THE UNIVERSITY 
 OF ILLINOIS 
 
 LIBRARY 
 
 From the collection of 
 
 Julius Doerner, Chicago 
 Purchased, 1918. 
 
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 F [ E L D-B K 
 
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 lAILllOAD ENGINEERS
 
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 FIE LD-BOOK 
 
 FOtt 
 
 RAILROAD ENGINEERS. 
 
 CO.\TAI.MN(J 
 
 F 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 = 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=^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 =-. R.
 
 4 CIRCULAR CURVES. 
 
 Solution. In tlie right triangle A 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 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 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 == 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 ^= 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 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 : 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 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, 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 
 
 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 = 
 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' = \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.'^^j cot 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 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 =^ 
 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( ^ : 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 be the centre of the curve, and draw the radii A 
 and C and the line CD parallel to A B. Then in the right triangle 
 COD we have C^ = CD"" + OD^ But 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 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 = 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 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 -- R. 
 
 Fig. 25 
 
 Solution. Measure or calculate the perpendicular CD. Then if CZ) 
 be produced to the centre 0, the right triangles AD C and CA 0, 
 having th3 jungle at G common, are similar, and give CD : A D = 
 AC: A 0, or 
 
 ^^A^XAC 
 CD 
 
 If it is inconvenient to measure the chord A B, a line E F, parallel 
 to it, may be obtained by laying off from C equal distances CE and 
 CF. Then measuring E G and G C, we have, from the similar tri- 
 
 GEXAC 
 %ng\esE GCand CAO, CG:GE =AC:AO,orR= — ^G — * 
 
 Example. Given ^ C = 246 and AD = 240, to find R. Here 
 
 240 X 246 
 VD = 54, and R = - ^'^= 1093.33.
 
 52 CIRCULAR CURVES. 
 
 74. Problem. Given the radius AO = R [foj 25), to find :ht 
 tangent A C = J- of a curve to unite two straight lines given on the ground 
 
 Solution. Lay off from the intersection C of the given straight lines any 
 equal distances CL and CF. Draw the pe7-pendictdar C G to the mid- 
 dle of E F, and measure G E and C G. Then the right triangles 
 E G Cand C A 0, having the angle at C common, are similar, and 
 give GE: CG = AO: AC, or 
 
 EF- r^__ CGx AO 
 
 GE 
 
 By this problem and the preceding one, the radius or tangent points 
 of a curve mav be found without an instrument for measuring angles. 
 
 Example. Given R = 1093|, G E = 80, and C G = 18, to find '/'. 
 
 18X1093^ 
 Here F = gQ = 246. 
 
 75. Problem* To find the angle of intersection I of two straight 
 lines, when the point of intersection is inaccessible, and to determine the tan- 
 gent points, when the length of the tangents is given. 
 
 Solution. I. To find the angle of intersection i L.ct A C and C I' 
 (fig. 26) be the given lines Sight from some point A on one line lo a 
 point B on the other, and measure the angles CAB and T B V. These 
 angles make up the change of direction in passing from one tangent to 
 the other. But the angle of intersection (§ 2) shows the change of di- 
 rection between two tangents, and it must, therefore, be equal to the 
 sum of C A B and T B V, that is, 
 
 t^ 1= CAB-^ TBV 
 
 But if obstacles of any kind render it necessary to pass from A C to 
 B Fby a broken line, as A D E F B, measure the angles C A D, N D E, 
 P E F, RFB, and S B V, observing to note those angles as mimts which 
 are laid off contrary to the general direction of these angles. Thus the 
 general direction of the angles in this case is to the right; but the 
 angle P EF lies to the left oi D E produced, and is therefore to be 
 marked minus. The angles to be measured show the successive changes 
 of direction in passing from one tangent to the other. Thus C A D 
 6hov/s the change of direction between the first tangent and A D, 
 ND E shows the change between A D produced and D E, P E F the 
 change between DE produced and E F, R F B the change between 
 £'F produced and FB, and, lastly, SB Fthe change between B F ])ro-
 
 MISCELLANEOUS PROBLEMS. 
 
 53 
 
 duccd and the second tangent. But the iing^lc of intersection (§ 2) 
 shows the change of direction in passing from one tangent to another, 
 and it must, therefore, be equal to the sum of the partial changes 
 naeasuved, that is, 
 
 13^ 
 
 / = CA D -\- y DE - PEF-^ II FB + SB V. 
 
 Fig. 26 
 
 II. To determine the tangent points. This will be done if we find 
 the distances .1 Cand B C; for then any other distances from Cmay 
 be found. It is supposed that the distance A B, or the distances A Z), 
 DE, E F, and FB have been measured. 
 
 Tf one line A B connects A and B. Jind A C and B C in the triangle 
 ABC. For this purpose we have one side A B and all the angles. 
 
 Jf a broken line A D E F B connects A and B, let fall a perpendicular 
 B G from B upon A C, produced if necessary, and find A G and B Q 
 hy the usual method of working a traverse. Thus, if A C is taken as a 
 meridian line, and D /v, E L, and FM are drawn parallel to A C, and 
 D H, E K, and FL are drawn parallel to B G, the difference of lati- 
 tude A G is equal to the sum of the partial differences of latitude A H. 
 D K, EL, and FM, and the departure B G h equal to the sum of the 
 partial departures D II, E K, F L, and B HI. To find these partial 
 differences of latitude and departures, we have the distances A I), DE, 
 E F, and F B, and tiie bearings may be obtained from the angles 
 already measured. Thus the bearing of yl Z) is C A D, the bearing of 
 DE is KDE = KDN+ NDE =^ C A D -\- NDE, the bearing 
 of jB F is LEF = LEP— PEF^ KDE— PEF, &nA the
 
 54 
 
 CIRCULAR CURVES. 
 
 bearing oi F B is MFB = MFR -{- RFB=^ LEF + RFB; that 
 is, the bearing of each line is equal to the algebraic sum of the preced 
 ing bearing and its own change of direction. The differences of lati- 
 tude and the departures may now be obtained from a traverse table, 
 or more correctly by the formulis : 
 
 DiiF. of lat. = dist. X cos. of bearing ; dep. = dist. X sin. of bearing 
 
 Thus, AH= AD cos. CAD, and DU=AD sin. CA D. 
 
 Having found A G and B G, we have, in the right ti'iangle B G C, 
 
 (Tab. X. 9) GC = B G cot. B C G, and BC = ^^^-q ■ But 
 BCG=180° — I. Therefore, cot. BCG = — cot. /, and sin.BCG 
 = sin. /. Hence G C =- — B G cot. 7, and BC = ^^^77 . Then, 
 since A C = A G -\- G C, we have 
 
 AC=AG — BG cot. /; 
 
 BC 
 
 BG 
 
 sm. 
 
 When /is between 90° and 180°, as in the figure, cot. /is negative, 
 and — B G cot. I is, therefoi-e, positive. When / is less than 90°, G 
 will fall on the other side of / ; but the same formula for A C wil still 
 apply ; for cot. / is now positive, and consequently, — B G cot. / is 
 negative, as it should be, since, in this case, A C would equal A G mi 
 mis G C. 
 
 Example. Given A D = 1200, DE = 350, E F ==^ 300, F B =^ 
 310, CAD== 20°, NDE = 44°, PE F =. — 25°, R FB = 31°. 
 and SB V ^ 30°, to find the angle of intersection /, and the distance? 
 A C and B C. 
 
 Here 7 = 20° + 44° — 25° + 31° + 30° = 100°. To find A G 
 and B G, the work may be arranged as in the following table : — 
 
 Angles to 
 the Right. 
 
 Bearings. 
 
 Distances. 
 
 N. 
 
 £. 
 
 
 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> give A : 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 : 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 ^ 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 C ^ 1 80° — i /, and the 
 angle 0FE=A0F— 0EF=^I-2Di, since 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. 
 
 
 
 n". 
 
 D'. 
 
 
 
 V. 
 
 Error. 
 
 n. 
 
 D. 
 
 
 
 «'. 
 
 
 
 V. 
 
 Error. 
 
 ; 
 
 i\ 
 
 1 
 
 n 
 
 1 
 
 
 
 5 
 
 1 
 
 4 10 
 
 0.9 
 
 1 
 
 3 
 
 4 
 
 3 
 
 7 12 
 
 27.2 
 
 1 
 
 
 
 5 
 
 3 
 
 12 30 
 
 25.3 
 
 2 
 
 3 
 
 4 
 
 3 
 
 4 
 
 23.5 
 
 2 
 
 
 
 3 
 
 3 
 
 5 24 
 
 22.1 
 
 3 
 
 3 
 
 4 
 
 3 
 
 1 42f 
 
 8.3 
 
 3 
 
 
 
 3 
 
 3 
 
 4 30 
 
 29.7 
 
 3 
 
 h 
 
 
 
 3 
 
 3 45 
 
 24.0 
 
 1 
 
 1 
 
 5 
 
 3 
 
 13 20 
 
 18.6 
 
 2 
 
 I 
 
 1 
 
 4 
 
 40 
 
 O.I 
 
 2 
 
 1 
 
 2 
 9 
 
 1 
 
 3 
 
 1 20 
 
 0.7 
 
 2 
 
 1 
 
 4 
 
 9 
 
 4 
 
 11.0 
 
 2 
 
 3- 
 
 3 
 
 7 48 
 
 15.0 
 
 1 
 
 6 
 
 2 
 
 6 
 
 4 
 
 23.5 
 
 
 
 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 
 
 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 - = 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 — 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 C and OC=EC— EO. 
 Thus if EO = 3 413, 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 
 
 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 
 
 
 
 
 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 ; 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, ^ =- 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 = 
 
 i r "o A ? = .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. 
 
 
 
 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. lo //^ ^= 12. 
 
 Sta. 
 
 
 
 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. 
 
 
 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. 
 
 
 
 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 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 = 
 
 . I f0(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 ( 
 
 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 
 
 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 
 
 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 
 
 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 
 
 (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 
 
 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 
 5 
 
 10 
 15 
 20 
 25 
 30 
 35 
 40 
 45 
 50 
 55 
 
 7 
 
 5 
 10 
 15 
 20 
 25 
 30 
 35 
 40 
 45 
 50 
 55 
 
 8 
 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. 
 
 10 573.69 
 
 .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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 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 
 
 .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 
 
 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 Cambridge, 1788, 6 38 
 
 8 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 
 
 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 
 
 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 ! 
 
 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 
 
 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 " 
 
 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 
 
 
 
 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 
 
 Roslyn, Peters- 
 
 
 
 
 
 
 burg, 
 
 37 14.4 
 
 77 23.5 
 
 Q. "W. Dean, C. S. 
 
 Aug., 1852 
 
 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 " 
 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 
 
 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^.^ 
 
 
 
 ./ 
 
 
 
 ? ! 
 
 
 ^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 
 
 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 
 
 
 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. 
 
 
 
 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 
 
 
 
 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 
 
 
 
 1 
 
 3 
 
 4: 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 -^1
 
 TABLE XII. LOGARITHMS OF NUMBERS. 
 
 159 
 
 No 
 
 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 
 
 
 
 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 
 
 
 
 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 
 
 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 
 
 
 
 
 1 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 i 6 
 
 7 
 
 18 19 
 
 Diff. 
 
 1
 
 TiiBLE XII. 
 
 LOGARITHMS OF NUMBERS. 
 
 163 
 
 No.| 
 
 
 
 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 
 
 
 , 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; 
 
 
 
 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 
 
 
 
 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 
 
 i lwo. 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; 
 
 
 
 
 
 .' , - 
 
 
 
 
 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. 
 
 
 
 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. 
 
 
 
 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<».! 
 
 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. 
 
 
 
 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 
 
 
 
 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. 
 
 
 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 
 
 
 M. 
 
 91'^ 
 
 889
 
 176 
 
 3=" 
 
 TABLE XIII. LOGAHITHMIC SINES, 
 
 173" 
 
 M. 
 
 
 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 
 
 
 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 
 
 
 M. 
 
 86<?
 
 178 
 
 40 
 
 TABLE XIII. LOGARITHMIC SINES, 
 
 175' 
 
 M. 
 
 
 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". 
 
 
 
 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 
 
 
 Tang. M- 
 
 84<3
 
 180 
 
 60 
 
 C- i y ^ <? 
 TABLE Xlll. LOGARITHMIC SINES, 
 
 173^ 
 
 M. 
 
 
 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 
 
 
 M 
 
 Tang. 
 
 oeo 
 
 830-
 
 COSINES, TANGENTS, AND COTANGENTS 
 
 181 
 
 M. 
 
 
 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 
 
 
 Tang. 
 
 M. 
 
 07- 
 
 8«-
 
 182 
 
 so 
 
 TABLE XIII. LOGARITHMIC SINES, 
 
 171<i 
 
 M. 
 
 
 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 
 
 
 
 Tang. 
 
 M. 
 
 as* 2 
 
 81
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 183 
 
 170^ 
 
 
 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 
 
 
 99^
 
 184 
 
 10^ 
 
 TABLE XIII. 
 
 LOGARITHMIC SINES, 
 
 169! 
 
 M. 
 
 
 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. 
 
 
 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. 
 
 
 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. 
 
 
 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 
 
 
 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 
 
 
 M. 
 
 Tang. 
 
 1040 
 
 T«i
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 189 
 
 M. 
 
 
 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. 
 
 
 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 
 
 
 Tang. 
 
 lor^ 
 
 4 
 3 
 2 
 1 
 
 
 M. 
 
 7a«
 
 192 
 
 183 
 
 TABLE XIII. LOGARITHMIC SINES, 
 
 161<; 
 
 M. 
 
 
 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. 
 
 
 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 
 
 
 D. 1". 
 
 Tang. 
 
 M. 
 
 1090 
 
 7©:
 
 194 
 
 TABLE XIII. 
 
 LOGARITHMIC SINES, 
 
 159<J 
 
 M. 
 
 
 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. 
 
 
 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 
 
 
 111 
 
 68^ 
 
 *-W9rr^^9^m^\^^ww-r #^f .A^V^^tV/jT^^AV
 
 196 
 
 933 
 
 TABLE XIII. LOGARITHMIC SI^'ES, 
 
 157' 
 
 M. 
 
 
 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. 
 
 
 
 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 
 
 
 Tang. M. 
 
 (QQC
 
 198 
 
 TABLE Xlll. LOGARITHMIC SINES, 
 
 155<. 
 
 M. 
 
 M. 
 
 isine. 
 
 
 
 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". 
 
 
 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 
 
 
 M. 
 
 1150 
 
 640
 
 200 
 
 TABLE Xlll. LOGAKITHMIC SINES, 
 
 153> 
 
 M. 
 
 
 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 
 
 
 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 
 
 
 M. 
 
 Tang. 
 
 1170 
 
 10 
 
 6a«
 
 202 
 
 280 
 
 TABLE XIII. LOGARITHMIC SINES, 
 
 131 
 
 M. 
 
 
 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. 
 
 
 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. 
 
 
 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". 
 
 
 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 
 
 
 D. 1", 
 
 Tang, 
 
 M 
 
 i»l^ 
 
 5.>i^
 
 206 
 
 33° 
 
 TABLE Xlil. LOGARITHMIC SINES, 
 
 M. 
 
 
 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 
 .7361 09 
 
 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 
 
 
 
 Tang 
 
 M.
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 207 
 
 1*1:0 
 
 M 
 
 
 
 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 
 
 
 M. 
 
 
 56
 
 i>08 
 
 340 
 
 TABLE XIII. LOGARITHMIC SINES, 
 
 1450 
 
 M. 
 
 
 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. 
 
 
 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 
 
 
 M.
 
 210 
 
 B60 
 
 TABLE XIII. 
 
 LOGARITHMIC SINES, 
 
 14:3 
 
 M. 
 
 
 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 
 
 
 
 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 
 12238 6 
 
 Tang. 
 
 ISd^ 
 
 63
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 211 
 
 1433 
 
 M. 
 
 Sine. 
 
 
 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. 
 
 
 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 
 
 
 l»8o 
 
 61
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 2Vc 
 
 14:0= 
 
 M. 
 
 
 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. 
 
 
 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. 
 
 
 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 
 
 
 Tang. M. 
 
 1310 
 
 403
 
 216 
 
 430 
 
 TABLE XIII. LOGrARlTHMlC SINES, 
 
 1370 
 
 M. 
 
 
 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 
 
 
 M 
 
 Tang 
 
 47'
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 430 
 
 2n 
 
 M. 
 
 
 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 
 
 
 M. 
 
 1 33 J 
 
 46C
 
 218 
 
 440 
 
 TABLE XIII. 
 
 LOGARITHMIC SINES, &C. 
 
 1354 
 
 M. 
 
 
 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 
 
 
 M. 
 
 134a 
 
 i'i
 
 TABLE XIV. 
 
 NATURAL SINES AND COSINES
 
 i-^{} 
 
 TABLE 
 
 XIV. 
 
 NATURAL SINES AND COSINES. 
 
 
 M. 
 
 
 
 , 
 
 
 
 = — 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. 
 
 
 M. 
 
 Cosin. 
 
 Sine. 
 
 Cosin. 
 
 Sine. Cosin. 
 
 8i 
 
 P 
 
 882 1 
 
 87^ 1 863 
 
 85° 1
 
 TABLE XTV. x^ATURAL SlIs'ES AND COSINES. 
 
 221 
 
 
 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. 
 
 
 
 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 
 
 
 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. 
 
 
 
 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. 
 
 
 M. 
 
 Sine. 
 
 Cosin. 
 
 Sine. 
 
 7 
 
 40 
 
 730 733 1 710 
 
 703
 
 5^24 
 
 TABLE XIV, 
 
 .NATURAL Sl^'ES AND COSINES. 
 
 
 M. 
 
 
 
 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. 
 
 
 M. 
 
 
 Cosin. 
 
 Sine. 
 
 Cosin. 
 
 
 693 1 
 
 683 1 
 
 673 1 
 
 663 1 
 
 653
 
 
 TABLE 
 
 XIV. 
 
 NATURAI 
 
 . SINES AND COSINES 
 
 >. 
 
 n 
 
 M. 
 
 
 
 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 
 
 
 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 
 
 
 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 
 
 
 
 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. 
 
 
 
 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 
 
 
 
 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. 
 
 
 
 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 
 
 
 M. 
 
 Tang. 
 
 Cotang. 
 
 Cotang. 
 
 Tang. 
 
 Cotang. 
 
 Tang. 
 
 i 
 
 93 
 
 8 
 
 §3 
 
 g 
 
 yo 
 
 g 
 
 60
 
 
 TABLt 
 
 , XV. 
 
 NATURAL TANGENTS 
 
 AND COTANGENTS. 
 
 231 
 
 M 
 
 
 
 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 
 
 
 
 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 
 
 
 
 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 
 
 
 ftl. 
 
 
 
 Cotang. 
 
 Tang. 
 
 Tang. 
 
 Cotang. 
 
 Tang. 
 
 Cotang. 
 
 Tang. 
 
 81° 1 
 
 803 1 
 
 793 1 
 
 783 1
 
 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 233 
 
 M. 
 
 
 
 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 
 
 
 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. 
 
 
 
 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 
 
 
 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 
 
 
 
 Cotang. Tang 
 663
 
 sat 
 
 ) TABLE XV. 
 
 NATURAL TANGENTS AND COTANGENTS 
 
 !• 
 
 M 
 
 
 
 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. ] 
 
 
 
 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 
 
 
 
 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 
 
 
 M. 
 
 
 Cotang. 
 6 
 
 Tang. 
 
 Cotang. 
 
 Tang. 
 
 Cotang. 
 
 Tang. 
 
 Cotang. 
 
 Tang. 
 
 
 602 
 
 5 
 
 .93 
 
 5 
 
 83 

 
 238 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 
 
 M 
 
 
 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 
 
 
 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 
 
 C otang. 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 
 
 Cotan g. 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 
 
 
 
 Cotang.^ Tang. M. 
 503 '1
 
 240 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 
 
 M. 
 
 
 
 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 
 
 
 
 
 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 
 
 
 M. 
 
 
 Cotang. 
 
 Tang. 
 
 Coteing. 
 
 Tang. 
 
 Cotang. 
 
 Tang. 
 
 
 453 
 
 450 
 
 450 
 
 
 V
 
 242 TABLE XVI. RISE PER MILE OF VARIOUS GRADES. 
 
 TABLE XVI. 
 
 RISE PER MILE OE VARIOUS GRADES. 
 
 Grade 
 
 per 
 Htatioa. 
 
 Rise per 
 
 Mile- 
 
 Grade 
 per 
 
 Station. 
 
 Rise per 
 Mile. 
 
 Grade 
 
 per 
 Station. 
 
 Rise per 
 Mile. 
 
 Grade 
 
 per 
 Station. 
 
 Rise per 
 Mile. 
 
 .01 
 
 .523 
 
 .41 
 
 21.643 
 
 .81 
 
 42.763 
 
 1.21 
 
 63.838 
 
 .02 
 
 1.0.56 
 
 .42 
 
 22.176 
 
 .52 
 
 43.296 
 
 1.22 
 
 64.416 
 
 .03 
 
 1.5S4 
 
 .43 
 
 22.701 
 
 .83 
 
 43..y2l 
 
 1.23 
 
 64.944 
 
 .04 
 
 2.112 
 
 .44 
 
 23.2.32 
 
 ..S4 
 
 44.3.52 
 
 1.24 
 
 65.472 
 
 .05 
 
 2.640 
 
 .45 
 
 23.760 
 
 .85 
 
 44.850 
 
 1.25 
 
 66.000 
 
 .06 
 
 3.163 
 
 .46 
 
 24.233 
 
 .86 
 
 45.403 
 
 1.26 
 
 66.523 , 
 
 .07 
 
 3.6S6 
 
 .47 
 
 24.816 
 
 .37 
 
 45.936 
 
 1.27 
 
 67.056 
 
 .OS 
 
 4.224 
 
 .43 
 
 2-5.344 
 
 .83 
 
 46.464 
 
 1.23 
 
 67.534 
 
 .09 
 
 4.752 
 
 .49 
 
 25.872 
 
 .89 
 
 46.992 
 
 1.29 
 
 63.112 
 
 .10 
 
 5.280 
 
 .50 
 
 26.400 
 
 .90 
 
 47.520 
 
 1.30 
 
 65.640 
 
 .11 
 
 5.803 
 
 .51 
 
 26.923 
 
 .91 
 
 48.043 
 
 1.31 
 
 69.163 
 
 .12 
 
 6.3.36 
 
 .52 
 
 27.4-56 
 
 .92 
 
 43.576 
 
 1.32 
 
 69.696 
 
 .1.3 
 
 6.S64 
 
 .53 
 
 27.934 
 
 .93 
 
 49.104 
 
 1.33 
 
 70.224 
 
 .14 
 
 7.392 
 
 .54 
 
 23.512 
 
 .94 
 
 49.632 
 
 1.34 
 
 70.752 
 
 .15 
 
 7.920 
 
 .55 
 
 29.040 
 
 .95 
 
 5OI60 
 
 1.35 
 
 71.230 
 
 .16 
 
 8.443 
 
 .56 
 
 29.563 
 
 .96 
 
 50.683 
 
 1.36 
 
 71.808 
 
 .17 
 
 8.976 
 
 .57 
 
 30096 
 
 .97 
 
 51.216 
 
 1.37 
 
 72.336 
 
 .13 
 
 9.504 
 
 .53 
 
 30.624 
 
 .93 
 
 51.744 
 
 1.33 
 
 72.S64 
 
 .19 
 
 10.032 
 
 .59 
 
 31.152 
 
 .99 
 
 52.272 
 
 1.39 
 
 73.392 
 
 .20 
 
 10. .560 
 
 .60 
 
 31.6S0 
 
 1.00 
 
 52.800 
 
 1.40 
 
 73.920 
 
 .21 
 
 11.083 
 
 .61 
 
 32.203 
 
 l.ni 
 
 53.323 
 
 1.41 
 
 74.443 
 
 .22 
 
 11.616 
 
 .62 
 
 32.738 
 
 IM 
 
 53.8.56 
 
 1.42 
 
 74.976 
 
 .23 
 
 12.144 
 
 .63 
 
 33.264 
 
 1.03 
 
 54.354 
 
 1.43 
 
 75.. 504 
 
 .24 
 
 12.672 
 
 .64 
 
 33.792 
 
 1.04 
 
 54.912 
 
 1.44 
 
 76.0.32 
 
 .25 
 
 13.200 
 
 .65 
 
 34.320 
 
 1.05 
 
 55.440 
 
 1.45 
 
 76.560 
 
 .26 
 
 13.723 
 
 .66 
 
 34.S43 
 
 1.06 
 
 55.963 
 
 1.46 
 
 77.038 
 
 .27 
 
 14.2.56 
 
 .67 
 
 35.376 
 
 1.07 
 
 56.496 
 
 1.47 
 
 77.616 
 
 .23 
 
 14.784 
 
 .63 
 
 35.904 
 
 1.03 
 
 57.024 
 
 1.43 
 
 78.144 
 
 .29 
 
 15.312 
 
 .69 
 
 36.432 
 
 1.09 
 
 57.552 
 
 1.49 
 
 78.672 
 
 .30 
 
 15.840 
 
 .70 
 
 36.960 
 
 1.10 
 
 53.030 
 
 1.50 
 
 79.200 
 
 .31 
 
 16.363 
 
 .71 
 
 37.483 
 
 l.Il 
 
 53.608 
 
 1.51 
 
 79.723 
 
 .32 
 
 16.896 
 
 .72 
 
 33.016 
 
 1.12 
 
 59.1.36 
 
 1.52 
 
 80.2.56 
 
 .33 
 
 17.424 
 
 .73 
 
 33.544 
 
 1.13 
 
 59.664 
 
 1.53 
 
 80.784 
 
 .34 
 
 17.952 
 
 .74 
 
 39.072 
 
 1.14 
 
 60192 
 
 1.54 
 
 81.312 
 
 .3.5 
 
 18.450 
 
 .75 
 
 39.600 
 
 1.15 
 
 60.720 
 
 1.55 
 
 81.840 
 
 .36 
 
 19.003 
 
 .76 
 
 40123 
 
 1.16 
 
 61.243 
 
 1.56 
 
 82.363 
 
 .37 
 
 19.536 
 
 .77 
 
 40.656 
 
 1.17 
 
 61.776 
 
 1.57 
 
 82.896 
 
 .33 
 
 20.0&4 
 
 .78 
 
 41.184 
 
 1.18 
 
 62.304 
 
 1.58 
 
 83.424 
 
 .39 
 
 20.592 
 
 .79 
 
 41.712 
 
 1.19 
 
 62.832 
 
 1.59 
 
 a3.952 
 
 .40 
 
 21.120 
 
 .80 
 
 42.240 
 
 1.20 
 
 63.360 
 
 1.60 
 
 &i.480
 
 TABLE XVI. RISE PER MILE OF VARIOUS GRADES. 243 
 
 Grade 
 
 Rise per 
 
 Grade 
 
 Rise per 
 
 Grade 
 
 Rise per 
 
 Grade 
 
 Rise per 
 
 per 
 Station. 
 
 Mile. 
 
 per 
 Station. 
 
 Mile. 
 
 per 
 Station. 
 
 Mile. 
 
 per 
 Station. 
 
 Mile. 
 
 1.61 
 
 S5.003 
 
 1.81 
 
 95.563 
 
 2.10 
 
 110.880 
 
 4.10 
 
 2I6.4S0 
 
 1.62 
 
 65.536 
 
 1.82 
 
 96.096 
 
 2.20 
 
 116.160 
 
 4.20 
 
 221.760 
 
 1.63 
 
 86.064 
 
 1.S3 
 
 96.624 
 
 2.30 
 
 121.440 
 
 4.30 
 
 227.040 
 
 1.64 
 
 86.592 
 
 1.84 
 
 97.152 
 
 2.40 
 
 126.720 
 
 4.40 
 
 232.320 
 
 1.6.5 
 
 87.120 
 
 1.85 
 
 97.630 
 
 2.50 
 
 132.000 
 
 4.50 
 
 237.600 
 
 1.66 
 
 87.643 
 
 1.86 
 
 98.208 
 
 2.60 
 
 137.280 
 
 4.60 
 
 242.880 
 
 1.67 
 
 88.176 
 
 1.87 
 
 93.736 
 
 2.70 
 
 142.560 
 
 4.70 
 
 243.160 
 
 1.63 
 
 88.704 
 
 I.S8 
 
 99.264 
 
 2.80 
 
 147.840 
 
 4.80 
 
 253.440 
 
 1.69 
 
 89.232 
 
 1.89 
 
 99.792 
 
 2.90 
 
 153.120 
 
 4.90 
 
 253.720 
 
 1.70 
 
 89.760 
 
 1.90 
 
 100.320 
 
 3.00 
 
 153.400 
 
 5.00 
 
 264.000 
 
 1.71 
 
 90.233 
 
 1.91 
 
 100.843 
 
 3.10 
 
 163.680 
 
 5.10 
 
 269.230 
 
 1.72 
 
 90.816 
 
 1.92 
 
 101.376 
 
 3.20 
 
 163.960 
 
 5.20 
 
 274.560 
 
 1.73 
 
 91.344 
 
 1.93 
 
 101.904 
 
 3.30 
 
 174.240 
 
 5.30 
 
 279.840 
 
 1.74 
 
 91.872 
 
 1.94 
 
 102.432 
 
 3.40 
 
 179.520 
 
 5.40 
 
 235.120 
 
 1.75 
 
 92.400 
 
 1.95 
 
 102.960 
 
 3 50 
 
 184.800 
 
 5.50 
 
 290.400 
 
 1.76 
 
 92.923 
 
 1.96 
 
 103.483 
 
 3.60 
 
 190.080 
 
 5.60 
 
 295.630 
 
 1.77 
 
 93.456 
 
 1.97 
 
 104.016 
 
 3.70 
 
 195.360 
 
 5.70 
 
 300.960 
 
 1.73 
 
 93.934 
 
 1.93 
 
 104.544 
 
 3.80 
 
 200.640 
 
 5.80 
 
 306.240 
 
 1.79 
 
 94.512 
 
 1.99 
 
 105.072 
 
 3.90 
 
 205.920 
 
 5.90 
 
 311.520 
 
 l.SO 
 
 95.040 
 
 2.00 
 
 105.600 
 
 4.00 
 
 211.200 
 
 6.00 
 
 316.800 
 
 THE mm
 
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 r 
 
 i
 
 L 
 
 /
 
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 ^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 ^
 
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 J
 
 UNIVERSITY OF ILLINOIS-URBANA 
 
 3 0112 084205183