, (i)
cos ^ sin ^ = cos 8 sin /.
When it is desired to compute both A and h by means of logarithms, the most
convenient formulas are,
m sin ^ = sin 8, tan 8
m cos M^=> cos 8 cos /, cos f
. • y . .^ ., tan / cos M , ^
sin Az=m cos (^ — Jf), tan A = ^-^ (^^AfY ^^^
cos Acqs A =im sin (ff> — J/), ^ __ cos ^ ^
cos A sin A ^= cos 8 sin /, tan (0 — Af}'
A > 180** when / > 180** and A < 180° when / < 180**.
For the computation of A and z separately, the following formulas are useful :
sin /
^ cot 8 cos /)
(3)
tznA— ^^g ^ ^^^ g ^ J _ ^^^ ^ ^^^ g ^^^ ^^
<7 sm /
1 — d cos /'
where
a = sec cot 8, ^ = tan ^ cot 8.
Formulas (3) are especially appropriate for the computation of a series of
azimuths of close circumpolar stars, since a and ^ will be constant for a given
place and date.
cos « = cos (0 '^ 8) — 2 cos 4^ cos 8 sin* i /,
sin« ^z = sin" H<^ '^ 8) + cos <^ cos 8 sin* i /,
(<^ ^ 8) = <^ — 8, for <^ >S ^"^^
= 8 — <^, for ff>< 8.
Digitized by VjOOQIC
ASTRONOMY. Ixiz
For logarithmic application of (4) we may write
n^ z= cos ^ cos 8, «* = sin* i (^ '^ 8),
tan N— - sin i /, (s)
/I
cos J\^ sm iV^''*" »
c. Declination and hour angle in terms of altitude and azimuth.
The fundamental relations for this case are
sin 8 = sin ^ sin A — cos ff> cos A cos A,
cos 8 cos / = cos ^ sin ^ -|- sin ^ cos A cos A, (i)
cos S sin / = cos A sin A,
For logarithmic computation by means of an auxiliary angle Mone may write
w sin M= cos A cos A, tan Jl/"= cot >4 cos A^
m cos il/rr: sin Af
sin 8= « sin (^-^, tan / = tan^sin^ ^^^
cos (9 — il/ )
COS S cos t-=.tn cos (<^ — J!/^,
cos 8 sin / = cos A sin -^, tan 8 = tan (^ — M) cos /.
d. Hour angle and azimuth in terms of zenith distance.
^^« * cos J? — sin <i sin 8
cos / = -^-— .
cos <^ cos 6
^^,^^^sin(^-.^)cos(o--8) ^ = j(^_|_a + ,).
cos <r cos (<r — «) a v-r I i /
COS ^_ sin<^cos^-sin8
COS ^ sin «
COS <r sm (<r — 6) ^ \^ \ 1 /
e. Formulas for parallactic angle.
cos = sin 8 sin <^ -^ cos 8 cos ^ cos /,
sin * cos q = cos 8 sin ^ — sin 8 cos ^ cos /,
sin ^ sin ^ = cos <^ sin /,
sin 8 = cos « sin ^ -|- sin z cos ^ cos /,
cos 8 cos ^ = sin s sin ^ -(" ^^^ s cos ^ cos -«4,
cos 8 sin ^ ^ cos <^ sin A,
(0
Digitized by VjOOQIC
IXX ASTRONOMY.
The first three of these are adapted to logarithmic computation as follows : —
n sin iV= cos ^ cos /,
n cos iV= sin ^,
cos ;5 = » sin (S -|- iVQ,
sin z cos q=^n cos (8 -|- iV),
sin 2 sin ^ = cos <^ sin //
whence
tan iV= cot ^ cos /,
4.«« . -;« tan / sin ^ / >,
tan if cos ^ = cot (8 + W).
A similar adaptation results for the last three of equations (i) by interchanging
S and jr. The equations (2) give both z and q in terms of ^ ^ and /, without
ambiguity, since tan z is positive for stars above the horizon.
If A^ Zy and q are all required from ^, 8, and /, they are best given by the
Gaussian relations
sin J ir sin ^A + ^) = sin J / cos i(^ + 8),
sin ^ z cos i{A + j^) = cos i / sin J(</> — 8), . .
cos ^ « sin ^A — ^) = sin J / sin J(^ + 8),
cos J z cos ^A ^ q) = cos J / cos i{</> — 8).
f. Hour angle, azimuth, and zenith distance of a star at elongation.
In this case the parallactic angle is 90^ and the required quantities are given by
the formulas
tan <t>
cos / =
tan 8'
cos z =
cos 8
cos <l>
sin <^
sin A = -— -T» (i)
cos 9 ^ ^
sm
When all of the quantities /, Ay and z are to be computed the following formulas
are more advantageous : —
JP = sin (8 + <t>) sin (8 - t^),
IT K K
sin t = -zZTTT^rv cos A = r— -p sin z = ^. ^» (2)
cos 9 sm 6 cos 9 sm 6 ^ ^
K , cos8 K
^VLt'=z , , ;. » tan ^ = — T?-» tan j?= 1^—7-
sm 9 cos o A. sm 9
g. Hour angle, zenith distance, and parallactic angle for transit of a
star across prime vertical.
In this case the azimuth angle is 90"^ and the required quantities are given by
the formulas
Digitized by VjOOQ IC
ASTRONOMY. Izzi
tan S
sin 8
cos « = ^. .> (i)
sin 9 ^ '
cos d} .
or, if all of them are to be computed, by the formulas
AT" = sin (^ + 8) sin (^ - 8),
^ . Jir K
sin / = -: — 1^ 5» sm z = -^ — ~j f cos ^ = m
sm </> cos o sm ^ ^ cos o
K ^ K cos«^
^"'= cos sins' ^*"^ = inr8' tan^ = -;g--
For special accuracy the following group will be preferred : —
sin (^ — S)
(2)
sin
T*+^
(3)
,,„a,, tan K<^ - 8)
tann^=tanK<A + «)
tan* (45 ° - i ^) = tan K<^ + 8) tan K* - 8).
h. Hour angle, and azimuth of a star when in the horizon, or at the
time of rising or setting.
In this case the zenith distance of the star is 90°, and the required quantities
are given by
cos / = — tan ^ tan 8,
sin 8
cos -4 = — —--7 ;
cos 9 '
or by '
cos
WV^I
tanH^-'-^-^^5£^^^^-±^.
On account of refraction, the values of / and A given by these formulas are
subject to the following corrections, to wit : —
^^ ~" cos <^ cos 8 sin/' ^ "~ sin ^ ^'
where R is the refraction in the horizon. Thus the actual values of the hour
angle and azimuth at the time of rising or setting of a star are
/ + A/ and ^ + A^.
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Izxii ASTRONOMY.
L Differential formulas.
The general differential relations for the altitude and azimuth and the declina-
tion and hour angle systems of coordinates are : —
^j? = — cos ^ </S + sin q cos 8 ^/ + cos A </^,
sin z dA-=^ sin ^ </8 -|- cos q cos h dt — cos * sin A dif>.
(0
</8 = — cos q dz -{- sin q sin z dA + cos / dtj^^ /^\
cos hdt=. sin q dz-^- cos ^ sin z dA '\- sin 8 sin / d<f>.
The following values derived from (i) are of interest as showing the dependence
of g and ^ on / in special cases : —
(§) m
cos 8
For a star in the meridian = o, = r[Z~i'
For a star in the prime vertical = cos ^ = sin ^
For a star at elongation = cos 8, = o.
3. Relations of Different Kinds of Time used in Astronomy.
a. The sidereal and solar days.
The sidereal day is the interval between two successive transits of the vernal
equinox over the same meridian. The sidereal time at any instant is the hour
angle of the vernal equinox reckoned from the meridian towards the west from o
to 24 hours. The sidereal time at any place is o when the vernal equinox is in
tiie meridian of that place.
The solar day is the interval between two successive transits of the sun across
any meridian ; and the solar time at any instant is the hour angle of the sun at
that instant. The solar day begins at any place when the sun is in the meridian
of that place.
The mean solar day is the interval between two successive transits over the
same meridian of a fictitious sun, called the mean sun, which is assumed to move
uniformly in the equator at such a rate that it returns to the vernal equinox at
the same instant with the actual sun.
Time reckoned with respect to the actual sun is called apparent time, while
that reckoned with respect to the mean sun is called mean time. The difference
between apparent and mean time, which amounts at most to about iG**, is called
the equation of time. This quantity is given for every day in the year in
ephemerides.
The sidereal time when a star or other object crosses the meridian is called the
right ascension of the object. The right ascension of the mean sun is also called
the sidereal time of mean noon. This time is given for every day in the year in
ephemerides for particular meridians, and can be found for any meridian by allow-
ing for the difference in longitude.
The time to which ephemerides and most astronomical calculations are referred
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ASTRONOMY. IxxiU
is the solar day, beginning at noon, and divided to hours numbered continuously
from o* to 24\ This is called astronomical time ; and such a day is called the
astronomical day. It begins, therefore, 12 hours later than the civil day.
b. Relation of apparent and mean time.
A = apparent time = hour angle of real sun,
M=^ mean time = hour angle of mean sun,
£ = equation of time.
M=A']'JS.
In the use of this relation, £ may be most conveniently derived (by interpola-
tion for the place of observation) from an ephemeris.
c. Relation of sidereal and mean solar intervals of time.
/= interval of mean solar time,
/' = corresponding interval in sidereal time,
r = the ratio of the tropical year expressed in sidereal days to the tropical
year expressed in mean solar days
= 3^^:2^^=1.002738.
365.2422
/' = rl= /-I- (r - i) 7= 7+ 0.002738 7
7= r-^ 7' = 7' - (i - r-0 7' = 7' - 0.002730 7'.
Tables for making such calculations are usually given in ephemerides (see, for
example, the American Ephemeris). Short tables for this purpose are Tables
34 and 35 of this volume.
Frequent reference is made to the relations
24* sidereal time = 23* 56"* o4.*o9i solar time,
24* mean time = 24* 03"' 56/555 sidereal time.
d. Interconversion of sidereal and mean solar time.
Ti, = mean time at any place,
Tg = corresponding sidereal time,
= right ascension of meridian of the place,
A = right ascension of mean sun for place and date,
= sidereal time of mean noon for place and date.
Tg= A -{^ T^ expressed in sidereal time.
Ti, = (7i — -^ expressed in mean time.
The quantity A is given in the ephemerides for particular meridians, and can
be found by interpolation for any meridian whose longitude with respect to the
meridian of the ephemeris is known. The formulas assume that A is taken out
of the ephemeris for the next preceding mean noon. Digitized byLjOOQlC
Ixxiv ASTRONOMY.
e. Relation of sidereal time to the right ascension and hour angle
of a star.
T, = sidereal time at any place,
= right ascension of the meridian of the place,
a = right ascension of a star,
/ = the hour angle of the star at the time Tg,
4. Determination of Time.
a. By meridian transits.
A determination of time consists in finding the correction to the clock, chro-
nometer, or watch used to record time. If To denote the true time at any place
of an event, T the corresponding observed clock time, and A 7' the clock correc-
tion,
To = 7"+ at:
The simplest way to determine the clock correction is to observe the transit of
a star, whose right ascension is known, across the meridian. In this case the
true time TJ = a, the right ascension of the star ; and if T is the observed clock
time of the transit,
Ar=a— T.
Meridian transits of stars may be observed by means of a theodolite or transit
instrument mounted so that its telescope describes the meridian when rotated
about its horizontal axis. The meridian transit instrument is specially designed
for this purpose, and affords the most precise method of determining time.*
Since it is impossible to place the telescope of such an instrument exactly in
the meridian, it is essential in precise work to determine certain constants, which
define this defect of adjustment, along with the clock correction. These con-
stants are the azimuth of the telescope when in the horizon, the inclination of the
horizontal axis of the telescope, and the error of collimation of the telescope.t
Let
a = azimuth constant,
d = inclination or level constant,
c = collimation constant.
a is considered plus when the instrument points east of south ; d is plus when
the west end of the rotation axis is the higher; and c is intrinsically plus when
the star observed crosses the thread (or threads) too soon from lack of collima-
tion. (The latter constant is generally referred to the clamp or circle on the
horizontal axis of the instrument.)
* The best treatise on the theory and use of this instrument is to be found in Chauvenet's
Manual of Spherical and Practical Astronomy^ which should be consulted by one desiring to go
into the details of the subject.
t Other equivalent constants may be used, but those given are most commonly employed.
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ASTRONOMY. IxXV
Also let
^ =: latitude of the place,
S = declination of star observed,
a = right ascension of star observed,
2"=: observed clock time of star's transit,
A 7"= the clock correction at an assumed epoch To,
r = the rate of the clock, or other timepiece,
^ = ^^"i'tT^^ = the " azimuth factor,"
cos '
^ = """it 7^^ = the " level factor,"
cos O '
C= « = the " collimation factor."
cos
Then, when a, d, c are small (conveniently less than lo' each, and in ordinary
practice less than i' each),
r-f AT-^- ^fl -[. ^* + a -[. r (7-- 71) = a.
This is known as Mayer's formula for the computation of time from star transits.
The quantity Bb is generally observed directly with a striding level. Assuming
it to be known and combined with T, the above equation gives
/^T'\-Aa'\-Cc'{-r(T-' TJ) = o - T. (i)
This equation involves four unknown quantities, A 7^ a, r, and r; so that in
general it will be essential to observe at least four different stars in order to get
the objective quantity ATI Where great precision is not needed, the effect of the
rate, for short intervals of time, may be ignored, and the collimation c may be
rendered insignificant by adjustment. Then the equation (i) is simplified in
Ar+ Aa = a- T. (2)
This shows that observations of two stars of different declinations will suffice to
give A 71 Since the factor A is plus for stars south of the zenith (in north lati-
tude) and minus for stars north of the zenith, if stars be so chosen as to make the
two values of A equal numerically but of opposite signs, A 7* will result from the
mean of two equations of the form (2). With good instrumental adjustments
(d and c small), this simple sort of observation with a theodolite will give A7' to
the nearest second.
A still better plan for approximate determination of time is to observe a pair of
north and south stars as above, and then reverse the telescope and observe an-
other pair similarly situated, since the remaining error of collimation will be partly
if not wholly eliminated. Indeed, a well selected and well observed set of four
stars will give the error of the timepiece used within a half second or less. This
method is especially available to geographers who may desire such an approxi-
mate value of the timepiece correction for use in determining azimuth. It will
suffice in the application of the method to set up the instrument (theodolite or tran-
sit) in the vertical plane of Polaris, which is always close enough to the meridian.
The determination will then proceed according to the following programme : — ,
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IxXVi ASTRONOMY.
1. Observe time of transit of a star south of zenith,
2. Observe time of transit of a star north of zenith.
Reverse telescope,
3. Observe time of transit of another star south of zenith,
4. Observe time of transit of another star north of zenith.
Each star observation will give an equation of the form (i), and the mean of
the four resulting equations is
^4*4^ 4 4
Now the coefficient of r in this equation may be always made zero by taking
for the epoch TJ the mean of the observed times T, Likewise, ^A and %C may
be made small by suitably selected stars, since two of the A% and Cs are positive
and two negative. The value \ 2(a — T^ is thus always a close approximation to
A 7* for the epoch 7J = J 27; when ^A and 2C approximate to zero. But if these
sums are not small, approximate values of a and c may be found from the four
equations of the form (i), neglecting the rate, and these substituted in the above
formula will give all needful precision.
For refined work, as in determining differences of longitude, several groups of
stars are observed, half of them with the telescope in one position and half in the
reverse position, and the quantities ^T^ a^ c^ and r are computed by the method
of least squares. In such work it is always advantageous to select the stars with
a view to making the sums of the azimuth and collimation coefficients approxi-
mate to zero, since this gives the highest precision and entails the siruplest com-
putations.*
b. By a single observed altitude of a star.
An approximate determination of time, often sufficient for the purposes of the
geographer, may be had by observing the altitude or zenith distance of a known
star. The method requires also a knowledge of the latitude of the place.
Let
Zx = the observed zenith distance of the star,
R = the refraction,
z = the true zenith distance of the star,
= ^1 + ^,
a, 8, = the right ascension and declination of the star,
/ = hour angle of star at time of observation,
T-=. observed time when Zy^ is measured,
AT= correction to timepiece,
</» = latitude of place.
Then the hour angle / may be computed by
* COS <r cos \<T — z)
* For details of theory and practice in time work done according to this plan see Bulletin 49,
U. S. Geological Survey. Digitized by LjOOg IC
ASTRONOMY. Ixxvii
Having the hour angle the clock correction A 7* is given by
Ar=a + /— 7;
in which all terms must be expressed in the same unit; /. ^., in sidereal or in mean
time.
The refraction R may be taken from Table 31.
The most advantageous position of the star observed, so far as the effect of an
error in the measured quantity ^i is concerned, is in the prime vertical, but stars
near the horizon should be avoided on account of uncertainties in refraction.
The least favorable position of the star is in the meridian.
Compared with the preceding method the present method is inferior in preci-
sion^ but it is often available when the other cannot be applied.
c. By equal altitudes of a star.
This method is an obvious extension of the preceding method, and has the
advantage of eliminating the effect of constant instrumental errors in the meas-
ured altitudes or zenith distances. Thus it is plain that the mean of the times
when a (fixed) star has the same altitude east and west of the meridian, whether
one can measure that altitude correctly or not, is the time of meridian transit
This method may, therefore, give a good approximation to the timepiece
correction when nothing better than an engineer's transit, whose telescope can
be clamped, is available. When the instrument has a vertical circle (or when a
sextant is used) a series of altitudes may be observed before meridian passage of
the star, and a similar series in the reverse order with equal altitudes respectively
after meridian passage. The half sums of the times of equal altitudes on the two
sides of the meridian will give a series of values for the time of meridian transit
from which the precision attained may be inferred.
This method is frequently applied to the sun, observations being made before
and after noon. For the theory of the corrections essential in this case on
account of the changing position of the sun, on account of inequalities in the
observed altitudes, etc., the reader must be referred to special treatises on prac-
tical astronomy.*
5. Determination of Latitude.
a. By meridian altitudes.
The readiest method of determining the latitude of a place is to measure the
meridian zenith distance or altitude of a known star. When precision is not re-
quired this process is a very simple one, since it is only essential to follow a (fixed)
star near the meridian until its altitude is greatest, or zenith distance least. Thus,
if the observed zenith distance is ^i, the true zenith distance z^ and the refrac-
tion R^
• The best work of this kind is Chauvenet*s Manual of Spherical and Practical Astronomy^ It
should be consulted by all persons desiring a knowledge of the details of praqtioal astmnoiiMf.^Q j p
igi ize y ^
Ixxviii ASTRONOMY.
and if the declination of the star is S and the latitude of the place ^
according as the star b south or north of the zenith.
A more accurate application of the same principle is to observe the altitudes
of a circumpolar star at upper and lower culmination (above and below the pole).
The mean of these altitudes, corrected for refraction, b the latitude of the place.
This process, it will be observed, does not require a knowledge of the star's
declination.
b. By the measured altitude of a star at a known time.
A = measured altitude corrected for refraction,
Tg = observed sidereal time,
a, 3 = right ascension and declination of star,
/ = hour angle of star,
^ = latitude of place.
Then ^ may be computed by means of the following formulas : —
tan^ = ^ cosy = 5!ll*i^,
cos / ' sm 8
In the application of these P may be taken numerically less than 90% and since
/ may also be taken less than 90°, P may be taken with the same sign as 5. y is
indeterminate as to sign analytically, but whether it should be taken as positive
or negative can be decided in general by an approximate knowledge of the lati-
tude, which is always had except in localities near the equator.
The most advantageous position of a star in determining latitude by this
method is in the meridian, and the least advantageous in the prime vertical.
When a series of observations on the same star is made, they should be equally
distributed about the meridian ; and when more than one star is observed it is
advantageous to observe equal numbers of them on the north and south of the
zenith.
The application of this method to the pole star is especially well adapted to
the means available to the geographer and engineer, namely, a good theodolite
and a good timepiece. In this case the following simple formula for the latitude
may be used : —
^ = ^ — / cos / -}- i/* sin i" sin* / tan ^,
where/ is the polar distance of Polaris in seconds (about 5400"), and the other
symbols have the same meaning as defined above. Tables giving the logarithms
of/ and ip* sin i" are published in the American Ephemerb.
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ASTRONOMY. Izxix
c. By the zenith telescope.
The zenith telescope furnishes the most precise means known for the deter-
mination of the latitude of a place. For the theory of the instrument and method
when applied to refined work the reader must be referred to special treatises.*
It will suffice here to state the principle of the method, which may sometimes be
advantageously applied by the geographer. Let z, be the meridian zenith distance
of a star south of the zenith, and Zj^ the meridian zenith distance of another star
north of the zenith. Let 8, and 8. denote the declinations of these stars respec-
tively. Then
z. = <t>''S„
whence
It appears, therefore, that this method requires only that the difference (z, — z^)
be measured. Herein lies the advantage of the method, since that difference
may be made small by a suitable selection of pairs of stars. With the zenith
telescope the stars are so chosen that the difference (z, — z^ may be measured by
means of a micrometer in the telescope.
The essential principles and advantages of this method may be realized also
with a theodolite, or other telescope, to which a vertical circle is attached, the
difference (z, — z^) being measured on the circle ; and a determination of latitude
within 5" or less is thus easy with small theodolites of the best class (/. e., with
those whose circles read to 10" or less by opposite verniers or microscopes).
6. Determination of Azimuth.
a. By observation of a star at a known time.
T, = sidereal time of observation,
a, 8 = right ascension and declination of star observed,
/ = hour angle of star,
= r. - a,
^ = latitude of place,
A = azimuth of the star at the time T, counted from the south around by the
west through 360^
The azimuth A may be computed by the formulas
a = sec ^ cot 8, 3 = tan ^ cot 8,
a sin / (i)
The angle A will fall in the same semicircle as /, and A is thus determined by its
tangent without ambiguity. The quantities a and ^ will be sensibly constant for
* Among which Chauvenet's Manual of Spherical and Practical Astronomy is the best.
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IXXX ASTRONOMY.
a given star and date ; and hence they need be computed but once for a series of
observations on the same star on one date.
The effects of small errors A/, A<^, and AS in the assumed time, latitude, and
declination are expressed by
cos S cos ^ . sin^^_
— .^ ^ A/, — sm A cot * A^ -v—^ A8,
sin z ' ^ sm z ^^
respectively, where z and g are the zenith distance and parallactic angle of the
star. Hence the effect of A/ will vanish for a star at elongation ; the effect of
^4> vanishes for a star in the meridian, and is always small (in middle latitudes)
for a close circumpolar star ; the effect of AS vanishes for a star in the meridian.
It appears advantageous, therefore, to observe for azimuth (in middle latitudes)
close circumpolar stars at elongations, since the effect of the time error is then
least, and the effects of errors in the latitude and declination are small and may
be eliminated entirely by observing the same star at both elongations.
The hour angle /«, the azimuth A„ and the altitude h^ of a star at elongation
are given by the formulas (2) of section 2,f. Those best suited to the purpose
are
K^ = sin (8 + 4>) sin (S - <^),
AT ^ ^ cos S , sin <j^ (2)
*'"<^ = sin^cos8 ' tan^. = -^. tan>». = -^- W
Knowing the time of elongation of a close circumpolar star, it suffices for many
purposes to observe the angle between the star and some reference terrestrial
mark at or in the vicinity of that time.
For precise determinations of azimuth it is customary to observe a star near
its elongation repeatedly, thus obtaining a series of results whose mean will be
sensibly free from errors of observation and errors due to instrumental defects.
The computation of the azimuth A may be made accurately in all cases by the
formulas (i) ; but when a close circumpolar star is observed near elongation, it
may be more convenient to use the following formulas : —
A/ = (/ — /,), or the interval before or after elongation at the time of
observation,
AA = (A — A^), or the difference in azimuths of the star at the time
of elongation and at the time of observation, (3)
_ (isV* sin S cos S Oi)!. sin S cos 8
^^ — 2p" sin/, cos ^^^^^^ =*= 2 (p'7 sin /. tan /. cos 1^ ^^^^
* To the same order of approximation one may write in the first term of this expression
which latter is the most convenient form when tables giving log — ^-^ ^„ — for the argument Af
in time are at hand. Such tables are given in Chauvenet's Manual of Sphitical and Practical
Astronomy (for full title see p. Ixxxii), and in Formeln und Hulfstafeln JUr Geographische Oris-
bcstimmungen^ von Dr. Th. Albrecht Leipzig: Wilhelm Engelmann, 4to, 2d ed., 1879.
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ASTRONOMY. Ixzzi
This last formula gives AA in seconds of arc when A/ is expressed in seconds
of time ; A/ is considered positive in all cases (in the use of the formula), and
with this convention the positive sign is used when the star is between either
elongation and upper culmination, and the negative sign when the star is between
either elongation and lower culmination. For a given star, place, and date the
coefficients of (A/^)' and (A/*)* will be sensibly constant and their logarithms will
thus be constant for a series of observations of a star on any date. By reason of
the large factors (p" = 206 264."8)' and tan /« in the denominator of the second
term, it will be very small unless A/* is large. Hence this term may often be
neglected. Using both terms, the formula will give XA for Polaris to the nearest
o."oi when A/ < 40"* and when observations are made in middle latitudes.
By reference to formulas (2) of section 2,/, it is seen that
sin
sin 8 cos 8 sin* S cos 8
sin /, cos <t> II *
sin S cos 8 sin* 8 cos* 8 sin <!>
/, tan /e cos <^"~ Z* *
^* = sin (8 + <t>) sin (8 - ^).*
b. By an observed altitude of a star.
A = true altitude of star observed ; /. ^., the observed altitude less the refrac-
tion,
^ = latitude of place,
p = polar distance of star,
A = azimuth of star.
tan» iA = sin(o--<A)sin(o-->4)
COS cr C0S(<7 — /)
The most advantageous position of the star, on account of possible error in the
observed value of A, is that in which sin A is a, maximum. This position is then
at elongation for stars which elongate, in the prime vertical for stars which cross
this great circle, and in the horizon for a star which neither elongates nor crosses
the prime vertical. A star will elongate when / < 90° — <^ ; it will cross the
prime vertical when/ lies between 90° — <;^ and 90° ; and it will neither elongate
nor cross the prime vertical when/ >9o°, or when the declination (8) of the star
is negative.
c. By equal altitudes of a star.
By this method, when a fixed star is observed first east of the meridian and
then west of the meridian at the same altitude, the direction of the meridian will
* In precise work the computed azimuth requires the following correction for daily aberration,
namely : —
A ^ ,/ cos ^
A^ =-0.-32 ^j^ cos ^,
where ^ is to be reckoned from the south by way of the west through 360^.
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IzXXii ASTRONOMY.
obviously be given by the mean of the azimuth circle readings for the two
observed directions. This process will thus give the direction of the meridian
free from the effect of any instrumental errors common to the equal altitudes
observed. Neither does it require any knowledge of the star's position (right
ascension and declination). It is therefore available to one provided with no-
thing but an instrument for measuring altitudes and azimuths, and is susceptible
of considerable precision when a series of such equal altitudes is carefully referred
to a terrestrial mark.
When the sun is observed, it is essential to take account of its change in
declination between the first and the second observation. Let Ai and A^ be the
true azimuths counted from the meridian toward the east and west respectively
at the times /i and /^ of the two observations. Also, let AS be the increase in
declination of the sun in the interval (/« — /i). Then
cos ^ sin ^/t — /i)
Calling the azimuth circle readings for the east and west observations J?i and J?t
respectively, the resulting azimuths are
References.
Many excellent treatises on spherical and practical astronomy are available.
Among these the most complete are the following : —
^' A Manual of Spherical and Practical Astronomy," by William Chauvenet.
Philadelphia: J. B. Lippincott & Co., 2 vols., 8vo, 5th ed., 1887. "A Treatise
on Practical Astronomy, as applied to Astronomy and Geodesy," by C. L. Doo-
little. New York: John Wiley & Sons, 8vo, 2d ed., 1888. "Lehrbuch der
Spharischen Astronomic," von F. Briinnow. Berlin : Fred. Diimler, 8vo, 185 1.
" Spherical Astronomy," by F, Briinnow. Translated by the author from the
second German edition. London : Asher & Co., 8vo, 1865.
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THEORY OF ERRORS.
I. Laws of Error,
The theory of errors is that branch of mathematical science which considers the
nature and extent of errors in derived quantities due to errors in the data on
which such quantities depend. A law of error is a relation between the magni-
tude of an error and the probability of its occurrence. The simplest case of a
law of error is that in which all possible errors (in the system of errors) are
equally likely to occur. An example of such a case is had in the errors of
tabular logarithms, natural trigonometric functions, etc. ; all errors from zero to
a half unit in the last tabular place being equally likely to occur.
When quantities subject to errors following simple laws are combined in any
manner, the law of error of the quantity resulting from the combination is in
general more complex than that of either component.
Let € denote the magnitude of any error in a system of errors whose law of
error is defined by <]^(€). Then if € vary continuously the probability of its
occurrence will be expressed by <fi(i)de. If c vary continuously between equal
positive and negative limits whose magnitude is a^ the sum of all the probabili-
ties ^(c)^€ must be unity, or
— a
For the case of tabular logarithms, etc., alluded to above, ^c) = r, a constant
whose value is 1/(2 a) = i, since a = 0.5.
For the case of a logarithm interpolated between two consecutive tabular
values, by the formula zr = t^i + (z'l — Vi) / = z^i (i — /) + z'j /, where Vi and
Vf are the tabular values, and / the interval between Vj and the derived value
V, ^(c) has the following remarkable forms when the extra decimals (practically
the first of them) in (v^ — z/j) / are retained : —
^*^ ^^ /. J7/\ f for values of c between — ^ and — (i — /),
= _ . for values of c between — (i — /) and + (J — /)> (^)
Google
= ^ \^ A . for values of c between + (i — /) and -|- J.
Digitized by^
IxXXiv THEORY OF KRRORS.
It thus appears that <]^c) in this case is represented by the upper base and the
two sides of a trapezoid.
When, as is usually the practice, the quantity (v^ — Vi) t is rounded to the
nearest unit of the last tabular place, ^(c) becomes more complex, but is still
represented by a series of straight lines. It is worthy of remark that the latter
species of interpolated value is considerably less precise than the former, wherein
an additional figure beyond the last tabular place is retained.
When an infinite number of infinitesimal errors, each subject to the law of con-
stant probability and each as likely to be positive as negative, are combined by
addition, the law of the resultant error is of remarkable simplicity and generality.
It is expressed by
where e is the Napierian base, ^ = 3. 141 59 -f-, and A is a constant dependent on
the relative magnitude of the errors in the system. This is the law of error of
least squares. It is the law followed more or less closely by most species
of observational errors. Its general use is justified by experience rather than
by mathematical deduction.
a. Probable, mean, and average errors.
For the purposes of comparison of different systems of errors following the
same law, three different terms are in use. These are ihtprobabU error* or that
error in the system which is as likely to be exceeded as not ; the nuan error^ or
that error which is the square root of the mean of the squares of all errors in the
system ; and the average error, which is the average, regardless of sign, of all
errors in the system. Denote these errors by ^ €^, €„ respectively. Then in all
systems in which positive and negative errors of equal magnitude are equally
likely to occur, and in which the limits of error are denoted by — a and -f- a, the
analytical definitions of the probable, mean, and average errors are : —
-a -cp o +€,
+ tf -\-a
a
• The reader should observe that the word probable is here used in a speciallj techiucal sense.
Thun, the probable error is not " the most probable error," nor " the most probable Talne of the
actual error," etc., as commonly interpreted.
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THEORY OF ERRORS. boXV
b. Probable, mean, average, and maximum actual errors of interpo-
lated logarithms, trigonometric functions, etc.
When values of logarithms, etc., are interpolated from numerical tables by means
of first differences, as explained above, the probable and other errors depend on
the magnitude of the interpolating factor. Thus, the interpolated value is
where Vi and v^ are consecutive tabular values and / is the interpolating factor.
For the species of interpolated value wherein the quantity (v2 — Vi) t is not
rounded to the nearest unit of the last tabular place (or wherein the next figure
beyond that place is retained) the maximum possible actual error is 0.5 of a unit
of the last tabular place, and formulas (i) and (3) show that the probable, mean,
and average errors are given by the following expressions : —
Cp = i (i — /) f or / between o and i,
= J — J V2/ (i— /) for / between \ and §,
= J / for / between | and i.
•-- I 96(1-/)/ \
I — (l — 2/)»
€« = J ^7 for / between o and J,
24 ^^i — I) I
I — (2/— 1)» ^ ^^ , J
^^ / _ A / for ^ between \ and i.
It thus appears that the probable error of an interpolated value of the species
under consideration decreases from 0.25 to 0.15 of a unit of the last tabular place
as / increases from o to 0.5. Hence such interpolated values are more precise
than tabular values.
For the species of interpolated values ordinarily used, wherein (z/j — v^ t is
rounded to the nearest unit of the last tabular place, the probable, mean, and
average errors are greater than the corresponding errors for tabular values. The
laws of error for this ordinary species of interpolated value are similar to but in
general more complex than those defined by equations (i). It must suffice here
to give the practical results which flow from these laws for special values of the
interpolating factor /.* The following table gives the probable, mean, average,
and maximum actual error of such interpolated values for /= i, i, J, . . . iV* It
will be observed that / = i corresponds to a tabular value.
* For the theory of the errors of this species of interpolated values see Annals of Mathematics^
▼oL ii. pp. 54-59.
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GooqIc
IxXXVi THEORY OF ERRORS.
Characteristic Errors of Interpolated Logarithms^ etc.
Interpolating
factor
/
Probable
error
Mean
error
Average
error
Maximam actual
error
I
0.250
0.289
0.250
\
.292
.408
•333
I
.256
•347
.287
»
.276
.382
•313
I
.268
•370
•303
ft
.277
.385
•315
I
.274
.380
•3"
H
1 •
.279
.389
.318
I
.278
.386
.316
H
tV
.281
•392
.320
I
2. The Method of Least Squares.
a. General statement of method.
When the errors to which observed quantities are subject follow the law ex-
pressed by
a unique method results for the computation of the most probable values of the
observed quantities and of quantities dependent on the observed quantities. The
method requires that the sum of the weighted squares of the corrections to the
observed quantities shall be a minimum,* subject to whatever theoretical condi-
tions the corrections must satisfy. These conditions are of two kinds, namely,
those expressing relations between the corrections only, and those expressing
relations between the corrections and other unknown quantities whose values are
disposable in determining the minimum. A familiar illustration of the first class
of conditions is presented by the case of a triangle each of whose angles is mea-
sured, the condition being that the sum of the corrections is a constant An
equally familiar illustration of the second class of conditions is found in the case
where the sum and difference of two unknown quantities are separately observed ;
in this case the two unknowns are to be found along with the corrections.
Mathematically, the general problem of least squares may be stated in two
* Hence the term least squares.
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THEORY OF ERRORS. IxXXVU
equations. Thus, let Xyy,z,. . . be the observed quantities with weights /, ^,
r, . . . . Let the corrections to the observed quantities be denoted by Ax, Ay,
A«, - . . ; so that the corrected quantities are x + Ajc, y -|- ^y, ;s + ^» • • • • Let
the disposable quantities whose values are to be determined along with the correc-
tions be denoted by ^, 17, {, . . . . Then, the theoretical conditions which must be
satisfied by jp + Ajc, _y -[" ^>'» ^ 4" ^> • • • ^^^ ^Y f » ^> f > • • • ^^V ^ symbolized
by
^niiyVfif'^ + ^fy-^ Ay, Z + ^Z,...) = 0. (4)
Subject to the conditions specified by the n equations (4), we must also have
/ (Ao:)* -[- ^ (Ayy + ^ i^y + • • • = * minimum (5)
= Uy say.
Equations (4) and (5) contain the solution of every problem of adjustment by
the method of least squares. Two examples may suffice to illustrate their use.
First, take the case of the observed angles of a triangle alluded to above.
Calling the observed angles x, y, z, we have
a: + Ajc-[-J' + A^ + ^ + ^= 180° -|- spherical excess,
or
Ao: + A^ 4- Ajj = 180° -|- spherical excess — (x -[" ^^ + *)
= Cy say.
This is the only condition of the form (4). The problem is completely stated,
then, in the two equations
Ax "I" Ay + ^ = ^
/ (Ax)* + ^ (A_y)* + r (A^:)" = a min. = «.
To solve this problem the simplest mode of procedure is to eliminate one of the
corrections by means of the first equation and then make u a minimum. Thus,
eliminating A5, there results
u =p (Ax)« + ^ (Aj/)« 4- r (^ - Ax - Aj^)«.
The conditions for a minimum of 2^ are : —
^ = 0> + r) Ax + r Ajf - rr = o,
9U A I / I \ A
5a^ '— I w I v-^
— rt.
V,
and these give, in connection with the value A« =
zc-
- Ax - Aj^,
where
Aj?
= 2.
r'
Q— '
" - + - + -'
When the weights are equal, or when / = ^ == r, the corrections are —
A* = Ay = A^ = i A Digitized byGoOQle
boxviii THEORY OF ERRORS.
Secondly, take the case, also alluded to above, of the observed sum and t±ie
observed difference of two numbers. Denote the numbers by ( and 17, the latter
being the smaller. Let the observed values of the sum (( -^ 17) be denoted
by ^1, ^ . . . x^ and their weights /i, /j, . . • /« respectively. Likewise, call
the observed values of the difference (£ — 17), yy, >'2, . . . y^ and their weights
^1, ^8 • . . ^» respectively. Then there will he m -{- n equations of the type (4),
namely : —
( + V-(^i + ^1) = o,
f + ^ — (^a + ^^2) = o,
i + v — (^«+ A««)= o,
(a)
i- ^-0» + Ay,) =0;
and the minimum equation is
«=A(^i)'+A(^^2)' + ... + ^i(Ari)' + ^i(Aj'2)" + ... = amin. (b)
The equations of group (a) give
A^l = f + 17 — dTi,
(c)
Ar2 = f - »; - j'a,
• • • i
and these values in (b) give
«=A(.i+r,-x,y-\-...-\-f,(i-r,-J>,y^... (d)
Thus it appears that all conditions will be satisfied if i and 77 are so determined
as to make u in (d) a minimum. Hence, using square brackets to denote sum-
mation of like quantities, the values of $ and 77 must be found from
|? = I> + ^]£ + |>-y],-|>* + iy] = o.
|^ = [j) - ^] £+ I> + y] ,-[/*- 4^] = 0.
Equations (e) give i and 17, and these substituted in (c) will give the corrections
to the observed quantities.
b. Relation of probable, mean, and average errors.
The introduction of the law of error (2) in equations (3) furnishes the following
relations, when it is assumed that the limits of possible error are —00 and -^ 00 :
€p = 0.6745 €« = 0.8453 'a- (6)
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THEORY OF ERRORS. Ixxxix
c. Case of a single unknown quantity.
The case of a single unknown quantity whose observed values are of equal or
unequal weight is comprised in the following formulas : —
XiyXi, . . . Xj^=z observed values of unknown quantity,
A A • • • /m = the weights of x^, x^, . . .
Vij Vf, . . . Vj^ = most probable corrections to Xi, x^ , , ,
X = most probable value of the unknown quantity,
m == the number of independent observations.
Then the conditional equations (4) are
X — j^a = ^«»
X — x^=.v^\
the minimum equation (5) is
P^\ -^rPt^t + . . . = {pv^ = lp(x — Xi)^ = a min.,
where / = i, 2, . . . »j, and
When the weights are equal, /i =A = . . , = A» ^^d
^~ IP
or the arithmetic mean of the observed values.
Weight of jc = [/] when the/'j are unequal,
= m when the/'j are equal.
Mean error of an observed value of weight unity = y *- ^^ for unequal weights,
=y _; • for equal weights.
Mean error of an observed value of weight/ = y > ^^ -'. for unequal weights.
Mean error of :c = y 7 — ^f_ Jr -. for unequal weights,
j::^ 4/ — li!L! — ^ for equal weights,
y pt {fn —■ 1}
The corresponding probable errors are found by multiplying these values by
0.6745. See equation (6). Digitized by LjOOQ IC
XC THEORY OF ERRORS.
A formula for the average error sometimes useful is
Average error = 4/ / _ i^ T^ ^^^ unequal weights.
= 7n^ ' — ^ for equal weights.
In these the residuals v are all taken with the same sign. A sufficient approzi-
mation in many cases of equal weights is ^^-^ ; but the above formulas dependent
on the squares of the residuals are in general more precise.
An important check on the computation of x is [/?'] = o ; /. ^., the sum of the
residuals v, each multiplied by its weight, is zero if the computation is correct
d. Case of observed function of several unknown quantities ^, 17, { - . . -
A case of frequent occurrence, and one which includes the preceding case, is
that in which a function of several unknown quantities is observed. Thus, for
example, the observed time of passage of a star across the middle thread of a
transit instrument is a function of the azimuth and collimation of the transit
instrument and the error of the timepiece used. In cases of this kind the con-
ditional equations of the type (4) assume the form
-F(fiiy,{.... ^ + A^) = o;
that is, each of them contains but one observed quantity x along with several
disposable (disposable in satisfying the minimum equation) quantities ^, 17, ([ . . . •
The process of solution in this case consbts in eliminating the corrections
Ajti, ^x^ . . . from the above conditional equations, substituting their values in
the minimum equation (5), and then placing the differential coefficients of u with
respect to f , 17, i . . . separately equal to zero. There will thus result as many
independent equations as there are unknown quantities of the class in which ^, 17,
C . . . fall, the remaining unknown quantities Ajc^, ^LXf, . . . , or the corrections to
the observed values, are then found from the conditional equations.
In many applications it happens that the conditional equations
^(ij ^, f, . . . :r + Aj:) = o,
are not of the linear form. But they may be rendered linear in the following
manner. First, eliminate the quantities jc -(- Ajp from the conditional equations.
The result of this elimination may be written
/(if 17, f ...) — «» — A^ = o.
Secondly, put
where fof ifc, • • • are approximate values of f, 17, ... , found in any manner, and
Af, Aiy, ... are corrections thereto. Then supposing the approximate values
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THEORY OF ERRORS. xa
^ 19^ . . . SO close that we may neglect the squares, products, and higher powers
of A^, A?;, . . . , Taylor's series gives
/(fo, -to £...•)+ ^Af + U ^-7 + If ^£+ •••-*- ^ = 0.
which is linear with respect to the corrections Af, A17, . . . . For brevity, and for
the sake of conformity with notation generally used, put
zi = Ar,
^_5/ ._5/ ,-^l
x = ^(, y = ^Vf ^ = Af,
Then the conditional equations will assume the form
ax -^ dy -^ cz '{' . . , — n=zv;
and if they are m in number they may be written individually thus : —
aix -\- ^ly + ^i« + . . . — «i = ^i»
W
am -{- ^m + r^ + . . . - n^=v^.
The minimum equation (5) becomes
« = \jv^ = [>(a:p + ^_y + ^ +...-«)*] ;
so that placing -^, -^v^, -^, . . . separately equal to zero will give as many
dx dy dz
independent equations as there are values oi x^y^ z, , . , . The resulting equa-
tions are in the usual (Gaussian) notation of least squares : —
[paa]x + [pad'jy + [P^c] ar + . . . — [pan] = o,
[pad] + [pdd] 4- O^^] + . . . - [pdn] = o, (b)
[pac] +[>^^] +|>^^] +...-[/^] = o,
The equations (a) are sometimes called observation-equations. The absolute
term n is called the observed quantity. It is always equal to the observed quan-
tity minus the computed quantity/ (fo, vot i -- •)» which latter is assumed to be
free from errors of observation. The term v is called the residual. It is some-
times, though quite erroneously, replaced by zero in the equations (a).
The equations (b) are called normal equations. They are usually formed
directly from equations (a) by the following process : Multiply each equation by
the coefficient of x and by the weight/ of the v in the same equation, and add
the products. The result is the first equation of (b), or the normal equation in x.
The normal equations in ^, ^r, . . . are found in a similar manner.
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ZCii THEORY OF ERRORS.
A noteworthy peculiarity of the normal equations is their symmetry. Hence in
forming equations (b) from (a) it is not essential to compute all the coefficients of
jc, ^, JBT, . . . except in the first equation.
Checks on the computed values of the numerical terms in the normal equations
are found thus : Add the coefficients a, ^, r, . . . of x, y, z^ . . . in (a) and put
«i + ^1 + ^1 + • • • = -^b
d^j + ^s + ^a + • • • = -^a*
Multiply each of these, first, by its pa; secondly, by its/^, etc., and then add the
products. The results are
Ipaa] + [pad] + |>^] + . . . = [pas]
[pad] + [pdd] + [pdc] -\-... = [pds]
These will check the coefficients of x,y,z,,.. in (b). To check the absolute
terms, multiply each of the above sums by its np, and add the products. The
result is
[pan] + [pdn] + [pen] + . . . = [psn],
which must be satisfied if the absolute terms are correct.
Checks on the computation of x, j', -sr, . . . from (b) and of Vi, v^ . . . from (a)
are furnished by
[pav] = o, [pbv] = o, [pcv] = o,
To get the unknowns x, y, z, and their weights simultaneously, the best method
of procedure is, in general, the following : For brevity replace the absolute terms
in (b) by A, B, C, . , . respectively. Then the solution of (b) will be expressed
by
^^ = oa + A + 72 + • • • » (c)
z = a^ +A +78 +.
9
in which aj, /Si, 71, . . . are numerical quantities ; and
weight oix = —y
weight of j^=-g»
weight of -gr == — >
7«
(d)
To compute mean (and hence probable) errors the following formulas apply : —
m = the number of observed quantities n
= number of equations of condition,
/JL = number of the quantities x^y, z, . . .
€^ = mean error of an observed quantity («) of weight unity,
€p = corresponding probable error = 0.6745 €„.
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THEORY OF ERRORS. XCIU
€^=zy L-^^ for unequal weights,
= i/ L^ for equal weights,
y m — fi
Mean error of any observed quantity (n) of weight/ = -j~*
Mean error of :p = €„ ^^,
Mean error of >' = €„ ^^
Mean error of 5 = €„ ^^
where a^, /S^, ya, • . . are defined by equations (c) and (d) above.
e. Case of functions of several observed quantities x, y, z, . . . .
This case is that in which the conditional equations (4) contain no disposable
quantities f , 17, i, . . . . It is the opposite extreme to that represented by the case
of the preceding section.* It finds its most important and extensive application
in the adjustment of triangulation, wherein the observed quantities are the angles
and bases of the triangulation, and the conditions (4) arise from the geometrical
relations which the observed quantities p/us their respective corrections must
satisfy.
An outline of the general method of procedure in this case is the following : —
The first step consists in stating the conditional equations and in reducing
them to the linear form if they are not originally so. The form in which they
present themselves is (4) with f , 1;, ^, . . . suppressed, or
wherein x^y, z, . . . of (4) are replaced by Xi, x^ Xg . . , for the purpose of sim-
plicity in the sequel. If this equation is not linear, Taylor's series gives
/^ (^1, ^ ^8 . . . ) + 5^ ^1 + -5^^ A^ = . . . = O,
since the method supposes that the squares, products, etc., of A:ri, ^x^ . . . may
be neglected. The last equation is then linear with respect to the corrections
AoTi, Aj^ . . . which it is desired to find.
For brevity put
F{xi, ^, ^3 . . . ) = ^1, a known quantity, .
9F 9F dF
Then the conditional equations will be of the type
• The middle ground between these extremes has been little explored ; indeed, most practical
applications fall at one or the other of the extremes.
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XCIV THEORY OF ERRORS.
There will be as many equations of this type as there are independent relations
which the quantities Xi -|- ^^u ^ -(~ ^^Sf • - • ii^ust satisfy. Suppose there are Jk
such relations, and let the differential coefficients 9l^/9xi, 9F/9x2, ... for the sec-
ond relation be denoted by ^i, ^a> ^s» • • • i ^or the third relation hy Ci^ Cf, c^ . . . ,
etc. Then all of the conditional equations may be written thus :
ai^Xi + 02^X2 + «3^^8 + • • • +^1=0,
^1 + ^2 + ^S + • • • + ^2 = 0» (a)
^l +^i + ^8 + • • • + ^8 = 0»
• • • >
the number of these equations being k.
Call the weights of the observed quantities jcj, :cs, . . . /i, /i, . . . . Then, sub-
ject to the conditions (a) we must have (in accordance with (5))
u = A(A*i)' +M^^ + . . . = [/(A*)*] (*)
a minimum.
Equations (a) and (H) contain the solution of all problems falling under the
present case. Obviously, the number of conditions (a) must be less than the
number of observed quantities x, or less than the number of ^x*s in (d) ; in other
words, if m denote the number of observed quantities, m > k,ioT ii m'^ k the
minimum equation (d) has no meaning.
The question presented by (a) and (ff) is one of elimination only. Two methods,
the one direct and the other indirect, are available. Thus, by the direct method
one finds from (a) as many Ax's as there are equations (a), or Jk such values, and
substitutes them in (^). The remaining (m — k) values of Ax in (d) may then be
treated as independent and the differential coefficients of u with respect to each
of them placed equal to zero. Thus all of the corrections Ax become known.
By the indirect process, one multiplies the first of equations (a) by a factor ^1,
the second by Q2, the third by ^5, . . . and subtracts the difiFerential (with respect
to the Ax's) of the sum of these products from half the differential of (d). The
result of these operations is
^du={fiiAx^ -(«i<2i + ^iG + ^i<23 + ..0}^^i
+ {/2^ -(«2Gl + ^2C2 + ^2G8+...)}^^^
+ ...
+ {/«AXm - (a^Ql + ^«G + ^mG + •••)} ^^^
Now we may choose the factors G> 02» • • • Gt i" such a way as to make k of the
coefficients of the differentials in this equation disappear ; and after thus elimi-
nating k of these differentials we are at liberty to place the coefficients of the
remaining (m — k) differentials equal to zero. Thus all conditions are satisfied
by making
^iQi + ^iG + ^iG + • • • — PA^i = o,
a% 4-^2 + ^2 + • • • — P^f^2 = o,
if)
«m + ^« +^m + • • • — Pm^^m = ©;
and the values of the corrections will be given by these equations when the fac-
tors G, Qa, . . . are known. To find the latter it suffices to substitute the values
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THEORY OF ERRORS.
XCV
of A^Ty Ajc^ . . . from (c) in (a), whereby there will result fc equations containing
the Q], Qi . . . Qk alone as unknowns. The result of this substitution is
[f]a+[^]a+[f]a+- ■+».=°>
■ab-
.P.
' ac~
Vp\
+
+
vbb-\
L/J
Vbc-\
+
+
Vbc-i
Vp.
' cc'
L/J
+ . . . + ^2 = O,
-f . . . 4- ^3 = O,
id)
These equations {d) are derived directly from (r) in the following manner : multi-
ply the first of (c) by ~> the second by % etc., sum the products, and compare the
sum with the first of {<£). The first of (^) is then evident \ the others are obtained
in a similar way.
The mean error of an observed quantity of weight unity is in this case given by
the formula
where k is the number of conditions {a) ; and the mean error of any observed
value of weight/ is
sTp
f. Compatation of mean and probable errors of functions of observed
quantities.
Let V denote any function of one or more independently observed quantities
4r, ^, 5, . . . ; that is, let
V=f{x,y,z...).
A question of frequent occurrence with respect to such functions is, What is the
mean * error of V in terms of the mean errors of x, j/, j?, . . . ? The answer to
this question given by the method of least squares assumes that the actual errors
(whatever they may be) of x^y^z^,,, are so small that the actual error of ^is a
linear function of the errors of x^ y^ z. In other words, if <« ^y, ^« . . . denote
the actual errors of x, y, z^ . . • , and A ^denote the corresponding actual error of
F, the method assumes that
(a)
wherein the squares, products, etc., of e„ e^ e„ , , . are omitted.
This condition being fulfilled, let c denote the mean error of V^ and e„ Cy, e, . . .
denote those of ^, j^, ^, . . . respectively. Then the law of error of least squares
requires that
-=(VQ'-'+(f)v+(lr7-'+- <*>
• Since the probable error is 0.6745 times the mean error the latter onlv need be (^^^f^^lp
XCVl THEORY OF ERRORS.
This equation includes all cases. Its analogy with {a) should be noted, since
the step from {a) to {b) is clear when the correct form of {a) is known. Mistakes
in the application of {b) are most likely to arise from a lack of knowledge of the
independently observed quantities jc, ^, jr, . , . or from a lack of knowledge of the
true form of {a). Hence,* in deriving probable errors of functions of observed
quantities attention should be given first to the construction of the expression for
the actual error {a),
A few examples may serve to illustrate the use of (a) and {b).
(i.) Suppose
Then
(2.) Suppose
Then
(3.) Suppose
Then
3V 3V ^ 5F ^ ,
^V=ae, + {b^a)e, + {b + c)e„
SV_ a dV_b dV ^by
V=z a log X •\- b %Vi y ■\- c log tan t.
and
5F_ a/tt 5r_ dV_ cy.
'd^—~^' "^ — ''cos^, 5, — sin z cos /
€* =
(f)V+(*»..)v+(i?^.)V.
(4.) Suppose the case of a single triangle all of whose angles are observed.
What is the mean error, ist, of an observed angle; 2d, of the correction to an
observed angle ; and 3d, of the corrected or adjusted angle ?
Let X, y, z denote the observed angles, /, q^ r their weights, and Aj:, Ay, A«
the corresponding corrections.
Then, as shown on p. Ixxxvii,
Aa: + ^J' + ^ = ^= 180° + sph. excess — {x + J' + «)
= error of closure of triangle,
C= '-
- + - + -
P^ q^ r
Ao: = ^, A^ = ^, Ajbt = ^.
p q r
• As remarked by Sir George Airy in his Theory of Errors,
t ^ ^ modulus of common logarithms. ^-^ ^
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THEORY OF ERRORS. ZCVU
For brevity, put
g = iSo*' + spherical excess, A = 1 •
-4-- + -
Then
:r + A* = 7-(^ - ^ -^ - ^) + *,
with similar expressions for the other two angles.
Now by the formula on p. xcv the square of the mean error of an observed
angle of weight unity is (since there is but one condition to which A^, A^, As are
subject),
Pit^r + q{Uyf + ritlzf = ^= ^.
Hence, the squares of the mean errors of the observed angles x^ y, z, their weights
being/, ^, r respectively, are
Ae^ h^ hc^
"jT* "ZT^ ""IT*
P q r
respectively.
To get the mean error of a correction, ^x for example, formula {a) gives
A V= A(A*) = - j(.. + ^, + O,
and the corresponding expressions for the actual errors of Aj/ and A? are found
from this by replacing phy q and r respectively. Thus by (b\ observing that
the mean errors of x^ y^ z are given above, there result
Square of mean error of Ar = {hcjff^
" t^y={hclq)\
Likewise, the formula for the actual error of jc -f- ^^ is
A F= A(^ + A:.) = (i -|)^. -|.
h
r
and the corresponding expressions for the actual errors of ^ -}~ A^ ^^^ z-^ ^
are found by interchange of q and r with/. Thus the squares of the mean errors
of the adjusted angles are : —
£or(, + A.). f(x-|).
forCy + A^), f(i-^).
<>«■(*+ ). — ^I--j- Digitized by GoOQIc
XCviii THEORY OF ERRORS.
In case the weights are equal, or in case p=^q =.r^ ^ = i» smd thefe
result, —
Square of mean error of observed angle = i ^t
" " " ** " correction to observed angle = J <r*,
" « " " " adjusted angle = * ^,
where c is the error of closure of the triangle ; so that in this case of equal weights
the three mean errors are to one another as ^^3, ^, and ^^2.
References,
The literature of the theory of errors, especially as exemplified by the method
of least squares, is very extensive. Amongst the best treatises the following are
worthy of special mention : Method of Least Squares, Appendix to vol. ii. of
Chauvenet's " Spherical and Practical Astronomy." Philadelphia : J. B. Lippin-
cott & Co., 8vo, 5th ed., 1887. " A Treatise on the Adjustment of Observations,
with Applications to Geodetic Work and Other Measures of Precision," by T. W.
Wright. New York : D. Van Nostrand, 8vo, 1884. " On the Algebraical and
Numerical Theory of Errors of Observation and on the Combination of Observa-
tions," by Sir George Biddle Airy. London : Macmillan & Co., i2mo, 2d ed.,
1875. " Die Ausgleichungsrechnung nach der Methode der Kleinsten Quadrate,
mit Anwendungen auf die Geodasie und die Theorie der Messinstrumente," von
F. R. Helmert. Leipzig : B. G. Teubner, 8vo, 1872.
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EXPLANATION OF SOURCE AND USE OF THE
TABLES.
Tables x and a are copies of tables issued by the Office of Standard Weights
and Measures of the United States, edition of November, 1891.
Table 3 is derived from standard tables giving such data. The arrangement
is that given in " Des Ingenieurs Taschenbuch, herausgegeben von dem Verein
* Hiitte'"* (nth edition, 1877). The numbers have been compared with those
given in the latter work, and also with those in Barlbw's ** Tables." The loga-
rithms have been checked by comparison with Vega's 7-place tables.
Table 4 is abridged from a similar table in the Taschenbuch just referred to.
Tables 5 and 6 are copies of standard forms for such table. They have
been checked by comparison with standard higher-place tables. The mode of
using these tables will be evident from the following examples : —
(i.) To find the logarithm of any number, as 0.06944, we look in Table 5
in the column headed N for the first two significant figures of the number, which
are in this case 69. In the same horizontal line with 69 we now look for the
number in the column headed with the next figure of the given number, which is
in the present case 4. We thus find .8414 for the mantissa of the logarithm of
the' number 694. To get the increase due to the additional figure 4, we look in
the same horizontal line under Prop. Parts in the column headed 4 and find the
number 2, which b the amount in units of the fourth place to be added to the
part of the mantissa previously found. Thus the mantissa of log (0.06944) is
.8416. The characteristic for the logarithm in question is —2 =8— 10. Hence
log (0.06944) = 8.841 6 —10.
(2.) To find the number corresponding to any logarithm, as 8.8416— 10, we
look in Table 6 in the column headed L for the first two figures of the mantissa,
which are in this case 84. In the same horizontal line with 84 we now look for
the number in the column headed by the next figure of the mantissa, which is in
this case x. We thus find 6394 for the number corresponding to the mantissa
8410. To get the increase due to the additional figure 6, we look in the same
horizontal line under Prop. Parts in the column headed 6 and find 10, which is
the amount in units of the fourth place to be added to the number previously
found. Thus the significant figures of the number are 6944, and since the char-
acteristic of the logarithm is8— io=— 2, the required number is 0.06944.
* Berlin : Verlag von Ernst & Kom. This work is an invaluable one to the engineer, archi-
tect, geographer, etc
Digitized by VjOOQIC
C EXPLANATION OF SOURCE AND USE OF TABLES.
Tables 7 and 8 are taken from " Smithsonian Meteorological Tables " (the
first volume of this series). Their mode of use will be apparent from the follow-
ing example : Required the sine and tangent for 28° 17'.
sine 28° 10', Table 7 0.4720. Tabular difference = 26.
Proportional part for 7' (7 X 2.6) . . 18.
sine 28^ 17' 0.4738.
tangent 28° 10', Table 8 0.5354. Difference for i' = 3.8.
Increase for 7' (7 X 3.8) 27.
tangent 28** if 0.5381.
Table 9 is a copy of a similar table published in " Professional Papers, Corps
Engineers," U. S. A., No. 12. It has been checked by comparison with other
tables in general use. This table is useful in computing latitudes and departures
in traverse surveys wherein the bearings of the lines are observed to the nearest
quarter of a degree, and in other work where multiples of sines and cosines are
required. Thus, if Z denote the length and B the bearing from the meridian of
any line, the latitude and departure of the line are given by
ZcosjB and Zsin^
respectively; the " latitude " being the distance approximately between the paral-
lels of latitude at the ends of the line, and the '' departure " being the distance
approximately between the meridians at the ends of the line. As an example, let
it be required to compute the latitude and departure for Z = 4837, in any unit;
and jB = 36*^ 15'. The computation runs thus : —
Latitude. Departure.
For 4000 3225.77 2365.23
800 645.16 473-05
30 24.19 17.74
7 5-63 4-14
4837 Zcos^ = 3900.77 Zsin-5= 2860.16
Tables 10 and 11 give the logarithms of the principal radii of curvature of the
earth's spheroid. They were computed by Mr. B. C. Washington, Jr., and care-
fully checked by differences. They depend on the elements of Clarke's spheroid
of 1866. The use of these tables is sufficiently explained on p. xlv-xlix.
Table 12 gives logarithms of radii of curvature of the earth's spheroid in sec-
tions inclined to the meridian sections. It is abridged to 5 places from a 6-place
table published in the " Report of the U. S. Coast and Geodetic Survey for
1876." Its use is explained on pp. bd-lxiv.
Tables 13 and 14 give logarithms of factors needed to compute the spheroidal
excess of triangles on the earth's spheroid. No. 13 is constructed for the Eng-
lish foot as unit, and No. 14 for the metre. These tables were computed by Mr.
Digitized byLjOOQlC
EXPLANATION OF SOURCE AND USE OF TABLES. a
Charles H. Kummell. Their use is explained on p. Iviii. The following example
will illustrate their use : —
Latitude of vertex A of triangle 48^ 08'
" B " 47 52
" " C " 47 04
Mean latitude 47 41
Angle C= 51** 22' ss" log sin C 9.89283 — 10
log a (feet) 5.64401
log ^ (feet) 5.58681
log factor, Table 13, for 47° 41' 0.37176
Spheroidal excess = 31. "290, log i. 49541
Tables 15 and 16 give logarithms of factors for computing differences of lati-
tude, longitude, and azimuth in secondary triangulatton whose lines are 12 miles
(20 kilometres) or less in length. These tables were computed by Mr. Charles
H. Kummell. Table 15 gives factors for the English foot as unit, and Table x6
for the metre as unit. The use of these tables is illustrated by a numerical exam-
ple given on pp. Ix and Ixi. For lines not exceeding the length mentioned, the
tables will give differences of latitude and longitude to the nearest hundredth of
a second of arc, using 5-place logarithms of the lengths of the lines.
Table 17 gives lengths of terrestrial arcs of meridians corresponding to lati-
tude intervals of 10", 20", . . . 60", and 10', 20', . . . 60', or lengths corresponding
to arcs less than i^. The unit of length is the English foot. The table was
computed by Mr. B. C. Washington, Jr.
The length corresponding to any latitude interval is the distance along the
meridian between parallels whose latitudes are less and greater respectively than
the given latitude by half the interval. Thus, for example, the length corre-
sponding to the interval 30' and latitude 37° (182047.3 feet) is the distance along
the meridian from latitude 36° 45' to latitude 37** 15'.
By interpolation, we may get from this table the meridional distance corre-
sponding to any interval. The following example illustrates this use : Required
the distance between latitude 41° 28' i7."8 and latitude 41° 39' 53."4. The
difference of these latitudes is 11' 35."6, and their mean is 41° 34' o5."6. The
computation runs thus : —
Latitude 410.
Tabular difference.
10'
60724.60 feet
10.70 feet
l'
6072.46 "
1.07 «
30"
3036.23 "
•54 "
s"
506.04 "
.09 "
o."6
60.72 "
.01 «
MM V-
12.41
7.05 "
sum,
12.41 «
Distance = 70407.10 **
When the degree of precision required is as great as that of the example just
Digitized byLjOOQlC
Cll EXPLANATION OF SOURCE AND USE OF TABLES.
given, it will be more convenient to use formulas (2) on p. xlvL Thus, in this
example, —
log.
A<^ = 6gS'% 2.8423596
<l> = 41° 34' os:'6, p^ (Table 10) 7.3196820
cons't 4-6855749
Length = 70407.10 feet 4.8476165
Table 18 gives lengths of terrestrial arcs of parallels corresponding to longi-
tude intervals of 10", 20", . . . 60", and 10', 20', . . . 60', or lengths corresponding
to arcs less than 1°. The unit is the English foot. This table was computed by
Mr. B. C. Washington, Jr.
The method of using this table is similar to that applicable to Table 17
explained above. For the computation of long arcs it will in general be less
laborious to use the formulas (i) on p. xlix than to resort to interpolation from
Table 18.
Tables 19-24 give the rectangular co-ordinates for the projection of maps, in
accordance with the polyconic system explained on pp. liii-lvi, for the following
scales respectively : —
Table 19, scale gl^j
21, • " gijg- (2 miles to i inch)
22, *' ^ (i mile to I inch)
^^9 sossoo
unit = English inch.
Tlunit = i
uoo5 J
^^ „ . . : millimetre.
24»
These tables were computed by Mr. B. C. Washington, Jr.
The use of these tables and their application in the construction of maps may
be best explained by an example. Suppose it is required to draw meridians and
parallels for a map of an area of 1° extent in longitude, lying between the paral-
lels of 34° and 35°. Let the scale of the map be one mile to the inch, or 1/63360,
and let the meridians and parallels be 10' apart respectively. Draw on the pro-
jection paper an indefinite straight line AB, Fig. 4, to represent the middle me-
ridian of the map. Take any convenient point, as Q on this line for the latitude
34°, and lay off from this point the meridional distances CD, CJS, CF^ , . . CI^
given in the second column of Table 22, p. 114.* Through the points D, E, /%
... I, thus found, draw indefinite straight lines perpendicular to AB, By means
of these lines and the tabular co-ordinates, points on the developed parallels and
meridians are readily found. Thus, for example, the abscissas for points ten
minutes apart on the parallel 34° 20' are 9.53, 19.06, and 28.59 inches. These
distances are to be laid off on JVJV in both directions from AB. At the points
Z, Mf N, Z', M\ N\ so determined, erect perpendiculars to NN' equal in
length, respectively, to the ordinates corresponding to the longitude intervals
* The meridional distances and the abscissas of the points on the developed parallels in Fig. 4
are one twentieth of the true or tabular values. The ordinates of points on the developed paral-
lels are the tabular values.
Digitized by V^OOQ IC
EXPLANATION OFV SOURCE AND USE OF TABLES.
cm
lo', 20', 30'. The curved line joinikg the extremities of these perpendiculars is
the parallel required. It may be drawn by means of a flexible ruler. The other
paraUels are constructed in the same manner. They are all concave towards the
north or south according as the map shows a portion of the northern or southern
hembphere. The meridians are drawn in a similar manner through the points
{e,g.y Pj Q, M, jR, S, 7; U"in Fig. 4) having the same longitude relative to the
middle meridian. All meridians are concave towards the middle meridian.
A test of the graphical work which should always be applied is the approxima-
tion to equality of corresponding diagonals in the various quadrilaterals formed.
Thus in Fig. 4, ^T should be equal to JVy, CN to CN\ EVXoEW, etc.*
1
}
as*
sor
Ati
X
I
V
y
e
s
4V
id
F
B.
li
M
E
i
M
N
D
Q
34'
w
C
-P
V
Tables 25-29 give areas of quadrilaterals, bounded by meridians and parallels,
of the earth's surface. They are taken from " Bulletin 50, U. S. Geological Sur-
vey." The unit of length used is the English mile, and the areas are thus given
in square miles. The method of using these tables is obvious.
Table 30 gives data for the computation of heights, from barometric meas-
ures, in accordance with the formula of Babinet.t This table is taken from the
" Smithsonian Meteorological Tables " (the first volume of this series). The
manner of using it is explained in connection with the table.
* It should be noted that CWis not equal to EV^ A^and F referring here to points on the
developed parallels.
t Compies Rendus, Paris, 1850, vol. xxv. p. 309. ^^ hkI OOoIp
Digitized by^
>8'
CIV EXPLANATION OF SOURCE AND USE OF TABLES.
Table 31 gives the mean astronomical refraction in terms of the apparent alti-
tude of a star or other object outside the earth's atmosphere. It is taken from
Vega's 7-place table of logarithms. Its use will be evident from the following
example : —
Apparent altitude of star =r 34° 17' n.";
Refraction = i' 24/'^ — » X i."i = i 24.1
True altitude of star =34 15 48.6
Tables 3a and 33 facilitate the interconversion of arc and time. They are
taken from the *' Smithsonian Meteorological Tables" (the first volume of this
series). The following examples illustrate their use : —
(i.) To convert 68® 29' 48."8 into time we have from Table 32 —
68** = 4^ 32™ GO-
29' = I 56
48" = 3-20
o/'8 = .OS
Equivalent in time = 4 33 59.25
(2.) To convert 5^ 43" 28.*8 into arc we have from Table 33 —
5 — 75 ^^^ 00
43" = 10 45 00
28' = 7 00
0/8 = 12
Equivalent in arc = 85 52 12
Tables 34 and 35 facilitate the interconversion of mean solar and sidereal
time intervals. They are taken from Vega's 7-place table of logarithms. The
mode of using them is explained in the tables themselves.
Tables 36 and 37 give the lengths of degrees of terrestrial arcs of meridians
and parallels expressed in metres,* statute miles (English), and geographic miles
(distance corresponding to i' on the earth's equator). These tables are taken
from the " Smithsonian Meteorological Tables " (the first volume of this series).
Table 38 facilitates the interconversion of statute (English) miles and nautical
miles. The nautical mile used is that defined by the U. S. Coast and Geodetic
Survey, namely : the length of a minute of arc of a great circle of the sphere
whose surface equals that of the earth (Clarke's spheroid of 1866). For formula
for radius of such sphere see p. lii. This table is taken from the '' Smithsonian
Meteorological Tables " (the first volume of this series).
Table 39 gives the English and metric equivalents of other standards of
length still in use or obsolescent. It is taken from the " Smithsonian Meteoro-
logical Tables " (the first volume of this series).
Table 40 gives values of the acceleration (g) of gravity, log ^, log (1/2^), log ^ig,
* It should be observed that the metric values given in these tables depend on Clarke's value
of the ratio of the yard to the metre, which is now known to be erroneous by about the 1/ loooooth
part.
Digitized by V^OOQ IC
EXPLANATION OF SOURCE AND USE OF TABLES. CV
and (^/w*) or the length of a seconds pendulum, for intervals of 5** of geograph-
ical latitude. It was computed by the editor, and is based on the formula for g
given by Professor William Harkness in his memoir '* On the Solar Parallax and
its Related Constants." *
Xable 41 gives the linear expansions of the principal metals. It was compiled
by the editor from various sources. The values given for the expansion per
degree Centigrade have been rounded (with one exception) to the nearest unit in
the millionths place, or to the nearest micron, since different specimens of the
same metal vary more or less in the ten-millionths place.
Table 42 gives the fractional changes in numbers corresponding to changes in
the 4th, 5tb, . . . 7th place of their logarithms. These fractions are often con-
venient in showing the approximate error in a number due to a given error in
its logarithm, or the converse. Thus, for example, referring to the remark in a
foot-note under explanation of Tables 36 and 37 above, the error in the loga-
rithm of Clarke's ratio of the yard to the metre is about 4 units in the sixth place
of decimals ; the Table 4a shows, then, that the metric equivalents in Tables
36 and 37 are erroneous by about i/ioo oooth part.
* Washington, Government Printing Office, 1891.
Digitized by
GooqIc
Digitized by VjOOQIC
GEOGRAPHICAL TABLES
Digitized by
GooqIc
Table 1 .
FOR CONVERTING U. S. WEIGHTS AND MEASURES.
CUSTOMARY TO METRIC.
a=
LINEAR.
CAPACITY.
1 =
2 =
3 =
4 =
1=
7 =
8 =
9 =
Inches to
milli-
metres.
25-4001
50*800 1
76*2002
10 1 '6002
127*0003
152-4003
177-8004
201*2004
228*6005
Feet to
metres.
0*304801
0-009601
0*914402
1*219202
I 1524003
1*828804
2*135604
2*430405
2743205
Yards to
metres.
0*91,
2*743205
3*657607
4572009
5-48641 1
6*400813
8*2296x6
Miles to
kilometres.
4-82804
6*43739
804674
9*65608
11*26543
12*87478
14*48412
9 =
Fluid
dnms to
millilitres
or cubic
centi-
metres.
3*70
7*39
11*09
18-48
22-18
25-88
29*57
33*27
Fluid
ounces to
milli.
litres.
29*57
mi
1 18*29
147-87
177*44
207*02
Quarts to
litres.
0*94636
1*89272
2*83908
3*78543
4*73»79
5*67815
6*624 u
7*57087
851723
Gallons to
litres.
3*78543
7-57087
"•35630
i5'I4«74
18*92717
22 7 1 261
26*40804
30*28348
34*06891
SQUARE.
WEIGHT.
1 =
2 =
3 =
4 =
7 =
8 =
9 =
Sauare
inches to
square
centi-
metres.
6452
12-903
19-355
25-807
32-258
38*710
45161
5A'^53
58-065
Square
feet to
square
deci-
metres.
Q.290
18-581
27*871
37161
46*452
55*742
65-032
74*323
83-613
Square
yards to
square
metres.
0*836
1-672
2508
3*344
4*181
5*017
^4
7*525
Acres to
hectares.
0*4047
0*8094
1*2141
1-6187
2-0234
2-4281
28328
3*2375
3-6422
1=
Grains to
milli-
grammes.
64.7989
129*5978
194*3968
259-1957
323*9946
3»87935
453*5924
518*3914
583-1903
Ayoirdu-
pois
ounces to
grammes.
28*3495
&
113*3981
141*7476
170*0972
198*4467
226*7962
255-1457
Avoirdu-
pois
pounds to
kilo-
grammes.
o*45359
0*90710
1-36078
1*81437
2-26796
2*72156
3*}75i5
3-62874
408233
Troy
ounces lo
31*103
62*:
93*31044
124-41392
155*51740
186*62088
21772437
24882785
279*93133
CUBIC.
2 =
3 =
4 =
1:
7 =
8 =
9 =
Cubic
inches to
cubic
centi-
metres.
Cubic feet
to cubic
metres.
16-387
32*774
49*161
65*549
81*936
98*323
114*710
13 « -097
147-484
002832
0-05663
0*08495
0*11327
0*14158
0*16990
0*19822
0*22654
0*25485
Cubic
yards to
cubic
metres.
0*765
1-529
2*294
3*058
3823
4*587
6-I16
6-881
Bushels to
hectolitres.
0*35239
07047?
1*05718
1-40957
1*76196
2*11436
246675
2-81914
3-'7i54
I Gunter*s chaun = 20*1168 metres.
1 sq. statute mile = 259-000 hectares.
I fathom = 1-829 metres.
I nautical mile = 1853*25 metres.
I foot = 0.304801 metre, 9*4840158 log.
1 avoir, pound = 453*5924277 gram.
15432*35639 grains = i kilogramme.
The only authorized material standard of customary length is the Trouehton scale belonging to this office, whose
length at ^9^.63 Fahr. conforms to the British sUndard. The yard in use m the United States is therefore equal to
the Britisn yard.
The only authorized material standard of customary weight is the Trov pound of the Mint. It is of brass of
unknown density, and therefore not suitable for a standard of mass. It was derived from the British standard Troy
pound of 175S by direct comparison. The British Avoirdupois pound was also derived from the latter, and contains
7t000 grains Troy.
l ne grain 1 roy b therefore the same as the grain Avoirdupois, and the pound Avoirdupois in use in the United
States is pqual to the British pound Avoirdupois.
The British gallon = 4-S4346 litres. The British bushel = 36.3477 litres.
The length of the nautical mile given above and adopted by the U. S. Coast and Geodetic Survey many yeare
ago is defined as that of a minute of arc of a great circle of a sphere whose surface equals tha| of the^cdi (Clarke's
Spheroid of 1866). iigitized by VjVjiJ V
* Issued by U. S. Office of Standard Weights and Measures, and republished here by permission of Superintendent
of Coast and Geodetic Survey.
Smithsonian Tables. 2
FOR CONVERTING U. 8. WEIGHTS AND MEASURES.
METRIC TO CUSTOMARY.
Table 2.
LINEAR.
CAPACITY.
Millilitres
Metres to
inches*
Metres to
fecL
Metres to
yards.
KUo-
meires to
miles.
or cubic
centi-
metres
to fluid
drams.
Centi-
litres to
fluid
ounces.
Litres to
quarts.
Deca.
litres to
gallons.
Hecto.
litres to
bushels.
I =
39.3700
6-56167
9-84250
1-003611
2-187222
0-62137
,=
0-27
0-338
1*0567
2-6417
2-8377
2 =
787400
1-24274
2 =
Jit
0-676
21134
52834
5-6755
8-5132
3 =
I18-1100
3-280833
186411
3 =
1-014
3*1700
7-9251
4 =
1574800
^3'^^333
4374444
sfMl
4 =
i*o8
»-353
4-2267
10-5668
14-1887
1 =
196*8500
16-40417
5-468056
6-561667
7-655278
5 =
1-35
1-691
5-2834
13-2085
236*2200
19*68500
3-72822
6 =
1-62
2-029
63401
IS'8502
18-4919
17-0265
7 =
275-5900
22-96583
4-3495?
Iz
1-89
2-367
7-3968
198642
8 =
314-9600
26-24667
8*748889
4-97096
216
2-705
8-4535
21-1336
22*7019
9 =
354-3300
29-52750
9.842500
559233
9 =
2-43
3-043
9*5101
23-7753
25-5397
SQUARE.
WEIGHT.
Square
cenn-
metrcsto
aquare
inches.
Square
metres
to square
fecL
Square
metres
to square
yards.
Hectares
to acres.
Mini-
grammes to
grains.
Kilo-
grainroes to
grains.
Hecto-
grammes
to ounces
avuirdu-
poU.
Kilo-
grammes
to pounds
avoirdu-
pois.
1 =
0-1550
10*764
21-528
1*196
2-471
I =
0-01515
0-03086
15432-36
30864-71
3-5274
7-0548
105822
2-20462
2 =
0-3100
rf
4-942
2 =
6*61387
8*81849
3 =
0-4650
32*292
l^l
3 =
0-04630
46297-07
6172943
77161-78
4 =
0.6200
43-055
4784
4 =
0*06173
0-07716
14-1096
1=
0-7750
53-819
64-583
5-980
12-J55
1=
17-6370
11*02311
0.9100
1-0850
7-176
8372
14-826
0-09259
92594-14
21-1644
2^-6918
28-2192
13*22773
15-43236
7 =
86*111
17-297
1=
0-1080^
108026-49
8 =
1-2400
9-568
19768
12JJ58-85
138891-21
17-63698
9 =
»-395o
96-875
10-764
22239
9 =
31-7466
19*84160
CUBIC.
WEIGHT — (continued).
Cnbic
centi-
metres to
cubic
inches.
CuWc
deci-
metres to
cubic
inches.
CoWc
metres
to cubic
feet.
Cubic
metres to
cubic
yards.
Quintals to
pounds aT.
Milliera or
tonnes to
pounds av.
Kilogrammes
to ounces
Troy.
I =
o-o6io
61*023
35*314
70-629
1-308
J —
220-46
2204-6
32-1507
2 =
0*1220
122-047
2-616
2 =
440-92
4409-2
64-3015
3 =
0-1831
183-070
105-943
141-258
3-924
3 =
661-39
881-85
n^Ps
96-4522
4 =
02441
244094
5-232
4 =
128-6030
5 =
t^l
305*117
366-140
IIX
5 =
1102-31
11023-1
160-7537
6 =
6 =
132277
132277
192-9044
7 =
0*4272
427-164
247-201
282*516
317-830
9-156
7 ^
1543-24
15432-4
225-0552
8 =
0*4882
488*187
10-464
8 =
1763-70
17637*0
257-2059
289-3567
9 =
0*5492
549*210
11-771
9 =
1984-16
19841-6
By the concurrent action of the principal rovemments of the world an International Bureau of Weights and
Measures has been established near Paris. Under the direction of the International Committee, two ingots were cast
of pore pdatinum-iridium in the proportion of 9 p<«rts of the former to 1 of the latter metal. From one of these a cer-
tain number of kilogrammes were prepared, from the other a definite number of metre bars. These standards of weight
and length were intercompared, without preference, and certain ones were selected as International prototype stand-
ards. The others were distributed by lot, in September, 1889, to the different governments and are called National
prototrpe standards. Those apportioned to the United Sutes were received in 1890 and are in the keeping of this office.
Tne metric system was legalized in the United States in 1866.
The International Standard Metre is derived from the M^tre des Archives, and its length is defined by the dis-
tance between two lines at aP Centigrade, on a platinum-indium bar deposited at the International Bureau of Weights
and Measures.
The International Standard Riloeramme is a mass of platinum-iridium deposited at the same place, and its weight
in Tacao is the same as that of the KuoKramme des Archives.
The litre is equal to a cubic decimetre, and it is measured by the quantity of distilled water which, at its maximum r
density, will counterpoise the sundard kilogramme in a vacuum, the volume of such a quantity qiwMcr being, as p
neariy as has been ascertained, equal to a cubic decimetre. igitizea Dy "^^ v^
SmTHSONiAN Tables. 3
Table 3.
VALUES OF RECIPROCALS, SQUARES. CUBES. SQUARE ROOTS, CUBE
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
n
1000}-
ffi
^
v«
»«
log.«
1
2
3
4
1000.000
Soaooo
333-333
25aooo
I
4
.1
I
8
I.OOOO
14142
I.732I
2.0000
1.0000
1.2599
1.4422
1.5874
OlOOOOO
a3oio3
a6o2o6
5
6
I
9
166.667
142^57
125.000
iii.iii
81
343
512
729
2.2361
3.0000
1.7 100
1.8171
1.9129
2.0000
2.0801
0.6^
0.7781S
0.84510
0.90309
0.95424
10
II
12
'3
14
loaooo
90.9091
83-3333
100
121
196
1000
'331
1728
2197
2744
3.6056
3.7417
2.1544
2.2240
2.2894
2.3513
2.4101
1.00000
1.04139
1.07918
i."394
1.14013
15
i6
;^
19
66.6667
62.1(000
58.8235
55.5556
52.6316
225
350
361
4913
6859
3.8730
4.0000
4.I23I
4.2426
4.3589
2.4662
2.5198
2.5713
2.6207
8.6684
1.17609
1.20412
1.23045
1.25527
1.27875
ao
21
22
23
24
47^6190
45-4545
400
8000
9261
10648
12167
13824
4.4721
4^990
2.7144
2.7589
2.8020
1.30103
1.32222
1.34242
1.36173
1.38021
25
26
%
29
40.0000
384615
37.0370
3448^
^6
729
15625
9,
21952
24389
5.0000
5.0990
5.1962
5.2915
5.3852
2.9240
2.962s
3.0000
3.0366
3.0723
1.39794
1.41497
M3136
144716
1.46240
30
3'
32
33
34
33.3333
32.2581
31.2500
30.3030
2941 18
IS
1024
1089
1 156
27000
3T6S
35937
39304
5-4772
5-8569
5.7446
5.8310
3.1072
3.1414
3.1748
3.2075
3.2398
147712
149136
1.50515
1.51851
1.53148
35
36
39
28.5714
27.7778
27.0270
26.3158
25.6410
1296
1369
1444
1521
4665!
50653
54872
59319
5.9161
6.0000
6.0828
6.1644
6.2450
3.27"
3.3019
3.3322
3.3020
3.3912
1.57978
1.59106
40
41
42
43
44
25.0000
24.3902
23.8895
23.2558
22.7273
1600
1681
1764
1849
1936
64000
68921
74088
6.3246
6.4031
6.4807
6.5574
6.6332
3.4200
3-4482
3.4760
3.5034
3.5303
1^)0206
1.61278
1.62325
«.63347
1.64345
45
46
%
49
22.2222
21.7391
21.2766
20.8333
2a4C»2
202q
21 16
2209
2304
2401
91125
97338
103823
1 10592
I 17649
6.7082
6.9282
7.0000
3.6342
3.6593
1.65321
1.66276
1.67210
1.68124
1.69020
50
SI
52
S3
S4
19.6078
18.5185
2809
2916
125000
I3265I
140608
148877
157464
7.0711
7.1414
7.2111
7.2801
7.3485
3.68JO
3-7084
3.7325
3.7563
3.7798
1.69897
1.70757
1.71600
1.72428
1.73239
Smitmsonian Tablks.
Digitized byLjOOQlC
Table 3.
VALUES OF RECIPROCALS, SQUARES. CUBES. SQUARE ROOTS, CUBE
ROOTSp AND COMMON LOGARITHMS OF NATURAL NUMBERS.
n
looa^^
ffi
«•
v«
v«
log. «
55
56
59
18.1818
17.8571
17-5439
17.2414
16.9492
3^3^
3249
3364
3481
175S16
185193
X95112
205379
74162
7.4833
7.5498
7.6158
7.681 1
3.8030
3-8709
38930
1.74036
1.74819
1.75587
*.78343
1.77085
GO
61
62
64
16.6667
16.3934
16.1290
15-8730
15.6250
3600
§4
216000
226981
238328
250047
262144
7.7460
7.8102
7.8740
7-9373
8.0000
3-9*49
3.9365
3-9579
3-979*
4.0000
1.77815
1.78533
1.79239
65
66
%
69
15.3846
151515
14.9254
14.7059
144928
4225
4350
4624
4761
287496
300763
314432
328509
8.0623
8.1240
8,3066
4.0207
4.0412
4-0615
4-0817
4.1016
1.81291
1.81954
1.82607
I'.838^S
70
71
72
73
74
14.2857
■a
13.5*35
4900
5041
5184
5329
5476
343000
35791 1
373248
389017
405224
8.3666
8.4261
8.4853
1-^
4.121J
4.i4<»
4.1602
4.»793
4.1983
1.84510
1.85126
1.85733
75
76
?i
79
13.3333
13.1579
12.0870
12.8205
12.6582
5625
5770
6241
438976
456533
474552
493039
8.6603
8.7178
5-Z750
8.8318
8.8882
4.2172
4-2358
4.2543
4.2727
4.2908
•3&,
1.88649
1.89209
1.89763
80
81
82
'4
12.5000
12.3457
12.19qi
I2/>482
11.9048
6400
6561
6724
7056
512000
531441
551368
571787
592704
8.9443
9.0000
90554
9.U04
9.1652
4.3089
4.3267
4.3445
4.3621
4.3795
1.90309
1.90849
1.9*38*
1.91908
1.92428
85
86
u
89
11.7647
11.6279
11.2360
7569
7744
7921
636056
6C8503
681472
704969
9.2195
9.2736
^3274
9.3808
94340
4.3968
4.4140
4.4310
4.4480
44647
1.92942
1.93450
1.93952
1.94448
1.94939
90
91
92
93
94
II.IIII
;^
ia7527
ia6383
8100
8281
IS
8836
729000
830584
9.4868
9.5394
9.5917
9.6437
9.6954
44814
4.4979
4.5*44
4.5468
1.95424
1.95904
1-973*3
95
96
99
10.5263
104167
10.3093
ia204i
laioio
9025
9216
9409
9604
9801
912673
941 192
970299
9.7468
9.8995
9.9499
4.5629
4.5789
4.5947
4.6104
4.6261
1.97772
1.98227
1.98677
1.99*23
1.99564
100
lOI
102
103
104
10.0000
9.90099
9.80392
9.70874
9.61538
lOOOO
I020I
I0816
lOOOOOO
I03030I
I06I208
\1X
IOlOOOO
iao499
10.0995
10.1489
10.1980
4.6416
4.6570
4.7027
2.00000
2.01284
2.01703
105
106
\%
109
9.52381
9-43396
9-34579
9.25926
9.17431
IIO25
II236
11881
1157625
II9IOI6
1225043
I2597I2
1295029
10.2470
10.2956
10.3441
10.3923
104403
4.7*77
4.7326
4.7475
4.7622
4.7769
2.021 19
2.02531
2.02938
2.03342
2.03743
SiimiaofiiA
Digitized byLjOOQlC
Table 3.
VALUES OF RECIPROCALS. SQUARES, CUBES, SQUARE ROOTS, CUBI
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
n
iooo.i-
n^
If*
V«
>
log. «
110
III
112
"3
114
9.09091
877193
12100
12321
12544
12769
12996
1331000
1367631
1404928
1442897
1481544
10.4881
10.5357
10.5830
10.6301
10.6771
4.7914
4.8059
4A»88
2.04139
2.04532
2.04922
2.05308
2.05690
115
ii6
119
8.69565
8.62069
8.54701
847458
840336
13225
13924
14161
1601613
1643032
1685159
ia7238
10.7703
10.8167
10.8628
10.9087
4.8629
4.8770
4.8910
4.9049
4.9187
2.06070
2.06446
2.0681Q
2.07188
2.0755s
120
121
122
"3
124
8.26446
8.19672
8.13008
8.06452
15129
15376
1728000
1860867
1906624
I0.954S
11.0000
11.0454
11.0905
"•1355
4.9324
4.9461
4.9597
2.07918
2.08279
2.08636
2.08991
2.09342
125
126
129
8.00000
7.93651
7.87402
7.81250
775»94
1 61 29
1953125
2000376
2048383
11.1803
11.2250
11.2694
"•3137
".3578
5.0000
5.0133
5.0265
2.09691
2.IOOy
2.10380
2.IO72I
2.1 1059
130
131
132
133
134
7.69231
7-63359
7-57576
7.46269
16900
17161
17424
17956
2352637
2406104
114018
"4455
11.4891
11.5326
11.5758
5.0916
5.1045
5.1 172
2.U394
2.11727
2.12057
2.12385
2.1 2710
135
136
139
7.40741
7.35294
7.19424
18225
1I769
19044
19321
2460375
2515456
2628072
2685619
1 1. 6190
11.6619
11.7047
5.1299
5.1426
5.»55i
5.1676
5.1801
2.13033
2.J3^
2.13988
2.I430I
140
141
142
143
144
7.14286
7.09220
7.04225
6.99301
6.94444
19600
19881
20164
20449
20736
2744000
2803221
2863288
2924207
2985984
11.8322
11.8743
11.9164
".9583
12.0000
5.1925
5.2048
5.2171
5.2293
5.2415
2.I46I3
2.14922
2.15229
2^15^36
145
146
\%
149
6.89655
6.84932
6.80272
6.75676
6.71 141
21023
21318
21609
21904
22201
3048625
3112136
3176523
3241792
3307949
12.0416
12.0830
12.1244
5.2656
in
5.3015
2.I6I37
2.1643s
2.16732
2.17026
2.I73I9
150
152
153
IS4
6.66667
6.62252
6.57893
6.53595
649351
22500
22801
23104
23409
23716
3375000
3442951
351 1808
3581577
3652264
12.3288
12.3693
12.4097
5.3*33
5.3485
5.3601
2.I760Q
2.17898
2. 181 84
2.18469
2.T8752
155
156
1 59
6.45161
6.41026
6.36943
6.3291 1
6.28931
2402J
24336
24649
25281
3723875
3796416
3869893
3944312
4019679
12.4499
12.4900
12:5698
12.6095
5.3717
5.3832
5.3947
5.4061
5.4175
2.19033
2.I9312
2.20140
160
161
162
163
164
6.25000
6.2U18
6.17284
6.13497
6.09756
25600
4096000
4173281
4251528
4330747
4410944
12.6491
12.6886
12.7279
5.4288
54401
5.4626
5.4737
2.20412
2.20683
2.20952
2.21 219
2.21484
Smithsonian Tablcs.
Digitized byLjOOQlC
Table 3.
VALUES OF RECIPROCALS. SQUARES, CUBES, SQUARE ROOTS. CUBE
ROOTS, AND COMMON \.OCARITHMS OF NATURAL NUMBERS.
165
i66
167
168
169
170
171
172
173
174
175
176
178
179
180
181
182
184
185
186
187
188
189
190
191
192
»93
194
195
196
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
l\l
219
lOOOs^
6.06061
6.02410
5.98802
5-95238
5.9I7I6
5-88235
5-84795
5-81395
5-78035
5-74713
5.71429
5.68182
5.64972
5.61798
5-58659
5-55556
5.52486
5-49451
5.46448
5-43478
5-40541
537634
5-34759
5-31915
5.29101
5.26316
5-23560
5-20833
S-«8i35
»i35
5464
5.12821
5.10204
5.07614
5.05051
5-02513
5.00000
4.97512
4.95050
4.9261 1
4.90196
4.87805
4-85437
4-83092
4.80769
4.78469
27225
^7550
27889
28224
28561
28900
29241
29584
29929
30276
30625
30976
32041
32400
32761
33124
33489
33856
34225
34596
34969
35344
35721
36100
36481
36864
37249
37636
38025
38410
39204
39601
40000
40401
40804
41209
41616
42025
42436
42849
43264
43681
44100
44521
44944
45369
45796
46225
46656
47089
47524
47961
4492125
4574296
4657463
4741632
4826809
4913000
50002 I I
5088448
51777 17
5268024
5359375
5451770
5545233
5639752
5735339
5832000
6128487
6229504
6331625
6434856
6539203
6644672
6751269
6859000
6967871
7077888
7189057
7301384
7414875
7529536
7045373
7762392
7880599
8000000
81 20601
8242408
8365427
8489664
8615125
874I8I6
8869743
8998912
9129329
9261000
9393931
9528128
9063597
9800344
9938375
10077696
I02I83I3
10360232
10503459
v«
12.8452
12.8841
12.9228
12.9615
13.0000
130384"
13.0767
i3-"49
13-1529
13-1909
13.2288
13-2665
13-3041
13-3417
>3-379i
13.4164
13-4536
13-4907
13-5277
13-5647
13.6015
13.6382
13.6748
i3-7"3
13-7477
13-7840
13.8203
13.8564
13-8924
13-9284
13.9642
14.0000
14.0357
14.0712
14.1067
14.1421
14.1774
14.2127
14.2478
14.2829
14.3178
14.3527
14.3875
14.4222
14.4568
14.4914
14.5258
14.5602
14.5945
14.6287
14.6629
14.6969
14.7309
14.7648
14.7986
9«
5.4848
5-4959
5.5060
5.5288
5-5397
5.5505
5.5013
5-5721
5.5828
5.5934
5.6041
5.6147
5.6252
56357
5.6462
5.6980
5-7083
5.718s
l:5g
5-7489
57590
5.7690
5.7790
5.7890
IP
5.8186
5!285
5.8383
5.8480
5.8578
5.8675
5.8771
5.8868
5-8964
5-9059
5.9155
5.9250
5-9345
5-9439
59533
5.9627
5.9721
5.9814
5-9907
6.0000
6.0092
6.0185
6.0277
log. «
2.21748
2.220Z z
2.22272
2.22531
2.22789
2.23045
2.23300
2.23553
2.23805
2.24055
2.24304
2.2455'
2.24797
2.25042
2.25285
2.25527
2.25768
2.26007
2.26245
2.26482
2.26717
2.26951
2.2718^
2.27416
2.27646
2.27875
2.28103
2.28330
2.28556
2.28780
2.29003
2.29226
2.29447
2.29667
2.29885
2.30103
2.30320
2.30535
2.30750
2.30903
2.31175
2-31387
2.31597
2.31806
2.32015
2.32222
2.32428
2.32634
2.32838
2.33041
2.33244
2-33445
2.33640
2.33846
2.34044
Digitized by V^OOQIC
SiirrNaoNiAN Tables.
Table 3.
VALUE8 OF RECIPROCALS, SQUARES, CUBES. SQUARE ROOTS, G
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS
CUBE
lOOO^
^n
J«
log*
220
221
222
223
224
225
226
227
228
229
230
232
234
235
236
*37
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
'4
259
260
261
262
263
264
265
266
267
26S
269
270
271
272
273
274
4.5454s
4.52489
4-50450
4.48431
4.46429
4.34783
4.32900
4.3»034
4.29185
4-27350
425532
4.23729
4.21941
4.201 68
4. 184 10
4.16667
4.14938
4.13223
4.H523
4.09830
4.08163
4.061504
4.04858
4.03226
4.01606
4.00000
3.98406
396825
3-95257
3-93701
3-92157
3-90625
3.89105
3-87597
3.86100
3-84615
3.83142
3.8022
3.78788
3-77358
3-75940
3-74532
3-73«34
3-71747
3-70370
3.69004
3.67647
3.66300
3.64964
48400
48841
49284
49729
50176
50625
51076
5' 529
S1984
52441
52900
53361
53824
54289
54756
56169
56644
57121
58081
58564
5904?
59536
60025
6051S
61009
61504
62001
62500
63001
635Q4
64009
64516
6502c
67600
68121
68644
69169
69696
70225
70756
71289
71824
72361
72900
73441
73984
74529
75076
0648000
0793861
0941048
1089567
1239424
1390625
1543176
1697083
'85235
2167000
2326391
2487168
2649337
2812904
2977875
3144256
3312053
3481272
3651919
3824000
3997521
4172488
4348907
4526784
4706125
4886936
5069223
5252992
5438249
5625000
5813251
6003008
6194277
)i94
>387^
6387064
6581375
6777216
6974593
7173512
7373979
7576000
7779581
7984728
8191447
8399744
8609625
8821096
9034163
9248832
9465109
9683000
9902511
20123648
20346417
20570824
14.8324
14.8661
14.8997
15.0000
15-0997
15-1327
15.1658
15.1987
15-2315
15.2643
15.2971
15-3297
15-3623
15-3948
15-4272
15-4596
15-4919
15.5242
\m
15.6205
15.7162
15.7480
15-7797
15.8114
15-8430
15.8745
15.9060
15-9374
16.0000
16.0312
16.0624
16.093s
16.1245
\tM
16.2173
16.2481
16.2788
16.3095
16.3401
16.3707
16.4012
16.4317
16.4621
16.4924
16.5227
16.5529
6.0368
6.0459
6.0550
6.0641
6.0732
6.0822
6x>9i2
6.1002
6.1001
6.1 180
6.1260
6.1446
6.1622
6.1710
tlWs
6^1972
6.2058
6.2145
6.2231
6.2317
6.2403
6.2488
6.2
6.;
6.2743
6.2828
6.2912
6.2996
6.3080
6.3164
6.3247
6-3330
6.3743
6.3825
5-3907
6.3988
64070
6.4151
64232
6-4312
6.4393
6.4473
6.4553
64633
64713
64792
6.4872
64951
2.34242
2.34439
234635
2.34830
235025
2.35218
2.35411
2.35603
2.35793
2.35984
2.36173
2.36361
2.36549
2.36736
2.36922
2.37107
2.37291
2.37475
2.37658
2.37840
2.38021
2.38202
2.38382
2.38561
2.38739
2.38917
2.39094
2.39270
2.39445
2.39620
2.39794
2.39967
240140
240312
240483
240654
240824
240993
2.41 162
241330
241497
241664
241830
241996
242160
242325
2.4248S
242651
242813
242975
243136
2.43297
243457
2.43616
243775
SmTHsoNiAN Tables.
Digitized byLjOOQlC
TABUC3.
VALUES OF RECIPROCALS, SQUARES. CUBES. SQUARE ROOTS, CUBE
ROOTS, AND COMMON LOCARITHWiS OF NATURAL NUMBERS.
looa-
v«
h
log. « -
275
276
%l
279
280
281
282
284
285
286
287
288
290
291
292
293
294
295
296
299
300
30«
303
304
305
306
309
310
3"
312
313
314
315
316
318
319
320
321
322
323
324
325
326
327
328
329
363636
3-62319
5.61 01 1
3-59712
3-58423
3-57143
3-55872
3.54610
3-53357
3-521 13
3-50877
3-49650
3-48432
3-47222
3.46021
344828
3-43643
3.42460
3-41297
3.40136
3-3
3.378.
3-36700
3.35570
3-34448
3-33333
3.32226
3.31 1 26
3-30033
3-28947
3.27869
326797
3-25733
3-24675
3.23625
3.22581
3.21543
3-20513
3-19489
3.18471
3.17460
3.16456
3-15457
3-14465
3.»348o
3.12500
3-" 527
3-I0559
3.09598
3.08642
3.07692
3-06748
3-05810
3-04878
3-0395*
76176
76729
77284
77841
78400
78961
79524
80656
81225
81796
82369
82944
83521
84100
8468Z
f5264
87025
87616
88209
88804
89401
90000
90601
91204
91809
92416
93636
94249
94864
95481
96100
96721
97344
&
100489
101124
101761
102400
X03041
103684
104329
104976
105625
106276
106929
107584
108241
20796875
21024576
21253933
21484952
21717639
2191J2000
22188041
22425768
22665187
22906304
23149125
23393650
23039903
23887872
24137569
24389000
24642 17 I
24897088
25153757
25412184
25672375
25934330
26198073
26463592
26730899
27000000
27270901
27543608
2781 81 27
28094464
28772625
28652610
28934443
29218112
29503629
29701000
- 0231
30371328
30664297
30959144
31255875
3» 554490
31855013
32157432
32461759
32768000
3307 61 61
33386248
33698267
34012224
34328125
34645976
34965783
35287552
3561 I 289
6.5831
6.6132
6.6433
6-6733
6.7033
6.7332
6.7631
6.7929
6.8226
6.8523
6.8819
6.91 1 5
6.941 1
6.9706
7.0000
73205
7-3494
7.3781
7.4069
7-4356
7.4642
7.4929
7.5214
7-5499
7.5784
7.6068
7-6352
76635
7.6918
7.7200
7.7482
7-7764
7.8045
7-8885
79165
7-9444
7.9722
8.0000
8.0278
i°555
8.0831
8.1108
8.1384
6.5030
6.5108
6.5265
6.5343
6.5421
6.5499
6.5654
6.5731
6.5808
6.5885
6.6039
6.61 1 5
6.61 91
6.6267
6.6343
66419
6.6494
6.6569
6.6644
6.6719
I®
6.6943
6.7018
6.7092
6.7166
6.7240
6.7313
6.7387
6.7460
tl^
6.7679
6.7752
67824
6.7969
6.8041
6.81 13
6.8185
6.8256
6.8328
6.8399
6.8470
6.8612
6.8683
6.8894
6.8964
6.9034
2.43933
2.44091
2.44248
2.44404
2.44560
2.44716
2.44871
2.45025
2.45179
2-45332
2.45484
2.45637
2-45788
2.45939
2.46090
2.46982
2.47129
2.47276
2.47422
2.47567
2.47712
2.47857
2.48001
2.48144
2.48287
2.48430
2.48572
2.48714
1.48855
;.48996
2.49136
2.49276
2.49415
2.49554
2.49693
2.49831
2.49909
2.50106
2.50243
2.50379
2.50515
2.50651
2.50786
2.50920
2.5^055
2.51 188
2.51322
2.51455
2.51587
2.51720
Ljoogle
SlIITMaONIAN TaSLBS.
Digitized by
Table 3.
VALUES OF RECIPROCALS, SQUARES, CUBES. SQUARE ROOTS, C
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
CUBE
v«
?«
log. If
331
332
333
334
335
336
337
3fi
339
340
341
342
343
344
345
346
348
349
350
3SI
352
353
354
355
356
359
360
361
362
363
364
365
3g
^S
369
370
371
372
373
374
375
376
377
378
379
380
382
^^
384
3.03030
3.021 1 5
3.01205
3.00300
2.99401
2.
2.971
.98507
.97619
2.96736
2.95858
2.9498s
2.941 18
2-93255
2.92398
2.91C4C
2.90698
2.89855
2.89017
2.88184
2.87356
2.86533
2^5714
2.84900
2.84091
2.83286
2.82486
2.81690
2.80899
2.801 1 2
2.79330
2.78552
2.77778
2.77008
2.76243
2.75482
2.74725
2.73973
2.73224
2.72480
271739
2.71003
2.70270
2.69542
2.68817
2.67380
2.66667
2.65957
2.65252
2.64550
2.63852
2.631
2.624(
2.61780
2.61097
2.60417
08900
09561
10224
10889
"556
12225
12896
13569
14244
I492I
15600
I628I
16964
17649
18336
19025
19710
20409
21 104
2180I
22500
23201
23904
24609
25316
2602c
26736
27449
28164
28881
29600
30321
31044
31769
32496
33225
35424
3616I
36900
37641
38384
39129
39876
40625
41370
42129
428S4
43641
44400
45161
47456
36594368
36926037
37259704
37595375
37933050
38272753
38614472
38958219
39304000
3965 I 82 I
40001688
40353607
40707584
41063625
41421736
41781923
42144192
42508549
42875000
43243551
43614208
43986Q77
44361864
44738875
45118016
45499293
45882712
46268279
46656000
47045881
47437928
47832147
48228544
48627125
49027896
49430863
49836032
50243409
50653000
51064811
51478848
51895117
52313624
52734375
53157376
53582633
5401 01 52
54439939
54872000
55306341
55742968
56181887
56623104
18.1659
18.1934
18.2209
18.2483
18.2757
18.3030
18.3305
'f-357o
18.3848
18.4120
18.4391
18.4662
18.4932
18.5203
18.5472
18.5742
18.601 1
18.6279
18.6548
18.68x5
18.7083
18.7350
18.7617
18.7883
18.8149
18.8414
18.8680
18.8944
18.9209
18.9473
18.9737
19.0000
19.0265
19.0520
19.0788
19.1050
19.1311
19.1572
19.1833
19.2094
19.2354
19.2614
19.2873
19.3132
19.3391
19.3649
19.3907
19.4165
19.4422
19.4679
19.4936
19.5192
19.5448
19.5704
19.5959
6.9104
6.9174
6.9244
6.9313
6.9382
7.0136
7.0203
7.0271
7.0338
7.0406
7.0473
7.0540
7.0607
7.0674
7.0740
7.0807
7.0873
7.0940
7.1006
7.1072
7.1138
7.1204
7.1269
7.1335
7.1400
7.1466
7.1531
7.1726
7.I79I
7.1855
7.1920
7.1984
7.2048
7.21 12
7.2177
7.2240
7.2304
7.2368
7-2432
7.2495
7.2558
7.2622
7.2685
2.518a
2.51983
2.521 14
2.52244
2.52375
2.52504
2.52634
2.52763
2.52892
2.53020
2.53148
2.53275
2.53403
2.5352?
2.53656
2.53782
2.53908
2-54033
2.54158
2.54283
2.54407
2.54531
2.54654
2.54777
2.54900
2.55023
2.55145
^•55267
2.55388
2.55509
2.55630
2.55751
2.55871
2.55991
2.561 10
2.56220
2.56348
2.56467
2.56585
2.56703
2.56820
2.56937
2.57054
2.5717I
2.57287
2.57403
2.57519
2.57634
2.57749
2.57864
2.57978
2.58092
2.58206
2.58320
2.58433
Smithsonian Taslcs.
Digitized byLjOOQlC
Table 3.
VAL.UE8 OF RECIPROCALS, 8QUARE8, CUBES. SQUARE ROOTS, CUBE
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
n
1000.^
««
if»
v«
>
log. If
385
389
259740
2^57732
2.57069
IS
149769
150544
15I32I
57066625
575^0
5841 1072
58863869
19.6214
19.6469
19.6723
19.6977
19.7231
7.2748
7.281 1
7.2874
7.2936
7.2999
2.58|46
2-58659
2.58995
390
391
392
393
394
2.56410
2-55754
2.55102
2.54453
2.53807
I 52100
I 52881
'53664
154449
155236
59319000
61 162984
19.7484
19-7737
19-7990
19.8242
19.8494
7.3061
7.3124
7.3186
7.3248
7-3310
2.59106
2.59218
2.59329
2.59439
2.59550
395
396
399
2.53165
2.52C25
2.51889
2.51256
2.50627
156025
I 56816
157609
158404
I5920I
61629875
62099136
62570773
63044792
63521 199
19.8746
19.8997
19.9249
J9-9499
19.9750
7.3372
7.3434
7-3496
7.3558
7.3619
2.59660
2.59770
2.59879
400
401
402
403
404
2.50000
2.49377
2.48756
2w|8i39
247525
160000
I6080I
161604
162409
I632I6
64000000
64481 201
64964808
65450827
65939264
20.0000
2ao25o
20.0499
20.0740
2ao998
7.3681
7-3742
7-3803
7.3864
7-3925
2.60206
2.60314
2.60423
2160^38
405
406
409
246914
246305
245700
245098
244499
164025
164836
;&'
I6728I
66430125
66923416
674I9143
679173I2
68417929
20.1246
20.1494
20.1742
20.1990
20.2237
7.3986
7.4047
74108
7.4169
74229
At
2.61 172
410
4"
412
413
414
2.43902
243309
242718
242131
241546
I68IOO
I6892I
169744
170569
I7I396
68921000
69426531
69934528
70444997
70957944
20.2485
20.2731
20.2978
20.3224
20.3470
74290
7-4350
7.4410
7.4470
74530
2.61278
2.61384
2.61490
2.61595
2.61700
415
416
419
2.40964
172225
174724
I7556I
71473375
71991296
725II713
73034632
73560059
20.3755
20.3961
20.4206
20.4450
20.4695
7.4590
7.4650
7.4710
7.4770
7.4829
2.61805
2.61909
2.62014
2.621 18
2.62221
420
421
422
423
424
2.38095
2.36967
2.36407
2.35849
176400
178084
178929
179776
74088000
74618461
76225024
20.4939
20.5183
20.5426
20.5670
20.5913
7-4880
7.4948
7.5007
7.5067
7.5126
2.62428
2.62531
2.62634
2.62737
-
425
426
429
2.35294
2.34742
2.34192
2.33645
2.33100
180625
I8I476
182329
I83I84
I8404I
7676^625
77854483
78402752
78953589
20.6155
20.6398
20.7123
7.5185
7-5244
7-5302
7-5361
7.5420
2.62839
2.62941
2.63043
430
431
432
433
434
2.32558
2.32019
2.31481
2.30947
2.30415
184900
I8576I
186624
;»
79507000
806215^8
81 182737
81746504
20.7364
20.7605
20.7846
20.8087
20.8327
7-5478
7-5537
7.5595
7.5654
7.5712
2.63347
2.63448
2.63548
2.63649
2.63749
435
436
439
2.29885
2.28311
2.27790
189225
190096
190Q69
191844
192721
82312875
82881856
83453453
84027672
84604519
20.8567
20.8806
20.9045
20.9284
20.9523
7^5828
7.5886
7.5944
7.6001
2.63849
2.63949
2.64048
2.64246
Taslks.
Digitize
d by V^OO^
le
II
Table 3.
VALUES OF RECIPROCALS, SQUARES, CUBES. SQUARE ROOTS, CUI
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
v«
?«
log. »
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
458
459
460
461
462
464
465
s
469
470
471
472
473
474
475
476
^77
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
2.27273
2.26757
2.26244
2-25734
2.25225
2.24719
2.24215
2.23714
2.23214
2.22717
2.22222
2.21730
2.21239
2.20751
2.2
2.20751
2.20264
2.19780
2.19298
2.18818
2.1834I
2.17865
2.I739I
2.16920
2.i64i;o
2.15983
2.155*7
2.15054
2.14592
2.14133
2.1367s
2.13220
2.12766
2.12314
2.1186A
2.11416
2.10970
2.10526
2.10084
2.09644
2.09205
2.08768
2.08333
2.07900
2.07469
2.07039
2.066x2
2.06186
2.05761
2-05339
2.04918
2.04499
2.04082
2.03666
2.03252
2.02840
2.02429
193600
194481
195364
196249
I97>36
198025
19801S
199809
200704
201601
202500
203401
204304
205209
2061 10
207025
209764
210681
21 1600
21252Z
213444
214369
215296
216225
217156
218089
219024
219961
220QO0
22 I 841
222784
223729
224677
225625
226578
227529
228484
229441
230400
231361
232324
233289
234256
235225
236196
237169
238144
239121
240100
241 08 I
242064
243049
244036
85184000
857661 2 I
86350888
86938307
87528384
8812112
89314623
8991;;
90518
887 16<
^
91 1 25000
91733851
92345408
92959677
93576664
94818816
95443993
96071912
96702579
97336000
97972181
98611128
99252847
99897344
100544625
loi 194696
101847563
102503232
1031 61709
103823000
104487111
I 05 I 54048
105821817
106496424
107171875
1 078501 76
108531333
109215352
109902239
I I 0592000
1 1 128464 1
111980168
I I 2678587
"3379904
I I 40841 25
I I 4791 256
"5501303
116214272
116930169
117649000
II 837077 I
I 19095488
"9823157
120553784
2a9762
21.0000
21.0238
2ix>476
21.0713
21.0900
21.1187
21.1424
21.1660
21.1896
21.2132
21.2368
21.2003
21.2838
21.3073
21.3307
21-3542
21.3776
214009
21.4243
214476
21.4709
214942
21.5174
21.5407
21.5639
21.5870
21.6102
21.6333
21/
£.6353
1-6564
21.6795
21.7025
21.7200
21.7486
21.7715
21.7945
21.8174
21.8403
21.8632
21.8861
21.9089
21.9317
21.9545
21.9773
22.0000
22.0227
22.0454
22.0681
22.0907
22.1133
22.1359
22.1585
22.1811
22.2036
22.2261
7.6059
7.6117
7.6174
7.6232
7.6289
7.6346
7-6403
7.6460
7.6517
7-6574
7-6631
7.6857
7.6914
7-6970
7.7026
7.7082
77138
7.7194
7.7250
7-7306
7.7362
7-7418
7.7473
7-7529
7.7584
7.7539
7.7695
7-7750
7.7805
7.7860
7-7915
7-7970
7.8025
7-8079
&
78243
7.8297
7-8352
7.8406
7.8460
7.8514
7.8568
7.8022
7.8676
7.87^
7-^37
7.8891
7!944
7-8998
7.9051
2.64345
2.64444
2.64542
2.64640
2.64738
2.64836
2.64933
2.65031
2.65128
2.65225
2.65321
2.65418
2.65514
2.65610
2.65706
2.65801
2.65896
la
2.66181
2.66276
2.66370
2.66464
2.66558
2.66652
2.6^39
2.66932
2.67025
2.67117
2.67210
2.67302
2.67^
2.67578
2.67669
2.67761
2.67852
2.67943
2.68034
2.68124
2.68215
2.68305
t&l
2.68574
2.68664
2^68842
2.68931
2.69020
2.69108
2.69197
2.69285
2.69373
Smithsonian Tablcs.
12
Digitized byLjOOQlC
VALUES OF RECIPROCALS, SQUARES. CUBES. 80UARE ROOTS, C
ROOTS, AND COMMON LOhARITHMS OF NATURAL NUMBERS.
Table 3.
CUBE
495
496
498
499
500
501
502
503
504
505
506
509
510
5"
512
5»3
514
515
S16
5»9
520
521
522
523
524
525
526
529
530
531
532
533
534
535
536
539
540
541
542
543
544
545
546
548
549
2.02020
2X)i6i3
2.01207
2.00803
2.00401
2.00000
.99601
3
98413
.98020
.97628
.97239
.968^0
.96484
.96078
•95695
•95312
•94932
•94553
.94175
•93798
•93424
•93050
.92678
.92308
•91939
.91571
.91205
.90840
.90476
.90114
89753
.89031
.88679
.88324
37970
.87617
.87266
.86916
.86567
.86220
.85874
•85529
.85185
.84843
.84502
.84162
.83824
.83486
•!3150
.82815
.82482
.82149
245025
246016
247009
248004
249001
250000
251001
252004
253009
254016
255025
256036
257049
258064
259081
260100
261 121
262144
263169
264196
265225
2662 JO
267289
268324
269361
270400
271441
272484
273529
274576
277729
278784
279841
280900
281961
283024
284089
285156
286225
287296
288369
289444
290521
291600
292681
293764
294849
295936
297025
2981 16
299209
300304
301401
121287375
122023936
122763473
123505992
124251499
125000000
I 25751 501
120506008
127263527
128024064
128787625
I 295542 I 6
X30323843
131090512
131872229
I 3265 I 000
133432831
134217728
X 35005697
135790744
138188413
138991832
139798359
140608000
141420761
142236648
143055667
143877824
144703125
145531576
146363183
147197952
148035889
148877000
149721291
150568768
X5'4i9437
152273304
153x30375
X 53990656
X 548541 53
155720872
I 56590819
157464000
1 58340421
160103007
160989184
161878625
X62771336
X 63667323
164566592
165469149
V«
22.2486
22.2711
22.2935
22.3159
22.3383
22.3607
22.3830
22.4054
22.4277
22.4499
22.4722
22.4944
22.5167
22.5389
22.5610
22.5832
22.6053
22.6274
22.6495
22.C
2.6716
22.6936
22.7156
22.7376
22.7596
22.7816
22.8035
22.8254
22.8473
22.8692
22.8910
22.9129
22.9347
22.9565
22.9783
23.0000
23.0217
23.0434
23.0651
23.0868
23.1084
23.130X
23x5x7
23-X733
23.1948
23.2164
232379
23-2594
23.2809
23.3024
23-3238
23.34|2
23.3666
23.3880
23.4094
23.4307
^n
7-9x05
7.9158
7.9211
7.9264
793x7
7-9370
7.9420
7-9476
7.9528
7.9581
^:»
7.9739
7-979X
7.9843
7.9896
7.9948
8.0000
8.0052
8.0104
8.0156
8.0208
8.0260
8.0311
8.0363
8.0415
8.0466
8.0517
8.0569
8.0620
8.0671
8.0723
8.0774
8.0825
8.0876
8.0927
8.0978
8.1028
8.1079
8.1130
8.1180
8.1231
8. 1 281
X332
1382
8.1483
8.
8.1583
8.1633
8.1683
8.1733
8.17
X833
8.1882
8.
.17»3
4^3
log. »
2.69461
2.69548
2.69636
2.69723
2.69810
2.69807
2.69984
2.70070
2.70157
2.70243
2.70329
2.704x5
2.70501
2.70586
2.70672
2.70757
2.70842
2.70927
2.71012
2.71096
2.71181
2.71265
2.7x349
2.7x433
2.715x7
2.71600
2.71684
2.71767
2.71850
2.7x933
2.72016
2.72009
2.72181
2.72263
2.72348
2.72428
2.72509
2.72591
2.72873
2.72754
2.72835
2.72916
2.72997
2.73078
2.73x59
2.73239
2.73320
2.73400
2.73480
2.73560
2.73640
2.73719
2.73799
2.73878
2-73957
Smitmsoiiian Tables.
13
Digitized by V^OOQ IC
Tablk 3.
VALUES OF RECIPROCALS, SQUARES. CUBES. SQUARE ROOTS, C
ROOTSp AND COMMON LOGARITHMS OF NATURAL NUMBERS.
CUBE
v«
v^
log. «
550
551
552
553
554
555
556
'3
SS9
560
563
S65
569
570
571
572
573
574
575
576
579
580
^^
584
5^
588
589
590
591
592
593
594
595
596
599
600
601
602
603
604
.81818
.81488
•f"59
.80832
.80505
.80180
.79856
•79533
.78571
•78253
•77930
.77620
•77305
.76678
•76367
.76056
•75747
•75439
•75131
.74825
.74520
.74216
•73913
.73611
•73310
.73010
.72712
.72414
.72117
.71821
•71527
•71233
.70940
.70648
■&
•69779
.69492
.69205
.68919
.68634
.68350
.68067
.67785
.67504
.67224
.66945
.66667
■^,
308025
309136
310249
3II364
31 2481
313600
31472I
318096
319225
320350
321489
322624
323761
324900
326041
327184
328329
329476
330625
33*770
332929
334084
335241
336400
337561
338724
339889
341056
342225
343390
344569
345744
346921
348100
349281
350464
351649
352836
354025
355216
356409
357604
358801
360000
361 201
362404
363609
364816
66375000
67284151
68196608
691 I 2377
70031464
72808693
7374" 1 2
74676879
75616000
76558481
77504328
78453547
79406144
80362125
81321496
82284263
83250432
84220009
85193000
86169411
87149248
88132517
891 19224
90109375
91 102976
92100033
93100552
94104539
951 1 2000
961 22941
97137368
98155287
99176704
200201625
201230056
202262003
203297472
204336469
205379000
206425071
207474688
208527857
209584584
2T0644875
21 1708730
2 I 27761 73
213847192
214921799
216000000
217081801
218167208
219256227
220348864
234521
234734
23-4947
23.5160
23-5372
23-5584
23-5797
23.6008
23.6220
236432
23.6643
23.6854
23.7065
23.7276
23^7487
23.7697
23.7908
23.8118
23.8328
238537
23-8747
23.8956
23.9165
23.9374
23-9583
23.9792
24.0000
24.0208
24.0416
24.0624
24.0832
24.1039
24.1247
24.1454
24.1661
24.1868
24.2074
24.2281
24.2487
24.2693
24.2899
24.3105
24-33"
24.3516
24.3721
24-3926
24-4131
24.4336
24.4540
244745
24.4949
24.5153
24.5357
24.5561
24-5764
§•'932
8.1982
8.2031
8.2081
8.2130
8.2180
8.2229
8.2278
8.2327
8.2377
8.2426
8.2475
8.2524
8.2573
8.2621
8.2670
8.2816
8.2865
8.2913
8.2962
8.3010
8.3059
8.3107
8.3155
8.3203
8.3251
§•3300
8.3348
8.3396
8.3443
8.3491
IS!
8.3730
5-3777
8.3825
8.3872
8.3919
8.3967
84014
8.406Z
8.4108
8.4155
84202
84249
8.4296
843*3
8.4390
8.4437
8.4484
8.4530
2.74036
2.741 1 5
2.74194
2.74273
2.74351
2.74429
2.74507
2.74586
2.74663
2.74741
2.74819
2.74896
2.74974
2.75051
2.75128
2.75205
2.75282
2.75358
2.75435
2-755"
2.75587
2.75664
2.75740
2.75815
2.75891
2.75967
2.76042
2.761x8
2.
2.71
2.76343
2.76418
2.76492
2.76567
2.76641
2.76716
2.76790
2.76864
2.76938
2.77012
2.77085
2.77159
2.77232
2.77305
2.77379
2.77452
2-77525
2-77597
2.77070
2.77743
2.77815
2.77887
2.77960
2.78032
2.78104
SmTHSONIAN TaBLKS.
H
Digitized byLjOOQlC
Table 3.
VALUES OF RECIPROCALS, SQUARES. CUBES. SQUARE ROOTS, CUBE
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
v»
v«
log. If
605
606
609
eio
611
612
614
ei5
616
617
618
619
620
621
622
624
625
626
627
628
629
630
632
633
634
636
53Z
638
639
640
641
642
645
646
648
649
650
652
654
655
656
t^
659
1.65289
1. 6501 7
1.64745
1.64474
1.64204
1-63399
1.63132
1.62866
1.62602
1.62338
1.62075
1.61812
1.61551
1. 61 290
1.61031
1.60772
1. 60514
1.60256
i-#730
1.^79
1.58228
1.57978
1-57729
1.57480
1.5723
1.56740
1.56495
1.56250
1.56006
1-55763
1.55521
1.55280
1-55039
1-54799
1.54560
1.54321
1-54083
1-5:^46
1.53610
1-53374
i-53«39
1.52905
1.52672
1-52439
1.52207
1.5x976
1-51745
366025
367236
368449
370881
372100
373321
374544
375769
376996
378225
379456
381924
383161
384400
388129
389376
390625
391876
393129
394384
395641
396900
398161
399424
400689
401956
403225
404496
405769
407044
408321
4^9600
410881
412164
413449
414736
416025
417316
418609
419904
421201
422500
423801
425104
426409
427716
429025
430336
4310
432, .
434281
8iim«oNiAii Tablks.
221445125
22254 coi 6
223648543
224755712
225866529
226981000
228099131
229220928
230346397
231475544
232608375
233744890
234885113
236029032
237176659
238328000
239483061
240641848
241804367
242970624
244140625
245314370
246491883
247673152
248858189
250047000
25123959'
252435968
253636137
254840104
256047875
257259456
258474853
259694072
260917 I 19
262144000
263374721
264609288
265847707
267089984
268336125
269586136
270840023
272097792
273359449
274625000
275894451
277167808
278445077
279726264
281011375
282300410
283593393
284890312
286191179
^5
24.5967
24.6171
24.6374
24.6577
24.6779
24.6982
24.7184
24.7386
24.7588
24.7790
24.7992
24.8193
24.8395
24.8596
24.8797
24.8998
24.9199
24.9399
24.9600
24.9800
25.0000
25.0200
25.0400
25.0599
25.0799
25-0998
25.1197
25-1396
25-1595
25.1794
25.1992
25.2190
25.2389
25.2587
25.2784
25.2982
25.3180
25-3377
25-3574
25.3772
25.3969
25.4165
25.4362
25-4558
25-4755
25.4951
25.5147
25-5343
25.5539
25-5734
25.5930
25.6125
25.6320
25.6515
25.6710
8.4577
8.4623
8.4670
8.4716
8.4763
8.4809
8.4856
8.4902
8.4948
8.4994
8.5040
8.5086
!'5i32
8.5178
8.5224
8.5270
8.5316
1-5362
8.5408
8.5453
8.5499
8.5544
8.5726
$•5772
8.5817
8.5862
8.5907
8.5952
8.5997
8.6043
8.6088
8.6132
8.6177
8.6222
8.6267
8.6312
8.6357
8.6401
8.6446
8.6490
8.6535
8.6579
8.6624
8.6668
8.6713
t^l
8.684s
8.6890
f^
8.7022
2.78176
2.78247
2.78319
2.78390
2.78462
2.78533
2.78604
2.78675
2.78746
2.78817
2.5
2.78958
2.79029
2.79099
2.79169
2.79239
2.79309
2.79379
2.79449
2.79518
2.79934
2.79657
2.79727
2.79796
2.79865
2.79934
2.80003
2.80072
2.80140
2.80209
2.80277
2.80346
2.80414
2.80482
2.80550
2.80618
2.80686
2.80889
2.80956
2.81023
2.81090
2.81158
2^1224
2.81291
2.81358
2.81425
2.81491
2.81558
2.81624
2.81690
^•5*257
2.81823
2.81889
Digitized by^OOQlC
Table 3.
VALUES OF RECIPROCALS, SQUARES. CUBES, SQUARE ROOTS. CUBE
ROOTS, AND COMMON \.OCARITHMS OF NATURAL NUMBERS.
looa-
v«
««
log. If
660
66i
662
663
664
665
666
667
668
669
670
672
674
675
676
677
678
679
680
681
682
683
684
685
686
688
689
692
694
695
699
700
701
702
703
704
705
706
707
708
709
710
7"
712
713
714
i!5i286
1.51057
1.50830
1.50602
1.50376
1. 501 50
M9925
1. 49701
M9477
1.49254
149031
m88io
148588
148368
1.48148
147929
147710
M7493
M727S
147059
146843
146628
M6413
146199
14598s
145773
145560
145349
145138
144928
144718
144509
1.44300
1.44092
143885
1.43678
143472
143266
143062
142857
142653
142450
1.42248
1.4204s
141844
141643
141443
141243
141044
140845
1.40647
140449
1.40252
140056
435600
436921
438244
4395'
442225
444^
446224
447561
448900
450241
451584
452929
454276
462400
463761
465124
466489
467856
469225
470590
471969
473344
474721
476100
477481
478864
480249
481636
483025
484416
485809
487204
4S8601
490000
491 401
492804
494209
495616
497025
498436
499849
501264
502681
504100
505521
506944
S08369
509796
287496000
288804781
2901 17528
291434247
292754944
294079625
295408296
296740963
298077632
299418309
300763000
3021 I 17 I I
303464448
304821 217
306182024
307546875
308915776
310288733
31 1665752
313040839
314432000
315821241
317214568
318611987
320013504
321419125
322828856
324242703
325660672
327082769
328509000
329939371
331373888
33281 25C7
334255384
335702375
340068392
341532099
343000000
344472101
345948408
347428927
348913664
350402625
351895816
353393243
354894912
356400829
357911000
359425431
360944128
362467097
363994344
256905
257099
257294
25.7488
25.7682
25.7876
25.8070
25.8263
25.8457
25.8650
25.8844
25-9037
25.9230
25.9422
25.9615
26.0768
26.0960
26.1151
26.1343
26.1534
26.2679
26.2869
26.3059
26.3249
26.3439
26.3818
26.4008
264197
264386
264575
264764
26.4953
26.5141
26.5330
26.5518
26.5707
26.6271
26.6458
26.6646
26.6833
26.7021
26.7208
8.7066
8.7 1 10
5-7'54
8.7198
87241
8.7^5
8.7329
87373
8.741D
8.7460
8.7503
8.7547
8.7590
8.7634
8.7677
8.7721
8.7850
87893
IE
8.8066
8.8108
8.8152
8.8194
8^82^
8.8323
8.8366
8.8408
8.8451
IS
8.8578
8.8621
8.8663
8.8706
8.8748
8.8790
8.8833
8.8875
8.8917
8.8959
8.900X
!-9°53
8.9085
8.9127
8.9169
8.921 1
8.9253
8.9295
5-9337
8.9378
2.81954
2.82020
2.82086
2.821 51
2.82217
2.82282
2.82347
2.82413
2.82478
2.82543
2.82607
2.82672
2-82737
2.82802
2.82866
2-82930
2.82995
2.83059
2-53123
2.83187
2^3251
2-53315
2.83378
2.83442
2.83506
2^3569
2.83632
2.83696
2.83759
2.83822
2.83885
2.83948
2.8401 1
2.8407;
2.84
2.84198
2.84261
2.84323
2.84386
2.84448
2.84510
2.84572
2A^634
2.84696
2.84757
2.84819
2.84880
2.84942
a.85003
2.85065
2.85126
2.85187
2.85248
2.85309
2^5370
^3!
Smithsonian Tables.
16
Digitized byLjOOQlC
Table 3.
VALUES OF RECIPROCALS. SQUARES, CUBES, SQUARE ROOTS, CUBE
ROOTS, AND COMMON \.OCARITHM8 OF NATURAL NUMBERS.
v«
v«
log. If
715
716
718
719
720
721
722
723
724
725
726
728
729
730
731
732
733
73*
735
736
737
738
739
740
741
742
743
744
745
746
748
749
750
751
752
753
754
755
756
759
760
761
762
763
764
765
766
7^
769
•39665
•39470
.39276
.39082
.38696
-38504
.38313
,38122
•3793«
•37741
37552
37303
37174
.36986
36799
.36612
.36426
.36240
.36054
.35501
■35318
■3513s
•34953
•34771
•3*S90
■34409
.34228
•34048
•33869
■33690
•335"
•33333
33156
32979
.32802
,32626
•32450
■32275
,32100
3*926
31752
31579
.31406
31234
.31062
.30890
■30719
30548
■30378
.30208
30039
5II22|
51 2656
514089
518400
5I984I
52x284
522729
524176
525625
527076
528529
529984
53*441
532900
534361
535824
537289
538756
540225
541696
543*69
544644
5461 21
547600
549081
550564
552049
553536
555025
556510
558009
559504
561001
562500
564001
565504
568516
570025
571530
573049
574564
576081
577600
582169
583696
586756
588289
589824
59*361
36552587s
367061696
368601813
370146232
371694959
373248000
374805361
376367048
377933067
379503424
381078125
382657178
384240583
385828352
387420489
389017000
390617891
392223168
393832837
395446904
398688256
400315553
401947272
4035834*9
405224000
4085^8488
410172407
41 1830784
413493625
41 5160936
416832723
418508992
420189749
421875000
42356475*
425259008
tx^
430368875
432081 216
433798093
4355*95*2
437245479
438976000
440711081
442450728
444194947
445943744
447697125
449455090
451 217663
452984832
454756609
26.7305
26.7582
26.7769
26.7955
26.8142
26.8328
26.8514
26.8701
26.8887
26^9072
26.9258
26.9444
26^9629
26.9815
27.0000
27.0185
27.0370
27.055s
27.0740
27.0924
27.1109
27.1293
27-*477
27.1662
27.1846
27.2029
27.2213
272397
27.2580
27.2764
27.2947
27-3*30
2733*3
27-3490
27-3679
27.3861
274044
274226
274408
27-459*
27-4773
27-4955
27-5*30
27-53*8
27.5500
27.5681
27.5862
27.6043
27.6225
27.6405
27.6586
27.6767
27.6948
27.7128
27.7308
8.9420
8.9462
8-9503
8.954s
8.9587
8.9628
8.9670
8.9711
8.9752
8.9794
8.9876
8.9918
8.9959
9.0000
9.00AI
9.0082
9.0*23
9.0164
9.0205
9.0246
9.0287
9.0328
9-0369
9.0410
9.0450
9.0491
9^0532
9.0572
9.0613
9.0654
9-0694
9-0735
90775
9.0816
9.0856
9^)896
9-0937
9-0977
9.1017
9.1057
9.1098
9,1138
9.1178
9.1218
9.1258
9.1298
9-*338
9-*378
9.1418
9.1458
9.1498
9-* 537
9-* 577
9.1617
2^543*
2.85491
2.85552
2.85612
2^5673
2.85733
2.85794
2.85854
2.85914
2.85974
2.86034
2.86094
2.86153
2.86213
2.86273
2.86332
2.86392
2.86451
2.86510
2.86570
2.86864
2.86923
2.86982
2.87040
2.87099
2.87157
2.87216
2.87274
2.87332
2.87390
2.87448
2.87506
2.87564
2.87622
2.87679
2.87737
2.87795
2.87852
2.87910
2.87967
2.88024
2.88081
2.88138
2.88195
2.88252
2.88309
2.88366
2.88423
2.88480
2.88536
2.88593
SniTHaONIAN TaSLBS.
17
Digitized by VjOOQlC
Tables.
VALUES OF RECIPROCALS, SQUARES. CUBES. SQUARE ROOTS. CUBE
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
v«
^
log. «
770
771
773
774
775
776
777
778
779
780
782
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
804
805
806
809
810
811
8l2
814
815
816
817
818
819
820
821
822
?^3
824
29870
29702
^^
29199
.28700
►28535
28370
28205
28041
27877
27714
27551
2738?
,27226
,27065
26904
26743
26582
,26422
26263
26103
25945
25786
25628
25471
25313
25156
25000
2484
24533
24378
24224
24069
23916
23762
23609
23457
23305
23153
23001
22850
22699
22549
22399
.22249
22100
21051
21803
.21655
.21507
21359
592900
594441
595984
597529
599076
600625
602 17D
603729
605284
606841
608400
609961
61 1 524
613089
614656
616225
617796
619369
620944
622521
624100
625681
627264
628849
630436
632025
633616
535209
636804
638401
640000
641601
643204
644809
646416
648025
649636
651249
652864
654481
656100
657721
"&
662596
672400
674041
675684
677329
678976
465484375
467288576
469097433
470910952
472729139
474552000
476379541
47821 1768
480048687
481890304
485587656
487443403
489303872
491 169069
493039000
494913671
496793088
498677257
500566184
50245987
5043583:
506261573
508169592
510082399
512000000
513922401
515849608
517781627
5197 18464
52x660125
523606616
525557943
527514112
529475129
531441000
533411731
535387328
537367797
539353144
541343375
543338490
545338513
547343432
549353259
551368000
553387661
555412248
557441767
559476224
27.7489
27.7669
27.7849
27.8029
27.8209
27.8388
27.8568
27.8747
27.8927
27.9106
27.9285
27.9464
27-9643
27.9821
28.0000
28.0179
28.0357
28.0535
28.0713
28.0891
28.1069
28.1247
28.1425
28.1603
28.1780
28.1957
28.2135
28.2312
28.2489
28.2666
28.2843
28.3019
28.3196
28.3373
28.3549
28.3725
28.3901
28.4077
28.4253
28.4429
28.4605
28.4781
28.4956
28.5132
28.5307
28.5482
^•5f57
28.5832
28.6182
28.6356
28.6531
28.6705
28.6880
28.7054
9.1696
91736
9-1775
9.1815
9.1855
9.1894
91933
9-1973
9.2012
9.2052
9.2091
9.2130
9.2170
9.2209
9.2248
9.2287
9-2326
9-2365
9.2404
9-2443
9.2482
9.2521
9.2560
92599
9-2638
9.2677
9.2716
9-2754
92793
9.2832
9.2870
9-2909
9.2948
9.2986
9-3025
9-3063
9.3102
9-3140
9-3179
9-3217
9-3255
9-3294
9-3332
9-3370
9-3408
9-3447
9-3485
9-3523
9-3561
9-3599
9-3§37
9-3675
9-3713
93751
2.88649
2.88705
2.88762
2.88818
2.88874
2^;
2.89042
2.89098
2^154
2.89209
2.89265
2.89321
2.89376
2.89432
2.89487
2.89542
2.89597
2.89653
2^708
2^73
2.89982
2.90037
2.90091
2.90146
2.90200
2.90255
2.90309
2-90363
2.90417
2.90472
2.90526
2.90580
2.90614
2.90687
2.90741
2.90795
2,90849
2.90902
2.90956
2.91009
2.91062
2.91 1 16
2.91 169
2.91222
2.91275
2.91328
2.91381
2.91434
2.91487
2.91540
2-91593
SmTHsoNiAN Tables.
18
Digitized byLjOOQlC
Tables.
VALUES OF RECIPROCALS, SQUARES, CUBES. SQUARE ROOTS, CUBE
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
000.^
v«
v«
log. «
825
826
827
828
829
830
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
P'
852
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
P'
872
P^
874
875
876
877
878
879
,21212
21065
20919
20773
20627
20482
20337
20192
20048
19904
19760
19617
19474
19332
19190
I8765
18624
18483
18343
18203
18064
17925
17786
17647
17509
17371
17233
17096
1^22
16686
16550
16414
16279
16144
16009
15875
1 5741
15607
15473
'5340
15207
15075
14943
14811
14670
14548
14416
14286
141 55
14025
680625
682276
683929
685584
6S7241
688900
690561
692224
693889
695556
700569
702244
703921
705600
707281
708964
710649
712336
71402c
715710
717409
719104
720801
722500
724201
725904
727609
729316
731025
732730
734449
736164
737881
739600
741321
743044
744769
746496
748225
749950
751689
753424
755161
756900
758641
760384
762129
763876
765625
767370
769129
770884
772641
561515625
563559970
565609283
567663552
569722789
571787000
573856191
575930368
578009537
580093704
582182875
584277056
586376253
588480472
590589719
592704000
599077107
60I2II584
6o335"2|
605495730
607645423
609800192
61 1960049
614x25000
61 629505 I
618470208
620650477
622835864
625026375
627222016
629422793
63 I 6287 I 2
633839779
636056000
638277381
640503928
642735647
644972544
647214625
649461896
651714363
653972032
656234909
658503000
600770111
663054848
665338617
667627624
669921875
672221370
6791 51439
28.7228
28.7402
28.7576
28.7750
28.7924
28.8097
28.8271
28.8444
28.8617
28.8791
28.8964
28.9137
28.9310
28.9482
28.9655
28.9828
29.0000
29.0172
29-0345
29.0517
29.0689
29.0861
29.1033
29.120^
29.1370
29.1548
29.1719
29.1890
29.2062
29.2233
29.2404
29.2575
29.2746
29.2916
29.3087
29.3258
29.3428
29.3598
29.3769
29-3939
29.4109
29-4279
29.4440
29.4618
29.4788
29.4958
295127
29.5296
29.5466
29-5635
29.5804
29-5973
29.6142
29.6311
29.6479
9-3789
9-3827
9-3865
9-3902
9-3940
93978
94016
9-4053
9.4091
9.4129
9.4166
94204
9.4241
9.4279
94316
9.4354
94391
9-4429
9.4466
9-4503
9.4541
9-4578
9.4615
9.4652
9.4690.
9.4727
94764
94801
9-4838
9-4875
9.4912
9.4949
9.4986
9-5023
9-5060
9.5097
9-5134
9-5171
9.5207
9-5244
9.5281
9.5317
9.5354
9-5391
9-5427
9-5464
9.5501
9-5537
9-5574
9.5610
9-5647
9.5683
9-5719
9-5756
9-5792
2.91645
2.91698
2.91751
2.91803
2.91855
2.91908
2.91960
2.92012
2.92065
2.921 17
2.92169
2.92221
2.92273
2.92324
2.92376
2.92428
2.92480
2.92571
2.(
2.C
2.92686
2.92737
2.92788
2.92840
2.92891
2.92942
2.92993
2.93044
2.93095
2.93146
2.93197
2.93247
2.93298
2.93349
2.93399
2.93450
2.93500
2.93551
2.93601
2.93651
2.93702
2.93752
2.93802
2.93852
2.93902
2.93952
2.94002
2.94052
2.94IOI
2.941 51
2.94201
2.94250
2.94300
2.94349
2.94399
Digitized by V^OOQ
SiiiTHSONiAN Tables.
19
Table;
VALUES OF RECIPROCALS, SQUARES, CUBES. SQUARE ROOTS, CUBE
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
880
88i
882
883
884
885
886
888
890
^'
892
593
894
896
899
900
901
902
903
904
905
906
909
910
911
912
913
9H
915
916
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
looa^
1.13636
1-13507
i-«3379
1.13250
1.13122
l!l2l
I.I2740
I.I26I3
I.I2486
I.I2360
I.I2233
I.I2I08
1.11982
I.I 1857
I.II732
I.I 1607
I.I 1483
i-"359
I.I 1235
i.iiiii
1.10988
1. 10865
1. 10742
1.10619
1.10497
I-I037S
1.10254
1.10132
I.IOOII
1.09890
1.09769
1.09649
1.09529
1.09409
1.08696
1.08578
1^)8460
1.08342
1.08225
I.08I08
1.07091
1.07875
1.07759
1.07643
1.07527
1. 0741 1
1.07296
I.07I8I
1.07066
774400
776I6I
777924
779689
781456
78322c
784996
786769
788544
790321
792100
793881
795664
797449
799236
801025
802816
804609
806404
808201
810000
811801
813604
815409
817216
819025
820836
822649
824464
826281
828100
829921
831744
833569
835396
837225
1^
842724
844561
846400
848241
850084
851929
853776
855625
857476
?|9329
861184
863041
864900
866761
868624
870489
872356
681472000
683797841
686128968
688465387
690807104
6931 54125
695506456
697864103
700227072
702595369
704969000
707347971
709732288
712121957
714516984
716917375
719323136
721734273
724150792
726572699
729000000
73432701
733870808
736314327
738763264
741 217625
743677416
746142643
748613312
751089429
753571000
756058031
758550528
761048497
763551944
766060875
768575296
771095213
773620632
776151559
778688000
781 229961
783777448
786330467
791453125
794022776
796597983
799178752
801765089
804357000
806954491
812106237
814780504
v»
29.6648
29.6816
29-6985
29-7153
29-7321
9.5828
9-5865
9-5901
9.5937
9-5973
29.7480
29.7658
29.7825
29-7993
29.8161
9.6010
9.61 18
9.6154
29.8329
29.8496
29.8664
29.8831
29.8998
29.9166
29-9333
29-9833
3aoooo
30.0167
30.0333
30.0500
30.0066
30.0832
30.0998
30.1164
30-1330
3ai496
30.1662
3ai828
30.1993
30-2159
30.2324
30.2490
30.2655
30.2820
30.2985
30-3150
30.3315
30.3480
30.3645
30.3809
30.3974
30.4138
30.4302
30.4467
30.4631
30.4795
30.4959
30-5123
30.5287
30,5450
30.5614
?«
9.6190
9.6226
9.6262
9.6298
9-6334
9.6370
9.6406
9.6442
9.6477
9-6513
9.6549
9.6585
9.6620
9.6656
9.6692
9.6727
9-6763
9.6799
9-6834
9.6870
9.6905
9.6976
9.7012
9-7047
9.7082
9.71 18
9.7188
9.7224
9.7259
9.7294
9.7329
9-7364
9.7400
9-7435
9-7470
9-7505
9.7540
9-7575
9.7610
9.7645
9.7680
9-7715
9-7750
log. H
2.94448
2.94498
2.94547
2.94596
2.94645
2.94694
2.94743
2.94792
2.94841
2.94890
2.94939
2.94988
2.95036
2.95085
2.95134
2.95182
2.95231
2.95270
2.95328
2.95376
2.95424
2.95472
2.95521
2.95569
2.95617
2.95665
2.95713
2.95761
2.95809
2.95856
2.95904
2.95952
2.95999
2.96047
2.96095
2.96142
2.96190
2.96237
2.96284
2.96332
2-9637?
2.96426
2.96473
2.96530
2.96567
2.96614
2.96661
2.96708
2.96755
2.96802
2.96S48
2.96895
2.96942
2.96988
2-97035
Smithsonian Tables.
20
Digitized by V^OOQlC
Table 3.
VALUES OF RECIPROCALS, SQUARES. CUBES. SOUARE ROOTS, CUBE
ROOTS, AND COMMON LOCARITHMS OF NATURAL NUMBERS.
lOOO^
v«
l«
log. «
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
9G0
964
965
966
969
970
971
972
973
974
975
976
979
980
^'
982
^3
984
986
i
989
1.06952
1.06838
1.06724
1.06610
1.06496
1.06383
1.06270
1. 061 57
1x3604s
1.05932
1.05820
1.05708
1.05507
1.05485
1-05374
1.05263
1-05152
1.05042
1.04032
1. 04022
1.047 1 2
1.04603
1.04403
1.04384
1.04275
1.04167
1.04058
1.03050
1.03832
1-03734
1.03627
1.03520
1.03413
1.03306
1-03199
1.02564
I-02459
1.02354
1.02249
ix)2i45
IU3204I
1.01037
1. 01833
I. 01 7 29
1. 01 626
1.01523
1. 01420
1.01317
1.01215
1^1112
874225
876096
877969
879844
881721
883600
885481
887364
889249
891 136
89302c
894Q16
898704
900601
902500
904401
906304
908209
910116
915849
917764
919681
921600
923521
925444
927369
929296
931225
933»S6
935089
937024
938961
942841
944784
946729
948676
950625
952576
954529
958441
960400
962361
964324
966289
968256
970225
972196
974169
976144
978121
SaiTNaoiiiAN Tables.
817400375
820025850
822658953
825293672
827936019
830584000
833237621
838561807
841232384
843908625
840590536
849278123
851971392
854670349
857375000
860085351
862801408
865523177
868250664
870983875
875722816
876467493
8792 I 791 2
881974079
884736000
887503681
890277128
893056347
895841344
898632125
901428696
904231063
907039232
909853209
91 267 wo
9I54986II
918330048
92II673I7
924010424
926859375
9297 141 76
932574833
935441352
938313739
941 192000
944076I4I
946066168
949862087
952763904
955671625
958585256
961504803
964430272
967361669
21
30-5778
30.5941
30.6105
30.6268
30.6431
30-6594
30.6757
30.6920
30.7083
30.7246
30.7409
30.7571
30.7734
30.7896
30.8058
30.8221
30.8383
30.8545
30.8707
30.8869
30.9031
30.9192
30.9354
30.9516
30.9677
30.9839
31.0000
3I.O161
31.0322
31-0483
31-0644
31.0805
31.0966
^;:;^
31.1448
31-1609
3i-»769
31.1929
31.2090
31.2250
31.2410
31.2570
31.2730
31.2890
31-3050
31.3209
31-3362
31-3847
31.4006
31-4166
31-4325
31.4484
9-7785
9.7819
^7854
9.7889
9-7924
9-7959
9-7993
9.8028
9.8063
9.8097
9.8132
9.8167
9.8201
9.8236
9.8270
9-8305
9-8339
9-8374
9.8408
98443
9-8477
9.8511
9.8546
9.8580
9.8614
9.8648
9-8683
9.8717
9-8785
9.8819
9§85
9.8922
9.8956
9-8990
9.9024
9.9058
9.9092
9.9126
9.9160
9.9194
9.9227
9.9261
9-9295
9-9329
99363
9-9390
•9-9430
9.9464
9-9497
99531
9-9565
9-9598
9-9032
2.97081
2.97128
2.97174
2.97220
2.97267
2.97313
2-97359
2.97405
2.97451
2.97497
2.97543
2.97589
2.97635
2.97681
2.97727
2.97772
2.97818
2.97864
2.97909
2.97955
2.98000
2.98046
2.98091
2.98137
2.98182
2.98227
2.98272
2.98318
t$S
2.98453
2.98498
2.i "
2.985s
2.9S032
2.98677
2.98722
2.98767
2.9881 1
2.98856
2.98900
2.98945
2.98989
2.99034
2.99078
2.99123
2.99167
2.992 1 1
2.99255
2.99300
2.99344
2.99388
2.99432
2.99476
2.99520
Digitized by LjOOQ
Table 3.
VALUES OF RECIPROCAL81 SQUARES. CUBES. SQUARE ROOTS. GUI
ROOTS, AND COMMON LOGARITHMS OF NATURAL NUMBERS.
n
looa^
««
««
v«
>
log. «
990
991
992
993
994
995
996
999
1000
1.01010
1.00908
1.00806
1.00705
1.00604
1.00503
1.00402
1.00301
1.00200
1.00100
1.00000
980100
982081
990025
992016
994009
996004
998001
lOOOOOO
970299000
973242271
976I9I488
979146657
982107784
^5074875
988047936
991026973
99401 1992
997002999
31-4643
314802
31.4960
3i-5"9
31.5278
31-5436
31-5595
31-5753
31-59"
31-6070
31.6228
9.9666
9-9699
9-9733
9.9766
9.9800
»
9.9900
9-9933
9-9967
10.0000
2.99564
2.99607
2.99651
2.99695
2-99739
2.99782
2.99826
2.99870
2.99913
2.99957
3.00000
lOOOOOOOOO
SiiiTH«ONiAN Tables.
22
Digitized by
GooqIc
CIRCUMFERENCE AND AREA OF CIRCLE
DIAMETER d.
IN
Table 4.
TERMS OF
d
Md
iw^
d
'Wd
\'wd^
d
wd
\'wd^
10
II
12
3I-4I6
34.558
37-899
78.5398
95-0332
113.097
40
41
42
125.66
128.81
131.95
1256.64
i32a25
1385.44
70
71
72
219.91
223.05
226.19
3848.45
3959.19
4071.50
'3
IS
40.841
43-982
47.124
132.732
153-938
176.715
43
44
45
141.37
1452.20
1520.53
1590.43
73
74
75
229.34
232.48
235.62
4185.39
4300.8A
4417.86
i6
50.265
53-407
56.549
201.062
226.980
254.469
46
144.51
147.65
150.80
1661.90
1809.56
76
238.76
241.90
245.04
4656!63
4778.30
19
20
21
65-973
283.529
346.301
49
50
51
153-94
157.08
160.22
1885.74
1963-50
2042.82
§2
81
248.19
251-33
254.47
4901.67
5026.55
5» 53-00
22
23
24
69.115
72.257
75.398
38ai35
415.476
452.389
52
53
54
163.36
166.50
169.65
2123.72
2206.18
2290.22
82
257.61
5281.02
5410.61
5541.77
27
84^23
490.874
530.929
572.555
57
172.79
175-93
179.07
2375.83
2463.01
2551.76
11
87
267.04
270.18
27332
5944.68
28
29
30
87.965
91.106
94.248
615.752
660.520
706^58
58
182.21
2642.08
2733-97
2827.43
88
89
90
^
6082.12
6221.14
6361.73
31
32
33
97.389
loaja
103.67
804.248
855.299
61
62
63
191.64
194.78
197.92
2922.47
3019.07
3"7.25
91
92
93
285.88
289.03
292.17
6503.88
6647.61
6792.91
34
106.81
109.96
113.10
907.920
1017.88
201.06
204.20
207.35
3216.99
3318.31
3421.19
94
295.31
298.45
30159
^1!
7238.23
39
116.24
119.38
122.52
1075.21
1134-11
1194.59
%
69
210.49
213.63
216.77
3739.28
99
307'i8
311.02
7389^
7097-09
BamtMHMN Tabus.
n
Digitized by
GooqIc
Table 5.
LOGARITHMS OF NUMBERS.
N.
12 3 4
5 6 7 3 9
Prop. Parts.
123 456 789
10
II
12
13
M
0000 0043 0086 0128 0170
0414 0453 0492 0531 0569
0792 0828 0864 0899 0934
"39 ^^73 1206 1239 1271
1461 1492 1523 1553 1584
0212 0253 0204 0334 0374
0607 0645 0682 0719 0755
0969 1004 10^8 1072 I 100
1303 1335 ^^7 1399 1430
1614 1644 1673 1703 1732
4 8 12
4 8 II
3 710
3 6 10
369
17 21 25
15 19 23
14 17 21
13 16 10
12 15 18
293337
263034
242831
23 26 29
21 2427
15
i6
\l
19
1761 1790 1818 1847 1875
2041 2068 2095 2122 2148
2304 2330 235s 2380 2405
2553 2577 2601 2625 2648
2788 2810 2833 2856 2878
1903 1931 1959 1987 2014
2175 2201 2227 2253 2279
2430 2455 2480 2504 2529
2672 2695 2718 2742 2765
2900 2923 2945 2967 2989
368
3 5 8
257
257
247
II 14 17
II 13 16
10 12 15
9 12 14
911 13
20 22 25
18 21 24
17 20 22
16 19 21
16 18 20
20
21
22
23
24
3010 3032 30S4 3075 3096
3222 3243 3263 3284 3304
3424 3444 3464 3483 3502
3617 3636 365s 3674 3892
3802 3820 3838 3856 3874
31 18 3139 3160 3181 3201
3324 3345 3365 3385 3404
3522 3541 3560 3579 3598
37" 3729 3747 3766 3784
3892 3909 3927 3945 3962
2 4 6
2 4 6
2 4 6
2 4 6
2 4 5
8
8
8
7
7
" 13
10 12
10 12
911
911
15 17 IQ
14 16 18
14 15 17
13 15 17
12 14 16
25
26
29
3979 3997 4014 4031 4048
4150 4166 418 J 4200 4216
4314 4330 4340 4362 4378
4472 4487 4502 4518 4533
4624 4639 4654 4069 4683
4065 4082 4099 4116 4133
4232 4249 4265 4281 4298
4393 4409 4425 4440 4456
4548 4564 4579 4594 4609
4698 4713 4728 4742 4757
235
2 3 5
235
2 3 5
I 3 4
7
I
6
6
1
8
8
7
10
10
9
9
9
12 14 15
"I315
II 13 14
II 12 14
10 12 13
30
31
32
33
34
4771 4786 4800 4814 4829
4914 4928 4942 4955 4969
5051 506? 5079 5092 5105
5185 5198 5211 5224 5257
53' 5 5328 5340 5353 5366
4843 4857 4871 4886 4900
4983 4997 50" 5024 5038
5119 5132 5145 5159 5'72
5250 5263 5270 5289 5302
5378 5391 5403 5416 5428
I 3 4
I 3 4
I 3 4
I 3 4
I 3 4
6
6
7
7
I
6
1
8
8
8
10 II 13
10 II 12
9 II 12
9 10 12
910 II
35
36
39
5441 5453 5465 5478 5490
5563 5575 5587 5599 56"
5682 5694 5705 S7J7 5729
5798 5809 5821 5832 5843
59" 5922 5933 5944 5955
5S02 5514 5527 5539 5551
5623 563s 5647 5658 5070
5740 5752 5763 5775 5786
5855 5866 5877 5888 C899
5966 5977 5988 5999 0010
I 2 4
I 2 4
I 2 3
I 2 3
I 2 3
6
6
6
6
5
9 10 II
8 ion
8 9 10
8 9 10
8 9 10
40
41
42
43
44
6021 6031 6042 6053 6064
6128 6138 6149 6160 6170
6232 6243 6253 6263 6274
6335 6345 6355 6365 6375
6435 6444 6454 6464 6474
<5o75 6085 6096 6107 6117
6180 6191 620X 6212 6222
6284 6294 6304 6314 6325
6385 6395 6405 641S 6425
6484 6493 6503 6513 6522
I 2 3
I 2 3
I 2 3
I 2 3
I 2 3
6
6
6
6
6
8 9 10
789
789
789
789
45
46
%
49
6532 6542 6551 6561 6571
6628 6637 6646 6656 6665
6721 6730 6739 6749 6758
6812 6821 6830 6839 6848
6902 6911 6920 6928 6937
6580 6500 6599 6609 6618
6075 6684 6803 6702 6712
6946 6955 6964 6972 6981
I 2 3
I 2 3
I 2 3
I 2 3
I 2 3
6
6
5
5
5
780
I 7 !
678
678
678
50
51
52
53
54
6990 6908 7007 7016 7024
7076 7084 7093 7101 7 no
7160 7168 7177 7185 7193
7243 7251 7259 7267 7275
7324 7332 7340 7348 7350
703; 7042 7050 70597067
7118 7126 7135 7143 7152
7202 7210 7218 7220 7235
7284 7292 7300 7308 7310
7364 7372 7380 7388 7396
I 2 3
I 2 3
122
122
I 2 2
3
3
3
5
5
5
5
5
678
678
^ I 7
667
667
N.
12 3 4
5 6 7 3 9
12 3
4
5
6
7 8 9
Shitn«onian Tables.
24
Digitized by
Google
Table 5.
LOGARITHMS OF NUMBERS.
55
56
59
60
61
62
64
65
66
67
68
69
70
71
72
73
74
75
76
7I
79
80
81
82
t3
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
7404 7412 7419 7427 7435
7482 7490 7497 7505 7513
7SS9 7566 7574 7582 7589
7634 7642 7649 7657 7664
7709 7716 7723 7731 7738
7782 7789 7796 7803 7810
7853 7860 -
7853 7§66 78^ 7875 7882
7924 7931 7938 7945 7952
7993 8000 8007 8014 8021
8062 8069 8075 8082 8089
8129 8136 8142 8149 8156
8195 8202 8209 8215 8222
8261 8267 8274 8280 8287
8325 8331 8338 8344 8351
8388 8395 8401 8407 8414
8451 8457 8463 8470 8476
8513 8519 8525 8531 8537
8921 8927 8932 8938 894;
8976 8982 8987 8993 899b
9031 9036 9042 9047 9053
9081; 9090 9096 9101 9106
9138 9143 9149 9154 9159
9191 9196 9201 9206 9212
9243 9248 9253 9258 9263
9294 9299 9304 9309 93JS
9345 9350 9355 93^© 93^5
9395 9400 9405 9410 9415
9445 9450 9455 946o 9465
9494 9499 9S04 95^9 95^3
9542 9547 9552 9557 95^2
9590 9595 9500 9605 9609
9S38 9043 9647 9652 9657
, ^5 9689 9694 9699 9703
9731 9736 9741 9745 9750
9777 9782 9786 9791 9795
9823 ^27 9832 9836 9841
98^ 9872 9877 9881 9886
9912 9917 9921 9926 9930
9956 9961 9965 9969 9974
8 9
7443 7451 7459 7466 7474
7520 7528 7536 7543 7551
7597 7004 7012 7619 7027
7672 7679 7686 7694 7701
7745 7752 7760 7767 7774
7818 7825 7832 7839 7846
7889 7896 7903 7910 7917
7959 7966 7973 7980 7987
8028 8035 8041 8048 8055
8096 8102 8109 8116 8122
8162 8169 8176 8182 8189
8228 8235 8241 8248 8254
8293 8299 8306 8312 8319
8357 8363 8370 8376 8382
8420 8426 8432 8439 8445
8482 8488 8494 8500 8506
8543 8549 8555 8j6i 8^67
8603 8609 8615 8621 8627
8663 8669 8675 8681 8686
8722 8727 8733 8739 8745
8779 8785 8791 8797 8802
8837 8842 8848 8854 8859
8893 8899 8904 8910 8915
8949 8954 8960 8965 8971
9004 9009 9015 9020 9025
9058 9063 9069 9074 9079
9112 9x17 9122 9128 9x33
9165 9170 9175 9180 9186
9217 9222 9227 9232 9238
9269 9274 9279 9284 9289
9320 9325 9330 9335 9340
9370 9375 9380 9385 9390
9420 9425 9430 9435 9440
9469 9474 9479 9484 948q
9518 9523 9528 9533 9538
9566 9571 9576 958 X 9586
96x4 96x9 9624 9628 9633
966X 9666 967X 9675 9680
9708 9713 97x7 9722 9727
9754 9759 9763 9768 9773
9800 9805 9809 9814 98x8
9845 9850 9854 9859 9863
9890 9894 9899 9903 9908
9934 9939 9943 9948 9952
9978 9983 9987 9991 9996
Prop. Farts.
12 3 4
3
3
3
3
3
2 2
2 2
2 2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3 4 4
3 4 4
3 3 4
3 3 4
3 3 4
3 3 4
3 3 4
3 3 4
3 3 4
234
8 9
6 7
4 5
4 5
4 4
4 4
4 4
6 6
6 6
6 6
5 6
4 4
4 4
4 4
4 4
4 4
12 3
6 7 8 9
123 456 789
8HmMoiiiAN Tables.
2S
Digitized by
Google
Table 6.
ANTILOCARITHMS.
.
12 3 4
5 6 7 8 9
Prop. Parts.
123 456 789
.00
1000 1002 looq 1007 1009
1023 1026 1025 1030 1033
1047 1050 1052 1054 ioi;7
1072 1074 1076 1079 looi
1096 1099 X102 I 104 I 107
10x2 1014 1016 1019 1021
]
2 2 2
.01
.02
•03
.04
1035 1038 1040 1042 1045
ioj9 1002 1064 1067 1069
1084 10S6 1089 1091 1094
1109 1112 1114 1117 1119
]
]
]
1 1
2 2 2
2 2 2
2 2 2
2 2 2
.05
.06
1122 1125 1127 1130 1132
1148 1151 1153 11156 IIW
1175 "78 "00 '^83 1186
1202 1205 1208 1211 1213
1230 1233 1236 1239 1242
1135 1138 1140 1143 1146
1161 1164 1167 1169 1172
2
2
2 2 2
2 2 2
•09
1189 1191 1194 1197 1199
1216 1219 1222 1225 1227
I24S 1247 1250 1253 1256
2
2
2
2 2 2
223
223
J.0
.11
.12
.14
I2j9 1262 1265 1268 1271
1288 I29I 1294 1297 1300
I3I8 I32I 1324 1327 1330
1349 1352 1355 1358 I36I
1380 1384 1387 1390 1393
1274 1276 1279 1282 1285
1303 1306 1309 1312 1315
1334 1337 1340 1343 1340
1365 1368 1371 1374 1377
1396 1400 1403 1406 1409
2
2
2
2
2
223
223
2 2 3
2 3 3
233
.15
.16
.19
1413 1416 1419 1422 1426
1445 1449 1452 I4S5 1459
1479 14^3 i486 1489 1493
1514 1517 1521 1524 1528
1549 1552 1556 1560 1563
1429 1432 1435 1439 1442
1462 1466 1469 1472 1476
1496 1500 1503 1507 1510
1531 1535 1538 1542 1545
1567 1570 1574 1578 1581
2
2
2
2
2
2
2
2
2
2
233
233
233
233
3 3 3
.20
.21
.22
•23
.24
1585 1589 1592 1596 1600
1622 1626 1629 1633 1637
1660 1663 1667 1671 1675
1698 1702 1706 1710 1714
1738 1742 1746 1750 1754
1603 1607 161 I 16x4 1618
1641 1644 1648 1652 X656
1679 1683 X687 1690 1694
1718 1722 1726 1730 1734
1758 1762 X766 1770 1774
[ I
[ 2
[ 2
[ 2
[ 2
2
2
2
2
2
2
2
2
2
2
3 3 3
3 3 3
3 3 3
3 3 4
3 3 4
.25
.26
.29
1778 1782 1786 1791 1795
1820 1824 1828 1832 1837
1862 1866 1871 1875 1S79
1905 1910 1914 1919 1923
1950 1954 1959 1963 1968
1799 1803 X807 x8ii 1816
1841 1845 X849 1S54 1858
1884 1888 1892 X897 1901
X928 1932 1936 X941 1945
X972 X977 1982 1986 1991
2
2
: 2
2
[ 2
2
3
3
3
3
3 3 4
3 3 4
3 3 4
3 4 4
3 4 4
.30
•31
•32
•33
•34
1995 2000 2004 2009 2014
2042 2046 2051 2056 2061
2080 2094 2099 2104 2109
2118 2143 2148 2153 2158
2188 2193 2198 2203 2208
20x8 2023 2028 2032 2037
2065 2070 2075 2080 20J4
2XX3 2118 2123 2128 2x33
2163 2168 2173 2178 2183
2213 22X8 2223 2228 2234
I
2
2
2
2
t 2
2
2
2
2
3
3
3
3
3
3
3 4 4
3 4 4
3 4 4
3 4 4
4 4 5
.35
.36
:^3^
•39
2239 2244 2249 2254 2259
2291 2296 2301 2307 2312
2344 2350 23SS 2360 2366
2399 2404 2410 2415 2421
2455 2460 2466 2472 2477
2265 2270 2275 228a 2286
2317 2323 2328 2333 ^339
2371 2377 2382 2388 2393
2427 2432 2438 2443 2449
2483 2489 2495 2500 2506
t 2
I 2
t 2
t 2
t 2
3
3
3
3
3
3
3
3
3
3
4 4 5
4 4 5
4 4 5
4 4 5
4 5 5
.40
.41
•42
•43
.44
2512 2518 2523 2529 2535
2570 2576 2582 2588 2594
2630 2636 2642 2649 2655
2692 2698 2704 2710 2716
2754 2761 2767 2773 2780
2000 2006 26x2 2018 2624
266X 266> 2673 2679 2685
2723 2729 2735 2742 2748
2786 2793 2799 2805 28x2
t 2
I 2
I 2
i 3
^ 3
3
3
3
3
3
4
4
4
4
4
4 5 5
4 5 f
4 5 6
4 5 6
.45
.46
its
.49
2818 2825 2831 2838 2844
2884 2891 2897 2904 291 1
2951 2958 2965 2972 2970
3020 3027 3034 3041 3048
3090 3097 3105 3"2 31 19
2851 2858 2864 2871 2877
29x7 2924 2931 2938 2944
2985 2992 2999 3006 30x3
305s 3062 3069 3076 3083
3126 3x33 3x41 3148 3155
i 3
*• 3
*' 3
i 3
*' 3
3
3
3
4
4
4
4
4
4
4
5 5 6
5 5 6
ut
566
la.
12 3 4
5 6 7 8 9
1
2 £
I 4
5
6
7 8 9
8mitn«onian Tabus.
26
Digitized by
Google
ANTILOGARITHM8.
Table 6.
.50
•51
•52
•53
•54
.55
.56
:^
•59
.60
.61
.62
.64
.65
.66
.67
.68
.69
.70
•71
•72
•73
•74
.75
.76
.78
79
30
.81
.82
.84
.85
.86
i
.89
30
•91
.92
•93
•94
.95
•96
•97
-98
•99
3162 3170 3177 3184 3192
3236 3243 32S1 3258 3266
33" 3319 3327 3334 3342
3388 3396 3404 3412 3420
3467 3475 3483 3491 3499
3548 3556 3565 3573 3581
3031 3039 3048 3050 3664
3715 3724 3733 3741 3750
3802 381 I 3810 3828 3837
3890 3899 3908 39»7 3926
3981 3900 3999 4009 4018
4074 4083 4003 4102 41 I I
4169 41 78 4188 4198 4207
4266 4276 4285 4295 4305
4365 4375 4385 4395 44o6
4467 4477 4487 4498 4508
4013
4571 4581 4i
4677 4688 4(
4786
4898
86 4797
192 4603
4710 4721
4819 4831
4909 4920 4932 4943
5012 5023 5035 5047 5058
512Q 5140 5152 5164 5176
5248 5260 5272 5284 5297
5370 5383 5395 5408 5420
5495 5508 5521 5534 5546
5623 5636 5649 5662 567s
5754 5768 5781 5794 5808
5888 5902 5916 5929 5943
6026 oo'j9 6053 6067 6081
6166 6180 6194 6209 6223
6310 6324 6339 6353 6368
6457 6471 6486 6501 6516
6607 6622 6637 6653 6668
6761 6776 6792 6808 6823
6918 6934 6950 6966 6982
7079 7096 7112 7129 7145
7244 7261 7278 7295 731 1
7413 7430 7447 7464 7482
7586 7603 7621 7638 7656
7762 7780 7798 7816 7834
7943 7962 7980 7998 8017
8128 8147 8166 8185 8204
8318 8337 8356 8375 8395
851 I 8531 8551 8570 8590
8710 8730 8750 8770 8790
8913 8933 8954 8974 8995
9120 91 41 9162 9183 9204
9333 9354 9376 9397 94i9
9550 9572 9594 9616 9638
9772 9795 9817 9840 9863
3199 3206 3214 3221 3228
3273 3281 3289 3296 3304
3350 3357 3365 3373 338i
3428 3436 3443 3451 3459
3508 3516 3524 3532 3540
3589 3597 3606 3614 3622
3673 3681 3690 3698 3707
3758 3767 3776 3784 3793
3846 3855 3864 3873 3882
3936 3945 3954 3963 3972
4027 4036 4046 4055 4064
41 2 I 4130 4140 4150 4159
4217 4227 4236 4246 4256
4315 4325 4335 4345 4355
4416 4426 4436 4446 4457
4519 4529 4539 4550 4560
4624 4634 4645 4656 4667
4732 4742 4753 4764 4775
4842 4853 4864 4875 4887
4955 4966 4977 4989 5000
5070 5082 5093 5T05 51 17
5188 5200 5212 5224 5236
5309 5321 5333 5346 5358
5433 5445 5458 547© 5483
5559 5572 5585 5598 5610
5689 5702 5715 5728 5741
6237 6252 6266 6281 6295
6383 6397 6412 6427 6442
65;}! 6546 6561 6577 6592
6683 6699 6714 6730 6745
6832 6855 6871 6887 6902
7015 7031 7047 7063
7161 7178 7194 7211 7228
7328 7345 7362 7379 7396
7499 7516
7674 7691 7709 7727 7745
7534 7551 7568
o «' 77097727 7745
7852 7870 7889 7907 7925
8035 8054 8072 8091 81 10
8222 8241 8260 8279 8299
8414 8433 8453 8472 8492
8610 8630 8650 8670 8690
8810 8831 8851 8872 8892
9016 9036 9057 9078 9099
9226 9247 9268 9290 931 I
9441 9462 9484 9506 9528
9661 9683 9705 9727 9750
9886 9908 9931 9954 9977
Prop. Farts.
12 3 4 5 6
1 2
2 2
2 2
2 2
2 2
2 2
2 3
2 3
2 3
2 3
2 3
2 3
2 3
2 3
2 3
2 3
2 3
2 3
2 3
2 3
3 4
3 4
3 4
3 4
3 4
246
246
2 4 7
247
257
456
456
456
4 5 6
456
456
456
^ k 7
467
567
6 7
6 7
6 7
6 8
6 8
8 10
8 10
8 10
9 10
911
911
7 911
8 911
8 10 12
8 10 12
8 10 12
8 10 12
8 II 13
9" 13
9" 13
911 14
789
7 910
8 9 10
8 9 10
8 9 10
8 911
8 10 II
9 10 II
9 10 II
9 10 12
9 10 12
9 II 12
10 II 12
10 II 13
10 II 13
10 12 13
11 12 14
II 12 14
11 13 14
" 13 15
12 13 15
121315
12 14 16
12 14 16
13 14 16
13 15 17
13 "517
14 15 17
14 10 18
14 16 18
15 17 19
15 17 19
15 17 20
16 18 20
16 18 20
I.. 012 34 5 6 78 9 123456789
8iiiTH«ONiAN Tables.
27
Digitized by
Google
Table 7.
NATURAL 8INE8 AND C08INE8.
Natural Sines.
MMg^
or
icr
2xy
3or
4or
SO'
€or
Aaate.
ter.,
(P
.002909
.0058 18
.008727
.011615
.02908
.014544
.0174 52
89»
2.9
I
.0174 52
.020-J6
.0378 1
.0232 7
.0261 8
•03199
.03490
88
2.9
2
^3490
.04071
.04362
.04653
•°i?J3
ll
2.9
3
4
^76
.07266
.05814
•07556
.06105
.07846
.08136
.06685
.08426
86
85
2.9
2.9
5
.08716
•09005
.09295
•09585
•09874
•'°i^5
.10453
.12187
84
2.9 !
6
.10453
.12187
.10742
.11031
.12764
.11320
.11609
.11898
l^
2.9
7
.1247 6
.13053
.1478
.1650
•1334
•'530
.1708
.1392
82
2.9
8
9
:;is
.1421
•1593
.1449
.1622
.1507
.1679
.1564
.1736
8i
80
2.9
2.9
10
.1736
.1765
.1794
.1822
.1851
.1880
.1908
79
2.9
II
.1908
•'937
.2108
\f^
•1994
.2022
.2051
.2079
78
:j
12
.2079
.2164
•2193
.2221
.2250
77
13
.2250
.2278
.2306
•2334
.2363
•2391
^^
76
2.8
14
.2419
.2447
.2476
.2504
.2532
.2560
75
2Z
15
i6
.2588
.2756
.2616
•2784
.2644
.2812
.2672
.2840
.2700
.2868
.2728
.2756
.2924
74
73
2.8
2.8
\l
.2924
.2952
.2979
.3007
.3035
.3062
.3090
72
2.8
•3090
.3118
.3145
:^33^
.3201
.3228
.3256
71
2.8
19
•3256
.3283
•33"
.3365
•3393
.3420
70
2.7
20
21
.3420
•3740
.3448
.3611
■m
:^5
•^9
•3557
•3584
.3746
69
68
2.7
2.7
22
•3773
.3800
.3827
^3854
.3907
^
2,7
23
•3957
•3934
•3961
•3987
.4014
.4041
•4067
66
2.7
24
.4067
•4094
.4120
•4147
•4173
4200
4226
65
2.7
25
.4226
•4253
•4279
•4305
:J^
•4358
.4384
64
2.6
26
4384
4410
.4436
.4462
'¥
.5000
63
2.6
^Z
'£
4566
•4592
.4617
•4643
62
2.6
28
29
4720
•4874
•4746
•4B99
.4772
4924
.4797
•4950
.4823
•4975
61
60
2.6
2.5
30
.5000
•5025
.5050
•5075
.5100
.5125
.5150
59
2.5
31
.5150
•517s
.5200
.5225
.5250
•5275
.5299
.5446
58
2.5
32
•5299
•5324
•53*8
•5373
•5398
•5422
57
2.5
33
.5446
.5471
•5495
:i^
:l^
.5568
.5592
56
24
34
•5592
.5616
.5640
.5712
•5736
55
2.4
35
•573f
.5760
.5783
.5807
.5831
•5854
.6157
54
2.4
36
.6018
*g?}
'K\
;^
•5972
.6111
•5995
.6134
53
52
2.3
2.3
.6157
!6i8o
.6202
.6225
.6248
.6271
g'J
51
2.3
39
.6293
.6316
■6338
.6361
•6383
.6406
50
2-3
40
41
.6428
.0091
.6820
M
S
.6494
.6626
s
.6670
.6561
.6691
49
48
2.2
2.2
42
43
•6734
.6862
:a
.6777
.6905
.6799
.6926
.6820
.6947
%
2.2
2.1
44
.6947
.6988
.7009
•7030
.7050
.7071
45
2.1
W
w
W
30^
20^
lor
a
Aigla
8HiTHa6NiAN Tables.
Natural Cosines.
28
Digitized by
Google
Tamx 7.
NATURAL SINES AND COSINES.
Natural Sines.
AaglA.
cr
lor
2or
ao'
4or
w
w
Aigia
furl/.
45<>
.7071
.7092
.7112
•7133
•7153
•7173
•7193
44«
2J0
46
•7193
.7214
.7234
•7254
•7274
.7294
.7314
43
2J0
^l
.7314
•7333
•7353
•7373
.7392
.7412
.7431
42
2,0
48
49
.7431
.7547
■^%
.7470
•7585
.7604
•7509
.7623
■i&
:^l£
41
40
1.9
1-9
50
51
.7660
•7ZZ'
.7679
•7^
.7808
:^^
7844
m
v&
39
38
;i
52
S3
.7880
.7986
.7916
A)2I
•7934
•8039
:gp
•7969
.8073
.7986
.8090
36
1.8
1-7
54
.8090
^107
.8X24
.8x41
.8158
.8175
^192
35
1-7
55
^192
SzdS
.8225
.8241
.8258
.8274
^290
-8387
34
1.6
56
.8290
•f?7
•8307
Sf^
•8339
.8355
•8371
33
1.6
59
.8434
.8450
.8465
^480
32
1.6
^480
^572
1^
•851 1
.8526
iei6
K
:S
&
31
30
1-5
15
eo
61
62
.8660
.8S43
.8689
^704
.8870
^718
^2
.8884
.8897
.8910
29
28
27
1.4
14
14
f3
^l^
.8923
.8936
•8949
^2
•8975
.89S8
26
13
64
.8988
.9001
•9013
.9026
•9038
•9051
•9063
25
13
65
•9063
•9075
.9088
.9100
.9112
.9124
•9135
24
1.2
66
•9135
•9147
•9159
.9228
.9171
.9182
.9194
.9205
23
1.2
g
•9205
.9216
•9239
.9250
.9261
.9272
22
I.I
.9272
.9283
•9340
•9293
.9358
•9304
•9315
.9325
•9336
21
I.I
69
•9336
•9367
•9377
•9387
.9397
20
1.0
70
•9397
.9407
.9417
.9426
•9436
•9446
•9455
19
1.0
71
•9455
.9465
•9474
.9483
.9492
.9502
•95"
18
0.9
72
.9511
.9520
.9528
•9546
•9^5
•9563
•9513
.9659
17
0.8
73
74
•9563
•9513
•9572
:^
:^
•9596
•9044
.'9652
16
15
75
•9659
.9667
•9^4
.9681
.9689
.9696
•9703
14
0.7
76
•9703
•9710
•9717
•9724
•97^
•9737
•9744
13
0.7
77
•9744
.97|o
.9757
•9763
•9775
•9781
12
0.6
78
79
tl
•9799
.9833
•983^
.9811
•9843
.9816
.9848
II
10
0.6
0.5
80
81
.9848
•9877
®?
:»
.9863
•9890
.9868
.9894
.9918
.9872
•9899
.9877
•9903
9
8
0.5
0.4
82
•9903
.9907
•99"
.9914
.9922
.9925
I
04
P
•9925
.9920
•9932
•9936
•9939
.9942
•9945
6
0.3
84
•9945
.9948
•9951
•9954
•9957
•9959
.9962
5
03
85
.9962
•9964
•9967
•9969
•9971
•9974
■^
4
0.2
86
«
•9978
.9980
.9981
•9983
•9985
3
0.2
fs
.9988
•9989
.9990
•9992
•9993
.9998
•9994
2
0.1
88
•9994
.9998
•9995
.9996
•9997
•9997
•9998
I
0.1
89
•9999
•9999
X.0000
1.0000
1.0000
1.0000
0.0
eor
50^
40^
30"
2xy
lO'
or
Aafto.
SamMONiAN Tables.
Natural Cosines.
29
Digitized by
Google
TAMX8.
NATURAL TANGENTS AND COTANGENTS.
Natural Tangents.
icr
W
da
w
5or
eor
Aagla
Pnp. Fnti
fori'.
29
IS
39
.00000
.01746
.03492
.0524 I
•06993
.08749
.1051 o
.12278
.1405
.1584
•1763
.1944
.2126
•2309
•2493
.2679
.2867
•3057
•3249
•3443
.3640
•3839
.4040
•4245
•4452
.4663
.4877
•5095
•5317
•5543
(774
.6249
.6494
.6745
.7002
.7265
.7530
•7813
•8^3
.9004
.00291
.020 J 6
•03783
•05533
•07285
GOT
•1793
•1974
.2156
•2339
•2524
•9713
SOT
Smithsonian Tabus.
.00582
.02328
.04075
.05824
.07578
•09335
1099
2869
1465
1644
1823
2004
2186
2370
2555
2742
2931
3121
3314
3508
3706
3906
.4108
43»4
•4522
4734
.4950
5169
5392
5619
•6330
:^^
7089
•7355
7627
•7907
•8195
.8491
.8796
.9110
•9435
.9770
4€r
.00873
U326i 9
.04366
.0611 6
.07870
JO9629
"394
I3>6 5
1495
1673
1853
2035
2217
2401
2586
2773
2962
3153
3340
3541
3739
3939
4142
4348
4557
,4770
4986
5206
5^30
5658
128
6171
,6619
.6873
7133
7400
7673
7954
8243
.9163
.9490
.9827
da
.01 16 4
.0291 o
.04658
.06408
.08163
.09923
.11688
.1346
.1524
•1703
.1883
.2065
.2247
.2432
.2617
.2805
.2994
.3185
.3378
•3574
•3772
•3973
.4176
•4383
•4592
.4806
.5022
•5243
.5467
.5696
m
.6412
.6661
.6916
•7177
•7445
.7720
.8002
.8292
•?59'
.9217
•9545
.0145 5
.0320 I
.04949
.06700
.0845 6
.1021 6
.11983
.1376
•«554
•1733
.1914
.2095
.2278
.2462
.2648
.2836
.3026
.3217
•34"
•3607
.40
.4210
.4417
.4628
4841
5505
5735
208
6453
.6703
,6959
7221
7490
7766
.8050
.8342
.8642
.8952
9271
9601
994a
.01746
•03492
.0524 I
.06993
•08749
1051 o
12278
1405
1584
1763
1944
2126
2309
2493
2679
2867
3057
3249
3443
,3640
3839
,4040
4245
4452
.4663
4877
5095
5317
5543
5774
.6009
.6249
6494
6745
7002
7265
753S
&
8391
8693
9004
9325
9057
87
86
85
84
P
82
81
80
79
78
7^^
75
74
73
72
71
70
68
67
66
65
64
62
61
60
59
58
II
55
54
S3
52
51
SO
49
48
47
46
45
2a
la
Aagtk,
2.9
2.9
2.9
2.9
2.9
2.9
2-9
>o
3-0
3^0
3-0
3-0
3-1
3-«
3-1
3-2
3^2
32
3-3
3-3
34
3-4
3-5
3-5
3^8
3-9
4.0
4.1
4-2
4-3
4-4
n
4-7
49
S^o
5-2
5^4
55
5-7
Natural CoUngentt.
30
Digitized by
Google
NATURAL TANGENTS AND COTANGENTS.
Natural Tangents.
Table S.
t^fit.
cr
lor
2or
dor
%or
50^
6a
PZ0P. Ptzti
45°
1.0000
1.0058
1. 0117
1. 01 76
1.0235
1.0295
1-0355
44«
^?
46
I-035S
1.0416
1.0477
1.0538
1.0599
1.0661
1.0724
43
%
1.0724
1.0786
1.0850
1.0913
1.0977
1.1041
1.1106
42
it
I.I 106
1.1171
1.1237
1.1303
1.1708
1.1369
1.1778
1.1436
1.1504
41
49
1.1504
1.1571
1. 1640
1.1847
1.1918
40
6.9
50
1.1918
1.1988
1.2059
1.2131
1.2203
1.2276
1-2349
39
7-2
51
1.2349
1.2876
1.2497
1.2572
1.2647
1.2723
1.2799
38
7-5
52
1.2799
1.2954
1.3032
1.311 1
;:i^
1.3270
37
1%
53
1.3270
\M
1-3432
1-3514
1-3597
1.3764
36
54
1-3764
1-3934
1.4019
1.4106
MI93
1.4281
35
8.6
55
1.4281
M370
1.4460
1.4550
1.4641
1-4733
1.4826
34
9.1
56
1.4826
1.4919
1.5013
1.5108
1.5204
1.5301
1-5399
1.6003
1.6643
33
9.6
P
1-5399
1.6003
1.5497
1. 61 07
1-5597
1.6212
1.5697
1.6319
1.6426
1.5900
1-6534
32
31
10.1
10.7
59
1.6643
1-6753
1.6864
1.6977
1.7090
1.7205
1.7321
30
"-3
60
1.7321
1.8 165
1.7556
1.8418
1.7796
1-7917
1.8040
29
12.0
61
1.8291
1,8546
1.8676
1.8807
28
12.8
62
\^
1.8940
1.9074
1.9210
1.9347
1.9486
1.9626
^l
13.6
§
1.9626
2.0503
1.9768
2.0655
1.9012
2.0809
2-0057
2.0965
2.0204
2.1123
2.0353
2.1283
2.0503
2.1445
26
25
14.6
1 5-7
65
2.1445
2.1609
2.I77S
2.2817
2.1943
2.2998
2.2113
2.2286
2.2460
24
16.9
66
2.2460
2.2637
2.3183
2.3369
2-3559
23
18.3
S
2.3559
2.37g
2.3945
2.4142
2.4342
2.5605
2.6985
2.4545
2.4751
22
19.9
68
2.4751
2.6051
2.5172
iS
2.5826
2.6051
21
21.7
69
2.6279
2.65 1 1
2.7228
2.7475
20
23.7
70
2-7475
2.7725
2.7980
ia
2.8502
2.8770
2.9042
19
71
2.9042
2.9319
2.9600
3.0178
3047s
3-0777
18
72
3-0777
3.1084
3-I397
3.I7I6
3.2041
3-2371
32709
»7
73
3-2709
3-3052
3-3402
3.3759
3.6059
3-4124
3-4495
3-4874
16
74
3-4874
3.5261
3-5656
3-6470
3.6891
3.7321
15
75
3-7321
3.7760'
3.8208
3.8667
3-9136
3.9617
4.0108
14
76
4.0108
4-0611
4.1126
4.1653
4-2193
4-5736
4-2747
4.33' 5
4.7046
13
7I
4.331 5
4.7046
4.3897
4-4494
4.5107
5-5764
12
4.7729
4-8430
4.9152
4-9894
5.1446
11
79
5.1446
5.2257
5.3093
5-3955
5.4845
5.6713
10
80
81
^"313^
7.2687
&
&
1^^
t3°
6.3x38
7.1154
9
8
82
7.1154
7-4287
nt
7-7704
7-9530
8.1443
7
53
8.1443
ir^.
8-5555
10.0780
9.0098
9-2553
9.5144
6
84
9.5144
10.3854
10.7119
11.0594
11.4301
5
85
11.4301
11.8262
12.2505
15.6048
21-4704
12.7062
13.1969
13.7267
18.0750
26.4316
14.3007
4
86
89
14.3007
19.081 1
28.6363
57.2900
14-9244
20.2056
16.3499
22.9038
38.1885
114.5887
17.1693
24.5418
19.0811
28.6363
3
2
31.2416
08.7501
^^9398
42.9641
171.8854
49.1039
343-7737
57.2900
QD
I
€or
50'
4€r
ao'
20^
jjor
0'
Aigla
SarmaoNiAii Tables.
Natural Cotangents.
31
Digitized by
Google
Table 9.
TRAVERSE TABLE.
DIFFERENCES OF LATITUDE AND DEPARTURE.
.1
8
i
S
OP
1
2°
8
1
1
^
Q
TAt.
Dep.
Lat.
Dep.
Lat
Dep.
P
i
1
1.00000
0.99984
0.01745
0.99939
1.99878
0.03490
0.06980
1
2
2.00000
aooooo
1.99969
0.03490
2
3
3.00000
0.00000
2-99954
ao5235
0.06980
2.99817
a 10470
3
4
4.00000
0.00000
3-99939
3-99756
a 13960
4
o
1
I
q.ooooo
6.00000
7.00000
0.00000
4-999^
0.08726
4.99695
0.17450
5
60
aooooo
0.00000
0.1047 1
0.12216
5.99634
6-99573
a2094o
0.24430
6
7
8.00000
0.00000
0.13961
7.99512
8.99451
a2792o
8
9
9.00000
0.00000
8.99862
0.15707
a3i4io
9
1
0.99999
aoo436
0.99976
ao2i8i
0.99922
2.99768
0.03925
007851
0.11777
1
2
3
1-99998
2.99997
aoo872
0,01308
1-99952
2.99928
0.04363
0.06544
2
3
4
399996
0.01745
t^
0.08725
3-99691
0.15703
4
«S
1
7
4.99995
0.02 18 1
0.10907
4.99614
019629
1
I
45
5-99994
0.99993
0.02617
0.03054
^•99833
0.13089
0.15270
6.99460
0.23555
0.27481
8
7.99992
0.03490
8199785
0.17452
0.19633
7.99383
0.31407
9
8.99991
0.03926
8.99306
0.35333
9
1
0.99996
0.00872
0.99965
ao26i7
0.99904
1.99809
004361
1
2
1.99992
0.01745
I-9993I
0.05235
0.08723
2
3
2.99988
0.02617
2.99897
0.07853
2.99714
0.13085
3
4
399984
0.03490
3.99862
0.10470
3.99619
0.17447
0.21809
4
30
1
7
4.99981
0.04363
4.99828
ai3o88
4.99524
1
7
30
5-99977
8.99973
0.05235
0.06108
5-99794
6.99760
0.15706
0.18323
5.99428
6.99333
7-99238
0.2617 1
0.30533
0.34895
8
8.99965
0.06981
0.07853
i:K
0.20941
8
9
0.23559
8.99143
0.39257
9
1
2
-999gi
0.01308
ao26i7
0.99953
0.03053
0.00107
0.99884
1.99769
0.04797
0.09595
1
2
3
2.99974
0.03926
2.99060
0.09161
2.99654
0.14393
3
45
4
I
9
399965
4-99957
S.99948
6.99940
0.05235
0.06544
0.07853
ao9i62
4.99766
5-99720
6-99673
7.99626
8.99580
ai22i5
a 15269
0.18323
a2i376
3-99539
4-99424
5-99309
6.99193
019191
0.28786
038382
0.43180
4
I
9
15
7-99931
8.99922
0.1047 1
0.1 1780
0.24430
a27484
1
•3
Dep.
T^it.
Dep.
Lat.
Dep.
Lat
g
5-
§
&
9
P
a
y>
a
3P
8
70
•
•
Smitnsonian Tables.
32
Digitized by
GooqIc
Table 9.
TRAVERSE TABLE* "■—— -»■
DIFFERENCES OP LATITUDE AND DEPARTURE. -Continued.
J
i
1
3°
4°
5°
i
1
Tat.
Dep.
Lat.
Dep.
Lat.
Dep.
o
IS
30
45
1
2
3
4
8^
9
1
2
3
4
I
I
9
1
2
3
4
I
9
1
2
3
4
1
I
9
0.99863
1.99726
2.99589
3-99452
4.99315
8.'98767
0.99839
1.99678
2.99517
3-99356
4.99195
5:98552
0.99813
1.99620
2.99440
3-99253
#
7.^507
8.98321
0.99785
1.99571
2.99357
3.99143
4.98929
S-9871S
6.98501
7.98287
8.98073
0.05233
0.10407
0.15700
0.26168
a3i40i
a4i868
a47i02
0.17007
0.22677
a28346
0.34015
0.39684
0.45354
0.51023
0.06104
ai2209
0.18314
0.24419
o.30|24
a36629
tm
0.54943
0^36540
0.13080
a 19620
a26i6z
0.32701
0.45782
a58^62
0.99756
1.99512
2.99269
i^ii
5.98538
6.^94
i'97807
0.99725
1.99450
2.99175
4^^625
6^98075
7.97800
8.97525
0.99691
1-99383
4.^458
C.98150
6.97842
7-97533
8.97225
0.99656
'M
3.98626
4.98282
5-97939
6.97595
2-97252
8.96908
0.06975
0.13951
0.20926
0.27Q02
0.34878
041853
0.48829
a6278o
0.07410
ai482i
a22232
0.29643
0.37054
0.44465
0.51875
0.07845
ai569i
0.23537
0.31383
a39229
047075
a5492i
a62767
a7o6i3
0.08280
0.16561
0.24842
0.33123
0.41404
049684
0.66246
0.74527
0.9961 Q
2.9I858
3-98477
4.98097
5-97716
6.97336
8.96575
0.99580
1.9Q160
2.98741
3-98321
4.97902
5.97482
6.97063
7.96643
8.96224
0.99539
^:»
3-98158
4-97698
6^96777
8.95856
0.99496
1.98993
2.98490
3-97987
^^^
6.96477
7.95974
8.95471
0.08715
0.17431
0.26146
0.34862
043577
0.52293
0.61008
0.60724
0.78440
0.09150
0.18300
0.27450
0.36600
045750
0.54900
a6405i
0.73201
a8235i
0.09584
0.19169
^3833^
a47922
0.57507
a67092
0.76676
0.86261
aiooi8
a20037
0.30056
040075
0.50094
0.601 1 2
0.70131
0.80150
0.90169
1
2
3
4
1
I
9
1
2
3
4
I
9
1
2
3
4
1
I
9
1
2
3
4
I
9
60
45
30
1
Dep.
Lat.
Dep.
Lat
Dep.
Lat
i
1
B6P
85°
84°
SMiTNaoiiiAN Tables.
33
Digitized by
GooqIc
■ ""■■ "■ TRAVERSE TABLE.
DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUCD.
1
.s
IS
in
s
6°
70
SO
8
c
ft
.3
1
Lat.
Dep.
TAt.
Dep.
Lat.
Dep.
1
0.99452
ai0452
0.99254
0.12186
0.99026
1-98053
2.97080
0.13017
0.27834
0.417^
1
2
3
2:98^6
3.97808
a20905
0.31358
0.41811
1.98509
2.97763
3.97018
0.24373
0.36560
2
3
4
0.48747
3.96107
a69586
4
o
1
I
4.97261
0.52264
0.62717
0.73169
4-96273
0.60934
4.95"34
5.94160
6.93187
1
7
60
6.96165
5-95519
6.94782
0.85308
0-83503
0.97421
7.95617
0.83622
7.94038
8.93291
ri^i
7.92214
1.11338
8
•
9
8.95069
0.94075
8.91 241
1.25255
9
1
r.p^??
0.10886
0.90200
1.98400
0.12619
0.9896s
1:11^
1
2
0.21773
0.25239
1.97930
2
3
2.98216
0.32660
2.97601
0.37859
2.96895
0.43047
3
4
3.97622
0.43546
3-96801
0.50479
0.63099
3-95860
0-57397
4
15
1
7
4.97028
0.54433
0.65320
0.76206
4.96002
4.94825
0.71746
1
I
45
6.95839
5.95202
6.94403
0.75719
0.88339
5-93790
6.92755
0.86095
1.00444
8
795245
0.87093
0.97980
7-93603
1.00959
7.91721
1.14794
9
8.94650
8.92804
1-13579
8.90686
1.29143
9
1
2
0.99357
X.98714
0.1 1 320
0.22040
r^iu
o.no52
0.20105
0.98901
1.97803
0.14780
0.29561
1
2
3
2.98071
0.33960
2.97433
0-39157
2.96704
3.95606
0.44342
3
4
3.97428
0.45281
0.56601
0.07921
0.79242
396577
0.52210
0.05263
0.78311
0.91368
0.59123
4
30
1
7
4.96786
4.95722
4.94508
0.73904
0.88685
1.0-1460
1.18247
1
I
30
5-96143
6.95500
7.94857
5.94866
6.9401 1
593409
6.9231 1
8
0.90562
1.01882
7-93' 55
1.04420
7.91212
8.901 14
9
8.94214
8.92300
I-I7473
X.33028
9
1
2
r^s
01 1753
0.23507
a99o86
1-98173
0.13485
0.26970
0.98836
1.97672
0.1 5212
0.30424
1
2
3
2.97920
0.35261
2.97259
0.40455
2.96508
0.45637
3
4
3.97227
0.47014
396346
0.53940
0.67425
3-95344
4
45
1
5
4-96534
5.95841
6.95147
0.58768
4-95432
4.94180
0.76061
i
15
0.70522
0.82276
6!936o6
0.80910
0.94395
1.07880
5-93016
6.9i8q3
8.89525
t»
8.93761
0.94029
7.92692
1.21698
9
1.05783
8.91779
1.21365
1.369"
9
S
r
?
1
Dcp.
Lat
Dep.
Lat.
Dep.
Lat
1
c
5
B3P
820
a
10
SMiTHaoMUM Table*.
34
Digitized by
GooqIc
Table Q.
TRAVERSE TABLE.
DIFFERENCES OP LATITUDE AND DEPARTURE. 'Continued.
ii
9^
lOP
110
Q
.1
IS
Lat.
Dcp.
Lat.
Dcp.
Lat.
Dep.
1
a98768
0.31286
0.98480
0.17364
0.98162
0.19081
0.38162
1
2
1-97537
1.96961
0.34729
219448^
2
3
2.96306
0.46930
2.95442
0.52094
0.69459
0.57243
3
4
3-95075
0.62573
393923
3.92650
0.76324
4
o
1
4.93844
q.92612
6.91 38 1
0.78217
0.93860
1.09504
0.86824
1.04188
II38918
6^87139
1
I
60
8.^19
1.25147
7;87M
7.85301
9
1.40791
8.86327
1.56283
8.83464
1.71729
9
1
2
a98699
i3
ai6o74
0.32148
t^
0.17794
0.35588
a98o78
1.96157
0.19500
0.30018
0.58527
1
2
3
0.48222
2.95212
0.53383
2.94235
3
4
3-94798
0.64297
3-93616
0-71177
3.92314
0.78036
4
IS
1
493498
a8o37i
4.92020
0.88971
6.8654Q
7.84628
0-97545
1
7
45
5.92197
6.00897
0.96445
1.12519
iiss^
1.06766
1.24560
1.17054
1-36563
7-89597
1.44668
8185636
1.42354
1.56072
8
9
8.88^
1.60149
8.82706
1.75581
9
1
a98628
0.16504
0-98325
a 18223
0.97992
1.95984
0.19936
0.39873
1
2
2*95885
0.33009
1.96650
0.36447
2
3
0-49514
2.94976
0.54670
2-93977
0.59810
3
4
394514
0.66019
3-93301
4.91627
0.72894
&
0.79747
4
30
1
4.93142
0.82521
a99028
I.I 5533
1.32038
0.91117
0.99683
1
I
30
5-9I77I
7.89028
8.84929
1.09341
"•^7564
1.45788
5.87954
6.85947
7.83939
1. 19620
1.39557
1.59494
9
8.87657
148542
1.64011
8.81932
1.79431
9
1
0-98555
0.16035
0.98245
0.18652
0.97904
1.95809
0.20364
1
2
1.97111
0.33870
1.96490
0.37304
0.40728
2
3
4
2.95666
3.94222
0.50805
0.67740
2-94735
3.92980
0-55957
0.74609
2.93713
3.91618
4.89522
a6i092
0.81456
3
4
45
1
7
4.92778
a84675
4.91225
5-89470
6.87715
0.93262
1. 01820
1
7
15
1.01610
1-18545
1.11914
1.30566
5.87427
0.85331
1.22185
1.42549
8
7.80444
1.35480
7.85960
1.49219
7.83236
1. 6291 3
8
9
8.87000
1.52415
8.84205
1.67871
8.81140
1-83277
9
3'
c
8
1
Dep.
Lat
Dcp.
Lat.
Dep.
Lat.
f
f
90P
790
7
dP
SWTIMOMIAM Ta.U..
35
Digitized by
GooqIc
Table Q.
TRAVERSE TABLE*
DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED.
i
3
•S
IS
1
12°
13°
14°
1
.a
1
a
.2
Lat
Dep.
Tat.
Dep.
Trfit.
Dep.
1
0.97814
0.20791
0.41552
0.97437
0.22495
0.97029
a48384
1
2
1.95629
1.94874
0.44990
0.67485
1.94050
2.01088
3.88118
2
3
2.93444
0.62373
2.Q2311
0.72576
3
4
3.91259
8.84703
7.82518
8.80332
0.83164
0.89980
a96768
4
5
I-03955
isf^s
1.12475
4.85147
1. 20961
1
60
6
1.24747
j.84622
6.82059
1.34970
6.79206
MSI 53
2
I-4S538
1.66329
1.57465
1.79960
1.69345
1.93537
I
9
1.87 1 20
2.02455
2.17729
9
1
0.97723
1.95446
0.21217
0.97337
1.94075
a22920
t&l
a246i5
1
2
0.42435
0.45840
0.49230
2
15
3
4
2.93169
^^
6.84061
0.63653
0.84871
1.06088
0.91680
1. 14600
3.p92
4|46ij
0.^461
1.23076
3
4
I
I
45
1.27306
1.48524
0SI365
1.77520
6.78461
1.47691
1.72307
7.81784
8.79507
1.69742
7.78703
ii^iS
7.75384
8.72307
1.96922
9
1.90959
8.76041
2.21537
9
1
a97629
0.97237
tM
a968i4
0.25038
1
2
1.94474
1.93629
a50O76
2
3
4
2.917 1 1
3.88948
0.93378
1. 16722
2.Q0444
3.87259
o.75"4
IX)OI52
3
4
30
1
I
9
1.08220
4.86185
6.77703
7.74518
8.71332
1.25190
1
I
9
30
5.85777
6.83407
1.29864
1.51508
6S0659
1.40067
i!8^56
2.10100
1.50228
1.75266
7.81036
8.78666
1.7315?
1.94796
7.77896
8.75133
2.00304
2.25342
1
0.97534
1.95068
a22o69
0.97134
Z.94268
0.23768
0.96704
a2546o
1
2
044139
0.47537
1.93409
0.50920
2
3
4
2.92602
3.90136
4.87671
a6620Q
0.88278
2.9x402
3.18536
4^5671
0.71305
3^\i
a7638o
1.01840
3
4
45
1
1
1.10348
4-83523
1.27301
1
I
15
5.85205
6.82739
1.32418
1.54488
5.82805
0.79939
1.42611
1.66380
6.76932
1.52761
1.78221
l^'^
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7.77073
8.74207
1.90148
7.73636
2.03681
9
2.13917
8.70341
2.29141
9
1
a
Dep.
Lat.
Dep.
Lat
Dep.
Lat
d
I"
770
76°
750
Smithsonian Tablcs.
36
Digitized by
GooqIc
Table Q.
TRAVERSE TABLE*
DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued.
s
3
C
.3
Q
15°
160
170
8
1
.a
Q
1
.s
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
1
2
3
0.96502
1.93185
2-89777
0.25881
0.51763
0.77645
0.96126
'•2"52
2.88378
0.27563
0.8269X
0.95630
1. 91 200
2.86891
0.29237
0.58474
0.877 II
1
2
3
4
1.03527
«
1.10254
3.82521
1.16948
4
1
I
4^82962
1.29409
1.37818
1.65382
1.92946
4.78i|2
6I69413
1.46185
1
7
60
mi'd
1.55291
1.81173
n%
1.75423
2.04660
7.72740
2.07055
7.69009
8.65135
2.20509
&l
2.33897
8
9
8.69333
2.32937
248073
2.63134
9
1
a96478
0.26303
a526o6
0.78909
0.96005
0.27982
0.95502
0.29654
1
2
3
IS
1.92010
2.88015
a«96q
0.83948
1.91004
2i65o6
2
3
4
3-85914
1.05212
3.84020
1.11931
3.82008
1.18616
4
15
1
4.82793
5.78872
6.75351
8.6830^
'•3i5«5
1.57818
4.80025
5.76030
6.7^
4.77510
mi
148270
X.77924
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45
1
9
1.84121
1.95800
2.07579
I
9
2136728
8164045
2.51846
7.64016
8.59518
t^^
1
2
3
0.96363
a26723
0.53447
0.801 7 1
0.95882
2.87646
0.28401
0J5204
0.95371
1.90743
2.86115
3.8i48§
0.30070
a6oi4i
0.902 1 1
1
2
3
4
3-55552
1.06895
3^3528
1.13606
1.20282
4
30
1
4.81815
Jg6i9
1^7^
4.79410
1.42007
4.76858
1.50352
1.80423
1
30
5.78178
6.74541
5.75292
6.71174
1.70409
6.67601
1
1.98810
2.10494
I
8.67267
2.13790
8.62938
2.27212
8:58345
2.40564
2.70635
9
2.40514
2.55613
9
1
0.96245
a54288
0.81432
0.95757
0.28819
0.95239
0.30486
0.00972
0.91459
1
2
3
3.84982
1.Q1514
2.87271
l'&
1.Q0479
2.85718
2
3
4
1.08576
3.83028
4.78785
1.15278
3-80958
1.21945
4
45
1
4.81227
1.35720
1.44098
4.76197
'•52432
1.82918
1
15
5.77473
6.73718
5.74542
6.70299
X.72917
i^l
1
1.90008
2.01737
2.13405
7
7.69964
8.66209
2.X7152
7.66057
2.30557
7.61916
2.43891
8
9
244296
8.61814
2.59376
8.57156
2.74377
9
1
5-
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
1
ff
ff
8
^
740
730
72
r>
i
8
BamwoNiAN Tables.
37
Digitized by
GooqIc
' *"** *■ TRAVERSE TABLE*
DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED.
j
i
lff>
13P
20P
5
i
i
Q
Lat.
Dep.
Lat.
Dep.
' Lat
Dep.
i
1
2
0.95105
1.00211
2.85316
3.80422
0.61803
0.94551
1.89103
0.32556
0.65113
1.87938
0.34202
a68404
1
2
3
4
?:f^
2.83655
3.78207
0.97670
1.62784
1.95340
2.27897
2.81907
3-75^77
1.02606
1.36808
3
4
o
1
I
475528
5.70633
0-65739
1.54508
1.8 5410
2. 1 631 1
4.72759
6!6i863
4.69846
5.63815
6.57784
1.71010
2.05212
2.39414
1
I
60
7-60845
2.47213
1:$^
2.60454
7.51754
8.45723
2.73616
9
8.55950
2.781 1 5
2.9301 1
3.07818
9
1
2
0.04969
1.89939
0.31316
0.62632
?«
0.32960
0.65938
0.98307
1.31876
1.64845
t&
ai46ii
a69223
1
2
3
4
2.84909
3-79»79
0.93949
1.25265
1.56581
1.87898
2.19214
2.83226
3.77635
2.81457
3.75276
i^3^6
3
4
15
1
7
4.74849
4.72044
4-69095
1.73058
1
7
45
6;64789
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1.97814
2.30783
2.65752
2.96721
5.62914
6.56733
2.07670
8
9
7-59759
8.54729
2.81847
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7-50553
8.44372
2.76893
3-11505
8
9
1
2
ris
0.31730
a6346o
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a6676i
0.93667
1.87334
a35020
0.70041
1
2
3
2.84497
0.95191
1.26921
2.82792
1. 00142
2.81001
ix>5o62
3
4
3-79329
377056
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3.74668
1.40082
4
30
1
I
9
4.74161
1.58652
1.90382
2.221 13
4-71320
4.68336
1.75103
7
30
5.65584
6.59849
2.00284
2.33664
2.67045
3.00426
5.62003
6.55670
2.10124
2.80165
3.15186
8.53491
2.53843
2.85574
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7.49337
8.43004
8
9
1
2
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0.32143
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a3379i
0.67583
0.93513
1.87027
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1
2
3
2.84079
0.96431
2.82352
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2.02750
2.36541
2.80540
3
4S
4
9
3-78772
4.7^65
1.28575
1.60719
1.92863
2.25007
3.76470
4.70588
3.74054
4-67 5S7
5.61081
0.545^
1.41716
1.77145
2.12574
2.48003
A
I
9
IS
7.57544
8.52237
2.57151
2.S9295
7.52940
8.47058
2.70333
3.04125
841621
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s
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Dep.
Lat.
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a
S
g
5
^
f
5
7
V*
70P
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f
Smithsonian Taslcs.
38
Digitized by
Google
Tabu Oa
TRAVERSE TABLE*
DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUCD.
8
.s
8
21°
220
23»
8
S
Minutes.
Lat
Dep.
Lat.
Dep.
Lat
Dep.
o
IS
30
45
3:
1
1
2
3
4
9
1
2
3
4
1
9
1
2
3
4
I
9
1
2
3
4
I
9
2.80074
4.66790
5.60148
840222
0.93200
1. 86401
3^72803
4.66004
5.59204
2.79125
&
5.58250
6.51292
7-44334
8.37375
0.02881
1.85762
2.78643
3-71524
4.64405
6.50167
7.43048
8.35929
0.35836
0.71673
1. 07 510
1-43347
1-79183
2.15020
i&
3-2253*
0.36243
0.72487
1.08731
x-44975
1.81219
2.17462
3.26194
0.36650
0.73300
;«
1.83250
2.19900
2.56550
2.93200
3.29851
0.37055
0.74111
1.11167
1.48222
1.85278
2.22334
2.59390
2.96445
3-33501
0.92718
1.85436
2.78155
3-70873
4.63591
5.5S310
6.49028
7.41747
8.34465
0.02554
1.85108
2.77662
3.70216
4.62770
6;47878
0.92388
ii4776
2.77164
3-69552
4.61940
6.46716
7-39104
8.31492
0.92220
3.68880
4.61 100
5-53320
6.45540
8.29980
0.37460
0.74921
1.12^81
1.87303
2.24763
2.62224
2-99685
3-37145
0.37864
0.75729
1-13594
1-S1459
1-89324
2.27189
2.65054
3.02918
3-40783
a38268
0.76536
X.14805
1-53073
1.91341
2.29610
2.67878
3.06146
3-44415
0.38671
0.77342
1.16013
1.54684
1-93355
2.32026
3-'^368
3-48039
0.02050
1. 84100
2.761 51
3.68201
4.60252
552302
6.44353
8!28454
3.67516
4.59395
5-51274
8-43153
8.26912
aoi7o6
1.83412
4-58530
5.50236
8-25354
5.49186
6.40718
7-32249
8.23780
0.7814S
1.17219
1.56292
1-95365
2.34438
2.735"
3.12584
3-51657
0.39474
o.789;|8
1.18423
1.57897
1-97372
2.36846
2.76320
3-15795
3-55269
0.39874
0.79749
1.19624
1-59499
x-99374
2.39249
2.79124
3-58874
0.40274
0.80549
1.20824
1. 61098
IW
2.81922
3.22197
3.62472
1
2
3
4
1
I
9
1
2
3
4
1
I
9
1
2
3
4
1
1
9
1
2
3
4
1
I
9
60
45
30
15
g
Dep.
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Dep.
Lat.
Dep.
Lat.
a
i
f
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BaiTHaoNM
39
Digitized by
GooqIc
Table 9«
^^^^ TRAVERSE TABLE*
DIFFERENCES OF LATITUDE AND DEPARTURE. 'CONTINUCD.
1
.s
8
240
a5<>
2GP
3
S
c
^
Q
Lat.
Dep.
Lat
Dep.
Lat
Dep.
Q
i
1
2
0.01354
1.82709
0.40673
0.81347
Tdi
0.42261
a84523
0^79
0.43837
0.87674
1
2
3
2.74063
3-65418
4.5^772
1.22020
2.71892
1.26785
2.69638
1.31511
3
4
1.62694
2.03368
3-62523
1.69047
3.59517
1.75348
4
o
1
9
4-53153
5.43784
6.34415
7.25046
8.15677
2.1 1309
4^9397
2.19185
1
7
60
5.48127
6.39481
2.44041
2.84715
2.53570
2.9^832
5.39276
6.29155
2.63022
3.0&59
7.30836
8.22190
l&
3^356
i-^i
3.50696
3-94533
8
9
1
2
a9ii76
1.82352
a4i07r
0.82143
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0.42656
0.85313
0.89687
1.79374
a44228
1
2
3
2.73528
1.23215
l^X
1.27970
2.60061
3.5»749
3
4
3.64704
1.64287
1.70627
1.76915
4
15
1
7
4.55881
2.05359
2.46431
2.87503
4.52227
2.13284
4.48436
2.21 144
1
9
45
5.47057
6.38233
6.331 18
2."9^598
5.38123
6.27810
2.65373
3.09002
8
9
7.29409
8.20585
1®
7.23564
8.14009
3.41254
3.8391 1
8.07185
3-53830
3.98059
1
2
0.90996
0.41469
0.82938
0.00258
1.80517
0.80102
^
a446i9
0.89239
1
2
3
2.72988
1.24407
2.70775
1.29153
2.68480
1.33859
1.78479
2.23098
2.67718
3.12338
3
4
3.63984
1.65877
3.61034
1.72204
3-57973
4
30
1
2
4.54980
5.4S976
6.36972
2.90285
4.51292
5.4i|5i
6.31869
2.58306
3.01357
4.47467
6.26454
1
30
i.l&s
3.31754
7.22068
8.12326
3.44408
7.15947
8.05440
3.56958
9
3-73223
3.87459
4.01578
9
1
2
0.90814
1.81628
041866
0.83732
0.90069
1.80139
0.8^8$
0.89297
1.78595
2.67893
0.45009
0.90019
1
2
3
2.72442
1.25598
2.70209
1.30333
1-73778
1.35029
1.80039
3
4
363257
1.67464
3-60279
4464^9
4
45
1
I
4.54071
2.09330
4.50349
6.30488
2.17222
2.25049
1
7
IS
5.44885
6.35700
8!i7328
2.51196
2.93062
2.60667
3.04111
6.25085
HTc^
3-34928
7.20558
8.10628
3.47556
7.14383
3.60078
8
9
3-76794
3.91000
8x>368i
4.05088
9
1
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
1
e
5«
&
ft«>
G3P
Smithsonian Tables.
40
Digitized by
GooqIc
Table 9«
TRAVERSE TABLE* "■•■— ^ «»■
DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued.
1
8
270
28»
2y>
.1
i
^
s
Lat
Dep.
Lat
Dep,
Lat
Dep.
Q
s
1
2
0.80100
1.78201
045399
090798
088294
1.76589
offiJ
0.87462
0.48481
0.96962
1
2
3
2.67301
1.36197
1.81596
2.64884
1.40841
2;62p6
145443
3
4
3-56402
3-53179
1.87788
3.49048
1.93924
4
o
1
9
4-45503
5-34003
6.23704
7.12805
8.01905
2.26995
2.72394
3-17793
3-6JI93
4.08591
4.41473
7.06358
7.94052
3.28630
3-75577
4.22524
4.37310
5.24772
0.12234
7.87156
2.42405
2.90886
4.36329
1
I
9
60
1
2
088901
1.77803
0.45787
0.91574
0.88089
1,76178
0.94664
1.41996
087249
2.61748
048862
0.97724
1
2
3
2.66705
3-55606
1^3149
2.28937
2.64267
1.46566
3
4
3-52356
1.89128
2.83992
348998
1.95448
4
»S
1
444508
4.40445
4.36248
244310
I
45
5.33410
2.74724
kfe^r,
5.23497
6.10747
2.93172
i
9
6.2231 1
3.20511
m
3.42034
I
9
7.11213
8x)oii5
&,
7.04712
7.92801
6.07996
7.85246
3.90896
4-39759
1
0.88701
046174
087881
0.47715
0.87035
0.45242
0.98484
1
2
1.77402
0.92349
1.75763
095431
1.74071
2
3
4
2.66103
354804
1.84699
2.63(S4S
3-51520
1*90863
2.61 106
3-48142
1.47727
1.96969
3
4
30
1
4-4350S
5.32206
6.20907
2.30874
4.39408
2.J8579
2.86295
4.35177
2.46211
1
30
2.77049
5.27290
6.15171
5.22212
^4^^
1
3-23224
3.3401 1
3-81727
I
7.09608
3.69398
7.03053
7.83320
3.93938
4.4318I
9
7.98309
4-15573
7.90935
4.29442
9
1
088498
046561
087672
048098
0.86819
049621
1
2
1.76997
0.93122
1-75345
2.63018
0.96197
1.73639
0.99243
2
3
2.65496
:«
1.44296
2.60459
i.i|864
3
4
3-53995
3.50690
1.92395
3.47279
1.98486
4
45
1
4.42493
2.32807
4.38363
5.26036
6.13708
2.40494
4.34099
2.48108
1
15
5-30992
6.19491
2.79368
2.88593
5.20919
6.07739
2.97729
I
9
3-25930
3.36692
3.4735*
I
9
l^
3-72491
4.19053
7.01 381
7.89054
tx
6.Q4559
7.81378
3.96973
446594
g
Dep.
LaL
Dep.
Lat.
Dep.
Lat
1
1
c
J?
&
8
^
G
2°
6
1°
6
ty»
^
•
Shitiwoiiian Tables.
41
Digitized by
GooqIc
Table 9«
TRAVERSE TABLE*
DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED.
i
6
30°
310
32P
g
8
1
c
1
1
1
ig
P
Lat.
Dep.
Lat
Dcp.
Lat.
Dep.
^
s
1
0.86602
a5oooo
0.85716
0.51503
a848o4
0.52991
J -0^3
1
2
1-73205
1.00000
1-71433
1.03007
1.69609
2
3
2.59807
1.50000
^
1.545"
2.54414
3
4
3.46410
2.00000
2.06015
3.39219
2.1 1967
4
o
1
4.33012
2.50000
4.28583
2.57519
4*24024
2.64959
i
60
6!o62i7
3.00000
5.14300
6.00017
3.09022
5.08828
317951
I
3.50000
3-60526
m
3-70943
i
6.92820
4.00000
6.85733
4-12030
4-23935
4.76927
9
7.79422
4.50000
7.71450
4-63534
7.63243
9
1
2
0,86383
1.72767
0.50377
1.00754
0.85431
1.70982
0.51877
1.03754
0.84572
1-69145
2.53718
0.53361
1.00722
1
2
3
2.59150
1.51132
2.56473
1. 55631
1.60084
3
4
3-45534
I:^;i8°?
3.41964
2.07509
3-38291
2.13445
4
^5
1
4.31917
4-27456
2.59386
4.22863
5.07436
2.6g8o7
1
45
5.18301
&04684
6.91068
3.02264
5.12947
3."263
3.20168
5
3.52641
4.03019
5.98438
0.83929
3-63141
4.6^5
6!76582
4.2^1
I
9
7-77451
4-53396
7.69420
7.61155
4.80253
9
1
0.86162
0.50753
0.85264
0.52249
2.53017
0.53730
1.07460
1.61190
1
2
3
1.7232c
1. 01 507
1.52261
1.70528
2.55792
1.04499
1.56749
2
3
4
3.44651
2.03015
3.41056
2.08999
3.37356
4.21695
2.14920
4
30
1
4.30814
2.53769
4.26320
2.61249
2.686qo
3.22380
1
30
5.16977
6.03140
3-04523
3-55276
4-06030
4.56784
S."584
3.13499
5-06034
5
9
6.821 12
7.67376
3-65749
4-1799S
4.70248
5-90373
6.74713
7.59052
3.76110
4.29840
4.83570
I
9
1
fytltr
0.51 120
0-85035
0.52621
a84io3
0.54097
1
2
1.022158
1.70070
1.05242
1.57864
1.68207
ix>8i94
2
3
2.57821
1-53387
2.55^05
2.5231 1
1.62202
2.16389
3
4
3.43762
2.55646
3-06775
3-57905
3.40140
2.10485
3-36415
4
45
1
g
4-29703
4.25176
2.63107
3-68349
4.20519
5.88827
6.72831
2.70487
1
7
15
6.01584
5.1021 1
6!8o28i
i*
6.87525
4.09034
4.20971
tm^'7
8
9
7.73465
4.60163
7.65316
4-73592
7-56935
9
1
Dcp.
Lat.
Dep.
Lat.
Dep.
Lat
f
1*
1
590
sep
570
t
SMiTHtoNiAN Tables.
42
Digitized by
Google
TRAVERSE TABLE. ^^""^ *'
DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued.
i
.S
i
33°
340
350
i
.S
IS
Q
Lat
Dep.
TAt.
Dep.
Lat.
Dep.
Q
^
1
2
0.83867
1.67734
0.54463
1.08927
0.82903
1.6C807
2.48711
0.55010
1.11838
0.81 QI 5
1.63830
0.57357
I. 14715
1
2
3
2.51601
1.6339'
2.17855
1-67757
2.45745
1.72072
3
4
335468
3.31615
4-14518
2.23677
327660
2!8l7^
4
o
1
4-19335
2.72319
2.79596
4.09576
^
60
5.03202
3-26783
4.Q7422
5.80326
6.63230
7.46133
3-35515
4.91491
3.44145
I
5^7069
6.70936
7.54B03
3.81247
4-357"
3-91435
4.47354
5.73406
6.55321
tw,
I
9
4-90175
5-03273
7.37236
5.I62I8
9
1
a83628
a54820
a8265o
1.65318
0.56280
0.81664
0.57714
1
2
l^l
1.09608
1.12560
1.68841
1.63328
1.15429
2
3
1.64487
2.47977
2.44992
1-73143
3
4
3-34514
4-18143
2.19317
3-30636
2.25121
3.26656
2.30S58
4
15
1
2.74146
4.13295
2.81402
4.08320
3-46287
1
45
5.01771
3-2%5
4-9S9S4
6.61272
3-37682
4.89984
5
5.69028
3-Mo5
393963
5.71649
6.53313
4.04001
7
4-93483
4-50243
4.6I7I6
8
9
752657
7.43931
5.06524
7.34977
5.19430
9
1
^
0.55193
1.10387
0.82412
0.56640
a8i4ii
0.58070
1
2
1.64825
1.13281
1.62823
1. 16140
2
3
2.50165
1-65581
2-47237
1.69921
3-25646
I.742IO
3
30
4
1
3-33554
4.16942
i,1SS
3-29650
4.12063
2.26562
2.83203
2.32281
2-90351
4
1
30
5-0033I
4-41549
6.59300
3-39843
3.48421
§
&67108
3-96484
4.53124
5.69880
6.51292
4.06492
4.64562
5.22632
I
9
7.50497
4.96743
741713
5.09765
7.32703
9
1
0.83147
1.66294
0.55557
0.82164
5.56999
0.81157
t.^ll
1
2
I.IIII4
1.64329
1.13999
1.62314
2
3
2.49441
1.66671
2.46494
1.70990
2.27998
2.43472
1.75275
3
4
332588
2.22228
3.28658
3.24629
2.33700
4
45
1
4.157^5
2.77785
4.92988
2.84998
3-41998
4.05787
4-86944
2.92125
3.50550
1
15
5
j.8^9
5.75152
6.57317
3-98997
5.68101
4.08975
I
6.6«76
7.48323
4.44456
4-55997
6.49260
4.67400
9
5.00013
7.39482
5-12997
7.30416
5-25825
9
d
Dcp.
Lat
Dcp.
Lat.
Dcp.
Lat.
1
r
^
8€P
5.
50
&
*o
1
SMmnoMMi Tabus.
43
Digitized by
GooqIc
Table 9*
TRAVERSE TABLE.
DIFFERENCES OF LATITUDE AND DEPARTURE. -CONTINUED.
1
t
3e»
3T>
38»
.s
i
9
.s
S
.S
s
Q
Tat.
Dcp.
Lat
Dcp.
Lat
Dep.
Q
;^
1
0.80Q01
0.58778
0.79863
0.60181
a788oi
a6i566
1
2
1.61803
i-*7557
1.59727
1.20363
1.57602
'•f3'32
2
3
ir^
1.76335
2.39590
1.80544
2.36403
1.84698
3
4
2.35"4
3.'9454
240726
3.15204
246264
4
o
1
4.04508
2.93892
3.99317
3.00907
4^^
3.07830
1
60
4.85410
5-6631 1
6.47213
7.281 15
3.52671
4-79181
3.61089
369396
5
4.1 1449
4.70228
m
4.21270
4-81452
5-41633
&
4.30963
4.92529
I
9
5.29006
7.18771
7.09209
5-54095
9
1
2
0.80644
1.61288
UU
a796oo
1.50200
a6o52o
1.21058
I-7853I
1.57063
a6ioo9
1.23818
1
2
3
2.41933
1.77392
2.38800
1.81588
2.35595
3.I4I26
1.85728
3
4
3.22577
2.36523
2.95654
3-54785
4.13916
4.73047
3.18400
2421 17
247637
4
IS
1
4.03222
3-98001
3.02647
3.92658
3.09547
1
45
4.83866
4.77601
3-63176
4.7 I 190
3-71456
4-33365
4.95275
5.57184
I
5.6451 1
6.45155
k'^.
4.23705
4.84255
5-44704
7-06785
I
9
7.25800
5.32178
7.16401
9
1
2
a8o385
1.60771
l\l&
t^i
0.60876
1.21752
0.78260
1.56521
a6225i
1.24502
1
2
3
2.41 1 57
1.78446
8.38005
1.82628
2.34782
1.86754
3
4
3.21542
2.37929
'^l
2.43504
3.J3043
249005
4
30
1
5
4.01928
2.9741 1
304380
3-91304
3." 257
3-735»
1
I
30
4.82U4
3.56893
4^75^5^
4.76011
3.65256
4.26132
4.87009
4-69564
IS
9
7-23471
5.35340
7.14017
5.47885
7-04347
5-60263
9
1
2
0:80125
1.60250
0.59832
1. 19664
a7oo68
1-58137
0.61 221
1.22443
0.77988
1.55946
0i»2502
I.25184
1
2
3
2.40376
1.79497
2.37206
1.83665
2.44886
2-33965
1.87777
3
4
4.00626
2.39329
3-J6275
3-"953
2.50369
4
45
1
2.90162
3-95344
3.06108
3-89942
312961
I
15
4.80752
4.74413
3-67330
4.67930
4^38146
§
5.60877
6.41003
7.21128
5.53482
6.3255"
4.28552
5-45919
6.23007
7.01896
I
4.78659
4-89773
5.00738
9
5.38492
7.1 1620
5-50995
5-63331
9
f
q
Dep.
Lat.
Dep.
Lat
Dep.
Lat
1
f
e
1
1
53»
52°
5.
l'^
c
2
Smithconian Tablcs.
Digitized by
GooqIc
Table Q«
TRAVERSE TABLE.
DIFFERENCES OF LATITUDE AND DEPARTURE. -Continued.
1
1
2
3y»
W
41^
1
(Q
.1
s
Q
Lat.
Dep.
Lat
Dep.
Lat
Dep.
Q
S
1
0.77714
0.6^
a766o4
0.64278
1.92836
0.75470
0.65605
1
2
155429
1.53208
1.50941
1.31211
2
3
3-10858
1.88796
2.29813
2.26412
1.96817
3
4
2.51728
3.06417
2.571 15
3.01883
2.62423
4
o
1
3.88573
3.14660
3-83022
3^85672
3-77354
4.52825
I.2829S
6.03767
6.79238
3.28029
1
60
4.66287
3.77592
4.59626
3-93635
7
6.21716
4.40524
5.36231
6.12835
449951
4.59241
I
8
5.14230
5.24847
9
6.99431
6.89439
5.78508
5.90453
9
1
2
?:»
0.65270
0.76121
a646i2
1.29224
a75i84
1,50368
\W
1
2
3
2-32317
2.28969
1.93837
2.58449
2.25552
2^63738
3
4
3.09757
2.53082
3.79623
4.42893
5.06164
3.05293
3.81616
3.00736
4
IS
1
5
3.87196
3.23062
3.75920
3.29672
1
I
45
4.64635
5-42074
6.19514
4-579^
SuIO^
S.16899
6.01472
6.76656
3.95607
4.61542
5.27476
9
6.96953
5.69434
6.86909
S^'S"
5934"
9
1
0.77162
a636o7
0.76040
?s^
0.74895
0.66262
1
2
1.54324
1.27215
1,52081
11^^
1.32524
2
3
2.31487
1.90823
2.28121
1.94834
1.98786
3
4
3.08649
2.5443«
3.04162
2.59779
2.99582
2.65048
4
30
1
3.85812
3.18039
3-80203
iJf^
374477
3.3>3io
1
30
4.62974
3.81646
4.56243
6.08324
449373
5.242^
5.30096
I
5.40137
6.17299
t^
4-54613
5.19558
I
9
6.94462
5.72470
6.84365
5.84503
5.96358
9
1
0.76884
»
0.75756
0.65276
0.74605
0.66588
1
2
1.53768
i.5'5i3
1.30552
1.95828
1.49211
1-33176
2
3
2.30652
1.91831
2.27269
2.21817
2.5^422
3.73028
1.99764
3
45
4
1
3.07536
3.84420
2.55775
3.19719
3.03026
3.78782
2.61 104
2.26380
3.91656
2.66352
3-32940
4
1
IS
4.61305
3.83663
4.54539
4.47634
399529
7
5.38189
6.15073
4-47607
5.30295
6.06052
4-56932
5.22240
4.661 17
I
8
5."55i
5.22208
5.96845
6.7145'
5.32705
9
6.91957
5.75495
6.81808
5.87484
5.99293
9
5"
S
B
Dep.
Lat.
Dep.
Lat.
Dep.
T4»t
q
1
&
c
s
5
OP
4
9P
4
8^
S
8
SMrmaoNiAN Tables,
45
Digitized by
GooqIc
Table 9.
TRAVERSE TABLE.
DIFFERENCES OF LATITUDE AND DEPARTURE. 'CONTINUCD.
i
.9
Q
420
43°
44°
8
1
.3
1
.s
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
1
2
1.48628
0.66913
1.33826
0.73135
1.46270
a68i99
1-36399
r^^
rp
1
2
3
2.22943
2.00739
2.19406
2.04599
2.1 5801
IfM
3
4
2.97257
2.67652
3-34585
4.01478
2.92541
2.72799
IS
4
1
3-71572
fM'
3.40999
3.47329
1
60
4.45886
4.09199
4.773^
4.31603
4.16795
7
5.20201
4.68391
5.11947
5.03537
4.86260
7
8
9
^1^^
5-35304
6.022x7
^5^218
6113798
5-75471
6.47405
5.55726
6.25192
8
9
1
0.74021
0.67236
0.72837
0.68518
0.71630
1.43260
1.39558
1
2
148043
1-34473
1.45674
2.18511
1-37036
2
3
2.22065
2.017 10
2.05554
2.14890
2.09337
3
4
2.96087
2.68946
3-36183
2.91348
2.74073
2.86520
2.791 16
3.48895
4
15
1
3.70109
3.64185
342591
3.58151
1
45
444130
5.18152
4.03420
4-37022
4.11109
4-7^28
^^6664
4.29781
4.18674
7
4.70656
5.09859
5.0141 1
4.88453
I
9
8
9
^6^196
5.37893
6.05130
5.82696
0.55533
6.44671
IS
1
0.73727
0.67559
i.35"8
0.72537
0.68835
0.71325
1.42650
0.700Q0
1.40181
1
2
1-47455
2.21x83
1.45074
1.37670
2
3
2.02677
2.17612
2.06506
2.13975
2.10272
3
4
^•S5?'2
2.70236
2.90149
2.75341
al^
2.80363
4
30
1
3.68638
4.42366
3-37795
3.62687
3.44177
3.50454
1
30
4.05354
4.35224
4.13012
4.27950
4.20545
4.90636
9
5.16094
4.72913
5.07762
4.81848
4.99275
I
9
I.89821
6.63549
6.08031
6.52836
5.50683
6.19519
5.70600
6.41925
kf^ll
1
a73432
0.67880
0.72236
0.60151
1.38302
O.7IO18
0.70401
1
. 2
1.46864
1.35760
1.44472
1.42037
140802
2
3
2.20296
2.03640
2.16709
2.07453
2.13055
2.11204
3
4
2.93729
2.71520
2.88945
2.76605
3-45756
2.84074
2.81605
4
45
1
3.67161
339400
3.61 182
3.55092
4.261 1 1
3.52007
1
15
4.40593
4.07280
4-33418
4.14907
4.22408
9
5.14025
4.75160
5.43040
6.10920
5.05654
5.77891
6.50127
4.84059
6.22361
6.39166
4.92810
1.6321X
8.33613
I
9
1
g
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
!
1'
t
^
470
46^'
45°
5
Smithsonian Tables.
46
Digitized by
GooqIc
Table 9«
TRAVERSE TABLEa ^^
DIFFERENCES OP LATITUDE AND DEPARTURE. -Continued.
a
Q
45«
.52
Q
Lat.
Dep.
1
0.70710
0.70710
1
2
1.41421
1.41421
2
3
2.12132
2.12132
3
4
2.82842
2.82842
4
5
3-53553
3-53553
5
6
4.24264
4.24264
6
7
4-94974
4.94974
7
8
5.65685
5.65685
8
9
6.36396
6.36396
9
q
Dep.
Lat.
1
4.
5«
SHmiaeMMii Tabus.
47
Digitized by
GooqIc
Table 10.
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE Pm IN ENGLISH
FEET.
[Derivation of table explained on p. xlv.]
Lat.
o«
1°
2°
3"
4^
f
6«
7**
SP
9°
IO«»
p. p.
I
a
3
4
1
I
9
10
II
13
«3
«4
:i
:i
«9
20
at
22
33
"4
30
31
3*
33
34
35
36
u
39
40
4«
4a
43
44
4S
46
ti
49
60
S»
5a
53
54
55
56
il
59
60
7.817
7379
7.817
739a
7.817
7433
7.817
7500
7.817
7593
7.817
77x4
7.817
7861
7.817
8034
7.817
8333
7.817
8458
7.817
8709
1
7379
7379
7379
7379
7379
7379
7379
7379
7379
739a
7393
7394
7394
7395
7395
7396
7396
7397
7434
-i
7438
7439
7440
7441
75o»
7503
7504
7506
7507
7508
7510
75«x
7513
7595
7597
7599
7603
7604
7606
7608
7610
7716
77x9
7731
7736
7738
7730
773a
7735
7873
7875
7877
7880
7883
7885
8037
8040
8043
8046
8050
8053
8056
8059
8063
8337
8340
8344
8347
8251
8a55
£2
8365
8463
8466
8470
8486
8490
8494
8733
8737
873 X
8735
8740
8744
8749
xo
30
50
s
.3
.3
.5
:l
t.o
7379
7397
744a
7514
761a
7737
78S8
8065
8369
8498
8753
7379
7379
7379
7380
7380
7380
7380
7J8o
7398
7398
7399
7399
7400
7401
7401
740a
740a
7443
7444
7445
7446
74*9
7450
745 »
75«5
75«7
7518
75ao
75a'
75aa
75a4
75*5
75*7
7614
7616
7618
7619
7611
7633
7635
7637
7639
7739
774a
7744
7746
7749
775X
7753
7755
7757
789X
7894
7896
7899
790a
7905
7908
7910
79x3
8068
8071
807s
8078
8081
8084
8087
809X
8094
S
8383
8387
8391
830X
8503
8506
85.0
P'i
8518
8533
8537
853X
8535
8794
a
10
ao
30
40
•3
.7
I.O
X.3
X.7
3.0
7380
7403
745a
75a8
7631
7760
79x6
8097
8305
8539
8798
7380
7380
7381
738.
7381
7381
7381
738a
738a
7404
7404
7405
7407
7408
7453
7454
7455
7456
7458
7459
7460
7461
746a
7530
753 X
7533
7534
7535
7537
7538
7540
754 «
7633
7635
7637
7638
7640
764a
B
7763
7765
7767
7770
777a
7774
7777
7779
778a
79x9
79aa
79a4
79a7
7930
7933
7936
7938
794X
8100
8104
8107
8txo
8114
81x7
8iao
8133
8137
8309
83x3
83x6
8330
8334
8337
8331
8543
8547
8551
8555
8559
8564
8568
857a
8576
88x3
8816
88si
8836
8830
8839
8
ID
20
30
40
£
.5
1.0
x-S
a.o
a. 5
30
738a
7409
7463
7543
7650
7784
7944
8x30
834a
8580
8844
738a
7383
7383
7383
7384
7384
7384
7384
7385
7410
7410
741 1
741a
74«3
7413
7414
7415
7415
7464
7466
7467
7469
7470
7471
747a
7473
7546
7548
7549
75SI
7553
7554
7556
7557
765a
7663
7665
7667
7669
7786
7789
779X
7794
7796
7799
7801
7947
7950
7953
7956
^?
7964
7967
7970
8133
8137
8x40
8x44
8x47
8x50
8154
8346
8350
8353
8357
8361
8365
8369
837a
8376
8584
8588
8593
8597
8601
8605
8609
8849
ia
8863
8867
8873
IS?
8885
4
10
30
y>
40
•7
1.3
a.o
a.7
3-3
4.0
7385
7416
7474
7559
7671
7809
7973
8x64
8380
8633
8890
7385
7386
7386
7386
7387
7387
7387
7387
7388
7417
7418
7418
74x9
74ao
74ai
74aa
74aa
7433
7475
748a
7483
7486
7561
756a
7564
7566
7569
7571
7573
7574
7673
7675
7677
7679
7683
7684
7686
7688
7690
781 1
7816
78x9
7831
7834
7836
7829
783 X
7976
799X
7994
7997
8000
8x67
8.7X
8x74
8l8t
8x84
8x88
819X
8x95
8384
8388
839a
8396
8400
8403
8407
84x1
84x5
8636
8631
8635
8643
8648
8653
St
Si!!
8904
8909
Sis
893a
6
xo
30
30
40
.8
t.7
as
3.3
4.a
5.0
7388
74^4
7487
7576
769a
7834
8003
8x98
84x9
8665
8937
7388
7389
7389
7390
7390
7390
739'
739'
739a
7425
7426
74a7
74a8
74*9
7439
7430
7431
743a
7488
7489
7490
749«
7493
7494
7496
11^
7578
7579
7581
7583
7588
7590
759»
7699
7701
7703
770s
7707
77x0
771a
7R37
7850
7853
7856
7858
8006
8009
. 8013
80x5
8019
8033
8035
8038
8031
830I
1^
83X3
82x5
83x9
8333
8336
8339
8433
8437
8431
8435
8439
844a
8446
8450
8454
8669
8674
8678
8683
8691
8696
8700
8705
8947
895X
8956
896X
8966
8971
739a
7433
7500
7593
77x4
786.
8034
8a33
8458
8709
8985
J
Smithsonian Tables.
48
Table 10.
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p^ IN ENGLISH
FEET.
[Derivation of table explained on p. xIt.]
Lat,
II«
I20
13°
I4«
IS''
16°
17°
l8«
19"
20°
P.P.
7.817
7.617
7.617
7.617
7.616
7.616
7.616
7.616
7.616
7.316
IF
I
a
3
8985
9385
961 1
9960
0333
0730
"49
i59«
ao54
3539
4
8995
8^99
9390
9396
9301
9617
9633
9638
9966
997a
9978
0340
«346
0353
0737
0744
0750
1156
1163
1171
1614
3063
2070
3<.78
a547
3556
3564
10
•7
4
I
9004
9009
9014
9306
931a
93x7
9633
9639
9645
9984
9990
9996
03 S9
0366
0373
0757
0764
0771
.178
X1M5
1x93
1631
1639
'637
3-86
ao94
3I03
3580
3589
so
30
40
«-3
3.0
a.7
3-3
4.0
I
9
10
II
la
9019
2SI
93aa
93a7
9333
9^50
9656
9661
*0003
^0008
*ooi4
oil?
039a
0778
0784
0791
"99
1207
1214
1644
1652
1659
3IIO
3Il8
3136
3614
6
9033
9338
9667
0398
0798
1331
1667
a 134
3623
9038
9043
9048
9343
9349
9354
s
«bo36
*0033
•0039
0404
0411
0418
08i3
0819
1338
1336
"43
1690
314a
3150
3158
3630
3639
3647
10
.8
«4
9053
9370
tss,
9701
J«45
•00s.
•0057
0424
0430
0437
0826
0833
0839
1350
1697
1705
I7»3
3166
?
3673
so
30
40
so
1.7
a-5
3.3
4.3
.'I
«9
20
ai
33
33
9067
907a
9077
9375
9380
9386
9707
9713
9718
•0063
•6076
0443
0450
0456
0846
1373
"79
1387
1720
1738
1735
3190
3^
3206
3680
3688
a697
1.
5.0
6
908a
939t
97a4
*00S3
0463
0867
"94
t743
3314
3705
9087
9093
9097
9396
940a
9407
9730
9736
974«
•bo88
•too94
•oioi
0470
0476
0483
0888
1301
1316
1751
3323
3330
3238
3713
3733
3730
10
X.O
*4
910a
9107
91 13
9413
9418
94a3
9747
9753
9759
•9107
•bii3
^119
0303
0895
0903
0909
'."3
1330
1338
1 781
1789
3346
a739
a747
a7S5
30
30
40
3.0
3.0
4.0
5-0
39
30
3«
33
33
9x17
9123
9127
9439
9434
9440
9765
9770
9776
^125
•0133
«bi38
0516
0523
0916
0933
0930
X345
i3Sa
1360
X8l2
3370
•3S?
3764
3781
6.0
7
9132
9445
978a
•b«44
0539
0937
1367
1830
3394
3789
9«37
914a
9i47
9450
9456
9461
9788
•0150
•0156
•bi63
0536
054a
0549
0944
0951
0958
1389
1838
1835
1843
3303
2310
3318
3814
10
1.3
34
9153
9467
947a
9477
9817
•0169
OS55
0563
0569
0965
0973
0979
1397
1404
14"
1851
3336
8334
a343
3823
3831
3840
30
30
40
50
a. 3
3.5
39
40
41
4*
43
9167
917a
9177
9*!2
9488
9494
9839
9835
•bi87
•bi94
•b200
0575
0583
0588
0986
0991
1000
I4t9
1426
14M
lis
.889
a35i
all?
3848
60
7.0
6
9182
9499
9841
•0306
0595
1007
1441
1897
a375
a874
9187
919a
9«97
9505
9510
95 «6
9847
9859
*b3I3
'^>3I9
•10335
0602
0608
0615
1014
I03I
I038
1448
I4S6
1463
1905
1913
1930
8383
a39«
3400
3883
,891
a899
to
1.3
44
45
46
9303
9307
9»i3
95a«
9527
9533
2?^
•6331
^>238
•6344
0633
0639
0635
1035
1043
XO50
«47i
1928
1936
«944
3408
3416
3434
390S
3916
3925
30
40
50
a.7
4-0
11
5
49
60
5«
53
93.8
9M3
9338
9538
9544
9549
9883
9894
•6350
•6356
•b363
0643
0649
0655
1071
X494
1501
1509
195a
1959
1967
3433
3441
3449
a933
394a
3950
60
8.0
9a33
9555
9900
^6369
0663
1078
1516
«975
a457
3959
9238
9243
9a49
9561
9566
9S7a
9906
99"
9918
0669
1085
1093
1099
1534
«53«
X539
1983
1991
>999
3465
"42
3968
10
'•5
3.0
1:1
7-5
54
55
56
9a54
r4
95«3
9589
9924
9930
9936
:^5
•6307
0696
0703
tio6
1113
II3I
«546
lilt
aoo7
3014
3033
3506
a993
3003
3011
30
40
1:
59
60
9369
9600
960s
994a
9948
9954
•0314
•6320
•0337
0710
0716
0733
XI38
"35
1143
»576
3030
3038
3046
3514
aw
3531
3019
3oa8
3036
9X>
9a85
9611
9960
•^333
0730
"49
I59«
ao54
a539
3045
e
SliiTMaoNiAN Tablcb.
49
Table 10.
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p« IN ENGLISH
FEET.
[Derivation of table explained on
p. xlv.]
Lat.
21°
22°
230
24°
25°
26°
27°
28<'
29«
30^
P.P.
I
a
3
4
1
I
9
10
II
la
13
11
17
i8
19
ao
ai
aa
33
34
39
30
31
33
S3
34
II
%
39
40
41
43
43
44
:i
%
49
50
5«
S3
53
54
M
59
60
7.818
3045
7.818
3570
7.818
4II5
7.818
4678
7.818
5359
7.818
5858
7.818
6474
7.818
7105
7.818
775 »
7.818
8413
8
306a
3070
3096
3105
3"3
3iaa
I'M
3597
3606
3614
3633
3633
3641
3650
4«34
4133
414a
4153
4161
4170
4«79
4189
4198
4688
4697
4707
4716
4736
4735
4745
5369
'^
5399
5309
5319
5338
5338
5348
5868
5^
5889
5899
5909
59«9
5939
5939
5949
6484
6494
6505
6515
6536
6536
6546
6PJ
7116
7136
7»37
7.48
7«58
7169
7180
7190
7301
7763
7817
7838
7839
7850
8433
US
8479
8490
850.
8513
10
ao
30
40
50
60
1.3
3.6
4.0
8.0
3131
3659
4307
4774
5358
5960
6578
7313
7860
8533
3x57
3165
IliJ
3191
3300
3309
3668
3695
3704
37»3
3733
3731
3740
4ai6
4336
433s
4344
4363
4391
4783
4831
4841
4851
4860
5368
54«7
5437
5437
5447
S970
5980
5990
6000
6011
6031
6031
6041
6051
6588
6630
6630
6640
6673
7333
7333
7344
7354
7376
7387
7''3S
7893
7904
79S9
8557
8568
8579
8591
86o3
8613
8634
9
10
30
30
40
50
60
1.5
3-0
ti
7-5
9-0
3317
3749
4300
4870
5457
606a
6683
7319
7970
8635
33a6
3335
3344
3370
3758
\^
37»5
ISJ
38.3
3823
3831
4310
Si?
4358
4356
4366
4899
4908
4918
4938
4937
4947
4957
5467
5497
5507
55«7
5537
5537
5547
6073
6083
6093
6io3
61 13
6133
6133
6143
6154
6703
6714
6734
6735
6745
6756
6766
6777
7339
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8680
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10
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8.3
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3840
4394
4966
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6787
7436
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8747
33 «3
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3340
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33 S7
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3867
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3904
3913
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4403
4413
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4431
444«
4450
4460
4469
4479
S005
5015
5035
5034
5044
5054
5567
fig
5W
5607
5617
5637
5637
5647
6195
6aos
6315
6aa6
6336
6346
6356
6798
6to8
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6839
6840
6851
6861
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7469
7480
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7503
7513
7533
809.
8103
81I3
8134
l:i:
8179
8759
IS
88x5
88a6
8849
11
10
30
30
40
1.8
3.7
5-5
7-3
9.3
11.0
3393
3931
4488
5064
5657
6367
6893
7534
8190
8860
340I
3410
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3940
3967
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3995
4004
4013
4498
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5717
5737
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6398
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763.
7633
8301
8313
8333
8357
8a68
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8883
8894
8938
8939
u
to
30
30
40
a.o
4-0
6.0
8.0
10.0
13.0
3481
4023
4583
5 161
5757
6370
6999
7643
8301
8973
3490
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3516
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4078
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4096
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4611
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5300
5310
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5330
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5767
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5848
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6391
6401
6411
6433
6433
6443
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7719
7739
7740
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8379
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4678
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5858
6474
7105
775«
8413
9086
Bmitma
lOMIAM 1
rABurs.
Jigltize
dbyV^
rro
^
50
Table 10.
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE Pm IN ENGLISH
FEET.
[Derivation of table explained on p. xlv.]
LaL
31°
32°
33°
34°
35°
36°
37°
38°
39°
400
p.p.
7.313
7.313
7.319
7.319
7.319
7.319
7.319
7.319
7.319
7.319
IF
I
a
3
9086
9773
0472
1x83
1903
363X
3369
4"4
4866
5623
11
9098
9109
9120
9785
9796
9807
0484
0495
0507
"94
I306
X3X8
X9X4
.926
1938
2643
3381
3394
3406
4x26
4139
4»5i
4878
489X
4904
5636
5649
5661
4
1
913a
9'43
9>54
9819
0519
o53«
054a
1330
124X
"53
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X962
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3680
3692
2705
34«8
3431
3443
4164
4x76
4189
49x6
4929
4941
5674
5687
5699
xo
X.8
3.7
5.5
7-3
9.a
X1.0
1
9
10
II
la
13
9166
9177
9189
9877
1%
0577
1265
1377
1389
1986
1999
201 1
3717
3729
2741
1
3480
4201
4214
4226
4954
4966
4979
S7xa
5725
5737
30
40
^JOO
9889
0590
X30X
2023
2753
349a
4239
4992
5750
9aii
9«3
9334
9900
99"
99a4
0601
0613
0635
t3'3
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1337
2035
2047
2059
2766
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3790
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to
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00
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9763
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6347
6360
6373
9773
•o47>
1X83
1902
2631
3369
4"4
4866
5623
6385
SiimiaoNiAN Tables.
51
Table 10.
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE pm IN ENGLISH
FEET.
[Deriyatioa of table explained on p. xlv.]
Lat.
41**
42°
43"
44"
45"
46°
47"
48"
49"
SO"
P.P.
7.319
7.318
7.818
7.318
7.319
7.8i0
7.320
7.820
7.320
7.320
I
a
3
638s
7«5a
7921
8693
9464
0236
1007
1776
2543
3306
6398
6411
6424
7164
7177
7190
?SI
7959
8704
8717
8730
9476
9489
9503
0248
0261
0374
1020
>033
1045
1789
1803
1815
a556
3569
3581
33<9
333 «
3344
4
5
6
6436
6449
6463
7228
B
8743
1^
95 '5
95a8
9541
0387
0300
0313
1058
,8,7
1840
1853
1^'
3619
3357
3369
338a
I
9
10
II
13
13
6500
7241
7a54
7267
8010
8023
8036
8781
9579
0336
0338
035«
1097
mo
1 122
1866
1879
1893
363a
^3^1
3395
3407
34ao
12
65'3
7280
8049
8820
959a
0364
"35
1904
3670
3433
6526
6538
6S5«
7292
730s
7318
8062
nil
8«»
9631
0377
0390
0403
,148
1 161
1174
1917
1930
1943
3709
3471
10
30
30
40
50
60
3.0
Vo
14
'5
i6
6564
7331
7344
7356
8100
8113
8126
lis
8897
9644
9669
0416
0439
0441
1187
1 199
I2I3
1981
2731
2734
3747
3496
3509
8.0
10.0
13.0
17
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20
ai
aa
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0020
7369
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8910
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6679
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8203
8216
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9747
9760
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1303
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3071
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3836
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3597
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as
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6717
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6832
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9927
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1456
1469
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3224
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3001
3736
3749
3763
10
30
30
40
3.2
B
10.8
13.0
11
39
40
41
4a
43
6858
6870
6883
7626
7638
765*
8396
8409
8422
9168
9180
9193
9940
0711
0724
0737
1482
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1507
2249
2262
a275
3014
3027
3039
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3787
3800
50
60
6896
7664
8434
9306
9978
0750
1530
3388
305a
38.a
6909
6921
6934
7677
7690
770a
8447
8460
8473
9219
9a3a
9a4S
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•0004
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0763
0776
0788
1546
1559
3301
3078
3090
3850
44
6947
6960
6973
77»5
77a8
7741
8486
8499
851a
9258
9270
9*83
•0030
•0043
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0801
0814
0827
1597
3339
3»03
3116
3138
3863
49
60
51
Sa
S3
701 1
7767
7779
8524
8537
8550
9296
9309
93aa
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0840
0853
0866
i6io
1623
1635
2377
3390
3403
3«4«
390«
39«3
39a6
7024
779a
8563
9335
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0878
1648
a4i5
3179
3938
7036
783X
8576
8589
8692
9348
9361
9373
•0120
•0133
•toi46
0891
0904
0917
1661
3438
3441
a454
3 '9a
3*05
3317
395«
54
II
7100
g
8614
9386
9399
9412
•bi58
•0171
•0184
0930
0943
0955
1699
1712
1725
3466
3479
3493
3a3o
3243
3a5S
3989
4009
4014
15
59
60
7113
7ia6
7139
7^>
7908
8679
9485
9438
9451
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•b323
0968
0981
0994
»738
1763
aso5
3517
a530
3368
3381
3293
4027
4039
4052
715a
7921
8693
9464
•0336
1007
1776
2543
3306
406s
iLir-ihin
Smithsonian Tables.
5*
Table 10.
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE Pm IN ENGLISH
FEET.
[Deriyadon of table explained on p. zlv.]
Lat.
51-
520
53"
54°
5!^
56-
57°
58O
59°
60°
P.P.
7.sao
7.sao
7.800
7.300
7.320
7.830
7.330
7.330
7.330
7.331
X
a
3
4o6s
48x7
5564
6303
7034
7756
8467
9168
9857
OS34
13
4077
4090
4102
48.9
484a
4854
5576
6315
6327
6340
7046
7058
7070
7768
7780
7792
8479
8491
8503
9180
9191
92P3
9868
9880
9891
0545
4
1
4"5
4127
4140
4867
4879
4892
5613
6352
6376
7082
7094
7x07
7804
78x5
7827
lilt
8538
92x4
9226
9238
9903
9914
9925
0578
I
9
10
XI
la
13
4152
4«6S
4*77
4904
49»7
4929
5675
6388
6401
6413
71 19
7'3'
7«43
7839
7851
7863
lis
8573
9249
926X
9272
9937
9948
9960
0612
0623
0634
30
30
40
IS
X3.0
4190
4942
5687
642s
7155
7875
8585
9284
9971
0645
4203
1^
4979
5699
5712
5724
6437
6449
6462
7167
7179
7191
7887
7899
79"
IIS
8630
9295
9307
9318
9982
9994
•0005
3
14
'5
i6
4240
4992
5004
50x7
5737
5749
5761
6498
7203
7215
7227
7923
7946
8632
8643
8655
9330
934«
9353
•0016
•0027
•0039
0689
0701
07x3
51
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ao
2X
33
23
4278
429X
4303
5029
5042
5054
5799
6510
6523
6535
7239
7251
7263
7958
7970
7982
8667
8679
8690
9364
9376
9387
•0050
•0061
•0073
0723
0734
0745
43 '6
5067
5811
6547
727s
7994
8702
9399
•0084
0756
4328
434«
4353
5079
5092
SX04
5823
5848
6559
6571
6584
7287
7299
73"
8006
8018
8030
8714
8725
8737
9410
9422
9433
JC095
•0107
•01 18
0767
0778
0789
24
4366
4378
439«
5x17
5129
5«4«
5860
6596
6608
6620
7323
7335
7348
8042
8053
8065
8749
8760
8772
B
*OI29
•6140
•0x52
0800
0812
0823
13
29
SO
31
32
33
4416
4428
5«79
5897
5909
5922
6632
6645
6657
7360
7372
7384
8x01
8784
8796
8807
9479
9491
9502
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0834
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0856
30
30
40
3.0
8.0
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5 '9'
5934
6669
7396
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11
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8196
8207
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9617
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61 18
613 X
6840
7576
7588
8278
8290
8302
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9697
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1.8
49
60
5«
52
53
4679
5403
5428
6143
6155
6168
6889
690X
7600
7612
7624
83x4
8325
8337
9017
9029
9040
9709
9720
9732
*o388
•6399
•64x1
iS
20
30
40
IS
3.7
5.5
7.3
9.3
IX.O
4692
5440
6180
6913
7636
8349
9052
9743
•6422
X087
4704
47«7
4729
1^5
5477
6192
6205
6217
6925
6937
6949
7648
7672
836X
9064
9087
97«>
9777
.098
X109
XX30
54
H
4742
5490
5502
55«4
6229
6241
6254
6961
7684
8420
9098
9XX0
9X23
9789
9800
981 X
^89
"3«
XI43
"53
59
00
4779
5527
5539
5552
6278
6391
699S
7010
7022
7720
7732
7744
8432
8443
845s
9«33
9M5
9156
9823
tt
•10500
•0512
•0523
XX64
48x7
5564
6303
7034
7756
8467
9x68
9857
•0534
"97
1
•■rmaoMiAN Tables.
S3
Table 10*
LOGARITHMS OW MERIDIAN RADIUS OF CURVATURE pm IN ENGLISH
[Derivation ol table ezphined on pi sir.]
Lat.
6i«
62O
63"
64°
6f
66°
67°
esp
69»
70°
p.p.
C
I
3
3
4
1
7
8
9
10
I.
la
«3
«4
ao
ai
aa
as
M
U
29
SO
s«
33
33
34
%
39
40
4«
42
43
44
49
60
5«
5*
53
54
59
60
7.821
"97
7.821
1845
7.821
2479
7.821
3097
7.821
3698
7.821
4282
7.821
4848
7.821
5396
7.821
5924
7.821
6432
11
iao8
1219
ia3o
1241
J25I
ia6a
"73
1384
1295
1856
1866
1877
1888
1898
1909
i9ao
193 1
1941
2489
2500
2510
2521
253 X
254X
2552
2562
2573
3x07
3XX7
3127
3x37
3147
3x58
3x68
3x78
3188
3708
37x8
3728
3738
3747
3757
3767
3777
3787
4292
430X
43XX
4320
4330
4340
4349
J5I2
4857
4867
4876
488s
4894
4904
49x3
4922
4932
5405
54x4
5423
5432
5440
5449
5458
5467
5476
5933
594X
5950
5958
5984
5993
6001
6440
6448
6457
6465
t^.
6498
6506
to
ao
30
40
(.8
3-7
5.5
7*3
9.2
11.0
1306
X952
2583
3x98
3797
4378
494X
5485
6010
65x4
13x7
1328
1338
1349
1360
i37«
138a
1392
1403
1963
«973
1984
1994
aoi6
aoa6
2037
2047
2614
2625
2635
2645
2677
32c*
33x8
3228
3238
3248
3259
3269
3807
llZ
11%
3856
3866
3875
3885
4387
4397
4406
44x6
4425
4435
4444
4454
4463
4950
4959
4969
4978
4987
4996
5005
50x5
5024
5494
5503
55x2
552 X
5529
5538
5556
5565
6018
6027
6035
6044
6061
OODO
6078
6086
6522
6530
6539
6547
6555
6563
6588
10
M14
2058
2687
3299
3895
4473
5033
5574
6095
6596
10
ao
30
40
«-7
3-3
1:;
10^0
1436
1447
1458
1468
1479
Z490
1501
151a
2069
2079
ao90
a 100
aiit
2Z22
2132
2143
2«53
2718
.728
2738
2749
2769
2780
3309
33x9
3329
3339
3349
3360
3380
3390
3905
39x5
3924
3934
3944
3954
3964
3973
3983
4482
4492
450«
45««
4520
4530
4539
:^J2
5042
5o5»
5060
5078
5088
5097
5106
SXX5
5583
5600
1^
5627
5636
5644
5653
6XOS
61 xa
6zao
6ia9
6x37
6146
t\n
6x71
6604
6612
662 1
6629
6637
6645
6670
9
1523
a 164
2790
3400
3993
45«
5x24
5662
6180
6678
1534
1545
'555
1566
1609
i6ao
2175
aiSs
2196
2206
2217
3228
2338
2249
2259
2800
28IX
383 1
383.
2841
2852
2862
2872
2883
34x0
3420
3430
3440
3460
3490
4003
4012
402a
4032
4041
405 X
4061
4596
4606
46x5
4624
4634
4643
4653
5x33
5x42
5'5x
5x60
5x69
5x79
5x88
5x97
5206
567 X
5680
5688
5706
57x5
5724
5732
574«
6188
6197
6205
6a.4
622a
6230
6239
6247
6256
6686
6710
6718
6727
6735
6743
6751
10
ao
so
40
S
«.5
3.0
tl
7.5
9.0
1631
2270
2893
3500
4090
4663
52x5
5750
6264
6759
S
164a
1663
9
1695
1706
1717
1727
2280
229X
2301
2312
2322
2333
2343
2364
2903
2913
2924
2934
2944
2954
2964
2975
298s
3510
3520
3530
3540
3549
3559
3569
3579
3589
4100
4x09
4119
4x28
4x38
4x48
4x67
4x76
4690
47x8
4727
4736
4746
5224
5233
5242
5260
5270
ns
5297
5759
5767
5776
5785
5793
5802
58x1
5820
5828
6272
628 X
6289
6298
6306
63x4
6323
633 X
6340
Si?
^1
6791
68x5
6823
6831
10
ao
30
40
1:
li
4.0
8.0
1738
2375
2995
3599
4186
4755
5306
5837
6348
6839
1749
«759
1770
1781
1791
1802
.8.3
1824
X834
2385
3396
2406
2417
2427
2437
2448
2458
2469
3005
3026
3036
3046
3056
3066
3077
3o«7
3f»9
3619
3629
3639
3648
3658
3668
3^^78
3688
4196
4205
42x5
4224
4234
4244
4253
42'^>3
4272
4764
4774
4783
4792
4801
4811
4820
4829
4839
53x5
5324
5333
5342
535X
5360
53^^
5378
5387
5846
5'^54
58^>3
5872
5880
5889
5898
5907
59x5
6356
636s
6373
6382
6390
6398
6407
64x5
6424
6847
6855
6863
687X
6879
6887
6895
6903
69x1
T
J845
2479
3097
3698
4282
4848
5396
5924
6432
69x9
Smithsonian Tables*
S4
Table 10«
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE pm IN ENGLISH
FEET.
[Derivation of table explained on p. zIt.]
Lnt.
71°
72°
73°
74°
7f
760
77°
78°
79"
80**
P.P.
C
I
a
3
4
1
i
9
10
II
la
13
M
:i
%
«9
20
ax
aa
Vi
M
U
a9
ao
3X
3*
33
34
u
39
40
4«
4a
43
44
%
49
60
5«
5a
5J
54
H
fi
59
00
7.821
6919
7.321
7385
7.821
7839
7.821
8351
7.821
8650
7.821
9035
7321
9377
7821
9704
T.aaa
0007
7322
0384
69*7
«935
69i3
s
6990
739a
7400
7407
7415
743a
7430
7437
7445
745a
7836
787a
7894
8358
836s
8371
13?
o3B5
8393
8399
8305
8313
8669
868a
8688
8695
8701
8708
903*
9037
9043
9049
9055
9061
9067
9073
9079
9383
9388
9394
9399
9405
94"
94x6
9433
9427
9709
97>4
9730
9785
9730
9735
9740
9746
975«
OOX3
00x7
003I
0036
0036
0041
0045
0050
0388
0393
0397
030a
0306
03x0
0315
0319
0334
10
To
40
1.3
a.6
4.0
8.0
6998
7460
7901
8319
87x4
9085
9433
9756
005s
0328
7
7006
7014
7021
7019
7037
7045
7^
7068
7467
7475
7483
7490
7497
7505
75"
7530
7527
7908
79»5
792a
7944
7951
8336
833a
8339
8346
8353
8359
8366
8373
8379
8730
8737
8733
8739
nil
8771
909X
9097
9«03
9x09
9"5
9X31
9x37
9'33
9139
-9438
9444
9449
9460
9466
947'
9477
9483
977X
9776
9781
9787
9793
9797
0060
0064
0069
0074
0088
0093
0097
033a
0337
o34«
0345
0349
0354
0358
0362
0367
xo
ao
30
40
x.a
a.3
3.5
ti
7.0
7076
7535
797a
8386
8777
9M5
9488
9807
0x03
037»
7084
709a
7099
7107
7««5
7133
7131
7138
7146
754a
7550
7557
7565
757a
7580
7587
7595
7003
7900
7993
8000
8007
80.4
8oai
8o88
8035
84x9
8436
8433
8440
8446
8783
8803
8808
88x5
8831
8837
8834
9151
9»57
9x63
9x69
9»74
9180
9x86
9XM
9198
9493
9499
9504
9510
95»5
95a X
9536
9533
9537
98x3
98x7
9833
9837
9833
9838
9853
0x07
OIIX
0x16
0I30
0x35
0X30
0134
0X39
0x43
0375
0379
0384
0388
039a
0396
0400
0405
0409
6
xo
ao
30
40
I.O
a.o
3.0
4.0
IS
7«54
7610
804a
8453
8840
9304
9543
9858
0148
0413
716a
7170
7«77
718s
7193
7aoi
7ai6
7aa4
7617
7635
763a
^l
7654
7661
7676
8091
8098
8105
8460
8466
8473
is?
8493
1^
8513
8846
8859
8865
8871
8877
8883
8890
8896
93x0
93x6
9331
9337
9333
9339
9*45
9350
9356
9548
9554
9559
9565
9570
9575
«?i
9586
959a
987s
9903
0x53
0x57
0x63
0166
0x71
0X76
0x80
0x85
0189
0417
0431
0436
0430
044a
0447
045 X
6
xo
ao
30
40
.8
1.7
a.5
3.3
4.3
50
7»3a
7683
8iia
85x9
8903
9363
9597
9908
0x94
<H55
7340
7M7
7*55
7»63
7370
7378
7386
7*94
7301
7690
7798
7705
7713
7719
77a7
7734
7741
7749
1:^6
8133
8140
8147
8154
8x6x
8168
8x75
8536
8533
8539
8545
8553
8559
8565
8578
8908
89x4
8931
8937
8933
8939
8945
8958
9368
9374
9279
9a8s
939X
9397
9303
9308
9314
9613
9619
9639
9635
99«3
99x8
99*3
9938
9933
9938
9943
9948
9953
0199
S3
03I3
0217
0333
0336
033X
0335
0467
0471
0493
4
xo
ao
30
40
■7
IS
3.0
a.7
3-3
40
7309
7756
8183
8585
8964
93ao
9651
9958
0240
0496
73 »7
7334
733a
7339
7347
7355
7363
7370
7377
7763
777>
7778
7785
7793
7800
7807
7814
7833
8189
8196
8303
8310
8316
8333
8330
8337
8344
IS!
86x1
86x7
8634
8630
8643
is2
0970
8983
8994
900X
9007
9013
90x9
9336
9331
9337
9343
9348
9354
9360
9366
937«
9002
9667
9673
9693
9699
9973
9978
9983
9987
999a
^0003
0344
0349
0353
0358
0363
0366
037 X
0300
0500
0508
05x3
0516
0530
0534
0528
0533
^
T
7385
7839
8351
8650
9035
9377
9704
*boo7
0384
0536
SMmtaoNiAH Table*.
55
TablcIO.
LOGARITHMS OF MERIDIAN RADIUS OF CURVATURE p» IN ENGLISH
FEET.
[DerivadoD of table eqjdained on p. zlv.]
Lat
Si*'
82°
83°
84^
85-
86°
87-
88°
89»
P.
P.
7.322
7.322
7.322
7.322
7.322
7.322
7.322
7.322
7.8tl
X
a
3
0536
0763
0963
.138
X285
1407
X50X
1569
1609
4
0540
0766
0770
0773
0966
0969
0972
1141
,287
1289
1293
1409
14 10
141a
X502
1504
«505
1570
'571
1571
1609
x6io
1610
4
1
055a
0556
0560
0777
0780
0784
097s
1148
1x51
"54
1296
1298
U'4
'415
1417
XS06
XS07
1509
«57a
«573
»574
x6xi
x6xi
x6xx
xo
ao
30
40
50
60
.7
1.3
a.o
a.7
I
9
10
II
la
«3
057a
0787
0791
0794
0991
1x56
"59
1161
1300
1303
1305
14x9
X42X
142a
1510
15"
i5»3
1575
«576
16x2
16x2
16x3
3.3
4.0
0576
0798
0994
1x64
>307
X424
1514
1577
16x3
0580
0801
0997
1000
1003
1x67
1x69
1x7a
1309
13"
1314
X426
1427
X429
«5i5
1518
1578
1579
«579
1613
x6i4
x6x4
»4
059a
0595
0599
o8ia
0815
0819
1006
1009
loia
"74
;;2
13 16
13x8
i3ao
143 «
'43a
1434
«5i9
1520
isaa
X580
X58X
1582
16x5
16x5
16x5
»9
SO
ai
aa
as
060s
0607
0611
o8aa
0826
0839
1015
1018
loai
1x82
1.8s
1x87
1322
13*5
1327
1436
1438
M39
«5a3
1534
1526
X583
1583
1584
x6x6
x6x6
1617
0615
0833
1024
1 190
1339
«44«
1537
1585
1617
10
ao
30
40
•S
i.o
1.5
aa>
a-S
S.O
0619
0836
1027
1030
«^3
1x92
"95
"97
«33i
1333
1335
1443
X528
1539
1530
XS86
1587
16x7
M
S
0630
0638
'Z
0853
XQ36
1039
104a
X200
1202
1205
'337
1339
X34I
«447
«449
M5«
1531
153a
1534
1588
1589
1618
x6i8
1618
a9
ao
33
33
064a
0645
0649
0856
1051
xao7
1210
X2I2
>343
«345
«347
145a
1454
MS5
1536
1537
1590
«59«
i59«
16x8
.6x9
16x9
1
0653
o8«6
1054
X3I5
1349
1457
1538
159a
X619
0664
1057
1060
io6a
X217
1220
122a
«35i
1353
«355
X460
146a
»539
1540
«54«
1593
"593
»594
1619
1619
i6ao
34
U
0668
0671
0675
0886
1071
1225
1227
X229
«357
1361
1463
1467
iS4a
«543
>545
"595
1596
1620
i6ao
10
.3
39
40
4«
4a
43
s
0889
0892
0896
X074
1076
1079
1232
«a34
«a37
1363
.365
1367
X468
X470
X471
1546
1548
1598
1598
i6ao
162X
1621
ao
30
40
.7
1.0
«.3
1.7
a.o
0690
0899
io8a
1239
1369
«473
«549
«599
1621
0694
0697
0701
0906
0909
1090
1241
"44
1246
1371
»373
1375
«477
1550
»55«
«55a
x6oo
1600
1621
x6at
X621
44
t
071a
091a
0915
0919
I093
1096
1099
1249
I2SX
"53
iii
X48X
«553
«554
1555
160X
1601
1602
1621
X621
X622
49
SO
s«
53
0716
07*3
09aa
0925
0929
txoa
1104
1107
1256
1483
X556
X602
X603
1603
1622
1622
1622
]
0727
0933
mo
X263
X388
1487
1559
x6o4
1622
XO
ao
30
40
.a
•3
•5
i
I.o
0731
0734
0738
0935
0938
0941
1113
11x6
1118
1267
X270
X39O
«39a
«394
X488
X490
X49X
X560
156X
1562
1604
1605
160S
1622
X622
X622
54
0741
0745
0749
0944
0947
095 «
X121
1 124
X127
1272
1396
1397
1399
1493
>494
1495
1563
1564
1565
x6o6
x6o6
X607
1622
1622
1623
1;
li
59
00
075a
0756
0759
0954
0957
X130
113a
"35
1278
X28I
.283
140X
1403
1405
1498
1500
1566
1567
1568
^608
x6oS
xfa3
1623
X623
T
0763
0963
1.38
1285
1407
X50X
•569
1609
1623
SliiTHaoNiAN Tabus.
S6
Table 11.
LOGARITHMS OF RADIUS OW CURVATURE OP NORMAL SECTION
p^ IN ENGLISH FEET.
[Derivatioii of table ejcpbined on p. xlv.]
Lat.
o''
1°
^
3**
4°
f
6*»
7^
8°
9°
Id*
P.P.
7.660
7.360
7.360
7.360
7.360
7.860
7.360
7.360
7.360
7.360
7.360
I
a
3
«75
6880
6893
6916
6947
6987
7036
7094
7160
7a35
73x9
6880
6880
6880
6893
6916
6917
6917
6948
6949
6949
6988
6989
6989
7038
7Q39
7096
7097
7161
7163
7x64
7336
7238
7339
7380
73»a
7383
4
1
6875
6880
6881
6881
689s
6918
6918
69x8
6990
J99«
6993
7040
704X
704X
7098
7099
7100
7x67
7340
7341
7a43
7387
I
9
10
II
xa
«3
687s
6881
6881
6881
SSI
69x9
6919
6953
6993
0993
6994
704a
7043
7044
710X
7103
7x03
7x68
7x70
7x71
7a44
7*45
7*47
7389
7330
733a
1
6875
6881
6896
6930
6953
6995
7045
7x04
7x73
7348
7333
S75
6881
6881
688a
6896
6930
6931
693X
6955
6955
6997
7046
7106
7107
7x73
7349
7351
7353
7334
7336
7338
»4
6876
688a
688a
688a
68,8
6933
^3
6956
6956
6957
6999
6999
7049
7050
7050
7108
7109
71XX
7179
7854
7339
734X
734a
xo
30
30
.a
•3
.5
iO
31
aa
as
6876
6876
6876
688a
6883
6883
6900
6924
6959
7000
700X
700X
7051
7053
7053
7x13
7"3
71x4
7180
7183
7183
7358
7343
7346
£
i
x.o
6876
6883
6900
69.S
6959
7003
7054
7115
7184
7a(n
7348
6876
6876
6876
6883
688i
6884
6900
6901
6936
6961
70Q3
7004
7004
7057
71x6
7x88
7>«3
7350
735«
7353
34
6876
6876
6876
6884
6^
6884
6901
69QS
6937
6963
6963
6963
7007
7058
7059
7119
7x30
7X33
7x89
7x90
7x91
7370
7358
2
a9
30
3S
3*
33
6S76
6876
6876
6884
6885
6885
69QS
6903
6938
6939
7009
7^
7063
7X33
7x34
7x35
719a
7x94
7«95
7a7a
7*73
7875
736X
736a
6876
6885
6903
6930
6966
7010
7064
7136
7196
7376
7364
^
6886
6904
6930
693 X
693 X
701 X
70x3
70x3
7067
7"7
7x38
7»a9
7197
7x99
7aoo
7377
7366
7IS
34
^
^
6886
6887
6887
6905
693a
693a
6933
6970
7014
7015
701S
7069
7070
7x30
7x31
7x33
730X
7303
7304
7383
7383
7384
7370
737X
7373
39
40
4«
4*
43
^
6887
6^
6907
6934
6935
6973
6973
70x6
7070
7071
7073
7x34
7x35
7x36
3
7a86
7387
7389
7374
7376
7377
6
6877
6888
6907
6935
^3
7019
7073
7x37
7ao9
7390
7379
10
30
30
40
•3
•7
1.0
X.3
6888
6888
6889
6909
6936
6936
6937
6974
6975
7030
703X
7031
7074
7075
7076
7x38
7x39
7140
7310
73X3
7ax3
7391
7893
7894
738X
7383
7384
44
49
60
5«
Sa
53
6878
6878
6878
6878
6889
6889
6^;
6889
6890
6890
6909
6910
6910
6910
69II
691 1
6937
6938
6939
6939
6940
6976
6977
6978
6979
7033
7033
7034
7035
7077
7078
7079
7081
714X
7x43
7«44
7«47
7316
73x7
73x8
73x9
733X
7896
7300
730*
73<*3
7387
7389
7390
739a
7393
£
x-7
3.0
6878
6890
6911
6941
6980
7037
708s
7M8
7333
7304
7395
6878
6878
6879
6890
6891
689X
691a
6913
6913
694a
694a
6943
698X
698.
6983
7038
7039
7Q30
!2«
7«49
7x50
7x5a
7333
7305
^3
7400
54
ft
«2
6891
689a
689>
69x3
69x4
69x4
6943
6944
6944
7031
703a
7Q3a
7090
7x53
7'54
7«5S
7338
7330
73x0
73XX
73«3
740X
7403
7405
59
60
6880
689a
689a
6893
69x5
6945
at
6986
7033
7034
7035
709X
7093
7093
7x56
7x58
7x59
7a3x
7333
7834
73x4
73x6
73x7
7406
7408
7409
6880
6893
69x6
6947
6987
7036
7«94
7x60
7a35
7319
74"
1
S u fTw aowiAii Tables.
57
Table 11.
LOGARITHMS OP RADIUS OP CURVATURE OP NORMAL SECTION
fK, IN ENGLISH PEET.
[DerivatioD of taUe escplained oo p. zIt.]
Lit.
lio
12^
13"
I4*>
If
l6<*
17**
18O
19"
20<»
p.p.
I
a
3
4
I
i
9
10
XI
xa
«3
«4
X5
x6
'A
«9
ao
ai
aa
as
24
U
^9
ao
3<
3a
33
34
'A
39
40
4«
4a
43
44
t
%
49
50
51
Sa
53
54
%
59
60
7.320
74"
7.310
75"
7.320
7619
7.820
7736
7.320
7860
7.320
799a
7.320
8x3 a
7.320
8279
7.320
8434
7.320
8595
1
74«3
7414
7416
74x7
74*9
74ai
74aa
74*4
74a5
7S«3
7516
75«8
75«9
75ai
75a3
75a6
76ai
7633
76a5
7630
763.
7738
7740
774a
"?
7748
7750
775a
7754
786a
7864
7867
7869
7871
7873
7994
7997
7999
8001
8«,3
8006
8008
8010
80,3
8x34
8137
8139
814a
\Z
8149
8151
8154
828a
8a84
8287
8289
829a
8395
8297
8300
830a
8437
8439
844a
8444
!^'
8450
845a
8457
Hoi
8603
8606
8609
86x3
861s
xo
ao
JO
40
£
•3
:J
1.0
74a7
75a8
7638
7756
788a
8015
8156
830s
8460
8633
74a9
7430
743a
7433
7435
7437
7438
7440
744«
7530
753a
7533
7535
7537
7539
7541
7S4a
7544
7640
764a
7644
7646
7647
7649
7651
7653
7655
7770
777a
7774
7888
7890
7901
8017
8oao
8oaa
£2
8oa9
8031
8158
8161
8163
8166
8168
8170
8.73
8307
8310
831a
83x5
8317
8330
83aa
Jsas
83a7
8463
8476
Its?
M4
86a6
8640
8643
1^
1
7443
7546
7657
7776
7903
8038
8180
8330
8487
86sx
74f8
7450
745 «
7453
7455
7457
745«
7548
7550
755«
7553
7555
7557
7560
7561
7663
7668
7670
767a
7674
7778
7780
778a
J|
7789
779*
7793
7795
7905
7907
7910
79"
79«4
7916
7918
79ai
79a3
8040
8043
8045
8047
8059
8x82
8x85
8187
8190
819a
8195
8x97
8aoo
82C2
8333
8340
i^
8348
8351
8353
8490
849a
8495
8498
8500
8503
8506
8509
851X
8654
Sf57
8659
866a
1^
8671
xo
ao
30
40
.3
.7
X.O
X.3
X.7
a.o
,^
7564
7676
7797
79a5
8061
8ao5
8356
8514
8679
746a
7463
7465
7466
7468
7470
747X
7473
7474
'55s
7568
7569
757«
7573
7575
768a
7^^
J688
7694
7803
7807
7810
7».a
79a7
79,9
793a
7934
7936
7938
7940
7943
7945
8068
8083
Jaoy
8a 10
821a
82x5
8217
82x9
8222
8«4
8327
836s
8366
8368
8371
8373
8378
85.7
8519
8533
8535
85a7
8530
ISI
8538
868a
8685
8687
8704
S
7476
758a
7696
7818
7947
8085
8239
838X
8541
8707
7478
7481
7483
7484
7486
7488
7490
749»
7584
7588
7590
759«
7593
7595
7597
7599
7698
7700
770a
7708
7710
77"
77*4
7830
78aa
7824
78a6
7828
7831
7833
7835
7837
7949
795a
7954
7956
7961
7963
8IOI
823.
8a39
8241
8244
8246
8249
8251
8389
839.
8394
8397
8399
840a
8404
8S44
8546
8549
855a
8554
8557
8560
8i63
8565
87.0
8713
8715
8718
873X
8734
8727
8729
873a
xo
30
30
40
S
•5
1.0
i.S
a.o
a.5
3.0
7493
7601
7716
7839
7970
8108
8254
8407
8568
8735
7495
7497
7498
7500
750a
7504
7506
7507
7509
7603
70cx>
7608
7610
76fa
7615
7617
7718
77ao
77aa
77a8
7730
773a
7734
784.
7843
7845
^7
7849
785a
7854
7858
797a
7974
7977
7<i90
8tio
8113
8115
8118
8120
813a
8125
8127
8130
8256
tx\
8266
8269
827X
8274
8276
8410
841a
8415
8418
8420
8433
84a6
8439
843 X
857.
f^
8584
8587
8590
859a
8738
874X
8743
8746
8749
875a
8755
75"
7619
7736
7860
799a
8.32
8279
8434
8595
8763
-■
SiiiTHaoNiAN Tabus.
S8
Table 11.
LOGARITHMS OF RADIUS OF CURVATURE OP NORMAL SECTION
p^ IN ENGLISH FEET.
[Derivation of table explained on p. jcIt.]
Lat.
2I<>
22«
23°
24«
2f
2&>
27^
28<»
29^
30^
P.P.
I
a
3
4
1
;
9
ID
XX
xa
»3
14
\l
\l
»9
SO
at
aa
23
24
U
29
ao
31
32
33
34
35
36
39
40
41
42
43
44
• %
49
60
5«
52
53
54
IS
u
59
00
7.sao
8763
7.320
8939
7.820
9120
7.820
9308
7.820
9502
7.820
9701
7.820
9907
7.821
0x17
7.821
0332
7.821
0553
2
8766
8769
8772
8780
8789
8942
8951
US
8965
9123
9x26
9129
9x32
9136
9<39
9142
9«45
9148
93"
9318
932X
9324
9327
9330
9333
9337
213
95«2
95«5
95i8
9S2I
21-1
9531
S3
97x2
97'S
97«8
9722
9725
9728
9732
9910
9913
99«7
99»
9924
9917
993«
9934
9938
OX2S
0124
0x28
013 1
0135
0x38
0x4a
0145
0149
0336
0340
0343
0347
o35«
0354
0358
0361
0365
0564
0567
0571
©575
0586
8792
8968
9>5»
9340
9535
9735
9941
0153
0369
0590
XO
ao
SO
40
.3
.7
x.o
1.3
x-7
a.o
E
8810
88x2
8815
8818
8971
8974
8977
8980
8983
8986
8992
8995
9«54
9«57
9160
9163
9167
9170
9»73
9176
9x79
9343
9346
9349
9353
9356
9359
9362
9365
9368
9538
954«
9545
9548
9551
9554
9558
956i
9739
9742
9745
9749
9752
9756
9759
9762
9766
9945
9948
9952
9955
9959
9962
9966
9V69
9973
0x56
OXS9
0163
0x67
0170
0174
0177
0181
ox8s
037s
%t
0383
0387
0391
0394
0398
0402
0594
060X
Si
0612
06x6
o6ao
0623
8
8821
8998
9x82
9372
9568
9769
9976
0188
040s
0627
8830
8833
8836
8839
^'
8847
9001
9004
9007
90x0
9020
90J3
9026
9188
919X
9«95
9198
9201
9204
9207
92x0
9375
9378
9391
9394
9398
940X
957«
9574
9578
958X
9584
9588
9591
9594
9598
9779
^U
9786
9790
9793
^¥>
9800
9980
9983
9987
9990
9994
9997
•toooi
0x92
0195
0199
02x0
0213
02x7
0220
0409
0413
04x6
0420
0424
0427
043X
0438
0631
^al
0642
0646
0649
0657
o66x
10
20
30
40
•5
x.o
»-5
2.0
25
SO
8850
9029
92x3
9404
960X
9803
•001 1
0224
0442
0664
nil
8859
8862
8877
9032
9035
9038
9041
9044
9047
9050
9216
9220
9223
9226
9229
9232
^11
9242
9407
94"
9414
94x7
9420
9424
9437
9430
9433
9611
9621
963 X
9807
98x0
9814
98x7
9820
9824
9827
9831
9834
•bois
•0018
•0022
•0029
•0032
•0036
•0039
•0043
0228
023X
0235
0238
0242
0246
0249
0256
0446
0449
0453
0464
0468
0471
0475
0668
0672
0676
068,
0691
4
8879
9059
924s
9437
9634
9838
•0046
0260
0479
0702
888a
8891
8897
8900
8903
8906
9062
9071
9074
9077
9086
9348
92SX
9254
9257
•9264
9267
9270
9273
9440
9443
9446
94SO
9453
9456
^'
9466
9638
9644
9648
9654
9658
9661
9664
984X
9858
9862
9865
9869
•0050
•0053
•0057
•0060
•0064
•0067
•0071
*oo74
•0078
0264
0271
0274
Si!
0285
0289
0293
0482
0486
0490
0493
0497
osox
0508
0512
0706
0710
0713
07x7
072X
0725
0728
0732
0736
XO
20
30
40
•7
1.3
a.o
2.7
3-3
4.0
8909
9089
9276
9469
9668
9872
•too82
oifj/b
0516
0740
8912
8921
8924
8927
8930
^^l
9093
9096
9099
9102
91"
91 14
91x7
9*89
9292
V295
9298
9302
9305
9472
9476
9479
9482
9485
94^
9492
9495
9498
9671
9681
969X
9695
9698
9575
9879
9882
9886
9889
9893
9896
9900
9903
•cx)«s
•0089
•0092
•0099
•0x03
•0106
•01 10
•01x3
0300
0303
0307
03x1
0314
0318
0322
0325
0329
0519
0523
0527
0530
0534
0538
0542
0545
0549
0743
0747
07SX
0755
0759
0766
0770
0774
8939
9x20
9308
9502
970X
9907
•01x7
0332
05S3
0777
■
SumtaoNiAM Tables.
59
Table 11.
LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION
p^ IN ENGLISH FEET.
[Derivation of tabk giphinrd on p. zIt.]
Lat.
31°
32°
33°
34°
35°
36°
37°
38°
39°
40**
p.p.
7.821
7.sai
7.8ai
7.821
7.821
7.821
7.321
7.821
7.821
7.821
I
a
3
0777
1006
"39
X476
X7x6
«959
aaos
2453
3704
2956
s
0781
0785
0789
lOXO
iois
xa43
X247
X251
X480
xMl
X7ao
X963
X967
X971
3209
32x3
32X7
2457
3463
3466
2708
37x2
37x6
396X
4
1
S
loaa
ioa6
1039
X255
1359
xa63
X500
X73a
X736
1740
«975
aaax
3336
3470
372X
2725
3739
S
xo
•5
i
9
10
IZ
la
13
0811
1033
«o37
1041
xa67
xa7x
X275
X5ia
'75*
X988
1996
3343
3483
3487
2491
2733
'737
2743
3986
3990
2994
_1j
1.0
x-5
3.0
a-5
3.0
081S
1045
xa79
1516
1756
aooo
3346
a495
2746
2999
0819
0833
0837
1049
«os3
1057
laSa
X386
xa90
xsao
'1^
30xa
3350
3354
3359
a499
3503
a507
2750
3003
3007
30x1
«4
0830
S060
x3oa
X536
X540
aox6
aoao
aoa4
3363
3367
337X
35x3
35x6
2530
11^
377X
3016
3030
3024
«9
ao
31
aa
as
0849
JOiJ%
1076
1080
.306
X3X0
X314
■55»
1793
ao38
ao33
ao37
3375
2532
«775
1SQ
3038
303a
3037
4
0853
1084
X3x8
•556
1797
3041
3387
2537
3788
304X
0865
1087
1091
1095
132a
X336
1330
x8oi
1809
ao45
ao49
ao53
3393
3396
3300
a54x
aS45
a549
379a
1&
3045
3049
3054
34
0869
087a
1099
1 103
1 107
X334
X337
X34X
X57a
>S>3
x8x7
x8ax
ao65
33x3
a553
2557
3563
1813
3066
10
•7
2
^9
ao
3X
3a
33
0880
IIZI
i.xS
Z118
X345
X349
X353
XS93
xjas
X839
x«33
ao69
ao73
ao77
33x6
3331
a3a5
3566
3570
a574
3823
3836
307X
307s
3079
30
30
40
««3
3.0
«.7
3.3
4.0
0891
1132
X3S7
1596
1837
ao8a
a3a9
a578
3830
3083
z
0903
1 136
XI30
XI34
.36X
X369
x6oo
184X
X845
X849
ao86
ao9o
ao94
a333
a337
a34x
3583
3587
3S9X
2843
3087
34
0907
0910
0914
XX38
1x4a
XX46
1373
x6xa
16x6
x6ao
X853
ao98
aio3
a 106
a345
a3SO
a354
2595
s
3x00
3104
3x09
39
40
4«
4»
43
0918
XX50
"53
x«57
X3«S
X389
«393
x63a
X865
X870
X874
axxo
axx4
31X9
3366
3608
36X3
3616
a
3"3
3"7
3X3X
6
0930
1 161
X397
X636
X878
2x23
3370
3630
3873
3X36
0933
0937
094X
1x65
X169
X173
X40X
X40S
1409
X640
i88a
x886
X890
3X37
ax3x
3X35
a374
3629
a633
:i2
3885
3'30
44
0945
0949
0953
XX77
xz8x
X185
Z4ia
X416
x4ao
x65a
!66o
x9oa
3139
ax43
2X47
a387
a39x
3395
2637
264X
2645
3889
2893
2897
3x43
3«47
3X5X
xo
30
30
40
S
.8
X.7
a.S
3.3
4.a
5.0
49
50
s«
53
0960
0964
>x89
XX92
X196
X43a
1673
X906
19x0
X9X4
axsx
3156
2160
a399
2649
2658
390a
3906
2910
3x64
0968
1200
1436
1676
X918
3x64
34x3
2662
29x4
3x68
097a
0976
0979
X3X3
1440
1680
1^
Z923
1936
X93«
3168
2x73
2x76
2416
2420
2424
3675
29.8
2923
3927
3x7a
3x81
54
0983
0987
0991
I3l6
X230
1334
145a
X456
1460
1693
X696
1700
X93S
X939
1943
2x80
2x84
ai88
2438
a433
a437
3687
293 X
2935
2940
3x85
3189
3193
11
59
60
0995
0999
1003
1228
X231
xa3S
147a
1712
X947
X95«
>9S5
ax93
a 197
2201
2441
2445
2449
269X
2696
2700
2948
2952
3x98
3206
too6
"39
X476
X716
X959
aaos
2453
2704
2956
33x0
■■
Smithsonian Tables.
60
Table 1 1 .
LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION
p» IN ENGLISH FEET.
[DeiivatioB of table explained on p. xhr.]
T^t.
41°
42-
43^
44^
45°
46*>
47**
48O
49°
50°
P.P.
7.011
7.311
7.811
7.311
7.311
7.311
7.811
7.311
7.311
7.311
z
a
3
saio
3466
3722
3979
4236
4494
475X
5007
5263
55x7
3215
3a 19
3M3
3470
3474
3479
3726
373 «
3735
399a
424X
4245
4249
4498
4S02
4507
1!^
4764
Soia
5016
5020
5267
527X
5276
5522
5526
5530
4
1
3**7
3^a
3236
3483
3*87
349«
3739
11%
3996
400X
4005
4254
4258
45XX
45x5
4520
4768
477*
4777
5024
5<M9
5033
5280
5284
5288
5534
5538
5543
i
9
lO
II
la
«3
3240
3244
3249
3496
3500
3504
3752
It
4009
4267
4271
4275
453a
478X
4785
4789
5<^7
5042
5046
5293
5297
530X
5547
555X
5555
4
3253
3508
3765
4022
4279
4537
4794
5050
5305
5560
3266
35»3
35«7
3521
37«9
4026
403X
4035
4a9S
454X
4545
4550
4&>7
5054
53x0
5572
»4
3270
3526
3530
3534
3786
379X
4039
4043
4048
4297
430X
4305
f
S067
5071
S076
53aa
5327
5331
If
5585
xo
.7
19
ao
31
aa
?f4
329*
3538
3543
3547
3795
Sol
4052
4309
43x8
4567
457«
4575
4832
5080
5S3S
5339
5344
5589
30
40
X.3
ajo
2.7
3-3
4.0
3295
3551
3808
¥f>S
4322
4580
4837
5093
5348
5602
3300
S3
1
3564
3812
38.6
3821
4327
433 X
4335
4588
459a
4845
4849
5097
5XOX
5x05
5356
5361
5606
5610
5614
24
33"
3317
3321
3568
3573
3577
3825
3829
3833
4091
4339
S3
460s
U
5x10
5369
5373
5619
55»3
5627
39
so
32
33
33*5
3329
3334
3581
3585
3590
3838
3846
4095
4099
4104
435a
4610
4866
487X
4875
5««3
5x27
5X3X
5386
5636
5640
333«
3594
3851
4108
4365
462a
4879
5x35
5390
5644
3342
3347
3351
S
S855
Jit?
411a
4116
4iai
4369
4378
45a7
463X
4635
2
4892
5140
5395
5399
5403
5648
5657
34
335S
f3S
3611
3615
3620
3868
4ias
4129
4«34
4382
4387
439X
4640
4896
4901
4905
5x52
5x57
5161
5407
54x2
54x6
5661
39
40
4«
4*
43
3368
3372
3376
363a
3889
4138
4142
4x46
4395
4399
4404
465.
1^1
4909
49x3
49x8
5x65
5x69
5x74
5420
g
10
ao
30
40
.8
«.7
a.5
3-3
4.2
5.0
3381
3637
3893
4>5»
4408
4665
4922
5x78
5433
5686
338s
3389
3393
3641
3645
3649
3898
3906
4x55
4164
44X2
44x7
442X
4670
4926
493 X
4935
5x8a
5.86
519X
5437
544X
5445
5690
5694
5699
44
1^
3398
3402
3406
3654
39"
39«5
39x9
4x68
4I7J
4x76
4425
4430
4434
468a
4691
4939
5x95
5x99
5203
5450
5703
5707
57X1
49
60
5»
53
34IO
3415
34«9
1^
3675
3932
4.8X
418s
4189
4438
4442
4447
469s
4700
4704
4952
5208
521a
52x6
547X
57x6
5720
5724
3423
3679
3936
4x94
445X
4708
4965
5220
5475
5728
3427
3432
3436
3684
3688
369a
394 >
3945
3949
4x98
4206
4464
47X2
47x7
4721
4969
5225
5229
5233
g
5732
5737
574X
54
3440
3445
3449
3697
3701
3705
B
42x1
42x5
4219
4468
4472
4477
4725
4730
4734
4990
5237
5246
S49J
5496
5500
5745
5749
5753
59
00
3453
3709
37'4
3718
3966
3971
3975
4228
4232
448.
4485
4490
4738
4742
4747
4995
4999
5003
5250
5254
5259
5505
5509
55x3
576a
5766
3466
3722
3979
4236
4494
475X
5007
5263
55x7
5770
>
SiuTNaoNiAfi Tables.
61
Table 11.
LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION
p^ IN ENGLISH FEET.
[Derivation of table espbdned oo p. xIt.]
Lit.
5iO
S^
53"
54**
55°
56O
57°
580
59°
6(f
P.P.
7.ttl
7.ttl
7.8il
7.8S1
7.321
7.aai
7.3S1
7.sai
7.321
7MX
I
a
3
5770
6Q3I
6370
6517
6760
7001
7a38
747a
7701
7987
5783
6oa9
6q34
3
653X
6535
6539
677a
7005
7009
70x3
7a4a
7a46
7350
It
7483
7705
7709
77«a
793«
4
1
5787
5791
5795
6qs8
604a
6046
6386
6390
6a9S
6533
6537
6541
6785
7017
7031
7oa5
7a54
7487
7491
7495
77«6
7730
7784
7948
7945
7949
6
g
I
9
10
II
13
«3
s
6050
6055
6059
6a99
6303
6307
6545
6549
6553
6789
6793
6797
7039
7033
7037
7365
7369
7a73
7499
75M
7506
77a8
7731
7735
7953
7960
ao
30
40
50
60
1.7
*.5
3.3
4-8
5.0
$8ia
6o(^
6311
6557
6801
704«
7377
75 «o
7739
7964
5816
58ao
58as
Si
6315
63 »9
6384
6561
6805
6809
68x3
7045
7049
7053
738X
7514
7518
7Saa
7743
7747
7750
7968
797«
7975
«5
i6
5839
III?
S
6338
633a
6336
6573
6817
6831
6835
7064
7300
75a6
7Sa9
7533
7754
7758
S
«9
ao
at
aa
n
5850
0096
6100
6340
6345
6349
6586
6590
6594
6839
6833
6837
7068
7072
7076
11^
73"
7537
7541
7545
7766
7769
7773
7990
7994
7997
5854
6104
6353
6598
684X
7080
73 «6
7549
7777
800X
5867
6108
6113
6117
6365
6606
6610
S*5
ss
7093
73ao
11^
7S5a
7560
778.
80x3
24
587.
6131
6135
6139
6369
till
6614
66x8
6633
6865
7096
7x00
7x04
733a
7335
7339
757a
779a
7000
8016
8019
8033
4
80
3>
Sa
33
589a
6133
6138
6143
6383
6386
6390
6637
6631
6635
6869
6877
7x08
71x3
7x16
7343
7347
735'
7576
7804
7807
781 1
803X
8034
xo
ao
30
40
.7
1.3
a.o
8.7
3-3
4.0
58g6
6146
6394
6639
6881
7130
7355
7587
78x5
8038
5900
5904
5909
6150
6398
6403
6406
6643
6647
665X
6885
6885
6893
l^
7132
7359
7363
7367
7591
l^
78x9
7833
7836
8043
£:5
34
59«3
59«7
59a«
6i6a
6166
6x71
64x0
64x4
64x9
6655
SJ
^
7136
7139
7'43
737«
7378
76x0
7830
7833
7837
§
IS
39
40
4«
4a
43
59»5
5930
5934
6175
64a3
6437
6431
6667
6671
6675
6909
69x3
69x7
7«47
7x51
7155
7386
7390
76x4
76x7
763X
784X
807X
S
5938
6x87
6435
6679
6931
7>59
7394
7635
785a
8075
594a
5946
5951
619 X
6195
6300
6439
J443
6447
6683
6687
669X
6935
6939
6933
7x63
7x67
7171
7398
7403
7406
7639
7636
7856
7860
7863
is?
8086
44
5955
63X3
6451
^15
6699
6704
6937
6941
6945
7175
7179
7x83-
74«o
7413
7417
7648
7867
787X
7875
80R9
8097
10
.5
49
60
5'
sa
53
5967
597a
5976
63x6
632 X
6225
6464
6468
6473
6708
6713
6716
6949
6957
7187
7191
7195
74ax
74a5
74a9
7653
7655
7659
7886
8x00
8104
8x07
ao
30
40
1:
X.O
«-5
a.o
8.S
3.0
5980
6329
6476
6730
6961
7199
7433
7663
7890
8xxx
599a
6233
6237
6341
6480
6734
6738
6733
6965
6969
6973
7203
7307
73XX
7437
744*
7445
7667
767 X
7674
7894
7897
790X
81x5
81x8
8X33
54
11
6005
6245
6249
6254
6493
6496
650X
6736
6740
6744
6985
73x5
7318
7323
7449
74Sa
7456
7678
7682
7686
79x3
8136
8x39
8133
IS
59
60
6013
6017
6258
6362
6266
6505
6509
65«3
6748
6752
6756
6989
6993
6997
7326
7*30
7234
7460
7468
7690
7693
7697
79x6
7930
7983
8137
8x4X
8144
602X
6370
63x7
6760
7cx>x
7a38
747a
770X
7987
8x48
Smitma
OMIAN 1
Digitize
rBy"tj
t!^t!^
^ '
62
Table 11.
LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION
p. IN ENGLISH FEET.
[DeriTation of table ezplainod on p. zIt.]
Lat.
6i*>
62«
630
64^
65"
66^
67°
68O
69-
70°
P.P.
a
3
4
1
I
9
10
II
13
»3
M
\l
:i
'9
SO
91
M
23
as
a6
%
a9
ao
j«
32
33
34
U
39
40
41
43
43
44
49
M
SI
S»
53
54
II
59
00
7.sai
8148
7.3S1
8364
7.821
8575
7.3ai
8781
7.3ai
8983
7.821
9176
7.821
9365
7.381
9548
7.381
9724
7.881
9893
4
815a
8155
8159
816a
8166
8170
8173
IS
8368
837«
8375
|J2
8i86
8389
8393
8396
8585
8589
8596
8603
8606
8791
8813
K5
901a
9179
9183
9186
9189
919a
9«95
9198
9ao3
9aos
9368
937«
9374
9380
9384
9387
9390
9393
955 >
9554
9557
9560
9562
9565
9568
957«
9574
9727
9730
973a
9738
974'
9746
9749
990Z
9906
9909
9913
99«5
99«7
8184
8400
8610
8815
9015
9308
9396
9577
9752
9920
10
ao
SO
40
•7
1.3
a.o
a-7
3.3
4.0
8188
8I9Z
8195
aao6
8ao9
8213
8ai6
8403
8407
8410
8431
8613
8617
863^
8634
8641
8818
8833
8835
8833
883s
8843
8846
9018
9031
9035
9038
9031
9034
9037
9041
9044
9311
9314
9318
9331
9334
9337
9330
9234
9*37
9399
9403
9405
9408
94"
9415
9418
9421
9434
9580
9589
9592
9595
9Cx>i
9604
9755
9758
9766
9769
9773
9775
9778
9936
9928
9931
9934
9937
9940
9942
9945
3
82ao
8435
8645
8849
9047
9340
9427
9607
9781
9948
82*4
8327
8231
pi
8243
8246
8250
8aS3
8438
8443
8445
8449
^5J
8456
8459
8648
8665
8669
8673
8676
8853
8856
8859
8863
8865
8869
8873
8879
9050
9054
9057
9060
9063
9067
9070
9073
9077
9346
9250
9»53
9*56
9»59
9363
9366
9369
9430
9433
9436
9439
9442
9445
9448
945«
9454
9610
9613
9616
96x9
9631
9634
9637
9630
9633
9784
9787
9789
9792
9801
9803
9806
995«
9956
9959
9961
9964
9967
9970
9972
xo
30
30
40
.5
I.O
1.5
3.0
2.5
3.0
8357
8470
8679
8883
9080
9273
9457
9636
9809
9975
8261
8^3^
8371
8375
8379
8383
8386
8389
8473
8491
850X
8683
8686
b689
1^
8699
8710
8885
8889
8893
8896
8913
9083
9086
9090
9093
9096
9099
910a
9106
9109
9«75
9378
9384
9387
9391
9»94
9397
9300
9460
9469
9472
9475
9484
9639
9643
9645
9648
9651
9654
9663
98,3
9815
9817
9830
9823
9836
9839
9831
9834
9978
9980
S^3
999«
9994
9997
9999
8
8»93
8505
8713
8916
91 13
9303
9487
9666
9837
•0003
8303
8307
8310
8314
8317
8331
8334
8508
8513
8515
8519
8533
8536
8539
8533
8536
8716
8730
87*3
8737
8730
8733
8737
8740
8744
8919
gl
8936
8939
8946
9115
91 18
9133
9135
9138
913I
9'34
9*38
9141
9306
9309
9312
9315
93 >8
93"
9325
93*8
933 «
9490
9493
9496
9499
9502
9506
9509
951a
9S«S
9669
9673
9675
9678
9680
9683
9686
9689
9693
9840
9843
9845
9848
9851
9854
9857
*ooo7
•0010
<^x>I3
^0015
^0O3I
10
ao
30
40
50
60
•3
.7
Z.O
i.S
"7
a.o
8338
8540
8747
8949
9144
9334
9518
969s
9865
*^39
833»
833s
8339
834*
8346
8350
8353
US
8543
8547
8550
8554
8571
8750
8754
8757
8761
V^
8767
877.
Hit
8956
8959
8963
li
8979
9M7
9150
9' 54
9157
9160
9»63
9166
9x70
9'73
9337
9340
9343
9346
9349
9353
9356
9359
9363
9521
9524
9527
9530
9533
9536
9539
9542
9545
9698
9701
9704
9707
9709
9713
97'5
9718
9721
9868
9871
9873
9876
gg
9885
9887
9890
•bosa
•>«>34
•0037
J»39
*0O43
•too45
JxH7
•0050
•0053
8364
8575
8781
8983
9176
9365
9548
9724
9893
•0055
r
^
e
SMiTHaoNiAii Tables.
63
Table 11.
LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION
p^ IN ENGLISH FEET.
PMvstioB of tabk apbiaed on p, xhr.]
Lat.
7i»
7^
73*
74°
7f
76«
7r
78°
79P
8o«
P.P.
7.ass
7.ass
7.SSS
7.ass
7.ass
7.SSS
7.ass
7.ass
7.ass
7.ass
I
a
3
OOS5
02 to
0359
0499
C6i2
0757
0875
0984
.08s
"77
oos8
0063
0213
02I(
0218
0361
0366
0501
0634
0636
0639
0761
0763
0877
S
0989
X090
:;2
1181
4
1
I
9
10
ti
la
S3
0066
0068
0071
0074
0077
0079
oaao
0231
0233
0369
0371
0373
0376
0378
0381
«>50»
0510
0513
0515
0517
osao
0641
0643
0645
0J47
0650
065a
077«
0773
0775
088a
0888
0891
0991
099a
0994
-1
0999
1091
1093
1095
1096
1098
1099
1183
II87
XI89
XX90
S
10
ao
30
40
£
•5
I.O
«.$
a^
a-5
3.0
008a
0236
0383
0522
0654
0777
0893
lOOI
IIOI
xi9a
0087
0090
0238
0241
0243
0390
0529
0656
0660
0783
0895
1003
1 102
1104
lies
"93
«4
3
0246
0248
0251
039a
0394
0397
0531
0533
o$35
066a
0785
0787
0789
0901
090a
0904
1008
1009
lOII
IIIO
1.98
"99
laoo
«9
SO
ai
aa
a3
0100
0103
oios
0256
0258
0399
0401
0404
0537
0540
054a
0673
o79(
0793
0795
5!S
0908
0910
IOI3
101$
1016
IIXI
III3
IX 14
xaoa
xao3
laos
0108
oa6i
0406
0544
067s
0797
091a
IOI8
III6
xao6
01 II
0113
0116
oa68
0408
0411
0413
0546
0549
0551
0677
3S?
080J
0916
0917
xoao
1021
loas
IXI8
1x19
IX2X
xao7
iao9
xaxo
S
•9
SO
SS
3*
33
0118
oiai
oia4
oia6
oiaQ
0131
0*71
0283
0416
0418
O4ao
04*3
SSI
0553
0565
06S3
0694
S
o8xx
08.3
0815
0919
09ai
09*3
SI
SI
1028
1030
1032
1033
1122
1 127
1129
XX30
taia
iai3
iai4
xat6
iai7
xax9
1
10
ao
30
40
50
to
>3
•7
1.0
«.3
X.7
0134
oa86
0430
0567
0696
0817
0930
X03S
xx3a
laao
0137
0139
014a
0288
0291
0293
043a
0435
<H37
0569
0571
0574
0698
0700
070a
cA.9
08a 1
o8a3
093a
0934
09S5
;SJ
1Q40
"33
IIJI
taai
iaa3
iaa4
34
0144
OI47
0150
SI
0300
0439
0441
0444
0576
B
SI
o8a8
0937
0939
0941
104a
1043
104s
1x38
1 139
1x41
I2a6
39
40
4«
4*
43
015a
0155
0157
0303
S3
0446
0448
0451
058a
0585
0587
0710
o7xa
0714
^90
^3a
0834
0943
0946
1047
1049
1050
"4*
"44
"45
1230
123 X
"33
0160
0310
0453
0589
07x6
0836
0948
1052
"47
1234
oi6a
0165
0167
031a
03«5
0317
0460
0591
0718
07ao
07aa
S^*
z
0950
095a
0953
xo$4
105s
«o57
1148
Its©
"5«
"35
44
49
00
51
5*
53
0x70
017a
0175
0177
0180
0x82
03ao
o3aa
©3*4
03a7
03*9
033a
046a
«^64
0467
0469
0471
0474
22
060a
0604
0607
0609
0739
073«
0733
0735
0848
0850
o8s2
0854
0955
0957
0959
0961
0962
0964
1062
.063
1065
1066
««53
"54
1x56
"57
1240
X24X
1242
«M4
"45
1247
1
to
ao
30
40
1
.a
•3
•5
i
1.0
0185
0334
0476
061X
0737
0856
0966
1068
1x62
1248
0187
0190
019a
0336
0339
0341
0478
0481
0483
0613
0615
0617
0739
0741
0743
0862
0968
0970
097X
1070
1071
1073
1x63
!:ti
1249
125 1
1252
54
0195
0197
0200
0344
0346
0349
0485
0487
0490
06x9
0621
0624
0745
0747
0749
0864
0865
0867
0973
097s
0977
X078
1x68
1x69
XX7X
"53
1256
11
59
60
oaoa
oaos
0207
035'
0354
0356
049a
0494
0497
0626
0628
0630
0751
0753
0755
0869
0871
0873
0979
0980
0982
1080
to82
1083
txTa
"74
•«75
1
0210
0359
0499
0632
0757
0875
0984
1085
XX77
126X
T
8mitn«onian Tables.
64
Table 11.
LOGARITHMS OF RADIUS OF CURVATURE OF NORMAL SECTION
p^ IN ENGLISH FEET.
[Derivation of table explained on p. zIt.]
Let.
8i*»
82°
830
84^
Zf
86°
87<'
88°
89^^
P.P.
7.381
7.3Sa
7.3Sa
7.3aa
7.823
7.333
7.333
7.333
7.333
t
3
3
ia6i
>337
1403
X461
"5"«
"55"
'583
1605
1619
ia6a
ia64
ia65
1338
'339
1340
1404
1462
1463
1464
"5«a
1512
"5«3
"55»
"55*
"553
1583
1584
'584
X605
1606
1606
x6x9
1619
Z619
4
1
ia66
1269
«34i
«34a
1344
;:3
X410
1465
h66
"5"4
»5"4
i5>5
"553
"554
"555
'585
1585
'585
1606
z6o6
.607
x6x9
1619
X620
I
9
10
la
«3
1270
ia7f
"73
1347
141 1
1412
I4«3
1469
1516
«5"7
«5"7
"556
"556
1586
X586
"587
'52
IS
1620
X620
X620
3
"74
1348
M«4
"470
1518
'557
"587
x6o8
x6ao
"75
1378
>349
«350
1353
1416
X417
"47"
"472
"473
"5«9
1519
1520
"558
"558
"559
1588
1608
1609
1609
X620
x6ao
«4
'.32
laSa
>353
1354
>355
1418
1419
1420
"474
"474
"475
1521
1521
XS22
156X
"589
X589
1589
1609
1609
x6xo
X620
i6ao
X62X
xo
•3
«9
10
ai
2a
33
'*!*
;SJ
1356
«358
"359
X42X
142a
1423
1476
"T8
"5»3
"5»4
"5*4
1561
1562
X590
1590
'59'
1610
z6io
x6ii
x6ax
i6ai
x6ai
20
30
40
•7
X.O
"3
"•7
2.0
1287
«36o
«4a4
"479
«5«5
"563
«59'
x6xx
i6ax
ia88
"90
"91
1361
X362
13^
1437
Z481
X526
X526
xsa7
X563
X5j4
1564
"59"
X592
X592
x6ix
161 z
1612
X62X
X62X
X62X
"9»
"93
Ka9S
1365
1367
1428
1429
1430
,482
1483
1484
X528
1528
1529
.565
1565
1566
"593
"593
"593
x6ia
z6ia
i6ia
Z621
X621
x62a
19
SO
31
3*
33
1296
"97
"99
1369
1370
143 «
1432
«433
.48s
"530
"53"
"53"
'567
"567
"594
'594
'595
Z6Z2
1613
1613
X622
X622
X622
1300
1371
M34
"487
"532
X568
'S9S
1613
1622
1301
130a
1304
1372
»373
«374
"437
1489
"489
"533
"533
"534
X569
X569
'596
1613
1613
1614
1622
1622
X622
34
^1
1307
1376
1378
«438
1438
X439
1490
"49"
1492
"535
1536
"570
"570
"57"
"597
"597
"597
x6z4
1614
1614
X633
x62a
1623
39
40
41
43
43
1308
1310
1311
1381
1440
1441
«44a
"493
"493
"494
-1
1538
«57«
1572
"57*
"599
x6z4
1615
Z615
1623
X623
1
xo
20
so
40
.a
*3
•S
■I
1.0
131a
1382
"443
"495
"539
'573
'599
X615
1623
1313
1316
.383
1384
1385
"444
"496
"497
"497
"540
"540
"54"
"573
"574
"574
'599
1600
z6oo
X615
1615
z6i6
1623
x6a3
1633
44
1317
1318
1320
1386
1387
1389
"447
1»
1498
"499
1500
"54"
"542
"543
'575
1576
x6oo
z6oo
z6oi
1616
x6x6
x6i6
1623
1623
X623
49
M
SI
53
1321
«322
1324
1390
1391
1392
"449
1450
"45"
iSoi
1501
1502
"543
"544
'544
XS76
'577
'577
z6oi
x6oi
1602
1616
1617
1617
1623
1623
1623
"335
«393
'45a
"503
"545
'578
1602
1617
1623
1326
1327
1329
«394
"453
"454
"455
"504
1505
"505
'546
'546
'547
"578
'579
'579
X602
1603
1603
x6i7
16x7
x6i8
X623
x6a3
1623
54
'330
«33«
1332
U99
"456
1456
"457
1506
"549
1580
1580
X581
X603
1603
X604
1618
1618
x6i8
X623
X623
1623
59
00
>333
1335
1336
1400
1401
1402
"458
"459
1460
1509
1509
1510
"549
"550
"550
.58«
1582
1582
1604
1605
x6i8
1619
1619
x6a3
1623
1623
«337
1403
1461
151X
"55"
'583
1605
16x9
x6a3
8HiTHaoNiAii Tables.
6s
Tabu 12.
LOGARITHMS OF RADIUS OF CURVATURE p« (IN METRES) OF SECTION
OF EARTH'S SURFACE INCLINED TO MERIDIAN AT AZIMUTH a.
[Fonmila for p« givei
1 on p. xIt.]
LATITUDE.
Azimath.
22°
^f
24**
2f
26*»
27°
28«
2^
30°
31°
&>
6A)237
6.80242
6.80248
6.80254
6.80260
6J80266
6.80272
6.80279
6.80285
6.80292
5
239
244
250
2|6
262
268
274
280
287
^
10
244
250
255
261
267
SI
^
285
292
^
IS
254
259
264
270
276
294
300
306
ao
266
^'
277
282
288
^
299
305
3"
3^7
25
282
287
292
297
302
313
319
325
^^l
30
300
305
309
314
319
324
330
335
340
346
3S
320
324
329
333
33»
343
f
353
358
3f3
40
341
345
350
354
358
^^
372
377
382
45
364
367
37'
375
379
383
387
391
396
400
SO
386
389
392
396
399
403
^
411
415
419
II
407
410
413
416
420
423
4*6
430
434
437
427
430
432
435
438
442
445
448
451
455
65
445
448
'4
478
tU
455
458
461
464
4^
470
70
461
.46J
476
T^
473
%l
478
481
484
75
473
480
484
489
492
494
80
483
485
487
489
491
tP
495
498
500
502
85
489
490
492
494
496
SOI
503
505
507
90
490
492
494
496
498
500
S02
504
507
509
LATIT
UDE.
Aximuth.
32°
33°
34°
35°
36°
37°
38°
39°
40«
41°
OO
6.80299
6.80306
6.80313
6.80320
6.80327
6.80335
6&)342
6A)35o
6.80357
6A>365
5
300
307
314
322
329
336
344
351
1
366
xo
305
312
31?
326
333
340
.348
W
370
15
3*3
320
326
333
340
348
355
369
376
20
324
330
337
343
ir.
3^1^
364
371
385
25
337
3^5^
349
355
'^
382
395
30
352
364
370
376
382
394
401
407
35
3^
374
380
385
391
397
402
408
414
420
40
386
392
397
402
407
412
418
423
429
434
45
405
410
414
419
424
429
434
439
444
449
50
423
428
432
436
441
445
1
480
484
459
464
II
441
458
III
449
465
t^
457
472
461
476
%^
478
491
65
V
476
480
483
486
489
493
496
500
503
70
486
489
492
495
^^0
501
504
507
510
514
75
497
SCO
502
505
508
510
5'3
516
5*9
522
80
85
505
507
510
512
515
517
520
523
525
528
510
512
5'1
5'Z
519
522
524
527
529
532
90
5"
514
516
518
521
523
526
528
531
533
Smithsonian Tables.
66
Table 12.
LOGARITHMS OF RADIUS OF CURVATURE pa, (IN METRES) OF SECTION
OF EARTH'S SURFACE INCLINED TO MERIDIAN AT AZIMUTH a.
[Fonmik for p« ghren
on p.
KlTj
LATITUDE.
Aamnth.
4^
43f
44''
45^
460
47'
48^
4<^
ScP
51'
o<»
6A)373
6A)38o
6.80388
6.80396
6.80404
6.80411
6.8041
9 6.80426
6&)434
6A>442
5
10
IS
i
3!»
38s
39«
389
393
399
397
400
406
4x3
4x2
415
420
42
4a
4a
!0 428
3 430
« 435
442
443
445
450
20
30
392
402
413
420
406
413
422
433
420
429
439
427
S2
4^
M
4!
A 441
^^ 445
;2 458
448
gi
471
35
40
45
426
440
454
432
446
459
438
444
457
470
475
i
480
4^
>2 468
^4 479
^5 490
495
480
490
500
SO
468
482
495
499
478
490
502
482
487
499
510
492
503
514
45
5«
51
)6 501
)8 512
8 522
S06
S16
526
510
520
530
65
70
75
507
517
525
510
520
528
514
523
530
534
520
524
532
539
52
5:
5^
^ 531
^ 539
^2 545
534
548
538
545
551
80
85
90
531
534
536
540
541
539
542
544
542
18
54^
5^
5!
SI
^7 55°
50 553
)« 554
553
559
LATITUDE.
Ajunuth.
520
S3?
54'
55^
56"
57'
58^
59'
6o<>
©*>
6A)449
&804S7
^80464
6.80471
6.80479
6.80486
6.80493
6A)5oo
6.80506
5
10
15
450
453
457
1
464
467
471
472
478
485
486
488
492
493
500
502
505
507
509
5"
20
25
30
469
477
1
490
483
489
495
502
496
508
514
509
514
519
5x5
520
525
35
40
45
496
505
492
501
506
515
503
512
520
509
517
525
515
522
530
520
527
534
525
532
539
531
537
543
50
5^5
524
533
520
528
537
524
533
541
528
537
544
533
IS
537
545
552
54J
548
555
546
558
55?
5^'
65
70
75
548
554
545
551
557
548
554
559
551
565
1
561
566
570
5^
569
573
567
572
575
80
85
90
1
S6i
563
564
IS
566
S
569
568
570
571
571
573
574
573
578
8hith«onian Tablcs.
67
Tablk 13.
LOGARITHMS OF FACTORS a^7;:FOR COMPUTING SPHEROIDAL
EXCESS OF TRIANGLES.
UNIT = THE ENGLISH FOOT.
[Derivaticm and ose of table explained on p. lyiiL]
log. factor and
log. factor and
log. factor and
log. factor and
^
change per
^
change per
^
change per
f
change per
minute.
minute.
minute.
minute.
<y>
0.37498
200
0.37429
400
0.3725s ^
W>
0.37056
— aoo
— 0.12
— 0.18
—0.15
I
498
21
422
41
244
61
047
— ao2
— 0.12
— 0.17
— ai5
2
497
22
415
42
234
62
038
— ao2
— ai2
— 0.17
— 0.13
3
496
23
408
43
224
63
030
— 0/)2
— ai2
— ai7
— 0.13
4
495
24
401
44
^'4 «
64
022
— ao3
— ai3
— ai8
— ai3
5
493
25
393
45
203
65
014
—0.03
— ai3
— 0.17
— ai3
6
491
26
385
46
193
66
006
— ao3
— ai3
— ai7
;"?"'3
7
489
27
377
47
183
67
0-36998
— 0.03
";S-'5
— ai7
— ai2
8
487
28
368
48
'73 ^
68
991
—0.05
— ai3
— ai8
~ai2
9
484
29
360
49
162
69
984
— 007
-ais
— ai7
~ai2
10
480
30
351
50
152
70
977
— 007
— ai5
— ai7
— aio
II
476
31
342
51
142
71
971
— ao7
-ais
— ai7
— ai2
12
472
32
333
52
132
72
^ «
— 0.07
— ai7
—0.17
^ao8
13
468
33
323
53
122
73
959
— ao8
—0.15
— ai7
— aio
14
463
34
314
54
112
74
953 „
— ao7
—0.17
-ais
— ao8
15
459
35
304
55
103
75
948 ,
— aio
— ais
—0.17
— ao8
i6
*S3 ,
36
295
56
093
76
^3
— ao8
— 0.17
—0.17
— ao8
17
448
37
28s
57
083
77
938
— aio
— 0.17
— ai5
—007
i8
442
38
27s
58
074
78
934
— 0.10
— ai7
—0.15
— 0.07
19
436
39
26s
59
065
79
930
—0.12
— O.I7
—CIS
— 0.07
20
4^9
— ai2
40
*5S „
— ai8
60
056
-ai5
00
926
SurmieiiiAM Tablu.
68
Digitized by
GooqIc
Tablk 14.
LOGARITHMS OF FACTOR8-a^ FOR COMPUTING SPHEROIDAL
EXCESS OF TRIANGLES.
UNIT=THE METRE.
[Derivation and use of table explained on p. Iviii.]
log. factor and
log. factor and
log. factor and
oiangeper
log. factor and
f
change per
^
change per
^
f
change per
minute.
minute.
minute.
minute.
0«>
14069s
200
1.40626
400
140452 ^
60<>
I402S3
— aoo
— O.X2
— ai8
— 0.1S
X
69s
21
619
41
441
61
244
— OX>2
— ai2
— ai7
— ais
2
694
22
612
42
431
62
235
— ao2
—0.12
— ai7
— ai3
3
693
23
60s
43
421
63
227
^ao2
— 0.13
— ax7
— 0.13
4
692
24
597
44
*" «
64
219
— ao3
— 0.12
— ai8
— 0.1S
5
690
25
590
45
400
65
2IO
— 003
— 0.13
— ai7
— ai2
6
688
26
582
46
390
66
203
— o/)3
— 0.15
— ai7
— ai3
7
686
27
573
47
^
67
«95
— 0.05
— ai3
— ai8
^ai2
8
683
28
565
48
369
68
188
—0.05
— ais
— ai7
— ai2
9
680
29
556
49
359
69
181
— 005
— ai3
—0.17
— ai2
10
677
ao
548
50
349
70
174
— ao7
— 0.1S
— 0.17
— aio
II
673
31
539
51
339
71
168
— 0^
— ais
— ai7
— 0.12
12
669
32
530
52
329
72
161
-0.07
— ai7
— ai7
—0.10
13
^s „
33
520
53
319
73
^55 Q
—0.08
— ais
— ai7
— 0.08
H
660
34
5"
54
309
74
ISO
—0.08
— ai7
— ai7
—0.10
15
655
35
501
55
299
75
'44 ^
— ao8
— ai7
—0.15
— ao8
i6
650
36
491
56
290
76
139
— aio
—ais
— ai7
— 0.07
17
^
37
482
57
280
77
'3^ Q
—0.08
— ai7
— 0.IS
— 0.08
i8
639
38
472
58
271
78
130
— 0.12
— ai7
— ais
— 0.07
19
632
39
462
59
262
79
126
— aio
— ai7
-ais
0.0s
20
626
— 0.12
40
— ai8
60
253
-ais
00
123
SumMONiAN Tamjw.
69
Digitized by
GooqIc
Table 15.
LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI-
TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION.
UNIT=THE ENGLISH FOOT.
[Derivation and uae of table explained on p. Ix.]
^
a\
^=^
Of
^
c%
^
a\
*i=fi
<H
^
c%
o^'
7.99669
7-99374
—00
^00
0.372
IO«00'
7.9965s
799369
9.621
9.926
0.398
10
^
374
7^39
8.140
s
0.372
10
655
369
9.628
9.933
0.399
20
669
374
0.372
20
654
369
9.636
9.941
0.400
30
669
374
8.316
8.614
0.372
30
654
369
9.643
9.948
0.401
40
669
374
8.441
i:ip
0.372
40
654
^
9.650
9-955
9.963
0402
50
669
374
8.538
0.372
SO
653
369
9.657
0.403
I 00
669
374
8.617
5-915
0.372
II 00
653
3f5
9.663
9.970
0404
10
^
374
8.684
8.982
0.372
10
652
3f5
9.670
9.977
0.404
20
668
374
8.742
9.040
0.372
20
652
368
9.677
9-983
0405
30
668
374
8.793
9.091
0.373
30
651
368
9.683
9-990
0406
40
668
374
5-539
9.137
0.373
40
651
3g
9.690
9-997
0407
0408
50
668
374
8.880
9-179
0.373
50
650
368
9.696
aoo3
200
668
374
8.918
9.216
0.373
1200
650
367
9.702
0.010
0.409
10
668
373
us?
9.2U
9.283
0.373
10
649
3^
9.708
0.016
0.410
20
668
373
0.373
20
649
367
9-714
0.023
a4i2
30
668
373
9.015
9.3M
0.374
30
648
367
9.720
0.029
0413
40
668
373
9.043
^3^
0.374
40
648
^^
9.726
0.035
0414
SO
668
373
9.069
0.374
50
647
367
9.732
ao4i
0415
300
668
373
9094
9.393
0.374
1300
646
3^
9.738
0.048
a4i6
10
667
373
9.1 18
9-417
0.375
10
646
3^!
9.744
0.054
ao6o
0417
20
667
373
9.140
9.439
0.375
20
64s
366
9.749
a4i8
30
667
373
9. 1 61
9.460
0.376
30
645
366
9-755
9.761
0.065
0.419
40
667
373
9.182
9.481
40
644
3^
0.071
a42o
so
667
373
9.201
9-500
0.376
50
644
365
9-766
0.077
0.422
400
667
373
9.<220
9-519
0.376
1400
643
365
9-771
ao83
ao88
0423
xo
666
373
9.237
9-537
0.377
10
642
365
9.777
a424
20
666
VZ
9.2S4
9.554
0.377
20
642
36s
9.782
0.094
0.425
30
666
373
9.271
9.287
9.570
0.377
30
641
36s
9.787
aioo
0.426
40
666
373
0.378
40
640
364
9.792
0.105
a428
50
666
373
9.302
0.378
50
640
364
9.798
am
0429
500
66s
373
9.317
9.617
0.379
1500
639
* 364
&
aii6
0430
10
^5
373
9.331
9.631
0.379
10
^39
364
ai2i
0.431
20
66s
372
9.3+S
9.645
0.379
20
638
363
9.813
0.127
0.433
30
66s
372
9.358
9.659
0.380
30
637
363
9.818
ai32
0434
40
664
372
9.372
9-^72
0.380
40
637
363
9.822
0.137
0.43s
50
664
372
9.384
9.685
0.381
SO
636
363
9.827
ai42
0.437
600
664
372
9.397
9-697
0.381
x6oo
635
363
9-832
0.147
a438
10
664
372
9.409
9-709
0.382
10
635
362
9.837
0.153
0439
20
663
372
9.420
9.721
0.383
20
634
362
9.841
0.158 a44i
ai63 a442
ai68 , a443
30
^3
372
9-432
9.732
0.383
30
633
362
9.846
40
663
372
9.443
9.744
0.384
40
632
362
9.851
50
662
372
9.453
9.755
0.384
50
632
361
9-855
0.173 0445
700
662
372
9.464
9.76s
^1
1700
631
361
9.860
ai78 0446
0.182 0448
10
662
371
9.474
9.484
10
630
361
9.864
20
662
Z7^
0.386
20
630
361
9.869
a 187
0.449
30
661
371
9.494
9.796
0.387
30
62!
360
9-873
0.192
0450
40
661
371
9.504
9.806
^•357
40
360
0.197
0452
so
661
Z7^
9.5^
9.816
0.388
50
627
360
9.882
0.202
0453
800
660
371
9.523
9.825
0.389
1800
627
360
9.886
a2o6
0455
0.456
10
660
371
9.532
9-834
0.389
10
626
359
9.890
0.211
20
659
371
9-541
9-843
0.390
20
625
359
9.89s
a2i6
0.458
30
6S9
371
9.558
^•t^'
0.391
30
624
359
9.899
0.220
0.459
0.461
40
§
370
9.861
0.392
40
624
9-903
0.225
SO
370
9.870
0.392
50
623
9.907
0.229
0.463
900
6s8
370
9.575
9-f78
0-393
1900
622
358
9-9"
0.234
0.464
10
657
370
9.583
9.886
0-394
10
621
358
9.915
0.239
0.466
20
657
370
9.591
9.895
0.39s
20
620
358
9.919
0.243
0.467
30
^57
370
t^
9.903
0.396
30
620
357
9-923
0.248
0.469
40
656
370
9.910
0.396
40
619
618
357
9-927
0.252
0.470
SO
6s6
369
9.614
9.918
0.397
50
357
9.931
0.256
0.472
1000
6S5
369
9.621
9.926
0.398
20.00
617
357
9-935
a26i
0474
SiiiTHaoNiAii Tables.
70
Table 16«
LOGARITHMS OF FAOTOR8 FOR COMPUTING DIFFERENCES OF LATI-
TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION.
UNIT=THE ENGLISH FOOT.
[Derivation and uae of table explained on p. Ix.]
f
a\
h^ci
0S
h
Ci
^
tfi
^ = r.
at
^
f%
20*»00'
7.99617
7-99357
9-935
a26i
0.474
3o''oo'
7.99558
7.99337
0.135
0.138
a496
0.593
10
616
356
9-939
a265
0.475
xo
557
337
0.500
0.595
0.598
20
61S
356
9-943
0.270
0.477
20
556
336
ai4i
0.503
30
615
356
9-947
0.274
a278
a282
0.479
30
555
336
0.144
0.507
0.600
40
614
355
9-951
0480
40
554
335
0.146
0.511
0.603
50
613
355
9-955
a482
50
553
335
0.149
0.514
0.605
21 00
612
355
9-958
9.962
0.287
0.484
31 00
552
335
0.152
0.518
0.607
10
611
355
0.291
0.486
10
550
334
0.155
0.158
0.522
a6io
20
610
354
9.966
0.295
0.487
20
549
334
0.525
0.612
30
^
354
9.970
0.299
0.489
30
548
333
ai6i
0.529
0.615
40
354
9.973
0.304
0.308
0.491
40
547
333
0.164
ai66
0-532
0.617
SO
608
353
9.977
0.493
SO
546
333
0-536
0.619
22 00
^
353
9.^1
0.312
0.494
0.496
3200
545
332
0.169
0.540
0.622
10
606
353
9-984
0.316
10
54*
332
0.172
0.543
0.624
20
605
353
9.988
a320
0.498
20
542
332
O.X75
0.547
0.627
30
604
352
9.991
^324
0.328
0.500
30
541
331
0.177
0.180
0.550
0.629
40
603
352
9995
9.998
0.502
40
540
33*
0.554
0.558
a632
SO
602
352
0.332
0.503
50
539
330
0.183
0.634
2300
601
351
0.002
0.336
0.505
3300
538
330
0.186
0.561
0.637
10
600
351
0.005
0.340
0.507
10
537
330
0.188
t'^
0.639
20
600
351
0.009
0.344
0.509
20
535
329
0.191
0.642
30
40
^
350
350
0.012
0.016
0.348
0-352
0.51 1
0.513
30
40
534
533
3?
0.194
0.197
0.572
0.575
0.644
0.647
SO
S97
350
0.0x9
0.356
0.515
so
532
328
0.199
0.579
0.650
2400
596
349
0.02Q
0.360
0.517
3400
531
328
0.202
in
0.652
10
595
349
0.364
0.518
10
IS
327
a205
0.208
0.655
20
594
349
0.029
0.368
0.520
20
327
0.590
0-657
30
593
348
0.033
0.036
0.372
0.522
30
5^J
326
0.210
0.593
a66o
40
592
3^5
a376
0.524
40
526
326
0.2x3
0.2x6
0.000
a663
50
591
348
0.039
0.380
0.526
50
525
326
a665
2500
500
347
0.043
0.046
°-3§4
0.528
3500
523
325
0.2x8
0.604
0.608
0.668
10
509
347
0.388
0.530
10
522
325
a22i
0.671
20
588
347
ao49
0.392
0.532
20
521
324
0.224
a6ii
0.673
30
587
346
0.052
0.396
0.534
30
520
324
0.226
0.615
0.618
0.676
40
^^
346
0.056
0.399
0.536
40
519
324
0.229
0.679
so
585
346
0.059
a403
0.538
50
517
323
0.232
0.622
0.681
2600
5!^
345
0.062
a407
0.540
3600
516
323
0.234
a625
0.684
10
5i3
345
0.065
ao68
0411
0.543
10
515
322
0.237
0.629
°-S7
20
582
345
0.415
0.545
20
514
322
0.239
0.632
0.689
30
5f^
344
0.072
0.418
0.547
30
512
322
0.242
0.636
0.692
40
580
344
0.075
0.078
0.422
0.549
40
5"
321
0.245
0.640
t^
SO
579
344
0426
0.551
50
510
321
0.247
0.643
2700
578
343
ao8i
0.430
0.553
3700
509
320
0.250
0.647
a7oo
10
577
343
ao84
0433
0.555
xo
5^2
320
0.253
0.650
0.703
0.706
20
576
343
0.087
0.437
0.557
20
506
320
0.255
0.654
30
575
342
0.090
0.441
0.559
0.562
30
505
319
0.258
IW.
0.709
40
574
342
0.093
0.096
0.44s
0.448
40
504
3'?
318
a26o
0.7x2
SO
573
342
0.564
50
503
0.263
0.665
0.715
2800
571
341
0.099
0.452
a566
3800
501
318
a266
0.668
0.717
10
570
341
0.10Z
0.456
0.460
0.568
10
500
317
0.268
a672
0.720
20
569
341
0.105
0.570
20
499
317
a27i
0.675
0.723
30
568
340
0.108
a463
0.573
30
498
317
0.270
0.679
0.726
40
5^
340
0.1 1 1
0.467
0.575
40
496
316
0.683
0.686
0.729
SO
566
340
0.114
0.471
0.577
so
495
316
0.278
0.732
2900
565
339
0.1 17
0.474
0.579
3900
494
31s
0.281
0.690
0.735
0.73?
10
564
l^
0.120
0.478
0.582
10
492
315
0.284
0.693
20
563
0.123
0.^182
0.584
20
491
315
0.286
0.697
0.741
30
562
338
0.126
0.485
°-S86
30
Ip
314
0.289
0.701
0.744
40
561
338
0.129
0.489
0.588
40
3M
0.291
0.704
0.747
SO
560
337
0.132
0.493
0.591
50
487
313
0.294
0.708
0.750
3000
558
337
0.135
0.496
0.593
4000
486
313
0.296
a7ii
0.753
SamiaoiiiAN Tables.
71
Table 16.
LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI-
TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION.
UNIT = THE ENGLISH FOOT.
[Dcriradon and use ci tabk etplainnH on p. Iz.]
f
ai
h^n
«s
At
c%
^
a\
h=ci
tfi
^
^2
AfPoo'
7.99486
7.99313
a296
a7ii
0.752
50°oo'
'"^
7.99287
a448
0.939
0^955
10
485
312
0.299
0.715
0.75s
10
^7
0450
0.944
0.958
20
484
3"
a30i
0.719
0.759
20
407
287
0453
a948
0.962
30
482
3"
0.304
a722
0.762
30
406
286
0.4S5
0.952
0.966
40
481
3"
0.307
a726
a;'^
40
404
286
0.458
0.956
a96o
0.970
50
480
3"
0.309
0.730
SO
403
285
a400
0.974
41 00
479
3"o
0.312
0.733
0.771
SI 00
402
^5
0463
a466
0.964
0.978
10
477
310
0.314
0.737
0.774
10
401
^
0.96S
0.982
20
476
309
o.3'7
a740
0.777
20
399
284
0468
0.972
0.985
30
47S
309
0.319
0.744
0.748
a78o
30
398
^
0^71
a976
0.989
40
473
^
0.322
S:S
40
397
^3
0.473
0476
a98i
0.993
so
472
0.324
0.75"
so
396
283
0.985
0.997
4200
471
308
0.327
0.755
0.789
52 00
394
282
0.478
0.989
1. 001
10
*?2
307
0.329
0.759
a762
0.792
10
393
282
0.993
0.998
1.005
20
468
307
0.332
0.796
20
392
281
0.484
1.009
30
467
306
0.334
0.766
0.799
0.802
30
301
281
0.486
1.002
1. 01 3
40
466
306
0.337
0.770
40
380
281
0.489
1.006
I.OI7
SO
464
306
0.339
0.774
0.805
SO
388
280
a49i
1.010
I. 021
4300
463
305
0.342
0.777
0.808
5300
3!7
280
0.494
1.015
1.025
xo
462
305
0.344
0.781
0.812
10
386
279
0.497
1.019
1.030
20
461
304
0-347
0.785
0.815
20
384
279
0.499
1.023
1.034
30
4^
304
0.349
0.788
0.818
30
3f3
^^
0.502
1.028
1.038
40
303
0.352
0.792
0.821
40
^o"
0.505
1.032
1. 042
so
4S7
303
0.354
0.796
0.824
50
381
278
0.507
1.036
1.046
4400
4SS
303
0.357
0.800
0.828
5400
379
378
277
0.510
1. 04 1
1.050
10
4S4
302
0.359
0.803
0.831
10
277
0.512
1.045
1.055
20
4S3
302
0.362
0.807
0.834
20
377
277
0.515
1.049
1.059
30
452
301
0.364
0.81 1
0.838
30
376
276
0.518
1.054
1.063
40
4SO
301
0.367
0.815
0.841
40
375
276
0.520
1.058
1.067
SO
449
300
0.370
0.818
0.844
SO
373
275
0.523
1.063
1.072
4500
448
300
0.372
0.822
0.848
55 00
372
275
0.526
1.067
1.076
xo
.446
300
0.375
0.826
0.851
10
371
275
0.528
1.072
1.080
20
44S
299
0.377
0.830
0.854
20
370
274
0.531
1.076
1.084
30
444
299
0.380
0.833
0.858
30
369
274
0.534
1.081
1.089
40
443
298
0.3S2
0.837
40
367
273
0.537
X.085
1.093
1.098
so
441
298
0.385
0.841
a865
SO
366
273
0.539
1.090
4600
440
297
0.387
0.845
0.868
5600
365
273
0.542
1.094
1. 102
10
439
297
0.390
0.849
0.872
10
364
272
0.545
1.099
1. 106
20
437
297
0.392
0.853
0.875
20
363
272
0.547
1. 104
I. Ill
30
436
'2^
0.395
0.856
0.860
°-!z^
30
361
271
0.550
1. 108
1. 115
40
43S
296
0.397
0.882
40
360
271
0.553
0.556
1.113
i.iiS
1. 120
SO
434
295
a40o
0.864
0.885
SO
359
271
I.I24
4700
432
295
a402
0.868
0.889
5700
358
270
0.558
1. 122
1. 129
10
43"
294
0.405
0.872
0.892
10
357
270
0.501
I.I27
1-134
20
430
294
0.407
0.876
0.896
20
356
269
0.564
I.I32
1. 138
30
428
294
0.410
0.880
0.900
30
354
269
0.567
I.I37
I.M3
40
^l
293
0.412
0.884
0.903
40
353
'^
0.569
I.I4I
1. 147
SO
426
293
0.415
0.888
0.907
50
352
0.572
1. 146
1.152
4800
42s
292
0.417
0.891
0.910
5800
351
268
0.575
0.578
I.I5I
1. 162
10
423
292
0.420
0.895
0.914
10
350
267
1. 156
1. 161
20
422
291
0.422
0.899
0.918
20
349
267
0.581
1.166
30
421
291
0.425
0.903
0.921
30
347
267
t^
I.166
X.171
40
420
291
0.427
0.907
0.925
40
346
266
1. 170
1. 176
50
418
290
0.430
0.91 1
0.929
SO
345
266
0.589
I.I75
1. 181
4900
417
f
0.432
0.915
0.932
59 00
344
266
0.592
1. 180
1. 185
10
416
0435
0.438
0.919
0.936
10
343
265
0.595
0.598
1. 185
1. 190
20
414
289
0.923
0.940
20
342
265
1.190
1.195
30
413
28Q
0.440
0.927
0.943
30
341
264
0.600
I.I95
1.200
40
412
288
0.443
0.931
0.947
40
264
0.603
1.200
1.205
SO
411
288
0.445
0.935
0.951
SO
264
0.606
1.205
1.210
SO 00
409
287
0.448
0.939
0.9SS
6000
337
263
0.609
1. 210
1.215
Smitmsonian Tables.
72
Table 16.
LOCARITHM8 OF FACTORS FOR COMPUTING DIFFERENCES OF LATI-
TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION.
UNIT = THE ENGLISH FOOT.
[DerivatioD and use ol table explained on p. Iz.]
^
tfi
h=ci
0S
h
<•«
^
«i
h=ci
fl«
At
^8
6o*>oo'
7-99337
7.99263
0.609
1.210
1.215
70**oo'
7.99278
799244
0.S09
1-575
;:?J
10
336
263
0.612
1.216
1.220
10
277
243
0.813
1.583
20
335
263
0.615
1.221
1.225
20
277
243
0.817
1.590
1.591
30
334
262
a6i8
1.226
1.230
30
276
243
0.821
1.598
1.605
\:m
40
333
262
0.621
1.231
1-235
40
275
242
0.825
so
332
261
a624
1.236
1.240
50
274
242
0.829
1. 61 3
I.6I4
61 00
331
261
a627
1.241
1.245
71 00
273
242
0.833
1. 62 1
1. 62 1
10
261
0.630
1.247
1.251
10
273
242
0.837
1.629
1.629
20
260
0.633
1.252
1.256
20
272
241
0.841
1.636
1.637
30
327
260
0.636
11
1. 261
30
271
241
0.845
1.644
1.645
40
326
260
0.639
1.266
40
270
241
0.849
1. 000
1.653
SO
325
259
0.642
1.272
50
269
241
0.854
I.66I
6200
10
324
323
259
^S^
1.273
1.279
1.284
1.282
7200
10
2
240
240
0.858
0.862
1.669
1.677
1.669
1.677
20
322
0.651
1.288
20
267
240
0.866
1.685
1.686
30
321
258
0.654
1.290
1-293
1.298
30
266
240
0.871
1.694
1.694
40
320
257
t^
1.295
40
266
239
f^
1.702
1.702
50
319
257
1.301
1.304
50
265
239
1.710
1.711
6300
318
257
0.663
0.666
1.306
1.309
7300
264
239
0.884
nit
1.720
10
3^7
256
1.312
1-3*5
10
264
239
0.889
1.728
20
316
256
0.669
1.318
1.320
20
263
238
0.893
1-737
1-737
30
31S
256
0.672
1-323
1.326
30
262
238
0.898
1.745
1.746
40
3H
255
0.676
1.329
1.332
40
261
^3f
0-903
1-754
1.703
1.764
50
313
255
0.679
1-335
1-337
50
261
238
0.907
6400
312
255
a682
1.341
1-343
7400
260
238
0.912
1.772
1-773
10
3"
254
0.685
0.688
1.346
1-349
10
259
237
0.917
1.782
1.782
20
310
254
1-352
1-355
20
259
237
0.922
1.791
1. 791
30
^
254
a692
1-358
1.360
30
258
237
0.927
1.800
1 .801
40
253
0.695
1.366
40
257
237
0.931
1.810
1.810
50
307
253
0.698
1.370
1.372
50
257
236
0.936
1.820
1.820
6500
306
253
0.701
1.376
1.378
7500
256
236
0.941
1.829
1.830
10
305
252
0.705
0.708
1.382
1.384
xo
255
236
0.946
1.839
1.839
20
304
252
1.388
1.390
20
255
236
0.952
1.849
1.849
30
303
252
0.7 1 1
1.394
1.396
30
254
236
0.957
0.962
\i^
;:ll|
40
302
251
0.715
0.718
1.400
1.402
40
254
23s
50
301
251
1.406
1.408
50
253
235
0.967
1.879
1.880
6600
300
251
a72i
1.413
1. 414
7600
252
23s
0.973
0.978
1.890
1.890
10
'^
250
0.725
a728
1.419
1.421
10
252
235
1.900
1. 901
20
250
1.425
1.427
20
251
235
0.984
1. 91 1
I.9II
30
297
250
0.732
1.432
1-433
30
250
234
0.989
1.922
1.922
40
296
249
0-735
1438
1.440
40
250
234
0.995
1-933
1-933
SO
29s
249
0.739
1.444
1.446
50
249
234
1.000
1.944
1.944
6700
294
249
0.742
1.451
1.452
7700
'^
234
1.006
;:^9^
\''J^
10
293
^
a746
1.484
1.459
1.465
10
234
1. 01 2
20
292
0.749
20
248
233
1.018
1.978
1.978
30
291
248
0-753
0.756
0.760
M70
1.472
30
247
233
1.024
1.989
1.989
40
248
1-477
1.478
1.485
40
247
233
1.030
2.001
2.001
SO
247
1.484
50
246
233
1.036
2.013
2.013
6800
280
247
0.763
1. 491
1.492
7800
245
233
1.042
2.025
2.025
10
288
247
0.767
1.497
1.499
10
245
233
1.048
2.037
2.037
20
287
246
0.771
1.504
1.505
20
244
232
1.054
2.050
2.050
30
286
246
0.774
1.511
1.512
30
244
232
1. 061
2.062
2.062
40
285
246
0.778
1.518
1.519
40
243
232
1.067
m
l-^
50
284
246
0.782
1-525
1.526
SO
243
232
1.074
6900
283
245
0.786
»-532
1-533
7900
242
232
1.081
2.I0I
2.101
10
282
245
0.789
1-539
1.540
10
242
232
1.087
2.1 14
2.114
2.128
20
282
245
0.793
1.546
1-547
20
242
231
1.094
2.128
30
281
244
0.797
1-553
1.561
^•554
30
241
231
I.IOI
2.142
2.142
40
280
244
0.801
1.562
40
241
231
1.108
2.156
2.156
SO
279
244
a8o5
1.568
1.569
50
240
231
X.116
2.170
2.170
7000
278
244
0.809
1-575
1.576
8000
240
231
1.123
2.184
2.184
Smitm«onian Taslcs.
73
Tablk 16.
LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI-
TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRi ANGULATION.
UNIT = THE METRE.
[Derivmtion and lue of table explained on p. be.]
«1
h=ci
Of
h
^«
^
tfl
«l=a
at
^
^
o°oo'
8.51268
8-S0973
— 00
— 00
1.404
io°oo'
8.51254
8.50968
t^
a958
1.430
10
268
973
8.871
9.169
1.404
10
254
^
1.431
20
268
973
9.172
9470
iu|04
20
253
968
0.668
0.973
1.432
30
268
973
9.348
9.646
1.404
30
253
^
°-f25
a98o
1.433
40
268
973
9.473
9.771
1.404
40
253
968
a682
0.987
M34
SO
268
973
9.S70
1.404
50
252
967
0.6S9
0.99s
M3S
I oo
267
973
9.649
9-947
1.404
II 00
252
967
0.695
1.002
1.436
lO
267
973
9.716
aoi4
1.404
10
251
967
a702
1.009
M36
20
267
973
9.774
ao72
1.404
20
251
967
0.709
1. 01 5
1.437
30
"&
973
9.825
0.123
1.405
30
250
967
0.715
1.022
1.438
40
267
973
9.871
0.169
1.405
40
250
^
0.722
1.029
M39
so
267
973
9.912
a2ii
1.405
SO
249
966
0.728
1.035
1440
200
267
972
9.950
9.985
a248
1.405
12 00
^
966
0.734
1.042
1441
10
267
972
0.283
1.405
10
^
0.740
1.048
1.442
20
267
972
0.017
0.315
1.405
20
248
966
0.746
1.055
1.444
30
266
972
0.047
0.346
1.406
30
247
9^
a752
i.o6i
1.440
40
266
972
0.075
0.374
1.406
40
246
^
0.758
0.764
1.067
so
266
972
aioi
0.400
1.406
SO
246
96s
1.073
1-447
300
266
972
0.126
0.425
1.406
1300
245
965
0.770
1.080
1.448
xo
266
972
0.150
0.449
1.407
10
245
?fs
a776
0.781
1.086
1-449
20
266
972
0.172
0.471
1.407
20
244
96s
1.092
1.450
30
266
972
0.193
0.492
1.407
30
244
96s
0.787
1.097
1.451
40
266
972
0.214
0.513
1.408
40
243
?f*
a792
1.103
1.452
50
266
972
0.233
0.532
1.408
SO
242
964
0.798
1.109
1.454
400
265
972
0.252
0.269
0.551
0.569
X.408
1400
242
964
0.803
1.115
1.456
10
265
972
1.409
10
241
964
0.809
1.120
20
265
972
a286
0.586
1.409
20
241
964
a8i4
1.126
M57
30
265
972
0.303
0.602
1.409
30
240
963
0.819
1.132
M58
1.460
40
265
972
0.319
0.618
X.410
40
239
963
a824
1.137
so
264
972
0.334
0.634
1.410
so
239
963
0.830
I.I43
1 .461
500
264
972
0.349
a649
1. 411
1500
238
963
a835
1.148
1^62
10
264
971
0.363
0.663
1.411
10
237
963
aS^o
1.153
M63
20
264
971
0.377
0.677
1.411
20
237
962
0.84s
I.I 59
1.465
30
264
971
0.390
a69i
1.412
30
236
962
0.850
1.164
1.466
40
263
971
a40^
0.416
0.704
1.412
40
235
962
a854
1.169
M67
so
263
971
0.717
1.413
so
235
962
0.859
1.174
M69
600
263
971
a^28
0.729
1.413
1600
234
961
o|64
1.179
1.185
1.470
10
263
971
a440
0.741
M14
10
233
961
0.869
1.471
20
262
971
0.4S2
0.753
1.415
20
233
961
0.873
1.190
1.473
30
262
971
a464
0-764
a776
0.787
MIS
1.416
30
232
961
0.878
°!!3
1.195
1474
40
262
971
0.47S
40
231
961
1.200
M7S
so
261
971
0.485
1.416
so
231
960
0.887
1.205
1-477
700
10
261
261
970
970
0.496
0.506
m
1.417
1.417
1418
1700
10
230
l§
0.892
0.896
1.210
1.214
1.478
1.480
20
260-
970
0.516
a8i8
20
960
0.901
1.219
1481
30
260
970
a526
a828
M19
30
328
959
0.905
1.224
1482
40
260
970
0.536
^^
X.419
40
227
959
a9io
1.229
1.484
so
259
970
0.545
0.848
1.420
50
226
959
0.914
1.234
1.485
800
10
259
970
970
0.S5S
0.564
&
1.421
1.421
1800
10
225
22s
959
958
a9i8
a922
1.238
i!248
1.487
1.489
20
970
0.573
0.875
1.422
20
224
958
0.927
1.490
30
^58
969
0.581
a884
X.423
30
223
958
0.931
1.252
1491
40
258
969
0.590
0.893
1.424
40
223
958
0.93s
1.2|7
1.493
so
257
969
0.598
0.902
1.424
so
222
957
0.939
1. 261
1.495
900
2S7
969
0.607
0.910
1.426
1900
221
957
0.943
1.266
1.496
10
256
0.615
0.918
10
220
957
0.947
1.271
1.498
20
256
969
a623
0.927
1.427
20
219
957
0.951
1.275
1.499
30
256
969
0.630
0.935
1.428
30
218
956
0.955
1-279
1.501
40
25s
0.638
0.942
1.428
40
218
956
0.959
0.963
1.502
50
25s
968
0.646
0.950
1.429
so
217
956
1.288
1.504
10 00
254
968
0.653
0.958
1.430
20 00
216
955
0.967
1-293
1.506
Smithsonian Tables.
74
Table 16.
LOGARITHMS OF FACTORS FOR COMPUTING DIFFERENCES OF LATI-
TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION.
UNtT = THE METRE.
[Derivation and oae of table explained on p. be]
f
tfi
h=ci
a%
h
c%
^
«i
^=a
As
^
ct
20«b(/
8.51216
8.50955
0.967
1.293
1.506
30^00'
8.51157
8.50936
I.167
1.528
1.625
10
215
955
0-97I
1.297
1.507
10
156
936
1.170
1.532
1.627
20
214
955
0-975
1.301
1.509
20
155
935
1.173
1.535
1.630
30
214
955
a979
a983
1.306
1.511
30
154
935
I.I76
1.539
1-632
40
213
954
1.310
X.512
40
153
934
1.178
I.18I
1.548
1.635
50
212
954
0.987
1.314
1.514
50
152
934
1.637
21 00
211
954
0.990
1-319
1.516
31 00
151
934
I.184
1.550
1.639
10
210
953
a994
1.323
1.518
10
\%
933
1.187
1-554
1.642
20
209
953
0.998
1.327
1.519
20
933
1.190
1.557
1.644
30
208
953
1.002
I.33J
1.521
30
147
933
1.193
1.561
1.646
40
207
953
1.005
1-336
1.523
40
146
932
1.195
'•564
1.649
SO
207
952
1.009
1.340
1.524
SO
145
932
1.198
1.568
X.651
22 00
206
952
1.0x3
1.010
1*348
1.526
3200
144
931
I.20I
1.572
;:§
10
205
952
1.528
10
143
931
1.204
1-575
20
204
951
1.020
1.352
1.530
20
141
931
1.207
1.579
1.659
30
203
951
IX>23
1-356
1.300
1.532
30
140
930
1.209
'•582
1.661
40
202
951
1.027
1.534
40
'39
930
1.212
1.586
l.DOO
SO
201
951
1.030
1.364
1.535
50
138
929
1.215
1.590
2300
200
950
1.034
1.368
1.537
3300
'?J
929
1.2X8
1-593
1.669
10
20
\^
950
950
1.037
1. 041
1.372
1.376
1.539
1.541
10
20
136
134
920
928
1.220
1.223
1. 000
1.671
1.674
30
197
949
1.044
i.^
1.543
30
133
928
1.226
1.604
1.676
40
197
949
ix>48
'•^5
1.545
40
132
927
1.229
1.607
1:^1
SO
196
949
i/)5i
1.388
1.547
SO
131
927
1.231
1.611
2400
»9S
948
1.058
I.05I
1.392
1.549
3400
'30
927
1.234
1.615
1.618
1.684
10
194
948
1.396
1.550
10
128
926
1.237
'•ff7
20
193
948
1.400
1.552
20
127
926
1.239
1.622
1.689
30
40
192
191
947
947
1.065
1.068
'•404
1.408
1.556
30
40
126
"5
925
925
1.242
1.248
1.625
1.629
1.692
1.695
SO
190
947
ijorji
1.412
1.558
50
124
925
1.632
1.697
2500
:i
946
1.075
1.078
1.081
1416
1.560
35 00
122
924
1.250
1.636
1.700
10
946
1.420
1.562
10
121
924
1.253
1. 256
1.639
1.702
20
187
946
1.424
1.564
20
120
923
1.643
1.705
30
186
945,
1.084
1.088
1.427
1.566
30
"?
923
1.258
1.647
1.708
40
*!s
945
M3I
1.568
40
118
923
1.261
1.650
1.711
SO
184
945
1.091
1.435
1.570
SO
116
922
1.264
1.654
1.713
2600
'|3
944
1.094
1.439
1.572
3600
115
922
1.266
i.6|7
1.716
10
183
944
1.097
1.443
1-575
10
114
921
1.269
1.661
1.719
20
181
944
I.XOO
1-447
1.577
20
113
921
1.271
1.664
1.721
30
180
943
I.I04
1.450
\W^
30
111
921
1.274
1.668
1.724
40
i;i
943
X.I07
1.454
1.458
40
110
920
1.277
1.672
1.727
SO
943
I.IIO
1-583
so
109
920
1.279
1.675
1.730
2700
'77
942
1. 116
1.462
1.585
3700
108
919
1.282
1.679
1.732
10
176
942
1.465
1.587
10
106
919
1.285
1.682
1738
20
175
942
1. 119
M69
1.589
20
105
919
1.287
1.686
30
174
941
1. 122
M73
1.591
30
104
918
1.290
1.689
1.741
40
172
941
1.128
1.477
1.594
40
103
918
X.292
1-693
1.744
50
171
941
1^80
1.596
SO
102
917
1.295
1.697
1.747
2800
170
940
1.131
\-&
1-598
1.600
3800
100
917
1.298
1.700
1.749
10
1§
940
1.134
10
^
916
1.300
1.704
1.752
20
940
1.137
1.492
1.602
20
916
1-303
1.707
1.755
30
167
939
1.140
1.495
1.605
30
097
916
::fJ
1.711
1.758
1.761
40
166
1. 146
1.499
1.607
40
095
915
1.7x5
1.718
50
i6s
938
1.503
1.609
SO
094
915
1.310
1.764
2900
164
938
1.149
1.506
1.611
3900
093
914
1.313
1.316
1.722
1-767
10
163
938
1.152
1.510
1.614
1.616
10
092
914
1.725
1.770
20
162
937
I.I55
1.514
20
090
914
1.318
1.729
1.773
30
40
161
160
937
937
\\t
1-517
X.52X
1.618
1.620
30
40
S§l
913
913
X.32I
1.326
1.736
1.776
1-779
1.781
SO
158
936
1. 164
1.525
1.623
50
086
912
1.740
3000
IS7
936
1.167
1.528
1.625
4000
085
912
1.328
1.743
1.784
•■mMeNMM Tamc*.
75
16.
LOGARITHMS OF FAOTOR8 FOR COMPUTING DIFFERENCES OF LATI-
TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION.
UNIT=THE METRE.
[Deiivmtioa aad om of table friplainwl on pw Iz.]
♦
«i
h=ci
Of
h
^2
4
«i
^=a
at
h
^
40l°bo'
8.51085
8.50912
1.328
1.743
1.784
5o**oo'
8.51008
^s°^
.^80
1.971
1.987
lO
084
911
1.33"
1.747
1.787
10
007
886
14S2
1.975
1.990
20
083
911
1-333
1.751
1.790
20
006
885
1.485
1.980
1.994
30
081
911
'•336
1-754
1.793
30
005
H5
1^87
^'^
1.9*
40
080
910
1.338
1-797
40
003
155
M90
1.988
2.002
50
079
910
1.341
1.800
SO
002
884
1492
1.992
2X)06
4100
078
909
1-344
1-765
1.803
1.806
5100
001
S*
:ip
1.996
2jOIO
10
076
^
1.346
1.769
10
000
883
2UX)0
2jOI4
20
07S
1.349
1.772
1.809
20
8.50998
883
1.500
24XH
2jOI7
30
074
9^
1.351
1.776
1.780
1.812
30
997
882
1.503
2.008
2.021
40
072
908
1354
1.815
i.8i8
40
996
882
\:^
2.013
2.025
SO
071
907
1-356
1.783
SO
994
882
2.017
2U)29
4200
070
907
1.359
1.787
1. 821
5200
993
88x
1.510
2U:>2I
2.033
10
069
906
1.361
1.791
1.824
10
992
881
1.513
1.516
2u:>25
2.037
20
067
906
1.364
1.794
1.828
20
991
880
2.030
2a>4i
30
066
905
1.366
1.798
1.802
1-831
30
P?
880
1.518
2.034
2.045
40
065
905
1.369
1.834
40
880
1.521
2/)38
2.049
50
063
905
1.371
1.805
1.837
SO
987
879
1.523
2.053
4300
062
904
1.374
1.809
1.840
5300
986
U
1.526
2.047
tsj
10
061
904
1.376
1.813
1.843
10
985
1.529
2.051
20
060
903
1.379
1.817
iA»7
20
983
878
1.531
2.055
2.066
30
058
903
1.381
1.820
1.850
30
982
Vf
I.S34
2J060
2.070
40
057
902
)^
1.824
1.828
1-853
1.856
40
981
!"
1537
*-^
2^4
50
056
902
SO
980
877
I-S39
2.068
2.078
4400
054
902
1.389
1.832
1.860
54 00
978
!'!
1.542
2.073
2.082
10
053
901
1391
1-835
1.863
1.866
10
977
876
1544
2,077
2.081
2.086
20
052
901
1.394
1.839
20
970
87s
1.547
2.091
30
051
900
1.396
1.843
1.870
30
97S
f'S
1.550
2.086
2.095
40
50
^
^
1-399
I^OI
1.847
1.850
\%i
40
SO
973
97a
874
1.552
1.555
2.090
2.095
2.099
2.104
45 00
047
fw
1.404
'•f54
1.880
55 00
97»
|7*
1.5S3
1-500
1563
2.099
2.108
xo
20
045
044
§8
1.407
1.409
1.858
1.862
1.883
X.886
10
20
970
969
§73
873
2.104
2.108
2.112
2.1x6
30
043
898
1.412
1.865
1.890
30
967
§73
1.566
2.113
2.121
40
042
897
1.414
1.869
1.893
40
966
872
X.568
2.1 17
2.125
50
040
897
1417
1.873
1-897
SO
96s
872
1.571
2.122
2.130
4600
S^
^
1.419
;l5?
1.900
5600
964
|7'
1-574
2.126
2.134
10
896
1422
1.903
10
963
§7'
1.577
2.131
2.138
20
036
896
1.424
1.885
1.907
20
961
871
1.579
2.136
2.143
30
03s
!9S
1427
1.888
1.910
30
960
870
1.582
2.140
2.147
40
034
195
1.429
^^l
1.914
40
'S
870
!:iy
2.145
2.152
y>
033
894
1.432
1.896
1.917
SO
869
2.150
2.156
4700
031
894
1434
1.900
1.921
57 00
957
i§
1.590
2.154
2. 1 61
10
030
593
1-437
1.904
1.924
10
956
IS
1.593
1.596
t\^
2.166
20
029
893
1439
1.908
1.928
20
954
2.170
30
027
593
1.442
1. 91 2
1-932
30
953
868
I.S99
2.169
2.x 75
40
026
892
1.444
1. 91 6
1.935
40
952
^7
1. 601
2.173
2.178
2.179
2.184
so
025
892
1-447
1.920
1.939
SO
951
867
1.604
4800
024
!9'
1.449
1.923
1.942
5800
950
867
1.607
2.183
2.188
2.189
10
022
!9'
1.452
1.927
1.946
10
949
866
1.610
2.193
2.1^
20
021
890
1.454
1.931
1.950
20
947
866
1.613
2.193
30
020
^
1457
1.935
1.953
30
946
866
1.615
1.618
X.621
2.197
2.203
2.208
2.213
40
SO
019
017
^
1.459
1.462
1.939
1-943
1-957
1.961
40
SO
945
944
86s
865
2.202
2.207
4900
016
889
1.464
1.947
''^
5900
943
864
1.624
2.212
2.217
10
015
880
1467
1.951
1.968
10
942
!*♦
1.627
2.217
2.222
20
013
888
1.469
1.95s
1.972
20
941
864
1.630
2.222
2.227
30
012
888
1.472
1.959
i-9§3
1975
30
l^
?*3
1.632
2.227
2.232
40
on
if7
1-475
1.979
1-983
40
5f3
I:g
2.232
2.237
SO
010
887
1.477
1.967
SO
937
863
2.237
2.242
5000
008
886
1.480
1-971
1.987
6000
936
862
1.641
2.242
«.247 1
8iiiTH«oNiAN Tabus.
76
Table 16.
LOGARITHMS OP FACTORS FOR COMPUTING DIFFERENCES OF LATI-
TUDE, LONGITUDE, AND AZIMUTH IN SECONDARY TRIANGULATION.
UNIT = THE METRE.
[Derivatum and use of table ezphined on p. be]
f
a\
*i=^i
<t
^
^«
^
tfi
*l=a
«i
h
c^
6<A)o'
8.50936
8.S0862
I.64I
2.242
2.247
70^00'
8.50877
S.5<^42
1.841
2.607
2.608
lO
93S
862
1.644
2.247
2.252
10
876
842
1.845
2.615
2.616
20
934
861
1.647
2.253
2.257
20
875
842
1.849
2.622
2.623
30
933
861
1.650
2.2C8
2.2S3
2.268
2.262
30
f75
842
•'•!53
2.630
2.631
40
SO
932
• 931
861
860
1.653
1.655
2.267
2.272
40
SO
874
873
841
1.857
2-637
2.64s
2.638
2.646
6100
10
928
860
860
1:^'
2.273
2.279
2.284
2.277
2.283
2.288
71 00
10
872
871
841
1.865
1.869
^^
2:^?
20
927
859
X.665
20
871
840
1-873
2.668
2.669
30
40
926
925
859
858
1.668
1.671
2.289
2.295
2.29J
2.298
30
40
870
86^
840
840
'lis?
2.676
2.684
l%\
50
924
858
1.674
2.300
2.303
50
840
1.886
2.692
2.693
6200
923
858
1.677
2.30s
2.309
7200
868
839
1.890
2.701
2.701
10
922
!S7
1.680
2.311
2.314
10
^l
839
;a
2.709
2.709
2.7x8
20
921
8S7
1.683
2.316
2.320
20
866
839
2.717
30
920
!s2
1.686
2.322
2.325
30
2^5
i
1.903
2.725
2.726
40
9x0
918
fsf
1.689
2.327
2.330
40
1^5
1.907
2.734
2.734
SO
856
1.692
2.333
2.336
SO
864
838
1 912
2.742
2.742
6300
917
856
;:|i
2.338
2.341
7300
5^3
§35
1. 916
2,760
2.700
xo
916
8SS
2.344
2.347
10
862
838
1.921
20
91 S
85s
1. 701
2.350
2-352
20
862
837
1.925
2.769
2.769
30
9'3
5ss
1.704
2.3S5
2.301
2.358
2.364
30
861
!37
1.930
2.786
2.778
2.787
40
912
8S4
1.708
40
860
837
1.935
so
911
854
I.7II
2.367
2.369
SO
860
837
1.939
2-795
6400
910
854
I.7I4
2.373
2.378
2.375
7400
li
836
1.944
2.804
2.80s
10
^
853
1.717
2.381
10
836
1.949
2.814
2.8x4
20
8S3
1.720
2.384
2.387
20
858
836
1-954
2.823
2.823
30
907
853
1.724
2.390
2.392
30
V>1
836
1-958
i!96^
2.832
2.833
40
906
852
1.727
2.396
2.398
40
!s6
836
2.842
2.842
SO
90s
852
1.730
2.402
2.404
SO
856
835
2.851
2.852
6500
904
852
1-733
2.408
2.410
75 00
Sss
83s
1.973
1.978
2.861
2.861
10
903
851
1-737
2.414
2.416
10
854
53s
2.8^1
^'IV
20
902
851
1.740
2.420
2.422
20
854
83s
1.984
2.881
30
90X
2S'
1.743
2.426
2.428
30
fS3
834
1.989
2.891
2.89X
40
900
850
1.747
2.432
2434
40
|5*
!34
1-994
2.901
2.90X
SO
900
850
1.750
2.438
2^40
SO
852
834
1-999
2.9II
2.912
6600
g
850
1.753
2445
2.446
7600
.85.
834
2.005
2.922
2.922
10
849
1.760
2.451
2.453
10
fs'
§34
2.010
2.932
2.933
20
897
849
2457
2.459
20
850
833
2.015
2.943
2.943
30
596
849
848
1.764
2.464
2.465
30
849
533
2.021
2.954
2.976
2.954
2.965
2.976
40
!9S
1.767
2.470
2472
40
its
233
2.027
SO
894
848
1.771
2.476
2.478
50
833
2.032
6700
893
848
1.774
2.483
2.484
7700
848
833
2.038
2.987
2.998
2.987
2.998
10
^
847
\lt
2.489
2491
10
5^7
!3*
2.044
20
891
847
2.496
2.497
20
847
83*
2X>50
3.010
3.010
30
890
f-*'
1:7^11
2.502
2.504
30
846
f3^
2.056
2.0S2
3.021
3.02X
40
889
f^z
2.509
2,510
40
845
53*
3.033
3-033
SO
888
846
1.792
2.516
2.517
50
84s
83*
2.068
3.045
3-045
6800
^
!t6
I.79S
2.522
2.524
7800
844
832
2.074
2.080
3.057
3-057
10
887
846
1.799
2.529
2.531
10
844
83'
3.069
3-069
20
886
84s
1.803
2.536
2.537
20
843
831
2.086
3.082
3-082
30
Ifs
84s
X.806
2.543
2.544
30
843
f3'
2.093
3.094
3-094
40
884
84s
1.810
2.550
2.551
40
842
83«
2.099
3.107
3.107
50
883
844
1.814
2.557
2.558
SO
842
831
2.100
3.120
3.120
6900
883
844
1.818
2.564
2.565
7900
b'
831
2.1 13
3133
3.146
3133
3.146
10
881
844
X.821
2.571
2.572
xo
841
830
2.1X9
20
880
844
1.825
2.578
2.579
20
840
830
2.126
3.160
3160
30
880
843
1.829
2.585
2.586
30
840
830
2.133
f;y
v:^
40
'^
843
1.833
2.000
2-594
40
839
830
2.140
SO
843
1.837
2.601
SO
839
830
2.148
3.202
3.202
7000
877
842
1.841
2.607
2.608
8000
839
830
2.155
3.216
3.216
SMimaomAM Tabic*.
77
Table 1 7.
LENGTHS OF TERRESTRIAL ARCS OF MERIDIAN.
P>rinaioo of tiUe wrplainwl on p. zlvi]
Latitnde
Ladtnde.
Latitode.
Latitude.
Ladtnde.
Latitude.
IntemL
o<»
1°
2°
3"
4"
F4tt,
F«tt,
FeH.
FtH.
F€€i,
lO"
1007.66
1007.66
lOOJ.t/J
1007.68
1007.71
ao
3015.31
3015.33
•015.34
aoi5-37
3015.41
y>
3022.97
4030.6J
3022.98
J033.01
3023.06
3033.13
40
4030.64
4030.68
4030.74
4030.83
t
5038.38
6045.94
^*
S52J5
5038.4a
6046.11
^ti
lO'
60459-4
60459.6
60460.3
60461.1
60463.^
130934.^
30
130918.8
130919.3
130930.4
130932.3
30
181378-3
181378.8
181380.6
181383.3
181387.3
40
341837.7
341838.4
341840.8
a4»844.4
341849.7
g
302397.1
903398.0
302301.0
362766.6
903311.1
362756.5
363757.6
362761.3
363774.5
f
(P
7^
^
9°
low
1007.73
1007.77
1007.81
1007.86
1007.01
3015.83
30
aoi5.47
3015.54
3015.63
3015.71
30
3033.30
3023.31
3033.43
3033.56
3033.73
40
4030.94
4031.08
4031.34
4031.43
4031.63
£
5038.67
5038.84
5089.O4
5039.a8
503954
6046.61
6046.85
6047.13
6047.45
to'
130930.3
60466.1
60468.5
60471.3
60474.5
30
"0933.3
130937.1
130943.6
130949.0
30
2E1
181398.4
181405.6
341874.'
181413.9
181433.4
341897.9
40
341864.6
341885.3
S
363784.6
303350.7
363796.8
JSIS:J
»l
r^v^
I0«
1I«>
I2«
13°
14^
\t/t
1007.97
1008.03
1008.10
1008.18
1008.36
30
«M5-93
3016.06
3016.30
3016.35
3016.51
30
3033.90
3034.09
3034.30
3oa4.5a
3034.77
40
4031.86
4033.13
4033.40
6049.05
4033.03
s:
5039-83
6047.80
1^:1
es:s
5041.38
6049.54
lO'
60478.0
60481.8
60486.0
60400.5
130981.0
60495.4
30
120955.9
130963.6
130972.0
\^i
30
181433-9
34i9«i.8
181445.4
181458.0
181471.5
40
341927.3
341944.0
341963.0
YA
F
JSI^I
302409.0
302430.0
302453.5
60
362890.8
362916.0
363943.0
S««97»-»
IS-
i6«
17«
i8«
I9P
10"
1008.34
1008.44
1008.53
1008.63
1008.74
3017.48
30
3016.69
S016.87
3017.06
3017.37
30
3025.03
3025.30
3025.60
3025.90
3026.33
40
4033.37
4033-74
5042.18
6050.61
4034.13
4034.54
4034.97
1:
5041.7*
6050.06
5043.66
6051.19
6051.81
5043.71
6053.45
.O'
6o5oa6
60506.1
60511.0
131023.8
60518.1
60534.5
30
121001.3
121012.3
12 1036.3
121049.0
30
181501.7
181518.3
181535.8
181554.3
181573.6
40
342002.3
342024.4
342047.7
342072.4
343O9S.I
5**
302502.9
363036.6
302559.6
302590.5
302623.6
60
363003.5
363071.5
363108.6
363147.1
20«
2I«
22°
23"
24°
j&t
1008.86
1008.97
1009.10
1009.32
1000.35
3018.70
ao
2017.71
aoi7.95
2018.19
3018.44
30
3026.56
3026.92
3027.28
3037.66
3038.06
40
4035.4a
4035.89
4036.38
4036.88
4037.41
S
5044.a8
5044.86
5045.48
5046.10
5046.76
6053.13
6053.84
6054.57
6055.33
6056.lt
1 '^
60531.3
60538.4
60545.7
60553.3
60561. I
1 90
1 2 1063.6 •
I 2 1076.8
121091.4
121 106.5
181659.8
131123.3
W s""
181593.9
181615.1
181637.1
181683.4
II ^
'*^i'!'
a42i53.5
242182.8
242213.0
343244.5
II ^
302656.5
302691.9
302728.5
302766.3
303805.6
I
••
ho
363187.8
363330.3
363274.3
363319.6
363366.7
n^^^
IAN TaBLI«
.
I.Q
Digit
ized by V^jC
'
>8i
Table 17.
LENGTHS OF TERRESTRIAL ARCS OF MERIDIAN.
[Derivation of table exidained on p. zlvi.]
Latitude
Latitude.
- ■
Latitude.
Latitude.
Latitude.
Latitude.
Interval.
250
26°
2t
28°
29^
Fe€t.
Ft€t.
F€*U
Feet.
Feet,
lO"
1000.49
aoi8.97
1009.63
1009.77
1009.93
3019.83
1010.07
ao
^^
3019.54
3030.13
30
3038.46
3039.31
3039.75
3030.30
40
4037.9s
4038.51
S'S
4039.67
4040.37
1;
5047.44
6056.9a
i^.^
5049.58
6059.50
irji
lO'
60569.3
131138.5
60577.6
60586.3
60595.0
60604.0
13 1308.0
30
131 155.3
131173.3
131 190.0
181785.0
30
181707.7
181733.7
181758.5
i8i8i3.o
40
343376.9
343310.3
343344.7
342379.9
343416.0
50
303846. 1
303887.9
303930.9
303974.9
303019.9
60
3634«5-4
363465.S
3635171
363569.9
363633.9
300
31"
320
33"
34"
to'/
ioio.aa
ZOIO.38
ZOI0.54
1010.70
ioia86
ao
3030.44
aOaO.75
aoal.07
3031.40
ao3i.73
30
3030.66
303».»3
3031.61
3033.10
3033.59
4043.46
40
404a88
4041.51
4043.1J
6063.3a
4043.80
S
I2;:S
6063.36
S^5S
5054.33
6065.19
lO'
60613.3
60633.6
60633.3
60643.0
131383.9
60651.0
131303.8
ao
131336.4
131345-3
z8l896!6
30
181839.7
181867.9
181935.9
181955-7
40
343453.9
a4a49o.5
343538.8
343567.9
343607.6
50
303066.1
303113.3
303«6i.i
303309.9
303359.4
60
3636793
363735.8
363793.3
363851.8
36391 1.3
35"
36"
37"
38"
39°
lO''
1011.03
loii.ao
1011.37
1011.55
1011.73
ao
ao33.o6
3033.40
aoa3.75
a033.o9
3033.44
30
3033.10
3033.61
3034.1a
3034.64
3035.17
40
4044. >3
4046.19
4046.89
U
^%
Sg:«
5057.74
6069.39
5058.61
6070.34
vJ
60661.9
60673.1
60683.4
181385.7
60703.4
ao
131333.9
IliJttJ
13 1364.9
131406.7
30
181985.8
183047.3
1831 10. 1
40
343647.8
34a688.5
343739.?
343771.4
343813.4
£
303309.7
30336a6
303413.3
303464.3
303516.8
363971.7
364033.8
364094.6
364«57.«
364330.3
40°
41"
42-
43"
44"
10"
1011.90
3033.80
1013.08
1013.35
1013.43
1013.61
ao
3034.15
3034.51
3034.87
3035.33
30
3035-70
3036.33
3036.77
303730
3037-84
40
4047.60
4048.31
4049.03
4049.74
4050.46
1;
5059-50
5060.38
5061.38
5063.17
5063.07
6071.39
6073.46
6073.53
6074.61
6075.69
lo'
60713.9
60734.6
60735.3
60746.1
60756.9
ao
131437.9
183141.8
131449.3
I3i47a6
13 1493.3
131513.7
30
183173.8
183306.0
183338.3
183370.6
40
343855.8
343898.4
34394«.3
343984.3
243037.4
50
303569.7
303633.0
303676.6
303730.4
303784.3
60
364283.7
364347.6
364411.9
364476.5
364541.3
45"
46°
47°
48"
49^
10"
IOIa.79
1013.97
1013.15
1013.33
1013.51
ao
3035.59
3038.38
3035.95
3038.93
3036.31
3036.67
3037.03
30
303946
3040.00
3040.54
40
4051.18
4051.90
4053.63
4053-34
4054.05
50
5063.97
5064.87
5065.77
5066.67
5067.56
60
6076.77
6077.8s
6078.93
6080.00
6081.08
to'
60767.7
60778.5
60789.3
183367.8
60800.0
60810.8
ao
131535.3
131556.9
I3160O.I
131631.5
30
183303.0
183335.4
183400. 1
183433.3
40
343070.6
343113-9
343157.0
343300.1
343343.0
S
303838.3
J»
303946.3
304000.1
304053.8
364606.0
364735-5
364800.3
364864.5
.MtTHSOWAtt TaSLC*.
J by
Google
79
Table 17.
LENGTHS OF TERRESTRIAL ARCS OF MERIDIAN.
[Derivation of taUe explained on p. zlvi]
Latitude
Latitude.
Latitude.
Latitude.
Latitude.
Latitude.
Ijatitnde*
Interval.
Soo
51"
520
53**
54^
55^
Fttt.
Fett.
Ftet.
Feet.
Fut.
Ftet.
lO".
1013.69
3027.38
1013.87
1014.04
10x4.33
Sajl
1014.56
ao
3037.74
3038.09
3028.44
. ao39.ia
30
3041.07
3041.60
3043.13
3043.65
3043.17
3043.68
4058.34
40
S
4055.47
4056.17
4056.87
4057.56
1^
^VL
ISJ:n
K
«
v/
60831.5
60833.x
60843.6
13x685.3
60853.1
60863.5
60873.7
30
131643.9
13 1664.3
Z3I706.S
X3I 736.9
131747.3
30
»J
183537.7
183559.3
183590.4
343453.8
183631.0
40
a433a8.3
343370.3
343413.3
343494.6
£
JSJiatI
304x60.4
36499a.5
304313.9
S65055.5
365118.5
365x80.1
304368.3
365343.0
56O
S?*'
S8o
59°
60O
61O
«o"
1014.73
X014.90
1015.06
X015.33
1015.38
1015.53
ao
3039.46
3039.79
3030.13
3030.44
3045.66
3030.76
»Q3i.o7
30
3044. »9
3044.69
4059.58
3045.18
4060.34
3046.14
3046.60
40
4058.9a
4060.88
4061.53
4063.14
^
l^^i
«
IS:|2
5076.10
6091.33
5076.90
6093.37
5077.67
6093.30
lo'
ao
60S83.8
131767.6
60803.8
131787.5
60903.6
131807.3
131836.5
183739.8
;Si
60933.0
X3ii64.i
30
183651.4
18368Z.3
183710.8
183796.1
40
'43535'a
JS^.t
343614.4
3045 '80
343653.0
343691.0
343738.3
50
304419.0
304566.3
3046x3.7
304660.3
60
3653<».8
365363.6
365431.6
365479.6
365536.4
365593.3
62O
63^
64^
65°
66°
67O
10"
1015.69
10x5.83
X015.98
1016.13
1016.26
X016.39
ao
ao3»-37
ao3x.67
3031.96
3032.34
3033.5 X
3033.78
50
3047.06
3047.50
3047.94
3048.36
3048.77
4065.03
3049.16
4065.5s
40
4063.74
4063.34
4063.93
5080.60
50
5078.43
5079.17
5079.90
6095.87
5081.38
5081.94
60
6094,13
6095.00
6096.71
6097.54
6098.33
It/
131883.3
60950.0
60958.7
60967.x
60975.4
^^1
ao
13x900.x
X83850.1
;ii2J?:J
13 1934.3
121950.7
131966.6
30
183833.5
J,w
183936.1
X83949.8
40
343764-6
343800.3
343835.0
343901.4
304876.8
343933.1
50
304705.8
304750.3
304793.7
304815.7
J?3CJ
60
365647.0
365700.3
365753.4
365803.8
365852.3
680
690
700
71°
720
73*
10"
1016.5a
ioi6.6«
»o33.a8
X016.76
1016.87
1016.98
10x7.09
ao
3033.03
3033.53
3033.75
3033.96
3034-17
30
40
3049.55
4066.07
5083.58
"^^
3050.38
406704
3050.63
4067.49
3050.95
4067.93
^^
50
5083.30
5083.80
5084.36
6101.89
5085.43
6103.53
60
6099.10
6X99.84
6x00.55
6XOX.34
lO'
60991.0
12x983.0
6X998.4
6x005.5
6x0x3.4
61018.9
1330S7.8
61035.3
ao
12 1996.8
X330IX.X
122024.8
132050.3
30
183973.1
182995.2
X83016.6
X83037.X
183056.8
183075.5
40
a43964.«
a43993.6
244032.3
244049.5
344075.7
344100.6
50
304955.'
304992.0
305027.7
305061.9
305125.8
60
365946.1
365990.4
366033.3
366074.3
366113.5
366151.0
74^
75^
76O
JT"
780
790
loff
1017.18
1017.28
1017.37
1017.45
1017.53
10x7.60
ao
ao34.37
3034.56
3051.84
3034.73
3034.90
3035.05
3053.58
3035.19
30
3051.56
3052. xo
"^^
3053.79
40
4068.74
4069.13
4069.46
4070.10
5087.63
50
5085.93
5086.40
5086.83
S087.34
60
6x03.11
6103.67
6104.30
6104.69
6105.16
6.05.58
lo'
6x031.1
61036.7
61043.0
X 33083.9
6x046.9
6I05I.6
6.055.8
ao
123062.3
X33073.5
123093.9
X83I40.8
133 103.1
X33IXX.5
30
183093.3
183x10.3
183x25.9
344167.8
183154.7
.83x67.3
40
244x34.4
344«470
305183.7
344187.8
344306.3
SI
50
^li:il:l
305209.8
305334.7
305357.8
60
366330.4
366251.8
366281.6
366309.4
V
Smithsonian Tables.
80
Digitized by^
Table 18.
LENGTHS OF TERRESTRIAL ARCS OF PARALLEL.
[DeriTatioQ of table explained on p. zlix.]
Longitude
Latitude.
Latitnde.
Latitude.
Latitude.
Latitude.
Interval.
OO
I«
2«
f
4^
Ftei.
Fnt.
/W/.
Put.
Fttt,
lO"
1014.5a
10x4.37
3028.74
XOX3.9X
3027.82
XOX3.14
X012.07
ao
3029.05
ao36.39
2024.14
30
3043.57
3043. u
3041.73
3039-43
3036.2X
40
4058.10
4057.48
4055.64
U
«
K
ffi
S^;JJ
x</
6087..4
60863.3
60834.6
60788.6
60724.2
X21448.4
182 I 72.6
ao
X21742.9
'^^\
1 21669.2
12x577.2
30
182614.3
183503.8
X82365.7
40
•43485.8
343449.0
243338.4
243154.3
343896.8
£
304357.2
304311.3
304173.0
303943.9
303621.0
365228.6
365173.6
365007.6
364731.S
364345.2
5°
6°
t
8«
9"
low
10x0.69
2021.38
X009.00
X007.0X
X004.73
I003.X3
ao
3018.0X
8014.03
2009.43
3004.33
3006.3s
30
303a.o7
3037.01
302x04
4018.87
40
4043.76
4036.03
4028.05
503506
4008.47
50x0.58
50
£U;tJ
504503
5023.58
60
6054.0a
6043.08
6038.30
60x3.70
ly
.JSI.1
60540.3
60430.8
60383.0
60137.0
ao
12x080.5
X3084X.6
x8olw.x
X303C4.0
18038X.X
30
181924.3
X81630.7
18x262.3
40
343565.6
34316X.0
24x683.1
341x33.1
340508.x
50
303207.0
302701.2
302x03.9
30x4x5.x
36x698.1
300635.x
60
363848.4
363341.4
362524.7
300703.x
10°
11°
12°
13^
14°
loff
999-31
'998.43
3997.64
996.0X
992.50
1985.00
98860
1977.38
984.58
ao
':^^
1969. X7
30
297730
3966.07
2953.75
40
3996.85
3984.03
3970.00
395476
3938.34
&
4996.06
4980.04
4962.50
4943.46
4922.9a
S995.a8
5976.04
S955.0O
5933.X5
5907.50
xt/
599Sa.8
59760.4
X 19530.8
X7938X.3
59550.0
S933X.5
59075-0
ao
30
1«
X 19 100.0
X78650.0
X 18643.9
237285.8
X18150.X
177225. X
40
239811.1
358563.5
338300.0
236300.2
£
•99763-9
3597*6.7
397750.0
3S730O.0
355928.8
295375.2
3S4450.2
i!f
\(P
I70
180
Xff
10"
980. x8
975-47
970.48
965. x8
959.60
ao
«96o.35
1950.95
X940.9S
1930.36
3895.55
3838.38
30
3940.53
3926.42
^:.l?
40
3930.71
4900.88
390X.90
3860.73
S
4877.37
'^t
4825.91
4797.98
5881.06
5852.84
579J-09
5757.58
xo'
58810.6
58528.4
58338.5
.f??:!:i
57575-8
ao
X1762X.3
XX 7056.9
175585.3
234113.8
1x6457.0
"Si5».S
30
X76431.9
174685-5
173732.8
172727.3
40
335»43.S
3329x4.0
33x643.7
230303.0
50
394053.1
292643.2
39XX43.5
289554.6
287878.8
60
353863.7
35 "70.6
34937«o
347465.5
345454-6
20°
21°
22^
23°
24°
10//
ao
953.7a
947.5s
X895. xo
1883. 19
.^t
1854.67
30
2842.66
2823.29
3764.38
8803.07
a782xx>
40
3814.87
3790.2X
3737.43
3709-33
SO
4768.59
4737.76
4705.48
4671.78
4636.66
60
5733.3 X
5685.3«
5646.58
5606. X4
5564.00
It/
57323.x
56853.1
56465.8
56061.4
X 13X22.8
55640.0
ao
"4446.2
1 13706.3
"3931.5
X 1x280.0
30
I7I669.2
"70559.4
169397.3
168184.3
X66919.9
40
228892.3
227412.5
225863.0
224245.7
222559.9
£
286x15.4
343338.S
284265.6
341118.7
282328.8
280307.1
278199.9
338794.6
336368.5
333839.9
8mit
81
/Google
Table 18.
LENGTHS OF TERRESTRIAL ARCS OF PARALLEL.
[DeriTatioa of table explained on p. zlix.]
Longitude
Latitude.
T.«^rit»iiU
Ladtude.
Latitude.
Latitude.
Interval.
250
260
270
28O
290
Ft€t.
FMt.
FttU
Ftti.
F*€t.
xo"
930103
,0
X809.X6
^<&
888.03
ao
1840.05
X 776.06
3o
vjbfi.%&
3737.33
3618.32
4533.89
2689.32
2664.09
40
36S0.XX
3649.77
3585.76
3S53.X2
1:
4600.14
4562.21
4482.20
SSII
SSao. 17
5474.65
5437.47
5378.64
lO'
55aox.7
54746.5
Si
53786.4
S338X.8
ao
109493.0
107573.9
106563.S
30
165605.0
164339.5
161359-3
159845.3
40
aao8o6.6
218986.1
2x7099.0
Si^*l:J
V^l
SO
a76oo8.3
373733.6
371373.7
60
33iao9.9
328479.1
325648.4
322718.6
319690.6
30^
31"
32"
33"
34"
xo"
879.35
1758.70
870.40
1740.80
861.18
851-71
1S3.94
ao
1733.37
1703.41
30
a638.o4
261X.20
3583.55
3555.13
3367.88
40
3517.39
3481.59
3444.74
3406,83
1;
4396.74
4351.99
4305.93
4358.53
4309.85
5276.09
5332.39
5x67. xo
5110.24
5051.8a
xo'
53760.0
xo5sai.8
52223.9
5x671.0
51 102.4
X02204.8
505x8.2
ao
103343. 1
101036.4
151554.6
3©
X58a8a.6
155013.1
153307.3
40
aiio43.5
308895.7
206684.2
304409.7
ao2072.8
S
363804.4
3611x9.6
358355.3
310026.3
355513.1
252591.0
316565.3
313343.5
306614.S
303109.3
3f
36"
37"
38"
39"
toff
ao
831.98
X663.95
Si:?
8x1.33
800.48
X600.97
is^t?
30
a495.9J
3a86.9t
3433^69
3401.45
2368.48
40
3327.91
4x59.88
4991.86
3344.92
320X.93
3157.97
S
4108.64
4930.37
Sl^isi
4002.42
4802.90
3947.46
4736.95
xo'
49918.6
49303.7
48673.8
48029.0
47369.5
ao
99837.3
98607.4
97347-6
96058.0
94739.1
30
149755-8
X4791X.2
146031.4
X44087.0
142x08.6
40
199674.3
^3i
295822.3
194695.3
192116.0
189478.2
£
a4959».9
399511.5
3433690
292042.8
240145.0
288174.0
284217.2
40°
41"
42O
43"
44"
x&f
778.26
766.79
1533.58
755.08
7il'S
730.98
20
1556.53
1510.17
X486.29
X46X.96
30
3334.78
3300.37
2265.25
2229.44
3x92.95
40
3113.04
3067.X6
3020.33
3973.59
3933.93
£
«
3833.94
46oa73
377543
4530.50
JJJi:§
Sitt;
xo'
*S«9S.6
46007.3
45305.0
44588.8
43858.9
87717.9
ao
93014.7
90610.0
89177.6
30
140086.7
138022.0
184029.3
i359»5o
133766.4
131576.8
40
186782.3
i8i22ao
178355.3
175435.8
so
333477-9
380173.5
230036.7
226525.0
271830.x
222944-0
219294.7
60
276044.0
267532.8
263153.6
45"
46°
47"
48"
49"
lO"
718.59
705.99
693. x6
680. X2
666.87
ao
1437.19
3155.78
1411-97
X386.32
1360.24
133375
30
2117.96
2079.48
2040.36
2000.62
40
3874.38
3823.94
3773.64
2720.49
2667.50
50
3593.97
3529.93
4158.96
3334.37
60
4311.56
4335.91
4080.73
4001.25
xo'
ao
SlsJit
84718.2
41589.6
.S;s:7
40807.3
8x614.6
40012.5
80024.9
30
129346.9
127077.3
122421.9
130037.4
40
173462.5
169436.5
166358.3
163229.2
X60049.9
SO
315578.2
358693.8
2x1795.6
207947.9
204036.4
200062.3
240074.8
60
354154.7
349537-5
344843.7
Smithsonian Tables.
82
Digitized by
r^oogle
Table 18.
LENGTHS OF TERRESTRIAL ARCS OF PARALLEL.
[Derivadon of tmble ocpfadiied on p. zlbc]
Longitude
Latitude.
Latitude.
Latitude
Latitude.
Latitude.
Latitude.
Interval.
50^
510
520 •
53^
54"
55"
Fett.
Fwt.
FmL
Ftet,
F*€t,
F*tt.
lO"
653.4a
i3o6.5s
639.77
635.92
1251.84
611.88
597-65
583.23
30
"7954
1233.76
1195.30
1x66.47
30
1960.27
«9«9.3«
1877-76
1835.63
1792.94
1749.70
40
3613.69
'559.08
3198,85
3503.68
2447.5"
3*JSJ2
2332.93
50
3367.13
3139-60
3059.39
3916.16
60
39»o.54
3838.63
3755.52
3671.37
3585.89
3499-40
lO'
78410.8
117616.1
38386.3
37555.2
367J2.7
35858.9
7x7x7.8
?m:
ao
76772.4
75"0.4
73425.4
110x38.0
30
XX5158.6
113665.6
107576.6
X0498X.9
40
156831.S
«53544.8
150220.8
187776.0
146850.7
183563.4
143435.5
139975.9
P
196036.9
19x931.0
179294.4
174969.9
60
a35a3a-3
330317.3
325331.2
330376.1
215153-3
209963-9
56-
57^
58-
59"
6o<»
610
!<//
568.64
553.87
538.93
523.82
508.55
493.13
ao
Ii37.a8
XX07.74
X077.86
1047.65
10x7. XI
986.36
30
1705.93
1661.61
16x6.79
I57I.47
1535.66
1479.38
40
3374.56
3315.48
2155.72
3095.39
3034.33
1972.52
3465.64
3958.77
S
a843.ao
2769.35
2694.64
3619.13
2542.77
34«i.83
3333.33
323357
3142.94
3051.33
lO'
ao
tt
33232.3
^1
V^W
mi
VSSLl
29587.7
3©
1O33SS-0
97007.3
94388.1
91539.9
40
«36473-4
133938.8
"29343.0
"57175
133053.3
X1835X.0
so
170591.7
i66t6x.o
161678.7
ISlJtl
147938.7
60
304710.0
199393.2
194014.4
177526.4
ea*'
63"
64"
65°
66*»
67"
i&i
477-55
461.83
445.96
413.83
397.55
ao
30
955.10
1433.66
1847.31
891.^
1337.88
837.63
1241.44
795.10
1192.64
40
1910.31
1783.84
1719.81
1655.26
1590.19
1987.74
SO
V^t
3309.14
3149^76
ao69.o8
60
3770.96
2675.75
2579.72
3482.89
238529
lo'
38653.1
37709.6
26757.S
25797.2
S
33853.0
ao
30
57306.3
85959.4
Sits;
s;i:4
5»594.4
103x88.7
40
114613.5
143365.6
17x918.7
•10838.5
107030.3
133787.7
99315.6
95411.5
1^
138548.1
138985.9
124144.5
119364.4
166357.7
160545-2
154783.1
148973.4
143117.3
68«
69°
700
7i«
720
73"
xdf
381.16
364.65
348.P3
331.30
JXJJ
297.54
ao
763.33
729-30
696.06
663.60
30
"43-47
;X4
1044.09
993.90
1357.88
40
«5«4-63
1392.12
1335.30
tX9ai6
£
iS-^'
1833.35
2187.90
'740. »4
3088.17
!*4^S
:si:it
148770
1785.23
lo'
33869.5
3x879.0
65637.0
ao88x.7
19878.1
18868.1
17852.3
ao
^:
4«763.S
39756.1
37736.3
35704.7
30
63645.3
59634.2
56604.4
53557.0
40
91477.9
87516.0
83537.0
795"-2
75472.6
i^:J
so
"43474
109395.0
104408.7
99390.3
ii33o8i8
60
X37>«6.9
X3 1274.0
"5290.4
1x9268.4
107x14.0
74^
75^
76O
77"
78°
79"
lo'/
380.53
263.41
346.33
338.96
31X.63
1§J'4J
ao
561.04
536.83
492.44
*^j&
423.24
30
841.56
790.33
738.66
634.85
5*^Sl
40
XX33.08
1053-64
984.88
915.83
846.47
776.86
t.
X403.60
1317.06
1331.10
1144.78
X058.09
971.08
X683.II
1580.47
«477.33
1373.73
1369.71
1165.39
td
X6831.I
15804.7
'4773.3
13737.3
13697.1
11653.0
23305.8
34958.7
ao
33663.3
3*609.3
29546.5
44319.8
27474-6
25394.2
30
67324.6
63218.6
4121X.9
38091.3
40
5(9093.0
50788.3
4661 X.6
S
loo^is
79023.3
94828.0
73866.3
88639.6
68686.5
8342 j.§
63485.4
76183.5
58164.5
69917.4
Amitusam
AM TaBI^VS
'ig
tized by V3V.
83
>8i
Table 19.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iW^rrr-
[Derivation of table explained on pp. Uii — Ivi.]
•8 .
Meridional dis-
tances from
even degree
paraUels.
CO-ORDINATES OF DEVELOPED PARALLEL FOR- 1
15' longitude.
Zd loBgitnde.
45' longitude.
lO lon^tude. 1
X
y
X
y
X
y
X
y
Inehtt,
IncJuu
ImcJut,
Incfus.
Inck4».
Inck4t.
Inckts.
Inches.
Inckts.
0<»00'
45
13-059
4.383
4.383
4.383
4.382
.000
.000
.000
.000
8.766
8.766
8.76s
8.765
JQOO
.000
.000
.001
13.148
13.148
13.148
13.147
.000
.000
.001
.001
17.531
17.531
17.530
17.530
.000
.001
.001
.002
I 00
IS
30
45
17.412
4.382
4.382
4.381
4.381
.000
.000
.000
.000
8.764
8.764
8.763
8.762
.001
.001
.001
.001
13.146
13-145
13-144
13.142
.001
.002
.002
.003
17.528
17.527
17.525
17-523
X)03
.003
.004
.005
13-059
200
30
45
17.412
4.380
4.379
4.379
4.378
.000
.000
.000
.000
8.760
8.759
8.757
8.755
.001
.001
JOOl
.002
13.141
13-138
I3.»36
13.133
.003
.003
.004
.004
17.521
17.518
17.514
175"
JOdS
.007
.007
13059
300
IS
30
45
17-413
4.377
4.376
4.375
4.373
.001
.001
.001
.001
8.753
8.751
8.749
8.747
.002
.002
.002
.002
13.130
13.127
13.124
13.120
.004
.005
17.507
17.503
17-498
17.494
.008
.008
.009
.009
13.060
400
IS
30
45
17.413
4.372
4.371
.001
.001
.001
.001
8.744
8.742
8.739
8.736
.003
.003
.003
.003
13.116
13.112
13.108
13.104
.006
.006
.007
.007
17488
17.483
17.478
17.472
.010
.Oil
.012
.013
4.353
8.707
13.060
500
15
30
45
17.413
4.366
4.364
4.363
4.361
.001
.001
.001
.001
8.732
8.729
8.72s
8.722
.003
.003
.004
.004
13.099
\\^
13.082
•007
.008
.008
.008
17.465
17.458
17.451
17.443
.013
.014
x>i4
.015
4-353
8.707
13.060
600
15
30
45
17-414
4.359
4.357
4.355
4-353
.001
.001
.001
.001
8.718
8.714
8.710
8.705
.004
.004
.004
.004
13.076
13.071
13.064
13-058
.009
.009
.010
.010
17.435
17.428
17419
17.410
.016
.017
.017
.018
4.354
8.707
13.061
700
'5
30
45
17.414
4.350
4.348
4.346
4.343
.001
.001
.001
.001
8.701
8.696
8.691
8.686
.005
.005
.005
.005
13.051
13044
13.036
13.029
.010
.011
•Oil
X>1I
17.401
17-392
17.382
17.372
.019
x>i9
.020
JQ20
4.354
8.707
13.061
800
15
30
45
17.415
4.340
4.338
4.335
4.332
.001
.001
.001
.002
8.681
8.67s
8.670
8.664
.005
.006
13.021
13-013
13.005
12.996
.012
.012
.013
.013
17.362
17.351
17-340
17.328
.021
.022
.022
.023
13.062
900
15
30
45
17.416
4.329
4.326
4.323
4.320
.002
.002
.002
.002
8.658
8.652
8.646
8.640
.006
.006
.006
12.987
12.979
12.969
12.960
.013
.014
.014
.014
17.316
17.305
17.292
17.280
.024
.024
.020
13.062
1000
17.417
4.317
.002
8.633
.006
12.950
.015
17.266
.026
Smithsonian Tables.
84
Table 19.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE jTzhv-
[Derivation of table explained on pp. liii — Ivi.]
'S .
Meridional dia.
Unces from
even degree
parallel*.
CO-ORDINATES OF DEVELOPED PARALLEL FOR —
IS' longitude.
y/ longitude.
45' longitude.
i^ longitude.
X
y
z
y
X
y
X
y
Inck4s,
Inches.
Inches.
Inches.
Inches.
Inches,
Inches.
Inches,
Inches,
loPbo'
30
45
4-354
8.709
13-063
4.317
4.313
4.3<o
4.306
.002
.002
.002
.002
8162^
8.620
8.613
.006
.007
.007
.007
12.950
12.940
12.930
12.919
.015
.015
.Ol|
.016
17.266
«7.253
17.240
17.226
.026
.027
!o28
II 00
30
45
17.418
4.303
4.299
4.295
4.292
.002
.002
.002
.002
8.606
8.598
8.591
8.583
.007
.007
12.908
12.875
.016
.016
.017
.017
17.211
17.IQ6
17.182
17.166
.029
.029
.030
..031
4.355
8.709
13.064
1200
15
3D
45
17.419
4.288
4.284
4.280
4.275
.002
.002
.002
.002
8.575
8.567
8.559
8.55*
.ooR
.008
.008
.008
12.863
12.851
12.839
12.826
JOI7
.018
.018
.019
17.150
17.134
17.118
17.102
.031
.032
.032
.033
4.355
8.710
13.065
1300
15
30
45
17.420
4.271
4.267
4.262
4.258
.002
.002
.002
.002
8.542
8.534
8.516
.008
.009
.009
.009
12.813
12.800
12.787
12.774
.019
.019
.020
.020
17.084
17.067
17.050
17.032
.034
.034
•035
•035
4.355
8.7 1 1
13.066
1400
>5
30
• 45
17.421
4.253
4.249
4.244
4.239
.002
.002
.002
.002
8.507
It
8.479
.009
.009
.009
.009
12.760
12.746
12.732
12.718
.020
.021
.021
.021
17.013
16.976
16.957
.036
.036
.037
.038
4.356
8.7 1 1
13.067
1500
>5
30
45
17.423
4.234
4.229
4.224
4.219
.002
.002
.002
.002
8.469
8.459
8.449
8.439
.010
.010
.010
.010
12.703
12.688
12.673
12.658
.022
.022
.022
.022
16.938
16.918
16.898
16.877
.038
.039
.039
.040
4.356
8.712
13.068
1600
'5
30
45
17.424
4.214
4.209
4.204
4.198
.003
.003
.003
.003
8.428
8.417
8.407
8.396
x>io
.010
.010
.oil
12.642
12.626
12.610
12.594
.023
.023
.023
.024
16.856
16.835
16.814
16.792
.041
.041
.042
.042
4.356
8.713
13.069
1700
"5
30
45
17.426
4.192
4.187
4.181
4.175
.003
.003
.003
/•OO3
8.385
5-374
8.362
8.351
.oil
.011
.oil
.oil
12.577
12.561
12.544
12.526
.024
.024
.025
.025
16.770
16.748
16.725
16.702
.043
.043
.044
.044
4.357
8.714
13.071
1800
15
30
45
17.427
4.170
4.152
.003
.003
.003
.003
8.339
f'327
8.316
8.303
JOll
.oil
.012
.012
12.509
12.491
12473
12.455
.025
.026
.026
16.679
16.631
16.606
.045
.046
.046
4.357
8.715
13.072
1900
15
30
45
17.429
4-145
4.139
4.133
4.127
.003
.063
.003
.003
8.291
8.278
8.266
8.253
.012
X)I2
.012
.012
12.436
12.418
12.309
12.380
.026
.027
.027
.027
16.582
16.557
16.506
.047
x>48
.048
13.073
2000
17.43'
4.120
.003
8.240
.012
12.360
.028
16.480
.049
8mitm«oiiiaii Tables.
8s
Tablk 19.
CO-ORDINATES FOR PROJECTION OF MAPS.
[DeriTStioB <d tabia espUaad on pp. UB-ItL]
SCALE ttAvy^
IS
30
45
21 00
15
30
45
2200
IS
30
4S
2300
IS
30
45
2400
IS
30
45
2500
15
30
45
2600
15
30
45
2700
IS
30
45
2800
IS
30
45
2900
IS
30
45
3000
Jnekgt,
4.358
8.717
13^75
17433
8.718
13-076
17.435
4.359
8.719
13-078
17.437
4.360
8.720
13.080
17-439
4.360
8.721
i3»o8i
17.442
4.361
8.722
13-083
17444
4.362
8.723
52
.723
13.085
17.446
1-362
8.724
13.087
17-449
13.088
17.451
4.363
8.727
13.091
I74S4
CO-ORDINATES OF DEVELOPED PARALLEL FOR—
is' longitiMle.
Jmhe*.
4.120
4.II4
4.107
4.100
4.094
4.087
4.080
4-073
4.066
4.058
4.051
4.044
4.036
4-029
4.021
4.014
4.006
3-998
3990
3.982
3-974
3.966
3-958
3-950
3-942
3-933
3-925
3.916
3-908
3.899
3.873
3.863
3.854
3.845
3836
3-827
3'5'7
3.808
3-799
.003
.003
.003
.003
.003
.003
.003
.003
•003
.003
.003
.003
.003
.003
.003
.004
.004
.004
.004
.004
•004
.004
.004
4XH
.004
.004
.004
.004
.004
.004
J0O\
«H
.004
.004
.004
.004
.004
.004
.004
.004
.004
so' kmptnde.
Jnckgt.
8.240
8.227
8.214
8.200
8.187
8.173
8.IS9
8.145
8.131
8.117
8.102
&088
8.073
8.058
8.028
8.012
7.997
7.^1
7.965
7.949
7.933
7.916
7.900
7.883
7.866
7.849
7.833
7.816
7.798
7.780
7.763
7.745
7.727
7.709
7.691
7.673
7.654
7-635
7.616
7.598
Inckgt*
.012
.012
.013
.013
.013
.013
.013
.013
.013
^13
.014
.014
.014
.014
.014
.014
.014
.014
.014
.015
•015
.015
.015
^15
.015
.015
.015
.015
.015
.016
.016
.016
.016
.0x6
.016
.016
.016
.016
.016
.016
.017
45^ longitade.
Imckes.
2.360
2.340
2.321
2.301
2.280
2.260
2.230
2.218
2.197
2.175
2.154
2.132
2.109
2.087
2.064
2.041
2.018
1.99s
I.971
1.948
1.923
1.899
1.874
1.850
1.825
1.800
1.774
1.749
1.723
1.697
1.67 1
1.644
I.618
1.591
1-563
1.536
1509
1. 48 1
^453
1.425
11.396
.028
X>28
.028
.029
.029
.029
.029
.030
x>2P
.030
^30
•031
.031
.031
.031
.032
.032
.032
.032
.033
•033
.033
.033
.034
^34
.034
.034
.035
.035
.035
.036
.036
.036
.036
.036
.036
•037
.037
.037
.037
>loi«itiide.
64S0
as
[6401
6.374
6.346
6.318
6.291
6.262
6.234
6.205
61176
6.146
6.1 16
6.086
6.055
6.024
5-993
5.962
5.930
5JJ98
5.865
5.832
5Ax>
5.767
5-733
5.699
5-665
5.631
5.596
5-561
5.526
5490
5454
5418
5.382
5-345
5.308
5.270
5.233
15.195
Inches.
.049
x>50
.050
.051
.051
.052
.052
.053
.053
-054
-054
.05s
■OSS
.055
•056
.056
.057
.057
J058
.058
•059
.059
.060
X36l
X36l
X36l
.062
.062
1^
.063
.064
.065
.065
!o66
.066
.066
SmTHSONIAN TaBUB.
86
Digitized byLjOOQlC
TABUE19.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iW^inr*
[Derivation of table explained on pp. liii-lvi.]
•3
•S2
30*00'
IS
30
45
31 00
15
30
45
3200
15
30
45
3300
>S
30
45
3400
15
30
45
3500
15
30
45
3600
15
30
45
3700
«5
30
45
3800
15
30
45
3900
»5
30
45
4000
Jncks$.
4-364
8.728
13.092
17457
4-365
8.730
13095
17.460
5-73'
13^7
17462
4.366
8-733
'3099
17.465
4.367
8.734
13.101
17.468
4.368
8.735
i3-'03
17.471
4.368
8.736
13.105
17-473
13.108
«7-477
4.370
8.740
13.110
17.480
4.371
8.741
13.112
17.483
CO-ORDINATES OF DEVELOPED PARALLEL FOR—
15' longitude.
jHchg*.
V^
3-779
3-770
3760
3.750
3-740
3-730
3-720
3-7 >o
3-700
3-690
3-679
3^648
3637
3.626
3.616
3.605
3-594
3.583
3572
3-561
3-550
3-539
3-527
3-5>6
3504
3470
3-458
3.446
3434
3.422
34"
3.386
3-374
3-362
Jmcktt.
.004
.004
.004
.004
.004
-004
.004
.004
JOOA
.004
.004
.004
.004
.004
.004
.004
.004
.004
.004
.004
.004
.004
.004
.005
.005
.005
.005
.005
.005
.005
.005
.005
.005
.005
.00$
xyos
•005
•005
.005
.005
.005
y/ longitude.
IncJus.
7.598
7.578
7.559
7.540
7.520
7.5^0
7480
7.460
7.441
7.420
7.400
7-379
7.359
7.338
7.317
7.296
7.27s
7.253
7.231
7.210
7.188
7.166
7.»44
7.122
7.100
7.077
7.054
7.032
7.009
6.986
6.963
6.939
6.916
6.892
6.869
6.845
6.821
6.797
6^74^
6.724
Juckes.
.017
.017
.017
.017
.017
.017
.017
.017
.0x7
.017
.017
.017
.017
.018
X)i8
.018
.018
.018
.018
.018
.018
.018
.018
.018
.018
.018
.018
.018
x>i8
.018
.018
.018
.019
.019
.019
.019
.019
.019
.019
.019
.019
45' longitude.
I ticket
'.396
1.3^
>.3:
«.309
1.280
1.250
1.22 1
1. 191
i.z6i
1.130
1. 100
f.069
1.038
1.007
0.975
0.943
0.912
0.879
0847
0.815
0782
0749
0.716
a6S3
a65o
0.616
0.582
0.547
0.513
0.479
0.444
0409
0.374
0.339
0.303
0.267
0.232
0.195
0.159
0.123
iao86
Inches.
.037
.037
.038
x>38
.C38
.038
.038
.039
.039
.039
.039
.039
•039
.040
.040
.040
.040
.040
.040
.040
.041
.041
-041
.041
.041
.041
.041
.041
.041
.042
.C42
.042
.042
.042
.042
.042
.042
.042
.042
1^ longitude.
'ncheu
5-»9S
5.156
5.1 18
5-079
5.040
5.001
4.961
4.921
4.881
4.840
4-799
4-758
4.718
4.676
4-633
4.591
4-549
4-506
4-463
4.420
4.376
4332
4.288
4.244
4.200
4.154
4.109
4.063
4.018
3.972
3.925
3.879
3-832
.3-785
3.737
3690
3642
3.594
3-545
3-497
.042 13448
Imche*,
.066
.067
.067
.067
.068
.c68
x)68
.068
.069
.069
.069
.070
X70
J070
.070
.071
.071
.071
.071
.072
.072
.072
.072
.073
.073
.073
.073
.073
.074
.074
.074
.074
.074
.074
.075
.075
.075
.075
.075
.075
.075
Smithsonian Tables.
87
.•igitizedbyVjODgtC
Tablk 19.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE irAlT
[Derivatum of table explained on p, Hia-lvi.]
Meridional dis.
tauces from
even degree
parallels.
CO-ORDINATES OF DEVELOPED PARALLEL FOR- 1
15' longitude.
SC longitude.
45Monjdtude.
lO longitude. II
z
y
z
y
z
y
X
y
/t$cJk4*.
IncJkM.
Inck4s.
JncJus,
Inches,
iMckes.
Imck4».
Inchu.
lnck€U
40O00'
IS
30
45
4^371
8743
13.114
3-362
3-350
3.337
3.325
.005
.005
.005
.005
6.724
^^
6.675
6.650
.019
X>19
.019
/)I9
10.086
iox>49
iaoi2
9-975
X>42
.042
.043
.043
13-448
13.399
13.349
13-300
•075
.070
.076
41 00
30
45
17.486
3.312
3.300
3.287
3-275
.005
.005
.005
.005
6.625
6.600
6.575
6.549
.019
.0x9
.0x9
/>I9
9.937
9-824
-043
.043
.043
.043
13.250
13-200
13149
13.098
.076
.076
.076
•O76
4.372
8.744
131 17
4200
15
45
17.489
3.262
3-249
3.236
3-223
•005
.005
.005
.005
6.524
6.498
6.472
6.447
.019
.019
.019
.0x9
9.786
9-747
9.709
9.670
.043
.043
.043
.043
13.048
12.996
12.045
I2i93
.076
x>76
.076
/176
13.119
4300
15
30
45
17.492
3.210
3.197
3.184
3.170
.005
.005
.005
.005
6.421
^^
6.342
.0x9
.019
.019
.019
9-631
9.592
9-552
9-5«3
.043
.043
.043
.043
12.842
12.789
.076
xyj6
4.374
8.747
13.121
4400
15
30
45
17495
3-158
3-144
3.131
3.1 18
.005
.005
.005
.005
6.316
6.289
6.262
6.235
.019
.019
X)I9
.019
9-473
9.433
9-393
9.353
-043
.043
.043
• .043
12.631
12.578
X 2.524
I247I
.077
.077
.077
.077
4.375
8.749
13.124
4500
15
30
45
17.498
3.104
3-091
3-077
3-063
.005
.005
.005
.005
6.209
6.181
6.154
6.127
.019
.0x9
.019
.019
9-313
9.272
9.231
9.190
.043
.043
.043
.043
12.417
12.363
12.308
12.254
.077
.077
.077
.077
4.375
8.751
13.126
4600
15
30
45
17.501
3.050
3.036
3.022
3.008
.005
.005
.005
.005
6.100
6.072
6.044
6.017
.019
.019
.019
.019
9.150
9.108
9.067
9.025
.043
.043
.043
.043
12.200
I2.X44
12.089
12.033
.077
.077
.077
.077
4.376
8.752
13.128
4700
15
30
45
17.504
2.994
2.980
2.966
2.952
.005
.005
.005
.005
5.989
5.961
5-933
5.904
.019
.019
.019
.019
8.983
us
8.857
.043
.043
.043
.043
n.978
11.922
IX.865
IX.809
.076
.076
.076
.076
4-377
8.754
4800
15
30
45
17.508
2.938
2.924
2.009
2.895
.005
.005
.005
.005
5.876
5.848
5.819
5.790
.019
.019
.019
x)X9
8.814
8.771
8.728
8.686
.043
.043
.043
.043
11.752
11.638
II.581
.076
.076
.076
.076
4.378
8.755
13.133
4900
15
30
45
17.511
2.881
2.866
2.852
2.837
.005
.005
.005
.005
5.762
5.733
5.704
5.675
.019
.019
.019
.0x9
8.643
8.599
8.555
8.512
-043
.043
.043
X)42
IX.524
X 1.465
11.407
11.349
.076
.076
.076
.076
4.378
8.757
13.13s
5000
17.514
2.823
•005
5646
.0x9
8.468
.042
1 1. 291
.076
SlIITMaOIIIA
M Tablci
u
Digitized
bytjOl
L/V
88
Tablk 19.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE mAnnr-
[Derivation of taUe expbined on p. liii-Ivi.]
^-i
Meridional dis-
tances from
parallels.
CO-ORDINAT£S OF DEVELOPED PARALLEL FOR— 1
1$' longitude.
30^ longitude.
45^ longitude.
1° longitude: 1
z
y
z
y
z
y
z
y
ImcJk^s.
iHcJkes.
Inck4U
Jnckgs,
InchMt,
Incfut.
Inches.
Inck4s.
Inches,
SoPbo'
IS
45
2io8
2.793
2.779
.005
.005
.005
.005
S.646
5.616
5-587
5-557
.019
.019
.019
.019
8.468
8.424
8.380
8.336
.042
.042
.042
.042
1 1. 291
IX.232
11.174
XI.II4
.076
.075
.075
.075
5100
'S
30
45
17-517
2.764
2.749
2.734
2.719
•005
.005
•005
•005
5-5*8
5.438
.019
.019
x>i9
.019
8.291
8.247
8.202
8.157
.042
.042
.042
XH2
11.055
10.996
10.036
10.876
.075
.075
.075
.075
13.140
5200
>5
30
45
17.520
2.704
2.689
2.674
2.659
•005
.005
.005
.005
5408
5.378
5-347
5-317
.019
.019
.018
8.II2
8x)67
8.021
7.976
.042
.042
.041
ia8i6
10.756
ia695
10-634
.074
.074
.074
.074
13.142
53 00
15
30
45
17.523
2.643
2.628
2.613
2.597
•005
.005
.005
•005
5.287
5.256
5.225
5195
.018
.018
.018
.018
7.930
7.792
.041
.041
.041
10.573
10.512
10.451
10.389
.074
.074
.073
.073
13-144
5400
'5
30
45
17.526
2.582
2.566
2.551
2.535
.005
.005
.005
.005
5.164
5.133
5.102
5.070
x>i8
.018
.018
.018
7.745
7.699
.041
.041
.041
.041
10.327
10.266
10.203
10.X41
•073
^3
.073
-072
13-147
5500
15
3D
45
17529
2.520
2472
•005
.004
.004
.004
5.039
5.008
4.976
4.945
.0x8
.0x8
.018
.018
7.559
7.512
7.465
7.417
.041
x>40
x>40
.040
iao78
10.016
9.953
9.890
^2
xyj2
.072
x)7i
13.149
5600
15
30
45
17.532
2456
2.441
2.425
2.409
.004
.004
.004
.004
4.913
4.881
4.849
4.817
.018
.018
.018
.0x8
7.370
7.322
7.274
7.226
.040
.040
.040
x>40
9.826
9-763
9.699
9.635
.071
.071
.071
.070
tf4
13-151
57 00
15
3D
45
17.535
2.393
2.377
2.361
2.344
.004
.004
.004
.004
4.785
4-753
4.721
4.689
.018
.017
.017
.017
7.178
7.130
7.082
7.033
.039
.039
.039
.039
9.571
9.507
9.442
9-378
xyjo
.070
.070
.069
13-153
5800
IS
30
45
17.537
2.328
2.296
2.279
.004
.004
.004
.004
4.656
4.624
4591
4.559
.017
.017
.017
.017
6.985
6.838
•039
.039
.038
.038
9.313
9.248
9-183
9.117
.069
.068
4-385
8.770
13-155
5900
IS
30
45
17.540
2.263
2.246
2.230
2.214
.004
.004
.004
.004
4.526
4.493
4.460
4-427
.017
.017
.017
.017
6.789
^^
6.690
6.641
.038
.038
.038
.038
9.052
8.986
8.920
8.S54
.068
.068
4.386
8.772
13-157
6000
17.543
2.197
.004
4-394
.017
6.591
•037
8.788
Xfy,
SMiTMaoNUN Tables.
89
Tablk 19.
CO-ORDINATES FOR PROJECTION OP MAPS. SCALE ifAir
(DerivBtioo of tabk aplaioed o« pp. Un-IrL]
60POO'
30
45
61 00
IS
30
45
6200
>5
30
45
6300
15
30
45
6400
15
30
45
6500
15
30
45
6600
15
30
45
67 00
»5
30
45
6800
»5
30
45
6900
>5
30
45
7000
ImcJktt,
4.386
8.773
13' 59
17.546
4.387
8.774
13-161
17.548
4.388
8.776
13163
»7.55i
8.777
13.165
17.554
m
13.167
17.556
4.390
8.779
13.169
17.559
4390
8.780
I3-I71
17.561
4-39»
8.782
13.172
»7.563
8.783
13.174
17.565
4-392
8.784
13.176
17.568
CO-ORDINATfiS OP DK\fiLOP£D PARALLEL FOR-
ly longitttde.
/t$cJUs.
2.107
2.1S0
2.164
2.147
2.130
2.II4
2.097
2.080
2.063
2.046
2.029
2.012
1-995
1.978
I. 961
1.944
1.926
1.909
10^92
1-875
1.857
1.840
1.823
1.S05
1.788
1.770
1-753
1-735
1.7x7
1.700
1.6S2
1.664
1.647
1.629
1. 61 1
1-593
1-575
1-557
1.540
1.522
1.504
Imckn.
.004
.004
.004
.004
•004
.004
.004
.004
■004
-004
-004
.004
.004
.004
.004
■004
■004
■004
■004
.004
•004
.004
•004
.004
■004
-004
.004
^3
.003
.003
.003
.003
■003
.003
.003
.003
x)03
.003
.003
•003
•003
jy kmgiiode.
Imeket
4.394
4.361
4.327
4-294
4.261
4.227
4.194
4.160
4.126
4-092
4.058
4-024
3990
3956
3-853
3.819
3.784
3-749
3-71 5
3.680
3-645
3.610
3-575
3.540
3-505
3-470
3-435
3400
3.364
3329
3293
3-258
3.222
3.186
3.151
3."5
3-079
3-043
3-007
Jucktt.
J017
.016
.016
woi6
.016
.016
.016
woi6
.016
.016
woi6
•oiS
-015
•015
•015
.015
.015
.015
•OIS
^15
^15
.014
.014
^14
.014
.014
-014
■014
.014
.014
.013
.013
.013
.013
•013
•013
.013
-013
.012
.012
45'
IttcAes,
6.591
6.541
6491
6441
6.391
6.340
d290
6.240
6.189
6.138
6.036
5-985
m
5-831
s-780
5-728
5-676
5.624
5-572
5-468
5415
S-363
5.310
5-258
5.205
5.152
509?
5.046
4.993
4.940
4.886
4.873
4-780
4.726
4.672
4.618
4564
4.510
ImcAm,
•037
•037
•037
•037
•037
.036
.036
.036
•036
WO36
•035
•035
•035
•035
•034
•034
■034
-034
•034
•033
^33
•033
•033
JC32
.032
•032
.032
.031
-031
•031
-031
.030
.030
-030
.029
X>29
.029
J02Q
.028
.028
.028
1^ longitude.
8.788
8.722
8.521
8.320
8.252
8.184
8.117
8/>48
7.980
7.012
7-844
7-775
7.706
7499
7430
7-360
7.290
7.220
7.151
7J0S0
7.010
6.940
6.870
&
6.658
6.586
6.515
6.444
6-373
6.301
6.230
6.158
6.086
6.014
ImcAn.
-067
.066
.066
i066
x)65
-064
^064
^064
-063
X)63
.068
X360
J060
W060
■059
•059
.og
.058
-057
-056
^56
■055
•055
•054
.054
•053
•053
•052
^52
•051
•051
•051
-050
■049
Smithsonian Tjislcs.
90
Digitized byLjOOQlC
Tablk 19.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE thW
[Derivation of uble explained on pp. Uii-lvL]
"8
Mend onal dis
tances from
even decree
parallels.
CO-ORDINATES OF DEVELOPED PARALLEL FOR —
isMongiiude. .
30^ longitude.
45' longitude.
1° longitude.
X
y
X
y
X
y
X
y
Inck4s,
IncAt*.
ImcJUs,
Inckts.
Inekes.
tnchts.
IncJus,
Incfus,
Jnckgt,
45
1.467
1.449
.003
.003
•003
•003
2.971
2.935
2.899
.012
.012
.012
.012
4.510
4456
4402
4.348
.028
.028
.027
.027
6.014
5.942
5-870
5-797
.049
.045
71 00
45
17.570
1431
1413
1-395
1-377
.003
.003
.003
.003
2.862
2.826
2.790
2.753
.012
.012
.Oil
.011
4.294
4.2J9
4.185
4.130
.027
.026
.026
.026
5.725
5.652
5.580
5.507
.047
.047
.046
.046
13-179
7200
IS
30
45
17-572
1.358
i.3to
1.322
1.304
.003
.003
.003
.003
2.717
2.681
2.644
2.607
.Oil
.Oil
.Oil
.Oil
4.075
4.021
3.966
3.911
.025
.025
.025
.024
5434
5-361
5.288
5.215
.045
.045
.044
.044
13.180
7300
IS
30
45
17.573
1.285
1.267
1.249
1.230
.003
.003
.003
.003
2.571
2.534
2.497
2.461
.Oil
.011
.010
.010
3.856
3.801
3.746
3.691
.024
.024
.024
.023
5.142
5.068
4.994
4.921
.043
•043
.042
.041
13-181
7400
15
30
45
17-575
1.2x2
1-193
.003
.003
.002
.002
2424
2.387
2.350
2.313
.010
.010
.010
.010
3.636
3.580
3.525
3.470
.023
.023
.022
.022
4.848
4.774
4.700
4.626
.041
x>40
.040
.039
13-183
7500
17-577
1.138
XX>2
2.276
.010
3414
.022
4.552
.038
15
30
45
13.184
1. 119
I.IOI
1.082
.002
.002
.002
2.239
2.202
2.165
.009
.009
.009
3.358
3303
3-247
.021
.021
.021
4478
4.404
4.329
.038
.037
.037
7600
«5
30
45
17.579
1.064
1.045
1.026
1.008
.002
.002
.002
.002
2.127
2.090
2.053
2X>l6
x)09
,009
.009
.009
3.191
3-135
3.079
3.023
.020
.020
.020
.019
4.255
4.X80
4.106
4.031
.036
.036
•035
•034
4-395
8.790
13.185
7700
15
30
45
17.580
0.989
0.970
0.952
0.933
.002
.002
.002
.002
1.978
1.941
.008
.008
xx)8
.008
2.967
2.QII
2.855
2.799
.019
.010
.0X8
.0X8
3.956
3.882
3.807
3.732
.034
•033
•033
.032
4.395
8.791
13.186
7800
15
30
45
17.582
0.914
0.895
0.877
0.858
.002
.002
.002
XX>2
1.828
1.791
1.753
1.716
.008
.008
2.743
2.686
2.630
2.573
/>i8
.017
.017
.017 '
3.657
3.582
3.506
3-43'
.031
.031
.030
.030
4.396
8.7QI
13.187
7900
IS
30
45
17.583
0.839
0.820
0.801
0.782
.002
.002
.002
XX>2
1.678
1.640
1.603
1.565
.007
.007
.007
.007
2.517
2461
.016
.016
.016
.0x5
3.356
3-281
3.205
3.130
.029
/)28
.028
.027
4.396
8000
17.584
0.764
.002
1.527
.007
2.291
•015
3-054
.026
Smithsonian Tables.
i
91
Tablk 20.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iWinnr
[Derivation ci table explained on ppL liii~ Ivi.]
^
tances from
even decree
parallels.
ABSCISSAS
OF DEVELOPED
PARALLEL.
^W%V%V^V A fWVStfV ^^«B
ORDINATES OF
DEVELOPED
s-
10'
15-
20^
25^
30'
PARALLEL.
longitude.
longitude.
longitude.
longitude.
longitude.
longitude.
lMek4*.
Inck0t.
Imk4*,
Jmckgs.
Inck4s.
Inches.
InclUs.
It
cPoc/
2.922
5.844
8.765
11.687
14.609
14.608
17-531
o«
I**
10
"5.804'
2.922
5-843
8.765
11.687
17-530
j5
20
11.608
2.922
2.922
5.843
5.843
8.765
8.765
II 686
14.608
14.608
17530
17-530
30
17.412
11.686
Imckes.
ImcAet.
40
23.216
2.922
5.843
8.764
11.686
14.608
5'
10
0.000
0.000
50
29.020
2.921
5-843
8.764
11.686
14.607
.000
.000
I 00
2.921
5-843
8.764
11.685
14.606
17.528
15
20
.000
.000
.000
JOOl
10
"V-f4o'
2.921
5-842
8.763
11.684
14.606
17.527
25
30
.000
.001
20
11.608
2.921
5.842
8.763
11.684
14.604
17.525
000
.001
30
17.412
2.921
5.841
8.762
n.683
14.604
17.524
40
23.216
2.920
5.841
8.761
11.682
14.602
17.522
so
291020
2.920
5.840
8.761
11.681
14.601
17.521
200
* "5.804'
2.920
5.840
S-839
8.760
5-759
8.758
11.680
11.678
14.600
14-598
17.520
17.518
2*
3^
10
2.920
20
11.608
2.919
5.83S
11.677
14.596
17.516
5
0.000
0.000
30
17.412
2.919
2.918
5-757
11.676
14-594
17.513
10
.000
.000
40
23.216
5-!37
8.756
11.674
14.592
17.511
15
.001
X)OI
SO
29.020
2.918
5.836
8.755
11.673
14.591
17.509
20
.001
.002
300
2.918
5.836
8.753
11.671
14.589
17.507
25
30
.0Q2
.003
.003
.004
10
Vi^'
2.917
5-835
8.752
11.669
14.586
17.504
20
11.608
2.917
2.916
5.834
5.832
8.750
8.749
11.667
11.665
14.584
17.501
30
1 7413
14-581
17.497
40
23.217
2.916
5^3'
8.747
11.663
14.578
17.494
4*
f
50
400
29.021
2.915
2.915
5.830
11^2?
8.746
8.744
11.661
11.659
14.576
14-574
17.491
17488
5
0.000
0.000
10
"Vioi'
2.914
8.742
11.656
14.570
17484
10
.001
.001
20
11.609
2-913
5.827
8.740
11.654
14.567
17480
15
.001
.002
30
17-413
2-913
5.825
8.738
II. 651
14.564
17.476
20
.002
.003
40
23.217
2.912
5.824
8.736
11.648
14.560
VM
25
.004
.005
50
29.022
2.911
5.823
8.734
11.646
14-557
30
.005
.007
Soo
2.911
5.822
8.732
11.643
H-554
17465
10
"Vi^"
2.910
5-?^S
8.730
11.640
14.550
17-459
6«
7^
20
11.609
2.900
2.908
2.908
2.907
5.818
8.727
11.636
14.546
17455
30
40
so
I7414
23.218
29.022
5.817
5.815
5.813
8.725
8.722
8.720
"•633
11.630
11.627
14.542
14.538
14.534
17450
17.445
17.440
5
10
0,000
.001
0.000
.001
600
2.906
5.812
8.718
11.624
14.530
17.435
15
20
25
30
.002
.002
10
'"i^s
2.905
5.810
8.715
11.620
14.524
17.429
^
■^
20
11.609
2.904
5.808
8.712
11.616
14.520
17.424
17418
.008
.009
30
I74I4
2.903
5.806
8.709
11.612
14515
40
23.219
2.902
5.804
5.802
8.706
8-703
11.608
11.604
14.510
14.506
174»3
50
29.024
2.901
17.407
700
2.000
5.800
8.701
1 1. 601
14.501
17.401
8^
10
"5*805*
2.099
5.798
8.697
11.596
14.496
17.395
20
II.6I0
2.890
5.796
8.694
11.592
14.490
14484
^7-3^7
5
0.000
30
17415
2.897
5794
8.690
8.687
11.587
17.381
10
JOOl
40
23.220
2.896
5-791
11.583
11.578
14.478
17.374
15
.003
so
29.025
2.89s
5.789
8.684
14-473
17.368
20
25
30
.005
x)07
010
800
2-894
5.787
8.680
11.574
14.468
17.361
Smithsonian Tables.
92
Digitized by V^OO^ltT
Table 20*
CO-ORDINATES FOR PROJECTION OP MAPS. SCALE rwifWV'
[Derivation of table explained on pp. liii — IvL]
8°oo'
lO
20
30
40
50
900
10
20
30
40
SO
1000
10
20
30
40
50
11 00
10
20
30
40
50
12 00
10
20
30
40
50
1300
10
20
30
40
50
1400
10
20
30
40
50
1500
10
20
30
40
50
1600
ill.
Inches.
5.80s
II. 610
17.416
23.221
29.026
S&Q6
II.6II
I74I7
23.222
29.028
5.806
II.6I2
17.417
23.223
29.029
5^)6
11.612
17.419
23.225
29^)31
5-807
11.613
17420
23.226
29^33
5.807
1 1. 614
17421
23.228
29-035
5.808
11*615
17422
23.230
29.038
5.808
II. 616
17424
23.232
29.040
ABSCISSAS OF DEVELOPED PARALLEL.
longitude.
Inches.
2.894
2.892
2.891
2.890
2.888
2.887
2.886
2.885
2.883
2.882
2.881
2.879
2.878
2.876
2.875
2.S73
2.872
2.870
2.869
2.867
2.865
2.864
2.862
2.860
2.858
2.857
2|55
2.853
2.851
2.849
2.847
2.846
2.844
2.842
2.840
2.838
2.836
2.834
2.831
2.829
2.827
2.825
2.823
2.821
2.818
2.816
2.814
2.812
2.809
10'
loDgitade.
Inches*
5.787
5.784
5.782
5-779
5-777
5-775
5-772
5-769
5-767
5764
5.761
5-758
5.755
5-752
5-749
5.746
5-743
5.740
5.737
5-734
5730
5.727
5.724
5.720
5.717
5.713
5.709
5.706
5.698
5.695
5-53'
5.687
5.683
5.679
5.675
5.671
5.667
5.654
5.650
5.646
5.641
5637
5.628
5.623
5.619
15-
longitude.
Inches,
8.680
8.677
8.673
8.669
8.666
8.662
8.658
8.654
8.650
8.646
8.642
8.637
8.628
8.624
8.619
8.614
8.610
8.606
8.601
8.596
If
8.580
8.575
8.570
8.564
8.559
8.548
8.542
8.536
8.530
8.524
8.519
8.513
8.507
8.500
8.494
8.488
8.481
8475
8.469
8462
8.448
8.441
8.435
8.428
20^
longitude.
Inches.
.569
-.S64
•559
.554
.549
•544
-539
.522
.516
.511
.504
498
492
.486
.480
474
468
461
.454
.447
•440
434
426
419
.412
.404
•397
■374
.366
■358
350
342
334
326
3'Z
.308
.300
.202
,282
.274
,264
.246
11.237
25'
longitude.
Inches,
14.468
14.461
14455
14.448
14442
14.436
14430
I442A
I44IO
I44IO
14402
14.396
14.388
14.380
\w>
14.358
14.350
14.342
14.334
14.326
14.318
14.309
14.300
14.292
14.282
14.274
14.264
14.256
14.246
14.237
14.228
14.218
14.208
14.198
14.188
14.178
14.168
14.157
14.146
14.136
14.125
I4.II4
14.103
14.092
14.000
14.069
14.058
14.046
30'
longitude.
Inches,
.026
16.856
ORDINATES OF
DEVELOPED
PARALLEL.
8<>
Inches,
0.000
.001
.003
.005
.007
.010
10"
0.000
.001
•009
^13
I2*»
0.000
.002
.004
.007
.Oil
.016
14°
5
10
0.000
.002
15
20
■:^
25
30
.012
.018
16°
aooi
.002
.005
.009
.014
.020
Inches,
aooo
.001
.003
.008
.012
aooo
.002
.006
.010
.014
13"
OJOOO
J0O2
•004
.007
J012
.017
15^
0.001
.002
•005
.009
.013
x>i9
Digitized by V^iJO^lC
Smithsonian Tables.
93
Taslc 20.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ttAvt
[Derivation of table explained on pp. liii-lvi.]
II
lO
20
30
40
50
17 00
10
20
30
40
SO
1800
10
20
30
40
50
1900
10
20
30
40
50
2000
10
20
30
40
50
21 00
10
20
30
40
50
22 00
10
20
30
40
50
2300
10
20
30
40
50
2400
ineJUs.
5-fo9
11.617
17.426
23234
29-043
ii.6i<
17-427
23.236
29.046
5.810
11.619
17.429
23239
29.049
5.810
11.621
X7-43I
23.242
29.052
5.811
11.622
"7433
23244
29^55
5.812
11.623
»7.43S
23-247
29.058
5.812
11.625
17.437
23.250
29002
11.626
17.439
ABSCISSAS OP DEVELOPED PARALLEL.
longitode.
2.809
2.807
2.804
2.802
2.800
2.797
2.795
2.792
2.790
2.787
2.785
2.782
2.777
2.774
2.772
2.769
2.766
2.764
2.761
2.758
2.755
2.752
2.750
2.747
2.743
2.741
2.738
2.735
2.732
2.729
2.726
2.723
2.720
2.717
2.714
2.710
2.707
2.704
2.701
2.697
2.694
2.691
2.688
2.684
2.681
2.677
2.674
2.671
10'
Imhit,
5.619
5.614
5.609
5.604
5-599
5-595
5-590
5.5I5
5.580
5-575
5570
5-564
5-559
5-554
5-549
5-543
5538
5-533
5-527
5.522
5.516
5.510
5-505
5-499
5-493
5-487
5482
5.476
5.470
5-464
5-458
5-452
5-445
5-439
5-433
5-427
5.421
5.414
5.408
5.401
m
5-382
5.362
5-355
5-34»
5-341
15-
longitude.
Inchts.
8428
8.421
8414
8406
8.399
8.392
8.385
8.369
8.362
8.354
8.347
8-339
8.331
8.323
8.315
8.307
8.299
8.291
8.282
11^
8.257
8.249
8.240
8.231
8.222
8.213
8.204
8.196
8.187
8.177
8.168
8.159
8.150
8.141
8.131
8.122
8.112
8.102
8.092
8.083
8.073
8.063
&053
8.042
8.032
8.022
8.012
20^
longitude.
Smithsonian Tables.
1.237
1.228
I.218
1.208
1. 109
1.189
1.180
1. 170
1.159
1.149
i-'39
1. 129
W^
1.007
1.087
1.076
1.065
1.054
1-043
X.032
1. 02 1
1.009
0.998
0.987
0.975
0.963
0.9SI
0.930
0.928
[a9i6
0.003
0.891
0.878
0.866
0.854
a842
0.829
0.816
0.802
0.790
0.777
0.764
0.750
0.737
0.723
0.710
0.696
10.683
94
25'
kmgitttde.
lMch4$.
4.046
4.034
4.022
4.010
3
y.
3-974
3962
3-94?
3-936
3-924
3-9"
3-872
3.859
3-845
3-832
3.818
3804
3-790
3-776
3-762
3-748
3-734
3-719
3-704
3-689
3-674
3-650
3-645
3.629
3-614
3-598
3-568
3-552
3-536
3-520
3-503
3-487
3-47 »
3-45:
3-4:'
3-421
3-404
3-3^7
3-371
13-354
30'
longitode.
Inckr*.
6.582
6-565
6.548
6.531
6.514
6.497
6.480
6.462
6.445
6427
6.409
6.391
6.373
6.318
6.300
6.281
6.262
6.243
6.223
6.204
6.184
6.165
6.145
6.125
6.105
6.085
6064
6.045
16.024
ORDINATES OF
DEVELOPED
PARALLEL.
It
10
»5
20
25
30
i6*»
0.001
.002
.005
•009
X)I4
.020
i8«
aooi
.002
.006
.010
.016
.022
aooi
1^
.011
.017
.025
5
0.001
10
.003
'5
.007
20
.012
2S
.0f8
30
.027
24"
o.ooi
.003
X)07
.013
.020
.028
It
Inches.
OOOI
J0O2
.005
.010
.015
.021
19-
aooi
:^
.010
.016
X)24
0.001
.006
.01 X
.026
23O
aooi
.003
.007
.01 2
.0X0
.028
Digitized by V^OOQlC
CO<^RDINATE8 FOR PROJECTION OF MAPS.
[Derivation of uble explained on pp. Uii-lvi.]
Table 20*
SCALE TltWff-
"3 .
ill.
ABSCISSAS OP DEVELOPED PARALLEL.
S'
longitude.
10'
longitude.
IS'
longitude.
20'
longitude.
*s'
longitude.
3°'
longitude.
ORDINATES OF
DEVELOPED
PARALLEL.
24*^00'
10
20
30
40
50
2500
ID
20
30
40
50
2600
10
20
30
40
SO
27 00
10
20
30
40
50
2800
10
20
30
40
SO
2900
10
20
30
40
SO
3000
10
20
30
40
SO
3« 00
10
20
30
40
50
3200
InchM
5.814
11.628
17-442
23.256
5.815
11.629
17.444
23-259
29.074
5.816
II.63I
17.446
23.262
29.077
5.816
»«-633
17-449
23.265
29.082
5.817
11.634
29.086
5.818
11.636
17.454
23.272
29.090
11^638
17-457
23.276
29.094
5.820
11.640
17460
23.280
29.100
2.671
2.667
2.664
2.660
2.657
2.653
2.650
2.646
2.642
2.639
2.635
2.631
2.628
2.624
2.620
2.616
2.613
2.609
2.605
2.601
2.597
2.593
2.586
2.582
2.578
2.574
2.566
2.562
2.558
2-553
2.549
2.54s
2.541
2-537
2.533
2.528
2.524
2.520
2.5x5
2.51 X
2.507
2.502
2498
2493
2489
2.485
2.480
Imckts.
5-341
5-334
5327
5-320
5-3»3
5306
5-299
5.202
5.2§q
5-278
5.270
5-263
5.256
5.248
5.240
5233
5.22q
5.218
4.960
Inckit.
8.012
8.002
•970
.960
1^
.916
>5
.883
.872
.861
.84Q
.838
.827
.816
.804
792
.780
.768
•757
.745
733
.721
.709
:^s
;^
.648
63s
.622
.610
^f
S85
572
546
533
520
507
494
.480
467
.454
7-441
ImksM.
0.683
0.669
0.655
0.641
0.627
0.613
0.509
0.584
0.570
0.555
0.540
0.526
0.511
0.496
0.481
0.466
0451
0.436
0.421
0.405
0.390
0.3 4
0.358
0.342
0.327
0.3 1 1
0.294
0.278
0.262
0.246
0.130
0.113
0.096
0.078
ao6i
0.044
0.027
aoo9
9.992
9-974
9.956
9-938
9.921
Inckts.
3-354
3-336
3-3'9
3-301
3-284
3.266
3-249
3-231
3.212
3194
3-176
3-157
3-»39
3.120
3.101
3-082
3-063
3045
3.026
3.006
2.987
2.967
2.947
2.928
2!889
2.868
2.848
2.828
2.808
2.788
2.767
2.746
2.725
2.704
2.683
2.662
2.641
2.620
2.598
2.577
2.555
2.534
2.512
2490
2467
2.445
2423
1 2401
inchts.
6.024
6w003
5.982
5.961
5.940
5-9»9
5.898
5-877
5.854
5-833
5.811
5-788
5-767
5-744
5.721
5.676
5-654
5.608
5-584
5.560
5-537
5-5M
5.490
5.466
5-442
5-418
5-394
5-369
5-345
5-320
5-295
5.270
S-245
5.220
5-^95
5.169
5-M3
5.1 18
5.066
5-040
5.014
4.987
4.960
4-934
4.908
14.881
24-
Inches,
aooi
.003
.007
.013
.020
.028
26°
0.001
:^
.013
.021
.030
28<>
.014
.022
.032
30°
s
0,001
10
IS
■1^
20
.015
25
30
.023
-033
320
0.001
.004
.009
.015
.024
-034
250
Inch**,
O.OOX
.003
.007
.013
.020
.029
27°
0.001
.003
.008
.014
.022
•031
29O
0.001
.014
.023
.032
31°
.015
.023
.034
SMmwoNiAii Tables.
I(jlll2^d by •
95
Tablk 20.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE yW^vr
[DwiTmtiott of table expUined oo pp. Hu-Iyl]
il^i
ABSCISSAS OF DEVELOPED
PARALLEL.
*8
/^OT\fWAnrL
E»0 ^W
VIKlilNAlbo vrr
DEVELOPED
^i
jjSii
5'
l</
IS-
20'
25^
30'
PARALLEI.
^
longitude.
loQgUude.
lo&tKodo.
langitnd*-
loositude.
longitode.
Inches.
Jmk0M.
Imek»9,
ImktM.
ImeAet,
InckM,
Imkes.
i-i
32^00'
2.480
4.960
7441
9.921
1 2401
14.881
fi
32^
3^
10
9r\
5.821
11.642
2476
2471
4.951
4.942
7427
7413
Wa
12.379
12.355
14.854
14.827
_Q.S
VJ
2f>
17462
2467
4.933
7400
^
12.333
14.800
ImeJkts.
inekts.
40
23-283
2462
4.924
7.386
9.848
"•3'0
^^772
10
o,noi
0.001
50
29.104
2458
4.915
7-373
9-830
12.288
14.745
.004
.004
33 «)
2448
4.Q06
7-359
9.812
12.265
'^2o7
15
20
.009
.015
.024
.034
^16
10
5^22*
4.896
73*5
9.793
12.241
14.689
25
30
.024
-03s
20
30
11.643
17465
2444
2439
^.87^
7.331
7.316
9.774
9-755
9.736
9.718
12.218
12.194
14.661
14.633
40
50
23.287
29.109
2.434
2429
4.868
4.859
7-302
7.288
12-171
14.605
14.570
12.147
3400
10
" 5-823'
2425
2420
4.850
4.840
7.260
^1
12.124
12.100
14.549
14.520
34<»
35*^
ao
Vi%
2.415
4.830
7.246
9.661
12.076
14491
s
aooi
0.001
30
2410
4.821
7.231
9-642
12.052
14462
10
.004
.004
40
23.291
2.406
4.811
7.217
9.622
12.028
14434
15
-009
^16
SO
29.113
2401
4.802
7.203
9.604
12.004
14.405
20
.016
3500
2396
4.702
4.782
7.188
9.584
11.980
14.376
25
30
.036
-.^3^
10
"Viii'
^•^§i
7.174
9.565
11.956
14.347
'4-318
14.288
20
30
11.647
17471
2.386
2.381
4.773
4.763
7.159
7.144
9545
9.526
11.932
11.858
Ills
40
23.294
29.118
2.377
4.753
7.130
9.506
14.259
36**
37**
50
3600
2.372
2.367
4.743
4.733
7.115
7.099
7.085
9486
9.466
14.230
14.200
5
aooi
aooi
10
"V-sU'
2.362
4.723
9.446
14.170
10
.004
.004
20
11.649
2.357
4.713
7.070
9426
11.783
14.139
15
xxy9
.016
30
17.473
2.351
4703
7.05s
9.406
11.757
14.100
14.078
20
.016
40
23.297
2.346
'^\
7.039
9.386
11.732
25
.036
.026
50
29.122
2.341
7.024
9.366
11.707
14.048
30
.037
37 «>
2.336
4.673
7*000
9.345
11.682
14.018
10
"5.8^6'
2331
4.662
6.978
7 j^j
9325
11.656
13.987
380
39^
20
11.651
2.326
4.652
9304
11.630
13.956
30
17.477
2.321
4.642
6.963
9.284
11.605
40
23.302
2.316
4.631
6.947
9.263
11.579
5
10
0.001
0.001
50
29.128
2.311
4.621
6.932
9.242
11.553
13.864
.004
.004
3800
2.305
4.611
6.916
9.222
11.527
13.832
15
20
.009
.017
.026
.009
.026
10
' "5-8^7*
2.300
4.600
6.900
6.884
6.869
9.200
11.501
13.801
25
30
20
30
11. 6 S3
17480
2.295
2.200
2.284
4.590
4.558
9.179
9.158
11.474
11.448
13.769
» 3.737
.037
.037
40
23.306
6.837
9137
9.116
11.421
13705
13-673
50
29133
2.279
11-395
3900
10
"V-Sis'
11^
4.548
4.537
6.821
6.805
9095
9-073
11.369
11.342
13.642
13.610
40"
20
11.655
17483
2.263
2.258
4.526
6.789
9.052
W'M
13-577
5
0.001
30
4-5' 5
p
9.030
13.545
10
.004
40
23.3'o
2.252
4.504
9.008
8.987
11.261
13.513
15
.009
50
29.138
2.247
4493
6.740
11234
13.480
20
.017
25
.026
4000
2.241
4.483
6.724
8.96s
11.207
13448
30
.038
Smithsonian Tables.
96
Digitized byLjOOQlC
Table 20.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE inAnnr-
[Derivation of table explained on pp. liii-lvi.]
h
Meridional dia-
tances from
even degree
parallels.
ABSCISSAS
OF DEVELOPED
PARALLEL.
ORDINATES OF
DEVELOPED
n
s-
10'
IS'
20'
25'
30'
PARALLEL
longitude.
longitude.
longitude.
longitude.
longitude.
longitude.
JncJUs.
IfU^I,
/meJke*.
IhcA^s.
ImcJUs.
Inc^s.
iMche*.
Js
40*»oo'
2.241
4-483
6.724
8.965
11.207
13.448
400
41°
10
2.236
4472
4.461
6.691
8.943
8.921
f|99
11.179
11.152
134IS
13-382
20
2.230
39
2.225
4.450
6.674
6.658
II. 124
13.349
Inches.
InclUs.
40
23.3H
2.219
4.439
H77
11.097
13.316
5'
10
0.001
0.001
50
29-M3
2.214
4.428
6.641
8.855
11.069
13.283
.004
.004
41 00
2.208
4.417
6.625
6.608
H34
11.042
13.250
XS
20
.017
.009
10
"5-830*
2.203
4.406
8.81 1
11.014
13.217
25
.026
.026
20
30
17.489
2.197
2.192
2.186
2.180
4.383
6.591
6.558
6.541
8.788
8.766
10.985
10.958
"3.183
13.149
30
.038
^38
40
23-319
4-372
4.360
8.744
8.721
10.929
13.1x5
13.081
50
29.149
10.901
42 00
10
"5.831*
2.169
4.349
4.338
6.524
6.507
8.698
8.676
10.873
10.844
13.048
X30I3
42O
43°
20
1 1. 661
2.163
4.326
6.490
8.653
10.816
12.979
S
aooi
aooi
30
17.492
2.157
4.31 5
6472
8.630
10.787
12.945
10
.004
.004
AD
23323
2.152
4.303
t^
8.607
10.759
12.010
12^76
15
.010
.010
y>
29.154
2.146
4.292
8.584
10.730
20
.017
.017
4300
2.140
4.281
6421
8.561
10.702
12.842
25
30
.026
x)38
^38
10
20
30
>749S
2.135
2.129
2.123
4.269
4.257
4.246
6.368
8.538
8.514
10.672
10.643
10.614
12.807
12.772
X 2.737
40
23-327
2.1 17
4.234
6.35X
8.46S
10.585
10.556
10.526
12.701
44^
45^
so
4400
29.159
2.1 1 1
2.105
4.222
4.210
6.333
6.316
8444
8421
12.667
X 2.631
5
aooi
aooi
10
20
'■^"
2.099
2.003
2.087
4.199
4.187
6.298
6.280
8.397
8.373
10.496
ia467
12.596
12.560
10
XS
.004
.010
.004
.010
30
17.498
4.17s
6.262
8.350
10437
12.524
20
.017
.017
40
23-33'
29.164
2X>8l
4.163
6.244
8.326
ia407
12.489
25
.038
:og
so
2.076
4.151
6.227
8.302
10.378
X 2.453
30
4500
10
20
2.070
2.064
2.057
4.139
4.127
4.1 1 5
6.209
6. 1 91
6.172
6.154
8.278
8.254
8.230
8.256
10.348
"^•317
10.288
10.257
I24I7
"Vfiti"
X2.38I
X2!308
460
47^
30
17.501
2.051
4.103
40
23-335
29.169
2.045
4.091
^'36
8.181
10.226
12.272
5
10
aooi
aooi
so
2.039
4.079
6.118
8.157
10.197
12.236
.004
.004
4600
2.033
4.067
6.100
lis
10.166
12.199
XS
20
.010
x>i7
.010
.017
10
"5-835'
2.027
4.054
6.081
10.136
X2.I63
25
30
.027
.027
20
11.670
2.021
4.042
6.063
8.084
10.104
12.125
.03S
.038
30
17.504
2.015
4.030
6.044
8.059
10.074
12.089
40
50
23-339
29.174
2.009
2-003
4.017
4-005
6.026
6.008
8.034
8.010
10.043
10.013
12.052
12.015
480
4700
10
"5.836*
1.996
1.9QO
1.984
3.992
3.980
5989
5-970
7.985
7.960
9.981
9.951
11.978
1 1. 941
20
11.672
3.968
5-95X
7.935
^212
\m
5
aooi
30
17.508
X.978
3-955
5-933
7.910
9.888
10
.004
40
23-344
1.97 1
3-943
iUt
H^l
11.828
«5
.010
SO
29.180
1.965
3930
7.860
9.826
11.791
20
25
.017
.026
4800
1-959
3-9x7
5-876
7.835
9.794
11.752
30
.038
SMiTMSOiiiAN Tables.
97
Taslc 20.
CO-ORDINATES FOR PROJECTION OP MAPS. SCALE ttAtt-
[Derivatkm of table ezplaiaed on pp. Uii-lvi.]
•3
48^00'
10
20
30
40
50
4900
10
20
30
40
50
5000
10
20
3^
40
50
51 00
10
20
30
40
SO
5200
10
20
30
40
50
5300
10
20
30
40
50
5400
10
20
30
40
50
5500
10
20
30
40
50
5600
-IS
Inches,
S-f37
11.674
17-5"
23-34''
29.185
11.676
23352
29.190
5.839
11.678
'7-517
23356
29.194
5-840
11.6S0
17-520
23.360
29.200
5.841
r 1.682
17.523
233^
29.204
5.842
11.6S4
17.526
23.368
29.210
5.843
11.686
17.529
23.372
29.214
17.532
23.376
29.220
ABSCISSAS OF DEVELOPED PARALLEL.
5'
longitude.
10'
longitude.
Inches.
1.882
1.869
1.862
1.856
1.849
1.842
1.836
1.829
1.823
1.816
1.809
1.803
1.796
1.789
1.782
1.776
1.769
1.762
i.'748
1.742
1.728
1.721
1.714
1.707
1.700
1:^
1.680
\^
1.659
1.652
1.645
1.638
Inches,
3-9»7
3.905
3.S92
3.879
3.867
3.854
3.841
3-828
3.815
3.803
3.790
3-777
3.764
3-75^
3-737
3.724
Z'7^^
3.698
3.685
3-^72
3.658
3.645
3-!32
3.618
3.605
3.592
3.578
3.565
3-55'
3.538
3.524
35"
3-497
3.483
3.470
3.456
3.442
3.429
3.415
3.401
3-387
3.373
3.359
3-345
3.33'
3-317
3.303
3.289
3.275
15^
longitude.
Inches,
5-876
5.838
5.819
5.800
5.781
5.762
5-743
5-723
5-704
5.684
5.665
5.646
5.626
5.606
5.587
5.567
5.547
5.528
5-527
5.48S
5.468
5.448
5428
5.408
5.388
5.367
5.347
5.327
5.307
5.287
5.266
5.246
5.225
5.205
5.184
5.164
5.143
5.122
5.101
5.080
5.060
5.039
5.018
4.997
4.976
4-955
4-934
4.9^3
20'
longitude.
longitude.
Inches.
7.835
7.810
7.784
7-759
7.733
7.708
7.682
7.657
7.631
Inches.
9.794
9.762
9.730
9.699
9.667
9.635
9.603
9.571
9.539
30'
longitude.
7.605
9.507
7.579
9474
7.553
9.442
7.527
9.409
7.501
9376
7.475
9-344
7.449
9.3^
7.422
9-278
7.396
9.245
7.370
9.212
7.343
9.179
7.317
9.146
7.290
9."3
7.264
9.080
7.237
9.046
7.210
7.184
t^
7.156
8.946
7.»3o
8.912
8.878
7-103
7.076
8.844
7.049
8.81 1
7.022
8.777
6-9!?4
8.742
6.967
8.708
6.940
8.674
6.912
8.640
6.885
8.606
6.8S7
8-572
6-830
8-537
6.802
8.502
61774
8.468
6.746
8.433
6.719
8-398
6.691
8.364
6.663
8.328
6.63s
8.294
6.607
8.258
6^579
8.224
6.551
8.188
Inches.
11.752
II.714
1 1.677
11.638
11.600
11.562
11.523
11.485
11.440
11.408
11.369
»'.330
11.291
11.251
II. 212
11.134
n.094
11.055
11.015
10.975
10.036
10095
10.855
10.8 16
10.775
10.734
10.694
10.654
10.6(3
'0.573
'0.532
10.491
10.450
10.409
10.368
10.327
ia286
10.244
10.202
10.161
10.120
loxfjZ
10.036
9-994
9-952
9.010
9.826
ORDINATBS OF
DEVELOPED
PARALLEL.
A^
50°
OXX>l
.004
.009
.017
.026
.038
49"
Inches.
Inches.
0.001
aooi
X)04
JQ\Q
.004
JQIO
.026
.038
.017
.026
.038
52"
0.001
.004
.009
.017
.026
.037
0.001
.004
.009
.016
.025
.036
560
0.001
-004
.009
.010
-025
.030
aooi
.004
.009
.026
.037
53"
0.001
.004
.009
x>i6
X)26
•037
55"
aooi
.004
.009
.016
.025
.03
025
030
Digitized by^L^^^vlc
Smithsonian Tables.
98
Table 20*
CO<^RDINATE8 FOR PROJECTION OF MAPS. SCALE Twivww
[DtfiTrntioa of table ocplained on pp. liii-4n.]
1
lO
20
3°
40
SO
5700
10
20
30
40
50
5800
10
20
30
40
SO
5900
10
20
30
40
SO
6000
10
20
30
40
50
61 00
10
20
30
40
SO
6200
10
20
30
40
50
6300
10
20
30
40
50
6400
lit
ImcJUs.
5^45
11.690
^7.535
23-3»o
29.324
5.846
11.692
17.537
23.383
29.229
S.847
11.694
17.540
23.387
29.234
5.848
11.695
17.543
23.39^
29.238
5.849
11.697
17.546
23.394
29-243
5.850
11.699
17.549
233^
29.248
5.850
1 1.701
'7.55«
23.402
29.252
5.851
11.702
17-554
23.405
29256
ABSCISSAS OF DEVELOPED PARALLEL.
.638
631
.624
.616
.603
S8i
1^
559
552
f^
530
509
.501
.494
.487
479
.472
.465
.457
450
.442
1^
.420
413
390
383
360
353
330
322
315
307
300
292
1.284
10'
loDgitiidc.
Imckts.
yvs
3.261
3.247
3.233
3.219
3.204
3.«90
3.176
3.162
3.147
3.133
3. "9
3-I04
3090
3075
3.061
3.046
3.032
3.017
2973
2.959
2.944
2.929
2.914
2.900
2.885
2.870
2.855
2.840
2.825
2.810
2.795
2.781
2.766
2.751
2.736
2.720
2.705
2.690
2.675
2.660
2645
2.630
2.614
2.569
loBgitnde.
Incks*.
4.913
4.892
4.870
4.840
4.828
4.807
4.785
4-764
4.742
4.721
t§l
4.656
4.634
4613
4-591
4.569
4.547
4.526
4.504
4.482
4.460
4.438
4.4x6
4.394
4.372
4.349
4.327
4.305
4.283
4.261
4.238
4.216
4- 103
4.171
4148
4.126
4.103
4.081
4.058
4-035
4.013
3.990
3.967
3.944
3921
3-!99
3.876
3-853
20'
longitude.
Imchn,
6.551
6.522
6.494
6466
6^437
6.409
6.380
6.352
6.323
1^
6.237
6.208
6.179
6.150
6.122
6.092
6.063
6.034
6.005
5.976
5.946
5-913
Smithsonian Tables.
5.858
5.829
5.799
5-770
5.740
5.710
5.681
5.651
5.621
5-5?i
5.561
5-531
5.501
5.47"
5.441
5.410
5.380
S350
5.320
5.290
5259
5228
5.168
5-^7
99
25'
longitude
Inehtt.
8.188
f'53
8.1 18
8.082
8.046
8.011
.976
940
832
96
760
724
688
652
616
579
470
360
HI
249
212
138
xoi
.064
.026
6.988
6.952
6.914
6.877
6839
6.801
^763
6.726
6.688
6.650
6.612
6.536
6.498
6.460
6.422
30;
longitude.
Inchtt,
9.826
9.784
9-741
9.656
9-613
9.571
9.527
9.485
9-442
9-398
9.356
9.313
9.269
9.226
9.182
9.>39
9.095
9.052
0.008
8.963
8.920
8.876
8.831
8.788
8-743
8.699
8.654
8.610
8.566
8.521
8.476
i%
8.342
8.297
8.252
8.207
8.T61
8.1 16
8.071
8.026
7.980
7.934
7*843
7.797
7.751
7.706
ORDTNATES OF
DEVELOPED
PARALLEL.
^•2
10
15
20
25
30
Inckts,
0.001
.004
jo\6
.02 c
.038
58**
0.001
.004
.009
.015
.024
.034
6o«
.004
.008
.015
.023
.033
62«
aooi
.004
.008
.014
.022
.032
64°
aooi
.ooj
.013
.021
.030
0.001
.C04
.009
.016
.024
•03s
59"
aooi
:^
.015
.024
.034
6i«»
.014
.023
.033
63**
aooi
.ooj
.014
.022
.031
)igitized by Lj^^^^vlC
Tablc 20«
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE hAtt
[Derivadoo of table explained on pp. liii-lvL]
I
10
20
30
40
50
6500
10
20
30
40
SO
6600
10
20
30
40
50
6700
10
ao
30
40
SO
6800
10
20
30
40
50
6900
10
20
30
40
50
7000
10
20
30
40
SO
71 00
10
20
30
40
SO
7200
/mcAm.
5-852
IX.704
17.556
23.408
29.260
5.853
11.706
17-558
23.411
29.264
5.854
11.707
17.561
23-4»4
29.268
5.854
11.709
17.563
23.418
29.272
5.855
11.710
17.565
23.420
29.276
5.856
11.712
17.567
23.423
29.279
5.856
".713
17.570
23.426
29.282
5.857
1 1.7 14
17.572
23.429
29.286
ABSCISSAS OF DEVELOPED PARALLEL.
lonptnde.
Inck€t.
284
277
.261
.254
246
238
231
223
215
207
.200
168
161
153
145
129
121
\^
JO^
.000
.082
.074
.066
.058
.050
.042
•034
.026
.018
.010
1.002
.978
.970
.962
.954
.946
.938
•930
.922
.914
.906
10'
loQgitttde.
Inchtt.
2.569
2.553
2.538
2.523
2.507
2.492
2477
2^6l
2.446
2.430
2.415
2.399
2.352
2.337
2.321
2.305
2.290
2.27J
2.258
2.243
2.227
2.21 1
M64
M48
2.1
2.
2.132
2.II6
2.T00
2.084
2.068
2.052
2.037
2.021
2.005
1.989
1.972
1.956
1.940
1.924
I.
i.2$92
1.876
X.860
1.844
1.828
I.811
IS-
longitude.
Inckt*.
3-853
3.830
3-807
3.784
3.761
3.738
3.7x5
3-692
3.668
3.645
3.622
3.599
3-575
3.552
3.529
3.505
3.482
3.458
3-435
3.388
3.364
3-340
3-317
3293
3.269
3.246
3.222
3.198
3.X74
3.151
3.127
3-103
3.079
3.055
3031
3007
2.983
2-959
2.935
2.91 1
2i86
2.862
2.838
2.814
2.790
2.765
2.741
2.717
20'
longitude.
Inches.
5.137
5.106
5.076
5-045
5.014
4.984
4.953
4.022
4.891
4.860
4.820
4.798
4.767
4.736
4.705
4.673
4.642
4.61 1
4.580
4.548
4.517
4.485
4.454
4422
4.391
4.359
4.328
4.296
4.264
4.232
4.201
4.169
4.137
4.105
4.073
4.041
4.009
3-977
3.945
'^.
3.848
3.816
3-784
3-752
3-720
3-687
3-655
3-623
25^
longitude.
Inches,
6.422
6.383
6-345
%^
6.230
6.192
6.153
6.114
6.075
6.037
5.998
5-959
5.842
5-803
5.764
5.646
5.607
5.567
5.528
5.489
5449
5.410
5.370
5.330
5.291
5.251
5.21 1
5.171
5."3i
5.092
5.052
5.012
4.972
4.931
4.891
4.851
4.81 1
4.771
4730
4.690
4.650
4-609
4.569
4.529
longitude.
Inches.
7.706
7.660
'■^
7.522
7.476
7430
7.384
7.337
7.290
7.244
7.198
7.151
7.104
7.057
7.010
6.910
6.869
6.822
6.680
6.634
6.586
6-539
6.491
6.443
6.398
6-349
6.30X
6.253
61205
6.157
6.110
6.062
6.014
5.966
!|^
5.821
5.773
5.676
5.628
5-579
5.531
S.483
5-434
ORDINATES OF
DEVELOPED
PARALLEL.
5-
10
15
20
25
30
Inches.
0.00X
.003
.008
.013
.021
.030
66«>
aooi
.003
.007
.013
.020
X>29
68»
0.001
.003
•007
.012
.019
.027
700
aooi
.003
.006
.011
.017
.024
72°
aooi
.006
.010
.016
.023
6^
Inches.
OXX)X
.003
.007
.013
.020
.029
67O
OJOOl
.003
.007
.012
.010
.028
69^
aooi
Si
JQW
.018
.026
71'
.003
.006
.010
.016
.024
Digitized by Lj^^^V Iv^
Smithsonian Tablks.
100
Tablc 20«
CO-ORDINATES FOR PROJECTION OF MAP8. SCALE iWm-
[Derivation of table explained on pp. liii-lvi.]
72**Oo'
lO
20
30
40
50
73 «>
10
20
39
40
SO
7400
10
20
30
40
50
7500
10
20
30
40
50
7600
10
20
30
40
SO
7700
10
20
30
40
SO
7800
10
20
30
40
50
7900
10
20
30
40
SO
8000
Inches.
5^58
II.716
'7-573
23.431
29.289
5-858
11.717
17-575
23-434
29.292
5-859
11.718
17-577
23-436
29.29s
5.860
11.710
17.578
23.438
29.298
5.860
11.720
17.580
23440
29.300
5.860
11.721
17.582
23.442
29.302
5.861
11.722
17.583
23.444
29-304
5.861
11.723
17.584
23-445
29.306
ABSCISSAS OF DEVELOPED PARALLEL.
5'
longitude.
Iiteket,
-717
longitude.
Inchtt,
.811
795
779
763
746
730
,714
!64^
■632
.616
i
550
534
5«7
-484
.468
•451
•435
418
«402
:^
•352
•335
•3"9
.302
.269
.252
-235
.219
.202
.169
.152
•135
.119
.102
:^
.052
•035
1.018
'5-
longitude.
2.717
\^
2.644
2.620
2.595
2.571
2.546
2.522
2.497
2-47'
2^
2.473
2448
2424
2.399
2.374
2.350
2.325
2.300
2.276
2.251
2.226
2.201
2.177
2.152
2.127
2.102
2.078
2.053
2.028
2.003
1.978
1-953
1.928
1.853
1.828
1.803
1.778
1-753
X.728
1.703
X.678
l!62^
1.602
1.577
1.552
1.527
20'
longitude.
SMmraoNiAN Tablcs.
Inchtt.
3-623
3-590
3-558
3-525
3-493
3-460
3-428
3-395
3-362
3-330
3-297
3264
3232
3-»?9
3.160
3-133
3.100
3067
3-034
3.002
2.968
2-935
2.002
2.870
2.836
2.803
2.770
2.737
2.704
2.671
2.638
2.604
2.571
2.538
2.504
2.471
2.438
2.404
2.371
2.338
2.304
2.270
2.237
2.204
2.170
2.136
2.103
2.070
2.036
lOI
25-
longitude.
Inches,
4.488
4-447
4-407
4-366
4.325
4.285
4.244
4.203
4.162
4.121
4.081
4.040
3-999
3-957
3.016
3-875
3-834
3-793
3752
3-7II
3-628
3587
3-546
3-504
3-463
3.421
3.380
3-339
3297
3-256
3-214
3-172
3-131
3.089
3-047
3-005
2.964
2.922
2.880
2.838
2.797
2.755
2.713
2.671
2.62^
2.587
2.545
30'
longitude.
Inckts,
m
\^
5-239
5.190
5.141
5.092
5.044
4-994
4.945
4.897
4^47
4-798
4.748
4.699
4.650
4.601
4-552
4.502
4.453
4-403
4-354
4.304
4.255
4.205
4.155
4.105
4-056
4.006
3-956
'1
3.806
3-757
3-706
^^
3-606
3-556
3.506
3456
3-406
3-356
3-305
3-255
3-205
3-155
3.104
3-054
ORDINATES OF
DEVELOPED
PARALLEL.
s
0.001
10
.002
15
•005
20
.009
25
.014
30
.020
720
Inches.
0.001
:^
.010
.016
.023
74°
760
0.001
.002
:^
.013
.018
780
0.000
.002
.004
.007
.on
.016
73"
Inches,
0.001
.002
.005
.010
.015
.021
75°
aooi
.002
•005
.009
.013
.019
77'
0.000
.002
.004
.007
.012
.017
79^"
0.000
.002
.00^
.006
.010
.014
Digitized by VjUuQIC
Tablc 21.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE n^rir
[DerivatiuQ of table «xpIaiBed oo pp. liii-hri.]
•8
s5«2-
CO-ORDINATES OF DEVELOPED 1
PARALLEL FOR-
1
IS' lonfcitude.
so' longitude.
45' longitude.
lO longitude. 1
X
y
X
y
X
y
X
y
Inckt*.
Ineius.
inches.
lH€kt*.
Inckss.
Inches.
Inches.
Inches.
Inches.
45
17.176
25.764
8.647
8.646
8.646
8.646
.000
.000
.000
.000
17.293
17-293
17.292
17.291
.000
.001
.001
.001
25940
25-939
25-938
25-937
.000
.001
.001
.002
34.586
34-585
34.584
34-582
.000
JOOl
.003
.004
I 00
15
30
45
34.352
8.645
8.644
8.643
8.642
.000
.000
.000
.001
17.291
17.289
17.287
17.285
.001
.002
.002
.002
25-936
25933
25-930
25.927
.003
.003
.004
-005
34.581
34.577
34.573
34.569
.005
•009
8.58S
17.176
25.764
200
15
30
45
34-352
8.64X
8.640
8.638
8.636
.001
.001
.001
.001
17.283
17.279
17.276
17.273
.003
.003
.003
.004
25-924
25.919
25.914
25.909
.006
.007
34.565
34.559
34552
34.546
JQll
X>I2
.014
.015
8.588
17.176
25765
300
34.353
8.635
.001
17.270
.004
25.904
.009
34.539
.016
15
30
45
8.588
17-177
25.765
8.633
8.630
8.628
.001
.001
.001
17.200
17.256
.004
.005
.005
25.898
.009
.010
.Oil
34-530
34.521
34.512
.018
.019
.oao
400
34-353
8.626
.001
17.251
.005
25.877
.012
34.502
.021
15
3^
45
8.589
25.766
8.623
8.620
8.617
.oot
.001
.002
17.245
17.240
17.234
.006
.006
.006
25.868
25.859
25.850
.012
-013
.014
34.491
34.479
34-467
.023
/>24
.025
500
34-354
8.614
.002
17.228
.007
25.842
.015
34.456
.026
15
30
45
8.589
17.177
25.766
8.610
8.607
8.603
.002
.002
.002
17.221
17.200
.007
25.831
25.820
25.809
.016
.016
.017
34.441
34.427
34.412
.028
X)29
.030
600
34.355
8.600
.002
17.199
.008
25-799
.018
34.398
•031
15
30
45
8.589
17.178
25.767
8.595
.002
.002
.002
17.191
17.182
17.174
.008
.008
.009
25.786
25-773
25.760
.019
.020
.021
34.381
34.364
34.347
■033
■034
•035
700
34356
8.SS3
.002
17.165
.009
25-748
.021
34.330
.037
15
30
45
8.589
251768
8.578
.002
.003
.003
17.155
17.145
17.136
.009
.009
.010
25-733
25.718
25.704
.022
.022
.023
34.310
34.291
34.272
.038
.040
.041
800
34.358
8.563
.003
17.126
.0x0
25.689
.023
34.252
.042
15
30
45
.^:?8^
25.769
8.558
8.552
8.546
.003
.003
.003
17.115
17.104
17.093
.010
.oil
.oil
25-673
25.656
25639
.024
.024
.025
34.230
34.208
34-186
.044
.046
900
15
30
45
34359
8.541
l:fj
8.522
.003
.003
.003
.003
17.082
17.069
17.057
17.045
.012
JQ12
.012
■o'3
25.622
25.604
25-585
25.567
.026
.027
34.163
34-138
34-114
34.089
.047
.048
.050
.051
8.500
17.180
25-771
1000
34.361
8.516
.003
17.032
.013
25.548
.029
34.064
.052
Smithsonian Tables.
Digitized by
102
LjUO^
Tjuux 21 .
CO-ORDINATES FOR PROJECTION OP MAPS. SCALE inW
[Derivation of table explained on pp. liiHvL]
•8^
Meridional di».
tances from
even decree
paraUeU.
CO-ORDINATES OF DEVELOPED PARALLEL FOR -- 1
IS' longitude.
so' longitode.
45' longitude.
lO longitude. 1
X
y
X
y
X
y
X
y
ImJus.
Inches.
Inchtt,
Inches.
Inches,
Inches,
Inches,
Inches.
Inches,
id'bo'
15
30
45
I7.181
25-772
8.516
8.509
8.502
8.496
•003
.003
.003
.003
17.032
17.019
17.005
16.991
.013
.013
.013
.014
25.548
25.528
25.507
25.487
.029
.030
.031
.032
34064
34.037
34.010
33.982
X>52
.054
.056
II 00
15
30
45
34.363
8^89
8481
14^
.004
.004
.004
X)04
16.977
16.962
16.947
16.933
.014
.014
.015
.015
25.466
25.444
25.421
25399
.032
•033
•033
.034
33-955
.057
.058
8.591
17.183
25-774
1200
'5
30
45
34.365
8.459
8.451
8.443
8.434
.004
.004
.004
.004
16.918
16.901
i6i85
16.869
X)I5
.016
.016
.016
25.376
25.352
25.328
25.304
•03s
.036
33-835
33803
33-770
33-738
x6i
•06s
8.502
17.18^
25.776
1300
IS
30
45
34.368
&426
8418
8.409
8.400
.004
.004
.004
.004
16.853
i6.8?8
16.800
.017
.017
.017
.018
25.279
25-253
25.227
25.201
.037
.038
•039
.040
33-706
33.671
33-636
33.601
«66
XfJO
i?:?g]
25.778
1400
15
30
45
34.370
8.391
8.382
5-373
8.363
.004
•005
.005
.005
16.783
16.764
16.720
.018
.018
.019
25.174
25.146
25.118
25.090
.040
.041
.041
.042
33.566
33.528
33-490
33.453
.071
.072
.073
.074
25.780
1500
'5
30
45
34.373
8.354
8.344
8.334
8.324
•005
.005
.005
.005
16.708
16.688
161668
16.647
.019
.019
.019
X>20
25.061
25.031
25.001
24.971
X)42
.043
.044
.045
33-415
33-375
33.335
33-295
.075
.078
•079
3
25.782
1600
15
30
45
34.376
8.314
8.303
8.202
8.282
•005
.005
■005
.005
16.627
16.606
16.564
.020
X>20
.020
.021
24.941
24.909
24.877
24.845
•045
.046
33-255
33.212
33-170
33-127
.080
.081
.082
.083
8.595
17.1Q0
25.784
1700
15
30
45
34.379
8.271
8.260
8.249
8.237
.005
.005
16.542
16.520
16.497
16.475
.021
.021
.021
/>22
24813
24-779
24.746
24.712
.047
.048
.049
.050
33-084
33039
32.994
32.949
.084
8.596
17.191
25.787
1800
15
30
45
34.382
8.226
8.214
8.202
8.190
.006
.006
.006
.006
16.452
16.428
16.404
16.381
.022
.022
.023
.023
24.678
24.642
24.607
24.571
.050
.051
.051
.052
32.904
32.856
32.809
32.761
.090
.091
X>92
8.596
17.193
25.790
1900
15
30
45
34.386
8.178
8.166
8.153
8.141
.006
.006
.006
.006
16.357
16.332
16.307
16.282
.023
.023
.024
.024
24.535
24.498
24.460
24.422
.052
.053
.054
.055
32.714
32.664
32.614
32.563
■093
.094
8.597
17.195
25-792
2000
34.390
8.128
.006
16.257
X)24
24.385
.055
32.513
.097
SniTNaoNiAN Tables.
e
103
Table 21 .
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iWnr-
(Derivaiioii of taUe oyptoinrd on pp. liU-lvi]
•«
I
15
45
21 00
15
30
45
15
30
45
2300
15
30
45
2400
»5
30
45
2500
15
30
45
2600
15
30
45
27 00
15
30
45
2800
IS
30
45
2900
15
30
45
3000
Inckts.
8.598
X7-I97
25-795
34.394
8.599
17.199
25-798
34.398
8.600
17.201
25.801
34.402
8.632
17.203
25.804
34406
8.603
17.20^
25.808
34.410
8.604
17.207
25.811
34.4x5
8.605
17.210
25.814
34.419
8.606
17.212
25.818
34.424
8.607
17.215
25.822
34430
8.609
17.217
25.826
34435
CO-ORDINATES OF DEVELOPED PARALLEL FOR —
ly longitude.
Inclut.
8.128
8.115
8.102
8.089
8.076
8.062
8.048
8.035
8.021
8.006
.992
.978
.963
.948
.933
.918
904
.888,
872
.857
.841
.825
.809
•793
.776
.760
.743
.726
.709
,692
,675
.657
.640
.622
550
•531
7.494
Ineke*.
.006
.006
.006
.006
.006
.006
.006
•007
.007
.007
joorj
,cxyj
.007
.007
.007
.007
.007
.007
.007
.007
.007
.007
.007
.007
^
.008
.008
.008
.008
.008
.008
.008
.008
.008
.008
.008
.008
.008
.008
3c/ longitude.
Inches.
16.257
16.230
16.204
16.178
16.152
i6lI24
16^)69
16.042
16.013
15.984
X5-955
15.927
15.897
15.867
15-837
15.807
15.776
15.745
15-713
15.682
15.650
15.617
15-585
15-553
15.519
15.486
X5452
15.419
384
350
315
280
X73
137
100
5.065
.026
989
InckMt.
.024
.024
.025
.025
.025
•02c
.020
.026
.026
.026
.027
.027
.027
.027
.028
X>28
^28
.028
.029
.029
^29
.029
.029
.030
.030
.030
.030
.030
.031
.031
.031
.031
.031
.031
.032
.032
.032
.032
.032
.033
.033
45' longitude.
Inckts.
24-385
24.340
24.306
24.267
24.227
24.186
24.145
24.104
24.062
24.019
23976
23.933
23.890
23.845
23.800
23.756
23.711
23.664
23.617
23.570
23.524
23-475
23.426
23.378
23.329
23279
23.229
23.179
23.128
23.076
23.024
22.972
22.920
22.866
22.813
22.759
22.705
22.650
22.594
22.539
22.483
Inches.
.055
•050
.056
-057
-057
.058
.058
.059
.059
.060
.060
.061
.061
X)62
.062
.063
.063
.064
.064
.065
.065
.065
.066
x)67
.068
.069
.069
.070
.070
.071
.071
.072
.072
.072
-073
.073
.074
i^ longitude.
Inches.
32.513
32461
32408
32.356
32.303
32.248
32.193
32.138
32.083
32.026
31.968
31.911
31-853
3X-794
31.734
3X-674
31.614
3X-552
31.489
3M27
3x365
3x300
3X.235
31.170
31.106
3X.039
30.972
30.905
30.838
30.769
30.699
30.630
30.560
30.489
30.4x7
30.345
30.274
30.200
30.125
30.05X
29.978
Inches.
.099
.100
.101
.102
.103
.104
.105
.106
:;s
.109
.109
.110
.III
.112
•1x3
.114
.115
.116
.117
■\\l
.119
.120
.121
.121
.122
-X23
.124
.124
.125
.126
.127
.127
.128
.129
.130
.130
X3I
Smithsonian Tabucs.
104
Digitized by V^OO^K^
Table 21.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE inVlT-
[Derivation of table explained on pp. liii-lvi.]
•8
Meridional dis-
tances from
even degree
parallel^
CO-ORDINATES OF DEVELOPED PARALLEL FOR- 1
isMonsitude.
so' longitude.
45' longitude.
lO longitude. 1
X
y
X
y
X
y
X
y
Inchet.
Inchet.
Incfu$.
titCmtt,
Imhes,
Inelu*,
Inehis.
Intlut.
Incfus.
3<A»'
15
45
8.610
17.220
25^30
7.494
7.47s
7.450
7437
.008
.008
.008
.008
14.989
14.951
14.913
14.874
•033
.033
.033
•033
22.483
22.426
22.369
22.312
.074
.074
.074
.075
29.978
29.QO2
29.825
29.749
.>3«
•X3X
.132
.133
3100
IS
30
45
34-440
7.398
7.379
7.359
.008
.008
.008
.008
14.836
14.797
14.758
14.7X8
•033
•033
.034
.034
22.254
22.195
22.137
22.078
.075
.076
29.672
29-594
29.515
29-437
•X33
.134
•X35
.135
8.61 1
17.213
25.834
3200
15
30
45
34-446
7.340
7.319
7.299
7.279
.008
.008
.009
.009
14.679
14.630
14.5^
14.558
.034
.034
.034
.034
22.019
21.837
.076
.077
.077
.077
29.358
29.278
29.197
29.116
.136
.136
.137
'^Z1
8.613
25-83^
3300
«S
30
45
34.451
7.259
7.238
7.217
7.»97
xx)9
.009
.009
.009
14.518
14.476
M.43S
M.393
.034
•035
.035
.035
21.777
21.714
21.652
21.590
.078
.078
.078
.078
29.036
28.786
.138
.138
.X39
.139
8.614
17.228
25.842
3400
15
30
45
34.456
7.176
7.154
7.112
.009
.009
.009
.009
X4.352
14.309
14.266
14.224
.035
•035
.035
•035
21.527
21.464
21.400
21.336
.079
.079
28.703
28.618
^•533
28.44^
.140
.141
.141
.142
8.615
17.231
25.846
3500
15
3°
45
34.462
7.091
7.069
7.047
7.025
.009
x)09
.009
.009
14.181
14.138
14.094
14.050
•035
.036
.036
.036
21.272
21.207
21. 141
21.076
.080
.oSo
.080
.080
28.362
28!?8l
28.101
.142
.142
.143
.H3
8.617
17.234
25.851
3600
15
30
45
34.468
7.003
6.981
.009
.009
.009
.009
14.007
13.962
13.917
13.873
.036
.036
.036
21.010
20.876
2a8o9
.081
.081
.oSi
.081
28.014
27.924
27-835
27.745
.144
.144
.144
.145
8.618
17.237
25.855
3700
15
30
45
34.474
6.914
6.845
.009
.009
.009
.009
13.828
13.782
13.736
13.690
.036
.036
.036
.037
20.742
20.673
20.604
20.536
.082
.082
.082
.082
27.655
27.564
27.472
27.38X
.145
■'\
.146
8.620
3800
15
30
45
34.480
6.822
6.799
6.775
6.752
.009
.009
.009
.009
X3.645
13.598
X3.504
.037
.037
.037
.037
20.467
20.397
2a326
20.256
.082
.083
.083
.083
27.289
27.196
27.102
27.008
.147
.147
.147
•M7
8.621
3900
15
30
45
34.485
6.729
6.705
6.68?
6.657
.009
•009
.009
.009
'3-457
13.409
I3-36X
X3.3X4
.037
.037
.037
.037
20.t86
2a 1 14
2ao42
19.970
.083
.083
.083
.084
26.914
26.819
.148
.148
8.623
4000
34.491
6^633
•009
13.266
-037
19.899
.084
26.532
.149
8mith«onian Tables.
loS
Tabu 21.
CO-ORDINATES FOR PROJECTION OF MAP8. SCALE rwifwv
[Derivalion of table nplAiiied on pp. liii-M.]
40*W
15
30
45
4100
15
30
45
4300
«5
30
45
4300
IS
30
45
4400
15
30
45
45 00
15
30
45
4600
IS
30
45
4700
15
30
45
4800
IS
30
45
4900
15
30
45
5000
ImcJUt.
8.624
17.249
25-873
34.497
8.625
17.250
25.875
34.500
8.627
34.510
8.629
34.515
8.630
17.201
25.891
34.522
34-528
8.633
17.267
25.901
34.534
8.635
17.270
25-905
34.540
8.637
17273
25.910
34.546
8.638
17.276
25.914
34.552
C(M)RD1NAT£S OP DEVELOPED PARALLEL FOR—
15' longitude.
Inehea*
6.e
6.584
6.560
6.535
6.510
6.485
61460
6435
6.410
6^385
6.359
6.334
6.308
6.282
6.256
61230
6.203
6.177
6.151
61I24
6.097
6.071
6.044
6.017
5-990
5.962
5-935
5.908
5.880
5.852
5.824
5796
S-768
5.740
5.712
5.684
r.6^1
5.598
5.569
.009
.009
.009
.009
.009
.009
.009
.009
•009
.009
.009
-009
.009
.009
.009
•009
.009
xx)9
.009
.009
JQO^
.009
•009
•009
.009
.009
•009
.009
.009
.009
.009
.009
.009
.009
.009
.009
•009
.009
.009
.009
.009
3</ loniptade.
3.266
3."9
3-070
3.020
2.970
2.920
2.871
2.820
2.760
2.718
2.667
2.615
2.563
2.512
2.460
2^407
2.354
2.301
2.249
2.195
2.141
2.088
2.034
1.979
1.025
1.870
1.815
1.760
1.704
1.648
1-593
M24
1-367
1.310
1-253
1.195
11.138
.037
.037
.037
.037
037
.037
.037
.037
.037
.037
.038
-038
.038
.038
.038
-038
.038
.038
.038
.038
.038
.038
^38
x>38
.038
.038
.038
.038
.038
.038
.038
.038
.037
.037
.037
.037
.037
.037
45^ longitude.
9-899
9.825
9-752
9.679
9-605
9.530
9.456
9-381
9.306
9.230
9-154
9.077
9.001
8.689
8.610
8.531
8452
8.373
8.292
8.212
8.131
8.051
7.805
7.723
7.640
7556
7.473
7-389
7-305
7.220
7-135
7.051
6.065
6.879
6.793
16.707
Inches,
.084
.084
.084
.084
.084
.084
.084
.085
.085
.085
.085
.085
.085
.085
.085
.085
.085
.085
.085
^^
.085
.085
^^
.085
.085
.085
.085
.085
.085
.085
.085
/)84
.084
.084
.084
.084
.084
.084
I® longitude.
25.041
5.841
fmcJUt.
26.532
26.434
26.336
26.238
26.140
26.041
25-j
2SI
25-741
25.640
25.538
25.436
25-335
25-231
25.127
25.023
24.919
24.8IA
24.708
24.603
24497
24.390
24.283
24.175
24.068
23.959
23.849
23.740
23.631
23.520
23.408
23.297
23.186
23*073
22.960
22^47
22.734
22.620
22.505
22.391
22.276
Inekn.
.149
.149
.149
•149
•ISO
.150
.150
•150
.150
.150
.151
.151
.151
•151
.151
.151
.151
.151
.151
.151
.151
.151
.151
.151
.151
.151
.151
.151
•151
.151
.151
.151
.150
.150
.150
.150
.150
.150
.150
.150
.150
Digitized by V^OO^ltT
Smithsonian Tables.
106
TABLK21.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE rvMv
[Derivmdon of table explained on pp. liii-lvL]
15
30
45
5100
IS
30
45
5200
15
30
45
53 00
IS
30
45
54 00
IS
30
45
55 00
IS
30
45
5600
15
30
45
5700
15
30
45
5800
15
30
45
5900
15
30
45
6000
Ineke*.
8.640
17.279
25.919
J4.S58
8.641
17.282
25.924
34.565
8.643
17.285
25.928
34.571
8.644
17.288
25-932
34.576
8.646
17.291
25.937
34.582
8.647
17.294
25.941
34.58s
8.648
17.297
25.946
34.594
8.650
17-300
25.950
34.600
8.651
17.303
25.954
34.605
8.653
17.305
25.958
34.611
CO-ORDINATES OF DEVELOPED PARALLEL FOR —
is' longitude.
5-569
5.540
5.511
5.482
5453
5423
5.394
5.364
5-334
5-305
5.275
5-245
5.215
5.185
5.154
5.124
5.094
5.063
5.032
5.OP2
4.971
4.940
t^
4.846
4.815
4.784
4.752
4.720
4.689
4.657
4.625
4.593
4.561
4.529
4.497
4464
4432
4.399
4.367
4.334
Inches,
.009
.009
•009
.009
.009
•009
•009
•009
.009
-009
ux)9
•009
.009
.009
.009
•009
ux)9
ux)9
.009
•009
.009
ux)9
.009
xx>9
.009
.009
.009
.009
.009
.009
.009
.009
•009
.008
.008
.008
.008
.008
.008
.008
.008
yif longitude.
Imckes.
I.138
1.080
1.022
0.963
0.905
0.846
0.787
a728
0.669
a6o9
0.549
0490
0.430
0.369
0.300
a248
0.187
0.126
ao64
aoo3
9.942
9.879
9.817
9.755
9.693
9.630
9-567
9.504
9.441
9.377
93M
9.250
9.186
9.122
^•993
8.929
8.864
8.799
8.734
8.669
Ittches.
.037
•037
.037
.037
.037
•037
.037
.037
-037
.036
.036
.036
.036
.036
.036
.036
.036
.036
•036
^36
.036
.035
•035
.035
.035
•035
.035
.035
.035
.035
.034
.034
.034
.034
.034
•034
.033
.033
•033
-033
.033
45' longitude.
'ttckts.
6.620
6.532
6.445
6^358
6.269
6.181
6.092
6.004
5.01-
5.82
U
24
5-734
5-645
5.554
5.463
5.372
5.281
5.189
5-097
5.004
4.912
4.819
4.726
4.633
4.539
4445
4.3SJ
4.256
4.162
4.066
3-970
3-875
3-779
3-490
3.393
3.296
3.198
3.100
13.003
Incket.
.084
.084
.084
.083
-083
.083
.083
.083
.083
.082
.082
.082
.082
.o8[
X)8i
.081
.081
.080
.080
.080
.080
.079
.079
.079
X>7Q
.078
.078
.078
.077
■077
.077
.076
.076
.076
.075
•075
.075
-075
■074
.074
i^ longitude.
Imket,
22.276
22.160
22.043
21.927
21. 8X0
21.692
21.574
21.456
21.338
21.218
21.099
20.979
20.860
20.738
20.617
2a496
2a374
20.252
2ai29
20.006
19.883
19-759
19-634
19.510
19.386
19.260
19-134
19.008
18.882
18.754
18.627
18.500
18.372
18.244
18.115
17.986
17.858
17.728
»7.597
17.467
17-337
Imkts.
.150
.149
.149
.149
.148
.148
.148
•147
•147
.146
.146
.145
•145
.145
.144
.144
.144
.143
.143
.142
.142
.141
.141
.140
.140
.140
.139
.139
.138
.^37
-137
-136
•'35
-135
.134
.134
•133
•133
•>32
•131
SMITNaOfllAN TaBLCS.
107
Tabuc 21.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ii^rir
[Deri^tion of Cable explained on pp. liii-hri.]
I
3^
CO-ORDINATES OF DEVELOPED PARALLEL FOR-
iS' longitude.
i</ loQgitude.
4S' longftude.
6o«o</
IS
30
45
61 00
IS
30
45
6200
'5
30
45
6300
15
30
45
6400
15
30
45
6500
15
30
45
6600
IS
30
45
6700
IS
30
45
68 00
IS
30
45
6900
IS
30
45
7000
8,6w
17.308
25.962
34.616
8.655
25.966
34.621
8.657
i7-3«3
25.970
34.626
8.658
17.316
25.974
34.632
8.659
17.318
25-977
34636
8.660
17.321
25.981
34.641
8.661
17.323
25.984
34.646
8.663
34.650
8.664
17-327
25.991
34.655
8.665
17-329
25-994
34.659
l^
ItuJUs.
4.334
4.30'
4.269
4.236
4.203
4.170
4.136
4.103
4.070
4.036
4.003
3970
3-936
\^
3-835
3.801
3767
3664
3630
3-596
3-561
3-527
3.492
3-458
3-423
3.388
3-353
3.318
3-283
3.248
3-213
3.178
3.143
3.108
3-072
3.037
3.002
2.966
JncktM.
.008
.008
.008
jocA
JOC&
.008
.008
jocA
.008
xxA
.008
.008
.008
.007
.007
.007
.007
.007
.007
.007
.007
.007
JOO7
.007
.007
.007
.007
.007
.007
.006
.006
.006
.006
.006
.006
•O06
Inckts.
8.669
8.603
8.537
8.471
8.406
8.339
8.273
8.207
8.140
IS?
7-939
7.872
7.804
V^
7.602
7.533
7.465
7.397
7-329
7.260
7.191
7-123
7.054
6.984
6J46
6.776
6.706
1%
6.497
6427
l^
6l2i6
6. MS
6.074
6.003
5-932
Jneket.
.033
.032
.032
.032
.032
.032
.032
.031
.031
.031
.031
.031
.031
.030
.030
.030
.030
.029
X)29
.029
.029
.028
J07&
.028
.028
.028
.027
.027
.027
.027
.026
.026
.026
.026
.025
.025
.025
.025
.024
.024
.024
JmcUs.
3-003
2.(
2.707
2.608
2.509
2.410
2.310
2.210
2.1 10
2.009
1.909
1.808
1.707
1.605
1.504
1.402
1.300
1.198
1.096
0.993
a684
a58i
0477
0.373
a269
0.165
ao6o
9-955
9.850
9.746
9.640
9-535
9-429
9-323
9.217
9.111
9-005
8.899
InektM.
•074
.074
.073
-073
.072
.072
.072
.071
.071
.071
.070
XfJO
-069
.068
.067
.067
.066
.066
.065
.065
.064
.064
.063
.062
X)62
.061
.061
.060
x)6o
.059
.050
.058
.058
.057
.057
.056
.056
.055
Jncku.
17.337
17.206
17.074
16.943
id8ii
\^
16413
16.280
16.146
16.012
15.878
15-744
15.609
15-474
15-338
15.203
15.067
14-930
14.794
14.658
14.520
14.383
14.245
14.108
13-969
13-830
13.692
13.553
13-413
13-273
13-134
12.994
12.854
12.713
12.572
12.431
12.290
12.148
12.006
11.865
.131
.131
.130
.129
.128
.128
.126
.125
.125
.124
.123
.122
.122
.121
.120
.119
.110
.118
.117
.116
.lis
.114
.113
.112
.III
.III
.110
.109
.108
.106
.105
.104
.103
•102
.101
.100
.097
gitizedby^OO g=
8iirTN«ONiAN Tables.
108
Tablk 21 .
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iWttt-
[Derivation of Uble explained on pp. liii-lvi.]
1^
7cA)o'
15
30
45
71 00
'5
30
45
7200
IS
30
45
7300
15
30
45
7400
15
30
45
7500
15
30
45
7600
15
30
45
7700
15
30
45
7800
15
30
45
7900
IS
30
45
8000
I ill
iHCMtS.
8.666
17-331
25997
34-663
8.667
17.333
26.000
34.667
8.668
17.33s
26.003
34-670
8.668
26.006
34.674
8.669
»7.
34.677
8.670
'7.340
26.010
34.680
8.671
17.342
26.013
34.684
8.672
17.343
26.015
34.686
8.672
17.344
26.017
34.689
8.673
17.340
26.018
34-691
CO-ORDINATES OF DEVELOPED PARALLEL FOR-
15' longitude.
lucktt.
2.966
2.030
2.89s
2.859
2.824
2.788
2.752
2.716
2.680
2.644
2.608
2.572
2.536
2.500
2463
2.427
2.391
2.3S4
2.318
2.281
2.245
2.208
2.172
2.135
2.098
2.062
2.025
1.988
1.951
I.0I4
1.877
1.840
1.804
1.766
1.729
1.692
1.655
1.618
1.581
1.544
1.506
Incki*.
.006
.006
.006
.006
.006
.006
.006
.006
.006
.006
.005
•005
x»5
•005
.005
•005
.005
.005
.005
•005
.005
.004
.004
.004
.004
,004
.004
.004
.004
.004
.004
.004
.004
.004
.004
.004
.004
.003
.003
.003
.003
30^ longitude.
Inchti.
5-790
5.718
5.647
5.576
5-504
5-432
5360
5.288
5.216
5-144
5.072
4.999
4.927
i854
4.782
4.636
4.563
4490
4.417
4-343
4.270
4.197
4123
4.050
3.976
3.903
3.829
3-5
3.^
3.607
3-533
%
3.310
3.236
3.162
3.087
3.013
^?^?
3.459
3.38!
Inchts.
.024
X>24
.023
.023
.023
.023
.022
.022
.022
.022
.021
.021
.021
.021
.020
.020
.020
.020
.019
.019
.019
.010
.018
.018
.018
.018
.017
.017
.017
.017
.016
.016
.015
.015
.015
.014
.014
.014
.013
.013
.013
45' longitude.
Indus.
8.899
8.7Q2
8.685
8.578
8.471
8.361
8.256
8.148
8.040
7.932
7.824
7.716
7.608
7.499
7.390
7.281
7.172
7.063
6.054
6.844
6^735
6.625
6.515
6.405
6.296
6.185
6.075
5.964
5-854
5-743
5632
5.522
5.41 1
5-322
5.188
5.077
4.966
4.854
4.742
4.631
4.519
Jncktt.
.055
.055
.054
-053
.052
.052
.051
.051
.050
.050
.049
.049
.048
.048
.047
.046
.045
.044
•044
.043
.043
.042
.042
.041
.040
.040
.039
.0^8
.037
.036
.036
•035
.034
.034
•033
.032
.031
.030
.030
.029
i<> longitude.
Jnekts,
11.865
11.722
11.580
"•437
11.294
II. 151
11.008
10.864
10.720
10.576
10.144
9.098
9.854
9.708
9563
9-417
9.272
9.126
8.980
8.834
8.687
8.540
8.394
8.247
8.100
7.952
7.805
7.658
7.510
7.362
7.214
7.066
6.918
6.769
6.621
6.472
6.323
6.174
6.026
Inches,
.096
•095
.094
•093
.092
.091
.090
.089
.088
.087
.086
.085
.084
.083
.081
.080
.070
.078
-077
.076
.074
.073
.072
.071
.067
.066
.064
.063
.062
.060
-057
-055
.054
•053
.052
SMiTHaoNiAN Tables.
109
Tablk 22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE iiItt-
[DerivatioD of tablo explained on pp. liii-lvL]
"3
lo
20
30
40
50
I 00
10
20
30
40
50
10
20
30
40
SO
300
10
20
30
40
SO
400
10
20
30
40
SO
500
10
20
30
40
SO
600
10
20
30
40
SO
7 00
Imcku.
1 1.45 1
22.901
34.3S2
45-803
S7.2S4
68.704
11.451
22.901
34.352
45-803
57-254
68.704
11.451
22.902
34.353
45.804
57-2S4
68.705
11.451
22.902
34.353
45.804
57.255
68.706
11.451
22.903
34.354
45-805
57.256
68.708
11.452
22.903
57.258
68.710
U.452
22.904
45^08
57.260
68.712
ABSCISSAS OF DEVELOPED PARALLEL.
S'
looj^tude.
Inchts.
5764
5-764
5764
5764
5.764
5-764
5.764
5-763
5-763
5.762
5.762
5.761
5.761
5.760
5-759
5.759
5.758
5.757
5-756
5756
5754
5753
5752
575"
5-750
5-749
5748
5.746
5745
5-744
5-743
5-741
5-739
5-738
5-736
5-735
5-733
573"
5729
5-727
5-726
5-724
5.722
10'
longitude.
Inches.
11.529
5^
528
528
527
527
526
525
524
524
523
522
520
5i9
516
5'4
5"3
5"
509
507
505
503
SOI
498
496
493
490
488
485
482
479
1.476
1.472
1.469
1.466
1.462
1.458
"•455
1. 451
1.447
".443
"5'
longitude.
Inches.
17.293
7.293
7.292
7.292
7.291
7.291
7.291
7^287
7.285
7.284
7.283
7.281
7.278
7.276
7.274
7.272
7.270
7.267
7.264
7.260
7-257
7.254
7.251
7.247
7.243
7.240
7.236
7.232
7.228
7.22J
7.218
7-213
7.209
7.204
7.199
7.182
7.171
17.165
20'
longitude.
Inches.
23.058
23057
23.056
23.056
23-055
23-054
23.054
23.052
23.050
23.049
23.047
23045
23-044
23.041
23.038
23.035
23.032
23.029
23.026
23.022
23.018
23.014
23.010
23.006
23.002
22.996
22.991
22.9S6
22.981
22.976
22.970
22.964
22.958
22.951
22.945
22.938
22.932
22.924
22.917
22.910
22.902
22.894
22.887
25'
longitude.
Inches.
28.822
28.821
28.821
28.820
28.819
28.818
28.818
28.816
28.813
28.811
28.809
28.807
28.805
28.801
28.797
28.794
28.783
28.778
28.773
28.767
28.762
28.757
28.752
28.746
28.739
28.733
28.726
28.720
28.713
28.705
28.697
28.689
28.681
28.673
28.665
28.656
28.646
28.637
28.628
28.618
28.609
30'
longitude.
Inches.
34.586
34.585
34.585
34.583
34.583
34.582
34.581
34.579
34.576
34.573
34-571
34.568
34.565
34-561
34.556
34.552
34.548
34.543
34.539
34.533
34.527
34.520
34 5"4
34.508
34-502
34.495
34.487
34.479
34.471
34.463
34.456
34.446
34.436
34.427
34.417
34.408
34.398
34.387
34.375
34-364
34.353
34-342
34.330
ORDINATES OF
DEVELOPED
PARALLEL.
5'
10
"5
20
25
30
Inches.
0.000
.000
.000
.000
.000
.000
aooo
.001
.001
.002
.004
•005
0.000
.001
.003
.005
.007
.oil
e"
0.000
.002
.004
.007
.oil
.016
Inches.
aooo
.000
.001
X>01
.002
•003
0.000
.001
.002
•003
.005
0.000
.001
.009
•013
aooo
.002
.013
.018
Smithsonian Tables.
no
Digitized by V^OOQ IC
Table 22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE vTifir-
[Derivation of table explained on pp. liii-lvi.]
n
.S.-SjS
ABSCISSAS OF DEVELOPED PARALLEL.
5'
longitode.
longitude.
longitude.
longitude.
longitude.
30'
longitude.
ORDINATES OF
DEVELOPED
PARALLEL.
7«bo'
10
20
30
40
50
800
10
20
30
40
9>
900
10
90
30
40
SO
xooo
10
20
30
40
50
10
20
30
40
so
10
20
30
40
SO
1300
10
20
30
40
SO
1400
Jtukts.
68.712
11.452
22.901c
34.358
45.810
57.262
68.715
"•453
22.906
34.359
45.812
57-265
68.718
11.454
22.907
33-361
V^
68.722
U.454
22.909
34.263
45.817
57.272
68.726
11.455
22.910
57-275
68.730
11.456
22.912
34.367
45-823
57-279
68.735
IM57
22.913
34-370
45.827
57.284
68.740
Incktt,
5.722
5.720
5-717
5-715
5.713
5.7 U
5.709
5.706
5.704
5.701
5.696
5-694
5.686
5.680
s-677
5-674
5.671
5.668
5.665
5.662
5.659
5.656
5-652
5.646
5.642
5-639
5.636
5*^32
5.628
5-625
5.621
5.618
5.614
5.610
5.606
5.602
5-598
5-594
IncluM,
"443
IM39
"-435
U.430
11.426
11.422
n.417
11.412
11.407
1 1. 40 J
11.3^
"•393
11.388
11.382
".377
".371
".366
11.360
"-355
"349
"343
"-33'
11.324
11.318
11.312
ii.3oq
11.298
IT.2Q2
11.285
11.278
11.264
11.257
11.250
11.242
"-235
11.227
11.220
II.2T2
11.204
II. 196
II.188
Imchts,
17.165
7.159
7.152
7.146
7.139
7.132
7.126
7-119
7.1 1 1
7.104
7.006
7.089
7.082
7.073
7.065
7.057
7.049
7.040
7.032
6.978
6.968
6.958
6.948
6.938
6.928
6.918
6.853
6.841
6.829
6.818
6.806
6.794
16.783
Smithsonian Tanlbs.
IncJUt,
22.887
22.878
22.869
22.861
22.852
22.843
22.834
22.825
22.815
22.805
22.795
22.786
22.776
22.764
22.754
22.742
22.732
22.720
22.710
22.608
22.685
22.673
22.661
22.649
22.637
22.624
22.610
22.507
22.584
22.570
22.557
22.542
22.528
22.514
22.499
22.485
22.470
22.455
22.439
22.424
22.408
22.392
22.377
III
28.609
III!
28.554
28.543
28.531
28.519
28.507
28.4<
28.470
28.456
28.442
28.428
28.415
28.401
28.387
28.372
28.357
28.342
28.327
28.311
28.296
28.280
28.263
28.246
28.230
28.213
28.196
28.178
28.160
28.142
28.124
28.106
28.088
28.069
28.049
28.030
28.010
27.991
27.971
Incktt.
34.330
34.317
34.304
34.291
34-278
34.265
34.252
34.237
34.222
34.208
34.193
34.178
34.163
34.147
34-130
34.114
34.097
34-081
34.064
34.046
34.028
34.010
33.992
33-973
33-955
33.935
33-915
33-895
33-875
33-855
33-835
33-814
33-792
33-770
33-749
33-727
33-706
33.682
33-659
33-635
33.612
33-589
33-565
•S^
10
15
20
25
30
5
0.001
10
IS
•^
20
.010
25
x>i6
30
.023
Inchts.
aooo
.002
1^
.013
.018
0.001
.003
.007
•01 3
.020
.028
13"
0.001
.004
XX&
.015
.023
.033
8»
Inekgt,
0.001
.002
.005
.009
X)i4
.021
0.001
.018
.026
OJOOl
:^
X)i4
.021
.031
14**
0.001
.004
.009
.016
•03s
Tabuc 22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE irW
[DeriTatioa of table explained on pp. Uii-lvi.]
"S^.
Meridional die
tances from
even degree
paiallels.
ABSCISSAS OF DEVELOPED PARALLEL.
ORDINATES OF
DEVELOPED
PARALLEL.
5-
longitode.
10'
IS-
l.n.«mtmtm
lon^iRMBe.
20'
25-
lopgitnde.
30-
loQsitiide.
Kinycoae.
i4*'oo'
10
20
30
40
50
1500
10
20
30
40
50
1600
10
20
30
40
SO
1700
10
20
30
40
SO
1800
10
20
30
40
SO
1900
10
20
30
40
50
2000
10
20
30
40
SO
21 00
JnektM.
68.740
IncMts.
5-594
w
5.582
5-578
S-S73
5-569
5-565
5-560
5-556
5-55»
5-547
S-542
5-538
5-533
5.528
5-524
5-5"9
5-514
5-509
5504
5-499
5-494
5-489
5484
5-479
r^
5463
5-458
s-452
5447
5.441
5-436
5-430
5-424
5.419
5413
5-407
5.401
5396
5-390
5-384
JnchtM.
II.I88
II.I80
II. 172
II.I63
II.I55
II.I47
II.I38
II.I30
II. 121
II. 112
1 1. 103
11.094
11.085
11.076
iix)66
11.057
11.047
11.038
11.028
II.0I8
iixx>8
ia978
ia968
10.957
10.947
10.936
ia926
10.915
10.905
10.871
10.860
10.849
10.838
10.826
ia8i4
ia8o3
10.791
10.779
10.768
Imeket.
16.783
16.770
16.758
16.745
16.733
16.720
16.708
16.667
16.654
16.641
16.628
16.613
16.599
16.585
16.571
16.556
16.542
16.527
16.512
16.467
16.452
16.436
16420
16.404
16.389
16.373
16.357
16.340
16.324
16.307
16.290
16.274
16.257
16.239
16.222
16.204
16.187
16.169
16.151
Jncku,
22.377
22.360
22.344
22.327
22.310
22.294
22.277
22.259
22.241
22.223
22.206
22.188
22.170
22.151
22.132
22.113
22.094
22.075
22^)56
22.036
22.016
21.996
21.976
21.956
21.936
21.015
2i|94
21.872
21.852
21.830
21.809
21.787
21.765
21.742
21.720
21.698
21.676
21.652
21.629
21.605
21.582
21.558
21.53s
Indus.
27.971
27.950
27.930
27.867
27.846
27.824
27.802
27.779
27.757
27.735
27.713
27.689
27.665
27.642
27.618
27.594
27.571
27.546
27.521
27.495
27470
27.445
27420
27-394
27.367
27.341
%-'^
27.262
^7.234
27.206
27.178
27.150
27.123
27.095
27^)65
27.036
26.948
26.919
Inckts.
33-565
33-540
33.515
33-490
33.465
33-440
3341S
33-389
33-362
33-335
33-308
33.282
33-255
33-227
33.198
33-'70
33-'42
33-"3
33-085
33.05s
33-025
32.994
32.964
32.934
32.904
32.872
32.840
32,809
32.777
32.746
32.7H
32.680
32.647
32.614
32.580
32.547
32.513
32.478
32.443
32.408
32.373
32.338
32.303
14**
^f
11.458
22.915
34-373
68.746
5-
10
15
20
25
30
Indkts.
0.001
.004
JO16
•025
•03s
Jnckis.
aooi
x)04
.009
.026
.038
11.459
22.917
34.376
45-834
57-293
68.752
16°
It
11.460
22.919
57.298
68.758
5
10
IS
20
25
30
aooi
.004
^10
.018
.028
.040
aooi
.005
.011
.019
.029
.042
1 1. 461
22.921
34.382
45-843
57.304
68.764
i8«
19"
11.462
m
57.310
68.771
5
10
IS
20
25
30
aooi
.005
.011
.020
.031
.044
aooi
.005
.012
.021
.032
.046
1 1463
22.926
57.316
68.779
20°
210
5
10
15
20
25
30
aooi
.005
.012
.022
.034
.049
aooi
.006
•013
.022
.03s
.051
11.464
22.929
4t*^58
57-322
68.787
SniTHaoNiAN Tables.
112
Digitized by V^OOQ IC
Tamje 22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE vrhv-
[Derivation of table eacplaiaed on pp. liii-lvi]
ABSCISSAS OF DEVELOPED PARALLEL.
5'
longitude.
longitude.
IS'
longitude.
2&
longitude.
longitude.
30'
longitude.
ORDI NATES OF
DEVELOPED
PARALLEL.
10
20
30
40
SO
22 00
10
20
30
40
50
10
20
30
40
50
2400
10
20
30
40
SO
2500
10
20
30
40
SO
2600
10
20
30
40
SO
2700
10
20
30
40
SO
2800
68.787
11.466
22.932
34.397
45.863
57.329
68.795
II467
«2.934
34401
45.868
57.336
68:803
11.469
22.937
34.406
45.874
57.343
68.812
11.470
22.940
34.410
45.880
57.350
68.821
11.472
22.943
34.415
45^86
57.358
68.830
"•473
22.946
34-419
45.892
57.365
68.838
11.475
22.950
34.424
45.899
57.374
68.849
Inckts.
5384
5-378
5.366
5.359
5-353
5-347
5-341
5-334
5.328
5-322
5-315
5-309
S-302
5.206
5.289
5.282
5.276
5.269
5.265
5-256
5.249
5.242
5.235
5.227
5.220
5-213
5.206
5-199
5.191
5.184
5-177
5.169
5,162
5-154
5-M7
5.140
5.132
5.124
5.116
5.109
5. 101
5093
Inckts,
10.768
0.755
0.743
0.731
0.719
0.707
0.694
0.682
0.669
0.656
0.643
a63i
0.618
0.604
a59i
0.578
0.565
0.551
0.538
0.526
0.512
a4Q8
a483
0.469
0.455
0.441
0.426
0.412
0.397
0.383
0.369
0.354
0.339
0.324
0.309
0.294
0.279
0.264
0.248
0.23;
0.2
0.202
?i
10.187
6.I5I
6.133
6.II5
6.078
6.060
6.042
6.022
6.003
5-984
5965
5.946
5-927
5.867
5.847
5.827
5.807
5.789
5.767
5.746
5-725
5.704
5.682
5.661
\^
5-596
5-575
5-553
5.531
5-5^
5.486
5463
5.441
5.419
5-396
5-373
5-349
5-326
5-303
15.280
Inckts.
21.535
2I.5II
21.486
21.462
21.438
21.413
21.389
21.363
21.338
21.312
21.287
21.261
21.236
21.209
21.182
21.156
21.129
21.102
21.076
21.052
21.023
20.995
20.967
20.938
2a9io
20.881
20.852
20.824
20.795
20.766
20.737
20.708
20.678
2a648
20.6r8
20.588
20.558
20.528
20.497
20.466
20.435
20.404
20.374
IncJUt,
26.919
26.889
26.858
26.828
26.797
26.767
26.736
26.704
26.672
26.641
26.609
26.577
26.545
26.511
26.478
26445
26412
26.378
26.345
26.315
26.279
26.244
26.209
26.173
26.137
26.101
26.065
26.029
25.922
25.884
25.847
25.810
25.772
25-735
25.698
25.659
25.621
25-582
25.544
25.505
25.467
Inches.
32.303
32.266
32.230
32.193
32.156
32.120
32.083
32.045
32.006
3'.969
31.930
31.892
31.853
3^.813
31.774
3^.733
31.694
31.654
31.614
31-577
31-535
31.493
3'.450
31408
31-365
31-322
31.279
3<-235
31.192
3'.M9
31.106
31.061
31.017
30.972
30.927
30.882
30.838
30791
30.745
30.699
30.653
30.607
30.560
2I
21"
Inches.
aooi
.006
.013
.022
.035
.051
23"
0.001
.006
.014
.024
.038
.054
25«
0.002
.006
.014
.026
.040
.058
27«
0.002
.007
.015
.027
.042
.061
22*>
Inches.
aooi
.006
.013
.023
.036
.052
24'
aoo2
.006
/>I4
.025
•039
.056
26*^
aoo2
.007
.020
.041
.059
28«
.007
.016
.043
.063
Smithsonian Tables.
"3
T^itizeTbTtjCTO^
.Tablk 22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE jwhw-
[Derivaticm of table wplaincid on pp. liii-lvL]
ABSCISSAS OF DEVELOPED PARALLEL.
5-
longitude.
10'
longitude.
IS'
longitude.
2or
longitude.
25'
longitude.
30'
longitude.
ORDINATES OF
DEVELOPED
PARALLEL.
28^'
xo
20
30
40
9>
2900
10
90
30
40
SO
3000
10
20
30
40
50
3100
10
20
30
40
50
3200
10
20
30
40
50
3300
10
20
30
40
so
3400
10
20
30
40
so
3S0O
68.849
n.476
22.953
34.430
45.906
57.383
68.859
11.478
22.957
34.435
45-9»3
57.391
68.870
11.480
22.960
34.440
45.920
57400
68.880
11.482
22.964
34446
45.927
57409
68.891
11484
22.967
3445*
45.934
57418
68.902
11485
22.971
34456
45-942
57427
68.913
1 1487
22.975
34462
45-949
57437
68.924
Inckts,
5.093
5.085
S-077
5.069
5.061
5.054
5.046
5.037
5.029
5.021
S-0I3
5.004
4.996
4.988
4.979
4.971
4.962
4.954
4-945
4.937
4.928
4.919
4.910
4.902
4.893
4.884
IS
4.848
4.839
4.830
4.821
4.812
4.802
4.793
4.784
4.774
4.765
4.755
4.746
4.737
4.727
iai87
10.171
10.155
10.139
10.123
iai07
10.091
iao7ij
10.058
10.042
10.025
iaoo9
9^993
9.976
9-959
9-942
9.925
9.908
9.891
^n
9-556
9-538
9.821
9.804
9.786
9.768
9.750
9732
9.696
9.679
9.660
9.642
9-623
9.605
9.586
9.568
9-549
9530
9.511
9492
9473
9454
Inckit,
15.280
5.256
5.232
5.208
5.185
5. 1 61
5.137
5.112
5.087
5.063
5.038
5.013
4.989
4.963
4.938
4.862
4.836
4.810
4.784
4.758
4.731
4.705
4.679
4.652
4.625
4.598
4.572
4.545
4.518
4490
4.462
4.435
4.407
4.379
4.352
4.323
4.295
4.267
4.238
4.210
14.181
/ncJUt.
20.374
20.342
20.310
20.278
2a246
20.214
2ai82
2ai5o
20.117
20.084
2ao5i
2aoi8
19.985
9-95»
9.849
9.815
9.782
9.747
9.712
9.677
9.642
9.607
9-572
9.536
9.500
9.465
9429
9-393
9-357
9.320
9.283
9.246
9.210
9-173
9.136
9.098
9.060
9.022
8.984
8.946
18.908
25.467
25427
25.387
25.347
25.308
25.268
25.228
25.187
25.146
25.105
25.064
25X>22
24.981
24.854
24.812
24.769
24.727
24.683
24.640
24.596
24.552
24-509
24465
24.420
24.376
241286
24.241
24.196
24.150
24.104
24.058
24.012
23-966
23.920
23.872
23.825
23-778
\Wz
23.636
IncMgt.
30.560
30.513
30.465
30.417
30.369
30.32'
30.274
30.224
30.175
3ai26
30.076
30.027
39.978
29.027
29^76
29.825
29.774
29.723
29.672
29.620
29.568
29.515
29.463
2941 1
29.358
29.305
29.251
29.197
29.143
29.089
29.036
28.980
28.870
28.814
28.759
28.704
28.647
28.590
28.533
28476
28420
28.363
•1
I.
a002
.016
.028
yp
jo\6
3«»
OJOQ2
.017
.030
34°
«f
Inchn,
OJ0O2
.016
.028
31"
O1OO2
.017
•030
zf
0.002
.008
.017
.031
^8
.069
3!>"
0.002
aoo2
.008
.008
.017
.018
.031
.031
.049
.049
.070
.071
Smitnbonian Taslcs.
114
Digitized by V^OOQlC
Tablb 22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE jtim>
[Derivation of tablo explained on p|». liii-IvL]
•8
II
ABSCISSAS OF DEVELOPED PARALLEL.
S'
longitude.
longitude.
'5'
longitude.
20'
longitude.
longitude.
30'
longitude.
ORDINATES OF
DEVELOPED
PARALLEL.
3focr
lO
20
y>
40
50
3600
10
20
30
40
so
3700
30
40
50
3800
ID
20
30
40
50
3900
10
20
30
40
50
4000
10
20
30
40
50
41 00
10
20
30
40
50
4200
Inchg*.
68.924
11.489
22.978
34.468
45-957
57446
68.935
11.4Q1
22.983
34474
45-965
57457
68.948
34.480
45.973
57466
68.959
"495
22.900
34.455
45,980
57475
68.970
"497
22.994
4s'9p
57485
68.982
11490
22.998
34497
45996
57495
68.994
11-501
23.002
34-503
46.004
57.506
69.007
IneMfs,
4.727
4.717
4.708
4.(
4.679
4.669
4.659
4.649
4-639
4.629
4.619
4.609
4.599
^579
4.5^
4.558
4.548
4.538
4.527
4.517
4.506
4496
4486
4.475
4464
4454
4.443
4433
4422
4.41 1
4.400
4.389
4.378
4.368
4.357
4.346
4.335
4324
4.312
4.301
4.290
Jnckts.
9454
9435
9415
9.396
9.377
9.357
9.338
9.318
9.298
9.278
9.258
9-238
9.219
9.198
9.178
9.157
9.137
9.117
9.096
9.076
9.055
9^34
9.013
8.992
8.971
8.844
8.822
8.800
8.779
8.757
8.735
8.713
8.691
8.669
8.647
8.625
8.603
8.581
JfcJUs.
X4.181
4.152
4.123
4.094
4.061
4.0:
s
4.007
3.977
3-947
3.828
3-797
3-767
3.736
3-706
3.675
3.645
3.613
3.582
3-551
3-520
3.488
3-457
3425
3.393
3-361
3-330
3.298
3.266
3.233
3.201
3.168
3-135
3.'03
3070
3037
3.004
2.971
2.937
2.904
12.871
'itches.
8.908
8.870
8.83'
8.792
8.753
8.714
8.676
8.636
8.596
8.556
8.517
8.477
8.437
8.396
8.356
8.315
8.274
8.234
8.193
8.1 51
8.fo9
8.068
8.026
7.984
7.943
7.Q00
7.858
7.815
7.773
7.730
7.688
7.644
7.601
7.557
7.5M
7470
7.427
7.383
7.338
7.294
7.250
7.205
7.161
Imckts.
23.636
23-587
23.539
23.490
23.442
23.393
23.345
23.295
23.245
23.195
23.146
23.096
23.046
22.995
22.944
22.894
22.843
22.792
22.741
22.689
22.637
22.585
22.533
22.
2.533
2481
22429
22.375
22.322
22.269
22.210
22.163
22.110
22.055
22.001
21.047
21.892
21.838
21.784
21.728
21.673
21.618
21.562
21.507
21.451
Jnche*.
28.363
28.305
28.246
28.188
28.130
28.072
28.014
27.954
27.894
27.835
27.775
27.715
27.656
27.594
27.533
27.472
27.411
27.350
27.289
27.227
27.164
27.102
27.039
26.977
26.914
26.851
26.787
26.723
26.659
26.595
26.532
26466
26401
26.336
26.271
26.206
26.140
26.074
26.007
25.941
25-742
•|5
5-
10
15
20
25
30
Zf
Inches.
aoo2
.008
m%
.031
.049
.071
37"
aoo2
.008
.018
.032
.050
.073
39°
0.002
.ooS
.018
•033
.051
.074
41"
aoo2
.008
.019
.033
.052
.075
Inckts,
0.002
.C08
.018
.032
.050
.072
aoo2
.008
.018
•033
.051
.073
40P
aoo2
.008
.019
•033
•052
.074
0.002
.008
.019
.033
.052
.075
SniTHaoNiAN Tables.
"S
Tablk 22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE vtW
[Derivation of table enrfainfd on pp. liii-lTi.]
uncct from
even degree
ponllelB.
ABSCISSAS OF DEVELOPED PARALLEL.
"S .
ORDINATES OP
DEVELOPED
PARALLEL.
5-
xo'
'5-
20'
25-
30'
JB-
longitude.
longitude.
loogitade.
longitttde.
longitude.
longitude.
Inctu*.
Incktu
Jnck4t.
Inches.
lneJk*9.
Inches.
Inches.
I-,
42<>oo'
lO
20
30
40
50
69.007
4.290
4.256
4.245
4.234
8.s8x
8.558
8-535
8.513
8.490
8.467
12^71
12.837
X2.803
12.769
".735
X2.7OX
X7.X61
X7.I16
17.071
17.025
16.980
16.935
2x451
21.39s
2X.338
21.282
21.225
21.169
25.742
25-538
25470
25.402
!-H
420
4^
11.501
23.006
34.510
46.0x3
5-
xo
15
Inches.
OXX>2
.008
.019
Inches.
aoo2
.ooR
JOX9
43 «>
lO
20
30
40
SO
4400
69.019
4.222
4.21 1
4.176
4.165
4.153
8445
8.422
8.399
8.376
8.353
8.330
8.307
8.283
8.260
8.236
8.213
8.189
12.667
X2.633
12.598
12.564
12.529
12494
12.460
16.890
16.844
16.798
16.751
16.705
16.659
16.6x3
16.566
16.519
16426
16.379
2I.XI2
2X.054
20.997
2a882
2a824
20.767
2a7o8
2a649
20.591
2a532
20.473
25-334
25.265
25.196
25.127
25.058
24.989
24.920
24.849
24-779
24.709
20
25
30
.033
.052
.075
.033
.052
.075
11.505
23.010
34.5>5
46.020
57-525
69.030
44**
4S^
10
20
30
40
SO
11.507
23.014
34-522
46.029
57536
4.142
4.130
4.1 18
4.106
4.095
X 2.425
X2.390
".354
X 2.319
12.284
5
xo
«5
20
0.002
.008
.019
.034
aoo2
.008
.019
•034
4500
10
20
30
40
SO
69-043
4.083
4.071
4.059
4.047
4.035
4.023
8.166
8.X42
8.118
8.094
8.070
8x>46
X2.249
X2.2X3
12.177'
I2.I4X
X2.I05
X 2.070
16.332
16.284
16.236
X6.188
16.141
16.093
20415
20.355
2a295
20.236
2a 176
20.116
24498
24.426
24-354
24.283
24.211
24.139
25
30
.052
.075
.076
11.509
23.018
34.528
46.037
57.546
46P
47**
4600
10
20
30
40
SO
4700
10
20
30
40
SO
4800
69-055
4.0x1
3-999
3-987
3-975
3963
3-951
3.938
3.926
3-914
3.877
3.864
8.023
7-998
7.974
7.950
7.925
7-901
7.877
7.852
7.827
7-803
7-778
7.753
7.729
X2X)34
IX .852
IX.8I5
11.778
11.741
\\]^
XX.630
"•593
16.045
15:851
15.802
15.754
15.704
\\^
15.556
15.507
15457
20.056
19.996
19-935
19-974
19-813
19-753
19.692
19.610
19.569
19.507
19-445
19-383
19.322
24.068
23.995
23.922
23-776
23.703
23.630
23.556
23.482
23408
23.334
23.260
23.186
11.511
23023
34.534
46.045
57.557
69.068
5
xo
'5
20
25
30
aoo2
.008
.019
.034
.076
aoo2
.008
.019
.034
.052
.075
"•513
23.027
34.540
46.053
57.587
69.080
5
10
15
20
25
30
480
49^
0.002
0.002
10
20
30
40
SO
11.516
23.035
57.577
3.852
3-839
3.827
3.814
3.802
7.704
7.679
7.603
"•555
11.518
11.480
X 1.442
".405
15.407
15.357
15.307
15-257
15.206
19.259
19.196
19.134
19.071
19.008
23.111
23.035
22.960
22^85
22.8X0
.008
.0x9
.033
.052
.075
.008
.019
.033
.052
.075
4900
69.093
3.789
7.578
11.367
15.156
18.945
22.734
^^^
Smith«onian Tabi.cs.
116
Digitized by V^OOQ IC
Table 22.
CO-ORDINATES FOR PROJECTION OF MAP8. SCALE wiiir-
[Derivation of table explained on pp. liii-lvi.]
49^00'
10
20
30
40
50
5000
10
20
30
40
SO
51 00
10
20
30
40
50
5200
10
20
30
40
SO
5300
10
20
30
40
50
5400
10
20
30
40
50
5500
10
20
30
40
50
5600
ABSCISSAS OF D£V£LOP£D PARALLEL.
5'
longitude.
Inckts.
69-093
11.517
23-035
34.552
46.070
S7.5«7
69-'oS
11.520
23039
34.55«
40.070
57.598
69.117
11.521
23043
34.564
46.086
57-607
69.12S
11.523
23047
34.570
46.094
57.617
69.140
11.525
23.051
34.576
46102
57.627
69.152
11.527
23-0'5S
34.582
57.636
69.164
11.529
^5^
57.646
69.176
ImJUt.
3-789
3-776
3-764
3-75i
3.738
3-725
3-713
3-700
3.687
^•^^
3-661
3648
3635
3.622
3.609
3-596
3-583
3-570
3-556
3-543
3-530
3-516
3-503
3-490
3-477
3463
3-450
3-436
3-423
3-409
3-396
3-382
3.368
3-355
3-341
3-327
3-3>4
3300
3-286
3.272
3258
3-245
3-231
10'
longitude.
IhcU*.
7.578
7.553
7-527
7-502
7.476
7.451
7-425
7-399
7-374
7.348
7-322
7-296
7-270
7-244
7.218
7.191
7.165
7-139
7-"3
7.086
7.060
7-033
7.006
6.980
6.953
6.926
6.872
6w845
6.818
6.791
6.764
6.737
6.709
6.682
^655
6.628
6.600
6.572
6.545
6.489
6462
IS'
loQgitude.
20'
longitude.
Incht*.
11.367
1-329
1. 291
1-253
I.214
1. 176
I.138
1.099
1.060
1. 021
0.983
0.944
0.905
0.866
0.827
0.787
0.748
0.709
0.669
0629
0.589
0.550
0.510
0.470
0.430
0.389
0.349
a228
ai87
0.146
0.105
0.064
0.023
9.982
9.941
9.000
9.859
9.817
9-776
9-734
9-693
'nekes.
5.156
5- 105
5054
5-003
4.952
4.901
4-850
4.799
4.747
4.695
4.644
4.592
4.540
4.488
4.436
4.383
4.330
4.278
4.226
4.172
4. 1 19
4.066
4.013
3960
3906
3.852
3.798
3-745
3-691
3-637
3.583
3-528
3-474
3-419
3-364
3-310
3-255
3-200
3-145
3-089
3-034
2.979
12.924
longitude.
Jnck4S,
18.945
18.882
18.818
18.754
18.690
18.627
18.563
18.499
18.434
18.369
18.305
18.240
18.176
18.IIO
18.045
17-979
I7!848
17.782
17.716
17.649
17.583
17.516
17.450
17.383
17.316
17.248
17.181
17.114
17.046
16.979
16.QIO
16.842
16.774
16.706
16.637
16.569
16.500
16^131
16.362
16.293
16.224
16.155
30'
longitude.
22.734
22.6158
22.581
22.505
22.429
22.352
22.276
22.198
22.121
22.043
V.^
2I.8II
21.732
21.653
21-574
21.496
21.417
21.338
21.259
21.179
21.099
21.019
20.939
20.860
20.779
23.698
20.617
20.536
20.455
20.374
20.292
20.210
20.128
20.047
19.964
19.883
19.800
19.717
19.634
19.468
19-385
ORDINATES OF
DEVELOPED
PARALLEL.
Imcke*.
<
0.002
10
.008
15
.019
20
.033
25
.052
30
.075
49"
51°
0.002
0.002
.008
.008
.019
x>i8
-033
•033
.051
.051
.074
.073
53^^
0.002
.008
.018
.032
.050
-073
55"
aoo2
.008
x>i8
.032
.049
.071
SO"
Inckts.
0.002
.008
.019
•033
.052
•075
54"
0.002
.008
.018
.032
.050
.072
0.002
.008
.018
.031
-049
.070
kJOO^lt
SiiiTHaoiiiAii Tablcs.
iigiTizea oy
117
Table 22«
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE irW
[Derivation of table explained on pp. Uii-lTi.]
I!
ABSCISSAS OF DEVELOPED FARALLEU
5'
loQgitude.
lo'
longitude.
IS-
lonfitnde.
longitude.
30'
longitude.
ORDINATES OF
DEVELOPED
PARALLEL.
56*»oc/
10
20
30
40
50
57 00
10
20
30
40
SO
5800
10
20
30
40
so
S90O
10
20
30
40
SO
6000
10
20
30
40
50
61 00
10
20
30
40
50
6200
10
20
30
40
50
6300
Indut.
69.176
"SJI
23^3
34-594
57.656
69.188
34.599
69.199
"•S3S
23.070
34.605
46.140
57675
69.210
"•537
23-074
34-610
46.147
57.684
"•539
23-077
34.616
46.154
57.693
69.232
11.540
23.081
34.621
46.162
57.702
69.242
11.542
23.084
34.626
46.168
57.710
69-253
Inches.
3.23'
3-217
3.203
3.189
3->75
3.161
3-147
3-»33
3-"9
3-»04
3-090
3-076
3.062
3-048
3-034
3.019
3-005
2.991
2.976
2.962
2.947
2-933
2.918
2.904
2.890
2.860
2.846
2.831
2.816
2.802
2.787
2.772
2.758
2-74^
2.;
2.743
2.728
2.713
l^
2.669
2.654
2.639
2.624
Indus.
6.462
6-378
6.350
6.322
6.294
6.266
6.237
6.209
6.181
6.152
6.124
6.096
6.067
6.038
6.010
5.981
5-9S3
5-924
ilii
5-779
S-750
5.721
5-633
5-604
5-574
5-545
5.48I
5-456
5-427
5-397
5-367
5-337
5-308
5.278
5.248
Inckts.
9-693
9.651
9.609
9-567
9-525
9-483
9»44i
9-398
9-356
9-314
9.271
9.229
9.186
9-M3
9.101
9.058
0.015
8.972
8^29
8.885
8.842
8.799
8.755
8.712
8.669
8.625
8.581
8.537
8.493
8450
8.406
8.361
8.317
8.273
8.229
8.184
8.140
8.096
8.051
8.006
7.961
7-917
7-872
Inches.
12.924
12.868
12.812
12.756
12.700
12.644
12.588
12.531
12.475
12.418
12.362
12.305
12.248
I2.19I
12.134
12.077
12.020
11.962
n.905
11.847
11.790
11.732
11.674
II.616
11.558
11.500
II.44I
"-383
11.324
11.266
11.208
II.I48
11.090
11.030
10.972
ia9i2
10.854
10.794
10.734
10.675
1061S
10.556
ia496
Inches,
16.155
6.085
6.015
5-945
5-875
5-805
5-735
5.664
5-594
5-523
5-452
5-381
5-3"
"CM
5-096
5.025
4.953
4.882
4.809
4-737
4.665
4.592
4.520
4.448
4-375
4-302
4.229
4.156
4.083
4.010
3-788
3-715
3-641
3-567
3.493
3.418
3-344
3-269
3-195
13.120
Inches.
19-385
19.301
19.217
19-134
r
18.882
18.797
18.712
18.627
18.542
18.457
18.373
18.287
18.201
18.115
18.029
17-944
17-858
':^
17.597
17.510
17.424
17-337
i6l8ii
16.723
16.634
16.546
i6ut57
16.369
16.280
16.191
16.102
16.012
15-923
15-833
15-744
se'
Inches.
aoo2
.008
.018
.031
.049
.070
580
aoo2
.008
.017
.030
•^
.000
6o<»
.007
.016
.029
-045
.065
62<»
0.002
.007
.016
.028
.044
.063
57"
Inches.
aoo2
.008
.017
.031
.048
.069
59"
aoo2
.007
.017
x)3o
.0^6
.067
61"
aoo2
.007
.016
.029
6f
0.002
.007
.015
J02y
-043
.061
Digitized by LjL^V^V
Smithsonian Tables.
118
CO-ORDINATES FOR PROJECTION OF MAPS.
[Derivatioa of Uble eaipUdned on pp. liii-lvi.]
Table 22.
SCALE irhir-
ABSCISSAS OF DEVELOPED PARALLEL.
longitude.
10'
longitude.
longitude.
20'
longitude.
longitude.
30^
longitude.
ORDINATES OF
DEVELOPED
PARALLEL.
63O0(/
10
20
30
40
SO
6400
10
20
30
40
50
6500
10
20
30
40
50
6600
10
20
30
40
50
6700
10
20
30
40
50
6800
10
20
30
40
SO
6900
10
20
30
40
SO
7000
Inches.
69-253
"•S44
23.087
34.631
46.175
57.718
69.262
11.54s
34.636
46.182
S7.727
69.272
11.547
23094
34-641
46.188
57.735
69.282
11.548
34.646
46.194
57-742
69.291
11.550
23.100
34.650
46.200
57.750
69.300
11.552
23.103
46.206
57.758
69.309
"•553
23.106
34.659
46.212
57.764
69-317
InchtM.
2.624
2.609
2.594
2579
2.564
2.549
2.534
2.519
2.488
2473
2458
2.443
2428
2412
2.397
2.382
2.366
2.351
2.336
2.320
2.305
2.290
2.274
2.259
^^
212
2.107
2.181
2.166
2.24;]
2.22
2.212
2.1(
2.:
2.150
2.134
2.1 19
2.101
2.072
2.056
2.040
2.025
2.009
1.993
1.977
Inch€S.
5.248
5.218
5.188
5.158
5.128
5.098
5.068
5-037
5.007
4.977
4.947
4.916
4.886
4.855
4.825
4.794
4.764
4.733
4.702
4.672
4.641
4.610
4.579
4.548
4.518
4487
4455
4.393
4.362
4.331
4.300
4.269
4.237
4.206
4.175
4.144
4.112
4.081
4.049
4.018
3.986
3.95s
Inch**.
7.872
7.827
7.782
7.737
7.692
7.647
7.602
7.556
7.511
7.465
7.420
7.374
7.329
7.283
7.237
7.191
7.145
-7.100
7.054
7.007
6.961
6.823
6.776
6.637
6.590
6-543
6.497
6450
6.403
6.356
6.263
6.216
6.169
6.121
6.074
6.027
5.980
5.932
SnrrHsoNiAN Taslcs.
IncktM,
10496
10436
10.376
10.316
10.256
XO.I96
iai36
10.075
10.014
9-954
9-893
9.832
9.772
9.711
1%
9-405
9-343
9.282
9.220
9.158
9.097
9.035
8.973
8.011
8.849
8.787
8.724
8.662
8.600
8.538
8475
8.412
8.350
8.288
8.225
8.162
8.099
8.036
7.973
7.910
"9
Inckts.
13.120
3045
2.070
2|95
2.820
2.745
2.670
2.594
2.518
2.442
2.367
2.291
2.215
2.139
2.062
1.986
1.909
1.833
1.756
1.679
1.602
1.448
I-37I
1.294
1.217
1. 061
0.984
0.906
0.828
0.750
a672
0.594
0.516
0.438
a36o
0.281
a 202
0.124
ox)45
9.966
9.888
Imcket.
15744
5473
5383
5.293
5.203
5.1 12
5.022
4.930
4.840
4.749
4.658
4.566
4.474
4.383
4.291
4.199
4.107
4.015
3.022
3.830
3-738
3.645
3-553
3.366
3-273
3.180
3.087
2-994
2.900
2.806
2.712
2.6x9
2.525
2431
2.337
2.242
2.148
2.054
1.959
11.865
^•a
5-
10
15
20
25
30
63"
Inches.
0.002
.007
.015
.027
-043
.061
65°
aoo2
.006
.01^
X)2D
.040
.058
67**
0.00 X
.006
.014
.024
.038
.054
69°
aoox
.006
.013
.022
.035
.05 X
64^
Inches.
0.002
.007
X>26
66*»
0.002
.006
.025
.039
x>56
68<»
0.00X
x)o6
.0x3
.023
.036
.053
70°
0.001
.005
.0X2
.022
.034
.049
t
TABLK22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE vrkr
(Derivatkn ol tible inrpbinad on p. Ifii4vi.]
70®00'
lO
20
30
40
50
71 00
10
20
30
40
50
78 00
10
90
30
40
SO
73 «>
10
90
30
40
50
7400
10
20
30
40
SO
7500
10
20
30
40
SO
7600
10
20
30
40
SO
7700
m
M fl « Bi
69-317
n.554
23.109
34-663
46.217
S7772
69.326
11.556
23.UI
34.667
46.222
S7778
69.334
"•5S7
23.114
34670
46.227
S7784
69.341
U.558
23.116
34.674
46.232
S7.790
69.348
11.550
23.118
34.677
46.236
S7.796
69.3SS
11.560
23.120
34.681
46.241
57A)i
69.361
1 1. 561
23.122
34-683
46.244
57.806
69.367
ABSCISSAS OF DEVELOPED PARALLEL.
s-
loDgitade.
Imck€$.
'•977
.946
.930
.866
.850
.835
.819
A>3
787
77«
75S
739
723
707
.691
.674
.658
.642
.626
.610
•S94
.562
•MS
.529
•SI3
497
.480
.448
432
415
•399
:S
•350
•334
•317
1.301
10'
3.9SS
3.923
3.860
3.828
3796
3765
3.733
3-637
3.605
3S74
3^M2
3.S09
3477
344S
3413
3-381
3.349
3.3*7
3.284
3.25a
3.220
3.188
3.ISS
3"3
3.091
3-058
3.026
2-993
2.961
2.928
2i96
2.863
2.831
2.798
2.765
2733
2.700
2.667
2.634
2.602
loogitiide.
IncMtt.
S-932
5.885
S.837
S-790
S.742
S.695
S-647
5.600
SSS2
S.S04
5.456
5-408
5.360
S.3'2
5.264
5.216
5.168
5.120
S-072
5.024
4.97s
4.830
4.782
4.S87
4-539
4.490
4.392
4.344
4.295
4.246
4.197
4.148
4.099
4.050
4.001
3-9S2
3903
20^
loi«itiidA.
SMmnoNiAir Tablcs.
Jnckeu
7.910
7A|6
7.783
7.656
7.S93
7.530
7466
7.402
7.338
7-275
7.211
7.147
7.083
7.019
6.555
6^91
6.826
6762
6.698
6.614
6.589
6.504
6440
6.376
6.31 1
6.246
6li8i
6.116
6.052
5.987
m
5.792
5.726
5.661
5.596
5.530
5465
5.400
5-334
5.269
5.204
120
25-
kmgitade.
/mAm.
9.808
9.729
9.650
9-571
9.491
9412
9.333
9.253
9.173
9-094
9-014
8.934
8.854
\^
8.614
8-533
8453
8-373
8.292
8.21 1
8.131
8.050
7-970
\^
7727
7-645
7.565
7484
7.402
7.321
7.240
7.158
7.077
6.995
6.586
6.505
Imckes,
11.865
11.770
11.675
11.579
11485
11.389
11.294
11.199
11.101
11.008
iaoi2
ioii6
ia72i
1 0.621c
ia528
ia432
10.336
ia240
10.144
10XH7
9.050
9-853
v&
9.563
9466
9.369
9.272
9-«75
9-077
8.980
8.882
8.785
8.687
8.590
8492
8.394
8.296
8.198
8.099
8.002
7.903
7.805
ORDINATES OP
DEVELOPED
PARALLEL.
II
ImOut,
s'
aooi
10
.005
IS
.012
20
jai2
25
.034
30
.049
5
0.001
10
•005
M
X>II
20
.020
25
•031
30
.044
70P
7«"
74»
aooi
.004
x>i6
.02
.0:
036
71'
OOOl
■005
X>12
.021
^32
•047
73^
aooi
■005
X>1I
.019
.029
.043
7S^
aooi
0.001
.004
X>10
.018
.028
.004
.009
>020
x>40
-038
77^
aooi
.004
.008
-015
.023
.033
bigitized byVjl50g
Taux 22.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE irW
[DerivetioD of table explained on p. liii-lTl]
"8
III
ABSCISSAS OP DEVELOPED PAKALEL.
S'
f.ii-i-8|.uf-
MWHHiwe.
10'
longitude.
IS'
longitude.
20'
lonsinKie.
25'
longicuae.
30-
longitode.
DEVELOPED
PARALLEL.
7/W
lO
20
30
40
50
7800
10
20
30
40
SO
7900
10
20
30
40
SO
8000
Inches.
69.367
Inches.
I.3OI
1.268
1.252
1.235
1.219
1.202
1.186
I.169
I.I53
I.I36
I.I20
1.104
1.087
1.070
1.054
1.037
X.02I
i/»4
Inches.
2.602
2.503
2470
2438
2405
2.373
2.'273
2.240
2.207
2.174
2.I4I
2.108
2.075
2.042
2j0O9
Inches.
3.903
3.854
3.804
3-755
3.706
3.656
3-607
3.558
3.508
3459
3.410
3360
3-3"
3.261
3*211
3.162
3.1 12
3.062
3-013
Inches.
5.204
5.138
5.072
5.006
4^1
4.^5
4.810
V^
4.612
4.546
4480
4.414
4.348
4.282
4.216
4.150
4.083
4.017
Inches.
6505
6.423
6.258
6.176
do94
6.012
i
5.600
5.518
5.435
5.352
5.270
5.187
5.104
5.022*
Inches.
7S0S
7.707
7.609
7.510
74"
7.313
7.214
7.IIS
7.016
6.918
6iE9
6.720
6.621
6.522
6422
6.323
6.224
6.125
6.026
!l
77^
78°
11.562
23.124
46:248
57.810
69.373
5-
10
15
20
25
30
Inches.
OjOOI
:^
•015
.023
.033
Inches.
OjOOI
.014
.021
•031
11.563
23.126
34.689
46.252
57.814
69.377
79**
8o«
11.564
23.127
34-691
5
10
15
20
25
30
aooi
.003
.007
.013
X>20
X>28
0.001
.018
.026
lax
Digitized by
GooqIc
Tablk 23.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE oAyt
[Deriiradon of table exptained on pp. liti-lYi]
•8 .
CO-ORDINATES OF DEVELOPED PARALLEL FOR--
lo' longitude.
ao' longitude.
30^ longitude.
40^ longitude.
so' longitude.
lOkmgitode.
X
y
X
y
X
y
X
y
X
y
X
y
mm.
mm.
mm.
mm.
tnm.
mm.
mm.
mm.
mm.
tmm.
$mm.
fWM.
mum.
6^00'
92.8
X>
185.5
.0
278.3
.0
371.I
.0
463.8
.0
556.6
.0
10
§
92.8
X>
185.5
.0
278.3
.0
37I-I
.0
463.8
.0
556.6
.0
20
92.8
.0
185.5
JO
278.3
.0
37 1. 1
.0
463.8
.0
556-6
.0
30
92.8
.0
185.5
.0
278.3
.0
371.0
.0
463.8
X>
556.6
.0
40
92.8
.0
185.5
JO
278.3
.0
371.0
.0
463.8
JO
556.6
.0
SO
460!?
92.8
.0
185.5
.0
278.3
.0
371-0
.0
463.7
JO
556.5
.1
I 00
92.8
.0
185.5
.0
278.3
.0
37 1 -o
.0
463.7
.1
556.5
,1
10
i^t3
92.7
.0
185.5
.0
278.2
.0
371-0
.0
463.7
.1
5564
.1
20
92.7
.0
185.5
.0
278.2
.0
371.0
.0
463.7
.1
5564
.1
30
368^6
92.7
X)
185.5
.0
278.2
.0
370.9
JO
463.7
.1
5564
.1
40
92.7
.0
1854
.0
278.2
JO
370.9
JO
463-6
.1
556.3
.1
so
460.7
92.7
.0
185.4
.0
278.2
X)
370.9
.1
463.6
.1
556.3
.2
200
92.7
»o
185.4
.0
278.1
jO
370.8
463.6
.1
556.3
.2
10
92.1
92.7
JO
1854
x>
278.1
.0
370.8
463.5
.1
556.2
.2
20
184-3
92.7
.0
185.4
.0
278.1
.0
370.8
4634
.1
556.1
.2
30
270.4
92.7
jO
>85-3
x>
278.0
.0
370.7
4634
.1
556.0
.2
40
368.2
92.7
.0
185.3
x>
278.0
370.6
463.3
.2
556.0
.2
so
460.7
92.7
.0
185.3
.0
278.0
I
370.6
463.2
.2
555-9
.2
300
92.6
.0
185.3
.0
277.9
370.6
463.2
.2
555-8
.2
10
i
92.6
.0
185.2
.0
277-9
277.8
370.5
463.1
.2
555-7
•3
20
92.6
JO
185.2
.0
3704
463.0
.2
555.7
.3
30
92.6
.0
185.2
.0
277.8
370.4
463.0
.2
555-5
.3
40
92.6
.0
185.1
.0
277.7
370.3
462.8
.2
5554
.3
so
460^7
92.6
.0
185.1
.0
277-7
370.2
462.8
.2
555-4
•3
400
92.5
.0
185.1
.0
277.6
370.2
.2
462.7
.2
555-2
-3
10
92.1
92.5
.0
185.0
.0
277.6
370.1
.2
462.6
.2
555-'
.3
20
104.3
92.5
.0
185.0
.0
277-S
370.0
.2
462.5
.2
555.0
•3
30
270.4
92.5
.0
185.0
.0
277.4
H
.2
4624
.2
l^
.3
40
368.0
92.5
JO
184.9
.0
277-4
.2
462.3
•3
-4
so
460.7
92.4
.0
184.9
x>
2773
369^
.2
462.2
•3
554.6
4
500
92.4
.0
'H
.0
277.3
369-7
.2
462.1
.3
554.5
4
10
104-3
92.4
.0
184.8
.1
277.2
369-6
.2
462.0
.3
554.3
4
20
924
.0
184.7
.1
277.1
369-3
.2
461.8
.3
554.2
4
30
276.4
92.3
.0
184.7
277.0
369-4
.2
461.7
.3
554.0
4
40
368.6
92.3
.0
184.6
.1
276.9
369-*
.2
461.6
.3
553-9
.5
so
460.7
92.3
.0
184.6
.1
276.9
369-2
.2
4614
•3
553-7
•S
600
92.3
.0
184.5
276.8
369.0
.2
461.3
.4
553-6
-5
10
itj.3
92.2
.0
184.5
276.7
^■t
.2
461.2
4
5534
•5
20
92.2
.0
184.4
.1
276.6
.2
461.0
4
553-2
.5
30
276.4
92.2
.0
184.3
276.5
3«-7
.2
460.8
4
553-0
i
40
368.6
92.1
.0
184.3
.1
276.4
368.6
.2
460.7
4
552.8
50
460.7
92.1
.0
184.2
.1
276.3
368.4
.2
460.6
4
552.7
.6
700
92.1
.0
184.2
,1
276.2
3^-3
•3
4604
4
552-5
.6
10
184.3
92.0
.0
184.1
.1
276.1
3ff-*
.3
46a2
4
552.2
.6
20
92.0
.0
184.0
.1
276.0
368.0
•3
46ao
4
552.1
.6
30
276.4
92.0
.0
184.0
.1
275-9
275-8
II
•3
459.9
4
551.9
.6
40
368.6
91.9
.0
11^
.1
-3
459.7
4
551.6
.6
so
460.7
91.9
.0
•*
275-7
.1
367.6
•3
459-5
•5
55M
.7
800
91.9
.0
183-7
.1
275.6
.2
367.5
•3
459.4
•5
551.2
.7
ft M 1 VUMAH 1 A M
—
^"^^
""
^^^
3iTizea D
7^-
122
Table 23.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE TV^hv-
[DerivatioD of table explained on pp. liii-lirL]
•8 .
Meridional dit-
unces from
even degree
parallels.
CO-ORDINATES OF
DEVELOPED PARALLEL FOR—
i</ longitude.
»of longitude.
3</ longitude.
40^ longitude.
50^ longitude.
xo longitude.
X
y
X
y
X
y
X
y
X
y
»
y
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
MM.
mm.
8^'
10
20
40
so
184.3
460^
91.Q
91.8
91.8
91.8
91.7
91.7
.0
.0
.0
.0
.0
.0
183.7
183.6
183.5
183.4
183.3
•'
275.6
275.5
275.4
275.2
275.1
275.0
.2
.2
.2
.2
.2
.2
3^-5
367.3
367.2
367.0
366.8
366.7
•3
•3
•3
:^
•3
•459.4
459-2
m
458.6
4S84
551.2
551.0
550.7
550.5
550.3
550.0
•7
900
10
20
30
40
50
184.3
368.i
460.8
91.6
91.6
91.5
91.5
91.5
91.4
.0
•P
.0
.0
.0
.0
183.3
183.2
183.1
183/3
182.9
182.8
.1
274.Q
274.8
274.6
274.5
2744
274.2
.2
.2
.2
.2
.2
.2
366.4
366.2
366.0
365.6
•3
.3
.3
•3
4
4
458.2
458.0
457-7
457.5
457.3
457.0
.6
549.8
549.5
5492
548.5
.8
.8
.8
.8
.8
.8
1000
10
20
30
40
50
184-3
276.5
914
913
913
91.2
91.2
9I.I
.0
.0
.0
.0
.0
.0
182.7
182.6
182.5
182.4
182.3
182.2
•«
274.1
274.0
273.8
273.7
273-5
2734
.2
.2
.2
.2
.2
.2
365.5
365.3
365.1
364.9
364.7
364.5
4
.4
4
4
4
4
456.8
456.6
456.4
456.1
455.9
455-6
.6
.6
.6
.6
.6
.6
548.2
547.9
547-6
547.3
547.0
546.7
.8
.8
.9
-9
.9
.9
II 00
10
20
30
40
50
184^3
276.5
9I.I
91.0
91.0
90.9
90.0
90.8
.0
.0
.0
.0
.0
.0
182.1
182.0
1 81. 9
181.8
181.7
181.6
.1
273.2
273.1
272.9
272.7
272.6
272.4
.2
.2
.2
.2
.2
.2
364.3
364.1
363.6
3634
363.2
•4
4
4
4
4
.4
455-4
455-»
454.8
454-6
454.3
454.0
.6
.6
.6
.7
546.4
546.1
545-8
545-5
545-2
544.8
•9
.9
.9
1.0
1.0
1200
10
20
30
40
5P
3p*.7
460.9
90.8
90.7
90.6
90.6
90.5
90.5
.0
.0
.0
.0
.0
.0
181.5
181.4
181.3
181. 1
181.0
180.9
•I
272.2
272.1
271.9
271.7
271.6
2714
.2
.2
.2
•3
•3
•3
363.0
362.8
362.5
362.3
362.1
361.8
4
4
4
453.8
4534
453-2
452.8
452.6
452.3
,7
544.5
544-1
543.8
543-4
543-1
542.8
1.0
1.0
I.O
1.0
1.0
I.I
1300
10
20
30
40
50
368.8
461.0
904
903
90.3
90.2
90.2
90. 1
.0
.0
.0
.0
.0
.0
180.8
180.7
180.6
180.4
180.3
180.2
.1
271.2
271.0
270.8
270.6
270.4
270.3
•3
•3
•3
.3
•3
•3
361.6
3614
361.1
360.8
360.6
3604
452.0
451.7
4514
451.0
450.8
450.4
.7
%
.8
.8
5424
542.0
541.7
541.3
540.9
540.5
I.I
I.I
I.I
I.I
I.I
I.I
1400
10
20
30
40
SO
276.6
368.8
461.0
90.0
89.8
89.7
.0
.0
.0
.0
.0
JO
180.1
179?
179.8
179.7
179.5
1794
• I
270.1
269.9
269.7
269.5
269.3
269.1
•3
•3
•3
•3
•3
•3
360.1
359.8
359-6
359-3
'3i
450.2
449.8
449.5
448.5
.8
.8
.8
.8
.8
.8
54a2
539.8
5394
538.2
I.I
1.2
1.2
1.2
1.2
1.2
1500
10
20
30
40
SO
1844
276.6
368.8
461.0
89.6
89.6
89.5
89.4
89.3
89.3
.0
.0
.0
.0
.0
.0
179.3
179.1
178.7
178.5
•I
268.9
268.7
268.5
268.3
268.0
267.8
•3
•3
•3
•3
•3
•3
358.5
358.2
358.0
357.7
3574
357.1
•5
:l
.6
.6
.6
448.2
447.8
4474
447.1
446.7
4464
.8
.8
.8
.9
•9
.9
537.8
5374
536.9
536.5
536.0
535.6
1.2
1.2
1.2
1.2
1.3
1-3
1600
89.2
.0
1784
.1
267.6
•3
356.8
.6
446.0
.9
535-2
1.3
Smithsonian Tables.
123
TAnK23.
CO-ORDINATBS FOR PROJECTION OF MAPS. SCALE jv^BWW'
[DtriTitko of tabk czphiaedon pp^ ffiMvL]
CO-ORDINATES OF DEVELOPED PARALLEL FOR—
■</ loQghnd*.
ao' loi^itods.
4V WHUIiiwc*
Sf/loaptadt.
i6«bor
10
20
30
40
50
1700
10
20
30
40
SO
1800
10
ao
30
40
50
1900
10
20
30
40
50
2000
10
20
30
40
50
21 00
10
20
30
40
50
22 00
10
20
30
40
50
2300
10
20
30
40
50
2400
02.2
368.9
461. 1
368.9
461.2
02.2
276.7
369.0
461.2
in
369-0
461.3
92.3
276S
369-1
461.4
184.6
276.8
369-1
461.4
89.0
89.0
88.9
88i
88.7
88.7
88.6
88.5
884
88.3
88.3
88.2
88.1
88x>
87.Q
87i
87.6
87.6
87.5
874
87.3
87.2
87.1
87.0
86.9
86i
86.7
86.6
86.5
86.4
86.3
86.2
86.1
86.0
in
III
85.5
85.4
85.3
85.2
85.1
85.0
84.9
84.8
784
78.2
78.1
77.9
77.6
77.5
77.3
77-*
7618
76.7
^P
76.0
75.8
75-6
75-5
75-3
75-1
in
74.6
74.4
74.2
74X)
7>8
73-7
73-5
73.3
73-1
72.9
7^7
7^S
72-3
72.1
71.9
71.7
71.5
71.3
71.1
70.9
70.7
70.4
7a 2
70.0
69.8
169.6
267.6
2674
267.2
266.9
266.7
266.5
266.2
26610
265.7
265.5
265.2
265.0
264.8
264.5
264.2
264.0
263.7
263.5
263.2
263.0
262.7
262.4
262.1
261.9
261.6
261.3
261/)
260.8
26a5
260.2
259-9
259.6
259.3
2584
258.2
257.8
257.6
257.2
256.6
256.3
256.0
255-7
255-3
255.0
254.7
254.4
35^
356.5
356.2
355'?
355.6
355.3
355-0
354-6
354.3
354.0
353.6
353.3
353^
352.6
352.3
352-0
35«.6
35«.3
35«-o
350.6
350.2
349.9
349-5
349-2
348.8
3484
348.0
347.7
347.3
346.9
346.6
346.2
345.8
3454
345-0
344-6
344.2
343.8
3434
343.0
342.6
342.2
341.8
341.3
340.9
340.4
340.0
339-6
339.2
446.0
445-6
445.2
444.8
444.1
443-7
443.3
442.9
442.5
442/>
441.6
441.2
440-8
439.6
439.1
438-7
438.2
437.8
4374
436.9
4364
436.0
435-6
435.0
434.6
434.2
433.6
433.2
432.7
432.2
431.7
431.2
430.8
430-2
429.8
:^
428.2
427.7
426.6
426.1
425.6
425.0
424.5
424.0
535.2
534-7
534.3
533-8
533.3
532.9
5324
532.0
531.5
531.0
530.5
530.0
529.5
520.0
545
528.0
527.5
526.9
5264
525.9
525-4
524.8
524.3
523.7
523.2
522.7
522.1
521.5
S21J0
5204
519-8
518.6
518.0
517.5
516.9
516.3
5' 5.7
515.1
514.5
513.8
513.2
512.6
512X)
5".3
510.7
510 1
5094
508.7
«.3
«.3
1.3
1.3
"3
14
M
M
14
14
1.4
14
M
14
i-S
«.5
IS
IS
"S
"S
i-S
IS
\i
1.6
1.6
1.6
1.6
1.6
1.6
1.6
1.6
1.6
1.7
1.7
1.7
1.7
1-7
1-7
1.7
1.7
1.7
\i
1.8
1.8
1.8
1.8
1.8
Ljouyit
Smithsoniaii Tables.
Digitized by
124
Tamx 28.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ttAtt-
[Derivation of taUe explaiiwd on pp. liii-lvL]
24^00'
10
20
30
40
SO
2500
10
20
30
40
SO
2600
10
20
30
40
50
2700
10
20
30
40
50
2800
10
20
30
40
50
2900
10
20
30
40
50
3000
10
20
30
40
SO
3100
10
20
30
40
50
3200
184.6
276.9
369-2
461.5
277-0
Si
92.3
184.7
277-0
369*3
461.0
02.4
184.7
277.0
02.4
184.7
277.1
461^
184.8
277.1
369-5
461.9
024
184.8
369.6
462.0
COORDINATES OF DEVELOPED PARALLEL FOR—
lo' lonfitnde.
84.8
84.6
84.4
84.2
84.1
84.0
111
Ilk
83.4
83.3
83.2
83.1
82.9
82i
82.7
82.6
82.5
82.3
82.2
82.1
82.0
8Z.8
81.7
81.6
81.5
81.3
8z.2
8I.I
8a9
8oi
80.5
8a4
80.3
8ai
8ao
79-9
79-7
79.6
79-4
79-3
79.2
79.0
78.9
78.8
9of lonSitniM.
169.6
1694
169.1
168.9
168.7
168.5
168.3
168.0
167.8
167.6
'67.3
167.I
i66l9
166.6
166.4
166.1
165.9
165.7
165.4
165.2
164.9
X64.7
1644
164.2
163.9
163.7
163.4
163.2
162.9
162.7
162.4
162.1
1 61.9
161.6
161.3
161.Z
i6a8
i6a5
i6a3
i6ao
159-7
159.5
159.2
158.6
158.3
1 58. 1
157.8
157.5
30^ longitode.
Smithsonian Tables.
2544
254.0
253-7
253-4
253-0
252.7
252.4
252.0
251.7
251-3
25IX>
25a6
250-3
249.9
249.6
248^8
248.5
248.1
247.8
247.4
247-0
246.7
246.3
245,9
245-5
245.1
244.7
244.4
244X)
243.6
243-2
242.8
242.4
242.0
241.6
241.2
240.8
240.4
240.0
239.6
239.2
238.8
238.4
2379
237.5
237.1
2367
236.2
"5
40/ longitude.
332;*
337
337-4
337.0
336.5
336.0
335-6
335-J
334.6
334.2
333.7
333-2
332-8
332.3
331.8
331-3
330.8
330.4
329.8
320.4
349
328.4
327.9
r^
326.3
325.8
325-3
324.8
324.3
323.8
323.2
322.7
322.2
321.6
321.1
320.6
320.0
3^9-4
318.9
318.4
317.8
3*7.2
316.7
316.1
3156
3150
so' longitude.
424.0
4234
422.8
422.3
421.8
421.2
42a6
420.0
419.5
418.9
418.3
417.8
416.6
416.0
415-4
414.8
414.2
413.6
41.^.0
412.3
411.7
411. 1
410.4
409.8
4^6
407.9
406.0
405.4
404.7
404.0
403.4
402.7
402.0
401.4
400.7
4oao
398.0
396.6
395-8
395-2
3944
393.8
«-3
»-3
i-3
1.3
1-3
1.3
1-3
1.3
1.3
1.3
1.3
1-3
1-3
1-3
1-3
1-3
14
1-4
14
14
1.4
1.4
M
1.4
14
14
1.4
M
14
14
14
14
14
1.4
14
X.5
1-5
1-5
1-5
'.5
1.5
1.5
>.5
1-5
1-5
1-5
1-5
'5
i-S
508.7
508.1
^A
506.8
506.1
5054
504.8
504.1
5034
502.7
502.0
S01.3
500.6
488x>
487.2
486.4
4f5.0
484.8
484.0
483.2
482.5
481.6
480.8
48ao
479.2
478.4
477.5
476.7
475.9
475.0
474.2
473.3
472.5
.8
.8
.8
.8
.8
•9
•9
.9
-9
-9
-9
•9
•9
.9
•9
-9
2.0
2J0
2X>
2J0
2J0
2X}
2jO
2J0
2.0
2.0
2.0
2.0
2.0
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.2
2.2
2.2
2.2
2.2
^igilizedby^OQ gfe
Table 23.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE TinAnnr-
[DeriYttion of taUe eiq>]aixied on pp. liii-lvL]
COORDINATES OF
DEVELOPED PARALLEL FOR— |
•8*
^11
II
IV IOQ2|it1Kl6.
ac/ loDgitiide.
y/ lon^tude.
4X/ lonxitade.
5c/ longitude.
1° longitude.
It
jjSIs,
'x
y
X
y
X
y
z
y
X
y
X
y
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
32*»oo'
7f-!
157.5
.2
236.2
315.0
1.0
393.8
1.5
472.5
471.6
2.2
10
^m
78.6
.1
157.2
.2
235.8
3144
1.0
393.0
1.5
2.2
20
78.S
.1
.2
235-4
313.8
1.0
392.3
391.6
1-5
470.8
2.2
30
277.2
7|-3
.1
156.0
.2
235.0
313-3
1.0
1-5
469.9
2.2
40
369.6
78.2
.1
156.3
.2
234.5
312.7
1.0
390.8
1.5
X
2.2
SO
462.0
78.0
156.0
.2
234.1
312.1
1.0
390.1
1-5
2.2
33 00
77-9
,1
155.8
.2
233.6
j6
311.5
I.O
Si
1-5
467.3
2.2
10
'\&
77-7
.1
155.5
.2
233.2
.6
310-9
1.0
1.5
4664
2.2
20
77.6
155.2
.2
232.7
.6
310-3
1.0
3«7-9
\i
465.5
2.2
30
277.3
77-4
.1
154.9
.2
232.3
.6
309-7
1.0
^l-^
464.6
2.2
40
3697
77-3
154.6
.2
231.9
.6
^:l
1.0
3864
1.6
s
2.2
50
462.1
771
•I
154.3
.2
231.4
.6
1.0
385-7
1.6
2.2
3400
7?i
,1
154.0
.3
231.0
.6
308.0
1.0
^4.9
1.6
461.9
2.3
10
184-9
7&8
.1
153-7
.3
230.5
.6
3074
1.0
At.2
1.6
461.0
2.3
20
76.7
.1
153-4
•3
230.0
.6
306.7
1.0
383-4
1.6
460.1
2.3
30
277-3
76.S
.1
i53-i
•3
229.6
.6
306.1
1.0
382^
1.6
450.2
458.3
2.3
40
369-7
76.4
.1
152.8
•3
22Q.1
228.7
.6
3055
1.0
381.9
1.6
2.3
SO
462.1
76.2
•'
152-4
•3
.6
304-9
1.0
381.1
1.6
457.3
2.3
3S00
76.1
1 52.1
•3
228.2
.6
304.3
1.0
3804
1.6
456.4
2.3
10
184.9
"1
151-8
.3
227.8
.6
3037
1.0
fy^l
1.6
455-5
454.6
2.3
20
.1
151.5
.3
I7d
.6
303-0
1.0
1.6
2.3
30
^7Jii
75.6
.1
151.2
-3
.6
302.4
1.0
378.0
1.6
453-6
2.3
40
369-8
75-4
.1
150.9
.3
226.4
.6
301.8
1.0
377-2
1.6
452.7
2-3
SO
462.2
75-3
•'
I5a6
.3
225.9
.6
301.2
1.0
376.5
1.6
451.8
2.3
3600
75.1
,1
150.3
•3
225.4
.6
300.6
1.0
375.7
1.6
450.8
2.3
10
i84;9
75.0
150.0
•3
224.9
.6
299.9
1.0
374.9
1.6
449-9
448.9
23
20
74.8
149-6
•3
224.5
.6
1^:1
1.0
374.1
1.6
2.3
30
277.4
74.7
.1
149-3
•3
224.0
.6
1.0
373.3
1.6
448.0
2.3
40
369-8
74.5
.1
I4Q.0
148.7
.3
223.5
.6
298.0
1.0
372.5
1.6
447.0
2.3
50
462.3
74.3
•*
.3
223.0
.6
297.4
1.0
371.7
1.6
446.0
2.3
3700
74.2
.1
148.4
•3
222.5
.6
296.7
1.0
370.9
1.6
445.1
2.3
10
185.0
74.0
.1
148.0
.3
222.1
.6
296.1
1.0
370.1
1.6
444-1
2.3
20
73.8
.1
147-7
.3
221.6
.6
295.4
294.8
1.0
a
1.6
443.1
2.3
30
277.4
73.7
.1
147.4
.3
221. r
.6
1.0
1.6
442.1
2.3
40
3699
73-5
.1
X47.1
.3
22a6
.6
294.1
1.0
3f7-6
1.6
441-2
2.4
y>
462.4
734
•»
146.7
.3
22a I
.6
293.4
1.0
366.8
1.6
440.2
2.4
3800
732
,j
146.4
•3
219.6
.6
292.8
1.0
366.0
1.6
439-2
438.2
24
10
185.0
73-0
.Z
146.1
•3
210.1
218.6
.6
292.1
1.0
365.1
1.6
2.4
20
72-9
145.7
•3
.6
291.4
I.I
364-3
1.6
437-2
24
30
277-S
72-7
.1
145-4
•3
218.Z
.6
290.8
I.I
P.
1.6
436.2
24
40
370.0
72.5
.1
145.1
•3
217.6
.6
200.1
289.4
I.I
1.6
435.2
24
so
462.5
72.4
•>
144.7
•3
217.1
.6
I.I
361.8
1.6
434.2
2.4
3900
72.2
,1
1444
.3
216.6
.6
288.8
I.I
361.0
1-7
433.1
24
10
185.0
72.0
.1
144.0
.3
216.1
.6
288.1
I.I
360.1
1.7
432.1
24
20
71.8
.1
143.7
•3
215.6
.6
287.4
I.I
359-2
3584
1.7
431.1
24
30
277.5
71-7
.1
143-4
.3
215.0
.6
^•7
I.I
1-7
430.1
24
40
370.0
71-5
.1
143.0
.3
214.5
.6
286.0
I.I
I'^i
1.7
429.0
2.4
50
462.6
71-3
142.7
•3
214.0
.6
285.3
I.I
1.7
428.0
24
4000
71.2
.1
142.3
•3
213.5
.6
284.6
I.I
355-8
1.7
427.0
24
Smithsonian Taslcs.
126
Digitized by V^OO^ ItT
Table 23.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE imAnnF-
[DerimdoD of table explained on ppb liii-lvi.]
CX>-ORDINATES OF DEVELOPED PARALLEL FOR— |
•s .
Meridional die-
ttncea from
even degree
parallels.
11
lo' longitode.
ao' longitude.
so' longitude.
40^ longitude.
so' longitude.
|0 longitude.
z
1
z
1
X
1
X
y
X
y
X
J
mm.
mm
mm.
mm.
mm.
M0f.
mm.
MM.
mm.
mm*
mm.
mm.
mm.
4cA)0'
71.2
.1
142.3
•3
2135
.6
284.6
I.l
355-8
>.7
427.0
24
lO
"'rf^'
71.0
.1
142.0
•3
212.9
.6
283.9
l.I
354.9
1-7
425-9
24
20
70.8
.1
I4I.6
.3
212.4
.6
283.2
I.I
354.0
1.7
424.0
423.1
24
30
277.6
70.6
.1
I4I.3
•3
21 1.9
.6
282.6
I.I
353-2
1-7
24
40
370.1
70.5
.1
140.9
•3
21 1.4
.6
281.8
I.l
352.3
1.7
422.8
24
SO
462.6
70.3
•I
140.6
•3
210.8
.6
28I.I
I.I
3514
1-7
421.7
2.4
41 00
70.1
.1
140.2
•3
210.3
209.8
.6
2804
l.I
350.6
1.7
420.7
24
10
ifil
i
139.9
•3
.6
279.7
I.l
^l
1.7
410.6
418.5
24
20
.1
139.5
•3
208.7
.6
279.0
278.3
277.6
l.I
1.7
24
30
277.6
69.6
.1
'32-2
•3
.6
l.I
347.9
1.7
417.5
24
40
370.2
69.4
.1
138.8
.3
208.2
.6
l.I
347.0
1.7
4164
24
50
462.7
69.2
•'
138.4
•3
207.7
.6
276.9
l.I
346.1
1.7
415.3
24
4200
69.0
i-9
,1
138.1
•3
207.1
.6
276.2
l.I
345-2
1-7
414.2
24
10
02.6
I85.I
.1
137.7
•3
206.6
.6
275-4
l.I
344-3
1-7
413.2
24
20
^•7
.1
137.4
•3
206.0
.6
274.7
l.I
3434
1-7
41 2.1
2.4
30
277-7
^•5
.1
137.0
•3
205.5
.6
274.0
I.l
3424
1-7
410.9
24
40
370.2
^•3
.1
136.6
•3
204.9
.6
273-2
I.l
340.6
1-7
'^%
2.4
SO
462.8
68.1
•>
136.3
•3
204.4
.6
272-5
I.l
1.7
24
4300
68.0
135.9
•3
203.8
.6
271.8
l.I
f^l
x-7
407.7
24
10
02.6
185.2
67.8
.1
135-5
.3
203.3
.6
271.0
I.l
1-7
406.6
2.4
20
67.6
.1
1352
.3
202.7
.6
268.8
l.I
337-9
«-7
405.5
24
30
277.7
67.4
.1
134.8
•3
202.2
.6
l.I
337.0
1-7
404.4
24
40
370.3
67.2
.1
134.4
•3
201.6
.6
l.I
336-0
1-7
403.3
24
SO
462.9
67.0
•'
134.0
•3
201. X
.6
26S.1
I.l
335-1
1-7
402.1
24
4400
66.8
,1
133.7
•3
200.5
.6
2674
l.I
334.2
1.7
401.0
24
10
Q2.6
185.2
66.6
.1
133.3
.3
200.0
.6
266.6
I.l
333-2
1-7
m
24
20
66.5
.1
132.9
.3
;p.i
.6
265.8
1.1
332.3
1-7
24
30
277.8
66.3
.1
132.6
.3
.6
265.1
I.l
33 '4
1-7
397-7
24
40
3704
66.1
.1
132.2
.3
198-3
.6
264.4
l.I
330-4
1-7
396-5
24
50
463.0
65.9
•I
131.8
•3
197-7
.6
263.6
l.I
329.5
1-7
3954
24
45 00
65.7
,1
131.4
•3
197.1
.6
262.8
l.I
328.6
1.7
394-3
24
10
92.6
185.2
65.5
.1
131.0
•3
196.6
.6
262.1
l.I
327.6
1.7
393-1
2.4
20
65.3
.1
130.6
.3
196.0
.6
261.3
l.I
326.6
1-7
39»-9
24
30
277.8
65.1
.1
130.3
.3
195-4
.6
260.5
2598
l.I
325.6
1.7
2.4
40
370.4
64.9
.1
129.9
•3
194.8
.6
l.I
324-7
1-7
309.6
24
50
463^
64.7
•*
129.5
.3
194.2
.6
259.0
l.I
323.7
1.7
3884
24
4600
64.6
,1
I2Q.I
128.7
.3
193.6
.6
258.2
I.l
322.8
1.7
357-3
24
10
02.6
185.3
64.4
.1
.3
193.^
.6
2574
I.l
321.8
1.7
386.2
24
20
64.2
128.3
.3
192.5
.6
256.6
l.I
320.8
1.7
3?S-o
24
30
277.9
^•2
.1
127.0
•3
191.9
.6
255-9
I.l
319.8
318.9
1-7
383-8
24
40
370.S
63.8
.1
127.6
•3
i9'.3
.6
255-1
I.l
17
382.7
24
SO
463.1
63-6
•'
127.2
•3
190.7
.6
254-3
l.I
317.9
1-7
381-5
24
4700
63.4
126.8
•3
100. 1
.6
253-5
l.I
316.9
1-7
380.3
24
10
92.6
185-3
63.2
126.4
.3
^§2-5
.6
252-7
l.I
315-9
1-7
379-1
24
20
$3-2
126.0
•3
'55-9
.6
251.9
l.I
3»4.9
1-7
377-9
24
30
277.9
62.8
125.6
.3
188.3
187.8
.6
251.1
l.I
313-9
1-7
376.7
2.4
40
370.6
62.6
125.2
.3
.6
250.4
I.l
313.0
1-7
375-5
24
SO
463.2
62.4
•*
124.8
•3
187.2
.6
249.6
l.I
312.0
1.7
374.3
24
4800
62.2
.1
124.4
.3
186.6
.6
248.8
l.I
311.0
1.7
3731
24
8«rrH«0MiAN Tablcs.
127
TAMX23.
CO*ORDINATBS FOR PROJBCTION OF MAPS. SCALB nAvr
[D«ivatlM of labk czphiaed on pp. Hii-M.]
CO-ORDINATSS OF DKVSXX)PSD PARALLEL FOR—
tylo^todo.
oe'loivltade.
So'lotvitade.
4o'kM«itade.
syioogitade.
lO
20
30
40
so
4900
10
20
30
40
50
5000
10
20
3Q
40
SO
5100
10
20
30
40
SO
poo
10
30
30
40
SO
53 00
10
30
30
40
SO
54 00
10
30
30
40
50
55 00
10
20
30
40
50
5600
18C3
278.0
370.6
463.3
02.7
18-54
278.0
370.7
4634
02.7
I8C4
278.1
370.8
4634
02.7
1854
278.1
463.6
02.7
1854
278.2
i8cs
278.2
371.0
463.7
02.8
i8cs
278.3
463.8
02.8
I85.S
278.3
463.8
62.2
62.0
61.8
61.6
614
61.2
61JO
60.8
6a6
604
6a2
6ao
59.8
59.S
S9-3
- -9
58-7
S^S
55-3
58.1
57.9
57.6
574
57-2
52-2
5618
5616
564
56.2
56.0
55.7
555
553
5S-»
549
54.6
54.4
54.2
54.0
53.8
53-6
53-3
53.1
52.9
52.7
52.4
52.2
52.0
1244
124.0
123.6
123.2
122.8
1224
122.0
121.6
121.1
I2a7
I2a3
1 19.9
1 19.5
iiai
1 18.7
1 18.2
117.8
1 174
1 17.0
1 16.6
11612
115.7
"5-3
1 14.9
114.5
1 14.0
113.6
1 13.2
112.8
1 1 2.3
1 1 1.9
111.5
111.0
iia6
iia2
109.7
100.3
108.9
1084
108.0
107.5
107.1
106.7
106.2
105.8
105-3
104.9
104.4
104.0
86.6
86.0
854
84.7
84.1
83.5
[82.9
82.3
81.7
81.1
8a5
79-9
in
78.0
7^
76.1
755
74.9
74.2
73-6
73.0
72.3
71.7
71.1
67.9
67.2
66.6
65.9
65.2
64.6
64.0
62.0
61.3
60.6
6ao
58.0
56.7
156.0
248.8
248.0
247.2
246.3
245-5
244.7
243-9
243.'
242.3
2414
240.6
239.8
238.2
237.3
236.5
235.7
23i8
234.0
233-2
232.3
231-5
230.6
229.8
228.9
228.1
227.2
826^4
225.5
224.6
223.8
222.9
222.1
221.2
22a3
219-5
218.6
217.7
2i6w8
216.0
21 5. 1
214.2
213-3
21 2. A
2II.0
2ia:
•9
208.0
311JO
3x0.0
30^
307.9
306.9
305.9
304-9
30;
301.8
300.8
299-8
2^.8
3966
295.6
294.6
293.6
.0
.0
X}
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
o
,0
.0
.0
.0
.0
.0
1.0
273-2
272.1
271.0
2<
267.7
266.6
265.6
264.4
263.4
262.2
261. 1
26ao
373-1
371.9
370.7
367.1
365-9
364.7
363-4
362.2
361.0
359-8
358.5
356^0
354.7
353-5
352.3
3SI-0
349.7
348.5
347-2
345-9
344-6
343-4
342.1
340.8
339-5
338.3
337.0
335-7
3344
333-i
331-8
330.5
329-2
326.6
325-3
323.?
322.6
321.3
320X)
3«8.7
3>7-3
316.0
3x4.6
3«3-3
312.0
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
2.4
2.4
24
24
24
24
24
24
24
24
24
24
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2-3
2.3
2.3
2.3
2-3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
8MITH80NIAII Tables. -
128
Digitized by V^OOQIC
Table 28.
OO-ORDINATBS FOR PROJECTION OF MAPS. 8CALB itAtv
p)efhPBtbB of tabla aplained on pp. Hii-lTi]
56^00'
10
20
30
40
SO
5700
10
20
30
40
50
5800
10
20
30
40
50
59 00
10
ao
30
40
50
6000
10
30
30
40
50
6100
10
20
30
40
50
6200
10
20
30
40
50
6300
10
20
30
40
50
€400
92.8
185.6
278.4
371.2
464^
92.8
i8c6
27I4
371.2
464.0
185.6
278.5
371.3
464.1
Q2J3
18C.7
278.5
464.2
02^
278.6
37M
464.2
02.9
37M
464-3
02.9
278.6
371.5
4644
278.7
371.6
4644
CO-OROINATBS OF DSVSLOPED PARALLEL FOR—
M/lM«hiid«.
52.0
51.8
51.6
51.3
51.I
50.9
50.6
5a2
50.0
49-7
49.S
49.3
48.6
48.4
48.1
47.9
47.7
474
47.2
47.0
46.7
gs
46.0
45.8
45-6
45-3
44.8
44.6
444
44-1
43.9
43-7
434
43.2
43-0
42.7
42.5
42.2
42X>
41.7
41.5
41.3
41^
4a8
M^koiitada.
104.0
103.6
103.1
102.6
102.2
I0I.8
IOI.3
loaS
ioa4
99.9
99-5
99.0
98.6
97.6
97.2
96.3
95.8
95.3
94.9
94.4
93.9
93.5
93-0
92.5
92.1
91.6
9a6
1.2
„>7
89.2
88.8
88.3
87^
86.9
864
85.9
85.4
84.9
84.S
84.0
83.S
83.0
82.5
82.0
81.6
so' longitucle.
156.0
I5S3
154.6
154.0
153.;
153.3
152.6
152.0
151.3
I5a6
149.9
140.2
148.5
147.8
147.2
146.J
145-8
145. 1
1444
143.7
143.0
142.3
141.6
140.9
I4a2
138.1
137.4
136.7
136.0
135.3
134.6
133-9
133.1
132.4
131.7
131.0
130.3
129.6
128.8
128.1
1274
126.7
126.0
125.2
124.5
123.8
123.1
122.3
40^10110111116.
1.0
1.0
I.O
1.0
1.0
1.0
■1.0
1.0
1.0
1.0
1.0
IJO
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
.9
.9
.9
•9
9
•9
•9
.9
.9
.9
.9
.9
.9
.9
•9
•9
•9
.9
.9
.9
.9
.9
.9
.9
.9
.9
so' loDgitodc
26ao
258.0
256.6
255.5
2544
253.2
252.1
251.0
245.8
248.7
247.6
2464
245.2
244.1
243.0
241.8
24a6
2:
237.2
236.0
234.8
233.6
232.5
231.3
230.2
229.0
227.8
226.6
225.4
224.2
223.1
221.9
22a7
219.6
218.4
217.2
216.0
214^
213.6
2124
211.2
210.0
208.8
207.5
206.3
205.1
203.9
.6
.6
.6
.6
5
5
5
5
5
5
-5
•5
5
'5
5
5
5
5
-5
5
5
'5
•5
5
5
'5
4
4
.4
.4
4
•4
4
4
•4
4
•4
4
4
.4
4
.4
4
4
4
3
3
3
1-3
lOloocitDde.
312.0
310.7
309.3
^1
3053
303.9
302.5
301.1
2908
2984
297.1
295.7
294.3
292.9
291.5
290.2
2^.8
2874
28do
284.6
283.2
281.8
2804
279.0
277-6
276.2
274-8
2734
271.9
270.5
269.1
266.3
264.8
263.5
262.0
260.6
259.1
257.7
256.3
254-8
2534
251.9
250.5
249.0
247.6
246.1
244.7
2-3
2.3
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.0
2jQ
2.0
2.0
2jO
2.0
2.0
2JO
2.0
2U>
2.0
2.0
1.9
1.9
1.9
1.9
l (j l i l 2t^d bV <
•MmMONiAN Tables.
129
Table 23.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE jvMv
[Derivation of table explained on pp. liii.-lviiL]
1*1
CO-ORDINATES OF DEVELOPED PARALLEL FOR- |
7!
1& longitude.
apf longitude.
30^ longitude.
40^ longitude.
5^ longitude.
1° longitude.
X
y
X
7
mm.
X
7
X
y
X
y
X
y
mm.
mm.
mm*
mtm.
MM.
mm.
MM.
MM.
mm.
mm.
mm.
64*»oo'
40.8
81.6
.2
122.3
I2I.6
120.9
163.1
1
.8
203.9
1-3
244.7
1-9
lO
20
278.7
40.5
40.3
81.I
80.6
.2
.2
162.2
161.2
202.7
2014
1.3
1.3
243.2
241.7
1-9
1.9
30
40.0
80.1
.2
1 20. 1
160.2
.8
2oa2
1-3
24a 2
1.9
40
371-6
39.8
79.6
.2
ii8!7
159.2
158.2
.8
199.0
1-3
238.8
1.9
50
464.S
39-6
79.1
.2
.8
197.8
1-3
2374
1-9
6500
39-3
78.6
.2
1 17.9
157.2
.8
196.6
1.3
2359
1.9
10
278.7
^§1
78.1
.2
1 17.2
156.2
.8
'95-3
1-3
234-4
11
20
4i
77.6
.2
116.5
'S5-3
.8
194.1
1-3
232.9
30
38-6
77.2
.2
115.7
'54-3
.8
192.9
1.3
231.5
1.8
40
371.6
38.3
76.7
.2
115.0
1 53-3
.8
191.6
«-3
230.0
1.8
SO
464.6
38.1
76.2
.2
1 14.2
1S2.3
.8
190.4
1-3
228.5
1.8
6600
37.8
.1
75-7
.2
1 13.5
1 1 2.8
•s
151.4
.8
189.2
188.0
1.3
227.0
1.8
10
'"02.9*
37-6
.0
75-2
.2
•4
150.4
.8
13
225.5
1.8
20
37.3
.0
74.7
.2
tl2.0
•4
143.4
148.4
.8
186.7
1.2
224.0
1.8
30
37-i
.0
74.2
.2
III.J
1 10.0
.4
.8
185.4
1.2
222.5
1.8
40
3717
36.8
.0
73-7
.2
•4
147.4
.8
1842
1.2
221.1
1.8
SO
464.6
36.6
.0
73.2
.2
109.8
.4
146.4
.8
183.0
1.2
219.6
1.8
6700
36.4
.0
72.7
.2
lOQ.O
108.3
107.6
.4
145.4
.8
181.8
1.2
218.I
1.8
10
02.9
36.1
.0
72.2
.2
.4
144.4
.8
180.5
1.2
216.6
1.7
20
35-8
.0
71.7
.2
.4
M3-4
.8
179.2
178.0
1.2
215.1
1.7
30
35.6
.0
71.2
.2
106.8
.4
142.4
J&
1.2
213.6
1.7
40
371-8
3S-4
.0
70.7
.2
106.0
.4
141-4
.8
176.8
1.2
212.1
1.7
SO
464.7
3SI
.0
70.2
.2
105.3
.4
140.4
.8
I7S-5
1.2
210.6
1.7
6800
34-8
.0
69.7
.2
104.6
.4
13^4
.8
174.2
1 2
209.1
17
la
93-0
in
34.6
.0
1'
.2
103.8
.4
•7
1730
1.2
207.6
1.7
20
34.4
.0
.2
103.0
•4
137.4
171.8
1.2
206.1
1-7
30
34-1
.0
68.2
.2
102.3
.4
1364
170.4
l.I
204.5
1.7
40
37'f
33.8
.0
67.7
.2
101.5
100.8
.4
1354
169.2
168.0
1.1
203.0
1.7
SO
464.8
33.6
.0
67.2
.2
•4
1344
I.I
201.5
1.6
6900
33-3
.0
f^7
.2
100.0
.4
133-4
166.7
I.I
200.0
1.6
10
93.0
185.9
278.0
^H
.0
66.2
.2
^^5
.4
1324
165.4
T.l
198.5
1.6
20
32.8
.0
65-7
.2
.4
131-3
164.2
l.I
197.0
1.6
30
32.6
.0
65.2
.2
97.7
•4
130.3
162.9
I.l
195-5
1.6
40
32.3
.0
64.7
.2
97.0
.4
120.3
128.3
161.6
1.1
194.0
1.6
SO
464.8
32.1
.0
64.1
.2
96.2
.4
160.4
l.I
192.4
1.6
7000
H
.0
63.6
.2
9S.S
.4
127.3
159.1
l.I
iQa9
189.4
1.6
10
93.0
185.9
278.9
31.6
.0
63.1
.2
94.7
.4
126.2
157.8
l.I
1.6
20
31-3
.0
62.6
.2
93-9
.4
125.2
156.6
1.1
187.9
1.6
30
3*i
.0
62.1
.2
93.2
.4
124.2
155.3
1.1
186.4
1.5
40
371-9
30.8
.0
61.6
.2
92.4
.4
123.2
154.0
I.l
184.8
1-5
SO
464.9
30.S
.0
61.1
.2
91.6
4
122.2
152.7
1.0
183.2
1-5
7100
30.3
.0
60.6
.2
90.9
.4
121.2
1 51.4
1.0
181.7
1-5
10
"m
30.0
.0
60.1
.2
•4
120.2
150.2
1.0
180.2
1.5
20
29.8
.0
59.6
.2
ii
.4
119.1
118.1
148.9
I.O
178.7
15
30
278.9
29.5
.0
Ts
.2
.4
147.6
1.0
177.1
1.5
40
371.9
29-3
.0
.2
87.8
.4
117.1
1
1464
1.0
175-6
^'S
SO
464.9
29.0
.0
58.0
.2
87.1
.4
116.1
.6
145.1
1.0
174.1
14
7200
28.8
.0
S7'S
.2
86.3
.4
115.0
.6
143-8
1.0
172.6
14
Smithsonian Tablcb.
130
TAMK28.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE itAtt-
[Darivmtion of table esplained on pp. IHi-lvL]
CO-ORDINATES OF DEVELOPED PARALLEL FOR— |
TJ .
Meridional di»
tanoea from
even degree
parallels.
i</ loogitode.
M/ Umgituda.
so' loogitnde.
^ longitude.
y/ longitude.
lOkmghada.
X
y
X
y
X
y
X
y
X
y
X
J
mm.
mm.
mm.
mm.
mm.
mm.
MM.
mm.
mm.
mm.
mm.
mm.
mm.
7^00'
28.8
.0
57.5
.2
!^3
•4
II5.0
.6
143.8
1.0
172.6
1.4
lO
"rUi'
^•5
.0
57.0
.2
85.5
.4
1 1 4.0
.6
142.5
1.0
17 1.0
14
20
28.2
.0
56.5
.2
84.7
•3
113^
.6
I4I.2
1.0
1694
14
30
279.0
28.0
.0
56.0
.2
839
.3
1 1 1.9
.6
1^1
1.0
167.9
1.4
40
372-0
277
.0
SS-5
.2
83.2
•3
1 10.9
.6
1.0
166.4
14
SO
465^
27.5
.0
54-9
.2
82.4
•3
109.9
.6
1374
1.0
164.8
14
7300
27.2
.0
S4.4
.2
81.6
•3
108.8
.6
136.0
•9
163.3
14
10
"m
27.0
.0
53-9
.1
8a8
.3
107.8
.6
134.8
•9
161.7
1.4
20
26.7
X)
53-4
.1
80.1
•3
106.8
.6
1334
.9
160.1
1-3
30
279.0
26.4
.0
52.9
.1
78*5
•3
105.7
.6
132.2
•9
158.6
»-3
40
372.0
26.2
.0
sii
.1
•3
104.7
.6
130.8
•9
157.0
1-3
SO
465.0
25.9
.0
.1
77-7
•3
103.6
.6
129.6
•9
155-5
1-3
7400
25.6
.0
5a8
.1
77.0
•3
Z02.6
.6
128.2
•9
153-9
1-3
10
"m
25.4
.0
.1
76.2
•3
10 1. 6
.6
127.0
•9
150.8
1-3
20
25.1
.0
503
.1
lit
•3
100.5
.6
125.6
.9
1-3
30
279-0
24-9
.0
49-7
.1
•3
^
.6
124.4
•9
149-2
1-3
40
372.0
24-6
.0
tS7
.1
73.8
•3
.6
123.0
•9
1477
1.2
SO
465.0
244
.0
.1
73-0
•3
97.4
•5
121.8
•9
1 46. 1
1.2
7500
^^l
.0
48.2
.1
72.3
•3
96^4
i2a4
.8
144.5
1.2
10
"m
23.8
.0
47.7
.1
71-5
.3
95-3
1 19.2
.8
143.0
1.2
20
23.6
.0
47.1
.1
70.7
•3
94.2
.5
117.8
.8
1414
1.2
30
279.1
23-3
.0
46.6
.1
69.9
•3
93-2
1 16.5
.8
138.2
1.2
40
372.1
23,0
.0
46.1
.1
t;
•3
92.2
115.2
.8
1.2
50
465.1
22.8
.0
45-S
.1
•3
91.1
113.8
.8
136.6
I.I
7600
22.5
.0
45.0
.1
^
•3
oao
89.0
112.6
.8
I35-I
I.I
10
"^r^'
22.2
.0
44.5
.1
•3
ni.2
S
>33-5
I.I
20
22.0
.0
44.0
.1
65.9
.3
87.9
im
.8
131-9
I.I
30
279.1
21.7
.0
43-4
.1
65.2
•3
86.9
85.8
.8
m
I.I
40
37*1
21.5
.0
42.9
.1
64.4
•3
107.3
.8
I.I
SO
465.1
21.2
.0
42.4
.1
63.6
•3
84.8
106.0
7
127.1
I.I
7700
2a9
.0
41.9
.1
62.8
•3
83.7
104.6
125.6
I.I
10
"\r.'
20.7
.0
4o!8
.1
62.0
•3
i^i
103.4
124.0
I.I
20
20.4
.0
.1
61.2
.3
81.6
102.0
122.4
1.0
30
279-1
20.1
.0
^
.1
6a4
•3
80.6
• J
100.7
120.8
1.0
40
372.2
19.9
.0
.1
^.1
•3
m
A
98.0
"9-3
1.0
SO
465.2
19^6
.0
39-2
.1
•3
.4
117.7
1.0
7800
19.4
.0
3|-7
.1
58/)
.2
77.4
.4
96.8
116.1
1.0
10
"\ii'
\ti
.0
38.2
.1
57.2
.2
76.3
•4
954
114.5
1.0
20
.0
37.6
.1
56.5
.2
75-3
4
94.,
112.9
I.O
30
279.1
18.6
.0
37'J
.1
55-7
.2
74.2
.4
92.8
HI.4
1.0
40
372.2
18.3
.0
36.6
.1
54.9
.2
73-2
•4
91.4
.6
10Q.7
108.1
.9
SO
465.2
18.0
JO
36.0
.1
54.1
.2
72.1
•4
90.1
.6
•9
7900
17.8
.0
35-5
.1
53-3
.2
71.0
•4
88.8
.6
106.6
•9
10
"'^^^'
t7-S
.0
350
.1
52.5
.2
70.0
•4
ll'^
.6
104.9
•9
20
17.2
.0
34.5
.1
5*7
.2
68.9
A
86.2
.6
103.4
•9
30
279.2
17.0
.0
33-9
.1
50.9
.2
4
84.8
.6
101.8
1
40
372.2
i6.7
.0
33-4
.1
50.1
.2
66.8
4
834
.6
loai
50
465.2
16.4
.0
32.9
.1
49.3
.2
65.7
4
82.2
jS
98.6
S
8000
16.2
.0
32.3
.1
48.5
.2
64.6
4
80.8
.6
97-0
£
Smitnsoiiian Tables.
131
Tablk 24.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ivW
[Derivation of table «xpfauoed en pp. Uli-lvi.]
ABSCISSAS OF DEVELOPED PARALLEL.
Iff
i-r
2Xf
lonsitode.
»s'
30'
longitiide.
ORDINATES OF
DEVELOPED
PARALLEL.
OnOO
10
20
30
40
SO
I 00
10
20
30
40
SO
200
10
20
30
40
50
300
10
20
30
40
50
400
10
20
30
40
SO
500
10
20
30
40
50
600
10
20
30
40
50
700
10
20
30
40
SO
800
2304
4«>-7
691.0
9214
1151^
230.4
460.7
691.0
9214
230.4
400.7
691.1
921.^
11S1.0
A60.8
691. 1
921.5
1151.9
691. 1
921.5
1151.9
ido
16.0
16.0
16.0
16.0
iS-9
15.9
15.9
15.9
15.9
15.9
«5-9
15.9
15.9
15.8
15.8
15.8
15.8
15.8
15.8
157
'57
157
»S-7
157
15.6
15.6
15.6
15.6
»5-5
»5-5
»5-5
>5-4
154
15.4
15-3
»5-3
15.2
15.2
15.2
»S-i
15.1
15.1
15.0
15.0
14.9
14.9
114^
2319
23«-9
2319
2319
2319
231.9
231-9
231.9
231.8
231.8
231.8
231.8
231.8
231.8
23«7
2317
23«7
231.6
231.6
231.6
231.5
231-5
23M
2314
2314
231.3
231.3
231.2
231.1
231.1
231.0
231.0
23ao
230.8
230.8
230.7
230.7
23a6
230.5
2304
2304
230.3
230.2
230.1
23ao
229.9
229.0
229.8
229.7
347.9
347.9
347-8
347-1
347.8
347-8
347-8
347-8
347.8
347-7
347-7
3477
347-7
347-6
347.6
347-5
347.5
347.5
3474
347.3
347-3
347.2
347.2
347.1
347.0
347.0
346.6
346.5
346.4
346.3
346.2
346.1
346.0
345.9
345.8
345-7
345-5
3454
345.3
345-2
345-0
344.9
344.5
344.6
344.5
SlllTN«OMIAII TaSLCS.
463.8
463.8
463.8
463.8
463.8
463.8
463.7
463.6
463.6
463.5
4634
4634
463.3
463.3
463.2
463.1
463.0
463,0
&
462.6
462.5
4624
462.3
462.2
462.1
462.0
461.8
461.7
461.6
4614
461.3
461.2
461.0
46a9
460.6
4604
46a2
460.0
459-9
459-7
459-5
4594
'32
579-8
579-8
579|
579-8
579-8
579-7
579-7
579-6
579-6
579-6
579-6
579-5
5794
5794
579.3
579-2
579-2
579-1
IL
578.5
5784
578.2
578.2
578.0
577-8
577-8
577.6
5774
577.3
577.1
577-S
576.8
576.6
5764
576.2
576.1
575.9
5757
575-5
575-3
575.0
574.8
574.6
5744
574.2
695-8
6957
695.7
695.6
695.6
695.5
695.5
695-5
6954
695-3
695-3
695.2
695.0
695.0
694.9
694.8
^l
694.6
6944
694.3
694.2
694.1
693.6
6934
693.3
693."
692.1
692.5
692.3
692.2
692.0
691.7
691.5
691-3
691.1
690.8
69a6
6904
69ai
689.8
689.6
689.3
689.0
5-
ox>
10
OjO
15
ao
20
ox>
25
ao
30
0.0
5
ox>
10
ox>
»5
0.0
20
ao
25
ai
30
ai
4^
ao
ao
ai
ai
0.1
a2
0.0
ao
0.1
0.1
a2
0.3
8<»
ao
ao
ai
0.2
0.3
a4
ao
ao
ao
ao
ao
ai
ao
0.0
ao
ai
ai
a2
ao
ao
ai
0.1
a2
0.3
ao
ao
ai
a2
0.3
04
ngilizedby^O ^=
Table 24.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE Tvkvw-
[DoiTation of table ocplained on pp. liii-lTL]
ABSCISSAS OF DEVSLOPSD PARALLEL.
kwcitada.
1&
IS'
loQgUade.
2&
25-
^
ORDINATES OF
DEVELOPED
PARALLEL.
8<^'
lO
20
30
40
50
900
10
20
30
40
50
1000
10
20
30
40
SO
II 00
10
20
30
40
SO
1200
10
20
30
40
SO
1300
10
20
30
40
SO
1400
10
20
30
40
SO
1500
10
20
30
40
SO
1600
a304
4608
6913
921.7
1152.1
2304
)i.
921
1 1 52.2
i
230.5
461.0
691.5
922.0
1152.4
14^
14.8
14.7
14.6
14.6
14.S
14.5
144
144
14.3
14.3
14.2
14.2
14.1
14.0
I4X>
13.9
13^
13.6
13.6
13s
134
134
13-3
13.2
13.2
13.X
13-0
12.9
12.8
12.8
12.6
12.5
12.5
124
"•3
12.2
12.1
12.0
;;i
11.8
11.6
111.5
229.7
229.6
229.5
2294
229.3
229.2
229.1
2200
228.9
228.7
228.6
228.5
228.4
228.3
228.2
228X>
227.0
227.8
227.7
227.5
2274
227.3
227.1
227.0
226.9
226.7
226.0
226.4
226^3
22d2
226.0
225.9
225.7
225.6
2254
225.2
225.1
224.9
224.7
224.6
224.4
224.2
224.1
223.9
223.7
223.5
223.3
223.2
223.0
344.5
344-4
344-2
344-1
343-9
343-S
343-6
343-4
343.3
343-1
343-0
342.8
342.6
342.4
342.3
342.1
341.9
341-7
341.S
341.3
341.1
340.9
340.7
340.5
340.3
340.1
339.9
339-7
3394
339-2
338-6
333.3
333.1
337-9
337-6
337.4
336.6
3364
336-1
335-f
335-6
335-3
335-0
334-7
334-5
4S94
459-2
4^S
458.6
4584
458.2
457.9
457-7
457.S
457.3
4S7.0
456.6
4564
456.1
455-8
455-6
4554
455-J
454-8
4S4.6
454.3
4S4-0
453-8
453-5
453-2
452.9
452.6
452.3
452.0
4S»-7
4514
451.1
450.8
450.5
450.2
449-8
449-S
448.8
448.5
448.1
447-8
4474
447-0
446.7
446.3
446.0
574-2
574-0
573-7
5734
573-2
S73-0
572.7
572.4
572.2
571.8
571.6
571-3
571.0
570.8
5704
57^i
569.8
569-5
'^
568.6
568.2
567.6
567.2
566.8
566.5
566.1
565.8
5654
564.6
564.2
563.9
563-5
563.*
562.7
561.8
5614
561.0
560.6
560.2
559-7
W^
5584
S57-9
5574
68ao
6887
688.4
688.1
687.8
687.S
687.2
686.9
686.6
686.2
685.9
685.6
S5-3
g4.9
684.1
683.8
6834
683.0
682.6
682.3
681.8
681.4
681.1
68a6
680.2
6798
67I9
678.5
678.1
677.6
676.2
675.7
57H
674.8
674.2
673.7
673-2
672.7
672.2
671.6
67 I.I
670.6
670.0
669.5
668.9
i
5
10
0.0
ai
IS
20
0.1
a2
25
30
0.4
0.5
*»
oo
ao
ai
a2
03
04
id»
12*
0.0
ai
a2
0.3
I4»
s
0.0
10
ai
IS
0.2
20
0.3
25
0.5
30
0.7
i6«
OuO
ai
a2
0.8
SP
ao
ai
ai
a2
0.3
0.5
II'
oo
ai
ai
a2
13"
ao
0.1
a2
0.3
0.5
0.7
IS"
OX}
ai
a2
0.3
SurmaoMiAii Tasuw.
133
Table 24.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE virW
[Derivatkm of table expUined od pp. liii-lvi.]
III
Sl^
ABSCISSAS OF DEVELOPED PARALLEL.
longitude.
longitude.
IS'
2</
^fT
ORDINATES OF
DEVELOPED
PARALLEL.
10
20
30
40
SO
1700
10
20
30
40
SO
1800
10
20
30
40
SO
1900
10
20
30
40
50
2000
10
20
30
40
SO
21 00
10
20
30
40
SO
2200
10
20
30
40
SO
2300
10
20
30
40
SO
2400
230.5
461. 1
691.6
922.1
1 1 52.6
230.6
461.1
691.6
922.2
1 1 52.8
230-6
461. 1
691.7
922.3
1 152.8
230.6
401.2
691.8
922.4
U53.0
23a6
461.2
691.9
922.5
"S3-I
230.6
461.3
692.0
922.6
"S3-2
230.7
461.4
692.0
922.7
"S34
230.7
461.4
692.1
922.8
"53-6
iii.S
1 1 1.4
1 1 1.3
111.2
III. I
1 1 1.0
iiao
iiaS
110.7
X10.6
1 10.5
1 10.4
1 10.3
iia2
1 10. 1
1 10.0
109.9
109.8
109.7
109.6
109.5
109.4
X09.2
109.1
X00.0
108.Q
108.8
108.7
108.5
1084
108.3
108.2
108. 1
107.9
107.8
107.7
107.6
107.4
107.3
107.2
107. 1
106.9
106.8
106.7
106.5
1064
106.3
106.1
106.0
223.0
222.8
222.6
222.4
222.2
222.0
221.8
221.6
221.4
221.2
221.0
220.8
22a6
220.4
220.2
220X>
219.8
219.6
2194
2iax
218.9
218.7
218.5
218.2
2l8.0
217.8
217.5
217.3
2I7.I
216.8
216.6
216.4
2I6.I
215.0
215.6
215.4
2I5.I
214.9
214.6
214.4
2I4.I
213.9
213.6
213.3
2I3.I
212.8
212.5
212.3
334.S
334-2
333-?
333.6
333.3
333-«
332.8
332.S
332.2
33>.9
33>.6
33«-3
331-0
330.6
330.3
330.0
329.7
3294
320.0
347
328.4
328.0
327.7
327.4
327.0
326.7
326.3
326.0
325.6
32S.3
324.9
324.S
324.2
323.8
323.4
323.1
322.7
322.3
321.9
321.6
321.2
320.8
320.4
32ao
319.6
318.8
3184
318.0
446.0
445.6
445.2
444.8
444.1
443.7
443-3
442.9
442.5
442.1
441.7
441.3
440.8
440.4
440.0
439.6
439.2
438.7
438.3
437.8
437.4
436.9
436.5
436.0
435-6
435.J
434.6
434-2
433.7
433-2
4327
432-2
43'.7
431-2
430.8
430.3
429.8
420.2
428.8
428.2
427.7
426.6
426.1
425.6
425-0
424.5
424.0
5574
557-0
556.5
556.0
555.6
SSS-i
SS4.6
SM-I
553-6
553.3
552.6
552.1
551-6
SS'-o
550.6
550.0
549-4
549.0
5484
547.8
546^^
546.1
545.6
545-0
544.4
543-8
543-3
542.7
542.1
541.5
540.9
540.5
539.6
ir.
537.8
536.6
536.0
535.3
534.6
534.0
533.3
532.0
532.0
S3'.3
530.0
530.0
668.9
668.3
667.8
667.2
666.7
666.1
664.9
664.3
663.7
663.1
662.5
661.9
661.3
66a7
66ao
tm
658.1
656!8
656.1
6554
654.7
654.1
652.6
652.0
651.2
650.5
649.8
t$4
647.6
646.9
646.1
644.6
6439
643.1
6424
641.6
640.8
640.0
636^
636.0
5-
10
15
20
25
30
i6*»
ao
ai
0.2
04
a6
o^
idP
0.0
0.1
0.2
0.6
a9
2XP
OJO
0.1
a2
04
0.7
1.0
22<»
5
ao
10
0.1
IS
0.3
20
0-5
25
0.7
30
I.I
ao
ai
0.3
x.i
17''
ao
ai
a2
SI
a8
19**
ao
0.1
0.2
04
0.6
0.9
OjO
0.1
0.3
o-S
0.7
1.0
23'
0.0
0.1
0.3
I.I
Digitized by LjUU^
SmTHsoNiAii Tablks.
134
TauM«
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE i^lvr
[Dtriratioo of tabte wplaiMd on pp. UU-ItL]
10
20
30
40
SO
2500
ID
20
30
40
SO
2600
10
20
30
40
50
27 00
10
20
30
40
SO
2800
10
20
30
40
SO
2900
10
20
30
40
SO
3000
10
20
30
40
SO
31 00
10
20
30
40
SO
3200
230.7
461.5
692.2
923.0
"537
692.3
923-1
"S3-8
230.8
461.6
692.4
923-2
1154.0
230.8
401.7
692.5
9233
1154.2
46li
692.7
923.6
"S4-S
231.0
461.9
692.9
923-9
1 154.8
ABSCISSAS OF DEVELOPED PARALLEL.
06.0
05.9
05.7
05.6
05^
05-3
05.2
05.0
04.9
04.7
04.6
04.4
04-3
04.1
04.0
03.8
03.7
03-5
03.4
03.2
03.1
02.0
02.S
02.6
02.5
02.3
02.1
02.0
01.8
01.7
01.5
01.3
01.2
01 .0
00.8
oa7
00.5
00.3
00.2
00.0
99-8
99.6
995
99-3
99.1
98.6
98.4
lO'
loDfitnde.
212.0
21 1.7
21 14
211.2
2ia9
210.6
2ia3
210.0
209.7
209.4
20Q.2
208.9
208.6
208.3
208.0
207.7
207.4
207.1
206.8
206.5
206.2
205.8
205.5
205.2
204.9
204.0
204.3
204.0
203.6
203.3
203.0
202.7
202.3
202.0
201.7
201.4
201.0
200.7
200.3
200.0
199.6
»99.3
T^
198.3
»97.9
197.6
197.2
196.9
IS-
longitude.
318.0
317.6
3*7.2
316.7
3«6.3
3«5-9
315-5
315-0
3«4.6
3<4.2
313-7
3>3-3
312.9
3«2.4
312.0
3"-5
3U.I
310.6
310.2
309-7
308.3
307.9
3074
306.9
306.4
3059
305.5
305.0
304.5
304.0
303-5
303-0
302.5
302.0
301.5
301.0
300.5
300.0
2995
299.0
298.4
297.9
297.4
296.9
296.3
295.8
2953
20^
longitude.
424.0
423.4
422,9
422.3
421.8
421.2
42a6
420.0
419.5
418.9
418.3
417-7
417.2
416.6
416.0
415-4
414.8
414.2
413.6
413.0
4' 2.3
41 1.7
411.1
410.5
409.8
4^6
407.9
407.
406.0
4054
404.7
404.0
403.4
402.7
402.0
4014
400.7
400.0
397.9
397.2
396.5
395-8
395-1
394.4
393-7
25'
kmgitade.
130.0
27-9
127.2
;26i5
25.8
125.0
24.4
23.6
22.9
;22.2
214
120.7
i20.0
I&4
17-7
17.0
16.2
15-4
14.6
13.8
»3-»
12.3
11.5
ia7
509.9
507.5
506.7
505-8
505.0
504.2
5034
502.6
501.7
500.8
500/}
490.1
498.2
4974
496.5
495.6
494.8
493-9
493-0
492.2
30-
loQgitode.
636^)
6352
634.3
632.6
631.8
631.0
63ai
629.2
628.3
^d
624.8
623.9
623.0
622.1
621.2
62a3
6i8!5
61 7-5
616.6
615.7
614.8
613.8
612.8
611.9
61 a9
610.0
609.0
608.0
607.0
606.0
605.0
604.1
603.1
602.0
601.0
599-9
598.9
597.9
59S-i
594.8
593-8
592.7
591-6
590.6
ORDINATSS OF
DEVELOPED
PARALLEL.
u
10
IS
20
25
30
24'
ao
ai
0.3
tk
I.I
26»
OX)
ai
0.3
tk
1.2
28»
0.0
ai
0.9
1-3
30*^
ao
ai
0.9
1-3
32'
ao
a2
tl
0.9
1.4
^t
oo
0.1
0-3
tk
1.2
oo
ai
0.3
«f
OuO
SI
«-3
31"
ao
ai
0.6
0.9
1-3
Smitimgmiaii Tables.
^2^^
Table 24.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE |i»^.
CDtrlTBdon of uUe ftiplainwi oa pp. Uii-lvL]
l|
ABSCISSAS OF DEVELOPED PARALLEL.
1
*!
ORDINATES OF
Jl
5-
lO'
'S'
20'
25-
30^
DEVELOPED
PARALLEL.
^
loogitade.
longitnde.
UM^tade.
j___ii_j_
longitade.
loodtiidft.
mum.
MM.
MM.
MM.
MM.
MM.
MM.
fl
lO
211.0
984
196.9
196.5
295-3
294.8
393-7
393-0
492.2
491.2
S?5
32«
3f
20
98.1
97.9
I96.Z
195.8
294.2
293-7
392.3
391.6
30
093.0
4^4
5f7-3
MfW.
MM.
40
924.0
97.7
«95-4
293-»
390.8
488.0
586.3
s-
10
0.0
oo
y>
1155.0
97.5
195.1
292.6
390.1
487.6
585.2
a2
a2
33 «>
10
23IX>
^02.1
093-2
974
97.2
194.7
194.3
292.1
291.5
^
486.8
455-8
584.1
583.0
15
20
25
30
0-9
M
1.0
ao
30
H
194.0
193.6
290.9
^^
289.3
#7-9
387-2
484.9
484.0
§il
14
40
SO
924.2
1155.2
96.6
964
193-2
192.S
386.4
385.7
483.0
482.1
579-7
578.5
3400
^^
192.5
^•7
^5.0
481.2
5774
34^
35"
10
2|I.I
96.0
192.1
288.2
384.«
48a2
576.3
""^^
30
462.2
95-9
191.7
287.6
P
479.3
478.3
575-2
5
ao
ao
30
^3-2
95-7
i9»-3
287.0
574-0
10
a2
a2
40
924.3
95-5
190.9
286.4
285^
381.9
4774
572.8
15
ti
tt
so
11554
95-3
190.6
381.1
4764
571.7
20
25
1.0
1.0
3500
10
23I.I
95.1
94.9
i9a2
189.8
285.3
284.7
37^8
378.0
4754
474.5
570.5
567.0
30
1.4
M
20
30
693.4
94.7
94.5
1894
284.1
283.5
473-5
472.S
471.6
40
924.5
1 1 55.6
94.3
188.6
282.9
377.2
565.9
36*^
37**
SO
3600
94.1
93-9
188.2
187.8
2824
281.8
376.5
375-7
470.6
564.7
563.5
5
ao
oo
10
231.2
462.3
693-5
924.6
93-7
1874
281.2
374.9
562.3
10
0.2
a2
20
935
187.0
280.6
374-1
467.6
561.1
»5
a4
ti
30
93-3
186.6
280.0
373-3
466.6
559.9
558.7
20
a6
40
93-1
186.2
'm
372.5
465.6
25
1.0
IjO
SO
II55.8
92.9
185.8
371-7
464.6
557.5
30
1.4
i-S
3700
10
92.7
92.5
1854
185.0
278.2
277.6
370.9
370.1
462.6
556.3
555-1
4624
6936
38^
39^
20
30
923
92,1
184.6
184.2
276.9
276.3
1
461.6
460.5
552.6
40
924.8
91.9
183.8
275-7
459-5
458.5
55<4
5
10
ao
ao
SO
1156^
91.7
1834
275.1
366^
550.2
a2
a2
3800
91.5
183.0
274.5
273.8
366.0
4574
548.9
15
20
04
0.7
1.0
04
0.7
1.0
10
231.2
91-3
182.6
365-1
4564
547.7
25
20
91.1
1 82. 1
273.2
364.3
4554
5464
30
«-5
1-5
30
6937
90.9
18 1. 7
272.6
362.6
361.8
4544
545-2
40
925.0
1 1 50.2
90.7
181. 3
180.9
272.0
453-3
544.0
50
904
2714
452.2
542.7
. _o
3900
10
20
30
693.8
90.2
90.0
89.8
89.6
180.5
180.1
179-6
270.7
270.1
361.0
360.1
451-2
450.2
449.0
448.0
5414
540.2
538.9
537.6
4o'»
5
10
0.0
a2
•
40
925.1
11564
894
268.2
357-6
447.0
536.3
15
0.4
50
89.2
178.3
267.5
356.7
445-8
535.0
20
25
30
0.7
I.O
1.5
4000
89.0
177-9
266.9
355.8
444.8
533.8
Smithsonian Tables.
136
Table 24.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ivirv
[DariTition of table ezplaanad on pp. Uii^Ti.]
i
ABSCISSAS OF DEVELOPED PARALLEL.
"S .
111'
ORDINATES OF
DEVELOPED
PARALLEL.
5-
lO'
IS-
20'
25-
30-
^
jgasl
looghndo.
loofltiido.
longitiicM*
loapiod.
loiVitude.
loDsitiide.
mm.
HMM.
m$m.
mm.
mm.
mm.
«•««.
!l
4cA»'
lO
20
30
40
SO
IIS&8
88.5
88-3
88.1
87.9
177.9
177.5
176^6
176.2
1757
265.6
264.9
264.2
263.6
355-8
355-0
354.1
353.2
352.3
351.4
444.8
443.7
442.6
441.5
4404
439.3
533.8
5324
S3I.J
520.8
528.5
527.2
4rf>
4i*
10
0.0
a2
mm.
ao
a2
4100
10
ao
30
40
50
87.6
874
87.2
87.0
86.8
86.5
175-3
174.8
174.4
173-9
173-5
173-0
262.9
262.3
261.6
260.9
26a2
259.6
350.6
347.9
347.0
346.1
438.2
437.1
436.0
434.8
433.8
432.6
525.8
524.5
523.1
521.8
520.5
519.1
15
20
25
30
04
0.7
IJO
1-5
04
0.7
1.0
4200
10
20
30
40
50
••••••••
694.2
925.6
1 1 57.0
86.3
86.T
85.8
85.6
854
85.2
172.6
172.1
171.7
171.2
1708
170.3
258.9
258.2
257.6
256.9
256.2
255.5
345.2
344.3
343-4
342.|
340.7
431-5
4304
420.2
428.1
427.0
425.8
517.8
5164
515.1
513.7
512.3
511.0
42"
if
5
10
15
20
ao
a2
04
0.7
ao
0.2
0.4
0.7
4300
10
20
30
40
50
4400
10
20
30
40
50
2314
925-8
1157.2
463,0
694.4
925.9
1 157.4
84.9
84.7
84.5
84.2
84.0
83^
83.6
83.3
83.1
82.8
82.6
824
169.9
169.4
169.0
168.5
168.0
167.6
167.1
166.6
166.2
165.7
165.2
164.7
254.8
254.1
253-4
252.8
252.0
251-3
25a6
249.9
249.2
248.5
247.8
247.1
337.9
337.0
336.0
335.1
334.2
333.2
332.3
331-4
330.4
329-5
424.7
423.6
422.4
421.2
42ao
418.9
Hi
416.6
415.4
414.2
413.0
41 1.8
506.9
505.5
504.1
502.7
S01.3
499-9
498.5
497.0
495-6
494-2
-^5
30
'5
I.I
1.5
44**
4!^
5
10
15
20
25
30
ox>
0.2
04
0.7
I.I
ao
a2
04
0.7
I.I
1-5
45 00
10
20
30
40
SO
&
1 1 57.6
82.1
81.9
81.6
814
81.2
80.9
162.3
161 .9
246.4
245-7
245.0
244.2
243-5
242.8
328.5
326.6
3256
324.7
323-7
410.6
40Q.4
408.2
407.0
405.8
404.6
492.8
491.3
k
487.0
485.6
5
10
460
47^
0.0
a2
ao
a2
4600
10
20
30
40
50
231.6
4531
926.3
80.7
804
8a2
8ao
79-7
79-5
161.4
160.9
160.4
159.9
242.1
240.6
239-9
230.2
238.4
322.8
321.8
320.8
3ia.8
318.9
317.9
4034
402.2
401.0
39Q.8
398.6
397-4
484.1
482.7
481.2
479.8
15
20
25
30
0.4
0.7
1.1
1-5
0.4
0.7
1.1
1.5
48°
4700
10.
20
30
40
50
231.6
g3-2
1158.0
79.2
78.0
1585
158.0
157.5
157.0
156.5
156.0
237.7
236.9
236.2
235.5
234.7
234.0
316.9
315-9
314.9
314.0
3130
312.0
396.2
394-9
3936
392.4
391.2
390.0
475-4
473-9
472.4
470.9
469.4
467.9
5
10
»5
20
25
30
ao
0.2
a4
0.7
1.0
4800
77-7
155.5
233.2
311.0
388.7
4664
1-5
Smithsonian Tables.
137
Table 24.
CO-ORDINATES FOR PROJECTION OP MAPS. SCALE viiinr-
[Derlvrndon of table explamed on pp. lilMvi.]
48*'oo'
10
20
30
40
50
4900
10
20
30
40
SO
5000
10
20
30
40
50
51 00
10
20
30
40
SO
5a 00
10
20
30
40
50
5300
10
20
30
40
50
5400
10
20
30
40
SO
5500
10
20
30
40
SO
5600
If!
231.6
463-3
92&6
1 1 58.2
2317
463-^
926!8
1158-4
^5-1
2317
463.5
695-2
926.9
1x53.6
231.8
463-5
6953
1158.8
3.6
231.8
695-4
927.2
1159.0
463.8
695-7
927.6
1 1 59.4
231.9
927.7
1 159.6
ABSCISSAS OF DEVELOPED PARALLEL.
5'
77-7
77-5
77.2
77.0
76.7
76.5
76.2
76.0
75-7
75-4
75-2
74.9
74.7
74-*
74.2
73-9
73-6
734
73-1
72.9
72.6
72.3
71.8
71-5
71-3
71.0
70.7
70.5
70.2
69.9
69.7
694
^i
68.6
68.3
68.0
67.8
67.2
66.9
66.7
66.4
6di
65.8
65.6
65-3
65.0
10'
kmgitade.
155-5
155.0
154.5
154.0
153-5
152.9
152.4
151.9
151-4
150.9
150.4
149.9
148.8
148.3
147.8
!t^
146.2
145-7
145.2
144.7
144.1
143-6
143-1
142.5
142.0
141.5
140.9
1404
139-9
m
138.3
1377
137.2
136.6
136.1
135-5
135-0
134.4
»33-9
«33-3
132.8
132.2
"317
131.1
130.5
X30.0
15'
233-2
232.5
2317
230.9
230.2
229.4
228.7
227.9
227.1
226.4
225.6
224^
224X>
2233
222.5
221.7
22a9
22ai
2IQ.4
218.6
217.8
2x7.0
2X6.2
2154
214.6
213.8
213.0
2X2.2
21X4
3ia6
209.8
20Q.0
208.2
206.6
205.7
204.9
204.1
203.3
202.4
201.6
200.8
200.0
:p:3
195.8
195.0
20'
longitude.
3"-o
310.0
308.9
307-9
306.9
305-9
270.8
278.7
277.6
276.5
275-4
274.3
273-2
272.2
271.0
2(
267.8
266.6
265.5
264.4
263.3
262.2
261. X
260.0
25-
longitude.
388.7
357-4
386.2
384.9
383.6
3824
381.1
378.6
377-2
376.0
374.7
373-4
372.x
370.8
366.9
365-6
3643
^P%
361.6
3604
359-0
3577
356.4
355-0
3537
3524
35'i-o
349.7
347-0
345-6
344.2
342.9
341.6
340-2
338.8
3374
336.0
334.7
333-3
33»-9
330.5
329.2
327.8
326.4
3250
30'
loqgitode.
466.4
464.9
4634
461.9
St
457-3
455-8
454.3
452.7
451.1
449-6
448.1
446.5
445-0
443-4
441.8
440.3
4387
437-2
435-5
434-0
432.4
430.8
429.2
427.6
426.1
424-4
422.8
421.3
4x3.6
418.0
4x6.41
414.8
413-1
4XX.5
409.9
408.2
406.6
404.9
403.3
401.6
400.0
i$i
395-0
393-3
391.6
389.9
ORDINATES OF
DEVELOPED
PARALLEU
<
ao
10
0.2
«5
04
20
0.7
25
1.0
30
1-5
48**
ao
0.2
04
0.7
1.0
i-S
520
ao
a2
04
0.7
1.0
1-5
ao
a2
ti
1.0
14
ao
a2
04
a6
x.o
14
4<f
ao
a2
04
0.7
1.0
'•5
5i»
ao
o^
04
0.7
i-S
sf
oo
o^
IJO
i-S
Sf
OlO
a2
IJO
14
811ITNS011IAN Tables.
138
Tabu 24.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE j^hj-
[Derivmtifm of taUe tzplained on pp. liii-lTi.]
lO
20
30
40
50
57 00
10
20
30
40
SO
5800
10
20
30
40
50
S9 0O
10
20
30
40
SO
6000
10
20
30
40
SO
61 00
10
20
30
40
SO
6200
10
20
30
40
50
6300
10
20
30
40
50
6400
ABSCISSAS OF DEVELOPED PARALLEL.
5'
loogttade.
232.0
403-9
695-9
927.9
"59-8
232.0
404.0
696UO
928.0
ii6ao
928.8
1161.0
928.9
1161.1
65.0
64.7
64^
64.2
63.6
$3.3
63.0
62.7
62.5
62.2
61.9
61.6
6..3
60.2
SM
S9.6
59-3
S8-4
57.8
S7-5
57-2
57.0
56.7
56.4
55.8
55-5
55.2
54.9
54-6
54.3
54.0
53-7
53-4
53-1
52^
52.S
52.2
51.6
5«-3
Si-o
loi^tnde.
130.0
1204
128.9
128.3
127.7
127.2
126.6
126.0
125.5
124.9
124.3
123.8
123.2
122.6
122.0
I2I.5
I2a9
I2a3
X19.7
119.2
118.6
1 18.0
1;^
1 16.3
1 1 5.7
115.1
1 14.5
"3-9
"3-3
1 1 2.7
IX 2.1
IX1.5
1x0.9
1 10.3
109.8
109.2
108.6
108.0
107.4
X06.8
I06l2
105.6
105.0
X04.4
103^
103.x
102.5
IOI.9
longitude.
20'
longitude.
95.0
94.1
93-3
92.4
9X.6
90^
89.9
111
85.6
84.8
83.9
83.1
82.2
81.4
8a5
70.6
78.7
77.9
77.0
76.x
75.3
74.4
72.6
7oi
70.0
^^
67.3
66.4
62.8
61.9
61.0
:60.x
59.2
58.3
57.4
55,6
X52.9
26ao
258.8
256.6
255.5
25M
253.2
252.1
251.0
240^
248.7
247-5
246.4
245.2
244.1
242.0
241.8
240.6
239.5
238.3
237.2
236.0
234-8
233.7
232.5
231.4
23a2
229.0
227.8
22d6
225.4
224.2
223.x
221.9
22a7
2x9.5
218.3
2x7.x
2x5.9
214.7
213.5
2x2.3
21X.X
200.9
208.7
207.5
206.3
205.x
203.9
25'
longitude.
325.0
323.6
322.2
320.8
319.4
318.0
316.6
315.I
3"3-7
3ioi
3094
308.0
306.6
305.1
303.6
302.2
300.8
299.4
297.9
296.4
295-0
293.6
292.1
2Qa6
209.2
286.2
284.8
283.3
281.8
28a3
278.8
277.4
275.8
274.4
272.9
27X.4
269.9
268.4
26619
2654
263.9
262.4
260.9
259-4
257.8
256.4
254.8
30'
longitude.
'3
386.6
384.9
383.2
381.5
379-9
378.1
3764
374.8
373.0
371.3
369-6
367.9
365.1
3644
362.7
361X)
359.2
357.5
355-7
354.0
352.3
3505
348.8
347.0
345-2
3434
341.7
340.0
338.2
3364
334.6
332.8
33'.o
329-3
327-5
325-7
3239
322.1
320.3
318.5
3'6.7
3 '4.9
313-1
3"-3
309.4
307.6
305.8
ORDINATES OF
DEVELOPED
PARALLEL.
5-
xo
15
20
25
30
S6o
0.0
0.2
tt
1.0
14
580
ao
0.2
IjO
14
6o<>
ao
0.1
0.0
0.9
1-3
620
ao
0.x
tl
0.9
».3
64^
0.0
0.1
0.3
ti
X.2
57*^
0.0
a2
t.l
1.0
14
59^
oo
ax
tl
0.9
1-3
6i«
ao
ai
U
0.9
1-3
63"
0.0
0.1
0.3
0.5
0.9
x.2
•MmMOHiAII Tablss.
139
:izedbyG00g ¥^
Table 24.
CO-ORDINATES FOR PROJECTION OF MAPS. SCALE ttW
[Derivation of taUe explained on pp. liii-lTi.]
ABSCISSAS OF DEVELOPED PARALLEL.
longitude.
longitude.
i-r
20'
longitude.
longitude.
3^
longitude.
ORDINATES OF
DEVELOPED
PARALLEL.
64O00'
10
20
30
40
50
6500
10
20
30
40
SO
6600
10
20
30
40
50
6700
10
20
30
40
50
6800
10
20
30
40
50
6900
10
20
30
40
50
7000
10
20
30
40
50
71 00
10
20
30
40
50
7200
232.2
929.0
1161.2
696.9
929.1
II6I.4
464^6
697.0
1161.6
464.8
697.1
929.5
1161.9
232-4
464.9
697.3
929.7
1 162.2
232.5
464.9
697.4
929.8
1162.3
51.0
50.7
50.4
49-8
49-4
4O
48.5
48.2
47.9
47.6
47-3
47.0
46.7
46.4
46.1
45-8
45-4
45.1
44-8
445
44.2
43-9
43-6
43-2
42.9
42.6
42.3
42.0
417
41.4
41.0
40.7
40.4
4ai
39.8
39-5
%
38.2
37-9
37.6
37-2
S?
36.3
35-9
101.9
101.3
100.7
100. 1
9^-9
98.3
97.7
96.4
95.8
95.2
94.6
940
93-4
92.7
92.1
91.5
90-9
89.0
88.4
87.7
87.1
86.5
85.9
85.2
84.6
84.0
83.4
82.7
82.1
8o!8
80.2
73.6
78.9
78.3
77.6
77.0
76.4
75-7
751
73-2
72.S
71.9
152.9
152.0
151.1
150.2
149.2
148.3
1474
146.5
145.6
1447
1437
142.8
141.9
141.
i4ao
I39-I
138.2
137.2
136.3
1354
1344
133.5
132.6
I3».6
130.7
129.8
128.8
127.9
126.9
126.0
125.0
1 24. 1
123.2
122.2
121.2
120.3
1 19.3
1 18.4
1 17.4
1 16.5
ii5.|
1 14.6
113.6
1 1 2.6
111.7
iia7
107.8
203.9
202.6
20X.4
200.2
199.0
197.8
196.6
1953
194.1
192.9
191.6
190.4
188.0
186.7
185.5
184.2
183.0
18X.8
180.5
170.2
178.0
176.8
174.2
1730
171.7
170.5
169.2
168.0
166.7
165.4
164.2
162.9
16X.6
i6a4
1 59. 1
'57-S
156.6
155.3
154.0
152.8
151.5
150.2
148.9
147.6
14613
1450
143.8
254.8
253.3
251.8
*5o-3
248.8
247.2
245.7
244.2
242.6
241. 1
230.6
238.0
236.5
235-0
233-4
231.8
228.8
227.2
225.6
224.0
222.5
221.0
219.4
217.8
216.2
214.6
213.1
21 1.6
208.4
206.8
205.2
203.6
202.0
20a 5
198.9
197-3
1957
194. 1
192.6
1 91.0
186.2
184.5
182.9
181.3
179.7
305.8
304.0
302.2
3004
296.D
294.8
293.0
291.2
289.3
287.5
285.7
283.8
281.9
28ai
278.2
2764
274.5
272.6
270.8
268.9
267.0
265.1
263.2
261.4
259-5
257.6
2557
2539
251.9
25ai
248.2
246.3
244.4
242.5
24a6
238.7
236.8
234.8
232.9
231.1
229.1
227.2
225.3
2234
221.4
219.5
2x7.6
215.6
5-
10
15
20
25
30
Smithsonian Tablin.
0.0
0.1
0.3
Si
1.2
66P
0.0
at
0.3
tl
I.I
68«
ao
at
0.3
0.5
0.7
I.I
70°
5
ao
xo
o.x
15
0.2
20
04
25
0.7
30
1.0
720
0.0
ai
a2
St
0-9
6f
ao
ai
0.3
6f
OjO
ai
0.3
6sP
ao
ai
0.3
o-S
0.7
1.0
71°
ao
ai
a2
04
0.7
0.9
140
gitized by ^30t?^"
CO-ORDINATES FOR PROJECTION OF MAPS.
[Derivaticm of table explained on pp. Iiut-ItL]
Tabu 24
SCALE TvHf.
i..
ABSCISSAS
OF DEVELOPED
PARALLEL.
1
•g
llfi
ORDINATES OP
•s^
DEVELOPED
3^
J3.
Sfl^a
5'
longitude.
10'
longitude.
longitude.
20'
longitude.
25^
longitude.
30'
longitude.
PARALLEL.
MTM.
mm.
mm.
mm.
mm.
mm.
mum.
^^
7jA)o'
35-9
719
107.8
143-8
1797
1 78.1
176.5
2x5.6
II
720
7f
10
"'2V2.5'
35-6
71.2
70.6
106.9
142.5
2x3.7
2II.8
2^
20
35-3
105.9
141.2
30
350
70.0
104.9
\m
1749
209.9
mtm.
mum.
40
929.9
34.6
&^
104.0
173-2
207.9
s'
ao
ao
50
1 1 62.4
34.3
103.0
137.3
X7I.6
206.0
xo
ax
ax
7300
34.0
68.0
102.0
136.0
170.0
204.1
IS
20
a2
a2
10
"2V2.5'
465.0
337
67.4
IOI.0
1347
'Si
202.x
25
20
30
33.4
330
^I
100. 1
97.1
1334
132.2
166.8
165.2
2oa2
X98.2
30
0.9
0.9
40
5P
1 162.6
327
32.4
§i
130.8
129.5
163.6
1 61 .9
196.3
194.3
74^
7f
7400
10
T"?
32.1
317
64..
96.2
95.2
128.2
127.0
160.3
X58.7
192.4
20
31.4
94.2
125.6
157.0
5
ao
0.0
30
697.6
3"i
62.2
93-2
124.3
155-4
153-8
X86.S
X84.6
10
ax
ax
40
1 162.6
30.8
61.5
92.3
123.0
IS
0.2
a2
50
7500
30.4
301
60.9
60.2
91-3
1 21. 8
120.4
152.2
150.6
182.6
X80.7
20
25
30
a8
0.3
10
M
29.8
50.6
56.9
89.3
1 19. 1
148.9
178.7
20
29.4
11
P
H7.8
1 16.5
147.2
145.6
176.7
30
.74.S
76°
t^iO
40
ii&is
86.4
1 1 5.2
X44.0
172.8
77
50
28.5
56.9
85.4
"39
142.4
X7a8
7600
28.1
Si
84.4
1 1 2.6
140.7
X68.8
s
xo
ao
ax
ao
0.1
10
465^1
27.8
83.4
1 1 1.2
1390
X66.9
IS
20
25
30
a2
0.2
20
30
27.5
55.0
54.3
82.4
81.4
m
137.4
135-8
164.9
X62.9
0.3
0.5
0.7
0.3
0.5
0.7
40
SO
ii62i
26.8
26.5
26.2
537
530
80.5
79-5
78.5
107.3
106.0
134.2
132.5
130.8
1 61.0
159.0
77 00
523
104.7
157.0
10
232.6"
25.8
517
77.5
103.4
X29.2
155.0
78**
79**
20
30
25-5
25.2
51.0
50.4
76^5
755
74.6
102.0
100.7
127.6
125.9
153.1
151.X
40
9304
1 163.0
24.8
497
r^
X24.2
X49.1
5
ao
0.0
50
24.5
49.0
73.6
122.6
147.1
xo
ai
0.1
X5
0.2
ax
7800
24.2
48.4
72.6
96.8
121.0
145.1
20
0.3
0-3
10
20
"232.6'
465.2
239
23s
477
47.1
71.6
70.6
95-4
94.1
"9-3
117.6
143.2
141.2
25
30
ti
S2
30
697,8
23.2
46.4
60.6
68.6
67.6
92.8
1 16.0
139.2
40
50
9304
1 163.0
22.9
22.5
457
45.1
91.4
90.x
"4-3
1 1 2.6
137.2
X35-2
7900
10
232.6
465.2
22.2
21.9
44.4
437
66.6
65.6
88.8
52-5
IXI.O
109.4
107.6
«33-2
i3'-2
800
20
21.5
431
64.6
86.x
X29.2
5
ao
30
6979
21.2
42.4
63.6
84.8
106.0
127.2
xo
O.I
40
930.5
1163.1
20.9
41.7
62,6
83.5
104.4
102.6
125.2
15
O.X
50
20.5
41.1
6i.6
82.x
X23.2
20
0.2
25
a4
8000
20.2
40.4
60.6
80.8
XOIX)
121.2
30
0.5
81IITHSOMIAN Tablbb.
141
Tablc 25.
AREAS OP QUADRILATERALS OP EARTH'S SURFACE OF lO"" EXTENT
IN LATITUDE AND LONGITUDE.
[Derivrntion of taUe explained oa pp. 1-lfi.]
Middle
Latitude of
Quadrilateral.
Area in
Square Miles.
o*>
474653
5
472895
10
467631
«5
458891
20
446728
as
431213
30
412442
35
390533
40
365627
45
337890
SO
3075U
55
274714
6o
239730
65
202823
70
164279
75
124400
80
83504
85
41924
SiirrHaoNiAii Tables.
142
Digitized by
GooqIc
Digitized by VjOOQIC
Tabuc 26.
AREAS OF QUADRILATERALS OP EARTH'S SURFACE OF 1<
LATITUDE AND LONGITUDE.
EXTENT IN
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
o£ quadrilateraL
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
oooo'
3f>
1 oo
I 30
4752.33
4752.16
4751-63
4750-75
26P0O'
26 30
27 00
27 30
4282.50
4264.51
4246.20
4227.56
52^ oo'
52 30
53 00
53 30
2950.58
2851.68
2 00
2 30
3 00
3 30
4749-52
4747.93
4746^
4743-7'
28 00
28 30
29 00
29 30
4208.61
4189.33
54 00
54 30
55 00
55 30
2818.27
2784.^
^^•^
2716.67
4 00
4 30
5 00
5 30
4738^
4734.74
473»-04
30 00
30 30
31 00
31 30
4129.60
4109.06
4088.21
4067.05
56 00
56 30
57 00
57 30
2647*^5
2613.13
2578.19
6 00
630
7 00
7 30
4727.00
4722.61
4717.86
4712.76
32 00
32 30
33 00
33 30
4045.57
4023.79
4001.69
3979.30
58 00
58 30
59 00
59 30
2543.05
2507.70
2472.16
243642
8 00
8 30
9 00
9 30
4707.32
34 00
34 30
35 00
35 30
3956.59
3933-59
3910.28
3886.67
60 00
60 30
61 00
61 30
2364.34
2338.02
2291.51
10 00
10 30
IX 00
u 30
4667.32
465943
36 00
3630
37 00
37 30
3862.76
3814.06
3789.26
62 00
62 30
63 00
63 30
2254.82
2217.04
2143.66
12 00
12 30
13 00
13 30
4651.20
4642.63
4633-71
4624.44
38 00
38 30
39 00
39 30
3764.X8
3738A)
3687.18
64 00
64 30
65 00
65 30
2106.26
2068.68
2030.94
X993.04
14 00
14 30
15 00
"5 30
4614.82
4604.87
4583^92
40 00
40 30
41 00
41 30
366a95
3634-42
3607.62
3580.54
66 00
66 30
67 00
67 30
1954.97
'?*^75
16 00
16 30
17 00
17 30
4561.61
4549.94
4537.93
42 00
42 30
43 00
43 30
3553."7
3525.54
3497.62
3469.44
68 00
68 30
69 00
69 30
1801.16
X 762.33
1723.38
1684.24
18 00
18 30
19 00
19 30
4525.59
4512.QO
44 00
44 30
45 00
45 30
3440.98
3412.26
3383.27
3354.01
70 00
70 30
71 00
71 30
1645-00
1605.62
1566.10
1526.46
20 00
20 30
21 00
21 30
4472.81
4458.78
4444.41
4429.71
46 00
46 30
47 00
47 30
332449
3264.68
3234.39
72 00
72 30
73 00
73 30
X446ii
1406.81
X366.69
22 00
22 30
•23 00
23 30
44x4.67
4399.30
4383.80
4367.57
48 00
48 30
49 00
49 30
3203.84
3x73.04
3x41.99
3xia69
74 00
74 30
75 00
75 30
132646
I28dl2
1245.68
1205.13
24 00
24 30
25 00
25 30
4351.21
4334.52
4317.51
4300.17
50 00
50 30
51 00
51 30
3079.15
3047.37
2983^08
76 00
76 30
77 00
77 30
1 16449
1x23.75
1082.91
1041.99
Smitnsonian Tables.
144
Digitized byLjOOQlC
AREAS OP QUADRILATERALS OF EARTH'S SURFACE OF V
^ LATITUDE AND LONGITUDE.
[DeiivatioD of table ezpbiocd oo pp. 1-liL]
Tabu 26.
EXTENT IN
Middle latitude
of quadrilateraL
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
78^00'
78 30
79 00
79 30
80 00
80 30
81 00 •
81 30
iooa99
959-90
918.73
877.49
836.18
794.79
753.34
7".83
82O0O'
82 30
83 00
83 30
84 00
84 30
85 00
85 30
67a27
628.64
586.97
545-24
??^
419.81
377.93
86*>oo'
8630
87 00
87 30
88 00
88 30
89 00
89 30
336.02
294X>8
252.x I
2iai2
168.12
126.10
84/7
42.04
••■niMeiiMN Tabvu.
US
Digitized by
GooqIc
Tabuc 27.
AREAS OF QUADRILATERALS OP EARTH'S SURFACE OF SC EXTENT IN
LATITUDE AND LONGITUDE.
[DenYatum of table explained oo pp» HiL]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
d°oo'
o IS
30
45
1 188.10
Ii88x)8
ii88u)S
ii88xx>
13° 00'
13 15
13 30
13 45
115844
1157.29
1 1 56.1 2
"S4-93
26° 00'
26 IS
26 45
I07a64
1068.40
1066.14
1063.86
I 00
I »5
I 45
1187.92
1187^2
1187.70
1187.56
14 00
M 15
14 30
14 45
1153-72
115248
1151.23
1149.95
27 00
27 15 .
27 30
27 45
1061.56
1059.24
1056^
1054.54
2 00
2 IS
a 30
a 45
"87.39
1187.20
1186.99
1186.76
15 00
15 15
15 30
15 45
1148.65
"4733
"45-99
1144.63
28 00
28 IS
28 30
28 45
1052.16
1049.76
1047.34
1044.90
3 00
3 IS
3 30
3 45
1186.S1
1186.24
1185.62
16 00
16 IS
16 30
16 45
"43.25
1 141.84
1 140.41
1138.96
29 00
29 15
29 30
29 45
104244
1039.97
103747
1034.9s
4 00
4 15
4 30
4 45
1185.28
1184.92
»i|4.53
1184.13
17 00
17 15
17 30
17 45
"37.50
1136.00
"34.49
1132.9S
30 00
30 15
30 30
30 45
103241
1029.85
1027.27
1024.68
5 «>
5 15
5 30
5 45
1183.70
"83.24
1182.28
18 00
18 15
18 30
18 45
1131.^1
1120^3
1128.24
1126.63
31 00
3" 15
31 30
3» 45
lOt2J06
101943
1016.77
IOI4.IO
6 00
6 IS
6 30
6 45
1 181.76
1 181.22
1180.66
ii8ox)8
19 00
19 15
19 30
19 45
1124.98
1123.32
II2I.64
" 19.93
32 00
32 15
32 30
32 45
IOII40
1008.69
1005.90
1003.20
7 00
7 15
7 30
7 45
"7948
1178^5
1178.20
"7753
20 00
20 IS
20 30
20 45
11 18.21
111647
III4.7I
1 1 12.92
33 00
33 15
33 30
33 45
100043
997.64
994.83
992.00
8 00
f '5
8 30
8 45
1176.84
1 176.13
"75-39
"74^3
21 00
21 IS
21 30
21 45
IIII.II
1109.28
1107.44
1105.57
34 00
34 15
34 30
34 45
98341
980.50
9 00
9 15
9 30
9 45
1173.86
"73-06
1172.23
"71-39
22 00
22 IS
22 30
22 45
1103.68
1101.77
1099.84
1097.88
35 00
35 15
35 30
35 45
977.58
974.64
968,70
10 00
IP 15
10 30
10 45
1160.63
23 00
23 15
23 30
23 45
1095.91
1093-92
IOOI.QO
108957
36 00
36 15
36 30
36 45
^5-70
962.68
959.65
956.60
II 00
II IS
II 30
" 45
1166.84
1165.86
116^.86
1163.85
24 00
24 15
24 30
24 45
1087.81
1085.74
1083.64
1081.52
37 00
37 15
37 30
37 45
953.52
95043
947.32
944.21
12 00
12 IS
12 30
12 4S
1 162.81
1161.75
1 160.67
1159.56
25 00
25 15
25 30
25 45
1079.39
1077-23
1075.05
1072.85
38 00
38 IS
38 30
38 45
941. o<
937.88
934.71
931-51
Smithmnian Tables.
146
Digitized byLjOOQlC
Tablc 27.
AREAS OF QUADRILATERALS OP EARTH'S SURFACE OF 3<y EXTENT IN
^ LATITUDE AND LONGITUDE.
[Derivation of table explained on pp. 1-lii.]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
39-00'
39 15
39 30
39 45
928.29
925.06
921.80
9»8-53
52*^ oo'
52 IS
52 30
52 45
737.65
733-57
729.47
725.36
6foo'
65 15
65 30
65 45
507.74
493.51
40 00
40 15
40 30
40 45
915.25
911.94
908.61
905.27
53 00
53 15
53 30
53 45
721.23
717.08
11^^
66 00
66 5
66 30
66 45
488.75
483.97
479.19
474.40
41 00
41 IS
41 30
41 45
895.14
891-73
54 00
54 IS
54 30
54 45
704.57
700.38
696.16
691.94
67 00
67 15
67 30
67 45
469.60
464.78
459-96
455.13
42 00
42 15
42 30
42 45
888.30
881.39
877-9«
55 00
55 15
55 30
55 45
687.70
683.44
679.17
674.89
68 00
68 15
68 30
68 45
450.29
445-45
440.59
435.72
43 00
43 »5
43 30
43 45
874.41
870.90
56 00
56 15
56 30
56 45
670.60
666.29
69 00
69 15
69 30
69 45
430^4
425.96
421.06
416.16
44 00
44 15
44 30
44 45
860.25
856.67
853.07
84946
57 00
57 15
57 30
57 45
64^*93
644.55
640.17
70 00
70 IS
70 30
70 45
411.25
406.34
401.41
39647
45 00
45 15
45 30
45 45
|45^2
842.18
58 00
58 15
58 30
58 45
635.77
^'/^
626.93
622.49
71 00
71 15
71 30
71 45
381.62
376.65
46 00
46 15
46 30
46 45
831.13
823.18
819.94
59 00
59 15
.59 30
59 45
618.05
613.59
609.11
604.62
72 00
72 IS
72 30
72 45
36^70
361.71
356.71
47 00
47 15
47 30
47 45
816.18
8x2.40
808.60
804.79
60 00
60 15
60 30
60 45
6oai3
595.62
73 00
73 15
73 30
73 45
336^65
48 00
48 15
48 30
48 45
800.97
797.13
79327
789.39
61 00
61 15
61 30
61 45
582.01
568.30
74 00
74 15
74 30
74 45
331.62
326.58
316^48
49 00
49 »5
49 30
49 45
^78?:S
777.68
773.74
62 00
62 15
62 30
62 45
563.71
559."
549^
75 00
75 15
75 30
75 45
306.36
301.28
296.21
50 00
50 15
50 30
50 45
769.79
765.83
761.85
757.85
63 00
63 15
63 30
63 45
545-23
540.58
535.92
531.25
76 00
76 15
76 30
76 45
291.12
286.04
280.04
275.84
51 00
5» IS
51 30
51 45
7S3.84
749.82
745.78
741.72
64 00
64 15
64 30
64 45
517.17
51246
77 00
77 15
77 30
77 45
265.62
260.50
255-38
SliiTHaoNMN Tables.
147
Tabu 27.
AREAS OP QUADRILATERALS OF EARTH'S SURFACE OF 30" EXTENT IN
LATITUDE AND LONGITUDE.
[DerivatioD of table explained oo pp. HiL]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
78^00'
78 IS
78 30
78 45
2sa25
245.12
239.98
23i83
♦ 82^00'
82 15
82 30
82 45
167.57
162.37
157.16
151.9s
86*>oo'
86 IS
86 30
8645
84.01
78.76
79 00
79 '5
79 30
79 45
229.68
224.53
219.37
214.21
83 00
83 IS
83 30
83 45
146.74
141.53
136.31
'31*09
87 00
87 15
87 30
87 45
63^3
57.78
47.28
80 00
80 15
80 30
80 45
203.88
198.70
193-52
84 00
84 15
84 30
84 45
125.87
I2a64
115.42
1 10.18
88 00
88 45
26.27
81 00
81 15
81 30
81 45
188.34
183.11
177.96
172.77
85 00
85 15
8s 30
85 45
104.95
99-72
89 00
89 15
89 30
89 45
21.02
15.76
laSi
5.26
SnrrNaoHiAii Tables.
14S
Digitized by
GooqIc
Digitized by
GooqIc
Table 28.
AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 1B^ EXTENT IN
LATITUDE AND LONGITUDE.
[Derivadoii ol table esplaioed on pp. 1-liL]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
o°07'3o"
15 00
22 30
30 00
297^2
297X>2
297.02
297.01
6° 3/30"
6 45 00
6 52 30
7 00 00
295.09
295.02
294.95
294.87
13" 07-30"
13 15 00
13 22 30
13 30 00
28947
289x53
37 30
45 00
52 30
1 00 00
297.01
297.00
7 07 30
7 15 00
7 22 30
7 30 00
294.79
294.71
294.63
294.55
13 37 30
13 45 00
13 52 30
14 00 00
288^
288.7J
288.58
288.43
I 07 30
I 15 00
I 22 30
I 30 00
296.94
296.93
7 37 30
7 45 00
7 52 30
8 00 00
294.47
294.39
294.30
294.21
14 07 30
14 15 00
14 22 30
14 30 00
288.28
288.12
I 37 30
I 45 00
I 52 30
3 00 00
SOI
8 07 30
8 15 00
8 22 30
8 30 00
294-12
294.03
293-94
293.85
14 37 30
14 45 00
14 52 30
15 00 00
287.65
28749
287.33
287.17
2 07 30
2 15 00
2 22 30
2 30 00
296.82
296.80
296.77
296.75
8 37 30
8 45 00
8 52 30
9 00 00
293-66
293.56
293.47
15 07 30
15 15 00
15 22 30
15 30 00
287^00
28d83
28667
286.50
2 37 30
2 45 00
2 52 30
3 00 00
^72
296.66
296.63
9 07 30
9 15 00
9 22 30
9 30 00
29337
293.27
293.16
293.06
15 37 30
15 45 00
15 52 30
16 00 00
28633
286.I6
3 07 30
3 15 00
3 22 30
3 30 00
296.56
296.53
29649
9 37 30
9 45 00
9 52 30
10 00 00
292.05
292.85
292.74
292.63
16 07 30
16 15 00
16 22 30
16 30 00
285.64
*!546
285.28
285.10
3 37 30
3 45 00
3 52 30
4 00 00
29645
296.36
296.32
10 07 30
10 15 00
10 22 2f>
10 30 00
292.52
29241
292.30
292.19
16 37 30
16 45 00
16 52 30
17 00 00
284.92
284.38
4 07 30
4 15 00
4 22 30
4 30 00
296.28
296.23
296.1I
296.13
10 37 30
10 45 00
10 52 30
11 00 00
292.07
291.05
291^3
291.71
17 07 30
17 15 00
17 22 30
17 30 00
284.19
284.00
283.81
283.62
4 37 30
4 45 00
4 52 30
5 00 00
296.08
296.03
295.98
295.93
II 07 30
II 15 00
II 22 30
II 30 00
291.59
29147
291.34
291.22
17 37 30
17 45 00
17 52 30
18 00 00
28343
283.24
5 07 30
5 15 00
5 22 30
5 30 00
295.87
295.81
295.75
295.69
" 37 30
II 45 00
11 52 30
12 00 00
29I.Og
290.70
18 07 30
18 15 00
18 22 30
18 30 00
282.66
28246
282.26
282.06
5 37 30
5 45 00
5 52 30
00 00
295-63
29557
295.51
29544
12 07 30
12 15 00
12 22 30
12 30 00
290-57
290.44
290.30
290.17
18 37 30
18 45 00
18 52 30
19 00 00
281.86
281.66
28145
281.25
6 07 30
6 15 00
6 22 30
6 30 00
295-37
295-31
295.24
295.17
12 37 30
12 45 00
12 52 30
13 00 00
289.75
289.61
19 07 30
19 15 00
19 22 30
19 30 00
281.04
280.83
280.62
28041
Smithsonian Tables.
150
Digitized by VjUUV It:
Table 28.
AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 1B^ EXTENT IN
^ LATITUDE AND LONGITUDE.
[Dertvmtion of table explained oo pp. 1-Ui.]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
19^ 37' 30"
19 45 00
19 52 30
20 00 00
280.20
279-99
279-77
279-55
26<>o7'3o''
26 15 00
26 22 30
26 30 00
267.38
267.10
266.82
266.54
32° 37' 30"
32 45 00
32 52 30
33 00 00
251.15
25a8o
250.45
250.11
20 07 30
20 15 00
20 22 30
20 30 00
279-34
273.12
26 37 30
26 45 00
26 52 30
27 00 00
266.25
265.39
33 07 30
33 15 00
33 22 30
33 30 00
249.76
249.41
249.06
248.71
20 37 30
20 45 00
20 52 30
21 00 00
278.46
278.23
278xx>
277.78
27 07 30
27 15 00
27 22 30
27 30 00
265.10
264.81
264.52
264.23
33 37 30
33 45 00
33 52 30
34 00 00
248.36
248.00
247.65
247.29
21 07 30
21 15 00
21 22 30
21 30 00
277.55
277-32
27 37 30
27 45 00
27 52 30
28 00 00
|r4
263.34
263.04
34 07 30
34 15 00
34 22 30
34 30 00
246.93
246.57
246.21
245-85
21 37 30
21 45 00
21 52 30
22 00 00
276.63
276.39
276.16
275.92
28 07 30
28 15 00
28 22 30
28 30 00
262.74
262.44
262.14
261.84
34 37 30
34 45 00
34 52 30
35 00 00
245-49
245-»3
244.76
244.40
22 07 30
22 15 00
22 22 30
22 30 00
275.68
275-44
275.20
274.96
28 37 30
28 45 00
28 52 30
29 00 00
261.53
261.23
260.61
35 07 30
35 15 00
35 22 30
35 30 00
243.29
242.92
22 37 30
22 45 00
22 52 30
23 00 00
274.72
27447
274.22
273-98
29 07 30
29 15 00
29 22 30
29 30 00
260.30
259-99
259.68
259.37
35 37 30
35 45 00
35 52 30
36 00 00
242.55
242.18
241.80
241.43
23 07 30
23 15 00
23 22 30
23 30 00
273-73
273.48
273-23
272.^
29 37 30
29 45 00
29 52 30
30 00 00
250.05
258.74
258.42
258.X0
36 07 30
36 15 00
36 22 30
36 30 00
241.05
240.67
240.29
239-91
23 37 30
23 45 00
23 52 30
24 00 00
272.72
272,47
272.21
271.95
30 07 30
30 15 00
30 22 30
30 30 00
257.78
257.46
257.14
256.82
36 37 30
36 45 00
36 52 30
37 00 00
239.53
239.15
238.38
24 07 30
24 15 00
24 22 30
24 30 00
271.69
271.44
271.17
270.91
30 37 30
30 45 00
30 52 30
31 00 00
256.49
256.17
255.84
255.52
37 07 30
37 15 00
37 22 30
37 30 00
237.99
237.61
237.22
236.83
24 37 30
24 45 00
24 52 30
25 00 00
TJOM
270.38
27a 1 1
269.85
3« 07 30
31 15 00
31 22 30
31 30 00
254.53
254.19
37 37 30
37 45 00
37 52 30
38 00 00
236.44
236.05
235.66
235.26
25 07 30
25 15 00
25 22 30
25 30 00
269.58
269.31
31 37 30
31 45 00
31 52 30
32 00 00
25386
253-53
25319
252.85
38 07 30
38 15 00
38 22 30
38 30 00
234.87
234-47
233-68
25 37 30
25 45 00
25 52 30
26 00 00
32 07 30
32 15 00
32 22 30
32 30 00
252.51
252.17
251-83
251.49
38 37 30
38 45 00
38 52 30
39 00 00
233.28
232.88
23248
232.07
SnrrNaoNiAN Tables.
151
Tablk 28.
AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF IS' EXTENT IN
LATITUDE AND LONGITUDE.
[Derivatioo of table ex
plained OQ pp. 1-ifi.]
Middle latitude
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
39^0/30"
39 '5 00
39 22 30
39 30 00
231.67
231.27
230^
230^5
45° 3/30"
45 45 00
45 52 30
46 00 00
200.17
208.71
208.25
207.78
S2*>07'30"
52 15 00
52 22 30
52 30 00
183.90
182.37
39 37 30
39 45 00
39 52 30
40 00 00
230.04
229.63
220.22
228^1
46 07 30
46 15 00
46 22 30
46 30 00
206.39
205.92
52 37 30
52 45 00
52 52 30
53 00 00
180.82
180.31
40 07 30
40 15 00
40 22 30
40 30 00
228.40
227.99
227.57
227.15
46 37 30
46 45 00
46 52 30
47 00 00
205.45
204.99
204.52
204.05
53 07 30
53 15 00
53 22 30
53 30 00
179-79
179.27
178.75
178.23
40 37 30
40 45 00
40 52 30
41 00 00
226.73
226.32
225.90
225.48
47 07 30
47 15 00
47 22 30
47 30 00
203.57
203.10
202.63
202.15
53 37 30
53 45 00
53 52 30
54 00 00
177.71
176.67
176.14
4» 07 30
41 15 00
41 22 30
41 30 00
225.06
224.64
224.21
223.79
47 37 30
47 45 00
47 52 30
48 00 00
201.67
201.20
200.72
20a 24
54 07 30
54 15 00
54 22 30
54 30 00
175.62
175.10
174.57
174^
41 37 30
41 45 00
41 52 30
42 00 00
223.36
222.93
222.50
222.08
48 07 30
48 15 00
48 22 30
48 30 00
199.76
198.32
54 37 30
54 45 00
54 52 30
55 00 00
173.51
172.99
17246
171-93
42 07 30
42 15 00
42 22 30
42 30 00
221.65
221.21
220.78
22a35
48 37 30
48 45 00
48 52 30
49 00 00
197.83
196.38
55 07 30
55 «5 00
55 22 30
55 30 00
170M
170.33
169.79
42 37 30
42 45 00
42 52 30
43 00 00
219.91
21948
210.04
218.60
49 07 30
49 15 00
49 22 30
49 30 00
195.89
195.40
194.91
194.42
55 37 30
55 45 00
55 52 30
56 00 00
169.26
168.72
168.19
167.65
43 07 30
43 »5 00
43 22 30
43 30 00
218.16
217.73
217.28
216.84
49 37 30
49 45 00
49 52 30
50 00 00
193-93
193-44
192.94
192.45
56 07 30
56 15 00
56 22 30
56 30 00
167. II
166.03
16549
43 37 30
43 45 00
43 52 30
44 00 00
216.40
215.96
215,51
2tsJo6
50 07 30
50 15 00
SO 22 30
50 30 00
191.95
I9M6
190.96
19046
56 37 30
56 45 00
56 52 30
57 00 00
163^
163.32
44 07 30
44 15 00
44 22 30
44 30 00
214.61
214.17
213.72
213.27
50 37 30
50 45 00
50 52 30
51 00 00
189.96
18Q.46
188.96
188.46
57 07 30
57 15 00
57 22 30
57 30 00
162.78
162.23
161.68
161.14
44 37 30
44 45 00
44 52 30
45 00 00
212.82
212.37
211.9Z
211.46
51 07 30
51 »5 00
51 22 30
51 30 00
187.96
187.46
186.95
186.45
57 37 30
57 45 00
57 52 30
58 00 00
160.59
i6ao4
159.49
158.94
45 07 30
45 15 00
45 22 30
45 30 00
2tI.OO
210.55
210.09
209.63
51 37 30
51 45 00
51 52 30
52^ 00 00
185.94
185.43
184.92
18441
58 07 30
58 15 00
58 22 30
58 30 00
158.39
157^4
157.29
156^73
Smithsonian Tanlcs.
152
Digitized byLjOOQlC
Table 28.
AREAS OF QUADRILATERALS OP EARTH'S SURFACE OF 16' EXTENT IN
LATITUDE AND LONGITUDE.
[Derivation of table explained on pp. 1-lii.]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
58** 37' 30''
58 45 00
58 52 30
59 00 00
59 07 30
59 15 00
59 22 30
59 30 «>
59 37 30
59 45 00
39 52 30
00 00 00
60 07 30
60 15 00
60 22 30
60 30 00
60 37 30
60 45 00
60 52 30
61 00 00
61 07 30
61 15 00
61 22 30
61 30 00
61 37 30
61 45 00
61 52 30
62 00 00
62 07 30
62 15 00
62 22 30
62 30 00
62 37 30
62 45 00
62 52 30
63 00 00
$3 07 30
63 15 00
63 22 30
63 30 00
53 37 30
63 45 00
63 52 30
64 CO 00
64
64
07 30
15 00
22 30
30 00
37 30
45 00
52 30
65 00 00
156.18
155.62
155-07
154.51
15396
153-40
152.84
152.28
151.72
151-16
150.60
.150-03
14947
148.91
148.34
14777
147.21
146.64
146.07
145.50
144-93
14430
143-79
143.22
142.65
142.08
141-50
140.93
140.35
I39-78
13C62
13804
137.47
136.89
136.31
135.73
»35-iS
134.56
133-98
133-40
132.81
132.23
131.64
131-06
'3°-47
129.88
129.29
128.70
128.12
127-53
126.94
65** 07' 30"
65 15 00
65 22 30
65 30 00
65 37 30
65 45 00
65 52 30
66 00 00
66 07 30
66 15 00
66 22 30
66 30 00
66 37 30
66 45 00
66 52 30
67 00 00
67 07 30
67 15 00
67 22 30
67 30 00
67 37 30
67 45 00
67 52 30
68 00 00
68 07 30
68 15 00
68 22 30
68 30 00
68 37 30
68 45 00
68 52 30
69 00 00
69 07 30
69 15 00
69 22 30
69 30 00
69 37 30
69 45 00
69 52 30
70 00 00
70 07 30
70 15 00
70 22 30
70 30 00
70 37 30
70 45 00
70 52 30
71 00 00
71 07 30
71 15 00
71 22 30
71 30 00
26.34
25.75
25.16
24.57
2397
23.38
22.78
22.19
21.59
20.99
2040
19.80
19.20
18.60
18.00
1740
16.80
16.20
15.59
14.99
14.39
13.78
13.18
12.57
11.97
11-36
10.76
10.15
09.54
493
08.32
07.71
07.10
fsM
05-27
04.65
04.04
03.43
02.81
02.20
0159
00.97
00.35
99-74
99.12
97.88
96.65
96.03
95.41
710 37' 30-
71 45 00
71 52 30
72 00 00
72 07 30
72 15 00
72 22 30
72 30 00
72 37 30
72 45 00
72 52 30
73 00 00
73 07 30
73 15 00
73 22 30
73 30 00
73 37 30
73 45 00
73 52 30
74 00 00
74 07 30
74 15 00
74 22 30
74 30 00
74 37 30
74 45 00
74 52 30
75 00 00
75 07 30
75 15 00
75 22 30
75 30 00
75 37 30
75 45 00
52 30
00 00
^
76 07 30
76 15 00
76 22 30
76 30 00
76 37 30
76 45 00
76 52 30
77 00 00
77 07 30
77 15 00
77 22 30
77 30 00
77 37 30
77 45 00
77 52 30
78 00 00
94.78
94.16
93-54
92.92
f.M
91.05
90.43
89.80
89.18
88.55
87.93
87.30
86.67
86.05
85.42
84-79
84.16
8353
82.91
82.28
81.65
81.01
80.38
79.75
70.12
7849
77.86
77.22
76.59
75-95
75.32
74.69
74.05
7342
72-78
72-14
71.51
70.87
70.24
69.60
68.96
(8.32
67.&
67.04
66.41
65.77
65.13
63-85
63.20
62.56
SiimMONiAN Tables.
153
Tam^ 28.
AREAS OP QUADRILATERALS OP EARTH'S SURPACE OP 16' EXTENT IN
LATITUDE AND LONGITUDE.
a
»eriTmtion ol taUe txplaiMd on pp. 1-liL]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateraL
Areafai
square milts.
78^07' 30"
78 15 00
78 22 30
78 30 00
61.92
61.28
60.64
6aoo
8/>o7'30"
82 15 00
82 22 30
82 30 00
41.24
40.59
39.94
39-29
86^ 07' 30"
86 15 00
86 22 30
86 30 00
20.35
19.69
78 37 30
78 45 00
78 52 30
79 00 00
59-35
58.06
57^
82 37 30
82 45 00
82 52 30
83 00 00
38.64
37.99
86 37 30
86 45 00
86 52 30
87 00 00
17.72
17.07
1641
15.76
79 07 30
79 15 00
79 22 30
79 30 00
56.78
56.13
55-49
54-84
83 07 30
83 15 00
83 22 30
83 30 00
36.03
35-38
34.08
87 07 30
87 IS 00
87 22 30
87 30 00
15.10
14-44
13.79
13.13
79 37 30
79 45 00
79 52 30
80 00 00
54.20
53.55
52.91
52.26
83 37 30
83 45 00
83 52 30
84 00 00
3342
32.77
32.12
3*47
87 37 30
87 45 00
IJ 52 30
88 00 00
Il!82
11.16
ia5i
80 07 30
80 15 00
80 22 30
80 30 00
51.62
50.97
84 07 30
84 15 00
84 22 30
84 30 00
30.81
30.16
88 07 30
88 15 00
88 22 30
88 30 00
1^85
80 37 30
80 45 00
80 52 30
81 00 00
47.08
84 37 30
84 45 00
84 52 30
85 00 00
28.20
26.24
f! 37 30
88 45 00
88 52 30
89 00 00
7.»a
6-S7
S-?i
81 07 30
81 15 00
81 22 30
81 30 00
4644
4579
45.14
44-49
85 07 30
85 15 00
85 22 30
85 30 00
25.58
24.93
24.27
23.62
89 07 30
89 15 00
89 22 30
89 30 00
4.60
2.63
81 37 30
81 45 00
81 52 30
82 00 00
43.84
43.19
41^9
85 37 30
85 45 00
85 52 30
86 00 00
22.97
21.00
89 37 30
89 45 00
89 52 30
1.97
•■mMOMAN Taslm.
154
Digitized by
GooqIc
Digitized by
GooqIc
Table 29.
AREAS OF QUADRILATERALS OP EARTH'S SURFACE OF lO" EXTENT IN
LATITUDE AND LONGITUDE.
[Derivatkxi of table expUined on pp. l-UL]
Middle latitude
of quadii]ateral.
Area in
square miles.
Middle latitude
uf quadrilateral.
Area in
square mUes.
Middle latitude
of quadrilateral.
Area in 1
square miles.
0^05'
15
25
35
132.01
132.01
132.01
132.00
8-45'
8 55
9 OS
9 15
130.SJ
130.46
13040
130.34
17^ 25^
17 35
17 45
17 55
126111
126100
125.88
125.77
45
55
1 OS
> IS
132.00
i3'-99
9 25
9 35
9 45
9 55
I3a28
I3a22
130.15
130.09
18 OS
18 IS
18 2S
18 35
125.65
125.54
125.42
125.30
I 25
« 35
I 45
I 55
131-97
131.96
131-95
i3«-94
10 OS
10 IS
10 2S
10 35
130.02
129.06
129.89
129.82
18 45
18 5S
19 OS
19 IS
125.18
125.06
124.94
124.81
2 OS
2 IS
2 2S
« 35
i3»-93
13^-91
10 45
10 S5
11 OS
II IS
129.76
129.61
129.54
19 2S
19 35
19 45
19 55
124.69
124.56
124.44
124.31
« 45
2 55
3 OS
3 >5
131-86
131-84
131.82
131-80
II 2S
" 35
" 45
" 55
129.47
129.39
129.32
129.24
20 OS
20 IS
20 2S
20 35
124.18
124.05
123.92
123.79
3 25
3 35
3 45
3 55
131.78
131.76
i3'-74
i3'-7i
12 OS
12 IS
12 25
12 35
129.16
129.08
129.00
128.92
20 45
20 55
21 OS
21 IS
123.66
123.52
123-39
123.25
4 OS
4 15
4 25
4 35
131.68
131.66
131-63
131.60
12 45
12 ss
13 OS
13 15
128.84
128.76
128.67
I28.S9
21 2S
21 35
21 45
21 SS
123.12
122.98
I22i4
12270
4 45
4 55
5 OS
5 15
131-57
131-54
131.SO
i3«-47
13 25
13 35
13 45
13 55
128.50
128.41
128.33
128.24
22 OS
22 IS
22 2S
22 35
122.56
122.42
122.28
122.13
5 25
5 35
5 45
5 55
131-44
131.40
131.36
131.33
14 OS
14 15
14 2S
M 35
128.14
1 28.0 J
127.06
12737
22 45
22 55
23 OS
23 15
121.09
121.84
121.69
121.5s
6 OS
6 2S
6 35
131.29
131.25
131.21
131. 16
14 45
14 55
15 OS
15 »5
127.77
127.67
127.S8
127.48
23 2S
23 35
23 45
23 55
121.40
121.25
I2I.I0
I2a94
6 45
6 55
7 OS
7 15
131.12
131.07
131.03
130.98
IS 2S
15 35
»5 45
»5 55
127.38
127.28
127.18
127.08
24 OS
24 15
24 25
24 35
I2a79
I2a64
I2a48
'2a33
7 2S
7 35
7 45
7 55
130.84
130.79
16 OS
16 IS
16 2S
16 35
126.08
126.87
126.77
126.66
24 45
24 55
25 05
25 15
120.17
120.01
"9-85
119.69
8 OS
f '5
8 2S
8 35
130.63
130.57
16 45
16 SS
17 OS
17 IS
126.SS
126.44
126.33
126.22
25 2S
25 35
25 45
25 55
"9-53
"9.37
1 19.21
119.04
•igitizea Dy ^
156
Tablk 29«
AREAS OP QUADRILATERALS OP EARTH'S SURPACE OP lO" EXTENT IN
^ LATITUDE AND LONGITUDE.
[Derinuion of table explained on pp. 1-liL]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
26^05'
26 15
26 25
26 35
118.87
.118.71
118.54
118.37
34^45'
34 55
35 05
35 IS
108.94
108.73
108.51
108.29
43^*25-
43 35
43 45
43 55
96.50
96.24
95.98
9571
26 45
26 55
27 OS
27 1$
1 18.21
118.04
"7^7
117.69
35 25
35 35
35 45
35 55
108.07
107.85
107.63
107.41
44 05
44 15
44 25
44 35
95.45
95.19
94.92
94.65
27 25
27 35
27 45
27 55
117.52
"7-35
117.17
116.99
36 OS
36 IS
36 25
36 35
106.74
106.51
44 45
44 55
45 05
45 IS
94.38
94.11
93.84
93.58
28 05
28 IS
28 25
28 35
116^2
116.64
116.46
116.28
3645
36 55
37 05
37 IS
106.29
106.06
105.83
105.60
45 25
45 35
45 45
45 55
93.30
9303
92.78
9248
28 45
28 55
29 05
29 15
116.10
115.92
"5-73
"5-55
37 25
37 35
37 45
37 55
105-37
105.14
46 05
46 15
46 25
46 35
92.21
91.38
29 25
29 35
29 45
29 55
"5.37
115.18
114.99
114^1
38 05
38 IS
38 25
38 35
10444
104.21
103.97
103.74
46 45
46 55
47 OS
47 IS
91.10
9a82
90.55
90.27
30 OS
30 IS
30 25
30 35
114.62
114-43
114.24
H4.04
3845
38 55
39 OS
39 15
103.50
103.26
103.02
102.78
47 25
47 35
47 45
47 55
89.99
89.70
89.42
89.14
30 45
30 55
31 05
31 15
"347
113.27
39 25
39 35
39 45
39 55
102.54
102.30
102.06
101.82
48 05
48 15
48 25
48 35
88.85
88.00
31 25
31 35
3« 45
31 55
HIS
112.68
112.^8
40 OS
40 15
40 25
40 35
101.57
101.33
101.08
100.83
48 45
48 55
49 OS
49 15
87.71
87.42
32 05
32 15
32 25
32 35
112.28
112.08
1 1 1.87
1 1 1.67
40 45
40 55
41 OS
41 15
100.59
ioa34
100.09
99.84
49 25
49 35
49 45
49 55
till
32 45
32 55
33 05
33 15
111.47
1 1 1.26
111.06
110.85
41 25
41 35
41 45
41 55
99.59
99.33
9^:83
50 05
50 15
50 25
SO 35
85.39
8+80
84.50
33 25
33 35
33 45
33 55
110.64
110.43
110.22
iiaoi
42 05
• 42 15
42 25
42 35
98.57
98.06
97.80
50 45
50 55
51 05
51 15
84.21
83.31
34 OS
34 15
34 25
34 35
109.80
109.59
109.37
109.16
42 45
42 55
43 05
43 15
97.55
97.29
97.03
96.77
51 25
51 35
51 45
51 55
83.01
82.71
82.41
82.11
SMrrNSONiAN Tasixs.
157
Tam^ 29.
AREAS OF QUADRILATEIIALS OP EARTH'S SURFACE OF lO" EXTENT IN
LATITUDE AND LONGITUDE.
Middle latitude
afqaadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miks.
52^05-
52 15
52 2S
52 35
81.81
81.51
81.20
8a90
60^45-
60 55
61 05
61 IS
f5-i7
64.84
^^
64.16
69^25-
69 35
69 45
69 55
52 45
52 55
53 OS
S3 15
8060
8a20
79.98
61 25
61 35
61 45
61 55
63.82
6348
63.14
62.80
70 05
70 15
70 25
70 35
45.51
44-78
44.42
S3 25
53 35
S3 45
S3 55
79.37
7Q.06
7|75
78^
62 05
62 15
62 25
62 35
62.46
62.12
61.78
61.44
70 45
70 55
71 OS
71 15
445s
4^69
43-32
42.9s
54 OS
54 15
54 2S
54 35
7813
77.82
77.51
77.19
62 45
62 55
63 05
63 15
61.10
60.75
60.41
60.06
71 25
71 35
71 45
71 55
42.58
42.22
41.85
41.48
54 45
54 55
55 OS
55 IS
7^88
76.57
76.25
75-94
63 25
^3 35
63 45
63 55
59.72
59.37
72 05
72 15
72 25
72 35
41.11
40.74
40.37
55 25
55 35
55 45
55 55
75.62
7530
74.99
74.67
64 OS
64 15
64 25
64 35
58.33
57.99
57.64
57.29
72 45
72 55
73 05
73 15
38.52
56 OS
56 IS
56 2S
56 35
74.35
74.03
73.71
73-39
64 45
64 55
65 05
65 15
56.94
56.59
56.24
55.89
73 25
73 35
73 45
73 55
38.15
37.78
37.41
37.03
56 45
56 55
57 05
57 15
7307
72.75
72.43
72.10
65 25
65 35
65 45
65 55
55-54
55-19
54-83
54.48
74 OS
74 15
74 25
74 35
36^
36.29
35.91
35-54
57 25
57 35
57 45
57 55
71.78
71.46
7113
7a8o
66 05
66 15
66 25
66 35
54-13
53.78
53.42
53.06
74 45
74 55
75 05
75 15
35-17
34-79
34-42
3404
58 OS
58 15
58 25
58 35
70.48
69.82
69.49
66 55
67 OS
67 15
52.71
52.35
52.00
51.64
75 25
75 35
75 45
75 55
33-66
3329
32.91
32.53
58 45
58 55
59 OS
59 15
68.51
68.18
67 25
67 35
67 45
67 55
51.28
50.93
50.57
50.21
76 05
76 15
76 25
76 35
32.16
3>-78
31.40
31-03
59 25
59 35
59 45
59 55
67.84
f7-5'
67.18
66.85
68 OS
68 IS
68 25
68 35
49.85
49-49
49-13
48.77
76 45
76 55
77 05
77 15
30.65
30.27
29.89
29.51
60 05
60 15
60 25
60 35
66.51
66.18
65.84
65.51
^ ^5
68 55
69 05
69 15
48.41
48.05
47.69
47.33
77 25
77 35
77 45
77 55
mi
28.37
27.99
Smithsonian Taslcs.
158
Digitized byLjOOQlC
Table 29.
AREAS OF QUADRILATERALS OF EARTH'S SURFACE OF 10" EXTENT IN
LATITUDE AND LONGITUDE.
[Derimtion of uble explaioed on pp. 1-lii.]
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
Middle latitude
of quadrilateral.
Area in
square miles.
yfos-
78 15
78 35
27.62
27.24
26.85
26.47
82° OS'
82 15
82 25
82 35
18.43
18.04
17.65
17.27
86^05'
86 15
86 25
8635
Q.14
h
7-97
78 45
78 55
79 05
79 15
26.09
25.71
25-33
24.95
82 45
82 55
83 OS
83 15
16.88
16.50
16.11
15-73
8645
86 55
87 05
87 15
7.59
7.20
6^1
6.42
79 25
79 35
79 45
79 55
24.18
23.80
2342
83 25
83 35
83 45
^3 55
15-34
14.95
14-57
14.18
87 25
87 35
87 45
87 55
6.03
5.64
80 05
80 15
80 25
80 35
23.04
22.65
22.27
21.89
84 05
84 15
84 25
84 35
'3-79
13.40
13.02
12.63
55^5
88 15
88 25
88 35
447
4-09
3-70
3-31
80 45
80 55
8i 05
8i 15
21.50
21.12
20.73
20.35
84 45
84 55
85 OS
85 15
\ts6
11.47
11.08
5! *5
88 55
89 05
89 15
2.92
2.53
2.14
1-75
81 25
81 35
81 45
81 55
1997
19-58
19.20
18.81
8s 25
85 35
85 45
85 55
ia69
10.30
9.92
^53
89 25
89 35
89 45
89 55
1.36
0.19
SMrTNaoNiAN Tables.
IS9
Digitized by
GooqIc
Table 30.
DETERMINATION OP HEIGHTS BY THE BAROMETER.
Formula of Babinot.
Bo + B
C(in feet) = 52494 [i + ?a±I=l§4l — English Measures.
C (in metres) = 16000 I i H '^^^^^ I — Metric Measures.
In which Zss Difference of height of two stations in feet or metres.
^o, i9= Barometric readings at the lower and upper stations respectively, corrected for
all sources of instrumental error.
tot /= Air temperatures at the lower and upper statbns respectively.
Values of C.
ENGLISH MEASURES.
METRIC MEASURES.
H'o+O-
logC.
C.
F.
Feet.
icy>
4.69834
49928
'5
.70339
505"
20
.70837
5*094
25
.71330
51677
30
.71818
52261
39
4.7*300
52844
40
.72777
53428
45
.73248
54OII
50
.73715
54595
55
.74177
55178
eo
4.74633
55761
65
.75085
56344
70
'75532
56927
75
.75975
575"
80
.76413
58094
85
4.76847
58677
90
.77276
59260
95
.77702
59844
100
.78123
60427
*fe+0.
logC
C.
c
Metres.
— icy>
— 8
— 6
— 4
— 2
4.18639
.19000
.19357
.19712
.20063
'5360
15616
15744
15872
+2
t
8
4.20412
.20758
.21 lOI
.21442
.21780
16000
16128
16256
16384
16512
10
12
18
4.2211 C
.22448
.22778
.23106
.23431
16640
16768
16896
17024
17152
ao
22
'A
28
4.23754
.24075
.24393
.24709
.25022
17280
17408
17792
30
32
J8
4.25334
.25843
.25950
.26255
18176
18304
8iirrN«oiiiAN Taslks.
160
Digitized by
Google
MEAN REFRACTION.
Table 31 •
Refraction.
P
Refraction.
Refraction.
CO -t^
Refraction.
Refraction.
lO
20
40
10
20
30
40
50
30
40
10
20
30
40
4 o
ID
20
30
40
10
20
30
-2L
60
10
20
30
40
7 o
34 $4-1
3249.2
30523
29 3-5
27 22.7
25 49-8
2424.6
23 6.7
21 55.6
20 50.9
51.9
58.0
8-^:6
723.0
640.7
6 0.9
5234
4 47-8
4146
343-7
3'S-o
248.3
223.7
2 0.7
l^±
I 18.3
058.6
039.6
021.2
o 3-1
«i
94^.5
930.9
916.0
848.4
835.6
IlU
8 1 1.6
8 0.3
7 49-5
7 39^
729.2
719.7
134-9
1 16.9
108.8
100.8
9»-9
85.a
77-9
71. 1
64.7
59.0
53-9
7 o
10
20
30
40
JO
8 o
45-6
4>.3
39-8
37-5
35-6
33.a
30.9
a8.7
36.7
34.6
33.0
31.8
30.6
19.7
19.0
18.4
17.9
16.8
15.6
14.9
14. 1
X3.5
13.8
13.3
11.7
it.3
ie.8
10.3
XOiA
95
10
20
30
40
J2-
90
10
20
30
40
10
20
30
40
10
20
30
40
50
10
20
30
40
J^
13 o
10
20
30
40
J2.
14 o
7 '9-7
710.5
653.3
645.1
637.2
629.6
622.3
615.2
6 8.4
6 1.8
549:1
543.3
537.6
532.0
526.5
516.2
511.2
5 6.4
5 1.7
457.2
452.8
448.5
444-3
4 4a2
436.3
4324
428.7
425c
421.4
4 18.0
414.6
4 "3
4 8.1
4 4-9
4 1.8
358.8
3 55-9
353.0
IiO:2
347-4
14 o
20
40
347.4
342.1
1 .37-0
20
40
16 o
20
40
17 o
20
40
20
40
20
40
20 o
20
40
21 O
20
40
20
40
23
20
40
24 O
20
40
20
40
26 O
40
27 o
20
40
28 o
3 32-1
327-4
322.9
318-6
314.5
Ijoj
6.6
3 2.9
2 59:3
155:8
252.5
2 49-3
246.1
243.1
240.2
237.3
234.5
^29.3
226.8
2_2_4i3
2 21.9
2 19.6
2174
i^
213.0
2 10.9
2 8.9
2 7.0
2 14
'59.6
157.8
I 56.1
I 52.8
I 51.2
'49.7
148.2
28 o
20
29 o
20
40
30 o
20
40
20
40
il.
20
40
33 o
20
40
JHJ
20
40
35 o
20
40
36 o
20
40
iZ_
20
40
380
20
— ±'1
390
20
40
40 o
20
40
4' o
20
40
42 o
48.2
46.7
^3:3
43-8
42.4
41.0
397
38.4
iii
34.5
22±
3O.J
29.8
28.7
27.6
26.5
25-4
24.3
12:3
22.3
21.3
20.3
19.3
18.3
174
26,5
15.6
14.7
13^
12.9
12.0
10.3
_SJ
JiZ
7.9
7-1
5.5
4.7
4.0
64.0
61.8
59.7
57.7
53.8
51.9
CO. 2
i2
51
52
54
5Z
^^ 33.3
33.3
32.0
30.7
294
28.2
26.9
£iZ.
El
45.1
43-5
41.9
40.4
36.1
24.5
23.3
22.2
21.0
17.7
7S5"
15-5
14-S
134
12.3
1 1.2
ia2
I:;
4-1
ao
1.9
t.9
t.7
1.8
t-7
t.6
1.6
1.6
-5
.5
.0
4.0
4.«
SiimiaoiiiAN Tablkb.
161
Digitized by
GooqIc
Tablc 32.
FOR CONVERSION OP ARC INTO TIME.
o
h. m.
h. m.
h. m.
h. m.
h. m.
h. m.
/
m. s.
*f
s.
60
4
120
8
180
12
240
16
300
20
oooo
I
It
61
ti
121
5 i
181
12 4
241
16 4
16 8
301
20 4
20 8
I
4
I
ao67
2
62
122
8 8
182
12 8
242
302
2
8
2
0133
^
012
63
412
123
812
183
12 12
243
16 12
,303
2012
3
012
3
a2oo
4
016
64
416
124
816
^
12 16
244
1616
i^
2016
4
016
4
0.267
5
020
65
420
125
820
1220
245
1620
2020
5
020
5
0-333
6
024
66
424
126
824
186
1224
246
1624
306
2024
6
024
6
a40o
I
028
%
428
127
828
\'^
1228
247
1628
307
2028
I
028
\
0.467
032
432
128
S32
1232
248
1632
308
2032
032
0.600
II
036
69
436
129
836
189
190
1236
249
1636
W
20 3C
-rl
036
9
040
70
440
130
840
1240
250
1640
310
2040
040
10
0.007
044
7>
444
131
844
191
1244
1248
2SI
1644
1648
3"
2044
II
044
II
0-733
12
048
72
448
132
848
192
252
312
2048
12
048
12
0.800
13
052
73
452
133
5sf
193
1252
253
1652
313
2052
«3
052
13
0^67
U
056
I
^5
4 5^
5
I'S
856
9
1%
1256
13
2^
1656
17
^'5'
2056
21
1'^
056
I
il
0-933
1.000
i6
\i
5 4
•36
9 4
196
'3 i
256
'7 i
17 8
316
lit
16
\i
16
1^)67
\l
77
5 8
9 8
\t^
13 8
257
3'7
\l
«7
1.133
1 12
78
S'2
138
912
13 12
2S8
17 12
318
21 12
1 12
18
1.200
19
116
79
5^6
139
916
199
13 16
259
17 16
319
21 16
21 20
2124
-^
116
I 20
^
1.267
20
21
I 20
124
80
520
140
920
200
1320
2&)
261
17 20
1724
320
321
1-333
81
S24
141
924
201
1324
21
\U
21
1.400
22
128
82
528
142
928
202
1328
262
1728
322
2128
22
22
1.467
23
132
53
532
143
932
203
1332
263
1732
323
21 32
23
132
23
1.600
24
136
«4
536
144
9 3<>
^
^33^
264
1736
324
2136
^
136
ii
25
140
85
540
145
940
1340
265
1740
325
21 40
140
25
1.667
26
144
86
S44
146
9':^
206
1344
266
1744
326
2144
26
144
26
1-733
27
148
|g
548
'47
207
1348
267
'748
327
2148
27
148
27
1.800
28
I S2
552
148
952
208
«3 52
268
'752
328
2152
28
152
7&
1.867
29
156
89
556
1 49
956
10
209
1356
269
1756
18
330
331
21 56
22
22 4
29
156
29
'•933
2.000
30
2
90
6
150
210
14
270
30
31
2
30
31
2 4
91
6 4
'5*
10 4
211
14 4
14 8
271
'5 i
2 i
31
2.067
32
2 8
92
6 8
152
10 8
212
272
18 8
332
22 8
32
2 8
32
2-133
33
2 12
93
612
I S3
10 12
213
14 12
273
18 12
333
2212
33
2 12
33
2.200
^
216
220
^
616
620
1'^
10 16
1020
214
215
14 16
1420
27^5'
18 16
1820
^
22 16
2220
i^
216
220
^
2.267
2.333
36
224
96
624
156
1024
216
1424
276
1824
336
2224
36
lU
36
2.400
37
228
97
628
'57
1028
^^l
1428
277
1828
337
2228
37
^l
2.467
38
232
98
632
IS8
1032
218
1432
278
1832
3.38
2232
,38
232
38
2.600
2.667
2-733
_39
40
236
99
636
159
1036
1040
1044
219
220
M3<'
1440
279
280
281
1836
1840
^
223b
2240
2244
39
40
236
39
240
100
lOI
640
160
240
40
41
244
644
161
221
1444
1844
.341
41
244
41
42
248
102
648
162
1048
222
1448
282
1848
342
2248
42
248
42
2.800
43
252
103
652
163
10 52
223
1452
283
1852
343
2252
43
252
43
2.867
44
2S6
104
656
164
1056
224
1456
284
1856
.344
22 56
44
256
44
2-933
45
3
105
7
165
II
225
IS
2B5
19
345
23
45
3
45
3.000
46
3 4
106
7 i
166
II 4
226
'5 4
286
19 4
346
"3 4
46
3 i
46
3.067
47
3 «
107
7 8
167
II 8
227
15 8
^SZ
19 8
347
23 8
47
3 8
47
3133
48
312
108
712
168
II 12
228
IS12
288
19 12
348
2312
48
312
48
3.200
49
50
316
109
716
720
169
II 16
229
15 16
289
19 16
1920
1924
349
350
351
23 16
2320
2324
49
50
'5'
316
320
324
49 3-267
50 3.333
320
110
III
170
171
II 20
II 24
230
23*
1520
1524
290
291
SI
324
51
3.400
S2
328
112
728
172
II 28
232
1528
292
1928
352
2328
52
328
52
3-467
S3
332
"3
732
^73
II 32
233
1532
293
'932
353
2332
53
332
53
3.600
^5'
336
114
736
174
II 36
2^^
1536
2^15'
1936
3^5^
233^
^
33^
^5^
340
115
740
175
II 40
IS40
1940
2340
340
3.667
S6
344
116
7 44
176
1144
236
1544
296
1944
356
2344
56
344
56
3-733
34«
"7,
748
177
II 48
237
1548
297
1948
357
2348
57
348
57
3.800
58
3S2
118
752
178
1152
238
1552
298
1952
35«
2352
58
352
58
3.867
59
60
356
119
756
179
11 56
12
239
240
1556
299
J9J?
20
359
360
23 S^'
24
59
60
4
59
60
3i933_
4.000
4
120
8
IBO
16
300
Smithsonian Tables.
162
Digitized byLjOOQlC
Tabu 33.
FOR CONVERSION OP TIME INTO ARC.
Hours of Time into Arc.
Time.
Arc.
Time.
Arc
Time
Arc
Time.
Arc
Time.
Arc
Time
Azc
krs.
hrs.
Ars.
hrs.
hrs.
hrs.
1
15
5
75
9
135
13
195
17
255
21
315
2
30
6
90
10
:is
14
210
18
%
22
330
3
45
7
105
11
15
225
19
23
SI
4
60
8
120
12
180
16
240
20
300
24
Minut
es of Time int
Arc.
Seconds of Time into Arc.
Dl.
'
m.
'
m.
/
s.
f /f
s.
/ tf
8.
f tt
1
015
21
5»S
41
10 15
1
015
21
515
41
10 15
2
030
22
530
42
1030
2
030
22
530
42
1030
3
04s
23
VI
43
1045
3
045
23
\'l
43
1045
4
I
24
44
II
4
I
24
44
II
5
I 15
25
6 IS
45
II IS
5
I 15
25
615
45
" 15
6
»30
26
630
46
II 30
6
130
26
630
46
II 30
I
MS
2
%
645
7
%
"45
12
\
145
2
%
645
7
%
11 45
12
9
215
29
715
49
12 15
9
215
29
715
49
12 15
10
230
30
730
50
1230
10
230
30
730
50
1230
II
2 45
31
I'l
5'
1245
II
2 45
31
VI
5»
1245
12
3
32
52
13
12
3
32
52
13
13
315
33
V^
53
13 '5
13
3*5
33
v^
53
13 15
14
330
34
830
54
1330
14
330
34
830
54
1330
15
3 45
35
845
55
1345
15
3 45
35
845
55
1345
i6
4
36
9
56
14
16
4
36
9
56
14
15
415
37
915
14 15
^7
415
37
9»S
14 15
430
3»
930
58
1430
18
430
3»
930
1430
19
4 45
39
9 45
59
1445
19
4 45
39
9 45
59
1445
20
5
40
10
60
15
20
5
40
10
60
15
Hundr
edths of a Sec
ond of
Time into Arc.
Hundredths
of a Sec-
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
ond of Time.
s.
//
//
//
tf
f*
//
//
tf
tt
tt
0.00
0.00
0.IS
?iS
0.45
0.60
0.75
a9o
1.05
1.20
ai
.ID
1.50
1.6S
1.95
2.10
2.25
2.40
2-55
2.70
.20
3-00
3.15
l^
3-45
3.60
3-75
3-90
4.05
4.20
t^i
•30
4.50
4.65
4.95
6!6o
6.75
&90
5-55
5.70
.40
6.00
6.15
6.30
6.45
7.05
7.20
7.35
0.50
7.50
7.65
7.80
7.95
8.10
8.25
8.40
8.55
8.70
8.85
.60
9.00
9.15
9-30
laSo
9-45
9.60
9.75
9.90
10.05
10.20
11^5
■^
10.50
10.65
10.95
II. 10
11.25
11.40
"55
11.70
12.00
12.15
13-80
1245
12.60
12.75
12.90
1305
13.20
13-35
14^5
^
»3-5o
1365
1395
14.10
14-25
14.40
M-55
14.70
8iirrH«ONiAii Taslcs.
163
Digitized byLjOOQlC
Take 84.
CONVERSION OP MEAN TIME INTO SIDEREAL TIME.
m
m
m
m
8
I
2
3
h m s
h m t
h m t
h m •
s
m s
s
m s
O
000
6 S15
12 10 29
18 1544
oxx>
a5o
3 3
I
065
6 II 20
12 1634
18 21 49
aoi
4
0.51
3 6
2
12 10
61725
12 2240
182754
0.02
7
0.52
310
3
018 16
62330
122845
'8 33 59
0^3
Oil •
0.53
3 '4
4
02421
62936
"34 50
1840 5
ao4
018
022
0.54
3 '7
1
03026
03631
^35 4«
64146
124055
1247 I
18 46 10
18 52 15
ao6
a56
321
I
04237
6475J
1253 6
185820
0.07
026
0.57
04842
65356
12 59 II
19 426
ao8
029
0.58
332
9
054 47
702
'3 5«6
19 10 31
0.09
033
0.59
3 35
10
I 052
7 6 7
13 II 21
19 16 36
0.10
037
0.60
3 39
II
I 658
7 12 12
13 17 27
19 22 41
0.1 1
040
0.61
Vd
12
'*3 3
1 19 8
7 18 17
13 23 32
192847
ai2
044
0.62
«3
72423
73028
13 29 37
'93452
0.13
047
0.63
350
14
I 25 13
13 35 42
19 40 57
ai4
051
0.64
354
15
131 19
1$'^
13 41 48
1947 2
0.16
058
0.65
0.66
3 57
i6
I 3724
13 47 53
135358
14 3
'9 53 7
4 I
\l
14329
14934
74844
7 54 49
8 054
'9 59 '3
20 5 18
0.17
ai8
I 2
I 6
0.67
0^
U
«9
I 5540
14 6 9
20 1 1 23
20 17 28
ai9
I 9
a69
412
20
2 I 45
8 659
1412 14
0.20
' '3
0.70
416
21
2 750
813 5
14 18 19
202334
0.21
' '7
0.7'
4 '9
22
2135s
8 19 10
142424
202939
0.22
1 20
0.72
423
23
2 20 I
82515
143030
203544
a23
I 24
0.73
427
24
226 6
83120
'43635
20 4' 49
0.24
I 28
0.74
430
^
*3J"
83726
14 42 40
204755
tU
'3'
a76
434
23816
8433'
144845
2054
'35
438
S
24422
84936
14 54 5"
21 5
a28
'39
0.78
441
25027
85541
15 056
21 6 10
142
4 4S
29
2S6^2
9 147
IS 7 I
21 12 16
0-29
146
0.79
449
30
3 237
9 752
15 13 6
21 18 21
0.30
I 50
OJtiO
452
3«
3 843
3M48
9 '3 57
151912
21 24 26
0.31
' 53
0^1
456
32
920 2
15 25 17
21303'
0.32
'57
0^2
4 59
33
l^^
926 8
15 31 22
21 36 37
0.33
2 I
a83
5 3
34
9 32*3
938 «»
153727
21 42 42
0.34
^ i
o|4
5 7
11
3 33 3
154333
154938
21 48 47
a^
2 8
0^5
0^
S'o
3 39 9
94423
9 50 28
21 5452
2 II
5 '4
?
3 45M
16 I 48
22 058
a38
215
0^7
518
3 5« 19
95634
22 7 3
2213 8
219
0^
521
39
3 57 24
10 239
16 7 54
0.39
2 22
0^
525
40
4 330
10 844
16 13 59
22 19 13
0.40
226
a90
529
41
4 9 35
10 14 49
16 20 4
22 25 19
041
230
0.91
5 32
42
41540
10 20 55
1626 9
22 31 24
042
233
a92
536
43
42145
1027
16 32 14
223729
0-43
237
0-93
540
44
42751
1033 5
16 38 20
22 43 34
0^
241
0.94
5 43
*d
43356
440 I
10 39 10
10 45 16
164425
165030
22 49 39
22 55 45
t^
ajt
0.95
a95
5 47
5 5'
47
446 6
10 51 21
165635
23 ' 50
047
252
a97
5Si;
48
45212
10 57 26
*7 24'
23 7 55
o^8
255
0.98
1^
49
45817
" 331
17 846
23 '4
0.49
2 59
0.99
«;o
5 422
" 9 37
17 14 51
2320 6
0.50
3 3
1.00
6 5
5>
51027
II 1542
17 20 56
23 26 II
52
5>6 33
52238
II 21 47
1727 2
23 32 '6
Exa
iiiiple : Let
the given
32'.S6.
L mean
S3
II 27 52
1733 7
233821
time 1
bt 14^ 57-
54
52843
53448
"3358
17 39 '2
234427
The
i table give
»
II
II 40 3
II 46 8
17 45 17
235032
first f
or 14^54-
51" 2-2
7*
54054
17 5' 23
17 57 28
18 938
23 56 37
thenf
or 2
41 <
344
p
54659
II 52 13
*4 2^2
2 2
7-44
5 53 4
II 5819
24 848
Th«
( sum
59
5 59 9
12 424
24 '4 S3
"i^
32'.56+2-
required &
ide^lfdi
15* OP OP
ne.
60
6 515
12 10 29
18 15 44
242058
Smithsonian Tanlks.
164
Digitized byLjOOQlC
Tablk 86.
CONVERSION OP SIDEREAL. TIME INTO MEAN TIME.
m
m
m
m
8
I
2
3
h m s
h m t
h m s
h m s
•
m s
s
m •
O
000
6 615
12 12 29
18 1844
0.00
0.50
3 3
I
066
6 12 21
12 18 35
18 24 50
0.01
4
0.51
3 7
2
12 12
61827
122442
18 30 56
0.02
7
0.52
310
3
018 19
•6 24 33
123048
1837 2
0.03
II
0.53
l\t
4
02425
63040
123654
1843 9
0.04
^^5
018
0.54
1
03031
63646
1243
184915
0.05
0.00
0.56
321
03637
64252
1249 7
18 55 21
022
325
7
04244
64858
12 55 13
19 1 27
0.07
026
0.57
329
8
048 50
655 4
13 1 19
19 7 34
0.08
29
0.58
332
9
05456
7 I II
13 725
19 13 40
0.09
033
0.59
336
lO
I I 2
7 7 17
131331
19 19 46
0.10
037
0.60
340
II
1 7 9
7 1323
13 19 38
19 25 52
0.11
040
0.61
343
12
I 13 IS
71929
132544
19 31 59
0.12
044
0.62
3 47
13
I 19 21
72536
13 31 SO
1938 5
0.13
048
0.63
351
14
12527
73142
13 37 56
194411
0.14
051
0.64
3 54
;i
I 31 34
73748
1344 3
19 50 17
0.15
0.16
055
0.65
0.66
358
13740
7 43 54
1350 9
19 56 23
059
4 2
"2
14346
750 I
13 56 15
20 230
0.17
0.18
I 2
^•57
4 5
i8
14952
756 7
8 2 13
14 2 21
20 836
I 6
0.68
4 9
19
» 55 59
14 828
20 14 42
0.19
I 10
0.69
413
20
2 2 s
8 8 19
141434
20 20 48
0.20
1 13
0.70
416
21
2 8 II
8 1426
14 20 40
20 26 55
0.21
1 17
0.71
4 20
22
2 14 17
82032
14 26 46
2033 1
0.22
I 21
a72
424
23
22024
82638
14 32 53
2039 7
0.23
^^i
0.73
427
24
22630
83244
143859
20 45 13
0.24
I 28
0.74
431
y
23236
83851
1445 5
20 51 20
0.26
132
0.76
:p
23842
84457
14 51 11
20 57 26
135
^
24449
851 3
14 57 18
21 332
0.28
139
0.78
442
25055
857 9
15 324
21 938
146
446
29
257 I
Q ^16
IS 9.10
21 1545
0.29
0.79
449
lo
H ^ 7
9 922
MIS 36
21 21 51
0.30
I 50
0.80
4 53
31
3 914
91528
152143
21 27 57
0.31
154
0.81
4 57
32
31520
921 34
15 27 49
2134 3
0.32
157
0.82
5
33
321 26
92741
15 33 55
21 40 10
0.33
2 I
0.83
5 i
34
32732
9 33 47
'540 I
21 46 16
0.34
u
0.84
5 8
P
33338
9 39 53
1546 8
21 52 22
0.36
0.85
0.86
511
3 39 45
9 45 59
15 52 14
21 5828
2 12
515
37
3 45 5*
952 5
155820
16 426
22 4 35
0.37
2 16
^8?
519
38
3 5* 57
95812
22 10 41
0.38
219
5 22
39
358 3
10 4 18
16 10 33
22 16 47
0.39
223
0.89
526
40
4 410
10 10 24
16 16 39
22 22 53
0.40
226
0.90
530
41
4 10 16
10 16 30
16 22 45
. 22 29
0.41
230
0.91
5 33
42
4 1622
10 22 37
1628 51
2235 6
0.42
234
0.92
5 37
43
42228
10 28 43
163457
2241 12
0.43
237
0.93
541
44
42835
103449
16 41 4
22 47 18
0.44
241
0.94
5^
:i
43441
10 40 55
16 47 10
22 53 24
0.45
0.46
It^
0.95
548
44047
1047 2
16 53 16
22 59 31
0.96
552
%
44653
1053 8
16 59 22
23 5 37
047
252
0.97
5 55
4 53
10 59 14
17 529
23 11 43
0.48
256
0.98
6 3
4Q
4 59 6
II 5 20
17 II 35
23 17 49
0.49
259
0.99
SO
5 512
II II 27
17 17 41
23 23 56
0.50
3 3
1. 00
6 6
5«
5 II 18
11 17 33
17 23 47
2330 2
52
S3
S4
^1
51725
52331
52937
5 35 43
541 50
11 2339
II 2945
113552
II 41 58
II 48 4
172954
1736
1742 6
17 48 12
175419
2336 8
23 42 14
23 48 21
23 54 27
24 033
Exs
Th«
first £<
theni
anple: Gi
: table giv<
5r 14*57"
or 2
ven 15*0
i8* 2-
i? _
27-
0.44
57
54756
II 5410
18 025
24 639
5 diftereno
2 i
7-44
S3
554 2
608
12 17
18 631
241246
TTm
t
59
12 623
18 12 :^7
24 18 52
iq^O-C
uthe
)• — 2-27'..
required e
♦4 = 14*5
)7-32".S6
60
6 615
12 12 29
18 18 44
24 24 58
tiean time
.
8mith«omian Tables.
16s
Digitized byLjOOQlC
Table 36.
LENGTH OF ONE DEGREE OP THE MERIDIAN AT DIFFERENT
LATITUDES.
[DeriTftdon of table expbbaad on pp. xM-slvm.]
Latitude.
Metres.
Statute
MUes.
I'of theEq.
L«titB<fe
Metres.
Statute
Miles.
Gegg^hic
i' of the Eq.
I
110568.5
I 10568 J
68.703
68.704
59-594
59-594
49»
46
111132.1
111151.9
P
59.898
59.908
2
3
1 10569.8
110571.5
68.705
68.706
59-595
59.598
%
111171.6
111191.3
69-079
69-091
59.9«9
59-929
4
"05739
68.707
59-597
49
Iii2ia9
69.103
59-940
5
110577.0
1 10580.7
68.709
59.598
59.600
50
1 1 1230.5
69.115
59.951
6
68.711
51
69.127
59.961
I
110585.1
1 10590.2
68.714
68.717
^^
52
53
1112^13
69.139
69.163
59^9^2
9
1 10595.9
68.721
59.609
54
"1307-3
59.992
10
1 10602.3
68.725
59.612
55
111326.0
^•'25
60.002
II
1 10609.3
68.729
59.616
56
1 1 1344.5
69.18I
6aoi2
12
110617.0
68.734
59.620
P
"1362.7
69.198
60.022
13
1 10625.3
68.739
59.625
" 1380.7
69.209
6ox>32
14
1 10634.2
68.745
59.629
59
111398.4
69.220
6ao4i
15
16
iiHi
68.751
59-634
59.640
60
61
111415.7
111432.7
69.230
69.241
00.G00
17
59.646
62
1 1 14494
69.261
60.069
18
1 10675.7
68.770
59.652
!3
1 1 1465.7
^
19
1 10687.5
68.778
59.658
64
II 1481.5
69.271
20
1 10699.0
1 107 1 2.8
68.786
59.665
65
1 1 1497.0
69.281
60.094
21
68.794
59.672
66
111512.0
69.290
60.102
22
1 10726.2
68.862
%
1 1 1526.5
^^
60.110
23
ii074ai
68.810
59.6S6
"1 540.5
60.118
24
1 10754.4
68.819
59-694
69
111554.1
69.316
6ai25
25
110769.2
68.829
68.838
59.702
70
111567.1
69-324
6ai32
26
110784.5
59.710
71
" 1579.7
69-332
60.139
27
1 10800.2
68.848
59-719
72
111591.6
1 1 1603.0
69.340
60.145
28
110816.3
1 10832.8
68.858
59-727
73
69.347
60.151
29
68.868
59736
74
111613.9
69.354
60.157
30
1 10849.7
68.879
59.745
75
111624.1
69.360
60.163
6ai68
31
110866.9
68.889
59-755
59-764
76
" 1633.8
69.366
32
1 10884.4
68.900
77
1 1 1642.8
69.372
60.173
33
1 10902.3
68.911
59-774
59.784
78
111651.2
69.377
6ai77
34
1 10920.4
68.923
79
1 1 1659.0
69.382
6ai82
35
36
H0938.8
110957.4
iss
59-794
59.804
80
81
1 1 1666.2
1 1 1672.6
69.386
69.390
60.186
60.189
1 10976.3
lis
59.814
82
1 1 1688.1
69.394
60.192
"0995-3
59.824
53
69.397
60.195
39
111014.5
68.981
59-834
84
69.400
6ai97
40
IH033.9
68.993
59-845
85
111691.9
69402
60.199
41
1 1 1053.4
69.005
tl^
86
11 1695.0
69.404
6a20i
42
111073.0
69.017
§7
1116974
69.405
6a202
43
111092.6
69.029
59^76
88
1 1 1699.2
69.407
60.203
44
III 1 12.4
69.042
59^7
89
1 1 1700.2
69.407
6a204
45
II 1 132.1
69.054
59^
90
1 1 1700.6
69.407
6a204
8lllTM«0IIIAN TaBUCS.
166
Digitized by
Google
Table 37.
LENGTH OP ONE DEGREE OF THE PARALLEL AT DIPPBRENT
LATITUDES.
[Deriiration of taUe explained on
p.xUx.1
Latitude.
Metres.
Statute
MUes.
Geo^phic
I' of the Eq.
Latitude.
Metres.
Statute
Miles.
I' of the Eq.
OP
111321.9
69.171
60.000
45»
'^s
48.995
42.498
I
111305.2
69.162
59.991
46
48.135
47.261
46.372
41.753
2
3
1 1 1254.6
111170.4
69.130
69.078
59.964
59.018
59-855
%
76050.2
74628.5
40.994
40.223
4
1 1 1052.6
69.005
49
73«74.9
45.469
39.440
5
110901.2
68.911
59.773
50
71698.0
702oa8
44.552
43.621
38.644
6
1 107 16.2
f?-7§^
59673
59.550
51
37.837
37X>i8
I
9
1 10497.7
1 1 0245.8
109960.5
68.660
52
68681.1
42.676
&
ir4?
53
54
&
41.719
40.749
36.187
35.346
10
109641.9
68.128
li
55
63997.1
^t
34.493
33.630
iz
109290.1
108905.2
67.670
56
62395.7
12
58.697
P
60775.1
37.764
32.757
13
108036.6
67.411
58.472
59«35.7
57478.1
36.745
31.873
14
67.131
58.229
59
35.715
30.979
15
107553.1
66.830
57.969
60
55802,8
34.674
30.076
i6
1070370
1064^8.5
66.510
57.690
61
54110.2
33.622
20.164
28.243
17
66.169
65.808
57.395
57.082
62
52400.9
32-560
i8
105907.7
63
50675.4
31.488
27.313
19
105294.7
65.427
56.751
64
48934.3
30.406
26.374
20
104649.8
65.026
56.404
65
47178.0
20.315
28.215
25.428
21
103973.2
64.606
56.039
66
45407.1
24473
22
103265.0
64.166
55.657
^l
43622.2
27.106
23.511
23
24
102525.4
IOI754-6
63.706
63.227
55.259
54.843
68
69
41823.8
400124
24!862
If^
25
100953.0
62.729
54411
70
38188.6
23.729
40.583
26
iooi2a6
62.212
53-963
53.498
53016
71
3635^.0
22.589
»7.597
^
&
61.676
61.121
72
73
21.441
20.287
29
97441.9
6a548
52.519
74
3078a9
19^126
16.590
30
96489.3
59.956
52.006
75
28903.6
'^960
15.578
3«
95507.3
59-345
58.717
51.476
76
27017.4
25122.8
16^788
14.562
32
94496.2
50.931
11
15.611
13-541
33
93456.3
92387.9
58.071
50.371
232204
21310.8
14.428
\m
34
57.407
49.795
79
13.24a
35
QOi66!i
89014.8
87835.6
56.726
48.598
80
19394.6
12.051
10.453
36
56.027
81
174724
10.857
nil
7.337
P
55-3"
54.578
53.829
47.977
47.341
82
83
15544-7
13612.2
&
39
86629.6
46.691
84
11675.5
7.255
6.293
40
85397.0
84138.4
53.063
46.027
85
9735-1
6.049
5.247
41
52.281
45-349
86
7791-7
4.841
4.200
42
82854.0
51-483
44.656
u
»3
3.632
3.«5i
43
81544.2
50.669
43.950
2422
2.101
44
80209.4
49.840
43.231
89
1949.4
1.211
1.051
45
78850.0
48.995
42.498
90
0.0
0.000
aooo
6iiiTM«oiiiAN Tables.
167
Digitized by
Google
Tabu 88.
INTERCONVER8ION OP NAUTICAL AND STATUTE MILES.
I nudcal inile*=6o8o.37 feet
Nautical Miles.
Sutute Milea.
Statute Miles.
Nautical MUes.
1
2
3
4
5
6
i
9
1.1516
2.3031
5-7578
6.9093
8.0609
9.2124
ia3640
1
2
3
4
5
6
I
9
0.8684
1-7368
2.6052
3-4736
4.3420
&0788
6.0472
7.8155
8iiiTNaoNiAii Tabucs.
» Aa ddined hf the Umted States Coast and Geodetic Survey.
Table 39.
CONTINENTAL MEASURES OP LENGTH WITH THEIR METRIC AND
ENGLISH EQUIVALENTS.
The asterisk (^ indicates that the measure is obsolete or seldom need.
Measure.
£1, Netherlands
Fathom, Swedish = 6 feet
Foot, Austrian,*^
old French*
Russian
Rheinlandisch or Rhenish (Prussia,
Denmark, Norway) *
Swedish*
Spanish *=:^ vara
*Klafter, Wiener (Vienna)
*Line, old French =r\x 'oot
Mile, Austrian post* =24000 feet. . . .
German sea
Swedish ^ 36000 feet
Norwegian = 36000 feet
Netherlands (mijl)
Prussian (law of 1868)
Danish
Palm, Netherlands
♦Rode, Danish
*Ruthe, Prussian, Norwegian ......
Safene, Russian
♦Toise, old French = 6 feet
♦Vara, Spanish
Mexican
Werst, or versta, Russian = 500 sagene
Metric Equiyalent.
English Equivalent.
I
metre.
3.2808 feet
1.7814
i«
5.8445 "
0.31608
u
1.0370 "
1.0657 "
0.32484
u
0.30480
u
I
^Vf^
u
«
1.0297 «
0.9741 "
0.2786
<i
0.9140 "
1.89648
M
6.2221 "
0.22558
cm.
0.0888 inch.
7-58594
1.852
km.
it
4.714 statute miles.
1.1508 "
10.69
<4
6.642 "
11.2986
U
7.02 " "
I
M
0.6214 " «
7.500
U
4.660 **
7.5324
M
4.6804^ «
0.1
metre.
0.3281 feet.
3.7662
<i
12.356 «
37662
u
12.356 "
2.1336
«
7
1.9490
0.83W
0.8380
«
ti
6.3943 «
2.7424 "
2.7293 "
1.0668
km.
3500
Smithsonian Tables.
z68
Digitized by
Google
Tablk 40.
ACCBLBRATION ig) OP GRAVITY ON 8URPACE OF EARTH AND
DERIVED FUNCTIONS.
r= 9.77989 + o»o5«« «in«4
= 9.80599— o4)a6io 00* a^ OMlm.*
4 = seogimphical latitude.
f
g
log^
logi
logVi?
5^
AUirws.
Mtin*.
oP
9.7798
0^9033
8.70864-10
0.64568
0-99090
S
.7803
035
863
569
095
10
.7814
040
8S7
57*
106
'S
.7834
049
848
576
127
20
.7859
060
837
582
152
25
.7893
075
832
589
186
30
•7929
091
806
597
222
35
.7969
109
788
€06
264
40
.8014
129
768
616
309
45
.8060
149
748
026
355
. SO
.8105
169
738
636
401
55
^150
189
708
646
447
60
.8191
207
690
655
488
65
.8227
223
674
663
525
70
.8261
238
659
670
559
75
.8286
249
648
676
• 584
80
^306
258
639
680
60s
85
^317
263
634
683
616
90
^322
26s
632
684
621
8MiTMaoNiAii Tables.
• From Tke Solar ParaUojc and its R§laUd C^iisUmts, hf Wm. HarimoM, Prof eitor of
WaaliiiiKtoo : GoTernment Printing Office, 1891.
t This it length 61 leoondi pendulum.
169
U.S.N.;
Digitized by
GooqIc
Table 41 .
LINEAR EXPAN8ION8 OP PRINCIPAL METAL8. IN MICRON8 PER
METRE (OR MILLIONTH8 PER UNIT LENGTH).
Name of metal.
Aluminum . .
Brass ....
Copper ....
Glass ....
Gold ....
Iron, cast . . .
Iron, wrought
Lead ....
Platinum . . .
Platinum-iridium ^
Silver ....
Steel, hard . .
Steel, soft . . .
Tin
Zinc
Expansion per
aegree C.
20
19
17
9
'S
II
12
28
19
12
II
19
29
Expansion per
degree F.
ii.i
10.5
9-4
6.1
6.7
4^
6.1
16.1
tMITHSONIAN TaBLCS.
1 Of Intenatioial Pi otatj pe Metm.
Tabu 42.
FRACTIONAL CHANGE IN A NUMBER CORRESPONDING TO A CHANCE
IN ITS LOGARITHM.
Computed from the formula,
0k = modulus of common logarithms = 0^3439448-
For
AlogiV
s= I unit in
AAT
AT
For
AlogA^
= 4 units in
AAT
N
(in round numbers)
4th place
Sth "
6th «
7th '*
Twin
4th place
Sth "
6th «
7th "
tAtf
Smithsonian Tanlcn.
170
Digitized by
GooqIc
APPENDIX.
CONSTANTS.
Numerical Constants.
NsMbw.
LiUfMim.
Base of natural (Napierian) logarithms,
= tf = 2.7182818
04342945
Log /, modulus of common logarithms,
= M-= 0434294s
9.6377843-
-10
Circumference of circle in degrees.
360
2.5563025
« « « in minutes.
a 21600
4.3344538
" " « in seconds.
= 1296000
6. 1 1 26050
Circumference of circle, diameter unity,
= w = 3.14159265
04971499
Nimbtr. LsftrHhm.
2» = 6.2831853 a798i799
i/«a = 01013212
9.0057003-
-10
— = 1^71976 00200286
!=» 0.3183099 9,5028501 — 10
V» = 1-7724539
02485749
^=0.5641896
9.7514251-
-10
«a =3 9-8696044 0-9942997
V 7=14142136
01505150
V 7-1.7320508
02385607
The arc of a circle equal to its radius is
in degrees, p^ » i8o/v
= 57-29578*
1.7581226
in minutes, p' ^6op^
- 3437.7468'
3.5362739
in seconds, p" =» 60 p'
- 206264^"
5.3144251
For a circle of unit radius, the
arcofi« =i/p*»
-ox)i74533
8.2418774-
-10
arc of i' « i/p'
64637261 -
-10
arc (or sine) of r= i/p''
>boxxxx)0485
4.6855749-
-10
Qaodetieal Constants.
Dimensions of the earth (Clarke's ^heroid, 186Q and derived quantities.
Equatorial semi-axis in feet,
s= a B 20926062.
7.320687s
in miles.
-«- 3963.3
3-5980536
Polar semi-axis in feet.
»^n 20855121.
7.3192127
in miles,
-*- 3949.8
3.5965788
(Eccentricity)*-'*"";^
^ = 000676866
7.8305030-
•10
Flattening-*^*
-/= 1/294.9784
7.5302098-
•10
Perimeter of meridian ellipse.
s 24859.76 miles.
Circumference of equator.
«= 24901.96 '
<
Area of earth's surface.
« 196940400 square miles.
Mean density of the earth (Ha&kniss)
=»5.576± 0.016.
Surface density " « "
«= 2.56 ±0.16.
Acceleration of gravity (Harknbss) :
^ (cm. per second) = 98060 (i —0002662
cos 2^) for latitude ^ and sea level
^, at equator = 977-99 5 ^. at Washington
= 980.07;.^, at Paris =
980.94;
^, at poles = 983.21 ; ^, at Greenwich
« 981.17.
Length of the seconds pendulum (Harkness)
:
/a. 39.01 2540 + 0.2082688m* finches «
0.990910 + 0005290 8in> ^ metres.
SiirrHSONiAii Tabus.
171
Digitized by V^OOQIC
APPENDIX.
CON8T ANT8. -ContiniMd.
Attronomioal Constants (HAXKims).
Sidereal jesr s= 565.256 357 8 mean solar days.
Sidereal day — 25* 56^ 4.^100 mean solar time.
Mean solar day = 24A 3M 56^546 sidereal time.
Mean distance of the earth from the smi » 92 800 000 miles.
Physical Constants.
Velocity of light (Harkness) » 186 537 miles per second » 299 878 km. per second.
Velocity of sound through dry air -^ 1090 \/i + ojoojffj fi C. feet per second.
Weight of distilled water, free from air, barometer 30 inches :
Wdgbt ia gnuBs. W«ght in gnauBai.
Volume. 6aO^. 4® C. 6a*>^. 4® C.
I cubic inch (determination of 1890) 252.286 252.568 16.3479 16.3662
I cubic centimetre (1890) 15*3953 154125 0.9976 019987
I cubic foot (1890) at 62^ F. 62.2786 lbs.
A standard atmosphere is the pressure of a vertical column of pure mercury whose
height is 760 mm. and temperature o^ C under standard gravity at latitude 45°
and at sea level.
I standard atmosphere = 1033 gnunmes per sq. cm. »= 14.7 pounds per sq. inch.
Pressure of mercurial column i inch high = 34.5 grammes per sq. cm. ss 0491
pounds per sq. inch.
Weight of dry air (containing aooo4 of its weight of carbonic add) :
I cubic centimetre at temperature 32° F. and pressure 760 mm. and under the
standard value of gravity weighs aooi 293 05 gramme.
Density of mercury at o^ C. (compared with water of maximum density under atmos>
pheric pressure) » I3«5956l
Free2ing point of mercury = — 38.''5 C (Rxgnault, 1862.)
CoefBcient of expansion of air (at const, pressure of 760MM) for i^ C. (do.) : aoo367a
Coefficient of expansion of mercury for Centigrade temperatures (Bkoch) :
A^ Ao(i —0.000181792/ — aooo 000 000 175/3 — .000000000035 116 i*).
Coefficient of linear expansion of brass for i^ C, /3 = aooooi74 to aooooi90.
Coefficient of cubical expansion of glass for i^ C, 7 = ojooo 021 to 0.000 028.
Ordinary glass (Recknagel) : at lo*^ C, 7 = aoooo255 ; at loO®, 7 = oxxx) 0276k
Specific heat of dry air compared with an equal weight of water :
at constant pressure, A> = 0.2374 (from o^ to 100° C*., Regnault).
at constant volume, Kv = 0.1689.
Ratio of the two specific heats of air (Rontgen) : A> fKv = 1*4053.
Thermal conductivity of air (Geabtz) : k = 0.000 0484 (i + aooi 85 f*, C.) g""^
[The quaadty of heat that paaiet hi anit tune through unit area of a plate of unit tfaJdcoeas, when ita
opposite faces differ in lemperatnre bj one degree.]
Latent heat of liquefaction of ice (Bunsen) == 804)25 mass degrees, C.
Latent heat of vaporization of water =: 606.5 — o*^5 ^ ^-
Absolute zero of temperature (Thomson, Heat, Encyc, Brit.) : — 273.% C. = — 459^^4 F.
Mechanical equivalent of heat : *
I pound-degree, F, (the British thermal unit) = about 778 foot-pounds.
I pound-degree, C. == 1400 foot-pounds.
I calorie or kilogramme-degree, C » 3087 foot-pounds = 426.8 kilogram-
metres =* 4187 joules (for ^ «= 981 cm.).
SiirrHSONiAN Tables. _ i r\nkr%\c>
Digitized by VorOOv IC
* Baaed on Praf. RowUmd^s detenmnationa. (Proc. Am, Acad. ArU amd Set., x88o.) ^
172
APPENDIX.
8YNOPTIC CONVERSION OF ENGLI8H AND METRIC UNIT8.
Englith to Matrio.
Units of length.
I inch.
I foot
I yau-d.
I mile.
Units of arsa.
I square inch.
I square foot.
I square yard.
I acre.
I square mile.
2.54000
a3048oi
0.914402
1-60935
929.034
0.836131
4687
2.59000
259.000
Unite of
I cubic inch. 16.3872
I cubic foot 0.028317
I cubic yard. 0.764559
Units of eapaoity.
I gallon (U. S.) s 231 cubic inches.
I quart (U. S.).
I Imperial gallon (British).
277-463 cubic inches (i89o]|.
I bushel (U. S.) = 215042 cubic inches.
I bushel (British).
centimetres,
metre.
u
kilometres.
645163 square centimetres.
square metre.
hectares.
square kilometres.
hectares.
cubic centimetres,
cubic metres or steres.
cubic metres or steres.
3.78544 litres.
0.946^6 litres.
4.54603 litres.
35-2393 litres.
36.3477 htres.
0404835
9484016 — 10
9.961 137 — 10
0.206 650
0.800669
2.968 032
9.922 274 — 10
9.607 120 — 10
0413300
M13300
1.214 504
8-452 047 — 10
9^883411 — 10
a 578 116
9.976056—10
0-657 709
1.547027
1.560477
Unite of mass.
I grain. 64.7990 milligrammes.
I pound avoirdupois. 0453593 kilogrammes.
I ounce ayoirdupois. 28.3496 grammes.
I ounce troy. 3i*ioj5 grammes.
I ton (2240 lbs.). ix>i6o5 tonnes.
I ton (2000 lbs.). a907io6 tonnes.
Unite of velocity.
I foot per sec ^0.6818 miles per hr.) s= 0.30480 metres per sec <
I mile per hr. (1.4667 feet per sec) «= 044704 metres per sec 3
Unite of force.
I poundal.
weight of I grain (for^= 981 cm.).
Weight of I pound av. (for ^«= 981 cm.).
' 10973
1.6093
1.811 568
9.656066—10
1452546
1492 810
0.006914
9-957696—10
km. per hr.
km. per hr.
4.140682
1.803 ^37
5-648335
]^^Z
S.624698
13825.5 dynes.
63.57 dynes.
4.45 X io» dynes.
Units of stress— In gnvKitlM matsura.
I pound per square inch — 70.^07 grammes per sq. centimetre.
I pound per square foot = 4^24 kilogrammes per sq. metre.
Units of work— in ibtolirts maatufs.
I foot-poundal. 421 403 ergs.
^in grivitiUon msMWS.
I foot-pound (for ^=s 981 cm.) = 1356.3 X 10* ergs = a 138255 kilogram-metres.
Units of activity (rate of doing workX
I foot-pound per minute (for>f = 981 cm.) = ao226o5 watts.
I horse-power (33 000 foot-pounds per mm.) — 746 wa 8 » 1.01387 force de chevaL
Units of heat
I pound-degree, F.
I pound<degree, C.
B 252 small calories or
ss lis pound-degrees,
(,C.
Smithsonian Tables.
173
APPENDIX.
SYNOPTIC CONVERSION OP ENGLISH AND METRIC UNITS.
Metric to English.
Units of lengtii.
cngim aquvinnii.
UgniWims.
I metre (ic^ microns).
39-3700
inches.
1.595 165
«
3.28083
feet.
0.515984
M
1. 09361
yards.
0^)38863
I kilometre.
a62i37
miles.
9-793350—10
Units of area.
I square centimetre.
0.15500
square inches.
9-19033' — 10
I square metre.
10.7639
square feet.
1.031 968
(( «4
I-I9599
square yards.
ao77 726
I hectare.
2.47104
acres.
a392 88o
I square kilometre.
0.38610
square miles.
9.586701 — 10
Units of voiume.
I cubic centimetre.
ao6i0234 cubic inches.
8.785496—10
I cubic metre or stdre.
35-3145
cubic feet.
1-547 953
(t <l M
1.30794
cubic yards.
aii6589
Units of capacity.
I litre (61.023 cubic inches).
0.26417
gallons (U. S.).
9421 884 — 10
u
1.05668
quarts (U. S.).
0.023944
u
0.21993
Imp. gallons (British).
9.342291 — 10
I hectolitre.
2^3774
bushels (U. S.).
0452973
«
2.751 21
bushels (British).
0.439523
Units of mass.
I gramme.
15-4324
grains.
1.188433
I kilogramme.
2.20462
pounds avoirdupois.
0.343334
u
35-2739
ounces avoirdupois.
1-547454
u
32.1507
ounces troy.
1.507 190
I tonne.
a9842i
tons (2240 lbs.).
9.993086—10
•(
1.10231
tons (2000 lbs.).
ao42 304
Units of velocity.
I metre per second.
3.2808
feet per second.
0.515984
tf « M
2.2369
miles per hour.
0.349653
I km. per hr. (a2778 m. per sec.).
0.62137
miles per hour.
9-793350 — 10
Units of force.
I dyne (weight of (981)-^ grammes, for ^= 981 cm.) = 7.2330 X icr^ poundals.
UnKs of stress ~" in gmvHitloii moasure.
I gramme per square centimetre.
0.014223 pounds per sq. inch.
I kilogramme per square metre.
0.204817 pounds per sq. foot.
I standard atmosphere.
14.7
pounds per sq. inch. (See def. p. 172.)
I erg.
23730 X 10-* foot poundal.s.
I megalerg = loP ergs ; i joule = lo^ ergs.
-in gnvitBtkn mnaau
re.
I kilogramme-metre (for ^ = 981 cm
.) = 981 X 10^ ergs = 7-2330 foot-pounds.
Units of activity (rate of doing work).
I watt = I joule per sec. (= 44-2385
foot-pounds per mmute, for ^ = 981 cm.) = aioi94
kilogramme-metre per sec., for ^^^ 981 cm.
I force de cheval = 75 kilogranune-metres per sec. — 735} watts = 0.98632 horse-power. |
Units of heat.
I calorie or kilogramme-degree = 3.968 pound-degrees, K = 2.2046 pound-degrees, C
I small calorie or therm, or gramme-degree = aooi calorie or kilogramme-degree.
SiirrHaoNiAN Tables.
174
APPENDIX.
DIMENSIONS OF PHYSICAL QUANTITIES.
L == length ; M » mass i T — time.
Area.
M
Volume.
ivq
Mass.
[M]
Density.
[ML-*]
Velocity.
[LT-i]
Acceleration.
ILT-«1
Angle.
[0]
Angular Velocity.
rr-»i
Qiimtity.
MomentuDL
Moment of Inertia.
Force.
Stress (per unit area).
Work or Energy.
Rate of Working (Power).
Heat
Thermal Conductivity.
[L M T-i]
[ML2]
[LMT-«]
[L-i M T-^
[L2MT-*]
[L^MT-*]
[L2 M T-2]
[L-A M T-i]
In Electrostatics.
Quantity of Electricity.
Surface Density: quantity per unit area.
Difference of Potential: quantity of work required
to move a quantity of electricity ; (work done) -7- (quan-
tity moved).
Electric Force, or Electro-motive Intensity:
(quantity) -r- (distance^).
Capacity of an accumulator : e-r- £.
Specific Inductive Capacity.
In Magnetics.
Quantityof Magnetism, or Strength of Pole.
Strength or Intensity of Field:
(quantity) -7- (distance^).
Magnetic Force.
Magnetic Moment: (quantity) X (length).
Intensity of Magnetization: magnetic moment per
unit volume.
Magnetic Potential: work done in moving a quantity
of magnetism ; (work done) -i- (quantity moved).
Magnetic Inductive Capacity.
SymlMl. electrwtatlc sjfstom.
e [L* M* T-i]
a [L-* M» T-i]
£ [L* M* T-i]
Cot q
k
m
S
«
ml
I
[L-*M»'r-i]
[L]
[o]
electro-magnetlG
syttMR.
£L» M* T~i]
[L-* M* T-i]
[L-»M»T-i]
£L» M* T-i]
[L-* M» T-i]
In Eieetro-magnetics. SymlMl.
Intensity of Current i
Quantity of Electricity conveyed by current : e
(intensity) X (time).
Potential, or di£Eerence of potential: (work
done) -=- (quantity of electricity upon which
work is done).
Electric Force: the mechanical force act-
ing on electro-magnetic unit of quantity;
(mechanical force) -7- (quantity).
Resistance of a conductor: E-r-i. R
Capacity: quantity of electricity stored up q
per unit potential-difference produced by it
Specific Conductivity: the intensity of
current passing across unit area under the
action of unit electric force.
Specific Resistance: the reciprocal of r
specific conductivity.
Vox a [L»M*T~i]
Dlmanilons in Name sf
electro-magnetic practical unit
[L* M* T-i] Ampdre.
£L* M*] Coulomb.
E [Ll M* XT
£ [L» M* T-«l
[LT-i]
[L-iT«]
Volt
OhnL
Farad.
[LaT-i]
igitized by
Go
Smithkihiah Tabu..
>7S
Digitized by
GooqIc
INDEX.
FAGB
Acceleration, dimensions of 175
of gravity, formula for 171
table of values of 169
Air, cubical expansion, specific heat, thermal
conductivity, and weight of 172
Airy, Sir George, treatise cited xcviii
Albrecht, Dr. Th., treatise cited Ixzx
Algebraic formulas ziii-xv
Alignment curve Ivi
Aluminum, linear expansion of 170
Ampere, dimensions of 175
Angles, equivalents in arcs xviii
sum of, in spheroidal triangle Ivii
Angular velocity, dimensions of 175
Annulus, circular, area of xxx
Antilogarithms, explanation of use of xcix
4-place table of 26, 27
Appendix 171-175
Arcs, equivalents in angles xvii
of meridians and parallels xlvi-l
table of lengths of meridional 7S-80
table of lengths of parallel 81-33
table of time equivalents. 162
Are xli
Area, of circle xxx
table of values of 23
of surface of earth 1-lii
Areas, of continents Ixv
of oceans Ixv
of plane and curved surfaces xxix-xxxi
of zones and quadrilaterals of the
earth's surface 1-lii
tables of values of 142-159
of regular polygons xxx
Arithmetic means, progression, and series, .xiii
Astronomical constants 172
co-ordinates Ixvii
latitude xliv
time Ixxii
Astronomy Ixvii-bcxxii
references to works on Ixxxii
Atmosphere, mass of earth's Ixvi
standard pressure of 172
weight of unit of volume of 172
Average error, definition of Ixxxiv
Azimuth, astronomical and geodetic Ivii
computation of differences of Iviii-lxi
determination of ixxix
Babinet, barometric formula of 160
Barometer, heights by 160
Binomial series xiv
Brass, linear expansion of 170
Briinnow, F , treatise cited Ixxidi
Bushel, Winchester zxzv
equivalent in litres 2
Cable length xxxviii
Calorie, value of 172
Capacity, measures of, British xxxviii
Metric xli
Centare xli
Chauvenet, Wm., treatise cited Ixxxii
Circumference, of circle xxviii
table of values of 23
of earth xlix, 171
of ellipse xxix
C. G. S. system of units xlii
Clarke, General A R., spheroid of xliii
treatise cited Ixvi
Coefficient, of cubical expansion of air and
mercury 172
of linear expansion of metals 170
of refraction ..... v Ixiii
Compression, of earth xliii
Computation, of di£Eerences of latitude, lon-
gitude, and azimuth Iviii
of mean and probable errors xcv
Conductivity, thermal, of air 172
Cone, surface of xxxi
volume of xxxii
Constants, astronomical 172
geodetical 171
numerical 171
of earth's spheroid xliv
Continental measures (table of British and
Metric equivalents) 168
Continents, areas of Ixv
average heights of Ixv
Conversion, of arcs into angles and angles
into arcs xvii
of British and Metric units. . .2, 3, 173, 174
Co-ordinates, astronomical Ixvii
for projection of maps liii-lvi
table of, scale 1/250000 84-91
table of, scale 1/125000 92-101
Digitized byLjOOQlC
X78
INDEX.
Co-ordinates {conHnued).
table of, scale 1/126720 I02>i09
table of, scale 1/63360 1 10-121
table of, scale 1/200000 122-131
table of, scale 1/80000 132-141
of generating ellipse of earth's spheroid . . xliv
Copper, linear expansion of 170
Cord (of wood), volume of zzxix
Correction, for astronomical refraction, table
of mean values of 161
to observed angle for eccentric position
of instrument bdii
to reduce measured base to sea level. . .bdv
Cosines, table of natural 28, 29
use of table explained c
Cotangents, table of natural 30> 3'
use of table explained c
Coulomb, dimensions of 175
Cubature, of volumes xxxii
Cubes, table of 4-22
Cube roots, table of 4-22
Cylinder, surface of xxxi
volume of xxxii
Day, sidereal and solar Ixxii, 172
Degrees, number of, in unit radius xviii
of terrestrial meridian xlvi, 166
of terrestrial parallel xlix, 167
Density, mean, of earth Ixv
mean, of superficial strata of earth Ixv
of mercury 172
Departures (and latitudes), uble of 32-47
mode of use of table explained c
Depths, average, of oceans Ixv
Determination,, of azimuth Ixxix
of heights, by barometer 160
by trigonometric leveling Ixi
of latitode Ixxvii
of time Ixxiv
Difference, between astronomical and geo-
detic azimuth Ivii
of heights, by barometer 160
by trigonometric leveling Ixi
Differences, of latitude, longitude, and azi-
muth, on earth's spheroid Iviii
table for computation of 70*77
Differential formulas xxi
Dimensions, of earth xliii, 171
of physical quantities 175
Dip, of sea horizon , Ixiii
Distance, of sea horizon Ixiii
of sun from earth 172
Doolittle, Prof. C. L., treatise cited Ixxxii
Earth, compression of xliii, 171
Earth (continued).
density of Izv
dimensions of xliii, 171
ellipticity of xliii, 171
energy (of rotation) of Ixvt
equatorial perimeter of xliii, 171
flattening of xliii, 171
mass of Ixvi
meridian perimeter of xlix, 171
moments of inertia of Ixvi
shape of xliii
surface area of Hi
volume of Ixv
Eccentricity, of ellipse xliii
of earth's spheroid xliv, 171
El, value of 168
Electric quantities, dimensions of 175
Electro-magnetic quantities, dimensions of . 175
Ellipse, area of xxz
equations to xliv
length of perimeter of xxiz
Ellipsoid, volume of (see Spheroid) xxxiii
Ellipticity, of earth xliii, 171
Energy, dimensions of 175
oi rotation of earth Ixvi
Equations, of ellipse xliv
of Prototype Kilogranmies xl
of Prototype metres xl
Error, in ratio of English yard to Metre, .zxxvii
Errors, probable, mean, average . .Ixxxiv, Ixxxviii
table of, for interpolated quantities. .Ixxxvi
theory of IxxxiU
Everett, J. D., treatise cited xlii
Excess, spherical or spheroidal Iviii
Expansion, cubical, for air and mercury .... 172
linear, of principal metals 170
Farad, dimensions of 175
Fathom, length of xxxviii
Swedish 168
Flattening, of earth xliii, 171
Foot, Austrian 168
British xxxvii
French, Rhenish, Spanish, Swedish. . . . 168
Force, dimensions of 175
Formulas, algebraic xiii-xv
for differentiation xxi
for integration xxiii
for solution of plane triangles xviii
for solution of spherical triangles xx
trigonometric xv
Freezing point of mercury 172
Functions, trigonometric, of one angle xv
of two angles xvi
special values of xv
values in series xvii
Digitized by V^OOQ IC
INDEX.
179
Gallon, British and wine zzxviii
Gauss's formulas for spherical triangles nd
Geocentric latitude zliv
Geodesy zliii-lzvi
references to works on Izvi
Geodetic azimuth Ivii
Geodetic differences of latitude, longitude,
and azimuth Iviii
Geodetic line Ivii
Geodetical constants 171
Geographical latitude xliv
Geographical positions, computation of.lviii-lxi
Geoid, definition of xliii
Geometric means, progression xiii
Glass, linear expansion of 170, 172
Gold, linear expansion of 170
Gravity, acceleration of, formula for 171
table of values of 169
Gunter's chain, length of xxxviii
Harkness, Proi Wm., memoir dted
Ixv, 169^ 171, 172
Heat, dimensions of 175
latent, of liquefaction of ice 172
of vaporization of water 172
mechanicad equivalent of 172
Hectare xli
Heights, average, of continents Ixv
determination of, by barometer 160
trigonometrically bd
Hdmert, Dr. F. IL, treatise on geodesy
dted Ixvi
treatise on theory of errors dted . . . .xcviii
Horizon, dip of sea bdii
Imperial pound and yard xxxiv
Integrals, definite xxvi
indefinite xxxiii
Interconversion, of English and Metric
onits «f 3.i73i»74
of sidereal and solar time • Ixxiii
tables for 164, 165
Iron, linear expansion of 170
Joule, value of 1 74
Kilogramme, Prototype
equations of xl
relation to pound • zzxvi, xli
Kinetic energy, dimensions of 175
of rotation of earth Ixvi
Klafter, Wiener, in terms of foot and
metre 168
Latitude, astronomical, geocentric, and re-
duced xliv
determination of Ixxvii
Latitudes and departures, table of • -32-47
mode of use of table explained c
Lead, linear expansion of 170
Least squares, method of Ixxxvi
references to works on xcviii
Legendre's theorem for solution of sphe-
roidal triangles Ivii
Length, of arc of meridian .xlvi
of arc of parallel xlix
of equator of earth 171
of meridian circumference of earth 171
of perimeter of ellipse xxix
of Prototype Metres Nos. 21 and 27 xl
of seconds pendulum, formula for 171
table of values of 169
Leveling, trigonometric M
Line (French), value of 168
Lines, lengths of xxviii
on a spheroid Ivi
Linear measures, British xxxvii
Metric xli
tables for interconversion of . .2, 3, 175, 174
Litre xli
Logarithms, anti-, 4-place table of 26b 27
explanation of use of xcix
4-plaoe table of common 24, 25
of natural numbers, table of 4-22
relations of different xv
series for xiv
Maclaurin's series xxii
example of xxiii
Magnetic quantities, dimensions of 175
Maps, co-ordinates for projection of (see
Co-ordinates for projection of maps) .liii
projection of di
Mass, of earth Ixv
of earth's atmosphere .Ixvi
of Prototype Kilogrammes Nos. 4 and
20 xl
Mayer's formula for transit instrument • . . .Ixxv
Mean, arithmetic and geometric xiii
Mean distance of earth from sun 172
Mean error, definition of Ixxxiv
computation of xcv
Mean time Ixxii
table for conversion to sidereal tune 164
Measures xxxiv
of capacity, British xxxviii
Metric xli
of length, British xxxvii
Continental 168
Metric xli
Digitized byLjOOQlC
i8o
IlfDBX.
Measures {camtinuid),
of surface, British zzzviit
Metric zli
tables for interconversion of . .2, 3, 173, 174
Mechanical equivalent of heat 172
Mechanical units, dimensions of 175
Mensuration zxviii-zzxiii
Mercury, density and cubical expansion of. . 172
Meridian, arcs of terrestrial xlvi
Uble uf lengths of 7^-^
drcuniference of earth zliz, 171
Method of least squares Izzxvi
Metre, Prototype xxziv
equations of Nos. 21 and 27 xl
relation to British yard xxxvi, xli
Metric system xl
Bfile, Austrian 168
British (sUtnte) xxxvii
Danish, German sea, Netherlands, Nor-
wegian, Prussian, Swedish 168
Nautical 168
Modulus of common logarithms xv
Moivre's formula xvi
Moment of inertia of mass, dimensions of . . . 175
Moments of inertia of earth Ixvi
Momentum, dimensions of 175
Napierian base (of logarithms) xiv, 171
Napierian logarithms xiv
Napier's analogies xx
Natural logarithms xiv
Nautical mile, table of equivalents in statute
miles 168
Numerical constants 171
Ohm, dimensions of 175
Palm, length of, English xxxviii
Netherlands 168
Parallel, arcs of terrestrial xlix
table of lengths of 81-83
Pendulum, length of seconds 171
table of lengths of 169
Perch (of masonry) volume of xxxix
Perimeter, of circle xxviii
of ellipse xxix
of regular polygon xxviii, xxx
Physical constants 172
Physical geodesy, salient facts of Ixv
Physical quantities, dimensions of • • • I75
Platinum, linear expansion of 170
Platinum iridium, linear expansion of 170
Polyconic projection of maps liii
graphical process of, explained cii
Polygons, regalar, areas of
lengths of lines of
Potential (electric), dimensions of 175
Pothenot's problem bdy
Pound, imperial, avoirdupois xxxiv
Power, dimensions of 175
Pressure, of atmosphere 172
Prism, volume of xxxii
Probable error, definition of Ixxxiv
computation of zcy
Projection of maps liii, cii
Prototype Kilogrammes and Metres xxziv
equations of xl
Quadrilaterals, of earth's surface, areas of . ... .1
tables of areas of 142-159
Quantity, of electricity, dimensions of 175
Radii, of curvature xiv
Radius of curvature, of meridian, table of
logarithms of 48-56
of section normal to meridian, table of
logarithms of 57*^5
of section oblique to meridian, table of
logarithms of 66, 67
Radius vector of earth's surface I
Rate of working (power), dimensions of . . . . 175
Ratio, of pound to kilogramme xxxvi
of specific heats of air 172
of yard to metre xxxvi
Reciprocals, of natural numbers, table of. .4-22
Reduced latitude xiiv
Reduction to sea level of measured base line. Ixiv
References, to works on astronomy Ixxxii
to works on geodesy Ixvi
to works on the theory of errors xcviii
Refraction, astronomical, table of 161
example of computation of dv
coefficients of terrestrial Ixiii
Right ascension Ixxii
Rode, Danish 168
Ruthe, Prussian, Norwegian 168
Sagene, Russian 168
Sea level (see Geoid)^ reduction of measured
base line to bdv
Sea surface, area of Ixv
Secondary triangulation, differences of lati-
tude, longitude, and azimuth in 1x
Series, binomial xiv
logarithmic xiv
of Madaurin and Taylor xxii
trigonometric xvii
Sidereal day and year, length of 172
Digitized byLjOOQlC
INDEX.
l8l
Sidereal time Izxii
table for conversion to mean time.. 165
Signs, of trigonometric functions zv
Silver, linear expansion of 170
Sines, table of natural 28, 29
explanation of use of c
Solar time •• Ixxii
table for conversion of mean solar
to sidereal 164
Solution, of plane triangles zviii
of spherical triangles xx
of spheroidal triangles Ivii
Span, length of xxxviii
Specific heat of air 172
Sphere, equal in surface with earth Hi
equal in volume with earth Hi
Slurface of xxxi
volume of xxxii
Spherical excess (see Spheroidal excess) . . . .Iviii
Spheroid, representing the earth xliu
surface of xxxi
volume of xxxiii
volume of earth's Ixv
Spheroidal excess Iviii
example of computation of ci
Spheroidal triangle Ivii
Square roots, table of 4-22
Squares, table of 4-22
Standards, of length and mass xxxiv
Steel, linear expansion of 170
St^re xH
Stress, dimensions of 175
units of 173, 174
Sums, of arithmetic and geometric progres-
sion, and special series xiii
Surfaces (see Areas) xxix
Surface measures, British xxxviii
Metric xli
tables forinterconversion of . . .2, 3, 173, 174
Surface, of continents Ixv
of earth's spheroid Hi
of oceans Ixv
of sphere and spheroid xxxi
Surveyor's chain, length of xxxviii
Table for conversion of arc into time 162
conversion of mean into sidereal time . . 164
conversion of sidereal into mean time. . 165
conversion of time into arc 163
determination of heights by barometer. . 160
interconversion of British and Metric
units 2, 3, 1 73» 1 74
interconversion of nautical and statute
miles 168
Table of acceleration of gravity and derived
quantities 16^
Table of {contimud).
antilogarithms, 4-place 2^ 27
areas of quadrilaterals of earth's surface
of lo^* extent In latitude and longi-
tude 142
i^ extent in latitude and longi-
tude 144,145
30' extent in latitude and longi-
tude 146-148
15^ extent in latitude and longi-
tude 150-154
lo' extent in latitude and longi-
tude 156-159
areas of regular polygons xxx
circumference and area of circle 23
constants, astronomical 172
geodetical 171
numerical 171
for interconversion of English and
Metric units 2, 3»i73» ^74
Continental measures of length 168
co-ordinates for projection of maps —
scale 1/250000 84-91
scale 1/125000 92-101
scale 1/X26720 102-109
scale 1/63360 1 10-121
scale 1/200000 122-131
scale 1/80000 132-141
departures and latitudes 32-47
dimensions of physical quantities 175
errors of interpolated values from nu-
merical tables Ixxxvi
expansions (linear) of principal metals. . 170
formulas for solution of plane triangles, .xix
fractional change in number due to
change in its logarithm 170
latitudes and departures 32-47
lengths of arcs of meridian 78-80
of arcs of parallel 81-83
of 1° of meridian 166
of i^ of parallel 167
linear expansions of metals 170
logarithms, 4-place 24, 25
anti-, 4-place 26, 27
of factors for computing spheroidal
excess 68,69
of factors for computing differences
of latitude, longitude, and azi-
muth 70-77
of meridian radius of curvature. •48-55
of radius of curvature of normal
section 56~65
of radius of curvature of obHque
sections 66^ 67
mean astronomical refraction 161
measures and weights —
British, of capacity. xxxix
Digitized by V^OOQIC
l82
INDEX.
Table of (conHnMed),
British, of length . . . • • zzzviii
British, of surface xxxviii
British, of weight zzxix
Metric xli
tables for interconversion of
natural cosines 28, 29
natural tangents 3^ 3^
radii of curvature, logarithms of, for
meridian section 4^55
for normal section 5^*^5
for oblique section 66, 67
reciprocals, squares, cubes, square roots,
cube roots, and logarithms of natural
numbers 4-22
refraction, mean astronomical 161
signs of trigonometrical functions xy
values for computing areas and dimen-
sions of regular polygons zxx
for computing perimeter of ellipse zxix
of log i (i — 2«) and log (i — m)
used in trigonometric leveling . . .Ixii
weights and measures (see Table of
measures and weights) 2, 3, 173, 174
Table, traverse (see TVaverse table) 33*47
Tangents, natural, table of 30, 31
use of table explained c
Taschenbuch, Des Ingenieurs xdx
Taylor's series xxil
Temperature, absolute zero of 172
of freezing mercury 172
Theory of errors Ixxxiii-xcviii
references to works on xcviii
Thermal conductivity, dimensions of 175
of air 172
Three-point problem Ixiv
Time, determination of Ixxiv
equivalents in arc, table of 163
example of use of table civ
interconversion of sidereal and solar,
tables for 164, 165
Tin, linear expansion of 170
Toise, value in feet and metres 168
Ton, long and short xxxix
Tonne 173, 174
Tonneau xli
Trapezoid, area of xxix
Traverse table 32-47
explanation of use of c
Triangles, plane, solution of xviii
Triangles {ccnHnuict),
spherical, solution of zx
spheroidal, solution of .* Ivii
Triangulation, primary and secondary, differ-
ences of latitude, longitude, and azimuth
in Iviii-lx
Trigonometric functions, of one angle xv
of two angles xvi
series for xvii
Trigonometric leveling Ixi
Units, British System xxxvii
C. G. S. System xlii
Metric System xl
standards of length and mass xxxiv
tables for interconversion of British and
Metric 2, 3. » 73. 1 74
Useful formulas xiii-xxvii
Vara, Mexican and Spanish 168
Velocity, dimensions of 175
of light and sound 173
Versta, Russian 168
Vertical section curve on spheroid Ivi
Volt, dimensions of 175
Volume, of earth Ixv
of solids xxxii
Weight, of distilled water 172
Weights and measures (see Measures and
weights)^ tables for interconversion of
British and Metric 2, 3, 173, 174
Werst, Russian 168
Work, dimensions of 175
Wright, Prof. T. W., treatise cited xcviii
Yard, imperial xxxiv
ratio of, to metre xxxvi, xxxvii
Zachariae, G., treatise cited xlvi
Zenith distances, use of, in trigonometric
leveling Ixi
Zenith telescope, use of Ixxix
Zero, of absolute temperature 172
Zinc, linear expansion of 170
Zones, of earth's surface, area of 1
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Th-ifl book should he returned to
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A flue of five oente a day ia mcmred
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