THE UNIVERSITY OF ILLINOIS
LIBRARY
From the collection of
Julius Doerner, Chicago Purchased, 1918.
H26f
--/^*^/V>\y^ ^/jj
93 ^ 3J^, 2/J?
_ <>. ? 9 ? * ^^^
=- ..^ i
^9V -
7 a
Z
ii n ^i ic c<f s±
9
Co -2/ <'^ ? 0 / i> 3 (>
^ ,9? 47 r/ 2. ? i' i i- ^ i
F [ E L D-B 0 0 K
H R
lAILllOAD ENGINEERS
.(■
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.
50
CIRCULAR CURVES.
Example. Given 7=27°, T = 600 or Here 6 = 600 tan. 6° 45' = 71 01, or tan. 13° 30' = 71.01.
R = 2499 15, to an.! ( li = 2499.18 ti»n. 6» 45
72. Problem* Given the angle of intersection / of two tangenti A C and B C {Jig- 24)^ to find the tangent point A of a curve, that ehaU pass through a point E, given by C D = a, D E =: b, and the angle CD E
= i/.
£ Ig. 'I
Solution. Produce DE to the curve at G, and draw C 0 to the cen- tre 0. Denote DFhyc. Then in the right triangle CDF we have (Tab. X. U) DF= CD cos. CDF, or
c = a COS. ^ I.
Denote the distance A D from D to the tangent point by x. Then, by Geometry, x^ = D E X D G. BntDG = DF-]-FG = DF-\- EF=2DF— DE = 2c — b. Therefore, x^ = b{2c^ b), and
x = ^b{2c — b).
Having thus found A Z), we have the tangent AC = AD -\- DC = x -{■ a. Hence, R or D may be found (^ 5 or ^ 11 ).
If the point E is given by JEJfTand Ci?^ perpendicular to each other, 0 and 6 may be found from these lines. For a = C H -\- D H
CH + EH cot. 1 1 (T.nb. X. 9). mdb=^DE
Hfi^
cV.
deproaaj]
"JT"-
Ifili.
EH
na<
^•.f.#.
>%%Jt.
MISCELLANEOUS PROBLEMS.
51
''■iiijai^
Example. Given I ^ 20° 16', a = 600, and 6 = 80, to find x and R. Here c = 600 cos. 10° 8' = 590.64, 2 c — 6 = 1101. 28, and x = ySO^X^ 1101.28 = 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 B, uniting the tangent points A and Z>, to Jind the radius A 0 ^ R.
Fig. 25
Solution. Measure or calculate the perpendicular CD. Then if C D be produced to the centre 0, the right triangles ADC and CA 0, having tho ^ngle at C common, are similar, and give CD : AD =■ AC'. A 0,0V
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- angles E GCand CA 0, C G : G E = A C: A 0, or R ^ ^^^g^ •
Example. Given ^ C = 246 and AD = 240, to find R. Here
240 y 246 CD = 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
"I
r
i
L
/
^
^j„^-. Efc— ifc^^ .
^ ? ^ Y ^ ^
7 ^ c r -J 9
9 6 o
r?.
"^Vo
7 ? i~ 7
7 y ^ t-^c
1^9 Z.S ^
m^
- iL- 2
J
UNIVERSITY OF ILLINOIS-URBANA
3 0112 084205183
|
^v^^l// |
|
IvN^^PV^ |
|
A'fv^: |
V,.''
'..'1
It
tS^t-tt'