THE UNIVERSITY
OF ILLINOIS
LIBRARY
From the collection of
Julius Doerner, Chicago
Purchased, 1918.
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lAILllOAD ENGINEERS
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FIE LD-BOOK
FOtt
RAILROAD ENGINEERS.
CO.\TAI.MN(J
F 0 R M U L /E
IfOIi LAYING OUT CURVES, DETERMINING FROG ANGLES, LEVELLING,
CALCULATING EARTH-WORK, ETC., ETC.,
TOGETUER WITH
TABLES
OF )L\»II, ORUINATE.S, DEFLECTIONS, LONG CHORDS, MAGNETIC VAEIA
TION, LOGAKlTII.Mis, LOGARITHMIC AND NATURAL SINES,
TANGENTS, ETC., ETC.
BY
JOHN B. HENCK, A.M.,
CIVIL ENGINEER.
NEW YORK:
D. APPLETON & COMPANY, 549 & 551 BROADWAY.
LONDON: IG LITTLE BKITAIN
1877.
EsTEiiF-D, according to Act of Congress, in the year 1854,
By D. APPLETON & CO.,
In the Clerk's Office of the District Court of the United States
for the Southern District of Xow York.
6>Z o . /
/81^
PREFACE.
The object of the present work is to supply a want very
i^enerally felt by Assistant Engineers on Railroads. Books
of convenient form for use in the field, containing the ordi
nary logarithmic tables, are common enough ; but a book
combining with these tables others peculiar to railroad
work, and especially the necessary formulse for laying out
curves, turnouts, crossings, &c., is yet a desideratum.
These formuke, after long disuse perhaps, the engineer is
often called upon to apply at a moment's notice in the
field, and he is, therefore, obliged to carry with him. in
manuscript such methods as he has been able to mvent or
collect, or resort to what has received the very appropriate
name of " fudging." This the intelligent engineer always
considers a reproach; and he will, therefore, it is hoped,
receive with favor any attempt to make a resort to it inex-
cusable.
Besides supplying the want just alluded to, it was thought
that some improvements upon former methods might be
made, and some entirely new methods introduced. Among
the processes believed to be original may be specified
those in §§41 — 48, on Compound Curves, m Chapter II.,
on Parabolic Curves, in §§ 106 - 109, on Vertical Curves,
and in the article on Excavation and Embankment. It is
4694 4?
V] PREFACE.
but just to add, that a great part of what is said on Reversed
Curves, Turnouts, and Crossings, and most of the Miscel-
laneous Problems, are the result of original investigations.
In the remaining portions, also, many simplifications have
been made. In all parts the object has been to reduce the
operation necessary in the field to a single process, inil;-
cated by a formula standing on a line by itself, and distin-
guished by a ly . This could not be done in all cases, as
will be readily seen on examination. Certain preliminary
steps were sometimes necessary, and these, whenever it
was practicable, have been indicated by words in italics.
Of the methods given for Compound Curves, that in
§ 46 will be found particularly useful, from the great variety
of applications of which it is susceptible.
Methods of laying out Parabolic Cui-ves are here given,
that those so disposed may test their reputed advantages.
Two things are certainly in their favor ; they are adapted
to unequal as well as equal tangents, and their cuiTature
generally decreases tov/ards both extremities, thus making
the transition to and from a straio-ht line easier. Some
labor has been given to devising convenient ways of laying
out these curves. The method of determinins; the radius
of curvature at certain points is believed to be entirely
WQW. Better processes, however, may already exist, par-
ticularly in France, where these curves are said to be in
general use.
The mode of calculating Excavation and Embankment
here presented, will, it is thought, be found at least as sim-
ple and expeditious as those commonly used, with the ad-
vantage over most of them in point of accuracy. The usual
Tables of Excavation and Embankment have been omitted.
To include all the varieties of slope, width of road-b^d, and
depth of cuttmg, they must be of great extent, and uiitiued
H
PREFACE. ri:,
tor a field-book. Even then they apply only to ground
whose cross-section is level, though often used in a mannei
shown to be erroneous in § 128. When the cross-section
of the ground is level, the place of the tables is supplied by
the formula of § 119, and when several sections are calcu-
lated together, as is usually the case, and the work is ar-
ranged in tabular form, as in § 120, the calculation is be-
lieved to be at least as short as by the most extended tables.
The correction in excavation on curves (§ 129) is not
known to have been introduced elsewhere.
In a work of this kind, brevity is an essential feature.
The form of "Problem" and "Solution" has, therefore,
been adopted, as presenting most concisely the thing to be
done and the manner of doing it. Every solution, how-
ever, carries with it a demonstration, which is deemed an
equally essential feature. These demonstrations, with a
few unavoidable exceptions, principally in Chapter II., pre-
suppose a knowledge of nothing beyond Algebra, Geome-
try, and Trigonometry. The result is in general expressed
by an algebraic formula, and not in words. Those familiar
with algebraic symbols need not Jje told how much more
uitelligible and quickly apprehended a process becomes
when thus expressed. Those not familiar with these sym-
bols should lose no time in acquiring the ready use of a
language so direct and expressive. It may be remarked
that it was no part of the author's design to furnish a col-
lection of mere " rules," professing to require only an abil-
ity to read for their successful application. Rules can sel-
iom be safely applied without a thorough understanding of
llie principles on which they rest, and such an understand-
ing, in the present case, implies a knowledge of algebraic
(ormulse.
The tables here presented will, it is hoped, prove relia
VUl PREFACE.
ble. Those specially prepared for this work have been
computed with great care. The values have in some cases
been carried out farther than ordinary practice requires, in
order that interpolated values may be obtained from them
more accurately. For the greater part of the material
composing the Table of Magnetic Variation the author is
indebted to Professor Bache, whose distinguished ability ir
conducting the operations of the Coast Survey is equalled
only by iiis desire to diffuse its results. The remaining
tables have been carefully examined by comparing them
with others of approved reputation for accuracy. Many
errors have in this way been detected in some of the tables
of corresponding extent in general use, particularly in the
Table of Squares, Cubes, &c., and the Tables of Logarith-
mic and Natural Sines, Cosines, &c. The number of tables
might have been greatly increased, but for an unwillingness
to insert any thing not falling strictly within the plan of th?
work or not resting on sufficient authority.
J. B. 11.
Boston, February, 1854.
TABLE OF CONTENTS.
CHAPTER I.
CIRCULAR CURVES.
Article I. — Simple Curves.
2. Definitions. Propositions relating to the circle . . 1
4. Angle of intersection and radius given, to find the tangent 3
5. Angle of intersection and tangent given, to find the radius 3
6. Degree of a curve 4
7. Deflection angle of a curve ♦
A. Method by Deflection Angles.
9. Radius given, to find the deflection angle .... 4
10. Deflection angle given, to find the radius . . , 4
11. Angle of intersection and tangent given, to find the deflection
angle . 5
12. Angle of intersection and deflection angle given, to find the
tangent
13 Angle of intersection and deflection angle given, to find the
length of the curve 6
U. Deflection angle given, to lay out a curve .... 7
.16. To find a tangent at any station 8
B. Method 1)1/ Tangent and Chord Dejlections.
17. Definitions ... .... .8
18. Radius given, to find the tangent deflection and chord deflection 9
19. Deflection angle given, to find the chord deflection . . 9
21. To find a tangent at any station 9
22. Chord deflection given, to lay out a curve . . . . 10
S TABLE OF CONTENTS.
C. Ordinatcs.
24. Definition • • • H
25. Deflection angle or radius given, to find ordinatcs . 11
26. Approximate value for middle ordinate . . . ■ l-^
27. Method of finding intermediate points on a curve approxi-
mately . . • . . 14
D. Cui~ving Rails.
29. Deflection angle or radius given, to find the ordinate for curv-
ing rails . • ^'^
Article II. — Reveesed and Compound Ccrtes,
30. Definitions • • • .15
31. Radii or deflection angles given, to lav out a reversed or com-
pound curve ^^
A. Reversed Curves.
32. Reversing point when the tangents are parallel . . 16
33. To find the common radius when the tangents are parallel 1 6
34. One radius given, to find the other when the tangents are par-
allel .... "
35. Chords given, to find the radii when the tangents are parallel 18
36. Radii given, to find the chords when the tangents are parallel 18
37. Common radius given, to run the curve when the tangents are
not parallel ^^
38. One radius given, to find the other when the tangents are not
parallel *^
39. To find the common radius when the tangents are not parallel 21
40. Second method of finding the common radius when the tan-
gents are not parallel 22
B. Compound Curves.
41. Common tangent point .... .23
42. To find a limit in one direction of each radius . . 24
44. One radius given, to find the other 25
45. Second method of finding one radius when the other is given 26
46. To find the two radii 2V
47. To find the tangents of the two branches .... 29
48 Second method of finding the tangents of the two branches . 30
TABLE OF CONTENTS. B
Article III. — Turxouts and Crossings.
HECT. PAQl
i9. Dcliiiitions '^1
A. Turnout from Straight Lines.
50. Radius given, to find the frog angle and the position of the frog 32
51. Frog angle given, to find the radius and the position of the frog 33
52. To find mechanically the proper position of a given frog . 34
53. Turnouts that reverse and become parallel to the main track 34
54. To find the second radius of a turnout reversing opposite the
frog ....... ... 35
B, Crossings on Straight Lines.
55. Kcferences to proper problems 36
56. Radii given, to find the distance between switches . 36
C. Turnout from Curves.
57. Frog angle given, to find the radius and the position of the frog 38
58 To find mechanically the proper position of a given frog . 41
59 Proper angle for frogs that they may come at the end of a rail 41
60 Radius given, to find the frog angle and the position of the frog 42
62 Turnout to reverse and become parallel to the main track. . 44
D. Crossings on Curves.
63. References to proper problems • ^^
64. Common radius given, to find the central angles and chords 45
Article IV. — Miscellaneous Problems.
65. To find the radius of a curve to pass through a given point 46
66. To find the tangent point of a curve to pass through a given
point 47
67. To find the distance to the curve from any point on the tan-
gent 47
68 Second method for passing a curve through a given point . 47
69. To find the proper chord for any angle of deflection . . 4*
70. To find the radius when the distance from the intersection
point to the curve is given 48
71 To find the distance from the intersection point to the curve
when the radius is given ... ... 49
Xll TABLE OF CONTENTS.
SECT. PAai
72. To finil the ta\igent point of a curve that shall pass through a
given point .... . 5C
73. To find the radius of a curve without measuring: angles . 51
74. To find the tangent points of a curve without measuring an-
gles . , . ... 5?
75. To find the angle of intersection and the tangent points when
the point of intersection is inaccessible .... 52
76. To lay out a curve when obstructions occur . . 5.t
77. To change the tangent point of a curve, so that it may pass
through a given pomt 50
78. To change the radius of a curve, so that it may terminate in
a tangent parallel to its present tangent . . . .57
79. To find the radius of a curve on a track alreadv laid . . 5;^
80. To draw a tangent to a given curve from a given point . . 59
81. To flatten the extremities of a sharp curve .... .tj
82. To locate a curve without setting the instrument at the tan-
gent point . . . .... 60
'*.'?. To measure the distance across a river . 6.H
CHAPTER II.
PARABOLIC CURVES.
Article I. — Locating Parabolic Clkvls.
84. Fropo.>itions relating to the parabola ... .65
85. To lay out a parabola by tangent deflections ... 66
36. To lay out a parabola by middle ordinates . . . .67
87. To draw a tangent to a parabola 67
89. To lay out a parabola by bisecting tangents • - . .68
90. To Iny out a parabola by intersections ... 69
Ai;tict.e II. — Radius of Curvature.
9^. Definition .... ... .71
9-3. To find the radius of curvature at certain stations . . .71
95. Simplification when the tangents are equal . . . 7«
TABLE OF CONTENTS. XIH
CHAPTER III.
LEVELLING.
AnriCLE I. — Heights and Slope Stakes.
»»JT. PAGE
96. Definitions 78
97. To find the diflovence of level of two points . . . .78
98 Datum plane 79
99. To find tlic heights of the stations on a line . . . . 8C
100. Sights denominated jo/ms and m««Ms 81
101. Form of field notes 82
102. To set slope stakes 82
AuTiCLE II. — Correction for the Earth's Curvature and
FOR Refraction.
103. Earth's curvature 84
104. Refraction 84
105. To find tlie correction for curvature and refraction . . 85
Article III. — Vertical Curves.
106. Manner of designating grades . 86
107. To find the grades for a vertical curve at whole stations 86
109. To find tlie grades for a vertical curve at sub-stations . 88
Article [V. — Elevation of the Outer Rail on Curves.
110. To find the proper elevation of the outer rail 89
.11. Coning of the wheels 89
CHAPTER IV.
EARTII-WORK,
Article I. — Prismoidal For.mula.
.12 Definition of a prismoid 92
[13. To find the solidity of a piismoid 92
Article II -Borroav-Pits.
114. Manner of dividing the ground 93
XIV T..i5LE OF CONTENTS.
SECT. PAOa
115. To find the solidity of a vertical prism whose horizontal sec-
tion is a triangle 93
116. To find the solidity of a vertical prism whose horizontal sec-
tion is a parallelogram 94
117. To find the solidity of a number of adjacent prisms having
the same horizontal section f '^
\rticle III. — Excavation and Embankment.
A. Centre Heights alone given.
119. To find the solidity of one section 97
120. To find the solidity of any number of successive sections . 98
B. Centre and Side Heights given.
121. Mode of dividing the ground 9^
122. To find the solidity of one section lUO
123. To find the solidity of any number of successive sections . 104
125. To find the solidity when the section is partly in excavation
and partly in embankment .... . . 105
126. Beginning and end of an excavation ... . 107
C. Ground very Irregular.
127. To find the solidity when the ground is very irregular . 108
128. Usual modes of calculating excavation 109
D. Correction in Excavation on Curves.
129. Nature of the correction 110
130. To find the correction in excavation on curves . . . 112
132. To find the correction when the section is partly in excava
tion and partly in embankment -113
TABLES.
RO. PAOB
I. Radii, Ordinates, Tangent and Chord Deflections, and Or-
dinates for Curving Rails 115
U. Long Chords 119
TABLE OP CONTENTS. X^
NO. PAGE
HI. (correction for the Earth's Curvature and for Rcfract'uin . 120
IV. Elevation of the Outer Rail on Curves . . . . I'iO
V. Frog Angles, Chords, and Ordinates for Turnouts . .121
VI. Length of Circular Arcs in Parts of Radius . . . 121
VJI. Expansion by Heat 122
VIII. Properties of Materials 123
IX. Magnetic Variation 126
X. Trigonometrical and Miscellaneous Ft (-mulie . . 13'i
XI Squares, Cubes, Square Roots, Cube Roots, and Recip-
rocals ....... . . 137
XII. Log Arithms of Numbers . . .... 155
XIII. Logarithmic Sines, Cosinee Tangents, and Cotangents 171
XIV. Natural Sines and Cosines 219
XV. Natural Tangents and Cotangents . . . 229
XVL Rise per Mile of Various Grades .... MJ
EXPLANATION OF SIGNS.
The sign + indicates that the quantities between which it is placed
ire to be added together.
The sign — indicates that the quantity before which it is placed
.s to be subtracted.
The sign X indicates tliat the juantities between which it is placed
are to be midtiplied together.
The sign -r- or : indicates that the fust of two quantities between
which it is placed is to be divided by the second.
The sign — indicates that the quantities between which it is placed
are equal.
The sign oo indicates that the difference of the two quantities be-
tween which it is placed is to be taken
The sign .• . stands for the word "hence " or " therefore."
The ratio of one quantity to another may be regarded as the quo-
tient of the first divided by the second. Hence, the ratio of a to 6 is
expressed by a : h, and the ratio of c to d by c : (/. A proportion ex
presses tlie equal it 1/ of two latios. Hence, . proportion is rcjiresented
by placing the sign — between two ratios ; as, a ■ b = c : d
In the text and in the tables the foot has been taken as the unit gi
measure when no other unit is specified.
FIELD-BOOK.
CH/VPTER I.
CIRCULAR CURVES.
Article I. — Simple Cuka'es
1. The railroad curves here considered are eitlier Circular or Para
holic. Circular curves are divided into Simple, Reversed, and Com
j)Ound Curves. We begin with Simple Curves.
2. Let the arc ADEFB (fig. 1) represent a railroad ciu've, unit
Fig. \.
2 CIRCULAR CURVES.
ing the straight lines GA and B FT. The lengtli of sudi a curve is
measured by cliords, each 100 feet long.* Tlius, if the chords AD^
DE, E F, and FB are each 100 feet in length, the whole curve is
said to be 400 feet long. The straight lines GA and BH are always
tangent to the curve at its extremities, which are called tangent points.
U GA and BH are produced, until they meet in C, ^ C and B C
are called the tangents of the curve. If ^ C is produced a little beyond
Cto /v, the angle KGB, formed by one tangent with the other pro-
duced, is called the angle of intersection, and shows the change of direc-
tion in passing from one tangent to the other.
The following propositions relating to the circle are derived from
Geometry.
I. A tangent to a circle is perpendicular to the radius drawn through
the tangent point. Thus, A C is perpendicular to A 0, and B C to
BO.
II. Two tangents drawn to a circle from any point are equal, and it
a chord be drawn between the two tangent points, the angles between
this chord and the tangents are equal. Thus AC— B C, and the
angle B A C =^ A B C.
III. An acute angle between a tangent and a chord is equal to half
the central angle subtended by the same chord. Thus, C A B —
hAOB.
IV. An acute angle subtended by a chord, and having its vertex in
the circumference of a circle, is equal to half the central angle sub-
tended by the same chord. Thus, D AE = i D OE.
V. Equal chords subtend equal angles at the centre of a circle, and
also at the circumference, if the angles are inscribed in similar seg-
ments. Thus, AOD = DOE, and D A E = E A F.
VI. The angle of intersection of two tangents is equal to the cen-
tral angle subtended by the chord which unites the tangent points.
Thus, KGB = AO b'
3. In order to unite two straight lines, as GA and B H, by a curve,
the angle of intersection is measured, and then a radius for the curve
may be assumed, and the tangents calculated, or the tangents may be
assumed of a certain length, and the radius calculated.
* Some engineers prefer a chain 50 feet in length, and measui'e the length cf :i
enrve by chords of 50 instead of 100 feet. The chord of 100 feet has been adopteii
throughout this article ; but the formulae deduced may be very readily modified t(.
Buit chords of any length. See also ^ 13.
SIMPLE CURVES.
ti
4. Pro'bleni. Given the angle of intersection K C B — 1 fjig \)
and the radius A 0 = R, tojind the tangent A C = T.
1-iy I.
Solution. ])niw CO. Then in the right triangle AOC we lia«'',
iTab. X. 3) 4-;- = tan. AO C, or, since A 0 0=^1 a 2, VI.)
A O
- = tan. 2 /;
T = R tan. ^ /.
Example. Given 7 = 22== .52', and /? = 3000, to find T. Here
A' = 3000 3.477121
^7=11° 26' tan. 9.305865
T= 606 72
2.7829»0
.5. Problem. Given the angle of intersection KCB = I {fg. I ),
ind the tangent A C — '1\ to find thp radius A 0 =-. R.
4 CIRCULAR CURVES.
Solution. In tlie right triangle A 0 C we have (Tab. X. 61
— = cot. A O C. ov — = cot. h i ;
AC ' r ^ '
!^= ,'. R== Tcot. i/.
Example. Given 7 = 31° 16' and r= 950, to find 72. Here
r=950 2.977724
^1= 15° 38 cot. 0.553102
R = 3394.89 3.530826
6. The decree of a curve is determined by the angle subtended at
its centre by a chord of 100 feet. Thus, if A 0 D = 6° (fig. 1),
ADEFB is a 6° curve.
7. Tlie deflection angle of a curve is the acute angle formed at any
point between a tangent and a chord of 100 feet. The deflection angle
is, therefore (^ 2, III ), half the degree of the curve. Thus, CAD or
CBF is the deflection angle of the curve A D E F B, and is half
A OD or half F 0 B.
A. Method by Deflection Angles.
8. The usual method of laying out a curve on the ground is by
means of deflection angles.
9. Problem. Given the radius A 0 == R {fig. \), to find the de-
flection angle C B F = D.
Solution. Draw OL perpendicular to B F. Then the angle BOL
= hBOF= D, and BL = hBF=50. But in the right triangle
OBL yve have (Tab. X. 1 ) sin. BOL = ^;
IW sin. Z) = — .
J. L
Example. Given R = 5729.65, to find D. Here
50 1.698970
72 = 5729.65 3.758128
D = 30' sin. 7.940842 .
Hence a curve of this radius is a 1° curve, and its deflection angle is
30'.
10. Problem. Given the deflection angle C B F = D (fig. 1), «»
find the radius A 0 ^= R. ■
METHOD BY DEFLECTION ANGLES. 5
Solution. By the preceding section we have sin. Z)= — , whence
R
fi sin. D=^ 50;
50
'. A' =
sin. D
By this formula the radii in Tahle I. are calculated.
Erampk. Given D = 1", to find R. Here
50 1.698970
■^=1'' sin. 8 241S.')5
i^= 2864.93 3.457115
1 1 . Problem. Given the angle of intersection KCB = I (Jig. 1 ),
and the tangent AC = T, to find the deflection angle CA D = D.
Solution. From § 9 we have sin. D = —, and from ^ 5, R =
7' cot. .^7. Substituting this value of 72 iv the first equation, we get
sm. D = ;
rcot. i /'
r5s« • T-. 50 tan. i /
ts^ . • . sm. D = L_ .
Example. Given 7 = 21° and T = 424.8, to find D. Here
50 1.698970
^7=10° 30 tan. 9.267967
0.9669S7
7' =424 8 2.628185
7) = 1° 15' sin. 8.338752
12. Problem. Given the angle of intersection KCB ^ I {fig. \)
and the deflection angle CAD = D, to find the tangent AC= T.
Solution. From the preceding section we have sin. D = - ^°' ^-\
T
Hence, Tsin. 7) = 50 tan. i 7;
j^=» . rp 50 tan. «i 7
sin. D
Example. Given 7 = 28° and D = 1°, to find T. Here
„ 50 tan. 14°
T= -~r~Tr- = 714.31.
Bin l""
b CIRCULAR CURVES.
13. Problem. Given the angle of intersei tion K CB = I {Juf. 1),
and the deflection angle C A D = D, to find the length of the curve.
Solution. By § 2 the length of a curve is measured by chords of 100
feet applied around the curve. Now the first chord A D makes with
the tangent A C oxi angle C A D =^ D, and each succeeding chord
DE,EF,&c. subtends at u4 an additional angle DAE, EAF, &c.
each equal to D; since each of these angles (§ 2, IV.) is half of a
central angle subtended by a chord of 100 feet. The angle CAB =
i A 0 B = ^ I is, therefore, made up of as many times Z), as there are
chords around the curve. Then if n represents the number of chords,
we have n D = ^ I',
hi
,• . n = - — .
D
If D is not contained an even number of times in ^ /, the quotient
above will still give the length of the curve. Thus, in fig. 2, suppose
D is contained 4| times in ^ /. This shows that there will be four
whole chords and | of a chord around the curve from A to B. The
angle GAB, the fraction of D, is called a sub deflection angle, and
G B. the fraction of a choi'd, is called a sub-chord*
The length of the curve thus found is not the actual length of tlie
arc, but the length required in locating a curve. If the actual length
of the arc is required, it may be found by means of Table VI.
Example. Given / = 16° 52' and D = \° 20', to find the length of
JL J- g3 9gl 506'
the curve. Here n = '^ = £5^ "=80^"" ^•^-^' ^^^^ ^^' *® ^"^^®
is 6.32.5 feet long.
To find the arc itself in this example, we take from Table VI. the
length of an arc of I60 52', since the central angle of the whole curve
is equal to /(§ 2, VI ), and multiply this length by the radius of the
curve.
Arc 10° = .1745329
" 6° = .1047198
« 50' = .0145444
« 2' = .0005818
" 16° 52' = .2943789
• This method of finding the length of a sub-chord is not mathematically accu-
rate ; for, by geometry, angles inscribed in a circle are proportional to the arcs on
which they stand ; whereas this method supposes them to be proportional to the
chords of these arcs. lu railroad curves, the error arising from this supposition m
too small to be regari'ed.
METHOD BY TiiniENT AND CHORD DEFLECTIONS. 9
o»rt% B 11 and C K of tlie same length as the chords. Draw O/i
nnd D K. B G is called the tangent deflection, and C H or D K the
du>nl deflection.
18. Problem. Given the radius AO = R (flg. S), to flnd the
tangent deflection B G, and die chord deflection C H.
Solution. The triangle C B II is similar to BOC; for*thc angle
BOC= 180= - {OBC-\- B CO), or, since BCO = ABO, BOC
= 180= — {0 BC -{- ABO) = CB H, and, as both tiie triangles are
isosceles, the remaining angles are equal. The homologous sides are.
therefore, proportional, that is, B 0 : B C = B C : C II, or, represent-
ing, the chord by c and the chord deflection by d, R : c =^ c : d\
c^
^ .-. d = -.
R
To find the tangent deflection, draw BM to the middle of 6*7/,
bisecting the angle C B H, and making i3il/C a right angle. Then
the right triangles B M C and AGS are equal ; fovBC=A B, and
the angle CBM=hCBII=iBOC=^AOB = BAG (§2,
III.). Therefore B G = CM= h OH = ^d, that is, the tangent de-
flection is half the chord deflection.
19. Pr61>!eill. Given the deflection angle D of a curve, to flnd the
chord deflection d.
Solution. By the precedin;; section we have d^= -77, and by \ 10,
tl = , — ^ Substituting this value of R in the first equation, we find
c^ sin. D
^ d =
50
This formula gives the chord deflection for a chord c of any length
though D is the deflection angle for a chord of 100 feet (^^ 7). When
c = 100, the formula becomes d= 200 sin D, or for the tangent de-
flection hd = 100 sin. D. By these formulte the tangent and chord
deflections in Table I. may be easily obtained from the table of natural
sines
20. The length of the curve may be found by first finding Z) (§ 9 or
J U), and then proceeding as in § 13.
21. Probleifla To drcntJ a tangent to the cun^e at any station,
HS B {Jig. 3).
Solution. Bisect tne chord deflection II 0 of the next station in M.
2
10 CIRCULAR CURVES.
A line drawn through B and 31 will be the tangent required ; foi it
has been proved (§ 18) that the angle C B M is in this case equai to
i B 0 0, and B J/ is consequently (§ 2, III.) a tangent at B.
If B is at the end of the curve, the tangent at B may be found with-
out first laying off // C. Thus, if a chain equal to the chord is extend-
ed to H on A B produced, the point H marked, and the chain ihon
swung ronnd, keeping the end at B fixed, until II M = h d, fJ M will
he the direction of the re(iuired tangent.*
22. ProtoleilS. Giveii the chord deflection (/, to lay .nil a curcc
from a given tangent point.
Solution. Let A (tig. 3) be the given tangent point, and suppose '/
has been calculated for a chord of 100 feet. Stretch a cbain of li'i;
feet from A to G on the tangent EA produced, and mark the poini
G. Swing the chain round towards AB, keeping the end at A fixed
until B G \s equal to the tangent deflection i c/, and B will be the first
station on the curve. Stretch the chain from B to H on AB pro
duced, and having marked this point, swing the chain round, until U C
is etpial to the chord deflection d. Cis the second station on the curve
Continue to lay off the chord deflection from the preceding chord pro
duced, until the curve is finished.
Should a sub-chord DF occur at the end of the curve, find the tan
gent DL at D (§ 21), lay off from it the proper tangent deflection Lf
for the given sub-chord, making DF of the given length, and F will
be a point on the curve. The proper tangent deflection for the sub-
chord may be found thus. Eepresent the sub-chord by c', and the cor-
responding chord deflection by d', and we have (§ 18) 5 c/' = — ; but
since hd = — ' we have ^ c/' : 2 cZ = c'- : c^. Therefore ^d' = ,^d(-]
Example. Given the intersection angle I between two tangents
equal to 16° 30', and R = 12.')0, to find T, c/, and the length of the
curve in stations. Here
(§4) T=R tan. j^ /= 1250 tan. 8° 15' = 181.24 ;
c'i 100*2
* Tlie distance B M is not exactly equal to the chord, but the error arising from
taking it equal is too .small to be regarded in any curves but those of very small
radius. If necessary, the true length of B M may be calculated ; for B M =:
ORDINATES. , 11
0 9) sin. L> = -f == -|?- = .04 .= nat. sin. 2° 17^';
, r , ox M 8 ' 15' 495'
(6 13) ;z = — = = = 3.60.
^^ ^ n 2J17J' 137.5'
These results show, that the tangent point A (fig. 3) on the first taii
gent is 18124 feet from the point of intersection, — that tlie tan<.en\
deflection G B=^ld= A feet, — that the chord deflection //Cor K D
= 8 feet, — and that the curve is 360 feet long. The three whole sta-
tions B^ C. and D having been found, and the tangent D L drawn, the
tangent deflection for the sub-chord of 60 feet will be, as shown above,
h cV = 4 C"- ) = 4 X .62 = 4 X .36 = 1 44. LF= 1.44 feet being
laid off from DL, the point F will, if the work is correct, fall upon
the second tangent point. A tangent at F may be found (§ 21) by
producing DF to P, making FP= DF= 60 feet, and laying ofl
PN = 1.44 feet. FN will be the direction of the required tangent,
which should, of course, coincide with the given tangent.
23. CurA^es may be laid out with accuracy by tangent and ch.ord
deflections, if an instrument is used in producing the lines. But if an
instrument is not at hand, and accuracy is not important, the lines may
be produced by the eye alone. The radius of a curve to unite two
given straight lines may also be found without an instrument by § 73,
or, having assumed a radius, the tangent points may be found by § 74.
C Ordinates.
24. The preceding methods of laying out curves determine points
100 feet distant from each other. These points are usually sufficient
for grading a road ; but when the track is laid, it is desirable to have
intermediate points on the curve accurately determined. For this pur-
pose the chord of 100 feet is divided into a certain number of equal
parts, and the perpendicular distances from the points of division to
the curve are calculated. These distances are called ordinates. If the
chord is divided into eight equal parts, we shall have points on the
curve at every 12.5 feet, and this will be often enough, if the rails,
which are seldom shorter than 15 feet, have been properly curved
(§ 28).
25. Problem. Given the dpflection angle D or the radius R of a
came, to Jind the ordinates for any chord.
Solution. I. To find the middle ordinate. Let AEB (fig. 4) be
ft portion of a curve, subtended by a chord A B, which may be de-
i'^
CIRCULAR CURVES.
noted by c. Draw the middle ordinate ED, and denote it by m. Pro-
duce ED to the centre F, and join A F and A E. Then (Tab. X. 3'«
I Xu
ED
Id
= tan. E A D, or E D
But, since the angle
E AD is measured by half the arc BE, or by half the equal arc AE^
we have EAD=hA FE. Tlierefore E D = AD tan. ^ A FE, ox
^ m^ hciVin-^AFE.
When c = 100, A FE = /) (§ "), and m = 50 tan. 5 /), whence 7/)
may be obtained from the tabic of natural tangents, by <liA-iding tan
4 Z) by 2, and removing the decimal point two places to the right.
The value of m may be obtained in another form thus. In the
triangle ADF we have DF= ^A F^ — A if- = ^72^ _ ^ ^2. Then
m = EF— DF= R — DF, or
7)1
= R — s/R-
4 ^ •
II. To find any other ordinate, as i?iV, at a distance DN =h from
ihe centre of the chord. Produce RN until it meets the diameter
parallel to ^ ^ in G, and join R F. Then RG= ^R F^ — F G* =
y^-ZTp; andRN= RG — XG= RG— DF. Substituting the
value o? RG and that of D F found above, we have
RN = ^R^ — V' - ^R^ — i c2.
ORDmATES. 13
By these fcrmulaj the ordinates in Table I are calculated.
The other ordinates may also be found from the middle ordinate by
jie following shorter, but not strictly exact method. It is founded on
the supposition, that, if the half-chord B D he divided into any number
of equal parts, the ordinates at these points will divide the arc E B into
the same number of equal j)arts, and upon the further supposition, that
the tangents of small angles are proportional to the angles themselves.
These suppositions give rise to no material error in finding the ordi-
nates of railroad curves for chords not exceeding 100 feet. Making,
for example, four divisions of the chord on each side of the centre, and
joining A B, AS, und A T, we have the angle RAN=^EAD,
since R B is considered equal to % E B. But EAD= iAFE.
Therefore, B. A N= | .1 FE. In the same way we should find SAO
-= ^ A FE, and TA P = ^ A FE. We have then for the ordinates,
R N=^ AN tan. RAN = ^c tan. | A FE, SO=AO tan. SA 0 =
I c tan. i A FE, and TP = AP tan. TAP ^Ic tan. J A FE.
But, by the second supposition, tan. %AFE = | tan. ^ AFE,
tan. \AFE = ^ tan. i A FE, and tan. ^AFE^\ tan. i A FE.
Substituting these values, and recollecting that §■ c tan. ^ A FE = m,
rte have
f 72iV= |g X i c tan. I ^ jP^ = jgwi,
SO^\x^ctaxi.^AFE = \ vi,
7 7
TP = jg X i c tan. ^ A FE = ^g m.
In general, if the number of divisions of the chord on each side of
the centre is represented by n, we should find for the respective ordi-
. (n + l)(n-l)m (n+2)(7t-2)m
nates, begmnmg nearest the centre, ■ — ■ :^ , ^^^ ?
;n + 3){n — 3)wz
«2
, &C.
Example Find the ordinates of an 8° curve to a chord of 100 feet.
Here wi = 50 tan. 2° = 1.746, TZiV^ ^ w = 1.637, 6' 0 = \m ^ 1..310,
7
and TP = ^ w = 0.764.
26. An approximate value of m also may be obtained from the for-
mula m = R — ^R^ — \ c^ This is done by adding to the quantity
under the radical the very small fraction g, ^j •> making it a perfect
CIRCULAR CURVES.
f quare, the root of which will he R — 5-5 . Wc have, then, n. «=> fi
i^-n-J-
8R
SB.)
8 R
27. From this value of m we see that the middle ordinates of any
two chords in the same curve are to each other nearly as the squares
of the chords. If, then, A E (fig. 4) be considered equal to ^ y4 S. its
middle ordinate C // == {ED. Intermediate points on a curve m;iy,
therefore, be very readily obtained, and generally with sufficient accu-
racy, in the following manner. Stretch a cord from A to B, and Ijy
means of the middle ordinate determine the point E. Then stretch
the cord from A to E, and lay off the middle ordinate C 11 = \ ED,
thus determining the point C, and so continue to lay off from the .'^■i;--
ressive half-chords one fourth the preceding ordinate, until a sufficicru
number of points is obtained.
D. Curving Rails.
28. The rails of a curve are usually curved before they are V.vkx To
do this properly, it is necessary to know the middle ordinate of the
curve for a chord of the lenjith of a rail.
29. Problem. Given the radius or deflection angle of a curve., to
find the middle ordinate for curving a rail of given length.
Solution. Denote the length of the rail by Z, and we have (§ 25)
the exact formula m = R — ^/FC^ — 4 ^'> and (§ 26) the approximate
formula
m — ^
2R
This formula is always near enough for chords of the lengtli of a rail
50
If we substitute for R its value (§ 10) R — sin^ ' ^^® have,
100
Example. In a 1° curve find the ordinate for a rail of 18 feet m
length. Here R is found by Table I. to be 5729.6.5, and therefore.
KliVERSED AND COMPOUND CURVES.
13
9-
by the first foi-mula, m -- 11459.3 = .00707. By the sccorul forniula,
m = .81 sin. 30' = .00707. The exact formula would give the same
result even to the fifth decimal.
By keeping in mind, that the ordinate for a rail of 18 feet in a 1=^
curve is .007, the corresponding ordinate in a curve of any other de-
gree may be found with suflficient accuracy, by multiplyiug tliis deci-
mal by the number expres.sing tlio degree of the curve. Thus, for a
curve of 5'^ 36' or 5.6°, the ordinate would be .M7 X •'>-6 = .0."9 ft. =-
468 in.
For a rail of 20 feet we have ^ /^ = 100, and, consequently, ?h =-
sin. D. This gives for a 1° curve, m = .0087. The corresponding or-
dinate in a curve of any other degree may be found with sufficient
accuracy, l^y multiplying this decimal by the number expressing the
degree of the curve.
By the above formula for m, the ordinates for curving rails in Table
I, are calculated.
Article II. — Reversed and Compound Curves.
30. Two curves often succeed each other having a common tangeni
at the point of junction. If the curves lie on opposite sides of the com-
mon tangent, they form a reversed curve, and their radii may be the
!,ame or different. If they lie on the same side of the common tangcTit
tney have different radii, and form a compound curve. Thus A B C
'fiff. .5") is a reversed cirve, and .1 B D % comoound curve.
16
CIRCULAR CURVES.
31, ProbleiJl. To lay out a reversed or a compound cun>e, tufien.
the radii or dejiection anyles and the tangent points are known.
Solution. I/ay out the first portion of the curve from A to B Cfig. 5),
by one of the usual methods. Find B F, the tangent to A B at the
point B (§ 16 or ^ 21). Then B F will be tlie tangent also of the sec-
ond portion B C oi a reversed, or Zi D of a compound curve, and from
this tangent cither of these portions may be laid ofl' in the usual man
ner
A. Reversed Curves.
32 I'SieOJ'CRi. Tlie reversing point of a reversed curve letwces
parallel tangents is in the line joining the tangent points.
Fig. 6.
t\
Demonstration. Let A CB (fig. 6) be a reversed curve, uniting tin
parallel tangents HA and B K, having its radii equal or unequal, and
reversing at C. If now the chords A Cam] CB are drawn, we have
to prove that these chords are in the same straight line. The radii
E C and C F, being perpendicular to the common tangent at C (§ 2, 1.),
are in the same straight line, and the radii A E and B F, being per-
pendicular to the parallel tangents HA and B K, are parallel. There-
fore, the angle AE C= CFB, and, consequently, E C A, the half
supplement of A E C, is equal to F C B, the half supplement of CFB;
but these angles cannot be equal, unless A Cand C B are in the same
straight line.
33. Proljlem. Given the perpendicular distance between two par-
alM tangents B D =^ b {Jig 6), and the distance between the two tangeni
voints A B = a, to determine the reversing point C and the common radnti
E C ^ C F = R of a reversed curve uniting the tangents HA and B K.
Solution. Let ACB be the required curve. Since the radii are
REVERSED CURVES.
n
equal, and the angle AE C = B F C, the triangles AE C and B FC
are equal, and A C = CB ^ ^a. The reversing point C is, therefore,
the middle point of A B.
To find R, draw E G perpendicular to A C. Then the right tri-
itngles AEG and BAD are similar, since (§ 2, III.) the angle
BAD = hAEC^ AEG. Therefore A & -. A G ^. AB : BD,
or ii : ^ a = a : 6 ;
46
Corollary. If R and h are given, to find a, the equation 72 = j^
gives a' = 4 Rb;
a
2 JR b.
Examples. Given 6 = 12, and a =^ 200, to determine R. Here
2002 10000
12 ~ 833|.
/i' =
4X12
Given R = 675, and b = 12, to find a. Here a = 2^675 X 12 =
2y8T00 == 2 X 90 = 180.
34. Protolem. Given the perpendicular distance between two par-
allel tangents B D = b {fig- 7), the distance between the two tangent points
A B = a, and the first radius E C = R of a reversed curve uniting the
tangents HA and B K. to find the chords A C ~ a' and C B = a", and
the second radius CF = R'.
Solution. Draw the perpendiculars E G- and FL. Then the right
triangles A B D and E A G are similar, since the angle B AD ^
i8 CIRCULAR CURVES.
iAEC= AE G. Therefore AB : B D = E A : A G, or a : b
2Rb
a
Since a' and a" are (§ 32) parts of a, we have
a" = a — a'.
To find R' the similar triangles A B D and F B L give A B : B D
= F B : B L,ox a :b = R' '. ^ a" ;
a a"
Example. Given 6 = 8, a = 160, and R = 900, to find a', a", and
/?'. Here a' = ^ = 90, a" = 160 — 90 = 70, and R< =
160 X 70 _^^
-2X8 =700.
35. Corollary 1. If 6, a', and a" are given, to find a. A', and A' ,
we have (§ 34)
^ a = a' + a" ; R=—; R' = 1^.
2 6 26
Example. Given 6 = 8, a' = 90, and a" = 70, to find a, A, and R
Here a = 90 -f 70 = 160, A = -g j< 8 = 900, and A' = ^xS =
700.
36. Corollary 2. If A, A', and h are given, to find a, a', and a'\
c have (§ 35), A •
B«= 26 (A + ^');
V r> I -ni aai + aa" a (a' -fa'') a2
wc have (§ 35), R -j- R' = ^b — ~ — 2b — ~ 2b- Therefore
.'.a = y2 6(A-t-A').
Having found a, we have (§ 34)
a a
Example. Given A = 900, A' = 700 and 6 = 8, to find a, a', aim
a". Here a = ^2 X 8 (900 + 700) = ^16 X 1600 ^- 160, a' =-
8 X 900 X 8 ^^ - ,, 2X700X8 _
— jg^j = 90, and a" = ^ = 70.
REVERSED CURVES.
19
37. Problem. Given the angle A K B = K, which shows the
change of direction of two tangents HA and B K {fig. 8), to unitr. these
tangents by a reversed curve of given common radius R, starting from a giv-
en tangent point A.
3l
B K
^^ Fig. 8.
Solution. With the given radius run the curve to the point Z), where the
tangent D N becomes parallel to D K The point D is found thus. Since
the angle N G K, which is double the angle II A D {(j 2, II.), is to be
made equal to A KB = K, lay off from FIA the angle HA D=\E
Measure in the direction thus found the chord AD = 2R sin. ^ 75:
This will be shown (§ 69) to be the length of the chord for a deflection
angle ^ K. Having found the point D, measure the perpendicular dis-
tance D M = b between the parallel tangents. ■
The distance DB = 2DC = a may then be obtained from the for-
muln r§ 3.3, Cor.)
l^ a = 2 ^ITb .
The second tangent point B and the reversing point Care now ue-
tcrniined. The direction o( D B or the angle B DNmnj also I)e ob-
tained ; for sin BDN
sin, DBM = TTiF,, or
sin. BDN
b
a
38. Problem. Given the line A B = a {fig. 9) which Joins the
fixed tangent points A and B, the angles HAB = A and ABL = B,
tnd the first radius A E = R, to find the second radius B F = R of a
Teversfd curve to unite the tangents H' A and B K.
First Solution. With the given radius run the carve to the point Z),
ohere the tangent D N becomes parallel to B K. The point D is found
20
CIRCULAR CURVES.
thus. Since the angle H G N, which is double HA D (§ 2, II.), is
equiil to J. CO S, lay off from HA the angle HA D — ^ (At^ B), and
measure in this direction the chord A D = 2 R sin. ^ (A&o^) (§ 69)
Setting the instrument at Z), run the curve to the reversing point C in the
line from D to B {^ 32), and measure D C and C B. Then the similar
triangles DEC and BFC give DC:DE =^ CB : B F, or D C : Ji
^ CB:R';
CB
.R'^.
DC
X R-
Second Solution. By this method the second radius may bt founu
by calculation alone. The figure being drawn as above, we have, in
the triangle A B D, AB = a, AD = 2R sin. ^ (A — B), and the
included angle DAB = HA li ~ HAD = A — h (A — B) ^
^ {A -\- B). Find in this triangle (Tab. X. 14 and \2) B D and the
angle ABD. Find also the angle DBL^B-\-ABD.
Then the chord C B = 2 R' sin. hBFC =2R' sin. D B L, and
the chord D G
CB = BD —
DBL,
= 2R sin. ^DE C = 2R sin. DBL (§ 69). But
D C; whence 2 R' sin. D B L = B D — 2 R sm
.R>
BD
2 sin. DBL
— R.
"When the point D falls on the other side of A, that is, when the
angle B is greater than jl, the solution is the same, except that the
mgle DAB is then 180° — ^(A -\- B), and the angle DBL= B —
ABD.
REVERSED CURVES.
21
39. Probloiia. Given the length of the common tangent D G — a^
and the angles of intersection I and I' (Jig. 10), to determine the common
radms C E = C F = li of a reversed curve to urate the tangents II A
rtnn B L.
Fig. 10.
T-
Solution. By § 4 wc have DC = R tan. | /, and CG= R tan. | /' .
whence R (tan. ^ / + tan. hi') = D C -{- C G = a, or
R =
tan. ^ / + ti^n- k ^'
This formula may be adapted to calculation by logarithms ; for we
have (Tab. X. 35) tan. ^7+ tm.^P = ^T.'^^jcot fj- Substituting
this value, we get
rw R - «gos. ^7cos. ^7^
sin.i(^+/0
The tangent points A and B are obtained by measuring from D a
iistance J. Z) = 72 tan |- 7, and from G a distance B G — R tan. \ I',
Example. Given a = 600, 1 = 12°, and F = 8° to find R. Here
a = 600 2.778151
i7=6° cos. 9.997614
f 7' = 4° cos. 9.998941
R = 3427.96
2.774706
sin. 9.239670
3.535036
22
CIRCULAR CURVES
40. Problem. Given the line AB = a {fig- 10), which jchis the
fixed tangent points A and B, the angle DAB = A, and thr angle
A B G = B,io Jind the common radius E C = CF = Rof ar, versed
■:urre to unite the tangents HA and B L.
Solution. Find Jirst the auxiliai-y angle A K E = B KF, ivhich inmj
be denoted by K. For this purpose the triangle A E K gives A E: E K
= sin. K : sin. E A AT. Therefore E K sin. K = A E sin. E A K ~
R cos. A, since EAK = 90^ — A. In like manner, the triangle
BFK gives FK sin K= BF sin. FBK = R cos. B. Adding
these equations, we have {E K-\- FK) sin. K= R (cos. J. -\- cos. B),
or, since E K + FK = 2 R, 2 R sin. K = R (cos. A + cos. B)
Therefore, sin. K = ^ (cos. A -\- cos. B). For calculation by loga-
rithms, this becomes (Tab. X. 28)
sin K = cos. i(A-\- B) cos. ^(A — B).
Having found K, we have the angle AE K = E = 18(P — K —
EAK= \%QP — K — (90^ — yl) = 90° + ^ — Z; and the angle
BFK= F= 180°— K— FBK = 180° — TT— (90=— J5) = 90-
-{- B — K Moreover, the triangle A E K gives Ah A K =
sin. K: sin. E,or R sin. E= J..K'sin. K and the triangle B F K gives
BF:BK = s\n.K: s'm.F, or R sin. F = B K sin. K. Adding these
equations, we have R (sin. E -f- sin. F) = (A K -j- B K) sin. K —
a sin. K. Substituting for sin. E 4- sin. Fits value 2 sin. ^ {E -j- l^
COMPOUND CURVES. 2S
(^g_ ^ (E — F) (Tab. X. 26), we have 2 li sin. .^ (A' -|- F) cos.
i a sin. K „.
^(£'_F)=asin.A:. Therefore R = ^1^77(5-4. f)^os. U^- -F) ' *'*
nally, substituting for A' its value 90° -f ^ — A', and for Fits value
grjo Lj. B — A', we get h {E + F) = 90° — [A' — ^ (.1 + 13)1 an J
,^ (A' — F) = i (yl — /}) ; whence
COS. [K- ^ {A 4- Z^)J cos ^(A — B)
Eximple. Given a =1500, A = 18°, and B = 6°, to find /.'. Here
^ (^ 4- C) = 12° cos. 9.990404
i (^1 — i?) = 6° cos 9 997614
A' = 76° 36' 10" sin. 9.988018
Aa = 750 2.87.^)001
K -^{A-^- B) ^ 64° 36' 10 COS. 9.632347
J(^ — /?) = 6*' COS. 9.9976 U
2.863079
9.629961
/{= 1710.48 3.233118
B. Compound Curves.
41. irhcorem* If one branch of a compound curve he produced^
HJilil the tangent at its extremity is parallel to the tangent at the extremity
of the second branchy the common tangent point of the tico arcs is in the
straight line produced, which passes through the tangent points of these par-
allel tangents.
Demonstration. Let A CB (fig. 11) be a compound curve, uniting
the tangents HA and B K. The radii C'F and C'F, being perpen-
dicular to the common tangent at C (§ 2, 1.), are in the same straight
line. Continue the curve A C to Z>, where its tangent OD becomes
parallel to B K, and consequently the radius DE parallel to B F.
Then if the chords CD and CB be drawn, we have the angle CE D
= CFB; whence E CD, the half-supplement of C E D, is equal to
F CB, the half-supplement of CFB. But E CD cannot be equal to
F C B, unless CD coincides Avith CB. Therefore the line B D pro-
inced passes through the common tangent point C
24
42. Problem.
compound curve.
CIRCULAR CURVES.
To find a limit in one direction of each radius of ol
!S)lution. Let A I and Bl (fig. 11) be the tangents of the curve.
Through the intersection point 7, draw IM bisecting the angle A IB.
Draw A L and B M perpendicular respectively to A I and B /, niect-
ing 1 M in L and M. Then the radius of the branch commencing on
the shorter tangent A /must be less than AL^ and the radius of the
branch commencing on the longer tangent B I must be greater than
BM. For suppose the shorter radius to be made equal to A L^ and
make IN = A I, and join L N. Then the equal triangles A IL and
NIL give A L = L N; so that the curve, if continued, will pass
through iV, where its tangent will coincide with IN. Then (§ 41) the
common tangent point would be the intersection of the straight line
through B and iVwith the first curve; but in this case there can be no
intersection, and therefore no common tangent point. Suppose next,
that this radius is greater than A L, and continue the curve, until its
tengent becomes parallel to BI. In this case the extremity of the
COMPOUND CURVES. 25
curve will fall outside the tangent BIm the line A iV produced, and a
straight line through D and this extremity will again fail to intersect
the curve already drawn. As no common tangent point can be found
when this radius is taken equal to A L or greater than A L, no com-
pound curve is possible. This radius must, therefore, be less than A L.
In a similar manner it might be shown, that the radius of the other
branch of the curve must be greater than B M. If we suppose the tan-
gents A I and B J and the intersection angle / to be known, we have
{^ 5) A L = A I cot. ^ /, and B M = B I cot. ^ 7. These values are
therefore, the limits of the radii in one direction.
43. If nothing were given but the position of the tangents and the
tangent points, it is evident that an indefinite number of different com-
pound curves might connect the tangent points ; for the shorter radius
might be taken of any length less than the limit found above, and a
corresponding value for the greater could be found. Some other con-
dition must, therefore, be introduced, as is done in the following
problems.
44. Problem. Given the line AB = a {Jig. 11), which joins Oie
fixed tangent points A and B, the angle B A I =^ A, the angle AB I =
D, and the first radius A E — B, to find the second radius B F = R' of
a compound curve to unite the tangents HA and B K.
Solution. Suppose the first curve to be run with the given radius
from A to Z), where its tangent DO becomes parallel to BI^ and
the angle IAD = i (^ -f- B). Then (§ 41) the common tangent
point C is in the line B D produced, and the chord CB = CD -j-
B D. Now in the triangle AB D we liave A B ^ a^ AD = 2 R
sin. ^ {A -\- B) (§ 69), and the included angle D A B =^ I A B —
IAD = A — ^ (A -{- B) ^ ^ {A — B). Find in this triangle
(Tab. X. 14 and 12) the angle A B D and the side B D. Find also the
angle CBI=B — ABD.
Then (^ 69) the chord CB ^ 2 R' sin. CB Z, and the chord CD =
2 R sin. CD0=2R sin. CBI. Substituting these values of CB
and CD in the equation found above, C B = CD -\- B D, we have
2/^' sin. CZ3/= 2 R sin. CBI+BD;
l^ -.R' = R+ ^^
2 sin. CBI
When the angle B is greater than A, that is, when the greater radius
»8 given, the solution is the same, except that the angle D A B ^
26 CIRCULAR CURVES.
J (D — A), and C BI'is found by snbtracting the supplement oi A Li IJ
from B. We shall also find CB — CD — B D^ and conscauenilv
^'^ ~ ^^ 2 sin. CBI'
If more convenient, the point D may be determined in the field, by
laying otf the angle I A D = ^ [A -j- B), and measuring the distance
A D -^ 2 R sin. i ( J. -j- B). BD and CB I may then be measured,
instead uf being calculated as above.
Example. Given a = 950, ^ = S^', /5 = 7^, and R = 3000, to find
A''. Here AD = 2 X 3000 sin. h (S^ + 7°) = 783.16, and DA B -=
^ (8° — 7°) = 30'. Then to find .1 B D we have
AB — A D ^ 166.84 2.222300
i (J DB -\- A B D) = 89° 45' tan. 2.360180
4.582480
A B -j- AD = 1733.16 3.238831
I (.1 Z) L.' — .1 Z; D) = 87° 24' 17" tan. 1.343641
.'.ABD = 2° 20' 43"
Next, to find B D,
A D == 783.16*
2.893849
DAB = 30'
sin
7.940845>
0.834691
ABD = 2° 20' 43"
sin
8.611948
jBZ) = 167.01
2.222743
B-
-ABD= CBI = 4^S9' 17"
2 (/2' — R) ^ 2058.03
sin.
8.909292
3.313451
.-.R' — R = 1029.01
R' ^
= 3000 4- 1029.01 =- 4029.01
To find the central angle of each branch, we have CI B — 2 C B 1
= 9° 18' 34", which is the central angle of the second branch; and
AEC=AED-CED = Ai-B — 2CBl = 5° 41' 26", which
is the central angle of the first branch
45. Problem. Given (Jig. 11) the tangents Al= T, B I = T',
the angle of intersection = /, and the first radius A E = R, to find the
second radius B F = R'-
Solution. Suppose the first curve to be run with the given radius
from A to D, where its tangent D 0 becomes parallel to Bl. Through
COMPOUND CURVKS. 27
D draw D P parallel to .4 7, and v/c have IP = DO = AO =
R tan. ^ 7 (M)- Then in the triangle D P B vre have D P = 1 0 =^
A1—A0= T - R tan. ^I, BP=BI— IP=T'— R tan. ^ 7,
and the included angle DPB = AIB = 180' — 7. T'/nc/ m //ijs ^v-
angle the angle C B I, and the side B D. The remainder of the solution
is the same as in § 44. The determination of the point D in the field
is also the same, the angle IAD being hei'e = ^ 7. When BU
gi cater than A, that is, when the greater radius is given, the solution is
d'.c same, except that D P = R tan. ^ I — T, and B P = R tan. 1 1
Example. Given T= 447 32, T' = 510. 84, 7 = 15° and R = 3000,
to find R'. Here 7^1 tan. | 7= 3000 tan. 7^° = 394.96, DP = 447.32
— 394.96 = 52.36, BP = 510.84 — 394.96 = 115.88, and DP B =
180^ — 15° = 165°. Then (Tab. X. 14 and 12)
SP — 7)P = 63.52 1.802910
|(SDP + P^Z)) = 7=30' tan. 9.119429
0.922339
.BP + 7)P = 16824 2.225929
^{BDP — PBD) = 2° 50' 44" tan. 8696410
.'.PBD= C23 7 = 4"^ 39' 16"
Next, to find B 7),
Z)P= 52.36 1.719000
DP B = lb° sin. 9.412996
1.131996
P Z^ D = 4° 39' 16" sin. 8.909266
2^7) =167.005 2.222730
Ibe tangents in this example were calculated from the example in
^ 44. The values of CB I and B D here found differ slightlv from
those obtained before. In general, the triangle DBP is of better
form for accurate calculation than the triangle AD B.
46. If no circumstance determines either of the radii, the condition
may be introduced, that the common tangent shall be parallel to the
line joining the tangent points.
Problem. Given the line AB = a (Jig. 12), which unites the
fixed tangent points A and B, the angle I A B = A, and the angle
A B I = B, to find the radii A E = R and B F = R' of a compound
'.urve^ having the common tangent I) G parallel to A 12
28
CIRCULAR CURVES.
Solution. Let A C and B C be the two brai^ches of the required
curve. ar:d draw the chords A C and B C. These chords bisect the
fig 12
angles A and B ; for the angle D A C = ^ ID G = ^ I A B,sluA the
angle G B C =--- ^ D G 1 = h A B I. Then in the triangle A C B wo,
have AC\AB^ sin. A BC : sin. A C B. But ACB= 180^ —
(C^ Z5 + CB A) = 180" — 4 (^1 4- B), and as the sine of the sup-
plement of an angle is the same as the sine of the angle itself.
sin. A CB = sin. \ {A -^ B). Therefore A C : a = sin. ^ B : sin.
^ (A -f- B), or A C = sin" 1^4, B) • ^^ ^ similar manner we should
a sin. ^ A
iAC
^^^ I^C= ,in. i (A% B) ' ^'o->^ ^^e have (§ 68) 72 = -^j^-p; , and
j^ , or, substituting the values of ^ Cand B Cjust found,
U' =
jB C
sin.
i a sin. ;V 5
a sin.
sin. ^ A sin. ^ (^ + 5) ' sin. ^ B sin. ^ ( A + i^)
Example, Given a = 950, ^ = 8°, and B = 7°, to find R and /2'
Here
COMPOUND CURVES'. 29
i a = 475 2.676 '.94
^ B ^ 3° 30' sin. 8. 785675
1.462369
1^ = 4° sin. 8.843585
i (A + ^) = 7^ 30' sin. 9.115698
7.959283
R = 3184.83 3.503086
i ransposing these same logarithms according to the formula for R
«e hare
? « = 475 1.676694
•M = 4° sin. 8.843585
h B = 3° 30' sin. 8.785675
^ (^ + Z?)-= 7-= 30' sin. 9 115698
1.520279
7.901373
R' = 4158.21 3.618906
47. Problem. Glveji the line AB = a {Jig. 12), wkich unites the
fixed tangent points A and B, and the tangents AI = T and BI = T',
Mjind the tangents AD = x and B G = y of the tico branches of a com-
pound curve, having its common tangent D G parallel to A B.
Solution. Since D C = A D = x, and C G =^ B G = y, we have
fjQ = x-j~jj. Then the similar triangles IDG and lAB give
f D : lA = D G : A B, or T - X - T ■= X -\- y : a. Therefore
aT — ax = Tx + Ty (1). Als( ^ 0 : A I = B G : B I, or
T:T = y:T'. Therefore Ty = T r (^'). Substituting in (1) the
ralue of Ty in (2), we have a T— ax ■. 7 r + 2'' :r, or a a: + Tor -|-
T'x = aT;
a-\-T-\-T''
T'x
and, since from {2),y = -y- ,
a-\-T-\-T'
The intersection points D and G and the common tangent pomt C
are now easily obtained on the ground, and the radii may be found by
the usual methods. Or, if the angles TAB = A and A B I -^ B
30
CIRCULAR CURVES.
have been measured or calculated, we have (§ 5) R =^ x cot. ^ A, and
R' = y cot. ^ B. Substituting the values of x and y found above, wa
have R = q^y^r' ' ^^"^ ^ = M=^M= T< •
Exampie. Given a = 500, T = 250, and T' = 290, to find x and y
Here a + 7 + I' = 500 + 250 + 290 = lOtO ; whence a: = 500 ><
250 -r 1040 = 120.19, and ^ = 500 X 290 -r- 1040 = 139.42.
48. Probleaea. Given the tangents Al = T, Bl =T', and tfu
angle of intersection /, to unite the tangent points A and B (Jig. 13) hy a
compound curve, on condition that the tivo branches shall have their angles
of intersection IDG and I GD equal.
Fig 13
^ututiirn. feince IDG = lGD = hl^yf& have I D = 1 G. Rep
'■escnt the line Ih ~ 1 Ghy x. Then if the perpendicular IHhe let
♦ The radii of an oval of given length and breadth, or of a three-centre arch of given
epan and rise, may also be found from these formulae In these cases A-^ B = 90'-,
and the values of R and R' may be reduced to R = — ; — ^^, 7;;^ and R' =
aTi
a+ T— Ti
calculated
a+T' — T
. These values admit of an easy construction, or they may be readily
TURNOUTS AND CROSSINGS. 31
fall fiom /, we !iave (Tab. X. U) D H = I D cos. IDG = x cos. ^ i,
sxiUDG^'lx COS. ^ /. But DG = DC^CG = AD-\-BG==
7' ~ 2 + T' — X = r + Ti — 2x. Therefore 2 a: cos. \l =
r + T' — 2 .r, or 2 T + 2 .r cos. i / =- T -\- T' ; whence jt =
j^:^^^;j^,or(lab.X.25)
I^' 2r =
^{T+rO
CO
s.^i/
The tangents AD = T— x and B G =- T' — x are now readilj
(bund. With these and the known angles of intersection, the radii oi
deflection angles may be found (§ 5 or § 11) This method answers
very well, when the given tangents are nearly equal ; but in general
• he preceding method is preferable.
Example. Given T =r 480. T' == 500, and 7=18=, to find :r. Here
^ (T -\- T') = 2-L5 2.3891 G6
^7=4=^ 30' 2 COS. 9.997318
X = 246.52 2.391848
Then AD = 480 — 246.52 = 233.48, and B G =- 500 — 246.52 -=
253.48. The angle of intersection for both branches of the curve being
y°, we find the radii AE = 233.48 cot. 4^ 30' = 2P66.65, and B F ^=
2'>3.t8 cot. 4° 30' = 3220.77.
Article III. — Turnouts and Crossixgs.
49. The Uaual mode of turning off from a main track is by switch-
ing a pair of rails in the main track, and putting in a turnout curve
tangent to the switched rails, wiih a frog placed where the outer rail
-^f the turnout crosses the rail of the main track. A B (fig. 14) repre-
sents one of the rails of the main track switched, B /''represents the
outer rail of the turnout curve, tangent to A B, and E shows the posi-
lion of the frog The switch angle, denoted by S, is the angle DAB,
^urnled by the switched rail A B with A D, its former position in the
main track. The frog angle, denoted by E, is the angle G EM made
Ijy the crossing rails, the direction of the turnout rail at 7^ being the
tangent EM at that jjoint. In the problems of this article the gauge
ot the track D C. denoted by g, and the distance D B, denoted by d
are supposed to be known. The switch angle S is also supposed to
bo known, since its sine (Tab. X. 1) is equal to d divided by the lengtu
Ori
CIRCULAR CURVES.
of the switched rail. If, for example, the rail is 18 feet in lengih and
i = .42, we have S == P 20'.
A. Turnout from Straight Lines.
50. ProfolCBll. Given the radius R of the centre line of a tur*-oui
(JiQ. U), to fold the frog angle G FM = F and the chard B F.
Solution. Through die centre E draw E K parallel to the n :
track. Draw Ci/and FK perpendicular to E K, and join L F.
Then, since E Fis perpendicular io F?,J and F K is perpendicular to
FG, the angle E rK = G FIvl = F; and since E B and B H are
respectively perpendicular to A B and A D, the angle E B H ^ DAB
= S. Now the t'ianglc E F E gives (Tab. X. 2) cos. E F K = f-^
But E F, the radius of the outer rail, is equal to R -\- ^ g, and
FK=CH=Bn— BC=B E cos. E B H — B C = ,R-\- ^ g)
cos. S — (g — d). Substituting these values, we have cos. E FK ~
IK + iff) COS. 5 -{g — d)
B + is
IS^
,or
cos. F = COS. S —
9 — ^
RTT9
From thin formula Fmay be found by the table of natural cosines
To adapt it to calculation by logarithms, we may consider^ — d to be
equal to (g — d) cos. S, which will lead to no material error since
TURNOUT FKOM STRAIGHT LINES. 33
^ — rf is very small, and cos. S almost equal to unity The value of
COS. F then becomes
1^ COS. F = (^ — ? .9 + c?) COS. S
To find BF, the right triangle BCF gives (Tab. X. 9) BF =
BC
Bin. BFC- ^"^ BC = y — d and the angle BF C = B FE
CFE -_ (900 _ LBEF) - (90° - F) = F - i BEF But
BEF - Z?/.F - EBL = F - S. Therefore BFC = F-
■? ^-^ ~ ^) = 2 (^+ ^)- Substituting those values in the formula
/or B F, \vt have
sin. '^{F+S)'
By the above formula; tlie columns headed /"and i^i^in Table V
are calculated.
Example. Given g = 4.7, d = .42, S = 1° 20', and R = .500, to
find /"and Z3 F. Here nat. cos. S = .999729, g — d = 4.28, /2 + ^^
= .^Oo.SS, and 4.28 -^ 502 35 = .008520. Therefore nat. cos. F =
999729 — .008520 = .991209, which gives F=r 36' 10" Next to
liud OF,
g — d = 4.2S 0.631444
H^+ S) =4°28'5" sin. 8.891555
, 25 F=^ 54.94 1.739889
M ProblCBta. Given the frog angle GFM = F {Jig. 14)^ to
find^ the radius R of the centre line of a turnout, and the chord B F.
Solution. From the preceding solution Ave have cos. F =
j-h2g)co3- S—(g~d)
K +Ti • T^fjerefore (R + ^ g) cos. F = {R + ^ g)
COS. ^ — (g — d), or
^ R-^lg= 9-d
cos. aS' — COS. F
For calculation by logarithms this becomes (Tab. X. 29)
£^= 72 + 1^ = h i9~d)
sin.i(i^4- ^')sin. i(/^— ^y
Having thus found R + ^ g, we find R by subtracting ^ g. B F u
found, as in the preceding problem, by the formula
7^ E = fJ — d
3 sin. !(/''+ 6') ■
S4
CIRCULAR CURVES.
Example, Given g = 4.7, d = .42, S = 1'^ 20', and F = 7^ to find
«. Here
i (^ —c/) = 2.14 0.330414
i (F-f 5) = 4'^ 10'
I (Z'— 5) = 2° 50'
sin 8.861 283
sin 8.G93998
R + ^g ^ 595. 85
.-. R = 593.5
7.555281
2.775133
52. Problem. To find mechanically the proper position of a given
frog.
Solution. Denote the length of the switch rail by /, the length of the
frog by/, and its width by w. From B as a centre with a radius
BH= 2/, describe on the ground an arc G H K {fig. 15), and from
the inside of the rail at G measure G H = 2 d, and from H measure
HK such that HK : B H = ^ ic : f ov H K: 21 ^ ^ tv : f; that is,
HK = y- . Then a straight line through B and the point K will
■-trike the inside of the other rail at F, the place for the point of the
tTOg. For the angle HB K has been made equal to h ^' and if B M
be drawn parallel to the main track, the angle MBH is seen to be
equal to h S. Therefore, MBK = BFC = ^ (F -[• S), and this
was shown (§ 50) to be the true value of B F C.
53. If the turnout is to reverse, and become parallel to the main
track, the problems on reversed curves already given will in general
be sufficient. Thus, if the tangent points of the required curve are
fixed, the common radius may be found by § 40 If the tangent point
at the switch is fixed, and the common radius given, the reversing
oint and the other tangent point may be found by ^ 37, the change
)f direction of the two tangents being here equal to S. Bur. when the
TURNOUT i^ROM STRAIGHT LINES.
35
frog angle is given, or determined from a given first radius, and the
point of the frog is taken as the reversing point, the radius of the sec-
ond portion may be found by the following method.
54. Problem. Given the frog anjle F and the distance H B = b
(Jig. 16) between the main track and a turnout, tojind the radius R' of the
second branch of the turnout, the reversing point being taken opposite F, the
fM}int of the frog.
Fig. 16
Solution. Let the arc FB be the inner rail of the second branch,
FG = R' — ^g its radius, and B the tangent point where the turnout
becomes parallel to the main track. Now since the tangent FK is one
side of the frog produced, the angle HFK= F, and since the angle
of intersection at iTis also equal io F, BF K= ^ F {(^2, II.) : whence
BFH=hF Then (^ es) F G = ^r^^^^ , or R' -
:^-^^ (Tab. X. 9), or i Bi^= ^-^
iBF
But BF
sin. iF- sin
6tituting this value of ^ B F, we have
^9 -
Sub
R'
^ ^ sin.2 1 F
In measuring the distance 11 B = b, it is to be observed, that tb«
leidths of both rails must be included.
36 CIRCULAR CURVE'3.
Example. Given 6 = 6 2 and i^ = 8^, to find R'. Here
16 = 3.1 0.491362
1 F = 4^ sin. 8.84358.5
1^^/.^= 44.44 1.647777
i F= 4' sin. 8.843585
/?' — i^r = 637.03 2.804192
•.R' = 639.43
B. Crossinrjs on Straight Lines.
55. When a turnout enters a parallel main track by a second switcn
it becomes a crossing. As the switch angle is the same on both tracks
a crossing on a straight line is a reversed curve between parallel tar.
gents. Let H D and iV/v (fig. 17) be the centre lines of two parallc
tracks, and HA and B /v the direction of the switched rails. If now
the tangent points A and D are fixed, the distance A B ^ a may be
measured, and also the perpendicular distance B P = b between ?.;■?
tangents // P and B K. Tlicn the common radius of the crossing
A C B may be found by ij 33 ; or if the radius of one part of the cross-
ing is fixed, the second radius may be found by § 34. But if both frog
nngles are given, we have the two nidii or the common radius of a
crossing given, and it will then be necessary to determine the distance
A B between the two tangent points.
56. Problem. Given the perpendicular distance G N= b (Jig. 17)
between the centre lines of two parallel tracks, and the 7Xidii E C =-^ R and
CF ^ R' of a crossing, to find the chords A C and B C
Solution. Draw E G perpendicular to the main track, and A L
CM, and B D parallel to it. Denote the' angle A E C by E. Then,
since the angle A E L = AUG = S, we have CE L = E -\- S,
and in the right triangle C E jV (Tab. X. 2), CE cos. OEM =
R cos. {E -]- S)=^ EM= EL — L M. But EL = AE cos. A EL
= R cos. S, and L M : L' M = A C : B C Now AC: B C ^
E C: CF= R: R>. Therefore, L M : L'M = R: R\ or L M : LM
■\- L'M= R: R + R'; that is, L M : b — 2d = R : R -\- R', whence
L M = "j^ , „, - . Substituting these values of E L and L Mm the
equation for R cos. {E + S), we have R cos. {E -\- S) = R cos. S — ■
R{b — 2d)
' R4- R' ■>
CROSSINGS ON STRAIGHT LINES.
S"?
G^
/ n 1 e\ c b — 2d
COS. {L + S) = COS. o — —
A' + R'
Having thus found jE + S, we have the angle E and also its equal
VFB. Then (§ 69)
irr- ^C= 2i2sin. iJE;; Z5 C = 2 72' sin. ^ ^.
We have also A D = A C -\- B C, since .4 C and Z? C are in the
Ecme straight line (§ 32), or .d C = 2 (i? + 72') sin ^ ^.
Fig. 17.
Whcu the two radii are equal, the same formulae apply by making
R' = R. In this case, we have
COS. (E-\-S) = COS. S — ~ '^ ;
2 72
AC= BC= 2Rs\n.^E.
Example. Given d = .42, g = 4.7, 5=1° 20', 6 = 11, and the an-
gles of the two frogs each 7°, to find A C = B C =^A B. The
common radius 72, corresponding to F = 7°, is found (^ 51) to be
593..5. Then 2 72= 1187, 6 — 2 (/ = 10.16, and 10.16^-1187 =
.008.56. Therefore, nat. cos. {E -\- S) = .99973 — .00856 = .99117 ;
whence E -^ S = 1°Z1' 15". Subtracting S, we have E = 6° 17' 15"
Next
2 72 = 1187 3.074451
i i?; = 3° 8' 37^" sin. 8.7.39106
^ C= 65.1
! 813557
38
CIRCULAR CURVES.
C. Turnout from Curves.
57. Problem. Given the radius R of the cadre line of the mair
track and the frog angle F, to determine the position of the frog by means
of the chord B F {figs. 18 and 19), and to find the radius R' of the cen
tre line of the turnout.
Fig. 18.
Solution, I. When the turnout is from the inside of the cunrt
(fig. 18). Let A G and CF be the rails of the main track, AB the
switch rail, and the arc ^i^the outer rail ot the turnout, crossing the
inside rail of the main track sliF. Then, since the angle E FK has its
sides perpendicular to the tangents of the two curves at F, it is equal to
the acute angle made by the crossing rails, that is, E F K = F. Also
E B L ^ S. The first step is to find the angle B KF denoted by K.
To find this angle, we have in the triangle B FK{Tab. X. 14), BK-\-
KF:BK—KF= tan ^ (B FK -{- FB K) : tan. ^ (B FK— F B K).
But B K = R -\- ^ g — d, and K F =^ R — ^ g. Therefore, B K -^
KF = 2R — d, and BK — KF =■- g - d. Moreover, B FK =
BFE + EFK= BFE + F, and FBK= EBF—EBK =
BFE — S. Therefore, BFK—FBK = F-^ S. Lastly, BFK
-f- FBK= 180° — K. Substituting these values in the preceding
I roportion. we have 2R — d:g —d^ tan. (90° — ^K): tan. | (F-
S),
TURNOUT FROM CURVES. 39
or tan. (90^ - i K) = il?.=3^^^ill±^ . But .an. (90» - J K)
= cot. iii = i^rA'*'
l^' • . tan. h K= - — = .^ ~ , ,r, , o> •
^ {2 11 — d) tan. J (F+ /S)
Next, to find the chord B F, we have, in the triangle B F C
{T:ah.X.\2),BF=^/'^j.H^. But B C = g - d, and BCF^
180° — FCK = 180° — (90° — h K) = 90° + ^ A', or sin. B C F
= COS. I K. Moreover, B F C =^ hA^ -\- S) ', for B F K = KFC
-f B f''c, and F B K = K C F —BFC = KFC — BF C. There-
fore, B F K - FBK^2B F C. But, as shown above, B FK —
FB K= F+ S. Therefore, 2 5 FC= F+ 5, or Z?FC= ^ (F+ 5).
Substituting these values in the expression for B F, we have
r^ ^ ,, ^ jg — d) COS. |i^:
•^ sin.H^+'5>') '
Lastly, to find R', we have {k ^^) R' -^ \g = E F = ^^J^ ^EF
But BE F = BLF — EBL, and BLF = L FK + L ^F =
F + TT. Therefore, BEF =F -\- K — S, and
sin. ^(F+A^— 6^)
II. When the turnout is from the outside of the curve, the preceding
solution requires a few modifications. In the present case, the angle
EFK' = F (fig. 19) and EB L = S. To find K, we have in the
triangle B F K, K F -\- B K : KF — B K = tan. ^ (FB K +
B F K) : tan. i (F C A' — i5 F A^. But KF= R-{-lg, and B K
= R — i g + d. Therefore, A"F + B K =-- 2 R + d, and KF —
BK = g — d. Moreover, F B A' = 180° — F B L = 180° —
(EBF—EBL) = 180°— (E BF — S), and BFK = 180° —
BFK' = 180^ — (BFE -\- EFK') = 180° — {E B F + F).
Therefore, FBK—B FK = F + ^. Lastly, Fi3 K -\- B FK =
180 — K Substituting these values in the preceding proportion, we
have 2R-\-d: q — d= tan. (90° — ^K) : tan. i (F + S), or
,an. (90° - i A^) =. (2A±^^i^±^ . But tan. ('90° - 1 /T) =
tan. h K
2
g — d
(2R -\-d) tan. ^ (F+ .S)
40
CIRCULAR CURVES.
Next to find B F, we have, in the triangle B T -^ 3 F ^
B C sin. B CF
sin. BFC
But BC = g ~ d, and B CF = 90^
En
AA, or
Fig. 19.
sin. BCF = COS. ^K. Moreover, BFC=^(F+ S); for BFK
= KFC—BFC,and FB K= KC F-\- B F C = KF C + BF C.
Therefore, FBK— B FK=2B F C. But, as shown above, FBK—
BFK= F+ S. Therefore, 2 BFC = F-}- S,ov B F C=^ {F-\- S).
Substituting these values in the expression for B F, we have, as before.
BF= (ff — ^) cos. hK*
},BF
Sin. 1(^+5)
Lastly, to find R', we have (§ GS) R' -\- ^ fj =r E F =
sin. i BEF
Since ^ Z is generally very small, an approximate valu iof B F may be obtained
By making cos. ^ K = 1. Tliis gives B F = —
g-d
- — , ; T-, r— c> 1 wbich is identical
sm. i (F+ 5) '
with the formula for BF'm^ 50. Table V. will, therefore, give a close approxima-
4on to the value of .B F on curves also, for any value of F contained in the table
TURNOUT FROM CURVES. 41
Bvit BEF = BLF - EBL, and BLF^LFK — LK F =
p _ A-. Therefore, DEF=F—K— 5, and
sin.^iF— K—S)
Example. Given g = 4.7, d = .42, 5 = 1° 20', R - 4583.75, and
F = 7^, to Hnd the chord B Fund the radius R' of a turnout from the
miside of the curve. Here
q — cl = 4.28 0.6.31444 0.631444
2/2 + (/= 9167.92 3.962271
1 (/.^_}_ S) = 4° 10' tan. 8.862433 ' sin- 8.861283
2.824704 1.770161
1 7^- ^ 22' 1.8" tan. 7.806740 cos. 9.999991
GF= 58.905 1.770152
2 0.301030
|(/^ _ 7v — ^') = 2^ 27' 58.2" sin. 8.633766
8.934796
/i' 4- -I ^ = 684.47 2.835356
.-.R' = 682.12
58. Problem. To Jind mechanically the proper position of a given
frog.
■Solution. Tlie niotliod here is similar to that ah-eady given, when
the turnout is from a straight line (§ 52). Draw B .l/(figs. 18 and 19)
parallel to /•' C, and we have FBM = B F C = h {F + S), as just
shown (§ 57). This angle is to be laid off from B M ; but as F is the
point to be found, the chord F C can be only estimated at first, .and
B M taken parallel to it, from which the angle ^ (F -]- S) mi\y be
laid off by the method of § 52. In this case, however, the first meas-
ure on the arc is t/, and not 2 rf , since we have here to start from B i\f,
and not from the rail. Having thus determined the point F approxi-
mately, B M may be laid off more accurately, and F found anew.
59. When frogs are cast to be kept on hand, it is desirable to have
them of such a pattern that they will fall at the beginning or end of a
certain rail; that is, the chord B F is known, and the angle F is re-
quired.
l2 CIRCULAR CURVES.
Problem* Given the position of a frog by means of the chord B F
[figs. 14, 18, and 19), to determine the frog angle F.
g — d
Solution. The formula B F = gin ^^(F -\- S) ' ^^^^^ ^^ exact on
straight lines (§ 50), and near en'jugh on ordinary curves (§ 57, note),
gives
1^ sin.^(F+^)=5:^.
By this formuUi ^ {F -\- S) may be found, and consequently F.
60. Problem. Gii^en the radius R of the centre line of the main
tracks and the radius R' of the centre line of a turnout, to find the frog
angle F, and the chord B F {figs. 18 and 19).
Solution. I. When the turnout is from the inside of the curve
(fig. 18). In the triangle BE Kfind the angle B E K and the side E K.
For this purpose we have B E = R' + h g, B K = R -\- ^ g— d, &nd
the included angle E BK = S. Then in the triangle E FK we have
E K, as just found, E F = R' -{- ^ g, and F K = R— ^ g The frog
angle EFK = F .nay. therefore, be found by formula 15, Tab. X.,
which gives
tan. A F =
_ l(s-6)(5-c)
V
s {s — a)
where s is tiie half sum of the three sides, a the side E K, and b and c
the remaining sides.
Find also in the triangle EFK the angle F E K, and we have the
angle BE F = BEK - FEK. Then in the triangle B E F we
have (§69)
1^^ BF=2{R' + ^g) s'm.^ BE F*
II. AVhen the turnout is from the outside of the curve (fig. 19). In
the triangle B E K find the angle BEK and the side EK For this
purpose we have B E = R' -\- ^ g, B K = R — ^ g -\- d, and the in-
cluded angle E BK= 180= — aS. Then in the triangle E FK vff
Iiave E K, as just found, E F = R' -\- ^ g, and F K = R+ ^ g. The
angle EFK may, therefore, be found by formula 15, Tab. X., which
gives tan. ^EFK = V^' 7(5-0]^^ • ^"^ ^^'° '^"^'^^ ^ ^^' = ^
* The value of B F maj' be more easily found by the approximate formula B F =
, and generally with sufficient accuracy. See note to § 57. This re-
nin. i{F+ S)
mark applies also to B F in the second part of this solution.
TURNOUT FROM CURVES. 43
^ ISO'' — EFK. Therefore ^F = 90° — ^EFK, and cot ^ F =•
tm. ^EFK', .
t^ . • . cot. ^F= \ 5^ — ^— — T — ' »
•^ ^ ^ s (s — a)
where s is tlie half sum of the three sides, a the side ^ K, and 6 and c
the remaining sides.
Find also in the triangle EFK the angle FE K, and ive have the angle
BE F= FE K — BE K. Then in the triangle BE F we have (§ 69)
13^ BF^2{R' + ^g)sm.^BEF
Example. Given g = 4.7, d = .42, 5=1° 20>, R = 4583.75, and
/{> r= 682.12, to find F and the chord Z? Fof a turnout from the outsida
of the curve. Here in the triangle i3 £ /v (fig. 19) we have BE =
^, ^ i ^ ^ 684.47, BK=R — kf} + d = 4581 82, and the angles
BEK+ BKE = S=l° 20'. Then
BK— BE = 3807.35 3.590769
^{BEK+BKE)= 40' tan. 8.065806
1.656575
BK-\- BE = 5266.29 3.721505
^ [BEK— BKE)* = 29.6029' tan. 7.935070
.'. BEK= l"" 9.6029'
„ ^ BK sia EBK . „ „
EK\s now found by the formula EK= sin BEK^ ' ^^' '^S- ^ ^
= log. 4581.82 + log. sin. 178° 40' — log. sin. 1° 96029' = 3.721491,
whence £ir= 5266.12.
Then to find F, we have, in the triangle EFK, s = ^ (5266.12 -f-
684.47 + 4586.10) = 5268.34, s — a =^ 2.22, s — 6 = 4583.87, and
s- c = 682.24.
s_6 = 4583.87 3.661233
s — c = 682.24 2.833937
s = 5268.34 3.721674
— a = 2.22 0.346353
6.495170
4.068027
2)^27T43
^F=3° 30' cot. 72135 71
.•.F= 7°
• This angle and the sine of 1° 9 6029' below, are found by the method given in
•onnection with Table XIII. If the ordinary interpolations had been used, wa
should have found F = 7'^ 7', whereas it should be 7^, since this example is tha
•inverse of that in § 57.
14
CIRCULAR CURVES.
To find FEK, we have s as before, but as a is here the side FR
opposite the angle sought, we have s — a = 682.24, s — h = 458.'? 87,
and s — c = 2. 22. Then bv means of the logarithms just used, we
find ^FEK= 3^ 2' 45". Sul)tnicting ^ B E K = W 48", we have
^BEF ^ 2° 27' 57". Lastly. BF = 1368 94 sin. 2^ 27' 57" =
58.897.
The formula ^ J^ = sm.t{F+ S) (§ 5"' "ote) would give BF =
58 906, and this value is even nearer the truth than that just found,
owing, however, to no eiTor in the formulfe, but to inaccuracifs inci-
dent to the calculation.
61. If the turnout is to reverse, in order to join a track parallel to
the main track, as A CB (fig. 20), it will be necessary to determine
the reversing points C and B. These points will be detennined, if we
find the angles A E C and B F C, and the chords A C and CB.
62 Problem. Given the radius D K = R {Jig 20) of the centrt
line of the main truck the common radius E C = CF = R' of the centre
line of a turnout, and the distance B G = b between the centre lines of the
^parallel tracks, to find the central angles A E C and B F C and the chorda
A C and EC.
Solution. In the triangle A E K fitrd the angle AEK and the side
CROSSINGS ON CURVES. i5
e K For tliis purpose we have AE = R', A K = R — d, and tlic
included angle E A K =- S. Or, if the frog angle has been previously
calculated by § GO, the values of A E K and E K are already known.*
Find in the triangle EFK the amjles E FKand F E K For this
purpose we have E K^ as just found, E F ^ 2 A", and FK = A -^-
R' — h. Then AE C = AEK — FEK, and BFC ^ E FK.
Lastly, (§69)
^^ AC^2Rs\n^AEC; C B = 2 R' sin. ^ B F C.
This solution, with a few obvious modifications, will apply, when
the turnout is from the outside of a curve.
D. Crossings on Curves.
63. When a turnout enters a parallel main track by a second switch,
■ t becomes a crossing. Then if the tangent points A and B (fig. 21)
are fixed, the distance A B must be measured, and also the angles
which A B makes with the tangents at A and B. The common ra-
dius of the crossing may then be found by § 40 ; or if one radius of the
crossing is given, the other may be found by \ 38. But if one tangent
point A is fixed, and the common radius of the crossing is given, it
will be necessary to determine the reversing point C and the tangent
point B. These points will be determined, if we find the angles AEC
vind B F C, and the chords A Cand C B.
64. Problem. Given the radius DK= R {Jig- 21) of the cetitte
line of the main track, the common radius E C = C F = R' of the centre
line of a crossing, and the distance D G = b between the centre lines of the
parallel tracks, to find the central angles AE C and B F C and the chords
A Cand CB.
Solution. In the triangle AEK find the angle AE K and the side
E K. For this purpose we have A E = R', A K = R — d, and the
included angle E Ax K = S.
Find in the triangle B FK the angle B F K and the side F K. For
this purpose we have B F ^ R', B K= R — h + d, and the included
&ng\QFBK= 180=^ — 6'.
Find in the trianale EFK the angles F E K and EFK. For this
* The triangle AEK does not correspond precisely with BEKm^ ^, A being
on the centre line and B on the outer rail ; but the difference is too slight to affect
the calculations.
16
CIRCULAR CURVES.
purpose we have E K and FK a.s just found, and E F —- 2 W. rhet>
AEC =^ AEK— FEK, and BFC^EFK—B FK. Lastlv
(§ 69,)
AC=^2R< sm.hAE C', CB == 2 R' sin. ^ BF C.
D
Fig. 21.
Article IV, — Miscellaneous Problems.
65, Problem. Given A B = a [Jig. 22) and the perpendicular
B C = b, to Jind the radius of a curve that shall pass through C and the
tangent point A.
Solution. Let 0 be the centre of the curve, and draw the radii A 0
and C 0 and the line CD parallel to A B. Then in the right triangle
COD we have 0 C^ = CD"" + OD^ But 0 C = R, CD = a, and
OD = AO — AD = R — b. Therefore, R"" = a"" -{- {R — 6)» =
a^ + R^ — 2 Rb -\- b\ or 2 Rb = a^ -{- b^ ;
2 b
Example. Given a = 204 and b = 24, to find R. Here R »-
204-2 24
2X-24 + 2 = «67 + 12 = 879.
iillSCELLANEOUS PROBLEMS.
47
C6. Corollary 1. If R and b are given to find A B = a, that
vs, to determine the tangent point from which a curve of given radius
most start to pass through a given point, we have (§65) 2Rb =
fl«-f i^ora' = 2Rb — b^;
.'.a = ^b {2R — b).
Example. Given 6 = 24 and 72 = 879, to find a. Here o =-
/94 (1758 — 24) = ^ 41616 = 204.
67. Corollary 2. If R and a are given, and b is required, we
have (§65) 2 Rb = a^ + 6^ or 6« — 2Rb = —a}. Solving this
equation, we find for the value of b here required,
b = R — ^R- — a\
68. Problem. Given the distance AC = c [Jig. 22) and the an-
gle B A C ^ A, to find the radius R or deflection angle D of a curve, that
fhall pass through C and the tangent point A.
Solution. Draw 0 E perpendicular to A C Then the angle AOE
^^A0C = BAC=A{(j2, III.), and the right triangle A OEgWos
(Tab.X.9)^0 = 3j^^i^;
• R- ^^
Sin. A
To find Z), we have (§ 9) sin. D =
•nst found, we have sin. Z) = 50 -^•
^ . Substituting for R its value
he
sin. A '
48 CIRCULAR CURVES.
c
Example. Given c = 2S5.t and ^l = 5°, to find R and D. Heix.
^, 142.7 ,^„,„ , . ^ iOOsin. 50 sin. 5-^
^' = ^75"^ = 163/. 3 ; and sin. D = -^g^- = 2So4 = s'"- ^ "^^
or D = 1 o 45'.
69. Problem. Given the radius R or the deJiecUon amjle D of a
curve, and the angle B A C = A {Jig. 22), made by any chuid with the
tangent at A^ to find the length of the chord A C ■= c.
he
Solution. If R is given, we have (§ 68) R = ^^— j ;
.- .c = 2 R sin. ,1.
Ti» -n, • • 1 ,, ^^v • r> 100 sin. A
n D IS given, we have (§ 68) sin. D ^ —
100 sin. A
c ==
sin. D
This formula is useful for finding tlic length of chords, when a curve
is laid out by points two, three, or more stations apart. Thus, suppose
that the curve ^ Cis four stations long, and that we wish to find the
length of the chord A C. In this case the angle A = A D and c =
100 sin. 4 D
sin. D
Bv this method Table II. is calculated.
Example. Given R = 2455.7 or Z> = 1° 10', and .1 = 4° 40', to
find c. Here, by the first formula, c =^ 4911.4 sin. 4° 40' = 399.59.
,^ , , ^ , 100 sin. 43 40'
Isy the second formula, c — gin \o iq' = 399.59,
70. ProblCDll. Given the angle of intersection K C B = 1 [fig. 23),
and the distance CD = h from the intersection point to the curve in the
direction of the centre., to find the tangent A C = T, and the radius A G
= R.
Solution. In the triangle ^ D C we have sin. CA D : sin. A D C =^
CD: AC. Bnt CAD = ^AOD = ili^ 2, III. and VI.), and as
the sine of an angle is the same as the sine of its supplement,
sin. A D C == sin A D E = cos. DA E = cos 4 /. Moreover, CD
= b and A C = T. Substituting these values in the prectrding pro-
b cos, -^ ^
portion, we have sin. ^ I : cos. ^ I = b : T, or T = ^.^ \*j^ ; whence
(Tab. X. 33)
MISCELLANEOUS PROBLEMS.
19
^- T =h cot. \ I.
To find R, we have (§ 5) R = T cot. ^ I. Substit iting for T ifc
falue just found, wc have
^" R = b cot. ^ 7 cot. ^ 2
Fig. 23.
hxample. Given 7 = 30°, 6 = 130, to find Tan! R. Here
h = 130
^7=7° 30'
7' = 987.45
17= 15°
72 = 368.5.21
2.113943
cot. 0.880571
2.994514
col. 0.571948
3.566462
7 1 . Problem. Given the angle of intersection KC B = 1 [Jig. 23 ).
%nd the tangent A C = T, or the radius A 0 = R, to find C D -^ b.
Solution. If T is given, we have (§ 70) T = h cot. ^ 7, or 6 =
T
lot i/'
.•.h= r tan. 17.
If R is given, we have (§ 70) R = b cot. ^7 cot. |^7, or 6
R
eot ^ Jcot. i / '
.'.b = R tan. ;J 7 tan. ^ 7.
50
CIRCULAR CURVES.
Example. Given /= 27°, T= 600 or 7^ = 2499 lb, to fin.l I
Here b = 600 tan. 6° 45' = 71 01, or i = 2499.18 tan. 6° 45
tan. 13° 30' = 71.01.
1
72. Problem. Given the angle of intersection I of two tangent
A C and D C (fg. 24) to find the tangent point A of a curve, that shed
pass through a point E, given by C D == a, D E =^ b, and the angle CD E
Eig. 24
Solution. Produce DE to the curve at G, and dra^7 C 0 to the cen-
tre 0. Denote DFbyc. Then in the right triangle CDF we have
(Tab. X. U) DF= CD cos. CDF, or
c = a cos.
Denote the distance A D from D to the tangent point by x. Then, by
Geometry, x^ = D E X D G. But D G = D F -\- FG = DF +
EF=2DF— DE = 2c — b. Therefore, x^ = b{2c — b), and
5^" x = ^b{2c — b).
Having thus found A Z), we have the tangent AC = AD -{• DC
= X -\- a. Hence, R ox D may be found (^ 5 or § 11).
If the point E is given by £^^and Ci/ perpendicular to each other,
a and b may be found from these lines. For a = C H -\- DH ^
(75"+ JE;77cot. iZ(Tab. X. 9). and6 =^DE = ^^i-
MISCELLANEOUS PROBLEMS.
5i
Example. Given I = 20° 16', a = 600, and 6 = 80, to find x and
H. Here c = 600 cos. 10° 8' = 59064, 2 c - 6 = 1101. 28, and x =
ySO X 110^28 = 296.82. Then T = 600 + 296.82 = 896.82, and
R = 896.82 cot. 10° 8' = 5017.82.
73. Problem. Given the tangent A C {Jig. 25), and the chora
A By uniting the tangent points A and B, to Jind the radius A 0 -- R.
Fig. 25
Solution. Measure or calculate the perpendicular CD. Then if CZ)
be produced to the centre 0, the right triangles AD C and CA 0,
having th3 jungle at G common, are similar, and give CD : A D =
AC: A 0, or
^^A^XAC
CD
If it is inconvenient to measure the chord A B, a line E F, parallel
to it, may be obtained by laying off from C equal distances CE and
CF. Then measuring E G and G C, we have, from the similar tri-
GEXAC
%ng\esE GCand CAO, CG:GE =AC:AO,orR= — ^G — *
Example. Given ^ C = 246 and AD = 240, to find R. Here
240 X 246
VD = 54, and R = - ^'^= 1093.33.
52 CIRCULAR CURVES.
74. Problem. Given the radius AO = R [foj 25), to find :ht
tangent A C = J- of a curve to unite two straight lines given on the ground
Solution. Lay off from the intersection C of the given straight lines any
equal distances CL and CF. Draw the pe7-pendictdar C G to the mid-
dle of E F, and measure G E and C G. Then the right triangles
E G Cand C A 0, having the angle at C common, are similar, and
give GE: CG = AO: AC, or
EF- r^__CGx AO
GE
By this problem and the preceding one, the radius or tangent points
of a curve mav be found without an instrument for measuring angles.
Example. Given R = 1093|, G E = 80, and C G = 18, to find '/'.
18X1093^
Here F = gQ = 246.
75. Problem* To find the angle of intersection I of two straight
lines, when the point of intersection is inaccessible, and to determine the tan-
gent points, when the length of the tangents is given.
Solution. I. To find the angle of intersection i L.ct A C and C I'
(fig. 26) be the given lines Sight from some point A on one line lo a
point B on the other, and measure the angles CAB and T B V. These
angles make up the change of direction in passing from one tangent to
the other. But the angle of intersection (§ 2) shows the change of di-
rection between two tangents, and it must, therefore, be equal to the
sum of C A B and T B V, that is,
t^ 1= CAB-^ TBV
But if obstacles of any kind render it necessary to pass from A C to
B Fby a broken line, as A D E F B, measure the angles C A D, N D E,
P E F, RFB, and S B V, observing to note those angles as mimts which
are laid off contrary to the general direction of these angles. Thus the
general direction of the angles in this case is to the right; but the
angle P EF lies to the left oi D E produced, and is therefore to be
marked minus. The angles to be measured show the successive changes
of direction in passing from one tangent to the other. Thus C A D
6hov/s the change of direction between the first tangent and A D,
ND E shows the change between A D produced and D E, P E F the
change between DE produced and E F, R F B the change between
£'F produced and FB, and, lastly, SB Fthe change between B F ])ro-
MISCELLANEOUS PROBLEMS.
53
duccd and the second tangent. But the iing^lc of intersection (§ 2)
shows the change of direction in passing from one tangent to another,
and it must, therefore, be equal to the sum of the partial changes
naeasuved, that is,
13^
/ = CA D -\- y DE - PEF-^ II FB + SB V.
Fig. 26
II. To determine the tangent points. This will be done if we find
the distances .1 Cand B C; for then any other distances from Cmay
be found. It is supposed that the distance A B, or the distances A Z),
DE, E F, and FB have been measured.
Tf one line A B connects A and B. Jind A C and B C in the triangle
ABC. For this purpose we have one side A B and all the angles.
Jf a broken line A D E F B connects A and B, let fall a perpendicular
B G from B upon A C, produced if necessary, and find A G and B Q
hy the usual method of working a traverse. Thus, if A C is taken as a
meridian line, and D /v, E L, and FM are drawn parallel to A C, and
D H, E K, and FL are drawn parallel to B G, the difference of lati-
tude A G is equal to the sum of the partial differences of latitude A H.
D K, EL, and FM, and the departure B G h equal to the sum of the
partial departures D II, E K, F L, and B HI. To find these partial
differences of latitude and departures, we have the distances A I), DE,
E F, and F B, and tiie bearings may be obtained from the angles
already measured. Thus the bearing of yl Z) is C A D, the bearing of
DE is KDE = KDN+ NDE =^ C A D -\- NDE, the bearing
of jB F is LEF = LEP— PEF^ KDE— PEF, &nA the
54
CIRCULAR CURVES.
bearing oi F B is MFB = MFR -{- RFB=^ LEF + RFB; that
is, the bearing of each line is equal to the algebraic sum of the preced
ing bearing and its own change of direction. The differences of lati-
tude and the departures may now be obtained from a traverse table,
or more correctly by the formulis :
DiiF. of lat. = dist. X cos. of bearing ; dep. = dist. X sin. of bearing
Thus, AH= AD cos. CAD, and DU=AD sin. CA D.
Having found A G and B G, we have, in the right ti'iangle B G C,
(Tab. X. 9) GC = B G cot. B C G, and BC = ^^^-q ■ But
BCG=180° — I. Therefore, cot. BCG = — cot. /, and sin.BCG
= sin. /. Hence G C =- — B G cot. 7, and BC = ^^^77 . Then,
since A C = A G -\- G C, we have
AC=AG — BG cot. /;
BC
BG
sm.
When /is between 90° and 180°, as in the figure, cot. /is negative,
and — B G cot. I is, therefoi-e, positive. When / is less than 90°, G
will fall on the other side of / ; but the same formula for A C wil still
apply ; for cot. / is now positive, and consequently, — B G cot. / is
negative, as it should be, since, in this case, A C would equal A G mi
mis G C.
Example. Given A D = 1200, DE = 350, E F ==^ 300, F B =^
310, CAD== 20°, NDE = 44°, PE F =. — 25°, R FB = 31°.
and SB V ^ 30°, to find the angle of intersection /, and the distance?
A C and B C.
Here 7 = 20° + 44° — 25° + 31° + 30° = 100°. To find A G
and B G, the work may be arranged as in the following table : —
Angles to
the Right.
Bearings.
Distances.
N.
£.
0
20
44
—25
31
N. 20 E.
64
39
70
1200
3.50
300
310
1127.63
153.43
233.14
106.03
410.42
314.58
188.80
291.30
1620.23
1205.10
The first column contains the observed angles. The second contains
the bearings, which are found from tne angles of the first column, iv
MISCELLANEOUS PROBLEMS.
55
the manner already explained. A Cis considered as running north
from A, and the bearings are, therefore, marked N. E. The other col-
umns require no explanation. "We find A G = 1620.23, and B G =
1205.10. Then GC = — BG cot. I = — 1205.1 X cot. 100° =-
212.49. This value is positive, because it is the product of two nega-
tive factors, cot. 100° being the same as —cot. 80°, a negative quanti-
ty. Then AC= AG + GC= 1620.23 + 212.49 = 1832.72, and
BC = -. — ^bn = 1223 69. Having thus found the distances of A
sin. 1UU-' °
and B from the point of intersection, we can easily fix the tangent
points for tangents of any given length.
76. Problem. To Uuj out a curve, when an obstruction of any kind
prevents the use of the ordinarij methods.
^ig. 27
Solution. First Method. Suppose the instrument to be placed at
A (fig. 27), and that a house, for instance, covers the station at B, and
also obstructs the view from A to the stations at D and E. Lay off
from A C, the tangent at yl, such a multiple of the deflection angle Z),
iis will be sufficient to make the sight clear the obstruction. In the
figure it. is supposed that 4 Z) is the proper angle. The sight will then
pass through F, the fourth station from A, and this station will be de-
termined by measuring from A the length of the chord A F, found by
56
CIRCULAR CURVES.
§ 69 or by Table II. From the station at i^ the stations at D and E
may afterwards be fixed, by laying off the proper deflections from the
tangent at F.
Second Method. This consists in running an auxiliary curve paral
lei to the true curve, either inside or outside of it. For this purpose
lay off perpendicular to A C, the tangent at A, a line A A' of any con
venient length, and from A' a line A' C parallel to A C. Then A' C'
is the tangent from which the auxiliary curve A< E' is to be laid off.
The stations on this curve are made to correspond to stations of 100
feet on the true curve, that is, a radius through B' passes through Zj, a
radius through D' passes through D, &c. The chord .4' B' is, tlicre-
fore, parallel to A B, and the angle C A' B' = CAB; tliat is, the de-
flection angle of the auxiliary curve is equal to that of the true curve
It remains to find the length of the auxiliary chords A' B', B' D', &c
Call the distance A A' = h. Then the similar triangles ABO and
A' B> 0 give A 0 : A' O = A B : A' B', or R : R — b = 100 : A' B>.
Therefore A< B< - ^^^<^~'^ _ i no ^^^ * tp .i -r
j-ueicrore, ^ ij — ^ = 100 — — ^ . If the auxihary curve
were on the outside of the true curve, we should find in the same way
.-l' B' ^ 100 4- -^ . It is well to make h an aliquot part of R ; foi
the auxiliary chord is then more easily found. Thus, if n is anv
whole number, and we make 6 = - , we have A' B' = 100 ± ^%^
= 100 ± — . If, for example, ^ = Jq^ , we have ?? = 100, and .1 ' B
= 100 ± 1 = 101 or 99. When the auxiliary curve has been run,
the corresponding stations on the true curve are found, by laying off
in the proper direction the distances B B', D D', &c., each equal to b.
77. Proljlcm. Having run a curve A B [Jig. 28), to change the
tangent point from A to C, in such a way that a curve of the same radius
may strike a given point D.
Solution. Measure the distance B D from the curve to D in a direction
parallel to the tangent C E. This direction may be sometimes judged
of by the eye, or found by the compass. A still more accurate way is
to make the angle DBE equal to the intersection angle at E, or to
twice BAE, the total deflection angle from A to B; orif^ can be
seen from B, the angle DBA may be made equal to BAE.
Measure on the tangent (backward or forivard, as the case may be) a dis
lance A C — B D, and C will be the 7iew tangent point required. For. if
rfl"be drawn equal and parallel to A F, we have Fi7 equal and par
MISCELLANEOUS PROBLEMS. 5/
uUel to AC, and therefore equal and parallel to B D. Hence D H ==
B F.= AF= CH, and D /7 being equal to C H, a. curve of radios
07 i^ from the tangent point C must pass through D.
78 ProblenB. Having run a curve A B (Jig. 29) of radius li <n
deflection angle Z>, terminating in a tangent B D, to Jind the radius IV or
deflection angle D' of a curve A C, that shall terminate in a given parallel
tangent CE.
Fig. 29.
A K
iSolution. Since the radii Z? F and CG are perpendicular to the par-
allel tangents CE and B L>, they are parallel, and the angle A GG =
Therefore, A C G, the half-supplement of A G C, is equal t«
4
4.Fb
m
CIRCULAR CURVES.
A B F, the half-supplement of A F B. Hence A B and B C are in the
same straight line, and the new tangent point C is the intersection ol
A B produced with C E.
Represent AB by c, and A C = c -\- B C by c'. Measure B C, or, if
more convenient, measure D C and find B C by calculation. To calculate
D C
B C from D C, we have B C =^ ^-^^ j^^^ (Tab. X. 9), and the angle
DBC = ABK= BAK, the total deflection from .4 to B. Then
the triangles AFBandAG C give A B : AC = BF : C G,oy c : c'
= R:R';
,'.R' = -R.
c
50
50
Sub-
To find Z)', we have (§ 10) /vl' = ^^^, , and R = ^^^ -
sdtuting these values in the equation for R', we have gj^ jy, =
50
TX
50
sin. D '
. sin. D' = -, sin. D.
79. Problem. Given the length of tico equal chords A C and B C
[Jig. 30), and the perpendicular CD, to find the radius R of the curve.
Fig. 30
Solution. From 0, the centre of the curve, draw the perpendicular
OE. Then the similar triangles QBE and BCD give B 0 : B E
^ BC: CD.orR:hBC=E C: CD. Hence
7? =
BC^
2 CD
MISCELLANEOUS PROBLEMS.
59
This problem serves to find the radius of a curve on a track already
laid. For if from any point C on tlie curve we measure two equal
.-hords .1 Cand B C, and also the perpendicular CD from Cu2)on the
whole chord A B, we have the data of this problem.
80. Prot>l.(3lll. To draw a tangent F G {Ji<j. 30) to a given curve
from a given point F.
Solution. On any straight line F/1, ichich cuts the curve in two points,
measure F C arid FA, the distances to the curve. Then, by Gcometrv,
FG =yFCx FA.
This length being measured from F, will give the point G. When
FG exceeds the length of the chain, the direction in which to measure
it, so that it will just touch the curve, may be found by one or two trials.
8\. Problem. Having found the radius A 0 ^ E of a curve
(fg. 31 ), to substitute for it tico radii A E = R^ and D F = A'o , (he,
'ongcr of vhich A E or B E ' is to be used for a certain distance only ai
mrh end of the curve.
>Jolution. Assume the longer radius of any length ivhich mat/ be thought
60 CIRCULAR CURVES.
proper, and find (§ 9) the corresponding deflection angle D^. Suppose
that each of the curves A D and B D' is 100 feet long. Then drawing
CO, we have, in the triangle FOE,OE:FE = s'm.OFE : sin. FOE.
But the side OE = AE— AO = Ri — R, F E = D E — D F ==
Z?i — /?<. , the angle FOE = \S0° — A 0 C ^ 1 80° — i /, and the
angle 0FE=A0F— 0EF=^I-2Di, since 0 E F = 2 D,
(§ 7). Substituting these values, and recollecting that sin. (180° — ^7)
= sin. ^ /, we have R^ — R\R^ — R. = sin. (i / — 2 Z), ) : sin. ^ 1
Hence
' sin.(i7-2Z)J
^2 is then easily found, and this will be the radius from D to D\ or
until the central angle DFD' = I— 4 D^.
The object of this problem is to furnish a method of flattening the
extremities of a sharp curve. It is not necessary that the first curve
should be ju'st 100 feet long ; in a long curve it may be longer, and in
a short curve shorter. The value of the an^le at E will of course
change with the length of A D, and this angle must take the place of
2 Di in the formula. The longer the first curve is made, the shorter
the second radius will be. It must also be borne in mind, in choosing
the first radius, that the longer the first radius is taken, the shorter will
be the second radius.
Example. Given R — 1146. 28 and 7= 45°, to find i?2> if ^i is as-
sumed = 1910.08, and A D and B D> each 100. Here, by Table I.,
Dj = 1° 30'. Then
A', —R = 763.8 2.8829S0
i / = 22° 30' sin. 9..582840
2.465820
i/— 2D,-= 19° 30' sin. 9.523495
Ri — R^ = 875.64 2.942325
.-. /?2 = 72i — 875.64 = 1034.44
82. Problem. To locate the second brcrch of a compound or re-
versed curve from a station on the first branch.
Solution. Let J. B (fig 32) be the first branch of a compound curve^
and D its deflection angle, and let it be required to locate the second
branch AB\ whose deflection angle is Z)', from some station B
unA B.
MISCELLANEOUS PROBLEMS. 61
Let n be tfie number of stations from A to B, and n' the number of sta-
lions from A to any station B' on the second branch. Represent by Vtht
%ngle A B B', which it is necessary to lay off from the chord B A to strike
B>. Let the correspondinj:; ande A B' B on the other curve be repre-
Fig. 32
rented by V. Then we have F+ F' = 180° — BAB'. But if
T T' be the common tangent at A, we have TA B + T' A B' = nD
J^ n' D' = 180° — BAB'. Therefore, V-{- V = nD -{• n' D'.
Next in the triangle AB B' we have sin. V : sin. V= AB : AB'.
But A B : A B' = n :n', nearly, and sin. V : sin. V = V : V, near-
n
ly. Therefore we have approximately F' : F = n : n', or F' = -, F.
Substituting this value of F' in the equation for F+ F', we have
r+ J V=nD-\-n'D'. Therefore, n' F+ n F= ?i' (nZ) + n'Z)'), or
n -\- n'
The same reasoning will apply to reversed curves, the only change
being that in this case F+ V = nD — n' D', and consequently
V= ^' i»^ — ^'D')
n -{• n'
When in this formula n' D' becomes greater than n D, V becomes
minus, which signifies that the angle Fis to be laid off above B A in-
stead of belov/.
This problem is particularly useful, when the tangent point of a
curve is so situated, that the instrument cannot be set o\cr it. The
same method is applicable, when the curve A B' starts from a straight
line ; for then we may consider A B' as the second branch of a com-
pound curve, of which the straight line is the first branch, having its
radius equal to infinity, and its deflection angle D = 0. Making
D = 0, the formula for F becomes
62
CIRCULAR CURVES.
n -\- )i'
When n and 71' are each 1, the formula for Fis in all cases exact,
for then the supposition that V : V = 71 : n' is strictly true, since AB
will equal A B', and Fand F', being angles at the base of an isosceles
triangle, will also be equal. Making n and 71' equal to 1, we have
When the curve starts from a sti-aight line, this formula becomes, by
making Z) = 0,
We have seen that when n or n' is more than 1, the value of Fis
only approximate. It is, however, so near the truth, that when nei-
ther n nor n' exceeds 3, the error in curves up to 5° or 6° varies from
a fraction of a second to less than half a minute. The exact value of
F might of course be obtained by solving the triangle ABB', in
which the sides AB and AB' may be found from Table II., and the
included angle at A is known. The extent to which these formnlte
may be safely used may be seen by the following table, which gives
the approximate values of Ffor several different values of n,n',D^
and />', and also the error in each case.
Compound Curves.
Reversed Curves.
n.
D.
0
n".
D'.
0
V.
Error.
n.
D.
0
«'.
0
V.
Error.
0 ;
i\
0 1
n
1
0
5
1
4 10
0.9
1
3
4
3
7 12
27.2
1
0
5
3
12 30
25.3
2
3
4
3
4 0
23.5
2
0
3
3
5 24
22.1
3
3
4
3
1 42f
8.3
3
0
3
3
4 30
29.7
3
h
0
3
3 45
24.0
1
1
5
3
13 20
18.6
2
I
1
4
0 40
O.I
2
1
2
9
1
3
1 20
0.7
2
1
4
9
4 0
11.0
2
3-
3
7 48
15.0
1
6
2
6
4 0
23.5
0
2
4
3
10 40
24.7
1
5
3
5
7 .'U)
51.8
3
3
3
4
10 30
54.0
2
3
5
3
0 25f
52.8
As the given quantities are here arranged, the approximate values
of Fare all too great ; but if the columns n and n' and the columns D
and D' were interchanged, and F calculated, the approximntc values
of F would be just as much too small, the column of cnoi> rcniaiuing
the same.
MISCELLANEOUS PROBLEMS.
63
83. Problem. To measure the distance across a river on a given
Uraight line.
D
Fig. 3.3.
Solution. First Method. Let A B (fig. 33) be the required distance
Measure a line A C along the bank, and take the angles B A C and
ACB. Then in the triangle ^1 C Cwe have one side and two angles
to nnd A B.
1( A Cis of such a length that an angle A C B = ^D A C can he
laid off to a point on the farther side, we have ABC=^DAC=^
ACB. Therefore, without calculation, AB = AC.
Fig. 34.
Second Method. Lay off ^ C (fig. 34) perpendicular to A B. Meas-
ure xi C, and at Clay off CZ) perpendicular to the direction CB, and
meeting the line of /I B in D. Measure A D. Then the triangles
A CD and ABC are similar, and give AD : A C =- A C : AB.
Therefore, AB ^ -^ .
If from C, determined as before, the angle A C B' be laid off equal
to yl CB, we have, without calculation, A B = AB'.
Third Method. Measure a line A D (fig. 35) in an oblique direction
from the bank, and fix its middle point C From any convenient
point E in the line of A B, measure the distance E C, and prodiue
64
MISCELLANEOUS PR0BLE3IS.
E C until CF= Ea Then, since the triangles A CE and D CF
are similar by construction, we see that DF is parallel to E B. Find
Fig. 35
now a point G, that shall be at the same time in the line of CB and
of D F, and measure G D. Then the triangles ABC and D G C sre
equal, and G D is equal to the required distance A B.
As the object of drawing E Fis to obtain a line parallel to A B, this
line may be dispensed with, if by any other means a line GFhe drawn
through D parallel to AB. A point G being found on this parallel in
the line of C B, we have, as before, GD = AB.
PARABOLIC CURVES.
65
CHAPTER II.
PARABOLIC CURVES.
Article I. — Locating Parabolic Curves.
84. Let AEB (fig. 36) be a parabola, A C and B C its tangents,
iiid .1 B the chord uniting the tangent points. Bisect A B in D, and
oin CD. Then, according to Analytical Geometry, —
Fig. 36.
L CD is a diameter of the parabola, and the curve bisects CDinE-
II. If from any points T, T', T", &c., on a tangent A F, lines be
a.-awn to the curve parallel to the diameter, these lines T M, T' M ,
1 "M" &c., called tangent deflections, will be to each other as the
Benares' of the distances AT, A T>, A T'\ &c. from the tangent
ptint A.
III. A line F D (fig. 37), drawn from the middle of a chord A Bio
the curve, and parallel to the diameter, may be called the middle ordi
nate of that chord ; and if the secondary chords A E and B E he drawn,
the middle ordinates of these chords, K G and /. H. are each equal to
{ED. In like manner, if the chords A A', KE,EL, and LB he
drawn, their middle ordinates will be equal to \KG or \L H.
\V. K tangent to the curve at the extremity of a middle ordinate,
is parallel to the chord of that ordinate. Thus MF, tangent to the
cur\ e at E, is parallel to A B.
rs
PARABOLIC CURVES.
V. If any two tangents, as yl C and B C, be bisected in M and /
ihe line il/F, joining the points of bisection, will be a new tangent, ita
middle point E being the point of tangency.
85. I*rol>leill. Given the tangents A C and B C, equal or unequal^
{Jig. 36,) and the chord A B, to lay out a parabola hy tangent deflections.
Fig. 36
Soluticm. Bisect A B in A and measure CD and the angle A CD^
or calculate CD* and A CD from the original data. Divide the tan-
gent A C into any number n of equal parts, and call the deflection
JM/for the first point a. Then {§ 84, II.) the deflection for the sec-
ond point will be T' M' = 4 a, for the third point T" M" = 9 a, and
60 on to the nth point or C, where it will be n^a. But the deflection
at this last point \sGE = ^CD{^ 84, I). Therefore, n^ a = C E.
and
CE
a =
n*
Having thus found a, we have also the succeeding deflections 4 a, 9 a.
16 a, &c. Then laying ofl^ at T, T', &c. the angles A T M, A T' M>,
&c. each equal to A CD, and measuring down the proper deflections,
just found, the points M, il/', &c. of the curve will be determined.
The curve may be finished by laying off on -4 C produced n parts
equal to those on A C, and the proper deflections will be, as before, a
multiplied by the square of the number of parts from A. But an
* Since C D is drawn to the middle of the base of the triangle ^ iS C, we have, hj
Rwmetrj-, C D'^ = ^ (A C^ + B C^) — A D"-.
LOCATING PARABOLIC CURVES.
67
PaMcr way generally of finding points beyond E is to divide the sec-
ond tangent B Cinto equal parts, and proceed as in the case of ^ I.
If the number of parts on B C be made the same as on A (7, it is obvi-
ous that the deflections from both tangents will be of the same length
for corresponding points. The angles to be laid off from B C must,
Df course, be equal to BCD.
The points or stations thus found, though corresponding to equal
distances on the tangents, are not themselves equidistant. The length
of the curve is obtained by actual measurement.
86. Problem. Given the tangents A C and B C, equal or unequal,
[fig. 37,) and the chord A B, to lay out a parabola by middle ordinates.
Solution. Bisect A B in D, draw CD, and its middle point E will
oe a point on the curve (§ 84, L). D E is the first middle oi^.nate,
and its length may be measured or calculated. To the point E draw
t>-.e chords A E and BE, lay off the second middle ordinates G K and
HL, each equal to \DE{^ 84, III), and K and L are points on the
curve. Draw the chords A K, K E, E L, and L B, and lay oft third
middle ordinates, each equal to one fourth the second middle ordi-
nates, and four additional points on the curve will be determined.
Continue this process, until a sufficient number of points is obtained
87. Prol>leiIl. To draiv a tangent to a parabola at any station.
Solution. I. If the curve has been laid out by tangent deflections
(^ 85). let M"' (fig. 36) be the station, at which the tangent is to be
drawn. From the'' preceding or succeeding station, lay off, parallel to
CD, a distance M"NoxEL equal to a, the first tangent deflection
(§ 85), and M'" N or M'" L will be the required tangent. The same
thing may be done by laying off from the second station a distance
j^, 7^/ ^ 4 „ or at the third station a distance GP = ^a; for the
BS PARABOLIC CURVES.
required tangent will then pass through T' or G. It will be seen,
also, that the tangent at M'" passes through a point on the tangent at
A corresponding to half the number of stations from A to 31'" ; that
is, M'" is four stations from A, and the tangent passes through T',
the second point on the tangent A C. In like manner, M'" is six sta-
tions from Z?, and the tangent passes through G, the third point on the
tangent B C
II. If the curve has been laid out by middle ordinates (§ 86), the tan-
gent deflection for one station is equal to the last middle ordinate made
use of in laying out the curve. For if the tangent A C (fig. 37) were
divided into four equal parts corresponding to the number of stations
from A to E^ the method of tangent deflections would give the same
points on the curve, as were obtained by the method of § 86. In this
case, the tangent deflection for one station would be a =^ i\ C E ^
jg DE., but the last middle ordinate was made equal to ^ G K or
ie D E. Therefore, a is equal to the last middle ordinate, and a tan-
gent may be drawn at any station by the first method of this section.
A tangent may also be drawn at the extremity of any middle ordi-
nate, by drawing a line through this extremity, parallel to the chord
of that ordinate (§ 84, IV.).
88. In laying out a parabola by the method in § 85, it may some-
times be impossible or inconvenient to lay off all the points from the
original tangents. A new tangent may then bo drawn by § 87 to any
station already found, as at M'" (fig. 36), and the tangent deflections
a, 4 a, 9 a, &c. may be laid off from this tangent, precisely as from the
first tangent. These deflections must be parallel to CD, and the dis-
tances on the new tangent must be equal to 7'' iV or iViV", which
may be measured.
89. Problem. Giveii the tangents A C and B C, equal or uneqiml,
[Jiy 38,) to lay out a parabola by bisecting tangents.
Solution. Bisect A C and B C in D and F, join D F, and find £", the
middle point of D F. E will be a point on the curve (§ 84, V.). We
have now two pairs of what may be called second tangents, A D and
I) E, and E F and F B. Bisect A Din G and D E in H, join G H,
and its middle point ilf will be a point on the curve. Bisect £" F and
F Bin K and L, join KL, and its middle point iVwill be a point on
the curve. We have now four pairs of third tangents, A G and G M,
M H and U E, E K and KN, and N L and L B. Bisect each pair in
turn, join the points of bisection, and the middle points of the joininj;
LOCATING PARABOLIC CURVES.
69
lines will be four new points, il/', M", iV", and N'. The same methcx?
may be continued, until a sufficient number of points is obtained.
Fig. 38.
90. Problem. Given the tangents A C and B C, equal or unequal
Hg. 39,) and the chord A B, to lay out a parabola by intersections.
Fig. 39
Solution. Bisect A B in D, draw CD, and bisect it in E. Divide
the tangents A Cand B C, the half-chords A D and D B, and the line
CE, into the same number of equal parts ; five, for example. Then
the intersection M of A a and F G will be a point on the curve. For
FM = I Ca, and Ca = i CE. Therefore. FM= 55 CE, which is
the proper deflection from the tangent atFto the curve (§ 8.5). In
like manner, the intersection N of Ab and II K may be shown to be a
point on the curve, and the same is true of all the similar intersections
indicated in the figure.
If the line DE were also divided into five equal parts, the line A a
would be intersected in il/on the curve by a line drawn from B through
a', the line A b would be intersected in iVon the cur\'e by a line drawn
70
PARABOLIC CURVES.
from B through 6', and in general any two lines, drawn from A and B
through two points on CD equally distant from the extremities Cand
D, will intei-sect on the curve. To show this for any point, as x)/, it is
sufficient to show, that B a' produced cuts F G on the curve ; for it
has already been proved, that A a cuts F G on the curve. Now
Da':MG^BD:B G = b:^,or M G =lDaK But Da' = \ C E.
Therefore, MG = h C E. Again, F G : CD =^ A G : A D = I ■:>.
Therefore, FG = \CD = lCE. We have then FM = F G —
MG = f CE — ii C E = is C E. As this is the proper deflection
from the tangent at F to the cm-ve (§ 85), the intersection of B a' with
F G is on the curve. This furnishes another method of laying out a
parabola by intersections.
91. The following example is given in illustration of several of the
preceding methods.
Example. Given AC = B C ^ 832 (fig. 40), and -1 B = 1536 to
lay out a parabola A E B. We here find CD = 320. To begin with
the method by tangent deflections (§ 85), divide the tangent A C into
C E ^(\0
eight equal parts. Then a = —^ = -wr = 2.5. Lay off from the
divisions on the tangent Fl = 2.5, G2 =4 X 25 = 10, ^3 =
9X25 = 22.5, and /v 4 = 16 X 2.5 = 40. Suppose now that it is
inconvenient to continue this method beyond K. In this case we may
Fig. 40
find a new tangent at E, by bisecting A Cand B C {^ 89), and draw-
ing KL through the points of bisection. Divide the new tangent
KE =^ ^ AD ^ 384 into four equal parts, and lay oflT from KE the
RADIUS OF CURVATURE.
71
same tangent deflections as were laid off from .fi iiT, namely, 3/5 -
22.5 A^6 = 10, and 07 = 2.5. To lay off the second half of the
curve by middle ordinates (§86), measure EB= 784.49. Bisect
EB in P, and lay off the middle ordinate P R = ^D E ^ AQ.
Measure ER^ 386.08, and BR = 402.31, and lay off the middle or-
dinates S T and V IF, each equal to ^ P /2 = 10. By measuring the
chords ET, TR, R TF, and WB, and laying off an ordinate fron'
each, equal to 2 5. four additional points might be found.
Article II. — Radius of Curvature.
92. The curvature of circular arcs is always the same for the same
arc, and in different arcs varies inversely as the radii of the arcs.
Thus, the curvature of an arc of 1,000 feet radius is double that of an
arc of 2,000 feet radius. The curvature of a parabola is continually
changing. In fig. 39, for example, it is least at the tangent point A,
the extremity of the longest tangent, and increases by a fixed laAv, un-
til it becomes greatest at a point, called the vertex, where a tangent to
the curve would be perpendicular to the diameter. From this poin;
to B it decreases again by the same law. We may, therefore, con-
sider a parabola to be made up of a succession of infinitely small cir-
cular arcs, the radii of which continually increase in going from the
vertex to the extremities. The radius of the circular arc, correspond-
ing to any part of a parabola, is called the radius of curvature at that
point.
If a parabola forms part of the line of a railroad, it will be necessa-
ry, in order that the rails may be properly curved (§ 28), to know
how the radius of curvature may be found. It will, in general, be
necessary to find the radius of curvature at a few points only. In
short curves it may be found at the two tangent points and at the mid-
dle station, and in^onger curves at two or more intermediate points
besides. The rails curved according to the radius at any point should
be sufficient in number to reach, on each side of that point, half-way to
the next point.
93. Problem. To find the radius of curvature at certain stations
on a parabola.
Solution. Let AEB (fig. 41) be any parabola, and let it be re-
quired to find the radii of curvature at a certain number of stations
72
PARABOLIC CURVES.
fron. A to E. Tliese stations must be selected at regular interral
from those determined by any of the preceding methods. Let n de
note the number of parts into which ^ £ is divided, and divide CL
into the same number of equal parts. Draw lines from A to the points
of division. Thus, if n — 4, as in the figure, divide CD into four
equal parts, and draw A F, A E, and A G. Let A D = c^ A F = Ci
A E = C2, A G — C3, and A C = T. Denote, moreover, C D hy d
and the area of the triangle A C B hy A. Then the respective radii
for the points .£,1,2, 3, and A will be
R = 2, /?, =
A
II
V2
A
A*3
A '
Ra =
A
The area A may be found by form. 18, Tab. X.; c and T are known ;
and Ci, Co, c^ may be found approximately by measurement on a figure
carefully constructed, or exactly by these general formulae : —
&c.
7^2 _c2 {n~\)d^
n
f2
—
c2
n
j'i
—
C2
n
'fi
—
c2
n'
[n
-3)
d^
n2
[n
-5)
f/2
n2
[n_
-7)
«2
d^
&c.
It will be seen, that each of these values is formed from the preceding,
by adding the same quantity — - — , and subtracting ^ multiphed in
STiccess-lorj hr w — 1, n - Z -n - 5, v^ ^flaking: ^> ~ i, we have
RADIUS OF CURVATUKb.
ra
c^^ = c^ 4-^(r2_c«)-i'gcr',
C2'' = c,^-hUT'-c-')-ud\
Ca^ = c
i' -\- ^ {T^ - c'') + ud'.
A.11 the quantities, wliicb enter in* j tlic expressions for the radii, are
now known, and the radii may, therefore, be determined. The same
method will apply to the other half of the parabola.
The manner of obtaining the preceding formulte is as follows. The
radius of curvature at any given point on a parabola is, by the Differ-
ential Calculus. R = 2^i^^.3 E ' ^" which p represents the parameter of
I lie parabola for rectangular coordinates, and E the angle made with
a diameter by a tangent to the curve at the given point. First, let the
middle station E (fig. 42) be the given point. Then the angle E is the
Fig. 42
angle made with E Dhy n tangent at E, or since A B is parallel to
the tangent at E (§ 84, IV.), sin. E = sin. ADE = sin. BDE. Let
p' be the parameter for the diameter E D. Then, by Analytical Ge
ometry, f
p' 8in.2 E
2 8in.3 E ^
c3
p' sin 2 E. Therefore, at this point R =
2 8in.3 E ~
2sihE ■ ^^^ P'-^^ = Vd' Therefore, R = j^
--= . . = — : since A ^^ cd sin. E (Tab. X. 17).
c d sin. E A ^ ^ '
Next, to find 7?i , or the radius of curvature at H, the first station
from E. Through ff draw EG parallel to CD, and from Fdraw the
tangent EK. Join A K, cutting C Dm L. Then from what has just
been pioved for the radius of curvature at E, we have for the radius
of curvature at //. A', = a F K' ^^^^ A G • A L = A F : A C =
74
PARABOLIC CURVES.
n~ I : n, or A G = - ~ x A L. But A L = c, For, Miice A F -
—^ X AC, the tangent deflection FH = ^" ~/^" . ^ (§ 84, II.), and
FG = 2FH=^-^^^^d. Then, since CL:FG = AC:AF =
n:n-l,CL = ^^X FG='^d. Hence L D = d - '^ d
= - c7, thut is, .1 L = Ci . Substituting this value in the expres-
sion for A G above, we have A G = -^— c^ . Moreover, since
A F = — - — X A C, a/id because similar triangles are to each other a?
the squares of their homologous sides, we have the triangle A F G =
^" ~ ^^' X A CL. But ACL:ACD=^CL:CD = n — l: n, or
ACL=^ "^ X A CD. Therefore, A F G '= ^-^^^~^ X A C D, and
AFK = 2AFG = ^^^^^ XACB = ^^.'^ A. Substituung
these values of A G and A F K in the equation R^ = j^p^ , and re-
ducing, we find 7?j = — . By similar reasoning we should find /?2 =
It remains to find the values of Cj , c. , &c. Through A draw J ili
pei-pendicular to CD, produced if necessary. Then, by Geometry, we
have AD"^ = A L" + L D" — 2 L D X LM, and AC = A L^ -{-
CU + 2 CL X L M. Finding from each of these equations the
value of 2 L M, and putting tliese values equal to each other, we have
zT^ = CL • ^"^ AL = Ci,LD=-d,
n 1
A D = c, A C =^ 2\ and CL = -^ — d. Substituting these values
in the last equation, and reducing, we find
r^ (» — l)c2 [n — \)d^
^^ - » + n ~ n^
By similar reasoning we should find
2 7^2 (u — 2)c2 2{n — 2)d
«
c,^= -:r +
s
n n n
3 r« (tt — 3)c« 3(n — airf"
&c. &c.
RADIUS OF CURVATURE.
75
From tlicsc equations the values of c,S Co', Cj"^ , &c. given on page 72
arc readily obtained. That given for Cj' is obtained from the first ol
these equations by a simple reduction ; that given for Cj- is obtained
by subtracting the first of these equations from the second, and reduc-
ing ; that given for c^^ is obtained by subtracting the second equation
from the third, and reducing ; and so on.
94. Example. Given (fig. A\) A C ^ T ^ 600, B C == T< ^ 520,
and AD = c = 550, to find R, R^ , H, , R3 , and R^ , the radii of cur-
vature at .E, 1 , 2, 3, and A.
To find CD = d, we have, by Geometry, d^=^[T- -{- 7'' ^j — c«
■
which gives d- = 12700.
To find the area of .1 CB = A, we have (Tab. X. 18) A =
./sis —a) (s —6) is—c) .
*^ ^ ' s = 1110 3.045323
c — a = 590 2.770852
s — 6 = 510 2.707570
s — c = 10 1.000000
2)9.523745
lojr. A 4.761872
■'to
Next ^ (r^ - 0') = i (r + c) (r- c) = ii5!^ = 14375, and
t „lf «L = 793.75. Then
•* lb
c^ = 550- = 302500
Cj^ = 302500 + 14375 — 3 X 793.75 = 314493.75
Co^ == 314493.75 + 14375 — 793.75 = 328075
C32 = 328075 + 14375 + 793.75 =- 343243.75
C3
To find /?, we have /2 = ^ , or log. R = 3 log. c — log. A.
c = 550 2.740363
c^ 8.221089
A 4.7618^
22 = 2878.8 3.459217
To find Rj, , we have Ri == ^ > or log. Ri =-2-log Cj^ — log. A.
Cj^ = 314493.75 5.49761
c,3 8.246418
A 4.76 872
i?, == 3051.7 3.484546
76 PARABOLIC CURVES.
In the same way we should find i?2 = 3251.5, R^ = 3479.6, R^ ^
3737.5.
To find the radii for the second part E B o( the parabola, the same
formulse applv, except that T' takes the place of T. We have then
l(r- - c',"= UT' + c) ,r - 0 = 15™^^ = _su.5
Hence
Ci*'' = 302.500 — S025 — 2381.25 = 292093.75
C2^ = 292093.75 — 8025 — 793.75 = 283275.
C32 = 2S3275 — 8025 + 793.75 := 276043.75
C 3 3
To find Ri , we have /?i = -y , or log. Ri = 5 log Ci"-' — log. A
c 2 = 292093.75 5.465523
c^ 8.198284
A 4.761872
/?, = 2731.6 3.436412
In the same way we should find R<i_ = 2608.8, R^ = 2509.5, R^ -=>
2433.
It will be seen, that the radii in this example decrease from one tan-
gent point to the other, which shows that both tangent points lie on
the same side of the vertex of the parabola (§ 92). This will be tho
case, whenever the angle BCD, adjacent to the shorter tangent, ex-
ceeds 90°, that is, whenever c' exceeds T'^ -\- d}. If B CD = 90°.
the tangent point B falls on the vertex. If BCD is less than 90°,
one tangent point falls on each side of the vertex, and the curvature
will, therefore, decrease towards both extremities.
95. If the tangents T and T' are equal, the equations for c,', Co', &c.
will be more simple; for in this case d is perpendicular to c, and T'
— c^ = d^. Substituting this value, we get
d^
3d^
Co = Cj -4- -^ ,
5d^
&C. &C.
example. Given, as in § 91, T ^ T' = 832, c = 768, and d =
RADIUS OF CURVATURE.
n
320, to find the radii /?, Ri , and R^ at the points E, 4, and A (fig. 40)
Here A = cd = 245760, n = 2, and c,' = c^ + |£/2 = 615424
c3 c2 7G82 _ C,3
Then /? ^. ^
C2 7G82 C,3 . _ r3
^' '''i = 75 ' ''""
c,2 = 615424
5.789174
cd = 245760
8.683761
5.390511
/?! = : 964.5
r= 832
3.293250
2.920123
23
erf = 245760
8.760369
5.390511
R.. = 2343.5
3.369858
W is the radius at the point R also, and 7?, the radius at the point B
78 LEVELLING.
CHAPTER IIL
LEVELLING.
Article I. — Heights and Slope Stakes.
96. The Level is an instriiinent consisting essentially of a telesco]>e.
.supported on a tripod of convenient lieight, and capable of being so
adjusted, that its line of sight shall be horizontal, and that the tel-
escope itself may be turned in any direction on a vertical axis. The
instrument when so adjusted is said to be set.
The line of sight, being a line of indefinite length, may be made to
describe a horizontal plane of indefinite extent, called the plane of Om
lei-el.
The levelling rod is used for measuring the vertical distance of any
point, on which it may be placed, below the plane of the level. Thi?
distance is called the sight on that point.
97. Pro1>lcill. To Jind the difference of level of two points, as A
and B [fig. 43).
Solution. Set the level between the two points,* and take sights on
both points. Subtract tlie less of these siglits from the greater, and
the difference will be the difference of level required. For i( F P rep-
resent the plane of the level, and A G he drawn through ^4 parallel to
FP, A F will be the sight on A, and B P the sight on B. Tiien the
required difference of level B G = BP - ^^G = BP — AF.
If the distance between the points, or rue nature of the ground,
makes it necessary to set the level more than once, set down all the
backward sights in one column and all the forward sights in another.
Add up these columns, and take the less of the two sums from, the
greater, and the difference will be the difference of level required.
Thus, to find the difference of level between A and D (fig. 43), the
level is first set between A and .B, and sights are taken on A and B ;
the level is then set between B and C, and sights are taken on B and
* The level should be placed midway between the two points, when practicable,
In order to neutralize the effect of inaccuracy in the adjustment of the instrument,
And for the reason given in § i05.
TIEir.IITS AND SLOPE STAKES.
79
C, lastly: the level is set
pa o
usually divided into regular
the datum plane is required
between C and D, and sights are taken
on 6' and D. Then the ditlcrence of
level between ^1 and D \s E D =
{BP+ KC-\- OD) — [AF-VBJ-\-
NC). For E D == no - LC ^
tl M -\-MC—L C. llutllM = h G
= BF- AF, MC =^KC - D I,
and L C =^ N C — 0 D. Sal)stituting
these values, we have ED = BP —
AF-\- KG -BI - iVC+ 0D =
(BP-]- KG + CD) — {AF+ Bl
-^ NC).
98. It is often convenient to refer all
heights to an imaginary level plane
called the Jalum plane. This plane
may be assumed at starting to pass
through, or at some fixed distance above
or below, any permanent o1)ject, called
a bmch-mark, or simply a bench. It is
most convenient, in order to avoid mi-
nus heights, to assume the datum plane
at such a distance below the bench-
mark, that it will pass below all the
points on the line to be levelled. Thus
if A F> (tig. 44) were part of the line to
be levelled, and if A were the starting
point, we should assume the datum
plane GD at such a distance below
some permanent object near A, as
would make it pass below all the points
on the line. If, for instance, we had
reason to believe that no point on this
line was more than 15 or 20 feet below
A, we might safely assume G D to be
25 feet below the bench near A, in
which case all the distances from the
line to the datum plane would be posi-
tive. Lines before being levelled are
stations, the height of each of which above
80
LEVELLING.
^9. Prol>!eill. To find the heights above a datum plane of the sev
eral stations on a given line.
Solution. JjetA B (fig. 44) represent
a portion of the line, divided into regu
lar stations, marked 0, 1,2, 3, 4, 5, «Sbc
and let CD represent the datum plane,
assumed to be 25 feet below a bench-
mark near .1. Suppose the level to be
set first between stations 2 and 3, and a
sight upon the bench-mark to be taken,
and found to be 3.125. Now as this
sight shows that the plane of the level
E F'ls 3.125 feet above the bench-mark
and as the datum plane is 25 feet bo
low this mark, we shall find the height
of the plane of the level above the da
turn plane by adding these heights,
which gives for the height of E F 25 -\-
3.125 = 28.125 feet This height mav
for brevity's sake be called the height
of the instrument, meaning by this the
height of the line of sight of the instru
ment.
If now a sight be taken on station 0,
vcQ shall obtain the height of this sta-
tion above the datum plane, by sub-
tracting this sight from the height of
the instrument ; for the height of this
station is 0 C and OC=EC— EO.
Thus if EO = 3 413, 0 C = 28.125 —
3.413 = 24.712. In like manner, the
heights of stations 1, 2, 3, 4, and 5 may
be found, by taking sights on them in
succession, and subtracting these sights
from the height of the instrument.
Suppose these sights to be respective-
ly 3.102, 3.827, 4.816, 6.952, and 9.016,
and we have
= 28.125 — 3.413 = 24.712,
height of station 0
1 = 28.125 — 3.102 = 25.023,
HEIGHTS AND SLOPE STAKES. 81
height of station 2 = 28.125 — 3.827 = 24.298,
" " " 3 := 28.125— 4.816 = 23.309,
'' " " 4 = 28.125 — 6.952 = 21.173,
« *' " 5 = 28.125 — 9.016 = 19.109.
Next, set tlie level between stations 7 and 8, and as the height of sta-
tion 5 is known, take a sight upon thTs point. This sight, being added
to the height of station 5, will give the height of the instrument in its
new position ; for G K = 6' 5 + 5 K. Suppose this sight to be G 5
= 2.740, and we have GK= 19.109 + 2.740 = 21.849. A point
like station 5, which is used to get the height of the instrument after
resetting, is called a turning point. The height of the instrument being
found, sights are taken on stations 6, 7, 8, 9, and 10, and the heights
of these stations found by subtracting these sights from the height of
the instrument. Suppose these sights to be respectively 3.311, 4.027,
3.824, 2.516, and 0.314, and we have
height of station 6 = 21.849 — 3.311 = 18.538,
■' " " 7:^-21.849 — 4.027 = 17.822,
" " " 8 = 21.849 — 3.824 = 18.025,
« » « 9 = 21.849 — 2.516 = 19.333,
" " " 10 = 21.849 — 0.314 = 21.535.
The instrument is now again carried forward and reset, station IC
IS used as a turning point to find the height of the instrument, and
every thing proceeds as before.
At convenient distances along the line, permanent objects are se
lected, and their heights obtained and preserved, to be used as starting
points in any further operations. These are also called benches. Let
us suppose, that a bench has been thus selected near station 9, and
that the sight upon it from the instrument, when set between stations
7 and 8, is 2.635. Then the height of this bench will be 21.849 —
2.635 = 19 214.
100. From what has been shown above, it appears that the first
thing to be done, after setting the level, is to take a sight upon some
point of known height, and that this sight is always to be added to the
known height, in order to get the height of the instrument. This first
sight may therefore be called a p///s sight. The next thing to be done
is to take sights on those points whose heights are required, and to
subtract these sights from the height of the instrument, in order to get
the required heights. These last sights may therefore be called mimia
sights
82
LEVELLING.
101. The field notes are kept in the following form. The first col
umn in the table contains the stations, and also the benches marked
B., and the turning points marked t. p., except when coincident wuli
a station. The second column contains the plus sights ; the third col-
umn shows the height of the instrument ; the fi)urth contains the ininus
sights ; and i\iQ fifth contains the heights of the points in the first column.
Station
+ s.
H.I.
— S.
1
n.
B.
3.125
25.000
0
28.125
3.413
24.712
1
3.102
25.023
2
3 827
24.298
3
4.816
23.309
4
6.952
21.173
5
2.740
9.016
19.109
6
21.849
3311
18.538
7
4.027
17.822
8
. 3.S24
18.025
9
2.516
19.333
B.
2.635
19.214
10
0.314
21.535
The height of the bench is set down as assumed above, namely, 25
feet; the first plus sight is set opposite B., on which point it was
taken, and, being added to the height in the same line, gives the height
of the instrument, which is set opposite 0 ; the minus sights are set
opposite the points on which they are taken, and, being subtracted
from the height of the instrument, give the heights of these points, as
set down in the fifth column. The minus sights are subtracted from
the same height of the instrument, as far as the turning point at station
5, inclusive. The plus sight on station 5 is set opposite this station,
and a new height obtained for the instrument by adding the plus sight
to the height of the turning point. This new height of the instrument
is set opposite station 6, where the minus sights to be subtracted from
it commence. These sights are again set opposite the points on which
they were taken, and, being subtracted from the new height of the in-
strument, give the heights in the last column.
102. Problem. To set slope stakes for excavations and embank-
ments.
Solution. Let A B H K C (fig. 45) be a cross-section of a proposed
excavation, and let the centre cut A M = c, and the width of the road
HEIGHTS j'ND SLOPE STAKES.
83
fM}d II K = b. The slope of the sides B H or C Kis usually given by
the ratio of the base K Nto the height E N. Suppose, in the present
case, that KN : E N ^ 3 : 2, and we -have the slope = I . Then if
the ground were level, as D A E, it is evident that the distance from
Fig. 45
the centre A to the slope stakes at D and E would be yl Z) = A E —
M K -\- KN=^b + I c. But as the ground rises from A to C
t!i rough a height C G = g, the slope stake must be set farther out a
distance E G = ^ g ; and as the ground falls from A to B through a
height B F =^ g, the slope stake must be set farther in a distance D F
3
= 2 9-
To find B and C, set the level, if possible, in a convenient position
for sighting on the points A^ B, and C. From the known cut at the
centre find the value oi AE = ^h -{-^c. Estimate by the eye the
rise from the centre to where the slope stake is to be set, and take this
as the probable value of g. To A E add | g, as thus estimated, and
measure from the centre a distance out, equal to the sum. Obtain
now by the level the rise from the centre to this point, and if it agrees
with the estimated rise, the distance out is correct. But if the esti-
mated rise prove too great or too small, assume a nev.^ value for g,
measure a corresponding distance out, and test the accuracy of the
estimate by the level, as before. These trials must be continued, until
the estitnated rise agrees sufficiently well with the rise found by the
level at the corresponding distance out. The distance out will then be
hb -\- 2*^ -{•% g- The same course is to be pursued, when the ground
falls from the centre, as at Z? ; but as g here becomes viinns. the dis-
tance out, when tlie true value of g is found, will hQ A F = A D —
DF- ^h-^lc-lg.
For embankment, the process of setting slope stakes is the same as
for excavation, except that a rise in the ground from the centre on
embankments corresponds to a fall on excavations, and vice vcrsd.
This will be evident by inverting figurd 45, which will then represent
84 LEVELLING.
an embankment. AMiat was before ^ fall to Z?, becomes now a rwe,
and what was before a rise to C, becomes now a fall.
WHien tlie section is partly in- excavation and partly in embankment,
the method above applies directly only to the side which is in excava
lion at the same time that the centre of the road-bed is in excavation,
or in embankment at the same time that the centre is in embank-
ment. On the opposite side, however, it is only necessary to make c
in the expressions above minus, because its effect here is to diminish
the distance out. The formula for this distance out will, therefore, be-
come ^b — 2*^ -^ 2 y-
Article II. — Correction for the Earth's Curvature and
FOR Refraction.
103. Let A C (fig. 46) represent a portion of the earth's surface.
Then, if a level be set at A, tlie line of sight of the level will be the tap-
gent A D, while the true level will be A C. The difference Z) C be-
tween the line of sight and the true level is the correction for the
earth's curvature for the distance ^1 D.
104. A correction in the opposite direction arises from refraction.
Refraction is the change of direction which light undergoes in passing
from one medium into another of different density. As the atmos-
phere increases in density the nearer it lies to the earth's surface, light,
passing from a point B to a. lower point ^4, enters continually air oJ
greater and greater density, and its path is in consequence a curve
concave towards the earth. Near the earth's surface this path may be
taKen as the arc of a circle whose radius is seven times the radius of
the earth.* Now a level at A, having its line of sight in the direction
A D, tangent to the curve A B, is in the proper position to receive the
light from an olyect at B ; so that this object appears to the observer
to be at D. The effect of refraction, therefore, is to make an object
appear higher than its true position. Then, since the correction foj
the earth's curvature D C and the correction for refraction D B aie in
opposite directions, the correction for both will ha B C = D C — D B.
* Peirce's Spherical Astronomy, Chap. X., § 125 It should be observed, how-
ever, that the effect of refraction is verj' uncertain, varjing with the state of the
atmosphere Sometimes the path of a r.i}- is even made convex towards the earthy
«nd sometinies the rays are refracted horizontiUy a^ well as yertically.
I
earth's curvature and refraction.
P5
This correction must be added to the height of any object as deter-
mined by the level.
105. Prol>leill. Given the distance AD = D [Jig. 46), the radim
of the earth A E = R, and the radius of the arc of refracted light = 7 R,
'<) find the correction BC = dfor the earih's curvature and for refraction.
Solution. To find the correction for the earth's curvature D C, we
have, by Geometry, D C {D C -{• 2E C) = A D^ or D C {D C + 2 R)
= D^. But as Z) Cis always very small compared with the diameter
of the earth, it may be dropped from the parenthesis, and we have
D C X 2 72 = D-, or Z) C = .y-^ . The correction for refraction D B
may be found by the method just used for finding D C, merely chang-
ing R into 7 R. Hence D B =^ ^-j. . We have then d = B C ^
DC- DB^ ^
J2L
UR
or
d =
3D^
7R
By this formula Table III. is calculated, taking R = 20,911,790 ft,
as given by Bowditch. The necessity for this correction may be
avoided, whenever it is possible to set the level midway between the
points whose height is required. In this case, as the distance on
each side of the level is the same, the corrections will be equal, and
will destroy each other.
66
LEVELLING.
Article III. — Vertical Curves.
106. Vertical curves are used to round off the angles fonaed b^
the meeting; of two grades. Let A Cand CB (fig. 47) be two grades
meeting at C. These grades are supposed to be given by the rise per sta-
tion in uoing in some particuU^r direction. Thus, starting from ^1. the
grades of A Cand (^ B may be denoted respectively by^ and 9'; that
is, (J denotes what is added to the height at every station on A C, and
ij' denotes what is added to the height at every station on CB],hui
since CB is a descending grade, the C[uantity added is a minus quan-
tity, and (/' will therefore be negative. The parabola furnishes a very
simple method of putting in a vertical curve.
107. Problem. Given the grade g of A C [fig. 47), the grade a
of C B, and the number of stations n on each side of C to the tangent points
A and B, to unite these points by a parabolic vertical curve.
Fig. 47
Solution. Let A E B he the required parabola. Through B and C
draw the vertical lines FK and C H, and produce A C to meet FK
in F. Through .1 draw the horizontal line A K, and join A B, cut-
ting C H in D. Then, since the distance from C to A and B is meas-
ured horizontally, we have A H =^ H K. and consequently AD =
D B. The vertical line CD is, therefore, a diameter of the parabola
(§ 84, L), and the distances of the curve in a vertical direction from
the stations on the tangent A i^are to each other as the squares of the
number of stations from A (^ 84, II.). Thus, if a represent this dis-
tance at the first station from A, the distance at the second station
would be 4 a, at the third station 9 a, and at B^ which is 2 n stations
FB
from xV, it would be 4ii^a; that is, FB = 4n^a, or a = ^^ . To find
a, it will then be necessary to find FB first. Through Cdraw the
horizontal line C G and we have, from the equal triangles C F G and
VERTICAL CURVES. ^'
ACH, FG = C II But C II is the rise of the first grade g in the n
stations from A to C; that is, 0 ^ =- n <j, or F G = n ,j. G B is also
the rise of the second grade g' in n stations, but since r/' is negative
(§ 106),weinustpat G'/->^ = -?ii/'- Tlicrefore, FZ^ = F G ■{- GB
= ng - ng'. Substituting this value of FB in the equation for a
ns — n
:ri
ive have a = — -^: , or
9—9'
a =
4 n
Tlie value of n being thus determined, all the distances of the curve
from the tangent .1^; viz. a, 4 a, 9 «, 16 a, &e, are known. Now
if ran«i '/'' be the first and second stations on the tangent, and verti-
cal lines IP and 2'' P' be drawn to the horizontal line J /if, the
height TP of tlie first station above A will be//, the height 7''P' of
the^sccond station above ^ will be 2g, and in like manner for suc-
ceeding stations we should find. the heights 3</, 4(/, &c As we have
already found TM = a, T' M' = 4 a, &c., we shall have for the
heights of the carve above the level of A, MP = T P — I'M =
g (^ ]ji pi ^ ']'! pi _ T' M' = 2g — 4 a, and in like manner for
the succeeding heights 3 ^r — 9 a, 4^ — 16a, &c. Then to find the
grades for the curve at tlie successive stations from A, that is, the rise
of each height over the preceding height, we must subtract each
height from the next following height, thus: {g — a) —0 = g — a,
{2g-4a)-{g-a) = g - 3 a, {3 g - 9 a) -(2^ -4 a) =g-5a,
(4 <7 — 1 6 a) — (3 ^ — 9 a) = g — 7 a, &c. The successive grades for
the vertical curve are, therefore,
^ S' — «5 g — 3a, g — 5a, g — 7 a, &c.
In finding these grades, strict regard must be paid to the algebraic
signs. The results are then general ; though the figure represents
but one of the six cases that may arise from various combinations of
ascending and descending grades. If proper figures were drawn to
represent the' remaining cases, the above solution, with due attention
to the signs, would apply to them all, and lead to precisely the same
formuloe.
108. Examples. Let the number of stations on each side of Che 3,
and let A C ascend .9 per station, and CB descend .6 per station. Here
S-s' .9- (-.6) 1.5
n ^ 3, g = .9, and g' = —.6. Then, a = -^- =- 4x3 ~ 12
v_ .12.'>, and the grades from A to B will be
88 LEVELLING.
g — a = .9 — .125 = 775,
g — 3 a = .9 — .375 = .525,
g — 5 a = .9 — .625 = .275,
g — 7 a = .9 — .875 = .025,
g — 9 a = .9 — 1.125 = — •.225,
^ ~ 11 a = .9 — 1.375 = — .475.
As a second example, let the first of two grades descend .8 per s'a
tion, and the second ascend .4 per station, and assume two stations on
each side of C as the extent of the curve. Here g = — .8, g' = A,
and n = 2. Then a = ^'2 — — s" ~ — '^^' ^"^ ^^^® ^^^^ grades
required will be
g—a = — .8— (— .15) = — .8 + .15 = — .65,
^ — 3 a = — .8 — (— .45) = — .8 + .45 = — .35,
g — 5a = — .8 — ( — .75) == — .8 + .75 = — .05.
^ — 7a = — .8 — (— 1.05) = — .8 + 1.05 = + .25.
It will be seen, that, after finding the first grade, the remaining grades
may be found by the continual subtraction of 2 a. Thus, in the first
example, each grade after the first is .25 less than the preceding grade,
and in the second example, a being here negative, each grade after
the first is .3 greater than the preceding grade.
109. The grades calculated for the whole stations, as in the fore-
going examples, are sufficient for all purposes except for laying the
track. The grade stakes being then usually only 20 feet apart, it will
be necessary to ascertain the proper grades on a vertical curve for
these sub-stations. To do this, nothing more is necessary than to let g
and g' represent the given grades for a sub-station of 20 feet, and n the
number of sub-station.s on each side of tlie intersection, and to apply the
preceding formulae. In the last example, for instance, the first grade
descends .8 per station, or .16 every 20 feet, the second grade ascends
.4 per station, or .08 every 20 feet, and the number of sub-stations io
200 feet is 10. We have then ^ = — .16, g' = .08, and n = 10
— -16 — .08 — .24 - rr.1 ^ . 1 • ^T,
Hence a = ^ -^q = —^ = — .OOt. The first grade is, there
fore, g — a = — .16 + .006 = — .154, and as each subsequent grade
increases .012 (§ 108), the whole may be written down without farther
trouble, thus: —.154, —.142, — .130, — .118, — .106, —.094, —.082,
— .070, —.058, —.046, —.034, —.022, —.010, + 002, -f .014, +.O^fi.
+ .038 -h .050, + 062, + .074.
ELLVATION OF THE OUTEPt RAIL ON CUKV£ES.
91
^ "'^ ft.,
Articlk IV. — Elevation of the Outer Eail on Curves. ^
110. Problem. Giveti the radius of a curve R, the gauge of the
track g, and the velocity of a car per second v, to determine the proper ele-
vation e of the outer rail of the curve.
Solution. A car moving on a curve of radius /?, with a velocity per sec-
ond = r, lias, by Mechanics, a centrifugal force -= j. ■ To counteract
this force, the outer rail on a curve is raised above the level of the
inner rail, so that the car may rest on an inclined plane. This eleva-
tion must be such, that the action of gravity in forcing the car down
the inclined plane shall be just equal to the centrifugal force, which
impels it in the opposite direction. Now the action of gravity on a
body resting on an inclined plane is equal to 32.2 multiplied by the
ratio of the height to the length of the plane. But the height of the
plane is the elevation e, and its length the gauge of the track g. This
action of gravity, which is to counteract the centrifugal force, is, there-
fore, = ^^ . Putting this equal to the centrifugal force, we have
322e
ifi.2 « 1
g ~ 1^
Hence
qv*
e = ^
32.2 R
50
If we substitute for R its value (§ 10) R = ^j^;^ , we have e =
ir "oA ? = .000G2112 7^2 sin. D. If the velocity is given in miles
^^ ^ ^-^ ' Jfx5280
per hour, represent this velocity by M, and -vve have v = gQ ^ qq ■
Substituting this value of y, we find e = .0013361 g M^ sin. D. When
g = 4 7, this becomes e = .00627966 M^ sin. D. By this formula
Table IV. is calculated. In determining the proper elevation in any
given case, the usual practice is to adopt the highest customary speed
of nassenger trains as the value of M.
111. Still the outer rail of a curve, though elevated according to the
preceding formula, is generally found to be much more worn than the
inner rail On this account some are led to distrust the formula, and
to give an increased elevation to the rail. So far, however, as the
centrifugal force is concerned, the formula is undoubtedly correct, and
the evil in question must arise from other causes, — causes which are
not counteracted by an additional elevation of the outer rail. The
principal ofthe.se causes is probably improper " coning" of the wheels.
Two wheels, immovable on an axle, and of the same radius, must, iC
90 LE /ELLING.
no slip is allowed, pass over equal spaces in a given number of revo-
lutions. Now as the outer rail of a curve is longer than the inner rail,
the outer wheel of sucli a pair must on a curve fall behind the inner
wheel. The first effect of this is to bring the flange of the outer wheel
against the rail, and to keep it there. The second is a strain on the
axle consequent upon a slip of the wheels equal in amount to the dif
ference in length of the two rails of the curve. To remedy this, con-
ing of the wheels was introduced, by means of which the radius of the
outer wheel is in effect increased, the nearer its flange approaches the
rail, and this wheel is thus enabled to traverse a greater distance than
the iTiner w^heel.
To find the amount of coning for a play of the wheels of one inch,
let r and r' represent the proper radii of the inner and outer wheels
respectively, when the flange of the outer wheel touches the rail. Then
r' — r will be the coning for one inch in breadth of the tire. To ena-
ble the wheels to keep pace with each other in traversing a curve, their
radii must be proportional to the lengths of the two rails of the curve,
or, which is the same thing, proportional to the radii of these rails. If
7t be taken as the radius of the inner rail, the radius of the outer rail
will be 72 + ^, and we shall have r : r' ^ R '. R -\- g. Therefore, r R
-\- r g ^^ r' R, or
r — r = _£, .
R
As an example, let R = 600, r = 1.4, and g = 4.7. Then we have
1.4 X 4-7
r' — r — gQQ " — Oil ft. For a tire 3.5 in. wide, the coning
would be 3. .5 X .011 = .038.5 ft., or nearly half an inch. Wheels
coned to this amount would accommodate themselves to any curves
of not less than 600 feet radius. On a straight line the flanges of the
two wheels would be equally distant from the rails, making both
wheels of the same diameter. On a curve of say 2400 feet radius, the
flange of the outer Avheel would assume a position one fourth of an
inch nearer to the rail than the flange of the inner wheel, which would
increase the radius of the outer wheel just one fourth of the necessary
increase on a curve of 600 feet. Should the flange of the outer wheel
get too near the rail, the disproportionate increase of the radius of this
wheel would make it get the start of the inner wheel, and cause the
flange to recede from the rail again. If the shortest radius were taken
1.4X 4.7
as 900 feet, r and g remaining the same, we should have ?' — r — — 900""
ELEVATION OF THE OUTER RAIL ON CURVES.
91
x= .0073, and for the coning of the whole tire 3.5 X -0073 - .0256 ft.,
or about three tenths of an inch. Wheels coned to this amount would
accommodate themselves to any curve of not less than 900 feet radius.
If the wheels are larger, the coning must be greater, or if the gauge of
the track is wider, the coning must be greater. If the play of the
wheels is greater, the coning may be diminished. Hence it might be
advisable to increase the play of the wheels on short curves, by a slight
increase of the gauge of the track.
Two distinct things, therefore, claim attention in regard to the mo-
tion of cars on a curve. The first is the centrifugal force, which is
generated in all cases, when a body is constrained to move in a cur-
vilinear path, and which may be effectually counteracted for any given
velocity by elevating the outer rail. The second is the unequal length
of the two rails of a curve, in consequence of which two wheels fixed
on an axle cannot traverse a curve properly, unless some provision is
made for increasing the diameter of the outer wheel. Coning of the
wheels seems to be the only thing yet devised for obtaining this in-
crease of diameter. At present, however, there is little regularity
either in the coning itself, or in the distance between the flanges of
wheels for tracks of the same gauge. The tendency has been to di-
minish the coning,* without substituting any thing in its place. If the
wheels could be made to turn independently of each other, the whole
difficulty would vanish ; but if this is thought to be impracticable, the
present method ought at least to be reduced to some system.
* Bush and Lobdell, extensive wheel-makers, say, in a note published in Apple-
tons' Mechanic's Magazine for August, 1852, that wheels made by them fcr the New
York and Erie road have a coning of but one sixteenth of an inch. This coning on
% track of six feet gauge with the c .her data as given above, would suit no ciirva
•f less than a mile radius.
^..
92
KARTH-WORK.
CHAPTER IV.
EAKTH-WORK.
Akticlk I. — Prisjioidal Formula.
112. Earth-work includes the regular excavation uirI tinbank
ment on the line of a road, borrow-pits, or such additional excavations
as are made necessary when the embankment exceeds the regular ex
cavation, and, in general, any transfers of earth that require calcula-
tion. We begin with the prismoidal formula, as this formula is fre-
quently used in calculating cubical contents both of earth and masonry.
A prismoid is a solid having two parallel faces, and composed of
prisms, wedges, and pyramids, whose common altitude is the perpen-
dicular distance between the parallel fiices.
113. Problem. Given the areas of the parallel faces B and B ,
the middle area 21, and the altitude a of a prismoid, to find its solidity S.
Solution. The middle area of a prismoid is the area of a section
midway between the parallel faces and parallel to them, and the alti-
tude is the perpendicular distance between the parallel faces. If now
b represents the base of any prism of altitude a, its solidity is ab. If 6
represents the base of a regular wedge or half-parallelopipedon of alti-
tude a, its solidity is kab. Kb represents the base of a pyramid of
altitude a, its solidity is ^ a 6. The solidity of these three bodies ad
mits of a common expression, which may be found thus. Let m rep-
resent the middle area of either of these bodies, that is, the area of a
section parallel to the base and midway between the base and top. In
the prism, m = b, in the regular wedge, m = ^b, and in the pyramid,
m = ^b. INIoreover, the upper base of the prism = b, and the upper
base of the wedge or pyramid = 0. Then the expressions a b, ha &,
and kab may be thus transformed. Solidity of
prism = ab =- X &b =-ib -\-b -{: Ab) =-{b-\-b-{- 4m),
6 6 6
wedge =ia6 = -X36 = f.(0 + 6-f2 6) =-(04-6+4 m),
6 6 6
pyramid =^ab = -X2b=-{0-\-b-^b) =f(0 + 6-1-4 '»j,.
6 6 6
EORROW-PITS.
93
Hence, the solidity of either of these bodies is found by adding togeth-
er the area of the upper base, the area of the lower base, and four
times the middle area, and multiplying the sum by one sixth of the
altitude. Irregular wedges, or those not half-parallelopipedons, may
be measured by the same rule, since they are the sum or difference of
a regular wedge and a pyramid of common altitude, and as the rule
applies to both these bodies, it applies to their sum or difference.
Now a prismoid, being made up of prisms, wedges, and pyramids of
common altitude with itself,.will have for its solidity the sura of the
solidities of the combined solids. But the sum of the areas of the
upper and lower bases of the combined solids is equal to 5 + B\ the
sum of the areas of the parallel faces of the prismoid ; and the sum of
the middle areas of the combined solids is equal to J/, the middle area
of the prismoid. Therefore •
5 = ^(S + 5' + 4 37).
6
AUTICLE II. — BORROW-PlTS.
114. For the measurement of small excavations, such as borrow-
pits, &c., the usual method of preparing the ground is to divide the
surface into parallelograms * or triangles, small enough to be consid-
ered planes, laid off from a base line, that will remain untouched by
the excavation. A convenient bench-mark is then selected, and levels
taken at all the angles of the subdivisions. After the excavation is
made, the same subdivisions are laid off from the base line upon the
oottom of the excavation, and levels referred to the same bench-mark
are taken at all the angles.
This method divides the excavation into a series of vertical prisms,
generally truncated at top and bottom. The vertical edges of these
prisms are known, since they are the differences of the levels at the
top and bottom of the excavation. The horizontal section of the
prisms is also knoAvn, because the parallelograms or triangles, into
which the surface is divided, are always measured horizontally.
11.5. Problem. Given the edges h, hi , and ho , to find the solidity
• If the ground is divided into rectangles, as is generally done, and one side b«
made 27 feet, or some multiple of 27 feet, the contents may be obtained at once in
rubic yards, by merely omitting the factor 27 in the calculation.
94
EARTH-WORK.
S of a veitical prism, whether truncated or not. whose horizontal section ti
o triangle of given area A.
Fig. 48
Solution. "Wlicn the prism is not truncated, we have h = h^ = k^'
The ordinary mle for the solidity of a prism gives, therefore, S = Ah
■^ A X b {h + hi -{- hr,). When the prism is truncated, let ABG-
F G H {i\g. 48) represent such a prism, truncated at the top. Through
the lowest point A of the upper face draw a horizontal plane A D E
cutting off a pyramid, of which the base is the trapezoid B D E C, and
the altitude a perpendicular let fall from A on D E. Represent this
perpendicular by p, and we have (Tab, X. 52) the solidity of the pyra-
mid = ^px BDEC ==\pxDExh{BD^ C E) = ^pX
DE X ^ {BD -\- CE) = A X h [BD + CE), since hp X DE
= A D E = A. But I {BD -\- CE) is the mean height of the verti-
cal edges of the truncated portion, the height at A being 0. Hence
the formula already found for a prism not truncated, will apply to the
portion above the plane ^ Z> £", as well as to that below. The same
reasoning would apply, if the lower end also were truncated. Hence,
for the solidity of the Avhole prism, whether truncated or not, we have
S=AXhih + h,+ h.).
116. Problem. Given the edges h, h^, hn, and A3, to Ji7id tU
solidity S of a vertical prism, ivheiher truncated or not, whose horizoUat
section is a parallelogram, of given area A.
BORROW-PITS.
9fi
Solution. Let B H (fig. 49) represent such a prism, whether trim
cated or not, and let the plane BFHD diviie it into two triangular
Fig. 49
prisms AFH and C F H. The horizontal section of each of these
prisms will be ^ A, and if A, h^ , h^ , and h^ represent the edges to which
they are attached in the figure, we have for their solidity (§ 115)
A FH =^A X k i^i-^ h + h). and CFH = ^A X ^ (^i + h +
^g). Therefore, the whole prism will have for its solidity S = ^ A X
^ {h + 2/tji + 112 + 2 A3). Let the whole prism be again divided b}
the plane AE G C into two triangular prisms BEG and D E G
Then we have for these prisms, B E G = hA X ^ {^^ + ^h + h)^
and D E G = h A X J (^ + ^'2 + '^3)5 and for the whole prism, S —
^A X ^ (2 A + /ij + 2 /<2 + h). Adding the two expressions found
for S, we have 2 S = ^ A {h -^ h^ + h^ -\- Jh), or
^ S=A X i{h-{-h, + h. + h,).
It will be seen by the figure, that h {h + ho) = KL = h {K + fh),
or h -\- kz = hi -{- h^ . The expression for S might, therefore, be re-
duced to S = A X k i^ + h), or S = A X ^ {hi + h^). But as
the ground surfaces A B CD and E F GHare seldom perfect planes,
it is considered l>etter to use the mean of the four heights, instead of
the mean of two diagonally opposite.
117. Corollary. When all the prisms of an excaA-ation have
ilic same horizontal section A, the calculation of any number of them
06
KARTH-WOKK
may be performed by one operation. Let figure 50 be a plan ot such
an excavation, the heights at the angles being denoted by a, Oi , Oo, ^
a,
«*«
\h3
d*
bs
\c
C/
r^
C3
Pd. Ca
d
d>
ds
a>
Fig. 50.
6i , &c. Then the solidity of the whole will be equal to \A multi
plied by the sum of the heights of the several prisms (§ 116). Into
this sum the corner heights a, Oo , h^h^, Cj,, </, and d^ will enter but
once, each being found in but one prism ; the heights 01,^4, c, di, do,
and rfj will enter ticice, each being common to two prisms ; the heights
fe. , bj, and t'4 will enter three times, each being common to three
prisms; and the heights ioj^ijCo, and c^ will enter four times, each
being common to four prisms. If, therefore, the sum of the first set of
heights is represented by Si , the sum of the second by So , of the third
by S3 , and of the fourth by s^ , we shall have for the solidity of all tho
pnsms
>S = I J. (si + 2 So + 3 S3 + 4 S4).
Article III. — Excavation and Embankment.
118. As embankments have the same general shape as excavations,
it will be necessary to consider excavations only. The "simplest case
is when the ground is considered level on each side of the centre line.
Figure 51 represents the mass of earth between two stations in an ex-
cavation of this kind. The trapezoid G B F H is a section of the
mass at the first station, and Gi Bi F^ H^ a section at the second sta-
tion; AE \s the centre height at the first station, and A^ E^ the centre
height at the second station ; HffiFiFis the road-bed, G Gi B^ B the
CENTRE HEIGHTS ALOJSE GU^EN.
9*:
surface of the ground, and G Gi H^ 11 and BB^F^F the planes form-
ing the side slopes. This solid is a prismoid, and might be calculated
bv^the prismoidal formula (§ 113). The following metaod gives the
same result.
A. Centre Ileifjhts alone given.
119. Problem. Given the centre heiyhts c and Cj , the width of the
road-hed 6, the slope of the sides s, and the length of the section I, to find
*he soliditij S of the excavation.
Fig. 51.
}sol-iiion. Let c be the centre height at A (lig. 51) and Cj the height
af, X. . The slope s is the ratio of the base of the slope to its perpen-
dicular height (§ 102). We have then the distance out ^ B = ^6 +
sc, and the distance out A^B^ ^ \h -\-sci{\ 102). Divide the whole
mass into two equal parts by a vertical plane A Ai E^ E drawn
through the centre line, and let us find first the solidity of the right-
hand half. Through B draw the planes BEE^, BA^Ei, and
B^jFi, dividing the half-section into three quadrangular pyramids,
having for their common vertex the point Z5, and for their bases the
planes AA^EiE, E Ey Fi F, and AiBiF^Ei. For the areas of these
bases we have
Areaof ^ Ji^i^
" " EEiFiF
" " A,B,F,E,
= iEEi X {AE-{- A^E,) =
^EFx EE, =
=^^A,E,X{E,F,+A,Bi):^^{bc,^sc,*);
^/(c + Ci),
hbl.
and lor tlie perpendiculars from the vertex B on these bases, produced
when necesyarv.
98 EARTH-WORK.
Perpendicular on A A^E^E = A B — 1 6 -f o c,
'' EExF^F = AE ^ c,
" « A^B^F.Ei = EEi = I.
Then (Tab. X. 52) the solidities of the three pyramids are
B-AA,E,E =|(i6 + sc) X ^/(c + cO=|/(i6c-f-^6c,-^
B-EE^l\F =\cY.\hl ^'llbc,
B-A^B,F,E,= II X h (^Ci + sO =U(6ci+sci2).
Their sum, or the solidity of the half-section, is
LS = \l[lh{c-\- Ci) + s (c^ + Ci2 + cci)l.
Therefore the solidity of the whole section is
^S- - i / [i Mc + cx) -f s (c^ + c,-' + cc,)J,
or
^ 5 = i / [6 (c + c) 4- I s (c' + Ci^ -f c c,) J
When the slope is 1^ to 1, s = i, and the factor fs = I may be
dropped.
120. Problem. To Jind the solidity S of any number n of succes-
sive sections of equal length.
Solution. Let c, Ci,C2,C3, &c. denote the centre heights at the suc-
cessive stations. Then we have (§ 119)
Solidity of first section = ^l[b {c + Ci) -f f s (c^ + Cj^ + c c^)],
" « second section = ^ Z [6 (ci + Co) + | s (cj* + Co- + c^ Co)],
« " third section = |/ [6 (cg + Cg) + | s (ca^ + Ca^ + C0C3)],
&c. &c.
For the solidity of any number n of sections, we should have ^l mul-
tiplied by the sum of the quantities in n parentheses formed as those
iust given. The last centre height, according to the notation adopted,
will be represented by c, and the next to the last by c„_i. Collect-
ing the terms multiplied by b into one line, the squares multiplied by
I s into a second line, and the remaining terms into a third line, we
have for the solidity of n sections
^^ S=hl 6 (c4- 2f: -f 2r, -f 2c3 + 2c„_i + c„)
4. |S (c2+2Ci2 4-2C22 + 2C32.... +2c2„_l + C»„)
+ I S (C Ci + Ci Co + C2 C3 + ^a C4 + c„- 1 On).
When s = I , the factor f s = 1 may be dropped.
CENTRE AND SIDE HEIGHTS GIVEN.
99
Example. Given / = 100, 6 = 28, s = i , and the stations and cen-
tre heights as set down in the first and second columns of the annexed
table. ''The calculation is thus performed. Square the heights, and
set the squares in the third column. Form the successive products
c ci , Ci C2 , &c., and place them in the fourth column. Add up the last
three columns. To the sum of the second column add the sum itself,
minus the first and the last height, and to the sum of the third column
add the sum itself, minus the first and the last square. Then 86 is the
multiplier of b in the first line of the formula, 592 is the second line,
since § s is here 1, and 274 is the third line. The product of 86 by b
= 28 is 2408, and the sum of 274, .592, and 2408 is 3274. This mul-
tiplied by |/ --= 50 gives for the solidity 163.700 cubic feet.
Station.
c.
c-i.
CCi.
0
9
4
1
4
16
8
2
7
49
28
3
6
36
42
4
10
100
60
5
1
49
70
6
6
36
42
7
4
16
24
46
306
274
40
286
592
86
592
2408
28
2)3274
2408
163700.
B. Centre and Side Heights given.
121. When greater accuracy is required than can be attained by Ae
preceding method, the side heights and the distances out (§ 102) are
introduced. Let figure 52 represent the riglit-hand side of an excava
tion between two stations. AAi By B is the ground surface ; AE =^ c
and A^Ei = Ci are the centre heiglits ; B G = h and C, Gi = hi , the
side heights ; and d and d^ , the distances out, or the horizontal distan-
ces of B and Bi from the centre line. The whole ground surface
may sometimes be taken as a plane, and sometimes the part on each
side of the centre line may be so taken ; * but neither of these suppo-
* It is easy in any given case to ascertain whether a surface like A Ai Bi £ is a
iOO EARTH-WORK.
sitions is sufficiently accurate to serve as the basis of a general mciiiod.
In most cases, however, we may consider the surface on each side of
the centre line to be divided into two triangular planes by a diagonal
passing from one of the centre heights to one of the side heights. A
ridge or depression will, in general, determine which diagonal ought
to be taken as the dividing line, and this diagonal must be noted in
the field. Thus, in the figure a ridge is supposed to run from B to
^4.1, from which the ground slopes downward on each side to A and
Bi . Instead of this, a depression might run from A to B^ , and the
ground rise each way to A^ and B. If the ridge or depression is very
marked, and does not cross the centre or side lines at the regular sta-
tions, intermediate stations must be introduced to make the triangular
planes conform better to the nature of the ground. If the surface
happens to be a plane, or nearly so, the diagonal may be taken in
either direction. It will be seen, therefore, that the following method
is applicable to all ordinary ground. When, however, the ground is
very irregular, the method of § 127 is to be used.
122. Problem. Given the centre heights c and c^ , the side heights
on the right h and h^ , on the left h' and h\ , the distances out on the right
d and d^ , on the left d' and d'l , the icidth of the road-bed b, the length of the
section /, and the direction of the diagonals, to find the solidity S of the
excavation.
Solution. Let figure 52 represent the right-hand side of the excava-
tion, and let us suppose first, that the diagonal runs, as shov,n in the
figure, from B to Ai- Through B draw the planes B E E^, B A^Ei,
and BEiFi, dividing the half-section into three quadrangular pyra-
mids, having for their common vertex the point B, and for their bases
the planes A A^ E^E, E E^ F^ F, and A, B, F^ E, . For the areas
of these bases we have
Areaof^^i^iJ^; = ^ E E, x{AL-]-A^E,) -|/(c-fc,),
" ^'EE.FiF =EFxEEi =^l^h
" » A, B, F, E, = ^ A, E^xdi + k ^i F^Xh, = ^d,c, -\- ibh, ,
and for the perpendiculars from the vertex B on these bases, produced
when necessary,
plane ; for if it is a plane, the descent from A to B will be to the descent from Ai to
Bi , as the distance out at the first station is to the distance out at the second sta-
tion, that is, c — h:ci — hi = d:di. K we had c = 9, A = 6, fi = 12, «! = 8,
d = 24, and di = 27, the formula would give 3 : 4 = 24 : 27 which shows that tho
lurface is not a plane.
CENTRE AND SIDE HEIGHTS GIVEN
Perpendicular on A A^^ E^ E — E G = d,
" E E, F, F =^ BG ^K
" A,B,F,E, = EE, -/.
10)
A I
Fig. 52.
Then (Tab. X. 52) the solidities of the three pyramids arc
B-AA,E,E = :^d X ^-Mc + ci) = |/ (c?c + c/c,),
B-EE,F,F = I A X ^ '^^ =llbh,
B-A,B,F,E, =kl X h{dic, + ^bh,) = U(^iCi+^6A,).
Their mm, or tlic solidity of the half-section, is
ll{dc-\- d,c^ + dc, + hh + hhK). (1)
Next, suppose that the diagonal runs from A to B^ . In this case,
through B, draw the planes B, E, E, B, A E, and B^EF {not rep-
resented in the figure), dividing the half-section again into three
quadrangular pyramids, having for their common vertex the point
Bi , and for their bases the planes A A, E^ E, E E^ F^ F, and A B FE
For the areas of these bases -vve have
Area of ^ ^1 , ^, ^ = U^^ ^: X {A E -{- A^E^) ^ ^l {^ + c^),
" '' EE^FiF =EFx EE^ =h^h
» '' ABFE =^AExd-{-^EFxh =^dc-\- ^bh;
and for the perpendiculars from Bi on these bases, produced when
necessary,
102 EARTH- WORK.
Perpendicular on A Ai E^ E = E^ G^ = d^
« « ABFE = E El = I.
Tiin {Tab. X. 52) the solidities of the three pyramids are
Bi-AAiEiE= ^di X hl{c + ci) =hl{chc-\-diCi).
Bi-EEiF^F = ^hi X k^'l = lib hi,
Bi- ABFE =11 X ^{dc + ^bh) = \l{dc-\- \bh).
Their sum, or the solidity of the' half-section, is
\l{dc + diCi + dic -\-bhi + hbh). (2)
We have thus found the solidity of the half-section for both direc
tions of the diagonal. Let us now compare the results (1) and (2),
and express them, if possible, by one formula. For this purpose let
(1) be put under the form
ll[dc + diCi-^dci J^lb[h+hi 4-^)1,
and (2) under the form
il[dc + d,ci + dic-^\b [h + hi + hi)\.
The only difference in these two expressions is, that dci and the last
h in the first, become di c and Aj in the second. But in the first case,
c, and h are the heights at the extremities of the diagonal, and d is the
distance out corresponding to h ; and in the second case, c and hi are
the heights at the extremities of the diagonal, and di is the distance
out corresponding to hi. Denote the centre height touched bij the diagonal
by C, the side height touched by the diagonal by H, and the distance out cor-
responding to the side height H by D. We may then express both c/c,
and dichy D C, and both h and hi by //; so that the solidity of the
half-section on the right of the centre line, whichever way the diago-
nal runs, may be expressed by
\l[dc^diCi -^DC-\-^b[h-^hi + H)\. (3)
To obtain the contents of the portion on the left of the centre line,
we designate the quantities on the left by the same letters used for cor-
responding quantities on the right, merely attaching a (') to them to
distinguish them. Thus the side heights are h' and h'l , and the dis-
tances out d' and d'l , while Z), C, and H become Z)', C, and H'.
The solidity of the half-section on the left may therefore be taken di-
rectly from (3), which will become
CENTRE AND SIDE HEIGHTS GIVEN.
io;j
Finally, by uniting (3) and (4), ^vc obtain ilie following formula for
the solidity of the whole section between two stations
j^ ^-^ U{{d-\-d')c-^r{d,^cl\)c,^DC-\-D'C<-\-'^h{h-{-
Example. Given / = 100, 6 = 18, and the remaining data, as ar
langcMl in the first six columns of the following tabic. The first col-
i.nn gives the stations ; the fourth gives the centre heights, namely,
c -- 13.6 luul ci =- 8 ; the two columns on the left of the centre heights
give the side heights and distances out on the left of the centre line of
tlie road, and the two columns on the right of the centre heights give
the side heights and distances out on the right. The direction of the
diagonals is marked by the oblique lines drawn_^from h' = 8 to Cj =- 8
and from c =^ 13. 0 lo //^ ^= 12.
Sta.
0
I
d'.
21
15
8\
4
c.
h.
10
^^12
d.
24
27
' d + d'.
(d + d^)c.
D' C
1G8
DC.
13.6 \
^ 8.0
45
42
612
336
367.2
12
12 168
20 367.2
54 X 9 = 486
•
6)1969.2(
3
32820.
To apply the formula, the distances out at each station are added
together, and their sum placed in the seventh column ; these sums,
multiplied by the respective centre heights, are placed in the eightli
column ; the product off/' == 21 (which is the distance out correspond-
inc^ to the side height touched by the left-hand diagonal) by c, = 8
(which is the centre height touched by the same diagonal) is placed
m the ninth column, and the similar product of c/j = 27 by c = 13.6
is placed in the last column. The terms in the formula multiplied by
^ b are all the side heights, and in addition all the side heights touched
by diagonals, or 8 + 4 + 10 + 12 + 8 + 12 = 54. Then by sub-
stitution in the formiila, we have S == h X 100 (612 + 336 + 168 +
867.2 + 9 X 54) =- 32,820 cubic feet.*
* The example here given is the same as that calculated in Mr. Borden's " Sya-
104 EARTH -WORK.
By applying the rule given in the note to § !'21, we see that the sar-
face on the left of the centre line in the preceding example is a plane •
since 13.6 — 8 : 8 — 4 = 21 : 15. The diagonal on that side might,
therefore, be taken either way, and the same solidity would be ob-
tained. This may be easily seen by reversing the diagonal in this ex-
ample, and calculating the solidity anew. The only parts of the for-
mula affected by the change are D' C and ^b H'. In the one case
the sum of these terms is 21 X 8 + 9 X 8, and in the other 15 X 13.6
+ 9X4, both of which arc equal to 240.
123 Problem. To find the solidity S of any number n of succes-
sive sections of equal length.
Solution. Let c, Cj , Co , c^, &c. be the centre heights at the succes-
sive stations; /(. lii , h., , h^ , &c. the right-hand side heights; h', li\ , A'o ,
Zi'o , .fcc. the left-hand side heights ; (/, t/j , c/., , d^ , &c. the distances out
on the right ; and t/', d\ , d'^ , d'^ , &c. the distances out on the left.
Then the formula for the solidity of one section (§ 122) gives for thp
solidities of the successive sections
\l[{d-\-d')cJr{<-h +^'i)c, ^DC+D' C'-\-hb{h + h,-^ H-\.
h[-\-h\^H%
\l[[d^J^d\)c, ^{d.-\-d>.)c. + D, Ci + D\C\-^^b{h^ +A2-H
ZTi + A'. + A'o + H'OJ,
G I \{dn + J'o) c, + ((/3 + c/'a) C3 + D. a + Z)'. C'2 + i 6 {h. + A3 -f
H.-i-h', + h>,.^H'.)l
"^nd so on, for any number of sections. For the solidity of any num-
ber n of sections, we should have g / multiplied by the sum of n paren-
theses formed as those just given. Hence
^ a5- I / (c?+ cZ') c + 2 {d,-\- d\)cy-\-2 {d., + f/'^) Co . . . -f {d„ + d'„) c„
+ DC+ D'C> + Z)iCi -I- D\ C\ + B.C. + D'.C. + &c.
4- ^ 6 i /i + 2 Ai + 2 /?., + Ih, + ^+ i/i + ZTo + &c.
I + /i'+ 2 /t'i+ 2 A'o . . . + It'n + H'-\-H\-\-H'. + &.C.
tem of Useful Fonnulne, &c ," page 187. It will be seen, that his calculation make?
the solidity 32,460 cubic feet, which is 360 cubic feet less than the result above.
This difference is owing to the omission, by Mr. Borden's method, of a pyramid in-
closed by the four pyramids, into which the upper portion of the right-hand hall
section is by that method divided.
CKNTRE AND SIDE HEIGHTS GIVEN.
105
Example. Given / = 100, b = 28, and the remaining data as given
in the first six columns of the following table.
'Sta.
0
d'.
k'.
c.
h.
d.
17
2 2
2
17
1
18.5
3 >4_
5
21.5
2
20
4-^ ^5^
^6
23
3
23
6 -^^6 ..^
'"•s
26
4
21. .5
5-^0,^0
>7
24.5
5
20
4 -^U^ G /
A
20 ,
6
15.5
1-^
i^
3
18.5
d + d'
25
22
90
69
102
171 X 14
35
30
37
T02
2394
2394
6)6212
103533 cubic feet.
The data in this table are arranged precisely as in the example for cal-
culating one section (§ 122), and the remaining columns are calculated
as there shown. Then, to obtain the first line of the formula, add all
the cumbers in the column headed {d-\- d') c, making 1389, and after-
wards all the numbers except the first and the last, making 1185.
The next line of the formula is the sum of the columns D' C and
D C, which give respectively 605 and 639. To obtain the first line of
the quantities multiplied by \b^ add all the numbers in column A,
making 35, next all the numbers except the first and the last, making
30, and lastly all the numbers touched by diagonals (doubling any one
touched by two diagonals), making 37. The second line of the quan-
tities multiplied by ^6 is obtained in the same way from the column
marked A'. The sum of these numbers is 171, and this multiplied by
16=14 gives 2394. "We have now for the first line of the formula
1389 + 1185, for the second 605 + 639, and for the remainder 2394.
100
By adding these together, and multiplying the sum by 5/ = -g- , we
get the contents of the six sections in feet.
124. When the section is partly in excavation and partly in embank-
ment, the preceding formula? are still applicable ; but as this applica-
tion introduces minus quantities into the calculation, the following
method, similar in principle, is preferable.
125. Problem* Given the ividlhs of an excavation at the road-bed
6
106
EARTH-WORK.
AF = w and Ai F, = Wi {Jig. 53), the side heights h and h^.the lenfftk
of the section /, arid the direction of the diagonal, to find the solidity S of
the excavation, when the section is partly in excavation and partly in em-
bankment.
Fig. 53
Solution. Suppose, first, that the surface is divided into two trian
gles by the diagonal B A^. Through B draw the plane BA^F,,
dividing that part of the section which is in excavation into two pyra-
mids B-AAiFiF and B-AiB^ F^ , the solidities of which are
B - A Ai F, F = I h X k ^ {lo + ivi) = ll{ioh -\- wi h),
B-AiBiFi =^lx^ioihi =llwihi.
The whole solidity is, therefore,
S = kl {wh -\- ivi Aj, + it'i h).
Next, suppose the dividing diagonal to run from Ato Bi. Through
Bi draw a plane BiAF (not represented in the figure), dividing the
excavation again into two pyramids, of which the solidities are
Bi-AAiF^F^^hi X hl{io-\-Wi) = \l{ioh + ^o^h)y
Bi-ABF =^lxh^h =11 wh.
The whole solidity is, therefore,
S = ll{wh + Wihi + lohi).
The only diff'erence in these two expressions is. that iVj h in the first
becomes v;/«i in the second. But in the first case the diagonal touch-
es io\ and h, and in the second case it touches iv and h^. If, then, we
designate the width touched by the diagonal by W, and the height
touched by the diagonal by H, we may express both Wi h and tv h^ by
WH; so that the solidity in either case may-be expressed by
CENTRE AND SIDE HEIGHTS GIVEN.
lOT
S^ll{ivh + iv,h, + WII).
Corollary. When several sections of equal length succeed one
another, the whole may be calculated together. For this purpose, the
preceding formula gives for the solidities of the successive sections
ll{ivh + it'iAj + IF//),
ll(w,h, + H',/2o+ TF1//1),
and so on for any number of sections. Hence for the solidity of any
number n of sections we should have
E^ S=ll{ivJi + 2ii\ /ii -f- 2 1^3 /to .... 4- Wn hn -f WH -\- Wi H^ -I-
WzH.^-^- &c.)
Example. Given I = 100, and the remaining data as given in the
irst three columns of the following table.
Station.
10.
h.
ich.
WH.
0
2
/l
2
1
8<
6
48
8
2
10.^
^7
70
56
3
13^
■^7
91
70
4
9
"^4
36
52
247
209
186
186
6)642
10700.
The fourth column contains the products of the several widths by
the corresponding heights, and the next column the products of those
widths and heights touched by diagonals. The sum of the products
in the fourth column is 247, the sum of all but the first and the last is
209, and the sum of the products in the fiftli column is 186. These
three sums are added together, multiplied by 100, and divided by 6,
according to the formula. This gives the solidity of the four sections
= 10700 cubic feet.
126. When the excavation docs not begin on a line at right angles
lo the centre line, intermediate stations are taken where the excava-
tkn b'^gins on each side of tlie road-bed, and the section may be calcu-
I Ob
EARTH-WORK.
[ated as a pyramid, having its A'ertex at the first of these points, and
for its base the cross-section at the second. The preceding method
gives the same result, since w and h in this case become 0, and reduce
;he foraiula to S ^^ i I w^ h^ . The same remarks apply to the end of
an excavation.
C. Grou7id very Irregular,
127. Prol>l€*m. To find the solidity of a section^ when the ground
is very irregular.
Fig. 54.
^ution. Let A HE FE - Ar CD Bi F^ Ei (fig. 54) represent one
side of a section, the surface of -which is too irregular to be divided
into two planes. Suppose, for instance, that the ground changes at
H^ C, and Z), making it necessary to divide the surface into five trian-
gles running from station to station.* Let heights be taken at /7, C,
and Z), and let the distances out of these points be measured. If now
we suppose the earth to be excavated vertically downward through
the side line B B^ to the plane of the road-bed, we may form as many
vertical triangular prisms as tliere are triangles on the surface This
iviM be made evident by drawing vertical planes through the sides
* It will often be necessary to introduce intermediate stations, in order to make
*he subdivision into triangles more conveniently and accurately.
GROUND VERY IRREGULAR. 109
A C, H C\ FID, and HB^ . Then the solidify of the kiJf-section will be
equal to the sinn of these prisms, minus the triangular mass BFG-
BiFi Gi .
The horizontal section of tlic prisms may be found from the distan-
ces out and the length of the section, and the vertical edges or heights
are all known. Hence tl>e solidities of these prisms may be calculated
by § 115.
To find the solidity of the portion BFG-B^ F^ Gx , which is to
be deducted, rci)resent the sloi)e of the sides by s {^ 102), the heights
at B and B^ by h and h^ , and the length of the section by I Then
we have F G ^ s /t, and Fi Gi = shi. Moreover, tlie area of B F G
- j s /r, and that of B^F^G^^^s h^^. Now as the triangles B F G
and L'l F, Gi are similar, the mass required is the frustum of a pyra-
mid, and the mean area is yj s /t^ x i s /'i^ = 3 '^^ ^' ^'i • '^'^^^"
(Tab. X 53) the solidity is B F G - B^ F^ G^^ U s (//-' + h^^ + h h^).
Example. Given Z = 50, 6 =18, s = i , the heights at .1, //, and B
respectively 4, 7, and 6, the distimces yl i/ = 9 and HB = 9, the
heights at A^ , C, D, and B^ respectively C, 7, 9, and 8, and the distan-
ces ^li C =4, CD =^ 5, and Z)/ii = 12 Then the horizontal sec-
tion of the first prism adjoining the centre line is ^ / X A^C, since the
distance ^i C is measured horizontally ; and the mean of the three
heighta is ^4 + 6 + 7) = ^ X 17. The solidity of this prism is
therefore ^ / X ^li C X ^ X 17 = b ^ X 4 X 17, that is, equal to \l
multiplied by the base of the triangle and by the sum of the heights.
In this way we should find for the solidity of the five prisms
1/(4 X 17 + 9 X 18 + 5 X 23+ 12 X 24 + 9 X 21)= 1/ X 822.
For the frustum to be deducted, we have
^/ X 1(62 + 8^ + 6X8) =U X 222.
Hence the solidity of the half-section is
\l (822 — 222) = g X 50 X 600 = 5000 cubic feet.
128. Let us now examine the usual method of calculating excava-
tun, when the cross-section of the ground is not level. This method
consists, first, in finding the area of a cross-seetion at each end of the
mass ; secondly, in finding the height of a section, level at the top,
equivalent in area to each of these end sections ; thirdly, in finding
from the average of these two heights the middle area of the mass ;
110 EARTH-WORK.
and, lastly, in applying the prismoidal formula to find the contents
The heights of the equivalent sections level at the top may be found
approximately by Trautwine's Diagrams,* or exactly by the following
method. Let A represent the area of an irregular cross-section, 6 the
width of the road-bed, and s the slope of the sides. Let x be the re-
quired height of an equivalent section level at the top. The bottom
of the equivalent section will be b, the top 6 -f 2 s ar, and the area will
be the sum of the top and bottom lines multiplied by half the height o
^.r (2 6 + 2st) = s X- -\- b X. But this area is to be equal to A
Therefore, s x- -\- b x -^ A, and from this equation the value of a: may
be found in any given case.
According to this method, the contents of the section already calcu-
lated in § 122 will be found thus. Calculating the end areas, we find
the first end area to be 387 and the second to be 240. Then as s is
here i and 6=18, the equations for finding the heights of the equiva-
lent end sections will be ia:^ + 18x = 387, and lx^-\- ISx = 240
Solving these equations, we have for the height at the first station
x = 11.146, and at the second, x = S. The middle area will, there-
fore, have the height ^ (11.146 + 8) = 9.573, and from this height the
middle area is found to be 309.78. Then by the prismoidal formula
(t 113) the solidity will heS^l X 100 (387 + 240 + 4 X 309.78)
— 31102 cubic feet.
But the true solidity of this section was found to be 32820 cubic
feet, a difference of 1718 feet. The error, of course, is not in the pris-
moidal formula, but in assuming that, if the earth were levelled at the
ends to the height of the equivalent end sections, the intervening earth
might be so disposed as to form a plane between these level ends, thus
reducing the mass to a prismoid. This supposition, however, may
sometimes be very far from correct, as has just been shown. If the
diagonal on the right-hand side in this example were reversed, that if
if the dividing line were formed by a depression, the true solidit}
found by § 122 would be 29600 feet ; whereas the method by equiva-
lent sections would give the sam.e contents as before, or 1502 feet too
much.
D. Correction in Excavation on Curves
129. In excavations on curves the ends of a section are not parallel
* A New Method of Calculating the Cubic Contents of Excavations and Embank
ments by the aid of Diagrams. By John C. Trautwine
CORRECTION IN EXCAVATION ON CURVES.
IIJ
to each other, but converge towards the centre of the curve. A section
between two stations 100 feet apart on the centre line will, therefore,
measure less than 100 feet on the side nearest to the centre of the
curve, and more than 100 feet on the side farthest from that centre.
Now in calculating the contents of an excavation, it is assumed thai
the ends of a section are parallel, both being perpendicular to the chord
of the curve. Thus, let figure 55 represent the plan of two sections ol
Fig. 55.
an excavation, EF G being the centre line, AL and Cil/the extreme
side lines, and 0 the centre of the curve. Then the calculation of tlie
Qrst section would include all between the lines .4 1 Ci and B^Di\
^-hile the true section lies between A C and B D. In like manner, the
calculation of the second section would include all between HK and
NP , while the true section lies between BD and L M. It is evident,
therefore, that at each station on the curve, as at jP, the calculation is
too great by the wedge-shaped mass represented hy KFD^, and too
Fig. 56
^n
■mull by the mass represented by BiFB These masses balance
112 EARTH-WORK.
each other, when the distances out on each side of the centre line are
equal, that is. when the cross-section may be represented hy AD F RE
(fig. 56). But if the excavation is on the side of a hill, so that the
distances out differ very much, and the cross-section is of the shape
AD FEE, the difference of the wedge-shaped masses may require
consideration.
130. Problem. Given the centre height c, the greatest side height h,
the least side height h', the greatest distance out d, the least distance out d',
and the ividlh of the road-bed b, to find the correction in excavation C, at
any station on a curve of radius R or defection angle D.
Solution. The correction, from what has been said above, is a trian-
gular prism of which B FR (fig. 56) is a cross-section. The height of
this prism at B (fig. 55) is Bi H, the height at A' is R^ S, and the height
at F is 0. Bi 11 and R^ S, being veiy short, are here considered
straight lines. Now we have the cross-secticn B FR = FB E G —
Fr'^EG = i^cd + ibh) - iUd' + ibh') = hc{d - d<) -f
ih{h — h'). To find the height Bi H, we have the angle B F 11 =
B FBi = D, and therefore Bi H = 2 HF sin. D = 2d s\n. D. In
like manner, R^ S = KD^ = 2KF sin. D =^ 2d' sin. D. Then
since the height at Fis 0, one third of the sum of the heights of the
prism will be f (d + fZ')sin. D, and the correction, or the solidity ol
the prism, will be (§ 115)
^ C=[hc{d- d') + ib{h-h')] X f(fZ-fcZ')sin. Z).
When R is given, iind not D, substitute for sin. D its value (§9)
50
Bin. D =^ jf . The correction then becomes
^ C^[U{d-d')-^-\bih-h')]x'^^^^±^.
This correction is to be added, when the highest ground is on the
convex side of the curve, and subtracted, when the highest ground is on
the concave side. At a tangent point, it is evident, from figure 55, that
the correction will be just half of that given above.
Ercanple. Given c = 28, h ^ 40, h< = 16, f? = 74, d' = 38, b = 28,
and R. = 1400, to find C. Here the area of the cross-section BFR -=
- (7-i — 38) 4- - (40 — 16) = 672, and one third of the sum of the
. 100(74 + 38) 8 ^ fi7o V - «
heights of the prism is 3 ^ ^^qq -= 3 • Hence C = 672 X 3 «
• 792 cubic feet.
CORRECTION IJ\ EXCAVATION ON CURVES. 113
131. When the section is partly in excavation and partly in em-
bankment, the cross-section of the excavation is a triangle lying
tvlioUy on one side of the centre line, or partly on one side and partlj
on the otlier. The surface of the ground, instead of extending from
B to D (fig. 56), will extend from B to a. point between G and E, or
to a point between A and G. In the first case, the correction will be
a triangular prism lying between the lines B^ /'and fl F (fig. 55), but
not extending below the point F. In the second case, the excavation
extends below F, and the correction, as in § 129, is the difference be-
tween the masses above and below F. This difference may be ob-
tained in a very simple manner, by regarding the mass on both sides
of i^as one triangular prism the bases of which intersect on the line
G F (fig. 56), in whicli case the height of the prism at the edge be-
low /'""must be considered to be ininus, since the direction of this edge,
referred to either of the bases, is contrary to that of the two others.
The solidity of this prism will then be the difference required.
132. Prol>8eill. Given the width of the excavation at the road-bed
w, the ividth of the road-bed 6, the distance out d, and the side height A, to
find the correction in excavation C, at any station on a curve of radius R
or deflection angle D, when the section is partly in excavation and partly in
embanlcinent.
Solution. When the excavation lies wholly on one side of the centre
line, the correction is a triangular prism having for its cross-section
the cross-section of the excavation. Its area is, therefore, ^ iv h. The
licight of this prism at B (fig. 56) is (§ 130) B^ IT = 2 H F s\r\. D =
2 d sin. D. In a similar manner, the height at E will be 2 G E sin. D
= b sin. Z>, and at the point intermediate between G and j5J, the dis-
tance of which from the centre line is ^t — ly, the height will be
2 {^b — 16') sin. D = (b — 2 iv) sin. D. Hence, the correction, or the solid-
ity of the prism, will he {^ 115) C = ^whxh {2d-i-b-{-b — 2iv) sin. Z)
-= ^loh X i {d -\- b — lo) sin. D.
When the excavation lies on both sides of the centre line, the cor-
rection, from what has been said above, is a triangular prism having
also for its cross-section the cross-section of the excaration. Its area
will, therefore, be ^ivh. The height of this prism at Bis also 2dsin.D,
and the height at E, b sin. D ; but at the point intermediate between A
and G. the distance of which from the centre line \s w — ^b, the height
will be 2 (iv — ^b) sin. Z) = (2 lo — b) sin. D. As this height is to
be considered minus, it must be subtracted from the others, and the
coriection required will be C=^wkxhi2d-\-b — 2w-\-b) sin. D
114 EAETH-WORK.
^ ^wh X I (^ + t — 't') sin. D. Hence, in all cases, when the sec
tion is partly in excavation and partly in embankment, we have the
formula
1^- C=^'u;hX ^ {d-\-b— iv) sin. D.
When R is given, and not D, substitute for sin. D its value (§ 9)
50
sin. D = -^ . T^e correction then becomes
This correction is to be added, when the highest ground is on the
convex side of the curve, and subtracted when the highest ground is on
the concave side. At a tangent point the correction will be just half
of that given above.
Example. Given if; = 17, 6 = 30, c? = 51, A = 24, and 22 = 1600,
to find C. Here the area of the cross-section is ^wh = \7 % 12 =
. If0(d+b—w)
204-. and one third of the sum of the heights of the prism is g^j
..^ ^^"^^^^^ = l Hence C = 204 X | = 272 cubic feet.
1.33. The preceding corrections (§130 and ^32) suppose the length
of the sections to be 100 feet. If the sections are shorter, the angle
B FH (fig. 55) may be regarded as the same part of D that FG is ol
100 feet, and Sj FB as the same part of D that jEJFis of 100 feet
The true correction may then be taken as the same part of C that the
mm of the lengths of the two adjoining sections is of 200 feet.
TABLE I.
UADII, ORDINATES, DEFLECTIONS,
AND
ORDINATES FOR CURVING RAILS.
Jroraiiila for Radii, ^ 10 ; for Ordinates, § 25 ; for Dcflectlong, $ 1*J
for CuiTiug Riiils, § 29.
lib
TABLE
I. RADII
, ORDINATES, DEFLECTIONS,
Degree.
Radu.
Ordinates.
Tangent
Deflec-
Chord
Deflec-
Ordinates for
Rails.
12^
25.
37i.
50.
tion.
tion.
18.
20.
O (
0 5
6S754.94
.008
.014
.017
.018
.073
.145
1
.001
.001
10
34377.48
.016
.027
.034
.036
.145
.291
.001
.001
15
22918 33
.024
.041
.051
.055
.218
.436
.002
.002
20
171SS.76
.032
.055
,063
.073
.291
.582
.002
.003
251
13751.02
mo
.063
.085
.091
.364
.727
.003
.004
30
11459.19
.013
.032
.102
.109
.4.36
.873
.004
004
35
9322. 1-?
.056
.095
119
.127
.509
1.013
.004
.005
40
8594.41
.064
.109
.136
.145
.532
1.164
.005
.006 1
45
7639.49
.072
.123!
.153
.164
.654
1.309
.005
.007 1
50
6375.55
.080
.136
.170
.132
.727
1.454
.006
.007
55
6250.51
.037
.150
.187
.200
.800
1.600
.006
.008
1 0
5729.65
.095
.164
.205
.218
,873
1.745
.007
.009
5
523S.92
103
.177
.222
.236
,945
1.891
.008
.009
10
4911.15
.111
.191
.239
.255
1,018
2.036
.008
.010
15
45S3.75
.119
.205
.256
.273
1.091
2.182
.009
.011
20
4297. 2S
.127
.218
.273
.291
1.164
2.327
.009
.012
25
4044.51
.135
.232
.290
.309
1 .236
2.472
.010
.012
30
33 19. S3
.143
.245
.307
.327
1.3(19
2.613
.011
.013
35
36 IS. SO
.151
.259
..324
.345
1 332
2.763
.011
.014
) 4C
3437.87
.159
.273
..341
.364
1.454
2.909
.012
.015
45
3274.17
.167
.236
.358
.332
1.527
3.054
.012
.015
50
3125.36
.175
.300
.375
.400
1.600
3.200
.013
.016
55
2939.43
.133
.314
.392
.418
1.673
3.345
.014
017
9 0
2S64.93
.191
.327
.409
.436
1.745
3.490
.014
.017
5
2750.35
.199
.341
.426
.455
1.818
3.636
.015
.013
in
2644.53
.207
.355
.443
.473
1.391
3.781
.015
.019
15
2546.64
.215
.363
.460
.491
1.963
3.927
.016
.020
20
2455.70
.223
.3c2
.477
.509
2.036
4.072
.016
.020
25
2371.04
.231
.395
.494
.527
2.109
4.218
.017
.021
30
2292.01
.239
.409
.511
.545
2.1S1
4.363
.018
.022
35
2213.09
.247
.423
..528
.564
2.251
4.503
.018
.023
40
2143.79
.255
.436
..545
.582
2.327
4.654
.019
.023
45
2033.6S
.263
.450
.562
.600
2.400
4.799
.019
.024
50
2022.41
.270
.464
.530
.613
2.472
4.945
.020
.025
55
1664 64
.278
.477
.597
.636
2.545
5.090
.021
.025
3 0
1910.03
.286
.491
.614
.655
2.6 IS
5.235
.021
.026
5
1358.47
•294
.505
.631
.673
2.690
5..381
.022
.027
10
1309.57
.302
.518
.643
.691
2.763
5.526
.022
.028
15
1763 13
.310
..532
.665
.709
2.336
5.672
.023
! .023
20
1719.12
.318
.545
.682
.727
2.908
5.817
.024
1 .029
25
1677.20
.326
.559
.699
.745
2.9S1
5.962
.024
1 .030
30
1637.28
.3-34
.573
.716
.764
3.054
6.108
.025
j .031
35
1599.21
.342
.536
.733
.782
3.127
6.2.53
.025
.031
40
1 562.SS
.3.50
.600
.750
.800
3.199
6.398
.026
, .032
45
1523.16
.353
.614
.767
.818
3.272
6.544
.027
I .033
50
1494.95
.366
.627
.784
.8.36
3.345
6.639
,027
j .033
55
1463 16
.374
.641
.801
.855
3 417
6,835
.028
.034
4 0
(432.69
.332
.655
.818
.873
3.490
6.930
.028
.035
5
1403 46
.390
.663
.835
.891
3.563
7.125
.029
.036
10
1375.40
.398
.632
.852
.909
3.635
7.271
.029
; .036
15
1343.45
.406
.695
.869
.927
3.703
7.416
.030
.037
20
1.3.22. .53
.414
.709
.836
.945
3.731
7..561
.031
.033
25
1297.53
.422
,723
.903
,964
3.3.53
7.707
.031
! .039
30
1273.57
.430
.736
.921
.932
3.926
7.352
.032
.039
35
12.50.42
.438
.750
.933
1.000
3.999
7.997
.032
.040
40
1223.11
.446
.764
.955
1.018
4.071
8.143
.033
.041
45
1206.57
.454
.777
.972
1.036
4.144
8.2.8S
.034
.041
50
1185.78
.462
.791
.939
1.055
4.217
8,4:33
.034
.042
55
1165.70
.469
.805
1.006
1.073
4.239
8.579
.035
.043
5 0
1146.23
.477
.818
1.023
1.091
4.362
8.724
.035
.044
AND^ORDINATES FOR CURVING RATI S.
117
Degree.
Radii.
o /
5 5
10
15!
201
25
30
35
40
45
50
55
6 0
5
10
15
20
25
30
35
40
45
50
55
7 0
5
10
15
20
25
30
35
40
45
50
55
8 0
5
10
15
20
25
30
35
40
45
50
55
9 o!
5
10
15
20
25
30
35
40
45
50
55
Ordinates.
12i.
i 127.50
1 [09.33
1091.73
1U74.68
1058.16
1042.14
1026.60
1011.51
996.87
982.61
968.81
955 37
912.29
929 57
917.19
905.13
893.39
SSI. 95
870.79
859.92
849.32
838.97
828.88
819.02
809.40
800.00
790.81
781. S4
773.07
764.49
756.10
747.89
739.86
732.01
724.31
716.78
709.40
702.18
695.09
688.16
681.35
674.69
66S.15
661.74
655.45
619.27
643.22
637.27
631.44
625 71
620.09
614. r,6
609.14
603.80
598.57
593.42
588.36
583.38
578.49
25.
.4;
.493
501
.509
.517
.525
.533
.541
.549
.557
.565
37*.
60.
10 0 573.69
.581
.589
..597
.605
.613
.621
.629
.637
.645
.653
.66
.669
.677
.685
.693
,701
.709
.717
.725
.733
.740
.748
.756
.764
.772
.780
.788
.796
.804
.812
.820
.828
.836
.844
.852
.860
.868
.876
.884
.892
.900
.908
.916
.924
,932
.940
.948
.956
.832
.846 1
,859
.873!
.887
.900
,914
.928
,941
.955
.96^
,982
,996
1,009
1,023
1.037
1,050
1.061
1.078
1.091
1.105
1.118
1.132
1.146
1.1.59
1.173
1.187
1.200
1.214
1.228
1.242
1.255
1.269
1.283
1.296
1.310
1,324
1,337
1,351
1.365
1.378
1.392
1,406
1.419
1.433
1.447
1,460
1.040
1,057
1,074
1.091
1.108
1.125
1.142
1.1.59
1.176
1.193
1.210
1.228
1.24.''i
1.262
1.279
1.296
1.313
1 .330
1.347
1.364
1.381
1.398
1.415
1.432
1.449
1.466
1.483
1.501
1.517
1.535
1.552
1.569
1.586
1.603
1.620
1.637
1.6.54
1.671
1.688
1.705
1.722
1.739
1.757
1.774
1.791
1.808
1.825
Tangent
Petiec-
.tion.
1.109
1.127
1.146
1.164
1.182
1 200
1.218
1.237
1.255
1 .273
1.291
1.309
1.327
1.346
1.364
1.382
1.400
1.418
1.437
1 .455
1.473
1.491
1.510
1 .528
1.546
1.564
1 .582
1 .600
1.619
1.637
1,655
1,673
1.691
1.710
1.728
1.746
1.764
1.782
1.801
1.819
1.8.37
1.8.55
1.873
1.892
1.910
1.928
1.940
1.474
1.488
1.501
1.515
1,529
I, .54 2
1,.556
1,570
1.583
1.597
1.611
1.624
1 .638
1.842
1.859
1.876
1.893
1.910
1.927
1.944
1.961
1.979
1.996
2.013
2.030
2.047
1.965
1.983
2.001
2.019
2.037
2.056
2.074
2.092
2.110
2.128
2.147
2.165
2.183
Chord
Ufllcc-
tion.
Oldir.ates fen-
Rails.
4.435
4.507
4.580
4.653
4.725
4.798
4.870
4.943
5.016
- 5.088
5.161
5.234
5.306
5..379
5.451
5.524
5..597
5.669
5.742
5.814
5.8S7
5.960
6.032
6.105
6.177
6.250
6.323
6.395
6.468
6.540
6.613
6.685
6,758
6.831
6.903
6.976
7.048
7.121
7.193
7.266
7,338
7.411
7.483
7,556
7,628
7.701
7.773
.7.846
7 918
7.991
8.063
8.136
8.208
8.281
8.353
8.426
8.49S
8.571
8.643
8.716
8.869
9.014
9.160
9.305
9.450
9.596
9.741
9.8.-^6
10.031
10.177
10.322
10.467
10.612
10.758
10.903
11.048
11.193
11.339
11.484
11.774
11 919
r2.u65
12.210
12.355
12.500
12.645
12.790 j
12.936
13.081
13.226
13.371
13.516
13.661
13.806
13.951
14.096
14.241
14.387
14.532
14.677
14.822
14.967
15.112
15.257
15.402
15.547
15.692
15.837
15.9S2
16.127
16.272
16.417
16.562
16.707
16.852
16.996
17.141
17.286
17.431
18.
.036
.037
.037
.038
.038
.039
.039
.n4()
.041
.(!41
.0-/2
.042
.043
.044
.044
.04
.04
.046
.047
.047
.(48
.048
.049
.049
.050
.051
.051]
.052
,052
,053
.054
,054
,055
.055
.056
20.
'I
.057
.057
.058
.058
.059
.0591
.060
.061
.061
.062
.062
.063
.064
.064
.065
.065
.066
.0661
.0671
.068
.0681
.069
.069
,070
.044
,045
,046
,047
,047
.048
.049
.049
.050
,051
,052
,052
.053
.054
.055
.055
.056
.057
.057
.058
.059
.060
.060
061
.062
.063
.063
.064
.065
.065
.066
.067
.068
.068
.069
070
,070
,071
,072
.073
.073
.074
.075
.076
.076
.077
.078
.078
.073
.080
,081
.081
.082
.083
.084
.084
.085
.0>6
,086
.071 .087
118
TABLE I. RADII, ORDINATES, DEFLECTIO.NS, i^C.
r
Degree.
Radii.
Ordinates.
Tangent
Deflec-
tion.
' Chord
Deflec-
tion.
Ordinates for
Rails.
12^.
25.
37*.
50.
18.
20.
o /
lU IJ
564.31
.97-2
1.665 2.031
2.219
8.S6C
17.721
.072
.039
2]
555. '23
.933 1.693 2.115 2.2.56
9.OO0
13.01 1
.073
.090
33
546.44
l.OM l.720l 2.149; 2.292
9.150
13.300
.074
.092
40
537.92
1.020 1.743
2.131
2.329
9.295
13.590
.075
.093
50
529.67
1.036
1.775
2.213
2.355
9.440
13.330
.076
.094
11 0
521.67
1.052
1.302
2.252' 2.402
9.535
19.169
.073
.098
10
51.3.91
I mi
1.S30
2.236: 2.4.33
9.729
19.459
.079
.097
20
506.33
1.0S4
1 .857
2.320
2.475
9.374
19.743
.030
.099
30
499.136
l.ldOl 1.334
2.3>1
2.511
10.019
20.0:33
.031
.100
40
491.96
l.llG 1.912
2.339
2.;547
10.164
20.327
.032
.102
50
4S5.05
1.132 1.9:33
2.423
2.531
10.:303
20.616
.034
.103
12 0
47S.ai
1.143
1.967
2.457
2.620
10.453
20.906
.035
.105
10
471.31
1.164
1.994
2.491
2.657
IO..597
21.195
.036
.106
20
465.46
1.130
2.021
2.525
2.693
10.742
21.434
.087
.107
30
459. 2S
I.I96I 2.049
2.560
2.730
10.337
21.773
.088
.109
40
4-53.26
1.212 2.076
2.594
2.766
11.031
22.063
.039
.110
50
447.40
1.223 2.104
2.623
2.303
11.176
22.-352
.091
.112
13 0
441.63
1J244 2.131
2.662
2.839
11.320
22.641
.092
.113
10
436. 12
1.260 2.159
2.697
2.376
11.465
22.930
.093
.115
20
430.69
1.277
2. 1 36
2.731
2.912
11.609
23.219
.094
.116
30
425.40
1.293
2.213
2.765
2.949
11.754
2:3.507
.095
.113
40
420.23
i.:3a9
2.241
2.799
2.935
11.393
23.796
.096
.119
50
41.5.19
1.325
2.263
2.3.3:3
3.022
12.013
24.035
.093
.120
14 0
410.23
1.341
2.296' 2.363
3.053
12.137
24.374
.099
.122
10
40.5.47
1.357
2.323
2.902
3.095
12.331
24.663
.100
.123
20
400.73
1.373
2.351
2.9.361 3.131
12.476
24.951
.101
.125
30
396.20
l.:3S9
2.373
2.970
3.163
12.620
25.240
.102
.126
40
391.72
1.405
2.406
3.005
3.204
12.761
25.523
.103
.123
50
337.34
1.421
2.4.33
3.039
3.241
12.903
25.317
.105
.129
15 0
333.06
1.4:37
2.461
3.073
3.277
1:3.0.53
26.105
.106
.131
10
373.33
1.4.53
2.4 S3
3.107
3.314
1:3.197
26.394
.107
.1.32
20
374.79
1.469
2.515
3.142
3.350
I3.:341
26.632
.103
.133
30
370.73
1.436
2..543
.3.176; 3.337
13.435
26.970
.109
.135
40
366.36
1.502
2.570
3.210 3.423
13.629
27.253
.110
.136
50
363.02
1.513
2.593
3.245
3.460
13.773
27.547
.112
.133
16 0
3.59.26
1..5.34
2.625
3.279
3.496
13.917
27.335
.113
.139
10
355. 59
1.550
2.6.53
3.313
3.5:33
14.061
23.123
.114
.141
20
351.93
1.566
2.630
3.317
3.569
14.205
23.411
.115
.142
30
a43.45
1.532
2.703
3.332 3.606
14.349
23.699
.116
.143
40
344 99
1.593
2.7.36
3.416 3.643
14.493
23.936
.117
.145
50
ail. 60
1.615
2.763
3.450
3.679
14.637
29.274
.119
.146
17 0
33S.27
1.631
2.791
3.435
3.716
14.731
29.562
.120
.143
10
335.01
1.617
2.313
3.519
3.7.52
14.925
29.3-50
.121
.149
20
3:31.82
1.663
2.346
3.-5.53
3.739
15.069
30.137
.122
.151
30
323.63
1.679
2.373
3.583
3.325
15.212
30 425
.123
.152
40
32-5.60
1.695
2.901
3.622
3 362
15.356
30.712
124
.154
50
322.59
1.711
2.923
3.656
3.393
15.500
31.000
.126
.155
18 0
319.62
1.723
2.956
3.691
3.935
15.643
31.237
.127
.1-56
10
316.71
1.744
2.933
3.725
3.972
1-5.737
31.574
.123
.153
20
313.36
1.760
3.011
3.7.59
4.003
15.931
31.361
.129
.159
30
311.06
1.776
3.0.39
3.794
4.045
16.074
32.149
.130
.161
40
303.30
1.792: 3.066 i
3.523
4.081
16.213
32.436
.131
.162
50
305.60
1.309
3.094
3.362
4.113
16.361
32.723
.133
.164
19 0
302.94
1.325
3.121
3.397
4.1.55
16.-505
33.010
.134
.165
10
3D0..33
1.341
3.149
3.931
4.191
16.643
33.296
.135
.166
20
297.77
1.357
3.177
3.965
4.223
16.792
a3.533
.1-36
.163
30
295.25
1.373
3.204
4.000
4.265
16.9-35
33.870
.137
.169
40
292.77
1.390
3.232
4.034
4.301
17.073
a4.157
.1-33
.171
50
290.33
1.906 3.2.59 j
4.069
4.333
17.222
34.443
.140
.172
20 0
237.91
1.922; 3.2371
4.103
4.374
17.365
34.730
.141
.174
TABLE II. LONG CHORDS.
119
TABLE II.
LONG CHORDS. § 69.
Degree of
Civrve.
2 Stations.
3 Stations.
4 Stations.
5 Stations.
.. .,
6 Stations. !
o t
0 10
200.000
299.999
399.993
499.996
599.993
20
199.999
.997
.992
.953
.970
30
.993
.992
.931
.962
.933
40
.997
.936
.966
.932
.832
50
.995
.979
.947
.894
.815
1 0
199.992
299.970
399.924
499.S43
599.733
10
.990
.959
.896
.793
.637
20
.956
.946
.865
.729
.526
30
.9S3
.932
.829
.657
.401
40
.979
.915
.789
.577
.260
oO
.974
.893
.744
.483
.105
2 0
199.970
299.378
399.695
499.-391
595.934 i
10
.964
.857
.643
.255
.7.50 1
20
.959
.SM
.5^36
.171
.5:50
30
.9.')2
.810
.524
•ai9
.336
40
.916
.733
.459
498.913
.106
50
.939
.756
.389
.778
597,662
3 0
199.931
299.726
399.315
49S.630
597.604
10
.924
.695
.2.37
.474
.331
20
.915
.652
.154
.309
.043
30
.907
.627
.Oft3
.136
596.740
40
.893
.591
398.977
497.955
.423
50
.833
.553
.882
.765
.091
4 0
199.S73
299.513
393.732
497.566
595.744
10
.563
.471
.679
.360
.353
20
.857
.423
.571
.145
.007
30
.846
.333
.459
496.921
594.617
40
.834
.337
.343
.639
.212
50
.822
.239
.223
.449
593.792
5 0
199.810
299.239
393.099
496.200
593.353
10
.797
.157
397.970
495.944
592.909
20
.733
AM
.837
.678
446
30
.770
.079
.709
405
591.963
40
.756
.023
.559
.123
.476
50
.741
293.9&1
.413
494.832
590.970
6 0
199.726
293.904
397.264
494.5^4
590.449
10
.710
.843
.110
.227
589.913
20
.695
.779
396.952
493.912
.364
30
.673
.714
.790
.553
533.300
40
.662
.643
.6-23
257
.221
50
.644
.579
453
492.917
537.623
7 0
199.627
298.509
396.278
492.563
537.021
10
.609
433
099
.212
536.400
20
.591
.3&1
395.916
491.347
535.765
30
.572
.239
.729
.474
.115
40
•553
.212
.533
.093
584.451
50
.533
.134
.342
490.701
533.773
8 0
.513
293.054
395.142
490.306
553.051
120
TABLE III. TABLE IV.
TABLE III.
CORRECTION FOR THE EARTH'S CURVATURE AND
FOR REFRACTION. § 105.
D.
J.
D.
d.
D.
d.
D.
d.
303
.002
ISOO
.066
3300
.223
4--'00
.472
400
.0ft3
1900
.074
3 J 00
.237
4900
.492
500
.005
2CH30
.0S2
3^500
.25!
5000
.512
600
.007
2100
.090
36Q0
.266
5100
.533
700
.010
2200
.099
37(10
.2S1
5200
,554
800
.013
2300
.lOS
3S00
.2i;6
1 mUe
.571
900
.017
2400
.113
3900
.312
2 «♦
2.235
1000
.020
2500
.123
4000
.328
3 »<
5.142
1100
.025
2600
.139
4100
.345
4 «
9.142
3200
.030
2700
.149
4200
.362
5 «
14.284 {
1300
.035
2SitO
.161
4300
.370
6 "
20.563
1400
.040
2900
.172
4400
.397
7 "
27.996
1500
.046
3000
.1S4
4501
.415
8 "
36.566
1600
052
3100
.197
460 J
.434
9 «
46.279
1700
.059
3200
.210
4700
.453
10 "
57.135
TABLE IV.
ELEVATION OF THE OUTER RAIL ON CURVES.
§ 110.
Degree.
RT = 15
M = 20.
M = 26.
M = 80.
M = 40.
M = 50.
o
1
.012
.022
034
.049
.088
.137
2
.025
.044
.068
.099
.175
.274
3
.037
.066
.103
.143
.263
.411
4
.049
.033
137
.197
.351
.543
5
.062
.110
.171
.247
.433
.685
6
.074
.131
.205
.296
.526
.822
7
.0S6
.153
.240
.345
.613
.953
8
.099
.175
.274
.394
.701
1.095
9
.111
.197
.303
.443
.788
1.232
10
.123
.219
.342
.493
.876
L36S
TABLE V. TABLE VI.
121
TABLE V.
FROG ANGLES, CHORDS, AND ORDINATES FOR
TURNOUTS.
This table is calculated for g = 4.7, d =- .42, and S = 1° 20'. For
mula for frog angle F, and chord B F, § 50 ; for m, the middle or-
dinate of B F, § 25 ; for ?/i', the middle ordinate for curving an 18 ft
rail, § 29.
R.
imo
F.
BF.
in.
niK
R.
600
F.
BF.
m.
m'
g 27 ik
72.22
.651
.041
O 1
6 57
48
59.17
.727
.068
975
5 31 39
71.53
.655
.012
575
7 6
26
58.16
.7.33
.070
950
5 35 44
70.S3
.659
,043
550
7 15
40
57.12
.739
.074
925
5 39 59
71.11
.663
,044
525
7 25
33
56.05
,745
,077
900
5 44 24
69.3S
.667
,045
500
7 36
10
54.94
.752
,081
875
5 49 1
68.64
.671
.046
475
7 47
37
53.79
.758
.085
850
5 53 50
67.88
.676
,01S
450
8 0
1
52.61
.765
.090
825
5 53 52
67.10
.680
,049
425
8 13
30
51. 3r
.773
.095
ST)
6 4 9
66.30
.685
,051
400
S 23
14
50.09
.780
.101
775
6 9 41
6."'.49
.690
.052
375
8 44
26
48.75
.788
.103
750
6 15 30
64.65
.695
.054
350
9 2
20
47.35
.796
.116
725
6 21 37
63.80
.701
.056
325
9 22
16
45.88
.805
.125
700
6 28 4
62.92
.705
.058
300
9 44
39
44.34
.814
.135
675
6 34 52
62.02
.710
,060
275
10 10
1
42.72
.824
.147
650
6 42 4
61.09
.716
,062
250
10 39
6
41.00
.834
.162
625
L— ... . .
6 49 42
60,14
.721
.065
225
11 12
55
39.16
.845
.180
TABLE VI.
LENGTH OF CIRCULAR ARCS IN PARTS OF RADIUS
o
1
,01745
32925
19943
1
.00029
08882
08666
//
1
,00000
48481
36811
9.
.03490
65850
39.S87
2
.00058
17764
17331
2
,00000
96962
73622
3
.05235
98775
59830
3
.00087
26646
25997
3
.00001
45444
10433
4
,(;69S1
31700
79773
4
.00116
35528
34663
4
.00001
9-3925
47244
ri
.03726
64625
997 1 6
5
.00145
44410
43329
5
,00002
42406
84055
6
,10471
9755 1
19660
6
.00174
53292
51994
6
.00002
90.888
20,867
7
.12217
.30476
39603
7
,00203
62174
60660
7
.00003
39369
57678
8
.13962
63401
59546
8
,00232
71056
69326
8
,00003
87850
94489
9
.15707
96326
79190
J_
,00261
79933
77991
9
,00004
36332
31300
122
TABLE VII. EXPANSION BY HEAT.
TABLE VII.
EXPANSION BY HEAT.
Bodies.
323 to 2123.
lO.
Authority.
Platina,
.0003S42
.000004912
Ilassler
Gold,
,001466
.000003141
((
Silver,
.001909
.000010605
((
Mercury,
.01S013
.0001001
((
Brass,
.00189163
.000010509
((
Iron,
.00125344
,000006964
((
^V'ater,
.0466
not uniform.
((
Granite,
.00036350
.0000(MS25
Prof. Bartlett.
Marble,
.00102024
.00000566.3
((
Sends tone,
.00171576
.000009532
u
TABLE VIII, PROPERTIES OF MATEUIALS.
123
TABLE VIII.
PROPERTIES OF MATERIALS.
The authorities referred to by the capital letters in the table are : —
B Barlow, On the Strength of
Materials.
Bevan.
Lieut. Brown.
Couch.
Franklin Institute, Report on
Steam Boilers.
Gordon, Eng. Translation of
Weisbach.
Hodgkinson, Reports to Brit.
Association.
Ha, Hassler, 2\ibles.
Be
Br
C.
F.
G.
H.
L. Lame.
M. Musschenbroek, Int. to Nat
Phil.
R. Rennie, Pliil. Trans.
Ro. Rondelet, Vxirt de Batir.
T. Telford.
Ta. Taylor, Statistics of Coal.
W. Weisbach, Mech. of Machin-
ery and Engineering.
The numbers without letters ar«
taken from Prof Moseley's En-
gineering and Architecture
In finding the weights, a cubic foot of water has, for convenience,
been taken at 62.5 lbs.
The numbers for compression taken from Hodgkinson were ob-
tained by him from prisms high enough to allow the wedge of rupture
to slide freely off. He shows that this is essential in experiments on
rompression.
The modulus of rupture *S is the breaking weight of a prism 1 in
broad, 1 in. deep, and 1 in. between the supports, the weight being ap-
plied in the middle. To find the corresponding breaking weight I^of
a rectangular beam of any other size, let / = its length, b =: its breadth,
2 b d'i
and d = its depth, all in inches. Then W = -or X 'S.
The numbers in the last three columns express absolute strength
For safety, a certain proportion only of these numbers is taken. The
divisors for wood may be from 6 to 10, for metal from 3 to 6, for stone
10, and for ropes 3.
When double numbers are used in the column headed " Crushing
Force per Square Inch in lbs.," the first applies to specimens moder-
ately dry, the second to specimens turned and kept dry in a warm
place two months longer. In the case of American Birch, Elm, and
Teak, the numbers apply to seasoned specimens.
134
T.ABLE VIII.
'ROPERTIES OF MATERIALS.
Materials.
it
Metals.
Ccppe.', oapt, . . ,
rLllcd, . ,
r'iie-firawTi,
GoH,
Iron, cast,
Canou Xo. 2, cold
" •' hot
Devon No. ?, ccld
' hoi
Butlery yo. 1, ci la"
'' " hoi
Iron, wroughS,
Encjlish bar,
Welsh "
Swedish " . .
Lancaster v'o , /
Tenness'ie
Missouri
Iron wire,
Enslish, a'an
rhmipsb'g, ra.
blast,
((
u
kC
u
Lead, cast, . .
Lead wir.3, ....
Mercury, ....
Platina,
Silver,
Steel, s<.fi, ....
•' razor-teaporYMi,
Tin, caet, ....
Zinc, fused, . .
" roUed, . . .
S33
Ash, English, . .
Birch, English,
" Americf n, .
Box,
Cedar, Canadian, .
Chestnut. . . .
Deal, Christiania mi(f i' »,
" Memel "
" Norwav Spruce,
" English. ....
Elm. seasoned, . . .
Fir, New England, . .
" Riga, . . .
Lignum-vitse, . .
Mahogany, Spanish,
Specific
Gravity.
8.399
8.6' )7
S.S64 F.
8.37S
19.2.3>Ha
19.361 Ua
7.066 H
7.046 H.
7.29.5 H.
7.2:id H.
7.079 H.
6.99S H.
7.700
7.473 F.
7.740 F.
7.S0.5 F.
7.722 F.
7.727 F.
II 446 M.
P.:J17
]l3.5i>S W.
l.Ta'XilLv
1^2.669 Ua
l0 474H.i
7.7S0
7.840
7 050 TV.
'.'.540 W
AV'eight
per
Cubic
Foot
in lbs
.760 B
.792 B.
.&iS B.
.960 B
909 C
.6-57 Ro.
.69S B.
.590 B
.340
.470
,553 B
.553 B.
.753 B.
1.220
.800
Tensile
I Strength
per Square
ilnchinlbs.
524.94
537.94
554.00
554.87
1203.62
1210.06
441.62
440.37
455.94
451, SI
442.44
437.37
431.25
467.37
4S3.75
4S7.S1
432 62
482.94
715.37
707.31
S49.S7
'218.75
lli6.81
65L62
43G.25
490.')0
451 63
1^0 JO
47i 2.>
17963 R.
19072
32S26 F.
6122S
16653 II.
13505 H.
21907 H.
17466 H.
13434 n.
57120 L.
61960 T.
64960 T.
5S134F.
5S661 F.
52099 F.
47909 F.
80214 T.
S41S6 F.
733-8 F.
89162 F.
1324 R.
2531 M.
40902 M.
120000
IrOOOO
5322 M.
47.53'
4'^.50;
4n.50'
60.00'
56.81'
41.06!
43.62
36.87
21.25
29.37
34.5G
34.-56
47.06
76.25
50.J0
2GC<JG ■»
1I40CB^.
1 3.300 Ro.
12400
17600
7000
13439 M.
12000 B.
IISOOM.
16500
Crushing
Force per
Square
Inch
in lbs.
10637511.
103540 U.
14.54.3511.
93335 II.
86397 H.
56000 ?G.
fS633H.)
\ 9363 H. )
f 3297 II. »
J6102H.(
11 663 II.
S771 H.
1033i B
Moduliis
of Rup-
ture S
in lbs.
38556 H.
37503 H.
36288 H.
43497 H.
37503 H.
35316 H.
54000 G.
12156 B
10920 B.
9624 B
9364 B.
10386 B.
i S^7S B.
' e612B.
{Il^,jl «-"■
(8193 F.
i 8193 H.
I
TABLE VIII. PROPERTIES OF MATERIALS.
125
Miiterials.
Specific
Gravitj'.
Woods.
Oak, English, . .
" Canadian,
Fine, pitch, .
red, . ...
American, white,
" Southern
Poplar,
ITeak, .
Other Materials.
Brick, red, . .
" Dale red.
Chalk,
Coal, Penn. anthracite,
" " semi-bituminous,
" Md. "
Penn. bituminous,
Ohio "
" English "
Earth,
loamy hard-stamped, fresh,
" " dry,
garden, fresh, . .
" dry, . .
dry, poor, . . .
Glass, plate, . . .
Gravel,
Gi-anite, Aberdeen, .
Ivory,
Limestone, ....
Marble, white Italian,
black Gal way,
Masoni-y, quarry stone, dry,
sandstone, "
" brick, dry, . |
Ropes,
hemp, under 1 inch diam.,
'* from 1 to 3 in. "
" over 3 inches "
Sand, river,
Sandstone, |
" Dundee, ....
" Derby, red and friable,
Slate, Welsh,
" Scotch,
.931 B.
.872 B.
.6S0 B.
.657 B.
.455 Br.
.372 Br.
.333 M.
.745 B.
2. 1 GSR.
2.0:^5 R.
2.7,S4
1.S69
1..327Ta
1.700 Ta
1.453 Ta.
1..552Ta.
1.312 Ta.
1.270 Ta.
1.2.59 Ta.
2.060 W.
1.930 W.
2.05 ) W.
1.630 W.
1.340 W.
2.453
1.920
2.625 R.
1.826
2.400 W.
2.S60 W.
2.63S H.
2.695 H.
2.400 W.
2.050 W.
1.470 W.
1.590 W.
1.8S6
1.900 W.
2.700 W.
2.530 R.
2.316 R.
2.S83
Weight
per
Cubic
Foot
iu lbs
58.3/
54.50
41.25
41.06
2S.44
54.50
23.94
46.56
135..50
130.31
174.00
116.81
82.94
106.25
90.81
97.00
82.00
79.3
73.69
!2>.Vo
12 ).62
128.12
101.87
83.75
153.31
1 20.no
164.06
114.12
1.50.00
173.75
164.87
168.44
150.00
128.12
91.87
99.37
Tensile
Strength
per Square
Inch in lbs.
117.87
118.75
168.75
1.58.12
144.75
180.50
10000 B.
10253
7318 M.
7200 Be.
15000 B.
280
300
9420
16626
Crushing
Force per
Square
Inch
in lbs.
(6184 II.)
1 1005-n)
14231 II. i
) 5982 II. j
6790 H. 1
6790 II. )
(5395H.)
\7518U.J
(310711.
I 5124 II.
12101 II.
SOSR.
562 R.
501 R.
9230 W.
7213 W.
5156 W.
10914 R.
1500 W.
6000 W.
9583 G.
Modulus
of Rup-
ture S
in lbs.
10032 B.
10596 B.
9792 B
8046 B.
7829 Br.
1.39S7Br.
14772 B.
340 W.
ISO w.
700 W.
1700 W
1062
2664
12800
9600
1400 W.
600 W.
13000 W.
800 W.
6630 R.
3142 R.
126 TABLE IX. MAGNETIC VARIATION.
TABLE IX.
MAGNETIC VARIATION.
The following table has been made up from varioi^ sources, prin-
cipally, however, from the results of the United States Coast Survey,
kindly furnished in manuscript by the Superintendent, Prof A. D
Bache. '• These results," he remarks in an accompanying note, " are
from preliminary computations, and may be somewhat changed by the
final ones." Among the other sources may be mentioned the Smith-
sonian Contributions for 1852, Trans. Am. Phil. Soc. for 1846, Lond.
Phil. Trans, for 1849, Silliman's Journal for 1838, 1840, 1846, and
1852, and the various American, British, and Russian Government
Observations. The latitudes and longitudes here given are not always •
to be relied on as minutely correct. Many of them, for places in the
Western States, were confessedly taken from maps and other uncer-
tain sources. Those of the Coast Survey Stations, however, as well
as those of American and foreign Government Observatories and Sta-
tions, are presumed to be accurate.
It will be seen that the variation of the magnetic needle in the
United States is in some places west and in others east. Tlie line of no
variation begins in the northwest part of Lake Huron, and runs through
the middle of Lake Erie, the southwest corner of Pennsylvania, the
central parts of Virginia, and through North Carolina to the coast.
All places on the east of this line have the variation of the needle
west, — all places on the west of this line have the variation of the
needle east ; and. as a general rule, the farther a place lies from this
line, the greater is the variation. The position of the line of no varia-
tion given above is the position assigned to it by Professor Loomis for
the year 1840. But this line has for many years been moving slowly
westward, and this motion still continues. Hence places whose varia-
tion is west are every year farther and farther from this line, so th&t
the variation west is constantly increasing. On the contrary, places
whose variation is east are every year nearer and nearer to this line,
so that the variation east is constantly decreasing. The rate of this
increase or decrease, as the case may be, is said to average ab:3Ut 2' for
the Southern States, 4' for the Middle and Western States, and 6' for
the New England States.* The increase in "Washington in 1840-2
was 3' 44.2"; in Toronto in 1841-2 it was 4' 46 2". The changes in
• Prof Loomis in Silliman's Journal. Vol. XXXIX.. 1R40.
TABLK IX. MAGNETIC VARIATION.
127
Cambridge, 1708,
1742,
1757,
1761,
1763,
1780,
1782,
1783,
u
{(
((
6 22
7 30
8 51
9 18
Cambridge, Mass. maybe seen from tbe following determinations of the
variation, taken from the Memoirs of the American Academy for 1846.
9 0 Cambridge, 1788, 6 38
8 0 Boston, 1793, 6 30
7 20 Salem, 1805, 5 57
7 14 " 1808, 5 20
7 0- " 1810,
7 2 Cambridge, 1810,
6 46 " 1835,
6 52 '' 1840,
But besides this change in the variation, which may be called secu-
lar, there is an annual and a diurnal change, and very frequently there
are irrc^-ular chanires of considerable amount. With respect to the
annual change, the variation west in the Northern hemi.>pbere is gen-
erally found to be somewhat greater, and the variation east somewhat
less, in the summer than in the winter months. The amount of this
change is different in different places, but it is ordinarily too small to
be of any practical importance. The diurnal change is well deter-
mined. At Washington in 1840-2, the mean diurnal change in the
variation was,* —
Summer, 10 4.1
Autumn, 6 21.2 Winter, 5 9.1 Spring, 8 10.7
At Toronto the means were, t —
t
1841.
6.67
9.46
12.38
1843.
1845.
1847.
1849.
1850.
1851.
Winter,
Spring and Autumn,
Summer,
5.64
9.36
11 70
5.73
9.15
13.36
7.28
10.08
13.84
8.25
12.25
14.80
8.01
10.90
13.74
7.01
10.82
12.61
The diurnal change in the variation is such that the north end of the
needle in the Northern hemisphere attains its extreme westerly posi-
tion about 2 o'clock, P. M., and its extreme easterly position about
8 o'clock, A. M. In places, therefore, whose variation is west, the
maximum variation occurs about 2 P. M., while in places whose vari-
ation is east, the maximum variation occurs about 8 A. M. In Wash-
ington, according to the report of Lieutenant Gilliss, the maximum va-
riation, taking the mean of two years' observations, occurs at l*^- 33'""
P. M., the minimum at s'^- 6"- A. M.
The determinations of the Coast Survey are distinguished by the
letters C. S. attached to the name of the observer. In some instances
the name of the nearest town has been added to the name of the Coast
Survey station.
* Lieut. Gilliss's Report, Senate Document 172, 1845
'■ London Philosophical Transactions. 1852
1-^8
TABLE IX.
MAGxXETlC VARIATIUJN.
Place.
Maine.
Agameuticus,
Bethel,
Bowdoin Hill, Port-
land,
Cape Neddick,York
Cape Small,
Kennebunkport,
Kittery Point,
Mt. Pleasant,
Portland,
Richmond Island,
Neiv Hampshire.
Fabyan's Hotel,
Hanover,
Isle of Shoals,
Patuccawa,
Unkouoouuc,
Vermont.
Burlington,
Ma.'i.'^ac/uisetts.
Annis-squani,
Baker's Island,
Blue Hill, Milton,
Cambridge.
Chapp.'iquidick.Ed-
gartown,
Coddonsimi,Mar-
blehead,
Copecut Hill,
Dorchester,
Fort Lee, .Salem,
Ilyaunis.
Indian Hill,
Little Xahant,
Nantasket,
Nantucket,
New Bedford,
ShootHying Hill,
Barnstable,
Tarpaulin Cove,
Rhode Island.
Beacon-pole Hill,
McSparran Hill,
Point Judith,
Spencer Hill,
Connecticut.
Black Rock, Fair-
field,
Bridgeporc,
Fort Wooster,
Groton Point, New
London,
Lati-
tude.
Longi-
tude.
Authority.
Date
A '
o *
43 13.4
70 41.2
T. J. Lee, C. S.
Sept., 1817
44 2S.0
70 51.U
J. Locke,
•lune, 1S45
43 33.8
70 16.2
J. E. Ililgard. C S.
Aug., 1S51
43 11.6
70 36.1
J. E. Hilgard; C. S.
Aug., 1851
43 46.7
69 50.4
G. \V. Dean, C. S.
Oct., 1851
43 21.4
70 27.S
J. E. Ililgard, C. S.
Aug., 1851
43 4.S
70 43.3
J. E. Ililgard, G. S.
Sept., 1850
44 1.6
70 49.0
G. W. Dean, 0. S.
Aug.. 1 851
43 41.0
70 20.5
J. Locke,
June, 1 "45
43 32.4
70 14.0
J. E. Hilgard, C. S
Sept., 1S50
44 16.0
71 29.0
J. Locke,
June, 1>45
43 42.0
72 10.0
Prof Young,
1 S3 J
42 59.2
70 36.5
T. J. Lee. C. S.
Aug , 1847
43 7.2
71 11.5
G. W. Dean, C. S.
Aug., 1849
42 59.0
71 35.0
J. S. Ruth, C. S.
Oct , 1848
44 27.0
73 10.0
J. Locke,
June, 1845
42 .39.4
70 40.3
G. W. Keely, C. S.
Aug., 1849
42 32.2
70 46.8
G. W. Keely, C. S.
Sept., 1S49
42 12.7
71 6.5
T. J. Lee, 0. S. j
Sept. and )
Oct., 1845 J
42 22.9
71 7.2
W. C. Bond,
1352
41 22.7
70 23.7
T. J. Lee, C. S.
July, 1816
42 31.0
70 .50.9
G. W. Keely, C. S.
Sept, 1^49
41 43.3
71 3.3
T. J. Lee, C. S. {
Sept and I
Oct , 1844 1
42 19.0
71 4.0
W. C. Bond,
1839
42 31.9
70 52.1
G. W. Keely, C. S.
Aug., 1849
41 3S.0
70 IS.O
T. J Lee, C. S.
Aug., 1S46
41 25.7
70 40.3
T. J. Lee, C. S.
Aug., 1S46
42 26.2
70 55.5
G. AV. Keelv, C. S.
Aug., 1-^49
42 18.2
70 54.0
T J. Lee, C. S.
Sept., 1847
41 17.0
70 6.0
T J. Lee, C. S.
July, 1346
41 33.0
70 54.0
T. J. Lee, C. S.
Oct., 1845
41 41.1
70 20.5
T. J. Lee. C. S.
Aug., 1 846
4i 23.1
70 45.1
T. J. Lee, C. S.
Aug., 1846
41 59.7
71 26.7
T. J Lee, C. S. {
Oct. and )
Nov., 1844}
41 29.7
71 27.1
T. J. Lee, C. S.
July, 1844
41 21.9
7) 28.9
R.H.Fauntleroy,C.S.
Sept , 1847
41 40.7
71 29.3
T. J. Lee, C. S {
July and )
Aug. 1844 j
41 S.6
73 12.6
J. Renwick, C. S.
Sept., 1845
4i 10.0
73 11.0
J. Renwick, C. S.
Sept., 1845
41 16.9
72 53.2
J. S. Ruth, C. S.
Aug., 1843
41 18.0
72 0.0
J. Renwick, C. S.
Aug., 1845
Variation.
o
10
11
II
11
12
11
10
14
11
12
iO.OW.
50.0 "
41.1
9.0
5.5
23.6
30.2
32.0
28.3
17.9
11 32.0 W.
9 15.0 "
10 .3.4 "
10 42.9 "
9 5.6 "
9 22.0 W.
11 36.7 W.
12 17.0 "
9 13.8 «
10 8.0 "
8 47.7 »
49.8
12.1
2.0
14.5
22.0
49.3
40.9
9 33.5
9 14.0
8 54.6 "
9 40.1
9 10.1
9 29.8 W.
8 53.3 "
8 59.4 "
9 11.9 »
6 53.5 W.
6 19.3 "
7 26.4 "
7 29.5 "
TABLE IX.
MAGNETIC VARIATION.
129
Place.
Lati-w^
tude.
Longi-
tude.
Authority.
Date.
4
Variation
O (
O (
o
1
Milfovd,
41 IG.O
73 1.0
J. Renwick, C S.
Sept , 1S45
6
.3'-.3 W
New llaveu, Pavil-
ion,
41 18.5
72 55.4
J. S. Ruth, C. S
Aug., 1848
6
37.5 "
New Haven, Yale
College,
41 1S.5
72 55.4
J. Renwick, C. S.
Sept., 1845
6
17.3 "
Nojwalk,
41 71
73 24.2
J. Renwick, C. S.
Sept., 1S44
6
46.3 "
1 Oyster Point, New
i Haven,
41 17.0
72 55.4
J. S. Ruth, 0. S.
Aug., 1843
6
32.3 "
'■jachenrs Head,
Guilford,
41 17.0
72 43.0
J. Renwick, C. S.
Aug., 1S45
6
15.2 "
Sawpits,
40 59 5
73 o9.4
J. Renwick, C. S.
Sept., 1344
6
1.6 "
Say brook.
41 16.0
72 20.0
.1. Renwick, C. S.
Aug., 1845
6
49.9 "
Stamford,
41 3.5
73 32 0
J. Renwick, C. S.
Sept., 1844
8
40.4 "
Stouiugton,
41 20.0
71 54.0
J. Renwick. C. S.
Aug., 1845
7
3^.2 "
Netv York.
Albany,
42 39.0
73 44.0
Regents' Report,
1836
6
47.0 W.
lllooiuingdale Asy-
hnii,
40 43.8
73 57,4
J. Locke, C. S.
April, 1846
5
10 9 "
Cole, Staten Island,
10 31.8
74 13.^
J. Locke, 0. S.
April, IS46
5
33.8 "
! Drowned Meadow.
i L. I.,
40 .56.1
73 3.5
J. Renwick, C. S.
Sept., 1845
6
3.6 "
Flatbush, L. L,
40 40 2
73 57.7
J. Locke, C. S.
April, 1 846
5
54.6 "
Greenport, L. 1.,
41 6.0
72 21.0
J. Jlenwick, C. S.
Aug., IS45
7
14.6 "
Leggett,
40 4^9
73 53 0
R.H. Fauntleroy,C.S.
Oct., 1847
5
40.6 "
Lloyd's Harbor,
L. I.,
40 55.6
73 24. S
J. Renwick, C. S
Sept., 1844
6
12.5 "
New lloehelle,
40 52.5
73 47.0
J. Renwick, C. S.
Sept., 1844
5
31.5 "
New York,
40 42.7
74 0 !
J. Renwick; C. S.
Sept., 1845
6
25. n »
Oyster Bay, L. I.,
40 52.3
73 31 3
J. Renwick, C. S.
Sept., 1344
6
53 G "
L'ou.^e's Point,
Sand.s Lighthouse,
45 0.(1
73 21.0
Boundary Survey,
Oct., 1845
11
2S.0 "
i
L. I.,
40 51.9
73 4.3.5
R.H. F:uintlcrov,C.S.
Oct., 1847
6
9.7 " 1
Sands Point, L. I.,
40 .52.0
73 43.0
J. Renwick, C. "S.
Sept., 1845
7
14.6 " i
\^'atchhill. Fire Isl-
li
and,
40 41.4
72 53 9
R.H. Fauntlcroy,C.S.
Oct., 1847
7
33 5 <^ ii
West Point.
41 25.(1
73 56 0
Prof. Davies,
Sept., 1835
6 32.0 "
Neiv Tcrstij.
Oape 5Iay Light-
' house.
38 55 8
74 57.6
J. Locke, C. S.
June, 1346
3
3.2 AY.
('Iiew,
39 43.2
75 9 7
J. Locke, 0. S.
July, 1316
3
20.4 "
Oiiurch Landing,
39 40 9
75 30.3
J. Locke, 0. S.
June, 1346
*5
45.8 «
Egg Island,
39 10.4
75 7.8
J. Locke, 0. S.
June, 1346
3
13.2 "
Hawkins,
.39 25.5
75 17.1
J. Locke, C. S.
June, 1346
2
.53.7 "
Mt.Piosc, Princeton,
40 22.2
74 42.9
J. E. Hilgard, C. S.
Aug., 1852
5
31.8 »
Newark,
40 44. '^
74 7.1)
■T. Locke, C. S-
April, 1346
5
32.7 "
Pine Mountain,
39 25.0
75 19 9
J. Locke, C. S.
June, 1346
2
52.0 »
Port Norris.
39 14.5
75 1.0
J. Locke. (". S.
June, 1346
.3
6.5 «
Sandy llDok,
40 28.0
73 59. S
J. Renwick, C. S
Aug., 1344
5
54 0 "
Town Bank, Cape
May,
39 .58.6
74 57.4
.7. Locke. C. S.
June, 1846
3
3 2 "
Tucker's Island,
39 30. S
74 16.9
T. J. Lee, 0. S.
Nov., 1846
4
23.8 "
White Hill, Bor-
■* '
dentown,
40 8.3
74 43 8
J. Locke, C S.
April, 1846
4 22.5 "
Pennsylvania.
Girard College,
Philadelphia,
39 58.4
75 9.9
J. Locke, C. S.
May, 1346
3
50.7 W.
Pittsburg,
40 26.0
79 .53.0
J. Locke,
May, ls45
0
33.1 "
Vauuxeni, Bristol, |40 5.9
74 52.7
J. Locke, C. S.
July, 1346
4
20.5 " 1
* Loeal ittrictinn exi.5t.=? here, according to Prof. Locke.
7
130
TABLE IX. MAGNETIC VAEIATION.
Place.
Lati-
tude.
Longi-
tude.
Authority. "^
Date.
Variation.
Delaivare.
Bombay Hook
o /
O 1
o
Lighthouse,
39 21.8
75 30.3
J. Locke, C. S
June, 1846
3 17.9 W
Fort^Delaware, Del-
aware River,
39 35.3
75 33.8
J. Locke, C. S.
June, 1846
3 16.0 "
Lewes Lauding,
3S 48.8
75 11.5
J. Locke, C. S.
July, 1846
2 47.7 "
Pilot Town,
.33 47.1
75 9.2
J. Locke, C. S.
July, 1346
2 42.2 »
Sawyer,
.39 42.0
75 .3.3.5
J. Locke, C. S.
June, 1346
2 47.8 "
Wilmingtv.n,
.39 44.9
75 33.6
J. Locke, C. S.
May, 1S46
2 31.8 «
Manjlnnd.
Annapolis,
33 56.0
76 35.0
T. J. Lee, C. S.
June, 1845
2 14.0 W.
Bodkiu,
39 8.0
76 25.2
T. J. Lee, C. S.
April, 1817
2 2.6 «
Finlay,
39 24.4
76 31.2
J. Locke, C. S.
AprU, 1846
2 19.5 "
Fort McIIenry,
Baltimore,
39 1.5.7
76 .34.5
T. J Lee, C. S.
April, 1347
2 13.0 »
Hill,
35 53.9
76 52.5
G. W. Deau, C. S.
Sept., 1850
2 1.5.4 "
Kent Island,
39 1.8
76 18.8
J. Ileustou. C. S.
July, 1349
2 30.5 "
Marriott's,
33 52.4
76 36.3
T J Lee, C. S. -
June, 1549
2 5.2 "
North Point,
•39 11.7
76 26.3
T J. Lee. C. S.
July, 1846
1 42.1 "
Osborne's Ruin,
39 27.9
76 16.6
T J. Lee, C. S.
June, 1845
2 32.4 «
Poole's Island,
39 17.1
76 15.5
T J. Lee, C. S.
June,- 1847
2 23.5 «
llosaune.
39 17.5
76 42.8
T. J. Lee. C. S.
June, 1815
2 12,0 "
Soper,
39 5.1
76 56.7
G. W. Deau, C. S.
July, 1350
2 7.0 "
South Base, Kent
Islaud,
33 53.S
76 21.7
T. J. Lee, C. S.
June, 1845
2 26.2 "
SusquehannaLight-
house, Havre de
Grace,
39 32.4
76 4.8
T J. Lee, C. S.
July, 1817
2 51.1 «
Tavlor,
33 59. S
76 27.6
T J. Lee, C. S.
May, 1347
2 18.4 "
Webb,
39 5.4
76 40.2
G W. Dean, C. S.
Nov., 1350
2 7.9 '
District of Colmn-
bia.
Oausten, George-
town,
33 5.5.5
77 4.1
G. W. Dean, C. S.
June, 1351
2 11.3 W.
Washington,
33 53.7
77 2.8
J. M. Gilliss,
June, 1342
1 26.0 «
Virginia.
Charlottesville,
33 2.0
73 31.0
Prof. Patterson,
1835
0 0.0
Roslyn, Peters-
burg,
37 14.4
77 23.5
Q. "W. Dean, C. S.
Aug., 1852
0 26.4 w^
Wheeling,
40 8.0
80 47.0
J. Locke,
April, 1345
2 4.0 E.
North Carolina.
Bodie's Island,
35 47.5
75 31.6
C. 0. Boutelle, C. S.
Dec., 1846
1 1.3.4 W.
Shellbank,
3. .3.3
75 44.1
C. 0. Boutelle, C. S.
Mar., 1847
1 44.8 "
Stevenson's Point,
.36 6.3
76 11.0
C 0. Boutelle, G. S.
Feb., 1847
1 39.7 "
South Carolina.
Breach Inlet,
.32 46.3
79 48.7
C. 0. Boutelle. C. S.
April, 1849
2 16.5 E.
Charleston,
32 41.0
79 53.0
Capt. Bamett'
May, 1341
2 24.0 "
Ri.st Base, Edisto,
.32 33.3
80 10.0
G. Davidson, C. S.
April, 1350
2 53.6 "
Georgia.
Atliens,
.34 0.0
33 20.0
Prof. McCay,
18.37
4 31.0 E.
Cohuubus,
.32 2S.0
85 10.0
Geol. Survey,
1839
5 30.0 "
Milledgeville,
.33 7.0
83 20.0
Geol. Survey,
1833
5 51.0 "
Savannah,
1
32 5.0
31 5.2
J. E. IlJlgard, ?■. S.
April, 1852
3 4.5.0 "
TABLE IX. MA
GNETIC VARIAT
ION.
m\
r
Place.
Lati-
tude.
Longi-
tude.
Authority.
Date.
Variation.
Florida.
O 1
4 25.2 E.
5 20.5 "
5 29.2 «
5 29.0 »
Cape Florida,
Cedar Keys,
St. Marks Light,
Saud Key,
o /
25 39.9
29 7.5
iO 4.5
21 27.2
SO 9.4
S3 2.8
84 12.5
81 52.0
J. E. Ililgard, C S.
J. E. Hilgard, C. S.
J. E. Hilgard, C. S.
J. E. Hilgard, C. S.
Feb., 1850
Mar., 1852
April, 1852
Aug., 1849
Alabama.
Fort IMorgan, Mo-
bile Bay,
Tuscaloosa,
30 13.8
33 12.0
SS 0.4
87 42.0
R.H.Fauntleroy,C.S.
Prof. Barnard,
May, ]3!7
1839
7 3.8 E.
7 28.0 "
Mississippi.
East Pascagoula,
30 20.7
88 31.4
R.II. Fauntleroy,C.S.
June, 1847
7 12.4 E.
Texas.
j
Dollar roint, Gal-
veston,
Mouth of Sabme,
29 2G.0
29 43.9
94 53.0
93 5L5
R.II. Fauntleroy,C.S.
J. D. Graham,
April, 1848
Feb., 1840
8 57.2 E.
8 40.2 "
Ohio.
Carrolton.
Cincinnati,
Columbus,
Hudson,
Mai-ietta,
Oxford,
St. Mary's,
39 33.0
39 6.0
39 57.0
41 15.0
.39 26.0
.39 .30.0
40 32.0
84 9.0
84 22.0
83 3.0
81 26.0
81 29.0
84 33.0
si ly.c
J Locke,
J. Locke,
J. Locke,
E. Loomis,
J. Locke,
J. Locke,
J. Locke,
Sept., 1845
April, 1845
July, 1845
1S4M
April, 184.5
Aug., 1845
Sept., 1345
4 45.4 E.
4 4.0 "
2 29.3 "
0 52.0 "
2 25.0 "
4 50.0 "
3 4.0 "
Tennessee.
\
Nashville,
36 10.0
86 49.(
Prof. Hamilton,
1835
7 7.0 B.
Michigan.
,
Detroit,
42 24.0
82 58.0
Geol. Report,
1840
2 0.0 E.
Indiana.
Richmond,
South Hanover,
39 49.0
33 45.0
&4 47.0
85 23.0
J Locke,
Prof. Dunn,
Sept., 1845
1837
4 52.0 E
4 35.0 "
1
Illinois.
Alton,
38 52.0
90 12.0
H. Loomis,
1840
7 45.0 E.
Missouri.
.
St. Louis,
33 36.0
89 36.0
Col. NicoUs,
1835
8 49.0 E.
Wisconsin.
^
Madison,
Prairie du Chien,
43 5.0
43 1.0
89 41.0
91 8.0
U. S. Surveyors,
U. S. Surveyors,
Nov., 1839
Oct., 1839
7 30.0 E.
9 5.0 "
loioa.
Brown's Settlement
Davenport,
Farmer's Creek,
42 2.f
41 30.C
42 13.C
91 J8.0
90 34.0
1 90 39. C
J. Locke,
U. S. Surveyors,
J. Locke,
Sept., 1839
Sept., 1839
Oct., 1839
9 4.0 E.
7 50.0 "
9 11.0 "
1
Wapsipinnicon
River,
41 44.C
1 90 39.C
J. Locke,
Sept., 1839
8 25.0 «
Cnlifornia.
Point Conception,
34 26.C
I 120 26.f
) G. Davidson, C. S.
Sept., 1850
113 49.5 E.
b.
.
15!^
TABLE
IX. MAGNETIC VARIATION.
Place.
Lati-
tude.
Longi-
tude.
Authority.
Date.
Variation.
Point Pinos,
O 1
o /
o
/
Monterey,
36 33.0
121 54.0
G. Davidson, C. S.
Feb., 1351
14
53.0 E.
PresiLlio, San
Francisco,
37 47.8
122 27.0
G. Davidson. C. S.
Feb., IS52
15
26.9 "
San Diego,
32 42.0
117 14.0
G. Davidson, C. S.
May, 1351
12 29.0 «
Oregon.
Cape Disap-
pointment,
46 16.6
124 2.0
0. Davidson, G. S.
July, 1351
20
45.0 E.
Ewing Harbor,
42 44.4
124 21.0
G. Davidson, C S.
Nov., 1351
13
29.2 «'
Washington
Territory.
Scarboro' Har-
bor,
I
43 21.3
124 37.2
G. Davidson, C.S.
Aug., 1852
21
.30.2 E.
BRiTisa Amer-
ica.
Montreal,
45 30.0
73 35.0
Capt. Lefroy,
1342
8
.53.0 W.
Quebec,
46 49.0
71 16.0
Capt. Lefroy,
1342
14
12.0 "
St. Johns, C. E.
45 19.0
73 13.0
Capt. Lefroy,
1842
11
22.0 "
StansteaJ,
45 0.0
72 1.3. Q
Boundary Survey,
Nov., 1345
11
33.0 *'
Toronto,
43 39.6
79 21.5
British Govern.,
Sept., 1344
1
27.2 "
New Grenada
Panama,
8 57.2
79 29.4
\V H. Emory,
Mar., 1349
6
54.6 S.
Eastern Hemi-
sphere.
Green\vich,Eng-
land.
51 23.0
0 0.0
Prof. Airy,
1841
23
16.0 W.
Makei-stoun,
Scotland,
55 35.0
2 31.0 \Y.
J. A. Broun,
1342
25
2=!.0 «
Paris, France,
43 50.0
2 20.0 E. 1
Paris Observatory
Nov., 1851
20
25.0 «
Munich, Bara-
ria,
43 9.0
11 .37.0 "
1842
16
43.0 "
St. Peter.^burg,
1
1
Russia.
59 56.0
30 19.0 «
Russian Govern.,
1842
6 21.1 " II
Catherineuburg
Siberia.
■56 51.0
60 ai.O " ;
Russian Govern.,
1842
6
33.9 B
Xertchiusk, Si-
beria.
51 56.0
116 31.0 "
Russian Govern.,
1342
3
46.9 W.
St. Helena,
15 .56.7 S.
5 40.5 W.I
British Govern.,
Dec., 1845
23
36.6 "
Cape of Good
Hope,
33 56.0 '■'
18 23.7 E.
British Govern ,
.July, 1346
29
8.0 «
Hobarton, Van
1
Diemen-s Ld.,
42 .52.5 •
147 27.5 " :
British Govern.,
Dec., 1343
10
8.01.
TABLE X. TRIGONOMETRICAL FORMULA.
133
TABLE X.
TRIGONOMETRICAL AND MISCELLANEOUS FORMULA
Let a (fig. 57) be any acute angle, and let a perpendicular B Che
irawn from any point in one side to the other side. Tlien, if the sidea
Fig. 57.
>f the right triangle thus formed are denoted by letters, as in the fig
arc, we shall have these six formula : —
1. sin, A =
2. COS. A = - .
3. tan. A =
4.
cosec.
-^-l
5.
sec.
■^-l
6.
cot.
a
Given-
a. c
a, b
J., a
A,b
10
11 :.4. c
Solution ofRi'jht Triangles (fig. 57).
Sought.
A,B,l
A, B, c
B,b,c
B, a, c
B,a,b
Formulae.
a
sin.^=-, cos. C = -, b=-^ic-\-a){c — a)
c c
tan. A = :^ , cot. B = -^ , c = ya* -f b*.
B =.90° — A, b = a cot. A, c =^
B = 90o — A. a = 6tan. -1. c =
sin. A
b
COS. A '
B=--90° — A. a = csin..4, i = c cos. .4
134
TABLE X. TRIGONOMETRICAL AND
Solution of Oblique Triangles (fig. 58).
Fig. 58
12
13
14
15
16
17
18
Given.
A, B, a
A, a, b
a,b, C
a, 6, c
A,B,C,a
A, b, c
a, b, c
Sought. I
b
B
b =
sin. B
a sin. B
sin. A '
b sin. A
a
Formulae.
A — Btan.^ {A — B)
ja — b) can, i (A + B)
•Ifs=i(a + 6 + c), sm. ^A=^l^^'l^.
cos4^= J^\ tan.i^= J(f^i^>,
•^ y/ be ^ > 5(5 — O)
. sin. A =
2 .^A* (5 — a){s — b) (s — c)
be
a- sin. B sit C
area
area
area
area =
2 sin. A
area = hbc sin. A.
s=i (a 4- 6 + r,, area=ys {s—a) {s—b) {«- «).
General Trigoriometri<v*' Fomnilce.
19!
20
21
22
23
24
25
26
27
28
29
•30
!3I
sin.'
J. + cos.'^ yl = 1.
.= 9
sin. (J. ± B) = .sin. ^1 cos. 75 ± sir h cos. J..
COS. {A da B) = COS. ^ COS. B rp sin. .^ "in. B.
sin. 2 A = 2 sin. ^ cos. ^.
COS. 2 J. = C0S.2 A — sin,2 J^ == i _ o siii i
sin." A = h — k ^os- 2 ^•
COS.- ^ ^ i + ^ COS. 2 ^.
sin. J. + sin. B = 2 sin. ^ (^ + S) cos. ^ (^ ZJ).
sin. .-1 — sin. B = 2 cos. ^ (^ + 5) sin.^ [A B).
COS. ^ + cos. B =-2 cos. |(A + JB) cos. ^ (J. • fJ).
cos. B — cos. ^ = 2 sin. ^ [A -\- B) sin. ^ (^ — P)
sin .2 A — sin.2jB= cos.= J5 -cos.^^ = sin. (^ + ^) sir i
COS.- /I — sin.'-' B = COS. (A + S) cos. (4 — B).
COS.* ^ — I-
MISCELLANEOUS FORMULJE.
195
, I Bin. A
132 tan. A = ^^;—^
COS. A
sin. A
33
cot. A
tan 4 ±jan B
tan ^ ±jan /J
34 tan. (^ ± i) = 1 q: t^. J^tan. B
I sin (A ±B)
35|tan. A ± tan. B = ^^^ ^ cos. B '
36 cot. A ± cot. /i
jsin ^ + sin B
38
sin. A — sin. B
sin A 4- sin- >S
COS. A -\- COS. .B
sinj^±^)
-^ sin j4 tin. B
tan ^{A^- B)
tan. i (4 — B) ■
tan. H^ + ^)
sin A -\- sin J5 ^ \ i a n\
391 _,-* T = cot. ^ (A — -U).
icos B — COS. A - ^
sin. -4. — sin. B ^ „ 1 / /i R\
40 rn u = tan. f ( A — /j •
;co.-^ .4 + cos B -^ ^
'sin. A — sin. B
I cos B — COS. A
sin A
42 tan. ^ A = 1 + cos. i
cot. ^(^ + 1^1-
43
cot. h A = ^
sin. .4
— cos A
Miscellaneous Formidai.
Sought. 1
Given.
Formom.
Area of
44
Circle
Radius = r
71 r^.
45
Ellipse
Semi-axes == a and b
nab.
46
Parabola
Chord = c, height = h
%ch*
47
Regular Polygon
Surface of
( Side = a, number of )
1 sides = « )
180°
\ or n cot. ^ •
48
Sphere
Radius = r
4 n r"'.
,'9
Zone
Radius = r, height =^ h
2 71 r h.
M^adiusof sphere=r )
S— (Ji - 2)180'-
50
Spherical Polygon
Solidity of
) sum of angles = ^^ (
( number of sides = n)
■;i/''X 180 D
.51
Prism or Cylinder
Base = b, height = k
bk.
52
Pyramid or Cone
Base = b, height = h
^bh.
53
Frustum of Pyr- )
amid or Cone )
( Bases = b and ftj , )
1 height = h )
kh{b-{-b, + ybb,)
* The area of a circular segment on railroa^l curves, where the chord is very long
m proportion to the height, may be found with great accuracy by the above formula
f36
TA.BLE X. JIISCELLANEOUS FOUMULiE.
54
55
Sough.,
Solidity of
Sphere
Given.
Radius
c; 1 • le J i TJi^dii of bases = r )
'■ "I and /-, , height = fi )
-^ T5 1 ^ o 1 -1 f Semi-transverse axis "
o6 Prohite Spiicroul ,. ,,.
' ■ J or ellipse = a
I Semi conjugate ax
Formulae.
4 -i
3 T r\
58
Oblate Spheroid
Paraboloid
ixis
[ of ellipse
j Kadi us of base = ?•, I
1 heiixht ^ /i (
(
3 71 a^ b.
* ;r r^ h.
TT. = .3.U159 265.35 89793 23846 26433 83280.
Log. 71 = 0.49714 98726 94133 85435 12682 88291
United States Standard Gallon = 231 cnb. in. = 0.133681 cub. ft
" " " Bushel = 21.50.42 "
British Imperial Gallon = 277.27384 "
According to Ilassler.
French Metre, = 3.2817431 ft.,
Litre, = 61.0741569 cub. in.,
Kilogram, = 2.204737 lb. avoir..
Weight of Cubic Foot of Water,
Barora. 30 inches. Therm. Falir. 39.83°,
(C
= 1.244456 "
= 0.160459 "
As usually given.
= 3.280899 ft.
= 61.02705 cub. in.
= 2.204597 lb. avoir
= 62.379 lb. avoir.
= 62.321 "
Length of Seconds Pendulum at Xcw York = 39.10120 inches.
'' " " " " London = 39.13908 "
" Paris = 39.12843 "
Equatorial Radius of Earth according to Bessel = 20,923.597.017 feet
Polar " •' « '■ = 20,853,654.177 ^
TABLE XI.
SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS,
AND
RECIPROCALS OF NUMBERS
T&OM 1 TO 1054.
138 TABLE XI. SqUAKES, CUBES, SQUARE ROOTS,
No.
Squares.
Cubes.
Square Roots.
Cube Roots.
Reciprocals.
1
1
1
1.0000000
1.0000000
1.000000000
2
4
8
1.41421.36
1.2599210
.500000000
3
9
27
1.73205:J8
1.44224<.6
.3333.333.33
4
16
64
2.0000000
1.. 53740 11
.250000000
5
25
125
2.2360680
1.7099759
.200000000
6
36
216
2.4494897
1.3171206
.166666667
7
49
343
2.6457513
1.9129312
.142857148
8
64
512
2.82S4271
2.0000000
.12.5000000
9
81
729
3.0000000
2.0800337
.111111111
10
100
1000
3 1622777
2.1544347
.100000000
11
121
1331
3.3166243
2.2239801
.090909091
12
144
1723
3.4641016
2.2894286
.033.333333
13
169
2197
3.605.5513
2.3513347
.076923077
14
196
2744
3.7416574
2.4101422
.071428571
15
225
3375
3.8729833
2.4662121
.066666667
16
256
4096
4.0000003
2.5198421
.062500000
17
239
4913
4.1231056
2..57128I6
.053823529
13
324
5332
4.2426107
2.6207414
.0555.55556
19
361
6859
4.3588989
2.6634016
.052631579
20
400
8000
4.4721360
2.7144177
.050000000
21
441
9261
4.5325757
2.7589243
.047619048
22
434
10643
4.6904153
2.3020393
.045454545
23
529
1216/-
4.7953315
2.8433670
.04347326
24
576
13324
4.3939795
2.8344091
.041666667
25
625
15625
5.0000000
2.9240177
.010000000
26
676
17576
5.0990195
2.9624960
.033461533
27
729
19633
5.1961524
3.0000000
.037037037
23
784
21952
5.2915026
3.0365339
.035714236
29
841
5^4339
5.3851643
3.0723163
.034482759
30
900
27000
5.4772256
3.1072.325
.033333333
31
961
29791
5.5677644
3.1413806
.0.32253065
32
1024
32763
5.6563542
3.1743021
.031250000
33
1039
35937
5.7445626
3.2075313
.030303030
34
1156
39304
5.S309519
3.2396113
.029411765
35
1225
42875
5.9160793
3.2710663
.028571429
36
1296
46656
6.0000000
3.3019272
.027777778
37
1369
506.53
6.0527625
3.-33222 13
.027027027
33
1444
54372
6.1644140
3.3619754
.026315739
39
1521
59319
6.2449930
3.3912114
.025641026
40
1600
64000
6.3245553
3.4199519
.025000000
41
1631
63921
6.4031242
3.4432172
.024390244
42
1764
74033
6.4307407
3.4760266
.023809524
43
1349
79507
6.5574335
3.503.3931
.0232.55314
44
1936
85134
6.6332496
3.5303483
.022727273
45
2025
91125
6.7032039
3.5568933
.022222222
46
2116
97336
6.7323300
3.583)479
.021739130
47
2209
103823
6.3556.546
3.6038261
.021276600
43
2304
110592
6.9232032
3.6:342411
.020333333
49
^01
117649
7.0000000
3.6593057
.020403163
50
2500
125000
7.0710673
3.6340314
.020000000
51
2601
132651
7.141-4234
3.7084298
.019607843
52
2704
140603
7.2 LI 1026
3.7.325111
.019230769
53
2309
143377
7.2801099
3.7562858
.018367925
51
2916
157464
7.3484692
3.7797631
.013518519
55
3J25
166375
7.4)61935
3.3029525
.013131818
56
3136
175616
7.4833143
3.3258624
.017357143
57
3249
185193
7.5493344
3.8485011
.017543360
53
3364
195112
7.0157731
3.8703766
.017241379
59
3481
205379
7.6311457
3.8929965
.016949153
60
3600
216000
7.7459667
3.9143676
,016666667
61
3721
226931
7.8102497
3.9364972
.016393443
62
3344
233323
7.3740079
3.9573915
.016129032
CUBE ROOTS, AND HECirilOCALS.
139
lU
63
64
65
66
67
6S
6^
70
71
72
73
74
7o
76
77
73
79
30
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
lUl
102
103
104
105
106~
107
103
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
3969
4016
4225
4356
4439
4624
4761
4900
504 1
5184
5329
5476
5025
57 76
5929
6034
6241
6400
6561
6724
6339
7056
7225
7396
7569
7744
7921
8100
8231
8464
8619
8336
9025
9216
9409
96:)4
10000
10201
10404
10609
10316
11025
11236
11449
11664
1 1331
12100
12321
12544
12769
12996
13225
13456
13639
13924
14161
14400
14611
14334
15129
15376
250047
262144
274625
2o7496
300763
314432
323509
343000
357911
373248
339017
405224
421375
433976
456533
474552
493039
512000
531441
551368
57 1 737
592704
614125
636056
653503
631472
704969
729000
i OOOi I
773633
804357
830534
857375
834736
912673
911192
970299
1000000
1030301
1061203
1092727
H24.S64
1157625
1191016
1225043
1259712
1295029
1331000
1367631
1404928
1442897
1431 544
1520375
1560896
1601613
1613032
1635159
1723000
1771561
1815848
1860367
1906624
7.9372539
8.0000000
8.0622577
8.1240334
8.1853528
8.2462113
8.3066239
8.3666003
8.4261498
8.4352314
3.54 10U37
8.6023253
8.6602S40
8.7177979
8.7749644
8.831 7 609
8.8831944
8.9442719
9.0000000
9.05.53351
9.1104336
9.1651514
9.2195445
9.2733185
9.3273791
9.3303315
9.4339311
9.4363330
9.5393920
9..59 10630
9.6436508
9.6953597
9.7467943
9.7979590
9.8483578
9.8994949
9.949o744
lo.onooooo
10.0493756
10.0995049
10.1433916
10.1930390
10.2469508
10.2956301
i0.34i'h04
10.3923048
10.4403065
10.4330035
10.5356538
10.5330052
10.6301458
10.67707S3
10.7233053
1 0.7703296
10.8166538
10.8627805
10.9087121
10.9544512
11.0000000
11.0453610
11.0905365
11.1355237
3.9790571
4.0000000 ■
4.02()72.'6
4.0112401
4.0615130
4.0816551
4.1015661
4.121 23.'i3
4.14U3173
4,1601676
4.1793390
4.1933364
4.2171633
4.2358236
4.2543210
4.2726536
4.2903404
4.3083695
4.32674b7
4.3444815
4.3620707
4.3795191
4.3963296
4.4140049
4.4310476
4.4479602
4.4647451
4.4814047
4.4979414
4.5143574
4.5306549
4.54633.59
4.5629026
4.5738570
4.5947009
4.6104363
4.6260650
4.6415888
4.6570095
4.6723287
4.6375482
4.70^6694
4.7176940
4.7326235
4.7474594
4.7622032
4.7763562
4.7914199
4.895t955
4.3202345
4.8345331
4.848^076
4.8629442
4.8769990
4.8909732
4.9043631
4.9186347
4.9324242
4.9460374
4.9596757
4.9731898
4.9366310
.015873016
.015625000
,015384615
.015151515
.014925373
.014705332
.0144927.54
0142N5714
.0140^4507
.(0 38^3889
.01369.'-630
.013513514
.013333333
.013157895
.012987013
.012820513
.012653223
.012500000
.012345679
.012195122
.012048193
.011904762
.011764706
.011627907
.011494253
.011363636
.011235955
.011111111
.010939011
.010^69565
.010752638
.01063.>298
.010526316
.010416667
.010309278
.0102(14032
.010101010
.010000000
.009900990
.009i03922
.00970i-738
.001,61.53'?5
.009523810
.009433962
.009345794
.009259259
.009174312
.O090909'i9
.009009009
.00>:923571
.003349.5.58
.Olte771930
.003695652
.003620690
.003547009
.003474576
.008403361
.008333333
.003261463
.008196721
.008130081
.008061516
14U
TABLE XI. SQUARES, CCBES,
SQUARE ROOTS,
No.
Squares.
Cubes
Square Roots.
Cube Roots.
Reciprocals.
125
1.5625
1953125
11.1303399
5.0000000
.O030JO000
126
15S76
2tr))376
11.2249722
5 0132979
.007936503
127
16129
2)43333
11.2694277
5 0265257
.007874016
12S
163S4
2)97152
11.3137035
5.0396342
.007812;500
12J
16611
21463^9
11.3573167
5.0527743
.0077519.33
133
16900
21 970 JO
11.4017543
5.0657970
.007692303
131
17161
2243091
11.4155231
5.0737.531
0)7633533
132
17424
2299963
11.4391253
5.0916134
.01)757.5753
133
176S9
2352637
11. .532.5626
5.10446^7
.007513797
i;i4
179.56
2106104
11.5753369
5 1172299
.0)7462637 ;
13.5
18225
2460375
11.6139.500
5 1299273
.037407407
136
13496
251.54.56
11.6619033
5 1425632
.0073.52941
137
13769
2.571353
11.7046999
5.1-551367
.0[)7299270
13S
19044
2623072
11.7473444
5 167^193
.007246377
139
19321
263.5619
11.7393261
5.1301015
.007194245
140
19600
2744010
11.3.321596
5.1924941
.007142357 i
141
19331
23f)322[
1 1.3743121
5.2043279
.007092199 1
142
20161
2363233
11.916.3753
5 2171034
.0070422^54 ;
143
20449
2921207
11.9532607
5.2293215
.006993007
144
20736
2935934
12.0'JOOOOO
5.2414S23
.006944444
14o
21 025
.3013625
12.041.5946
5.2.535379
.006>96552
146
21316
3II2136'
12.0330460
5.2656374
.0OG349315
147
21609
3176523
12.1243557
5.2776.321
.00630272 i
U3
21904
32 U 792
12.16-55251
5.2395725
.00075675? '
149
22201
3307949
12.2065558
5..3014592
.00671 1409 1
150
22500
3375000
12 2474437
5.31.32923
.006666667
151
22301
3142951
12 23320.57
5.3250740
.0)6622517
1 152
23101
.3511303
12.3233230
5 3363033
.006573947 |
153
23109
3531577
12 3693169
5.3434312
.006535943
154
23716
36.52264
12.4096736
5..3601034
.006493506
155
24025
3723375
12.4493996
5.37163.54
.006451613
156
21.3.36
3796416
12.4509960
5.3332126
.006410256
157
21619
3369393
12..5293641
5.3946207
.006369427
15S
24961
3944312
12..5693051
5.4051202
.006329114
159
252S1
4019679
12.6095202
5.4175015
.006239303
160
25600
4096000
12.6491106
5.4233-3^52
.006250000
161
2.5921
4173231
12.6335775
.5.4401218
.006211130
162
26244
4251528
12.7279221
5.4513618
.006172-40
163
26.569
4330747
12.7671453
5.46255.56
.006134969
164
26596
•4410944
12.3062435
5.4737037
.006097561
165
27225
4492125
12.3452326
5.4343066
.006)60606
166
27556
4574296
12.3310937
5.49-53647
.006024096
167
27339
4657463
12.9223430
5.5063784
.01.5933024 i;
163
23221
4741632
12.9614314
5.5173434
.00-59.52331 j
169
23561
4326309
13.0000000
5..5237748
.035917160
170
23900
4913000
13.0334043
.5.5396533
.005332.353
171
29241
5000211
I.3.0r66963
5.5.504991
.00.53479-53
172
29534
5033443
13.1143770
5.5612973
.00.53139.53
173
29929
5177717
13.1529464
5.5720.546
.005730347
174
30276
5263024
13.1909060
5.5327702
.00.5747126
175
30625
5359375
13.2237566
5..5934447
.005714236
176
30976
5451776
13.2661992
5.6040737
.00.5631318
177
31329
.5545233
13.3011347
5.6146724
.005649713
178
316S4
5639752
13.3416641
5.6252263
.00.561797.^
179
32)41
573.5339
13.3790332
5.6357403
.005536592
130
32400
.5332000
13 4164079
5.6462162
.005555-556
181
32761
5929741
13.4536240
5.6566.523
.005524362
132
33124
6023563
13.4907376
5.6G705!]
005494505
133
.33439
6123437
13.5277493
5.6774114
.005161431
134
.333.56
6229.504
13.5646600
5.6377:340
.005434733
13.5
34225
6331625
13.6014705
5.6930192
.ftO54O.540E
186
34596
d434356
1.3.6331317
5.7032675
.005376344
CUBE KOOTS, AND RECII EOCALS.
141
T-
No.
1S7
158
ISO
190
191
192
193
194
195
196
197
198
199
210
211
212
213
214
215
216
217
213
219
220
221
222
22-3
224
225
226
227
223
229
240
241
242
243
244
245
246
247
243
Squares.
■'A'. 69
:i5:{l4
35721
36100
;-i(.4-i
36364
37249
37636
38025
38416
38S09
39204
39601
200
40000
201
40401
202
40304
203
41209
204
41616
205
42025
206
42436
207
42349
203
43264
209
43631
Cubes.
Square Roots.
41100
44521
44944
45369
45796
46225
46656
47039
47524
47961
48400
43841
49284
49729
50176
50625
51076
51529
5 1934
52441
230
52900
231
53361
232
53824
233
54289
234
54756
235
55225
236
55696
237
56169
238
56644
2.39
57121
57600
58081
5S5M
59049
59536
60025
60516
61009
61504
6539203
6644672
6751269
6859000
6967871
7077838
7139057
7301384
7414875
7529536
7645373
7762392
7330599
8000000
8120601
8242408
836542?
8489664
8615125
8741816
8369743
8993912
9129329
9261000
9393931
9523128
9663597
9S00;344
9933375
10077696
1021S313
10360232
10503459
10648000
10793361
10941048
11039567
11239424
11390625
11543176
11897083
11852352
12008989
12167000
1232S391
1 2437 1 68
12649337
12812904
12977875
13144256
13312053
13431272
13651919
13324000
13997521
14172438
14343907
14526784
14706125
14336936
15069223
15252992
Cube Roots.
13 6747943
13.7113092
13.7477271
1.3.7840138
13.8202750
13.8564065
13.8924440
13.9233383
13.9642400
14.0000000
14.03.56688
14.0712473
14.1067360
14.1421356
14.1774469
14.2126704
14.2473068
14.2828569
14.3173211
14.3527001
14.3374946
14.4222051
14.4568323
14.4913767
14..5258390
14.5602198
14.5945195
14.6237388
14.6623733
14.6969385
14.7309199
14.7643231
14.7986488
14.8323970
]4.866(i637
14.8996644
14.9331345
14.9666295
15.0000000
15.0332964
15.0665192
15.0996639
15.1327460
15.1657509
15.1936342
15.2315462
15.2643375
15.2970535
15.3297097
15,3622915
15,3943043
15,4272486
15.4596248
15.49193.34
15.5241747
1.5.5563492
15..58S4573
15.6204994
15.6524758
15.6343371
15,7162336
15.7480157
Reciprocalfi.
5.71S4791
5.7236543
5.7387936
5,7438971
5,7539652
5.7639932
5.7739966
5.7889604
5.7983900
5 8037857
5.8136479
5.8284767
5.3382725
5.8480355
5.8577660
5.3674643
5.8771307
5,83676.53
5.8963685
5 9059406
5.91.54817
5.9249921
5.9344721
5.9439220
5.953.3418
5.9627320
5.9720926
5.9314240
5.9907264
6.0000000
6.00924.50
6.0184617
6.0276502
6.0368107
6,0459435
6 0550489
6.0641270
6.0731779
6.0822020
6.0911994
6.1001702
6.1091147
6.1180332
6.12692.57
6.1357924
6.1446337
6.1534495
6.1622401
6.1710058
6.1797466
6,1884628
6.1971544
6.20.58218
6,2144050
6.2230843
6.2316797
6,2402515
6.2487998
6.2573248
6,2653266
6,2743054
6,2327613
.005347591
.00.5319149
005291 U05
.005263153
.005235602
.005208333
.005181347
.0051.54639
.005128205
.005102041
.005076142
.005050505
005025126
.005000000
.004975124
,004950495
.004926108
,004901961
.004378049
.004354369
.004830918
.004807692
.004784639
,004761905
,0047393:-6
.0047169S1
.004694336
,004672397
.004651163
,004629630
.004603295
,004587156
.004566210
,004545455
.004524387
,004504505
,004434305
.004464236
,004444444
,004424779
35S
;5£
.004366812
.001347826
.004329004
.004310345
.004291845
.004273504
.0042,55319
,0CI42372S3
,004219409
.004201681
.004184100
.004166667
.004149378
,004132231
.004115226
.004098361
.0040816.33
004065041
.004043583
.004032258
142
TABLE XI. SQUARES, CUBES,
SQUARE R«'ul&,
! -1
No.
Squares.
Cubes.
Square Roots
Cube Roots.
Reciprocals.
249
62001
154.38249
15.7797333
6.2911946
.004016064
250
62500
15625000
15.8113383
6.2996053
.004000000
251
53001
15813251
15.3429795
6.3079935
.00.3934064
252
63504
16103003
15.3745079
6.3163.596
.0039632.54
253
64009
16194277
15.9059737
6.3247035
.0039-52569
251
64516
16387064
15.9373775
6.3330256
.003937003
255
65025
16581375
15.9637194
6.34132.57
.003921569
256
65536
16777216
16.0000000
6.3196042
.00-3906250
257
66049
16974593
16.0312195
6.3573611
.003391051
25S
66564
17173512
16.0623734
6.3663963
.00387.5969
259
670S1
17.37.3979
16.0934769
6.. 37431 11
.003361004
260
67600
17576000
16.12451.55
6.3325043
.00.3346154'
261
68121
17779581
16.1.5.54944
6.-39)6765
.00.3831418
26-2
68644
17931723
16.1864141
6.. 3938279
.03.3316:5:94
263.
69169
18191447
16.2172747
6.4069535
.003302231
264
69696
18.399744
16.2430763
6.41.50637
.003787879
265
70225
18609625
16.2783206
6.4231533
.003773585
266
70756
18321096
16.309.5064
6.4312276
.003759398
267
71289
19034163
16.3401346
6.4392767
.00374.5318
203
71824
19243332
16.3707055
6.4473057
.003731.343
269
72361
19165109
16.4012195
6.4553143
.003717472
270
72900
1938.3000
16.4316767
6.4633041
.003703704
271
73441
19902511
16.4620776
6.4712736
.003690037
272
73984
20123643
16.4924225
6.4792236
.003676471
273
74529
20346417
16.5227116
6.4371541
.003663004
274
75076
20570324
16.0.529454
6.49506.53
.003649635
275
75625
20796375
16.5831240
6.. 5029572
.003636364
276
76176
21024576
16.6132477
6.5103300
.003623133
277
76729
212539.33
16.6433170
6.5186339
.00.3610108
278
77284
21434952
16.6733.320
6..5265139
.003597122
279
77841
21717639
16.7032931
6.. 5343351
.003534229
230
78400
219.52000
16.7332005
6.5421326
.00.3571429
2S1
73961
22188041
16.7630.546
6. .54991 16
.003553719
282
79524
22425768
16.7923.5.56
6.5576722
.003.546099
283
80039
22665] 37
16.3226033
6.. 56.54 144
.003.5.3.3569
284
80656
229(16304
16.3-522995
6.5731335
.00.3521127
285
81225
23149125
16.8819430
6.5303443
.003503772
286
81796
23393656
16.9115.345
6.5385323
.003496503
287
82369
23639903
16.9410743
6.5962023
.003434321
288
82944
23887372
16.970.5627
6.6033545
.003472222
289
83521
241.37569
17.0000300
6.6114890
.0034613203
290
84100
24339000
17.0293=64
6.6I910S0
.00.3443276
291
84631
24642171
17.0.537221
6.62670.54
.00:}4-36426
292
85264
243970 S8
17.0380075
6.6342874
.003424653
293
85849
251537.57
17.1172123
6.6113.522
.033412969
294
86436
25112184
17.1464232
6.6193993
.003401-361
295
87025
25672375
17. 175.56 to
6.6569302
.003339831
296
87616
25931336
17.2046.505
6.6644437
.003378378
297
38209
26193073
17.2.33G879
6 6719403
.003367003
298
88304
26163.592
17.2626765
6.6794200
.003355705
299
89101
26730399
17.2916165
6.6863831
.003344432
300
90000
27000000
17.320.5081
6.694-3295
.00.3333333
301
90601
27270901
17.349.3516
6.7017593
.003322259
302
91204
27543603
17.3781472
6.7091729
.0033112.58
303
91309
2781S127
17.4Q68952
6.7165700
.003300330
304
92416
23094464
17.435.59.53
6.7239503
.003289474
305
93025
23372625
17.4642492
6.73131.55
.00.3278639
306
93636
236.52616
17.4928557
6.7336641
.003267974
307
94249
23934443
17.5214155
6.7459967
.00.3257329
308
94864
29213112
17.&499283
6.7.533134
.003246753
309
9.5481
29503629
17.. 578.39.53
6.7606143
.003236246
310
96100
29791000
17.6063169
6.7673995
.00322.5806
CUBE ROOTS, AND KECIPROCALS.
143
No.
311
312
313
314
315
316
317
318
319
320
321
322
323
324
32.5
326
327
325
329
330
331
332
333
334
335
336
337
33S
339
340
341
342
343
344
345
346
347
343
349
350
351
3.52
353
351
355
3.56
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
IL.
Squares.
96721
97344
97969
9S596
99225
99S56
100489
101124
1U1761
102100
103041
10.36S1
104329
104976
10.5625
] 06276
1(16929
1075S4
103241
108900
109561
110224
11 0339
111556
112225
112S96
113.569
1 14244
114921
115600
116231
116964
117619
113336
119025
119716
120409
121104
121301
122500
12-3201
12.3904
124609
125316
126025
126736
127449
128164
128881
129600
130.321
131044
131769
132496
133225
1339.56
134639
13.5424
136161
136900
137641
133384
Cubes.
Square Koots.
Cube Roots.
30080231
30371328
30664297
309.59144
31255875
31554496
31855013
321574.32
32461759
32763000
33076161
3;53S6243
33698267
34012224
34323125
34615976
34965783
352-57552
3-561 1239
35937000
38264691
36594363
36926037
37259704
37595375
37933056
38272753
336H472
33953219
39.304000
39651821
40001688
40353807
40707584
4106.3625
41421736
41781923
42144192
42508549
42375000
4.3213551
43614208
439S6U77
44361364
44733375
45118016
45499293
458S2712
46263279
466.56000
47045831
47437923
47832147
48228544
48627125
49027396
49430S63
49836032
50243409
5065.3000
51064811
51478S48
17.6351921
17.6635217
17.6918060
17.7200451
17.7432393
17.7763388
17.8044933
17.8325,545
17.fc605711
17.888.54.38
17.9164729
17.9443.534
17.9722008
18.0000000
18.0277564
18.0554701
18.0.-31413
18.1107703
18.138.3571
18.1659021
18.1934054
13.2203672
1S.24S2876
18.2756669
18.3030052
18.3303023
18.3575598
18.3347763
18.4119526
18.4390889
18.46618.53
18.4932420
18.5202592
18.5472370
13.5741756
18.80107.52
18.6279360
18.6.547581
18.6315417
18.7082S69
18.7.349940
18.7616630
18.7882942
13.8143877
18.8414437
18.8679623
18.8944436
18.9203879
18.9472953
18.9736660
19.0000000
19.0262976
19.0.52.5589
19.0787840
19.1049732
19.1311265
19.1.572441
19.1833261
19.2093727
19.2353841
19.2613603
19.2873015
Reciprocals.
6.7751690
6.7324229
6.7396613
6.7963344
6.8040921
6.8112347
6.8184620
6.8256242
6.8327714
6.8399037
6.8470213
6.8.541240
6.8612120
6.8632355
6.8753443
6.8323888
6.8894188
6.8964.345
6.9034359
6.91042.32
6.917.3964
6.9243556
6.9313008
6.9.332321
6.9451496
6.9520533
6.9589434
6.9658198
6.9726S26
6.9795321
6.9S63631
6.99319(16
7.0000000
7.0067962
7.013.5791
7.0203490
7.02710.58
7.03.33497
7.0405806
7.0472987
7.0.540041
7.0606967
7.0673767
7.0740440
7.0806988
7.0873411
7.0939709
7.1005SS5
7.10719.37
7.1137866
7.1203674
7.1269360
7.1334925
7.1400370
7.146.5695
7.1530901
7.1.595938
7.1660957
7.1725809
7.1790544
7. 1855 162
7.1919663
.003215434
.003205128
.003194388
.003134713
.003174603
.003164557
.003154574
.003144654
.003134796
.003125000
.003115265
.00310.5590
.003095975
.003036420
.003076923
.003067435
.0030.58104
.003048780
.003039514
.003030303
.003021148
.003012048
.003003003
.002994012
.002935075
.002976190
.002967359
.0029585^(1
.002949353
.002941176
.002932551
.002923977
.002915452
.002906977
.002898551
.002890173
.002381844
.002873563
.002865330
.002357143
.002849003
.002840909
.002832861
.002824859
.002816901
.002808989
.002301120
.002793296
.002785515
.002777773
.002770083
.002762431
.002754321
.002747253
.002739726
.002732240
.002724796
.002717391
,002710027
.0027Lr2703
.002695418
.002688172
I
11
TABLE XI. SQUARES, CUBES,
SQUARE ROOTS,
No.
Squares.
Cubes.
Square Roots.
Cube Roots.
Reciprocal*.
373
139129
51395117
19.3132079
7.19340.50
.002630965
371
139376
52313624
19.. 3390796
7.204332 a
.002673797
375
140325
52734375
i9..3649167
7.2112479
.002666667
376
141376
53157376
19.3907194
7.2176522
.0026.59574
377
142129
53532633
19.4164373
7.2240450
.0026.52.520
373
142334
54010152
19.4422221
7.2304263
.002645503
379
143641
54439939
19.4679223
7.2367972
.002633522
330
144400
54372000
19.4935337
7.2431565
.002631579
331
145161
55306:M1
19.5192213
7.2495045
.002621672
332
145924
55742963
19.5443203
7.2.553415
.002617301 1
333
146639
56131337
19.5703353
7.2621675
.002610966
334
147456
56623104
19..5959179
7.2GS4-24
.0026 m 67 I
335
143225
57066625
19.6214169
7.2747^64
.002597403
336
143993
57512456
19.6463327
7.2310794
.002590674
337
149769
57960603
19.67231.56
7.2373617
.00253:3979
333
150544
53411072
19.69771-56
7.29.36330
.002577320
339
151321
53363369
19.72.30329
7.2993936
.002570694
390
152100
59319000
19.7434177
7.30314.36
.002564103
391
152331
59776471
19.7737199
7.3123^23
.002557545
392
153664
60236233
19.7939399
7.3136114
.002.551020
393
154449
60693457
19.3242276
7.-3243295
.002.544529
394
155236
61162934
19.^494332
7.3310369
.002533071
395
156025
61629375
19.8746069
7.3372339
.002531646
396
156316
62099136
19.3997437
7.34.34205
.00252.52.53
397
157609
62570773
19.9243533
7.3495966
.002518392
393
153404
63044792
19.9499373
7.3557624
.002512563
399
159201
63521199
19.9749344
7.3619178
.002506266
400
160000
64090000
20.0000000
7.-3630630
.002.500000
401
160301
64431201
20.0249344
7.3741979
.002493766
402
161604
61964303
20.0499377
7.-3303227
.002437562
403
162409
6.5450327
20.0743599
7.3364373
.002431390
404
163216
65939264
20.0997512
7.3925418
.00247.5243
405
164025
66431125
20.1246113
7.-3936363
.002469136
406
161S36
66923416
20.1494417
7.4047206
.00246:30.54
407
165649
6741*143
20.1742410
7.4107950
.002457002
403
166464
67917312
20.1990099
7.4163595
.0024.50980
409
167231
63417929
20.2237434
7.4229142
.002444938
410
163100
63921000
20.2434.567
7.4239.539
.002439024
411
163921
69426531
20.2731349
7.4:349933
.00243-3090
412
169744
699:34523
20.2977631
7.4410139
.002427184
413
170569
70444997
20.3224014
7.4470-342
.002421.303
414
171396
70957944
20.3469399
7.4530.399
.00241:5459
415
172225
71473375
20.371.5433
7.4590-3.39
.002409639
1 ■■■*■-'
. 416
173056
71991296
20. .3960781
7.4650223
.002403346
417
173339
72511713
20.4205779
7.4709991
.002393032
413
174724
73031632
20.4450433
7.4769664
.002392.344
419
175561
73560059
20.4694395
7.4529242
.002356635
420
176400
74033003
20.4939015
7.4833724
.002330952
421
177241
74613461
20.5132345
7.4943113
.002375297
422
17S034
75151443
20..5426336
7.5007406
.002369663
423
173929
75636967
20.5669633
7.5066607
.002364066
424
179776
76225024
20.5912603
7.5125715
.002-353491
425
139625
76765625
20.615.5231
7.5134730
.002:3.52941
426
181476
77303776
20.6397674
7.5243652
.002-347418
427
132329
77S;54433
20.66397S3
7.5302432
.002-341920
423
133134
78402752
20.6331609
7.5361221
.002336449
429
184041
78953539
20.71231.52
7.5419367
.002-331002
430
184900
79507000
20.7.364414
7.5473423
.002325581
431
185761
80062991
20.7605395
7.55.36333
.002:320186
432
1S6624
80621563
20.7846097
7.5595263
.002314315
433
137439
81132737
20.8036.520
7.5653.543
.002.309469
434
1S3356
81746504
20.8326667
7.5711743
1 .002.304147
CUBE ROOTS, AM> R KCll'ROCALS.
145
No.
43.3
4:3fi
•137
■i.-'S
439
410
4-;i
442
413
444
445
440
447
44S
449
. 450
451
4.52
453
4.54
455
456
457
45S
459
460
461
462
463
46!
465
466
4/57
463
469
470
471
472
473
474
475
476
477
478
479
4S0
481
482
483
484
435
4 So
487
488
439
490
491
492
493
494
495
496
\...
Squares.
189225
1 9U0.;6
1 '.K)ii69
191844
192721
193600
194481
195364
196249
197136
19-025
19^916
I 99S09
200704
201601
202500
203101
2043111
205209
2061 16
207025
207936
208849
209764
210681
211600
212521
21.3444
214369
215296
216225
217156
218089
219024
219961
220900
221841
222784
223729
224676
225625
226576
227529
228484
229441
2.30400
231361
232324
233289
234256
235225
236196
237169
233144
239121
240100
241081
242061
243049
244036
245025
246016
Cubes
Square Roots.
82312875
82881856
83 i^' 3453
Sl(l:;7672
f46('4519
S51e84000
85766121
86350888
8693;307
S752>^3>4
88121125
88716536
89314623
89915392
90518849
9112.5000
91733851
92345403
92959677
93576664
94196375
948 188 16
95443993
96071912
S6702579
97336000
97972181
98611128
99252847
99897344
100.544625
101194096
101847.563
102503232
103101709
103823000
104487111
105154048
10.5823S17
106496424
107171875
1078.50176
1035313.33
10921.53.52
109902239
110592000
1H2S4641
111930168
112678.587
113379904
1140-^4125
114791256
11.5501303
116214272
116930169
117649000
118370771
119095488
119^23157
120553784
121287375
12202.3936
Cube Roots.
20.8566536
20.8806130
20.904.5450
20.9284495
20.9523263
20.9761770
21.0000000
21.0237960
21.0475652
21.0713075
21.0950231
21.1187121
21.142.3745
21.1660105
21.1896201
21.2132034
21.2367606
21.2602910
21.2837967
21.3072758
21.3307290
21.3541565
21.3775583
21.4009346
21.4242853
21.4476106
21.4709106
21.4941853
21.5174348
21.5406592
21.. 5638.587
21.5870331
21.6101828
21.6333077
21.0564078
21.0794834
21.7025344
21.72.55610
21.748,5632
21.7715411
21.7944947
21.8174242-
21.8403297
21.8632111
21.8500636
21.9089023
21.9317122
21.9544984
21.9772610
22.0000000
22.02271.55
22.04.54077
22.0680765
22.0907220
22.11.33444
22.13594.36
22.1.585193
22.1810730
22.2030033
22.2261103
22.2485955
22.2710575
Reciprocals.
7.5769849
7.5827865
7.5885793
7.5943633
7.6001385
7.60.59049
7.6116626
7.6174116
7.6231519
7.6288837
7.6346067
7.6403213
7.0460272
7.6517247
7.0574138
7.6630943
7.6687665
7.6744303
7.6800857
7.6857.323
7.6913717
7.6970023
7.7026246
7.7082388
7.7138448
7.7194426
7.7250325
7.7306141
7.7361877
7.7417532
7.7473109
7.7523606
7.7584023
7.7639261
7.7694620
7.7749301
7.7804904
7.7859928
7! 79 14875
7.7909745
7.80245.38
7.80792.54
7.8133392
7.8188456
7.8242942
7.8297353
7.8351638
7.8405949
7.8460134
7.8514244
7.8568281
7.8622242
7.8676130
7.8729944
7.S7830S4
7.8837352
7.8890940
7.8944403
7.8997917
7.9051294
7.9104.599
7.9157832
.002298851
.002293578
002288330
.002283105
.002277904
.002272727
.002267574
.002262443
.002257330
.002252252
.002247191
.002242152
.002237130
.0022.32143
.002227171
.002222222
.002217285
.002212389
.002207506
.002202643
.002197802
.002192982
.002188184
.00218.3406
.002178649
.00217.3913
.002109197
.002164502
.002159827
.002155172
.002150538
.00214.5923
.002141328
.0021.30752
.002132196
.002127660
.008123142
.002118644
.002114165
.002109705
.002105263
.002100840
.002096436
.002092050
.002087633
.002083333
.002079002
.002074689
.002070393
.002066116
.0020618.56
.002057613
.002053388
.002049180
.002044990
.002040816
.002036660
.002032520
.002028398
.002024291
.002020202
.002(00129
4(5
TABLE Xf. SQUARES, CUBES,
SQUARE ROOTS,
No.
Squares.
Cubes.
Square Roots.
Cube Roots.
Reciprocal*.
497
247009
122763473
22.29:34963
7.9210994
.002012072
493
243001
123505992
22.3159136
7.9264035
,002008032
499
249001
124251499
22.3333079
7.9317104
.002004003
sao
250000
125000000
22.3606793
7.9370053
.002000000
501
2510J1
125751501
22.3330293
7.9422931
.001996003
502
252004
126506033
22.4053565
7.9475739
.001992032
533
253009
127253527
22.4276615
7.9.523477
.001938072
504
254016
12S024064
22.4499443
7.9581144
.001934127
505
255025
123787625
22.4722051
7.9633743
.001930193
506
256036
129554216
22.4944433
7.9636271
.001976285
507
257049
130323343
22.5166605
7.9733731
.001972337
503
253064
131096512
22.5333553
7.9791122
.001963534
509
259031
131372229
22.5610233
7.9343444
.001964637
510
260100
132651030
22. .533 1796
7.9395697
.031960734
511
261121
133432331
22.6353091
7.9947833
.001956947
512
262144
134217723
22.6274170
8.0300000
.0019.53125
513
263169
135005697
22.6495033
8.0352049
.001949318
514
264196
135796744
22.671.5631
8.0104032
.00194.5525
515
265225
136590375
22.6936114
8.0155946
.001941748
518
266256
137333096
22.71.563.34
8.0207794
.001937934
517
267239
133133413
22.7376340
8.0259574
.001934236
518
263324
133991 S32
22.75961.34
8.0311287
.001930502
519
269361
139793359
22.7815715
8.0362935
,001926732
520
270400
140603030
22.8035035
8.0414515
.00192.3077
521
271441
141420761
22.82:54244
8.0466030
.001919386
522
272434
142236643
22.8473193
8.0517479
.001915709
523
273529
143)55667
22.8691933
8.0568862
.001912046
5^4
274576
143377324
22.8910463
8.0620180
.001903397
525
275625
144703125
22.9123785
8.06714.32
.001904762
526
276676
145531576
22.9346399
8.0722620
.001901141
527
277729
1463631 S3
22.9564806
8.0773743
.0018975.33
523
27S734
147197952
22.9732506
8.0324300
.001893939
529
279S41
143035S39
23.0000000
8.0375794
.001390359
530
230900
148377000
23.0217239
8.0926723
.001386792
531
231961
W 972 1291
23.0434372
8.0977.539
.0013332.39
532
233024
150563763
23.0651252
8. 1023390
.001379699
533
234039
151419437
23.0867923
8.1079123
.001876173
534
235156
152273304
23.1034400
8.1129303
.031872659
535
236225
153130375
23.1300670
8.1180414
.001869159
536
237296
153990656
23.15167.33
8.12.30962
.00186.5672
537
23S369
154354153
23.1732605
8.1281447
.001362197
533
239444
155720372
23.194-270
8.1331370
.001353736
539
290521
156590319
23.2163735
8.1332230
.001855288
540
291600
157461000
23.2379031
8.14.32.529
.001851852
541
292631
153340421
23.2594067
8.1432765
.00184*429
542
293764
159223333
23.2303935
8.1532939
.001845018
543
294349
160103007
23.30236134
8.1533051
.031841621
544
295936
163939134
23.3233076
8.1633102
.001833235
545
297025
161378625
23.3452351
8.1633092
.001834362
546
293116
162771336
23.3666429
8.17.33020
.001831502
547
299209
163667323
23.33S0311
8.17S2833
.0018231.54
543
300304
164566592
23.4093993
8.1832695
.001324818
549
301401
165469149
23.4307490
8.1382441
.001821494
550
302500
166375000
23.4520733
8.1932127
.001818182
551
303601
167234151
23.4733392
8.1931753
.001814832
552
304704
163196603
23.4946302
8.2031319
.001811594
553
305309
169112377
23.5159520
8.2030325
.001303318
554
306916
170031464
23.5372348
8.21.30271
.00180.5054
555
303025
170953375
23..55S4330
8.2179657
.001801302
556
309136
171879616
23.5796.522
8.2223935
.001793.561
557
310249
172303693
23.6033474
8.2278254
.031795.3.32
553
311364
173741112
23.6220236
8.2327463
.001792115
CUBE ROOTS, AMD RECIPROCALS.
Ul
No.
559
560
561
562
563
564
565
566
567
563
569
570
571
572
573
574
575
576
577
578
579
580
581
582
5S3
534
585
586
587
588
539
590
591
592
593
594
595
596
597
593
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
Squares.
312481
313600
314721
315844
316969
318096
319225
320356
321439
322624
323761
324900
326)11
327184
32S329
32^)476
330625
331776
332929
334084
335241
336400
337561
338724
339339
341056
342225
343396
344569
345744
346921
aisioo
349231
350464
351619
352836
354025
355216
356409
357604
358801
360000
361201
362404
363609
364316
366025
367236
363449
369664
370S81
372100
373321
374544
375769
376996
37^225
379456
3306S9
331924
333161
3S4400
Cubes
Square Roots.
174676879
175616000
176558481
177504328
178453547
179406144
180362125
181321496
182234263
183250432
1842200JJ
185193000
1S6169411
187149248
183132517
189119224
190109375
191102976
192100033
1 93 1 00552
194104539
195112000
196122941
197137368
193155237
199176704
200201625
201230056
202262003
203297472
204336469
205379000
206425071
207474638
203527357
209534534
210644875
21170S736
212776173
213.347192
214921799
216000000
217031801
218167208
219256227
220343864
221-445125
222545016
223643543
224755712
225366529
226931000
225099131
229220923
230346397
23147.5544
232608375
233744396
234335113
236029032
237176659
23S32S000
Cube Roots.
23.6431803
23.6643191
23.6854336
23.7065392
23.7276210
23.7486842
23.7697236
23.79117545
23.8117618
23.8327506
23.8537209
23.8746723
23.8956063
23.9165215
23.9374184
23.9582971
23.9791576
24.0000000
24.0208243
24.0416306
24.0624183
24.0S31S91
24.1039416
24.1246762
24.1453929
24.1660919
24.IS67732
24.2074369
24.2230329
24.2487113
24.2693222
24.2S991.56
24.3104916
24.3310501
24.3515913
24.3721152
24.3926213
24.4131112
24.4335334
24.4540335
24.4744765
24.4943974
24.5153013
24.5.3.56383
24.5560583
24.5764115
24.5967473
24.6170673
24.6.373700
24.6576560
24.67792.54
24.6981781
24.7184142
24.7386333
24.7.583363
24.7790234
24.7991935
24.3193473
24.3.394347
24.3596058
24.8797106
24.3997992
Reciprocals.
8.2.376614
S. 2425706
8.247474')
8.2523715
8.25726.33
8.2621492
8.2670594
8.2719039
8.2767726
8.2316355
8.2S64928
8.2913444
8.2961903
8.3010304
8.3053651
8.3106941
8.3155175
8.3203353
8.3251475
8.3299542
8.3347553
8.3395509
8.344-3410
8.^4912.56
8.3539047
8.3536734
8.3634466
8.3632095
8.3729668
8.3777188
8.3324653
8.3372065
8.3919423
8.3966729
8.4013981
8.4061180
8.4103326
8.4155419
8.4202460
8.4249448
8.4296333
8.4343267
8.4390098
8.4436377
8.4433605
8.4530231
8.4576906
8.4623479
8.4670001
8.4716471
8.4762892
8.4309261
8.4355579
S.4901S43
8.4943065
8.4994233
8.5ai0.350
8.5086417
8.5132435
8.5173403
8.-5224321
8.5270139
.001783909
.001785714
.001732531
.001779359
.001776199
.001773050
.001769912
.001766784
.001763663
.001760563
.001757469
.001754336
.001751313
.001748252
.001745201
.001742160
.0017391-30
.001736111
.001733102
.001730104
.001727116
.001724138
.001721170
.001718213
.001715266
.001712329
.001709402
.001706135
.001703.578
.001700630
.001697793
.001694915
.001692047
.001639189
.001636.341
.001633502
.001630672
.001677852
.001675042
.001672241
.001669449
.001666667
.001663394
.001661130
.0016.53375
.001655629
.001652393
.0016-50165
.001647446
.001644737
.001642036
.001639344
.0016.36661
.001633987
.001631321
.001623664
.001626016
.001623377
.001620746
.001613123
.001615509
.001612303
L4W
TABLE XI. SQUARES, CUBES,
SQUARE ROOTS,
No.
Squares.
Cubes.
Square Roots
Cube Roots.
1
Reciprocal*.
621
3 356 11
239433061
24.9193716
8.5316309
.031610306
622
336334
240641343
24.9399273
8. 5361 730
.031607717
623
333129
241304367
24.9599679
8.5407501
.001605136
624
339376
242970624
24.9799920
8. .54531 73
.001602564
625
39J625
244140625
25.0330000
8.5493797
.001600000
626
391376
245314376
25.0199920
8.5.544372
.001597444
627
393129
246491333
25.0.399631
8.-5539399
.001.594396
62S
394334
217673152
25.0599232
8.563.5377
.001592:357
629
395641
243353139
25.0793724
8.5633307
.001.539325
630
396903
250047000
25.0993003
8. -5726 139
.001537302
631
393161
251239591
2.5.1197134
8.577152:3
.001.534736
632
399424
252435963
25.1.396102
8-5316309
.001.532273
633
400639
253636137
25.1594913
8.5362047
.001.579779
634
401956
254340104
25.1793566
8.59t)7233
.0)1.577237
635
403225
256047375
2.5.1992063
8.5952330
.001574303
636
404495
257259456
25.2193404
8.5997476
.001-572327
637
405769
253474353
25.2333539
8.6342.525
.001.5693-59
63 S
407044
259694072
25.2536619
8.6037526
.001.567:393
639
403321
■260917119
2-5.2734493
8.61.32430
.001.564945
61)
409690
262144030
25.2932213
8.6177333
.001-562-500
641
410331
253374721
25.3179773
8.6222243
.001560062
612
412164
264639233
25.3377139
8.6267063
.0015-576:32
613
413149
265347707
25.3574447
8.6311330
.001-5-5-5210
614
414736
267039934
25.3771551
8.6-3.56551
.001-552795
615
416925
263336125
25.3963502
8.6401226
.001-553333
646
417316
269536136
2.5.4165301
8.6445355
.001547938
647
413609
270340023
25.4361947
8.6493437
.001-545595
643
419904
272097792
2.5.4553441
8.65:34974
.00154-3210
649
421201
273359449
25.47.54734
8.6579465
.00] 540-^.32
650
422503
274625000
25.4950976
8.6623911
.0015:33462
651
423301
275394451
25.5147016
8.6663310
.0015:36093
652
425104
277167303
25.5342907
8.6712665
.00l5:-;3742
653
426409
273445077
25.5533647
8.G756974
.001.531394
654
427716
279726264
25.5734237
8.63012:37
.001529(152
655
429025
231011375
25.-5929673
8.634;54.56
.001526713
656
433336
232300416
25.6121969
8.6339633
.001.524.390
657
431649
233593393
25.632flll2
8.693:3759
.001-522070
653
432951
234390312
2.5.6515107
8.6977343
.031519757
659
434231
236191179
25.6709953
8.7021332
.001517451
660
435630
237496000
25.69346-52
8.7065377
.001515152
661
436921
233304731
25.7099203
8.7109327
.001512359
662
433244
293117523
25.729.3607
8.71.5-37^4
.001510574
663
439569
291431247
25.7437364
8.7197596
.001503296
664
440396
292754944
25.7631975
8.7241414
.001.506024
655
442225
294079625
25.73759-39
8.7235187
.001503759
666
443556
295403298
25.3069753
8.7.323913
.031501502
657
444399
296740963
25.8263431
8.7372604
.0014992.50
66S
446224
293077632
25.3456960
8.7416246
.001497006
669
447561
299413309
25.3650343
8.7459346
.001494763
670
443903
300763000
25.8343-532
8.7503401
.001492537 \
671
450241
302111711
25.9036677
8.7.546913
.001490313
672
451534
3)3164443
25.9229623
8.7.590333
.001433095
673
452929
304321217
25.9422435
8.76-33309
.001435334
674
454276
306132024
2.5.9615100
8-7677192
.001433630
675
455625
307546375
25.9307621
8.7720532
.001431431
676
456976
303915776
26.0300300
8.77633.30
.001479290
677
453329
310233733
26.0192237
8.7307034
.001477105
673
4596 34
311665752
26.03S433I
8.73-50296
.001474926
679
461041
313346339
26.0576234
8.789-3466
.001472754
630
462400
314432000
26.0763096
8.7936.593
.001470.533
631
453761
315321241
26.09-59767
8.7979679
.03146^129
632
465124
317214563
26.1151297
— ' —
8.3022721
.001466276
CUBE ROOTS, AND IIECIPROCALS.
149
No.
6-3
6-4
6 So
6S6
6S7
6S3
659
690
691
692
693
694
695
696
697
69S
699
700
701
702
703
704
705
706
707
703
709
710
711
712
713
714
715
716
717
71S
719
IL
720
721
722
723
724
725
726
727
723
729
730
731
732
733
734
735
736
737
733
739
740
741
742
743
744
Squares.
4G64S9
467S.36
469225
470596
471969
473344
474721
476100
477431
473S64
4S0249
4S1636
4S3025
434416
4S5309
437204
433601
490001
491401
492304
4942'i9
495GI6
497025
49S436
499S49
501264
502631
504100
505521
506944
503369
509796
511 225
512656
514039
515524
516961
Cubes.
Square Roots.' Cube Roots. Reciprocals.
513 100
519341
521234
522729
524176
525625
527076
523529
529934
531441
532900
534361
535324
5372S9
533756
540225
541696
543169
544614
5-16121
547600
549)31
550564
552049
553536
31S611937
320013504
321419125
322323356
324242703
325660672
327032769
323509000
329939371
331373333
332312557
334255334
335702375
337153536
33S60S373
340063392
341532099
343000000
344472101
34594340S
34742-927
343913664
350402625
351-95316
353393243
354394912
356400329
357911000
359425431
360944123
362467097
363994344
365525375
367061696
363601313
370146232
371694959
373245000
374305361
376367048
377933067
379503424
331073125
332657176
354240533
33532,3352
337420439
3^9017000
390617591
392223163
393532537
395446904
397065375
39563,3256
400315553
401947272
403533419
405224000
406>69021
40>5134S8
41(»172407
411530734
26.1342687
26.15:3.3937
26.1725047
26.1916017
26.2106543
26.2297541
26.2433095
26.2678511
26.256.3739
26.30.53929
26.3245932
26.3435797
26.3623527
26.331S119
26.4tM375r6
26.4196396
26.4356031
26.4575131
26.4764046
26.49.52526
26.5141472
26.5329933
26.551,5361
26.571 6605
26..5594716
26.60-2694
26.6270539
26.64-552.52
26.6645333
26.6533231
26.7020593
26.7207784
26.73945.39
26.7551763
26.7765557
26.7955220
26.81417.54
26.8.323157
26.85144.32
26.8700577
26. ,8536593
26.9072481
26.9255240
26.9443572
26.9629375
26.9514751
27.0000000
27.0185122
27.0370117
27.0.5.549-5
27.0739727
27.0924344
27.1105334
27.1293199
27.1477439
27.1661554
27.184.5544
27.2029410
27.22131.52
27.2.396769
27.2550263
27.2763634
8.8065722
8.8108631
8.8151598
8.819^1474
8.8237307
8.8250099
8.8322550
8.8365559
8.84(15227
8.3450554
8.8493440
3.85359.35
8.8575489
8.8620952
8.8663375
8.8705757
8.8748099
8.8790400
8.6532661
8.8874582
8.3917063
8.8959204
8.9001304
8.9043:^6
8.9035337
8.9127369
8.9169311
8.9211214
8.925.3073
8.9294902
8.9336687
8.9375433
8.9420140
8.9461509
8.9503433
8.9545029
8.9556581
8.9623095
8.9669570
8.9711007
8.9752406
8.9793766
8.9335089
8.9876373
8.9917620
8.9953329
9.0000000
9.0041134
9.0052229
9.0123233
9.0164309
9.0205293
9.0246239
9.0287149
9.0325021
9.0365357
9.0409655
9.04.50419
9.0491142
9.0531831
9.0572482
9 0613098
.ft01464129
.001461933
.001459854
.001457726
.001455604
.001453483
.001451379
.001449275
.001447173
.001445087
.00144.3001
.001440922
.00143-.549
.001436782
.0014:34720
.0014:32665
.0014.3C615
.001423571
.001426534
.001424501
.001422475
.001420455
.001418440
.001416431
.001414427
.001412429
.001410437
.001403451
.001406470
.001404494
.001402525
.001400560
.00139860!
.001396648
.001:394700
001392758
.001390821
.001388889
.001356963
.00135.5042
.001333126
.001331215
.001379310
.001.377410
.00137.5516
.001373626
.001371742
.001369363
.001:167959
.001366120
.001364256
.roi 362398
.001360544
.001355696
.001356352
.001355014
.001353180
.001351351
.001349528
.001347709
.001:345.-95
.001344036
15U
TABLE XI SQUARE
S, CUBES,
SQUARE R(
)OTS,
No.
Squares.
Cubes.
Square Roots.
Cube Roots.
Reciprocala.
745
555025
4134936i5
27.2946331
9.0653677
.001342232
746
556516
415160936
27.3130006
9.0694220
.001340433 .
747
55S039
416332723
27.3313007
9.0731726
.001333638
74S
559504
413503992
27.3495337
9.0775197
.001336-93
749
561031
4201 39749
27.3673644
9.0315631
.001335113
750
562500
421375000
27.3361279
9.0356030
.001333333
751
564001
423564751
27.4043792
9.0396.392
.001331553
752
565501
425259003
27.4226134
9.0936719
.001329737
753
567009
426957777
27.4403455
9.0977010
.001323021
754
563516
423661064
27.4590604
9.1017265
.001326260
755
570025
430363375
27.4772633
9.1057435
.001324503
756
571536
432031216
27.4954542
9.1097669
.001322751
757
573 )49
433793093
27.5136330
9.1137818
.001321004
75S
574564
435519512
27.5317993
9.1177931
.031319261
759
576031
437245479
27.5499546
9.1213010
.001317523
7^,0
577600
433976000
27.5630975
9.1253053
.001315739
761
579121
440711031
27.5362234
9.1293061
.001314060
76-2
5S0644
442450723
27.6343475
9.13.330a4
.001312336
763
532169
444194947
27.6224546
9.1377971
.001310616
764
583696
445943744
27.6405499
9.1417374
.001303901
765
535225
447697125
27.6536334
9. 1457742
.001307190
766
536756
449455096
27.6767050
9.1497576
.001305483
767
533239
451217663
27.6947643
9.1537375
.001-303781
763
539324
452934332
27.7123129
9.1.5771.39
.001302033
769
591361
454756609
27.7303492
9.1616369
.001300390
770
592900
456533000
27.74337.39
9.16.56565
.001293701
771
594441
453314011
27.7663363
9.1696225
.001297017
772
595934
463399648
27.7843330
9.17353.52
.001295337
773
597529
461839917
27.3023775
9.1775445
.00129:^661
774
599 i-e
463634324
27.8203555
9.1315003
.001291990
775
603625
465434375
27.8333218
9.18.54527
.001290323
776
602176
467233576
27.3567766
9.1394018
.001233660
777
603729
469397433
27.8747197
9.1933474
.001237001
773
605234
470910952
27.3926514
9.1972397
.00123.5.347
779
636341
472729139
27.9105715
9.2012236
.001233697
780
633400
474552000
27.9234301
9.2051641
.001282051
73 1
609961
476379541
27.9463772
9.2090962
.001230410
732
611521
473211763
27.'J042629
9.21302.50
.031278772
783
613039
430043637
27.9321372
9.2169505
.001277139
784
614656
431590304
28.00313030
9.2203726
.001275510
733
616225
433736625
23.0178515
9.2247914
.001273385
736
617796
435537656
23.0.356915
9.2237063
.001272265
737
619369
43744:J403
23.0535203
9.2.326189
.001270648
733
620944
4393)3372
23.0713377
9.236.5277
.001269036
789
622521
491169069
23.03914.33
9.2404333
.001267427
790
624100
493039000
23.1069336
9.244.3355
.001265323
791
625631
494913671
23.1247222
9.2432344
.001264223
792
627264
496793033
28.1424946
9.2.521300
.001262626
793
623S49
493677257
23.1602.557
9.2560224
.0012610:J4
794
630436
503566134
23.17303-56
9.2599114
.0312.59446
795
632325
502459375
23.1957444
9.2637973
.001257362
796
633616
504353336
23.2134720
9.2676793
.001256281
797
635209
506261573
28.2-311834
9.2715592
.001254705
793
636304
503169592
23.2433933
9.2754.3.52
.0012.5313?
799
633401
510032399
23.266.5831
9.2793031
.001251564
300
640000
512000000
28.2342712
9.2831777
.001250000
801
641601
513922431
23.3019434
9.2370440
.0012434.39
802
643204
515349603
23.3196045
9.2909072
.001246333
803
644309
517731627
23.3372546
9.2947671
.0012453.30
sai
646416
519713464
23.3.543933
9.2936239
.001243781
805
643025
521660125
23.372.5219
9.3024775
.001242236
806
649636
523636616
23.3901391
1 9.3063273
.001240695
CUBE ROOTS, AND RECIPROCALS.
151
No.
Squares.
Cubes. i
Square Hoots.
Cube Roots.
Reciprocals.
807
651219
.525557943
23.4077454
9.3101750
.001239157
803
652364
527514112
28.4253403
9.3140190
.001237624
809
6.54431
529475129
23.44292.33
9.3178599
.001236094
810
656100
531441000
23.4604939
9.3216975
.001234563
811
657721
533411731
23.4780617
9.325.5320
.001233046
812
6593 14
5353S7323
23.4956137
9.3293634
.001231527
813
660369
537367797
23.5131549
9.3331916
.001230012
814
662596
.539353144
23.5.3063.52
9..3370167
.001223.301
815
661225
541343375
23.. 5432043
9.3403336
.001226994
816
665356
543338496
23.5657137
9.3146575
.00122.3490
817
6674S9
545333513
23. .533211 9
9.3434731
.001223990
81^
66 J 124
547343432
23.6006993
9.3522357
.001222494
819
670761
549353259
23 6131760
9.3560952
.001221001
820
67240)
5513630X
23.63.56421
9.3599016
.001219512
821
674041
553337661
23.6530976
9.3637049
.001213027
822
675634
555412213
23.6705424
9.3675051
.001216545
823
677329
557441767
23.6S79766
9.3713022
.00121.3067
824
673976
559476224
23.7054002
9.3750963
.001213.592
825
6S0625
561515625
23.7223132
9.3733373
.001212121
826
632276
563559976
23.7402157
9.33267.32
.001210654
827
6S3929
565609233
23.7576077
9.3364600
.001209190
82S
635534
567663552
23.7749391
9.3902419
.001207729
829
6S7241
569722739
23.7923601
9.3940206
.001206273
830
633900
571737000
23.8097206
9..3977964
.001204319
831
690561
573356191
23.8270706
9.4015691
.001203369
832
692224
575930363
23.3444102
9.40533S7
.001201923
833
693339
578009537
23.3617394
9.4091054
.001200430
834
695556
580093704
23.3790532
9.4123690
.001199041
835
697225
532132375
23.S963666
9.4166297
.001197605
836
693396
534277036
23.9136646
9.4203373
.001196172
837
700569
536376253
23.9309523
9.4241420
.001194743
S3S
702244
533430472
23.9432297
9.4278936
.001193317
839
703921
590:89719
23.96.54967
9.4316423
.001191395
840
705600
592704000
23.93275.35
9.4353380
.001190476
841
7072SI
594323321
29.0000000
9.4.391.307
.001139061
842
703964
596947633
29.0172.363
9.4123704
.001187643
843
710349
599077107
29.0344623
9.4466072
.001136240
844
712336
601211534
29.0516731
9.450.3410
.001184334
845
714025
603331125
29.0633337
9.4510719
.001183432
846
715716
605495736
29.0360791
9.4577999
.001132033
847
717409
607645423
29.1032644
9.4615249
.001130633
848
719104
609300192
29.1204396
9.46.52470
.001179245
849
720301
611960049
29.1376046
9.4639661
.001177856
850
722500
614125000
29.1.547595
9.4726324
.001176471
831
724201
616295051
29.1719043
9.4763957
.001175033
852
725904
613470203
29.1390390
9.4301061
.001173709
853
727609
620650477
29.2061637
9.4333136
.001172333
854
729316
622335364
29.2232734
9.4375182
.001170960
855
731025
625026375
29.2403331
9.4912200
.001169.591
856
732736
627222016
29.2574777
9.4949133
.001163224
857
734449
629122793
29.274.5623
9.4936147
.001166361
853
736164
• 63162>712
29.2916370
9.5023073
.001163.501
859
737331
633339779
29.3037018
9.50.59930
.001164144
860
739600
636056000
29.3257566
9.5096354
.001162791
861
741321
633277331
29.3423015
9.51.33699
.001161440
862
743044
610503923
29.3593365
9.5170515
.001160093
863
744769
642735647
29.3763616
9.5207303
.001153749
864
746 196
6 14972544
29.3933769
9.5244063
.001157407
865
743225
617214625
29.4103323
9. .5230794
.001156069
866
749956
619161896
29.4273779
9.5317197
.001154734
867
751639
651714363
29.4443637
9.53.34172
.001153403
863
,
753424
' 653972032
29.4613397
9.5390318
.001152074
152
TABLE XI.
SQUARES, CUBES, SQUARE KOO/S,
No.
S69
870
871
872
873
874
875
876
877
878
879
880
88 1
882
8S3
8.S4
SSo
856
837
8.38
SS9
890
891
892
893
894
895
896
897
893
899
900
901
902
903
9m
9135
906
907
90S
909
Squares.
920
921
922
923
924
925
926
927
923
929
930
755161
756900
7;:?&41
7603S4
762129
76;JS76
765625
767376
769129
770S34
772641
774400
776161
777924
7796S9
781456
7S3225
784996
7S6769
78S.544
790321
792100
793>S1
795664
797449
799236
801f!25
802S16
804609
806404
80S20I
810000
81IS01
813604
815409
817216
819025
820S36
822649
824464
826231
Cubes.
910
S2S10V-)
911
829921
912
S3 1 744
913
83:3569
914
835396
915
837225
916
839056
917
&103S9
918
842724
919
844561
S46400
84S241
850054
851929
853776
855625
857476
859329
861154
863041
864900
Square Roots.
656234909
65.S503000
660776311
6630.54S43
665335617
667627624
669921575
672221376
674526133
676-36152
679151439
651472000
6S3797S41
6-612S965
65S4653S7
69OS071O4
693154125
695506456
697864103
7ai227072
702595369
704969000
707347971
709732258
712121957
714516954
716917375
719:323136
7217:34273
724150792
726572699
729000000
731432701
73-35705(:«3
7:36314:327
7-3576:3264
741217625
74:3677416
746142643
74561:3312
751059429
753571000
756055031
75555052S
761045497
76:3551944
766360575
76-575296
771095213
77:362(;'632
776151559
77S65.5000
781229961
7.53777445
756:330467
785559024
791453125
794022776
796597S53
79917S752
801765059
804357000
Cube Roots.
29.4788059
29.4957624
29.5127091
29.5296461
29..S1657.^
29.56:34910
29.580:3959
29.5972972
29.6141555
29.6310&JS
29.rA79M2
29.6647939
29.6516442
29.6934545
29.7153159
29.7:321:375
29.7459496
29.7657521
29.732.54.52
29.799:3259
29.3161030
29.5:325678
29.5496231
29.566:3690
29.5531056
29.3995.328
29.916-5506
29.93:32-591
29.9499-533
29-9666431
29.95:3.3257
30.0000000
30.0166620
30.03:33143
30.0499554
3(». 0665923
30.03:32179
30.099-:339
30.1164407
30.1:3:30:353
30.1496269
30. 1662063
30.1527765
30-199:3-377
30.21-55599
30.2-324:329
30.2459669
3<t. 26.549 19
30.2320079
30.29-5143
30.31-50123
30-3-31-5013
30.^479313
30.3644529
30..3309151
30.397:3653
30.41-35127
30.4302451
30.4466747
30.46-30924
30.4795013
30.4959014
Reciprocals.
9.5427437
9.5464027
9.5500539
9.5-537123
9.55736:30
9.5610103
9. -5646.559
9.5652932
9.5719377
9.5755745
9.5792055
9..552>:397
9.5564632
9.5900939
9.5937169
9.-5973373
9.6009.545
9.604-5696
9.6051517
9.6117911
9.615:3977
9.619f)017
9.6226030
9.6262016
9.6297975
9.6:3:3:3907
9.6369512
9.64(t5690
9.&44h542
9.6477:367
9.65131C6
9.6.5459.33
9.6-534654
9-6620403
9.66-5G096
9.6691762 '
9.6727403
9.6763017
9.6795604
9.6534166
9.6.569701
9.690521 1
9.6940694
9.6976151
9.701 1;583
9.7046959
9.7052-369
9.7117723
9.71^3051
9.713-:354
9.722:3631
9.7255853
9.7294109
9.7329:309
9.7364454
9.7399634
9.74:^753
9.7469557
9.7.504930
9.7539979
9.7575002
9.7610001
.001 1.50743
.001149425
.001145106
.001146759
.001145475
.001144165
.001 142357
.001141553
.001140251
.001 13-952
.0011:37656
.00113n:i64
.0011.3.5074
.0011:3.3757
.001132503
.001131222
.001129944
.001123663
.001127396
.001126126
.001124559
.00112.3596
.■001122:3:34
-001121076
.001119521
.001113568
.001117313
.001116071
.001114527
.00111.3.556
.001112:347
.001111111
.001109578
.001105&47
.001107420
.0(01106195
.001104972
.00110.3753
.001102536
.001101.322
.001100110
.001095901
.001097695
.001096491
.001095290
.001094092
.001092396
.001091703
.001090513
.001059:325
.001055139
.001056957
.001055776
.001084.599
.0010S:M23
.001052251
.0010810.31
.001079914
.001078749
.001077556
.001076426
.00107.5269
CUBE ROOTS, iND RECIPROCALS.
153
No.
Squares.
Cubes.
Square Roots.
Cube Roots.
Reciprocals.
931
866761
806954491
30.5122926
9.7644974
.001074114
932
S6S624
809557563
30. .5236750
9.7679922
.001072961
933
870439
812166237
30.5450437
9.7714345
.001071811
934
872356
814730501
30.5614136
9.7749743
.001070664
935
874225
817400375
30.5777607
9.7734616
.001069519
933
876 J36
820025356
30..5941171
9.7820466
.001063376
937
877969
822656953
30.6104.557
9.7854233
.001067236
93S
879344
825293672
30.6267357
9.7339037
.001066098
939
831721
827936019
30.6431069
9.7923361
.001064963
940
833600
833534000
30.6594194
9.7953611
.001063330
941
835481
833237621
30.6757233
9.7993336
.001062699
942
8^7364
835396333
30.6920185
9.30230.36
.031051571
943
8^9249
833561307
30.7033051
9.8062711
.001060445
944
891136
841232334
3' (.7245330
9.8097362
.0010.59322
945
893')25
843903625
30.7403523
9.8131989
.001053201
. 946
894916
S46590536
30.7571130
9.8166591
.001057082
947
896309
849273123
30.7733651
9.8201169
.001055966
943
893704
85197 L392
854670349
30.7896036
9.8235723
.0010.54852
949
9)0601
30.8053436
9.8270252
.0010.53741
950
902500
857375000
33.3223700
9.8.304757
.0010.32632
951
901401
860035351
30.3332379
9.3339233
.001051525
952
9063 )4
862301403
30.8544972
9.8373895
.0010.50420
953
903209
865523177
30.8706931
9.3403127
.001049313
954
910116
863250664
.30.3863904
9.8442536
.001048213
955
912025
870933375
.30.9030743
9.8476920
.001047120
956
913936
S73722316
30. 9 1'j24 97
9.8511230
.031046025
957
915349
876467493
30.93.54166
9.8545617
.001044932
958
917764
879217912
30.9515751
9.8579929
.00104.3341
959
919631
831974079
30.9677251
9.8614218
.031042753
960
921600
834736000
30.9333663
9.8643433
.001041667
961
923521
837503631
31.0000000
9.8632724
.001043533
962
925444
890277123
31.0161243
9.8716941
.0010.39501
963
927369
89305G347
31.0322413
9.8751135
.0010.33422
964
929296
895341344
31.0433494
9.8785305
.001037344
965
931225
893632125
31.0644491
9.8319451
.001036269
966
933156
901423696
31.0835405
9.8353574
.001035197
9o7
935039
904231063
31.0966236
9.8337673
.001034126
96^
937024
907039232
31.1126934
9.8921749
.001033353
969
933961
909353209
31.1237643
9.89.55301
.001031992
970
940301
912673000
31.1443230
9.8939330
.001030928
971
942-^11
915493611
31.1633729
9.9023S35
.001029366
972
9447S4
913333043
31.1769145
9.9057317
.001023307
973
946729
921167317
31.1929479
9.9091776
.001027749
974
943676
924010424
31.2039731
9.912-3712
.001026594
j 97c
950625
926359375
31.2249900
9.91.59624
.001023641
J
976
952576
929714176
31.2409937
9.9193513
.001024590
977
954529
932574333
31.2560992
9.9227379
.001023.341
973
956434
935441352
31.2729915
9.9261222
.001022495
979
953441
93S313739
31.2339757
9.9295042
.001021450
930
960400
941192003
31. .304951 7
9.9323339
.001020403
931
962361
944076141
31.3209195
9.9362613
.031019363
932
964324
946066! 63
31.3363792
9.9396.363
.001018330
933
9662S9
949;:62337
31.3523303
9.94.30092
.001017294
94
96325'6
952763904
31.3637743
9.9463797
.001016260
935
970225
955671625
31.3347097
9.9497479
.001015223
'■j-6
972196
9535^52.56
31.4006369
9.9.531133
.031014199
9S7
974169
951504303
31.4165.361
9.9.564775
,001013171
93S
976144
964430272
31.4324673
9.9593389
.001012146
939
978121
967361669
31.4433704
9.9631931
.001011122
990
930100
970299000
31.4642654
9.9665549
.001010101
991
932031
973242271
31.430152:5
9,9699095
.001009082
992
934064
976191433
31.4960315
9.9732619
.001008065
1D±
TABLE XI. SQUARES, CUBES, &C.
No.
Squares.
Cubes.
Square Roots.
Cube Roots.
Reciprocals.
993
956049
9791466.57
31. .51 19025
9.9766120
.001007049
994
933036
982107784
31.5277655
9.9799.599
.001006036
99.5
990025
935074375
31.54.36206
9.93.33055
.001005025
996
992016
933047936
31.5594677
9.9&66488
.001004016
997
994)09
991026973
31.5753063
9.9899900
.001003009
993
996004
994011992
31.5911380
9.9933289
.001002004
999
993001
997002999
31.606.613
9.9966656
.001001001
1000
1000000
1000000000
31.6227766
10.0000000
.001000000
1001
1002001
1003J0.3001
31.6.33.5340
10.0033322
.0009990010
1002
1004004
1006012003
31.6.543836
100066622
.0009980040
1003
10i'6!09
1009027027
31.67017.52
10 0099599
.0009970090
1 1004
100S0I6
1012043064
3 1.63 59:' 90
lOO 1.331 55
.0009960159 1
1005
1010025
101.5075125
31.7017319
100166339
.0009950249 !
! 1006
1012036
1018103216
31.7175030
10.019C60]
.0009940358 i
1007
1014049
1021147.343
31.7332633
10.02.32791
.0009930487 j
1003
1016064
1024192512
31.74901.57
IO02659.53
.0009920635
1009
1018031
1027243729
31.7647603
100299104
.0009910803
1010
1020100
1030.301003
31.7804972
10.0.332228
.0009900990
j 1011
1022121
1033.364331
31.7S^62262
10.0.3653.30
.0009391197
i 1012
1024144
10.36433723
31.8119474
100393410
.0009881423
1013
1026169
1039.509197
31.8276609
1O0431469
.0009371668
1014
1023196
1W2590744
31.84.3.3666
100464.506
.0009561933
1015
1030225
104.5678375
31.8590646
10.0497521
.0009852217
1016
1032256
1048772096
31.8747.549
1O0.530514
.0009-842520
1017
10342S9
1051871913
31.8904374
10.0563435
.0009832842
1018
1036324
10.54977332
31.9061123
10.0596435
.0009323133 ■
1019
1033361
1053039359
31.9217794
100629364
.0009813543
1020
\yinioz
r06 1203000
31 9374388
10.0662271
.0009803922
1021
i'34244I
1064332261
31.9.530906
1006951.56
.0009794319
1022
1044434
1067462643
31.9637347
10.072-020
.0009784736
1023
1046529
1070599167
31.984.3712
100760-63
.0009775171
1024
1013576
1073741324
32.0000000
10.0793634
.0009765625
1025
1050625
1076^90625
32.0156212
10.0326434
.0009756098
iOv6
1052676
1030045576
32.0.3*2:343
100-59262
.0009746589
1 1027
10.54729
1083206633
32.0463107
10.0392019
.0009737098
1023
10.56784
10S6373952
32.0624391
10.0924755
.0009727626
1029
10.58341
1039.547389
32.0730293
10.0957469
.0009718173
■ 1030
1060900
1092727000
32.09.36131
10.0990163
.000)9708738
1031
1062961
109.5912791
32.1091337
10.10228.35
.0009699321
1032
106.3024
1099104763
.32.1247563
lO105r>187
.0009639922
1033
1067039
1102.3029.37
32.1403173
10 10381 17
.0009680542
1031
1069156
1105.507.304
.32.1.5.58701
10.1120726
.0009671180
1035
1071225
1108717375
32.1714159
1011.5.3314
.0009661836
1036
1073296
1111934656
32.18695.39
10.118.5832
.0009652510
I 1037
1075.369
11151.576.53
32.2024344
10.1218428
.0009643202
1033
1077444
11133>56872
32.2180074
10.12.509.53
.00096.3391 1
1039
1079.521
1121622319
32.2335229
101283457
.0009624639
1040
I0316!J0
1124364000
32.2490310
10.131.5941
.000961.5335
H"41
1033631
1123111921
32.264.5316
101.343403
.0009606143
li42
1035761
1131366033
32.2800248
1O1.3S0345
.0009596929
if 43
1037349
1134626507
32.2955105
lO 1413266
.0009587738
1044
1039936
1137393134
32.3109338
10.1445667
.0009578.514
1045
1092125
1141166125
32.3264598
101478047
.0009569378
1046
1094116
114444.5.336
32.3419233
101510406
.0009560229
1047
1096209
1 147730-23
32.3573794
101.542744
.0009551098
104S
1093.304
1151022592
32.3723231
10.1575062
.0009541985
1049
1100401
1154320649
32.3882695
10.1607.3.39
.0009532888
in50
1102.500
11.5762.50ao
32.4037035
lO 1639636
.0069.52.3810
1051
1104601
1160935651
32.4191.301
101671393
.0009514748
1052
1106704
1164252608
32.434.5495
10 1704 1 29
.0009505703
1053
1103309
1167575377
32.4499615
101736:M4
.00O94S6676 |
io.:4
1110916
117090.5464
32.4653662
10176^539
.00OP437666
f^.^
0 ./ 0
? !
^j ^, t.i^ V y bC
\
?
i
-
1-
A ^ TABLE XII.
,/. ^^^,. ..
■*
"
LOGARITHMS OF NUMBERi
— c; //
-*
FROM 1 TO 10,000
-..
^
4
1
\
156
TABLE XII. LOGARITHMS
Of
NUMBERS.
Ino.i
0 1 1 1
OOUOijG 000434]
3
000S63
3
001301
4.
5
6 1 7 i 8
9 iDiff.
100
001734
002166
002598003029 003461
003691
432
1
4321
4751
5181
5609
603-
6466 6394!
7321 1 7748
8174| 428
2
8600
90261
9451
9376 0103001
010724 011147
011570 011993
0124151 424
3'
012S37
013259
0136S0
014100:
4521
4940
5360
5779'
6197
6016
420
4
7033
74511
7868
82^41
8700
9110
9532
9947
020361
020775
416
5
021189
021603
022016
022423 022341
023252
023664
024075
4466
4896
412
6
5306
5715;
6125
6533; 6942
7350
7757i
8164
8571
8978
408
7
93S4
9789
030195
0.30600^
031(104
031408
031812'
032216 032619!
03:3021
404
8
033424
7426
03:3326
4227
4623
5029
5430
.5830
6230
6629
702ft
400
9
7825
8223
8620 ;
9017
9414
9811
040207
040602
040998
397
no
041393
041787
042182
1
042576
042969
043362
043755
044148
044540
044932
393
1
5323
5714
6105
6195'
63S5
7275
7664
8053
8442
6830
390
2
9218
9606
9993
050380
050766
051153
051538
051924
052309
052694
336
3
053076
053463
053^:46 4230
4613
4996
5378
5760
6142
6524
383
4
6905
72-6
7606 80^6
8426
8805
9185
9563
9942
060320
379
5
06069S
061075
061452 001829
002206
062582
062958
063333
063709
4083
376
6
445S
4S32
5206
55S0
5953
6326
6099
7071
7443
7815
373
7
8186
8557
892S
9293
9663
070038
070407
070776
071145
071514
370
8
0718S2
072250
072617
072985
073352
3718
4085
4451
4816
5182
3G0
9
5547
5912
6276
6640
7004
7303
7731
8094
8457
8819
363
120
079181
079543
079904
0S0266
030626
080987
081347
081707
082067
082426
360
1
0327S5
083144
0S3503
3361
4219
4576
4934
5291
5647
6004
357
2
6360
6716
7071
7426
7781
8136
8490
8845
9198
9552
355
3
9905
09025S
090611
090963
0913t5
091667
092018
092370
092721
093f!71
352
4
093422
3772
4122
4471
4820
5169
5518
6866
6215
6562
349
. 5
6910
7257
7604
7951
8293
86^14
8990
9335
9631
100026
340
6
100371
100715
1010.59
101403
101747
102091
102434
102777
103119
3462
;343
7
3S04
4146
4487
4828
5169
5510
5851
6191
6531
6371
341
8
7210
7549
7883
8227
8565
8903
9241
9579
9910
n0253
338
9
110590
110926
111263
111599
1119:34
112270
112605
112940
113275
3609
335
130
1139t3
114277
il4611
114944
115278
11.5611
11.5943
116276
11600ft
116940
333
1
7271
7603
7931
8265
8595
8926 9256!
9586
9915
120245
330
2
120574
120903
121231
121560
121838:12221'>|122544|
122871
123198
3525
328
3
3S52
4178
4504
4330
5156
.5481 5806
6131
6-356
6781
325
4
7105
7429
7753
8076
8399
8722 9045
9368
9690
130012
323
5
130334
130655
130977
131298
131619
131939 132260
132580
132900
3219
321
6
3539
3-^5S
4177
41P6
4314
5133 5451
5769
60S6
6403
318
7
6721
7037
7354 7671
7987
8.303 8r,18
8934
9249
9564
316
8
9S79
140194
14050S
140822
141136
141450 141763
142076
142369
14270:<;
314
9
143015
3327
3639
3951
4263
4574
4885
5196
5507
5818
311
140
14612S
14643S
146743
147058
147.367
147676
147985
148294
148603
148911
309
1
9219
9527
9835
150142
150149
150756
151063
151:370
151676
151952
307
2
1522SS
152594
1529(10
3205
3510
.3315
4120
4424
4728
5032
305
3
5336
.56411
5943
, 6246
6549
6352
7154
7457
7759
6061
303
4
8362
86f54
8965
9266
9567
936-^
160163
100469
160769
16106S
301
5
161 36S
161667
161967
1162266
162564
162363
3161
3460
37.58
4055
299
6
4353
4650
4947
: 5244
5.541
5833
61:34
6430
6726
7022
297
7
7317
7613
790s
i 8203
8497
8792
9086
9360
9674
9968
295
8
170262
170555
170848
171141
171434
171726
172019
172311
172603
172895
293
9
3186
3478
3769
4060
4351
4641
4932
5222
5512
5802
291
150
176091
1763^1
176670
176959
177248
177536
r7325
178113
178401
178689
289
1
8977
926 1
9552 9839
,180126
180413
180699
180986
181272
181558
287
2
181 844
182129
182415 18270!)
2985
3270
3555
3f339
4123
4407
285
3
4691
4975
.5259 5.542
5S25
6108
6.391
6674
6956
7239
283
4
7521
7S03
8n84 8366
8R47
8928
9209
9490
9771
190051
231
5
190332! 19061 2
190-^92 191171
191451
1917.30
192010
192289
192567
2S46
279
6
3125 3403
3681 3959
4237
4514
4792
5069
.5346
5623
278
7
5900 6176
6453 6729
7005
7231
7556
7832
8107
8382
276
8
8657' 8932
9206 94SI
9755
200029
200303
200577
200850
201124
274
9
{NO.
201397
0
201670
201943
20*22 16
I 3
; 2024 38
2761
3033
3305
3577
3848
272
Diff.
1
a
I *
5
6
7
8
9
TABLE XII. LOGAPJTHMS OF NUMBERS.
[j1
No.! O
1
2
3
GS26
9515
2I21S3
7434
22a 1 OS
2716
5309
7SS7
3
170
11
2;
3
4!
5
6
7
S
9
230149
299G
5o-23
8016
210ol9
30:}S
5513
7073
25012 »
2353
204391
7096'
9783
212154
5109
7747
220370'
2976|
5563
S144
230701
3250
130,
i!
2
3
4
5'
S\
7
8
9
255273
7679
260:J7l
2451
4313
7172
9513
271312
4153
6162
5731
3297
210799
3236
5759
R219
250661
3J96
2.)1663
7365
210)51
2720,
5373 1
soio;
220631:
32361
5326'
8409
230959
3501
6033
8543
241043
3531
6096
8461
250903
3333
2019311
7631
210319
2936
5633
8273
220392
3196
60311
8657
231215
3757
6235
8799
241^297
3782
6252
8709
251151
3530
205204
7901
210536
3252
5902
8536
221153
3755
6312
8913
20547
3173
210353
3513
6166
8793
221414
4015
66o;)
9170
8
205746
84411
211121'
3783
6130
9060
221675
4274
6858
9426
190 278754
1231033
2! 3301
7802
290035
2256
4466
6665
8353
255514
7913
260310
2633
5051
7406
9746
272074
4339
6692
278932
231261
3527
5782
8026
290257
2473
4637
6334
9071
2J0/0-J
8153
260513
2925
5290
7611
9930
272303
4620
6921
255996
8393
260787
3162
5525
7875
'270213
2533
4350
7151
231470
4011
6537
9049
211516
4030
6199
8951
25139
3322
256237
8637
261025
3399
5761
8110
270116
2770
5031
7330
^00 301030
3196
5351
7496
9639
311751
3367
5970
8)63
9,320146
210
322219
30124
3112
5566
7710
9343
311966
4073
6130
8272
320354
322426
1
4232
4433
2
6336
6541
3
8330
8533
4
330114
330617
5
2433
2640
6
4451
4655
7
6160
665)
8
8456 8656
9
310144 310612
No.
0
1
27921 1
231433
3753
6007
8219
290430
2699!
4907'
7104
9239
301464
3623
5781
7924
310056
2177
4259
6390
8131
32J562
322633
4691
6745
8787
330319
2312
4356
6360
8355
310311
3
231721
4261
6739
9299
241795
4277
6745
9193
251633
4064
256177
8877
261263
3636
5996
8344
270679
3001
279439
231715
3979
6232
8473
290702
; 2920
I 5127
7323
9507
279667
231942
4205
6456
8696
290925
3141
5347
7542
9725
5311
. 7609
279395
232169
4431
6631
8920
291147
3363
5567
7761
9943
231979
4517
7041
9550
242044
4525
6991
9443
251881
4306
256718
9116
261501
3373
6232
8578
270912
3233
554?
206016
8710
211333
4049
6691
9323
221936
4533
7115
9632
232234
4770
7292
9300
242293
4772
7237
9637
252125
4543
9
206236
8979!
211651
4314
6957
9535
222196
4792
7372
9933
Diff.
206556
9247
211921
4579
7221
9346
222456
5051
7630
230193
301631 301893
3344 4059
599Gi 6211
8 137 1 8351
3102631310431
256953
9355
261739
4109
6467
8812
271144
.3464
5772
8067
232488
5023
7541-
240050
2541
5019
7432
9932
252363
4790
257193
9591
261976
4346
6702
9046
271377
3696
6002
8296
230123
2396
4656
6905
9143
291369
3584
5787
7979
232742
5276
7795
240300
2790
5266
77231
250176
2610
5031
257439
9333
262214
4532
6937
9279
271609
3927
6232
8525
2339
4499
6599
8639
320769
322339
4399
6950
8991
.331022
.3011
5057
7060
9054
311039
2609
4710
6309
8393
320977
323046
5105
7155
9194
331225
3216
302114
4275
6425
8561
310693
2312
4920
7018
9106
321181
300161
302331
4491
6639
8778
310906
3023
5130
7227
9314
321391
230351
2622
4332
71.30
9366
29159!
3301
6007
8193
300373
302517
4706
6351
899
311113
3231
5340
7436
9522
321593
271
269
267
266
261
262
261
259
253
256
255
253
252
250
219
248
246
245
243
242
241
239
233
237
235
234
233
2.32
23C
229
5257
7260
9253
311237
323252
5310
7359
9393
.331427
3447
5458
7459
9451
3414.35
323453
5516
7563
9601
331630
3619
5653
7659
96.50
341632
230573
2349
5107
7354
9539
291813
4025
6226
8416
300595
302764
4921
7063
9204
311330
3445
5551
7616
973n
32130."
280806
3075
5332
7578
9812
292031
4246
6446
8635
300313
302930
5136
7232
9417
311542
3656
5760
7854
9933
.322012
323665
5721
7767
9305
331S32
3350
5859
7853
9349
341330
323371
5926
7972
330003
2034
4051
6059
8053
340047
2023
8
324077
6131
8176
3.30211
2236
4253
6260
8257
340246
2225
9
228
227
226
225
223
222
221
220
219
218
21i
216
215
213
212
211
210
209
203
207
206
205
204
203
202
202
201
200
199
otff.ji
158
TABLE XII. LOGARITHMS OF NUMBERS.
No.
220
0
1
342620
3
342317
3 I
4:
343212
5
6
7
343302
8
343999
9
Diff.
31^423
343014
343409
343606
344196
197
1
4392
4539
4785
4981
6178
5374
5570
5766
5S62
6157
196
2
6353
6549
6744
6939
7135
7330
7525
7720
7915
8110
195
3
8305
8500
8694
8889
9083
9278
9472
9666
9360
350054
194
4
350248
350442
350636
350829
351023
351216
351410
351603
351796
1939
193
5
2183
2375
2563
2761
2954
3147
3339
3532
3724
3916
193
6
4103
4301
4493
4635
4876
5063
5260
5452
6643
5834
192
7
6026
6217
64G3
6599
6790
6931
7172
7363
7554
7744
191
8
7935
8125
8316
8506)
8696
8336
9076
9266
9456
9646
190
9
9835
360025
360215
360404
360593
360783
360972
361161
361350
361539
1S9
230
361723
361917
362105
362294
362482
362671
362859
363048
363236
363424
188
1
3612
3S00
3933
4176
4363
4551
4739
4926
6113
5301
138
2
5483
5675
5862
6049
6236
6423
6610
6796
6983
7169
187
3
7356
7542
7729
7915
8101
8287
8473
8659
8845
9030
186
4
9216
9401
95S7
9772
9953
370143
370328
370513
370698
370383
185
5
371063
371253
371437
371622
371806
1991
2175
2360
2544
2728
184
6
2912
3096
3230
3464
3647
3831
4015
4198
4382
4565
184
7
4748
4932
5115
5298
5481
5664
5846
6029
6212
6394
133
8
6577
6759
6942
7124
7306
7438
7670
7852
8034
8216
132
9
8398
8580
8761
8943
9124
9306
9487
9668
9849
380030
181
240
380211
330392
330573
380754
330934
331115
331296
331476
381656
381837
181
1
2017
2197
2377
2557
2737
2917
3097
3277
3456
3636
ISO
2
3315
3995
4174
4353
4533
4712
4391
5070
5249
5423
179
3
5606
5735
5964
6142
6321
6499
6677
6356
7034
7212
178
4
7390
7563
7746
7923
8101
8279
8456
S634
8811
8989
178
5
9166
9343
9520
9693
9875
390051
390226
390405
390532
390759
177
6
390935
391112
391288
391464
391641
1317
1993
2169
2345
2521
176
7
2697
2873
3048
3224
3400
3575
3751
3926
4101
4277
176
8
4452
4627
4802
4977
5152
5326
5501
5676
5850
6025
175
9
6199
6374
6543
6722
6896
7071
7245
7419
7592
7766
174
250
397940
398114
398237
398461
393634
398808
393981
399154
399323
399501
173
1
9674
9347
400020
400192
400365
400533
40071 1
400383
401056
401228
173
2
401401
401573
1745
1917
2039
2261
2433
2605
2777
2949
172
3
3121
3292
3464
3635
3307
3978
4149
4320
4492
4663
171
4
4S34
5005
5176
5346
5517
6688
6858
6029
6199
6370
171
5
6540
6710
6381
7051
7221
7391
7561
7731
7901
8070
170
6
8240
8410
8579
8749
8918
9037
9257
9426
9595
9764
169
7
9933
410102
410271
410440
410609
410777
410946
411114
411283
411451
169
8
411620
1738
1956
2124
2293
2461
2629
2796
2964
3132
163
9
3300
3467
3635
3S03
3970
4137
4305
4472
4639
4806
167
260
414973
415140
415307
415474
415641
415808
415974
416141
416308
416474
167
1
6641
6307
6973
7139
7306
7472
7638
7804
7970
8135
166
2
8301
8467
8633
8798
8964
9129
9295
9460
9625
9791
165
3
9956
420121
420236
420451
420616
420781
420945
421110
421275
421439
165
4
421604
1763
1933
2097
2261
2426
2590
2754
2918
30S2
164
5
3246
3410
3574
3737
3901
4065
4228
4392
4555
4718
164
6
4332
5045
5203
5371
5534
5697
5860
6023
61S6
6349
163
7
6511
6674
6336
6999
7161
7324
7486
7648
7311
7973
162
8
8135
8297
8459
8621
8783
8944
9106
9268
9429
9591
162
9
9752
9914
430075
430236
430398
430559
430720
430881
431042
431203
161
270
431364
431525
431635
431846
432007
432167
432328
4324S8
432649
432809
161
1
2969
3130
3290
3450
3610
3770
3930
4090
4249
4409
160
2
4569
4729
4888
50-18
5207
5367
5526
5635
5844
6004
159
3
6163
6322
6481
6640
6799
6957
7116
7275
7433
7592
159
4
7751
7909
8067
8226
83S4
8542
8701
8359
9017
9175
153
5
9333
9491
9643
9806
9964
440122
440279
440437
440594
440752
158
6
440909
441066
441224
441381
441533
1695
1352
2009
2166
2323
157
7
2480
2637
2793
2950
3106
3263
3419
3576
3732
3889
157
8
4045
4201
4357
4513
4669
4825
4981
5137
5293
5449
156
9
No.
5694
5760
5915
6071
3
6226
6382
5
6537
6
6692
6348
7003
155
0
1
3
4:
7
8
9
Diff.
-^1
TABLE XII. LOGARITHMS OF NUMBERS.
159
No 0
1
2
3
4
290
1
2
3
4
5
447153
87061
450219
17S6
331S
4S15
6366
78S2
9392
46JS'JS
46239S
3393
53 S3
6S63
S347
, 9322
6 471292
a
2756
4216
5671
447313
8S61
450403
1940
3171
4997
651S
8033
9543
46104
462543
4042
5532
7016
8495
9969
47143S
2903
4362
5816
447463
90151
450557
2093
3624
5150
6670
8l8i
9694
461198
447623
9170
450711
2247 1
3777|
5302
0S21
8336
«S45
461348
:447778
1 9324
1450365
2100
I 3930
.300,477121
1 8566
2430007
462697
4191
5630
7164
8643
470116
1535
3019
450S
5962
3
4
5
6
7
8
9
310
1
2
3
4
5
6
7
8
9
320
1|
3
^1
6
7
8
9
462847
4340
5329
7312
8790
470263
1732
3195
4653
6107
5454
6973
6437
9995
461499
462997
4490
5977
7460
8933
470410
187S
3341
4799
6252
8
443242
9787
451326
2859
4387
5910
7428
8940
460447
1943
1413
2S74
4300
5721
7133
8551
9953
491362
2760
4155
5541
6930
831
95S7
501059
2127
3791
477266
871!
480151
15S6
3016
4142
5363
7230
8692
77411
4
«3oo
450294
1729
3159
4535
6005
7421
8333
463146
4639
6126
7603
9035
470557
2025
3437
4944
6397
490099 490239
491502
2900
4294
5633
7063
8443
. 9324
1501196;
2564
3927
477555
8999
430433
1872
3302
4727
6147
7583
8974
490330
477700
9143
430582
2016
3445
4369
6289
7704
9114
490520
491612
3040
4433
5322
7206
8586
9962
501333
2700
4063
505150
6505
785i
9203
510515
i 1833
3218
4513
5874
7196
463296
4783
6274
7756
9233
470704
2171
3633
5090
6542
463445
4936
6423!
7904
9330
470351
2318
3779
5235
6637
448397
9941
4-31479
3012
4540
6062
7579
9091
460597
2093
463594
448552
450095
1633
3165
4692
6214
7731
9242
460743
2248
463744
491782
3179]
4572
5960
7344
8724
500099
1470
2837
4199
491922
477844
9287
480725
2159
3587
5011
6430
784;
9255
490661
492062
3319
4711
6099
7483
8362
500236 1500374
1744
3453
4350
6233
7621
8999
505236
6640
7991
9337
510679
20171
3351
46S1
6006
7323
330
1
2
3
4
5
6
T
I
8
9
51S514
9323
521133
2444
3746
5045
6339
7630
3917
V30200
518646
9959
521269
2575
3376
5174
6469
7759
9045
505421
6776
8126
9471
510313
2151
3434
4313
6139
7460
518777
520090
1400
2705
4006
505557
6911
8260
9606
510947
2234
3617
4946
6271
7592
1607
2973
4335
505693
7046
8395
9740
511031
2418
3750
5079
6403
7724
No. O
5304
6593
7888
9174
530456
3
520221
1530
2335
4136
5434
6727
8016
9302
530534
519040
520353
3109
4471
505823
7181
8530
9374
511215
2551
3333
5211
6535
7855
519171
520434
477939
9431
430369
2302
3730
5153
6572
7986
9396
490801
192201
3597
4939
6376
7759
9137
590511
1830
3246
4607
478133
9575!
481012
2445
3872
5295
6714
8127
9537
490941
5035
6571
8052
9527
470993
2464
3925
5331
6332
478278
9719
481156
2588
4015
b3Jij
8269
9677
491031
5234
6719
8200
9675
471145
2610
4071
5526
6970
478422
9363
481299
2731
4157
5579
6997
8410
9318
491222
492341
3737
5128
6515
7397
9275
500643
2017
3332
4743
492481
3376
5267
6653
8035
9412
5007S5
2154
3518
4878
505964
7316
8664
510009
1349
2684
4016
5344
6663
7937
492621
4015
540e
6791
8173
9550
500922
2291
3655
5014
u06099
7451
8799
U10143
1482
2318
4149
5476
6300
8119
Diff. '
155
154
154
153
153
152
152
151
151
150
150
149
149
148
148
147
146
146
146
14;'
!45
144
144
143
143
142
142
141
141
140
140
139
139
1391
1381
I33i
1371
137
136
136
506234
7536
8934
510277
1616
2951
4282
5609
6932
8251
1661
1792
2966
3096
4266
4396
5563
5693
6356
6935
8145
8274
9130
9559
530712
530340
519303
520615
1922
3226
4526
5822
7114
8402
9637
530963
6
520745
2053
3356
4656
.5951
7243
8531
9815
531096
519566
520376
2183
3486
4785
6031
7372
8660
9943
531223
8
506370
7721
9063
510411
1750
3034
4415
5741
7064
8382
519697
521007
2314
3616
4915
6210
7501
8788
530072
135
136
135
135
134
134
133
133
133
132
132
13!
131
131
130
130
129
129
129
128
128
Diff.i
IbU
TABLE XII. LOGARITHMS OF
.NUMBERS.
No.
340
0
531479
1
531607
a
3
531862
4:
531990
5
6
7
532372
8
9
Diff.
123
531734
532117
532245
53250D
532627
1
2754
2332
3009
3136
3264
3391
3518
3645
3772
3S99
127
2
4026
4153
4230
4407
4634
4661
4787
4914
5041
5167
127
3
5294
5421
5547
5674
5800
5927
6053
6180
6306
6432
126
4
6558
6635
6311
6937
7063
7139
7315
7441
7567
7693
126
5
7819
7945
8071
8197
8322
8443
8574
8699
8325
8951
126
6
9076
9202
9327
9452
9578
9703
9829
9954
540079
540204
125
7
540329
540455
540530
r40705 540330 1
540955
541030
541205
1330
1454
125
8
1579
1704
1829
1953
2078
2203
2327
2452
2576
2701
125
9
2S25
2950
3074
3199
3323
3447
3571
3696
3820
3944
124
350
544063
544192
544316
544440
544564
544635
544812
544936
545060
545183
124
1
5307
5431
5555
5673
5302
5925
6049
6172
6296
6419
124
2
6543
6666
67S9
6913
7030
7159
7282
7405
7529
7652
123
3
7775
7898
8021
8144
8267
8389
8512
8635
8758
6881
123
4
9003
9126
9249
9371
9494
9616
9739
9S61
9934
550106
123
5
550223
550351
550473
550595
550717
550840
550S62
551034
551206
1328
122
6
1450
1572
1694
1316
1938
2060
2181
2303
2425
2547
122
7
2663
2790'
2911
3033
3155
3276
3398
3519
3640
3762
r^i
8
3383
4004
4126
4247
4368
4439
4610
4731
4852
4973
121
9
5094
5215
5336
5457
5578
6699
5820
5940
6061
6182
121
360
556303
556423
556544
556664
5567S5
556905
557026
557146
557267
5573S7
120
A
7507
7627
774«
7868
7988
8108
8228
8349
8469
8689
120
2
8709
8S29
S94S
9063
9188
9308
9423
9548
S667
9787
120
3
9907
560026
560146
560265
560385
560504
560624
560743
560363
560982
119
4
561101
1221
1340
1459
1573
1693
1817
1936
2055
2174
119 j
5
2293
2412
2531
2650
2769
2837
3006
3125
3244
3362
119
6
3431
3600
3718
3337
3955
4074
4192
4311
4429
4548
119
7
4666
4734
4903
5021
5139
5257
5376
5494
6612
5730
lis
8
5348
5966
6034
6202
6320
6437
6555
6673
6791
6S09
118
9
7026
7144
7262
7379
.7497
7614
7732
7849
7967
8084
lis
370
563202
568319
563436
568554
563671
568788
568905
569023
569140
569257
117
1
9374
9491
960*
9725
9342
9959
570076
570193
570309
570426
117
2
570543
570660
570776
570393
571010
571126
1243
1359
1476
1592
117
3
1709
1325
1942
2058
2174
2291
2407
2523
2639
2755
116
4
2S72
29.-;8
3104
3220
3336
3452
3563
3684
3800
3915
116
5
4031
4147
4263
4379
4494
4610
4726
4841
4957
5072
116
6
5183
5303
5419
5534
5650
5765
5880
5996
6111
6226
115
7
6341
6457
6572
6637
6302
6917
7032
7147
7262
7377
115
8
7492
7607
7722
7836
7951
8066
8181
8295
£410
8525
115
9
8639
8754
8868
8983
9097
9212
9326
9441
9555
9669
114
380
579784
579398
580012
580126
580241
580355
580469
580583
580697
580811
114
]
580925
581039
1153
1267
1381
1495
1608
1722
1836
19£0
114
2
206:i
2177
2291
2404
2518
2631
2745
2858
2972
3085
114
3
3199
3312
3426
3539
3652
3765
3879
39921 4105
4218
113
4
4331
4444
4557
4670
4783
4396
5009
51221 5235
5348
113
5
5461
5574
5636
5799
5912
6024
6137
6250
6362
6475
113
6
6537
6700
6312
6925
7037
7149
7262
7374
7486
7599
112
7
7711
7823
7935
8047
8160
8272
8334
8496
8608
8720
112
S
8332
8944
9056
9167
9279
9391
9503
9615
9726
9838
U2
9
9950
590061
590173
590284 590396
590507
590619
590730 590342
590953
112
390
591065
591176
5912S7
591399 591510
591621
591732
591.843 591955
592066
111
1
2177
2238
2399
2510
; 2621
2732
2343
2954; 3064
3175
111
2
3236
3397
3503
3613
: 3729
3340
3950
4(:6i: 4171
4232
111
3
4393
4503
4614
4724
4834
4945
5055
51651 6276
5336
110
4
5496
5606
5717
5827
5937
6047
6157
6267
6377
6487
110
5
6597
6707
6317
6927
1 7037
7146
7256
7366
7476
7586
i 110
6
7695
7805
7914
8024
8134
8243
8353
8462
8572
8631
1 no 1
7
8791
8900
9009
9119 9223
9337
9446
9566
9665
9774' 1091
8
9333
9992
600101
600210 600319
,600423 600537
600646 600755 eO0.-;64 liiy||
No
600972
601082
1191
1299 1403
1517
5
1625
6
1734
7
1843; 1951
109
0
I 1
a
3
4:
8 1 9
Diff
TABLE Xll. LOGAItlTHMS OF NUMBEUS.
161
I No.'
!4U0
1
2
3
4
5
6,
7i
0^1 1
6lv2()6U 602169
3M4
4226
53):')
6331
7455
8526
9194
3253
4334
54)3
6439
7562
6633
9701
S 610660 610767
a
9
410
I
2
3
4
1723
6127S4
3342
4897
5950
7000
8043
9093
620136
1176
2214
420 623249
4232
5312
6340
7366
8339
6j 9410
7 63 )ia»5
8 1444
2157
1629
612390
3947
5003
6055
7105
8153
9193
620240
1230
2313
623353
4335
5115
6443
7463
8491
9512
63053:)
1515
2559
602277
3361
4442
5521
6596
7669
8740
9803
610373
1936
602336
3469
4550
5623
6704
7777
8347
9914
610979
2012
612996
4053
5103
6160
7210
8257
9302
620344
1334
2421
623456
4438
5518
6516
7571
8593
9613
613102
4159
5213
6265
7315
8362
9406
62()443
1433
2525
602494
3577
4658
5736
6311
7834
8954
610021
1036
2143
613207
4264
5319
6370
7420
8466
9511
620552
1592
2623
5
602603
3636
4766
5344
6919
7991
9061
610123
1192
2254
613313
4370
5424
6476
7525
8571
9615
620656
1695
2732
6
602711
3794
4374
5951
7026
8093
9167
610234
1296
2360
«
613419
4475
623559 623663
430 633463
1 4477
5434
6438
7490
8439
9436
7613131
8i 1474
9 I 2455
440 613453
4591
5621
6643
7673
8695
9715
630631 630733
1647 1743
II
2i
4i
5i
61
4439
5422
6104
7353
8360
9335
633569
4578
5534
6533
7590
8539
9536
640531
1573
2563
643551
4537
5521
6502
7431
8453
9432
2660
633670
4679
8
602319 602926
4010
50^9
6166
7241
8312
9331
610447
1511
257-2
9 Diffi
7 65030S 650405
8
9
450
1
2
3
4
5
6
7
1375
2343
1276
2246
653213
4177
5133
6093
7056.
8011
8965
9916 660111
660365 0960
1813 1907
5635
6633
7690
8639
9636
64003)
1672
2662
643650
4636
5619
6600
7579
8555
9530
650502
1472
2440
2761
633771
4779
5735
6789
7790
8739
978^
610779
1771
2761
643749
4731
5717
6693
767^
8653
9627
650599
1569
2536
4695
5724
675
7775
8797
9317
630335
1849
2362
633372
4330
5529
6581
7629
8670
9719
620760
1799
2835
3902
4982
6059
7133
8205
9274
610341!
1405
2466
613525
4531
5634
6636
7734
8780'
9324
620364
1903
2939
5336
6339
7390
8333
9335
640379
1871
2360
623766
4793
5
6853
7373
8900
9919
630936
1951
2963
633973
4931
5936
6939
7990
8933
9934
640978
1970
2959
603036
108
4116
103
5197
103
6274
103
7313
107
8419
107
9438
1(17
610551
107
1617
106
2676
106
613630
463G
5740
6790
7839
8834
9923
62096-
201)7
3042
623869
4901
5929
6956
7930
9002
630021
1033
2052
3061
634074
5031
6037
7089
8090
9038
640034
1077
2069
3053
623973 621076
5107
6135
7161
8185
9206
630224
1241
2255
3266
653309
4273
5235
6194
7152
8107
9060
653405
4369
5331
6290
7217
8202
9155
660106
1055
2002
613847
4332
53 1 5
6796
7774
8750
9724
650696
1666
2633
643946
4931
5913
6894
7672
8343
9321
650793
1762
Na O
653502
4465
5127
6336
7313
8293
9250
660201
1150
2096
53593
4562
5523
6482
7433
8393
9346
66029G
1245
2191
653695
4653
5619
6577
7534
8133
9441
660391
1339
2236
5
644044
5029
6011
6992
7969
8945
9919
650390
1859
2326
653791
4754
5715
6673
7629
8534
9536
660436
1434
2330
6032
7053
8032
9104
630123
1139
2153
3165
63417;:>
5132
6137
7189
8190
9183
640183
1177
2163
3156
644143
5127
6110
7039
8067
9043
650016
0937
1956
2923
613736
4792
534."
6395
794;;
8989
620032
1072
2110
3146
624179
5210
6233
726:j
82S7
9306
630:326
1342
2356
3367
634276
5233
6237
7290
8290
9237
640233
1276
2267
3255
644242
108
106
105
105
105
105
104
104 I
1041
104
103
103
103
103
102
102
102
102
101
101
634376
533:
6333
7390
83^9
9337
540332
1375
2366
3354
644340
653S83
4350:
5310
6769
7725
8679
9631
660531
1529
2475
6
5226
6208
7137
816:)'
9140
650113
1081
2053
3019
653934
4946
5906
6364
7820
8774
9726
660676
1623
2569
8
6306.
7235
8262
923
6.50210
1161
2150
3116
6540^0
5042
6002
6960
7916
8870
9321
660771
1713
2663
101
101
100
100
100
100
99
99
99
99
93
93
93
93
98
97
97
97
97
97
96
96
96
96
96
95
95
95
95
9^.
9 iCiff.
Ib):^
TABLE XII. LOGARITHMS
> OF
NUMBERS.
No.
460
0 1
662753
1 1
662S52
3
3
■* 1
663135
5 6 7 \
663230 663324 663418
8 1
9
5636071
Di£L )
94
662947
563041
663512
1
3701
3795
3339
3983
^078
4172 42661
4360
4454
4543
94
2
4642
4736
4330 4924
5018
5112
5206
5299
5393
6487
&4i
3
5581
5675
5769 5862
5956
6050
6143
6237
6331
6424
94
4'
6518
6612!
6705 6799
6392
6986
7079
7173
7266
7360
94
5
7453
7546
7640
7733
7S26
7920
8013
8106
8199
8293
93
6
83S6
8479
8572
8665
S759
8852
8945
9033
9131
9224
93
7
9317
9410
9:503
9r:96
9639
9782
9875
9967;
670060
670153
93
8
670246
670339
670431 '670524
670617
670710.670302 670895
0938
1030
93
9
1173
1265
1353
1451
1543
1636 1723 1821
1913
2005
93
470
672098
672190
6722S3
672375
672467
672560 672652 672744
672836
672929
92
1
3021
3113
3205
3297
3390
3432
3574
3666
3758
3850
92
2
3942
4034
4126
4218
4310
4402
4494
45S6
4677
4J69
92
3
4361
4953
5045
5137
5223
5320
5412
6503
5595
6637
92
p
4
577S
5370
5962
6053
6145
6236
6323
6419
6511
6602
.92^
5
6694
6785
6576
6968
7059
7151
7242
7333
7424
7516
91
6
7607
7698
7789
7381
7972
8063
8154
8245
8336
8427
91
7
8518
8609
8700
8791
8832
8973
9064
9155
9246
9337
91
8
9423
9519
9610
9700
9791
9532
9973
680063,
630154
630245
91
9
630336
630426
630517 630607
680693
630789
680879
0970
1060
1151
91
' 480
681241
631332
681422
681513
681603
681693
681784
681874
631964
682055
90
! 1
2145
22:35
2326
2416
2506
2596
2636
2777
2567
2957
90
1
i 2
3047
3137
3227
3317
3107
3197
3587
3677
3767
3857
90
1
! 3
3947
4037
4127
4217
4307
4396
44.36
4576
4666
4756
90
4
4345
4935
5025
5114
5204
5294
5383
5473
5563
5652
90
5
5742
5331
5921
6010
6100
6189
6279
6368
6458
6547
89
6
6636
6726
6315
69(M
69M
7083
7172
7261
7351
7440
89
7
7529
7618
7707
7796
7836
7975
8064
8153
8242
8331
89
8
8420
S509
8593
8637
8776
8865
8953
9042
9131
9220
89
9
9309
9393
9436
9575
9664
9753
9841
9930
650019
690107
89
490
690196
690235
690373
690462
690550
690639 '690723
690318
690905
690993
89
1
1031
1170
12.53
IM7
1 1435
1524 1612
1700
1789
1877
88
2
1965
2053
2142
2230
2313
2406 2494
2533
2671
2759
•68
3
2347
2935
3023
3111
3199
3237
a375
3463
3551
3639
88
4
3727
3315
3903
3991
1 4078
4166
42^
4342
4430
4517
88
5
4605
4693
4731
4363
4956
5044
5131
5219
5307
5394
88
6
5432
5569
5657
5744
5332
5919
6007
6094
6162
6269
87
7
6356
6444
6531
6618
: 6706
6793
63.30
6963
7055
7142
87
8
7229
7317
7404
7491
^ 7578
7665
7752
7839
7926
8014
87
9
8101
8183
8275
8362
8449
1
8535 8622
8709
8796
8883
87
500
693970
699057
699144
699231 699317
699404 699491
699578
699664
699751
87
j ]
9333
9924
700011
700093 700134
700271 !700353;70O444
700531
700617
87
2
700704
700790
0377
0963 1050
1136
1222
1309
1395
1432
86
1 3
1563
1654
1741
1827 1913
1999
2036
2172
2253
2ai4
86
4
2431
2517
2603
2639 2775
2361
2947
30a3
3119
3205
86
5
3291
3377
3463
3;549 3635
3721
3507
3393
3979
4065
86
6
4151
4236
4322
4403 4494
45791 4665
4751
4337
4922
86
7
5003
5094
5179
5265 5350
5436 i 5522
5607
6693
1 5778
86
8
5864
5949
6035
6120 6206
6291
6376
6462
6547
6632
85
9
6718
6303
6338
6974 7059
1
7144
7229
7315
7400
7485
85
510
707570
707655
707740
707826 707911
707996
703031
703166
703251
703336
85
1
8421
8506
8591
S676 8761
8846! 8931
9015
9100
9185
85
2
9270
9355
9440
9524 9609
9694! 9779
9863
9943
710033
85
3
710117
710202
7102S7
710371 710456
710540 710625
710710 710794
0379
85
4
0963
1043
1132
1217 1301
1335; 1470
1554
1639
1723
84
5
1807
1892
1976
2060 2144
2229; 2313
2397
2481
2566
84
6
2650
2734
2313
2902 2936
3070 3154
3238
3323
3107
84
7
3491
3575
3659
3742 3326
39IOI 3994
4078
, 4162
4246
84
8
4330
4414
4497
4531 4665
4749 4333
4916
500C
5084
84
S
1 So
5167
5251
5335
5418 5502
5536 5669
5753
5336
592G
! 84
0
1 1
3
3
4
5
i 6
7
18 19
Diff.
1
TiiBLE XII.
LOGARITHMS OF NUMBERS.
163
No.|
0
i
3
3
4:
5
71G121
»
7
8
9
716754
Dili".
83
520 716003
716037
716170 716254
716337
7165041716588 71 6671
1
6S3-i
6921
7004
7088
7171
7254
7338
74211 7504
7587
83 1
2
7671
7754
7837
7920
8003
8036
8169
8253 83:56
8419
831
3
S502
8585
8668
8751
8834
8917
9000
9083 9165
9248
831
4
93311 9414
9497 95S0
9663
9745
9828
991 1
9994
720077
83
5
72!)15y' 720242
720325 720407
720490
720573
720655
720733
720321
0J03
83
6
09c)ii| 106S
1151 1233
1316
1398
1481
1563
1646
1728
82
7
1811
1893
1975 2058
2140
2222
2305
2337
2469
2152
82
8 263t
2716
2798 2881
2963
3145
3127
3209
3291
3374
82
9 34.-)6
353-
362 1 3702
3784
3866
3948
4030
4112
4194
S2
530 724276
724358
724 MO 724522
721604
724685
724767
724849
724931
725013
8;g
1
5095
5176
5258 5310
5422
5503
5585
5667
5748
5830
82'
2
5912
5993
6075' 6156
6233
6320
6101
6483
6564
6646
S'>l
3
6727
6309
G390: 6972
7053
7134
7216
7297
7379
7460
811
4
7541
7o23
77041 7785
7866
7948
6029
8110
8191
8273
81 !
5
8354
8135
8516] 8597
8678
8759
8841
8922
9003
9084
81!
G
916>
9246
9327 9403
9489
9570
9651
9732
9813
9893
81
7
9974
730055
730136 730217
73029S
730378
730459
730.540
730621
730702
81
S
7307S2
03G3
09441 1024
1105
1186
1266
1347
1423
1508
81
9
1589
1669
1750 1830
1911
1991
2072
2152
2233
2313
81
5-10
732394
732-174
732555 732G35
732715
732796
732376
732956
733037
733117
80
1
3197
3278
3353 3138
3518
3598
3679
3759
3839
3919
80
2
3999
4079
41 go; 4240
4320
4400
4480
4560
4610
4720
80
3
4S0O
4S30
4960: 5010
5120
5200
5-^79
5359
5139
5519
80
4
5599
5679
5759; 5833
5918
5998
G078
6157
6237
6317
80
5
6397
(M76
6556 6635
6715
6795
6371
6954
7034
7113
80
6
7193
7272
7352; 7431
7511
7590
7G70
7749
7829
7908
79
7
79S7
8067
8146; 8225
8305
8384
8463
8543
8622
8701
79
8
8781
8860
8939 9'I18
9097
9177
9256
9335
9414
949:i
79
9
9572
9651
9731 9310
98S9
9968
740047
740126
740205
740284
79
550
7403G3
740412
7 10521 1740600
740678
740757
740336
740915
740994
741073
79
I
1152
1230
1309! i:388
- 1467
1546
1624
1703
1782
186'1
79
2
1939
20!S
2036 ; 2175
2254
2332
2411
2489
2568
2647
79
3
2725
2304
2iSi> 2961
3039
3118
3196
3275
3353
3431
78
4
3510
35-;:
3567: 3745
3323
3902
3980
4058
413G
4215
78
5
4293
4 3/-i
4 119 4528
4606
4684
476:^
4340
4919
4997
78
G
5075
5153
5231
5309
5337
5465
5543
5621
5699
5777
78
7
5855
5933
6011
6089
6167
6245
6323
6401
6479
6556
78
8
6634
6712
6790
6363
6945
7023
7101
7179
7256
7334
78
9
7412
7489
7567
7645
7722
7800
7878
7955
8033
8110
78
"GO
748183
74326G
743343 748421
748493
748576
748653
748731
748808
74388."
77
1
89G3
9;>10
9118
9195
9272
9350
9427
9501
9582
r;650
77
2
9736
9811
9391
9968
750045
750123
750200
750277
750354
750431
77
375050S
75058G
750663 750740
0817
0894
0971
1048
U25
12il2
77
4
1279
1356
1433 1510
1537
1661
1741
1818
1895
1972
77
5
2018
2125
2202 2279
2356
2433
2509
258G
2663
2740
77
G
2S1G
2^93
2970 3047
3123
3200
3277
3353
343')
3506
77
7
3583
3660
3736 3313
3339
3966
4042
4119
4 1 95
4272
77
8
4313
4125
4501
4578
4654
4730
4807
4383
496 I
5036
76
9
5112
5189
5265
5341
5417
5494
5570
5646
5722
5799
76
570
755375
755951
756027
756103
756180
756256
756332
756108
756484
756560
76
I
G^13G
6712
6788
6361
6940
7016
7092
7163
724 1
7320
76
2
7396
7472
7548
7624
7700
7775
7851
7927
80' tt
8079
76
3
8155
8230
8306
8332
8458
8533
8609
8685
8761
8836
76
4
3912
8938
9063; 9139
9214
9290
9366
9441
9517
9592
76
5
9863
9743
9819 9391
9970
760045
7G0I2I
760 1 96
760272
760347
75
fi
761122
760493
760573 760619
760724
0799
0375
095!)
1025
1101
75
7
1176: 1251
1326
1402
1477
1552
1627
1702
177^
1853
75
8
9
\:)-i^\ 2003
2078
2153
2223
2303
2373
2453
2529
2604
75
2679
0
, 2754
2329
3
29ai
2978
3053
5
3123
6
3203
7
3278
8
3353
9
75
Dlff.
Na
1
3
4
164
TABLE Xll. LOGARITIOIS OF NUMBERS.
No.i
530;
0
76.:«2S
1 j
763503
3
3 i * 1
5
6
763877
7 ,
8
» :
Diff.
75
763573 763653 763727
763302
763952
764027:7641011
1
4176
42511
43261 4400^ 4475
45501 4624
4699
4774
43481
75
2
4923 4993
5072 5147 5221
5296: 5370
5445
5520
5594
75
3
5669 5743
5318 5892 5966
6041 6115
6190
6264
6333]
74
4
6413 64S7
6562
6636 6710
6785 6359
6933
7007
70821
74
5
7156 7230
7304
7379, 7453
7527 7601
7675
7749
7823
74
6
789S; 7972
8046 8I20; 8194
8268
8342
8416
8490
8564
74
7
S63SI 8712
8766i 8860 8934
9008
9082
9156i
9230
.9303
74
8
9377; 9451!
9525 9599 9673
9746
9820
98941
9968
770fi42
74
9
770115
770189;
770263 770336 7704101
1 1
770484
770557
770631
770705
0778
■ 74
590
770352
770926
770999
771073 771146
771220
771293
771367
771440
771514;
74
1
15S7
1661
1734
1803: 1331
1955
202.S
2102
2175:
2243'
73
2
2322
2395
2463
2542 2615
2688
2762
2835
2908 j
2981
73
3
3055
3123
3201
3274! 3348
3421
3494
3567
3640
3713
73
4
3786
3360
3933
40061 4079
4152
4225
4298
4371
4444
73
5
4517
4590
4663
4736! 4809
4882
4955
5028
5100
5173
73
6
5246
5319
5392
5465 5538
5610
5683
5756
5829
5902
73
7
5974
6047
6120
61931 6265
6338
&41I
6483
6556
6629
73!
8
6701
6774
6846
6919, 6992
7064
7137
7209
7282
7354
1-0
9
7427
7499
7572
7644' 7717
j
7789
7862
7934
.8006
8079
72
600
778151
773224
778296
778363 778441
778513
778585
778658
778730
778802
72
]
8374
8947
9019
9091 i 9163
9236
9303
9380
9452
9524
72
2
9596
9669
9741
9813' 9835
9957
730029
780101
780173
780245
72
3
780317 7S0aS9
730461
780533 780605
7S0677
0749
0821
0893
0965
72
4
1037; 1109. 1181
1253: 1324
1396
1468
1540
1612
1684
72
5
1755
1827
1899
1971 2042
2114
2186
2253
2329
2401
72
6
2473
2544
2616
2688, 2759
2831
2902
2974
3046
3117
72
7
3139
3260
3332
3403 3175
3546
3618
3689
3761
3832
71
' 8
3904
3975
4046
4113 4189
4261
4332
4403
4475
4546
71
9
4617
4639
4760
4331 4902
4974
5045
5116
5187
5259
71
610
735330
735401
785472
785543 785615
785686
785757
785828
785899
785970
71
1
6041
6112
6183
62^54 6325
6396
6467
6533
6609
6630
71
2
6751
6322
6393
6964 7035
7106
7177
7248
7319
7390
71
3
7460
7531
7602
7673 7744
7815
7885
7956
8027
8098
71
4
8168
8239
8310
83S1 8451
8522
8593
8663
8734
8804
71
5
SS75
8946
9016
9087 9157
9228
9299
9369
9440
9510
71
6
9581
9651
9722
9792 9363
9933
790004
790074
790144
790215
70
7
790235
790356
790426
790496 790567
790637
0707
0778
0848
0918
70
8
09SS
1059
1129
1199 1269
laio
1410
1480
1550
1620
70
9
1691
1761
1831
1901 1971
2041
2111
2181
2252
2322
70
620
792392
792462
792532
792602 792672
792742
792812
792^82
792952
793022
70
1
3092
3162
3231
330 r 3371
3441
3511
3531
3651
3721
70
2
3790
3860
39:30
4000 4070
4139
4209
4279
4349
4413
70
3
4483
455S
4627
4697 4767
4836
4906
4976
5045
5115
70
4
5185
5254
5324
5393 5463
5532
5602
5672
5741
5811
70
5
5830
5949
6019
6038 6158
6227
6297
6366
6436
6505
69
6
6574
6644
6713
6732 6352
6921
6990
7060
7129
7198
69
7
7268
7337
7406
7475 7545
7614
7683
7752
7821
73S0
69
8
7960
8029
8093
8167 8236
8305
8374
8443
8513
8582
69
9
8651
8720
8789
8858 8927
8996
9065
9134
9203
9272
69
630
799341
799409
799478
799547 799616
799685
799754
799823
799892
799661
69
1
3U0029
30009-
3001671300236 800305
800373 1800442
800511
800530
800643
69
2
0717
0786
0354
0923 0992
1061
1129
1198
1266
1335' 69
3
1404
1472
1541
1609 1673
1747
1815
1834
1952
2021
69
4
2aS9
2153
2226
2295 2363
2432
2500
2563
2637
2705
03
5
2774
2842
2910
2979 3047
3116
3184
3252
3321
3389
CS
6
34571 3525
3594
3662 3730
3793
3S67
3935
: 4003
4071
68
7
4139
4203
4276
4344 4412
4430
4548
4616
4635
4753
68
c
432 1
4339
4957
5025 5093
5161
5229
5297
! 5365
5433
63
ft
5501
5569
1
5637
5705 5773
5841
5908
5976
6044
j
6112
1 68
Na
0
2
3 4
5
6
7
1 8
9
Diff.
TABLE Xll. LOGARITHMS OF NUMBERS.
165
No.
640
1
2
3
4
5
6
7
8
9
8061801806218
6S58
7535
8211
8SS6
9560
810233
0904
1575
2245
69 6
7603
8279
8953
9327
810300
0971
1612
2312
650312913
8
9
660
1
2
3
4
5
6
7
3581
424S
4913
5578
6241
6904
7565
8226
8885
800:316
6994'
7670
8346
9021
9694
810367
1039
1709
2379
812980
3648
4314
49S0
5644
6303
6970
7631
82:)2
8951
819544
820201
0358
1514
2163
2322
3474
4126
4776
5426
819610
320267
0924
1579
2233
23S7
3539
419!
4841
5491
806451
7129
7806
84SI
9156
9829
810501
1173
1843
2512
806519
7197
7873
8549
9223
9896
810569
1240
1910
2579
813114
3781
4447
5113
5777
6440
7102
7764
8424
9083
IG70
1
2
3
4
5
6
81 9676
320333
0989
1645
2299
2952
3605
4256
4906
5556
813181
3348
4514
5179
5843
6506
7169
7830
8490
9149
826075
6723
7369
8015
8660
9304
9947
830539
1230
1870
819741
829399
1055
1710
2364
3013
3670
4321
4971
5621
806655
7332
8008
8634
9358
810031
0703
1374
2044
2713
81324
3914
4531
5246
5910
6573
7235
7896
8556
9215
_^1
806723
7400
8076
8751
9425
810093
0770
144
211
2780
9 Diff
813314
3981
4647
5312
5976
6639
7301
7962
8622
9281
806790
7467
8143
8818
9492
810165
0837
1508
2178
2847
819807
820464
1120
1775
2430
3083
3735
43S6
5036
5636
813381
4048
4714
5373
6042
6705
7367
8028
8638
9346
819873
820530
1186
1841
2495
3148
3300
4451
5101
5751
813448
4114
4780
5445
6109
6771
7433
8094
8754
9412
826140
6787
7434
8030
8724
93681
330014
0653
1294
1934
826204
6352
7499
8144
8789
1 9132
330075
0717
1358
1998
6SC
I
2
3
4
5
6
7
8
9
832509
3147
3734
4421
5056
5691
6324
6957
7533
8219
826269
6917
7563
8209
8853
9497
830139
0781
1422
2062
832573
3211
3343
4434
5120
5754
6337
7020
7652
8232
82G334
6931
7628
8273
8918
9561
830204
0845
1486
2126
832637
327."
3912
4548
5183
5317
6451
7033
77 Ir
8345
819939
820595
1251
1906
2560
3213
3865
4516
5166
5815
813514
4181
4347
5511
6175
6833
7499
8160
8320
9478
820004
0661
1317
1972
2626
3279
3930
4581
5231
5880
820070
0727
1332
2037
2691
3344'
3996
4646
5296
5945
826399
7046
7692
8333
8982
962
830263
0909
1550
2189
832700
3333
3975
4611
5247
5831
6514
7146
7773
8403
326464
7111
7757
8402
9046
9690
830332
0973
1614
2253
832764
3402
4039
4675
5310
5944
6577
7210
7841
8471
326528
7175
7821
8467
9111
9754
830396
1037
1678
2317
832323
3466
4103
4739
5373
6007
6641
7273
7904
8534
b90; 833849
1 1 9178
21840106
0733
1359
19S5
2609
3233
3355
4477
833912
9.; 41
840169
0796
1422
2047
2672
3295
3918
4539
ilwo. O
833975:
96041
840232
0359
1435
2110
2734
3357
3980
4691
3
83903S
9667
840294
0921
1547
2172
2796
3420
4042
4664
839101
9729
840357
0934
1610
2235
2859
3482
4104
4726
820136
0792
1448
2103
2756
3409
4061
4711
5361
6010
826593
7240
7886
8531
9175
9818
830460
1102
1742
2381
832892
3530
4166
4302
5437
6071
6704
7336
7967
8597
39164
9792
840420
1046
1672
2297
2921
3544
4166
4788
826658
7305
7951
8595
9239
9882
8305
1166
1806
2445
63
63
63
67
67
67
67
67
67
67
67
67
67
66
06
66
66
6G
66
66
66
€6
66
65
65
05
65
65
65
65
65
65
65
64
64
64
64
64
61
64
832956
3593
4230
4866
5500
6134
6767
7399
8030
8660
833020
3657
4294
4929
5564
619
6830
7462
8093
8723
839227
^55
840482
1109
1735
2360
2983
3606
4229
4850
839239
9918
840545
1172
1797
2422
3046
3669
4291
4912
833083
3721
4357
4993
5627
6261
6394
7525
8156
8786
8
839352 839415
9931
840608
1234
1860
2484
3108
3731
4353
4974
166
TABLE XII. LOGARITHMS OF KX'3IBEBS.
Sa O
6
8
700 S450&S Sioien S45222 S452S4 S45:i46
ll 571 S 5rS<J 5>i2 5&W 59G6
63&? &i6l 6523 6565'
S454C«S 54547' ^-"'^2 S45594 S
6337
6955
7573|
SI 59
7017,
7684
S251
S505; SS66
7tJ79 7141 7202
7696 775S 7519
8312* S374 S435
S92S S^9 9051
o-jo ci«wi fy;*
710^ri5S 351320 5513=1 =
1S70 1931
■24S» ^5*1
3695
4306
4913!
55191
3l5Ci
3759
4367
4974
555»:»
6124' 61S5
6729 67S9
If- -
2& 4
3211
3S2.J
4425
5* -
56-.
6245
6S5'J
....":
3272
35-51
6910
6025
6646
7264
75i51
84£'7
9112
C-l-AJ
6fr
67l-
7326
7943
S559
9174
: 6213
^77u 6532
73SS 7449
8004 S066
8620 8682
9235 9297
F^9 9911
" - 55C«524
- -c 1136
S4?656
6275
6594
7511
81 2S
8743
9355
9972
55C555
1!&7,
DiftM
i
3333 3394
39411
6970
2236
2546
3455
2297
23r5
2965
3577
2419
3029
£6?7
;;-—:»
-o o =.^74.53!s57513'S57574
5»:66 -5116. 5176
71 15341 1594 1654
81 2131! 2191 5251
9 2725 2757. 2547 j
Ai'. :
- ■•-->
4124
4!n5
424
5
4-
'.1
473!
47i-2
jc;
0
.' , -
64^7
7t>3l
64 ■r7
7091
OO-IS
7152
66i«5
7212
C'
<^-
-
S57fi"-'
i-.-c:i.<
^r"* . . r>-*
..-.,.
c -",*;"
5..
-
^
I
iiiO 1176. r^j<j li ' . "
1714, 1773! 153:3 1^. - - ,
2310, 2370 2430 245v 254yi 26tK?
2906, 29661 3025; 3055; 3144, 3204
2665
3263
llr.
3
-:0 1 1
5104
516:3
5-^
52-2
4,
5755
o^ -
"■"*■'
D
6257
6346
64
6
6575
e9:r
&-
. .-
^
7i«~
"?^*>^
75 r
~
5174
5i3o
5762
S521
_ _:
,1 ;/^->:;p, Qfi'w^' =636=0 863739 563799' S63555
4274 4333 4392: 4452
-.:-.- ^-r 4567 4926 49S5J f'"-"
5-541 5400 5459 5519 5575' c-
7521 75''0 79??
54 - - ■ " - "
S521 5579, S&S5, a^.
74/, i^
-r>"-*-y «..~ J-.
a-'i^ ^;?-J
! I "i I '"'1 "i
fe ^Q-'.- ico^^Li vcpjv c>'0'r.r;pQ-rj-i
-X, Ig^o, lt>o^
Si 2?56' 2215
I
25->5
Q10-
iii*,ii,
^^"'j'
I,
5«i j
25(^6
3055;
2564 26221
3146 3204
^07 crx: =
730 875061 875119 875177 875235 575293 875351 S75409
]i 5640 5695 5756
5640
^15
6795
2
51
6
7
8
9
Ko O
5695
G276
6553
69101
552^
■557^
-6:37
9096
915:3
921'
9669
9726
9:
550242
551299 350::"
5513
^91
6^5
7544
5119
5694
5571
6449
7026
76f^
5177
5752
5929
6507 i
7053*
7659
8234
55'!'9
62
62!
62
62
62
62
61
61
61
61
=51625551656 S51747'S51809! 61
61
611
61!
6II
61"
GO
60
60
60
59
59
59
59
- : _ .
^~ "^ -
-. --
■- - -
5409
87.=466
S75524
875.=,
..'!
5957
6045
6102
6
6564
6622
6650
6:.:
.'.-
7141
7199
7256
7314
58;
7717
7774
7532
7559
58i
^292
049
6407
8464
57 j
8566
=924
895!
9CS9
01 ■■
944*^'
9497
95 ~."
f^?!2
571
6
Diff.
TABLE XII. LOGABlTHi:
NUMBERS.
161
2!
4;
6;
S;
9
0_
ida5\
l&55i
3*51!
i7ii5
5361
I/ii
144-2
2)JI2
25^1
3i->J
371S
42>5
4>52
541S
5953
14:
at
3-.
37:
43-^
4^.>9
5474
603d
~ll-56^r.-
1727 ."
2^7 -
•25j->* -2:-
4i(e"i;
5531
50-^1
61-52'
3'-i7-
4->.>'
5135
57aj
G265
5?1:<2
57-57
6321
KS7;
^34 56 1
-47 SS66''4 ssScfifX 55671^:
5 S»2i 9355 9-
6 ; ~ -
7 H - - _
7955 9311' =*:«7
S516 -"" --^
?:'77
1705^ I7e&i ISlSi lS7:i 1225: laii:?
.£>:* s^
175? 5
3
4
5
6
7
9
-
- -
37621
S517
431^'
».-^— -
45:
- -
.54i-
;--. -
-5375
&iX>
65-26
65? i
7077
7132
^w-%— r» ~-~
1 ^ 59-^^ 59^2
-^:^- j
6^36
79f"; 597^7 '^765-2
1 Si
3 S7:
r297i 7352: 7iJ7 7-iS^ ■>
tax
• r-
14
2-
-- -
27641 *5iS' 2S73 29271 :^?.
3 471t
3i a
9^ 7949i 5j»2
1 '
■2 f
Sf"fc%5 -ii- 5163; 3217; 8270 53^^
1
^j5699.
o
6
9
11-5?
16^1'
2753
3254
O
1-.
17-.
2*75
2506
.3:»7
i4-2j
.:.; .r: ..--. 19^
23a^ 2351, ^33 24S5
2559 291^: 2966 ^319
3391.V 3143 3496 3549
3
36/2 :je'-5
6 7
3.<.«?
-i.Ci;
Mt'
168
TABLE Xll. LOGARITHMS OF NUMBERS.
No.
0
1
913567
3
3
913973
4t
5
6
914132
t
8
9142-37
9
DiB.]
820:913314
913920
914026
914079
914134
914290
53 1
1
4ai3
4396 4449
4502
4555
4603
4660
4713
4766
4819
53'
2
4S72
492r 4977
5030
5033
5136
5139
5241
5294
5347
53 i
3
5400
545b 5505
55-53
5611
5664 5716
5769
5822
5375
53 1
4
5927
5930 60a3
6035
6138
6191 6243
6296
6349
6101
53 :
5
fr454
6-507
6559
6612
6G64
6717 6770
6322
6875
6927
£3'
6
6930
7033
7035
7135
7190
7243
7295
7343
7400
7453
53; i
7
7506
7553
7611
7663
7716
7763
7320
7373
7925
7973
52:
8
8030
8083
8135
8163
8240
3293 8345
8397
8450
~ 8502
52
9
8555
8607
8&59
8712
8764
8316 8369
8921
8973
9026
52
830
919073
919130
919183
9192-35
919237
919310 919392
919444
919496
919549
62
1
9601
9653
9706
975-3
9310
9562 9914
9967
920019
920071
52
2
920123
920176 92022.S
920230
920-332
920334 920136
920439
0-541
0593
52
3
0645
0697
0749
0301
0353
09061 09-53
1010
1062
1114
52
4
1166
1213
1270
1322
1374
1426
1473
1530
1532
1634
52
5
1656
1735
1790
1342
1S94
1946
1993
2050
2102
21.54
52
6
2206
22-53
2310
2362
2414
2466
2513
2570
2622
2674
52
7
2725
2777
2329
2331
2933
2935
3037
3039
3140
3192
52
8
3244
3-296
3345
3399
3451
3-j03
355-5
3607
3658
3710
52
9
3762
3314
3365
3917
3969
4021
4072
4124
4176
4223
52
840
924279
924-331
924383
9^44-31
924486
924533
924539
924641
924693
924744
52
1
4796
4343
4399
4951
5003
5a54
5R6
5157
5209
5261
52
2
5312
5364
5415
5-167
5513
5570
5821
5673
5725
5776
52
3
5828
5379
5931
5932
6034
6035
6137
6188
6240
6291
51
4
6342
6394
6445
6497
6543
6600
6651
6702
6754
6305
51
5
6857
6903
6959
7011
7062
7114
7165
7216
7263
7319
51
6
7370
7422
7473
7524
7576
7627
7673
7730
7781
7332
51
7
7833
7935
7936
8037
8033
8140
8191
8242
8293
8345
51
8
8396
8447
8493
S549
S601
36-52
8703
8754
aso5
8357
51
9
8903
8959
9010
9061
9112
9163
9215
9266
9317
9363
51
850
929419
92^70
929521
929572
929623
&29674
929725
929776
929827
929379
51
i
9930
9931
930032
930033
930134
930135
930236
930287
930333
930339
51
2
930440
930491
0542
0592
0643
0694
0745
0796
0347
0393
51
3
0949
lOOO
1051
1102
1153
1201
1254
1305
1356
1407
51
4
14-53
1509
1560
1610
1661
1712
1763
1314
1365
1915
51
5
1966
2017
2065
2113
2169
2220| 2271
2322
2372
2423
51
6
2474
2324
2575
2626
2677
2727 2773
2329
2S79
29-30
51
7
2931
3031
3032
31-33
3133
32-31 3235
3335
3336
3137
51
8
»137
a533
3539
3639
3690
37401 3791
3341
3392
3943
51
9
3993
4014
4094
4145
4195
4246
4296
4347
4397
4443
51
860
934498
93i549 931599
931650
931700
934751
931801
934352 934902
931953
50
J
5003
5054 1 5104
5154
5205
5255
5306
5356 54C6
5457
50
•2
5507
55.53 5603
56-53
5709
5759
5309
5360 5910
5960
50
3
6011
60611 6111
6162
6212
6262
6313
6363
6413
6463
50
4
6514
6564
6614
6665
6715
6765
6315
6365
6916
6S66
50
5
7016
7066
•7117
7167
7217
7267
7317
7367
7413
7463
50
6
7518
7568
7613
7663
7718
7769
7819
7369
7919
7S69
50
7
8019
8069
8119
8169
8219
8269
8320
8370
8420
3470
50
S' 8520
8570' 8620
3670
3720
8770
8320
8870
8920
6970
50
9| 9020
9070 9120
9170
9220
9270
9320
9369
9419
9469
50
870
939519
9.39-569 939619
939669
939719
939769
939319
939869 939918
939963
50
1
i«0018
W0063 9401 13
940165
940213
940267 940317
940367 940417
940467
50
2
0516
0566 0616
0666
0716
0765
0315
0565 0915
0964
50
3
1014
1061
1114
1163
1213
1263
1313
1362 1412
1462
50
4
1511
1561
1611
1660
1710
1760
1809
13-59 1909
19.58
50
5
2003
2058
2107
21-57
2207
2256
2306
2355 2405
2455
50
6
2504
2-554
2603
2653
2702
2752
2301
■ 2S51 2901
29-50
50
7
3000
3049 3099
3143
3193
3247
3297
3-316 3396
3145
49
8
3495
3-544 3-593
3643
3692
3742 3791
3841 3390
3939
49
9
3939
40-3.3 4033
1 4137
1 3
4136
4236 4235
4335 4.384
44-33
49
Diff.
No.
0
1 1
i 3
4:
5
6
7 8
9
TABLE XII. LOGARITHMS OF x\U3IBtKS.
169
No,
550
1
2
3
4
5
6
7
8
9
O
9444S3
4976
."-169
5961
mo2
6'J43
7434
7924
>413
3902
S90 949390
H 9S7
502.3
551-
6'Jic
e^n
6992
7433
7973
-4^2
8951
t4r43-
9926
a
8
Di£f.
9443-1
5074
5567
6059
6551
7041
7532
8022
8511
8999
949485
9975
2 950363 950414 950462
0S51
13:?S
1S23
230S
2792
3276
3760
9ao'954243
6'
0900
13-6
1372
2356
2341
3325
3303
0949
143:
1920
24 13 1
2339
3373
3356
944631
5124
5616
(•103
6600
7090
7531
8070
8560
9048
949536
950024
051 1
0997
14S3
1969
2453
293S
3421
3905
9446-^!
5173
5665
6157
C649
7140
7630
8119
8609
9097
944729 944779 94432
5222]
5715|
6207
6693
7189
7679
8163
8657
9146
5272
5764
6256
6747
7233
7728
8217
87(16
9195
5321
5313
6305
6796
7287
7777
8266
8755
9244
954291 954339
4725
5207
5633
6163
6649
7125
7607
8036
8564
4773
5255
5736
6216
6697
7176
7655
8134
8612
4321
5303
5784
6265
6745
7224
7703
8131
8659
949535
950073
0560|
1046!
1532
2017
2502
2936
3470
3953
949634 919683
950121 950170
0603! 0657
1095'
1530,
2066
114
1629
•.ell!
2550 ! 2599
954337 954435
910 959011 959039 959137
]; 951^1 9566! 96!4
2 9995 960042 960090
3 960471
4
i
7
8'
9j
920
ll
2'
3
4
5
6
7
8
9
0946
1421
1395
2:369
2343
3316
963733
425 )
4731
5202
5672
6142
6S11
7<i30
7543
8016
0513
0994
1469
1943'
2417
2890
3363
963335
4307
477S
52491
5719!
6139j
6653 1
71-27
7595
8062
0566
1011
1516
199D
2464
2937
3410
4369
5a5i
5832
6313
6793
7272
7751
8-229
8707
959185
9661
960133
0613!
1039
1563
2033
2511
2935
3457
4918
5399
5380
6361
6310
7320
7799
8277
8755
3034
a513
4001
95^1434
4966
5447
5923
6409
6333
7363
7347
8325
8303
3083
3566
4049
954532
5014
5495
5976
6457
6936
7416
7894
&373
8850
949731
950219
0706
1192
1677
2163
2647
3131
3615
4093
944377
5370
5362
6354
6345
7335
7326
8315
8304
9-292
9497S0!
953267
0754
1240
1726
2211
2696
3180
3663
4146
939-232
9709
960133
06GI
li:i6
1 1611
' 20-5
2559
21)32
3504
9493-29
950316
0303
1-289
1775
2-260
2744
3223
3711
41^
930! 963433
8950
9416
93-2
970317
0812
1-276
1740
2203
2666
96333'^
4354
43-25
5296
5766
i 6236
6705
7173
7642
8109
959230
9757',
960233
0709!
11341
1653
2132
2606
3079
3552
9593-28
9304
960230
0756
1231
1706
2180
2G53
3126
3599
963929
4401'
4372
5343
5313
62^3
6752
7-220
7633
8156
963977
4443
4919
5393
5-60
9S3530
89951
9463;
9923'
970393
0353;
1322
1736'
-2249,
2712
954530
5062
5.543
6024
6505
6934
7464
7942
8421
8393
959375
9S52
9603-28
0304
1279
1753
2227
2701
3174
3646
95462
5110
5592
6072
6553
7032
7512
7990
8463
8946
959423 959471
9900 9947
960376 960423
N<».! 0
963576
90431
9509
9S75
970440
0901
i:369
1332
-2295
2753
3
9636-23
9090
9556
970021
04-6
0951
1415
1879
2:342
a304
6329
6799
7-267
7735
8203
963670
9136
9602
970063
964024
4495
4966
5437
5907
6376
6345
7314
7782
8249
963716
9133
9649
9&4071
4542
5013
5434
5954
6423
6392
7361
7S29
8296
964113
4590
5061
5531
6001
6470
6939
740S
7875
0351
1326
ISOl
2275
2743
3221
3693
0599
1374
1343
2322
2793
3-263
3741
964165 964212
8343 8390 8436 47
963763
9-2-291
96951
970114;970I61
05791 0626:
0997
1461
19-25
2-333
2351
1 (44!
1.50S
1971
2434
2397
1090,
1554
20131
2431
2943
963310
9276
9742
970207
0672
1137
1601
2064
1 252
2939
4637
5103
557S
6043
6517
6936
74^54
7922
4634
5155
56-23
6095
6364
7033
7501
7969
963355
9323
9739
970254
0719
1183
1647
2110
2573
3035
8
963903
9:369
9335
,970:300
0763
1-229
1693
2157
2619
3032
43
43
43
43
43
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
46
46
46
46
46
46
Diff.
170
TABLE XII. LOGARITHMS OF NUMBERS.
977724
8181
8637
9093
9548
980003
0458
0912
1366
1S19
5426
5875
6324
936772
7219
7666
8113
8559
9005
9450
9895
990339
0783
991226
1669
2111
2554
299.5
3436
3377
4317
4757
5196
K3.
0
1
1 ^
973220
3 ! 4
973174
9/3266
973313
3636
36S2
372>S
3774
4097
4143
4189
4235
4553
4604
4650
4696
5018
5064
5110
5156
5478
5524
5570
5616
5937
6983
6029
6075
6396
6142
6488
6533
6354
6900
6946
6992
7312
7358
7403
7449
977769
977815
977361
977906
8226
8272
8317
8363
8683
8728
8774
8819
9133
9184
9230
9275
9594
9639
9635
9730
930049
9S0094
930140
930185
0503
0549
0594
0640
0957
1003
I04S
1093
1411
1456
1501
1547
1864
1909
1954
2000
982316
932362
982407
982452
2769
2814
2.359
2904
3220
3265
3310
33L6
3671
3716
3762
3807
4122
4167
4212
4257
4572
4617
4662
4707
5022
5067
5112
5157
5471
5516
5561
5606
5920
5965
6010
6055
6369
6413
6458
6503
936817
986361
986906
9S6951
7264
7309
7353
7393
7711
7756
7800
7845
8157
8202
8247
8291
S604
8643
8693
8737
9049
9094
9138
9183
9494
9539
9583
9623
9939
9983
990023
990072
9903S3 99042S
0472
0516
0827
0871
0916
0960
991270
991315
991359
991403
1713
1753
1802
1846
2156
2200
2244
2288
2593
2642
26S6
2730
3039
3083
3127
3172
34S0
3524
3563
3613
3921
3965
4009
4053
4361
4405
4449
4493
4S01
4845
4889
4933
5240
5284
5328
5372
995679 995723
995767
995311
6117 6161
6205
6249
6555 6599
6643
6637
6993 7037
7030
7124
7430 7474
7517
7561
7867 7910
7954
7998
8303 8347
8390
8434
8739 8732
8826
8869
9174 9218
9261
9305
9609
9652
3
9696
3
9739
1
4
5
6
973359 973405
3-20
3t66
4281
4327
4742
4788
5202
5248
5662
5707
6121
6167
6579
6625
7037
7083
7495
7541
977952
977993
8409
8454
8865
8911
9321
9366
9776
9321
980231
980276
0635
0730
1139
1134
1592
1637
2045
2090
932497
982543
2S49
2994
3401
3446
3852
3897
4302
4347
4752
4797
5202
5247
5651
5696
6100
6144
6548
6593
986996
987010
7443
7488
7890
7934
8336
8381
8732
8826
9227
9272
9672
9717
990117
990161
0561
0605
1004
1049
991448
991492
1890
1935
2333
2377
2774
2819
3216
3260
3657
3701
4097
4141
4537
4581
4977
5021
5416
5460
995354
995398
6293
6337
6731
6774
7163
7212
7605
7648
8041
8035
8477
8521
8913
8956
9348
9392
9783
9826
6
5
7 8 1 9
DifF.
973451 973497 973543
46
3913 3959
4^05
46
4374
4420
4406
46
4834
4880
4926
46
5294
53-10
53;>o
16
6753
5799
5346
46
6212
6258
6304
46
6671
6717
67(j3
ir,
7129
• 7175
7^21)
46
7686
7632
7678
46
978043
978089
978.3;,
40
8500
8546
8591
46
8956
9002
9047
IG
9412
9457
9503
46
9367
9912
9958
46
950322
980367
930412
15
0776
0821
ose:
45
1229
1275
1320
45
1633
172,3
1773
15
2135
2181
22"G
45
982588
982633
9S2678
10
3040
3085
3i:-!P
45
3491
3536
3531
45
3942
3987
4032
45
4392
4437
44,-^
45
4842
4887
4932
45
5292
5337
63S2
45
5741
5786
6«;^"
45
6189
6234
6279
45
6637
6682
6727
45
987035
987130
987175
45
7532
7577
7622
45
7979
8024
800r!
■15
8425
8470
8514
45
8371
8916
8960
45
S316
9361
9405
45
9761
9306
98oO
44
9S0206
990250
990294
44
0650
0694
073S
41
1093
1137
Ii5^
44
991536
991580
991625
44
1979
2023
2067
44
2421
2465
2509
44
2863
2907
29^1
14
3304
3348
?o':Z
±4
3745
3789
3833
44
4185
4229
42"3
14
4625
4609
4-'-:
■A
5065
5108
5152
44
5504
5547
5591
44
995942
S959S6
996030
44
6330
6424
6468
44
6818
6862
69.. i^
• 4
7255
7299
7343
44
7692
7736
7779
44
8129
8172
82: ;i
»1
8564
8608
fc6u2
44
9000
9043
9087
44
9435
9479
95'32
.4:
9870
9913
995,
DiCfj
7
8
9
TABLE X 1 1 1 .
LOGARITHMIC SINES, COSINES, TANGENTS.
AND
0
OTANGENTS.
172 TABLE XIII. LOGARITHIVIIC SINES,
NOTE.
The table here given extends to minutes only. The usual methcd
of extending such a table to seconds, by proportional parts of the
difference between two consecutive logarithms, is accurate enough
for most purposes, especially if the angle is not very small. When
the angle is very small, and great accuracy is required, the following
method may be used for sines, tangents, and cotangents.
I. Suppose it were required to find the logarithmic sine of 5' 24"
By the ordinary meth'^i VQ should have
lo<x. sin. 5' = 7.162696
diff. for 24" = 31673
log. sin. 5' 24" -- 7.194369
Ttic more accurate method is founded on the proposition in Trigo
nometry, that the sines or tangents of very small angles are propor
tional to the angles themselves. In the present case, therefore, we
have sin. 5': sin. 5' 24' = 5' : 5' 24 ' = 300" : 324". Hence sin. .5' 24'
= '"' ^.^""' , or log. sin. 5' 24" = log. sin. 5' + log. 324 — log. 30ii
The difference for 24" wiU therefore, be the difference between tlie
logarithm of 324 and the logarithm of SCO. The operation will stand
thus : —
log. 324 = 2..510.145
locr. 300 =2.477121
diff. for 24 = 33424
los. sin. 5' = 7.162696
W. sin. 5' 24" = 7.196120
■'o
Comparing this value with that given in tables that extend to seconds
we find it exact even to the last figure
PI
II. Given log. sin. A = 7.004438 to find A. The sine next less
than this in the table is sin. 3 = 6.940817. Now we have sin. 3' : sin. A
= 3 . A. Therefore, A = "1]^7 > oi" log- ^ = ^^S- ^ + ^^o- ^^^- ^^
- log. sin. 3'. Hence it appears, that, to find the logarithm of A in
COSINES, TANGENTS, AND COTANGENTS. 173
minutes, we must add to the logarithm of 3 the difference oetween
lojr. sin, A and log. sin. 3*.
log. sin. ^1 = 7.004438
loiT. sin. 3' =- 6.940S47
G3591
W, 3 = 0.477121
A --= 3.473 0.540712
r,j. 4 ^ 3/ 28.38". By the common method we should have found
A = 3' 30. .54".
The same method applies to tangents and cotangents, except that in
the case of cotangents the differences are to be subtracted.
• * The radius of this table is unity, and the characteristics % 8, 7,
and 6 stand respectively for —1, —2, —3, and —4.
174
0^
TABLE Xlll, «f.OGARlTHMIC SINES,
179^
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
30
3r
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine.
Inf. neg.
6.463726
.764756
.940347
7.0657S6
.162696
.241877
.303324
.366S16
.417963
7.463726
.505113
.542906
.577663
.609353
.639316
.667345
.694173
.718997
.742478
7.764754
.735943
.806146
.325451
.3439.34
.361662
.S7S695
.895035
.910379
.926119
7.940342
.9.55032
.963370
.9S2233
.995193
3.007737
.020021
.031919
.043501
.054731
8.065776
.076500
.036965
.097133
.107167
.116926
.126471
.13.5310
.1449.53
.15-3907
8.162631
.171230
.179713
.137935
.196102
.204070
.211395
.219531
.2271.34
.2.34.557
.241855
Cosine.
D. 1 .
5017.17
2934.85
2052.31
1615.17
1319.69
1115.73
966.53
8.52. 54
762.62
639.33
629.81
579.37
536.41
499.33
467.14
433.31
413.72
391.35
371.27
353.15
336.72
321.75
303.05
295.47
233.33
27.3.17
263.23
253.99
245.33
237.33
229.80
222.73
216.03
209.81
203.90
193.31
193.02
133.01
133.25
173.72
174.42
170.31
166.39
162.65
159.03
155.66
152.33
149.24
146.22
14.3..33
140.54
137.36
135.29
132.80
130.41
123.10
125.87
123.72
121.64
D. 1".
Cosine.
0.000000
.000000
.000000
.000000
.000000
.000000
9.999999
999999
.999999
.999999
9.999993
.999993
.999997
.999997
.999996
.999996
.999995
.999995
.999994
.999993
9.999993
.999992
.999991
.999990
.999939
.999939
.999933
.999937
.999936
.999935
9.999933
.999932
.999931
.999930
.999979
.999977
.999976
.999975
.999973
.999972
9.999971
.999969
.999963
.999966
.999964
.999963
.999951
.999959
.999953
.999956
9.9999.54
.999952
.999950
.999943
.999946
.999944
.999942
.999940
.999933
.999936
.999934
Sine.
D. 1'
.00
.00
.00
.00
.00
.00
.00
.00
.01
.01
.01
.01
.01
.01
.01
.01
.01
.01
.01
.01
.01
.01
.01
.01
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.04
.04
.04
D. 1".
Tang.
D. 1".
Inf. neg.
6.463726
.764756
.940347
7.0657S6
.162696
.241373
.303525
..366317
.417970
7.463727
.505120
.542909
.577672
.609357
.639320
.667849
.694179
.719003
.742484
7.764761
.73.5951
.806155
.825460
.843944
.561674
.873703
.895099
.910394
.926134
7.94035S
.9.55100
.963589
.932253
.99.5219
S. 007809
.020044
.031945
.043527
.054309
8.06^5306
.07653'
.036997
.097217
.107203
.116963
.126510
.135351
.144996
.153952
8.162727
.171323
.179763
.183036
.196156
.204126
.211953
.219641
.227195
.234621
.241921
Cotang.
5017.17
29.34.85
2032.31
1615.17
1319.69
111.5.73
966.54
852.55
762.63
639.33
629.81
-579.37
536.42
499.39
467.15
433.82
413.73
391.36
371.23
353.16
336.73
321.76
.303.07
295.49
233.90
273.13
263.25
254.01
245.40
237. .35
229.32
222.75
216.10
209.83
203.92
193.33
193.05
183.03
183.27
173.75
174.44
170.a4
166.42
162.63
159.11
155.69
1.52.41
149.27
146.25
143.36
140.57
137.90
135.32
132.84
130.44
123.14
12.5.91
123.76
121.63
D. 1".
Cotang. I M.
Infinite.
3.536274
.235244
.0.591.53
2.934214
.337304
.753122
.691175
.633133
.5320.30
2.536273
.494830
.457091
.422323
.390143
.360180
.332151
.305321
.230997
.2.57516
2.23-5239
.214049
.19.3345
.174540
.156056
.133326
.121292
.104901
.039106
.073366
2.059142
.044900
.031111
.017747
.004781
1.992191
.979956
.963055
.956473
.945191
1.934194
.923469
.913003
.9027-33
.892797
.83.3037
.873490
.664149
.85.50f>4
.&46043
1.837273
.823672
.8202.37
.811964
.80-3344
.795874
.783047
.780359
.772305
.765379
.753079
Tang.
90O
89"
COSINES, TANGENTS, AND COTANGENTS.
175
1T83
M.
Sine
11
12
.3
14
15
.(]
17
13
.9
20
?1
>£;'2
23
24
iij
28
27
29
30
32
32
33
34
35
36
3'
38
39
40
41
42
43
41
45
Ifi
47
43
«
6i
o2
53
54
55
56
57
58
59
60
D. 1'
8,2418r)->
.213033
.256' 19-1
.263012
.2693S1
.276614
.2S3213
.2Si)773
.296^07
8.303794
.3149.54
.321027
.327016
.332924
.333753
.3-14504
.350131
.3.J5733
.361315
8.366777
.372171
.377499
.332762
.337962
.3931111
.393179
.4031':!9
.403161
.413063
S.417919
.422717
.427462
.432156
.436300
.441.394
.445941
.450440
.454393
.459301
8.463665
,467935
.472263
.476493
.480693
.4.34343
.483963
.493040
.497073
.501030
5.505015
.503974
.512^67
.516726
.520551
.524313
..523102
.531823
.535.523
.539136
.542319
Cosine.
119.63
117.69
115.30
113.93
11221
110.50
103. S3
107.22
105.66
104.13
102.66
101.22
99.82
93.47
97.14
95.86
94.60
93.38
92.19
91.03
89.90
83.80
87.72
86.67
85.64
84.64
83.66
82.71
81.77
80.36
79.96
79.09
78.23
77.40
76.53
75.77
74.99
74.22
73.47
72.73
72.00
71.29
70.60
69.91
69.24
63.-59
"67.94
67.31
66.69
66.03
65.43
61.89
64.32
63.75
63.19
62.65
62.11
61.53
61.06
60.55
— I
M.. I Cosine.
D 1''
9.999934
.999932
.999929
.999927
.999925
.999922
.999920
.999913
.999915
.999913
9.999910
.999907
.999905
.999902
.999399
.999397
.999394
.999391
.999333
.999335
9.999332
.999379
.999376
.999373
.999370
.999367
.999364
.999361
.999353
.999354
9.999351
.999348
.999344
.999341
,999333
.999334
.999331
.999327
.999324
.999320
9.999316
.999313
.999309
.999305
.999301
.999797
.999794
.999790
.999736
.999732
9.999773
.999774
.999769
.999765
.999761
.9997.57
.999753
.999743
.99974 4
.999740
.9997,35
D. 1".
Sine.
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.05
.06
.06
.06
.06
.06
.06
.06
.06
.06
.06
.06
.06
.06
.06
.06
.06
.07
.07
.07
.07
.07
.07
.07
.07
.07
.07
.07
.07
.07
.07
Tang.
D. 1".
D. 1".
3.241921
.249102
.256165
.263115
.269956
.276691
.283323
.239356
.296292
.302634
8.303834
.31.5046
.321122
.327114
.333025
.3338.56
.314610
.350289
.355895
.361430
8.366395
.372292
.377622
.332839
.338092
.39.3234
.398315
.403333
.408304
.413213
8.418063
.422369
.427613
.432315
.436962
.441.560
.446110
.450613
.455070
.459431
8.463349
.463172
,4724.54
.476693
.430-92
.485050
.489(70
.493250
.497293
.501293
8.505267
.509200
.513998
.516961
.520799
..524536
..523349
.532030
.535779
.5.39447
.543034
Cotang.
M.
119.67
117.72
115.84
114.02
112.25
110.54
103.87
107.26
105.70
104.18
102.70
101.26
99.87
93.51
97.19
95.90
94.65
93.43
92.24
91.08
89.95
88.85
87.77
86.72
85.70
84.69
83.71
82.76
81.82
80.9]
80.02
79.14
78.29
77.45
76.63
75.83
75.05
74.23
73.53
72.79
72.06
71.35
70.66
69.93
69.31
63.65
63.01
67.33
66.76
66.15
65. 55
64.96
64.39
63.82
63.26
62.72
62.18
m.65
61.13
60.62
1.758079
.750893
.743<35
.736335
,730044
.723309
.716677
.710144
.703703
.697366
1.691116
,6349.54
,673373
672S86
666975
.661144
.65.5390
.619711
.644105
.633570
1.6.33105
.627703
,622373
.617111
.611903
.606766
.601635
.596662
.591696
.536737
1.5319.32
,577131
.572332
.567635
.563033
.553440
,553890
.549337
.544930
.540519
1,5.36151
.531823
..527546
.523307
.519103
.514950
,510330
.506750
,.502707
.493702
1.494733
.490300
.436902
.483039
.479210
.475414
.471651
.467920
.464221
.460553
.456916
CoUin?. I D. 1".
Tang.
60
59
53
57
56
55
54
53
52
51
50
49
43
47
46
45
41
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
23
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
M.
91'^
889
176
3="
TABLE XIII. LOGAHITHMIC SINES,
173"
M.
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
IS
19
20
21
22
23
21
25
26
27
23
29
30
31
32
33
34
35
36
37
3S
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine.
8.542319
.546422
..549995
.553.539
,557054
,560540
,563999
.567431
.Ol
4214
8.577566
.580392
,584193
,587469
.590721
,593943
.597152
.6J0332
.603439
.606623
8.6097.34
.612323
.615391
.613937
.621962
.62^965
.627943
.631911
.6333.54
.636776
8.63G630
.642563
.64:5423
.643274
.651102
.653911
.636702
.659475
.662230
.664963
8.667639
.670393
.673030
.675751
.678405
.631043
.6-3665
.6-6272
.633-63
.6914.33
8.693998
.696.543
.699)73
.701539
.704090
.706.377
,709049
,711.507
.713952
.716333
.718300
Cosine.
D, 1".
60.04
59.55
59.06
53.53
53.11
57.65
57.19
56.74
56.30
55.57
55.44
55.02
54.60
54.19
53.79
53.39
53.00
52.61
52.2:3
51.86
51.49
51.12
50.77
50.41
50.06
49.72
49.33
49.04
48.71
43.39
43.06
47.75
47.43
47.12
46.32
46.52
46.22
4.5.93
45.63
45.35
45.07
44.79
44.51
44.24
43.97
43.70
43.44
43.18
42.92
42.67
42.42
42.17
41.93
41.63
41.44
41.21
40.97
40.74
40.51
40.29
D. 1".
Cosine. D. 1".
9.999735
.999731
.999726
.939722
.999717
.999713
.999703
.999704
.999699
.999694
9.999639
.999635
.999630
.999675
.999670
.999665
.999660
.9996.55
.999650
.999645
9.999640
.999635
.999629
.999624
.999619
.999614
.999608
.999603
.999.597
.999592
9.9995S6
.999581
.999575
.999570
.999564
.999.553
.999.553
.999.547
.999.541
.999535
9.999529
.999.524
.999513
.999512
.999506
.999500
.999493
.999437
.999431
.999475
9.999469
.999463
.999456
.999450
.999443
.999437
.999431
.999424
.999413
.999411
.999404
Sine,
.07
.07
.03
.03
.03
.08
.08
,08
.08
.08
,03
,03
.03
.03
,03
,03
.08
.03
.03
.09
.09
.09
.09
.09
.09
.09
.09
.09
.09
.09
.09
.09
.09
.09
.09
.10
,10
,10
,10
,10
,10
.10
.10
,10
,10
.10
.10
.10
,10
,10
,10
,1
,1
,1
,1
,1
,1
,1
,1
,1
D. 1".
Tang,
D. 1".
8.543034
.546691
.550268
.553317
.557336
.56J328
.564291
..567727
1137
.0/
.57
4520
8.577877
,531203
.584514
.537795
.591051
,594283
,597492
.600677
.603339
.606973
8.610094
.613189
.616262
.619313
.622343
.62-53.52
.623340
.631303
.634256
.637134
8.640093
.6429-2
.645353
.643701
.651.537
.654352
.6.57149
.659923
.662639
.66.5433
8.603160
.670370
.673563
.676239
.673900
.631544
.634172
.636784
.639331
,691963
8.694529
.697081
.699617
.702139
.704046
,707140
.709618
.712033
.714534
.716972
.719396
Cotang.
60.12
59.62
59.14
58.66
58.19
57.73
57.27
56.62
56.38
55.95
5.5.10
54.63
54.27
53.87
53.47
53.08
52.70
52.32
51.94
51.58
51.21
50.85
50.50
50.15
49.81
49.47
49.13
48.80
. 48.48
48.16
47.84
47. .53
47.22
46.91
46.61
46.31
46.02
45.73
45.45
45.16
44.33
44.61
44.34
44.07
4.3.30
43.54
43.23
43.03
42.77
42.52
42.23
42.03
41.79
41.55
41.32
41.08
40.85
40.62
40.40
D, 1".
Cotang.
M.
1.456916
60
.453309
59
.4497.32
58
.446183
.57
.442664
.56
,439172
55
.435709
54
,432273
53
.428863
52
,425480
51
1.422123
50
.418792
49
,41.5436
48
,412205
47
,408949
46
,405717
45
.402503
44
,399323
43
.396161
42
,39.3022
41
1.389906
40
,3.36311
39
.383738
38
330637
37
377657
36
,374648
.371660
.36-692
.365744
.362816
1.359907
.357018
,3.54147
..351296
.343463
.345643
.342851
..340072
.3.37311
.a34567
1.331840
.3291.30
.32&4.37
•323761
.321100
.313456
.315323
.313216
.310619
.308037
1.30.5471
.302919
.300383
.297861
,295354
,292360
.290382
,287917
.28.5466
.253023
,280604
Tang.
9«3
•il-'
COSINES, TAiMGENTS, AND COTAKGENTS.
n7
176^
M
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
13
19
20
21
22
83
24
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
Sine
D 1".
Cosine.
D. 1"
. '-.718300
.7212)4
.723595
.725972
.723337
.730633
.733027
.735354
.737667
.739969
S. 742259
.744536
.746302
.749055
.751297
.753523
.755747
.7579")5
.76)151
.762337
3.761511
.766675
.76S323
.770970
.773101
.77.5223
.777333
.779434
.781524
.733605
3.735675
.737736
.739737
.791323
.793359
.795331
.797394
.799397
.801392
.803376
8.80.5852
.807819
.809777
.311726
.81.3657
.81.5.599
.317522
.819436
.321313
.323240
8.325130
.327011
.823384
.8.30749
.832607
.834456
.836297
.8.33130
.339956
.341774
.843535
40.06
39. -^4
39.62
39.41
39.19
33.93
33.77
33.57
33.36
33.16
37.96
37.76
37.56
37.37
37.17
36.93
36.30
36.61
36.42
36.24
36.06
35.83
35.70
35.53
35.. 35
35.13
35.01
31.31
34.67
31.51
31.31
31.18
34.02
.33.36
33.70
33.54
33.39
.33.23
33.03
32.93
32.73
32.63
32.49
32.34
32.20
32.05
31.91
31.77
31.63
31.49
31.36
31.22
31.03
30.95
30.82
30.69
30.56
30.43
30.30
30.17
9.999404
.999393
.999391
.999334
.999378
.999371
.999364
.999357
.9993.50
.999313
9.999336
.999329
.999.322
.999315
.999303
.999301
.999294
.999237
.999279
.999272
9.999265
.999257
.9992.50
.999242
.999235
.999227
.999220
.999212
.999205
.999197
9.999189
.999181
.999174
.999166
.999153
.999150
.999142
.999131
.999126
.999113
9.999110
.999102
.999094
.999036
.999077
.999069
.999':)61
.999053
.999044
.999036
9.999027
.999019
.999010
.999002
.993993
.993931
.993976
.993967
.993953
.9939.50
.99-^941
Tang.
D. 1"
Cosine. I D. 1".
Sine
.11
.11
.11
.11
.11
.11
.11
.11
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.13
.14
.11
.14
.14
.14
.14
.14
.14
.14
.14
.14
.14
.14
.14
.14
.14
.15
.15
.15
.15
D. 1".
8. 71 9396
.721306
.724204
.726533
.723959
.731317
.733663
.735996
.733317
.740626
8.742922
.745207
.747479
.749740
.751939
.754227
.756453
.753663
.760372
.763065
8.765216
.767417
.769573
.771727
.773366
.775995
.773114
.730222
.732320
.784403
8.736436
.7335.54
.790613
.792662
.794701
.796731
.793752
.800763
.802765
.804753
8.806742
.803717
.810633
.812641
.3145^9
.816529
.813161
.320334
.822293
.824205
3.326103
.827992
.829374
.831743
.833613
.83^5471
.837.321
.839163
.840993
.842325
.844644
Cotang.
Cotang.
40.17
39.95
39.74
39.52
39.31
39.10
33.89
33.63
33.43
38.27
38.07
37.83
37.63
37.49
37.29
37.10
36.92
36.73
36.55
36.36
36.18
36.00
35.83
35.65
35.43
35.31
35.14
31.97
34.80
34.64
34.47
34.31
34.15
33.99
33.83
33.63
33.52
33.37
33.22
33.07
32.92
32.77
32.62
.32.43
32.33
32.19
32.05
31.91
31.77
31.63
31. .50
31.36
31.23
31.09
30.96
30.83
30.70
30.57
30.45
30.32
D. 1".
1.230604
.273194
.275796
.273412
.271011
.263633
.266337
.264004
.261633
.259374
1.257078
.254793
.252521
.250260
.243011
.245773
.243.547
.241332
.239123
.236935
1.234754
.232533
.230422
.223273
.226131
.224005
.221886
.219773
.2176i0
.215592
1.213514
.211446
.209337
.207333
.205299
.203269
.201243
.199237
.1972.35
.195212
1.193253
.191233
.189317
.137359
.135411
.133471
.131.539
.179616
.177702
.17.5795
1.173397
.172003
,170126
.163252
,166337
.164529
.162679
.1603.37
.159002
.157175
.1.55356
Tang.
M.
60
59
53
57
56
55
54
53
52
51
50
49
43
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
23
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
M.
86<?
178
40
TABLE XIII. LOGARITHMIC SINES,
175'
M.
0
1
2
3
4
5
6
7
8
9
10
il
12
13
14
15
16
17
13
19
20
21
22
23
24-
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
Sine.
8.S435S5
.S453S7
.847183
.348971
.850751
.852.525
.8-54291
.856049
.857801
.859546
8.S612S3
.863014
.864733
.866455
.863165
.S69S63
.871.565
.87.3255
.874933
.876615
8.878235
.879949
.831607
.333253
.834903
.836542
.838174
.889301
89 1 421
.893035
8.394643
.896246
.897342
.899432
.901017
.902596
.904169
.905736
.907297
.903353
8.910404
.911949
.913433
.91.5022
.916550
.918073
.919'.91
.921103
.922610
.924112
8.925609
.927100
.923587
.930063
.931544
.933015
.934431
.935942
.937393
.938350
.940296
1 M. Cosine.
^o
D. 1".
30.05
29.92
29.80
29.63
29.-55
29.43
29.31
29.19
29.03
28.96
23.84
28.73
23.61
23.50
23.39
28.23
23.17
23.06
27.95
27.34
27.73
27.63
27.52
27.42
27.31
27.21
27.11
27.00
26.90
26.30
26.70
26.60
26.51
26.41
26.31
26.22
26.12
26.03
25.93
25.34
25.75
25.66
25.56
25.47
25.38
25.29
25.21
25.12
25.03
24.94
24.86
24.77
24.69
24.60
24.52
24.43
24.-35
24.27
24.19
24.11
Cosine.
D. 1'.
9.99S94I
.993932
.993923
.993914
.993905
.993896
.998.387
.993378
.993369
.993360
9.993351
.993341
.993332
.993323
.993313
.993304
.993795
.993785
.993776
.993766
9.998757
.993747
.998733
.993723
.993718
.993703
.993699
.993639
.995679
.993669
9.998659
.993649
.993639
.993629
.993619
.99S609
.993599
.993.589
.993573
.993563
9.993553
.993-S43
.993537
.993527
.993516
.993506
.993495
.993485
.998474
.993464
9.993453
.993442
.993431
.993421
.993410
.993399
.993338
.993377
.993.366
.9933-55
.993344
D. 1".
Sine.
.15
.15
.15
.15
.15
.15
.15
.15
.15
.15
.15
.15
.15
.16
.16
.16
.16
.16
.16
.16
.16
.16
.16
.16
.16
.16
.16
.16
.16
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.18
.18
.18
.18
.18
.18
18
.18
18
.18
.18
.18
.18
.18
.18
Tang.
D. 1".
8.844644
.346455
.843260
.850057
.851846
.853628
.85.5403
.857171
.858932
.860636
8.S62433
.864173
.865906
.867632
.869-351
.871064
.872770
.874469
.876162
.877349
8.879529
.881202
.832369
.834530
.886185
.837833
.889476
.891112
.892742
.894366
8.895934
.897596
.899203
.900303
.902398
.90-3937
.905570
.907147
.903719
.910235
8.911346
.913401
.914951
.916495
.9130-34
.919563
.921096
.922619
.924136
.925649
8.927156
.923658
.9-301-55
.931647
.9-33134
.934616
.936093
.937565
.9390-32
.940494
.941952
D 1". Cotang.
30.20
30.07
29.95
29.83
29.70
29.53
29.46
29.35
29.23
29.11
29.00
23.88
23.77
23.66
23.55
23.43
23.32
23.22
23.11
23.00
27.89
27.79
27.63
27.-58
27.47
27.37
27.27
27.17
27.07
26.97
26.87
26.77
26.67
26.58
26.48
26.-39
26.29
26.20
26.10
26.01
2-5.92
25.83
25.74
25.65
25.56
25.47
25.38
25.29
25.21
25.12
25.04
24.95
24.87
24.78
24.70
24.62
24.53
24.45
24.37
24.29
Cotang. 1 D. 1".
1.1553S6
.153545
.151740
.149943
.143154
.146372
.144597
.142329
.141063
.139314
1.137567
.135327
.1-34094
.132363
.130649
• .123936
.127230
.125531
.123333
.122151
1.120471
.113793
.117131
.11.5470
.113315
.112167
.110524
.108383
.107258
.105634
1.104016
.102404
.100797
.099197
.097602
.096013
.094430
.092853
.091231
.039715
LOSS 154
.086599
.085049
.033505
.081966
.080432
.078904
.077381
.075864
.074351
1.072344
.07i:J42
.069345
.0633.53
.066366
.065384
.063907
.062435
.060968
.059506
.058048
Tang.
8»<3
COSINES, lANGENTS, AND COTANGENTS.
M.
Sine.
D. 1".
0
8.940296
1
.941733
2
.913174
3
.944606
4
.946034
5
.947456
6
.948374
7
.950237
8
.951696
9
.953100
10
3.954499
11
.955394
12
.957234
13
.953670
14
.960052
15
.961429
16
.962301
17
.964170
18
.965531
19
.933993
20
3.963249
21
.969600
22
.970947
23
.972239
24
.973623
25
.974902
26
.976293
27
.977619
23
.978941
.930259
3l
^ 8.931573
31
.932333
32
.934189
33
.93.5491
34
.936789
35
.933033
36
.939374
37
.990660
33
.991943
39
.993222
40
3.994497
41
.995763
42
.997036
43
.993299
44
.999560
45
9.000316
46
.002069
47
.003318
43
.004563
49
.005305
50
9.007044
51
.003278
52
.009510
53
.010737
54
.011962
55
.013182
56
.014400
57
.015613
53
.016324
59
.018031
60
.019235
M.
»5<3
Cosine.
24.03
23.95
23.87
23.79
23.71
23.63
23.55
23.48
23.40
23.32
23.25
23.17
23.10
23.02
22.95
22.83
22.31
22.73
22.66
22.59
22.52
22.45
22.33
22.31
22.24
22.17
22.10
22.03
21.97
21.90
21.33
21.77
21.70
21.64
21. .57
21.51
21.44
21.33
21.31
21.25
21.19
21.12
21.06
21.00
20.94
20.83
20.82
20.76
20.70
20.64
20.53
20.52
20.46
20.40
20.35
20.29
20.23
20.17
20.12
20.06
D. 1".
9.99S344
.993333
.993322
.998311
.993300
.998239
.993277
.993266
.993255
.993243
9.993232
.998220
.993209
.993197
.993186
.993174
.993163
.998151
.993139
.993123
9.99SI16
.993104
.993092
.993030
.993063
.993056
.993044
.993032
.993020
.993008
9.997996
.997984
.997972
.997959
.997947
.997935
.997922
.997910
.997897
.997885
9.997372
.997360
.997847
.997335
.997822
.997809
.997797
.997734
.997771
.997753
9.997745
.997732
.997719
.997706
.997693
.997630
.997667
.9976;j-l
.997641
.997623
.997614
Cosine. D. 1".
Sine.
.18
.19
.19
.19
.19
.19
.19
.19
.19
.19
.19
.19
.19
.19
.19
.19
.19
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.21
.21
.21
.21
.21
.21
21
.21
.21
.21
.21
.21
.21
.21
.21
.21
.22
.22
.22
.22
.22
.22
.22
.22
.22
.22
Tans.
8.941952
.94:^0-1
.944352
.946295
.947734
.949168
.950597
.952021
.953441
.954356
8.956267
.957674
.959075
.960473
.961866
.963255
.964639
.966019
.967394
.963766
8.970133
.971496
.972355
.974209
.975560
.976906
.978243
.979536
.930921
.932251
8.933577
.934899
.936217
.937532
.938342
.990149
.991451
.992750
.994045
.995337
8.996624
.997903
.999188
9.000465
.001733
.0^3007
.004272
.005534
.006792
.008047
9.009293
.010546
.011790
.013031
.014268
.015502
.016732
.017959
.019183
.020403
.021620
D. 1".
D. 1'.
24.21
24.13
24.05
23.97
23.90
23.82
23.74
23.67
23.59
23.51
23.44
23.36
23.29
23.22
23.14
23.07
23.00
22.93
22.86
22.79
22.72
22.65
22.58
22.51
22.44
22.37
22.30
22.24
22.17
22.10
22.04
21.97
21.91
21.84
21.78
21.71
21.65
21.59
21.52
21.46
21.40
21.34
21.27
21.21
21.15
21.09
21.03
20.97
20.91
20.85
20.80
20.74
20.63
20.62
20.56
20.51
20.45
20.39
20.34
20.28
Cotang.
179
174-0
M.
Cotang.
1.058048
.056596
.055148
.053705
.052266
.050332
.049403
.047979
.046559
.045144
1.0437.33
.042326
.040925
.039527
.038134
.036745
.035361
.033981
.032606
.031234
1.029367
.023504
.027145
.025791
.024440
.023094
.021752
.020414
.019079
.017749
1.016423
.015101
.013783
.012468
.011153
.009851
.008549
.007250
.005955
.004663
1.003376
.002092
.000312
0.999535
.993262
.996993
.995728
.994466
.993203
.991953
0.990702
.939454
.988210
.986969
.985732
.984498
.933268
.982041
.980817
.979597
.978380
D. 1".
6C
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Tang. M-
84<3
180
60
C- 0 i y ^ <?
TABLE Xlll. LOGARITHMIC SINES,
173^
M.
0
1
•2
3
4
Sine.
D. 1'
10
11
12
13
14
15
16
17
IS
19
20
21
22
23
24
25
26
27
2S
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.019235
.020435
.021632
.0228-25
.024016
.02.5203
.026366
.027567
.028744
.029918
9.0310-9
.032257
.033421
.034582
.0.35741
.036896
.038048
.039197
.040.342
.041485
9.042625
.013762
.044895
.046026
.047154
.048279
.049400
.050519
.051635
.0.52749
9.053859
.054966
.056071
.057172
.058271
.059367
.060460
.061.551
.062639
.063724
9.064-06
.065885
.066962
.06-036
.069107
.070176
.071242
.0723%
.07.3366
.074424
9.075460
.076533
.077583
.078631
.079676
.080719
.081759
.082797
.083832
.084864
.085894
M. ! Cosine.
20.00
19.95
19.89
19.84
19.78
19.73
19.67
19.62
19.57
19.51
19.46
19.41
19.36
19.30
19.25
19.20
19.15
19.10
19.05
19.00
18.95
18.90
18.85
18.80
18.75
18.70
18.65
18.60
18.. 55
18.50
18.46
18.41
18.36
18.31
18.27
18.22
18.17
18.13
18.08
18.04
17.99
17.95
17.90
17.66
17.81
17.77
17.72
17.68
17.64
17.59
17. .55
17.51
17.46
17.42
17.38
17.34
17.29
17.25
17.21
17.17
Cosine.
D. 1".
9.997614
.997601
.997588
.997574
.997.561
.997.547
.997534
.997520
.997507
.997493
9.997480
.997466
.997452
.997439
.997425
.997411
.997397
.997333
.997369
.997355
9.997341
.997327
.997313
.997299
.997285
.997271
.997257
.997242
.997228
.997214
9.997199
.997185
.997170
.997158
.997141
.997127
.997112
.997098
.997083
.997063
9.997053
.997039
.997024
.997009
.996994
.996979
.996964
.996949
.996934
.996919
9.996904
.996889
.996874
.996858
.996843
.996828
.996812
.996797
.996782
.996766
.996751
Sin*.
D. 1".
.22
22
22
.22
,22
22
.23
23
23
23
23
,23
.23
23
.23
.23
.23
.23
.23
.23
.23
.23
.24
.24
.24
.24
.24
.24
.24
.24
.24
.24
.24
.24
.24
.24
.24
.24
.25
.25
.25
.25
.25
.25
.25
.25
.25
.25
.25
.25
.25
.25
.25
.25
.26
.26
.26
.26
.26
.26
Tang.
D. 1".
D.l
9,021620
.022834
.024044
.025251
.026455
.027655
.028852
.030046
.031237
.032425
9.033609
.034791
.r,35C69
.337144
.038316
.039485
.040651
.M1813
.042973
.044130
9.0452^4
.046134
.047.582
.04S727
.049869
.051008
.0.52144
.053277
.054407
.055535
9.056659
.0.57781
.058900
.060016
.0611.30
.062240
.063.348
.064453
.065556
.066655
9.067752
.068846
.069933
.071027
.072113
.073197
.074278
.075.356
.076432
.077505
9.078576
.079644
.080710
.081773
.082S33
.08.3^91
.084947
.086000
.087050
.088098
.089144
20.23
20.17
20.12
20.06
20.01
19.95
19.90
19.85
19.79
19.74
19.69
19.64
19.. 58
19.53
19.48
19.43
19.38
19.33
19.28
19.23
19.18
19.13
19.08
19.03
18.98
18.93
18.89
18.84
18.79
18.74
18.70
18.65
18.60
18.56
18.51
18.46
18.42
IS.. 37
18.33
18.28
18.24
18.19
18.15
18.10
1S.06
18.02
17.97
17.93
17.89
17.84
17.80
17.76
17.72
17.67
17.63
17.59
17.55
17.51
17.47
17.43
Cotang I D. 1".
Cotang.
60
0.976380
.977166
59
.9759.56
58
.974749
57
.973545
56
.972315
55
.971148
54
.£699.54
53
.968763
52
.967575
51
0.966391
50
.965209
49
.964031
4L
.962856
47
.961684
46
.960515
45
.9.59349
14
.9.58187
43
.957027
42
.955870
41
0.9.54716
40
.95.3566
39
.952418
38
.951273
37
.950131
£6
.948992
35
.947856
34
.946723
33
.945593
,32
.944465
31
0.943341
l'.0
.942219
29
.941100
28
.939984
27
.938870
26
.937760
25
.936652
24
.935547
25
.934441
22
.933345
21
0.9.32248
20
.9311.54
19
.930(62
18
.928973
17
.927887
16
.926303
15
.925722
14
.924644
13
.92.3.-68
12
.922495
11
0.921424
10
.920356
9
.919290
8
.918227
7
.917167
6
.916109
5
.915053
4
.914000
3
.912950
2
.911902
1
.S10^C6
0
M
Tang.
oeo
830-
COSINES, TANGENTS, AND COTANGENTS
181
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
13
19
20
21
22
23
24
2;5
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
Sine.
D. 1".
M.
9.085394
.036922
.037947
.033970
.039990
.091008
.092024
.093037
.094047
.095056
9.096062
.097(165
.09S066
.099065
.100[)62
.101056
.102013
.103037
.104025
.105010
9.105992
.106973
.107951
.105927
.109901
.110373
.111842
.112309
.113774
.114737
9.115693
.116656
.117613
.113567
.119519
,120469
.121417
.122362
.123306
.124243
9.125187
.126125
.127060
.127993
.123925
.129354
.130731
.131706
.1.32630
.133551
9.134470
.13.53S7
.136303
.137216
.133123
.139037
.139944
.140350
.141754
142655
.143555
CoBine.
17.13
17.09
17.05
17.00
16.96
16.92
16.88
16.84
16.30
16.76
16.73
16.69
16.65
16.61
16.57
16.53
16.49
16.46
16.42
16.33
16.34
16.30
16.27
16.23
16.19
16.16
16.12
16.03
16.05
16.01
15.98
15.94
1.5.90
15.87
15.83
15.80
15.76
15.73
15.69
15.66
15.62
15.59
15.56
15.52
15.49
15.45
15.42
15.39
15.35
15.32
15.29
15.26
15.22
15.19
15.16
15.13
15.09
15.06
15.03
15.00
D. 1".
9.996751
.996735
.996720
.996704
.9966SS
.996673
.996657
.996641
.996625
.996610
9.996594
.996573
.996562
.996.M6
.996530
.996514
.996198
.996482
.996465
.996449
9.996433
.996417
.996400
.996334
.996363
.996351
996335
.996318
.996302
.996235
9.996269
.996252
.996235
.996219
.996202
.996135
.996163
.996151
.996134
.996117
9.996100
.996033
.996066
.996049
.996032
.996015
.995993
.995930
.995963
.995946
9.995928
.995911
.995394
.995876
.995359
.995341
.995323
.995306
.995783
.995771
.995753
Sine.
.26
.26
.26
.26
.26
.26
.26
.26
.26
.26
.27
.27
.27
.27
.27
.27
.27
.27
.27
.27
.27
.27
.27
.27
.27
.27
.23
.28
.28
.23
.28
.28
.28
.28
.28
.23
.23
.23
.23
.23
.23
.23
.28
.29
.29
.29
.29
.29
.29
.29
.29
.29
.29
.29
.29
.29
.29
.29
.29
.30
9.039144
.090137
.091223
.1)92266
.093302
.094336
.095367
,096395
.097422
.093446
9.099463
.100487
•101504
.102519
.103532
.104542
.105550
.1065.56
.107559
.108560
9.109559
.1105.56
.111551
.112543
.113533
.114.521
.115507
.116491
.117472
.118452
9.119429
.120404
.121377
.122343
.123317
.124234
.125249
.126211
.127172
.123130
9.129037
.130041
.130994
.131944
.132393
.133339
.1347^34
.135726
.136667
.137605
9.133542
.139476
.140409
.141310
.142269
.143196
.144121
.14.5044
.145966
.146335
.147803
D. 1",
17.39
17.35
17.31
17.27
17.23
17.19
17.15
17.11
17.07
17.03
16.99
16.95
16.91
16.83
16.84
16.80
16.76
16.72
16.69
16.65
16.61
16.53
16..54
16.50
16.47
16.43
16.39
16.36
16.32
16.29
16.25
16.22
16.18
16.15
16.11
16.03
16.04
16.01
15.93
15.94
15.91
15.87
15.84
15.81
15.77
15.74
15.71
15.63
1.5.64
15.61
15.58
15.55
15.51
1.5.43
15.45
15.42
15.39
1.5.36
15.32
15.29
0.910S56
.909S13
.908772
.907734
.906693
.905664
.904633
.903605
.902573
.901554
0.900532
.899513
.893496
.897451
.896463
.895453
.894450
.893444
.892441
.891440
0.890441
.839444
.833449
.837457
.836467
,835479
.884493
.883509
.832.523
,831543
0.880571
.879596
.878623
.877652
.876633
.875716
.874751
.873789
.872328
.871870
0.870913
.869959
.869006
.863056
.867107
.866161
.865216
.864274
.863333
.862395
Cotang.
D. 1'.
0.861453
.860.524
.859591
.853660
.857731
.856304
.85.5379
,854956
.854034
,853115
.852197
60
59
53
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
33
37 I
36 I
35
34
33
32
31
30
29
23
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Tang.
M.
07-
8«-
182
so
TABLE XIII. LOGARITHMIC SINES,
171<i
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine
9.143555
.144453
.145349
.146243
.147136
.143026
.143915
.149302
.150636
.151569
9.152451
.1.53.330
.1.34203
.1.55033
.1-559.57
. 1.56330
.1.57700
.153569
.159435
.160301
9.161164
.162025
.162335
.163743
.164600
.1654.54
.166307
.1671.59
163003
.1633.56
9.169702
.170.547
.171.339
.172230
.173070
.173903
.174744
.175573
.176411
.177242
9.173072
.173900
.179726
..130551
.181374
.132196
.13.3016
.133831
.184651
.135466
9.136230
.187092
.187903
.183712
.139519
.190-325
.191130
.191933
.192734
.193534
.194332
Cosine.
D. 1".
4.97
4.93
4.90
4.S7
4.34
4.31
4.78
4.75
4.72
4.69
4.66
4.63
4.60
4.57
4-54
4.51
4.43
4.45
4.42
4.39
4.. 36
4.33
4.. 30
4.27
4.24
4.22
4.19
4.16
4.13
4.10
4.07
4.05
4.02
3.99
3.96
3.94
3.91
13.83
-3.35
3. S3
3.80
3.77
3.75
3.72
3.69
.3.67
3.64
3.61
3.-59
3.56
3.54
3.51
3.48
3.46
3.43
3.41
3.33
.3.36
-3.33
3.31
D. 1".
Cosine.
9.995753
.995735
.995717
.99.5699
995631
.995664
.995646
.995628
.995610
.995591
9.995573
.99.5555
.995537
.995519
.995501
.995432
.995464
.995446
.995427
.995409
9.99-5390
.995372
.995353
.995334
.995316
.995297
.995273
.99.3260
.995241
.995222
9.995203
.995184
.995165
.995148
.995127
.995103
.995039
.995070
.995051
.995032
9.99.5013
.994993
.994974
.994955
.994935
.994916
.994396
.994377
.994357
.994333
9.994818
.994793
.994779
.994759
.994739
.994720
994700
.994630
.994660
.994640
■994620
Sine.
D.r
.30
.30
.30
.30
.30
.30
.30
.30
.30
.30
.30
.30
.30
.30
.30
.31
.31
.31
.31
.31
.31
.31
.31
.31
.31
.31
.31
.31
.31
.31
.31
.32
.32
.32
.32
.32
.32
.32
..32
.32
.32
.32
.32
.32
.32
.32
.a3
..33
.33
.33
.33
.33
.33
.33
.33
.33
.33
.33
.33
.33
D. 1".
Tang. D. 1". Cotang.
9.147803
.143713
.149632
.150544
.151454
.152363
.153269
.1.54174
.15.5077
.155978
9.156877
.157775
.158671
.159565
.1604.57
.161.317
.162236
.163123
.164003
.164892
9.16.5774
.166654
.167532
.168409
.169234
.170157
.171029
.171899
.172767
.173634
9.174499
.17.5362
.176224
.177084
.177942
.178799
.179655
.180508
.181360
.182211
9.183059
.18.3907
.184752
,185597
.186439
.187280
.188120
.188953
.139794
.190629
9.191462
.192294
.193124
.19.3953
.194730
.19.5606
.196430
.1972.53
.193074
.19S394
199713
Cuiang.
15.26
15.23
15.20
15.17
15.14
15.11
1.5.03
15.05
15.02
14.99
14.96
14.93
14.90
14.87
14.84
14.81
14.78
14.75
14.73
14.70
14.67
14.64
14.61
14..53
14.56
14.-53
14.50
14.47
14.44
14.42
14.39
14.36
14.33
14.31
14.28
14.25
14.23
14.20
14.17
14.15
14.12
14.09
14.07
14.04
14.02
1.3.99
13.97
13.94
13.91
13.89
13.86
13.84
1-3.81
13-79
1-3.76
1-3.74
13.71
1-3.69
13.66
13.64
D. 1".
0.852197
60
.851282
59
.850363
58
.849456
57
.848546
56
.847637
55
.846731
54
.845826
53
.844923
52
.844022
51
0.843123
50
.842225
49
.841329
43
.8404.35
47
.839543
46
.833653
45
.837764
44
.836377
43
.835992
42
.835103
41
0.834226
40
.8-33.346
39
S32463
38
.831591
37
.830716
36
.829343
35
.828971
34
.828101
33
.827233
32
.826366
31
0.825501
30
.824633
29
.823776
28
.822916
27
.822053
26
.821201
25
.820345
24
.819492
23
.818640
22
.817789
2i
0.816941
20
.816093
19
.81.5243
18
.814403
17
.813561
16
.812720
15
.811330
14
.811042
13
.810206
12
.809371
11
0.808538
10
.807706
9
.806376
8
.806047
7
.805220
6
.804394
5
.803570
4
.802747
3
.801926
2
.801106
1
.800237
0
Tang.
M.
as* 2
81
COSINES, TANGENTS, AND COTANGENTS.
183
170^
0
1
2
3
4
5
6
i
S
9
10
11
12
13
14
15
16
17
IS
19
20
21
22
23
24
2.",
26
27
23
29
30
31
32
33
34
35
36
37
3S
39
40
41
42
43
44
45
4(r
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.194332
.195129
.195925
.196719
.197511
.193.302
.199091
.199379
.200666
.201451
9.202234
.203017
.203797
.201577
.20.5351
.206 1 31
.2)6906
.207679
.2,03452
.209222
9.200992
.210760
.211.526
.212291
.2131.55
.213313
.214579
.215333
.216097
.216354
9.217609
.213363
.219116
.219363
.220613
.221367
.222115
.222361
.223396
.221349
9.225092
.225333
.226573
.227311
.223013
.223734
.229513
.230252
.230934
.231715
9.232114
.233172
.233399
.231625
.235319
.236073
,236795
.2.37515
.2332.-5
.233953
.2.396"1
13.28
13.26
13.23
13.21
13.18
13.16
13.13
13.11
13.03
13. 16
13.MI
13.01
12.99
12.96
12.91
12.92
12.89
12.37
12.85
12.82
12.80
12.73
12.75
12.73
12.71
12.63
12.66
12.64
12.62
12.59
12.. 57
12.55
12.53
12. .50
12.43
12.46
12.44
12.42
12.39
12.37
12..35
12.. 33
12.31
12.29
12.26
12.21
12.22
12.20
12.18
12.16
12.14
12.12
12.10
12.07
12.05
12.03
12.01
11.99
11.97
11.95
9.994620
.991600
.994-530
.991.560
.99454 )
.994519
.994190
.991479
.994459
.994433
9.994413
.994393
.991377
.994357
.991336
.991316
.994295
.994274
.9942.54
.994233
9.994212
.994191
.994171
.9941.50
.994129
.994103
.994037
.994060
.994045
.994024
9.994003
.993932
.993960
.993939
.993913
.993597
.993375
.903354
.993332
.993311
9.993739
.993763
.993746
.993725
.993703
.993631
.993660
.99.3633
.99.3616
.993594
9.993572
.993550
.993528
.993.506
.993434
.993462
.993140
.993413
.99.3396
.993374
.993351
.33
.33
.34
.31
.34
.34
.34
.34
.34
.34
.34
.34
.34
.31
.34
.34
.34
.34
.35
.35
.35
.35
.35
.35
.35
.35
.35
.35
.35
.35
.35
.35
.•35
..35
.36
.36
.36
.36
.36
.36
..36
..36
.36
.36
..36
.36
.36
.36
.36
.36
.37
.37
.37
.37
.37
.37
.37
.37
.37
.37
9.199713
.200529
.201315
.2021.59
.202971
.203732
.204592
.205400
.206207
.207013
9.207317
.203619
.209420
.210220
.211013
.211815
.212611
.213405
.214193
.214939
9.215780
.216563
.217356
.218142
.213926
.219710
.220492
.221272
.222052
.222830
9.223607
.224332
.225153
.225929
.226700
.227471
.22^239
.229007
.229773
.230539
9.231302
.232065
.232326
.233586
.234345
.235103
.235359
.236614
.237.363
.233120
9.233372
.239622
.240371
.211118
.211365
.242610
.243354
.244097
.244839
.245579
.216319
1.3.62
13.. 59
13.57
13.54
13.52
13.49
13.47
13.45
13.42
13.40
13.33
13.35
13.33
13.31
13.23
13.26
13.24
13.21
13.19
13.17
13.15
13.12
13.10
1.3.03
1.3.06
13.03
13.01
12.99
12.97
12.95
12.92
12.90
12.83
12.36
12.84
12.82
12.79
12.77
12.75
12.73
12.71
12.69
12.67
12.65
12.63
12.60
12. .58
12.56
12. .54
12.52
12.. 50
12.43
12.46
12.44
12.42
12.40
12.33
12.36
12.34
12.32
0.800237
.799471
.7936.55
.797341
.797029
.796213
.795403
.794600
.793793
.792937
0.792183
.791331
.790530
.739780
.788982
.788185
.787339
.736595
.785302
.78501 1
0.784220
.783432
.782644
.781858
.731074
.7^0290
.779503
.778723
.777943
.777170
0.776393
.775618
.774844
.774071
.773300
.772529
.771761
.770993
.770227
.769461
0.763693
.767935
.767174
.766414
.765655
.764397
.764141
.763336
.762632
.761880
0.761128
.760378
.759629
.753332
.758135
.757390
.756646
.75.5903
.755161
.754421
.753631
60
59
53
57
56
55
54
53
52
51
50
49
43
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
23
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
4
3
2
1
0
99^
184
10^
TABLE XIII.
LOGARITHMIC SINES,
169!
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
lo
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
4t
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine.
9.239670
.2403S6
.241101
.241814
.242526
.243237
.243947
.244656
.245363
.246069
9. 246 r 75
.247478
.24S181
.24S3S3
.219583
.250282
.250980
.251677
.252373
.253067
9.253761
.2.54453
.255144
.255834
.256523
.25721 1
.257893
.258583
.259263
.259951
9.260633
.261314
.261994
.262673
.263351
.264027
.264703
.26.5377
.266r)51
.266723
9.267395
.263065
.263734
.269402
.270069
.270735
.271400
.272064
.272726
.273338
9.274049
.274708
.275367
.276025
.276631
.2773.37
.277991
.278645
.279297
.279943
.230599
Cosino.
D. 1".
11.93
11.91
11.89
11.87
11.35
11.83
11.81
11.79
11.77
11.75
11.73
11.71
11.69
11.67
11.65
11.63
11.61
11.59
11.58
11.56
11.54
11. .52
11.50
11.43
11.46
11.44
11.42
11.41
11.39
11.37
11.35
11.33
11.31
11.30
11.23
11.26
11.24
11.22
11.20
11. L9
11.17
11.15
11.13
11.12
11.10
11.08
11.06
11.05
11.03
11.01
10.99
10.93
10.96
10.94
10.92
10.91
10.89
10.87
10.86
10.S4
D. 1".
Cosine.
9.993351
.993329
.99.3307
.99.3284
.993262
.993240
.99.3217
.993195
.993172
.993149
9.993127
.993104
.993081
.993059
.993036
.993013
.992990
.992967
.992944
.992921
9.992898
.992375
.992852
.992329
,992306
.992783
.992759
.992736
.992713
.992690
9.992666
.992643
.992619
.992596
.992572
.992549
.992525
.992501
.992478
.992454
9.992430
.992406
.992382
.992359
.992335
.992311
.992237
.992263
.9922.39
.992214
9.992190
.992166
.992142
.992118
.992093
.992069
.992044
.992020
.991996
.991971
.991947
D. 1".
.37
.37
.37
.37
.37
.37
.38
.38
.38
.33
.38
.38
.38
.33
.38
.38
.33
.38
.38
.33
.38
.38
.39
.39
.39
39
39
39
39
39
39
39
39
39
.39
.39
.39
.39
.40
.40
.40
.40
.40
.40
.40
.40
.40
.40
.40
.40
.40
.40
.40
.41
.41
.41
.41
.41
.41
.41
Sine. D. 1". Cotang
Tang.
9.246319
.247057
.247794
.243530
.249264
.249993
.250730
.251461
.252191
.252920
9.253648
.254374
.255100
.255824
.256547
.257269
.257990
,258710
.259429
,260146
9.260863
.261578
.262292
.263005
.263717
.264428
.265133
.265347
.266555
.267261
9.267967
,263671
.269375
.270077
.270779
,271479
,272178
.272876
.273573
.274269
9.274964
,275653
,276351
.277043
.277734
.278424
.279113
.279301
.230438
.281174
9.281858
.232542
.233225
.283907
.234.588
.235263
.235947
.236624
.287301
.237977
.2386.52
D, 1".
12.30
12.28
12.26
12.24
12.22
12.20
12.18
12.17
12.15
12.13
12.11
12.09
12.07
12.05
12.03
12.01
12.00
11.98
11.96
11.94
11.92
11.90
11.89
11.87
11.85
11.83
11.81
11.79
11.78
11.76
11.74
11,72
11,70
11.69
11.67
11,65
11.64
11.62
11.60
11.58
11.57
11.55
11.53
11.51
11.50
11.48
11.46
11.45
11.43
11,41
11,40
11,38
11.36
11,35
11.33
11.31
11.30
11.23
11.26
11.25
D, 1".
Cotang,
0.753681
,752943
.752206
.751470
,750736
,750002
.749270
,748539
,747309
,747080
0.746352
,745626
,744900
,744176
,743453
,742731
,742010
,741290
,740571
,7.39854
0,739137
,738422
,737708
,736995
,736283
.735572
,734862
,7341.53
,73.3445
,732739
0,732033
,731329
.7.30625
.729923
,729221
,728521
,727822
,727124
,726427
,725731
0.725036
,724342
,723649
,722957
,722266
,721576
,720887
.720199
,719512
,718826
0,718142
.717453
,716775
716093
,715412
,714732
.714053
,713376
,712699
.712023
,711348
Tang,
1003
»$»*
COSINES, TANGENTS, AND COTANGENTS.
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
l.j
16
17
IS
19
20
21
22
23
21
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
54
55
56
57
53
59
60
M.
185
168°
Sine.
9.230599
.231243
.231897
.232.544
.233190
.233336
.284480
.235124
.235766
.236103
9.237043
.237633
.238326
.283964
.239600
.290236
.290370
.291504
.292137
.292763
9.293399
.291029
.294658
.295286
.29.5913
.296.539
.297161
.297733
.293412
.299031
9.299555
.300276
.300395
.391514
..3021.32
.302743
.303364
..303979
.304593
.305207
9.305319
.3061.30
.307041
.307650
.3082,59
.3)8367
.309474
.310030
.310635
.311239
9.311893
.312495
.31.3097
.31.3693
.314297
.314397
.315495
.316092
.316039
.317234
.317879
D. 1".
10.82
10.81
10.79
10.77
10.76
10.74
10.72
10.71
10.69
10.67
10.66
10.64
10.63
10.61
10.59
10.53
10.56
10. .55
10.53
10.51
10.50
10.43
10.47
10.45
10.43
10.42
10.40
10.39
10.. 37
10.. 36
10.. 34
10.33
l(l.31
10. .30
10.23
10.20
10.25
10.23
10.22
10.20
10.19
10.17
10.16
10.14
10.13
10.12
10.10
10.09
10.07
10.06
10.04
10.03
10.01
10.00
9.93
9.97
9.96
9.94
9.93
9.91
Co-sine. D. 1"
Cosine.
9.991947
.991922
.991897
.991873
.99134^
.991823
.991799
.991774
.991749
.991724
9.991699
.991674
.991619
.991624
.991.599
.991574
.991549
.991524
.991498
.991473
9.99144^
.991422
.991397
.991372
.991346
.991321
.991295
.991270
.991244
.991218
9.991193
.991167
.991141
.991115
.991090
.991064
.991038
.991012
.9909S6
.990960
9.990934
.990908
.990332
.990355
.990829
.990303
.990777
.990750
.990724
.990697
9.990671
.990645
.990618
.990.591
.990565
.990538
.990511
.990485
.9904.58
.990431
.990404
Sine.
D. 1".
.41
.41
.41
.41
.41
.41
.41
.41
.41
42
.42
.42
.42
.42
.42
.42
.42
.42
.42
.42
.42
.42
.42
.42
.42
.43
.43
.43
.43
.43
.43
.43
.43
.43
.43
.43
.43
.43
.43
.43
.44
.44
.44
.44
.44
.44
.44
.44
.44
.44
.44
.44
.44
.44
.44
.44
.45
.45
D. 1".
Tang.
9.23^652
.239326
.239999
.290671
.291312
.292013
.292632
.293350
.29liM7
.294634
9.295349
.296 )13
.296677
.297339
.298001
.298662
.2993i!2
.299930
.300033
.301295
9.301951
.302607
.303261
.303914
..304.-)67
.305218
.305369
.306519
.307168
.307816
9.303463
.309109
.3097.54
.310399
.311042
.311635
.312327
.312968
.31.3603
.314247
9.314335
.315523
.316159
.316795
.3174.30
.318064
.313697
.319.330
.319961
.320592
9.321222
..321S51
.322479
.323106
.3237.33
.324358
.324933
.32.5607
.32H231
.3268.53
.327475
Cotang.
D. 1".
11.23
11.22
11.20
11.18
11.17
11.15
11.14
11.12
11.11
11.09
11.07
11. '^6
11.04
11.03
11.01
11.00
10.98
10.97
10.95
10.93
10.92
10.90
10.89
10.87
10.^6
10.84
10.^3
10.81
10.8')
10.78
10.77
10.76
10.74
10.73
10.71
10.70
10.68
10.67
10.65
10.64
10.62
10.61
10.60
10.58
10.. 57
10.55
10.54
10. .53
10.51
10.50
10.48
10.47
10.46
10.44
10.43
10.41
10.40
10.39
10.37
10.36
D. 1".
Cotang.
0.711343
.710674
.710001
.709329
.708658
.707987
.707318
.706650
.705933
.705316
0.704651
.703987
.703323
.702661
.701999
.701333
.700678
.700020
.699362
.698705
0.693049
.697.393
.696739
.6960*6
.6954.33
.691782
.694131
.693431
.692832
.692184
0.691537
.690591
.690246
.639601
.638953
.633315
.637673
.63703-2
.636392
.685753
0.635115
.684477
.633841
.633205
.632570
.631936
.631303
.630670
.630039
.679403
0.673778
.673149
.677521
.676>!94
.676267
.675042
.675017
.674.393
.673769
.673147
.672525
Tang.
lOlo
r8«
186
TABLE Xlll L05AR1TJMIC SINES,
16TC
M.
0
1
2
3
4
5
6
7
S
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
5S
59
60
M.
Si
ine
9.317879
.318473
.319066
.3196.0
.320249
.320840
.321430
.322019
.322607
.323194
9.323780
..324366
.324950
.32.5534
.326117
.326700
.327281
.327862
.328442
.329021
9.329.399
..330176
.330753
..331329
.331903
.332478
.333051
.333624
.334195
.334767
9.355337
.335906
.336475
.337043
.337610
.333176
.338742
.339-307
.339871
.340431
9.340996
.341558
.342119
.342679
.ai32.39
.343797
.3443.55
.344912
.34.5469
.346024
9.346579
.347134
.347637
.348240
.348792
.349343
.349393
.350443
.350992
.351540
.352088
Cosine.
D. 1".
Cosine.
9.90
9.83
9.87
9.86
9.34
9. S3
9.81
9.-0
9.79
9.77
9.76
9.75
9.73
9.72
9.70
9.69
9.63
9.66
9.65
9.64
9.62
9.61
9.60
9.53
9.57
9.. 56
9.54
9.53
9.52
9.50
9.49
9.43
9.46
9.45
9.44
9.43
9.41
9.40
9.39
9.37
9.36
9.35
9.34
9.-32
9.31
9.30
9.29
9.27
9.26
9.25
9.24
9.22
9.21
9.20
9.19
9.17
9.16
9.15
9.14
9.13
D. 1".
9.990404
.990378
.990351
.990324
.990297
.990270
.y9ri243
.990215
.990183
.990161
9.990134
.990107
.990079
.9900.52
.990025
.989997
.989970
.939942
.939915
.989837
9.989860
.989832
.989804
.939777
.939749
.989721
.989693
.939665
.939637
.989610
9.989-532
.989553
.939525
.989497
.939469
.989441
.939413
.939385
.9393.56
.989328
9.989300
.989271
.989243
.939214
.939186
.9391.57
.939123
.989100
.989071
.939042
9.989014
.988985
.988956
.938927
.933393
.933369
.938340
.933311
.988782
.988753
.'933724
Sine.
D. 1".
.45
.45
.45-
.45
.45
.45
.45
.45
.45
.45
.45
.45
.46
.46
.46
.46
.46
.46
.46
.46
.46
.46
.46
.46
.46
.46
.46
.47
.47
.47
.47
.47
.47
.47
.47
.47
.47
.47
.47
.47
.47
.47
.47
.48
.48
.48
.48
.48
.43
.48
.48
.48
.48
.48
.48
.48
.48
.43
.49
.49
D. 1".
Tang. I D. 1".
9.327475
.328095
.323715
.329334
.3299.53
.3.30.570
.331137
.331803
.332413
.333033
9.333646
.3.34259
.334871
.335482
.336093
.336702
.337311
.337919
.338527
.339133
9.3397.39
.340344
.340943
.341552
.342155
.3427.57
343358
3439.38
.3445.58
.345157
9.345755
..346.353
..346949
.347.545
.343141
.348735
.349329
..349922
.3.50514
.351 IG6
9.351697
..352287
.352376
.3.53465
.3540.53
..3:34640
.355227
.355813
..356398
.356982
9.357566
.3.58149
.358731
.3.59313
.3.59893
.360474
.361053
.3616.32
.362210
.362787
■ 363.364
Cotang.
10.35
10.33
10.32
10.31
10.29
10.23
10.27
10.25
10.24
10.23
10.21
10.20
10.19
10.17
10.16
10.15
10.14
10.12
10.11
10.10
10.03
10.07
10.06
10.05
10.03
10.02
10.01
10.00
9.98
9.97
9.96
9.95
9.93
9.92
9.91
9.90
9.88
9.87
9.66
9.85
9.84
9.82
9.81
9.80
9.79
9.78
9.76
9.75
9.74
9.73
9.72
9.70
9.69
9.63
9.67
9.66
9.65
9.63
9.62
9.61
D. 1".
Cotang.
0.672525
.671905
.671285
.670666
.670047
.669430
.663813
.663197
.667582
.666967
0.666354
.665741
.665129
.664518
.663907
.663293
.662689
.662031
.661473
.660867
0.660261
.659656
.6590.52
.6.5S448
.657845
.657243
.656642
.6.56042
.655442
.654843
0.654245
.6.53647
.6.5.3051
.652455
.651859
.651265
.6.50671
.650078
.649486
.648594
0.648303
.647713
.&47124
.646.535
,645947
.645360
.644773
.644187
.643602
.643018
0.6424:34
.641851
.641269
.640687
.640107
.639.526
.638947
.638368
.637790
.637213
.636636
Tang.
103?
77=
COSINES TANGENTS, AND COTANGENTS.
13^
M.
0
1
2
3
4
5
6
7
181
1663
Sine.
D. 1".
10
U
12
13
14
15
16
17
IS
19
20
21
22
23
24
25
26
27
2S
29
30
31
32
33
31
3o
3d
37
3S
39
40
41
42
43
41
45
46
47
4S
49
50
51
52
53
54
55
56
57
53
59
60
9.352'H8
.352635
.353181
.353726
.354271
.3.') IS 1 5
.355353
.355901
.3.56113
.356934
9.357524
.35SU64
.358603
.359141
.359673
.360215
.360752
.361287
.361822
.362356
9.362389
.353122
.363954
..364485
.365016
.365546
.366075
.3^)66)4
.367131
.367659
9. .368 1 85
.368711
.369236
369761
.370285
.370SOS
.371330
.371852
.372373
.372894
9..37;M14
.373933
.374452
.374970
.375487
.376003
,377035
.377549
.3780S3
9.373577
.379089
.379601
.380113
.380621
.381134
.381643
.382152
.3S2661
.3S3163
.383675
Cosine.
9.11
9.10
9.09
9.08
9.07
9.05
9.04
9.03
9.02
9.01
8.99
8.98
8.97
8.96
8.95
8.91
8.92
8.91
8.90
8.89
8.83
8.87
8.86
8.84
8.33
8.82
8.81
8.80
8.79
8.78
8.76
8.75
8.74
8.73
8.72
8.71
8.70
8.69
8.63
8.66
8.65
8.61
8.03
8.62
8.61
8.60
• 8.59
8.. 53
8.57
8.56
8.55
8.53
8.52
8.51
8.. 50
8.49
8.48
8.47
8.46
8.45
D.l".
9.938724
.988695
.938666
.938636
.938607
.933573
.988.548
.938519
.938489
.938460
9.988430
.9^3401
.988371
.933342
.933312
.913282
.9>3252
.938223
.938193
.938163
9.933133
.933103
.988073
.988043
.938013
.937933
.937953
.937922
.937892
.987862
9.9873.32
.937801
.987771
.937740
.937710
.937679
.937649
.937618
.937.588
.987557
9.937526
.937496
.937465
.937434
.957403
.937372
.937341
.987310
.937279
.987248
9.937217
.937186
.987155
.937124
.937092
.987061
.937030
.936998
.936967
.936936
.936904
M.
103 2
Cosine.
D. 1".
Tang.
Sine.
.49
.49
.49
.49
.49
.49
.49
.49
.49
.49
.49
.49
.49
.50
.50
..50
.50
.50
.50
.50
.50
.50
.50
.50
.50
.50
.50
.50
.50
.51
.51
.51
.51
.51
.51
.51
.51
.51
.51
.51
.51
.51
.51
.51
.51
.52
.52
.52
.52
.52
.52
..52
.52
.52
..52
.52
.52
52
.52
.52
D. 1"
9.363364
.363910
.361515
.365f)90
.365664
.366237
.366810
.367332
.367953
.363524
9.369094
.369663
•370232
.370799
.371367
.371933
.372499
.373064
.373629
.374193
9.374756
.375319
.375881
.376442
.377003
.377563
.378122
.373631
.379239
.379797
9.3803.54
.330910
.331466
.332020
.382575
.333129
.333632
.384231
.334786
.385337
9.335388
.386433
.386937
.387536
.333031
.383631
.339178
.339724
.390270
.390315
9.391360
.391903
.392447
.392939
.39.3531
.394073
.394614
.395154
.395694
.396233
.396771
Cotang.
9.60
9.59
9.58
9.57
9.. 55
9.54
9.53
9.52
9.51
9.50
9.49
9.48
9.47
9.45
9.44
9.43
9.42
9.41
9.40
9.39
9.33
9.37
9.36
9.-35
9.33
9.32
9.31
9.30
9.29
9.23
9.27
9.26
9.25
9.24
9.23
9.22
9.21
9.20
9.19
9.18
9.17
9.16
9.15
9.14
9.12
9.11
9.10
9.09
9.08
9.07
9.06
9.(je
9.04
9.03
9.02
9.01
9.00
8.99
8.93
8.97
0.636636
.636060
.635435
.634910
.0.34336
.633763
.633190
.632618
.632047
.631476
0.630906
.630337
.629768
.629201
.628633
.628067
.627501
.0269.36
.626371
.625807
0.625244
.624681
.624119
.623558
.622997
.6221.37
.621873
.621319
.620761
.620203
0.619640
.619090
.618.534
.617930
.617425
.616371
.616318
.615766
.615214
.614663
0.614112
.61.3562
.613013
.612464
.611916
.611369
.610822
.610276
.609730
.609185
0.6)8640
.608097
.607553
.607011
.6()6469
.605927
.605386
.604846
.604306
.603767
.6')3229
M.
60
59
53
I
D. 1". I Cotang. D. 1"
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35 !
34
33
32
31
30
29
23
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Tang.
M.
7Bi
1«8
140
TABLE XIII. LOGARITHMIC MNES,
165C^
M.
~0
1
2
3
4
5
6
7
8
9
]0
II
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine.
9.333675
.334182
.334637
.335192
.335697
.336201
.3S6704
.337207
.337709
.333210
9. .33 37 11
.339211
.33971 1
.390210
.390703
.391206
.391703
.392199
.392695
.393191
9.393635
..394179
.394673
.395166
.395653
..3961.50
.395641
.397132
.397621
.398111
9.398600
.399038
.399575
.400062
.400549
.401035
.401520
.402005
.402439
.402972
9.403455
.403938
.404420
.404901
.405382
.405362
.406341
.406320
.407299
.407777
9.403254
.408731
.409207
.409682
.410157
.410632
.411106
.411579
.412052
.412524
.412996
Cosine.
D. 1".
8.44
8.43
8.42
8.41
8.40
8.-39
8.38
3.37
3.36
8.35
8.34
8.-33
8.32
8.31
8.30
8.29
8.28
8.27
8.26
8.25
8.24
8.2:3
8.22
8.21
8.20
8.19
8.18
8.17
8.16
8.15
8.14
8.13
8.12
8.11
8.10
8.09
8.08
8.07
8.06
8.05
8.04
8.03
8.02
8.01
8.00
7.99
7.93
7.97
7.96
7.96
7.95
r.94
7.93
7.92
7.91
7.90
7.89
7.83
7.87
7.86
Cosine.
9.9869C4
.936873
.936841
.986809
.986778
.986746
.986714
.986633
.986651
.936619
9.9S6587
.986555
.986523
.936491
.936459
.936427
.936.395
.986-363
.986-331
.986299
9.9=6266
.986234
.986202
.986169
.9361.37
.986104
.936072
.936039
.986007
.93.5974
9.98.5942
,985909
.935876
.935843
.985811
.985773
.985745
.985712
.985679
.935646
9.985613
.985580
.98.5547
.98-5514
.98-5480
.935447
.935414
.985381
.98-5347
.985314
9.935230
.935247
.93-5213
.985130
.985146
.935113
.935079
.935045
.93501 1
.934978
.934944
Sine.
D. 1".
.53
.53
.53
.53
.53
.53
.53
.53
.53
.53
.53
.53
.53
.53
.53
..54
.54
.54
.54
.54
.54
.54
M
.54
.54
..54
.54
.54
.54
.54
.54
.55
..55
..55
.55
.55
.55
.55
.-55
.55
.55
.55
.55
.55
..56
.56
.56
.56
.56
.56
.56
.56
.5f
M
.56
..56
.56
.56
D. 1".
Tang.
9.396771
.397309
.397846
.393333
.398919
.399455
399990
.400524
.401058
.401591
9.402124
.402656
.403187
.403718
.404249
.404778
.405308
.405836
.406364
.406392
9.407419
.407945
.408471
.408996
.409521
.410045
.410.569
.411092
.411615
.412137
9.4126.53
.413179
.413699
.414219
.414738
.415257
.415775
.416293
.416810
.417-326
9.417842
.4183-58
.418873
.419337
.419901
.420415
.420927
421440
4219-52
.422463
9.422974
.423434
.423993
.424503
.425011
.425519
.426027
.426-534
.427041
.427.547
.428052
Cotang.
D. 1".
8.96
8.96
8.95
8.94
8.93
8.92
8.91
8.90
8.89
8.88
8.87
8. 86
8.85
8.84
8.83
8.82
8.81
8.80
8.79
8.78
8.77
8.76
8.75
8.75
8.74
8.73
8.72
8.71
8.70
8.69
8.63
8.67
8.66
8.65
8.65
8.64
8.63
8.62
8.61
8.60
8.59
8.58
8.57
8.56
8.56
8.-55
8.-54
8.53
8.52
8.51
8.50
8.49
8.49
8.48
8.47
8.46
8.45
8.44
8.43
8.43
D. 1".
Cotang
M.
60
0.603229
.602691
59
.602154
58
.601617
57
.601081
56
.600545
55
.600010
54
.599476
53
.593942
52
.598409
51
0.597376
50
.597344
49
.596313
48
.596232
47
.595751
46
.595222
45
.594692
44
.5941&4
43
.5936.36
42
.593103
41
0.592531
40
.5920.55
39
.591529
38
.591004
37
.590479
36
.589955
35
.539431
34
.588908
33
.583385
32
.587863
31
0.537342
30
.586821
29
.586301
28
.535781
27
.585262
26
.534743
25
.584225
24
.583707
23
..583190
22
.582674
21
0.582158
20
.581642
19
.581127
18
.-580613
17
.580099
16
.579535
15
..579073
14
.578560
13
.578043
12
.577537
11
0.577026
10
.576516
9
.576007
8
.57.5497
7
.574989
6
.574481
5
.573973
4
.573466
3
.572959
2
.572453
1
.571948
0
M.
Tang.
1040
T«i
COSINES, TANGENTS, AND COTANGENTS.
189
M.
0
1
2
3
4
5
6
7
Sine.
9.412996
.413467
.413933
.414408
.414S78
.415347
.415815
.416283
.416751
.417217
D.l"
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
.34
35
36
37
38
39
40
41
42
43
44
45
46
47
4S
49
50
51
52
53
54
55
56
57
58
ro
60
9.417634
.418150
.418615
.419079
.419544
.420007
.420470
.420933
.421395
.421857
9.422318
.422773
.4232.33
.423697
.424156
.424615
.425073
.425530
.425987
.426443
9.426899
.427354
.427809
.428263
.428717
.429170
.429623
.430075
.430.527
.430978
9.431429
.431879
.4.32329
.432778
.433226
.433675
.434122
.434569
.435016
.435462
9.43.5903
.436353
.436793
.437242
.437636
.438129
.43^572
.439014
.439456
.439397
.440338
Cosine.
7.85
7.84
7.84
7.83
7.82
7.S1
7.80
7.79
7.78
7.77
7.76
7.75
7.75
7.74
7.73
7.72
7.71
7.70
7.69
7.68
7.67
7.67
7.66
7.65
7.6-4
7.63
7.62
7.61
7.61
7.60
7.59
7.58
7.57
7.56
7.55
7.55
7.53
7.52
7.52
7.51
7.50
7.49
7.49
7.48
7.47
7.46
7.45
7.44
7.44
7.43
7.42
7.41
7.40
7.40
7.39
7.38
7.37
7.36
7 36
7.35
D. 1".
9.984944
.984910
.984876
.934842
.984308
.984774
.934740
.934706
.934672
.934638
9.984603
.984569
.984535
.984500
.984466
.984432
.934397
.934363
.984328
.934294
9.934259
.934224
.934190
.934155
.984120
.984085
.984050
.984015
.933931
.983946
9.9S3911
.983875
.933840
.933805
.983770
.933735
.983700
.983664
.983629
.983594
9.983553
.933523
.983487
.9834.52
.983416
.983381
.983345
.983309
.98.3273
.983238
M. Cosine. I D. 1".
9.98.3202
.983166
.983130
.983094
.983058
.933022
.982936
.982950
.932914
.932378
.982342
Tang.
.56
.57
.57
.57
.57
.57
.57
.57
.57
.57
.57
.57
.57
.57
.57
.57
.58
.58
.58
.58
.58
.58
.58
.58
.58
.58
.53
.53
.53
.58
.58
.58
.59
.59
.59
.59
.59
.59
.59
.59
59
.59
.59
.59
.59
.59
..59
.60
.60
.60
.60
.60
.60
.60
.60
.60
.60
.60
.60
.60
9.428052
.428558
.429062
.429566
.430070
.430573
.431075
.431577
.432079
.432580
9.433080
.433580
.434080
.434579
.435078
.435576
.436073
.436570
.437067
Sine.
D. 1".
.437563
9.438059
.4335.54
.439048
.439543
.440036
.440529
.441022
.441514
.442006
.442497
9.442988
.443479
.443968
.444458
.444947
.445435
.445923
.446411
.446898
.447384
9.447870
.443356
.443841
.449326
.449810
.450294
.4.50777
.451260
.451743
.452225
9.452706
.453187
,453668
.4.54148
.4.54628
.455107
.455586
.456064
.456542
.457019
.457496
D.r
Cotang.
8.42
8.41
8.40
8.39
8.38
8.38
8.37
8.36
8.35
8.34
8.33
8.33
8.32
8.31
8.30
8.29
8.28
8.28
8.27
8.26
8.25
8.24
8.24
8.23
8.22
8.21
8.20
8.20
8.19
8.18
8.17
8.16
8.16
8.15
8.14
8.13
8.13
8.12
8.11
8.10
8.09
8.09
8.08
8.07
8.06
8.06
8.05
8.04
8.03
8.03
8.02
8.01
8.00
8.00
7.99
7.98
7.97
7.97
7.96
7.95
M.
0.571948
.571442
.570933
.570434
.569930
.569427
.568925
.563423
.567921
.567420
0.566920
.566420
.565920
.565421
.564922
.564424
.563927
.5^3430
.562933
,562437
0.561941
.561446
.560952
.560457
.559964
.5.59471
.556978
.558486
.557994
.557.503
0.5.57012
.556521
.556032
.555542
.555053
.554565
.554077
.553589
.553102
.552616
0.552130
60
59
58
57
56
55
54
53
52
51
.fc)0
Cotang.
D. 1".
1644
.551159
.550674
.550190
.549706
.549223
.543740
.543257
.547775
0.547294
.546313
.546332
.545852
.545372
.544893
.544414
.543936
.543458
.542981
.542.504
Tang.
50
49
43
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
_0^
M.
105°
7*0
190
160
TABLE Xlll. LOGARITHMIC SINES,
163f
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
IS
19
20
21
22
23
24
25
26
27
23
29
30
3!
32
33
31
35
3G
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine.
9.440333
.440778
.441213
.4416.53
.442096
,442535
.442973
.41.3410
.443347
.444231
9.444720
.445155
.445590
.446025
.4464-59
.446393
.447326
.447759
.443191
.443623
9.4490.54
.449435
.449915
450345
.450775
.451204
.451632
.4.52060
.452483
.4.52915
9.4.53342
.453763
.451194
.454619
.45.5044
.45.5469
.455393
.4.56316
.456739
.457162
9.457534
.453006
.453427
.453S4S
.459263
.4.59633
.460103
.460527
.460946
.461364
9.461782
.462199
.462616
.463932
.463448
.463864
.454279
.464694
.465103
.465522
.465935
Cosine.
D 1".
7.34
7.33
7.32
7.31
7.31
7.30
7.29
7.23
7.27
7.27
7.26
7.25
7.24
7.24
7.23
7.22
7.21
7.20
7.20
7.19
7.18
7.17
7.17
7.16
7.15
.14
.13
.13
.12
.11
7.
7.
7.
7.
7.
7.10
7.10
7.09
7.08
7.07
7.07
7.06
7.05
7.04
7.04
7.03
7.02
7.01
7.01
7.00
6.99
6.98
6.98
6.97
6.96
6.96
6.95
6.94
6.93
6.93
6.92
6.91
6.90
6.90
6.S9
Cosine.
9.982842
.982305
.982769
.9327.33
.982696
.932660
.9326^4
.9»25'^7
.982551
.982514
9.9S2477
.932441
.982404
.932367
.932331
.982294
.9322.57
.932220
.982183
.982146
9.9>2109
.932072
.9820.35
.931998
.981961
.931924
.9818^6
.981849
.931812
.981774
9.931737
.931700
.931662
.931625
.981587
.931549
.931512
.981474
.931436
.981399
9.981361
.981323
.931235
.931247
.931209
.931171
.9311.33
.931095
.931057
.981019
9.980931
.930942
.930904
.930366
.930827
.930789
.930750
.930712
.930673
.930635
.980.596
D. 1". I Sine.
D. 1".
.60
.60
.61
.61
.61
.61
.61
.61
.61
.61
.61
.61
.61
.61
.61
.61
.61
.62
.62
.62
.62
.62
.62
62
.62
.62
.62
.62
.62
.62
.62
.62
.63
.63
.63
.63
.63
.63
.63
.63
.63
.63
.63
.63
.63
.63
.63
.64
.64
.64
.64
.64
.64
.64
.64
.64
.64
.64
.64
.64
Tang.
9.4574,16
.457973
.453449
.453925
.459400
.459875
.460.349
.460323
.461297
.461770
9.462242
.462715
.463b:6
.463658
.464128
.464.599
.46.5069
.465539
.466008
.466477
9.4R6945
.467413
.4678^0
.468347
.463314
.469280
.469746
.470211
.470676
.471141
9.471605
.472069
.472.532
.472995
.473457
.473919
.474381
.474342
.475303
.475763
9.476223
.476633
.477142
.477601
.478059
.473517
.478975
.479432
.479839
.430345
9.480301
.431257
.431712
.432167
.482621
.483075
.433.529
.433932
.434435
.434837
.435339
D. 1". Cotang.
D. 1".
7.94
7.91
7.93
7.92
7.91
7.91
7.90
7.89
7.83
7.83
7.87
7.>6
7.86
7.85
7.84
7. S3
7.83
7. 82
7.^1
7.81
7.30
7.79
7.78
7.73
7.77
7.76
7.76
7.7o
7.74
7.74
7.73
7.72
7.71
7.71
7.70
7.69
7.69
7.63
7.67
7.67
7.66
7.65
7.65
7.64
7.63
7.63
7.62
7.61
7.61
7.60
7.59
7. .59
7.53
7.57
7.57
7. .56
7.55
7.55
7.54
7.53
D. 1".
Cotang.
0.542504
.542027
.541.551
.541075
.540600
.540125
..539651
.539177
.538703
..533230
0.537758
.5372^5
.5.36314
.5.36;M2
.535372
.535401 '
.534931
.15.34461
.533992
.533523
0.533055
.532587
.5.32120
.5316.53
.531186
.530720
.530254
.529739
..529324
.528859
0.523395
.527931
.527463
.527005
.526-543
.526031
.52.5619
.525153
.524697
.524237
0.523777
.523317
..522353
.522399
..521941
..521433
.521025
.520.563
.520111
.519655
0.519199
.518743
.518283
.517833
.517379
.516925
.516471
.516018
.515565
.515113
.514661
Tang.
^060
73^
COSINES,
TANGENTS, AND COTANGENTS.
191
M.
a
6
7
8
9
in
11
12
13
14
1-3
16
17
18
19
20
21
22
23
24
25
26
27
2S
23
30
31
32
33
34
35
36
37
SS
39
40
41
42
43
44
4',
46
47
4S
49
50
51
Sine.
9.465935
.466348
.466761
.467173
.4675S5
.467996
.463407
.463317
.469227
.469637
9.470046
.470455
.470363
.471271
.471679
.472036
.472492
.472393
.473301
.473710
9.474115
.474519
.474923
.475327
.475730
.476133
.476.536
.476933
.477310
.477741
9.478142
.478542
.478942
.479342
.479741
.430140
.430539
.480937
.431334
.431731
•9.432128
.432525
.432921
.483316
.483712
.484107
.434501
.434395
.435239
.485632
9.436075
.436467
D 1". I Cosine. D- 1". Tang.
52
.436S60
53
.437251
54
.437643
55
.483034
56
.433424
57
.488314
53
.439204
59
.489593
60
.439932
M.
Cosine.
6 83
6.88
6.87
6.86
6.85
6.85
6.84
6.83
6.83
6.82
6.81
6.81
6.80
6.79
6.78
6.78
6.77
6.76
6.76
6.75
6.74
6.74
6.73
6.72
6.72
6.71
6.70
6.69
6.69
6.63
6.67
6.67
6.66
6.65
6.65
6.64
6.63
6.63
6.62
6.61
6.61
6.60
6.59
6.59
6.57
6.57
6.56
6.55
6.55
6.-54
6.. 54
6.53
6.52
6.52
6.51
6.50
6.50
6.49
6.48
9.950596
.930553
.930519
.930430
.980412
.980403
.930364
.930325
.930236
.930247
9.980208
.930169
.980130
,930091
.980052
.980012
.979973
.979934
.979395
.979855
9.979316
.979776
.979737
.979697
.979653
.979613
.979579
.979539
.979499
.979459
9.979420
.979330
.979310
.979300
.979260
.979220
.979130
.979140
.979100
.979059
9.979019
.978979
.978939
.978898
.978353
.978317
.978777
.973737
.978696
.978055
9.978615
.978574
.978533
.978493
.978452
.973411
.978370
.978329
.978233
.978217
.973206
D. 1"
D. 1". I Sine
9.485339
.485791
.436242
.4'^6693
.487143
.437593
.483043
.488492
.483941
.489390
9.439833
.490236
.490733
.491180
.491627
1 .492073
.492519
.492965
.493110
.493354
9.494299
.494743
.495136
.495630
.496073
.496515
.496957
497399
.497841
.493232
9.493722
.499163
.499603
.500042
.500431
.500920
.5013.59
.501797
..502235
.502672
9.503109
50.3546
503932
.504418
.504354
.505239
.50.5724
.506159
..506593
.507027
9.507460
.507893
.503326
.503759
.509191
.509622
.5100.54
.510435
.510916
.511346
.511776
Cotang.
D. 1". I Cotang.
7.53
7.52
7.51
7.51
7.50
7. .50
7.49
7.43
7.43
7.47
7.46
7.46
7.45
7.44
7.44
7.43
7.43
7.42
7.41
7.41
7.40
7.39
7.39 .
7.33
7.33
7.37
7.36
7.36
7.35
7.34
7.-34
7.33
7.33
7.32
7.31
7.31
7.30
7.30
7.29
7.23
7.23
7.27
7.27
7.26
7.25
7.25
7.24
7.24
7.23
7.23
7.22
7.21
7.21
7.20
7.20
7.19
7.18
7.18
7.17
7.17
0.514661
.514209
.513758
.513307
.512357
.512407
.5119.57
.511503
.5110.59
.510610
0.510162
.509714
.509267
.503320
.505373
.507927
.507431
.507035
.506590
.506146
0.505701
.505257
.504314
.504370
.50-3927
.503435
.50-3043
.502601
.5021-59
.501718
0.501278
.500^37
.500397
.499958
.499519
.499030
.493641
.493203
.497765
.497328
0.496391
.496454
.496018
.495532
.495146
.494711
.494276
.493341
.493407
.492973
M.
60
59
53
57
56
D. 1".
Ci492540
.492107
.491674
.491211
.490309
.490373
.4899 16
.439515
.489034
.483654
,438224
54
53
50
49
43
47
46
45
44
43
42
41
40
39
33
37 I
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
0
Tang.
lor^
4
3
2
1
0
M.
7a«
192
183
TABLE XIII. LOGARITHMIC SINES,
161<;
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine.
9.4S9932
.490371
.490759
.491147
.491535
.491922
.492303
.492695
.493031
.493466
9.493351
.494236
.494621
.495005
.49.!;333
.495772
.496154
.496.537
.496919
.497301
9.497632
.493064
.493444
.493325
.499214
.499534
.499963
.500342
.500721
.501099
9.. 50 14 76
.5015.54
..50-^231
..502607
.502934
.503360
.5037.3.J
.504110
.504435
.504560
9. .505234
.505603
..505931
..506354
..506727
.507099
.507471
.507343
..503214
.503535
9.503956
.509326
.509696
.510065
.510434
.510303
.511172
.511540
.511907
.512275
.512642
C:«ine.
D. 1".
6.43
6A7
6.46
6.46
6.45
6.45
6.44
6.43
6.43
6.42
6.41
6.41
6.40
6.39
6.. 39
6. .33
6.33
6..37
6.36
6.36
6.35
6.34
6.3^4
6.33
6.33
6.32
6.31
6.31
6.30
6.30
6.29
6.23
6.23
6.27
6.27
6.26
6.25
6.25
6.24
6.24
6.23
6.22
6.22
6.21
6.21
6.20
6.19
6.19
6.13
6.13
6.17
6.16
6.16
6.15
6.15
6.14
6.14
6.13
6.12
6.12
D. 1".
Cosine.
9.973206
.973165
.973124
.973033
.973042
.973001
.977959
.977913
.977377
.977835
9.977794
.977752
.977711
.977669
.977623
.977536
.977^544
.977503
.977461
.977419
9.977377
.977335
.977293
.977251
.977209
.977167
.977125
.977033
.977041
976999
9.976957
.976914
.976372
.976330
.976737
.976745
.976702
.976660
.976617
.976574
9.976532
.976439
.976446
.976404
.976:361
.976313
.976275
.976232
.976139
.976146
9.976103
.976060
.976017
.975974
.975930
.975357
.975344
.975300
.975757
.975714
■975670
Sine.
D. 1".
.63
.69
.69
.69
.69
.69
.69
.69
.69
.69
.69
.69
.69
.69
.69
.69
.70
.70
.70
.70
.70
.70
.70
.70
.70
.70
.70
.70
.70
.70
.70
.71
.71
.71
.71
.71
.71
.71
.71
.71
.71
.71
.71
.71
.71
.72
.72
.72
.72
.72
.72
.72
.72
.72
.72
.72
.72
.72
.72
.72
D. 1".
Tang.
9.511776
.512206
.512635
.513064
513493
.513921
.514349
.514777
.515204
.515631
9.516057
.516434
.516910
.517.335
.517761
.513156
.515810
.519034
.5194.53
.519382
9.520305
.520723
.521 151
.521573
..521995
.522417
.522333
.523259
.52.3630
.524100
9.524-520
.524940
.525359
.525778
.526197
..526615
.527033
.527451
.527863
.528285
9.525702
.-529119
..529.535
..529951
.5.30-366
..5-30781
..531196
..531611
.53202-5
.5-32439
9.-532353
.533266
.5.33679
.534092
.534-504
.53^916
.53.5323
.535739
.-5361.50
.-5-36561
.5-36972
Cotang.
D. 1".
7.16
7.16
7.15
7.14
7.14
7.13
7.13
7.12
7.12
7.11
7.10
7.10
7.09
7.09
7.03
7.03
7.07
7.07
7.06
7.05
7.05
7.04
7.04
7.03
7.03
7.02
7.02
7.01
7.01
7.00
6.99
6.99
6.93
6.98
6.97
6.97
6.96
6.96
6.G5
6.95
6.94
6.94
6.93
6.93
6.92
6.91
6.91
6.90
6.90
6.89
6.39
6.33
6.83
6.87
6.87
6.86
6.86
6.85
6. 85
6.34
D. 1".
Cotang.
0.438224
.437794
.437365
.456936
.486507
.456079
.435651
.435223
.434796
.434369
0.48-3943
.433516
.45-3090
.452665
.452239
.431514
.481.390
.430966
.430.542
.430113
0.479695
.479272
.478349
.473427
.478005
.477553
.477162
.476741
.476320
.475900
0.475450
.475060
.474641
.474222
.47-3303
.473355
.472967
.472;!^ 9
.472132
.471715
0.47129S
.47033!
.470465
.470049
.4696:34
.469219
.465504
.463389
.467975
.467561
0.467147
.466734
.466321
.465908
.465496
.465034
.464672
.464261
.46-33^50
.46:34:39
.463023
Tang
M.
1085
7J'
COSINES, TANGENTS, AND C0TANC4ENTS.
193
160C
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
lo
16
17
IS
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
51
55
56
57
5S
59
60
M.
Sine.
D. 1".
9.512642
.513009
.513375
.513741
.514107
.514472
.514837
.515202
.515566
.515930
9.516294
.516657
.517020
.517745
.518107
.518463
.513829
.519190
.519551
9.519911
.520271
.520631
.520990
.521349
.521707
.522066
.522424
.522781
.52:3133
9.523495
.523352
.524203
.524564
.524920
.525275
.525630
.525934
..526339
.526693
9.527046
.527400
.527753
.523105
.523453
.523310
.529161
.529513
.529364
.530215
9.530r565
.530915
.531265
.531614
.531963
.532312
.532661
.533009
.533357
.533701
.531052
Cosine.
Cosine.
6.11
6,11
6.10
6.09
6.09
6.03
6.03
6.07
6.07
6.06
6.05
6.05
6.04
6.01
6.03
6.03
6.02
6.02
6.01
6.00
6.00
5.99
5.99
5.93
5.98
5.97
5.97
5.96
5.95
5.95
5.94
5.94
5.93
5.93
5.92
5.92
5.91
5.90
5.90
5.89
5.89
5.88
5.88
5.87
5.87
.5.86
5.86
5.85
5.85
5.84
5.33
5.82
5.82
5.81
5.81
5.30
5.30
5.79
5.79
D. 1".
9.975670
.975627
.975533
.975539
.975496
.975452
.975403
.975365
.975321
.975277
9.975233
.975189
.975145
.975101
.975057
,975013
.974969
.974925
.974880
.974336
9.974792
.974748
.974703
.974659
.974614
.974570
.974525
.974481
.974436
.974391
9.974347
.974302
.974257
.974212
.974167
.974122
.974077
.974032
.973937
.973942
9.973397
.973852
.973307
.973761
,973716
.973671
.973625
.973530
.973535
.973489
9.973444
.973393
.973352
.973307
.973261
.973215
.973169
.973124
.973078
.973032
.972936
.73
.73
Tang.
D. 1".
Sine.
.73
.73
.73
.73
.73
.73
.73
.73
.73
.73
.73
.74
.74
.74
.74
.74
.74
.74
.74
.74
.74
.74
.74
.74
.74
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.75
.76
.76
.76
.76
.76
.76
.76
.76
.76
.76
.76
•.76
.76
.76
.77
.77
9.536972
.537382
.537792
,533202
.53861 1
.539020
.539429
.539837
.540245
D. 1"
.0
40653
9.541061
,541463
.541875
.542231
.542638
.543094
.543499
.543905
.544310
,544715
9.545119
,545524
.545928
,546331
.546735
,547138
.547540
.547943
.548345
,548747
9.. 549 149
.549550
..549951
,550352
.550752
.551153
.551552
.551952
.5523:51
.552750
9.553149
.553548
.553946
.554344
.554741
.555139
.555536
.555933
.556329
.556725
9.5.57121
.557^17
.557913
..558303
.558703
..559097
.559491
.559335
.560279
.560673
..561066
6.84
6.33
6.83
6.82
6.82
6.81
6.81
6.80
6.80
6.79
6.79
6.78
6.78
6.77
6.77
6.76
6.76
6.75
6.75
6.74
6.74
6.73
6.73
6.72
6.72
6.71
6.71
6.70
6.70
6.69
6.69
6.68
6.63
6.67
6.67
6.67
6.66
6.66
6.65
6.65
6.64
6.64
6.63
6.63
6.62
6.62
6.61
6.61
6.60
6.60
6.59
6.59
6.59
6.58
6.58
6.57
6.57
6.56
6.56
D. 1". Cotang.
Cotang.
0.463023
.462618
.462203
.461798
.461339
.460980
.460571
.460163
,459755
.459347
0.458939
.458532
,458125
.457719
.457312
.4.56906
.456501
.4.56095
.455690
.455285
0.454881
.454476
.4.54072
.4536P9
.453265
.452362
.452460
,452057
.451655
,451253
0.450351
.450450
.450049
.449643
.449248
.448847
.443443
,443018
.447649
.447250
0.446351
,446452
,446054
,445656
.445259
,444861
.444464
.441067
.443671
.443275
0.442379
.442483
.442037
.441692
.441297
,440903
,440509
.440115
.439721
.439327
.433934
M.
60
59
53
57
56
55
34
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
23
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
D. 1".
Tang.
M.
1090
7©:
194
TABLE XIII.
LOGARITHMIC SINES,
159<J
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
IS
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
4S
49
50
51
52
53
54
55
56
57
58
59
60
M.
Sine.
9.534052
..5-34399
.534745
.53.5092
.53.5438
.535783
.536129
.536474
.536318
.537163
9.537507
.537851
.533194
.538538
.533880
.539223
..5.39.565
.539907
.540249
.540590
9.540931
.541272
.541613
..541953
.542293
.5426.32
.542971
.543310
.543649
.543987
9.544325
.544663
.545000
545338
.545674
.546011
.546347
..546633
.547019
.547354
9.547639
.548024
.543359
.543693
..549027
.549360
.549693
.5.50026
.5.50359
.550692
9. .55 1024
.551356
.551687
..552018
.552349
.552630
.5.53010
.55.3341
.553670
.554000
.554329
Cosine.
D. 1".
5.78
5.73
5.77
5.77
5-. 76
5.76
5.75
5.75
5.74
5.74
5.73
5.73
5.72
5.71
5.71
5.70
5.70
5.69
5.69
5.68
5.63
5.67
5.67
5.66
5.66
5.65
5.65
5.64
5.64
5.63
5.63
5.62
5.62
5.61
5.61
5.60
5.60
5.59
5.59
5.58
5.58
5.57
5.57
5.56
5.56
5.55
5.55
5.55
5.54
5.54
5.53
5.53
5.52
5.52
5.51
5.51
5.50
5.50
5.49
5.49
D. 1' .
Cosine.
9.972986
.972940
.972394
.972348
.972302
.972755
.972709
.972663
.972617
.972570
9.972524
.972478
.972431
.972335
.972333
.972291
.972245
.972193
.972151
.972105
9.972053
.972011
.971964
.971917
.971870
.971323
.971776
.971729
.971632
.971635
9.971588
.971.540
.971493
.971446
.971398
.971351
.971303
.9712.56
,971203
.971161
9.971113
.971066
.971018
.970970
.970922
.970874
.970827
.970779
.970731
.970633
9.970635
,970586
.970538
.970490
.970442
.970394
.970345
.970297
.970249
.970200
.970152
Sine.
D. 1".
.77
.77
.77
.77
■ .77
.77
.77
.77
.77
.77
.77
.77
.73
.78
.78
.78
.78
.78
.78
.78
.78
.78
.78
.73
.78
.78
.78
.79
.79
.79
.79
.79
.79
.79
.79
.79
.79
.79
.79
.79
.79
.80
.80
.80
.80
.80
.80
.80
.80
.80
.80
.80
.80
.80
.80
.81
.81
.81
.81
.81
D. 1".
Tang.
9.561066
.561459
.561851
.562244
.562636
.563023
.563419
.563311
.564202
.564593
9.564933
.565373
.565763
.566153
.566542
.566932
.567320
.567709
.563093
.563486
9.563373
..569261
.569643
.570035
.570422
.570309
.571195
.571531
.571967
.572352
9.572733
.573123
.573507
.573392
,574276
.574660
.575044
.575427
.575310
.576193
9.576576
.576959
.5773-11
.577723
.578104
.578486
.578367
.579243
.579629
.530009
9.580339
.580769
.581149
.581523
.531907
.532236
.532665
.533044
.533422
.533300
.584177
Cotang.
D. 1".
6.55
6.54
6.. 54
6.54
6.53
6.53
6.52
6.52
6.51
6.51
6.. 50
6.50
6..50
6.49
6.49
6.48
6.48
6.47
6.47
6.46
6.46
6.46
6.45
6.45
6.44
6.44
6.43
6.43
6.43
6.42
6.42
6.41
6.41
6.40
6.40
6.40
6.39
6.39
6.38
6.33
6.37
6.37
6.37
6.36
6.36
6.35
6.35
6.34
6.34
6.34
6.33
6.33
6.32
6.32
6.32
6.31
6.31
6.30
6.30
6.30
D. 1".
Cotang.
0.438934
.4.33541
.433149
,437756
.437364
.436972
.436.-5S1
.436139
.435793
.435407
0.43.5017
.434627
.434237
.4.33847
.4.33453
.4.33068
.4.32680
.432291
.431902
.431514
0.431127
.430739
.430352
.429965
.429578
.429191
.428805
.428419
.423033
.427648
0.427262
.426377
.426493
.426103
.425724
.425340
.424956
.424573
.424190
.423307
0.423424
.423041
.422659
.422277
.421896
.421514
-.421133
.420752
.420371
.419991
0.419611
.419231
.418351
.418472
.413093
.417714
.417335
.416956
.416578
.416200
.415823
Tang.
IIOO
603
COSINES, TANGENTS, AND COTANGENTS.
1580
M.
Sine.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
4S
49
50
51
52
53
54
55
56
57
53
59
60
9.554329
.554653
.55411^7
.555315
.555643
. .555971
.556299
.r 56626
.556953
.557280
9.557606
.557932
..558253
.55858 !
.5.58909
.5592:34
.559558
.559883
.560207
.560531
9.560855
..561178
.561501
.561824
..562146
D. 1''.
.0
62468
.562790
.563112
.563433
.563755
9.564075
.564396
.564716
.565036
.565356
..565676
.565995
..566314
..566632
..566951
9.567269
.567587
.567904
.568222
.568539
.568856
.56917^
.569438
..569804
.570120
9.570435
.570751
.571066
..571.3S0
.571695
.572009
,572323
572636
572950
.573263
..573575
Cosine.
D. 1".
5.48
5.48
5.47
5.47
5.46
5.46
5.45
5.45
5.44
5.44
5.44
5.43
5.43
5.42
5.42
5.41
5.41
5.40
5.40
5.39
5.39
5.38
5.33
5.37
5.37
5.37
5.36
5.36
5.35
5.-35
5.34
5.33
5.33
5.32
5.32
5.32
5.31
5.31
5.30
5.. 30
5.29
5.29
5.28
5.28
5.28
5.27
5.27
5.26
5.25
5.24
5.24
5.24
5.23
5.23
5.22
5.22
5.21
9.970152
.970103
.970055
.970006
.969957
.969909
.969'-60
.969811
.969762
.969714
9.969665
.969616
.969567
.969518
.969469
.969420
,969370
.969321
.969272
,969223
9.969173
,969124
.969075
.969025
.968976
.968926
.968877
.968827
.968777
.968728
9.96S678
.968628
.968578
.968523
.968479
.968429
.968379
.963329
.963278
.963228
9.968178
.968128
.963078
.963027
.967977
.967927
.967876
.967826
.967775
.967725
9.967674
.967624
.967573
.967.522
.967471
.967421
.967370
.967319
.967268
.967217
.967166
M. I Cosine. I D. 1".
Tang.
D. 1".
Sine.
.81
.81
,81
.81
81
.81
.81
.81
.81
,81
,82
.82
.82
.82
.82
.82
.82
.82
.82
.82
.82
.82
.82
.82
.83
,83
.83
,83
,83
,83
.83
,83
,83
.83
.83
.83
.83
.83
.84
.84
84
.84
.84
.84
.84
.84
.81
.84
.84
.84
.84
.84
.85
.85
.85
.85
.85
.85
.85
9.584177
.584555
.584932
..585309
.585686
.586062
.536439
.5%815
.587190
.587566
9.587941
.588316
.588691
..589066
.589440
.589814
.590188
.590562
.590935
.591308
9.591681
.592054
,592426
.592799
.593171
.593.542
.593914
.594285
,594656
,595027
9.595393
,595768
.596138
..596508
.596878
..597247
I .597616
I .597985
.598354
.598722
9.599091
.599459
.599827
.600194
.600.562
.600929
.601296
.601663
.602029
.602395
9.602761
.603127
.603493
.603858
.604223
.604583
.601953
,605317
.605682
.606046
.606410
Cotaiig. M.
D. 1". Cotang.
6.29
6.29
6.28
0.28
6.28
6.27
6.27
6.26
6.26
6.26
6.25
6.25
6.24
6.24
6.24
6.23
6.23
6.22
6.22
6.22
6.21
6.21
6.20
6,20
6.20
6.19
6.19
6.18
6.18
6.18
6.17
6.17
6.16
6.16
6.16
6.15
6.15
6.15
6.14
6.14
6.13
6.13
6.13
6.12
6.12
6.12
6.11
6,11
6.10
6.10
6.10
6.09
6.09
6.09
6.08
6.03
6.07
6.07
6.07
6.06
0. 41.5^23
.415445
.41.5068
.414691
.414314
.413938
.413561
.413185
.412810
,412434
0.412059
.411684
,411309
,410934
.410560
.410186
.409812
.409438
.409065
.408692
0.408319
.407946
.407574
.407201
.406829
.406458
.406086
.405715
.405344
.404973
0.404602
.404232
.403^62
.4034'..2
.403122
.402753
.402384
.402015
.401646
.401278
0.400909
.400541
,400173
.399806
.399433
.399071
.398704
.398337
.397971
.397605
0.397239
.396873
.396507
.396142
.395777
.395412
.395047
,394683
.394318
.393954
.393590
1). 1".
Tang. M
60
59
68
56
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
111
68^
*-W9rr^^9^m^\^^ww-r #^f .A^V^^tV/jT^^AV
196
933
TABLE XIII. LOGARITHMIC SI^'ES,
157'
M.
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
2S
29
30
31
32
33
34
3-5
36
37
33
39
40
41
42
43
44
45
46
47
4S
49
50
51
52
53
54
55
58
57
53
59
60
Sine.
9.573575
.573333
.574200
.574512
.574S24
.575136
.57.5147
.575753
.576069
.576379
9.576639
.576999
.577309
.577613
.577927
.573236
.573545
.573853
.579162
.579470
9.579777
.530035
.530392
.530699
.531005
.531312
.531613
.531924
.532229
.532535
9.532340
.533145
.533449
.533754
.581053
.534361
..531665
.534963
..535272
.535574
9. 535S77
..536179
. .536132
.536733
,537035
.537336
.537633
.537939
.583239
.533590
9.533390
.539190
.539439
.539789
.590033
.590337
.593636
.500934
.591232
.591530
.591373
D. 1". Cosine.
M. Cosine. D. 1".
5.21
5.2(
5.2C
5.20
5,19
5.19
5.13
5.13
5.17
5.17
5.17
5.16
5.16
5. 15
5.15
5.14
5.14
5.14
5.13
5.13
5.12
5.12
5.11
5.11
5.11
5.10
5.10
5.09
5.09
5.09
5.03
5.03
5.07
5.07
5.06
5.06
5.06
5.05
5.05
5.04
5.04
5.04
5.03
5.03
5.02
5.02
5.01
5.01
5.01
5.00
5.00
4.99
4.99
4.99
4.93
4.93
4.97
4.97
4.97
4.96
9.967166
.967115
.967064
.967013
.966961
.966910
.966359
.966303
.966756
.966705
9.966653
.966602
.966550
.966499
.966447
.966395
.966iH
.966292
.966240
.966133
9.966136
.966035
.966933
.965981
.965929
.965376
.965324
.965772
.965720
.965663
9.96.5615
.965563
.965511
.965453
.96^5406
.965a53
-.965301
.965243
.965195
.965143
9.965090
.965037
.961934
.964931
.961379
.964326
.961773
.961720
.961666
.961613
9.961560
.964507
.964454
.964400
.964:347
.961294
.961240
.961187
.964133
.961030
.961026
D. 1".
Sine.
.85
.85
.85
.85
.85
.35
.86
.86
.86
.36
.86
.86
.86
.86
.86
.86
.56
.86
.86
.86
.87
.87
.87
.37
.87
.87
.87
.87
.87
.87
.87
.37
.87
.87
.83
.83
.83
.88
.83
.83
.83
.83
.83
.83
.83
.33
.33
.83
.89
.39
.89
.89
.89
.89
.39
.39
.89
.89
.89
.89
Tang.
D. 1".
9.605110
.606773
.607137
.607.500
.6' 17363
.603225
.6034533
.603950
.609312
.609674
9.610036
.610397
.610759
.611120
.611430
.611341
.6122)1
.612561
.612921
.613231
9.613641
.614000
.6143.59
.614713
.615077
.6154.35
.615793
.616151
.616509
.616367
9.617224
.617.532
.617939
.613295
.613652
.619033
.619364
.619720
.620076
.623432
9.620737
.621142
.621497
.621352
.622207
.622561
.622915
.623269
.623623
.623976
9.624330
.624633
.6250.36
.62-5333
.625741
.626093
.626445
.626797
.627149
.627-501
.627352
Cotang.
D. 1".
6.06
6.06
6.05
6.05
6.05
6.04
6.04
6.03
6.03
6.03
6.02
6.02
6.02
6.01
6.01
6.01
6.00
6.00
6.00
5.99
5.99
5.93
5.93
5.93
5.97
5.97
5.97
5.96
5.96
5.96
5.95
5.95
5.95
5.94
5.94
5.94
5.93
5.93
5.93
5.92
5.92
5.92
5.91
5.91
5.91
5.90
5.90
5.90
5.89
5.89
5.89
5.83
5.83
5.83
5.87
5.87
5.87
5.86
5.86
5.86
D. 1".
Cotang.
0.393590
.393227
.392363
.392500
.392137
.391775
.391412
..3910-50
.390633
.390326
0.339964
.339603
.339241
.333330
.333520
.338159
.337799
.337439
.387079
.336719
0.336359
..336000
.33.5641
.33-5232
.334923
.384565
.384207
.33:3349
.333491
.333133
0.332776
.332413
.332061
.331705
.331:343
.330992
.330636
.330230
.379924
.379563
0.379213
.378353
.373503
.373143
.377793
.377439
.377035
.376731
.376:377
.376024
0.375670
.375317
.374964
.374612
.3742.59
.373907
.373555
.373203
.372351
.372499
.372148
Tang.
iia^j
67"
COSINES, TANGENTS, AND COTANGENTS.
830
\91
15G3
i.
Sine.
0
9.591878
1
.59217G
2
..592-173
3
.592770
4
.5'.):{I67
5
..^.93363
6
.n936.j9
7
.593955
8
.594251
9
.591547
D. 1".
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2G
27
28
29
30
31
32
33
31
35
36
37
33
39 j
40
4r
42
43
44
45
46
47
4S
49
50
51
52
53
54
55
56
57
53
59
60
ri3o
9.594842
.595137
..59.5432
..595727
.596021
.596315
.5966; )9
.59(5903
.597196
.597490
9.. 597783
.593075
.598363
.593660
.598952
.599244
.599.536
.599827
.600118
.600409
9.600700
.601)990
.6012-0
.601570
.601860
.692 150
.602439
.602728
.603017
.603305
9.603594
.603382
.6:14170
.604457
.604715
.605032
.605319
.605606
.605892
.606179
9.60r.'65
.606751
.607036
.607322
.607607
.607892
.608177
.60-461
.603745
.609029
.609313
Cosine.
4.96
4.95
4.95
4.95
4.94
4.94
4.93
4.93
4.93
4.92
4.92
4.91
4.91
4.91
4.90
4.90
4.89
4.89
4.S9
4.88
4.83
4.83
4.87
4.S7
4.86
4.86
4.86
4.85
4.85
4.S4
4.84
4.84
4.83
4.83
4.83
4. 82
4.82
4.81
4.81
4.81
4.80
4.80
4.79
4.79
4.79
4.73
4.78
4.7.S
4.77
4.77
4.76
4.76
4.76
4.75
4.75
4.74
4.74
4.74
4 73
4.73
D. 1".
Cosine.
9.964026
.963972
.963919
.963365
.963811
.9(;37.j7
.963704
.9636.30
.963596
.963:542
9.9634-13
.963434
.963379
.963325
.96327!
'.■^3217
.9631ftJ
.9631(13
.96;',(l.54
.962999
9.962945
.962390
.962336
.962781
.932727
.962672
.9G2;i7
.962.562
.962.503
-.962453
9.962398
.962343
.962288
.962233
.962178
.962123
.962067
.962012
.9619.57
.961902
9.961846
.961791
.9617.35
.961630
.961624
.961569
.961513
.9614.53
.961402
.961346
9.961290
.961235
.961179
.96112!
.961007
.961011
.960955
.96:)V.)9
.960343
.96II7S6
.96)730
Sine.
D. 1"
Tang.
.89
.89
.90
.90
.90
.90
.90
.90
.90
.90
.90
.90
.90
.90
.90
.911
.91
.91
.91
.91
.91
.91
.91
.91
.91
.91
.91
.91
.91
.92
.92
.92
.92
.92
.92
.92
.92
.92
.'M
.92
.92
.92
.92
.93
.93
.93
.93
.93
.93
.93
.93
.93
.93
.93
.93
.93
.93
.94
.94
.94
1). 1"
9.627852
.628203
,<^28554
.623905
.629255
.629606
.629956
.630306
.630656
.631005
9.631355
.631704
.6.32053
.6321(12
.632750
.633099
.633447
.633795
.634143
.634490
9.634333
.635185
.635532
.635379
.636226
.636572
.636919
.637265
.637611
.637956
9.638302
.633617
.633992
.639337
.639632
.640027
.610371
.640716
.641060
.6414(04
9.641747
.642091
.642434
.642777
.643120
.643463
.643306
.644148
.644490
.644332
9.645174
.64-5516
.645S57
.646199
.616540
.646361
.647222
.647562
.617903
.643243
.643583
D. 1",
5.85
5.85
5.85
5. 84
5.54
5. 84
5.83
5.83
5.83
5.82
5.82
5.82
5.81
5.81
5.81
5.80
5.80
5.80
5.79
5.79
5.79
5.78
5.78
5.78
5.77
5.77
5.77
Cotang.
.76
6
o./
5.75
5.75
5.75
5.74
5.74
5.74
5.73
5.73
73
Cotang.
y,
5.73
5.72
5.72
5.72
5.71
5.71
5.71
5.70
5.70
5.70
5.69
5.69
5.69
5.69
5.68
5.68
5.68
5.67
5.67
5.67
-||
0.372148
.371797
.371446
.371095
.370745
.370394
.370044
.369694
.369344
.363995
0.36S645
.363296
.367947
.367593
.367250
.366901
.366553
.366205
.365857
.365510
0.365162
.364>15
.364468
.364121
.363774
.363423
.363081
.3627.35
.362339
.362044
0.361698
.361353
.361008
.360663
.360318
.3.59973
.359629
.359234
.358940
.358596
0.358253
.357909
D. 1".
.357223
.356380
.356537
,356194
,355852
.355510
.355163
0.354826
.3.544^4
.3.54143
.3533! II
.353460
.353119
.352778
.352433
.352097
.351757
.351417
M.
GO
.59
53
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
23
27
26
2-5
24
23
22
21
20
19
13
17
16
15
14
13
12
11
10
9
6
5
4
3
2
1
0
Tang. M.
(QQC
198
TABLE Xlll. LOGARITHMIC SINES,
155<.
M.
M.
isine.
0
9.609313
I
.609597
2
.609S80
3
.610164
4
.61/)147
5
.610729
6
.611012
7
.611294
8
.611576
9
.611853
10
9.612140
11
.612421
12
.612702
13
.612933
14
.613264
15
.61.3545
16
.613325
17
.614105
13
.614385
19
.614665
2D
9.614944
21
.615223
22
.615.502
23
.615731
24
.616030
25
.616333
26
.616516
27
.616394
23
.617172
29
.617450
30
9.617727
31
.618004
32
.618231
33
.618.553
34
.613334
35
.619110
36
.619336
37
.619662
38
.619933
39
.620213
40
9.620133
41
.620763
42
.621033
43
.621313
44
.621537
45
.621861
46
.6221.35
47
.622409
48
.622632
49
.622956
50
9.623229
51
.623502
52
.6^3774
53
.624047
54
.624319
55
.624591
56
.624363
57
.625135
58
.625406
59
.625677
60
.625948
D. 1".
Cosine.
4.73
4.72
4.72
4.72
4.71
4.71
4.71
4.70
4.70
4.69
4.69
4.69
4.63
4.63
4.63
4.67
4.67
4.67
4.66
4.66
4.65
4.65
4.65
4.64
4.61
4.64
4.63
4.63
4.63
4.62
4.62
4.61
4.61
4.61
4.60
4.60
4.60
4.59
4.59
4.59
4.53
4.53
4.53
4.57
4..57
4..57
4.56
4.56
4.56
4.55
4.55
4.54
4.54
4.. 54
4.53
4.53
4.53
4.52
4..52
4,52
D. 1".
Cosine.
D. 1".
9.9607.30
.960674
.960618
.96)561
.960505
.960443
.960392
.960335
.96)279
.960222
9. 96 T 165
.960109
.96)052
.959995
.959933
.9.59332
.959325
.9.59768
.959711
.959654
9 959596
,959539
,959432
,959425
,959363
,959310
.9.59253
.959195
.9.59133
.959030
9.959023
.953965
.958908
.9533.50
.953792
.9587.34
.953677
.953619
.953.561
.953503
9.958445
.953337
.953329
.953271
.9.53213
.9.53154
.9.58096
.958038
.957979
.957921
9.957863
.957804
.957746
.957637
.957623
,957570
,957511
.957452
.957393
.9573a5
,957276
Sine.
,94
,94
.94
,94
,94
.94
.94
.94
.94
.94
.95
.95
.95
.95
,95
.95
.95
.95
,95
.95
.95
.95
.95
.95
.96
.96
,96
.96
,96
,96
.96
.96
.98
.96
.96
.96
.96
.97
.97
,97
.97
.97
.97
.97
.97
.97
,97
.97
,97
.97
.97
.93
.93
.93
,98
.98
.93
.93
,98
.98
D. 1".
Tang,
9.648583
.643923
.649263
.649002
.649942
.650231
.650620
.650959
.651297
.651636
9.651974
.652312
.652650
.652933
.653326
.653663
.6.54000
,654337
.654674
.655011
9.655343
.65.5634
.656020
.656356
.656692
.6.57023
.657364
.657699
,653034
,653369
9.653704
.659039
,6.59373
.6-59703
.660042
.660376
.660710
.661043
.661-377
.661710
9.662043
.662376
.662709
.663042
.663375
.663707
,664039
.664371
.664703
.665035
9.66-5366
.66.5693
.666029
.666360
,666691
,667021
.6673^52
.667682
.663013
.663343
,668673
Cotang.
D. 1'.
5.67
5.66
5.66
5.66
5.65
5.65
5.65
5.64
5.64
5.64
5.64
5.63
5.63
5.63
5.62
5.62
5.62
5.62
5.61
5.61
5.61
5.61
5.60
5.60
5.60
5. .59
5.59
5.59
5.58
5.58
5.-58
5.58
5.57
5.57
5.57
5.56
5.56
5.56
5.. 56
5.55
5.55
5.55
5.54
5.54
5.54
5.54
5.53
5.-53
5.-53
5.53
5.52
5.52
5, .52
5.51
5.51
5.51
5.51
5.50
5.50
5.50
D. 1".
Cotang,
0.351417
.351077
.350737
.350398
.350058
.349719
.349380
.349041
,343703
.348364
0.343026
.347638
.347a50
.347012
.346674
.346337
.346000
.345663
,345326
.344939
0.344652
.344316
,34.3930
,343644
,313308
,342972
.31^636
.342301
.341966
.341631
0.3-11296
.340961
..340627
,340292
.3-39953
,a39624
,339290
.333957
,a33623
,333290
0.337957
,.337624
.337291
.336958
,3-36625
.336293
.3.35961
.335629
,3-35297
.33496-.
0.3-34634
.334302
.333971
.-333640
.-3-33309
.332979
.332643
.332318
.331987
.3316.57
,331327
Tang.
1140
690
COSINES, TANGENTS, AND COTANGENTS.
199
154ta
M. Sine.
D. 1".
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
M.
9.625948
.626219
.626490
.62(3760
.627030
.627300
.627570
.627840
.62S109
.628378
9.628647
.623916
.629185
,629453
.620721
.629989
.630257
.630524
.630792
.631059
9.631326
.631593
.631859
.632125
.632392
.6326.58
.632923
.633189
.633454
.633719
9.633934
.634249
.634514
.634778
.635042
.635306
.635570
.635834
.636097
.636360
9.636623
.636886
.637148
.637411
.637673
.637935
.638197
.638458
.638720
.63S081
9.639242
.639503
.639764
.610f/24
.640284
.frl0544
.640304
.641064
.641324
.&11583
.641842
Cosine.
4.51
4.51
4.51
4.50
4.50
4.50
4.49
4.49
4.49
4.48
4.48
4.48
4.47
4.47
4.47
4.46
4.46
4.46
4.45
4.45
4.45
4.44
4.44
4.44
4.43
4.43
4.43
4.42
4.42
4.42
4.41
4.41
4.41
4.40
4.40
4.40
4.39
4.39
4.39
4.33
4.38
4.38
4.37
4.37
4.37
4.36
4.36
4.36
4.35
4.35
4.35
4.34
4.34
4.34
4.33
4.33
433
4.32
4.32
4.32
D. 1".
Cosine.
9.957276
.957217
.957158
.957099
.957040
.956981
.956921
.956^62
.956S03
.956744
9. 9566.84
.956625
.956566
.956506
.956447
.956.387
.956327
.956268
.956208
.956148
9.956089
.956029
.955969
.95.5909
.955849
.955789
.955729
.955669
.955609
.955548
9.955488
.955428
.955363
.955307
.95.5247
.955186
.955126
.955065
.955005
.954944
9.954883
.954823
.954762
.954701
.954640
.954.579
.9.54518
.954457
.954396
.954335
9.954274
.954213
.954152
.954090
.954029
.953968
.953906
.953845
.953783
.953722
.953660
Sine.
D 1".
Tang.
D. 1".
.98
.98
.98
.98
.99
.99
.99
.99
.99
.99
.99
.99
.99
.99
.99
.99
.99
.99
1.00
1.00
1.00
1.00
1. 00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.03
1.03
1.03
D. 1".
9.66-:673
.669002
.669332
.669661
.669991
.670320
.670649
.670977
.671306
.671635
9.671963
.672291
.672619
.672947
.673274
.673602
.673929
.674257
.674.584
.674911
9.675237
.675564
.675890
.676217
.676543
.676869
.677194
.677520
.677846
.678171
9.678496
.678821
.679146
.679471
.679795
.680120
.680444
.680768
.681092
.681416
9.681740
.682063
.682387
.682710
.683033
.683356
.683679
.684001
.684324
.634646
9.684968
.685290
.68.5612
.6S5934
.686255
.686577
.686898
.687219
687540
687861
.688182
Gotang.
Cotang.
5.50
5.49
5.49
5.49
5.49
5.48
5.48
5.48
5.47
5.47
5.47
5.47
5.46
5.46
6.46
5.46
5.45
5.45
5.45
5.45
5.44
5.44
5.44
5.44
5.43
5.43
5.43
5.42
5.42
5.42
5.42
5.41
5.41
5.41
5.41
5.40
5.40
5.40
5.40
5.39
5.39
5.39
5.39
5.38
5.38
5.38
5.33
5.37
6.37
6.37
5.37
6.36
5.36
5.36
5.36
5.35
6.35
5.35
6.35
5.35
J).V.
M.
0.331327
.330998
.330668
.330339
.330009
.3296^0
.329351
.329023
.328694
.328365
0.328037
.327709
.327381
.327053
.326726
.326398
.326071
.325743
.325416
.325089
0.324763
.324436
.324110
.323783
.323457
.323131
.322806
.322480
.322154
321829
0.321504
.321179
.320854
.320529
.320205
.319880
.319556
.319232
.318908
.318584
0.318260
.317937
.317613
.317290
.316967
.316644
.316321
.315999
,315676
.315354
0.315032
.314710
.314388
.314066
.313745
.313423
.313102
.312781
,312460
.312139
.311818
Tuc.
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
6
4
3
2
1
0
M.
1150
640
200
TABLE Xlll. LOGAKITHMIC SINES,
153>
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
IS
19
20
21
22
23
21
2.-,
26
27
2S
29
.30
31
82
33
31
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
58
59
60
Sine.
9.641812
.642101
.642360
.64 26 IS
.642877
.643135
.613393
.6136.50
.643908
.644165
9.641123
.614'J80
.644,(36
.645193
.615150
.645700
.64-5962
.646218
.646474
.646729
9.646984
.647210
.647194
.647749
.648004
.64S258
.648512
.643766
.649020
.649274
9.649527
.649781
.650:)34
.65y2s7
.650.539
.650792
.651044
.651297
.651549
.651800
9.6.520.52
.652304
.652555
.652806
.653057
.653.303
.653553
.653303
.6.54059
.654309
9.654553
.6.54303
.6.550-58
.6.55307
.655556
.655805
.656054
.656302
.656551
.656799
.657047
D. 1".
4.32
4.31
4.31
4.31
4.30
4.30
4.30
4 29
4.29
4.29
4.28
4.28
4.23
4.27
4.27
4.27
4.26
4.26
4.26
4.26
4.25
4.25
4.25
4.24
4.24
4.24
4.23
4.23
4.23
4.22
4.22
4.22
4.22
4^21
4.21
4.21
4.20
4.20
4.20
4.19
4.19
4.19
4.18
4.18
4.13
4.18
4.17
4.17
4.17
4.16
4.16
4.16
4.15
4.15
4.15
4.15
4.14
4.14
4.14
4.13
M.
1163
Cosine. D. 1"
Cosine.
9.953660
.953599
.953537
.95.3475
.9.53113
.953352
.953290
.953228
.9.53166
.953104
9.9.53042
.9.52980
.9-52918
.952855
.952793
.9.52731
.952669
.9.52606
.9-52.544
.952481
9.952419
.952356
.952234
.952231
.952168
.952106
.952043
.951980
.951917
.9518-54
9.951791
.951723
.951665
.951602
.951539
.951476
.951412
.951319
.9.J1286
.951222
9.951159
.951096
.9510-32
.9-50963
.950905
.9-50841
.9-50778
.950714
.9506-50
.950586
9.950522
.950453
.950394
.9.50330
.950266
.950202
.9-50133
.9.50074
.9.50010
.949945
.949381
Sine.
D. 1".
1.03
1.03
1.03
1.03
1.03
1.03
1.03
I 03
1.03
1.03
1.03
1.04
1.01
1.04
1.04
1.04
1.04
1.01
1.04
1.04
1.04
1.04
1.04
1.01
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.07
1.07
1.07
1.07
1.07
1.07
1.07
1.07
1.07
1.07
D. 1".
Tang.
9.688182
.638502
.633823
.639143
.639463
.639783
.690103
.690423
.69 1742
.6J1062
9.691-381
.6 J 1700
.692019
.692333
.692656
.692975
.693293
.693612
.693930
.694248
9.694566
.694833
.69-5201
.695518
.69-5336
.6961.53
.696470
.696787
.697103
.697420
9.697736
.6:)8053
.693369
.693635
.699001
.699316
.699632
.699947
.700263
.700578
9.700393
.701208
.701.523
.7018.37
.7021.52
.702466
.702781
.703095
.7034.^9
.703722
9.704036
.7043-50
.704663
.704976
.705290
.70560-3
.70.5916
.706228
.706541
.7063-54
.707166
Cotang.
D. 1".
5.34
5.34
5.34
5.34
5.-33
5.33
5.33
5.33
5.32
5.32
5.32
5.-32
5.31
5.31
5.31
5.31
5.30
5.-30
.5.. 30
5.30
5.29
5.29
5.29
5.29
5.29
5.23
5.28
5.23
5.27
5.27
5.27
5.27
5.26
5.26
5.26
5.26
5.26
5.25
5.25
5.25
5.25
5.24
5.24
5.24
.5.24
5.23
5.23
5.23
5.23
5.22
5.22
5.22
5.22
5.22
5.21
5 21
5.21
5.21
D. 1".
Cotang.
M.
0.311813
60
.311498
59
.311177
58
.310S57
57
.310-5.37
56
.310217
55
.309397
54
..309577
53
J30925S
52
.308938
51
0..305619
50
.308300
49
.307981
48
.307662
47
.307344
46
.307025
45
..306707
44
.306338
43
.306070
42
.305752
41
0.30.5434
40
.305117
39
.304799
33
.304482
37
.301164
36
.303347
35
303530
34
.303213
33
.3 12397
32
.302580
31
0.302264
30
.301947
29
.3)1631
28
.301315
27
.30f)999
26
..3011634
25
.300363
24
.300053
23
.299737
22
.299422
21
0.299107
20
.298792
19
.298477
18
.293163
17
.297848
16
.297534
15
.297219
14
.296905
13
.296.591
12
.296278
11
0.295964
10
.295650
9
.295337
8
.295r.<24
7
.294710
6
.294-397
5
.294034
4
.293772
3
.293459
2
.293146
1
.2923.34
0
M.
Tang.
e3<
COSINES, TANGENTS, AND COTANGENTS.
201
153-
M.
-I-
Siiie.
10
II
12
13
14
15
16
17
13
19
20
21
22
23
24
2.3
26
27
23
29
30
31
32
33
36
37
33
39
10
II
12
13
14
15
16
17
IS
19
50
51
52
53
54
55
56
57
5S
5£
6C
D. 1".
9.637017
.657293
.637542
.657790
.653037
.6.58234
.6.53531
.653773
639025
.659271
G 659317
.639763
.660009
.660255
.660501
.660746
.660991
.66123G
.661-131
.6t)1726
9.661970
.6;2214
.6521.59
.662703
.662916
.663190
.663433
.663677
.663920
.664163
9.661106
.664643
.661391
.665133
.665375
.665617
.66.5859
.666100
.666342
.666533
9.666324
.667065
,667305
.667346
.667736
.663027
.663267
.663506
.663746
.663936
9.669225
.669464
.669703
.669942
.670131
.670419
,670653
.670396
.6711.34
.671372
.671609
4.13
4.13
4.12
4.12
4.12
4.12
4.11
4.11
4.11
4.10
4.10
4.10
4.10
4.09
4.09
4.09
4.03
4.03
4.03
4.03
4.07
4.07
4.07
4.05
4.06
4.06
4.05
4.03
4.03
4.05
4.04
4.04
4.04
4.03
4.03
4.03
4.03
4.02
4.02
4.02
4.01
4.01
4.01
4.01
4.00
4.00
4.00
3.99
3.99
399
3.99
3.93
3.9?
3.93
3.93
3.97
3.97
3.97
3.96
3.96
Cosiue.
D. 1"
9.919331
.919316
.949752
.919658
.949623
.949353
.949194
.919429
.949364
.949300
9.949235
.949170
.949105
.949040
.948975
.943910
.913345
,943730
.943715
,913650
9.943.531
.913319
.943454
.943338
.943323
,913257
,918192
.943126
.943060
.947995
9.947929
.947863
.947797
.947731
.947665
.917600
.947533
.917467
.917401
.947333
9.947269
.947203
.947136
.947070
.9170r)l
.916937
.946371
.946304
.946733
,916671
9,916604
.946533
.946471
.946404
.946337
.916270
,946203
.946136
.916069
.916(02
.9439.35
1.07
1.07
1,07
1,03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1,03
1.08
1,09
1.09
1.09
1.09
1.09
1,09
1,09
1,09
1,09
1,09
1,09
1.09
1.09
1.10
1.10
1.10
1.10
1. 10
1.10
1.10
1. 10
l.IO
1. 10
1. 10
l.!0
1.11
1,11
l.ll
1.11
1,11
1.11
1.11
l.U
1.11
1.11
l.U
1.11
1.11
1.12
1.12
1.12
1.12
1.12
1 12
Tang.
9.707166
.707478
.707790
.703102
.703414
.703726
.709037
.709349
.709660
.709971
9.710232
.710593
.710304
.711215
,71 1525
.711336
.712146
.712156
.712766
,713076
9,71.3336
,713696
,714005
,714314
,714624
,714933
,71.5242
.715551
.715360
.716168
9,716477
,716735
.717093
.717401
,717709
,713017
.718.325
,718633
.713940
.719213
9.719.555
.719362
.720169
.720176
.720783
.721039
,721396
.721702
,722009
,722315
9,722621
.722927
.723232
.723338
,723344
,724149
.7244.54
.724760
.725065
.723370
.725674
M. Cosine, D. 1", Sine, D, 1", Cotang, D, 1"
D. 1".
5.20
5.20
5.20
5,20
.5.20
5,19
5.19
5.19
5.19
5,13
5.18
5.18
5.13
5,17
5,17
5.17
5.17
.5,17
5,16
5,16
5,16
5.16
5.15
5.15
5.15
5.15
5.15
5.14
5.14
5.14
5.14
5.14
5.13
5,13
5,13
5.13
5.13
5.12
5.12
5.12
5.12
5.11
5.11
5.11
5.11
5.11
5.10
5.10
5.10
5.10
5.10
5.09
5.09
5.09
5.09
5.09
5,08
5,08
5.03
Cotang. M
0,292834
,292522
,292210
.291893
,291536
,291274
,290963
.290651
.290340
,290029
60
59
58
57
56
53
54
53
52
51
0.239718 .
50
.239407
49
.239096
48
.238785
47
.238475
46
.233164
45
.237854
44
.237514
43
.2372.34
42
.236924
41
0.2S6614
40
.236304
39
.235995
33
.235836
37
.285376
36
.235067
35
.234753
34
.234149
33
.281140
32
.233332
31
0.283523
30
.233215
29
.232907
23
.232.599
27
.232291
26
.231933
23
.231675
24
.231367
23
.231060
22
.230752
21
0.280445
20 '
.2301.33
19
.279331
13
,279324
17
,279217
16
.273911
15
,273604
14
.273293
13
,277991
12
.277635
11
0,277379
10
,277073
9
.276768
8
,276462
7
.276156
6
.275351
5
.275.546
4
.275240
3
.274935
2
.274630
1
.274326
0
M.
Tang.
1170
10
6a«
202
280
TABLE XIII. LOGARITHMIC SINES,
131
M.
0
I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
53
59
60
Sine.
9.671609
.671847
.672034
.672321
.672553
.672795
.673032
.673263
.673505
.673741
9.673977
.674213
.674443
.674634
.674919
.675155
.675390
.67.5624
.67.5359
.676094
9.676323
.676.562
.676796
.677030
.6772&4
.677493
.677731
.677964
.678197
.673430
9.673663
.673395
.679123
.679360
.679.592
.679324
.630056
.630233
.630519
.630750
9.630932
.631213
.631443
.631674
.631905
.6321.35
.632365
.632595
.632325
.6S3055
9.633234
.633514
.633743
.633972
.634201
.634430
.634653
.634337
.63.S115
.63.5343
.635571
M. Cosine.
D. 1".
3.96
3.96
3.95
3.95
3,95
3 94
3,94
3.94
3.94
3.93
3.93
3.93
3.93
3.92
3.92
3.92
3.91
3.91
3.91
3.91
3.90
3.90
3.90
3.90
3. 39
3.89
3.89
3.83
3.83
3.33
3.83
3.87
3.87
3.37
3.37
3.86
3.86
3.86
3.86
3.85
3.85
3.35
3.34
3.84
3.84
3.84
3.83
3.33
3.33
3.83
3.32
3.82
3.82
3.82
3.81
3.81
3.81
3.80
3.80
3.80
D. 1".
Cosine.
9.94.5935
.945363
.945800
.945733
.945666
.945593
.94.5.531
.94.5464
.945396
.945323
9.94.5261
.945193
.945125
.945053
.944990
.944922
.944354
.944786
.944718
.9446.50
9.944.532
.944514
.944446
.944377
.944.309
.944241
.944172
.944104
.944036
.943967
9.943S99
.9433.30
.943761
.943693
.943624
.943555
.943436
.943417
.94.3343
.94.3279
9.943210
.943141
.94.3072
.943003
.9429-34
.942364
.942795
.942726
.942656
.942.587
9.942517
.942443
.942373
.942.308
.942239
.942169
.942099
.942029
.941959
.941839
.941819
Sine.
D. 1".
,12
,12
,12
,12
,12
,12
,12
,13
,13
,13
,13
,13
,13
,13
,13
,13
,13
,13
,13
,13
,14
,14
,14
,14
,14
14
,14
,14
14
14
14
14
15
15
15
15
15
15
15
15
15
15
15
15
15
16
16
16
16
16
16
16
16
16
16
16
16
17
17
17
D. 1".
Tang.
9.725674
.725979
.726234
.726538
.726392
./27197
.727.501
.727805
.723109
.723412
9.723716
.729020
.729.323
.729626
.729929
.730233
.730535
.730338
.731141
.731444
9.731746
.732043
.732351
.732653
.732955
.733257
.733558
.733360
.734162
.734463
9.734764
.735066
.735367
.735663
.735969
.736269
.736570
.786370
,737171
.737471
9.737771
.733071
.733371
.733671
.735971
.739271
.739570
.739370
.740169
.740468
9.740767
.741066
.741365
.741664
.741962
.742261
.742559
.742858
.743156
.743454
.743752
D, 1".
Cotang.
5.08
5.08
5.07
5.07
5.07
5.07
5.07
5.06
5.06
5.06
5.06
5.06
5.05
5.05
5.05
5.05
5.05
5.05
5.04
5.04
5.04
5.04
5.04
5.03
5.03
5.03
5.03
5.03
5.02
5.02
5.02
5.02
5.02
5.01
5.01
5.01
5.01
5.01
5.01
5.00
5.00
5.00
5.00
5.00
4.99
4.99
4.99
4.99
4.99
4.93
4.98
4.98
4.93
4.93
4.93
4.97
4.97
4.97
4.97
4.97
D. 1".
Cotang.
0.274326
.274021
.273716
.273412
.273103
.272303
.272499
.272195
.271891
.271588
0.271234
.270930
.270677
.270374
.270071
.269767
.269465
.269162
.263859
.268556
0.2632.54
.267952
.267649
.267347
.267045
.266743
.266442
.266140
.265833
.265537
0.2652.36
.264934
.261633
.264-332
.264031
.263731
.263430
.263130
.262,329
.262529
0.262229
.261929
.261629
.261329
.261029
.260729
.260430
.260130
.2.59331
.259532
0.259233
.253934
.2.58635
.258336
.253033
.257739
.257441
.257142
.2.56344
.2.56.546
.256243
Tang.
118'
COSINES, TANGENTS, AND COTANGENTS.
M
Sine.
0
1
2
3
4
5
6
7
8
9
!0
11
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
58
59
60
D. 1".
9.635371
.6S579'J
.6S6027
.656254
.636432
.636709
.656936
.637163
.637339
.637616
9.657313
.653069
.633295
.655521
.633747
.655972
.659193
.6>9123
.6^9615
.659573
9.690095
.890323
.690543
.693772
.690996
.691220
.691444
.691665
.691892
.692115
9.692339
.692562
.692785
.693003
.693231
.693453
.693676
.693593
.694120
.694342
9.694.564
.694736
.69.5007
.695229
.695450
.69.5671
.695392
.696113
.696334
.696554
9.696775
.696995
.697215
.697435
.697654
.697874
.695094
.698313
.698532
.693751
.698970
Cosine.
D. 1".
M. Cosine.
3.80
3.79
3.79
3.79
3.79
3.78
3.78
3.78
3.73
3.77
3.77
3.77
3.77
3.76
3.76
3.76
3.76
3.75
3.75
3.75
3.75
3.74
3.74
3.74
3.74
3.73
3.73
3.73
3.73
3.72
3.72
3.72
3.72
3.71
3.71
3.71
3.71
3.70
3.70
3.70
3.70
3.69
3.69
3.69
3.69
3.63
3.63
3.63
3.63
3.67
3.67
3.67
3.67
3.66
3.66
3.66
3.66
3.65
3.65
3.65
9.941819
.911749
.941679
.911609
.941539
.941469
.941393
.941323
.941253
.941137
9.941117
.941046
.910975
,940905
.940334
.940763
.940693
.940622
.940551
.940480
9.940409
.940333
.940267
.940196
.940125
.940054
.939982
.939911
.939340
.939768
9.939697
.939625
.9.395.54
.939482
.939410
.939339
.939267
.939195
.939123
.939052
9.935930
.935908
.933336
.933763
.933691
.935619
.933.547
■ .933475
.933402
.933330
9.9382.58
.933185
.933113
.933040
.937967
.937895
.937822
.937749
.937676
.937604
.937531
Tang.
D. 1".
Sine.
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.18
1.19
1.19
1.19
1.19
1.19
1.19
1.19
1.19
1.19
1.19
1.19
1.19
1.19
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1 20
1.20
1.20
1.21
1.21
1.21
1.21
1.21
1.21
1.21
1.21
1.21
1.21
1.21
1.21
1.22
D. 1".
9.743752
.744050
.744343
.744645
.744943
.745240
.74.5533
.74.5335
.746132
.746429
£. r46726
.747023
.747319
.747616
.747913
.748209
.748505
.748801
.749097
.749393
9.749639
.749935
.750281
.750576
.750872
.751167
.751462
.751757
.752052
.752347
9.752642
.7.52937
.753231
.753526
.753320
.754115
.754409
.754703
.754997
.755291
9.7.5.5585
.755373
.756172
.756165
.756759
.757052
.757345
.757633
.7.57931
.758224
9.753517
.758810
.759102
.759395
.759687
.759979
.760272
.760564
.760856
.761148
.761439
Cotang.
D. 1". I Cotang.
4.96
4.96
4.96
4.96
4.96
4.96
4.95
4.95
4.95
4.95
4.95
4.95
4.94
4.94
4.94
4.94
4.94
4.93
4.93
4.93
4.93
4.93
4.93
4.92
4.92
4.92
4.92
4.92
4.92
4.91
4.91
4.91
4.91
4.91
4.91
4.90
4.90
4.90
4.90
4.90
4.89
4.89
4.89
4.89
4.89
4.89
4.88
4.83
4.83
4.88
4.88
4.88
4.87
4.87
4.87
4.87
4.87
4.87
4.86
4.86
0.256243
.255950
.255652
.255355
.255057
.254760
.25-1462
.2.54165
.253363
.253571
0.253274
.252977
.2.52031
.252334
.252037
.251791
.251495
.251199
.250903
.250607
0.2.50311
.250015
.249719
.249424
.249123
.243833
.248538
.248243
.247943
.247653
0.247358
.247063
.246769
.246474
.246180
.24.5835
.245591
.245297
.24.5003
.244709
0.244415
.244122
.243325
.243535
.243241
.242948
.242655
.242362
.242069
.241776
0.241483
.241190
.240398
.240605
.240313
.240021
.239723
.239436
.2.39144
.238852
.238561
D. 1". Tang.
1190
60<
204
30^
TABLE ^'III. LOGARITHMIC SINES,
M.
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
2:5
24
25
26
27
28
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine.
D. 1".
9.693970
.699139
.699407
.699626
.699844
.700062
.700230
.700493
.700716
.700933
9.701151
.701363
.701.585
.701302
.702019
.702236
.702452
.702669
.702335
.703101
9.703317
.7()3533
.703749
.703664
.704179
.704395
.704610
.704S25
.705040
.705254
9.705469
.705633
.705398
.706112
.706326
.706.539
.706753
.706967
.707130
.707393
9.707606
.707819
.703032
.703245
.7034.33
.703670
.703832
.709094
.709306
.709518
9.709730
.709941
.710153
.710364
.710575
.710736
.710997
.711208
.711419
.711629
.711839
3.65
3.64
3.64
3.64
3.64
3.63
3.63
3.63
3.63
3.62
3.62
3.62
3.62
3.61
3.61
3.61
3.61
3.60
3.60
3.60
3.60
3.59
3.59
3.59
3. .59
3.59
3.58
3. .58
3.53
3.. 53
3.57
3.57
3.. 57
3.57
3.-56
3.56
3.56
3.56
3.55
3.55
3.55
3.55
3.54
3.54
3.54
3.54
3.54
3.53
3.53
3.53
3.53
3.52
3.52
3.52
3.52
3.51
3.51
3.51
3.51
3.51
Cosine. D, 1".
Cosine.
9.937531
.937453
.937.335
.937312
.937233
.937165
.937092
.937019
.936946
.936872
9.9.36799
.936725
.936652
.936578
.936.505
.936431
.936-357
.936284
.936210
.936136
9.936062
.935938
.935914
.9-3-5340
.935766
.935692
.935618
.9-35543
.935469
.935395
9.93.5320
.935246
.9-35171
.935097
.935022
.934943
.934873
.934793
.9-34723
.934649
9.934574
.9.34499
.934424
.934349
.934274
.934199
.9-34123
.934043
.93-3973
.933S98
9.93-3822
.933747
.933671
.933596
933520
933445
933369
933293
933217
.933141
.933066
D. 1".
Sine.
1.22
1.22
1.22
1.22
1.22
1.22
1.22
1.22
1.22
1.22
1.22
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.24
1.24
1.24
1.24
1.24
1.24
1.24
1.24
1.24
1.24
1.24
1.24
1.25
1.25
1.25
!.!.5
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
I.2G
1.26
1.26
1.26
1.26
1.26
1.26
1.26
1.26
1.26
1.26
1.26
Tang.
D. 1".
9.761439
.761731
.762023
.762314
.762606
.762897
.763133
.76.3479
.763770
.764061
9.764.352
.764643
.764933
.765224
.765514
.765305
.766095
.766385
.766675
.766965
9.767255
.767.545
.7678.34
.768124
.768414
.768703
.763992
.769231
.769-571
.769560
9.770148
.770437
.770726
.771015
.771.303
.771592
.771830
.772163
.772457
.772745
9,7730-33
.773-321
.773608
.773896
.774184
.774471
.774759
.775046
.7753-33
.775621
9.775908
.776195
.776482
.776763
.777055
.777342
.777623
.777915
.773201
.773488
.778774
D. 1".
Cotang.
Cotang.
4.S6
4.86
4.S6
4.86
4.S6
4.85
4.85
4.85
4.85
4.85
4. 35
4.84
4.84
4.84
4.84
4.84
4.84
4.83
4.83
4.83
4.83
4.83
4.83
4.82
4.32
4.82
4.82
4.82
4.82
4.82
4.81
4.81
4.81
4.81
4.SI
4.81
4.80
4.80
4.S0
4.80
4.80
4.80
4. SO
4.79
4.79
4.79
4.79
4.79
4.79
4.78
4.78
4.78
4.78
4.78
4.78
4.78
4.77
4.77
4.77
4.77
D. 1".
0.238561
.238269
.237977
.237686
.237394
.237103
.236312
.2-36521
.236230
.235939
0.235643
.235357
.235067
.234776
.234486
.234195
.2339C5
.233615
.233325
.233035
0.232745
.232455
.232166
.231876
.231586
.231297
.231008
.230719
.230429
.230140
0.229S52
.229563
.229274
.228985
.223697
.228403
.228120
.227832
.2275-13
.227255
0.226967
.22G679
.226392
.226 1 ((4
.22.5316
.225.529
.22.5241
.224954
.224667
.224379
0.224092
.223305
.223513
.223232
.222945
.222658
.222372
.222035
.221799
.221512
.221226
Tang.
M.
60
59
58
57
56
55
54
53
62
51
50
49
43
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
IS
17
16
15
14
13
12
•1
10
9
8
7
6
5
4
3
2
I
_0
M.
laoo
COSINES, TANGENTS, AND COTANGENTS.
20e
148=
Sine.
D. 1".
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
ir>
1(3
17
18
19
2(3
21
22
23
24
25
25
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
9 711839
.712050
.712260
.712469
.712679
,712339
.713093
.713303
.713517
.713726
9.713935
.714144
.714352
.714561
.714769
.714978
.715186
.715394
.715602
.715309
9.716017
.716224
.716132
.710639
.716346
.7170.33
.717259
.717466
.717673
,717879
9.7 1 8035
.71S291
.713497
.718703
.718909
.719114
.719320
.719525
.719730
.719935
9.720140
.720345
.720549
.720754
.720953
.721162
.721366
.721570
.721774
.721978
9.722181
.722335
.722588
.722791
.722994
.723197
.723400
.723603
.723305
.724007
.724210
Cosiiie. D. 1"
3.50
3.50
3.50
3.50
3.49
3.49
3.49
3.49
3.43
3.43
3.43
3.43
3.43
3.47
3.47
3.47
3.47
3.46
3.46
3.46
3.46
3.46
3.45
3.45
3.45
3.45
3.44
3.44
3.44
3.44
3.43
3.43
3.43
3.43
3.43
3.42
3.42
3.42
3 42
3.41
3.41
3.41
3.41
3.41
3.40
3.40
3.40
3.40
3.39
3.39
3.39
3.39
3.39
3.38
3.38
3.33
3.33
3.37
3.37
3.37
9.933006
.932990
.932914
.932833
.932762
.932685
.932609
.932533
.932457
.932330
9.932304
.932228
.932151
.932075
.931998
.931921
.931845
.931763
.931691
.931614
9.931537
.931460
.931333
.931306
.931229
.931152
.931075
.930993
.930921
.930343
9.930766
.930638
.930611
.930533
.930456
.930378
.930.300
.930223
.930145
.930067
9.929939
.929911
.929333
.929755
.929677
.929599
.929521
.929442
.929364
.929266
9.929207
.929129
.929050
.923972
.923393
.923315
.923736
.923657
.923573
.923499
.923420
Cosine. D, 1"
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.23
1.28
1.28
1.28
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.30
1.30
1,30
1.30
1.30
1.30
1.30
1.31
1.31
1.31
1.31
1.31
1.31
1.31
1.31
1.31
1.31
1.31
1.31
1.32
Sine
Tang.
9.778774
.779060
.779346
.779632
.779918
.7«0203
.780489
.780775
.731060
.781346
9.781631
.781916
.782201
.782186
.782771
.783056
.783341
.783626
.783910
.734195
9 784479
784764
785043
.785332
.785616
785900
.786184
.786463
.786752
.787036
9.737319
.787603
.787886
.783170
.783453
.783736
.789019
.789302
.789535
.7893C8
9.790151
.790434
.790716
.790999
.791281
.791563
.791846
.792128
.792410
.792692
9.792974
.793256
.793533
.793819
.794101
.794333
.794664
.794946
.795227
,795508
.795739
D. 1",
D. 1"..
4.77
4.77
4.77
4.76
4.76
4.76
4.76
4.76
4.76
4.76
4.75
4.75
4.75
4.75
4.75
4.75
4.75
4.74
4.74
4.74
4.74
4.74
4.74
4.74
4.73
4.73
4.73
4.73
4.73
4.73
4.73
4.72
4.72
4.72
4.72
4.72
4.72
4.72
4.71
4.71
4.71
4.71
4.71
4.71
4.71
4.70
4.70
4.70
4.70
4.70
4.70
4.70
4.70
4.69
4.69
4.69
4.69
4.69
4.69
4.69
Cotang
Cotang.
0.221226
.220940
.220654
.220363
.220082
.219797
.219511
.219225
.218940
.218654
0.218369
218034
.217799
.217514
.217229
.216944
.216659
.216374
.216090
.215805
0.215521
.215236
.21 49-52
.214663
.214334
.214100
.21.3816
.213532
.213243
.212964
0.212681
.212397
.212114
.211830
.211547
211264
.210981
.210698
.210415
.210132
0.209849
.209566
.209284
.209001
.208719
.208437
.208154
.207872
.207590
.207308
0.207026
.206744
.206462
.206181
.205399
.205617
.20.3336
.205054
.204773
.204492
.204211
M.
60
59
58
57
56
55
54
53
52
51
50
49
43
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
D. 1",
Tang,
M
i»l^
5.>i^
206
33°
TABLE Xlil. LOGARITHMIC SINES,
M.
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
58
59
60
M.
l«20
Sine.
9.724210
.724412
.724614
.724S16
.725017
.725219
.725420
.725622
.725523
.726024
9.726225
.726426
.726626
.726327
.727027
.727228
.727423
.727623
.727823
.728027
9.72S227
.728427
.725626
.728325
.729024
.,_J29223
.729422
.729621
.729320
,730018
9.730217
.730415
.730613
.730811
.731009
.731206
.731401
.731602
.731799
.731996
9.732193
.732390
.732537
.732784
.732930
.733177
.733373
.733569
.733765
.733961
9.7.34157
.734353
.734549
.734744
.734939
.735135
.735330
.735525
.735719
.735914
.736109
Cosine.
D. 1".
3.37
3.37
3.36
3.36
3.36
3.36
3.36
3.35
3.35
3.35
3.35
3.34
3.31
3.34
3.34
3.3i
3.33
3.33
3.33
3.33
3.33
3.32
3.32
3.32
3.32
3.31
3.31
3.31
3.31
3.31
3.30
3.30
3.30
3.30
3.30
3.29
3.29
3.29
3.29
3.28
3.28
3.28
3.28
3.28
3.27
3.27
3.27
3.27
3.27
3.26
3.26
3.26
3.26
3.26
3.25
3.25
3.25
3.25
3.25
3.24
D. 1".
Cosine.
9.928420
.923342
.928263
.923153
.923104
.925025
.927946
.927567
.927787
.927708
9.927629
.927549
.927470
.927390
.927310
.927231
.927151
.927071
.926991
.926911
9.926331
.926751
.926671
.926591
.926511
.926431
.926351
.926270
.926190
.926110
9.926029
.925949
.925563
.925733
.925707
.925626
.925545
.925465
.925334
.925303
9.925222
.925141
.925060
.924979
.924397
.924816
.924735
.924654
.924572
.924491
9.924409
.924328
.924246
.924164
.924083
.924001
.923919
.923837
.923755
.923673
.923-591
Sine.
D. 1".
1.32
1.32
1.32
1.32
1.32
1.32
L32
1.32
1.32
1.32
L32
1.33
1.33
1.33
1.33
.33
1.33
1.33
1.33
1.33
1.33
1.33
1.33
1.34
1.34
1.34
1.34
1.34
1.34
1.34
1.34
1.34
1.34
1.34
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.36
1.36
1.36
1.36
1.36
1.36
1.36
1.36
1.36
1.36
1.36
1.37
1.37
1.37
D. 1".
Tang.
9.7957S9
.796070
.796351
.796632
.796913
.797194
.797474
.797755
.798036
.798316
9.798596
.798877
.799157
.799437
.799717
.799997
.800277
.800557
.800336
.801116
9.801396
.801675
.801955
.802234
.602513
.802792
.803072
.803351
.803630
.803909
9.804187
.804466
,804745
.805023
.805302
.805580
.805859
.806137
.806415
.806693
9.806971
.807249
.807527
.807805
.803033
.803361
.805633
.803916
.809193
.809471
9.809748
.810025
.810302
.810580
.810857
.811134
.811410
.811687
.811964
.812241
.812517
Cotang
D. 1",
4.63
4.68
4.68
4.68
4.68
4.63
4.68
4.68
4.67
4.67
4.67
4.67
4.67
4.67
4.67
4.66
4.66
4.66
4.66
4.66
4.66
4.66
4.66
4.65
4.65
4.65
4.65
4.65
4.65
465
4.65
4.64
4.64
4.64
4.64
4.64
4.64
4.64
4.64
4.63
4.63
4.63
4.63
4.63
4.63
4.63
4.63
4.62
4.62
4.62
4.62
4.62
4.62
4.62
4.62
4.61
4.61
4.61
4.61
4.61
D.l"
Cotang.
M.
0.204211
60
.203930
59
.203649
58
.203363
57
.203037
56
.202306
55
.202526
54
.202245
53
.201964
52
,201634
51
0.201404
50
.201123
49
.200843
48
.200563
47
.200283
46
.200003
45
.199723
44
.199413
43
.199164
42
.198884
41
0.195604
40
.19532.5
39
.195045
38
.197766
37
.197487
36
.197208
35
,196923
34
,196649
33
.196370
32
,196091
31
0.195813
30
.195534
29
,195255
28
,194977
27
,194698
26
.194420
25
,194141
24
.193363
23
.193555
22
.193307
21
0.193029
20
.192751
19
.192473
IS
.192195
17
.191917
16
.1916.39
15
.191362
14
.191084
13
.190807
12
.190529
11
0.190252
10
.189975
9
.189698
8
.189420
7
.189143
6
.188866
5
.188590
4
.188313
3
.188036
2
,187759
1
.187483
0
Tang
M.
COSINES, TANGENTS, AND COTANGENTS.
207
1*1:0
M
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
53
59
60
Sine.
D. 1".
9.736109
.736303
.736493
.736692
.736SS6
.737030
.737274
.737467
.737661
.737855
9.733043
.733241
.733434
.733627
.733320
.739013
.739206
.739393
.739590
.739783
9.739975
.740167
.740359
,740550
.740742
.740934
.741125
.741316
.741508
.741699
9.741389
.742080
.742271
.742462
.742632
.742S42
.743033
.743233
.743413
.743602
9.743792
.743932
.744171
.744361
.744550
.744739
.744928
.745117
.745306
.745594
9.745633
.745371
.746060
.746248
.746436
.746624
.746812
.746999
.747187
.747374
.747562
M.
Cosine.
Cosine.
3.24
3.24
3.24
3.23
3.23
3.23
3.23
3.23
3.22
3.22
3.22
3.22
3.22
3.21
3.21
3.21
3.21
3.21
3.20
3.20
3.20
3.20
3.20
3.19
3.19
3.19
3.19
3.19
3.18
3.18
3.18
3.18
3.18
3.17
3.17
3.17
3.17
3.17
3.16
3.16
3.16
3.16
3.16
3.15
3.15
3.15
3.15
3.15
3.14
3.14
3.14
3.14
3.14
3.13
3.13
3.13
3.13
3.13
3.12
3.12
D. 1".
9.923591
.923509
.923427
.923345
.923263
.923181
.923093
.923016
.922933
.922351
9,922768
.922636
.922603
.922520
.922433
.9223.55
.922272
.9221S9
.922106
.922023
9.921940
.9218.57
.921774
.921691
.921607
.921524
.921441
.921357
.921274
.921190
9.921107
.921023
.920939
.920856
.920772
.9206S3
.920604
.920520
.920436
.920352
9.920268
.920184
,920099
.920015
.919931
.919346
.919762
.919677
.919593
.919503
9.919424
.919339
.919254
.919169
.919035
.919000
.918915
.918830
.918745
.918659
.918574
Tang.
D. 1 '.
D. 1".
1.37
1.37
1.37
1.37
1.37
1.37
1.37
1.37
1.37
1.38
1.33
1.33
1.33
1.38
1.3S
1.38
1.38
1.38
1.33
1.33
1.39
1.39
1.39
1.39
1.39
1.39
1.39
1.39
1.39
1.39
1.39
1.39
1.40
1.40
1.40
1.40
1.40
1.40
1.40
1.40
1.40
1.40
1.40
1.41
1.41
1.41
1.41
1.41
1.41
1.41
1.41
1.41
1.41
1.41
1.42
1.42
1.42
1.42
1.42
1.42
Sine
9.312517
.812794
,813070
.813347
.813023
.813399
.814176
.814452
.814728
.815004
9.815230
.815555
.815-31
.816107
.816382
.816653
.816933
.817209
.8174.34
.817759
9.818035
.818310
.818585
.818360
.819135
.819410
.819634
.8199.59
.820234
.820503
9.820783
.8210.57
.821332
.821606
.821880
.822154
.822429
.822703
.822977
.823251
9.823524
.823793
.824072
.824345
.824619
.824893
.825166
.825439
.82.5713
.825936
9.8262.59
.826532
.826305
.827078
.827351
.827624
.827897
.828170
.828442
.823715
.823987
Cotang.
4.61
4.61
4.61
4.61
4.60
4.00
4.60
4.60
'1. 60
4.60
4.60
4.60
4.59
4.^9
4.59
4.59
4.59
4.59
4.59
4.59
4.59
4..53
4.53
4.. 53
4.58
4.58
4.-53
4.53
4.58
4.58
4.57
4.-57
4.57
4.57
4.-57
4.57
4.57
4.57
4.57
4.56
4.56
4.56
4.56
4.56
4.56
4.56
4.56
4.56
4.. 55
4.55
4.55
4.55
4.55
4.55
4.55
4.55
4.55
4.54
4.54
4.54
0.187483
.187206
.106930
.186653
.186377
.186101
.1S5>24
.185543
.185272
.184996
0.184720
.184415
.134169
.183893
.183618
.183342
.183067
.182791
.182516
.182241
0.181965
.181690
.131415
.181140
.180665
.180590
.180316
.180041
.179766
.179492
0.179r.l7
.178943
.178668
.173394
.173120
.177846
.177571
.177297
.177023
.176749
O.'l 76476
.176202
.175928
.17.5655
.175381
.175107
.174834
.174,561
.174287
.174014
0.173741
.173468
.173195
.172922
.172649
.172.376
.172103
.171830
.171558
.171235
.171013
D. 1". Cotang. I D. 1". Tang.
M.
60
59
53
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
23
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
U
10
9
8
7
6
5
4
3
2
1
0
M.
56
i>08
340
TABLE XIII. LOGARITHMIC SINES,
1450
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
l-J
14
15
16
17
13
19
20
21
22
23
24
25
26
27
2S
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
Sine.
9.747562
,747749
.747936
.743123
.743310
,743497
.743633
.743370
.749056
.749243
9.749123
.749615
.749301
.749937
.750172
.750353
.750543
.750729
.750914
.751099
9.751234
.751469
.751654
.751839
.752023
.752203
.752.392
.752576
.7.52760
.752944
9.753123
.753312
.753495
.753679
.753362
.754046
.75422J
.751412
.754595
.754778
9.754960
.755143
.755326
.755503
.755690
.755372
.756054
.756236
.756413
.756600
9.756782
.756963
.757144
.757326
.757507
.757633
.757869
.753050
.758230
.753411
.753591
M. Cosine.
D. 1".
3.12
3.12
3.12
3.11
3.11
3.11
3.11
3.11
3.K)
3.10
3.10
3.10
3.10
3.10
3.09
3.09
3.09
3.09
3.09
3.03
3.03
3.03
3.03
3.03
3.07
3.07
3.07
3.07
3.07
3.06
3.06
3.06
3.06
3.06
3.05
3.05
3.05
3.05
3.05
3.05
.3.04
3.04
3.04
3.04
3.04
3.03
3.03
3.03
3.03
3.03
3.02
3.02
3.02
.3.02
3.02
3.02
3.01
3.01
3.01
3.01
Cosine.
D. 1".
9.913574
.91-4^9
.918404
.913313
.913233
.913147
.913062
.917976
.917391
.917805
9.917719
.917634
.917548
.917462
.917376
.917290
.917204
.917118
.917032
.916946
9.916359
.916773
.916637
.916600
.916514
.916427
.916.341
.916254
.916167
.916031
9.915994
.915907
.91.5320
.915733
.915646
.915.5.59
.91.5472
.915335
.915297
.915210
9.915123
.915035
.914948
.914360
.914773
.914635
.914593
.914510
.914422
.914334
9.914246
.914153
.914070
.913932
.91.3394
.913366
.913718
9136-30
.913.541
.913453
.913365
D. 1".
Sine.
1.42
1.42
1.42
1.42
1.42
) 43
1.43
1.43
1.43
1.43
1.43
1.43
1.43
1.43
1.43
1.43
1.43
1.44
1.44
1.44
1.44
1.44
1.44
1.44
1.44
1.44
1.44
1.44
1.45
1.45
1.45
1.45
1.45-
1.45
1.45
1.45
1.45
1,45
1.45
1.46
1.46
1.46
1.46
1.46
1.46
1.46
1.46
1.46
1.46
1,46
1.47
1.47
1.47
1.47
1.47
1.47
1.47
1.47
1.47
1.47
Tang.
D. 1".
9.8239S7
.829260
.829532
.829 >05
.830077
.830349
.Sb'06-4!l
.830393
.831165
.8314.37
9.831709
.831931
.832253
.832.-,25
.832796
.833063
.833339
.8.3361 1
.833332
.834154
9.331425
.83 J 696
.8.34967
.8.35233
.835509
.835780
.836051
.836322
.836593
.836364
9.837134
.837405
.837675
.837946
.833216
.833487
.838757
.839027
.839297
.839563
9.839833
.840103
.840378
.840643
.840917
.841187
.8414.57
.841727
.841996
.842266
9.842535
.842305
.843074
.843343
.84.3612
.843332
.844151
.844420
.844639
.8449.58
.84.5227
D. 1".
4.54
4.. 54
4.54
4.54
4.54
4.54
4.53
4.53
4.53
4.53
4.53
4.53
4.. 53
4.53
4.53
4.53
4.52
4.52
4.52
4.52
4.52
4.52
4.52
4.52
4..52
4.52
4.51
4.51
4.51
4.51
4.51
4.51
4.51
4.51
4.51
4.51
4.50
4.50
4.50
4.50
4.50
4.50
4.50
4.50
4.50
4.49
4.49
4.49
4.49
4.49
4.49
4.49
4.49
4.49
4.49
4.49
4.48
4.48
4.48
4.43
Cotang.
M
0.171013
60
.170740
59
.170463
58
.170195
57
.169923
56
.169651
55
.169379
54
.169107
53
.163335
52
.163563
51
0.168291
50
.163019
49
.167747
48
.167475
47
.167204
46
.166932
45
.166661
44
.166339
43
.166113
42
.165846
41
0.165575
40
.165301
39
Cotang. : D. 1".
.165033
.164762
.164491
.164220
.16-3949
.163678
.16.3407
.163136
0.162S66
,162-595
.162-325
.162054
,161784
,161513
.161243
.160973
.160703
.160432
0.160162
.1-59392
.159622
.159352
.1.59083
.153313
,158543
,1.58273
.153004
.1577'^
0.157465
.1-57195
.1-56926
.156657
.156-333
.156118
.1-5.5349
.155530
,15.53!!
.154773
Tang. I M.
124c
553
COSINEb, TANGENT&, AND COTANGENTS.
209
14:43
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Sine.
9.758591
.758772
.758952
.759132
.7o9:n2
.759492
.759672
.759852
.760:131
.760211
9.760390
.760569
.760748
.760927
.761106
.761235
.761464
.761642
.761821
.761999
9.7C2177
.7623.")6
.762534
.762712
.762889
.763067
.763245
.763422
.763600
.763777
9.763954
.764131
.764308
,764435
.764662
.7648.33
.765015
.765191
.765.367
.765544
9.765720
.765896
.766072
.766247
.766423
.766593
.766774
.766949
.767124
.767300
D.l"
50
51
52
53
54
55
56
57
53
59
60 1
M. I
9.767475
.767649
.767824
.767999
.763173
.768348
.768522
.763697
.763371
.769045
.769219
Cosine.
3.01
3.00
3.00
3.00
3.00
3.01)
2.99
2.99
2.99
2.99
2.99
2.99
2.98
2.98
2.98
2.93
2.98
2.97
2.97
2.97
2.97
2.97
2.97
2.96
2.96
2.96
2.96
2.96
2.95
2.95
2.95
2.95
2.95
2.95
2.94
2.94
2.94
2.94
2.94
2.93
2.93
2.93
2.93
2.93
2.93
2.92
2.92
2.92
2.92
2.92
2.91
2.91
2.91
2.91
2.91
2.91
2.90
2.90
2.90
2.90
D. 1".
9.913365
.913276
.913Ib7
.9130'J9
.913010
.912922
.912833
.9127-14
.912655
.912566
9.912477
.912388
.912299
.912210
.912121
.912031
.911942
.911853
.911763
.911674
9.911584
.911495
.911405
.911315
.911226
.911136
.911046
.910956
.910866
.910776
9.910636
.91U596
.910506
.910415
.910325
.910235
.910144
.910054
.909963
.909873
9.9097S2
.909691
.909601
.909510
.909419
.909328
.909237
.909146
.909055
.903964
9.908873
.903781
.908690
.903599
.903507
.908416
.903324
.903233
.903141
.903049
.907958
Cosine. D. 1".
Tang.
D. 1".
1.47
1.48
1.43
1.48
1.48
1.48
1.48
1.48
1.48
1.48
1.48
1.48
1.49
1.49
1.49
1.49
1.49
1.49
1.49
1.49
1.49
1.49
1.49
1.50
1.50
1.50
1.50
1.50
1.50
1.50
I.. 50
1.50
1.50
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1.53
1.53
1.53
1..53
1.53
Sine.
9.845227
.845496
.845764
.846033
.846302
.846370
.846839
.847108
.847376
.847644
9.847913
.843181
.843449
.848717
.843986
.849254
.849522
.849790
.850057
.350325
9.850593
.850861
.851129
.851.396
.851664
.851931
.852199
.852466
.852733
.853001
9.853268
.853535
.853302
,854C69
.854336
.854603
.854870
.855137
.8.55404
.855671
9.855933
.856204
.8.56471
.856737
.857004
.857270
.857537
.857803
.858069
.858336
9.85S602
.858868
.859134
.859400
.859666
.859932
.860198
.860464
.860730
.860995
.861261
Cotang.
4.48
4.48
4.48
4.43
4.48
4.48
4.48
4.47
4.47
4.47
4.47
4.47
4.47
4.47
4.47
4.47
4.47
4.46
4.46
4.46
4.46
4.46
4.46
4.46
4.46
4.46
4.46
4.46
4.46
4.45
4.45
4.45
4.45
4.45
4.45
4.45
4.45
4.45
4.45
4.44
4.44
4.44
4.44
4.44
4.44
4.44
4.44
4.44
4.44
4.44
4.44
4.43
4.43
4.43
4.43
4.43
4.43
4.43
4.43
4.43
0.154773
.154504
.154236
.153967
.153698
153430
.153161
.152892
.152624
.152356
0.152087
.151819
.151551
.151283
.151014
.150746
.150478
.150210
.149943
.149675
0.149407
.149139
.148871
.148604
.148336
.148C69
.147801
.147534
.147267
.146999
0.146732
.146465
.146198
.145931
.145664
.145397
.145130
.144363
.144596
.144329
0.144062
.143796
.143.529
.143263
.142996
.142730
.142463
.142197
.141931
.141664
0.141398
.141132
.140866
.140600
.140334
.140063
.139802
.1395.36
.139270
.139005
.138739
D. 1". I Cotang. D. 1". I Tang.
M.
60
59
58
57
56
55
54
53
52
51
50
49
43
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
M.
210
B60
TABLE XIII.
LOGARITHMIC SINES,
14:3
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
53
59
60
M.
Sine.
9.769219
.769393
.769566
.769740
.769913
.770037
.770260
.770433
.770506
.770779
9.770952
.771125
.771293
.771470
.771843
.771815
.771937
.772159
.772331
.772503
9.772675
.772847
.773013
.773190
.773361
.773533
.773704
.773575
.774046
,774217
9.774338
.77455S
,774729
,774399
.775070
775240
.775410
.775530
.775750
.775920
9.776090
.776259
.776429
.776593
,776763
.776937
,777106
,777275
,777444
.777613
9.777731
.777950
.773119
.778237
.773455
.773624
.773792
.773960
.779123
.779293
.779463
Cosine.
D. 1".
2.00
2.90
2.39
2.39
2.39
2.89
2.89
2,33
2.33
2.83
2.88
2.33
2.33
2.37
2.37
2.37
2.37
2.37
2.37
2.S6
2.36
2.36
2.36
2.86
2.35
2.35
2.85
2.85
2.35
2.35
2.34
2.84
2.34
2.34
2.34
2.84
2.S3
2.33
2.33
2.83
2.83
2.83
2.32
2.32
2.82
2.82
2.82
2.82
2.81
2.81
2.81
2.81
2.81
2.81
2. SO
2.80
2.80
2.80
2.S0
2.79
D. 1".
Cosine.
9.907953
.907866
.907774
.907632
.907590
.907493
.907406
.907314
.907222
.907129
9.907037
.906945
.906352
.906760
.906667
.906575
.906432
.906339
.906296
.906204
9.906111
.906018
.905925
.90.5332
.905739
.905645
.905552
.905459
.905366
.905272
9.905179
.905035
.904992
.904893
.904304
.904711
.904617
.90^1523
,904429
.904335
9.904241
.904147
,904053
.903959
.903364
.903770
.90.3676
.903.581
.90.3487
.903392
9.903293
,903203
.903103
.903014
.902919
.902324
,902729
.9026.34
.902.539
,902444
.902349
Sine.
D. 1".
,53
,53
,53
,53
,53
,53
,54
,54
.54
54
,54
,54
.54
,54
54
.54
,55
55
,55
,55
,55
,55
,55
,55
,55
,55
,55
,56
,56
,56
,.56
,56
.56
56
56
,56
,56
,57
,57
57
,57
,57
,57
57
57
,57
,57
.57
,58
,58
.58
,58
,58
,53
,53
,.53
,58
,53
.59
,59
D. 1".
Tang.
9.861261
.861.527
.861792
.862058
,862323
.862.589
,862354
,863119
,863335
.86.3650
9.863915
.864180
,864445
,864710
.864975
.865240
,865.505
.865770
.866035
.866300
9.866564
,866329
.867094
,8673.58
.867623
.867337
.8681.52
.863416
.86S630
.863945
9.869209
.869473
.869737
.870001
.870265
.870529
,870793
,871057
,871321
.871585
9.871349
.872112
.872376
,872640
.872903
.873167
.873430
.873694
.873957
.874220
9.874434
.874747
.875010
.875273
.8755.37
.875300
.876063
.876326
.876589
,376352
■877114
Cotang,
D. 1".
4.43
4.43
4.43
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.42
4.41
4.41
4.41
4.41
4.41
4.41
4.41
4.41
4.41
4.41
4.41
4.41
4.41
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
. 4.40
4.39
4.39
4.39
4.39
4.39
4.39
4.39
4.39
4.39
4.39
4.39
4.39
4.39
4.33
4.33
4.33
4.33
4.38
4.33
D. 1".
Cotang.
0.133739
,133173
133203
,137942
,137677
,137411
137146
,136881
,136615
,136350
0
136085
135820
135555
135290
135025
134760
134495
1342.30
13.3965
133700
133436
133171
132906
132642
132.377
132113
131843
131534
131320
131055
130791
130527
130233
129999
129735
129471
129207
123943
123679
123415
128151
127883
127624
127360
127097
126833
126570
126306
126043
125780
125516
125253
124990
124727
124463
124200
123937
123674
12311 1
123143
122386
Tang.
ISd^
63
COSINES, TANGENTS, AND COTANGENTS.
211
1433
M.
Sine.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
[7
IS
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
3i
35
36
37
35
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
o7
5S
59
60
D. 1".
9.779463
.779631
.779798
.779966
.780 133
.780300
.78(H67
,780634
.780801
.780968
9.781134
,781301
.781463
.781634
.781800
.731966
.782132
.782293
.782464
.782630
9.782796
.782961
.783127
,783292
.783453
.783623
,783783
,783953
.784118
,784282
9.784447
.784612
.784776
.784941
.785105
.785269
.785433
.785597
.785761
.785925
9.736039
.7862.52
.786416
.786579
.786742
.736906
.737069
.787232
.787395
.787557
9.787720
.737833
.738045
.733208
.733370
.783532
.783694
.783856
.789018
.789180
.739^12
Cosine.
2.79
2.79
2.79
2.79
2.79
2.78
2.78
2.78
2.73
2.73
2.78
2.77
2.77
2.77
2.77
2.77
2.77
2.76
2.76
2.76
2.76
2.76
2.76
2.75
2.75
2.75
2.75
2.75
2.75
2.74
2.74
2.74
2.74
2.74
2.74
2.73
2.73
2.73
2.73
2.73
2.73
2.73
2.72
2.72
2.72
2.72
2.72
2.72
2.71
2.71
2.71
2.71
2.71
2.71
2.70
2.70
2.70
2.70
2.70
2.70
D. 1".
9.902349
.902253
.902158
.902063
.901967
.901372
.901776
.901681
.901585
.901490
9.901394
.901293
.901202
.901106
.901010
.900914
.900818
.900722
.900626
.900529
9.900433
.900337
.900240
.900144
.900047
.899951
.899354
.899757
.899660
.899564
9.899467
.899370
.899273
.899176
.899073
.893981
.893834
.893787
.898689
.893592
9.898494
.898397
.893299
.898202
.893104
.893006
.897908
.897810
.897712
.897614
9.897516
.897418
.897320
.897222
.897123
.897025
.896926
.896828
.896729
.896631
.896532
Tang.
1.59
I. .59
1.59
1.59
1.59
1.59
1..59
1.59
1.59
1.60
1.60
1.60
1.60
1.60
l.GO
1.60
1.60
1.60
1.60
1.61
1.61
1.61
1.61
1.61
1.61
1.61
1.61
1.61
1.61
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.64
1.64
1.64
1.64
1.64
1.64
1.64
1.64
1.64
1.64
M. Cosine. I D. 1". I Sine. D. 1". Cotang.
D. 1".
9.877114
.877377
.877640
.8779ft3
.873165
.878423
.878691
.878953
.879216
.879478
9.879741
.880003
.880265
.880528
.880790
.8.^5 1052
881314
.881577
.881839
.882101
9.8S23e3
.882625
.832887
.883143
.883410
.883672
.883934
.834196
.884457
.884719
9.884930
.835242
.885504
.835765
.886026
.886283
.886.549
.886811
.887072
.887333
9.887594
.837855
.883116
.883378
.838639
.888900
.889161
.889421
.839682
.889943
9.890204
.890465
.890725
.8909-^6
891247
.891507
.891763
.892023
.892239
.892549
.892310
Cotang.
4.38
4.38
4.38
4.38
4.38
4.38
4.38
4.33
4.. 37
4.37
4.37
4.37
4.37
4.37
4.-37
4.37
4.37
4.37
4.37
4.37
4.37
4.37
4.36
4. -36
4.36
4.-36
4.36
4.36
4.36
4.30
4.36
4.36
4.36
4.36
4.36
4.36
4.36
4.35
4.35
4.35
4.35
4.35
4.-35
4.35
4.. 35
4.35
4.35
4.35
4.35
4.35
4.35
4.35
4.34
4.34
4.34
4.34
4.34
4.34
4.34
4.34
M.
0.122886
.122623
.122360
.122097
.121835
.121572
.121309
.121047
.120784
.120522
0.1202.59
.119997
.119735
.119472
.119210
.118943
.118686
.118423
.118161
.117899
0.1176.37
.117375
.117113
.116852
.116590
.116328
.116066
.115804
.115543
.115281
0.115020
.114758
.114496
.114235
.113974
.11.3712
.113451
.113189
.112928
. 1 126G7
0.112406
.112115
.111 S84
.111(;22
.111361
.imon
.I1(K:!9
.11(1579
.11(1318
.1 10057
0. l(l'.)796
.l(i',).'):'.5
.l!n)275
.109014
.1087 .53
.108493
.108232
.107972
.107711
.107451
.107190
D. 1".
60
59
58
57
56
55
52
51
50
•49
48
47
46
^5
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
IS
17
10
15
14
13
12
II
1ft
9
8
Tang. M
laT'
5a<-
212
38°
TABLE XIIT.
LOGARITHMIC SINES,
14:1C
M.
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
31
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Sine.
D. 1".
9.789312
.789504
.789665
.789S27
.789983
.790149
.790310
.790471
.790632
.790793
9.790954
.791115
.791275
.791436
.791596
.791757
.791917
.792077
.792237
.792397
9.7925.57
.792716
.792376
.793035
.793195
.793354
.793514
.793673
.793832
.793991
9.794150
.794303
.794467
.794626
.794784
.794942
.795101
.795259
.795417
.795575
9.795733
.795891
.796049
.796206
.796.364
.790521
.796679
.796836
.796993
.797150
9.797.307
.797464
.797621
.797777
.797934
.793091
.798247
.793403
.798560
,798716
.793872
M. Cosine.
2.69
2.69
2.69
2.69
2.69
2.69
2.6S
2.63
2.68
2.63
2.63
2.63
2.67
2.67
2.67
2.67
2.67
2.67
2.67
2.66
2.66
2.66
2.66
2.66
2.66
2.65
2.65
2.65
2.65
2.65
2.65
2.64
2.64
2.64
2.64
2.64
2.64
2.64
2.63
2.63
2.63
2.63
2.63
2.63
2.62
2.62
2.62
2.62
2.62
2.61
2.61
2.61
2.61
2.61
2.61
2.61
2.61
2.60
2.60
2.60
Cosine.
D.l
9.896.5-32
.896433
.896335
.896236
.896137
.896038
.895939
.895840
.895741
.895641
9.895542
.89.5443
.895343
.895244
.895145
.895045
.894945
.894846
.894746
.894646
9.894.546
.894446
.894346
.894246
.894146
.894046
.893946
.893846
.893745
.893645
9.893544
.893444
.893343
.893243
.893142
.893041
.892940
.8923.39
.892739
.892633
9.892536
.892435
.892334
.892233
.892132
.892030
.891929
.891827
.891726
.891624
9.891523
.891421
.891319
.891217
.891115
.891013
.890911
.890809
.890707
.890605
.890503
D. 1".
Sine.
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.66
1.66
1.66
1.66
1.66
1.66
1.66
1.66
1.66
1.66
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.67
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.69
1.69
1.69
1.69
1.69
1.69
1.69
1.69
1.69
1.69
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
Tang.
9.892S10
.893070
.893331
.893591
.893851
.894111
.894372
.894632
.894892
.895152
9.895412
.895672
.895932
.896192
.8964.52
.896712
.896971
.897231
.897491
.897751
9.898010
.898270
.8985.30
.898789
.899049
.899308
.899563
.899827
.900087
.900346
9.900605
.900864
.901124
.901383
.901642
.901901
.902160
.902420
.902679
.902933
9.303197
.903456
.903714
.903973
.9042.32
.904491
.904750
.905003
.805267
.905526
9.905785
.906043
.906302
.906560
.906819
.907077
.907336
.907594
.907853
.903111
.908269
D. 1".
D. 1". Cotang.
4.34
4.34
4.34
4.34
4.34
4.34
4.34
4.34
4.33
4.33
4.33
4.33
4.33
4.33
4.33
4.33
4.33
4.33
4.-33
4.33
4.33
4.33
4.33
4.-33
4.33
4.32
4.32
4.32
4.32
4.-32
4.32
4.32
4.32
4.32
4.32
4.32
4.32
4.32
4.32
4.32
4.32
4.32
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
4.31
Cotang.
M.
0.107190
60
.106930
59
.106669
58
.106409
57
.106149
56
.105889
55
.105628
54
.10.5368
53
.105108
52
.104848
61
0.104583
50
.104328
49
.104063
48
.103808
47
.103548
46
.103283
45
.103029
44
.102769
43
.102509
42
.102249
n I ni nnn
41
D. 1".
.101730
.101470
.101211
.100951
.100692
.100432
.100173
.099913
.099654
0.099395
.099136
.098876
.098617
.098358
.098099
.097840
.097580
.097321
.097062
0.096S03
.096544
.096286
.096027
.095768
.095509
.095250
.094992
.094733
.094474
0.094215
.09.3957
.093698
.093440
.093181
.092923
.092664
.092406
.092147
.091889
.091631
Tang. M
10
9
8
7
6
5
4
3
2
1
0
l»8o
61
COSINES, TANGENTS, AND COTANGENTS.
2Vc
14:0=
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
53
57
5.3
59
60
Sine.
D. 1".
9.793372
.799023
.799134
.799339
.79949.5
.799651
.799306
.799962
.800117
.800272
9.300427
.800532
.S0J737
.800392
.801047
.801201
.8013.56
.801511
.801665
.801819
9.801973
.802123
.802232
,802436
.802589
.802743
.802397
.803050
.803204
.803357
9.803511
.803664
.803817
.803970
.804123
.804276
.804423
.804531
.804734
.804836
9.805039
.805191
.80.5343
.805495
.80.5647
.805799
.80.5951
.806103
.8062.54
.806406
9.806557
.806709
.806360
.80701 1
.807163
.8(37314
.807465
.807615
.807766
.807917
.803067
Cosine.
2.60
2.60
2.6')
2.59
2.59
2.59
2.59
2.59
2.59
2.59
2.58
2.53
2.58
2.58
2. .58
2.58
2.57
2.57
2.57
2.57
2.57
2.57
2.57
2.. 56
2.56
2.56
2.56
2.56
2.56
2.55
2.55
2.55
2.55
2. .55
2.55
2.. 55
2.54
2.54
2.54
2.54
2.54
2.54
2.54
2.53
2.53
2. .53
2. .53
2.. 53
2.53
2.52
2.52
2.52
2.52
2. .52
2.. 52
2.52
2.51
2.51
2.51
2.51
M.
LS9^
Cosine.
D. 1".
9.890503
.890400
.890293
.890195
.890093
.839990
.889333
.889785
.889632
.839579
9.839477
.839374
.889271
.839163
.889064
.838961
.833853
.833755
.888651
.838543
9.833444
.833341
.838237
.833134
.888030
.887926
.887822
.837718
.837614
.837510
9.837406
.837302
.837198
.837093
.836'989
.836385
.836780
.836676
.886571
.886466
9.836362
.836257
.836152
.836047
.835942
.835337
.835732
.835627
.835.522
.835416
9.83.5311
.83.5205
.835100
.884994
.834339
.834783
.834677
.834572
.834466
.834360
.884254
D. 1"
Tang.
D. 1".
1.71
1.71
1.71
1.71
1.71
1.71
1.71
1.71
1.71
1.71
1.72
1.72
1.72
1.72
1.72
1.72
1.72
1.72
1.72
1.72
1.73
1.73
1.73
1.73
1.73
1.73
1.73
1.73
1.73
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.74
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.76
1.76
1.76
1.76
1.76
1.76
1.76
1.76
1.76
1.77
1.77
Sine.
9.903369
.903628
.90S336
.909144
.909402
.909660
.909918
.910177
.910435
.910693
9.910951
.911209
.911467
.911725
.911932
.912240
.912493
.912756
.913014
.913271
9.913529
.913787
.914044
.914302
.914.560
.914317
.915075
.91.5332
.915.590
.915347
9.916104
.916362
.916619
.916877
.917134
.917391
.917648
.917906
.918163
.918420
9.918677
.918934
.919191
.919448
.919705
.919962
.920219
.920476
.920733
.920990
9.921247
.921503
.921760
.922017
.922274
.922530
.922787
.923044
.923300
.923557
.923314
Cotang.
4.30
4.30
4.30
4.30
4.30
4.30
4.30
4.30
4.30
4.30
4.30
4.30
4.30
4.. 30
4.30
4.30
4.30
4.30
4.30
4.30
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.29
4.23
4.23
4.28
4.28
4.23
4.23
4.23
4.28
4.23
4.28
4.28
4.23
4.28
4.28
4.28
4.23
4.23
4.28
4.23
4.23
D. 1". I Coteng.
0.091631
.091372
.091114
.090356
.090598
.090340
.090032
.089323
.039565
.039307
0.0S9049
.038791
.038533
.083275
.033018
.037760
.037502
.037244
.036936
.086729
0.036471
.086213
.0359.56
.085693
.085440
.035183
.084925
.084663
.084410
.034153
0.033396
.033633
.033331
.033123
.032366
.082609
.032352
.032094
.031337
.081580
0.031323
.031066
.080309
.030552
.030295
.030033
.079781
.079524
.079267
.079010
0.073753
.073497
.078240
.077933
.077726
.077470
.077213
.076956
.076700
.076443
.076136
M.
60
D. 1".
Tang.
59
53
57
56
55
.54
53
52
51
49
43
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
23
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
_0_
M.
tu^.
214
*0O
TABLE XIII. LOGARITHMIC SINES,
139"
M.
0
I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
2S
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
,48
49
50
51
52
53
54
55
56
57
58
59
60
M.
Sine.
9.808067
.808218
.803363
.808519
.808669
.808819
.808969
.809119
.809269
.809419
9.809569
.809713
.809363
.810017
.810167
.810316
.810465
.810614
.810763
.810912
9.311061
.811210
.8113.53
.811507
.811655
.811804
.811952
.812100
.812243
,812396
9.812544
312692
.812S40
.812933
.813135
.8132S3
.813430
.813573
.813725
.813872
9.814019
.814166
.814313
.814460
.814607
.814753
.814900
.815046
.815193
.815339
9.815185
.815632
.815773
.815924
.816069
.816215
.816361
.816.507
.816652
.816793
.816943
D. 1".
2.51
2.51
2.51
2.50
2.50
2.50
2.50
2. .50
2.50
2.50
2.49
2.49
2.49
2.49
2.49
2.49
2.43
2.43
2.43
2.43
2.43
2.43
2.43
2.47
2.47
2.47
2.47
2.47
2.47
2.47
2.46
2.46
2.46
2.46
2.46
2.46
2.46
2.45
2.45
2.45
2.45
2.45
2.45
2.45
2.44
2.44
2.44
2.44
2.44
2.44
2.44
2.43
2.43
2.43
2.43
2.43
2.43
2.43
2.42
2.42
Cosine
D. 1".
Cosine.
9.8342-54
.831143
.881042
.883936
.883329
.833723
.833617
.833510
.883401
.883297
9.883191
.883034
.882977
.882371
.882764
.882657
.882550
.882443
.8823:36
.882229
9 882121
.882014
.831907
.831799
.831692
.881534
.881477
.831369
.831261
.831153
9.831046
.880933
.830330
.830722
.850613
.830.505
.830397
.880239
.830180
.830072
9.379963
.879355
.879746
.879637
.879529
.879120
.879311
.879202
.879093
.873984
9.878375
.878766
.878656
.878547
.87,3433
.878323
.873219
.878109
.877999
.877890
.877780
Sine.
D. 1".
.77
.77
.77
.77
.77
.77
.77
.77
.73
.73
.73
.78
.78
.73
.78
.78
.73
.79
.79
.79
.79
.79
.79
.79
.79
.79
.79
.80
.80
.80
.80
.80
.80
.80
.80
.80
.31
.31
.81
.81
.81
.81
.81
.81
.81
.81
.82
.82
.82
.82
.82
.82
.82
.82
.82
.83
.83
.83
.83
.83
D. 1".
Tang.
9.92.3314
.921070
.924327
.921583
.924840
.925096
.925352
.925609
.92-5865
.926122
9.926373
.9266.34
.926890
.927147
.927403
.927659
.927915
.928171
.923127
.92?634
9.923940
.929196
.929452
.929703
.929964
.930220
.930475
.930731
.930937
.931243
9.931199
.931755
.932010
.932266
.932522
.932773
.933033
.933239
.93.3545
.933800
9.931056
.9.31311
.9.31567
.9.31322
.9.35078
.935.3.33
.935.539
.935314
.936100
.9363.55
9.936611
.936366
.937121
.937377
.937632
.937837
.933142
.938393
.9336.53
.933903
.939163
D. 1".
4.28
4.23
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.27
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
4.26
426
4.26
4.26
4.25
4.25
4.25
4.25
4.25
4.25
4.25
Cotang. i D. 1".
Cotang.
0.076186
.075930
.075673
.075417
.075160
.074901
.074613
.071391
.074135
.073878
0.073622
.073366
.073110
.072853
.072597
.072.341
.072135
.071329
.071573
.071316
0.071060
.070301
.070.548
.070292
.070036
.069730
.069525
.069269
.069013
.063757
0.068501
.063215
.067990
.067731
.067478
.067222
.066967
.066711
.06&155
.066200
0.06.5944
.065639
.0654.33
.065178
.061922
.061667
.064411
.0641.56
.063900
.063645
0.063.389
.063134
.062379
.062623
.062363
.062113
.061358
.061602
.061347
.061092
.060837
Tang. I M.
I9f%0
49^
COSlNEll, TANGENTS, AND COTANGENTS.
410
215
1383
M.
0
1
•2
3
4
5
6
7
S
9
10
U
12
13
14
15
16
17
IS
19
20
21
22
23
24
25
26
27
2S
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
54
55
56
57
53
59
60
M.
Sine.
9.816943
.817038
.817233
.817379
.817524
.817663
.817813
.817953
.818103
.818247
9.818392
.818536
.818631
.81S325
.818969
.319113
.819257
.319401'
.819545
.3196S9
9.819832
.819976
.820120
.820263
.820406
.820550
.820693
.S20S36
.820979
.821122
9.321265
.821407
.821550
.321693
.321335
.321977
.822120
.822262
.822404
.822546
9.822633
.322830
.822972
.323114
.823255
.82.3397
.823539
.82.3680
.823821
.823963
9.824104
.824245
.824386
.324527
.824668
.824303
.824949
.8-25090
.82.5230
.825371
.825511
Cosine.
D. 1".
2.42
2.42
2.42
2.42
2.42
2.41
2.41
2.41
2.41
2.41
2.41
2.41
2.40
2.40
2.40
2.40
2.40
2.40
2.40
2.39
2.39
2.39
2.39
2.39
2.39
2.39
2.33
2.-33
2.38
2.38
2..33
2.38
2.33
2.37
2.37
2.37
2.37
2.37
2.37
2.37
2.37
2.. 36
2.36
2.. 36
2.. 36
2.. 36
2..36
2.36
2.35
2.35
2.35
2.35
2.35
2.35
2.35
2.34
2.34
2.34
2.34
2.34
D. 1".
Cosine.
9.877780
.877670
.877560
.877450
.877340
.877230
.877120
.877010
.876399
.876739
9.376678
.876568
.876457
.876347
.876236
.876125
.876014
.875904
.875793
.875682
9.875571
.875459
.875.348
.8752:37
.875126
.875014
.874903
.874791
.874680
.874568
9.874456
.874344
.874232
.874121
.874009
.873396
.873734
.873672
.873560
.873443
9.873335
.873223
.873110
.872993
.872385
.872772
.872659
.872.547
.3724.34
.872321
9.872203
.872095
.871981
.871863
.871755
.871641
.871528
.871414
.871301
.871137
.871073
Sine.
D. 1".
1.83
1.83
1.83
1.83
1.84
1.34
1.84
1.84
1.84
1.34
1.84
1.84
1.84
1.84
1.85
1.85
1.85
1.85
1.85
1.85
1.85
1.85
1.85
1.86
1.86
1.86
1.86
1.86
1 .86
1.86
1.86
1.86
1.87
1.87
1.87
1.87
1.87
1.87
1.87
1.87
1.87
1.88
1.83
1.88
1.83
1.83
1.88
1.88
1.88
1.88
1.89
1.89
1.89
1.89
1.89
i.89
1.89
1.89
1.89
1.90
D. 1".
Tang.
9.939163
.939418
.939673
.939923
.940183
.940439
.940694
.940949
.941204
.941459
9.941713
.941968
.942223
.942478
.942733
.942988
.943243
.943493
.943752
.944007
9.944262
.944517
.944771
.94.5026
.945281
.945535
.945790
.946045
.946299
.946554
9.946S08
.947063
.947318
.947572
.947827
.948031
.948335
.948590
.948344
.949099
9.949353
.949603
.949862
.950116
.950371
.950625
.950879
.951133
.951383
.951642
9.951896
.952150
.952405
.952659
.952913
.953167
.953421
.953675
.953929
.954183
.954437
Cotang.
D. 1'.
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.25
4.24
4.24
4.24
4.24
4.21
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.24
4.23
4.23
4.23
D. 1".
Cotang.
0.060837
.060582
.060327
.060072
.059817
.059561
.059306
.059051
.058796
.058541
0.053287
.053032
.057777
.057522
.057267
.057012
.056757
.056.502
.056248
.055993
0.055733
.0.55483
.055229
.054974
.054719
.054465
.0.54210
.053955
.05.3701
.053446
0.053192
.052937
.052682
.052128
.052173
.051919
.051665
.051410
.051156
.050901
0.050647
.050392
.050138
.049834
.049629
.049375
.049121
.043367
.048612
.048358
0.043104
.047850
.047595
.047341
.047037
.046333
.046579
.046325
.046071
.045817
.045563
M.
60
59
53
57
56
55
54
53
52
51
50
49
43
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
13
17
16
15
14
13
12
ir
10
9
8
7
6
5
4
3
2
1
0
Tang. M.
1310
403
216
430
TABLE XIII. LOGrARlTHMlC SINES,
1370
M.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
23
29
30
31
i 32
33
34
35
3f3
37
3S
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
51
55
56
57
53
59
60
M.
i3a^
Sine.
9.825511
.825651
.825791
.825931
.826071
.826211
.826351
.826491
.826631
.826770
9.826910
.827049
.827189
.827328
.827467
.827606
.827745
.827384
.823023
.823162
9.S2S301
.828439
.823578
.823716
.828855
.828993
.829131
.829269
.829407
.829.545
9.S29633
.829321
.829959
.830097
.830234
830372
.830509
.830646
.83)784
.8.30921
9. 33! 058
.831195
.831.332
.831469
.831606
.831742
.831879
.832015
.832152
.832233
9.832425
.832561
.832697
.832333
.832969
.8.33105
.833241
.833377
.833512
.833643
.833783
Cosine.
D. v.
2.34
2.31
2.33
2.33
2.33
2.33
2.-33
2.-33
2.-33
2.33
2.32
2.32
2.32
2.32
2.-32
2.32
2.-32
2.31
2.31
2.31
2.31
2.31
2.31
2.31
2.31
2.30
2.30
2.30
2.30
2.30
2.30
2.30
2.29
2.29
2.29
2.29
2.29
2.29
2.29
2.29
2.23
2.23
2.23
2.23
2.23
2.23
2.23
2.27
2.27
2.27
2.27
2.27
2.27
2.27
2.27
2.26
2.26
2.26
2.26
2.26
D. 1".
Cosine.
9.871073
.870960
.870346
.870732
.870613
.870504
.870390
.870276
.870161
.870047
9.869933
.869818
.869704
.869539
.869474
.869-360
.869245
.8691-30
.869015
.863900
9-863735
.863670
.868555
.863440
.863324
.863209
.863093
.867978
.867862
.867747
9.867631
.867515
.867399
.867233
.867167
.867051
.8669-35
.866319
.866703
.866586
9.866470
.866353
.866237
.866120
.866004
.865387
.865770
.86-5653
.86.5536
.86-5419
9-86-5302
.865185
.865063
.8&4950
.864333
.864716
.864-593
.864431
.864363
.864245
.864127
Sine.
D. 1".
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.91
1.91
1.91
1.91
1.91
1.91
1.91
1.91
1.91
1.92
1.92
1.92
1.92
1.92
1.92
1.92
1.92
1.92
1.93
1.93
1.93
1.83
1.93
1.93
1.93
1.93
1.93
1.94
1.94
1.94
1.94
1.94
1.94
1.94
1.94
1 94
1.95
1.95
1.95
1.95
1.95
1.95
1.95
1.95
1.95
1.96
1.96
1.96
1.96
1.96
1.96
1.96
D. 1".
Tang.
9.9-54437
.9-54691
.9.54946
.9-55200
-9554-54
.955703
.9-55961
.9-56215
.956469
.9-56723
9.956977
.957231
.957435
.957739
.957993
.953247
.953500
.953754
.959003
.959262
9.9-59516
.9-59769
.960023
.960277
.960530
.960784
.961033
.961292
.961545
.961799
9.9620-52
.962306
.962560
.962313
.963067
.963320
.963574
.963323
.964031
.964335
9-964583
.964342
.96.5095
.965349
.965602
.965355
.966109
.966362
.9666.6
.966369
9.967123
.967376
.967629
.967333
.963136
.963339
.963643
.963396
.969149
.969403
.969656
Cotang.
D. 1".
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.23
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
D. 1'.
Cotang.
0.045563
.045309
.04.5054
.044800
.044546
.044292
.044039
.0^43735
.043531
.043277
0.04.3023
.042769
.042515
.042261
.042007
.041753
.041.500
• .041246
.040992
0.040484
41
40
.040231
39
.039977
33
.039723
37
.039470
36
.039216
35
.033962
34
.038703
33
.033455
32
.033201
31
0.037943
30
.037694
29
.037440
23
.037187
27
.036933
26
.036630
25
.036426
24
.036172
23
.035919
22
.035665
21
0.03-5412
20
.035153
19
.034905
18
.034651
17
,034393
16
.0-341-15
15
.0-33391
14
.033633
13
.033334
12
.033131
11
0-032377
10
.032624
9
.0.32371
8
.032117
7
.031864
6
.031611
5
.0313-57
4
.031104
3
.0-30351
2
.030597
1
.030344
0
M
Tang
47'
COSINES, TANGENTS, AND COTANGENTS.
430
2n
M.
0
1
2
3
4
5
e
7
Sine.
10
11
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
3S
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
GO
D. 1".
9.833783
.833919
.834054
.834189
.834325
.834460
.831595
.834730
.834365
.834999
9.835134
.835269
.835403
.835r,.3S
.8356:2
.835807
.835941
.836075
.836209
.836313
9.836477
836611
.836745
.836378
.837012
.837146
.837279
.837412
.837546
,837679
9.837812
.837945
.833078
.833211
.833344
.833477
.833610
.833742
.833375
.839007
9.839140
.839272
.839404
.839536
.839663
.839800
.839932
.840064
.840196
.840323
9.840459
.840591
.840722
.840854
.840985
.841116
.841247
.841373
.841509
.841640
.&il771
M.
Cosine.
Cosine.
2.26
2.26
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.24
2.24
2.24
2.24
2.24
2.24
2.24
2.23
2.23
2.23
2.23
2.23
2.23
2.23
2.23
2.22
2.22
2.22
2.22
2.22
2.22
2.22
2.22
2.21
2.21
2.21
2.21
2.21
2.21
2.21
2.21
2.20
2.20
2.20
2.20
2.20
2.20
2.20
2.19
2.19
2.19
2.19
2.19
2.19
2.19
2.19
2.18
2.18
2.18
2.18
D. 1".
9.864127
.864010
.863392
.863774
.863656
.863533
.863419
.863301
.863183
.863064
9.862946
.862827
.S627(.9
.662590
.862471
.862353
.862234
.862115
.861996
.861877
9.861758
.861638
^61519
.861400
.861230
,861161
.861041
,860922
,860302
.860632
9.860562
.860442
.860322
.860202
.860082
.859962
.859842
.859721
.859601
.859480
9.859360
.859239
.859119
,858998
.853877
,858756
.858635
.858514
.858393
,858272
9.858151
.858029
.857908
.857786
.857665
.857543
.857422
.857300
.857173
.857056
.856934
D. 1".
Tang.
1.96
1.97
1.97
1.97
1.97
1.97
1.97
1.97
1.97
1.97
1.93
1.93
1.93
1.93
1.98
1.98
1.98
1.98
1.98
1.99
1.99
1.99
1.99
1.99
1.99
1.99
1.99
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.01
2.01
2.01
2.01
2.01
2.01
2.01
2.01
2.02
2.02
2.02
2.02
2.02
2.02
2.02
2.02
2.02
2.03
2.03
2.03
2.03
2.03
2.03
2.03
Sine.
D. 1".
9.969656
.969909
.970162
.970416
.970669
.970922
.971175
.971429
.971682
,971935
9.972188
,972441
.972695
.972943
.973201
.973454
.973707
.973960
.974213
,974466
9.974720
,974973
,975226
,975479
.975732
.975935
.976233
.976491
.976744
.976997
9.977250
,977503
,977756
,978009
.978262
.978515
.978763
.979021
.979274
.979527
9.979730
.980033
,980286
.980533
.980791
.981044
.981297
,981550
,981803
,932056
9.982309
.9S2562
.932314
.933067
.983320
.933573
.983326
.934079
.984332
.984534
.934337
D. 1". Cotang.
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.22
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
Cotang.
0.030344
.030091
.029833
.029584
.029331
.029078
.028825
.028571
.028318
.028065
0.027812
.027559
,027305
,027052
,026799
.026546
.026293
.026040
.025787
.025534
0.025280
.025027
.024774
,024521
.024263
,024015
,023762
.023509
.023256
.023003
0.022750
.022497
.022241 ■
.021991
.021738
.021435
.021232
.020979
.020726
.020473
0.020220
.019967
.019714
.019462
.019209
.018956
.018703
,018450
.018197
,017944
0.017691
,017438
,017186
,016933
.0166-^0
,016427
,016174
.015921
.015663
.015416
.015163
M.
60
59
58
57
56
55
54
D. 1".
Taug.
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
M.
1 33 J
46C
218
440
TABLE XIII.
LOGARITHMIC SINES, &C.
1354
M.
0
I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
2S
29
30
31
32
33
34
35
36
37
33
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
53
59
60
Sine.
D. 1".
9.34] 771
.841902
.842033
.842163
.842294
.842424
" .842555
.842635
.842315
.842946
9.843076
.843206
.843336
.84.3466
.843595
.843725
.843355
.843934
.844114
.844243
9.844372
.844502
.844631
.844760
.844889
.845018
.845147
.845276
.84.5405
.84.55.33
9.845662
.845790
.84.5919
.846047
.846175
.846304
.846432
.846558
.846638
.846316
9.346944
.847071
.547199
.847327
.3474.54
.847532
.847709
.847836
.347964
.843091
9.848213
.843345
.843472
.843599
.813726
.843352
.843979
.849106
.8492-32
.849.359
.849435
2.13
2.18
2 18
2.13
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.16
2.16
2.16
2.16
2.16
2.16
2.16
2.16
2.15
2.15
2.15
2.15
2.15
2.15
2.15
2.15
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.13
2.13
2.13
2.13
2.13
2.13
2.13
2.13
2.12
2.12
2.12
2.12
2.12
2.12
2.12
2.12
2.11
2.11
2.11
2.11
2.11
2.11
2.11
2.11
Cosine.
9.3.56934
.856312
.856690
.856-568
.856446
.856323
.856201
.8.56078
.85.59.56
.855333
9.8.5.5711
.85.5533
.855465
.855342
.8-55219
.855096
.8.54973
.8543.50
.8.54727
.854603
9.8.54430
.8.543-56
.854233
.8-54109
.8.53936
.853362
.8.53733
.853614
.853490
.853366
9.35-3242
.8-53118
.852994
.352369
.352745
.852620
.852496
.852371
.852247
.852122
9.851997
.851372
.851747
.851622
.851497
.851372
.851246
.851121
.850996
.850370
9.850745
.850619
.3.50493
.850363
.8-50242
.8.50116
.849990
.849364
.849733
.849611
.849435
D. 1".
M. I Cosine. I D. 1". | Sine. D. 1". Cotang. | D. 1".
2.03
2.04
2.04
2.04
2.04
2.04
2.04
2.04
2.04
2.04
2.05
2.05
2.05
2.05
2.05
2.05
2.05
2.05
2.06
2.06
2.06
2.06
2.06
2.06
2.06
2.06
2.06
2.07
2.07
2.07
2.07
2.07
2.07
2.07
2.07
2.03
2.03
2.03
2.03
2.03
2.03
2.03
2.03
2.09
2.09
2.09
2.09
2.09
2.09
2.09
2.09
2.10
2.10
2.10
2.10
2.10
2.10
2.10
2.10
2.11
Tang.
9.934337
.985090
.935343
.93.5.596
.935343
.936101
.936-3-54
.936607
.936360
.937112
9.937365
.937618
.987871
.938123
.933376
.933629
.933332
.939134
.939337
.939640
9.939393
.990145
.990393
.990551
.990903
.9911.56
.991409
.991662
.991914
.992167
9.992420
.992672
.992925
.993178
.993431
.99.3633
.9939:36
.994139
.994441
.994694
9.994947
.995199
.995452
.995705
.995957
.996210
.996463
.996715
.996963
.997221
9.997473
.997726
.997979
.993231
.9934*4
.993737
.993939
.999242
.999495
.999747
0.000000
D. 1".
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.2)
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4.21
4-21
4.21
4.21
4.21
Cotang.
0.015163
.014910
.0146-57
.014404
.0141.52
.013399
.013646
.013393
.013140
.012388
0.012635
.0123S2
.012129
.011877
.011624
.011.371
.011118
.010366
.010613
.010360
0.010107
.009355
.009602
.009349
.009097
.003844
.003591
.003333
.003036
.007833
0.007530
.007323
.007075
.006322
.006569
.006317
.006064
.005811
.00.5559
.005306
0.005053
.004301
.004.543
.004295
.004043
.003790
.003537
.003235
.003032
.002779
0.002.527
.002274
.002021
.001769
.001516
.001263
.001011
.000758
.000.505
.000253
.000000
Tang.
M.
60
59
53
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
II
10
9
8
7
6
5
4
3
2
I
0
M.
134a
i'i
TABLE XIV.
NATURAL SINES AND COSINES
i-^{}
TABLE
XIV.
NATURAL SINES AND COSINES.
M.
0
,
= — il
00
i^
}c~^ \ a^
*
^
1
M.
60
Sine.
.00000
Cosin.
Sine.
Cosin.
Sine.
Cosin.
Sine.
Cosin.
Sine.
Cosin.
.99756
One.
.01745
.99985
.03490
.99939
.0.5234
.99863
.06976
1
.00029
One.
.01774
.999S4
.03519
.99933
.05263
.99-61
.07005
.99754
59
2
.00058
One.
.01303
.99934
.03-543
.99937
.0.5292
.99360
.07034
.99752
58
3
.0JJS7
One.
.01332
.99933
.03577
.99936
05321
.99358
.07063
.99750
57
4
.00116
One.
.01362
.99933
.03606
.99935
.05-350
.99357
.07092
.99748
56
5
.00145
One.
.01391
.99932
.03635
.99934
05379
.99355
.07121
.99746
55
6
.00175
One.
.01920
.99932
.C3661
.99933
.0-5403
.99354
.07150
.99744
54
7
.00204
One.
.01919
.99931
.03693
.99932
.05437
.99852
.07179
.99742
5c
8
.00233
One.
.01973
.99930
.03723
.99931
.0-5466
.99351
.07203
.99740
52
9
.00262
One.
.02097
.99930
.03752
.99930
.0.5495
.99349
.07237
.99738
51
10
.00291
One.
.02036
.99979
.03781
.99929
.05524
.99847
.07266
.99736
50
11
.00320
.99999
.02065
.99979
.03310
.99927
.05553
.99346
.07295
.99734
49
12
.00349
.99999
.02094
.99973
.0.38.39
.99926
.0-5532
.99344
.07324
.99731
48
13
.00373
.99999
.02123
.99977
.03363
.99925
.05611
.99342
.073.53
.99729
47
14
.00407
.99999
.021-52
.99977
.03397
.99924
.05640
.99341
.07332
.99727
46
15
.00436
.99999
.02131
.99976
.03926
.99923
.05669
.993.39
.07411
.99725
45
16
.00463
.99999
.02211
.99976
.03955
.99922
.05698
.99333
.07440
.99723
44
17
.00495
.99999
.0221'!
.99975
.0-3934
.99921
.05727
-99336
.07469
.99721
43
15
.03524
.99999
.02269
.99974
.04013
.99919
.0-5756
.99334
.07493
.99719
42
19
.00553
.99993
.02293
.99974
.04042
.99918
.05785
.998.33
.07527
.99716
41
20
.00532
.99993
.02327
.99973
.04071
.99917
.0-5814
.99831
.07556
.99714
40
21
.00611
.99993
.02356
.99372
.04100
.99916
.05344
.99329
.07535
.99712
39
22
.00640
.99993
.02335
.99972
.04129
.99915
.05873
.99827
.07614
.99710
33
23
.03660
.99993
.02414
.9p971
.04159
.99913
.05902
.99326
.07643
.99708
37
21
.00693
99993
.02443
.99970
.04183
.99912
.05931
.99-24
.07672
.99705
36
25
.00727
.99997
.02472
.99969
.04217
.99911
.05960
.99322
.07701
.99703
35
26
.00756
.99997
.02.501
.99969
.04246
.99910
.05989
.99321
.07733
.99701
^
27
.00785
.99997
.02530
.99963
.04275
.99909
.06018
.99319
.07759
.99699
33
2S
.00314
.99997
.02.560
.99967
.04804
.99907
.06047
.99317
.07783
.99696
32
29
.00314
.99996
.02589
.99966
.04333
.99906
.06076
-99315
.07817
.99694
SI
30
.O0S73
.99996
.02613
.99966
.04362
.99995
.06105
.9981-3
.07846
.90692
30
31
00502
.99996
.02647
99965
.04391
.99904
.061.34
.99312
.07875
.9:^i
29
32
.00931
.99996
.02676
.99964
.04420
.99902
.06163
.99310
.07904
.99687
28
33
.00960
.99995
.02705
.99963
.04449
.99901
.06192
.99333
.07933
.99635
27
34
.009S9
.99995
.02734
.99963
.04478
.99900
.05221
.99806
.07962
.99633
26
35
.0101.3
.99995
.02763
.99962
.04.507
.99393
.06250
.99804
.07991
.99680
25
36
.01047
.99995
.02792
.99961
.045.36
.99397
.06279
.99-03
.08020
.99678
24
37
.01076
.99994
.02321
.99960
.04-565
.99396
.06303
.99-01
.08049
.99676
23
3S
.01105
.99994
.02350
.99959
.04594
.99394
.06337
.99799
.03073
.99673
22
39
.01134
.99994
.02379
.999.59
.04623
.99393
.06.566
.99797
.03107
.99671
21
40
.01164
.99993
.02903
.99953
.04653
.99392
.06395
.99795
.08136
.99668
20
41
.01193
.93993
.02933
.99957
.04632
.99390
.06424
.99793
.03165
.99666
19
42
.01222
.99993
.02967
.999.56
.04711
-99339
.064-53
.99792
.08194
.99664
18
43
.01251
.99992
.02996
.99955
.04740
.99333
.06432
.99790
.03223
-99661
17
44
.01230
.99992
.03025
.99954
.04769
.99336
.06511
.99788
.08252
.99659
16
45
.01309
.99991
.03054
.99953
.04798
.99335
.06540
.99736
.08281
-99657
15
46
.01.333
.99991
.0.3033
.99952
.04327
.99333
.06569
.99734
.08310
-99654
14
47
.0136/
.99991
.03112
.99952
.048.56
.99332
.06598
.99782
.033.39
.99652
13
4S
.01396
.99990
.03141
.99951
.04335
.99331
.(,'6627
.99780
.03.363
.99649
12
49
.01425
.99930
.03170
.99950
.04914
.99379
.1)66-56
.99773
.03397
.99647
11
50
.01454
.99939
.03199
.99949
.04943
.99-73
.06635
.99776
-03426
.99644
10
5[
.01433
.99939
.03223
.99943
.04972
.99376
.06714
.99774
.08455
.99642
9
52
.01513
.99939
.03257
.99947
.0-5001
.99375
."6743
.99772
.08434
.996.39
8
53
.01542
.99933
03236
.99946
.05030
.99373
.06773
.99770
.08513
.99637
/
54
.01.571
.99933
.03316
.99945
.0.5059
.99372
.06-02
.99763
.03542
.99635
6
55
.01600
.99937
03345
.99944
.05083
.99370
M<31
.99766
.08.571
.99632
5
56
.01629
.99937
.03374
.99943
.05117
.99369
.06-^60
.99764
.08630
.99630
4
57
.01653
.99936
.0.3403
.99942
.05146
.99-67
.06339
.99762
.08629
.99627
3
5^
.01637
.99936
.03432
.99941
.05175
.99S6S
.06918
.99760
.08653
.99625
2
59
.01716
.99935
.03461
.99940
.05205
.99364
.06947
.997.53
.08687
.99622
1
60
M.
.01745
.99935
.03490
Cosin.
.99939
Sine.
.0-5234
.99363
-06976
.997.56
Sine.
.08716
Coein.
.99619
Sine.
0
M.
Cosin.
Sine.
Cosin.
Sine. Cosin.
8i
P
882 1
87^ 1 863
85° 1
TABLE XTV. x^ATURAL SlIs'ES AND COSINES.
221
0
I
2
3
6
7
8
9
10
11
12
13
14
15
16
17
IS
19
20
21
22
23
24
25
26
27
2S
29
30
31
32
33
34
35
36
37
3S
39
40
41
42
43
44
45
l!)
47
4S
49
50
51
52
53
54
55
56
57
5S
59
60
m7
Sine- Cosin.
.ostTo'
.0S745|
.0S771
.OSSO:}
.0.SS31
.OSSG )
.03S^9
.039 1 S
.OS947
.nS976
.09)1)5
.09[)3l
.09063
.09092
.09121
.091.50
63
.09179
.092)55
.092:^7
.(I926G
.09295
.09321
.09353
.093S2
.09411
.09440
.09469
.0949S
.09527
.09556
.09535
.09614
.09642
.09671
.09700
.09729
.0975S
.097S7
.09816
.09345
.09374
.09903
.09932
.09961
.09990
.10019
.99619
.99617
.99614
.99612
.99609
.99607
.99614
.99602
.99599
.99596
.99594
.99591
.9953 S
.99536
.9953 5
.99530
.99573
.99575
.99572
.99570
.99567
.99564
.99562
.99559
.99556
.99553
.99551
.99543
.99545
.99542
.99540
.99537
.99534
.99531
.99523
.99526
.99523
.99520
.99517
.99514
.99511
.99503
.99506
.99503
.99500
.99497
70
8^
Siue. Cosin. | Sine. : Cosin. Sine. Cosin
.10043
.10077
.10106
.10135
.10164
.10192
.10221
.10250
.10279
.10313
.10337
.10366
.10395
10424
10453
.99494-
.99491
.99433
.99435
.99432
.99479
.99476
.99473
.99470
.99167
.99461
.99461
.99453
.99455
.99452
10453
1043 i
10511
10'.4')
10569
10597
10626
10655
10634
10713
10742
.10771
.1030: 1
.10 52 J
.10 553
.10^37
.10916
.10945
.10973
.11002
.11031
.11060
.11039
.11113
.11147
.11176
.11205
.11234
.11263
.11291
.11320
.11349
.11373
.11407
. 1 1436
.11465
.11494
.11523
.115.52
.11530
.11609
.11633
.11667
.11696
.11725
.11754
.99452
.99119
.99446
.99143
.99440
.99437
.99434
.99131
.99128
.9.)l2l
.99121
.99113
.99415
.99112
.99409
.99406
.99402
.991H9
.99396
.9;) !9 !
.99390
.993>6
.99333
.99330
.99377
.99374
.99370
.99367
.99364
.99360
.99357
.99354
.99351
.99347
.99314
.99341
.99337
.99331
.99331
.99327
.99324
.99320
.99317
.99314
.99310
.99307
Cosin. Sine
8lo
11733
11312
1 1340
11369
11893
.11927
11956
11935
12914
.12013
.12071
.12100
.12129
.121.53
.12137
.99303
.99300
.99297
.99293
.99299
.99236
.99233
.99279
.99276
.99272
.99269
.99265
.99262
.99253
.992.55
Cosin. Sine.
833
12137
12216!
12245
12274
I23i)2
,12331
.12369
.12339
.12413
.12447
.12476
.12504
.12533
.12562
.12591
.12620
.12619
.12673
.12706
.12735
.12764
.12793
.12322
.12351
.12330
.12908
.12937
.12966
12995
.13024
.13053
.1303!
.13110
.13139
.13163
.13197
.13226
.1.3251
1.3233
,13312
.13311
.13370
.1.3399
.13427
.134.56
.13485
.13514
.13543
.13.572
.13600
.13629
.13653
.13637
.13716
.13744
.13773
.13302
13831
.13360
.13339
.J3927
Cosin. Sine
833"
.99251
.99243 ,
.99244 '
.99241) '
.99237
.99233
.99230
.99226,
.99222
.99219
.99215
.99211
.99208
.99204
.99200
.99197
.99193
.99139
.99136
.99132
.99173
.99175
.99171
.99167
.99163
.99160
.99156
.99152
.99143
.99144
.99141
.99137
.99133
.99129
.99125
.99122
.99113
.99114
.99110
.99106
.99102
.99098
.99994
.99091
.99037
.99033
.99079
.99075
.99071
.99067
.99063
.99059
.99055
.99051
.99047
.99043
.99039
.99035
.99031
.99027
13917
13946
13975
14i)'4
14033
14061
11119)
14119
1414s
,14177
,14205
,142.34
.14 263
.14292
.14320
.14349
.14373
.14407
.14136
.1446}
.14493
.14.522
.14551
.14.530
. 14603
.14637
.14666
.14695
.14723
.14752
.14731
.14310
.14333
.14367
.14396
14925
14954
14932
1.5011
1.5040
1.5069
,15097
,15126
,151.55
,15134
.15212
,1.5241
.15270
.15299
.15327
.1.53.56
.1.5335
.15414
.15442
.15471
15500
.15529
.155.57
.15586
.1.5615
.15643
90
.99027
.99023
.99019
.99015
.99)11
.990' )6
.99002
.93993
.9>994
.9>990
.9 -•9 -6
.9>9^2
.93973
.93973
.93969
.9^965
.93961
.93957
.93953
.9-^913
.9^941
.93940
.9-936
.93931
.9>927
.93923
.93919
.93914
.93910
.93906
.939:12
.93397
.93393
.93339
.933S4
.93330
.93376
.93871
.9,3867
.98363
.93.353
.988.54
.98.349
.93,345
.93,341
.93336
.93332
.93827
.93823
.9,3313
.98314
.93309
.93305
.93300
.93796
.93791
.98737
.987,32
.9,8773
.93773
.93769
Sine. I Cosin. M
J5643 .93769
.1.5672 .9,8764
.1.57111 .98760
.15730 .93755
.1575> .93751
.15737 .9.>746
.1.5316 .9,3741
.153451.93737
.1.5373 '.93732
.15902 .93723
.1.5931 .93723
.159.59 .93713
.1.5933 .93714
.16017 .93709
.16046 .93704
.16074 .93700
Cosin. Sine
8I0
16l!l3
16132
16160
16139
1621
,16246
,16275
,16304
.16333
,16361
,16.390
,16419
.16447
.16476
.16505
.16533
.16562
.16591
.16620
.16643
.16677
.16706
.167.34
.16763
.16792
.16320
.16349
.16873
.16906
.16935
.16964
.16992
.17021
.17050
.17073
.17107
,17136
.17164
.17193
.17222
.17250
.17279
.17.303
.17336
.17365
60
59
53
57
56
55
54
53
52
51
50
49
43
47
46
45
44
43
93695
93690
9,3636 42
.9,3631 41
.93676 40
.93671 39
.95667
.93602
.98657
.9,3652
.9.3613
.9,3643
.936:33
.93633
.93629
Cosin.
93624
93619
93614
93609
98604
93600
9,359.'
,9.8.590
,9353i:
,9,3580
,93575
.9,3570
.93565
,93561
,93556
.9,3551
.93.546
.93541
.93536
.93531
.93.526
.93521
.93516
.98511
.93.506
.98501
.9,8496
.93491
.93436
.93431
38
37
36
35
34
33
32
31
30
Sine.
803
29
23
27
26
25
24
23
22
21
20
!9
13
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
J)
M.
222
TABLE XK
i\ATU:iAL bi:sKS A.\D COSINES.
M.
0
103 1
110
130
133
140
M.
Sine.
.17365
Cosin.
.93431
Sine.
Cosin.
.93163
Sine.
Cosin.
.97815
Sine.
Cosin.
.97437
Sine.
Cosin.
.19031
.20791
.22495
.24192
.970:30 60
I
.17393
.93476
.19109
.93157
.20820
.97809
.22523
.97430
.24220
.97023 .59
2
.17422
.93471
.19133
.93152
.20348
.97803
.225.52
.97424
.24249
.97015
53
3
.17451
.93466
.19167
.93146
.20377
.97797
.22.530
.97417
.24277
.97008
57
4
.17479
.93461
.19195
.93140
.20905
.97791
.22603
.97411
.24305
.97001
56
5
.17503
.93455
.19224
.931.35
.20933
.97734
.22637
.97404
.24333
.98994
55
6
.17537
.934.50
.19252
.93129
.20962
.97778
.22665
.97393
.24-362
.96987
54
7
. 1 7565
.93445
.19231
.98124
.20990
.97772
.22693
.97.391
.24390
.96980
53
8
.17594
.93440
.19309
.93113
.21019
.97766
.22722
.97334
.24418
.96973
52
9
.17623
.93435
.193.33
.93112
.21047
.97760
.22750
.97373
.24446
.96966
51
10
.17651
.93430
.19-366
.93107
.21076
.97754
.22773
.97371
.24474
.96959
50
11
.17630
.93425
.19.395
.9310!
.21104
.97748
.22307
.97.365
.24503
.96952
49
12
.17703
.93420
.19123
.93096
21132
.97742
.22335
.97.3-58
.21531
.96945
48
13
.17737
.93414
.19452
.93090
.21161
.97735
.22363
.97351
.24559
.96937
47
14
.17766
.93409
.19431
.93034
.21189
.97729
.22^92
97345
.24587
.96930
46
15
.17794
.934 >1
.19509
.93079
.21218
.97723
.22920
.97:3:38
.24615
.96923
45
16
.17323
.98399
.19533
.93073
.21246
.97717
.22948
.97.331
.24644
.96916
44
U
.173.52
.9.3394
.19.566
.93067
.21275
.97711
.22977
.97325
.24672
.96909
43
13
.17330
.93339
.19.595
.93;i0l
.21303
.97705
.23005
.97318
.24700
.96902
42
19
.17909
.93333
.19623
.93056
.21331
.97693
.230.33
.97311
.24723
.96394
41
20
.17937
.93373
.19652
.93050
.21360
.97692
.23062
.97304
.24756
.96387
40
21
.17966
.93373
.19630
.93044
.21338
.97636
.23090
.97293
.24734
.96880
39
22
.17995
.93363
.19709
.93039
.21417
.97630
.23118
.97291
.24313
.96373
33
23
.13023
.93362
.19737
.930-33
.21445
.97673
.23146
.97234
.24841
.96866
37
21
.130.52
.93357
.19766
.93027
.21474
.97667
.23175
.97278
.24869
.96858
36
25
.13031
.93352
.19791
.93021
.21502
.97661
.23203
.97271
.24897
.96851
35
26
.13109
.93347
.19323
.93016
.215.30
.976.55
.2.3231
.97264
.24925
.96844
.34
27
.131.33
.93341
.19351
.93010
.21559
.97648
.23260
.97257
.24954
.96337
33
2S
.13166
.98336
.19330
.93004
.21587
.97642
.23233
.97251
.24982
.96329
32
1
29
.13195
.93.331
.19903
.97998
.21616
.976.36
.23316
.97244
.25010
.96322
31
30
.13224
.93.325
.199-37
.97992
.21644
.97630
.23:345
.972:37
.25033
.96315
30
31
.132.52
.93.320
.19965
.97937
.21672
.97623
.2.3373
.972.30
.25066
.96807
29
32
.13231
.93315
.19994
.97931
.21701
.97617
.23401
.97223
.25094
.96300
28
33
.13309
.93310
.20022
.97975
.21729
.97611
.2-3429
.97217
.25122
.96793
27
34
.13333
.93.301
.20051
.97969
.21758
.97604
.2:34-58
.97210
.25151
.96786
26
35
.13367
.9.3299
.20079
.97963
.21786
.97593
.23136
.97203
.25179
.96778
25
36
.13:395
.93294
.21103
.97953
.21814
.97592
.2:3514
.97196
.25207
.96771
24
37
.13424
.93238
.201.36
.97952
.21343
.97535
.2.3-542
.97189
.2.52.35
.96764
23
33
.1^.52
.93283
.20165
.97946
.21871
.97579
.2:3571
.97132
.2.5263
.96756
22
39
.13431
.93277
.20193
.97940
.21899
.97573
.2:3.599
.97176
.2.5291
.96749
21
40
.18509
.93272
.20222
.979.34
.21923
.97566
.23627
.97169
.2.5320
.96742
20
41
.13.538
.93267
.20250
.97923
.219.56
.97560
.2:3656
.97162
.25348
.96734
19
42
.18567
.93261
.20279
.97922
.21935
.97553
.23G34
.97155
.25376
.96727
18
43
.13595
.932.56
.20307
.97916
.22013
.97547
.2:3712
.97143
.2.5404
.96719
17
44
.13624
.93250
.20.3.36
.97910
.22041
.97541
.23740
.97141
.2.5432
.96712 16|
45
.13652
.93245
.20364
.97905
.22070
.97534
.23769
.971.34
.25460
.96705
15
46
.18631
.93210
.20393
.97399
.22093
.97523
.2.3797
.97127
.2-5483
.96697
14
47
.13710
.93234
.20421
.97893
.22126
.97521
.23325
.97120
.25516
.96690
13
43
.13733
.93229
.20450
.97337
.221.55
.97515
.2:3353
.97113
.2.5.545
.96632
12
49
.13767
.93223
.20478
.97831
.22183
.97503
2-3332
.97106
.25573
.96675
11
50
.18795
.93213
.20.507
.97375
.22212
.97502
.23910
.97100
.2-5601
.96667
10
51
.13824
.93212
.20.535
.97869
.22240
.97496
.2:3935
.97093
.25629
.96660
9
52
.188.52
.93207
.20.563
.97863
.22263
.97439
.2:3966
.97036
.25657
.96653
8
53
.18331
.93201
.20.592
.97357
.22297
.97433
.23995
.97079
.2.5635
.96645
7
54
.18910
.93196
.20620
.97851
.22325
.97476
.24023
.97072
.25713
.96633
6
55
.18933
.93190
.20649
.97345
.22.353
.97470
.21051
.97065
.25741
.96630
5
56
.18967
.98135
.20677
.97339
.22332
.97463
.24079
.9705S
.25769
.96623
4
57
.18995
.93179
.20706
.97833
.22410
.974.57
.24103
.97051
.25793
.96615
3
53
.19024
.93174
.207.34
.97827
.22438
.974.50
.24136
.97044
.25826
.96603
2
59
.19052
.93163
.20763
.97321
.22467
.97444
.24164
.97037
.25854
.96600
1
60
M.
.19031
.93163
.20791
.97815
.22495
.97437
.24192
.97030
.25832
.96593
0
M.
Cosin.
Sine.
Cosin.
Sine.
Cosin.
Sine.
Cosir.
Sine.
Cosin.
Sine.
1
793 1
780 1
770 1
76C 1
7S
P
1
1
TABLE
XIV.
NATURAL
SIxNES AND COSINES.
ft
<
wa
M.
0
150 1
163 1 170 1
183 1
19a 1
60
Sine.
.253 S2
Cosia.
Sine.
Ccsin. Sine. 1
Cosin.
.95630
Sine.
Cosin.
.95106
Sine.
.32557
Cosin.
.96593
.27.564
.96126
.29237
.30902
,94552
1
.25910
.96535
.27592
.96118
.29265
.95622
-30929
.95097
.32584
.94542
59
2
.25933
.96578
.27620
.96110
.29293
.95613
.30957
.95088
.32612
.94.533
58
3
.25960
.96.570
.27648
.96102
.29321
.9-5605
.30985
.9.5079
.32639
.94.523
57
4
.25994
.90.562
.27676
.96094
.29343
.9.5596
.31012
.95070
.32667
.94514
56
5
.26022
.96555
.27704
.960S6
.29376
.95538
.31040
.95061
.32694
.94504
55
^ 6
.26050
.96547
.27731
.90078
.29404
.95579
.31063
.9-5052
.32722
.94495
54
7
.26079
.96.540
.27759
.96070
.29432
.95571
.31095
.95043
.32749
.94435
53
8
.26107
.965.32
.27787
.96062
.29460
.9-5562
.31123
.95033
.32777
.94476
52
9
.26135
.96.524
.27815
.90f)54
.29487
.95554
.31151
.95024
.32804
.94466
51
10
.26163
.96517
.27843
.96046
.29515
.95545
.31178
.95015
.32332
.94457
50
11
.26191
.90.509
.27871
.96037
.29543
.95536
.31206
.95006
.32859
.94447
49
12
.26219
.96502
.27899
.96029
.29571
.95523
.31233
.94997
.32837
.94438
48
13
.26247
.96494
.27927
.96021
.29599
.95519
.31261
.94988
.32914
.94423
47
14
.26275
.96436
.279.55
.96013
.29626
.95511
.31239
.94979
.32942
.94418
46
15
.26303
.96179
.27983
.96005
.290-54
.95502
.31316
.94970
.32969
.94409
45
16
.26331
.90171
.2301 1
.95997
.29032
.95493
.31344
.94961
.32997
.94.399
44
17
.26359
.96463
.23039
.95939
.29710
.95435
.31-372
.949.52
.3-3024
.94390
43
13
.263S7
.96456
.23067
.9-5981
.29737
.95476
.31399
.94943
.3.3051
.94380
42
19
.20415
.96443
.23095
.95972
.29765
.9.5467
.31427
.94933
.33079
.94370
41
20
.26443
.96440
.23123
.9-5964
.29793
.9.54.59
.31454
.94924
.33106
.94.361
40
21
.26471
.95433
.23150
.95956
.29821
.954-50
.31482
.94915
.33134
.94-351
39
22
.26500
.96425
.28178
.9.5943
.29349
.9-5441
.31510
.94906
.33161
.94.342
33
23
.26523
.96417
.23206
.95940
.29876
.9.5433
.31.537
.94897
.33189
.94332
37
24
.26556
.96410
.232.34
.95931
.29904
95424
.31565
.94383
.33216
.94.322
36
25
.26534
.964 )2
.23262
.95923
.29932
.95415
.31593
.94878
.33244
.94313
35
26
.26612
.96394
.23290
.9.5915
.29960
.9-5407
.31620
.94869
.33271
.94303
34
27
.26610
.963S6
.23318
.95907
.29987
.95398
.31648
.94860
.33293
.94293
33
2S
.26603
.96379
.23346
.9.5393
.30015
.9-5339
.31675
.94851
.33326
.94234
32
29
.26696
.96371
.23374
.9-5390
.30043
.95330
.31703
.94842
.33353
.94274
31
30
.26724
.96363
.23402
.95382
.30071
.95372
.31730
.94832
.33381
; 94264
30
31
.26752
.96355
.28429
.9.5374
.30093
.95363
.31758
.94823
.33408
.94254
29
32
.26780
.96347
.23457
.9-5365
.30126
.95354
.31786
.94814
.33436
.94245
2.8
33
.26303
.96340
.23435
.95357
.30154
.9.5345
.31813
.94805
.3-3463
.94235
27
34
.26836
.96332
.23513
.95849
.30182
.9-5337
.31841
.94795
.3-3490
.94225
26
35
.26364
.96324
.28541
.9-5341
.30209
.95.323
.31868
.94786
.33518
.94215
25
36
.26392
.96316
.28-569
.9.5332
.30237
.9-5319
.31896
.94777
.33.545
.94206
24
37
.2692 J
.96303
.23597
.95324
.30265
.95310
.31923
.94763
.3.3573
.94196
23
3S
.26943
.96301
.23625
.9-5316
..30292
.95301
.31951
.94753
.33600
.94186
22
39
.26976
.96293
.23652
.95807
.30320
.95293
.31979
.94749
.33627
.94176
21
40
.27004
.96235
.23630
.95799
.30348
.95234
.32006
.94740
.33655
.94167
20
41
.27032
.96277
.23703
.9.5791
.30376
.95275
.32034
.94730
.33632
.941.57
19
42
.2706'!
.96269
.237.36
.95732
.30403
.95266
.32061
.94721
.33710
.94147
18
43
.27033
.96261
.23764
.95774
.30431
.95257
..32039
.94712
.33737
.94137
17
44
.27116
.96253
.23792
.95766
.30459
.95243
.32116
.94702
.33764
.94127
16
45
.27144
.96246
.28320
.'^rjlDl
.30436
.95240
.32144
.94693
.33792
.94118
15
46
.27172
.96233
.23847
.95749
.30514
.95231
.32171
.94634
.33319
.94108
14
47
.27200
.90230
.23375
.95740
.30.542
.9-5222
.32199
.94674
.33346
.94098
13
4S
.27223
.96222
.23903
.95732
.30570
.9-5213
.32227
.94665
.33874
.94088
12
49
.27256
.96214
.23931
.95724
..30597
.9.5204
32254
.94656
.3-3901
.94078
11
50
.27234
.96206
.23959
.9.5715
.30625
.95195
.32232
.94646
.33929
.94068
10
51
.27312
,96193
.23937
.95707
.30653
.95186
.32.309
.94637
.33956
.940-58
9
52
.27340
.96190
.29015
.95693
.30630
.95177
.32.337
.94627
.3.39-3
.94049
8
53
.27363
.96182
.29042
.9509)
.30703
.95163
.32364
.94618
.3401 1
.940,39
7
54
.27.396
.96174
.29070
.9-5631
.307.36
.951.59
.32392
.94609
.34038
.94029
6
55
.27421
.96166
.29093
.95673
.30763
.95150
.32419
.94599
.34065
.94019
5
56
.27452
.96153
.29126
.9-5664
..30791
.95142
.32447
.94.590
.34093
.94009
4
57
.27430
.96150
.29154
.95656
..30319
.95133
.32474
.94580
..34,120
.93999
3
5S
.27503
.96142
.29132
.9.5617
.30346
.95124
.32502
.94571
.34147
.9.3989
2
59
.27536
.961.34
.29209
.95639
.SO 574
.95115
.32.529
.94.561
.34175
.93979
1
60
M.
.27.561
Cosin.
.96126
Sine.
.29237
Cosin.
.95630
Sine.
.30902
Cosin.
.95106
.32557
.94.552
.34202
Cosin.
.93969
Sine.
0
M.
Sine.
Cosin.
Sine.
7
40
730 733 1 710
703
5^24
TABLE XIV,
.NATURAL Sl^'ES AND COSINES.
M.
0
303
310
233
333
34:3
Sine.
.31211 >
1 Cosin.
Sine.
.358:37
Cosin.
Sine.
i Cosin.
Sine.
Cosin.
.92050
Sine.
Cosin.
1.91355
M.
.9.3^6 J
.9:3338
.37461
I.9271>
.-39LI73
.40674
60
1
.34229
! .93959
.35564
.93:348
.37483
1.92707
-39100
.92039
.40701 1
.91343
59
2
.34257
1 .93949
.35891
.93:337
.-37515
1.92697
-39127
.92028
.40727
.91:331
58
3
.:342>4
i.9393D
.3.59l>
.93:327
.37.542
.926-6
-.391-53
.92016
.40753
.91319
57
4
34311
1.93929
.33945
.93316
.37.569
.92675
.:39180
.92005
.40780
.91:307
.^6
5
.34339
.93919
.-35973
.933;»6
.37.595
.92664
.39207
.91994
.40=06
.91295
55
6
..34366
.93909
.36000
.9-3295
.:37622
.92653
.:392:34
.919-2
.40-33
.91283
54
7
.31-393
.93899
.36027
.9:32-5
.:37649
.92642
.:-926:)
.91971
.40560
.91272
53
8
.34421
.93889
..360.54
.93274
.37676
.92631
.3;J257
.919:59
.40886
.91260
52
9
.34445
.93879
.36051
.93264
.37703
.9262 )
.39314
.91948
.40913
.91248
51
10
.34475
.93?6J
.3610-
.932-53
.37731
.92609
.3;«41
.919:36
.409-39
.912.36
50
11
.34503
.93559
.-36135
.9:3243
.37757
.92598
.:39.367
.91925
.40966
.91224
49
12
.34530
.93549
.:36162
.932:52
.37784
.92587
.39.394
.91914
.40992
.91212
4S
13
.34357
.93539
..36190
.93222
.:378ll
.92576
.:39421
.91902
.41019
.9120(»
17
14
.345S4
.93529
.:35217
.9321 [
.37-35
.92-365
.3944-
.91891
.41045
.9118>
46
15
.31612
.93519
.:36244
.93201
.37865
.925:54
.-39474
.91879
.41072
.91176
45
16
.34639
.93-09
.:3627l
.93190
.37892
.92.543
.-39501
.9186=
.4109=
.91164
44
17
.34666
.937^9
.:36298
.93150
.37919
.92.3:32
.:3952?
.918:36
.41125
.91152
43
IS
.34694
.937->>9
.36:325
.9316J
.37946
.92321
.-39553
.91845
.41151
.91140
42
19
.34721
.93779
..36:332
.93159
.37973
.92510
.:39-38i
.918:33
.41178
.91128
41
20
.3474S
.93769
.-36379
.93148
.37999
.92499
.39605
.91822
.41204
.91116
40
21
.34775
.937.59
.:36106
.931:37
.35026
.92455
.39635
.91810
.412:31
.91104
.39
22
.34S03
.93748
.:364 34
.93127
.33053
.92477
.-39661
.91799
,412.57
.91092
38
23
.;MS3)
.9373-^
.:334G1
.93116
.3305<i
.92466
.39688
.91787
.41254
.91080
37
1
21
.:34S37
.93728
.:3645-
.931(6
..38107
.924-35
..39715
.91775
.41310
.9106=
36
23
.34SS4
.93718
.36515
.9.3095
.3-134
.92444
.:39741
.917fr4
.413:37
.910:36
35
26
.34912
.93708
.:365I2
.9:3084
.351Gi
.924:32
.:39768
.91752
.41.363
.91044
:34
27
.3193.^
.93695
.-365^15
.9:3074
.3^188
.92421
.:39795
.91741
.41390
.910.32
.33
2S
.34956
.93658
.36536
.9:3!')63
.:38215
.92410
..39-22
.91729
.41416
.91020
32
29
.34993
.93677
;36'323
.93052
.:33241
.92:399
..39848
.91718
.41443
.91008
31
30
.35021
.93667
.:36650
.9.3042
.:38265
.92358
.:39875
.91706
.41469
.90996
30
31
.3504-
.93657
.36677
.93031
.3^^293
.92.377
.39902
.91694
.41496
.90934
29
'
32
.33075
.93647
.337 '4
.9:3020
.3 -322
.92366
.-3992?
.916=3
.41,-22
.90972
2=
33
.33102
.9:;637
.3573!
.9:3:>:il
.3?:349
.92:355
.:39955
.91671
.41549
.90960
27
34
.:!5I30
.93526
.36758
.92999
.3<3:6
.92:343
.39982
.91660
.41.575
.90948
26
i
33
.331-37
.93616
.337<3
.92955
.3-403
.92:3-32
.40005
.9164-
.41602
.909-36
25
1
3n
.33I>4
.9:3606
.:3681 i
.92978
.384:^0
.92:321
.40035
.916:36
.4162=
.90924
24
37
.33211
.93596
.365:3H
.92967
.:38156
.92310
.40062
.91625
.41655
.9091 1
23
3i
.35230
.9:3535
.:36567
.92956
.3?483
.92299
.40083
.91613
.41651
.90899
22
39
.-35266
.9:3-575
.:36594
.92915
.35510
.92287
.40115
.91601
.41707
.90337
21
40
.35293
.93565
.:36921
.929:J5
.:38537
.92276
.40141
.91-590
.4173-1
.90375
20
41
.35320
.9:3355
.3394N
.92924
.38.564
.92265
.4016=
.91:373
.41760
.90363
19
42
.35:347
.9:3.544
..36975
.9291:^
.35.591
.922.54
.40195
.91566
.41737
.90=51
18
43
.35375
.9.3->34
.37002
.92902
.35617
.92243
.40221
.91:555
.41313
.90839]
17
•
44
..35402
.93-524
.37029
.92-92
.35644
.92231
.40243
.91.543
.41840
.90826 16
45
.35429
.9:3514
.370.56
.92881
.:35671
.92220
.40275
.91-531
.41866
.908141 15
46
..35456
.93503
.37033
.92-70
.35698
.92209
.40.301
.91519
.41892
.903021
14
47
.3:5454
.93493
.37110
.92559
.38725
.92193
.40:325
.91:31)8
.41919
.90790!
13
4S
..35511
.93453
.:371.37
.92849
.337.52
.92186
.40:3.55
.91496
.41945
.90773!
12
49
.3553S
.93472
.37164
.92-38
.38778
.92175
4J:381
.914=4
.41972
.907661
11
50
.33565
.93462
.-37191
.92-27
.:38805
.92164
.40408
.91472
.41998
.907.53
10
51
.33592
.93452
.37215
.92.316
-388-32
.92152
.4114.34
.91461
.42024
.90741
9
52
.35619
.9:3411
.37245
.92805
-335.59
.92141
.40461
.91449
.42051
.90729
8
53
.33647
.93431
.37272
.92794
.38886
.92130
.40488
.914:37
.42077
.9(J717
7
54
.-35674
.93420
.37299
.92754
.33912
.92119
.40514
.91425
.42104
.90704
6
55
.35701
.93410
.37.326
.92773
.33939
.92107
.40541
.91414
.421.30
.90692
5
56
.3572S
.93400
.37353
.92762
.33966
.92096
.40567
.91402
.421:56
.90680
4
57
.■ir>/00
.a3389
.37:380
.92751
.-38993
.92055
.40594
.91390
.42133
.90663
3
53 1
.35782
.93379
.37407
.92740
.39020
.92073
.4)621
.91378
.42209
.9CK555
2
59
.35810
.9-3-363
..374.34
.92729
.39046
.92062
.40647
.91.366
.422:35
.£0643
I
60 ;
M. j
1
.35837
.933-58
Sine.
.:37461
Cosin.
.9271-
Sine.
.3£073
Cosin.
.920-50
.40674
.9ia35
Sine.
.42262
Cosin.
.90631
Sine.
0
M.
Cosin.
Sine.
Cosin.
693 1
683 1
673 1
663 1
653
TABLE
XIV.
NATURAI
. SINES AND COSINES
>.
n
M.
0
35^
3G3
27-
38-^
39^
M.
60
Sine.
.42262
Oosia.
Sine.
.43337
Ccsin.
■89379
Sine.
Gosin.
Sine.
Cosin.
.88295
Sine.
.48481
Cosin.
.90631
.43399
.89101
.46947
.87462
I
A22^S
.90618
.43363
.39S67
.4.5425
' .89087
.46973
.88281
.43506
.87443
59
2
.42315
.90306
.43389
.39354
.45451
.89074
.46999
.88267
.48532
.87434
53
3
.42311
.90594
.43916
.S9S41
.45477
.89061
.47024
.83254
.48557
.87420
57
4
.42367
.905S2
.43942
.89S23
.45503
.89048
.47050
.8321(1
.48583
.87406
56
5
.42394
.90569
.43963
.89316
.45529
.890.35
.47076
.88226
.43608
.87.391
55
6
.42120
.90557
.43991
.89803
.45.554
.89021
.47101
.83213
.43634
.87377
54
7
42446
.90545
.44020
.89790
.45530
.89003
.47127
.88199
.48659
.87363
53
8
.42173
.90532
.44046
.89777
.45606
.83995
.47153
.88185
.48634
.87349
52
9
.42199
.90520
.44072
.89764
.45632
.88981
.47178
.88172
.48710
.87335
51
10
.42525
.90507
.44093
.89752
.45658
.83968
.47204
.83158
.48735
.87321
50
11
.42.552
.90495
.44124
.89739
.45634
.88955
.47229
.88144
.48761
.87306
49
12
.42573
.90433
.14151
.89726
.45710
.88942
.47255
.88130
.48786
.87292
48
13
.42604
.90470
.44177
.89713
.45736
.88923
.47231
.88117
.4881 1
.87278
47
14
.42631
.90453
.44203
.89700
.45762
.88915
.47306
.83103
.43837
.87264
46
15
.42657
.90146
.44229
.89687
.45787
.88902
.47332
.88089
.48862
.87250
45
16
.42633
.904.33
.412.55
.89674
.45313
.88883
.47358
.88075
.48868
.87235
44
17
.42709
.90421
.44281
.89662
.45339
.88375
.47333
.88(162
.48913
.87221
43
IS
.4273'';
.90403
.44307
.89619
.45865
.88862
.47409
.88043
.43938
.87207
42
19
.42762
.9:)396
.44.333
.89636
.45891
.83848
.47431
.830.34
.43964
.87193
41
20
.427-:;S
.90333
.44359
.89623
.45917
.88835
.47460
.88020
.48989
.87178
40
21
.42?15
.90371
.44335
89610
.45942
.88822
.47486
.83006
.49014
.87161
39
22
.42341
.90358
.44411
.89.597
.45963
.83308
.4751 1
.87993
.49040
.87150
33
23
.42S67
90.346
.44437
.89584
.45994
.83795
.47.537
.87979
.49065
.87136
37
24
.42894
.90334
.44464
.89571
.46020
.88782
.47562
.87965
.49090
.87121
36
2.3
.42920
.90.321
.44490
.895.33
.46046
.83768
.47588
.87951
.49116
.87107
35
26
.42946
.90309
.44516
.89545
.46072
.88755
.47614
.87937
.49141
.87093
34
27
.42972
.90296
.44542
.89532
.46097
.88741
.47639
.87923
.49166
.87079
33
23
.42999
.90234
.44.368
.89519
.46123
.88728
.47665
.87909
.49192
.87064
32
29
.43025
.90271
.44594
.89506
.46149
.88715
.47690
.87896
.49217
.87050
31
30
.43051
.902.59
.44620
.89493
.46175
.88701
.47716
.87882
.49242
.87036
30
31
.4.3077
.90246
.44646
.89480
.46201
.88688
.47741
.87868
.49263
.87021
29
32
.43104
.90233
.44672
.89467
.46226
.83674
.47767
.878.54
.49293
.87007
28
33
.431.30
.90221
.44693
89454
.46252
.88661
.47793
.87840
.49318
.86993
27
34
43156
.90208
.44724
.89441
.46278
.88647
.47818
.87826
.49.344
.86978
26
35
.43182
.90196
.44750
.89423
.46301
.88634
.47844
.87812
.49.369
.86964
25
36
.4.3209
.90183
.44776
.89415
.46330
.88620
.47869
.87798
.49394
.86949
24
37
.43235
.90171
.44802
.89402
.46355
.88607
.47395
.87784
.49419
.86935
23
3S
.43261
.90153
.44323
.89339
.46381
.83593
.47920
.87770
.49445
.86921
22
39
.43237
.90146
.44354
.89376
.46407
.83580
.47946
.87756
.49470
.86906
21
40
.43313
.90133
.44380
.89363
.46433
.88566
.47971
.87743
49495
.86392
20
41
.43340
.90120
.44906
.89350
.46453
.88553
.47997
.87729
.49521
.86878
19
42
.43366
.90103
.44932
.89337
.46484
.88.539
.43022
.87715
.49.546
.86863
18
43
.43392
.90095
.44958
.89321
.46510
.83526
.43043
.87701
.49571
.86349
17
44
.43413
.90082
.44934
.89311
.465.36
.83512
.48073
.87687
.49596
.86834
16
45
.43145
.90070
.45010
.89293
.46561
.88499
.43099
.87673
.49622
.86320
15
46
.43471
.90057
.45036
.89235
.46587
.83435
.48124
.87659
.49647
.86305
14
47
.43197
.90045
.45052
.89272
.46613
.83472
.481.50
.87645
.49672
.86791
13
4S
.43523
.90032
.4:5038
.89259
.46639
.88453
.48175
.87631
.49697
.86777
12
49
.43549
.90019
.45114
.89245
.46664
.88445
.48201
.87617
.49723
.86762
11
50
.43575
.90007
.45140
.89232
.46690
.83431
43226
.87603
.49743
.86743
10
51
.4.3602
.89994
.45166
.89219
.46716
.88417
.48252
.87589
.49773
.86733
9
52
.43623
.89931
.45192
.89206
.46742
.88404
.48277
.87575
.49793
.86719
8
53
.436.54
.89963
.45218
.89193
.46767
.83390
.48303
.87.561
.49824
.86704
7
54
.4.3630
.89956
.45243
.89180
.46793
.88377
.48328
.87546
,49349
.86690
6
55
.43706
.89943
.4.5269
.89167
.46319
.83363
.483.54
.87532
.49374
.86675
5
56
.43733
.89930
.45295
.89153
.46844
.88349
.43379
.87518
.49899
.86661
4
57
.43759
.89918
.45321
.89140
.46870
.88336
.48405
.87504
.49924
.86646
3
63
.43785
.899*5
.4.5.347
.89127
.46896
.83322
.48430
.87490
.49950
.86632
2
59
.43311
.89392
.45373
.89114
.46921
.88308
.48456
.87476
.49975
.86617
1
60
.43337
.89379
SineJ
.45399
.89101
.46947
.88295
Sine.
.48481
.87462
Sine.
.50000
.86603
0
M.
Cosln.
Cosln.
Sine.
Cosln.
Cosin.
Cosin.
Sine.
640 1
030 1
eao I
610 1
603 1
11
226
TABLE XIV. NATURAL SINES AND COSINES.
~0
303 1
310 1
333 1
333 1
343
M.
60
Sine.
.50000
Cosin.
Si.:e.
Co-iii.
Slue.
Ccsin.
Sine.
Cosin.
Sine.
Cosin.
.86603
51504
.85717
.52992
.84305
.54464
.3:3367
.55919
.>29J4
I
.50J25
.86533
51529
.85702
.53017
.84789
.544-5
,83351
.55943
.82337
59
2
.5 J 150
.86573
51554
.85687
.53041
.84774
.-,4513
.83335
.55963
.82371
58
3
.50076-
.56559
.51579
.85(72
.53066
.^4759
.54537
.33319
.55992
.82355
57
-»
50101
.86544
.51604
.85657
.53091
.34743
.54561
.83304
..56016
.82539
56
.?
.50126
.86530
.51623
.-55642
.53115
.3472>
.."^45-6
.33788
.56040
,82322
55
6
.50151
.86515
.51653
.35627
.53140
.34712
.54610
.33772
.56064
,82306
54
7
..50176
.86501
.51678
.85612
.53h:4
.34fi97
.546:35
.83756
.56033
.82790
53
5
.50201
.86486
.517(3
.35597
.53189
.84631
.54659
.8:3740
.56112
.82773
52
9
.50227
.86471
.51728
.^55>i
5.3214
.54666
,54653
.33724
.56136
.82757
51
10
.50252
.86457
.51753
.'^."i-'ifl,
.53,-38
.34650
..54703
.8370-
.56160
.82741
50
11
50277
.86442
.51778
.85551
.53263
.^4635
.=547.32
.83692
..56134
.82724
49
12
.50302
.86427
.51803
.85536
.53288
.84619
.547.56
.8.3676
..56203
,82708
48
Vo
.50327
.86413
.51323
.85521
.53312
.3 16; 4
.54731
.-366 1
.562.32
.82692
47
14
.50352
.8639S
.513.52
.855 16
.53337
.8153-
.&4S05
.3:3645
.56256
.82675
46
15
.50377
.86334
.51377
.854^;
..53J61
.84573
.54829
.3:3629
.56280
.82659
45
16
.50403
.86.369
.51902
.85476
.53336
.-^15.'>7
.543.54
.s;3613
..56:3)5
.82643
44
17
.5042-
.863.54
.51927
.8.5461
..53411
.34rv42
.54? 78
.83597
.56:329
.82626
43
IS
.50453
.86340
.519.52
.35416
.m435
.34 '.if;
.54902
.S3531
.56353
.82610
42
19
.5047S
.86325
.51977
.3)4:1
.53460
.84511
.54927
.83565
.56377
.82593
41
20
.50503
.8o310
.52002
.85!: 6
.5.34>l
.84495
.54951
'.3:3.549
..56401
.82577
40
21
.5052S
.8rt-2:'5
.52026
.3.5401
.53509
.814-0
..54975
.8:3533
..56425
.82561
39
22
..50.5.53
.86231
..52)51
.35335
.53531
.34l6i
.54999
.83517
.56449
.82544
33
23
.50578
.86266
.52076
.85370
.53553
.3411-
.5.5024
.83501
.56473
.82523
37
21
.50603
.862,31
..52101
.85355
.53.533
.34433
.55043
..S34-5
.56497
.82511
36
25
..50623
.86237
.521 6
.8534 .
.53607
.84417
.5.5072
.33469
.56521
.82495
35
26
.506.54
.862i2
.52151
.85325
.53632
.344' 2
.55097
.3-3453
..56545
.82473
.34
27
.50679
.86207
..52175
.85310
..".3656
.-<!!- 6
.55121
.Si43r
.56569
.82462
33
2>
.50701
.86192
..52200
.35294
.53631
.S4370
.55145
.8:3421
.56593
.82446
32
29
.50729
.86178
.52225
.85279
.53705
.84:3.55
.55169
.3:3405
.56617
.82429
31
30
.50754
.86163
.52250
.85264
.53730
.84339
.55194
.83389
.56641
.82413
30
31
.50779
.86143
.52275
.85249
.537.->l
.34.324
.55218
.33373
.56665
.82396
29
32
..50304
.86133
.52299
.3.5231
.53779
.34313
.55242
.83356
.56639
.82330
23
33
.50329
.86119
.52321
.8.5213
.53>04
.84292
.5.5266
.83340
.56713
.82363
27
3;
.503:54
.86101
.5234:^
.85ai3
.5332-
.64277
.5529[
.83:324
.567.36
.82:347
26
35
.50379
.86039
.52374
.35133
.53353
.84261
.55315
.83303
.56760
.82330
25
36
.50904
.86074
.52.399
.s-)l7;i
.53377
.34245
.55:3:39
.83292
.56784
.82314
24
37
..50929
.86059
.52423
.85157
.53902
.^42:30
.55363
.33276
.56303
.82297
23
3>
.50954
.85045
.52443
.85142
.53926
.84214
.55333
.8.3260
.563.32
.82231
22
39
.50979
.85030
.52473
.35127
.53951
.81193
.55412
.3:3244
.56356
.82264
21
40
.51004
.86015
.5243>
.85112
.53975
.34132
.5.5436
.■^3223
.56330
.82243
20
41
.51029
.86000
.52.522
.85096
.54000
.81167
.5.5460
.8.32 U
.56904
.82231
19
42
.51054
.85935
.52547
.85081
.54024
.84151
.55434
:83195
.56923
.32214
18
43
.51079
.8597(1
.52572
.3.5006
.54049
.84135
.5.5509
.83179
.569.52
.8219?
17
44
.51104
.35956
..52597
.850.51
.54073
.84120
.5.55.3.3
.83163
.56976
.82181
16
45
.51129
.8594 1
.52621
.85035
.54097
.84104
.55557
.83147
.57000
.82165
15
46
.511.54
.85926
..52646
.35020
.54122
.84033
..5.5.531
.83131
.57024
.82143
14
47
.51179
.&5911
..52671
.85005
.54146
.84072
..5.5605
.83115
.57047
.82132
13
4S
.51204
.85396
..52696
.84939
.54171
.84057
.556:30
.83098
.57071
.32115
12
49
.51229
.85331
..52720
.34974
.54195
.84041
..55654
.83082
.57095
.82098
11
50
.51254
.35366
.52745
.84959
..54220
.34025
.55678
.3:3066
.57119
.82082
10
51
.51279
.3^5-51
.52770
.34943
..54244
.34009
.55702
.83050
.57143
.82065
9
52
.51304
.85336
.52794
.8492?
.54269
.33994
.55726
.8.30.34
.57167
.82043
8
53
.51329
.85321
..52319
.34913
..54293
.83973
.55750
.83017
.57191
.820.32
7
54
.51.354
.85306
..52344
.■34397
.54317
.83962
. .55775
.8.3001
.57215
.82015
6
55
.51379
.85792
..52369
.84332
.54342
.83946
..55799
.82935
.572.33
.81999
5
56
.51404
.85777
..52393
.84366
.54366
.839.30
.55823
.82969
.57262
.81932
4
57
.51429
.85762
.52918
.84351
.54391
.83915
.5.5347
.82953
.57286
.81965
3
53
.51454
.85747
..52943
.84836
..54415
.83399
.5=5871
.82936
.57310
.81&49
2
59
.51479
.85732
.52967
.84S20
..54440
.83383
.55895
.82920
.57334
.81932
1
60
M.
.51504
.85717
Sine.
.52992
Cosin.
.84805
Sine.
.54464
Coain.
.83367
.55919
.82904
.57353
.81915
0
M.
Cosin.
Sine.
Cosin.
Sine.
Cosin.
Sine.
593 1
5
33
'—^
573 <
563
553
TABLE XIV. NATURAL SINES AND COSINES.
227
M.
C
350
3GO
370 1 383
390
M.
) 60
Sine.
.573.jt:
Cosin
Sine.
Cosin.
.80902
Sine.
Cosin.
.79364
Sine.
Cosin.
.78301
Sine.
.62935
Cosin
.777K
.«1915
.5S77<J
.60185
.6156e
1
.57381
.81399
.58302
.80S8.')
.6020-
.79346
.6153£
.78783
.62955
.7769t
) 69
2
.57405
.81382
.5S32G
.80367
.6022.5
.79329
.61615
.78765
.62977
.77676
] 58
3
.57420
.81865
.58349
.80350
.60251
.79311
.61635
.78747
.63000
.7766t
) 57
4
.57453
.81843
.58873
.80333
.60274
.79793 J .61953
.78729
.63022
.77641
56
5
.57477
.81832
.58896
.80816
.60298
.7977G
.61631
.78711
.G3045
.77625
55
G
.57501
.81815
.58920
.80799
.60.321
.79758
.61704
.76694
.630G6
.7760£
54
7
Sural
.81793
.58943
.8078-2
.60344
.79741
.6172G
.78676
.6.3090
.7758C
53
8
.57:-i-
.81782
.58967
.30765
.G0367
.79723
.61749
.78653
.63113
.77563
52
9
.57572
.81765
.58990
.80743
.60390
.79706
.61772
.78640
.63135
.7755C
51
10
."j/o'.iG
.81748
.59014
.80730
.60414
.79638
.61795
.78622
.63158
.77531
50
11
.570 19
.81731
.59037
.80713
.60437
.79671
.61818
.78604
.63180
.77512
49
12
.57613
.81714
.59061
.80696
.60460
.796.53
.61841
.78586
.63203
.77494
48
13
.57667
.81698
.59084
.80679
.60483
.79635
.61864
.78563
.63225
.77476
47
14
.57691
.81631 S.. 59108
.80662
.60506
.79618
.61887
.78550
.63248
.77458
46
15
.57715
.81664
.59131
.80644
.60529
.79600
.62909
.78532
.63271
.77439
45
16
.57733
.81647
.59154
.80627
.60053
.79583
.61932
.78514
.63293
.77421
44
17
.57762
.81631
..59173
.80610
.60576
.79.565
.61955
.78496
.63316
.77402
43
13
.57786
.81614
.59201
.80593
.60599
.79547
.61978
.78478
.63333
.77384
42
19
.57^10
.81597
.59225
.80576
.60622
.79530
.62001
.764G0
.63.361
.77366
41
20
.575.33
.81580
.59248
.80553
.60645
.79512
.62024
.78442
.63333
.77347
40
21
.57857
.ai563
.59272
.80541
.60G63
.79494
.62046
.78424
.63406
.77329
39
22
.57881
.81546
.59295
.80524
.60691
.79477
.62069
.78405
.63423
.77310
38
23
.57904
.81530
.59318
.80507
.60714
.79459
.62092
.78337
.6.3451
.77292
37
24
.57923
.81513
.59312
.80489
.60738
.79441
.62115
.78369
.63473
.77273
36
25
.57952
.81496
.59365
.80472
.60761
.79424
.62138
.78351
.63496
.772-55
35
26
.57976
.81479
.59389
.80455
.60784
.79406
.62160
.78333
.63518
.77236
34
27
.57999
.81462
.59412
.80433
.60807
.79338
.62183
.78315
.63540
.77218
33
23
.58023
.81445
.594.36
.80420
.60830
.79371
.62206
.78297
.63563
.77199
32
29
.53017
.81423
.59459
.80403
.60853
.79353
.62229
.78279
.63535
.77181
31
30
.53070
.81412
.59482
.80336
.60876
.79335
.62251
.78261
.63608
.77162
30
31
.53094
.81395
.59506
.80363
.60899
.79318
.62274
.78243
.6.3630
.77144
29
32
.53118
.81378
.59529
.80351
.60922
.79300
.62297
.76225
.63653
.77125
28
33
.53141
.81361
.59552
.803.34
.60945
.79232
.62320
.78206
.63675
.77107
27
34
.58165
.81344
.59576
.80316
.60963
.79264
.62.342
.78158
.63693
.77088
26
35
.53189
.81327
.59599
.80299
.60991
.79247 .62365
.78170
.63720
.77070
25
36
.53212
.81310
.59622
.80282
.61015
.79229 .62-3881
.78152
.63742
.77051
24
37
.58236
.81293
.59646
.80264
.61033
.79211
.62411
.78134
.63765
.77033
23
33
.58260
.81276
.59669
.80247
.61061
.79193
.62433
.78116
.63787
.77014
22
39
.58283
.81259
.59693
.80230
.61084
.79176
.62456
.73093
.63310
.76996
2!
40
.53307
.81242
.59716
.80212
.61107
.79158
.62479
.78079
.G3S32
.76977,
20
41
.533.30
.81225
.59739
.80195
.61130
.79140
.62502
.78061
.63354
.76959
19
42
.58354
.81208
.59763
.80178
.61153
.79122
.62524
.78043
.63377
.76940
18
43
.58378
.81191
.59786
.80160
.61176
.79105
.62547
.78025
.6.3399
.76921
17
44
.53401
.81174
.59809
.80143
.61199
.79087
.62570
.73007
.63922
.76903
16
45
.58425
.81157
.59332
.80125
.61222
.79069
.62.592
.77983
.6.3944
.76884
15
46
.53449
.81140
.59856
.80108
.61245
.79051
.62615
.77970
.63966
.76366
14
47
.53472
.81123
.59879
.80091
.61263
.79033
.62633
.77952
.639>9
.76347
13
48
.53496
.81106
.59902
.80073
.61291
.79016
.62660
.77931
.64011
.7GS2S
12
49
.53519
.81089
.59926
.80056
.61314
.78998
62633
.77916
.64033
.76810
11
50
..58543
.81072
.59949
.80038
.61337
.78^30
.62706
.77897
.64056
76791
10
51
.53567
.81055
.59972
.80021
.61360
.78962
.62728
.77879
.64078
76772
9
52
.58590
.81038
.59995
.60003
.61333
.78944
.62751
.77661
.64100
76754
8
53
.58614
.81021
.60019
.79986
.61406
.78926
.62774
.77843
.64123
76735
7
54
.586.37
.81004
.60042
.79963
.61429
.78908
.62796
77824
.64145
76717
6
55
.58661
.80987
.60065
.79951
.61451
.78891
.62319
77806
.64167
76698
5
56
.58634
.80970
.60089
.79934
.61474
.78873
.62842
77788
.64190
76679
4
57
.58708
.80953
.60112
.79916
.61497
78855
.62864
77769
.64212
76661
3
58
.58731
.80936
.60135
.79399
.61.520
78837
.62887
77751
.64234
76642
2
59
58755
.80919
.60158
.79381
.61543
78819
.62909
77733
.64256 .
76623
1
60
M.
58779
CJosin.
80902
.60182
79864
.61566
78801
.62932
Cosin.
77715
.64279 .
Cosin.
76604
0
Sine. Cosin. 1
Sine.
Cosin.
Sine.
Sire.
Sine. I
540 1 530 1
sao r
510 1
500 1
228
TAB].E XIV. NATURAL SINES AND COSINES.
M.
0
4:03
4:10
4:30
4:33
4:40
M.
60
Sine.
.64279
Cosin.
Sine.
Cosin.
Sine.
Cosin.
Sine.
Cosin.
Sine.
Cosin.
.76604
.65606
.75471
.66913
.74314
.68200
.73135
.69466
.719.S4
1
.64301
.76586
.65623
.75452
.66935
.74295
.68221
.73116
.69437
.71914
59
2
.64323
.76567
.65650
.75433
.66956
.74276
.68242
.73096
.69503
.71894
53
3
.64346
.76543
.65672
.7.5414
.66978
.74256
.68264
.73076
.69.529
.71873
57
4
64363
.76.530
.65694
.75395
.66999
.74237
.63235
.73356
.69549
.71853
56
5
.64390
.76511
.65716
.75375
.67021
.74217
.68.306
.73036
.69570
.71833
55
6
.64412
.76492
.65733
.75.356
.67043
.74193
.68327
.73016
.69591
.71813
54
7
.64435
.76473
.65759
.75337
.67064
.74173
.68349
.72996
.69612
.71792
53
8
.64457
.76455
.65731
.75313
.67086
.74159
.68.370
.72976
.69633
.7F72
52
9
.64479
.76436
.65303
.75299
.67107
.74139
.68391
.72957
.69654
.71752
51
10
.64501
.76417
.6582.5
.75280
.67129
.7412)
.63412
.72937
.69675
.71732
50
11
.64524
.76393
.6.5347
.75261
.67151
.74100
.63434
.72917
.69696
.71711
49
12
.64546
.76330
.65369
.75241
.67172
.74080
.63455
.72897
.69717
.71691
43
13
.64568
.76361
.65391
.75222
.67194
.74061
.63476
.72377
.69737
.71671
47
14
.64590
.76342
.65913
.75203
.67215
.74041
.68497
.72857
.69753
.71650
46
15
.64612
.76323
.65935
.75134
.67237
.74022
.63513
.72837
.69779
.71630
45
16
.646.35
.76304
.65956
.75165
.67258
.74002
.685.39
.72317
.69300
.71610
44'
17
.64657
.76286
.65973
.75146
.67230
.73933
.68561
.72797
.69321
.71590
43
13
.64679
.76267
.66000
.75126
.67301
.73963
.68532
.72777
.69342
.71569
42
19
.64701
.76248
.66022
.75107
.67323
.73944
.68603
.72757
.69862
.71549
41
20
.64723
.76229
.66044
.75083
.67344
.73924
.68624
.72737
.69883
.71.529
40
21
.64746
.76210
.66066
.75069
.67366
.73904
.68645
.72717
.69904
.71503
39
22
.64763
.76192
.66088
.75050
.67387
.73835
.68666
.72697
.69925
.71483
38
23
.64790
.76173
.66109
.75030
.67409
.73865
.68683
.72677
.69946
.71463
37
24
.64812
.76154
.66131
.75011
.67430
.73346
.68709
.72657
.69966
.71447
36
25
.64834
.76135
.66153
.74992
.67452
.73826
.68730
.72637
.69937
.71427
35
26
.643.56
.76116
.66175
.74973
.67473
.73806
.68751
.72617
.70003
.71407
34
27
.64873
.76097
.66197
.74953
.67495
.73787
.68772
.72597
.70029
.71386
33
23
.64901
.76073
.66213
.74934
.67516
•73767
.68793
.72577
.70049
.71366
32
29
.64923
.76059
.66240
.74915
.675.33
.73747
.63814
.72557
.70070
.71345
31
30
.64945
.76041
.66262
.74896
.67559
.73723
.63835
.72537
.70091
.71325
30
31
.64967
.76022
.66234
.74876
.67530
.73703
.63357
.72517
.70112
.71305
29
32
.64939
.76003
.66306
.74857
.67602
.73683
.63373
.72497
.70132
.71234
23
33
.6.5011
.75984
.66-327
.74838
.67623
.7.3669
.63399
.72477
.70153
.71264
27
34
.6.5033
.75965
.66349
.74818
.67645
.73649
.68920
.72457
.70174
.71243
26
35
.65055
.75946
.66371
.74799
.67666
.7.3629
.68941
.72437
.70195
.71223
25
36
.6.5077
.75927
.66393
.74730
.67633
.73610
.63962
.72417
.70215
.71203
24
37
.65100
.75908
.66414
.74760
.67709
.7.3.590
.63933
.72397
.702.36
.71182
23
33
.65122
.75389
.66436
.74741
.67730
.73570
.69004
.72377
.70257
.71162
22
39
.65144
.75370
.66458
.74722
.67752
.73551
.69025
.72357
.70277
.71141
21
40
.65166
.75851
.66480
.74703
.67773
.73531
.69046
.723.37
.70293
.71121
20
41
.65133
.75832
.66501
.74683
.67795
.73511
.69067
.72317
.70319
.71100
19
42
.65210
.75813
.66523
.74664
.67316
.73491
.69033
.72297
.703.39
.71030
18
43
.65232
.75794
.66545
.74644
.67337
.73472
.69109
.72277
.70360
.71059
17
44
.652.54
.75775
.66566
.74625
.67359
.73452
.69130
.72257
.70381
.71039
16
45
.65276
.75756
.66533
.74606
.67330
.73432
.69151
.72236
.70401
.71019
15
46
.65298
.75733
.66610
.74536
.67901
.73413
.69172
.72216
.70422
.70993
14
47
.65320
.75719
.66632
.74567
.67923
.73393
.69193
.72196
.70443
.70978
13
48
.65.342
.75700
.66653
.74.543
.67944
.73373
.69214
.72176
.70463
.70957
12
49
.65364
.75680
.66675
.74523
.67965
.73353
.69235
.72156
.70434
.70937
11
50
.6.5336
.7.5661
.66697
.74509
.67987
.73333
.69256
.72136
.70505
.70916
10
51
.65403
.75642
.66718
.74439
.68008
.73314
.69277
.72116
.70525
.70396
9
52
.65430
.75623
.66740
.74470
.63029
.73294
.69293
.72095
.70546
.70875
8
53
.6.5452
.7.5604
.66762
.74451
.68051
.73274
.69319
.72075
.70567
.70355
7
54
.65474
.75535
.66733
.74431
.68072
.73254
.69340
.72055
.70537
.703.34
6
55
.65496
.75566
.66805
.74412
.68093
.73234
.69361
.72035
.70603
.70313
5
56
.65518
.75547
.66327
.74392
.63115
.73215
.69382
.72015
.70623
.70793
4
57
.65540
.75528
.66343
.74373
.63136
.73195
.69403
.71995
.70649
.70772
3
58
.65562
.75509
.66370
.74353
.68157
.73175
.69424
.71974
.70670
.70752
2
59
.65534
.75490
.66891
.74334
.63179
.73155
.69445
.71954
.70690
.70731
1
60
M.
.65606
.75471
.66913
.74314
.68200
.73135
.69466
.719^4
.70711
.70711
0
Cosin.
Sine.
Cosin. Sine.
Cosin. 1
Sine. Cosin. 1
Sine.
Cosin.
Sine.
493 1
4:83 473 1 4:63 |
4:53 1
Tvnr^^
TABLE XV.
NATURAL TANGENTS AND COTANGENTS
230 TABLE XV. NATURAL TANGENTS AMU COTANGENTS.
M.
0
03 1
1
.0
ao 1
30
M.
60
Tang.
Cotang.
Tang.
Cotang.
Tang.
Cotang.
23.6363
Tang.
Cotang.
.00000
Infinite.
.01746
57.2900
.03492
.0.5241
19.0811
1
.00029
3437.75
.01775 1
56.3506
.03521
23.3994
.05270
18.9755
59
2
.00053
1713.57
.01304 i
5.5.4415
.03550
28.1664
.'05299
18.8711
58
3
.00087
1145.92
.01333
54.. 56 13
.03579
27.9372
.05328
18.7678
57
4
.00116
859.436
.01362
53.7036
.03609
27.7117
.05357
18.66.56
56
5
.00145
637.549
.01391
52.8321
.03633
27.4899
05387
18.5645
55
6
.00175
572.957
.01920
52.0307
.03667
27.2715
/J5il6
.•e.4645
54
7
.00204
491.106
.01949 i
51.3032
.03696
27.0566
.05445
18.3655
53
8
.00233
429.713
.01978
50.5485
.03725
26.5450
.05474
18.2677
52
9
.00262
33L971
.02007
49.3157
.03754
26.6367
.05503
18.1708
51
10
.00291
343.774
.02036
49.1039
.03783
26.4316
.05533
13.0750
50
11
.00320
312.521
.(12066
43.4121
.03312
26.2296
.05562
17.9502
49
12
.00349
2-;6.473
.02095
47.7395
.0.3342
26.0.307
0.5591
17.5563
48
13
.00373
264.441
.02124
47.0353
.03371
25.3348
.05620
17.7934
47
14
.00407
245.552
.02L53
46.4439
.03900
25.6413
.05649
17.7015
46
15
.00436
229.132
.02132
45.3294
.03929
2.5.4517
.05678
17.6106
45
16
.00465
214.858
.O23I0
45.2261
.03958
25.2644
.05708
17.5205
44
17
.00495
202.219
44.6336
.03937
25.0798
.05737
17.4314
43
13
.00524
190.934
.02269
44.0661
.04016
24.8973
.05766
17.3432
42
19
.00553
130.932
.02298
43.5031
.04046
24.7135
.05795
17.2553
41
20
.00532
171. 3S5
.02323
42.9641
.04075
24.5413
.05824
17.1693
40
21
.00611
163.700
.02357
42.4335
.04104
24.3675
.0.5354
17.0537
39
22
.00640
] 56.259
02336
41.9153
.04133
24.1957
.05833
16.9990
38
23
.00669
149.465
.02415
43.4106
.01162
24.0263
.0.5912
16.9150
37
24
.00693
143.23?
.f'2444
40.9174
.04191
23.8593
.05941
16.8319
36
25
.00727
1^7.507
152.219
.02473
40.4358
.04220
23.6945
.05970
16.7496
35
26
.00756
.02502
39.9655
.042.50
2.3.5.321
.05999
16.6631
34
27
.00735
127. .321
.02531
39.5059
.04279
23.3718
.06029
16.5374
33
23
.00315
122.774
.02560
39.0563
.04.303
2.3.2137
.06053
16.5075
32
29
.00344
118.510
.02589
33.6177
.043.37
23.0577
.06037
16.4233
31
30
.00S73
114.5S9
.02619
38.1885
.04366
22.90.33
.06116
I6..3499
30
31
.00902
110392
.02643
37.7636
.04395
22.7519
.06145
16.2722
2J
32
.00931
107.426
.02677
37.3579
.04424
22.6920
.06175
16.19.52
28
33
.00960
104.171
.02706
36.9560
.04454
22.4541
.06204
16.1190
27
34
.00939
101.107
.02735
36.5627
.04483
22.3031
.06233
16.04-35
26
35
.01013
93.2179
.02764
36.1776
.04512
■22.1fr40
.06262
15.9637
25
36
.01047
95.4395
.02793
35.3006
.04541
22.0217
.06291
15.3945
24
37
.01076
92.90S5
.02322
35.4313
.04570
21.3813
.06321
15.5211
23
33
.01 lOo
90.4633
.02351
35.0695
.04599
21.7426
.06350
15.7433
22
39
.01135
83.14.36
.02881
^4.7151
.04623
21.6056
.06379
15.6762
21
40
.01164
35.9393
.02910
34.3678
.04658
21.4704
.06408
15.6043
20
41
.01193
83.3435
.02939
^4.0273
.04637
21.3369
.06437
15.5-340
19
42
.01222
31.3470
.02963
33.6935
.04716
21.2049
.06467
15.46.33
18
43
.01251
79.94:34
.02997
33.3662
.04745
21.0747
.06496
15.3943
17
44
.01230
73.1263
.03026
33.0452
04774
20.9460
.06.525
15.3254
16
45
.01309
76.3900
.03055
32.7303
.04303
2^3183
.06554
15 2571
15
46
.01333
74.7292
.03034
32.4213
.04333
20.6932
.06.584
1-5.1593
14
47
.01367
73.1390
.03114
32.1181
.04862
205691
.06613
15.1222
13
48
01396
71.6151
.03143
31.8205
.04891
20.4465
.06642
15.0557
12
49
.01425
70.1533
.03172
31.5234
.04920
20..3253
.06671
14.9398
11
50
.01455
63.7501
.03201
31.2416
.04949
20.2056
.06700
14.9244
10
51
.01434
67.4019
.0.32.30
309599
.04978
20.0872
.06730
14.8596
9
52
.01513
66.1055
.032.59
30.6333
.05007
19.9702
.06759
14.7954
8
53
.01542
64.3.530
.03233
30.4116
.05037
19.3546
.06788
14.7317
7
54
.01571
63.6567
.0-3317
30.1446
.05066
19.7403
.06317
14.6655
6
55
.01600
62.4992
.03346
29.3323
.05095
19.6273
.06847
14.6059
5
56
.01629
■ 61.3329
.0.3376
29.6245
.O0I24
! 19.5156
.06376
14.54-33
4
57
.016.58
60.3053
.03405
29.3711
.05153
19.4051
.06905
14.4523
3
53
.01637
59.2659
.03434
29.1220
.05182
19.29.59
.06934
14.4212
2
59
.01716
58.2612
.0.3463
23.3771
.05212
19.1579
.06963
14.3607
1
60
m:
.01746
Co tang.
57.29flCi
.03492
23.6363
Tang.
.05241
19.0311
.06993
14.. 3007
0
M.
Tang.
Cotang.
Cotang.
Tang.
Cotang.
Tang.
i
93
8
§3
g
yo
g
60
TABLt
, XV.
NATURAL TANGENTS
AND COTANGENTS.
231
M
0
4rO
50
60
70
M.
60
. TaDg
.06993
1 Cotang
14.3au7
Tang.
Cotang.
Taug.
Cotang.
9.51436
Tang.
.12273
Cotang.
8.14435
.03749
11.4301
.10510
1
.07022
14.2411
.08778
11.3919
.10540
9.4^731
. 1 2.303
8.12431
59
2
.07051
14.1821
.08807
11.3540
.10569
9.46141
.12333
8.10536
58
3
.07080
14.1235
.08837
11.3163
.10599
9.43515
.12367
8.08600
57
4
.07110
14.0655
.08366
11.27.39
.10623
9.40904
.12397
8.06674
56
5
, .07139
14.0079
.03895
11.2417
.10657
9.3^307
.12426
8.04756
55
6
J .07168
13.9.507
.03925
11.2 t4>
.10637
9.35724
.12456
8.02,348
54
7
.07197
13.8940
.0j954
11.1631
.10716
9.33155
.12435
8.00948
" ^ 1
53
8
• .07227
13.8378
.03933
11.1316
.10746
9.30599
.12515
7.99058
52
S
.072.56
1.3.7821
.09013
11.09.54
.10775
9.28058
.12544
7.97176
51
10
.072S5
13.7267
.09042
! 1.0594
.10305
9.25530
.12574
7.95302
60
11
.07314
13.6719
.09071
11.02.37
.10:^34
9.23016
.12603
7.93433
49
12
i .07314
13.6174
.09101
10.93^2
.10j63
9.20516
.12633
7.91582
48
13
1 .07373
13.5634
.091.30
10.9529
.10>93
9.13028
.12662
7.89734
47
14
i .07402
13.5093
.091.59
10.9178
.10922
9.15554
.12692
7.37895
46
15
1 .07431
13.4566
.09189
10.8329
10952
9.13093
.12722
7.S6C64
45
16
.07461
13.4039
.09218
10.8483
.10981
9.10616
.12751
7.84242
44
17
.07490
1.3.. 351 5
.09247
10.8139
.11011
9.03211
.12781
7.82428
43
18
.07519
13.2996
.09277
10.7797
.11040
9.05789
.12810
7.80622
42
19
.07548
13.2480
.09306
10.74.57
.11070
9.03379
.12840
7.78325
41
20
.07578
13.1969
.09335
10.7119
.11099
9.00953
.12369
7.770.35
40
21
.07607
13.1461
.09365
10.6783
.11128
8.93598
.12-99
7.75254
39
22
.076:^6
13.0953
.09394
10.64.50
.11158
8.S6227
.12929
7.73480
38
23
.07665
13.0458
.09423
10.6118
.11187
8.9.3367
.129.58
7.71715
37
24
.07695
12.9962
.09453
10.57^9
.11217
8.91520
.12938
7.69957
36
25
.07724
12.9469
.09432
10.. 5462
.11246
•8.39135
.13017
7.63208
35
26
.07753
12.8981
.09511
10.5136
.11276
8.36362
.13047
7.66466
1
34
27
.07782
12.3496
.09541
10.4313
.11305
8.34551
.13076
7.647.32
33
23
.07812
12.8014
•09570
10.4491
.11335
8.82252
.13106
7.6.3005
32
29
.07841
12.7536
.09600
10.4172
.11364
8.79964
.131.36
7.61287
31
30
.07870
12.7062
.09629
10.3354
.11394
8.77689
.13165
7.59575
30
31
.07^99
12.6.591
.09658
10. .3533
11423
3.7.5425
.13195
7.57372
29
32
.07929
12.6124
.09688
10.3224
.11452
8.73172
.13224
7.. 56 J 76
28
33
.07958
12.5660
.09717
10.2913
.11482
8.70931
.13254
7.54487
27
34
.07987
12.5199
.09746
10.2602
.11511
8.63701
.1.3234
7.52806
26
35
.08017
12.4742
.09776
. 10.2294
.11.541
8.66432
.1.3313
7.51132
25
36
.08046
12.4233
.09305
10.1938
.11570
8.64275
.13343
7.49465
24
37
.03075
12.3333
.09334
10.1683
.11600
8.62078
.1.3372
7.47306
23
33
.03104
12.. 3.390
.09364
10.1381
.11629
8.. 59893
.13402
7.46154
22
39
.081.34
12.2946
.09893
10.1080
.116.59
8.57718
.134.32
7.44509
21
40
.03163
12.2505
.09923
10.0780
.11633
8.55555
.1.3461
7.42871
20
41
.08192
12.2067
.09952
10.0433
.11718
8.53402
.13491
7.41240
19
42
.08221
12.1632
.09931
10.0137
.11747
8.512.59
.13521
7.39616
18
f.
.0823 I
12.1201
.10011
9.93931
.11777
8.49128
.13550
7.37999
17
44 1
.08230
12.0772
.10040
9.96G07
11806
8.47007
.13.580
7.36389
16
45
.08309
12.0346
.10069
9.93101
.11836
8.44396
.13609
7.34786
15
46
.08339
11.9923
.10099
9.90211
.11365
8.42795
.13639
7.33190
14
47
.08368
11.9504
.10123
9.87333
.11395
8.40705
.13669
7.31600
13
48
.03397
11.9037
.10153
9.S44S2
.11924
S..3362.5
.13693
7.30013
12
49
.08427
11.8673
.10187
9.81641
.11954
8.36555
.13728
7.23442
11
50
.03456
11.8262
.10216
9.78817
.11933
8.34496
.13758
7.26873
10
51
.08485
11.7353
.10246
9.76009
.12013
8.32446
.13787
7.25310
9
52
.08514
11.7448
.10275
9.73217
.12012
8.30406
.13817
7.23754
8
53
.08544
11.7045
.10305
9,70441
.12072
8.2-3376
.13346
7.22204
7
54
.03573
11.6645
.10.334
9.67680
.12101
8.26355
.13376
7.20661
6
55
.08602
11.6243
.10363
9.64935
.12131
8.24345
.13906
7.19125
5
56
.08632
11.5353
.10393
9.62205
.12160
3.22.344
.139,35
7.17594
4
57
.08661
11.5461
.10422
9.59490
.12190
8.20.352
.1.3965
7.18071
3
58
.08690
11.5072
.10452
9.56791
.12219
8.18370
13995
7.14553
2
59
.03720
11.4635
.10481
9.54106
.12249
8.16393
. 14024
7.13042
1
6ii .(Lsz-jy
11 4301
.10510 9.51436
.12278
8.144.35
Tang. (
.14054
7.11.537
0
1 i
M. Cotang.
Tang. (
Jotang. 1 Tang. (
[Jotang. J
Cotang.
Tang. 1
^_-.
w.:
i^
840
833 1 8JJ0
'46:<
; TAP
!LE XV.
I^JATURAL TANGENTS AND COTANGENTS
).
M
0
80
9^
lOO
110
1
M.
60
Tang.
.14054
CotaDg.
7.11537
Tang.
Cotang.
Tang.
Cotang.
5.67128
Tang.
Cotang.
5.144.55
.15333
6.31375
.176.33
.19438
1
.14084
7.10038
15868
6.30189
.17663
5.66165
.19468
5.13658
59
2
.14113
7.03546
.15398
6.29007
.17693
5.65205
.19498
5.12862
58
3
.14143
7.07059
.15928
6.27829
.17723
6.64248
.19529
5.12069
57
4
.14173
7.05579
.15958
6.26655
.17753
5.63295
.19559
5,11279
56
5
.14202
7.04105
.15988
6.25436
.17783
5.62344
.19539
5.10490
55
6
.14232
7.02637
.16017
6.24321
.17813
5.61397
.19619
5.09704
54
7
.14262
6.91174
.16047
6.23160
.17343
5.60452
.19649
5.03921
63
8
.14291
6.99713
.16077
6.22003
.17373
5.59511
.1L630
5.031.39
52
9
.14321
6.9326S
.16107
6.20351
.17903
5.58573
.19710
5.07360
51
10
.14351
6.96323
.16137
6.19703
.17933
5.57638
.19740
5.06584
50
11
.143S1
6.95335
.16167
6.18559
.17963
5.56706
.19770
5.0.5809
49
12
.14410
6.9.3952
.16196
6.17419
.17993
5.55777
.19801
5.05037
48
13
.14440
6.92525
.16226
6.16233
.13023
5.54851
.19831
5.04267
47
14
.14470
6.91104
.16256
6.15151
.18053
5.53927
.19861
5.03499
46
15
.14499
6.39683
.16236
6.14023
.18083
5.53007
.19391
5.02734
46
16
.14529
6.88278
.16316
6.12399
.18113
5.52090
.19921
5.01971
44
17
.14559
6.86374
.16346
6.11779
.18143
5.51176
.19952
5.01210
43
18
.14588
6.85475
.16376
6.10664
.13173
5.50264
.19952
5.00451
42
19
.14618
6.84032
.16405
6.09.552
.13203
5.49356
.20012
4.99695
41
20
.14643
6.82694
.16435
6.03444
.13233
5.48451
.20042
4.98940
40
21
.14678
6.SI3I2
.16465
6.07340
.13263
5.47548
.20073
4.98188
39
22
.14707
6.79936
.16495
6.06240
.18293
5.46643
.20103
4.97433
38
23
.14737
6.73564
.16525
6.05143
.13323
5.45751
.20133
4.96690
37
24
.14767
6.77199
.16555
6.04J51
.13353
5.44357
.20164
4.95945
36
25
.14796
6.75S33
.16535
6.02962
.18334
5.43966
.20194
4.9.5201
35
26
.14326
6.74433
.16615
6.01 37S
.18414
5.43077
.20224
4.94460
34
27
.14S56
6.73133
.16645
6.00797
.18444
5.42192
.20254
4.93721
33
23
.14336
6.71789
.16674
5.99720
.18474
5.41309
.20285
4.92934
32
29
.14915
6.70450
.16704
5.93646
.18504
5.40429
.20315
4.92249
31
30
.14945
6.69116
.16734
5.97576
.18534
5.39552
.20345
4.91516
30
31
.14975
6.67737
.16764
5.96510
.18564
5.33677
.20376
4.90785
29
32
.15005
6.66463
.16794
5.9.S448
.18594
5.37805
.20406
4.90056
28
33
.15034
6.65144
.16324
5.94390
.18624
5.36936
.20436
4.89330
27
34
.1.5064
6.6.3331
.16354
5.9a335
.18654
5.36070
.20466
4.SS605
26
35
. 1 5094
6.62523
.16381
5.92283
.18634
5.35206
.20497
4.87882
26
36
.15124
6.61219
16914
5.91236
.18714
5.»4345
.20527
4.87162
24
37
.15153
6.59921
.16944
5.90191
.18745
5.33437
.20557
4.86444
23
33
.15183
6.53627
.16974
5.89151
.18775
5.32631
.20588
4.85727
22
39
.15213
6.57339
.17004
5.88114
.18805
5.31778
.20618
4.85013
21
40
.15243
6.56055
.17033
5.87030
.18835
5.30928
.20648
4.84300
20
41
.15272
6.54777
.17063
5. 8605 1
.18865
5.30030
.20679
4.83590
19
42
.15302
6.53503
.17093
5.85024
.18895
5.29235
.20709
4.82382
18
43
.15332
6.52234
.17123
5.84001
.18925
5.23393
.20739
4.82175
17
44
.15362
6.. 50970
.17153
5.82982
.18955
5.27553
.20770
4.81471
16
45
.15391
6.49710
.17183
5.81966
.18936
5.26715
.20300
4.80769
15
46
.15421
6.48456
.17213
5.80953
.19016
5.25880
.20830
4.80063
14
47
.15451
6.47206
.17243
5.79944
.19046
5.25048
.20861
4.79370
13
48
.15481
6.45961
.17273
5.78938
.19076
5.24213
.20891
4.7S673
12
49
.15511
6.44720
.17303
5.77936
.19106
5.23391
.20921
4.77978
11
50
.15540
6.43434
.17333
5.76937
.19136
5.22566
.20952
4.77236
10
51
.15570
6.42253
.17363
5
75941
.19166
.5.21744
.20982
4.76595
9
52
.15600
6.41026
.17393
5
74949
.19197
5.20925
.21013
4.75906
8
53
.15630
6.39804
.17423
5
73960
.19227
5.20107
.21043
4.75219
7
54
.1.5660
6.33587
.17453
5
72974
.192.57
5.19293
.21073
4.74534
6
55
.15639
6.37374
.17483
5.71992
.19237
5.18480
.21104
4.73851
5
56
.15719
6.36165
.17513
5.71013
.19317
5.17671
.21134
4.73170
4
67
.15749
6.34961
.17543
5.70037
.19:347
5.16363
.21164
4.72490
3
58
.15779
6.33761
.17573
5.69064
.19378
5.16053
.21195
4.71813
2
59
.15309
6.32566
.17603
5.63094
.19403
5.15256
.21225
4.71137
1
60
1
.15838
6.31375
.17633
Cotang.
5.67123
.19433
5.14455
.21256
4.70463
0
ftl.
Cotang.
Tang.
Tang.
Cotang.
Tang.
Cotang.
Tang.
81° 1
803 1
793 1
783 1
TABLE XV. NATURAL TANGENTS AND COTANGENTS. 233
M.
0
130
130 1
1*0
150
M.
60
Tang.
Cotang.
Tang.
Cotang. ^
Cang.
Cotaug.
Tang.
Cotang.
21256
4.70463
.23087
4.33148 .
24933
4.01078
.26795
3.73205
1
.2I2S6
4.69791
.23117
4.32573 .
24964
4.00582
.26826
3.72771
59
2
.21316
4.69121
.23148
4.32001 .
24995
4.00086
.26857
3.72338
58
3
.21347
4.68452
.23179
4.314.30 .
25026
3.99592
.26883
3.71907
57
4
.21377
4.67786
.23209
4.30S60 .
25056
3.99099
.26920
3.71476
66
5
.21408
4.67121
.2.3240
4.30291 .
25087
3.93607
.26951
3.71046
55
6
.21433
4.66458
.23271
4.29724 .
25118
3.98117
.26982
3.70616
54
7
.21469
4.65797
.2.3301
4.29159 .
25149
3.97627
.27013
3.70188
53
8
.21499
4.65133
.23332
4.2S595 .
25180
3.97139
.27044
3.69761
52
9
.21529
4.64480
.23363
4.28032 .
25211
3.96651
.27076
3.69335
51
10
.21560
4 63825
.23393
4.27471 .
25242
3.96165
.27107
3.68909
60
11
.21.590
4 63171
.23424
4.26911 .
25273
3.95680
.27138
3.68485
49
12
.21621
4.62518
.23455
4.26352 .
25304
3.95196
.27169
3.68061
48
13
.21651
4. 6 1 868
.23485
4.25795 .
253.35
3.94713
.27201
3.67638
47
14
.21682
4.61219
.23516
4.252.39 .
25366
3.94232
.27232
3.67217
46
15
.21712
4.60572
.23547
4.24685 .
25397
3.93751
.27263
3.66796
45
16
.21743
4.59927
.23578
4.24132 .
25428
3.93271
.27294
3.66376
44
17
.21773
4.. 592-^3
.23608
4.23580 .
2.5459
3.92793
.27326
3.65957
43
IS
.21804
4.58641
.23639
4.23030 .
25490
3.92316
.27357
3.65538
42
19
.21834
4.55001
.23670
4.22481 .
25521
3.91839
.27388
3.65121
41
2()
.21864
4.57363
.23700
4.21933 .
25552
3.91364
.27419
3.64705
40
21
.21895
4.56726
.23731
4.21387 .
25583
3.90890
.27451
3.64289
39
22
.21925
4.56091
.23762
4.20842 .
2.5614
3.90417
.27482
3.63874
38
23
.21956
4.55458
.23793
4.20293 .
25645
3.89945
.27513
3.63461
37
24
.21986
4.. 54826
.23823
4.19756 .
25676
3.89474
.27545
3.63048
36
25
.22017
4.54196
.23854
4.19215 .
25707
3.89004
.27576
3.62636
35
26
.22047
4.53568
.23885
4.18675 .
25738
3.88536
.27607
3.62224
34
27
.22078
4.52941
.2.3916
4.18137 .
25769
3.88068
.27633
3.61814
33
28
.22108
4..52316
.23946
4.17600 .
25800
3.87601
.27670
3.61405
32
29
.22139
4.51693
.23977
4.17064 .
2.5831
3.87136
.27701
3.60996
31
30
.22169
4.51071
.24008
4.16530 .
25862
3.86671
.27732
3.605SS
30
31
.22200
4.50451
.24039
4.15997 .
25893
3.86208
.27764
3.60181
29
32
.22231
4.49832
.24069
4.15465 .
25924
3.85745
.27795
3..59775
28
33
.22261
4.49215
.24100
4.149.34 .
25955
3.85284
.27826
3.59370
27
34
22292
4.48600
.24131
4.14405 .
259S6
3.84824
.27858
3.58966
26
35
22322
4.47986
.24162
4.13877 .
26017
3.84.364
.27889
3.55562
25
36
22.353
4.47374
.24193
4.13350 .
26048
3.83906
.27921
3.58160
24
37
22383
4.46764
.24223
4.12825 .
26079
3.8.3449
.27952
3.57758
23
38
22414
4.46155
.24254
4.12301 .
26110
3.82992
.27933
3.-57357
22
39
22444
4.45548
.24285
4.11778 .
26141
3.82.537
.28015
3.56957
21
40
.22475
4.44942
.24316
4.11256 .
26172
3.82083
.28046
3.56557
20
41
.22505
4.44.3.38
.24347
4.10736 .
26203
3.81630
.28077
3.561.59
19
42
.22536
4.4.3735
.24377
4.10216 .
26235
3.81177
.28109
3.55761
18
43
.22567
4.43134
.21408
4.09699 .
26266
3.80726
.28140
3. .55364
17
44
.22597
4.42534
.244.39
4.09182 .
26297
3.80276
.28172
3.54963
16
45
.22623
4.41936
.24470
4.0S666 .
26328
3.79S27
.28203
3. .54573
15
46
.22658
4.41.340
.24.501
4.08152 .
26359
3.79.378
.23234
3..54179
14
47
.22689
4.40745
.24.532
4.076.39 .
26390
3.78931
.28266
3.5.3785
13
48
.22719
4.401.52
.24562
4.07127 .
26421
3.78485
.28297
3.. 53393
12
49
.22750
4.39560
.24.593
4.06616 .
26452
3.78040
.2S329
3.53001
11
50
.22781
4.38969
.24624
4.06107 .
26483
3.77.595
.2*360
3.52609
10
51
.22811
4.3S381
.246.55
4.05599 .
26515
3.771.52
.28.391
3. .522 19
9
52
.22842
4.37793
.246S6
4.05092 .
26546
3.76709
.28423
3.51829
8
53
.22872
4.37207
.24717
4.04586 .
26577
3.76268
.28454
3.51441
7
54
.22903
4..36623
.24747
4.04081 .
26608
3.75828
.28486
3.51053
6
55
.22934
4.36040
.24778
4.03578 .
26639
3.75388
.28517
3.50666
5
56
.22964
4.354.59
.24809
4.03076 .
26670
3.749.50
.28549
3.50279
4
57
.22995
4.34879
.24840
4.02574 .
26701
3.74512
.28580
3.49894
3
58
.23026
4.34300
.24871
4.02074 .
26733
3.74075
.28612
3.49509
2
59
.23056
4.. 33723
.24902
4.01576 .
26764
3.73640
.28643
3.49125
1
60
m:
.23(:87
4..3:3143
.24933
4.01078 .
26795
3.73205
.28675
3.48741
0
M.
Co tang.
Tang.
Cotang.
Tang. C
:tang.
Tang.
Cotang.
Tang.
i
TO
reo 1
750
7
4:0
u;j4
. TABLE XV.
NATl
URAL TANGENTS AND
COTANGENTS
•
M.
0
160
170
18^
190
M.
60
Tang.
.23675
Cotang.
Tang.
Cotang.
Tang.
.32492
I Cotang.
3.07763
Tang.
Cotang.
2.90421
3.43741
.30573
3.27035
-34433
1
.28706
3.43.359
.30605
3.26745
.32.524
3.07464
.34465
2.90147
59
2
.23738
3.47977
.-30637
3.26406
..32556
3.07160
.34493
2.89373
58
3
.23769
3.47596
.30669
3.26067
.32533
.3.06357
.34530
2.89600
57
4
.28800
3.47216
.30700
3.25729
.32621
3.06554
.34563
2.89327
56
5
.23832
3.46337
.-307-32
3.25392
-32653
3.062-52
-34596
2.89055
55
6
.23S&4
3.46453
..30764
3.25055
.32635
3.059-50
.34628
2.83783
54
7
.23S95
3.46030
.30796
3.24719
.32717
3.05649
.34661
2.8351 1
53
8
.23927
3.45703
.30823
3.24333
.32749
3.05349
.34693
2.33240
52
9
, .28953
3.45.327
.30360
3.24049
.32732
3.05049
.-34726
2.87970
51
10
.28990
3.44951
.-30391
3.2.3714
.32814
3.04749
.34758
2.87700
50
11
.29021
3.44576
.30923
3.23-331
.32346
3.04450
.34791
2.S7430
49
12
.29053
3.44202
.30955
3.2304S
.32373
-3.041.52
.34824
2.37161
43
13
.29034
3.43323
.3'i9';7
3.22715
.32911
3.0.3354
.34356
2.86392
47
14
.29116
3.434.56
.31019
3.22-334
.32943
3.03556
-343S9
2.86624
46
15
.29147
3.43084
.31051
3.22053
..32975
3.03260
.34922
2.S6356
45
16
.29179
3.42713
.31083
.3.21722
.a3007
3.02963
.349.54
2.86039
44
17
.29210
3.42343
.31115
,3.21-392
.33040
3.02667
.34987
2.85822
43
13
.29242
.3.41973
.31147
.3.21063
.33f)72
3.02372
.3-5020
2.85555
42 :
19
.29274
.3.41604
.31178
3.20734
-33104
3.02077
..35052
2.8.5239
41
20
.29305
3.412.36
.31210
3.20406
.331-36
3.01733
.3.5035
2.85023
40 1
21
.29337
3.40^69
.31242
3.20079
.-33169
3.01439
.35113
2.S475S
39 ;
22
.29363
3.40502
.31274
-3.19752
..33201
.3.01196
.351.50
2.84491
38
23
.29400
3.401.36
.31306
-3-19426
.3.3233
3.00903
.35133
2.8-1229
37 !
24
.29432
3.. 39771
.31333
3.19100
.-33266
3.00611
.3.5216
2.83965
36 i
25
.29463
3.. 39406
.31370
3.13775
.33293
3.00319
.3.5248
2.83702
35
26
.29495
3.39042
.31402
3.13451
.33330
3.00023
.35231
2.83439
34
27
.29.526
3.33679
.314-^
3 18127.
.-33.363
2.997.33
.35314
2.83176
33
23
.29558
3..38317
.31466
-3. i 7304
.33-395
2.99447
.3.5346
2.82914
32
29
.29590
3.37955
-31493
3.17431
.-3-3427
2.99158
.35379
2.82653
31
30
.29621
3.37594
.31530
3.17159
.33460
2.93363
..3-5412
2.82391
.30
31
.29653
3.-372.34
.31562
3.16333
.3-3492
2.93530
.35445
2.82130
29
32
.29635
3.-36375
.31.594
3.16517
.33524
2.93292
.35477
2.81870
23
33
.29716
3.36516
.31626
3.16197
.335-57
2.93004
.3.5510
2.SI610
27
34
.29743
3.36153
.31653
3.15377
.3-3559
2.97717
.-35-543
2.81350
26
35
.29730
3.35800
.31690
3.15558
-33621
2.974-30
.35576
2.81091
25
36
.29311
3.35443
.31722
3.15240
.-3-36.54
2.97144
..35603
2.60333
24
37
.29343
3. .3-5037
.31754
3.14922
.33636
2.963-58
.3.5841
2.S0574
23
3S
.29875
3.347-32
.31786
3.14605
.33718
2.96573
.35674
2.80316
22
39
.29906
3.34377
.31818
3.14288
.-33751
2.96283
..35707
2.80059
21
40
.29933
3.ai023
.31850
3.1-3972
.-33733
2.96004
..35740
2.79S02
20
41
.29970
3.33670
.31832
3.136-56
..3-3316
2.95721
.35772
2.79545
19
42
.30001
3.-3-3317
.31914
3.13-341
.3-3348
2.9-5437
..35305
2.79289
18
43
..30033
3.32965
.31946
3.1.3027
..3-3381
2.951-55
.35838
2.79033
17
44
.30065
3.32614
.31973
3.12713
-33913
2.94372
.35371
2.78773
16
45
..30097
3.-32264
.32010
3.12400
.33945
2.94591
.35904
2.78523
15
46
.30128
3.31914
.32042
3.12087
.33978
2.94309
.35937
2.78269
14
47
.30160
3.31-565
.32074
3.11775
.34010
2.94028
.35969
2.78014
13
48
.30192
3.31216
.32106
3.11464
.34043
2.9-3748
.36002
2.77761
12
49
.30224
3. -3036 3
.32139
3.11153
.34075
2.93463
.36035
2.77507
11
50
.30255
3.30521
.32171
3.10342
.34108
2.93189
.36068
2.772.54
10
51
.30237
3.-30174
..32203
3.10532
.34140
2.92910
..36101
2.77002
9
52
.30319
3.29329
..32235
3.10223
.34173
2.92632
.36134
2.76750
8
53
.30.351
3.29433
..32267
3.09914
.31205
2.92.3-54
.36167
2.76493
7
54
.30382
-3.291-39
.32299
3.09606
.34233
2.92076
.36199
2.76247
6
55
.30414
3.23795
.-32-331
3.09295
.34270
2.91799
.362-32
2.75996
5
56
.30446
3.234-52
.32.363
3.03991
.34303
2 91.523
..36265
2.75746
4
57
.30478
3.23109
.32396
3.03635
.343.35
2.91246
.36298
2.75496
3
58
.30509
3.27767
.32428
3.08-379
..34363
2.90971
.36331
2.75246
2
59
.30.541
3.27426
.32460
3.03073
.34400
2.90696
.36364
2.74997
1
60
M.
.30573
3.27035
.32492
3.07768
.34433
2.90421
.36397
2.74743
0
M.
Cotang.
Tang.
Cotang.
Tang.
Cotang.
Tang.
Cotang.
Tang.
T33 1
73° 1
71^ \
703 1
TABLE X\^
NATURAL TANGENTS AND COTA.'JGENTS. 235
M.
20^
31^
/
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
53
59
60
m;
T36397
.36430
.36463
.364%
.36529
.36562
.36595
.36628
.36661
.36694
.36727
.36760
.36793
.36S26
.36859
.36892
.36925
.36958
.36991
.37024
.37057
.37090
.37123
.37157
.37190
.37223
.37256
.37289
.37322
.37355
.37333
.37422
.37455
.37438
.37521
.37554
.37583
.37621
.37654
.37687
.37720
.37754
.37787
.37820
.37353
.37837
.37920
37953
.37936
.33020
.38053
.33036
33120
.33153
.33186
. 3^221)
.3=!2.-)3
.3-2-6
.3-320
.3-353
.38:386
Cotang.
Cotang. Tang.
2.74748
2.74499
2.74251
2.74004
2.73756
2.73509
2.73263
2.73017
2.72771
2.72526
2.72281
2.72036
2.71792
2.71548
2.71305
2.71062
2.70819
2.70577
2.70335
2.70094
2.69853
2.69612
2.69371
2.69131
2.68892
2.68653
2.63414
2.63175
2.67937
2.67700
2.67462
2.67225
2.66939
2.66752
2.66516
2.66231
2.66046
2.65811
2.6.5576
2.65342
2.65109
2.64875
2.64642
2.64410
2.64177
2.63945
2.63714
2.634S3
2.63252
2.63021
2.62791
2.62.561
2 62332
2.62103
2.61^74
2.61646
2.61418
2.61190
2 60963
2.60736
2.60509
Cotang.
.33336
.33420
.33453
.38487
.38520
.38553
.38587
.38620
.33654
.33687
.3^721
.33754
.38787
.38321
.33354
.33888
.33921
.33955
.38988
..39022
.390.55
.39089
.39122
.39156
.39190
.39223
.39257
.39290
.39324
.39357
.39391
.39425
.39453
.39192
.39526
.39559
.39593
..39626
.39660
.39694
.39727
.39761
.39795
.39829
.39362
.39896
..39930
.39963
.39997
.40031
.40065
.40093
.401,32
.40166
.40200
.40234
.40267
.40301
.40335
.40369
.40103
a^j
23C
Tang.
Tang.
69=
2.60509
2.60283
2.60057
2.59331
2.59606
2.593>1
2.59156
2.53932
2.58703
2.53434
2.53261
2.58038
2.57815
2.57593
2.57371
2.57150
2.56923
2.56707
2.56487
2.56266
2.56046
2.55827
2.55608
2.553S9
2.55170
2.54952
2.54734
2.54516
2.54299
2.b40S2
2.53365
2.53643
2.-53432
2.53217
2.53001
2.52786
2.52571
2.52357
2.52142
2.51929
2.51715
2.51502
2.51289
2.51076
2.50364
2.50652
2.50440
2.50229
2 50018
2.49807
2.49.597
2.49336
2.49177
2.4-!967
2.487.58
2.43549
2.48340
2.48132
2.47921
2.47716
2.47509
Cotang. Tang.
68=
.40403
.40136
.40470
.41)504
.40538
.40572
.40606
.40640
.40674
.40707
.40741
.40775
.40309
.40843
.40377
.4091 1
.40945
.40979
.41013
.41047
.41081
.41115
.41149
.41183
.41217
.41251
.41235
.41319
.41353
.41337
.41421
.41455
.41490
.41524
.41558
.41592
.41626
.41660
.41694
.41723
.41763
.41797
.41331
.41365
.41399
.41933
.41963
.42002
.420.36
.42070
.42105
.42139
.42173
.42207
.42242
.42276
.42310
.42345
.42379
.42413
.42447
Cotang.
■2.47509
2.47302
2.47095
2.46883
2.46632
2.46476
2.46270
2.46065
2.45860
2.45655
2.45451
2.45246
2.45043
2.44^39
2.44636
2.44433
2.44230
2.44027
2.43325
2.43623
2.43422
2.43220
2.43019
2.42819
2.42618
2.42418
2.42213
2.42019
2.41819
2.41620
2.41421
2.41223
2.41025
2.40827
2.40629
2.40432
2.40235
2.40033
2.39841
2.39645
2.39449
2.392.53
2.39053
2.33>63
2.3^66^
2.33473
2.38279
2.. 33034
2.. 37891
2.37697
2.37504
2.37311
2.37118
2.36925
2.367.33
2.36541
2.36349
2.36158
2.35967
2.35776
2.35585
Tang. I Cotang.
.42147
.424^2
.42516
.42.551
.425S5
.42619
.42654
.42638
.42722
.42757
.42791
.42826
.42-:'60
.42394
.42929
.42963
.42998
.43032
.43067
.43101
.431.36
.43170
.43205
.43239
.43274
.43308
.43343
.43378
.43412
.43447
.43431
.43516
.43550
.4.3535
.43620
.436.54
.43639
.43724
.43758
.43793
.43328
.43362
.43897
.43932
.43966
.44001
.44036
.44071
.44105
.44140
.44175
.44210
.44244
.44279
.44314
.44349
.44334
.44418
.44453
.444.83
.44523
Cotang. Tang
67=
2.35585
2.35395
2.35205
2.35015
2.34825
2.34636
2.34447
2..34258
2.34069
2.33881
2 33693
2 33505
2,33317
2.33130
2.32943
2.32756
2..32570
2.32383
2.32197
2.32012
2.31826
2.31641
2.31456
2.31271
2. 31 086
2.30902
2.30718
2.30.5.34
2.30351
2.30167
2.29984
2.29801
2.29619
2.29437
2.29254
2.29073
2.28S91
2.23710
2.28523
2.283-18
2.28167
2.27987
2.27506
2.27626
2.27447
2.27267
2.27033
2.26909
2.26730
2.265.52
2.26374
2.2615:6
2.26013
2.25840
2.25663
2.25436
2.2.5309
2.25132
2.24956
2.24780
2.24604
M.
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Cotang. Tang
663
sat
) TABLE XV.
NATURAL TANGENTS AND COTANGENTS
!•
M
0
1 340
353
2&0
370
1
Tang.
, .44523
Cotang.
Taog.
Cotang.
2.14451
Tang.
.43773
Cotang.
2.05030
Tang.
Cotang.
1.96261
M.
60
2.24604
.46631
.50953
1
.44553
2.24423
.46666
2. 142S3
.43309
2.04379
.50939
1.96120
59
2
.44593
2.24252
.46702
2.14125
.43345
2.04723
.51026
1.95979
58
3
.44627
2.24077
.46737
2.13963
.43Ssl
2.04577
.51063
1.953.38
57
4
.44662
2.23902
.46772
2.13<01
.43917
2.04426
.51099
]. 95698
56
5
.44697
2.23727
.46-03
2.136-39
.4^953
2.04276
.51136
1.95557
55
6
.44732
2.2.3553
.46>43' 2.1.3477
43939
2.04125
.51173
1.9.5417
54
7
.44767
2.233/->
.46379
2.1.3316
.49026
2.0.3975
.51209
1.95277
53
8
.44S02
2.2321)4
.46914
2.13154
.49062
2.03825
.51246
1.95137
52
9
■ .44337
2.2.3030
.46950
2.12993
.49093
2.0.3675
.51283
1-94997
51
10
.44372
2.22S57
.46935
2.12332
.491.34
2.03526
.51319
1.94358
50
11
.44907
2.22633
.47021
2.12671
.49170
2.03376
.51-356
1.94718
49
12
.44W2
2.22510
.47056
2.12511
.49206
2.03227
.51393
1.94579
43
13
'■ .44977
2.22337
.47092
2.123-50
.49242
2.03073
.514.30
1.94440
1 47
14
1 .45012
2.22164
.47123
2.12190
.49273
2.02929
.51467
1.94.301
46 ,
15
: .45047
2.21992
.47163
2.12030
.49315
2.027S0
.51503
1.94162
45
16
■ .450S2
2.21319
.47199
2.1 1371
.49351
2.02631
.51;540
1.94023
44
17
.45117
2.21647
.472-34
2.11711
.49337
2.02433
.51.577
1.9.3885
43
18
.45152
2.21475
.47270
2.11552
.49423
2.02335
.51614
1.9.3746
42
19
.45187
2.21:304
.47.305
2.11.392
.49459
2.02187
.51651
1.9.3608
41
20
.45222
2.21132
.47311
2-11233
.49495
2.02039
.51633
1.93470
40
21
.45257
2.20961
.47377
2.11075
.49532
2.01391
.51724
1.93-332
39
22
.45292
2.2)79:1
.47412
2.10916
.49-563
2.01743
.51761
1.93195
38
23
.45.327
2.20619
.47443
2.10753
.49604
2.01596
.51798
1.9.3057
37
24
.4.5362
2.20449
.47433
2.10600
.49640
2.01449
.51835
1.920-.40
36
25
.45397
2.20273
.47519
2.1W42
.49677
2.01302
.51372
1.92782
35
26
.454.32
2.20103
.47555
2.10234
.49713
2.01155
.51909
1 92&45
34
27
.45467
2.1993S
.47590
2.10126
.49740
2.01008
.51946
i. 92503
33
28
.4.5502
2.19769
.47626
2.09969
.49736
2.00=62
.51983
1.92371
32
29
.45533
2.19-599
.47662
2.09311
.49322
2.00715
.52020
1.92235
31
30
.45573
2.194.30
.47693
2.09654
.493.53
2.00569
..52057
1.92093
30
31
.45603
2.19261
.47733
2.0349S
.49394
2.004V3
.520G1
1.91962
29
32
.45643
2.19092
.47769
2.09.341
.49931
2.00277
.52131
1.91326
28
33
.45673
2.13923
.47305
2.09134
.49967
2.00131
.55163
1.91690
27
34
.45713
2.13755
.47^^0
2.09023
.50004
1.999-6
.52205
1.91554
26
35
.45743
2.13.537
.47876
2-03372
..50040
1.99341
.52242
1.91418
25
36
.45731
2.13419
.47912
2.03716
..50076
1.99695
.52279
1.91232
24
37
.45319
2.13251
.47943
2.03560
.50113
1.99.550
..52316
1.91147
23
38
.453:54
2.13034
.47934
2.03405
..50149
1.99406
.52353
1.91012
22
39
.45389
2.17916
.43019
2.03250
.50135
1.99261
.52-390
1.90376
21
40
.4.5924
2.17749
.430.55
2.03094
.50222
1.99116
.52427
1.90741
20
41
.45960
2.17.532
.43091
2.07939
.50253
1.93972
.52461
1.90607
19
42
.45995
2.17416
.43127
2.07785
.50295
1.93328
.52.501
1.90472
18
43
.46030
2.17249
.43163
2.076.30
..50331
I.986&}
.525.38
1.90.3.37
17
44
.46065
2.17033
.43193
2.07476
.50363
1.93540
.52575
1.90203
16
45
.46101
2.16917
.43234
2.07321
.50404
1.93396
.52613
1.90069
15
46
.461.36
2.16751
.43270
2.07167
.50441
1.932.53
.52650
1.89935
14
47
.46171
2.16535
.43306
2-07014
.50477
1.93110
.52637
1.89301
13
4S
.46206
2.16420
.43342
2 06360
.50514
1.97966
.52724
1.89667
12
49
.46242
2.162.55
.43378
2.06706
.50550
1.97323
..52761
1.89.533
11
50
.46277
2.16090
.43414
2-06-5.53
.50537 1
1.97631
.52793
1 .89400
10
51 1
.46312
2.15925
.43450
2.06400
.50623
1.97.5.33
.52836
1.89266
9
52
.46.343
2.15760
.43436
2.06247
.50660
1 97395
52373
1.89133
8
53
.46.333
2.15.596
.43521
2.06094
.50696
1.97253
.52910
1.89000
7
54
.46418
2.1.54.32
.4S557
2.05942
.50733
1.97111
.52917
1. 83367
6
55
.464.54
2.1.5263
.43593
2.05790
.50769
1.96969
.52935
1.83734
5
56
.46439
2.15104
.43629
2.05637
.50806
1.96327
.53022
1.88602
4
57
.46525
2.14940
.43665
2.05435
.50843
1.96635
.53059
1.88469
3
58
.46560
2.14777
.48701
2.05333
.50379
1.96544
.53096
1.83337
2
59
.46595
2.14614
.43737
2. 05 182
..50916
1.96402
.53134
1.88205
1
60;
.46631
2.14451
.43773
2.0.5030
Tang. (
.50953
IJotang.
1.96261
..53171
1.88073
Tang. ]
0
M. Cotang. !
Tang.
Cotang.
Tang. <
;:!otang.
i
■s::
6i
5C
64° 1
633 1
633 1
lABLE XV. NATURAL TANGENTS AND COTA/JGENTS. 23T
M
0
aso
393
30O
310
M.
60
Tang.
.53171
Cotang.
Tang.
Cotang.
Tang.
.57735
Cotang.
1.73205
Tang.
Cotang.
1.88073
.55431
1.80405
.60086
1.6642-5
1
.53208
1.87941
.55469
1.80231
.57774
1.73089
.60126
1.66318
59
2
.53246
1.87809
.55507
1.80158
.57813
1.72973
.60165
1.66209
58
3
.53283
1.87677
.55.545
1.80034
.57851
1.72357
.60205
1.66099
57
4
.53320
1.87546
.55583
1.79911
.57890
1.72741
.60245
1.65990
56
5
.53358
1.87415
.55621
1.7978S
.57929
1.72625
.602.34
1.65S81
55
6
.53395
1.87233
.55659
1.79665
.57968
1.72509
.60324
1.65772
54
7
.53432
1.87152
.55697
1.79542
..58007
1.72393
.60364
1.65663
53
8
..53470
1.87021
.55736
1.79419
.58046
1.72278
.60403
1.65554
52
9
.53507
1.86391
.55774
1.79296
.53035
1.72163
.60443
1.65-145
51
10
.53545
1.86760
.5.5812
1.79174
.581^
1.72047
.60483
1.65337
50
11
.53582
1.86630
.55850
1.79051
.58162
1.71932
.60522
1.65228
49
12
.53620
1.86499
.55838
1.78929
.58201
1.71817
.60562
1.65120
48
13
.53657
1.86369
.55926
1.78507
..58240
1.71702
.60602
1.6.5011
47
14
.53694
1.862.39
.55964
1.78635
.58279
1.71588
.00642
1.64903
46
15
.53732
1.86109
.56003
1.78563
.58318
1.71473
.60681
1.64795
45
1
16
.53769
1.85979
.56041
1.78441
.58357
1.71358
.60721
1.64687
44
17
.53307
1.85350
.56079
1.78319
.53396
1.71244
.60761
1.61579
43
18
.53844
1.85720
.56117
1.78198
..58435
1.71129
.60801
1.64471
42
19
.5.3882
1.85591
.561.56
1.78077
.58474
1.71015
.60841
1.64363
41
2n
.53920
1.85462
.56194
1.77955
.58513
1.70901
.60381
1.64256
40
21
.53957
1.85333
..56232
1.77^34
.58552
1.70787
.60921
1.64148
39
22
.53995
1.85204
..56270
1.77713
.58591
1.70673
.60960
1.64041
33
23
.54032
1.8.5075
.56309
1.77592
.58631
1.70560
.61000
1.63934
37
24
.54070
1.84946
..56347
1.77471
.58670
1.70446
.61040
1.63826
36
25
..54107
1. 84318
.56335
1.77.351
.58709
1.70332
.61080
1.63719
35
26
.54145
1.64683
.53424
1.77230
.58748
1.70219
.61120
1.63612
34
27
.51183
1.84561
.56462
1.77110
.58787
1.70106
.61160
1. 63505
33
23
.54220
1.84433
..56501
1.76990
.58826
1.69992
.61200
1.63398
32
29
.542:58
1.84305
.56539
1.76369
..58865
1.69379
.61240
1.63292
31
80
.54296
1.84177
.56577
1.76749
.58905
1.69766
.61280
1.63185
30
31
.54333
1.84049
.56616
1.76629
.58944
1.696.53
.61320
1.63079
29
32
.54.371
1.83922
.56651
1.76510
.58983
1.69541
.61360
1.62972
28
33
.54409
1.83794
.56693
1.76390
.59022
1.69423
.61400
1.62866
27
34
.54446
1.83667
..56731
1.76271
.59061
1.69316
.61440
1.62760
26
35
.54434
1.83540
.56769
1.76151
.59101
1.69203
.61480
1.62654
25
36
.54522
1.83413
.56303
1.76032
.59140
1.69091
.61.520
1.62.548
24
37
.54560
1.83286
.56346
1.75913
.59179
1.63979
.61561
1.624-12
23
38
.54597
1.83159
.56335
1.75794
.59218
1.63866
.61601
1.62.336
22
39
.54635
1.83033
.56923
1.75675
.59258
1.63754
.61641
1.62230
21
40
.54673
1.82906
.56962
1.75556
.59297
1.63643
.61681
1.62125
20
41
.5^1711
1.827S0
.57000
1.75437
.59336
1.68531
.61721
1. 6201 9
19
42
.54743
1.82654
.57039
1.75319
.59376
1.63419
.61761
1.61914
18
43
.54786
1.82523
.57078
1.75200
.59415
1.68308
.61801
1.61803
17
44
.54324
1.82402
.57116
1.7.5032
.59154
1.68196
.618-12
1.61703
18
45
.54362
1.82276
.57155
1.74964
.59494
1.63035
.61882
1.61593
15
46
.54900
1.82150
.57193
1.74346
.59533
1.67974
.61922
1.61493
14
47
.549:«
1.82025
.57232
1.74728
.59573
1.67863
.61962
1.61338
13
48
.54975
1.81899
.57271
1.74610
.59612
1.67752
.62003
1.61233
12
49
.5.5013
1.81774
.57309
1.74492
.59651
1.67641
.62043
1.01179
11
50
.55051
1.81649
..57343
1.71375
.59691
1.67530
.62033
1.61074
10
51
.55a39
1.81524
.57336
1.74257
.59730
1.67419
.62124
1.60970
9
52
.55127
1.81399
.57425
1.74140
.59770
1.67309
6216^1
1.60665
8
53
.55165
1.81274
.57464
1.74022
.59809
1.67198
.62204
1.60761
7
54
.5.5203
1.81150
.57503
1.7.3905
.59849
1.67088
.62245
1.60657
6
55
.55241
1.81025
.57541
1.73788
.59883
1.66978
.62235
1.60553
5
56
.55279
1.80901
.57580
1.73671
..59928
1.66867
.62.325
1.60449
4
57
.55317
1.80777
..57619
1.7.3555
.59967
1.66757
.62366
1.60345
3
58
.55355
1.80653
.57657
1.73138
.60007
1.66647
.62406
1.60241
2
59
.55393
1.80529
.57696
1.73.321
.600-16
1.66538
.62446
1.60137
1
60
M.
.55431
1.80405
.57735
1.73205
.60086
1.66423
.62487
1.60033
0
M.
Cotang.
6
Tang.
Cotang.
Tang.
Cotang.
Tang.
Cotang.
Tang.
602
5
.93
5
83
238 TABLE XV. NATURAL TANGENTS AND COTANGENTS.
M
0
323
33^ 1
34 ;
3
5^
M.
60
Tang.
.624S7
Cctang.
1.600.33
Tang.
Cotang.
Tang.
.67451
Cotang.
1.45-2-56
Tang.
Cotang.
1.4-2>15
.64941
1.5:39^6
.70021
1
.62527
I.. 59930
.649-2
1.53S5S
.67493
1.43 1 63
.70C64
1.42726
59
2
.62.563
1.59326
.65024
1.53791
.675:36
1.43070
.70107
1.426:33
58
3
.62603
1.59723
.65065
1.53693
.67573
1.47977
.70151
1.42550
57
4
.62649
1.596-20
.65106
1.. 53595
.67620
1.47335
.70194
1.42462
56
5
.62639
1.59517
.65143
1.53497
.67663
1.47792
.70233
1.42374
55
6
.62730
1.59414
.65159
1.534G0
.67705
1.47699
.70231
1.42236
54
7
.62770
1.59311
.65231
1.5:3:302
.67743
1.47607
.70325
1.42198
53
8
.62311
1.59203
.65272
1.53205
.67790-
1.47514
.70.363
1.42110
52
9
.62352
1.59105
.6.5314
1.53107
.67332
1.47422
.70412
1.42022
51.
10
.62392
1.59002
.65-3.55
1.53010
.67375
1.47.3:30
.70455
1.419:3-1
50'
11
.62933
1.53900
.6-5397
1.52913
.67917
1.47233
704S9
1.41647
49
12
.62973
1.53797
.65433
1.52316
.67960
1.47146
.705^42
1.417.59
48
13
.63014
1.53695
.65450
1.52719
.63002
1.470.33
.■70536
1.41672
47
14
.63055
1.53.593
.63521
1.52622
.63045
1.46932
.70629
1.41-5.34
46
15
.63095
.1.53490
.6-5563
1.52525
.63033
1.46370
.70673
1.41497
45
16
.6-3136
1.5S333
.65604
1.52429
.63130
1.46773
.70717
1.41409
44
17
.63177
1.53236
.65646
1.523:32
.63173
1.46656
.70760
1.41322
43
13
.63217
1.53134
.65633
1.522-35
.63215
1.46595
.70804
1.41-235
42
13
.63253
1.53033
.65729
1. 52139
.65-253
1 .46503
.70348
1.41148
41
20
.63299
1.57931
.65771
1.52043
.6530!
1.46411
.70391
1.41C61
40
. 21
.6-3:340
1.57379
.65313
1.51946
.65-343
1.46.3-2(J
.709.35
1.40974
39
; 22
.6-3-350
1.57773
.655.54
1.51350
.65.356
1.46229
.70979
1.40S37
33
23
.63121
1.57676
.65396
1.51754
.65429
1.46137
.710-23
1.40300
37
24
.6-3462
1.57575
.65933
1.51653
.63471
1.46046
.71066
1.40714
36
25
.6-3503
1.57474
.65950
1.51562
.65514
1.4.5955
.71110
1.40627
35
26
.6.3->14
1.57372
.66021
1.51466
.65557
1.4-5564
.71154
1.40-510
M
27
.63.534
1.57271
.60G63
1.51370
.63600
1.45773
.71193
1.4W54
.33
28
.63625
1.57170
.66105
1.51275
.63642
1.45632
.71242
1.40367
32
29
.63666
1.57069
.66147
1.51179
.63635
1.4-5592
.71235
1.40231
31
30
.63707
1.56969
.66159
1.51034
.63723
1.45501
.71329
1.40195
30
31
.63743
1.56563
.66230
1.50933
.63771
1.4.5410
.71373
1.40109
29
32
.63739
1.56767
.66272
1.50393
.63314
1.45320
.71417
1.4'X)22
28
as
.6:3330
1.56667
.66314
1.50797
.63357
1.45229
.71461
1.39936
27
34
.63371
1.56566
.66:356
1.50702
.63900
1.45139
.71505
1.39350
26
35
.63912
1.56466
.66393
1.-50607
.63942
1.4-5049
.71549
1.39764
25
36
.639.53
1.56-366
.66440
1.50512
.63985
1.44953
.71593
1.39679
24
37
.63994
1.56265
.664^2
1.50417
.69028
1.44563
.71637
1.39-593
23
33
.&4035
1.56165
.66-524
1.50322
.69071
1.44773
.71631
1.39507
22
39
.64076
1.56065
.66566
1.50223
.69114
1.44633
71725
1.39421
21
40
.&4117
1.. 5.5966
.66603
1.50133
.69157
1.44.593
71769
1.39336
20
41
.641-53
1.-55566
.66650
1.50033
.69200
1.44503
71813
1.392.50
19
42
.&4199
1.55766
.666.:.2
1.49944
.69-243
1.44413
.71857
1.39165
18
43
.64240
1.55666
.66734
1.49549
.69256
1.44.3-29
.71901
1.39079
17
44
.64231
1.55;'567
.66776
1 .49755
.69:329
1.442:39
.71946
1.35994
16
45
.643-22
1.55467
.66313
1.49661
.69:372
1.44149
.71990
1.33909
15
46
.61363
1.55363
.66360
1.49566
.69416
1.44060
.72034
1.33824
14
47
.64404
1.. 5.5269
.66902
1 .49472
.694-59
1.43970
.72073
1.357.38
13
48
.64446
1.55170
.66944
1.49373
.69-502
1.43531
.72122
1.35653
12
49
.64437
1.5.5071
.66936
1.49-234
.69-545
1.43792
.72167
1.33563
11
50
.64.528
1.-54972
.6702.3
1.49190
.69.533
1.43703
.72211
1.. 33134
10
51
.64-569
1.54373
.67071
1.49097
.69631
1.43614
.72255
1.33399
9
52
.64610
1.54774
.67113
1.49003
.69675
1.43:325
.72299
1.33314
8
53
.64652
1.54675
.67155
1.45909
.69718
1.434-36
.72.344
1.33229
7
54
.64693
1.54.576
.67197
1.43316
.69761
1.4.3347
.72.358
1.38145
6
55
.64734
1.54478
.672-39
1.48722
.69504
1.43253
.72432
1. 33060
5
56
.64775
1.54379
.67232
1.48629
.69347
1.43169
.72477
1.. 37976
4
57
.64SI7
1.54231
.67324
1.435.36
.69591
1.4.30SO
.72-521
1.37891
3
53
.643.53
1.54133
.67.366
1.45442
.69934
1.42992
.72565
1.37507
2
59
.64399
1.54035
.67409
1.43^9
.69977
1.42903
.72610
1.37722
1
j 60
f M.
1
.64941
1.. 5.3956
.67451
1.48-256
Tang. (
.7(021
1.42315
Tang.
.726.34
1.37638
0
M.
Gotang.
Tang.
Cotang.
Cotang.
Cotang.
5'
Tang.
to
5
r=
5
6= 1
5
53
TABLE XV. NATURAL TANGENTS AND COTANGENTS.
239
36
M.' Tang. Cotang.
1
21
3|
41
5 i
6i
rr
I
81
^1
10
11 1
12'
13 i
14 1
37^
10 '
16!
17 i
18!
19
20
21
22
2;i
2-1
25
26
27
23
29
30
.72654 ,
.72699 I
.72743 I
.72783
.72S32
.72S77
.72921
.72966
.73010
.7.3055
.73100
.73144
.731 S9
.73-234
.73278
.73323
.7336S
.73413
.73457
.73502
.73.547
.73592
\ .73637
\ .73631
;'. 73726
; .73771
\ .7.3S16
.7.3S61
.73906
.73951
.73996
31 .74041
32 I .74056
331 .74131
34 I .74176
.74221
.74267
.74312
.74357
.74402
.74447
.74492
Tang. Cotang.
35
36
37
33
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
53
59
60
.74533
.74-533
.74623
.74674
.74719
.74764
.74510
.74355
.74900
.74946
.74991
.75037
.75032
.75123
.75173
.75219
.75264
.75310
.75355
1 .37633
1.37554
1.37470
1.373S6
1.37302
1.3721S
1.37134
1.370^50
1.36967
1.36^33
1.36-00
1.. 367 1 6
1.36633
1.36549
1.3&166
1.36333
1.36300
1.36217
1.36134
1.36051
1.3.5963
1.3533^5
1.35302
1.35719
1.35637
1.3.5554
1.35472
1.353=9
1.35307
1.3.5-224
1.35142
1.3.5060
1.34973
1.34396
1.:34314
1.34732
1.34650
1.34563
1.34437
1.34405
1.34323
1.34242
1.34160
l.:31079
1.33993
1.33916
M.;Dotang.
1.33335
1.337.54
1.33673
1.33.592
1.33511
1.33430
1.33349
1.3.3263
1.33137
1.33107
1.33026
1.32946
1.32365
1.32735
1.32704
.75401
.7;5447
.75492
.75533
.75534
.75629
.75675
.75721_
.75767
.75312
.75353
.75904
.75950
.75996
.76042
.76033
.76134
.76130
.76226
.76272
.76313
.76364
.76410
.76456
,76502
.76543
.?6594
.76640
.76636
.76733
.76779
.76325
.76371
.76913
.76964
.77010
.770-57
.77103
.77149
.77196
.77242
.77239
.77335
.773S2
.77428
.77475
.77521
.77563
.77615
.77661
.77703
.77754
.77301
.77343
.77395
.77W1
.77933
.73035
.73082
.73129
38^
Tang.
1.32704
1.32624
1.32^544
1.32464
1.32.334
1.32.304
1.32224
1.32144
1.32064
1 31934
1.3190-1
1.31S25
1.31745
1.31666
1.31556
1.31507
1.31427
1.31343
1.31269
1.31190
1.31110
1.31031
1.30952
1.30373
1.30795
1.30716
1.30637
1.30553
1.30430
I.3<3401
1.30323
1.30244
1.30166
1.30037
1.30009
1.29931
1.293-53
1.29775
1.29696
1.29613
1.29.541
1.29463
1.29335
1.29307
1.29229
1.29152
Cotang.
39c
"
.73129
.73175
.78222
.78269
.78316
.76363
.73410
.78457
.76504
.73-551
.78598
.78645
.78692
.78739
.78786
.78834
.78831
.78928
.78975
.79022
.79070
.79117
.79164
.79212
.79259
.79-306 !
.79354
.79401
.79449
.79496
.79544
.79591
.796:39
.79636
.79734
.79781
.79329
.79377
.79924
.79972
.80020
.80067
.80115
.80163
.80211
.80253
1.27994
1.27917
1.27541
1.27764
1.27633
1.27611
1.27535
1.274-53
1.27,332
1.27396
1.27230
1.27 i53
1.27077
1.27C01
1.26925
1.26549
Tang. Cotang. ^M.
.60973
.Slu27
.51075
.51123
.81171
.81220
.81263
.81316
.81364
.81413
.81461
.81510
.81553
.81606
.81655
.81703
Tang.
53^
1.29074
1.2>997
1.23919
1.23342
1.23764
1.2.5637
1.25610
1.23533
1.23456
1.23379
1.28302
1.23-225
1.23143
1.23071
1 .27994
Cotang. Tang.
53=
1.26774
1.26693
1.266-22
1.26546
1.26471
1.26395
1.26319
1.26-244 I
1.26169
1.26093
1.26013
1.25943
1.25567
1.25792
1.25717
1.25642
1.25567
1.2-5492
1.2.5417
1.25343
1 .2-5-263
1.2:5193
1.25113
1.25044
1.24969
1.24595
1.245-20
1.24746
1.24672
1.24597
1.24523
1.24449
1.24375
1.24301
1.24227
1.24153
1.24079
1.24005
1.2-3931
1.23553
1.23784
1.23710
1.23637
1.23563
1.23490
Cotang. Tang.
513
.80306
.80354
.80402
.80450
.50493
.50546
.80594
.80642
.50690
.80733
.80736
.50334
.80552
.80930
.80973
.81752
.61300
.81849
.51593
.31946
.51995
.3-2044
.52092
.82141
.52190
.5-22.33
.R2287
.5^:336
.52335
.82434
.52453
.5-2.531
.5-2530
.32629
.5-2678
.52727
.3-2776
.82325
.32374
.52923
.5-2972
.53022
.83071
,53120
.53169
.83215
.53268
.83317
.83366
.83415
.83465
.53514
.53564
.83613
.53662
.33712
.83761
.8331 1
.83560
.33910
1.-23490
1.23416
1.2-3343
1.23270
1.23196
1.23123
1.2.30.50
1.22977
1.2-2904
1.22531
1.-2-2753
1.2-2635
1.-2-2612
1.2-2539
1.-2-2467
1.-2-23W
1.22321
1.22-249
1.22176
1.22104
1.22031
1.21S59
1.21586
1.21814
1.21742
1.21670
1.21593
1.215-26
1.21454
1.21352
1.21310
60
59
58
57
56
55
54
53
52
51
50
49
43
47
46
45
44
43
42
41
40
39
33
37
36
35
34
33
32
31
30
1.21-2.33
1.21166
1.21094
1.210-23
1.20951
1.20579
1.20503
1. -20736
1.20665
;. 20593
1.20522
1.20451
1.20379
1.20308
1.-20-237
1.20166
1.20095
1.20024
-..19953
1.19582
1.19511
1.19740
1.19669
1.19.599
1.19523
1.19457
1 19387
1.19316
1.19-246
1 19175
29
23
27
-26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Cotang.^ Tang. M.
503 '1
240 TABLE XV. NATURAL TANGENTS AND COTANGENTS.
M.
0
4:03
4:10 1
4:
20
433 1
M.
60
1
!
1
1
Tang.
.S39I0
Cotang.
Tang.
Cotang.
Tang.
Cotang.
1.11061
Tang.
Cotang.
1.19175
.86929
1.15037
.90040
.93252
1.072.37
1
.83960
1.19105
.S6930
1.14969
.90093
1.10996
.93306
1.07174
59
(
2
.84009
1.19035
.87031
1.14902
.90146
1.10931
.93360
1.07112
58
3
.84059
1.18964
.87032
1.14334
.90199
1.10367
.93415
1.07049
57
4
.84103
1.18394
.87133
1.14767
.90251
1.10302
.93469
1.06937
56
5
.84153
1.18324
.871.34
1.14699
.90304
1.10737
.93524
1.06925
55
j
6
.84203
1.18754
.87236
1.14632
.90357
1.10672
.93578
1.06362
54
1
7
.84258
1.1 8634
.b72S7
1.14565
.90410
1.10607
.93633
1.06300
53
1
8
.84307
1.18614
.87333
1.14493
.90463
1.10543
.93638
1.06733
52
1
9
.84357
1.13544
.87339
1.14430
.90516
1.10473
.93742
1.06676
51
1
10
.84407
1.13474
.87441
1.14363
.90569
1.10414
.93797
1.06613
50
11
.84457
1.18404
.87492
1.14296
.90621
1.10349
.93352
1.06551
49
12
.84507
1.18334
.87543
1.14229
.90674
1.10235
.93906
1.06439
48
13
.84556
1.18264
.87595
1.14162
.90727
1.10220
.93961
1.06427
47
14
.84606
1.13194
.87646
1.14095
.90731
1.10156
.94016
1.06365
46
15
.84656
1.13125
.87693
1.14023
.90334
1.10091
.94071
1.06.303
45
16
.84706
1.13055
.87749
1.13961
.90337
1.10027
.94125
1.06241
44
17
.84756
1.17936
.87801
1.13394
.90940
1.09963
.94180
1.06179
43
18
.84306
1.17916
.87352
1.13323
.90993
1.09399
.94235
1.06117
42
19
.84356
1.17346
.87904
1.13761
.91046
1.09834
.94290
1.06056
41
20
.84906
1.17777
.87955
1.13694
.91099
1.09770
.94345
1.05994
40
21
.84956
1.17703
.88007
1.13627
.91153
1.09706
.94400
1.059.32
39
22
.85006
1.17633
.83059
1.13561
.91206
1.09642
.94455
1.0.5370
33
23
.85057
1.17569
.83110
1.13494
.91259
1.09573
.94510
1.0.5309
37
24
.85107
1.17500
.83162
1.13423
.91313
1.09514
.94565
1.05747
36
25
.85157
1.174.30
.83214
1.13361
.91366
1.09450
.94620
1.05635
35
26
.8.5207
1.17351
.83265
1.13295
.91419
1.09336
.94676
1.05624
34
27
.85257
1.17292
.83317
1.13223
.91473
1.09322
.94731
1.05562
33
23
.85303
1.17223
.83369
1.13162
.91526
1.09253
.94736
1.05501
32
29
.85353
1.17154
.83421
1.13096
.91530
1.09195
.94341
1.05439
31
30
.85403
1.17035
.83473
1.13029
.91633
1.09131
.94396
1.05378
30
31
.Si>453
1.17016
.83.524
1.12963
.91637
1.09067
.94952
1.05317
29
32
.85509
1.16947
.83576
1.12397
.91740
1.09003
.95007
1.05255
23
33
.8.5559
1.16378
.88623
1.12331
.91794
1.08940
.95062
1.05194
27
34
.85609
1.16309
.83630
1.12765
.91347
1.03876
.95113
1.05133
26
35
.85660
1.16741
.88732
1.12699
.91901
1.03813
.95173
1.05072
25
36
.85710
1.16672
.83784
1.12633
.91955
1.08749
.9.5229
1.05010
24
37
.85761
1.16603
.88336
1.12567
.92003
1.03636
.95234
1.04949
23
38
.85311
1.16535
.83333
1.12501
.92062
1.0S622
.9.5340
1.04333
22
39
.85362
1.16466
.83940
1.124.35
.92116
1.03-559
.95395
1.04327
21
40
.85912
1.16.393
.83992
1.12369
.92170
1.03496
.95451
1.04766
20
41
.85963
1.16.329
.89045
1.12303
.92224
1.03432
.95506
1.04705
19
42
.86014
1.16261
.89097
1.122.33
.92277
1.03369
.95562
1.04644
18
43
.86061
1.16192
.89149
1.12172
.92331
1.03306
.95618
1.04.533
17
44
.86115
1.16124
.89201
1.12106
.92335
1.03243
.95673
1.04.522
16
45
.86166
1.16056
.892.53
1.12041
.92439
1.03179
.95729
1.04461
15
46
.86216
1.15937
.89306
1.11975
.92493
1.03116
95785
1.04401
14
47
.86267
1.1.5919
.893.53
1.11909
.92547
1.03053
.95341
1.04340
13
48
.86318
1.1.5351
.89410
1.11844
.92601
1.07990
.95397
1.04279
12
49
.86363
1.1.5733
.89463
1.11778
.92655
1.07927
95952
]. 04218
11
50
.86419
1.15715
.89515
1.11713
.92709
1.07864
.96003
1.041.53
10
51
.86470
1.15647
.89.567
1.11643
.92763
1.07301
.96064
1.04097
9
52
.86521
1.15.579
.89620
1.11532
.92817
1.07733
.96120
1.04036
8
53
.86572
1.15511
.89672
1.11517
.92372
1.07676
.96176
1.03976
7
54
.86623
1.1.5443
.89725
1.11452
.92926
1.07613
.96232
1.0.3915
6
55
.86674
1.15375
.89777
1.11.337
.92930
1.07550
.96238
1.03355
5
56
.86725
1.15303
.89330
1.11.321
.93034
1.07437
.96:344
1.03794
4
57
.86776
1.15240
.89SS3
1.112.56
.93038
1.07425
.96400
1.03734
3
58
.36327
1.15172
.89935
1.11191
.93143
1.07362
.964.57
1.03674
2
59
.86378
1.15104
.89938
1.11126
.93197
1.07299
.96513
1.0-3613
1
60
M.
.86929
1.15037
.90040
1.11061
.93252
1.07237
.96569
1.03553
0
Cotang.
Tang.
Cotang.
Tang.
Cotang.
Tang.
Cotang.
Tang.
4
9=
4
:8 =
4
.70
463
TABLE XV. NATURAL TANGENTS AND COTANGENTS. 241
M.
(J
440 1
M.
fiO
M.
20
440
M.
40
M.
40
440
20
Tang.
Cotang.
Tang.
Cotang.
1.02355
Tang.
Cotang.
.96569
1.03553
.97700
.98843
1.01170
1
.96625
1.03493
59
21
.97756
1.02295
39
41
.98901
1.01112
19
9
.96631
1.03133
58
22
.97813
1.02236
38
42
.989.58
1.01053
18
3
.96738
1.03372
57
23
.97870
1.02176
37
43
.99016
1.00994
1/
4
.96794
1.03312
56
24
.97927
1.02117
36
44
.99073
1.00935
16
5
.96350
1.03252
55
25
.97934
1.02057
35
45
.99131
1.00376
15
fi
.96907
1.03192
54
2(5
.93041
1.01998
34
46
.99189
1.00818
14
7
.96963
1.03132
53
27
.93093
1.01939
33
47
.99247
1.00759
13
8
.97020
1.03072
52
28
.93155
1.01879
32
48
.99304
1.00701
12
9
.97076
1.03012
51
29
.93213
1.01820
31
49
.99362
1.00642
11
10
.97ia3
1.02952
50
30
.93270
1.01761
30
50
.99420
1.00583
10
11
.97189
1.02892
49
31
.98327
1.01702
29
51
.99478
1.00525
9
P
.97246
1.02832
48
32
.93334
1.01642
28
52
.99536
1.00467
8
13
.97302
1.02772
47
33
.93441
1.01533
27
53
.99594
1.00403
"i
14
.97359
1.02713
46
34
.93499
1.01524
26
54
.99652
1.00350
6
15
.974)3
1.02653
45
ai
.93556
1.01465
25
55
.99710
1.00291
b
16
.974: 2
1.02593
44
36
.93613
1.01406
24
56
,99763
1.00233
4
17
.97529
1.02533
43
37
.93671
1.01347
23
57
.99326
1.00175
3
18
.97536
1.02474
42
38
.93728
1,01283
22
58
.99884
1.00116
2
19
.97643
1.02414
41
39
.98786
1.01229
21
59
.99942
1.00058
1
20
M.
.97700
1.02355
40
M.
40
M.
.93343
1.01170
20
M.
60
M.
1.00000
1.00000
0
M.
Cotang.
Tang.
Coteing.
Tang.
Cotang.
Tang.
453
450
450
V
242 TABLE XVI. RISE PER MILE OF VARIOUS GRADES.
TABLE XVI.
RISE PER MILE OE VARIOUS GRADES.
Grade
per
Htatioa.
Rise per
Mile-
Grade
per
Station.
Rise per
Mile.
Grade
per
Station.
Rise per
Mile.
Grade
per
Station.
Rise per
Mile.
.01
.523
.41
21.643
.81
42.763
1.21
63.838
.02
1.0.56
.42
22.176
.52
43.296
1.22
64.416
.03
1.5S4
.43
22.701
.83
43..y2l
1.23
64.944
.04
2.112
.44
23.2.32
..S4
44.3.52
1.24
65.472
.05
2.640
.45
23.760
.85
44.850
1.25
66.000
.06
3.163
.46
24.233
.86
45.403
1.26
66.523 ,
.07
3.6S6
.47
24.816
.37
45.936
1.27
67.056
.OS
4.224
.43
2-5.344
.83
46.464
1.23
67.534
.09
4.752
.49
25.872
.89
46.992
1.29
63.112
.10
5.280
.50
26.400
.90
47.520
1.30
65.640
.11
5.803
.51
26.923
.91
48.043
1.31
69.163
.12
6.3.36
.52
27.4-56
.92
43.576
1.32
69.696
.1.3
6.S64
.53
27.934
.93
49.104
1.33
70.224
.14
7.392
.54
23.512
.94
49.632
1.34
70.752
.15
7.920
.55
29.040
.95
5OI60
1.35
71.230
.16
8.443
.56
29.563
.96
50.683
1.36
71.808
.17
8.976
.57
30096
.97
51.216
1.37
72.336
.13
9.504
.53
30.624
.93
51.744
1.33
72.S64
.19
10.032
.59
31.152
.99
52.272
1.39
73.392
.20
10. .560
.60
31.6S0
1.00
52.800
1.40
73.920
.21
11.083
.61
32.203
l.ni
53.323
1.41
74.443
.22
11.616
.62
32.738
IM
53.8.56
1.42
74.976
.23
12.144
.63
33.264
1.03
54.354
1.43
75.. 504
.24
12.672
.64
33.792
1.04
54.912
1.44
76.0.32
.25
13.200
.65
34.320
1.05
55.440
1.45
76.560
.26
13.723
.66
34.S43
1.06
55.963
1.46
77.038
.27
14.2.56
.67
35.376
1.07
56.496
1.47
77.616
.23
14.784
.63
35.904
1.03
57.024
1.43
78.144
.29
15.312
.69
36.432
1.09
57.552
1.49
78.672
.30
15.840
.70
36.960
1.10
53.030
1.50
79.200
.31
16.363
.71
37.483
l.Il
53.608
1.51
79.723
.32
16.896
.72
33.016
1.12
59.1.36
1.52
80.2.56
.33
17.424
.73
33.544
1.13
59.664
1.53
80.784
.34
17.952
.74
39.072
1.14
60192
1.54
81.312
.3.5
18.450
.75
39.600
1.15
60.720
1.55
81.840
.36
19.003
.76
40123
1.16
61.243
1.56
82.363
.37
19.536
.77
40.656
1.17
61.776
1.57
82.896
.33
20.0&4
.78
41.184
1.18
62.304
1.58
83.424
.39
20.592
.79
41.712
1.19
62.832
1.59
a3.952
.40
21.120
.80
42.240
1.20
63.360
1.60
&i.480
TABLE XVI. RISE PER MILE OF VARIOUS GRADES. 243
Grade
Rise per
Grade
Rise per
Grade
Rise per
Grade
Rise per
per
Station.
Mile.
per
Station.
Mile.
per
Station.
Mile.
per
Station.
Mile.
1.61
S5.003
1.81
95.563
2.10
110.880
4.10
2I6.4S0
1.62
65.536
1.82
96.096
2.20
116.160
4.20
221.760
1.63
86.064
1.S3
96.624
2.30
121.440
4.30
227.040
1.64
86.592
1.84
97.152
2.40
126.720
4.40
232.320
1.6.5
87.120
1.85
97.630
2.50
132.000
4.50
237.600
1.66
87.643
1.86
98.208
2.60
137.280
4.60
242.880
1.67
88.176
1.87
93.736
2.70
142.560
4.70
243.160
1.63
88.704
I.S8
99.264
2.80
147.840
4.80
253.440
1.69
89.232
1.89
99.792
2.90
153.120
4.90
253.720
1.70
89.760
1.90
100.320
3.00
153.400
5.00
264.000
1.71
90.233
1.91
100.843
3.10
163.680
5.10
269.230
1.72
90.816
1.92
101.376
3.20
163.960
5.20
274.560
1.73
91.344
1.93
101.904
3.30
174.240
5.30
279.840
1.74
91.872
1.94
102.432
3.40
179.520
5.40
235.120
1.75
92.400
1.95
102.960
3 50
184.800
5.50
290.400
1.76
92.923
1.96
103.483
3.60
190.080
5.60
295.630
1.77
93.456
1.97
104.016
3.70
195.360
5.70
300.960
1.73
93.934
1.93
104.544
3.80
200.640
5.80
306.240
1.79
94.512
1.99
105.072
3.90
205.920
5.90
311.520
l.SO
95.040
2.00
105.600
4.00
211.200
6.00
316.800
THE mm
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