FOR FOR RAILROAD ENGINEERSo CONTAINING F 0 R Al U L iE FOR LAYING C UT CURVES, DETERMINING FROG ANGLES, LEVELLING, CALCULATING EARTII-WORKI ETC., ETC., TOGETHER WITH TABLE S OF RADII, ORDINATES, DEFLECTIONS, LONG CHORDS, BIAGNETIC VARIATION, LOGARITIMIS, LOGARITHMIC AND NATURAL SINES, TANGENTS. ETC., ETC. BY JOHN B. HEN CK, A.mO., CIVIL ENGINEER. NEW YORK: D. APPLETON AND COMPANY9 346 AND 348 BROADWAY. LONDON, 16 LITTLE BRITAIN. 1854. Entered according to Act of Congress, in the year 1854, by D. APPLETON AND COMPANY, in the Clerk's Office of the District Court for the Southern District of New Tc~ rh PR EFACE. THE object of the present work is to supply a want very generally 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 formula for laying out curves, turnouts, crossings, &c., is yet a desideratum. These formulae, 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 invent or collect, or resort to what has received the very appropriate name of" 1 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 inexcusable. 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, in Chapter II., on Parabolic Curves, in ~~ 106- 109, on Vertical Curves, and in the article on Excavation and Embankment. It is a'" vi PREPAACEo but just to add, that a great part of what is said on Reversed Curves, Turnouts, and Crossings, and most of the Miscellaneous 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, indicated by a formula standing on a line by itself, and distin. guished by a El. 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 Curves 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 curvature generally decreases towards both extremities, thus making the transition to and from a straight line easier. Some labor has been given to devising convenient ways of laying out these curves. The method of determining the radius of curvature at certain points is believed to be entirely new. 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 simple and expeditious as those commonly used, with the advantage 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-bed, and depth of cutting, they must be of great extent, and unfitted PREFACE V1ii for a field-book. Even then they apply only to ground whose cross-section is level, though often used in a manner 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 calculated together, as is usually the case, and the work is arrangfed in tabular form, as in ~ 120, the calculation is believed to be at least as short as by the most extended tables, The correction in excavation on curves (~ 129) is not lknown 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, however, carries with it a demonstration, which is deemed an equally essential feature. These demonstrations, with a few unavoidable exceptions, principally in Chapter II., presuppose a knowledge of nothing beyond Algebra, Geometry, and Trigonometry. The result is in general expressed by an algebraic formula, and not in words. Those familiar with algebraic symbols need not be told how much more intelligible 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 collection of mere "6 rules," professing to require only an ability to read for their successful application. Rules can seldom be safely applied without a thorough understanding of the principles on which they rest, and such an understanding, in the present case, implies a knowledge of algebraic formulae. The tables here presented will, it is hoped, prove relia Viii 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 in conducting the operations of the Coast Survey is equalled only by his desire to difflse its results. The remaining tables have been carefully examined by comparing them wvith 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 Logarithmic 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 the work or not resting on sufficient authority. J.oB. H. BOSTON, Februa/y, 1854. TABLE OF CONTENTS. CHAPTER I. CIRCULAR CURVES. ARTICLE I.- SIMPLE CURVES. 5ECT, PAGA 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.o. 4 A. Mlethod 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.. o.. 5 13. Angle of intersection and deflection angle given, to find the length of the curve 6..... 6 14. Deflection angle given, to lay out a curve.. 7 16. To find a tangent at any station... 8 B. 2Method by Tangent and Clord Deflections. 17. Definitions.... o o o. 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 X TABLE OF CONTENTS. C. Ordinates. BECT. PAGE 24. Definition. b.... 11 25. Deflection angle or radius given, to find ordinates.. 11 26. Approximate value for middle ordinate 1.. 13 27. Method of finding intermediate points on a curve approximately.... 14 D. Curving Rails. 29. Deflection angle or radius given, to find the ordinate for curving rails...... 14 ARTICLE IL. —REVEIRSED AND CO3IPOUND CURVES. 30. Definitions...... O 15 31. Radii or deflection angles given, to lay out a reversed or comnpound curve.... 16 A. Reversed Curves. 32. Reversing point when the tangents are parallel o.. 16 33. To find the common radius when the tangents are parallel 16 34. One radius given, to find the other when the tangents are parallel.... b O 17 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.. 19 38. One radius given, to find the other when the tangents are not parallel.. 19 39. To find the common radius when the tangents are not parallel 21 40. Second method of finding the common radius when the tangents are not parallel....... 22 B. Compountd Curves. 41. Common tangent point.. b. 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...... 27 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. xi ARTICLE III.- TURNOUT AND CROSSINGS. SECT. PAGE 49. Definitions o.... 31 A. Turnout fJiom 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 fiog. 34 53. Turnouts that reverse and become parallel to the main track 34 54. To find the second radius of a turnout rceversing opposite the frog..... 35 B. Crossings on Straight Lines. 55. References to proper problems... 36 56. Radii given, to find the distance between switches.. 36 C. Turnout.fiom7 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. 45 64. Common radius given, to find the central angles and chords 45 ARTICLE IV. - MISCELLANEOUS PROBLEM3S. 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 fiom any point on the tangent.... 47 68. Second method for passing a curve through a given point. 47 69. To find the proper (hord for any angle of deflection. 48 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 Xii TABLE OF CONTENTS, SECT. PAGE 72. To find the tangent point of a curve that shall pass through a given point...... ~.. 50 73. To find the radius of a curve without measuring angles. 51 74. To find the tangent points of a curve without measuring angles..... 52 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.. 55 77. To change the tangent point of a curve, so that it may pass through a given point.... 56 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 already laid. o 58 80. To draw a tangent to a given curve from a given point.. 59 81. To flatten the extremities of a sharp curve.,. 59 82. To locate a curve without setting the instrument at the tangent point..... 60 83. To measure the distance across a river... 63 CHAPTEIR II. PARABOLIC CURVES. ARTICLE I. — LOCATING PARABOLIC CURVES. 84. Propositions relating to the parabola... 65 85. To lay out a parabola by tangent deflections.. d 66 86. 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 lay out a parabola by intersections.. 69 ARTICLE II. —RADIUS OF CURVATURE. 92. Definition..7.1.. 71 93. To find the radius of curvature at certain stations 71 95. Simplification when the tangents are equal o o o 76 TABLE OF CONTENTS. Xiii CHAPTER XII. LEVELLING. ARTICLE I.- HEIGHTS AND SLOPE STAKIES. SECT, PAaE 96. Definitions... O n 78 97. To find the difference of level of two points.. 78 98. Datum plane.. 79 99. To find the heights of the stations on a line.. 80 100. Sights denominated plls and minus... 81 101. Form of field notes...... 82 102. To set slope stakes.. 8. 82 ARTICLE II.- CORRECTION FOR THE EARTH'S CURVATURE AND FOR REFRACTION. 103. Earth's curvature.,,,..... 84 104. Refraction,....,,. 84 105. To find the 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 the grades for a vertical curve at sub-stations.. 88 ARTICLE IV. —ELEVATION OF THE OUTER RAIL ON CURVES. 110. To find the proper elevation of the outer rail.. 89 1 11. Coning of the wheels. 89 CHAPTER IV. EARTIH-WORK. ARTICLE I. —PRISMOIDAL FORMIULA. 112. I)efinition of a prismoid....... 92 113. To find the solidity of a prismoid.. 92 ARTICLE II. — BORROW-PITS. 114. Manner of dividing the ground... 93 Xiv TABLE OF CONTENTS. SECT. PAGE 115. To find the solidity of a vertical prism whose horizontal sece tion is a triangle.. 93 116. To find the solidity of a vertical prism whose horizontal section is a parallelogram... 94 117. To find the solidity of a number of adjacent prisms having the same horizontal section O O O 95 ARTICLE III. — ExCAVATION AND EIMVIBANKrMENT. Ao Centrie 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 Heiglts given. 121. Mode of dividing the ground....... 99 122. To find the solidity of one section,,... 100 123. 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. Grou0nd 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... * e e o 110 130. To find the correction in excavation on curves. 112 132. To find the colrrection when the section is partly in excavation and partly in embankment... 113 TABLE S. NO. PAGE I. Radii, Ordinates, Tangent and Chord Deflections, and Or_ dinates for Curving Rails.. o115 II. Long Chords.. O... o 119 TABLE OF CONTENTSo XV NO. PAGE III. Correction for the Earth's Curvature and for Refraction o 120 IV. Elevation of the Outer Rail on Curves 1. l20 V. Frog, Angles, Chords, and Ordinates for Turnouts o o 121 VI. Length of Circular Arcs in Parts of Radius o. 121 VII. Expansion by Heat.. o 122 VIII. Properties of Materials. 123 IX. Magnetic Variation.....126 X. Trigonometrical and Miscellaneous Formuln. 133 XI. Squares, Cubes, Square Roots, Cube Roots, and Reciprocals..... 137 XII. Logarithms of Numbers.. 155 XIII. Logarithmic Sines, Cosines, Tangents, and Cotangents 171 XIV. Natural Sines and Cosines. 219 XV. Natural Tangents and Cotangents o 229 XVI. Rise per Mile of Various Grades.. 242 EXPLANATION OF SIGNS. THE sign + indicates that the quantities between which it is placed are to be added together. The sign - indicates that the quantity before which it is placed is to be subtracted. The sign X indicates that the quantities between which it is placed are to be multiplied together. The sign q- or: indicates that the first 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 Xc indicates that the dlftrenlce of the two quantities between which it is placed is to be taken. The sign.o. stands for the word " hence " or " therefore." The ratio of one quantity to another may be regarded as the quotient of the first divided by the second. Hence, the ratio of a to b is expressed by a: b, and the ratio of c to d by c: d. A proportion expresses the equality of two ratios. Hence, a proportion is represented 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 of measure when no other unit is specified. CHAPTER I. CIRCULAR CURVES. ARTICLE I. - SIMPLE CURVES. 1. THE railroad curves here considered are either Circular or Parabolic. Circular curves are divided into Simple, Reversed, and Compound Curves. We begin with Simple Curves. 2. Let the arc A DEFB (fig. 1) represent a railroad curve, unit~ E D / / I\ 0 2 CIRCULAR CURVES. ing the straight lines GA and B [1. Tile length of such a curve is measured by chords, each 100 feet long.* Thus, if the chords A D, DE, EF, and Bl13 are each 100 feet in length, the whole curve is said to be 400 feet long. The straight lines GA and B 11 are always tangent to the curve at its extremities, which are called tancget points. If GA and BRH are produced, until they meet in C, A C and B C are called the taigents of the curve. If A C is produced a little beyond C to K, the angle K CB, formed by one tangent with the other produced, is called the onyle of inltersection, and shows the chclrqe of direc.. tion in passing fiom 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 B 0. II. Two tangents drawn to a circle from any point are equal, and if a chord be drawn between the two tangent points, the angles between this chord and the tangents are equal. Thus A C — 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, CABR= A O B. 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 subtended by the same chord. Thus, D A = E D 0 E. V. Equal chords subtend equal angles at the centre of a circle, and also at the circumference, if the angles are inscribed in similar segments. Thus, AOD D = D O E, and D A E =- EAFo VI. The angle of intersection of two tangents is equal to the central angle subtended by the chord which unites the tangent points. Thus, K CB = A 0 B. 3. In order to unite two straight lines, as GA and B IH, 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 measure the length of a curve by chords of 50 instead of 100 feet. The chord of 100 feet has been adopted throughout this article; but the formula- deduced may be very readily modified to suit chords of any length. See also ~ 13. SIMPLE CURVES. 3 4. Problem. Given the angle of intersection C 3 = I (fig. 1), and the radius A 0 -= R, to find the tangent A C = T. /A / G R Solution. Draw CO. Then in the right triangle A 0 C we have AC (Tal. X. 3) A -O tan. A 0 C, or, since A 0 C-= I (~ 2, VI.), T = tan. i 7;. T = R tan. i L Example. Given 1 - 22~ 52', and R- = 3000, to find T. Here R - 3000 3.477121 1=- 11~ 26t tan. 9.305869 T- 606.72 2.782990 5. Problem. Given the angle of intersection K CB = I (fig. 1), and the tangent A (7= T, to find the radius A 0 = R. 4 CIRCULAR CURVES. Solution. In the right triangle A 0 C we have (Tab. X. 6) AO R A — = cot. A O C, or = cot.; 1 o.Oo R- T Tcot. i1. Example. Given 1= 310 16' and T= 950, to find R. Here T -= 950 2.977724 -= 15~ 381 cot. 0.553102 R -= 3394.89 3.530826 6. The degree of a curve is determined by the angle subtended at its centre by a chord of 100 feet. Thus, if A OD = 6~ (fig. 1), A D E FB is a 6~ curve. 7. The 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 CB F is the deflection angle of the curve A D E PB, and is half A OD or half FOB. A. M2ethod by Deflection Angles. 8. The usual method of laying out a curve on the ground is by means of deflection angles. 9. ProbeiemoI Given the radius A 0 =- (fig. 1), to find the deflection angle CB F = D. Solution. Draw OL perpendicular to BF. Then the angle B OL B 0 F= D, and B L BF= 50. But in the right triangle B BL OBL we have (Tab. X. 1) sin. B OL = -o 5O.'. sin. D-. Example. Given R 1- 5729.65, to find D. Here 50 1.698970 R _ 5729.65 3.758128 D - 30' sin. 7.940842 Hence a curve of this radius is a 10 curve, and its deflection angle is 301. 10. Probllemli Given the deflection angle B F'- = D (fi.. 1), to find the radius A 0 = R. METHOD BY DEFLECTION ANGLES. 5 50 Solution. By the preceding section we have sin. D -= whence R1 sin. D = 50; 50 sin. D.By this formula the radii in Table I. are calculated. Exampk. Given D - 10, to find R. Here 50 1.698970 D - 1~ sin. 8.241855 R = 2864.93 3.457115 11. Problem.o Given the angle of intersection K C B I (fig. 1), and the tangent A C = T, to find the defection angle CA D - D. Solution. From ~ 9 we have sin. D = R- and from ~ 5, R T cot. 4 I. Substituting this value of R1 in the first equation, we get 50 sin. D= Tcot. -I; 50 tan.' I a, sin. D tan Example. Given I = 210 and T = 424.8, to find D. Here 50 1.698970 = 100 301 tan. 9.267967 0.966937 T'- 424.8 2.628185 D Z 10 15' sin. 8.338752 12. Problem.o Given the angle of intersection K CB = I (fig. 1), and the deflection angle CA D = D, to find the tangent A C = T. Solution. From the preceding section we have sin. D tan. = T Hence, T sin. D 50 tan. I; 50 tan. _ sin. D Example. Given I — 28~ and D = 1~, to find T. Here 50 tan. 14; T — il.I 714.31. sin. 1~ 6 CIRCULAR CURVES. 13. Problemn. Given tle angle of intersection KCB = I (fi. 1), and the deflection angle CA D = D, to find the length of the curvle. Solution. By ~ 2 the length of a curve is measured by chords of 100 feet applied around the curve. Now the first chord AD makes with the tangent A C an angle CAD = D, and each succeeding chord D E, E F, &c. subtends at A an additional angle D A E, E A, &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 CA B - A 0 B = 1 is, therefore, made up of as many times D, as there are chords around the curve. Then if n represents the number of chords, we have n D = - I; i* D If D is not contained an even number of times in. 1, the quotient above will still give the length of the curve. Thus, in fig. 2, suppose D is contained 4A times in AI. This shows that there will be four whole chords and - of a chord around the curve fiom A to B. The angle GA B, the fraction of D, is called a subdleflection angle, and G B, the fiaction of a chord, is called a sub-chlord.",The length of the curve thus found is not the actual length of the are, but the length required in locating a curve. If the actual length of the are is required, it may be found by means of Table VI. E'xanple. Given I = 160 52' antd D = 1~ 20', to find the length of I 80 26' 506' the clurve. Here 2 20 80 = 6.325, that is, the curve D 10 20' 80' is 632.5 feet long. To find the are itself in this example, we take from Table VI. the length of an are of 160 52t, since the central angle of the whole curve is equal to I (~ 2, VI.), and multiply this length by the radius of the curve. Are 100 --.1745329 4" 6~ -.1047198 " 501.0145444 "' 2t -.0005818 " 160 52' -.2943789 * This method of finding the length of a sub-chord is not mathematically accurate; 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. In railroad curves, the error arising from this supposition is too small to be regarded. DIETHOD BY DEFLECTION ANGLES. 7 The radius of the curve is found from Table I. to be 2148.79, and this multiplied by.2943789 gives 632.558 feet for the length of the arc. 14. Problenl. Given the deflection angle D, to lay out a ctrve friom a given tangent p)oint. Fig. 2. i LN Solution. Let A (fig. 2) be the given tangent point in the tangent H C. Set the instrument at A, and lay off the given deflection angle D from A C. This will give the direction A D, and 100 feet being measured from A in this direction, the point D will be determined Lay off in succession the additional angles D A E, E A I, &c., each equal to D, and make DE, EBl, &c. each 100 feet, and the points E, F, &c. will be determined. The points D, E, 1, &c., thus determined, are points on the required curve (~ 7, and ~ 2, III., IV.), and are called stations. If there is a sub-chord at the end, as G B, the sub-deflectiolj angle GA B must be the same part of D that G B is of a whole chord (~ 13). 15. It is often impossible to lay out the whole of a curve, without removing the instrument fiom its first position, either on account of the great length of the curve, or because some obstruction to the sight ma'y be met with. In this case, after determining as many stations as possible, and removing the instrument to the last of these stations, we -qghlt to be able tq find the tnuig'nt ro the culrve at this station; for.. 8 CIRCULAR CURVES. then the curve could be continued by deflections from the new tangent, in precisely the same way as it was begun from the first tangent. 16. Plrob~lemo After running a curve a certain number of stations, to find a tangent to the curve at the last station. Solution. Suppose that the curve (fig. 2) has been run three stations to F, and that FL is the tangent required. Produce A F to IK, and we have the angle KFL =- A F C. But (~ 2, 11.) A F C — A C. Therefore KFL = FA C. Now PFA C is the sum of all the deflection angles laid off fiom the tangent at A, that is, in this case, FA C - 3 D, and the tangent FL is, therefore, obtained by laying off from A F produced an angle KTFL equal to the total deflection from the preceding tangent. If the curve is afterwards continued beyond F, as, for instance, to B, a tangent B N at B is obtained by laying off from FB produced an angle IB N_= L B F = L F B, the total deflection from the preceding tangent FL. B. Mlethod by Tangent and Chord Deflections. 17. Let A B CD (fig. 3) be a curve between the two tangents E A and D L, having the chords A B, B C, and CD of the same length. ifig. 3. A 0 Produce the tangent A, and from B draw B G perpendicular to 4 G. produce also the chords 4 B and B C, anj make the produced METHOD BY TANGENT AND CHORD DEFLECTIONS. 9 parts B H and CK of the same length as the chords. Draw CH and D K. B G is called the tangent deflection, and CH or D K the chord deflection. 18. Problenl. Given the radius A 0- R (fig. 3), to find the tangent dcflection B G, ancd the chord deflection C H. Solution. The triangle CB H is similar to B 0 C; for the angle B 0 C =- 1 -80 - ( 0 B C + B C 0), or, since B C 0 = A B 0, B 0 C - 180 - (O B C +- AB 0) -- CB H, and, as both the triangles are isosceles, the rernaining.angles are equal. The homologous sides are, therefore, proportional, that is, B 0: B C = B C: CH, or, representing the chord by c and the chord deflection by d, R: c =. c: d;., d To find the tangent deflection, draw B3M to the middle of C0,J bisecting the angle COB I, and making Bill C a right angle. Then the right triangles B1 1 C and A G B are equal; for B C - A B, and the angle CB=1cHS= B0o C= —AO B = BAG (~ 2, III.). Therefore B G- = C O-= COH = d, that is, the tangent deflection is helf the chord deflectioz. 19. Problem. Given the deflection angle D of a curve, to find the chord deflection d. Solution. By the preceding section we have d and by ~ 10,? - sin. = Substituting this value of R in the first equation, we find c2 sin. D d - O 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 deflection ~ d = 100 sin. D. By these formul:e 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 D (~ 9 or ~ 11), and then proceeding as in ~ 13. 21. Probaleull. To draw a tangent to the curve at any station, as B (fig. 3). Solution. Bisect the chord deflection HI C of the next station in 3l. 10 CIRCUILAR CURVE&h A line drawn through B and v will be the tangent required; tbr it has been proved (~ 18) that the angle CB N1 is in this case equal to i B 0 C, and B N1' is consequently (~ 2, III.) a tangent at B. If B is at the end of the curve, the tang6nt at B may be found without first laying off H C. Thus, if a chain equal to the chord is extended to H on A B produced, the point H marked, and the chain then swung round, keeping the end at B fixed, until HtJI = 2 d, B m will lbe the direction of the required tangent.* 22. Problem. Given the chorddeflection dc, to lay out a cuzrve ^fr0t a given tangent point. Solution. Let A (fig. 3) be the given tangent point, and suppose d has been calculated for a chord of 100 feet, Stretch a chain of 100 feet from A to G on the tangent B A produced, and mark the point G. Swing tie chain round towards A B, keeping the end at A fixed, until B G is equal to the tangent deflection 2 d, and B will be the first station on the curve. Stretch the chain from B to H1 on AB produced, and having marked this point, swing the chain round, until HC is equal to the cho'rd deflection d. C is the second station on the curve. Continue to lay off the chord deflection from the preceding chord produced, until the curve is finished. Should a sub-chord D F occur at the end of the curve, find the tangent 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 subchord may be found thus. Represent the sub-chord by ct, and the corresponding chord deflection by d', and we have (~ 18) a dt' = 1; but since a d -l we have d': 4 t= C2: C. Therefore A d' _ __ d t Exaniple. Given the intersection angle I between two tangents equal to 160 30', and R _ 1250, to find T, d, and the length of the curve in stations. Here (~ 4) T== R tan. Il 1250 tan. 80 15' 181.24 i ca 1002 (~ 18) d= - 2 8; Pt 1250 * The distance B III is not exactly equal to the chord, blut 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 31 may be calculated; for B 31 B/"5 -- H d:.i ORDINATES. 1t (~ 9) sn. D 1250 —.04 - nat. sin. 2~ 17It; IZ 1250 (~13) __ _ 815 49 =3.60. D 20 17l' 137.5' These results show, that the tangent point A (fig. 3) on the first tangent is 181.24 feet from the point of intersection, - that the tangent deflection G B = — =c- 4 feet, - that the chord deflection H C or fiKD = 8 feet, - and that the curve is 360 feet long. The three whole stations B, C, and D having been found, and the tangent DL drawn, the tangent deflection for the sub-chord of 60 feet will be, as shown above, 60 2 d' = 4 \(0) 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 lF may be found (~ 21) by producing D F to P, making FP -= D7-= 60 feet, and laying off N = 1.44 feet. FNV will be the direction of the required tangent, which should, of course, coincide with the given tangent. 23. Curves may be laid out with accuracy by tangent and chord 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 fiom 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 purpose the chord of 100 feet is divided into a certain number of equal parts, and the perpendicular distances firom 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, whichl are seldomn shorter than 15 feet, have been properly curved (~ 28). 25.. PoT' lento. Givena thle deflection ceagle D or the radius!R of a curvpe, to find the ordinatesfor any chord. Solution. I. To find the middle ordinate. Let AEB (fig. 4) be a portion of a curve, subtended by a chord A B, whicll may be de 12 CIRCULAR CURVES, noted by c. Draw the middle ordinate ED3 and denote it by m. Produce ED to the centre F. and join A F and A E. Then (Tab. X. 3) Fig. 4. BD tan. E A D, or E D = A D tan. E A D. But, since the angle AD EA D is measured by half the arc BE, or by half the equal arc A E, we have E A D = A FE. Thelretorie E D A D tan. A FE, or L91~s -- a c tanl. A FE. When c = 100, AFE = D (~ 7), and m = 50 tan. 1 D, whence 711 may be obtained from the table of natural tangents, by dividing tan. D 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 A D F we have D = F/A 172 - A D2> = /,2. Then =-E -DF= R - DF, or m=1- 112-. c!2 II. To find any other ordinate, as R IV, at a distance DV= b from the centre of the chord. Produce 1i N until it meets the diameter parallel to A B in G, and join R F. Then R G -= JR F2 _ 1 G7 = -/ _ -b2, and RP N-= - G - N G =. R G -- D F, Substituting the value of R1 G and that of D F found above, we have R N V= V122 _- 2 -. /22 _ C2. ORDINATES. 13 By these formulme the ordinates in Table I. are calculated. The other ordinates may also be found from the middle ordinate by the following shorter, but not strictly exact method. It is founded on the supposition, that, if the half-chord B D be divided into any number *of equal parts, the ordinates at these points will divide the are E B into the same number of equal parts, 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 ordinates 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, and A T, we have the angle RAAN-= E AD, since I B is considered equal to R E B. But E A D AFE. Therefore, ]? A NV= - A F.E. In the same rway we should find SA 0 = } A FE, and TA P A= 1R'E. We have then for the ordinates, I N=- A N tan. AN = A c taL. n A FE, S O = A O tan. S1 = c tan. I A FE1, and TP = AP tan. TA P= c tan. I A FE. But, by the second supposition, tan. A ARE = 4 tan. I A FE tan. 1 A FE tan. I- A RE, and tan. 4 A FE = 4 tan. - A RFEl. Substituting these values, and recollecting that I c tan. 2 A FE = m, we have 15, 15 F RN= 0 6 X I c tan. - A FE -,= -1m SO= T X c tan. - A E = 7 7 TP = X - c tan. 4 FE = 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 r 1) (n — 1) m (n + 2) (n -- 2) nz nates, beginning nearest the centle,' +2)(n-2 (n~ + 3) (n- 3) rn.'2, &c. Example. Find the ordinates of an 80 curve to a chord of 100 feet. 15 8 I-Iere m = 50 tan. 20 = 1.746, RN.J= j -6 = 1.637, S O = 4 -r -1.310) 7 and TP - 6- = 0.764. 26. An approximate value of?m also may be obtained fi-om the formula n = R -, R2 - 4 C'2. This is done by adding to the quantity under the radical the very smn-lll fi-action 64 C*, making it a perfect 2 14 CIRCULAR CURVES. ~2 squal'e, the root of which will be R --. We have, then, inm (R C2 ); c2 8 1 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. A B, its middle ordinate CH = E D. Intermediate points on a curve may, therefore, be very readily obtained, and generally with sufficient accuracy, in the following manner. Stretch a cord from A to B, and by means of the middle ordinate determine the point E. Then stretch the cord from A to E, and lay off the middle ordinate CH= 4 E~D, thus determining the point C, and so continue to lay off from the successive half-chords one fourth the preceding ordinate, until a sufficient number of points is obtained. D, Carving Rails. 28. The rails of a curve are usually curved before they are laid. To do this properly, it is necessaryl to know the middle ordinate of the curve for a chord of the length of a rail. 29. ProblemW E Given the rctdizts 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 l, and we have (~ 25) the exact formula mn = R - d/R2 - 4 1, and (~ 26) the approximate formula 211 1 m- 4X. This formula is always near enough for chords of the length of a rail. 50 If we substitute for R its value (~ 10) R = sin. D we have, X sin. D 100 E,7xample. In a 10 curve find the ordinate for a rail of 18 feet in length. Here R is found: by Table Is to be 5729.65, and therefore, REVERSED AND COMPOU ND CURVES. 10 by the first formula, ml = 1 1459.-3 -.00707. By the second formula, m. =.81 sin. 30f =.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 degree may be found with sufficient accuracy, by multiplying this decimal by the number expressing the degree of the curve. Thus, for a curve of 5~ 36' or 5,6%, the ordinate would be.007 X 5.6 =.039 ft..468 in. F or a rail of 20 feet we have /12 = 100, and, consequently, n = sin. D. This gives for a 10 curve, fn =.0087. The corresponding ordinate in a curve of any other degree may be found with sufficient accuracy, by multiplying this decimal by the number expressing the degree of the curve. By the above formula for mn, the ordinates for curving rails in Table X. are calculated. ARTICLE II.- RE1VERSrED AND COMIPOUD) CURVES. 30. Two curves often succeed each other having a common tangent at the point of junction. If the curves lie on opposite sides of the common tangent, they form a reversed curve, and their radii may be the same or different. If they lie on the same side of the common tangent, Fig. 5, AI they have different radii, and form a compoundll-l culre. Thuls A B C (fig. 5) is a reversed cv(., iun(! t md4 A P),t (onr)otln,,n (trv(e. 16 Ci, CU LA R CUt VS. 31. Problehm. ob lay out Wa reversed or a comnpoaund curve, when the radii or deflection anyles and the tangent points are known. Solution. Lay out the first portion of the curve fiom A to B (fig. 5), by one of the usual methods. Find B 1, the tangent to 1 B, at the point B (~ 16 or ~ 21). Then B F will be the tangent also of the second portion B C of a reversed, or B D of a compound curve, and firom this tangent either of these portions may be laid off in the usual manner. A. Reversed Curves. 32. Thol'eorel. T/e reversing point of a reversed curve between parallel tangents is in the line joining the tangent points. Fig. 6. F EB R Demlonstration. Let A CB (fig. 6) be a reversed curve, uniting the parallel tangents HA and B K, having its radii equal or unequal, and reversing at C. If now the cllhords A C and CB are drawn, we have to prove that these chords are in the same straight line. The radii B C and C F, being perpendicular to the common tangent at C (~ 2, I.), are in the same straight line, and the radii A E and B F, being perpendicular to the parallel tangents HA and B K, are parallel. Therefore, the angle A E C = CGFB, and, consequently, E CA, the half supplement of A E C, is equal to' C B, the half supplement of CiB i but these angles cannot be equal, unless A C and CB are in the same straight line. 33. Prot'blemo Given the perpendicular distance between twzo parallde tangents BD b= (fig. 6), and the distance betwveen the two tangent points A B a= a, to determine t/he reversing point C and thle common radius EB C = -CF _ R of a reversed curve uniting the tangents HA and B K. Solution. Let A CB be the required curve. Since the radii are REVEERSED CURVES. 17 equal, and the angle A E C -B F C, the triangles A E C and B F C are equal, and A C = CB=. a. The reversing point C is, therefore, the middle point of A B. To find IR, draw E G perpendicular to A C. Then the right triangles A E G and B A D are similar, since (~ 2, III.) the angle BAD = 2A EC= AEG. Therefore AE: AG AB: BD, or R': - a = a:b; 4b ag2 Corollariy. If R? and b are given, to find a, the equation R - 4b gives a2 = 4 R b; F1., o. a = 2,/ 1 -. Examrples. Given b = 12, and a = 200, to determine R. Here 2002 10000 - 4X 2 -- 12 =833g. Given R = 675, and b = 12, to find a. Here a = 2,/675 X 12 2,,,/8100 = 2 X 90 = 180. 34. ProbleLm. Given the perperndicular distance between tzco parallel tangents B D = b (fig. 7), the distance betzeen the two tangent points A B = a, and the first radius E C = R of a reversed curve uniting the tangents EHA and B K to find the chords A C = a' and CB - a", and the second radius CF -.:E A P B K Fig. 7. Solution. Draw the perpendiculars E G and FL. Then the right triangles ABD and EBA G are similar, since the angle BAD= is CIRCULAR CURV'ES. AEC = AEG. Therefore tAB: BD=EIA:AG, or a:b n, R: 2 a'; 2Rb..at _. —-. Since a' and alt are (~ 32) parts of a, we have 1j' al = a - at. To find Rl the similar triangles A B D and FB L give A B B D.- FB: B Lj, or a: b -= Rt:. all; a aft Example. Given b = 8, a = 160, and Ri = 900, to find a', a"l, and 2 X 900 X 8 Ri. Here at 160 = 90, a = 160- 90 = 70, and R1t' 160 X 70 = 700 2X8 700. 35. Corollary 1. If b, a', and all are given, to find a, R, and R', we have (~ 34) -at + ala'; -a R a — a aa 2h 2b Example. Given b = 8, at 90, and al' = 70, to find a, R, and RI. 160 X 90 160 X 70 Here a = 90 + 70 = 160, R -2 8= 900 and 1 = 2x 700. 36. Corollary 2. If R1, R11, and b are given, to find a, a', and all, we have (~ 35), 1 -t- I aa' 2b 2b -a Therefore a2 = 2b (R + 1R');.. a =- 2 b (R +'). Having found a, we have (~ 34) 2Rb a, 2 R1 b a a Example. Given R = 900, Rt = 700, and b = 8, to find a, at, and alt. Here a =,/2 X 8(900 + 700) =,,/16 X 1600 = 160, al = 2 X 900 x 8 2 X O70 8 16 - = 90, and a" l 10 160. REVERSED CURVES. 19 37. Problena. Given the angle A K B = -i, which shows the change of direction of tiwo tangents HA and B K (fig. 8), to unite these tangents by a reversed curve of given common radius R, starting fomnz a given tangent point A. 1-1 F/ D\ /A N M /B I E:1ny vFig. 8. Solution. With the given radius run the curve to the point D, where the tangent D N beconmes parallel to B K. The point D is found thus. Since the angle N G IC, which is double the angle HA D (~ 2, II.), is to be made equal to A KB -= I, lay off from HA the angle HA D = 2 IC. Measure in the direction thus found the chord A D = 2 R sin. 2 K. This will be shown (~ 69) to be the length of the chord for a deflection angle I K. Having found the point D, measure the perpendicular distance D 31 = b between the parallel tangents. The distance DB = 2 D C = a may then be obtained from the formula (~ 33, Cor.) = cza = 2 VR/ b. The second tangent point B and the reversing point C are now determined. The direction of D B or the angle B D N may also be obD 31 tained; for sin. B D N -= sin. D B D1 = -B or sin. BDN= b. a 38. ProLblem. Given the line A B = a (fig. 9), which joins the fixed tangent points A and B, the angles HA B = A and A B L = B, and the first radius A E =- R, to find the second radius B F = Rl of a reversed curve to ulnite the tangents I't A and B ICK. First Solution. WTVith the given radius run the curve to the point D, where the tangent D N beco7mes parallel to B K. The point D is found '0 CIRCULAR CURVES. thus. Since the angle 1 G N, which is double HA D (~ 2, II.), i, equal to A co B, lay off from lHA the angle HIA D = I (A vc B), ani measure in this direction the chord A D - 2 R sin. I (Ace B) (~ 69). /F Fig. 9. Setting the instrument at D, run the curve to the reversing point C in the linefrom D to B (~ 32), and mneasure D C and CBo Then the similar triangles DEG Cand B F C giveD C: DE= CB: BE, or D C: R CB: Rt; [~, RI: — ~BX Pi, D C Second Solution. By this method the second radius may be found by calculation alone. The figure being drawn as above, wve have, in the triangle A B D, A B a, A D 2 R sin. 2 (A - B), and the included angle DAB -- /lAB - HAED -= A - (A-B)-=. (A + B). Find in this triangle (Tab. X. 14 and 12) B D and the angle A B D. Find also the angle DBL = B + A B D. Then the chord CB- 2 R sin. A B F C = 2 Rt sin. D B L, and the chord D C= 2 R sin. D C 2 R sin. DBL (~ 69). But CB -BD - DC; whence 21' sin. DBL — BD 2R sin. DBL; BD 2 sin. D B L When the point D falls on the other side of A, that is, when the angle B is greater than A, the solution is the same, except that the angle DA B is then 1800 - 1 (A + B), and the angle DBL = B AB D. REVERSED CURVES. 21 39. ]Probl1em. Given, the length of the comamon tangent D G = a, and the angles of intersection I and PI (fig. 10), to determine the common radius CE == C F= R of a reversed cturve to unite the tangents HA and B L. Fig. 10. Solution. By ~ 4 we have D C = R tan. I l, and C G = R tan. 2P; whence R (tan. 1-+ tan. P') = D C + C G = a, or a tan. I- 1 + tan. I It This formula may be adapted to calculation by logarithms; for we sin. 1 (I+ T') have (Tab. X. 35) tan. -1 tan. - 2G =.i o Substituting this value, we get R a cos. I cos. I 1 sin. 3 (I + I') The tangent points A and B are obtained by measuring from D a distance A D = R tan. 1, anrl firom G a distance B G = R tan. 2 It. Example. Given a = 600, 1= 12~, and I' - 8, to find R. Here a = 600 2.778151 I = 60 cos. 9.997614 12 = 40~ cos. 9.998941 2.774706 (- I+ ) = 100 sin. 9.239670 R - 3427.96 3.535036 CIRCULAR CURVES. 40. Problem. Given the line A B = a (fig. 10), which joins the fixed tangent points A and B, the angle DAB = A, and the angle A B G = B, to find the common radius E C - CF = of a reversed curve to unite the tangents HA and B L. Fig. 10. Solution. Find first the auxiliayq angle A KE = B KF, which nab be denoted by Ik. For this purpose the triangle A E K gives A EB; E' = sin. K: sin. E A i. Therefore E K sin. K --- A E sin. E A K = IR cos. A, since E AK = 900 A. In like manner, the triangle B FK gives ~FK sin. i- =- BE sin. FB K = R cos. B. Adding these equations, we have (E K +- FK) sin. Ki = R (cos. A + cos. B), or, since E K + F K- = 2 R, 2 R sin. K = R (cos. A + cos. B) Therefore, sin. I = - (cos. A + cos. B). For calculation by logao rithms, this becomes (Tab. X.. 28) sin. KE= cos. - (A - B) cos. i (A - B). H-Iaving found K, we have the angle A E K = -= 180 - KEA K - 1800 - K - (900 - A) = 90~ + A - K, and the angle B FK1 F 1= 180 -- - FB K = 1800 - KE- (90~ - B) = 90g + B - K Moreover, the triangle A E K gives A E -A K = sin. K: sin. E, or R sin. E -- A K sin. IKt and the triangle B FK gives BiF: B F K = sin. K: sin. F, or R sin. F — B Ksin. KC Adding these equations, we have 12 (sin. E + sin. F) = (A K + B K) sin. K _ a sin. K. Substitnting for sin. E +- sin. F its value 2 sin. I (E + ~ ) COMPOUND CURVES. 23 COS, I (E - F) (Tab. X. 26), we have 2 R sin. 2 (E + F) cos. 2a sin. K (E - F) = a sin. IC. Therefore = — sin. (E tF)cos. 2 (-F)' inally, substituting for E its value 90~ + A - IC, and for F its value 900 - B+- IC, we get 2 (E + F) = 90~ - [KI- I (A + B)], and (E - F) -' (A - B); whence -, 2 ~ R =2a sin. K cos. [K- I (A + B)] cos, I (A - B) Example. Given a =1500, A = 180, and B = 6~, to find R. Here' (A + B) = 12~ cos. 9.990404, (A - B) = 6~ cos. 9.997614 EK= 76~ 36' 10,' sin. 9.988018 a = 750 2.875061 2.863079 K- (A + B) c 640 36' 10" cos. 9.632347 (A - B) = 6~ cos. 9.997614 9.629961 R2= 1710.48 3.233118 B. Comlapound Curves. 41. Tlehoreno. If one brasnch of a comnpound ciurve be prodtuced, ulntil the tangent at its extremity is parallel to the tangent at the extremity of the second branch, the common7e tangent point of the tiuo arcs is in the straiqht line produced, swhich passes throu/gh the tangenlt points of these parallel tangents. Demonstration. Let A CB (fig. 11) be a compound curve, uniting the tangents HA and BK. The radii CE and CF, b)eing perpendicular to the common tangent at C (~ 2, 1.), are in the same straight line. Continue the curve A Cto D, wlhere its tangent OD becomes parallel to B1I, and consequently the radius DE parallel to BF. Then if the chords CD and CB be drawn. we have the angle CED = CFB; whence E CD, the half-supplement of CE D, is equal to F CB, the half-supplement of CFB. But E CD cannot be equal to F CB, unless CD coincides with CB. Therefore the line BD produlced passes through the common tangent point C ~ —---- i"""" I " zn 24 CIRCULAR CURVES. 42. Probltema. To find a limit in one direction of each eradius of a compound curve. A Fig. I.X Solution. Let A I and B I (fig. 11) be the tangents of the curve. Through the intersection point I, draw IM bisecting the angle A IB. Draw A L and B M1 perpendicular respectively to A l and B I, meeting 1 M in L and M1. Then the radius of the branch commencing on the shorter tangent A I must be less than A L, and the radius of the branch commencing on the longer tangent B I must be greater than B M. 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 AL = L N; so that the curve, if continued, will pass through N, where its tangent will coincide with IN. Then (~ 41) the common tangent point would be the intersection of the straight line through B and Nwith 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 tangent becomes parallel to BI. In this case the extremity of the COMPOIND CURVES. 25 curve will fall outside the tangent B 1in the line A N produced, and a straight line through B 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 zA L or greater than AI L, no compound curve is possible. This radius Inust, 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 11i. If we suppose the tangents A I and B I and the intersection angle I to be lknown, wve have (~ 5) A L = A I cot. 1, and B JM = B I cot. I 1. These values are. therefore. the limits of the radii in one direction. 43. If nothing were given but the position of thle tangents and the tanllgent points, it is evident that an indefinite number of diffierent compound curves might connect tle tangent points; for the shorter radius might be taken of any length less thlan the liamit found above, and a corresponding value for the greater could be found. Somle other condition must, therefore. be introduced, as is done in the following problems. 44. Pll'obIlela. Givens the line A B = a (fig. 1I), swhich joins the fixed tanq/ent points A and B, the angle B A 1 = A, the anyle A B I = B, and the first radius A E =- R, tofind the second racdius B F = - Ri of a comipound curve to unite the tangents HL A and B K. Solution. Suppose the first curve to be run with the given radius from A to D, where its tangent DO becomes parallel to B l, and the angle IA D = 2 (A + B). Then (~ 41) the common tangent point C is in the line B D produced, and the chord CB CD + B D. Now in the triangle AB D we have AB a, A D = 2 R sin. 2 (A + B) (~ 69), and the included angle DAB =! AB - IA D = A - (A- B) (A +- 1B). Find in this triangle (Tab. X. 14 and 12) the angle A B D and the side B D, Find also the acngle CB L = B - A B D. Then (~ 69) the chord CB = 2't sin. CB I, and the chord CD 2 R sin. CD 0 = 2 R sin. CB I. Substituting these values of CB and CD in the equation found above, CB - CD ~ B D, we have 2 R1 sin. CBI = 2 R sin. CB 2+ B D; gal'==R~ BD 2 sin. CB I When the angle B is greater than A, that is, when the greater radiqs is given, the solution is the same, except that the angle D, A B 3 26 CIRCULAR CURVES. (B- A), and CB I is found by subtracting the supplement of A B D from B. We shall also find CB = CD - B D, and consequently = -2 sin. CBI' If more convenient, the point D may be determined in the field, by laying off the angle IA D = I (A + B), and measuring the distance A D = 2 R sin. 4 (A + B). BD and CB I may then be measured, instead of being calculated as above. Example. Given a = 950, A = 80, B = 70, and R = 3000, to find R'. Here A D = 2 X 3000 sin. 4 (80~ 70) = 783.16, and DA B = ~ (8~- 7) = 30'. Then to find A B D we have AB -A D == 166.84 2.222300 (A D B A B D) - 89~ 45' tan. 2.360180 4.582480 A B + A D = 1733.16 3.238839 4 (ADB- ABD) -- 87~ 24 17 tan. 1.343641.A BD -= 20~ 20' 43 Next, to find B D, A D 783.16 2.893849 D A B = 301 sin. 7.940842 0.834691 ABD - 20 201 43'1 sin. 8.611948 B D 167.01 2.222743 B D -D CB I = 40 391 171" sin. 8.909292 2 ('I - R) = 2058.03 3.313451.. I - R = 1029.01.. =B 3000 +- 1029.01 = 4029.01 To find the central angle of each branch, we have CF B = 2 C B 1 90 18' 3411, which is the central angle of the second branch; and AEC = AED - CED = A B - 2 CBI = 50 4126,which is the central angle of the first branch. 45. Problem. Given (.fig. 11) the tangents A I T, BI = T', the angle of intersection = — I, 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? 0 becomes parallel to B I. Through CO{MPOUND cURVES. 27 D draw D P parallel to A l, and we have IP = D0=A 0 R tan. I (~ 4). Then in the triangle D P B we have D P = I 0 AI —A O = — T-Jtan., 1 BP = BI-IP= Tt-R tan.Il, and the included angle DP B = A IB = 180~ —. Find in this triangle the angle CB I, and the side B D. The remainder of the solutionz is the same as in ~ 44. The determination of the point D in the field is also the same, the angle IA D being here = I 1 When B is greater than A, that is, when the greater radius is given, the solution is the same, except that D P = IR tan. I I — T, and B P -= tan. 1 t. Example. Given T= 447.32, Tt = 510.84, 1= 150, and R = 3000, to find Rt. Here R tan. I= — 3000 tan. 74~ = 394.96, DP -447.32 - 394.96 = 52.36, BP = 510.84 - 394.96 = 115.88, and DPB 180 - 150~ 1650. Then (Tab. X. 14 and 12) B P-D P - 63.52 1.802910 (B D P - P B D) =7~ 301 tan. 9.119429 0.922339 B P + D P = 168.24 2.225929 4 (B D P P BD) - 2~ 50' 44/" tan. 8.696410. PBD = CB I= 40 39t 16tt Next, to find B D, DP = 52.36 1.719000 D P B = 15~ sin. 9.412996 1.131996 P B D 40 39' 161 sin. 8.909266 b'D= — 167.005 2.222730 The tangents in this example were calculated fiom the example in ~ 44. The values of CBI and BD here found differ slightly 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 joinin(g the tangent points. Problem. Given the line A B = a (fig. 12), which unites the fixed tangent ploints A and B, the angle I A B = A, and the angle A B I = B, to fltdl the radii A F == I and B F = R' of a colmpoinqlc curve, hacving te common lta7n.qent i'? G cr'alel to A B. CIRCULAR CURVES. Solution. Let A C and B C be the two branches of the required curve, and draw the chords A Cand B C. These chords bisect the Fig. 12. F angles A and B; for the angle D A C = ~ ID G = ~ IA B, and the angle G B C - D G = I A B J. Then in the triangle A CB we have AC:AB -sin. ABC: sin. ACB. But ACB= 180s3(CA B ~- CB A) = 1800 - I (A + B), and as the sine of the supplement of an angle is the same as the sine of the angle itself, sin. A CB = sin. I (A + B). Therefore A C: a = sin. I B: sin. a sin. ~L B (A + B), or A C = sin ( B) In a similar manner we should a sin. ~A A A find B C a =sin. (A. _- _ Now wYe have (~ 68) R sin A, and sin. B, ol, substituting the values of A Cand B Cjust found, I Rasin. I B' a sin. IA sin. I A sin. ~ (A + B); sin. I B sin. (A + B)' Example. Given a = 950, A = 80, and B = 70, to find R and B'.'ere COMPOUND CURVES. 29 ~ a = 475 2.676694 I B = 30 30' sin. 8.785675 1.462369 2 A = 40 sin. 8.843585 2 (A + B) = 70 30' sin. 9.115698 7.959283 R = 3184.83 3.503086 Transposing these same logarithms according to the formula for r', wc have 2 a = 475 2.676694 - A = 40 sin. 8.843585 1.520279 1 B - 30 30t sin. 8.785675 2 (A + B)= 70 301 sin. 9.115698 7.901373 R' = 4158.21 3.618906 47. Problem. Given the line A B = Ca (fig. 12), which unites the fixed tangent points A and B, and the tangents A I = 1' and B I = T', to find the tangents A D _ x and B G = y of the two branches of a conmpound curve, having its common tangent D G parallel to A B. Solution. Since D C = A D = x, and CG = B G=- y, we have D G = x y. Then the similar triangles ID G and IA B give ID: IA = DG: AB, or T- x: T -= x + y: a. Therefore aT — ax T= Tx +- Ty (1). Also AD: AI = B G: BI, or x: T== y: Tl. Therefore Ty = Tzx (2). Substituting in (1) the value of Ty in (2), we have a'- a x = Tx ~ T' x, or a. + Tx + T'x = a T;.x = a T a+- T 2- TI T'x and, silnce firom (2),y - T - a T a+ T+'l The intersection points D and G and the common tangent point C are now easily obtained on the ground, and the radii may be found by the usual methods. Or, if the angles IA B = A and A B I = B 3* 30 CIRCULAR CURVES. have been measured or calculated, we have (~ 5) R = — x cot. I A, and R = — y cot. 2 B. Substituting the values of x and y found above, we T cott i A a T cot. I; have R 1 ~ T T, and f' -RT-.l at T1- T a + T + TO' E xample. Given a = 500, T -- 250, and T' _ 290, to find x and y. Here a + T + T' - 500 + 250 + 290 =- 1040; whence x = 500 X 250 —. 1040 = 120.19, and y = 500 X 290 -~ 1040 = 139.42. 48. Problem. Given the tangents A I =- T,. B I = T', and the angle of intersection 1, to unite the tangent points A and B (fig. 13) by a comnpound curve, on condition that the two branches shall have their angles of intersection ID G and I G D equal. Fig.13. m. SolutXon. Since ID G I GD we have ID G. Represent the line I J = I G by x. Then if the perpendicular IH be let * The radii of an oval of given length and breadth, or of a three-centre arch of given span and rise, may also be found from these formule In these ecases A +- B = 90:? a T' and the values of R and R' may be reduced to R = + T- and R'= a T' aT- - T" These values admit of an csy comstruction, or they may be readily a cul T-a T calculated. TURNOUTS AND CROSSINGS. 31 fall from l, we have (Tab. Xo 11 ) D H = I D cos. 1D) G = x cos. 1I, and D G-=2 x cos. B L 3ut D G == D C CG == AD) + B G=T- x + T' — x -= T- T' —2 x. Therefore 2 x cos. I -1 T + T' - 2 x, or 2 x + 2 x cos. I - = T -+ TI; whence x I (T+ TI) 1 - cos. ~, or (Tab. X. 25) Ep~ X = 41 (T + T') cos.2 1 1 The tangents AD = T — x and B G = T' - x are now readily found. With these and the known angles of intersection, the radii or deflection angles may be found (~ 5 or ~ 11). This method answers very well, when the given tangents ale nearly equal; but in general the preceding method is preferable. Exaciple. Given T= 480, Tl = 500, and I= 180, to find x. Here (T - T) = 245 2.389166 =- 4~ 30t 2 cos. 9.997318 x - 246.52 2.391848 Then A D = 480 -- 246.52 = 233.48, and B G 0= 500 - 246.52 - 253.48. The angle of intersection for both branches of the curve being 9~0, we find the radii A E = 233.48 cot. 40 30' -- 2966.65, and B F = 253.48 cot. 40 30' = 3220.77. ARTICLE III.- TURNOUTS AND CROssINGs. 49. THE usual mode of turning off from a main track is by switching a pair of rails in the main track, and putting in a turnout curve tangent to the switched rails, with a fiog placed where the outer rail of the turnout crosses the rail of the main track. A B (fig. 14) represents one of the rails of the main track switched, B Frepresents the outer rail of the turnout curve, tangent to A B, and F shows the position of the frog. The switch angle, denoted by S, is the angle D A B, formed by the switched rail A B with A D, its former position in the main track. The frog angle, denoted by 1, is the angle GF-7M made by the crossing rails, the direction of the turnout rail at F being the tangent F l -at that point. In the problems of this article the gauge of the track. D C. denoted by y. and the distance D B, denoted by d, are supposed to be known. The switch angle Sis also supposed to be known, since its sinP (Tab. X 1) is e(lual to dl divided by the length 32 CIRCULAR CURVES. of the switched rail. If, for example, the rail is 18 feet ia length and d =.42, we have S -= 1 20'. A. Tarnoit from traigllt Lines. 50. PD'obllems Given the roadius R of the centre line of a tur2'no (fig. 14), to find the f.og angle G F M F anEd the chord B' F. A.N D 11ig. 14. B H K Solution. Through the centre E draw E K parallel to the main track. Draw B H and FK perpendicular to E K, and join E F. Then, since E F is perpendicular to F2M and FK is perpendicular to F G, the angle E' FKi = G FilM F; and since E /3 and B H are respectively perpendicular to A B and A D, the angle E B H =H D A B - S. Now the triangrle KT gives (Tab. X. 2) cos. E F i-7 F But E F, the radius of the outer rail, is equal to IR - + g, and FK= CH= B - B C= 3 E cos. EBH C- B C = (+2) cos. S - (g - d). Substituting these values, we have cos. E FK == ( r- +g) cos. S -( - d) AS + g, 2or cos. = cos. S -- 9d. From this formula Fmay be found by the table of natural cosines. To adapt it to calculation by logarithms, we may consider g - d to be equal to (g- d) cos. S, which will lead to no material error, since TURNOUT FROSI STRAIGHT LINES. 33 g — d is very small, and cos. S almost equal to unity. The value of cos. F then becomes cos. - = (R -2.+ d) cos. S To find BEV, the right triangle B CF gives (Tab. X. 9) B F BC Bsn.BFC' But BC = - d and the angle BFCl = BFE CFE = 90~ - BEF) — (90 -; F) = F -- EF. But BE = BL F -EBL = E- S. Therefore BFC = -. (F - S) =- (F - S).- ubstituting these values in the formula for B F, we have 97-~~ BF — B- d sin..~ (F + S) By the above formulw the columns headed F and BF in Table V. are calculated. Example. Given g = 4.7, d =.42, S = 1~ 201,'and R - 500, to find F and B E. Here nat. cos. S =.999729, - d = 4.28, R + I g — 502.35, and 4.28 * 502.35 =.008520. Therefore nat. cos. F =.999729 -.008520 =.991209, which gives F =- 7 36' 10". Next, to find B F, g - d =-4.28 0.631444 T (F ~ S) = 40 28t 5" sin. 8.891555 BF = 54.94 1.739889 51. Problem. Given the frog angle G FM = F (fig. 14), to find the radius R of the centre line of a turnout, and the chord B Fo Solution. From the preceding solution we have cos. F = (R +g. Therefore (R + ~ g) cos. F= (R + I g) cos. S - (g - d), or R t- ~g = cos. F - For calculation by logarithms this becomes (Tab. X. 29) " (g-d) R g in. ( + S) s in. 2(F S) IHaving thus found Pi + I g, we find R by subtracting g. B F is found, as in the preceding problem, by the formula sin. 5 (F7 + S) 34 CIRCULAR CURVESd Example. Given g = 4.7, d =.42, S = 1~ 20', and P = 70, to find R. Here ~ (g - d) 204 0.330414 (F+ S) 4~0 10o sin. 8.861283 ~ (F — S) - 2~ 50' sin. 8.693998 7.555281 _ + 1 g 595.85 2.775133.o 1- 593.5 52. Problem. To find mnechainically the proper position of a given frog. rSolltion. Denote the length of the switch rail by 1, the length of the frog by f and its width by zv. From B as a centre with a radius _, H = 2 1, describe on the ground aIn arc G Kl(fig. 15), and fiom the inside of the rail at G measure G H = 2 d, and from H measure HK such that HKI: B H = I w': f, or fIf: 21 = w: f; that is, 1tK —'.-. Then a straight line through B and the point K will strike the inside of the other rail at F, the place for the point of the Fig. 1 5, frog. For the angle HB K has been made equal to I,' and if B 1M be drawn parallel to the main track, the angle M1 B H is seen to be equal to 2 S. Therefore,.MI fB K =B P C =' (IF + 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 commuon radius given, the reversing point and the other tangent point may be found by ~ 37, the changme of direction of the two tangel: ts bl)cing lere equal to S. But when the TURNOUT FROMI STRAIGHT LINES. 35 fiog 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 second portion may be found by the following method. 54. Problem. Given the frog angle Fand the distance HtB = b (,fiy. 16) betiween.the main track and a turnout, to find the radius R'I of the second branch of the turnout, the reversing point being taken opposite F, the point of the friog. C/ Fig. 16. _ _ _/II Solution. Let the are FB be the inner rail of the second branch, F G = RI - 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 K is also equal to F, B F = F (~ 2, II.); whence'-BF BFHI,= F. Then (~ 68) F G = sin F X or'BF HB Jb Bsi. t BF= sin. FH(Tab. X. 9), or BFsin F SubSin. IF 2 sin. BF stituting this value of 1 B F, we have R' — g- sin F In measuring the distance HB — b, it is to be observed, that the widths of both rails must be included. 36( CIRCULAR CUJRVIES, Examnple. Given b - 6.2 and F1= 8, to find It'. HIere -- b = 31 0.491362 ~ F= 43 sin. 8.843585 B F =- 44.44 1.647777' F = 4' sin. 8.843585'! -2 g = 637.08 2.804192..' 639.43 B. Crossings on Straiglt Lines. 55. WVhen a turnout enters a parallel main track by a second switch, 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 tan. gents. Let HD and 1VK (fig. 17) be the centre lines of two parallel tracks, and 11A and B K the direction of the switched rails. If now the tangent points A and B are fixed, the distance A B = a may be measured. and also the perpendicular distance B P = hb between the tangents HP and BIC. Then the common radius of the crossing A CB may be found by ~ 33; or if the radius of one part of the crossing is fixed, the second radius may be found by ~ 34. But if both fiog angles are given, we have the two radii 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. Problemn Given the perpeendicular distance G N= b (fig. 17) between the centre lines of tsvo parallel tracks, and the radii E C == R and CrF = Rl of a crossing, to find the chords A C and B C. Soluztion. Draw E G perpendicular to the main track, and A L, CMI, and B L' parallel to it. Denote the angle A E C by E. Then, since the angle A EL = A HG = S, we have CEL = E + S, and in the riglt triangle CEM (Tab. X. 2), CE cos. C EM = R cos. (E + S) = E ll = E L - L1f. But E L = A E cos. A E L - R cos. S, and LM': LtM = A C: B C. Now A C B C = E C: C:F = R:R'. Therefore, L M: LtM 1 = R: R', or LM A: LM + L' l= _ R: R + R ti; that is, L Ml: b- 2 d = R: + R', whence L (b - 2 d) L M R + R-, Substituting these values of E L and L Mir the equation for R cos. (E + S), we have R cos. (E - S = R cos. S (b - 2 d).+? CROSSINGS ON STRAIGHT LINES. 37 RF'I.. cos. (E S- ) = cos. S —b 2 Having thus found E + S, we have the angle E and also its equal CFB. Then (~ 69) A C= 2 R sin. ~ E; B C = 2 RI sin. ~ E. We have also A B = A C + B C, since A C and B Care in the same straight line (~ 32), or A B = 2 (R + R') sin. 2 E............ _ ~~~E'l,~~~~~~ ~Fig. 17. When the twzo radid are equal, the same formule apply by making Rt = R. In this case, we have cos. (E- +S) = cos. S- b - 2d. GY, A C = B C = 2R sin. I E. Example. Given d =.42, g = 4.7, S = 1~ 20', b = 11, and the angles of the two frogs each 70, to find A C = B C = ~ AB. The common radius R, corresponding to F = 7~, is fbound (~ 51) to be 593.5. Then 2R1 = 1187, b - 2 d 10.16, and 10.16 - 1187-.00856. Therefore, nat. cos. (E + S) =.99973 -.00856 =.99117 i whence E + S = 7~37t 15'1. Subtracting 5, we have E = 60 17' 15". Next 2R = 1187 3.074451 E- 3~ 8' 537~1 sin. 8.739106 A C 65.1 1.813557 4 38 CIRCULAR CURVES. C. Turnout firom Curves. 57. Problem. Given the radius R of the centre line of the nain 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 RI of the centre line of the turnout. D Fig. 18. Solution. I. When the turnout is from the inside of the curve (fig. 18). Let A G and CF be the rails of the main track, A B the switch rail, and the arc B F the outer rail of the turnout, crossing the inside rail of the main track at F. Then, since the angle E FKhas 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 FKI = F. Also E B L = S. The Jirst step is to find the alngle B KiF denoted by KC To find this angle, we have in the triangle B FK (Tab. X. 14), B K+ KF: B K- KF = tanI (B FK+ FBK) tan. (BFK- FB K). But B K- = - g - d, and KF = R - I/g. Therefore, B K + KF = 2R-d, and BK - KF.=:g- d. Moreover, BFK=BFE + EFK = BFE + F, and FBK -= EBF- EBK B FE —S. Therefore, B F K - FB K = F + S. Lastly, B FK + FBK I= 1800~- -.. Substituting these values in the preceding proportion, we have 2 ) -- d: -d = tan. (900 -- K): tan. 1. (F+ S), TURNOUT FROM CURVES. 39 (2 R - d) tan. I (F + s) or tan. (900 1 I (2 tn g- But tan. (90 -- ) = cot. I X tan. K ic K.tan. I K) —d.. tan. E(2 R d) tan. ~ (F+- S) Next, to find the chord B F, we have, in the triangle B F C B Csin. B CF' (Tab. X. 12), B F sin F C But B C= g - d, and B CF = 180-F CK = 1800 — (900 - K) = 900 + -K IC, or sin. B CF - cos. K. Moreover, BFC = (F+ S); for B F = KFC 4+ B F C, and FB K = KK CF -E BGF C F C -- B F C. Therefore, B FK - FB K 2B F C. But, as shown above, B FK - FB K'= F S. Therefore, 2 B F C F+ S or B F C - (F S), Substituting these values in the expression for B F, we have ~ F I= (g - d) cos. I K sin. ~ (F + S) Lastly, to find R', we have (~ 68) RI +'g = E F in._ BEEF But BEF = BLF - EBL, and BLF - LFK+ LKF-= F + K. Therefore, B E F -= F - IC- S, and BF sin. ~ (F K- S) II. When the turnout is from the outside of the curve, the preceding solution requires a few modifications. In the present case, the angle FKt =' F (fig. 19) and EB L-= S. To find IC we have in the triangle BFK, F + BK: KF- BK- tan. (FB K - B FK) tan. - (FB K - BF). But KF= — R + ~ g, and BK -R- -g - d. Therefore, KF + BK=- R + d, and KF BK= g - d. Moreover, FB K = 180~ - FB L = 180 (EBF- EBL) S10o - (EBF - S), and BFK = 180 - BFKt = 1800 - (BFE + EFKIt) = 180~ - (EBF + F). Therefore, FB - BF - F = F S. Lastly, FB K' + BFK = 180~ - K. Substituting these values in the preceding proportion, we have 2R + d: g - d = tan. (900 ~ — ): tan. I (F -- S), or (2 RL 3l) tan. i~ (F+- S) tan. (90 -- ) _ (2 d) tan.1(F + S) But tan. (900 - ) = cot.'K tKn=K; 2e.'. tan. K= - - (2 R +- () tan. - (F+ S) 40 CIRCULAR CURVES. Next to find BF, we have, in the triangle B F C, B F B Csin. B CF ]sins. FC But BC = g -- d,and BCF -- B 9o0~- K, or L \ \ Fig. 19. \ \ sin. B CF cos. K. Moreover, B F C = (F+ S); for BE eK =KFC —BFC, and FBK=KCF-+BFC=EKFC+BFCU Therefore, FB K- B FK= 2B F C. But, as shown above, FBK- BFK= F+ S. Therefore, 2 BFC= F+ Sor BFC= (F- S). Substituting these values in the expression for B F, we have, as before, 0X' F= (g - d) cos. K2 sin. I (F + S) Lastly, to find R', we have (~ 68) R' + -- = F = in. g1s ~-F * Since ~ K is generally very small, an approximate value of B F may be obtained by making cos. = 1. This gives BF - d which is identical sin. (F +- S) i with the formula for B F in ~ 50. Table V. will, therefore, give a close approximao tion to the value of B F on curves also, for any value of F contained in the table. TURNOUT FROM CURVES. 41 But BEF BLF - EBL, andBLF= LFK -LIKF F- I. Therefore, B E FF - K-F S, and RBF 2 in. ( - K- S) Exanmple. Given g 4.7, d =.42, S = 10 20', R = 4583.75, and F - 7*, to find the chord B F and the radius RI of a turnout from the outside of the curve. Here g - d = 4.28 0.631444 0.631444 2 1: - ci = 9167.92 3.962271 2 (F- S) = 4 10' tan. 8.862433 sin. 8.861283 2.824704 1.770161 I K 22o 1. 8t tan. 7.806740 cos. 9.999991 B F = 58.905 1.770152 2 0.301030 * (F- K- S) -= 2 27t 58.211 sin. 8.633766 8.934796 RI - + y = 684.47 2.835356.. = 682.12 58. Problem. To find mechanically tihe proper position of a given frog. Solution. The method here is similar to that already given, when the turnout is from a straight line (~ 52). Draw B M(figs. 18 and 19) parallel to F C, and we have FB = B F C -= 2 (F + S), as just shown (~ 57). This angle is to be laid off firom B Ml; 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 Nwhich the angle ~ (F + S) may be laid off by the method of ~ 52. In this case, however, the first measure on the are is d, and not 2 d since we have here to start firom B 1AT, and not from the rail. Having thus determined the point F approximately, B Al 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 suchl 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 required. 4,, 42 CIRCULAR CURVES. Problem. Given the position of a frog by means of the chord B j (figs. 14, 18, and 19), to determnine the fr-og angle F. ot - fl Solution. The formula B F = sin. (F S) which is exact oi straight lines (~ 50), and near enough on ordinary curves (~ 57, note) gives Kg~3 sin. 2 (F + S) - - d 2 BBF By this formula I (F + S) may be found, and consequently F. 60. P'roblem Given the radius R of the centre line of the maiai track, and the radius Rt of the centre line of a tiurnouit, to finld the fio09 angle F1 and the c]hord B E (figs. 18 and 19). Solution. I. When the turnout is from the inside of the curve (fig. 18). In the triangle B E Kfind the cngle B E K and the side E JI For this purpose we have BE =' R+ g, BE =K + 2 g - d, and the included angle E B K= S. Then in the triangle E FKwe have E IC, as just found, E = lg' -+ I g, and FK = R - - g. The frog angle E FK = F may, therefore, be found by formula 15, Tab. X., which gives t2. ~ FJ =(s -b) (s-) s (s-a) where s is the half sum of the three sides, a the side E I, and b and e the remaining sides. Find also in the triangle E FK the angle FE K', and we have the angle B E 1 B E I - FE IC. Then in the triangle BEF we have (~ 69) B F = 2 (R' + 1 g) sin. - B E F.* II. When the turnout is from the outside of the curve (fig. 19). Inz the triangle BE K find the angle B E K and the side E IC. For this purpose we have B E = RI + I g, B K = --- I g + d, and the included angle EB K = 1800 - S. Then in the triangle E Fi we have.E K, as just found, E F =Rt + yg, and F K = R+ - g. The angle E FK may, therefore, be found by formula 15, Tab. X., which gives tan. 2 EI sE= ( s(s -c). But the angle EFKIl --- F gives tan. ~ E FK 4(s - b) (s - c) * The value of B F may be more easily found by the approximate formula B F = g-d sin (F - S) i and generally with suffcient accuracy. See note to ~ 57. This remark applies also to B F in the second pa-rt of this solution. TURNOUT FROMI CURVES, 43 -1800 ~-E FK. Therefore F -= 900-~ E FICK, and cot. ~ _F-= tax. 2 E FK;.. cot. F= F (s - b) (s - c) s (s - a) where s is the half sum of the three sides, a the side E K, and b and c the remaining sides. Find also in the triangle E F K the angle FE KC, and we have the angle B E F FE K -B E K. Then in the triangle B E F we have (~ 69) BF = 2 ( it- ~ g) sin. ~ B E F. Example. Given g = 4.7, d.42, S = 1~ )20, I 4583.75, and RIt = 682.12, to find F and the chord B F of a turnout from the outside of the curve. Here in the triangle B EK (fig. 19) we have BE = Rl + 1 g 684.47, B KI = R - 2g - d - 4581.82, and the angles BEK+ IE- S- 10 20'. Then B -- B E- 3897.35 3.590769 ~ (B E K BKE) = 40t tan. 8.065806 1.656575 B K+ B E 5266.29 3.721505 (B E K B KE) = 29.6029' tan. 7.935070.o. B E 1C= 10 9.6029' B K sin.EBK E K is now found by the formula E IK - sin B EK, or log. E log. 4581.82 + log. sin. 1780 40' - log. sin. 1~ 9 6029' = 3.721491, whence E K -= 5266.12. Then to find F, we have: in the triangle E FI, s = ~ (5266.12 + 684.47 + 4586.10) = 5268.34, s - a = 2.22, s - b = 4583.87, and s - c = 682.24. s - b 4583.87 3.661233 s - c 682.24 2.833937 6.495170 s = 5268.34 3.721674 s - a - 2.22 0.346353 4.068027 2)2.427143 1 1F= 3~ 301 cot. 1.213571'. -= 7~ * This angle and the sine of 1o 9 6029' belowv, are found by the method given in connection with Table XTII. If the ordinary interpolations had been used, we should have founld F -- 7j 7, wlhereas it should be 7), since this exa.mple is the converse of tl 1t in ~ 57. 44 CIRCULAR CURVES. To find FE IC, we have s as before, but as a is here the side FK opposite the angle sought, we have s -- a = 682.24, s - b = 4583.87 and s - c = 2.22. Then by means of the logarithms just used, we find F2E IK = 33 2' 45I". Subtracting ~ B E K = 341 48It, we have 2BEF = 2~ 27t 57/". Lastly, BF - 1368.94 sin. 2~ 271 57t - 58.897. The formula BF= sin. g (F S) (~ 57, note) would give B F 58.906, and this value is even nearer the truth than that just found owing, however, to no error in the formulm, but to inaccuracies 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 C]B (fig. 20), it will be necessary to determine the reversing points C and B. These points will be determined, if we find the angles A E C and B F C, and the chords A C and CB. 62 Problemna, Givenr the radius D K = I (fig 20) of the centre line of the main track, the conimmon radius E C C F = RII of the centre line of a tournout, 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 chords A C and B C. D, Fig. 20. Solution. In the triangle A E K find the alngle A E K and the side CROSSINGS ON CURVES. 45 E K. For this purpose we have A E = t', A K = R - d, and the included angle E A K = S. Or, if the frog angle has been previously calculated by ~ 60, the values of A E K and E K are already known.* Find in the triangle E FK the angles E EK and FE IK. For this purpose we have E ]K, as just found, E F = 2 P1', and FK -~ RL - Rt - b. Thlenz A E C = AE K — 1FE K, and 1BFC = E K IC. Lastly, (~ 69) A C = 2 RI sin I A E C; C'B = 2 R sin. I B F C. This solution, with a few obvious modifications, will apply, when the turnout is from the outside of a curve. ID. Crossinys'on C'uves. 63. When a turnout enters a parallel main track by a second switch, it 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 mnakes with the tangents at A and B. The common radius 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 A E C and B F C, and the chords A C and CB. 64. Problemn. Given the radius D K - (Jiq. 21) of the centre line of the main track, the common radius E C = CF = R' of the centre line of a crossing, and the distance D G = between the centre lines of the parallel tracks, to finzd the central angles A E C and B F C and the chords A Cand CB. Solution. In the triangle A E K find the angle A E K and the side E K. For this purpose we have A E = R', A K = R - d, and the included angle EA K = S. TFind in the trianzgle B FK the angle B _FK and the side FI K. For this purpose we have BF= B', B-: = R - b + d, and the included angle F BK = 180~ - S. Find in the triangle E F K the angles FE K and E FIK. For this * The triangle A E K does not correspond precisely with B B K in ~ 60, A being on the centre line and B on the outer rail; but the difference is too slight to affect the calculations. 46 CIRCULAR CURVES. purpose we have E K and FK as just found, and E F = 2 t'. Then A E C=-A E K- FE IC, and B F C = E FK-B FKC. Lastly, (~ 69,) AC=2R'tsin.-AEC; CB=2R/'sin. BFC. D A Fig. 21. ARTICLE IV. -.MISCELLANEOUS PROBLEMIS. 65. Problrem. Given A B = a (fig. 22) and the perpendicular B C = b, to find the radius of a curve that s/hall pass tlhrought C and the tangent point A. Solution. Let 0 be the centre of the curve, and draw the radii A 0 and CO and the line CD parallel to A B. Then in the right triangle COD we have O C2 = CD2 + 0D2. But O C- R, CD == a, and OD = A 0 - A D R-b. Therefore, 122 = a2 + (R —b)2 a2 + R2 - 2 R b + b2, or 2 Rb = a2 + b2; e~. ri = a + 2 b 2b Exatmple. Given a = 204 and b = 24, to find R. Here R 2042 24 2x24 + 2 = 867 +- 12 = 879. MISCELLANEOUS PROBLEMS. 47 66. Corollary 1. If R and b are given to find A B = a, that is, to determine the tangent point fiom which a curve of given radius A TI D Fig. 22. 0 must start to pass through a given point, we have (~ 65) 2Rb = a2 + b2, or a2 = 2Rb b2; E ~ ~ *.. a = l6 (2 K - b). Example. Given b = 24 and RI = 879, to find a. Here a = V/24 (1758 - 24) =./ 41616 = 204. 67. Corollary 2. If ] and a are given, and b is required, we have (~ 65) 2lb =- as + b2, or b2 - 2Rb = —a2. Solving this equation, we find for the value of b here required, I~~n~'" 6 b = R - /I2 -- a. 68. Problem. Given the distance A C = c (fig. 22) and the angle B A C = A, to find the radius R or deflection angle D of a curve, that shall pass throzugh C and the tangernt point A. Solution. Draw 0 E perpendicular to A C. Then the angle A 0 E = A 0 C = BA C'= A (~ 2, III.), and the right triangle A OE gives AE (Tab. X. 9) A 0 = sin. A OE;..R / s= 2 sin. A To find D, we have (~ 9) sin. D =. Suhbstituting for R its value just found, we ha]ve sin D - 50 sin A; shin.A 48 CIRCULAR CURV'ES. 100 sin. A.. sin. D. C Exanmple. Given c = 285.4 and A - 50~, to find Rf and D. Here 142.7 100 sin. 50 sin. 5' -- sin. 5 -1637.3; and sin. D- 285.4 - 2.854 sin. 1~ 45', or D= 10451. 69. P'roblemlrn. Given the radits Pr or the deflection angle D of a curve, and the angle B A C A=:t (fil/. 22), made by anly chord with the tantgent at A, to find the length of thle chord A C = c. Solution. If f is given, we have (~ 68) 1- si,-t;.'. C = 2 R sin. A. 100 sin. A If D is given, we have (~ 68) sin. D --- s A; i 100 sin. A sin. D This formula is useful for finding the length of chords, when a curve is laid out by points two, three, or more stations apart. Thus, suppose that the curve A C is four stations long, and that we wish to find the length of the chord A C. In this case the angle A =4 D and c 100 sin. 4 D 0sin..D-. Bv this method Table II. is calculated. sin. D Example. Given t = 2455.7 or D = 10 10', and A 40 401, to find c. Here, by tile first formula, c = 4911.4 sin. 40 40' = 399.59. 100 sin. 40 40' By the second formula, c sin. 1 10' 399.59. 70. Problemino Given the anyle of intersection I CB = 1 (fig. 23), and the disteLnce CD -= from the intersection point to the curve in the direction of the centre, to Jind the tangent A C = T, cetd the raedias A 0 ft. Solution. In the triangle A D Cwe have sin. CA D: sin. A D C- CD: A C. Bult CA D A 0D 41(~ 2. 111. anti VI.), and as the sine of an angle is the same as the sine of its supplement, sin. A D C sin. A E = cos. DA E = cos. 41 1. Moreover, CD b and A C = T. Substituting these values in the precjding proportion, we have sin.: cos. 4 1 = b T, or T or T- =. t vence 4(Tab. ~ r-l X 33)sin. whnce (Tab. X. 33) MISCELLANEOUS PROBLEsS. 49 T= b cot. I. To find /2, we have (~ 5) R- = T cot. L. Substituting for T its value just found, we have R = b cot. 1 cot. ~I.L A - Fig. 23. Example. Given I = 30, b 130, to find T' and R. Here b 130 2.113943 4I - 7~ 30' cot. 0.88057.1 2' = 987.45 2.994514 I -=15~ cot. 0.571948 - 1 3685.21 3.566462 71. Plioblen. Given the angle of intersection - CB = I (fig. 23), and the tangent A C T= T, or the radius A 0 = R, to Jfid CD = b. Solution. If T is given, we have (~ 70) T = b cot. 1I, or b= oeat;,'...b = T tan. IL IfT R is given, we have (~ 70) R- = b cot. I cot. I or b = eot r Icot. ~ I'.. 5 -= R- tan. A I tan. I 7 5 50 CIRCULAR CURVES. Example. Given I-= 27~, T=- 600 or R= 2499.18, to find b. Here b - 600 tan. 6~ 45' - 71.01, or b- 2499.18 tan. 60 451 tan. 130 30' = 71.01. 72. Problem. Given the angle of intersection I of two tangents A C and B C (fig. 24), to find the tangent point A of a curve, that shall pass through a point E, given by CD = a, DE = b, and the angle CD E Fig. 24. A 0 Solution. Produce D E to the curve at G, and draw CO to the centre 0. Denote D F by c. Then in the right triangle CD F we have (Tab. X. 11) D F = CD cos. CD F~, or c= a cos. 1. Denote the distance A D fiom D to the tangenzt point by x. Then, by Geomety, x2- = D EX DG. But DG- DF -- G= —DF+ F-=2DF-D E- 2c-b. Therefore, x2 - b (2 c - b), and x = b (2 c — ). Having thus found A D, we have the tangent A C A D + D C X —- a. Hence, R or D may be found (~ 5 or ~ 11). If the point E is given by E I and CH perpendiculgr to each other, a and b may be found from these lines.;For a = CH +i pJgCH+ EHcot. I (Tab. X. 9), anrdb=DE D -- 2 " IC"'.3' t j,, V~F UIsin.4I MBISCELLANEOUS PROBLEMLS. 5 Example. Given 1_ 200 16', a = 600, and b 80, to find x and R. Here c = 600 cos. 100 8' = 590.64, 2 c b 1101.28, and x V/80 X 1101.28 = 296.82. Then 1'= 600 + 296.82 = 896.82, and R = 896.82 cot. 100 8' = 5017.82. 73. Problemn. (ivens the tangent A C (fig. 25), and the chord A B, uniting the tangent points A and B, to,find the radius A 0 - R. Fig. 25. Solution. Measure or0 calculate the perpendicular CD. Then if C D be produced to the centre 0, the right triangles A D C and CA O, having the angle at C common, are similar, and give CD: AD = A C: A 0, or E = A D X A C CD If it is inconvenient to measure the chord A B, a line E 1,E parallel to it, may be obtained by laying off from C equal distances CE and C-F. Then measuring E G and G C, we have, from the similar triGEXA C angles E G C and CA 0, C G: GE = A C: A O, orR= CR G' Example. Given A C= 246 and A D = 240, to find P. Here 240 < 246 CD = 54, and 54 -- 1093.33. 52 CIRCULAR CURVES. 74. Problenlo Given the radius A 0 = R (fig. 25), to find the tangent A C = T of a curve to unite two straight lines given on the ground. Solution. Lay off fjoi the intersection C of the given straight lines anal equal distanzces CE and CFo Draw the perpendicular C G to the mid~dle of E F, and nmeasure G E and C G. Then the right triangles G C and CA 0, having the angle at C common, are similar, and give GE: CG = A 0: A C, or T CGx A0. GE By this problem and the preceding one, the radius or tangent points of a curve may be found without an instrument for measuring angles. Exanmple. Given 12 = 1093~, CGE: = 80, and C G = 18, to find To. 18 x 10931 I-Iere T= 80- 246. 75. Problemo To find the angle of intersection I of two straiglht lines, wohen the point of intersection is inaccessible, and to determiane the tangent points, whenl the length qf thie tangents is given. Solution. I. To find the angle of intersection I. Let A C and C V (fig. 26) be the given lines. Sight fioom some point A on one line to a point B on the other, and measure the angles CA B andl T:B V. These angles make up the change of direction in passing firom one tangent to the other. But the angle of intersection (~ 2) shows the change of direction between two tangents, and it must, therefore, be equal to the sum of CA B and TB T, that is, I= CAB+ TB i. But if obstacles of any kind render it necessary to pass from A C to B Vby a broken line, as A D E FB, measure the anglles CA D, ND E, PE, RF 1B3 and SB V, observing to note those atngles as lzinns 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 E F lies to the left of 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 CA D shows 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 F7, RB F the change between EF produced and FB, and, lastly, SB Vthe change between BF pro MISCELLANEOUS PROBLEMS. 53 duced and the second tangent. But the angle 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 measured, that is, I = CAD - +NDIE - PEF+ RFB - SB V. Fig. 26. A II. To determine the tangent points. This will be done if we find the distances A C and B C; for then any other distances from C may be found. It is supposed that the distance A B, or the distances A D, D E, E F, and FB have been measured. If one line A B connects A and B;, find A C and B C in the triangle A B C. For this purpose we have one'side A B and all the angles. If a brolen line A D E F B connects A and B, let fcall a perpendicular B G firom B upon A C, prodsuced if necessary, and find A G and B G by the uzsual 7method of zcorlinzg a traverse. Thus, if A C is taken as a meridian line, and D IC, E L, and FMJ- are drawn parallel to A C, and D II, E K, and FL are drawn parallel to B G, the difference of latitude A G is equal to the sum of the partial differences of latitude A H, D K, E L, and Fl31, and the departure B G is equal to the sum of the partial departures D H, E C, FL, and B 1]. To find these partial diffelences of latitude and departures, we have the distances A D, Dl,. E F, and FB, and the bearings may be obtained from the angles already measured Thus the bearing of A D is CA D, the bearing of DE is KD1E =-D NV JNDE = CAD + NDE, the bearing of EF is LEF = LEP — PEFI= KDE - PEF, and the 5" 54 CIRCULAR CURVES. bearing of FB is 1FB = A2FR + R FB = L E + - FB; that is, the bearing of each line is equal to the algebraic sumt of the preceding bearing and its own change of direction. The differences of latitude and the departures may now be obtained frorn a traverse table, or more correctly by the formulae: Diff. of lat. = dist. X cos. of bearing; dep. -= dist. X sin. of bearing. Thus, A H A D cos. CA D, and DIi = AD sin. CA D. Having found A G and B G, we have, in the right triangle B G C, BG (Tab. X. 9) G C = B G cot. B C G, and B C sin. B C G C But B C G = 180 -I. Therefore, cot. B C G — cot. I, and sin. B C G BG =sin. I. Hence G C —B G cot. I, and B C i I' Then, since A C = — A G + G C, we have L A AC-A G-BG cot. I; B C =- B G silln. I When I is between 900 and 1800, as in the figure, cot. I is negative, and -B G cot. I is, therefore, positive. When I is less than 900, G will fall on the other side of I; but the same formula for A C( wilb still apply; for cot. I is now positive, and consequently, —B G cot. 1 is negative, as it should be, since, in this case, A C would equal A G nzinus G C. Example. Given A D = 1200, DE = 350, EF=-300, FB310, CA D-20~, NDE = 44o, P E F - 25O, RFB = 31o, and SB V — 30o, to find the angle of intersection I, and the distances A C and B C. Here I = 200~ + 440 - 250~ + 310~ + 300 1000. To find A G and B G, the work may be arranged as in the following table:Angles to. Bearings. Distances. N. the Right. 0 0 20 N. 20 E. 1200 1127.63 410.42 44 64 350 153.43 314.58 -25 39 300 233.14 188.80 31 70 310 106.03 291.30 1620.23 1205.10 The first column contains tle observed angles. The second contains the bearings, which are found fiom the angles of the first column, in MISCELLANEOUS PROBLEIS. 55 the manner already explained. A Cis considered as running north fiom A, and the bearings are, therefore, marked N. E. The other columns require no explanation. We find A G - 1620.23, and B G1205.10. Then G C - B G cot. I - 1205.1 X cot. 1000 212.49. This value is positive, because it is the product of two negative factors, cot. 1000 being the same as -cot. 800, a negative quantity. Then A C = A G + G C = 1620.23 4- 212.49 - 1832.72, and 1205.1 B C sin. 1005- 1223.69. Having thus found the distances of A and B from the point of intersection, we can easily fix the tangent points for tangents of any given length. 76. Problem. To lay out a curve, when an obstruction of any kind prevents the use of the ordinary methods. C Fig. 27. 0 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 firom A C, the tangent at A, such a multiple of the deflection angle D, as will be sufficient to make the sight clear the obstruction. In the figure it is supposed that 4 D is the proper angle. The sight will then pass through F, the fourth station from A, and this station will be determined by measuring firom A the length of the chord A F, found by 56 CIRCULAR CURVES. ~ 69 or by Table II. From the station at F the stations at D and E may afterwards be fixed, by laying off the proper deflections fiom the tangent at lF. Second ilMethod. This consists in running an auxiliary cnrve parallel to the true curve, either inside or outside of it. For this pr'pose, lay off perpendicular to A C, the tangent at A, a line A At of any convenient length, and from A' a line A' C' parallel to A C. Then At C' is the tangent from which the auxiliary curve A'tE is to be laid of'. 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 B, a radius through Dt passes through D, &c. Thlle chord At B is, therefore, parallel to A B, and the angle CI'' B' = CA B; that is, the deflection angle of the auxiliary curve is equal to that of the true curve. It remains to find the length of the auxiliary chords A' Bt, B' D', &c. Call the distance A A' = b. Then the similar triangles A B 0 and A' B' O give A 0: A' O = A B: A' B', or P:R -b = 100: A' B'. o 100 (R - b) 100 b Therefore, A' B' 100 —. If the auxiliary curve were on the outside of the trune curve, we should find in the same way 100 b ABt = 100 + -. It is well to make b an aliqnot part of?; for the auxiliary chord is then more easily found. Thus, if n is any R 100 b whole number, and we make b =, e have AlB' = 100 ~t 100 R =100 ~: -. If, for example, b =, we have = 100 and A'B' 100 ~ 1 l101 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 BB', D DI, &c., each equal to b. 77. Problenm Ilavii7gy run a ciurve A B (fig. 28), to chlaicqe the tanigent point fjrom A to C, in such a eway that a curve of the sname radiius may strikce a given point D. Solution. lMeasure the distance B D firom the curve to D in a directionn parallel to the tanigent CE. 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 D BE equal to the intersection angle at E, or to twiice B A E, the total deflection angle from A to B; or if A can be seen fiom B, the angle D B A may be made equal to B A E. fleazsuure on the tangent (baickcivard o,forwarcd, as the case nay be) (a clistanuce A C = B D, and C wvill be the nuew tagiyeat poiuzt requirled. For, if C'H be drawn equal and n)arallel to Al 1F, we hlave R;' equal arid par MISCELLANEOUS PROBLEMS. 57 allel to A C, and therefore equal and parallel to B D. Hence D H= B F - F= C H and D H being equal to C H a curve of radius CH from the tangent point C must pass through D. Fig. 280 C A 78. Problem. Having run a curve A B (fig. 29) of radius R or deflection angle D, terminating in a tangent B D, to find the radius R' or deflection angle DI of a curve A C, that shall terminate in a given parallel tangent CE. Fig. 29. Solution. Since the radii B F and C G are perpendicular to the parillel tangents CE and B D, they are parallel, and the angle A G C A FB. Therefore, A C G, the half-supplement of A G C, is equal to CIRCULAR CURVES. A B F1 the half-supplement of A FB. Hence A B and B C are in the same straight line, and the new tangent point C is the intersection of A B produced with CE. Represent.A B bty 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 B C fro6m D C, we have B C = D C (Tab. X. 9), and the angle D B C = A B IC = B A K, the total deflection from A to B. Then the triangles AFB and A G C give A B: A C-= BF. C G, or c:c' = R: R' us,~ ~ ~. oRl= C- IR. 50 50 To find D', we have (~ 10) RI Sill ) and Rn s. Subsn ^'' and -sin. D c1 60 C.. v sin. Dr = sin, D. 79. Probleml, Given the length of two equal chords A C and B C (fiy. 30), and the perpendicular CD; to find the radius R of the curve. A _ Fig. 30. Solution. From O, the centre of the curve, draw the perpendicular 0 E. Then the similar triangles 0 BE and B CD give B 0: BE B C CD,: or R: B C B C: CD. Hence 2 CD) MIISCEtLANEOUS PROBLEMISo 59 This problem serves to find the radius of a curve on a track already laid. For if from any point C on the curve we measure two equal chords A C and B C, and also the perpendicular CD from C upon the whole chord A B, we have the data of this problem. 80. ProblRl.e To draw a tangent F G (fig. 30) to a given curve from a given point F. Solution. On any straight line F A, which cuts the curve in two points, Mneasuyre F C and FA, the distances to thle curtve. Then, by Geometry, FG =C.,FC X Ao. This length being measured from F, will give the point G. When F' G 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. 81. Problbenat Having/ found the radiuzs A 0 = R of a curve (,fig. 31), to substitute for it two radii A E -= n1 acd D F = R2, the longer of which A E or B E t is to be used for a certain distance only at each end of the curve. Fig. 31. Solttion. Assumne thle lonzger radils of az,7 length which?,Coay be thought 60 CIRCULAR CURVES. proper, and find (~ 9) the corresponding deflection angle D5. Suppose that each of the curves A D and B Dt is 100 feet long. Then drawing C 0, we have, in the triangle F O, 0 E OFE: = sin. O0 FF: sin. F0 E. But the side OE = -A E - 0 = R -, FE = DE - FRI - R2, the angle F O = 1t80~ O C-180~ — ~ anc the angle OFE - A OF- OEF= 21 - 2D1, since OEF= 2D1 (~ 7). Substituting these values, and recollecting that sin. (1800~ - I) =sin. 1, we have RI -R/:RI- =sin. (1- 2DI):sin. L Hence R1 -R = (R - ) sin. ~1 sin. (I I- 2 DI) R2 is then easily found, and this will be the radius from D to D', or until the central angle D F D' - I 4 DI. 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 angle at E will of course change with the length of A D, and this angle must take the place of 2 D1 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. Exanmple. Given R -- 1146.28 and I- 450, to find R2, if R1 is assumed - 1910.08, and A D and B D' each 100. Here, by Table I., D1 = 10 30'. Then 1 R = 763.8 2.882980.I= 220 30' sin. 9.582840 2.465820 I 2 2 D1 = 190 30' sin. 9.523495 RI - R 875.64 2.942325.R2 = RI - 875.64 = 1034.44 82. ]ProblemI To locate the second branch of a com0pound or reversed curve fionm a station on the first branch. Solution. Let A B (fig..32) be the first branch of a compound curve, and D its deflection angle, and let it he required to locate the second branch A B', whose deflection angle is Dt, from some station B on A B. IISCELLANEOUS PROBLEMS. 61 Let n be the number of stations fiom A to B, and n' the number of stations from A to any station B' on the second branch. Represent by V the angle A B B', which it is necessary to lay off fSoin the chord B A to strike B'. Let the corresponding angle A B' B on the other curve be repreT A rt Fig. 32. I sented by Vt. Then we have V - VI' 180 -BA B'. But if TTt be the common tangent at A, we have TAB + T' A B =- n D + n'D' = 180~ -B BA B'. Therefore, V - V' = n D + nt D. Next in the triangle A B B' we have sin. V': sin. V= A B:AB. But A B: A B=: —:', nearly, and sin. V': sin. V- V': V, nearly. Therefore we have approximately V': V- n nt,/ or VI = nf V. Substituting this value of V' in the equation for V V', we have V+ V== nD + n D. Therefore,nt V-n V= n' (nD + - D),or' = n2 ( D +- n' DI') n +- nt The same reasoning will apply to reversed curves, the only change being that in this case V+ Vr = n D - nt D', and consequently = n (nD - nD') n A- n' When in this formula n' D becomes greater than n D, V becomes minus, which signifies that the angle Vis to be laid off above B A instead of below. This problem is particularly useful, when the tangent point of a curve is so situated, that the instrument cannot be set over it. The same method is applicable, when the curve A B' starts fionm a straight line; for then we may consider A B' as the second branch of a compound 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 V becomes 6 62 OeCIRCLLAR CURVESo - n2 ~Dl. n +-t t When n and n1 are each 1, the formula for Vis in all cases exact; for then the supposition that VI V- n: nt is strictly true, since A B will equal A B', and Vand VI, being angles at the base of an isosceles triangle, will also be equal. Making n and nt equal to 1, we have V=~-I(D+ D). When the curve starts fiom a straight line, this formula becomes, by making D - 0, 2 = D. We have seen that when n or nt is more than 1, the value of Vis only approximate. It is, however, so near the truth, that when neither a nor n' exceeds 3, the error in curves up to 5~ or 6~ varies from a fiaction of a second to less than half a minute. The exact value of V might of course be obtained by solving the triangle A B Bl, in which the sides A B and ABI may be found from Table II., and the included angle at A is known. The extent to which these formulae may be safely used may be seen by the following table, which gives the approximate values of Vfor several different values of sn, nr, D, and D', and also the error in each case. Compound Curves. IReversed Curves. it. D. a'. /D. V. Error... a'. lD. v Error. 0 0 it 0 0 0 I II 1 0 5 1 4 10 0.9 1 3 4 3 7 12 27.2 1 0 5 3 12 30 25.3 2 3 4 3 4 0 23.5 2 0 3 3 5 24 22.1 3 3 4 3 42q 8.3 3 0 3 3 4 30 29.7 3 3 3 3 3 45 24.0 1 5 3 13 20 18.6 2 1 1 4 0 40 0.1 2 1 3 1 20 0.7 2 1 4 2 4 0 11.0 2 2 3 3 7 48 15.0 1 6 2 6 4 0 23.5 2 2 4 3 10 40 24.7 1 5 3 5 7 30 51.8 3 3 3 4 10 30 54.0 2 3 5 3 625-1 52.8 As the given quantities are here arranged, the approximate values of V are all too great; but if the columns n and nt and the columns D and D' were interchanged, and Vcalculated, the approximate values of V would be just as much too small, the column of errors remaining the same. MItSCELLANEOTS PROBLEMS. 63 83. Problem. To nmeasure the distance across a river on a given straight line. Fig. 33, c\ Solution. First MIethod. 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 A CB. Then in the triangle A B C we have one side and two angles to find A B. If A C is of such a length that an angle A CB = ~ D A C can be laid off to a point on the farther side, we have A B C -= D A C A CB. Therefore, without calculation, A B A C. Fig. 34. Second Mlethod. Lay off A C (fig. 34) perpendicular to A B, Measure A C, and at C lay off CD perpendicular to the direction CB, and meeting the line of A B in D. Measure A D. Then the triangles A CD and AB C are similar, and give AD: A C =A C: AB. Therefore, A B = A D If from C, determined as before, the angle A CB' be laid off equal to A CB, we have, without calculation, A B =- A B'. Third iMethod. 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, mneasure the distance E C, and produce 64 IMISCELLANEOUS PROBLEMS. E C until CF = E C. Then, since the triangles A CE and D CF are similar by construction, we see that D F is parallel to E B. Find Fig. 35. E 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 A B C and D G C are equal, and G D is equal to the required distance A B. As the object of drawing EF is to obtain a line parallel to A B, this line may be dispensed with, if by any other means a line GF be drawn through D parallel to A B. A point G being found on this parallel in the line of CB, we have, as before, G D = A B. PARABOLIC CURVES. 65 CHAPTER II0 PARABOLIC CURVES. ARTICLE I.-LOCATING PARABOLIC CURVES. 84. LET A EB (fig. 36) be a parabola, A Cand B C its tangents, and A B the chord uniting the tangent points. Bisect A B in D, and oin CD. Then, according to Analytical Geometry,Fig. 36, A D B I. CD is a diameter of the parabola, and tle curve bisects CD in E II. If from any points T, Tt, T &c., on a tangent A F, lines be drawn to the curve parallel to tihe diameter, these lines TM, T' 3tJ, T2.ltt', &c., called tangent defections, will be to each other as the squares of the distances A T, A T', A T11, &c. fiom the tangent point A. III. A line ED (fig. 37), drawn fiom the middle of a chord A B to the curve, and parallel to the. diameter, may be called the niclddle ordinate of that chord; and if the secondary cllords A E and B E be drawn, the middle ordinates of these chords, K G and L I, are each equal to ~ED. In like manner, if' the chords A K, KE, EL, and LB be drawn, their middle ordinates will be equal to 1 K G or I L H. IV. A tangent to the curve at the extremity of a middle ordinate, is parallel to the chord of that ordinate. Thus iEf, tangent to the curve at E, is parallel to A B. 6 * 66 PARABOLIC CURVES. V. If any two tangents, as A C and B C, be bisected in M and F, the line MF, joining the points of bisection, will be a new tangent, its middle point E being the point of tangency. 85. Problem. Given the tangents A C and B C, equal or unequal, (fig. 36,) and the chord A B, to lay out a parabola by tangent deflections. Fig. 36., A D B Solution. Bisect A B in D, and measure CD and the angle A CD); or calculate CD * and A CD from the original data. Divide the tangent A C into any number n of equal parts, and call the deflection TM for the first point a. Then (~ 84, II.) the deflection for the second point will be T' Mt = 4 a, for the third point T" M" = 9 a, and so on to the nth point or C, where it will be n2 a. But the deflection at this last point is CE -= CD (~ 84, I.). Therefore, n2 a = CE, and CE at ='-"-. Having thus found a, we have also the succeeding deflections 4 a, 9 a, 16 a, &c. Then laying off at T, T', &c. the angles A TM, A Tt M1', &c. each equal to A C'D, and measuring down the proper deflections, just found, the points Mi, Mi', &c. of the curve will be determined. The curve may be finished by laying off on A Cproduced 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 A B C, we have, by G(eometry, C D2 = 1 (A C2 + B C2) - A Do. LOCATING PARABOLIC CURVES. 67 easier way generally of finding points beyond E is to divide the secand tangent B C into equal parts, and proceed as in the case of A C. If the number of parts on B C be made the same as on A C, it is obviDUS that thte deflections from both tangents will be of the same length for corresponding points. The angles to be laid off from B C must, of course, be equal to B CD. 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. Pr'oblem., Given the tangents A C and B C, equal or unequal, (fig. 37,) and the chord A', to lacy out a parabola by middle ordinates. Fig. 37. AA lb Solution. Bisect A B in D, draw CD, and its middle point E will be a point on the curve (~ 84, I.). D E is the first middle ordinate, and its length may be measured or calculated. To the point E draw the chords A E and BE, lay off the second middle ordinates G K and HL, each equal to I D E (~ 84, III.). and K and L are points on the curve. Draw the chords A K, ICE, E L, and L B, and lay off third middle ordinates, each equal to one fourth the second middle' ordinates, and four additional points on the curve will be determined. Continue this process, until a sufficient number of points is obtained. 87. Problemn. To draw a tanigent to a parabola at any station. Solution. I. If the curve has been laid out by tangent deflections (~ 85), let 1Mt (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 1iH" Nor EL equal to a, the first tangent deflection (~ 85); and M' N or J11"' L will be the required tangent. The same thing may be done by laying off fiom the second station a distance ll' T = 4 a, or at the third station a distance (C P 9 a; for the 68 PARABOLIC CURVES. required tangent will then pass through T' or G. It will be see also, that the tangent at 3J1111 passes through a point on the tangent A corresponding to half the number of stations from A to nt1"l; th is, Mill is fourt stations fiom A, and the tangent passes through I the second point on the tangent A C. In like manner, 111 is six st tions from B, and the tangent passes through G. the third point on t] tangent B C. II. If the curve has been laid out by middle ordinates (~ 86)7 the ta gent deflection for one station is equal to the last middle ordinate ma( use of in laying out the curve. For if the tangent A C (fig. 37) we: divided into four equal parts corresponding to the number of statiol fiom A to E, the method of tangent deflections would give the san points on the curve, as were obtained by the method of ~ 86. In th case, the tangent deflection for one station would be a =- G CE = G DE; but the last middle ordinate was made equal to;- GK c t_ D E. Therefore, a is equal to the last middle ordinate, and a tar 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 ord: nate, by drawing a line through this extremity, parallel to the chor 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 fiom th original tangents. A new tangent nmay then be drawn by ~ 87 to an; station already found, as at ili"' (fig. 36), and the tangent deflection a, 4 a, 9 a, &c. may be laid off from this tangent, precisely as from th, first tangent. These deflections must be parallel to CD, and the dis tances on the new tangent must be equal to Tl Nor Nlr1t1, whiel may be measured. 89. Problem. Given the tangents A C and B C, equal or unequal (fig 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 E? the middle point of D F. E wvill be a point on the curve (~ 84, V.). W{ have now two pairs of what may be called second tangents, A D an( D E, and E F and FB. Bisect AD in G and DE in IH join G / and its middle point 3iwnill be a point on the curve. Bisect BEFan( FB in RKand L, join KL, and its middle point N will be a point oI the curve. We have now four pairs of third tangents, A G and G 11 lMH and HE, E K and K N, and NL and LB. Bisect each pair it turn, join the points of bisection, and the middle points of the joining LOCATING PARABOLIC CURVES. 69 lines will be four new points, iit, il/~f, NVt, and NV. The same method may be continued, until a sufficient number of points is obtained. c Fig. 38. M/ N h I L A B 90. ProbIleni. Givent the tangents A C and B C, equal or unequal, (fig. 39,) and thle clord A B, to lay out a parabola by intersections. A g 39.K Solution. Bisect A B in D, draw CD, and bisect it in E. Divide the tangents A C and 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 31of A a and F G will be a point on the curve.:For F211 5 Ca, and Ca -- CE. Therefore, F311= zs CE, which is the proper deflection fiom the tangent at F7to the curve (~ 85). In like manner, the intersection N of A b and HKIK 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 Ml on the curve by a line drawn firom B through at, the line A b would be intersected in Non the curve by a line drawn 70 PARABOLIC CURVES. from B through b', and in general any two lines, drawn fiom A and L through two points on CD equally distant from the extremities C and D, will intersect on the curve. To show this for any point, as,Al, it is sufficient to show, that B a' produced cuts F G on the curve; for it has already been proved, that A ac cuts F G on the curve. Now D) a': M G = B D: G= 5:'9, or - G = Dat. But D a't - CE. Therefore, f G = CE. Again, F G CD = A G: AD = 1: 5. Therefore, F G = CD= CE. We have then M = F G - M111G C= 2 CE - CE C. As this is the proper deflection from the tangent at IF to the curve (~ 85), the intersection of Ba with F G is on the curve. This furnlishes another method of laying out a parabola by. intersections. 91. The following example is given in illustration of several of th( preceding methods. Example. Given A C = B C = 832 (fig. 40), and A B = 1536, tc lay out a parabola A EB. We here find CD = 320. To begin with the method by tangent deflections (~ 85), divide the tangent A C intc CE 160 eight equal parts. Then a t= t -- 64 2.5. Lay off from the divisions on the tangent F1 = 2.5, G2 = 4 X 2.5 = 10, 113 9 X 2.5 = 22.5, and K4 =- 16 X 2.5 = 40. Suppose now that it is inconvenient to continue this method beyond IC. In this case we may Fig. 40. " —DI A B find a new tangent at E, by bisecting A C and B C (~ 89), and draw. ing KL through the points of bisection. Divide the new tangenl KE - A D - 384 into four equal parts, and lay off fr'om KCE th( RADIUS OF CURVATURE. 7 same tangent deflections as were laid off from A xK, namely, M/5 = 22.5, N6 = 10, and 07 = 2.5. To lay off the second half of the curve by middle ordinates (~ 86), measure EBs= 784.49. Bisect E B in P, and lay off the middle ordinate P R =R D E 40. MBeasule E R =- 386.08, and Bi? = 402.31, and lay off the middle ordinates S T and V TV, each equal to P R =- 10. By measuring the chords E T, TR, R Wi, and TVB, and laying off an ordinate from each, equal to 2.5, four additional points might be found. ARTICLE II. -RADIUS OF CURVATURE. 92. TIIE 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 are of 1,000 feet radius is double that of an are of 2,000 feet radius. The curvature of a pl)al)ola 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 law, until it becomes greatest at a point, called the vertex, where a tangent to the curve would be perpendicular to the diameter. From this point to B it decreases again by the same law. We may, therefore, consider a parabola to be made up of a succession of infinitely small circular arcs, the radii of which continually increase in going from the vertex to the extremities. The radius of the circular are, corresponding to any part of a parabola, is called the rLadius of curvature at that point. If a parabola forms part of the line of a railroad, it will be necessary, 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 middle station, and in longer 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. Problemn. To f2dc the radius of curvature at certain stations on a parabola. Solution. Let A EB (fig. 41) be any parabola, and let it be required to find the radii of curvature at a certain number of stations $2'PARABOLIC CURiVES. from A to E. These stations must be selected at regular intervals from those determined by any of the preceding methods. Let n denote the number of parts into which A E is divided, and divide CD into the same number of equal parts. Draw lines from A to the points Fjiig. 41. A' DII 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 = c,, A E = c,, A G = C3, and A C = T. Denote, moreover, CD by d, and the area of the trianole A C13 by A. Then the respective radii for the points E, 1, 2, 3, and A. will be c3 c13 C C3.T3 A='i,,=R i, R2~-,:3e_, I=7 The area A may be found by form. 18, Tab. X.; c and T' are known; and ce, c2, c3 may be found approximately by measurement on a figure carefully constructed, or exactly by these general formulk': — 2 _- c (i - 1) d2 T2 - C2 (u2 - 3) d2 C,2 = C"2 + t, 2 it T2-_c2 (n - 5) ds T2 -C2 (?2- 7) d2 C42 = C32 + - 2 &c. &c. It will be seen, that each of these values is formed formed fiom the preceding, T2 - C2 d'z by adding the same quantity --, and subtracting -% multiplied in succession by n - 1, n - 3. n - 5, &c. Making na = 4, we have RADIUS OF CURVATURE. 7 c2 c2 + 4 (T2 _C2) - -6 d2, C22 C12 + x (T2 2) - t dS C32 Co 2 + x (Tf- _ C2) + -6L d2. All the quantities, which enter into the 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 formult is'as follows. The radius of curvature at any given point on a parabola is, by the Differential Calculu~s, R 2 sin.3 E, in which )p represents the parameter of the parabola for rectangular coordinates, and E the angle made with a diameter by a tangent to the curve at the given point. li'St, let the middle station E (fig. 42) be the given point. Then the angle E is the Fig. 42 angle made with E D by a tangent at E, or since A B is parallel to the tangent at E (~ 84, IV.), sin. E - sin. A DE = sin. B DE. Let p' be the parameter for the diameter E D. Then, by Analytical Geometry, p pt sin.2 E. Therefore, at this point R =-2 P3 p sin.2 E 2' AD 2 c2 C2 2 sin.3 E ='2si.l.E But E -ut. Therefore, P- i. c3 c3 cd sin. since A = d sin. E (Tab. X, 17). Next, to find RB, or the radius of curvature at H, the first station fiom E. Through H draw F G parallel to CD, and from F draw the tangent FK. Join A IC, cutting CD in L. Then fiom what has just bohe proved for the radius of curvature at E, we have for the radius A G3 0f Vct'rtureC at HI, R = —-* _' Now. A G: AL - AF: A C 7 74 PARABOLIC CURVES. n- 1:n, or A G- AL ButAL=c o For,sinceAF-= ~, -- 11 (n - -i)2 d -b X A C, the tangent deflection F 2 ) f (~ 84, II.), and FCG =2F11 (n -- )d. Then, since CL: FG=A C: AF= — n:n- 1, CL= - X F G / d. iHence LD = d — a d = d, that is, A L- = c, Substituting this value in the expression for A G above, wve have A G - c1. Moreover, since A = - na X A C, and because similar triangles are to each other as the squares of their homologous sides, we have the triangle A F G (, X ACL. But ACL: A CD = CL~ CD -n —l n, or A CL = I X A CD. Therefore, A F G - 1) X A CD, and AF = 2 A F G ( XA CB = A. Substituting A G3 these values of A G and A FIK in the equation R1, — A.G 9 and reducing, we find R A_ ~ By similar reasoning we should find Re = A, R3 — = &c. It remains to find the values of c, c,, &c. Through A draw A Nl perpendicular to CD, produced if necessary. Then, by Geometry, we have A D2 = A L2 -+ L DI -2 L D X L 31, and A C2 = AL2 + CL2 -+ 2 CL X L.L. Finding from each of these equations the value of 2 L 21, and putting these values equal to each other, we have AL2+LD- AD2 A C2-AL2 CL2 ButAL LD LD - CL Bt c,LD= at A D - c, A C = T, and CL- 1 d. Substituting these values in the last equation, and reducing, we find T2 (ia - l)c2 (a1- l)d2 C 12 _+- n2 By similar reasoning Mwe should find 2 T2 (1 - 2)c2 2( - 2)cld 2 - + 8Th (Z - )-c 3(n - 3)d2 &c. &e. RADIUS OF CURVATURE, 5 From these equations the values of c12, c22, C32, &c. given on page 72 are readily obtained. That given for c52 is obtained from the first of these equations by a simple reduction; that given for c22 is obtained by subtracting the first of these equations from the second, and reducing; that given for c32 is obtained by subtracting the second equation from the-third, and reducing; and so on. 94. Example. Given (fig. 41) A C= T= 600, B C= T' = 520, and A D = c = 550, to find Pt, R1, J2, R3, and R4 the radii of curvature at E, 1, 2, 3, and A. To find CD - d, we have, by Geometry, d2 = (T2 + T12) -c, which gives d2 12700. To find the area of A CB - A, we have (Tab. X. 18) A Vs (s - a) (s -- b) (s - c). s 1110 3.045323 s -a - 590 2.770852 s - b 510 2.707570 s - c- 10 1.000000 2)9.523745 log. A 4.761872 1 Tc)1150 x 50 Next (T - c2) — (T + c) (T - c) = - 14375, and d2 12700 12 = 16 - 793,75. Then C -2 5502 - 302500 c,2 = 302500 + 14375 - 3 X 793.75 = 314493.75 co2 314493.75 + 14375 - 793.75 = 328075 c32 328075 + 14375 + 793.75 = 343243.75 C3 To find R, we have R-, or log. -- = 3 log. c - log. A. c - 550 2.740363 c3 8.221089 A 4.761872 R 2878.8 3.459217 C13 3 To find R,, we have R = - A or log. IL1 = — log. c,2 - log. A. cl2 = 314493.75 5.497612 ca3 8.246418 A 4.761872 1, = 3051..7 3,484546 76 PARABOLIC CURVES. In the same way we should find R2 - 3251.5, R13 = 3479.6, R4 3737.5. To find the radii for the second part EB of the parabola, the same formul. apply, except that T' takes the place of To We have then I (T12-c2) -'(T' + c) (T' - c) 1070x-30 Hence C,' = 302500 - 8025 - 2381.25 - 292093.75 ca'2 292093.75 - 8025 - 793.75 = 283275. c32 = 283275 - 8025 + 793.75 = 276043.75 To find R1, we have R1 =', or log. R, - log. c1- log. A c2 292093.75 5.465523 ca3 8.198284 A 4.761872 1 1- 2731.6 3.436412 In the same way we should find 12 = 2608.8, R3 = 2509.5, 1R4 2433. It will be seen, that the radii in this example decrease from one tangent 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 the case, whenever the angle B CD, adjacent to the shorter tangent, exceeds 900, that is, whenever c2 exceeds T12 + d2. If B CD = 900, the tangent point B falls on the vertex. If B CD is less than 900, 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 T1 are equal, the equations for c12, c2, &c. will be more simple; for in this case d is perpendicular to c, and T2 - 2 = d2. Sutbstituting this value, we get d2 c2= C2 + ~2 3 d2 C22 C= C02 + 2 2 5d2 C32 = C,2 + _22 9 &c. &c. dxamnple. Given, as in ~ 91, T= Tt = 832, c- 768, and d RADIUS OF CURVATURE. 77 320, to find the radii R, R1, and R, at the points E, 4, and A (fig. 40). Here A c d = = 245760, n 2, and c2= c2 + Id2 = 615424. cS c2 7(682 e l3 T3 Then c d -— 20 =-1843.2, R1 =- d, and R? -- ca c2 - 615424 5.789174 c13 8.683761 cd -245760 5.390511 A. = 1964.5 3.293250 T 832 2.920123 T3 8.760369 c d= 245760 5.390511 R/ 2343.5 3.369858 R, is the radius at the point X also, and RB the radius at the point B. 78 LEVELLING. CHAPTER II. LEVELLING. ARTICLE I. - HEIGIITS AND SLOPE STAKIES. 96. THE Level is an instrument consisting essentially of a telescope, supported on a tripod of convenient height, and capable of being so adjusted, that its line of sight shall be horizontal, and that the telescope 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 the level. 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. This distance is called the sight on that point. 97. Problem. e To find the difelrence of level of tco points, as A and B (figq. 43). Solution. Set the level between the two points,* and take sights on both points. Subtract the less of these sights from the greater, and the difference will be the difference of level required. For if F P represent the plane of the level, and A G be drawn through A parallel to FP, A F will be the sight on A, and B P the sight on B. Then the required difference of level B G B= B P - P G = - B P — A F. If the distance between the points, or the 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 ~ 105. HEIGHTS AND SLOPE STAKIES. 79 C; lastly, the level is set between C and D, and sights are taken on C and D. Then the difference of level between A and D is ED =(BP+C- e OD) — (AF+BI+ NC). For D = ITIC — LC,= [1 11 + il C —LC. But 3-1== 1G = BP-Al, A = -KC —B1, and L Cu= NC- OD. Substituting these values, we have ED = B P - AlF+K C -BI — NC++ OD= (BRP + K C + OD) - (AF+- BI + NC). 98. It is often convenient to refer all heights to an imaginary level plane called the datlum plane. This plane may be assumed at starting to pass through, or at some fixed distance above or below, any permanent object, called a bench-mark, or simply a bench. It is most convenient, in order to avoid minus heights, to assume the datum plane at such a distance below the benchmark, that it will pass below all the points on the line to be levelled. Thus if A B (fig. 44) were part of the line to'h _ Xh W J m be levelled, and if A were the starting point, we should assume the datum plane CD 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 / 4 line was more than 15 or 20 feet below A, we might safely assume CD to be / S( 25 feet below the bench near A, in which case all the distances from the line to the datum plane would be positive. Lines before being levelled are usually divided into regular stations, the height of each of which above the datum plane is required. 80 LEVELLING. 99. Pr'oble.iDlO To find the heights above a datoum plane of the several stations on a givenz line. Solution. Let A B (fig. 44) represent a portion of the line, divided into regular stations, marked 0, 1, 2, 3, 4, 5, &e., and let CD represent the datum plane, assumed to be 25 feet below a bencht' mark near A. Suppose the level to be i~ JI set first between stations 2 and 3, and a sight upon the bench-mark to be taken, e,__- and found to be 3.125. IN!ow as this sight shows that the plane of the level E F is 3.125 feet above the bench-mark, and as the datum plane is 25 feet below this mark, we shall find the height of the plane of the level above the datum plane by adding these heights, which gives for the height of E F 25 + 3.125 = 28.125 feet This height may for brevity's sake be called the 7height of the instrumnent, meaning by this the height of the line of' sight of the instrument. If now a sight be taken on station 0, wve shall obtain the height of this station above the datum plane, by subtracting this sight from the height of the instrument; for the heiglht of this station is 0 C and 0 C = E C- E 0. Thus if E0 = 3.413, 0 C = 28.125 - 3.413 = 24.712. In like manner, the heights of stations 1, 2, 3, 4, and 5 may be found, by taking sights on them in succession, and subtracting these sights from the height of the instrument. Suppose these sights to be respectively 3.102, 3.827, 4.816, 6.952, and 9.016, and we have height of station 0 = 28.125 - 3.413 = 24.712, " " "' 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, "'"L " 4 = 28.125 - 6.952 = 21.173, t" "i 6 5 - 28.125 - 9.016 19.109. Next, set the level between stations 7 and 8, and as the height of station 5 is known, take a sight upon this point. This sight, being added to the height of station 5, will give the height of the instrument in its new position; for G I = G 5 + 5 IC Suppose this sight to be G 5 = 2.740, and we have G K.= 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,'4': " 7 _ 21.849 - 4.027 17.8229 " " a 8 21.849 - 3.824 - 18.025, "4'cc C 9 = 21.849 - 2.516 = 19.333,' " " 10 - 21.849 - 0.314 21.535. The instrument is now again carried forward and reset, station 10 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 selected, 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 plus sight. The next thing to be done is to take sights on those points whose heights are required, and to sutbtract these sights fiom the height of the instrument, in order to get the required heights. These last sights may therefore be called minus sighllts. 82 LEVELLINGo 101. The field notes are kept in the following form. The first colulmn in the table contains the stations, and also the benches marked B., and the turning points marked t. p., except when coincident with a station. The second column contains the plus sights; the third column shows the height of the instrument; the fourth contains the minus sights; and the fifth contains the heights of the points in the first column. Station. S.. I. I, S. H. B. 3.125 25.000 0 28.125 3.413 24.712 1 3.102 25.023 2 3.827 24.298 3. 4.816 23.309 4 6.952 21.173 5 2.740 9.016 19.109 6'21.849 3.311 18.538 7 4.027 17.822 8 3.824 18.025 9 2.516 19.333 3B.2.635 19.214 | 10 0.314 21.535 The height of the bench is set down as assumed above, namely, 25 feet; the first plus sight is set opposite B., on which point it was taken, and, being added to the height in the same line, gives the height of the instrument, which is set opposite 0; the minus sights are set opposite the points on which they are taken, and, being subtracted from the height of the instrument, give the heights of these points, as set down in the fifth column. The minus sights are subtracted from the same height of the instrument, as far as the turning point at station 5, inclusive. The plus sight on station 5 is set opposite this station, and a new height obtained for the instrument by adding the plus sight to the height of the turning point. This new height of the instrument is set opposite station 6, where the minus sights to be subtracted fiom it commence. These sights are again set opposite the points on which they were taken, and, being subtracted fiom the new height of the instrument, give the heights in the last column. 102. FroblemP. To set slope stakes for excavations and enbalnkments. Solution. Let A L IH K C (fig. 45) be a cross-section of a proposed excavation, and let the cen:tre cidt J[-f c: and the width of the road HEIGHTS AND SLOPE STAKES. 83 bed HIK = b. The slope of the sides B H or C K is usually given by the ratio of the base IN to the height E N. Suppose, in the present case, that KEN: EN =- 3: 2, and we have the slope -. Then if the ground were level, as D A E, it is evident that the distance from Fig. 45. e the centre A to the slope stakes at D and E would be A D A E MK K+ -KN - Ib + 3 c. But as the ground rises from A to C through 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 -'= g, the slope stake must be set farther in a distance D F sg. —-. 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 of A EB - b + - 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 3 g, as thus estimated, and measure from the centre a distance out, equal to the s um. Obtain now by the level the rise from the entre to th is tpoint, and if it agrees with the estimated rise, the distance out is correct. But if the estimated rise prove too great or too small, assume a new value for g,sition fmeasure corresponding dist ance out, and test the accuracy of the estimate by the level, as before. These trials must be continued, until the estimated rise agrees sufficiently well with the rise found by the level at the corresponding distance out. The distance out will then be ow by the level g. The same om thrse is to be pursued, when the ground falls from th e centre, as at B; but as g here becomes minus, the distance out, when the true value of g is found, will be acc F A Dof the D F- ~b b+ c - q. For em bankment, the process of setting slope stakes is the, same as for excstimation, except that a rise in the ground frbyom the centre on embankments corresponds to a fcel on Thexcavations, and vice versd. This will be evident eby inverting figure 45be, which will then represent This will be evident ~by inerstilg figure ZS, lvhic will then represent 834 LEVELLING. an embankment. What was before a fcall to B, becomes now a rise, and what was before a rise to C, becomes now a fall. When the section is partly in excavation and partly in embankment, the method above applies directly only to the side which is in excavation 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 embankment. On the opposite side, however, it is only necessary to make c in the expressions above minuts, because its effect here is to diminish the distance out. The formula for this distance out will, therefore, become lb - c +- c g. 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, the line of sight of the level will be the tangent A D, while the true level will be A C. The difference D C between the line of sight and the true level is the correction for the earth's curvature for the distance A 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 atmnosphere increases in density the nearer it lies to the earth's surface, light, passing from a point B to a lower point A, enters continually air of 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 arec 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 fiom an object at B; so that this object appealrs 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 for the earth's curvature D C and the correction for refraction D B ale in opposite directions, the correction for both will be B C = D C' - DB. - Peirce's Spherical Astronomy, Chap. X., ~ 125. It should be observedl, however, that the effect of refraction is very uncertain, varying with the state of the atmosphere. Sometimes the path of a ray is even made convex towards the earth, and sometimes the rays are refracted horizontally as well as vertically. EARTHI S CURVATURE AND REFRACTION. This correction must be added to the height of any object as determined by the level. 105. Problen.o Given the distance A D = D (fig. 46), the radius of the earth A E =, and the radius of the arc of refracted light = 7 R, o Jfind the correction B C = dfor the earth's ciurvature and for refraction. AD Fig. 46. Soluttion. To find the correction for the earth's curvature D C, we have, by Geometry, D C (D C + 2 E C) = A D2, or D C (D C + 2 1R) = D2. But as D C is always very small compared with the diameter of the earth, it may be dropped from the parenthesis, and we have D2 D C X 2 R = D2, or D C - 2R The correction for refraction D B may be found by the method just used for finding D C, merely chang-.D2 ing R into 7 R. Hence D B- 14. We have then d=B C=.D2 D2 D C - DB -= R- or 3 D2 7R 13y 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. 86 LEVELLING. ARTICLE III. —VERTICAL CURVES. 106. VERTICAL curves are used to round off the angles formed by the meeting of two grades. Let A C and CB (fig. 47) be two grades, meeting at Co These grades are supposed to be given by the rise per station in going in some particular direction. Thus, starting from A, the grades of A C and CB may be denoted respectively by g and g'; that is, g denotes what is added to the height at every station on A C, and g' denotes what is added to the height at every station on CB; but since CB is a descending grade, the quantity added is a minus quantity, and g' 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 g' of CB, and the numlber 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. * C / A P H? t Solution. Let A E B be the required parabola. Through B and C draw the vertical lines FK and C:FI, and produce A C to meet FKC in F. Through A draw the horizontal line A (K, and join A B, cutting C H in D. Then, since the distance from C to A and B is measured horizontally, we have A tI-= I-K, and consequently A D = D B. The vertical line CD is, therefore, a diameter of the parabola (~ 84, I.), and the distances of the curve in a vertical direction from the stations on the tangent A F are to each other as the squares of the number of stations from A (~ 84, II.). Thus, if a represent this distance at the first station fiom 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 from A, it would be 4 n2 a; that is, FB == 4 12 a, or a - 4 ore. To find a, it will then be necessary to find FB first. Through C draw the horizontal line C G, and we have, from the equal triangles C F G and VErRTICAL CubVES. 8~L A CH, F G = C tf. But C H is the rise of the first grade g in the n stations from A to C; that is, CH = nig, or F G = ng. GB is also the rise of the second grade gl in n stations, but since gf is negative (~ 106), we must put GB = -— ngt. Therefore, FB - F G -I GB =ng - ng'. Substituting this value of FB in the equation for ca it g - n gr we have a = A, — X o a = g - g_' 4 n The value of a being thus determined, all the distances of the curve ffrom the tangent A F, viz. a, 4 ac 9 a, 16 ae, &c., are known.g Now if T and Tt be the filst and second stations on the tangent, and vertio cal lines TP and T' P' be drawn to the horizontal line A I, the height TP of the first station above A will be g, the height Tl pt of the second station above A will be 2g, and in like manner for su~c ceeding stations we should find the heights 3g, 4g, S&c. As we have already found T_/k = a, Tf./l = 4a, &c., we shall have for the heights of the curve above the level of A, MIP = TP - T1M= g - a, Mtr --- Tt =t Tl Pl t = 2g 4 a, and in like manner for the succeeding heights 3g -- 9 a, 4g ~ 16a, &c. Then to find the grades for the curve at the 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 a) 0 O= - a, (2g 4 a) (g — a)-g -- 3 a, (3g 9 a) (2g — 4 a) = g 5a, (4 g ~ 16 a) ~ (3 g -- 9 a) = g - 7 a, &c. The successive grades for the vertical culr ve are, tlherefore, g-a, g 3 a go 5a g 7a, &C. v a n.idn ths_ 1rds Atic Ar gal m 4 st UA pAid w; _I;to A A A an @5 1.e^E.^.; LEVELLING. a =.9 -.125 =,775, g- 3 a.9 -.375 =.525, g- 5 a.9 -.625 =.275, g — 7a =.9 -.875 =.025, g- 9 =.9- 1.125 = —.225, g-11 at =.9- 1.375- -.475. As a second example, let the first of two grades descend.8 per station, and the second ascend.4 per station, and assume two stations on each side of C as the extent of the curve. H-Iere g = -.8, g'.4, -.8 -.4 - 1.2 and n =2. Then a — +4X 2 =- = -.15, and the four grades required will be g —a =-.8- (-.15) —.8 +.1 5=-.65, g- 3a = —.8- (-.45) =-.8 -+.45 =-.35, g - (-a -.75)-.8 -.75) =.75 = -.05, g - 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 foregoing 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-stations on each side of the intersection, and to apply the preceding formule. 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 in 200 feet is 10. ~We have then = -.16, gl =.08, and n = 10. -.16 -.08 -.24 Hence a= 4 x 10 - 40 = -.006. The first grade is, therefore, 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, +-.014, +.026, -+.038, +.050, -.062, --.074. ELEVATION OF THE OUTER RAIL ON CURVES. 89 ARTICLE IV. —ELEVATION OF THE OUTER RAIL ON CURVES. 110. Pl'oblelmo Givens 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 elevation e of the outer rail of the curve. Solution. A car moving on a curve of radius R, with a velocity per seeond - v, has, by Mechanics, a centrifugal force -=. 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 elevation 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, there32.2 e fore, g Putting this equal to the centrifugal force, we have 32.2 e v2 g - R.m Hence 32.2 R If we substitute for /R its value (~ 10) R si we have 6e eg v2 sin. D 50 X 32.2 --.00062112 y v2 sin. D. If the velocity is given in miles 11r X 5280 per hour, represent this velocity by 11i; and wre have v 60 x 60 Substituting this value of v, we find e =.0013361 g MI2 sin. D. When g = 4.7, this becomes e =.00627966 III2 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 passenger trains as the value of lM. 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 of these causes is probably improper " coning" of the wheels. Two wheels, immovable on an axle, and of the same radius, must, if 8 * 90 LEVELLING. no slip is allowed, pass over equal spaces in a given number of revolutions. Now as the outer rail of a curve is longer than the inner rail, the outer wheel of such 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 difference in length of the two rails of the curve. To remedy this, coning 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 inner wheel. To find the amount of coning for a play of the wheels of one inch, let r and rt represent the proper radii of the inner and outer wheels respectively, when the flange of the outer wheel touches the rail. Then.4 - r will be the coning for one inch in breadth of the tire. To enable 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 R be taken as the radius of the inner rail, the radius of the outer rail will be R + g, and we shall have 2r: r R = R: R + g. Therefore, r R + rg = r2 R, or -r - rg As an example, let R - 600, r = 1.4, and g = 4.7. Then we have 1.4 X 4.7 r/ 1.r = 600 =.011 ft. For a tire 3.5 in. wide, the coning would be 3.5 X.011 =.0385 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 wheel 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 4.79 as 900 feet, r and g remaining the same, we should have rl - r 4900 ELEVATION OF THE OUTER RAIL ON CURVES. 91.0073, and for the coning of the whole tire 3.5 X.0073 =.0256 ft., or about three tenths of an inch. WVheels 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 motion 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 curvilinear 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 increase 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 diminish 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 he 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 Appletons' Mechanic's Magazine for August, 1852, that wheels made by them for the New York and Erie road have a coning of but one sixteenth of an inch. This coning on a track of six feet gauge with the other data as given above, -would suit no curve of less than a mile radius. 92 EAIARTH-WORK. CHAPTER IV. EARTH-AVORIK. ARTICLE I. - PSRIS3IoIDAL FORIMULA. 112. EARTI-ur-wORK. includes the regular excavation and embankment on the line of a road, borrow-pits, or such additional excavations as are made necessary when the embankment exceeds the regular excavation, and, in general, any transfers of earth that require calculation. We begin with the prismoidal formula, as this formula is frequently used in calculating cubical contents both of earth and masonry. A prisuoid is a solid having two parallel faces, and composed of prisms, wedges, and pyramids, whose common altitude is the perpendicular distance between the parallel faces. 113. Probe]n1o Given the areas of the paracllel faces B and B', the middle area 11f, and the allitucle c of a prisasoicl, to find its solidity S. Solztion. The middle area of a prismoid is the area of a section midway between the parallel faces and parallel to them, and the altitude is the perpendicular distance between the parallel faces. If now b represents the base of any prism of altitude a, its solidity is a b. If b lepresents the base of a regular wedge or half-parallelopipecdon of altitude a, its solidity is a b. If b represents the base of a pyramid of altitude a, its solidity is I a b. The solidity of these three bodies admits of a common expression, which may be found thus. Let as represent 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, a?- = b, in the regular wedge, am = ~ b, and in the pyramid,?11 = b. Moreover, the upper base of the prism = b, and the upper base of the wedge or pyramid - 0. Then the expressions a b, I a b, and - a b may be thus transfornmed. Solidity of prism = a b X 6 b (b - b 4 b) a (b + b 4 ), 6 6 6 wedge = a b = 3 b =a (o + b 2 b) a (O b 4 m) 6 6 6 pyramid=Iab —aX' 2b a(O-b(+6 ) a (0+ + b+ 4m). 6 6 6 BORROW-PITS. 93 Hence, the solidity of either of these bodies is found by adding together 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 sum 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 B + 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 ii the middle area of the prismoid. Therefore 9T: S = a (B + S1 + 4 I1J). ARITICLE II. - BORROW-PITS. 114.'Fox the measurement of small excavations, such as borrowpits, &c., the usual method of preparing the ground is to divide the surface into parallelograms * or triangles, small enough to be considered 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 bottom of the excavation, and levels referred to tle same bench-rmark 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 kinown, because the parallelograms or triangles, into which the surface is divided, are always measured horizontally. 115. Pl'rodblei Given the edces h, 1i, and A2, to find the solidity * If the ground is divided into rectangles, as is generally done, and one side be made 27 feet, or some multiple of 27 feet, the contents may be obtained at once in cubic yards, by merely omitting the factor 27 in the calculation. 94 EARtTH-WOl1RK. S of a vertical prism, whether truncated or not, whlose horizontal section is a triangle of given area A. Fig. 48. / 1 Solution. When the prism is not truncated, we have h — h hI o The ordinary rule for the solidity of a prism gives, therefore, S A hi = A X ax (h +- h1 + h2). When the prism is truncated, let AB CF G H (fig. 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 DE. Represent this perpendicular by p, and we have (Tab. X. 52) the solidity of the pyramid = p X BDEC= p X DE x (BD + CE)= Ip X DE X ~ (BD +- CE) = A X ~ (BD + CE), since IP X DE = A D E A. But ~ (B D + CE) is the mean height of the vertical 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 A D E, as well as to that below. The same reasoning would apply, if the lower end also were truncated. Hence, for the solidity of the whole prism, whether truncated or not, we have F7- S = A X ~ (h - ( 1t h + h2). 116. Problemn Given the edges h, h,, h2, and h3, to find the solidity S of a vertical prismn, whether truncated or ^not, whlose horizontal section is a parallelogram of given area A. BORROW-PITS. 95 ASolution. Let B H (fig. 49) represent such a prism, whether truncated or not, and let the plane B FHD divide it into two triangular Fig. 49. h,'i I' prisms A FH and CFH. The horizontal. section of each of these prisms will be A, and if it, h, h,, and, rand epresent the edges to which they are attached in the figure, we have for their solidity (~ 115), AFH- = A X ~ (1 A- 11 + hK3), and CFH — = A X 4~ (Al + h,2 + h3). Therefore, the whole prism will have for its solidity S = -A X 4 (h + 2 hi + h1o + 2 h3). Let the whole prism be again divided by the plane A E G C into two triangular prisms B E G and D E G. Then we have for these prisms, B E G - A X 4 (h + hl + h-2), and DE G = A X ~ (h +- h2 + ha), and for the whole prism, S = 2 A X ~ (2 h + 1h + 2112 + 1h3). Adding the two expressions found for S, we have 2 S A= ( -1i - 1 + - Ah3), or s = A X 4 (h + 1 + -t- - A3). It will be seen by the figure, that I (It + It2) = = KL (hl + 1h3), or h + ih2 = fhl - hA. The expression for S might, therefore, be reduced to S-A X (+- h,), or S=A X 2 (/lt+h3). But as the ground surfaces A B CD and E F G fI are seldom perfect planes, it is considered better to use the mean of the four heights, instead of the mean of two diagonally opposite. 117. Col'rolm]i-y Wihen all the prlisms of an excavation have the same horizontal section A, the calculation of any number of them 96 EARTH-WORK. may be performed by one operation. Let figure 50 be a plan of such an excavation, the heights at the angles being denoted by a, a,, a,, b, b a, 2I Fig. 50. b,, &c. Then the solidity of the whole will be cqual to:1A multiplied by the sum of the heights of the several prisms (~ 116). Into this sum the corner heights a, c,, b,, c,, d, and (14 will enter but once, each being found in but one prism; the heights a,, b4 c, d,, and d3 will enter twice, each being common to two prisms; the heights bl, b3, and c4 will enter three times, each being common to three prismsi and the heights b5, c,, c2, 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 s,, the sum of the second by s,, of the third by s3, and of the fourth by s4, we shall have for the solidity of all the prisms S- I4 A (Si + 2 s + 3 3 - 4 s4). ARTICLE III. —EXCAVATION AND EInIANxIsENT. 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. 3Figure 51 represents the mass of earth between two stations in an excavation of this kind. The trapezoid GBFHIT is a section of the mass at the first station, and GB, F~, HI a section at the second station; A E is the centre height at the first station, and A, E, the centre iheight at the second station; HIH, F, F is the road-bed, G GI B, B the CENTRE HEIGHTS ALONE GIVEN. 97 surface of the ground, and G G1 H, H and B B1 F1 I7 the planes forming the side slopes. This solid is a prismoid, and might be calculated by the prismoidal formula (~ 113). The following method gives the same result. A. Centre HIeights alone given. 119. iPr'oblelm.i, Given the centre heights c aznd c,, the width of the road-becl b, the slolpe of the sides s, and the lenyth of the section 1, to J2nd the solidity S of the excavation. In Fig. 51. Solution. Let c be the centre height at A (fig. 51) and c, the height at A,. The slope s is the ratio of the base of the slope to its perpendicular height (~ 102). We have then the distance out AB = + b + sc, and the distance out A, B, - 2b + s cl (~ 102). Divide the whole mass into two equal parts by a vertical plane A A, E, E drawn through the centre line, and let us find first the solidity of the righthand half. Through B draw the planes B EE,, B A, El, and BEl FI, dividing the half-section into three quadrangular pyramids, having for their common vertex the point B, and for their bases the planes A A1 E, E, EE1, F, F and A, B1, F E,. For the areas of these bases we have AreaofAA,,EB = I BE1 X (AE + AE,) -= I(c + c),." E E,1FF B= E X EEl = blt, A, B1 FAI E- A, l X, ) (El,+ii, + l I(b c, +s C15); and for thle perpendiculars firom the vertex B on these bases, producedc wlhen necessary, 98 EARTHZWORCo. Perpendicular on AA1A E E -A B b + sc,.6 dE EE1 FF =AE =c,.... A5 B1.A l E1 = E E l = 1. Then (Tab. X. 52) the solidities of the three pyramids are B-AA2E5E - j-( brsc) X 1 (c c,-4c) =6l(Lbc+ bc5 SCO +- s cc,), B -E1E, F = c x b I = I bc, B -A B1 E5 =- F 1I El I (b cl + s c,2) - (bc ~ Their sum, or the solidity of the half-section, is S = 1 [2 b (C + c~) + s (Cc + C c + C C)]. Therefore the solidity of the whole section is S - 1 [~ b (c + c2) + S (c2 + C12 + C Cl)), or S - iz[ [b (c + C) + - S (c + C12 + C cC)]. 2 )= B \L3 cT a3 When the slope is 12 to 1, s =, and the factor 2 s = 1 may be dropped. 120. IEor asDbenoo To Afid tlhe solildity S of any ll5mber n of successive sections of equal length. Solution. Let c, C, c2, C, &c. denote the centre heights at the suce cessive stations. Then we have (] 119) Solidity of first section =2 1 [ b(c + c,) -+ 3- (c + c,2 + c Cc)],... second section = r [b (c, -- c2) + a S (c,2 + C22 + C1 C2)],... third section = l I [b (c2 + C3) + s (c2 -- C32 -c2 3)]j &e. &c. For the solidity of any number n of sections, we should have 1 1 multiplied by the sum of the quantities in n parentheses formed as those just given. The last centre height, according to the notation adopted, will be represented by c,, and the next to the last by c,,_ 1. Collecting the terms multiplied by b into one line, the squares multiplied by,2s into a second line, and the remainino, terms into a third line, we have for the solidity of n sections S= - I b (e + 2 c + 2 co, -+ 2 c....+ 2 c, _ + c,,) + a- s (c2 - 2 c2 + 2 C22 + 2 C32... 0 - C2 O + C2,) + s (e Cc + l C, C2C3 t C3C4. s. + m le roppe fPhen I-3, the factor s =1 may he dropped CENTRE AND SIDE HEIGHTS GIVEN. 99 Example. Given 1 = 100, b = 28, s = -, and the stations and centre 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 cc,, c1, 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 multiplied by 1 = 50 gives for the solidity 163,700 cubic feet. Station. c c2l. cC 0 2 4 1 4 16 8 2 7 49 28 3 6 36 42 4 10 100 60 5 7 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 Hteigyts given. 121. When greater accuracy is required than can be attained by the preceding method, the side heights and the distances out (~ 102) are introduced. Let figure 52 represent the right-hand side of an excava tion between two stations. A A, B, B is the ground surface; A E = c and A, E1 = cl are the centre heiglts; B G = h and B, G, =- hi, the side heights; and d and d,, the distances out, or the horizontal distances of B and B, 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 Al B, B is a o00 EARTHTWOR.Co sitions is sufficiently accurate to serve as the basis of a general method. 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 B3 to A1, from which the ground slopes downward on each side to A and Bo Instead of this, a depression might run fiom A to B3,1 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 stations, 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. LPrbFD@ e o Given the centre heights c and c1, the side heights 01n the right h and h1, o0Z thle left h' earrl h'1, the distances out 0o2 the e'iyht d and d,, on the left d' and d',,the width of the eoacd-bed b, the lenyth of the section 1, and the directionl of the diagonals, to filc the solidity S of the excavationz. Solution. Let figure 52 represent the right-hand side of the excavation, and let us suppose first, that the diagonal runs, as shown in the figure, from B to A1. Through B d aw the planes B E EE, B AL El, and B E1,F1 dividing the half-section into three quadrangular pyramids, having for their common vertex the point B, and for their bases the planes A A., E1 E, E E1 P'1 F, and A1 B 1 I5 E1. For the arens of these bases we have AreaofAAE1 E = El X (AE A 1E1) - l(c +c1), E6 iEl E1 I -E F17 a< E El -- 2bl) 1, " a' 8 C 1 2 = 2AB Fe X cl+ 1cll 1c1+ 4t1 and for the perpendiculals 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 A 1 to Bl, as the distance out at the first station is to the distance out at the second station, that is, c -i: cl - hll = d:.dl. If we had c = 9, I = 6, el = 12,;i = 8, d = 24, and dl= 27, the formula wouled give 3: 4- 24: 27, which shows that thle surface is not a plane. CENTRE AND SIDE HEIGHTS GIVEN. 101 Perpendicular on A A1 E = E G = d,.." EEl F = — B G =h1, " A B, F1 Eil =E EI=. Fig. 52. At Then (Tab. X. 52) the solidities of the three pypamids are B - A A1 E1 E d X I (c + cL) 1 (dc + dc), B -E E1 F1 F = 1 X I bi 1 b h, B-AB1,F1El E1=1 X I (di c, +2 b hi) 1 I (dl c, + I bhl). Their sum, or the solidity of the half-section, is -6l (tdc + drc, + de + + b + b). (I) Next, suppose that the diagonal runs fiom A to B1. In this case, through B1 draw the planes B1 E, E, B, A E, and B, E F (not represented in the figure), dividing the half-section again into' three quadrangular pyramids, having for their common -vertex the point B,, and for their bases the planes A A1E E, E EB1 F1., and A B FE. For the areas of these bases we have Arca of A A 1 E E EE X (AE+ A- l) - 1, (c c), " EELF,F E=EFX EEl -lbl, ". ABFE = -AE X d+ EFX h ='dcb+ ibh; and for the perpendiculars from B1 on these bases, produced when necessary, 9 02 EARTH-WORK, Perpendicular on A A, El = E, G d, 1 E El} F -- F B, Glhl,, 6 6 BFE -E i -go Then (Tab. X. 52) the solidities of the three pyramids are BI- A Al El E d1_ X I - I \ 2 ( C +C -6 i (di C + Cl), B1 - E El F -I hj X -1 -b I 19 B, -AB~FE = l X I (dC + bh) = l (dc+ I bh). Their sum, or the solidity of the half-section, is (dc 4- d, el + d,c + b l, + 2 bh) (2) We have thus found the solidity of the half-section for both direetions 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 1 [dc + dcl, + -t cl ~ b (h + I, + h)], and (2) under the form I [dc + dc, - dc +- b (I + h, + h1)]o The only difference in these two expressions is, that d c and the last h in the first, become d, c and It, 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 h, are the heights at the extremities of the diagonal, and d, is the distance out corresponding to hi. Denote the centre height touched by the diagonal by C, the side height touched by the diagonal by H, and the distance out corresponding to the side height IH by D. We may then express both d c and d, c by D C, and both h and h1 by H; so that the solidity of the half-section on the right of the centre line, whichever way the diagonal runs, may be expressed by 1H I[dc + dc, + D C- 2(h + h, + 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 corresponding quantities on the right, merely attaching a (1) to them to distinguish them. Thus the side heights are hl and h',, and the distances out d' and d',, while D, U, and IH become Dt, Ct, and Hto The solidity of the half-section on the left may therefore be taken directly from (3), which will become CENTRE AND SIDE HEIGHTS GIVEN. 103 l [dt' c + d'1 c, + D' Cl + I b (i' + h1 +', H')]. (4) Finally, by uniting (3) and (4), we obtain the following formula for the solidity of the whole section between two stations S= S- l [(d + d') c + (d- + d'l) cl + D C+- D C't + b (h + h, + H — It' -1- Iht l+- H')]. Example. Given I = 100, b = 18, and the remaining data, as arranged in the first six columns of the following table. The first column gives the stations; the fourth gives the centre heights, namely, c = 13.6 and cl- 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 the 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 ht _ 8 to c, 8 and from c = 13.6 to h =- 12. Sta. d'. ch. c. h. d. d +-d'. (d-~ dl)c. DI C'. D C. 0 21 8 13.6. 10 24 45 612 1 15 4 - 8.0 -12 27 42 336 168 367.2 12 12 168 20 367.2 54 X 9 - 486 6)1969.20 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 eighth column; the product of d' = 21 (which is the distance out corresponding 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 in the ninth column, and the similar product of dl = 27 by c =- 13.6 is placed in the last column. The terms in the formula multiplied by 2 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 substitution in the formula, we have S = ~ X 100 (612 +- 336 + 168 + 367.2 + 9 X 54) = 32,820 cubic feet.* " The example here given is tihe same as that calculated in Mr. Borden's " Sys 104 EARTH-WORK. By applying the rule given in the note to ~ 121, we see that the surface 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 obtained. This may be easily seen by reversing the diagonal in this example, and calculating the solidity anew. The only parts of the formula affected by the change are D' C' and I 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 + 9 X 4, both of which are equal to 240. 123. Probleo To find thle solidity S of any number n of successive sections of equal length. Solution. Let c, cl, c,, c3, &c. be the centre heights at the successive stations; h, hi, h2, h,, &c. the righlt-hand side heights; h', h',, ht2 hrl3, &c. the left-hand side heights; dl, cd1, (d, c13, &c. the distances out on the right; and cl', d,, dl', d'a, &c. the distances out on the left. Then the formula for the solidity of one section (~ 122) gives for the solidities of the successive sections 6 l[(d + C) c +t (d + d,) c, + D C + D' Ct + I b ( + h + H+ h' + th''+ I'), ], HI + h'1 + t + H)], - [(d2 + d'o) c.. + (3 + d'3) c3 + D2 G + DI, C'2 + I b (h11 + h3 qand so on, for any number of sections. For the solidity of any number n of sections, we should have 1 1 multiplied by the sum of I parentheses formed as those just given. HIence i- S- I (d + d) c + 2 (cdl+ cl') cl+ 2 (d,2 + dl2) c... (d +- dl) c,, +DC+D- DCt - DC1 + D1Ct1 + - DC2+ D'2, C',2 + &c. + I b + 2hl h - 2 h2..... + h,,+ H+- I1, +II h + &c. +'+ h 2 htl' 2 h'...-..+ h', + l{'+1l1+1i'2 + &C. tern of Useful Formulcm, &c.," page 187. It will be sen, that his calculation makes the soliclity 32,460 cubic feet, which is 300 cubic feet less than the result above. This difference is owing to the omissiou, by Mr. Borlen's method, of a pyramid inclosed by the four pyramics, into which the upper portion of the right-hand halfsection is by that ncetlsod divided. CENTRE AND SihE HEIGHTS GIVEN. 105 Example. Given I = 100, b 28, and the Remaining data as giveh in the first six columns of the following table. Sta. di. 1'. c. h. d. d +d c. (d+d')c.'C'. DC. 0 17 2.. 2 2 17 34 68 1 18.5 3 > 4 "-5 21.5 40 160 68 43 2 20 4 5 -6 23 43 215 80 92 3 23 6 6 8 26 49 294 115 130 4 21.5 5-i 6 >7 24.5 46 276 129 147 5 20 4 6 4 20 40 240 120 147 6 15.5 1- 4- 3 18.5 34 136 93 80 25 35 1389 605 639 22 30 1185 22 37 605 69 102 639 102 2394 171 X 14 = 2394 6)6212 103533 cubic feet. The data in this table are arranged precisely as in the example for calculating one section (~ 122), and the remaining columns are calculated as there shown. Then, to obtain the first line of the formula, add all the numbers in the column headed (d + dt) c, making 1389, and afterwards all the numbers except the first and the last, making 1185. The next line of the formula is the sum of the columns Dt C/ and D C, which give respectively 605 and 639. To obtain the first line of the quantities multiplied by I b, add all the numbers in column h, 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 quantities multiplied by ~ b is obtained in the same way from the column marked h/. The sum of these numbers is 171, and this multiplied by 2b = 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 I 1 -, we get the contents of the six sections in feet. 124.'When the section is partly in excavation and partly in embankment, the preceding formula are still applicable; but as this application introduces minus quantities into the calculation, the following method, similar in principle, is preferable. 125. P 1roblel21. Given the iwidtls of' on excavation at tAle road-bcd 106 EAlRT-ElWORK. A F = w and Al F1 = wl (fig. 53), the side heights h and h, the length of the section 1, and the direction of the diagonal, to find the solidity S of the excavation, when the section is partly in excavation and partly in embanlkment. Fig. 53. Al Solution. Suppose, first, that tile surface is divided into two triangles by the diagonal B AI. Through B draw the plane BAl, F, dividing that part of the section which is in excavation into two pyramids B - A A,1 F F and B - A1 B1F 11, the solidities of which are B-AAFyF= -h X,1(e+ w,)={l(wh+f, h), B-l Blt -~ I X I wi h, 1- wh, o The whole solidity is, therefore, S= 1l (wh + t1h1 + wt h). Next, suppose the dividing diagonal to run from A to B1. Through B, draw a plane B, A F (not represented in the figure), dividing the excavation again into two pyramids, of which the solidities are B1, -AA1F F1-h hi X nu) I (w hi -- n'w h1), B1-ABF --- X wtUh = -lwh. The whole solidity is, therefore, S =lI (wh + tv, h, + w h,). The only difference in these two expressions is, that to, h in the first becomes w h1 in the second. But in the first case the diagonal touches wzu and h, and in the second case it touches w and h1. If, then, we designate the width touched by the diagonal by WV, and the height toulched by the diagonal by H, we may express both wv, h and wv h1 by WH; so that the solidity in either case may he expressed by CENTRE AND SIDE HEIGHTS GIVEN. 107:= 1 IS== 1 (wt + ZoVIh, + Ht). 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 I (zv h, + w, h + WH), (i Zc1 A~+ u,2 h2 + T'V1 1), I (w2 h2 + w3 h3+ WT H2), and so on for any number of sections. Hence for the solidity of any number n of sections we should have Go S = (w l + 2 wl, h, + 2 w2k h2....+ v, hi, + TWH+ TV1 H1 + TWV2 ]o + &c.) Example. Given I = 100, and the remaining data as given in the first three columns of the following table. Station. W. h. | hu IVt 1. 0 2 -2 1 2 1 8 < 6 48 8 2 10 - 70 56 3 13,'7 91 t { 4 9 36 52 247 186 209 186 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 fifth 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 does not begin on a line at right angles to the centre line, intermediate stations are taken where the excavation begins on each side of the road-bed, and the section may be calcuw 108 EARTH-tWORK. lated as a pyramid, having its vertex 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 I in this case become 0, and reduce the formula to S =' 1 w1 hl. The same remarks apply to the end of ane excavation. C. Ground very Irregularo 127. Problem. To find the solidity of a section, when the ground is very irregular. C Fig. 54. Solution. Let A iB FE - AI COD BD F1 E1 (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 HI, C, and D, making it necessary to divide the surface into five triangles running from station to station."' Let heights be taken at IS, C, and D, 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 there are triangles on the surface. This will be made evident by drawing vertical planes through the sides * It will often be necessary to introduce intermediate stations, in order to make the subdivision into triangles more conveniently and accurately. GROUND VERY IRREGULAR. 109 A C, H C, HDD and HB1. Then the solidity of the hMi-section will be equal to the sum of these prisms, minus the triangular mass B F GB1 F1 G1. The horizontal section of the prisms may be found from the distances out and the length of the section, and the vertical edges or heights are all known. Hence the solidities of these prisms may be calculated by ~ 115. To find'the solidity of the portion B F G - B F1 GI, which is to be deducted, represent the slope of the sides by s (~ 102), the heights at B and B, by h and h1, and the length of the section by 1. Then we have F G = s h, and F1I G1 = s h1. Moreover, the area of B F G - s he, and that of B1 F1' G1 - s h12. Now as the triangles B KF G and BI F1 G1 are similar, the mass required is the frustum of a pyramid, and the mean area is /Jy s /2 X I s h12 = s hl. Then (Tab. X. 53) the solidity is ~B F G- B1 F G1 -- 1I s (h2 + 7h12 +i h l). Example. Given 1 = 50, b =18, s =, the heights at A, H, and B respectively 4, 7, and 6, the distances A II= 9 and REB = 9, the heights at Al, C, D, and B1 respectively 6, 7, 9, and 8, and the distances Al C = 4, CD = 5, and D B 12. Then the horizontal section of the first prism adjoining the centre line is 1I X A1 C, since the distance A1 C is measured horizontally; and the mean of the three heights is?3 (4 + 6 + 7) = X 17. The solidity of this prism is therefore I, X A, C X 1 X 17 = -l X 4 X 17, that is, equal to 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 (4 X 17 + 9 X 5 X 23 182 X 24 + 9 X 21) = 6 X 822. For the frustum to be deducted, we have 6i X 2(62 +82+6x8) =1 X 222. Hence the solidity of the half-section is 1 (822 - 222) = ~ X 50 X 600 _= 5000 cubic feet. 128. Let us now examine the usual method of calculating excavation, when the cross-section of the ground is not level. This method consists, first, in finding the area of a cross-section 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 averiage of these two heights the middle area of the mass y 10 110 ECARTH-IWORK. 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, b the width of the road-bed, and s the slope of the sides. Let x be the required height of an equivalent section level at the top. The bottom of the equivalent section will be b, the top b + 2 s x, and the area will be the sum of the top and bottom lines multiplied by half the height or r (2 b + 2)s ) = sx + bx. But this area is to be equal to A. Therefore, s x2 + b x = A, and from this equation the value of x may be found in any given case. According to this method, the contents of the section already calculated 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 3 and b = 18, the equations for finding the heights of the equivalent end sections will be 3 x2 + 18 x 387, and x2 + 18 x 240. Solving these equations, we have for the height at the first station x = 11.146, and at the second, x = 8. The middle area will, therefore, have the height I (11.146 + 8) = 9.573, and from this height the middle area is found to be 309.78. Then by the prismoidal formula (~ 113) the solidity will be S - 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 prismoidal 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 is, if the dividing line were formed by a depression, the true solidity found by ~ 122 would be 29600 feet; whereas the method by equivalent sections would give the same 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 Emabank ments by the aid of Diagrams. By John C. Tralutwine. CORRECTION IN EXCAVATION ON CURVES. 111 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 that 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 of B, I i A Fig. 55. an excavation, E F G being the centre line, A L and CM the extreme side lines, and 0 the centre of the curve. Then the calculation of the first section would include all between the lines A, C, and B1 D1; while the true section lies between A C and B D. In like manner, the calculation of the second section would include all between HE.and NYP; while the true section lies between BD and L 1. It is evident, therefore, that at each station on the curve, as at F, the calculation is too great by the wedge-shaped mass represented by KFDj, and too Fig. 56. l b P small by the mass represented by BP'H. These masses balance each other, when the distances out on each side of the centre line are equal, that is, when the cross-section may be represented by A DFRE (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 A DFBE, the difference of the wedge-shaped masses may require consideration. 130. ProbrblemBl Given tlhe centre height c, the greatest sidce height hI, the least side height h', the greatest dlistantce oEt d, the least distace out d', and the cvidth of the road-becl b, to find the correction in excavation C, at any station on a curve of radius R or dlflection antgle D. Solution. The correction, from what has been said above, is a triand gular prism of which B FR (fig. 56) is a cross-section. The height of this prism at B (fig. 55) is B, H, the height at _1 is 11 S, and the height at F is 0. B_1 H and R, S, being very short, are here considered straight lines. Now we have the cross-section B F1R = FB E G FRE G - (ecd +- bh) - (cd l bhf') =- Ic(d d') ~ b ( -/ h'). To find the height B, II, we have the angle B EL I B FB1 = D, and therefore B31 E =- 2 HFI sin. D = 2 d sin. D. In like manner, R, S -= KID1 = 21 K sin. D - 2 d sin. D, Then since the height at F is 0, one third of the sum of the heights of the prism will be. (d dcl') sin. D, and the coirection, or the solidity of the prism, will be (~ 115) C- [ C (d - d') + lb ( - h)] X - (d + d') sin. D. When R is given, and not D, substitute for sin. D its valtte (~ 9) 50 sin. D. The correction then becomes C- [ c (d d') + b (1- )] 100 ( + ) 3 ]' This correction is to be addled, 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, fiom figure 55, that the correction will be just half of that given above. Examiple. Given c-= 28, Ih = 40, h1- = 16, d - 74, d' 38, b - 28, and R = 1400, to find C. Here the area of the cross-section BFR - 28 28 (74 - 38) +- -, (40 - 16) = 672, and one third of the sum of the 100 74+ - 38) 8 8 heights of the prism is 13x140 - Hence C — 6 672 X 1 792 cubic feet. CORRECTION IN EXCAVATION ON CURVES. 113 131. When the section is partly in excavation and partly in embankment, the cross-section of the excavation is a triangle lying wholly on one side of the centre line, or partly on one side and partly on the other. 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 Go In the first case, the correction will be a triangular prism lying between the lines B, F and HF (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 between the masses above and below F. This difference may be obtained in a very simple manner, by regarding the mass on both sides of 1F as one triangular prism the bases of which intersect on the line C F (fig. 56), in which case the height of the prism at the edge below F must be considered to be nzints, since the direction of this edge, referred to either of the bases, is contrary to that of the two others. The solidity of thlis prism will then be the difference required. 132. Problemn. Given the width of the excavation at the road-bed w, the wzidth of the road-bed b, the distance out d, and the side height h, to find the correction in excavation C, at any station on a curve of radiuls R or deflection angle D, when thle section is partly in excavation and partly in embanklntent. 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, ~ w h. The height of this prism at B (fig. 56) is (~ 130) B1 H —= 2 HF sin. D = 2 d sin. D. In a similar manner, the height at E will be 2 G E sin. D = b sin. D, and at the point intermediate between G and E, the distance of which fiom the centre line is Ib - cw, the height will bhe 2 (I b - w) sin. D= (b- 2 wc) sin. D. Hence, the correction, or the solidity of the prism, will be (~ 115) C= wh X ~ (2 d-+ bd +b- 2we) sin. D = 2wh X (d +- b -w) sin. D. When the excavation lies on both sides of the centre line, the correction, from what has been said above, is a triangular prism having also for its cross-section the cross-section of the excavation. Its area -will, therefore, be I zw h. The height of this prism at B is also 2 d sin. 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 is w - Ib, the height will be 2 (so - I b) sin. D= (2 zv - b) sin. D. As this height is to be considered anints, it must be subtracted from the others, and the correction required will be C Ic v h X ~ (2d d+ b - 2 w + b) sin. D 10 * 114 EARTH-WORK. w I X ~ (d + b - wo) sin. D. Hence, in all cases, when the section is partly in excavation and partly in embankment, we have the formula 7 C z~C=.w X (d J + b- tw) sin. D. When R is given, and not D, substitute for sin. D its value (~ 9) sin. D = - The correction then becomes @?e" C= ~wa h X leo (d+b- - w) 3R This correction is to be added, when the highest ground is on the convex side of the curve, and slbtracted 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 w - 17, b = 30, d - 519 h =- 24, and R = 1600, to find C. Here the area of the cross-section is w h 17 X 12 -- 1500 (dqb-.bw) 204, and one third of the sum of the heights of the prism is R ) 100 (51 ~- 30 - 17) 4 C 4 = 31X 1600 7)3 Hence C- 204 X = 272 cubic feet. 133. The preceding corrections (~ 130 and ~ 132) 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 F G is of 100 feet, and B, FB as the same part of D that EFis of 100 feet. The true correction may then be taken as the same part of C that the sum of the lengths of the two adjoining sections is of 200 feet. TABLE I. RADII, ORDINATES, DEFLECTIONS, AND ORDINATES FOR CURVING RAILS. Formula for Radii, Q 10; for Ordinates, ~ 25; for Deflections, ~ 19 for Curving Rails, ~ 29. 116 TABLE I. RADII, ORDINATES, DEFLECTIONS, Ordinates. Ordinates for OJrdiinates. Tangent Chord Rails. De gree. JBadii. Defiec- Deflec12 25. 37~. 50. tion. tion. 18. 20. o0. o 5 68754.94.008.014.017.018.073.145.001.001 10 34377.4S.016.027.034.036.145.291.001.001 13 22918 33.024.041.031.055.218.436.002.002 20 17188.76.032.055.06.073.291.582.002.003 25 13751.029.040.068.085.091.364.727.003.004 30 11459.19.01S.0S2.102.109.436.873.004.004 35 9822.1S. 056.095.119.127.509 1.018.004.005 40 8594.41.064.109.136.145.582 1.164.005.006 45 7639.49.072.123. 153.164.604 1.309.005.007 50 6375.55.080.136.170.182.727 1.454.006.007 55 6250.51.^087.I150.187.200.0S0 1.600.006.008 I 0 5729.65.095.16-4.205.218.873 1.745.007.009 5 523. 92 103.177.222.236.945 1.891.008.009 10 4911.15.111.191.239.255 1.018 2.036.008.010 15 4583.75.119.205.256.273 1.091 2. 182.009.011 20 4297.28/.127.218.273.291 1.164 2.327.009.012 25 4044.51.135.232.290.309 1.236 2.472.010.012 30 3819.83.143.245.307.327 1.309 2.618.011.013 35 3618.80.151.259.324.345 1 382 2.763.011.014 40 3437.87.159.273.341.364 1.454 2.909.012.015 45 3274.17.167.286.355.382 1.527 3.054.012.015 50 3125.36.175.300.375.400 1.600 3.200.013.016 55 2989.48.183.314.392.418 1.673 3.345.014 017 2 0 2864.93.191.327.409.436 1.745 3.490.014.017 5 2750.35.199.341.426.455 1.818 3.636.015.018 10 2644.58.207.355.443.473 1.891 3.781.015.019 15 2546.64.215.368.460.491 1.963 3.927.016.020 20 2455.70.223.332.477.509 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 2.181 4.363.018.022 35 2218.09.247.423.528.564 2.254 4.508.018.023 40 2148.79.255.436.545.582 2.327 4.654.019.023 45 2083.68.263.450.562.600 2.400 4.799.019.024 50 2022.41.270.464.5s80.618 2.472 4.945.020.025 55 1964.64.278.477.597.636 2.545 5.090.021.025 3 0 1910.0S.286.491.614.655 2.618 5.235.021.026 5 1858.471 294.505.631.673 2.690 5.381.022.027 10 1809.57.302.518.64S.691 2.763 5.526.022.028 15 1763 18.310.532.665.709 2.836 5.672.023.023 20 1719.12.318.545.6S2.727 2.908 5.817.024.029 2.5 1677.20.326.559.699.745 2.981 5.962.024.030 30 1]637.28.334.573.716f.764 3.054 6.108.025.031 35 1599.21.342.586.733.782 3.127 6.253.025.031 40 1562.8S.350.600.750.800 3.199 6.398.026.032 45 1528.16.358.614.767.818 3.272 6.544.027.033 50 1494.95.366.627.784.836 3.345 6.689.027.033 55 1463.16.374 641.801.855 3.417 6.835.028 o.034 4 0 1432.69.382.655.818.873 3.490 6.980.028.035 5 1403.46.390.668.835.891 3.563 7.125.029.036 10 1375.40.398.682.852.909 3.635 7.271.029.036 15 1348.45.406.695.869.927 3.708 7.416.030.037 20 1322.53.414.709.886.945 3.781 7.561.031.038 25 1297.58.422.723.903.964 3.853 7.707.031.039 30 1273.57.430.736.921.982 3.926 7.852.032.039 35 1250.42.438.750.938 1.000 3.999 7.997.032.040 40 1228.11.446.764.955 1.018 4.071 8.143.033.041 45 1206.57.454.777.972 1.036 4.144 8.288.034.041 50 1185.78.462.791.989 1.055 4.217 8.433 O.034.042 55 1165.70.469.805 1.006 1.073 4.289 8.579.035.043 5 0 1146.28.477.818 1.023 1.091 4.362 8.724.035.044 _ _ _ _ _ ~~~~~~~.51 AND ORDINATES FOR CURVING RAILS. 117 Ordinates. Ordinates for Ordinates. Tangenti Chord Rails. Degree. Radii Deflec- i Deflec25. 37. 50. tion. tion. 18. 20. 5 5 1127.50.485.S32 1.040 1.109 4.435 S c69.036.044 10 1109.33.493.846 10.(57 1.127 4.507 9.014.037.045 15 1091 73 501.859 1.074 1.146 4.580 9.160.037.016 20 1074.68.509.873 1.091 1.164 1.653.305.038.047 25 1058.16.517.837 1.108 1.182 4.725 9.450.038.047 30 1042.14.525.900 1.125 1.200 4.798 9.5' 3.039.048 35 1026.60.533.914 1.142 1.2lS1 4.870 9.741.039.049 40 1011.51.541.928 1.159 1.237 4.943 9.SEG.044.049 45 996.87.549.941 1.176 1.2555 5.016 11.031.041.050 50 982.64.557.955 1.1931 1.2738 5.0 1 1 0.177.041.051 55 968.81.565.S78 1.210 1.291 5.161 10.322.042.052 6 0 955.37.573.982 1.228 1.309 5.234 10.467.042.052 5 942.29.581.996 1.2451 1.327 1.3C6 10.612.043.053 10 929.57.589 1.009 1.262 1.3461 5.379 10.758.04.054 1 5 917.19.597 1.023 1.279 1.364 5.4590 10.903.044.05. 20 905.13.605 1.037 1.296 1.3S2 5.524 11.04.048.055 2.7 893.39.613 1.050 1.313 1.400 5.597 1.190.04 056 30 881.95.621 1.0641 1.330 1.418 5.669 11.3391.046.057 35 870.79.629 1.078 1.347 1.437 5.742 1 1.414.047.057 40 859.92.637 1.091 1.364 1.455 5.,814 11.629.047.058 45 849.32.645 1.105 ].381 1.4736 5.8S7 11.774.04S.059 50 838.97.653 1.118 1.398 1.491 5.960 11.919.048.060 55 828.88.661 1.132 1.415 1.510 6.32 12.065.049.060 7 0 819.02.669 1.146 1.432 1.528 6.105 12.210.0149 061 5 809.40.677 1.159 1.449 1.546 6.1'7 12.355 1.050.062 10 700.00.4S5 1.173 1.466 1.564 6.25 12.500 1.051.063 15 790.81.693 1.187 1.483 1.5821 6.323 12.645.051.063 20 7S1.84.701 1.2)00 1.501 1.60( 6.395 12.790.052.061 25 773.07.709 1.214 1.517 1.619) 6.468 12.936.052.065 30 764.49.717 1.22S 1.535 1.637' 6..540 13.081.053.065 35 756.10.725 1.242 1.552 1.655 6.613 13.226 1.05;.066 450 747.89.733 1.255 1.569 1.67.3 6.6851 13.371].054.067 45 739.86.740 1.269 1.586 1.691 6.758 13.516 1.055.068 50 732.01.748 1.283 1.603 1.7101 6.833 13.661 7.055.068 55 724.31.756 1.296 1.620 1.728 6.9103 13.806.056.069 0 716.781.764 1.310 1.637 1.7461 6.976 13.951.057 070 5 709.407.772 1.324 1.654 1.764 7.048 14.096.057.070 10 702.184.780 1.337 1.671 1.7321 7.121 14.241.058.071 15 695.09.788 1.351 1.688 1.2801 7.193 14.3870.0582.072 20 683.162.796 1.365 1.705 1.919 7.266 14.532..059.073 25 681.35.804 1.378 1.722 1.8379 7.338 14.677.059.073 30 674.69.812 1.392 1.739 1.85.5 7.411 14.822.0601.074 35 668.15.820 1.406 1.757' 1.8731 7.431 14.96782.061.0 40 661.740.828 1.419 1.774. 1.892 7.556 15.1127.061.076 45 655.456.836 1.433 1.791 1.910 7.628 15.257.062.076 50 649.271.844 1.4472 1.908 1.923 7.7018 15.4021.062.077 55 643.228.852 1.460 1.82 1.946 7.773 15.5417.063.078 9 0 637.27].860 1.4740 1.842 1.9652 7.846 15.6927.064.078 5 631.44.868 1.488{ 1.S59 1.9831 7918 15.837.0641.079 10 625,71.876 1.501 1.876 2.001 7.9911 15.82.0651.0SO 15 620.091.884 1.515 1.893 2.019 8.063 16.127.065(.OS1 20 614.56.892 1.529 1.910 2.037 8.1361 16.272].066i.OS1 25 609.143.900 1.5429 1.927 2.0561 8.208 16.4179.066/.082 30 603.801.90S 1.556 1.944| 2.074 S.281 16.562.067j.083 35 598571.916 1.570 1.961 2.0921 8.353 16.707[.068.084 40 593.429.924 1.583 1.979 2.110 8.426 16.852/.0681.084 45 588.36i.932 1.597 1.996 2.128 8.498 16.996.069:.085 50 583.38.940 1.611 2.013 2.147 8.571 17.1411.0691.086 551 578.491.948 1.624 2.030 2.1651.643 17.286/.070.086 10 0 573.69!.956 1.638 2.0471 2.183 8.716 17.431].071.087 118 TABLE I. RADII, ORDINATES, DEFLECTIONS, &C. Ordinates for Ordinates. Tanoent Chord Rails. Degree. Radii. Deflec- Deflec124. 25. 371 50. tion. tion. 18. 20. 0 o 10 10 564.31,972 1.665 2.031 2.219 8.360 17.721.072.089 20 555.23.933 1.693 2.115 2.256 9.005 18.011.073.090 30 546.44 1.004 1.720 2.149 2.292 9.150G 18.300.074.092 40 537.92 1.020 1.74S 2.184 2.329 9.295 18.590.075.093 50 529.67 1.036 1.775 2.218 2.3653 9.440 18.880.076.094 11 0 521.67 1.052 1.302 2.252 2.402 9.585 19.169.078.096 10 513.91 1.063 1.S30 2.2S6 2.43S 9.729 19.459.079.097 20 506.33 1.084 1.857 2.320 2.475 9.874 19.748.080.099 30 499.06 1.100 1.83S4 2.354 2.511 10.019 20.033.OS1.100 40 491.96 1.116 1.912 2.389 2.547 10.164 20.327.0S2.102 50 4S5.05 1.132 1.938 2.423 2.584 10.308 20.616.084.103 12 0 478.34 1.14S 1.967 2.457 2.620 10.453 20.906.095.105 1 0 471.81 1.164 1.994 2.491 2.657 10.597 21.195.086.106 20 465.46 1.180 2.02 1 2.525 2.693 10.742 21.484.037.107 30 459.28 1.196 2.049 2.560 2.730 10.SS7 21.773.088.109 40 453.26 1.212 2.076 2.594 2.766 11.031 22.063.089.110 50 447.40 1.228 2.104 2.628 2.803 11.176 22.352.091.112 13 0 441.68 1 244 2.131 2.662 2.839 11.320 22.641.092.113 10 436.12 1.260 2.159 2.697 2.876 11.465 22.930.093.115 20 430.69 1.277 2.186 2.731 2.912 11.609 23.219.094.116 30 425.40 1.293 2.213 2.765 2.949 11.754 23.507.095.118 40 420.23 1.309 2.241 2.799 2.985 11.898 23.796.096.119 50 415.19 1.325 2.268 2.833 3.022 12.043 24.085.098.120 14 0 410.28 1.341 2.296 2.868 3.058 12.187 24.374.099.122 10 405.47 1.357 2.323 2.902 3.095 12.331 24.663.100.123 20 400.78 1.373 2.351 2.936 3.131 12.476 24.951.101.125 30 396.20 1.339 2.:378 2.970 3.163 12.620 25.240.102.126 40 391.72 1.405 2.406 3.005 3.204 12.764 25.528.103.123 50 387.34 1.421 2.433 3.039 3.241 12.908 2.817.105.129 15 0 383.06 1.437 2.461 3.073 3.277 13.053 26.105.106.131 I0 378.88 1.453 2.498 3.1071 3.314 13.197 26.394.107.132 20 374.79 1.469 2.515 3.142 3.330 13.341 26.6,2.108.133 30 370.78 1.486 2.543 3.176 3.387 13.485 26.970.109.135 40 366.86 1.502 2.570 3.210 3.423 13.629 27.258.110.136 50 363.02 1.518 2.593 3.245 3.460 13.773 27.547.112.138 16 0 359.26 1.534 2.62.5 3.279 3.496 13.917 27.835.113.139 10 355.59 1.550 2.653 3.313 3.533 14.061 28.123.114.141 20 351.98 1.566 2.680 3.347 3.569 14.205 28.411.115.142 30 348.45 1.582 2.708 3.382 3.606 14.349 28.699.116.143 40 344.99 1.598 2.736 3.416 3.643 14.493 28.986.117.145 50 341.60 1.615 2.763 3.450 3.679 14.637 29.274.119.146 17 0 333.27 1.631 2.791 3.485 3.716 14.781 29.562.120.148 10 335.01 1.647 2.318 3.519 3.752 14.925 29.850.121.149 20 331.82 1.663 2.846 3.553 3.789 15.069 30.137.122.151 30 323.6S 1.679 2.873 3.5388 3.S25 15.212 30 425.123.152 40 325.60 1.695 2.99 1 3.622 38362 15.356 30.712.124.154 50 322.59 1.711 2.923 3.656! 3.893 15.500 31.000.126.155 18 0 319.62 1.723 2.906 3.691 3.935 15.643 31.237.127.156 10 316.71 1.7441 2.933 3.725 3.972 13.787 31.574.128.158 20 313 s86 1.760 3.011 3.759 4.003 15.931 31.861.129.159 30 311.06 1.776 3.039 3.794 4.045 16.074 32.149.130.161 40 303.30 1.792 3.0636 3.328 4.031 16.218 32.436.131.162 50 305.60 1.809 3.094 3.862 4.1185 16.361 32.723.133.164 19 0 302.94 1.825 3.121 3.897 4.155 16.505 33.010 |.134.165 10 300.33 1.841 3.149 3.931 4.191 16.648 33.296.135.166 20 297.77 1.857 3.177 3.965 4.223 16.792 33.5S3.136,.168 30 295.25 1.8731 3.204 4.000 4.265 16.935 33.870.1371.169 40 292.77 1.890} 3.2321 4.034 4.301 17.078 31.157.133.171 50 299.33 1.906 3.259 4.069 4.33S 17.222 31.4430.140.172 20 0! 237.91 1. 922 3.2837 4.103 1.:374 17.365 34.730.141.174 101~ ~~~ —-------- TABLE II. LONG CHORDS. 119 TABLE I1. LONG CHORDS. ~ 69. Degree of 2 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. Curve. 0 10 200.000 299.999 399.993 499.996 599.993 20 199.999.997.992.933.970 30.998.992.9S1.962.933 40.997.986.966.932.832 50.995.979.947.894.815 1 0 199.992 299.970 399.924 499.848 599.733 10.990.959.896.793.637 20.986.946.865.729.526 30.983.932.S29.657.401 40.979.915.789.577.260 50.974.898.744.488.105 2 0 199.970 299.878 399.695 499.391 598.934 10.964.857.643.2;5.750 20.959.834.586.171.550 30.952.810.524' 049.336 40.946.783.459 498.918.106 50.939.756.339.778 597.862 3 0 199.931 299.726 399.315 498.630 597.604 10.924.695.237.474.331 20.915.652.154.309.043 30.907.627.06S.136 596.740 40.893.591 398.977 497.955.423 50.888.553.882.765.091 4 0 199.878 299.1513 398.782 497.5t66 595.744 10.868.471.679.360.333 20.857.428.571.145.007 30.846.383.459 496.921 594.617 40.834.337.343.689.212 50.822.289.223.449 593.792 5 0 199.810 299.239 393.099 496.200 593.358 10.797.187 397.970 495.944 592.909 20.783.134.837.678.446 30.770.079.700.405 591.968 40.756.023.559.123.476 50.741 298.964.413 494.832 590.970 6 0 199.726 298.994 397.264 494.534 590.449 10.710.843.110.227 589.913 20.695.779 396.952 493.912.364 30.678.714.790.588 533.800 40.662.648.623 257.221 50.644.579 453 492.917 537.628 7 0 199.627 298.509 396.278 492.56S 587.021 10.609.438.099.212 586.400 20.591.364 395.916 491.847 585.765 30.572.289.729.474.115 40' 553.212.533.093 584.451 50.533.134.342 490.704 583.773 8 0.513 293.054 395.142 i 4909.306 583.081 120 TABLE III. — TABLE IV. TABLE III. CORRECTION FOR THE EARTHS CURVATURE AND FOR REFRACTION. ~ 105. D. d. D. d. D. d. D. d. 300.002 1800.066 3300.223 4S00.472 400.003 1900.074 3400.237 4900.492 500.005 2000.082 3500.251 5000.512 600.007 2100.090 3600.266 5100.533 700.010 2200.099 3700.2S1 5200.554 800.013 2300.108 3800.296 1 m ile.571 900.017 2400.1 1 3900.312 2 2.285 1000.020 2500.123 4(100.328 3 5.142 1100.025 2600.139 4100.345 4 9.142 1200.030 2700.149 4200.362 5 14.284 1300.035 2300.161 4300.379 6 20.568 1400.040 2900,172 4400.397 7 27.996 1500.046 3000.184 4500.415 8 36.566 1600.052 3100.197 4600.434 9 46.279 1700.059 3200.210 4700.453 10 57.135 TABLE IV. ELEVATION OF THE OUTER RAIL ON CURVES. ~ 110. Degree. ZI= 15. M1- =2. 1 = 25. 51= 30. 31 40. II = 50. 1.012.022 03)4.049.OSS.137 2.025.0144.06 99. 175.27-1 3.037.066.103.148.263.4l 1 4.0419.0 137.197.3 1.548 5.062.110.171.247.138.685 6.074.131.205.296.526.822 7.086.153.240.345.613.958 8.099.17.5.274.394.701 1.095 9.1.197.308.443.788 1.232 10.123.219.342.493.876 1.368 TABLE V. - TABLE VI. 121 TABLE oV. FROG ANGLES, CHORDS, AND ORDINATES FOR TURNOUTS. This table is calculated for g -= 4.7, d =.42, and S - 10 20'. Formula for frog angle ], and chord B F, ~ 50; for 7m, the middle ordinate of B F ~ 25; for int, the middle ordinate for curving an 18 ft. rail, ~ 29. F. BF..n. In'. R. F. B F. in. i. 0 I II 0 I I/ 1000 5 27 44 72.22.651.041 600 6 57 48 59.17 727.063 975 5 31 39 71.53.655.042 575 7 6 26 58.16.733.070 950 5 35 44 70.83.659.043 550 7 15 40 57.12.739.074 925 5 39 59 70.11.663.044 525 7 25 33 56.05.745.077 900 5 44 24 69.38.667.045 500 7 36 10 54.94.752.031 875 5 49 1 68.64.671.0416 475 7 47 37 53.79.758.085 850 5 53 50 67.88.676.0813 4,50 8 0 1 52.61.765.090 825 5 58 52 67.10.680.019 425 8 13 30 51.37.773.095 800 6 4 9 66.30.685.051 400 8 23 14 50.09.780.101 775 6 9 41 65.49.690.052 375 8 44 26 48.75.788.108 750 6 15 30 64.65.695.054 350 9 2 20 47.35.796.116 725 6 21 37 63.80.700.056 325 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 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. 1.01745 32925 19943 1.00029 08882 08666 1.00000 48481 36811 2.03490 65850 39887 2.00058 17764 17331 2.00000 96962 73622 3.05235 98775 59830 3.00087 26646 25997 3.001)01 45444 10433 4.06981 31700 79773 4.00116 35528 34663 4.00001 93925 47244 5.08726 64625 99716 5.00145 44410 43329 5.00002 42406 84055 6.10471 97551 19660 6.00174 53292 51994 6.00002 90888 20367 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 79490 9.00261 79938 77991 9.00004 36332 313001 aE~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~s 122 TABLE VII. EXPANSION BY HEAT. TABLE VIIo EXPANSION BY HEAT. Bodies. 320 to 2123. 10. Authorityo Platina,.000SS42.0000004912 IIassler. Gold,.001466.00000S141 " Silver,.001909.000010605 ( Mercury,.0180C1S o0001001 Brass,.00189163.000010509 Iron,.00125344.000006964 VWater,.0466 not uniform. s Granite).000'6850.00000482.5 Prof. BarJlett. Marble,.00102024.000005668 " Sandstone,.00171576.000009532 _ _ - _-~~~~ TABLE VIII. PROPERTIES OF M[ATERIALS. 123 TABLE VIII. PROPERTIES OF MlATERIIALS. The authorities referred to by the caplital letters in the table are: — B Barlow, On the Strenlyth of L. Lamc. Mlaterials. M. Musschenbroek, Ilt. to Nlat. Be. Bcvan. Phil. Br. Lieut. Brown. R. Rennie, Phil. Trans. C. Couch. Ro. Roldelet, L'Art de Batir. F. Franklin Institute, Relport on T. Telford. Steamt Boilers. Ta. Taylor, Statistics of Coal. G. Gordon, ~Eig. Translatioz of W. Weisbach, Mlech. of 1MilchinWVeisbach. erg and Engineering. H. Hodgkinson, Reports to Brit. The numbers without letters are Associatiolz. taken from Prof. Moseley's EnHa. HIassler, Tables. 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 obtained by him from prisms high enough to allow the wedge of rupture to slide fieely off. Ile shows that this is essential in experiments on compression. The modulus of rupture S is the breaking weight of a prism 1 in. 1)road, 1 in. deep, and 1 in. between the supports, the weight being applied in the middle. To find the corresponding breaking weight T' of a rectangular beam of any other size, let I = its length, b = its breadth, and d = its depth, all in inches. Then TIV -3- 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 0, 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 moderately dry, the second to specimens turned and kept dry in a warm place two months longer. In the case of American Birch, Elm, and Tceak, the numbers apply to seasoned specimcns. 124 TABLE VIII. PROPERTIES OF MATERIALS. Meight.ese Crushing Moulus MI\aterials. Specific per IStength Force p 1er. Cubic tcgth Square of EupGtravity. eFoot per Squ Inch ture S in lbs. Inchinb. in lbs. in lbs. Mlietals. Brass cast...... 8.399 52-1.94 17963 R. Copper, cast,. 8.607 537.94 19072 " rolled,. 8.864 F. 554.00 32826 F. b; wire-drawn,. 8.878 554.87 61228 Gold,.;....{. 19.258 I1a. 1203.62 19.361 IIa. 1210.06 Iron, cast, Carron No. 2, cold blast, 7.066 11. 441.62 16633 II. 10637511. 3S556 II. hot' 7.046 It. 440.3'7 13505 II. 103540 II. 37503 II. Devon No. 3, cold " 7.29. 11. 4355.94 36288 II.'" C.hot " 7.229 H. 451.,1 21907 II. 14543511. 43497 lI. Buffery N'o.1, cold " 7.079 11. 442.44 17466 II. 93385 II. 37503 II. " hot " 6.998 HI. 437.37 13434 HI. 86397 H. 35316 II. Iron, wrought, English bar,.7.700 481.25 57120 L. 56000 G. 54000 G. WVelsh ",.,. 61960 T. Swedish ".... 64960 T... 44... [ 7.478 F. 467.37 58184 F. Lancaster Co., Pa. bar, 7.740 F. 483.75 58661 F. Tennessee'. 7.SO5 F. 487.81 52099 F. Missouri " 7.722 F. 432.62 47909 F. Iron wire, English, diam. =.100 in. 80214 T. Phillipsb'g, Pa.:".333 " 7.727 F. 482.941 84186 F. 1'.190, 73888 F.. ".156 " 89162 F. Lead, cast, I... 1.446 1. 715.37 1824 R. Lead wire,...... 11.317 707.31 2531 II. Mercury,... 13.598 W. 849.87 P19500OIIa. 1218.75 ~Pla~s~tina,. 22.669Ia. 1416.81 Silver,. 10.474 I-a. 654.62 40902 MI. Steel, soft,.... 7.780 486.251 120000 66 razor-tempered,.. 7.840 490.00 150000 Tin, cast,... 7.291 455.69 5322 31. Zinc, fused,. 7 050 W. 440.62'rolled,.540W. 471.25 Voods. Ash, English,.. 760 B 47.50 17000 B. f936311. 121-56 B. Birch, English,.792 B. 49.50 15000 02 H3 10920 B. " American,...648 B. 40.50 11 663 H. 9621 B. Box,....... 960 B. 60.00 20000 B. 9771 II. Cedar, Canadian,...909 C. 56.81 11400Be. 5363 II. Chestnut,..657 Ro. 41.06 1330011o. Deal, Christiania middle,.698 B. 43.62 12400 9361 B. Memel.590 B. 36.87 103S6 B. " Norway Spruce,.340 21.25 17600 ":lEnglish, 4..0 29.37 7000 Elm, seasoned,.553 B. 34.561 13489 1. 103311-I. 6078 B. Fir, New England,.5 53 B. 34.56 6612 B. 1igsa,........ 753 B. 47.06 12000 B. 6I586 II. G6 i D. Lignum-vit,.... 1.220 76.25 1180 IM. Mahogany, Spanish,.$ 03 50.00 16500 81989 II. 819S I1I. TABLE VIII. PROPERTIES OF BIATERIALS. 125'deIt Tensile MCshin odulus Specific per Streigtl cForce per Materials. Specifle Cubic Stei 1 Sgtr of Rup Gravity. Foot per Square nc ture S in ls. Inchinl bs. in lbs. Woods. Oak, English,.931 B. 58.37 10000 B. 10032 B.'G Catnadian,.....872B. 54.50 10253 10496 B. Pine, pitch,.....60 B. 41.25 7818. 6790 I. 9792 B. c red......657 B. 41.06 1 5395 E } 8046 E. " American, white,..455 Br. 2S.441 7829 Br. " " Southern,.872 Br. 54.50 13937 Br. Poplar,.......33 3I. 23.94 7200 Be. 31074 I. 5124 1-I. Teak,.745 B. 46.56 15000 B. 12101 1-1. 14772 B. Other 3iaterials. Brick, redc, 2.163. 135.50 2S0 803 R. 340 WV. pale red,.. 2.0 5 IR. 130.3L 300 562 R. 180 W. Chalk,...... 24 174.00 501 R. 1.869 116.81 Coal, Peenn. anthracite. 1. 327 Ta 82.9-4 1.700 Ta 106.25 semi-bitumlinous, 1.4;53 Ta. 90. 1 "; Mid. " 55.2 T-. 97.00 " Penn. bituminous,. 1.312 Ta. 82.00 " Ohio ~ 1.270 Ta. 79.37 " English " 1.2.9 Ta. 7.6 Earth, loamy hard-stamped, fresh, 2.060 W. 129.75 "; "; dry, 1.930 VW. 12.62 garden, fresh,.. 2.05). 128.12 " dry,.. 1.630 V. 101.87 dry, poor,..... 1.340 W. 83.75 Glass, plate,.... 2.4 3 153.31 9420 Gravel,....... 1.920 120.00 Granite, Aberdeen, 2.62- R.31 164.06 10914 R. Ivory...... 1.26 114.12 16626 2.400 W. 150,00 1500 W. 700 WV. Limestone, ne..,, 2.S60 IV 178.75 6000 WV. 1700 WV. Marble, white Italian,. 2.6:38 I. 164.87 953 G. 1062 " black Galvay, 2.690a II. 168.4-1 2664 MIasonry, quarry stone, dry, 2.400IV. 150.03) sandstone, 2.050 IV. 128.12 1.470 W. 91.87 " brick, dry, 1. 19 V. 99.37 Ropes, hemp, under 1 inch dianm., 9280 W. " fronm 1 to 3 in. " 7213 V. " over 3 inches " 5156 WV. Sand, river,.1... 1.86 117.87 Sandstone, 1.900 W. 118.75 1400 W. 600 IV. s e.... 2.700WV. 163.75 13000 IV. 8UV IV " Dundee, 2.530 R. 158.12 6630 R. " Derby, red and friable, 2.316 R. 144.75 3142 1t. Slatec Welsh.... 2.838 130.50 12300 Scotch, 9600 1_. ___ 1_, 1 — I _,_* 126 TABLE IX. MAGNETIC VARIATIONY, TABLE IXo MAGNETIC VARIATION. THE following table has been made up from varionus sources, prina cipally, however, from the results of the United States Coast Survey, kindly furnishedin manuscript by the Superintendent, Prof. A. D. Bache.; These results," he remarks in an accompanying note, are from preliminary computations, and may be sonmewhat changed by the final ones." Among the other sources may be mentioned the Smithsonian Contributions for 1852, Trans. Am. PI'il. Soc. for 1846, Lond. Phil. Trans. for 1849, Silliman's Journal for 1838, 1840. 1846, and 1852, and the various.American, Britisli, and Russian Government Observations. The latitudes and longitudes Ilere given are not always to be relied on,as minutely correct. 5Many of them, for places in the Western States, were confessedly taken fiom maps and other uncertain sources. Those of the Coast Survey Stations, however, as well as those of American and foreign Government Observatories and Stations, are presumed to be accurate. It will be seen that the variation of the magnetic needle in the United States is in some places vwest and in others east. Th/e line of no variation begins in the northwest palrt of Lake Iluron, 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 wvest of this line ha've the variation of the needle east; and, as a general rule, the farther a place lies fiom this line, the greater is the variation. The position of the line of no variation given above is the position assigned to it by PrIofessor Loomis for the year 1840. But this line has for many years been movinlg slowly werstwlard, and tliis motion still continues. Hence places whlose valriation is west are every year farther and farther from this line, so thlat the variation west is constantly increasing. On the contrary, places whose variation is east are every year nearer and ncearer to tlhis line, so that the variation east is constantly decreasin'g. The rate of this increase or decrease, as the case may be, is said to average about 2t for the Southern States, 4' for the MLiddle and Western States, and 6' for the New Engoland States.* The increase in Waslhington in 1840- 2 was B3 44.211; in Toronto in 1841 - 2 it lwas 4' 46 21". The c(lIang-es in Prof. Loom's in Stiina!l's.1tournal, Vol. X.XX., 184 TABLE IX. MIAGNETIC VARIATION. 127 Cambridge, Mass. may be seen from the following determinations of the variation, taken fiom the Memoirs of the American Academy for 1846. 0 I 0 1 Cambrid(e, 1708, 9 0 Cambridge, 1788, 6 38 1742, 8 0 Boston, 1793, 6 30 " a1757, 7 20 Salem, 1805, 5 57 " L 1761, 7 14 " 1808, 5 20 1763, 7 0 " 1810, 6 22 1780, 7 2 Carmbridge, 1810, 7 30 " 1782, 6 46 " 1835, 8 51 rr" 1783, 6 52 " 1840, 9 18 But besides this change in the variation, which may be called secular, there is an annual and a diurnal change, and very fiequently there are irregular changes of considerable anmount. AW-ith respect to the annual change, the variation west in the:Northern hemisphere is generally found to be somewhat greater, and the variation east somerwhat 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 ainy practical importance. The diurnal change is well determined. At Washington in 1840 - 2, the mean diurnal change in the variation was,*I " I ) IIi Summer, 10 4.1 Autumn, 6 21.2 Winiter, 5 9.1 Spring, 8 10.7 At Toronto the means were, f - 1841. 1843. 1846. 1847. 1849, 1850. 185l. Winter, 6.67 5.64 5.73 7.28 8.25 8.01 7.011 Splring and Autumn, 9.46 9.36 9.15 10.08 12.25 10.90 10.82 Summer, 12.38 11 70 13.36 13.84 14.80 13.74 12.61l The diurnal change in the variation is such that the north end of the needle in the Northern hemnisphere attains its extreme westerly position about 2 o'clock, P. MI., 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.'T., while in places whose variation is east, the maximum variation occurs about 8 A. M. In Washington, according to the report of' Lieutenant Gilliss, the maximum variation, taking the mean of two years' obse-rations, occurs at 1' 33'11. 1P. M1. the minimum at 811 61'- A. MI. The determinations of the Coast Survey are distinguished by the letters C. S. attalched 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 Iteport, Senate Document, 172, 1845. It lonldonl Plhilosophlical Tra.,nsactions, 1852. 128 TABLE IX. MIAGNETIC VARIATION. le tulde. Lontude. Authority. Date. Variation..lIaisze, o I o I o / Aganlenticus, 43 13.4 70 41.2 T. J. Lee, C. S. Sept., 1847 10 10.0 W. Bethel, 44 28.0 70 51.0 J. Locke, June, 1845 11 50.0 " Bowdoin IIill, Port-. land, 43 33.8 70 16.2 J. E. Ililgard, C S. Aug., 1851 11 41.1 CapeNeddick,York 43 11.6 70 36.1 J. E. Ililgard, C. S. Aug., 1851 11 9.0 " Cape Small, 43 46.7 69 50.4 G. IW. Dean, C. S. Oct., 1851 12 5.5 " KIennebunukport, 43 21.4 70 27.8 J. E. Ililgfard, C. S. Aug., 1851 11 23.6 6l Kittery Point, 43 4.8 70 43.3 J. 1E. Hilgard, C. S. Sept., SSO5 10 30.2 ]" Mlt. Pleasant, 44 1.6 70 49.0 G. W. Dean, C. S. Aug., 1851 14 32.0'1 Portland, 43 41.0 70 20.5 J. Locke, June, 1845 11 28.3 6[ Rlichmond Island, 43 32.4 70 14.0 J. E. Iilgard, C. S. Sept., 1850 12 17.9 o NTewv Htamvpshire. Fabyan's Hotel, 44 16.0 71 29.0 J. Locke, June, 1845 11 32.0 W. Hanover, 413 42.0 72 10.0 Prof. Young, 1839 9 15.0 46 Isle of Shoals, 42 59.2 70 36.5 T. J. Lee, C. S. Aug, 1847 10 3.4 "; Patuccawa, 4i3 7.2 71 11.5 G. WV. Dean, C. S. Aug., 1849 10 42.9 6" Unkonoonuc, 42 59.0 71 35.0 J. S. Ruth, C. S. Oct, 1848 9 5.6 a Versmsnt. Burlington, 4-4 27.0 73 10.0 J. Locke, June, 1845 9 22.0 W. llassachussetts. Annis-squam, 42 39.4 70 40.3 G. W. Keely, C. S. Aug., 1849 11 36.7VW. Baker's Island, 42 32.2 70 46.8 G. V. IKeely, C. S. Sept., 1849 12 17.0 Sept. and 1 Blue Hill, MIilton, 42 12.7 71 6.5 T. J. Lee, C. S. Oct., 1845 9 9 13.8 Cambridge, 42 22.9 71 7.2 W. C. Bolnd, 182 10 8.0 Chappaquidick,Ed- gartown, 11 22.7 70 23.7 T. J. Lee C. S. July, 186 S 47.7 Codlon's Hill, Marblelhead, 42 31.0 70 50.9 G. W. KIeely, C. S. Sept., 1849 11 49.S Copecut IIill, 11 43.3 71 3.3 T. J. Lee, C. S. Oc., n184 9 12.1 4' Dorchester, 42 19.0 71 4.0 W. C. Bond, 1839 9 2.0 Fort Lee, Salem, 42 31.9 70 52.1 G. V. IKeely, C. S. Aug., 1849 10 14.5 " Hyannis, 41 33.0 70 18.0 T. J. Lee, C. S. Aug., 1846 9 22.0 " Indian 51ill, 41 25.7 70 40.3 T. J. Lee, C. S. Aug., 18146 8 49.3 " Little Nahant, 42 26.2 70 55.5 G. WV. Keely, C. S. Aug., 1849 9 40.9 NaTntasket, 42 18.2 70 54.0 T. J. Lee, C. S. Sept., 1847 9 33.5 Nantucket, 41 17.070 6.0 T. J. Lee, C. S. July, 1846 9 14.0 C New Bedford, 41 38.0 70 54.0 T. J. Lee, C. S. Oct., 1845 8 54.6 Shootflying IIill, Barnstable, 41 41.1 70 20.5 T. J. Lee, C. S. Aug., 1846 9 40.1 a Tarpaulin Cove, 41 28.1 70 45.1 T. J. Lee, C. S. Aug., 1846 9 10.1 Rl1ode Islanld. Beacon-pole Ilill, 41 59.7 71 26.7 T. J. Lee, C. S. vt4ov.,44 9 29.3SW.I HMeSparran Hill, 41 29.7 71 27.1 T. J. Lee, C. S. July, 1844 8 53.3 " Point Judith, 41 21.9 71 28.9 RI.1 Fauntleroy,C.S. Sept, 1847 8 59.4' Spencer 3ill, 41 40.7 71 29.3 T. J. Lee, C. S. { A.ll1844 9 11.9' ColnnecticuLt. Black Itock, Fairfield, 41 8.6 73 12.6 J. Penwick, C. S. Sept., 1845 6 53,5 W. Bridgeport, 41 10.0 7.3 11.0 J. Renwick, C. S. Sept., 1845 6 19,3'[ Fort \VWooster, 41 16.9 72 53.2 J. S. Ruth, C. S. Aug., 1813 7 264 Groton Point, New London, 41 13.0 72 0.01J. Renwick, C. S. I Au., 134- 7 29.5 C L =_ _ _ _ _ _ _ _ _ = _= TABLE IX. MAGNETIC VARIATION. 129 Place. Lati- ouongi Authority. Date. Variation. tude. tude. o 10 o Milford, 41 16,0 73 1.0 JbRenwick, C. S Sept., 1845 6 38.3 W. New Haven, Pavilion, 41 18.5 72 55.4 J. S. Ruth, C. S. Aug., 1848 6 37.5 " New Haven, Yale College, 41 18.5 72 55.4 J. Renwick. C. S. Sept. 1845 6 17.3 66 Norwalk, 41 7.1 73 24.2 J. lenwick, C. S. Sept., 1844 6 46.3 Oyster Point, New Haven, 41 17.0 72 55.4 J. S. Ruth, C. S. Aug., IS48 6 32.3 6 Sachem's Iead, Guilfordc, 41 17.0 72 43.0 J. ]Renwick, C. S. Aug., 1845 615.2" Sawpits, 40 59.5 73 39.4 J. Renwick, C. S. Sept. 1844 6 1.6 Saybrook, 41 16.0 72 20.0 J. Renwick, C. S. AuC., 1845 6 499' Stamford, 41 3.5 73 32.0 J. Renwick, C. S. Sept., 1844 6 40.4 Stonington, 41 20.0 71 54.0 J. Renwick, C. S. Aug., 1845 7 38.2 66 NTew York. Albany, 42 39.0 73 44,0 Regents' Report, 1836 6 47.0 W. Bloomingdale Asylum, 40 48.8 73 57.4 J. Locke, C. S. April, 1846 5 10.9 6 Cole, Staten Island, 40 31.8 74 13.8 J. Locke, C. S. April, 1846 5 33.8 " Drowned Mleadow, L. I., 40 56.1 73 3.5 J. Renwick, C. S. Sept., 1845 6 3.6 6" Flatbush, L. I., 40 40.2 73 57.7 J. Locke, C. S. April, 1846 5 54.6 " Greenport, L. I., 41 6.0 72 21.0 J. Renwick, C. S. Aug., 1845 7 14.6 Leggett, 40 48.9 73 530R.0 I. Fauntleroy,C.S. Oct., 1847 5 40.6 6 Lloyd's IHarbor, L. I., 40 55.6 73 24.8 J. Renwick, C. S. Sept., 1844 6 12.5 New Rochelle, 40 52.5 73 47.0 J. Renwick, C. S. Sept., 1844 5 31.5 " New York, 40 42.7 74 0.1 J. Renwick, C. S. Sept., 1845 6 25.3 6 Oyster Bay, L. I., 40 52.3 73 31.3 J. Renwick, C. S. Sept., 1844 6 53.6 "6 Rouse's Point) 45 0.0 73 21.0 Bouncary Survey, Oct., 1845 11 28.0 66 Sands Lighthouse, L. I., [ 40 51.9 73 43.5 R.H, Fauntleroy,C.S. Oct,, 1847 6 9.7 Sands Point, L. I., 40 52.0 73 43.0 J. Renwick, C, S. Sept., 1845 7 14.6 Watchhill. Fire Island, 40 41.4 72 58.9 HI,. F'auntleroyC.S. Oct., 1847 7 33.5 " West Point, 41 25.0 73 58.0 Prof. Davies, Sept,, 1835 6 32.0 " New Jersey. Cape May Lighthouse, 33 558 74 57.6 J. Loclke, C. S. June, 1846 3 3.2W. Chew,:39 48.2 75 9.7 J. Locke, C. S, July, 1846 3 20.4 " Church Landing, 39 40.9 75 30.3 J. Locke, C. S. June, 1846:5 45.8 Egg Islandcl 39 10.4 75 7.8 J. Locke, aC.. June, 1846 3 18.2 " I-fawkins, 39 25.5 75 17.1 J. Locke, C. s. June, 1846 2 58.7 Mt.Rose,Princeton, 40 22.2 74 42.9 J. E. Hilgard, C. S. Aug., 1852 5 31.8 " Newark, 40 44.8 74 7.0 J,. Locke, C. S. A.pril, 1846 5 32.7: Pine Mountain, 39 25.0 75 19 9 J. Locke, C. s. June, 1846 2 52.0 Port Norris, 39 14.5 75 1.0 J. Locke,.C. S. June, 1846 3 6.5 41 Sandy Hookl, 40 28.0 73 59.8 J. Renwick, C... Aug., 1844 5 54.0 6 Town ]3ank, Cape May, 38 58.6 74 57.4 J. Locke, C. S. June, 1846 3 3.2 11 Tucker's Island, 39 30.8 74 16.9 T. J. Lee, C. s. NTo., 1846 4 23.8 " White I-Till, Eordentown, 40 8.3 74 43 8 J. Locke, C S. April, l146 4 22.5 " Penn sylvalnia. Girard College, Philadelphia, 39 53.4 75 9.9 J. Locke, C. S. May, 1846 3 50.7 V. Pittsbur^g, 40 26.0 79 53.0 J. Locke, May, 1845 0 33.1 Vanuxem, Eristol, 40 5.9 74 52.7 J. Locke, C. S. July, 1846 4 20.5 " Local attraction exists here, according to;Prof. Locke. 130 TABLE IX. e MAGi ETIC VARIATION, Place. Lati- t o,.-e. Place. tude. Lni Authority. Date. Variation. Deldaware. Bombay Hook o o o 1 o 1 Lighthouse, 39 21.8 75 30.3 J, Locke, C. 8. June, 1846 3 17.9 WV; Fort Delaware, Delaware River, 39 35.3 75 33.8 J. Locke, C. 8. June, 1846 3 16.0 " Lewes Landingt 38 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, 1846 2 42.2 " Sawyer, 39 42.0 75 33.5 J. Locke, C. S. June, 1846 2 47.8 ": Wilmington, 39 44.9 75 33.6 J. Locke, C. S. May, 1846 2 31.8 " llasrylarld. Annapolis, 33 56.0 76 35.0 T. J. Lee1 Ci S. June, 1845 2 14.0 IV, Bodkin, 39 8.0 76 25,2 T. J, Lee, C. S. April, 1847 2 2.6 ts Finlay, 39 24.4 76 31.2 J. Locke, C. S. April, 1846 2 19.5 " Fort McIlenry, Baltimore, 39 15.7 76 34.5 T. J. Lee, C. S. April, 1847 2 13.0 " Hill, 33 53.9 76 52,5 G. VW. Dean, C. S. Sept., 1850 2 15.4 6 Kent Island,:39 1.8 76 18.8 J. IIeuston, C. S. July, 1819 2 30.5 4 Marriott's, 33 52.4 76 36.3 T. J. Lee, C. S. June, 1849 2 5.2 " North Point, 39 11.7 76 26.3 T. J. Lee. C. S. July, 1846 1 42.1 4 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. Ji Lee, C, S. June, 18-17 2 28.5 4 Rosanne, 39 17.5 76 42.8 T1 J. Lee, C. S. June, 1845 2 12.0 " Soper, 39 5.1 75 56.7 G. W. Deal4, C. 8, July, 1850 2 7,0 "4 South Base, Kent Island, 38 53.8 76 21.7 T. J. Lee, C. S. June, 1845 2 26.2 6 SusquehannaLi ghthouse, HIavre do Grace, 39 32.4 76 4.8 T, J, Lee, C. S. July, 1847 2 51.1 " Taylor, 33 59.8 76 27.6 T. J. Lee, C. S. May, 1847 2 18.4 "; Webb, 39 5.4 76 40.2 G. W. Dean, C. S. Nov., 1850 2 7.9 Distsict of Cohslsenbia. Causten, Georgetown, 38 55.5 77 4.1 G. W. Dean, C. S. June, 1851 2 11.3W. Washington, 33 53.7 77 2.8 J. M. Gilliss, June, 1842 1 26.0' Virginia. Charlottesville, 38 2.0 78 31.0 Prof. Patterson, 1835 0 0.0 Roslyn, Petersburg, 37 14.4 77 23.5 G. W. Dean, C. S. Aug., 1852 0 26.4W. Wheeling, 40 8.0 80 47.0 J. Locke, April, 1845 2 4.0 E. Nl'orth Carolinea. Bodie's Island, 35 47.5 75 31.6 C. O. Boutelle, C. S. Dec., 1846 1 13.4 W. Shellbank, 36 3.3 75 44.1 C. O. Boutelle, C. S. Mar., 1847 1 44.8 " Stevenson's Point, 36 6.3 76 11.0 Co O. Boutelle, C.. Feb., 1847 1 39.7 "' Sozuth Caroltsa. Breach Inlet, 32 46.3 79 48.7 C, O. Boutelle. C. S. April, 1849 2 16.5 E. Charleston, 32 41.0 79 53.0 Capt. Barnett; MSay, 1841 2 24,0 " East Base, Edisto, 32 33.3 80 10.0 G. Davidson, C. S. April, 1850 2 53.6 s' Georeg'gea. Athens, 34 0.0 83 20.0 Prof. McCay, 1837 4 31.0 E. Columbus, 32 28.0 85 10.0 Geol. Survey, 1839 5 30.0 " Milledgeville, 33 7.0 83 20.0 Geol. Survey, 1838 5 51.0 " Savannah, 32 5.0 81 5.2 J. E. IIilgard, C. S. April, 1852 3 45.0 ~~~~~~~~~~~~~34.0 TABLE IX. MIAGNETIC VARIATION. 131 Place. Lati- LongP a. tude. te. Authority. Date. ariation. Florida. Cape Florida, 25 39.9 80 9.4 J. E. IIilgard, C S. Feb., 1850 4 26.2 E. Cedar Keys, 29 7.5 83 2.8 J. E. Hilgard, C. S. Mar., 1852 5 20.5 " St. Marks Light, 30 4.5 84 12.51J. E. Iiilgard, C. S. April, 1852 5 29.2" Sand Key, 24 27.2 81 52.0 J. E. HIilgard, C. S. Aug., 1849 5 29.0 " Alabama. Fort Morgan, Mobile Bay, 30 13.8 8S 0.4.I. Fauntleroy,C.S. May, 1817 7 3.8 E. Tuscaloosa, 33 12.0 87 42.0 Prof. Barnard, 1839 7 28.0" Mll ssisissppi. East Pascagoula, 30 20.7 88 31.4 R.It. FauntleroyC.S. June, 1847 7 12. 1 E. Texas. Dollar Point, Galveston, 29 26.0 94 53.0 R.I. Fauntleroy,C.S. April, 1848 8 57.2 E. M Iouth of Sabine, 29 43.9 93 51.5 J. D. Graham, Feb., 1840 8 40.2 " Ohio. C(arrolton, 39 33.0 84 9.0 3 ljocke, Sept., 1845 4 45.4 E. Cincinnati, 39 6.0 84 22.0 J. Locke, April, 1845 4 4.0 Columbus,:39 57.0 83 3.0 J. Locke, July, 1845 2 293 " Hudson 411 15.0 81 26.0 EI. Loonis, 1840 0 52.0" Marietta, 39 26.0 8t 29.0 J. Locke, April, 1845 2 25.0 Oxford, 39 30.0 81 33.0 J. Locke, Aug., 1845 4 50.0" St. Mary's, 40 32.01 1.1 F(J. Locke, Sept., 1845 3 4.0" Tennessee. Nashville, 36 10.0 8;G'1'?.! Prof. Ilamilton. 1835 7 7.0 E..Sichigalo. Detroit, 42 24.0 82 58.0 Geol. Report, 1840 2 0.0 E. Indianla. Richmond, 39 49.0 84 47.01J Locke, Sept., 1845 4 52.0 E. South Ianover, 3S 45.0 85 23.0 Prof. Dunn, 1s37 4 35.0 Illinois. Alton, 33 52.0 90 12.0 1-t. Loomis, 1840 7 45.0 E. 2li.ssosri. St. Louis, 33 36.0 89 36.0 Col. Nicollo, 1835 8 49.0 E. Wisconsint. Madison, 43 5.0 89 41.0.U. S. Surveyors, Nov., 1839 7 30.0 EI Prairie da Chien, 43 1.0 91 8.0 U. S. Surveyors, Oct., 18391 5.0 Ioswa. Brown's Settlement 42 2.0 91 18.0 J. Locke, Sept., 1839 9 4.0 E1 I Davenport, 41 30.0[ 90 34.0 U. S. Surveyors, Sept., 1839 7 50.0 Farmer's Creek, 42 13.0 90 39.01. Locke, Oct., 1839 9 11.0 " Wa psipinnicon K ltivoer, |41 44.01 90 39.0 J. Locke, Sept., 1839 8 25.0 (Carliforia. tPoint Cosception, 34 26.9 1120 26.0IG. Davidson, C. S. Sept., 1S50113 49.5 E. 132 TABLE IX. MlAG(TiNETIC VARIATION. Place. Authority. Date. Variation. tucle. tnde. Point Pinos, o 0 Monterey, 36 33.0 121 5-1.0 G. Davidson, C. S. Feb., 1851 14 58.0 E. Presidio, San F1rancisco, 37 47.8 122 27.0 G. Davidson, C. S. Feb., 1852 15 26.9 1" San Diego, 32 42.0 117 14.0 G. Daviclson, C, S.May 151 12 29.0'" Oregon. Cape Disappointinent, 46 16.6 124 2.0 i. Davidson, C. S. Fuly, 1551 20 45.0 B. Ewing Harbor, 42 44.4 124 21.0 G. Davidson, C. S. Nov., 1851 18 29.2" Washsintoen Territory. Scarboro' Harbor, 48 21.8 124 37.2 G. Davidson, C.S. Au-., 1852 21 30.2 E. BRITISH AaIEnICA. Miontreal. 45 30.0 73 35.0 Capt. Lefroy, 1842 8 58.0 W. Quebec, 46 49.0 71 16.0 Capt. Lefroy, 1842 14 12.0 " St. Johns, C. E. 45 19.0 73 18.0 Capt. Lefroy, 1842 11 22.0 "6 Stanstead, 45 0.0 72 13.0 Boundary Survey, Nov., 1845 11 33.0 ~ Toronto, 43 39.6 79 21.5 British Govern., Sept., 1844 1 27.2 NEW GRENADA. Panama, 8 57.2 79 29.4 WT. II. Emory, Mar., 1849 6 54.6 E. EASTERN HESIISPHERE. Greenwich,England, 51 23.0 0 0.0 Prof. Airy, 18411 23 16.0W. I-akerstoun, Scotland, 55 35.0 2 31.0 W. J. A. Broun, 1842 25 29.6 "6 Paris, France, 48 50.0 2 20.0 E. Paris Observatory Nov., 1851 20 25.0'" Munich, Bavaria, 48 9.0 11 37.0 " 1842 16 48.0 St. Petersburg, Russia, 3;9 56.0 30 19.0 "' Russian Govern., 1842 6 21.1 Catherinenburg Siberia, 56 51.0 60 34.0 "' Russian Govern., 1842 6 33.9 E. Nertchinsk, Siberia, 51 56.0 116 31.0 " Russian Govern., 1842 3 46.9W. St. RI-elena,, 15 56.7 S. 5 40.5 TV. British Govern., Dec., 1845 23 36.6' Cape of Good Hobarton,' Van Diemen's Ld., 42 52.5" 147 27.5 " British Govern., Dec., 1848 10 8.0 B. TABLE X. TRIGONOMIETRICAL FOREIULAE. 133 TABLE X. TRIGONOMETRICAL AND MISCELLANEOUS FORMULIE. LET A (fig. 57) be any acute angle, and let a perpendicular B Cbe drawn from any point in one side to the other side. Then, if the sides Fig. 57. of the right triangle thus formed are denoted by lettelrs, as in the figure, wve shall have these six formul: - 1. sin. A = 4. cosec. A =. C C b c 2. cos. A =. 5. sec. A== - a b 3. tan. A = 6. cot. A = b a Solutionz of Right Triaiygles (fig. 57). Given. Sought. Formulm. a aa 7 a, c' A, B b sin. A ccos. B= c, b= (c a) (ca) a a a, b A), ctan. A = cot. B- c a2+V a 9 A,a B, b, s c B -900- -, B = acot., cA qsin. A b toA, b BA, c =900 - A, a -btan. A, c os. l, c Ba b B goA90~ A a= c sin. A, b =c cos. A. 12 134 TABLE X. TRIGONOMETRICAL AND Solution of Oblique Triangles (fig. 58). Fig. 5 8. / a A \ A. \b (Given. Sought. j Formule. 12 A, B,a b b a sin. A [.A~a~b B si n.B~bsi A 1 3 A,)a, b B sins. zB =_ ---- A 14 a, b, C A-B tan. (A — B) = b) tan. (+'B) If S=I (a + b + c), sin. A= ( ) (s c) 15 a, b,c A cos. A= s(- -a) tan. (S -b) ((s c} i. 2 5 (5 - a) (s-b) (s = c) aS sin. B sin C 16 A, B,Ca arlea slea 2sin. A 17 A, b, c area area = be csin. A, 18 a, b, c area s=I (a + b + c), area=Vs (s-a) (s-b) (s-a) General Tirigortnometrlical For'lmiZlca.b 19 sin.2 A + cos.2 A = 1. 20 sin. (A: B) = sin. A cos. B: sin. B cos. A. 21 cos. (A: B) = cos. A cos. B: sin. A sin. B. 22 sin. 2 A = 2 sin. A cos. A. 23 cos. 2 = cos.2 A - sin.2 A 1 -2 sin.2 A =2 cos.2 A -- 24 sinl.2 A = - I cos. 2 A. 25 cos.2A -= + cos. 2 A. 26 sin. il + sin. B3 2 sin. I (A + B) cos. I (A -B). 27 sin. A - sin. B 2 cos. I (A + B) sin.- (A B). 28 cos. A + cos. B 2 cos. (A B) cos. ( -- B). 29 cos. B - cos. A = 2 sin. I (A + B) sin. I (A - B). 30 sin.2 A - sin. = cosB os.2 2 A n = sin. (A -+ B) sin. (A B). 31o Is. A-sin?. B cos. (A + B) cos, (A- B). MIISCELLANEOUTS FORM1ULIE 183 sin. A 32 tari. A A cos. A 33 cot. A cos sin. A tan. A ~ tan. B 34 tan. (A~ ri B) I tan. A tan.B' 35 tan. A i tan B cosin (A cos B) -- A cos. co 36 cot. A ~ cot. B i sin. (A ~ B) sin. A sin. B sin. A + sin. B tan. ~ (A + B) sin. A - sin B - tan. (A - B)' sin. A +- sin. B cos. A 4- cos. tan (A B) sin. A - sin. B 39 cot. 1 (A - B), cos. B - cos. A c sin. A - sin. B cos. A + cos. B = tan (A B) sin. A - sin..B 41cos. -cos. = cot. I (A + B). 42 tan. A sin. A tan. A = - 1 + cos. A' 43 cot.1A sin. A cot. -- cos. A IlIiscellaneouts Forsbmiule, Sought. Given. Formulh, Area of 44 Circle Radius =. r2. 45 Ellipse Semi-axes = a and b 7r a b, 46 Parabola Chord c, height = h c hi. Regular Pygon Side - a, numbcr of a n cot a 47 Regular Polygon sides n a2 cot.. Surface of 48 Sphere Radius r 4 aT r7.2 49 Zone Radius = r, height = I 2 a r 1 i. Radius of sphere~s ) S- (it-2)1800 50 SphericalPolygon sum of angles-S 7t 2X 80o number of sides n Solidity of 51 Prism or Cylinder Base = b, height = b h. 52 Pyramid or Cone Base b, height = h ~ b h. 53 Frustum ofPyr- Bases =b and b, i i (l 6 + 6 + b bh, amid or Cone height ( - ), The area of a circular segment on railroad curves, where the chord is very long in proportion to theilheight nml y be found with gre,:lt accuracy by theab)ove formlla. 136 TABLE X. MIISCELLANEOUS FOREIEJUL~. Sought. Given. Formul. Solidity of 54 Sphere Radius =2 - - Tr r.3 ~55 Sphecaemet Radii of bases = r I /(2+212). 55 SphericalSegment and1heght= } ( r )o Iand ri,, height I= i 56 Prolate Spheoi Semi-transverses axis 4 b2. 56,rolate Spheroid of ellipse a= a -a7a ~~57 ~~~ Semi-conjugate axis 4 2 b. Oblate Spheroid uis a2 b. Oblate Sphe of ellipse = b height = h I2 581 Paraboloid { Radius of base = ~: } 2 = 3.14159 26535 89793 23846 26433 83280. Log. - = 0.49714 98726 94133 85435 12682 88291 United States Standard Gallon 2 231 calb. in. = 0.133681 cub. ft. "( "r "6 ~Bushel = 2150.42 1.244456 6 British Imperial Gallon = 277.27384 6' = 0.160459'~ According to Hassler. As usually given. French Metre, _ 3.2817431 ft., 3.280899 ft. "6 Litre, = 61.0741569 cub. in., = 61.02705 cub. in. C6 Kilogram, = 2.204737 lb. avoir., - 2.204597 lb. avoirl Weight of Cubic Foot of W~ater, Barom. 30 inches, Therm. Fahr. 39.830, = 62.379 lb. avoir. * 620 - 62.321' Length of Seconds Pendulum at New York = 39.10120 inches. 6 ( " " " " London = 39.13908 " 4C " 6 66 " Paris - 39.12843 " Elquatorial Radius of Earth according to Bessel = 20,923,597.017 feet. Polar 66 I 66 6 I 20,853,654.177 " TAI3LE XI. SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND RECIPROCALS OF NUMBERS FROM 1 TO 105o. 12 * 138 TABLE XIl SQUARES, CUBES, St(UARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 1 1 1 1.0000000 1.000)000 1.00000300(0 2 4 8 1.4142136 1.0205J210.5(00(100() 3 9 27 1.73205 0 1.4-1221, 6.333333333 4 16 6-1 2.000030 1. 074011.2503000000 5rS 252 125 2o2360690 1.7099759.200000000 6 36 216 2.4194S97 1.8171206.166666667 7 49 3-13 2.6457513 1.1291292.142S57143 8 64 512 2.32 4271 2.0000t000.12o000000 9 t 1 729 3.0000000 2.0)300337.111111111 10 100() 1000 3 1622777 2. ]-544I7.100000(00 11 121 1331 3.3166243 2.223930L.09090991 12 14- 1723 3.1641016 2.2394123.0033333333 13 169 2197 3.6055513 2.35133-17.076923077 14 196 27-11 3.7416574 2.4101-22.07142571 15 225 3375 3.3729533 2.4662121.066666667 16 256 4036 4.009000) 2.5 13942.062500000 17 239 4913 4.1231056 2.5712 1 6.053323529 13 324 5332 4.2126107 2.6207414.055555556 19 361 6359 4.3585939 2.6634016.052631579 20 400 S000 4.4721360 2.7144177.05000(0000 21 4-11 926 4.5 3257577 2.753S)243.047619048 232 434 10343 4.6901153 2.302(:393.0154354545 23 529 12167 4.7953315 2. 43S670.043478261 24 576 13S24 4. 399795 2. 341-4991.0-11666667 25 625 1562. 5.00000000 2.9240177.04000000 26 676 17576 5.0990195 2.9621960.033461533 27 729 1963 5. 1961524 3.0))00000).0370370337 23 784 21952 5.2915026 3.0365439.035714236 29 841 21339 5.3S31613 3.07231 6.034482759 30 900 27000 5.4772256 3.1072325.033333333 3L 961 29791 5.5677644 3.1113 3(6.03225G(65 32 1024 32763 5.6063.542 3.1743021 ~031250000 33 1039 35937 5.7445626 3.2075313,030303030 34 1156 39304 5.8309519 3.2396113.029411765 35 1225 42375 5.9160793 3.2710663.02So71429 36 1296 466.56 6.0000001) 3.3019272.027777778 37 1369 50653 6.0827625 3.3322218.027027027 33 1 t1444 54872 6. 1644140 3.3619754.026315789 39 1521 59319 6.2449930 3.3912114.025641026 40 16399 61000 6.3245553 3.4199519 ~025000000 41 1631 63921 6.4031242 3.4482172.024390244 42 1764 74033 6. 4307407 3.4760266.0238309524 43 13849 79507 6.5574385 3.5033981.023255314 44 1936 8318-4 6.6332496 3.5303433.022727273 45 220 25 91125 6.7032039 3.5565933.022222222 46 2116 97336 6.7323300 3.5330479.021739130 47 2209 103323 6.355654-16 3.6095261.021276600 43 2301 110592 6.92320:32 3.632-111.0209)3339:(33 49 2101 117619 7.0000003 3 693057.020403163 50 2500 125009 7.0710673 3.6340314.n20000300 51 260(1 132651 7.141.4231 3.7034293.019607713 52 2701 140(;03 7.2 11026 3.7'325111.0192')0769 53 2309 14S377 7.2.3'01099 3.75/ 623)3.01)3S639 2,5 51 2916 15761 7.3 -13692 3.7797631. 01.51819 553 33025 166375 7.4161935 3. 3029525.0131>S113 56 3136 175616 7.4133143 3.325624.017357143 57 32-349 135193 7.5193344 3.4385011.017543360 553 336-1 19112 7.615773L 3.3709766.017241379 59 3131 205379 7.6311457 3.92965.01694319153 63 3600 216030 7.7459667 3.914S676.0166660G7 61 3721 22693 7.1 02497 3.936 1972 1. 93:1333 62 3 3314 2:323 3 7.3740)7 9 3.1i7 3915.l 6'12 9'2 CUBE ROOTS9 ANDI RECIPROCALS. 139 No. Squares. Cubes. Square Roots. Cube Roots. Rleciprocals. 63 3969 250047 7.9372539 3.9790571.015873016 64 40U6 2621 44 8.0000000 4.000UU0000.015625000 65 4225 274625 8.0622577 4.020(7256.015384615 66 4356 2S7496 8.1240384 4.0 12401 ~015151515 67 4489 300763 8.1853528 4.0615480.014925373 63 4624 314432 8.246 113 4.0816551.014705882 69 4761 328509 8.3066239 4.1015661.014492754 70 4900 343000 8.3666003 4.1212853.014285714 71 5041 357911 8.461498 4. 1408178.014084507 72 5184 373248 8.4835214 4 16G01676.013S8S889 73 5339 389017 8.5440Jl37 4.1793390.013698630 74 5476 405224 8.6023253 4.1983364.013513514 75 5625 421875 8.6602 40 4.2171633 (01 333333 76 5776 43S976 8.7177979 4.2358236.013157895 77 5929 456533 8.77419644-1 4.2543210.012987013 78 6084 474552 8.8317609 4.2726586.012820513 79 6241 493039 8.8881944 4.2908404.012658228 80 6400 512000 8.9442719 4.3088695.012500000 81 - 6561 531441 9.0000000 4.32674o7.012345679 82 6724 551368 9.0353o51 4.3444815.012195122 83 6889 571787 9.1104336 4.3620707.012048193 84 7056 592704 9.1651514 4.3795191.011904762 85 7225 614125 9.2194415 4.3968296.011764706 86 7396 636056 9.2736185 4.4140049.011627907 87 7569 658503 9.3273791 4.4310476.011494253 88 7744 681472 9.3808315 4.4479602.011363363 89 7921 704969 9.4339811 4.4647451.011235955 90 8100 729000 9.486S330 4.4814047.011111111 91 8281 753571 9.5393920 4.4979414.010989011 92 8464 7786S8 9.5916680 4.5143574.010869 65 93 8649 804357 9.6436508 4.5306v49.010752688 94 8836 8305S4 9.6953597 4.5468o59.010638298 95 9025 857375 9.7467943 4.5629026.010526316 96 9216 884736 9.7979590 4.5788570.010410667 97 9409 912673 9.8488578 4.5947009.010309278 98 96(04 941192 9.8994949 4.6104363.010204082 99 9801 970299 9.9498744 4.6260650.010101010 100 10000 1000000 10.0000000 4.6415888.010000000 101 10201 1030301 10.0498756 4.6570095.009900990 102 10404 1061208 10.0995049 4.6723287.009803922 103 10609 1092727 10.1488916 4.6875482.009708738 104 10816 112486- 10.198()0390 4.7026694.0096153c5 105 11025 1157625 10.2469508 4.7176940.009523810 1.06 11236 1191016 10.2956301 4.7326235.008433962 107 11449 1225043 10.3440804 4.7474594.0()934;57?'4 103 11661 1259712 10.3923048 4.7622032.009299259 109 1 1881 1295029 10.4403065 4.7768562.009174312 110 12100 1331000 10.4880085 4.7914199.000909099 111 12321 1367631 10.5356538 4.8058955.009009009 112 12544 1404928 10.5830052 4.8202845.008928571 113 12769 1442897 10.6301458 4.8345881.008849558 114 12996 1481544 10.6770783 4.8488076.008771930 115 13225 1520875 10.7238053 4.8629442.008695652 116 13-156 1560896 10.7703296 4.8769990.008620690 117 13689 1601613 10.8166538 4.8909732.008547009 118 13924 1643032 10.8627805 4.9048681.008474576 119 14161 1635159 10.9087121 4.9186847.008403361. 120 14400 1728000 10.9544512 4.9324242.008333333 121 14641 1 1771561 11.0000000 4.9460874.002644-163 122 14384 1815848 11.0453610 4.9596757.008196721 123 15129 1860367 11.0905365 4.9731898.008130081 124 15376 19106624 11.13j5287 4.9866310.008064516 140 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 125 15623 1953125 1 1.1303399 5. 0000000.0030o000 126 15376 2900376 11.2249722 5 0132979.007936o03 127 16129 2094333 11.2694277 5.0265257.007871016 123 16334 2097152 11.3137035 5.0396842.007812500 129 16611 2146639 11.3573167 5.0527743.007751938 130 16900 2197000 11.4017543 5.0657970.007G69230 131 17161 221iu91 11.45)5231 5.0787531.007633583 132 17421 2299963 11.4391253 5.0916134.00757o753 133 17639 23.2637 11.5325626 5.1044697.007518797 134 17956 2 06o10 11.5753363 5 1172299,007462637 135 18225 2460375 11.6189500 5.1299278.007407407 136 18496 2.31546 11.6619033 5.1425632.007352941 137 18769 2571353 11.7046999 5.1551367.007299270 133 19044 2623072 1 1.7473444 5 1676493.007246377 139 19321 2635619 1 1.793261 5.1801015.007194245 140 19600 2744090 11.8321596 5.1924941.007142357 141 193 1 23(3 221 1 1.8743421 5.2043279.007092199 142 20164 2363233 11.9163753 5 2171034.007042254 143 20449 2924207 11.9532607 5.2293215.006993007 144 20736 2935934 12.0000000 5.2414323.006944441 145 21025 30418625 12.0415946 5.2535879.006S96552 146 21316 3112136 12.0830460 5.2656374.006849315 147 21609 3176523 12.1243557 5.2776321.006302721 148 2 190- 3211792 12.1655251 5.2395725.006756757 149 22201 3307949 12.2065536 5.3014592.006711409 150 22500 3375003 12 2174487 5.3132923.006666667 151 22301 3442951 12.2332057 5.3250740.006622517 152 23104 3511803 12.32383230 5 3363033.006789417 153 23109 3531577 12 3693169 5.3144812.006535948 154 23716 3652261 12.4096736 5.3601034.006193506 155 21025 3723375 12.4493996 5.3716354.006451613 156 21336.3796416 12.4399960 5.3332126.006410256 157 21649 3369393 12.5299611 5.3946907.006369427 153 24961 3944312 12.5693()51 5.4061202.006329114 159 25231 4019679 12.6095202 5.4175015.006239308 160 25600 4096000 12.6491106 5.4238352.006250000 16L 25921 4173231 12.6335775 5.4401218.006211180 162 26214 4251523 12.7279221 5.4513618.006172340 163 26569 4330747 12.7671453 5.4625556.006134969 164 26896 4410944 12.8062485 5.4737037.006097561 165 27225 4492125 12.8452326 5.4348066.006060606 166 27556 4574296 12.8340937 5.4958647.006024096 167 278S9 4657463 12.9223480 5.5063784.005988024 163 23224 4741632 12.9614814 5.5173431.005952381 169 23561 4326309 13.0000000 5.5287748.005917160 170 23900 4913300 13.0334013 5.53965S3.005332353 171 29241 5000211 13.0766963 5.5504991.005847953 172 295334 5033443 13.1 143770 5.5612978.0051S3953 173 29929 5177717 13.1529461 5.5720546.005780317 174 30276 5269021 13.1909060 5.5827702.005747126 175 30625 5359375 13.2237566 5.5934447.005714256 176 39976 5451776 13.2661992 5.6040787.005631818 177 31329 5545233 13.3011347 5.6146724.005649718 173 31634 5639752 13.3116641 5.6252263.005617973 179 32041 5735339 13.3790382 5.6357403.005536592 180 32400 5332000 13 4161079 5.6162162.005355556 181 32761 5929741 13.45362 10 5.6566528.005524862 182 33124 6923563 13.4907376 5.6670511.005494505 183 33139 61234S7 13.5277493 5.6774114.005164481 134 33356 6229501 13.5646600 5.6377340.005434733 185 31225 6331625 13.6014705 5.6930192.005405405 136 31596 6131356 13.6331817 5.7032675.005376344 _.~~~~~~~~~~~~ CUBE ROOTS, AND RECIPROCALS. 141 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals, 187 34969 6539203 13.6747943 5.7184791.005347594 188 35344 6644672 13.7113092 5.7286543.005319149 189 35721 6751269 13.7477271 5.7387936 005291005 190 36100 6859000 13.7840488 5.7488971.005263158 191 364o1 6967871 13.8202750 5.7589652.005235602 192 36864 7077888 13.8564065 5.7689982.005208333 193 37249 7189057 13.8924440 5.7 789966.005181347 194 37636 7301384 13.9283S83 5.7889604.005154639 195 38025 7414875 13.9642400 5.7988900.005128205 196 38416 7529536 14.0000000 5.8087857.005102041 197 38809 7645373 14.0356688 5.8186479.005076142 198 39204 7762392 14.0712473 5.8284767.005050505 199 39601 7880599 14.1067360 5.8382725.005025126 200 40000 8000000 14.1421356 5.8480355.005000000 201 40401 8120601 14.1774469 5.8577660.004975124 202 40804 8242408 14.2126704 5.8674643.004950495 203 41209 8365427 14.2478068 5.8771307.004926108 204 41616 8489664 14.2828569 5.8867653.004901961 205 42025 8615125 14.3178211 5.8963685.004878049 206 42436 8741816 14.3527001 5 9059406.004854369 207 42849 8869743 14.3874946 5.9154817.004830918 208 43264 8998912 14.4222051 5.9249921.004807692 209 43681 9129329 14.4568323 5.9344721.004784689 210 44100 9261000 14.4913767 5.9439220.004761905 211 44521 9393931 14.5258390 5.9533418.004739336 212 44944 9528128 14.5602198 5.9627320.004716981 213 45369 9663597 14.5945195 5.9720926.004694836 214 45796 9800344 14.6287388 5.9814240.004672897 215 46225 9938375 14.6628783 5.9907264.004651163 216 46656 10077696 14.6969385 6.0000000.004629630 217 47089 10218313 14.7309199 6.0092450.004608295 218 47524 10360232 14.7648231 6.0184617.004587156 219 47961 10503459 14.7986486 6.0276502.004566210 220 48400 10648000 14.8323970 6.0368107.004545455 221 48841 10793861 14.8660687 6.0459435.004524887 222 49284 10941048 14.8996644 6.0550489.004504505 223 49729 11089567 14.9331845 6.0641270.004484305 224 50176 11239424 14.9666295 6.0731779.004464286 225 50625 11390625 15.0000000 6.0822020.004444444 226 51076 11543176 15.0332964 6.0911994.004424779 227 51529 11697083 15.0665192 6.1001702.004405286 228 51984 11852352 15.0996689 6.1091147.004385965 229 52441 12008989 15.1327460 6.1180332.004366812 230 52900 12167000 15.1657509 6.1269257.004347826 231 53361 12326391 15.1986842 6.1357924.004329004 232 53824 12487168 15.2315462 6.1446337.004310345 233 54289 12649337 15.2643375 6.1534495.004291845 234 54756 12812904 15.2970585 6.1622401.004273504 235 55225 12977875 15.3297097 6.1710058.004255319 236 55696 13144256 15.3622915 6.1797466.004237288 237 56169 13312053 15.3948043 6.1884628.004219409 238 56644 13481272 15.4272486 6.1971544.004201681 239 57121 13651919 15.4596248 6.2058218.004184100 240 57600 13824000 15.4919334 6.2144650.004166667 241 58081 13997521 15.5241747 6.2230843.004149378 242 58564 14172488 15.5563492 6.2316797.004132231 243 59049 14348907 15.5884573 6.2402515.004115226 244 59536 14526784 15.6204994 6.2487998.004098361 245 60025 14706125 15.6524758 6.2573248.004081633 246 60516 14886936 15.6843871 6.2658266.004065041 247 610(09 15069223 15.7162336 6.2743054.004048583 248 61504 r15252992 15.7450157 6.2827613.004032258 142 TABLE Xl. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 219 62331 1533S2 3 15.7797333 6.2911946.034016064 2)3 6'2590) 156250)) 15.8113333 6.2996053.004000000 25.4 63901 15313251 15.8429795 6.3079935.003934064 2 2 63501 16033')0 15.8745079 6.3163596.003963254 2-3 64309 16t94277 15.9059737 6.3247035.003952569 251 64516 16:37061 15.9373775 6.3330256.003937008 255 63 t;502.5 16.531375 15.9637194 6.3413257.003921569 2-6 63.5536 16777216 16.0050000 6.3196042.003906250 257 66949 16374,593 16.0312195 6.3573611.'003391051 253 66. 564 17173512 16.0623734 6.3660963.003875969 259 67031 17373979 16.0934769 6.3743111.003861004 263 67603 17576030 16.1245155 6.3325043.003346154 261 63121 17779531 16.1554944 6.3906765.003831418 262 ~ 6964-1 17931728 16.136414l 6.3938279.003816794 263 69169 18191447 16.2172747 6.4069585.003802281 261 4 69696 18:39744 16.2430763 6.4150687.003787879 265 70225 18609625 16.2783206 6.1231583.003773585 266 70756 18321036 16.:3095061 6.4312276.003759398 267 71239 19031163 16.3101346 6.4392767.003745318 263 71824 19213:32 16.3707055 6.4473057.00:3731343 269 72361 19465109 16.4012L95 6.4553148.003717472 270 72900 19633003 16.4316767 6.4633041.003703704 271 73441 19902511 16.4620776 6.4712736.003690037 272 73934 20123643 16.4924225 6.4792236.003676471 273 74529 20346417 16.5227116 6.4871541.003663004 274 75076 20570321 16.5529454 6.4950653.003649635 275 75625; 20796375 16.5331240 6.5029572.003636364 276 76176 21024576 16.6132477 6.5103300.003623188 277 76729 - 21253933 16.6433170 6.5186839.003610108 278 77284 21484952 16.6733320 6.5265189.003597122 279 77841 21717639 16.7032931 6.5343351.003584229 230 73400 21952000 16.7332005 6.5421326.003571429 231 78961 22188041 16,7630546 6,54991 16.003558719 282 7952t 22425763 16.792S8556 6.5576722.003546099 283 80039 22665137 16.8226033 6.5654144.003533569 23S 80656 22906304 16.8522995 6.5731385.003521127 235 81225 23149125 16.8319430 6.5S03443.003508772 286 81796 23393656 16.9115345 6.5885323.003496503 237 82369 23639903 16.9410743 6.5962023.003484321 233 82944 2338o7872 16.9705627 6.6038545.003472222 239 83521 24137569 17.0001)000 6.6114890.003460208 290 84100 24339000 17.0293864 6.6191060.003448276 291 81631 24642171 17.0587221 6.6267054.003436426 292 85264 24897038 17.0380075 6.6342874.003424658 293 85849 25)153757 17.1172428 6.6418522.003412969 294 86436 25412184 17.1464282 6.6493998.003401361 295 87025 25672375 17.1755640 6.6569302.003389831 296 87616 25934336 17.20465X05 6.6644437.003378378 297 88209 26198073 17.2336879 6.6719403.003367003 293 833804 1 26463592 17.2626765 6.6794200.003355705 299 89101 26730899 17.2916165 6.6868831.003344482 300 90000 27000000 17.320)5081 6.6943295.003333333 301 90601 27270901 17.3493516 6.7017593.003322259 302 91204 27543603 17.3781472 6.7091729.003311258 3!3 91809 27818127 17.4068952 6.7165700.003300330 304 92416 23094464 17.4355958 6.7239508.003289474 305 93025 23372625 17.4642492 6.7313155.003278689 306 93636 23652616 17.4928557 6.7336641.003267974 307 94249 23934443 17.5214155 6.7459967.003257329 303 91864 29218112 17.5499238 6.7533134.003246753 309 94381 29.303629 17.5783953 6.7606143.003236246 310 9610 29791900:3 17.6063169 6.7678995.003225806 CUBE ROOTS, AND RECIPROCALS. 143 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 311 96721 30080231 17.635192L 6.7751690.003215434 312 97344 30371328 17.6635217 6.78242 9.003205128 313 97969 39664-97 17.6918060 6.7396613.003194S88 314 98596 3095914-4 17.7200451 6.796844.003184713 315 99225 31255875 17.74 2:393 6. 040921.003174603 316 99856 31554496 17.77638 8 6.8112S47.003164557 317 100489 31855013 17.8044938 6.8184620.00315-574 318 101124 32157432 17.8325545 6.825624-.003144654 319 101761 32461759 17.S605711 6.8327714.003134796 329 102400 32768000 17.8365433 6.8399037.003125000 321 10301l 33076161 17.9164729 6.84 70213.003115265fi 322 1036,1 333S6248 17.9443584 6.8541240.003105590 323' 101329 33698267 17.972200)3 6.8612120.0030)5975 32 10(4976 31!)12'224 18.(0000000 6.86528>)5.003086-120 32 1056 5 34328125 18. 027756- 6.87.2;3443.00307 6923 326 105276 34645976 18. 0554701 6.8233 8 8.003067485 327 1 i6929 3496)5753 18.0 31413 6. 89634188.0030.38104 323 1075S8 35287552 18.1107703 6.S964315.(103048780 329 103241 35611259 1 8.1383571 6.9034359.003039514 333 103900 35937000 18. 1659021 6.9104232.003030303 331 109561 36261691 18.1934054 6.9173964.003021148 332 110224 36o94363 18.2208672 6.9243556.003012048 333 110889 36926037 18.2482876 6.9313008.003003003 334 111556 37259704 18.2756669 6.9382321.002994012 335 112225 37595375 18.3030052 6.9451496.002985075 336 112896 37933056 18.3303023 6.9520533.002976190 337 113569 38272753 18.3575593 6.9589434.002967359 338 114244 35614 —472 18.3847763 6.9658198.002958580 339 114921 33958219 18.4119526 6.9726826.002949853 310 115600 39304000 18.4390889 6.9795321.002941 176 341 116231 39651821 18.4661853 6.9863681.002932551 342 116964 40001688 18.4932420 6.9931906.002923977 343 117649 40353607 18.5202592 7.0000000.002915452 3t4 118336 407(17584 18.5472370 7.0067962.002906977 345 119025 41063625 18.5741756 7.0135791.002898551 346 119716 41421736 18.6010752 7.0203490.002890173 3-7 120409 41781923 18.6279360 7.0271058.002881844 348 121104 42144192 18.6547581 7.0338497.002873563 349 121801 42508549 18.6815417 7.0405806.002865330 350 122500 42375000 18.7082869 7.0472987.002857143 351 123201 43243551 18.7349940 7.0540041.002849003 352 123904 43614208 18.7616630 7.0606967.002840909 333 124609 43986977 - 18.7882942 7.0673767.002832861 354 125316 44361864 18.8148877 7.0740440.002824859 355 126025 44733875 18.8414437 7.0806988.002816901 356 126736 45118016 18.8679623 7.0873411.002808989 357 127449 45499293 18.8944436 7.0939709.002801120 358 128164 45882712 18.9208879 7.1005885.002793296 359 128881 46268279 18.9472953 7.1071937.002785515 360 129600 46656000 18.9736660 7.1137866.002777778 361 130321 47045881 19.0000000 7.1203674.0(12770083 362 131044 47437928 19.0262976 7.1269360.002762431 363 131769 47832147 19.0525589 7.1334925.002754821 361 132496 4S228544 19.0787840 7. 1400370.002747253 365 133225 48627125 19.1049732 7.1465695.002739726 366 133956 491)27896 19.1311265 7.1530901.002732240 367 134639 49430863 19.1572441 7.1595988.002724796 363 135421 49836032 19.1833261 7.1660957.002717391 369 136161 50243409 19.2093727 7.1725509.002710027 370 136900.50653000 19.2353341 7.1790544.002702703 371 137641 51064811 19.2613603 7.1855162.002695418 372 138384 51478848 19.28'73015 7.1919663 1.002688172 144 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. S uare Roots. Cube Roots. Reciprocals. 373 139129 51895117 19.3132079 7.1934050.002630965 374 139376 524313621 19.3390796 7.2048322.002673797 375 14062 ) 52734375 19.3649167 7.2112479.002666667 376 141376 53 L 57376 19.390719-t 7.2176;22.002659574 377 14i212J 53532633 19.4164373 7.2240450.002652520 378 ]14-24 1 5101015 19.4422221 7.2304263.002645503 379 143611 5 14439399 19.4679223 7.2367972.002633522 330 [ 14 1r~I3 t 5'1372000 19.4935387 7.2431565.002631579 341 1-4.516 55396:341 19.5192213 7.2495045.002624672 332 1 11 921.55742963 19.5444203 7.2558415.002617801 333 lti66 9 563148107 19.5703558 7.2621675.002610966 334 117156 56623101 19.59.59179 7.26134824.002604167 335 141225 57066625 19.6214169 7.2 47864.002597403 346 143994 575 12156 19.6464527 7. 281079-1.002590674 337 149769 57960603 19.6723156 7.2073617.002583979 339 15054 ) 53411072 19.6977156 7.2936330.002577320 339 15132l 54363463 19.7230829 7.299S936.002570694 390 152100 59319003 19.7404177 7.3061436.002564103 391 15233S 59776171 19.7737199 7.3123832.002557545 392 153661l 60236203 19.79594399 7.3186114.002551020 393 154449 60693457 19.8242276 7.3248295.002544529 394 155236 61162904 19.8494332 7.3310369.002538071 395 156025 61623375 19.8746069 7.3372339.002531646 396 156316 62099136 19.8997437 7.3434205.002525253 397 157609 62570773 19.9241850 7.3495966.002518892 398 150401 63041792 19.9-199373 7.3557624.002512563 399 159201 63521199 19.9749344 7.3619178.002506266 400 160000 64000000 20.0000000 7.3680630.002500000 401 16001 61481201 20.0249844 7.3741979.002493766 402 16160-t1 6 964003 20.0499377 7.3803227.002487562 403 162109 65450327 20.0748599 7.3864373.002431390 404 163216 65939264 20.0997512 7.3925418.002475248 405 164025 6643"3125 20. 1246118 7.3936363.002469136 406 164036 66923416 20.1494417 7.4047206.002463054 407 165649 6741l1.43 20.1742410 7.4107950.002457002 403 166464 67917312 20.1990099 7.4160595.002450900 409 167231 60417929 29.2237404 7.4229142.002444938 410 163100 63921000 20.2484567 7.4289589.002439024 411 16S921 69426531 20.2731349 7.4349938.002433090 412 169744 69934523 20.2977831 7.4410189.002427184 413 170569 70444997 20.3224014 7.4470342.002421308 414 171396 70957944 20.3469899 7.4530399.002415459 415 172225 71473375 20.3715438 7.4590359.002409639 416 173056 71991296 20.3960781 7.4650223.002403846 417 173089 72511713 20.4205779 7.4709991.002398082 418 174724 73034632 20.4450483 7.4769664.002392344 419 175561 73560059 20.4694895 7.4829242.002386635 420 176400 74080000 20.4939015 7.4888724.002380952 421 177241 74618461 20.5182345 7.4948113.002375297 422 1789034 75151448 20.5426336 7.5007406.002369668 423 178929 75606967 20.5669635 7.5066607.002364066 42-4 179776 76225024 20.5912603 7.5125715.002358491 425 180625 76765625 20.6155281 7.5184730.002352941 426 181476 77308776 20.6397674 7.5243652.002347418 427 182329 77854483 20.6639783 7.5302482.002341920 42S 183184 70402752 20.6881609 7.5361221.002336449 429 184041 7895353S9 20.7123152 7.5419367.002331002 430 184900 79507000 20.7364414 7.547,423.002325581 431 185761 80062991 20.7605395 7.5536388.002320186 432 186624 83621569 20.7846097 7.5595263.002314815 433 187489 81182737 20.8086520 7.5653548.002309469 434 180356 81746504 20.8326667 7.5711743.002304147 CUBE ROOTS, AN4D:tCIPR9SOCALS. 145 No. Squares. Cubes. | Scquare l'oots. Cube 16oots. IReciprocils. 4135 189225 8231275 i 20.56t5 6 7.769-.(02 9I1 436 19002 6 S231S56 20.IC61:3 7. Z,-(65.00(229:3)578 437 19263o (S4. 34;3 2f.904.50 1'73 63,s593. 022>392 14:2S 191844 L -!77672 2. 59261441;5 7,4 55363:3.o 0225'31 03 439 192721 S416(4519 20.9325236 7.600135~.002277904 410 193600 85184000 20.9761770 7.60,59049.002272727 4-:1 1944s3 83766121 21.0000000 7. l61 66.00)226774 442 193364 SG3603388 21.0237960 7. 6174116.002262-143 443 196249 8693o-307 21.,047T652 7.6231519.002257336 444 197136 87528334 21.0713075 7.6'2.SS37.002235252 44.5 198025 28121125 21.095023 7.63-16067.002247191 446 19596 t 89716:36 21. 1187121 L 7.6401321:'.102242152 447 1996(9 8931462: 21.112:.5745 7.6460272.002237136 4-18 200704 91)1 39)2 21.16)30105 7.6.;17i-7.002232141:3 449 201601 90518349 21.1896201 7.6574 156.002227171 450 202500 91125000 21.2132034 7.6630913.0022222222 4531 203401 91 73385 1 21.23676(16 7.6667665.00221 7295 452 204:204 923435420 21.2602916 7.67441303.0022123,-9 -453 205209 92959677 21.2>37967 7.6800S57 00220()7506 454 206116 93576664 2l.30 2758 7.6 5772S. 002202643 455 207025 94196375 21.330729!0 7.6913717.02(2197802 456 207936 9418I116 21.3541565 7..6970023.002192982 457 208819 95143993 21.3775583 7.7026246.0021S881'84 458 209764 96071912 21.4009346 7. 7082388.002183406 459 210681 96702579 21.4242S53 7.7138448.00217E649 460 211600 97336000 21.4476106 7.7194426.002173913 461 212521 97972181 21.4709106 7.7250325.002169197 462 21:3444 95611123 21.4911853 7.7306141.002164502 463 214369 99252847 21.517434S 7.7361877.002159827 464 215296 99897344 21.5406592 7.7417532.002155172 465 216225 100544625 21.5638587 7.7473109.002150)538 466 21.7156 101194696 21.5870331 7.752S606.002145923 467 218039 101847563 21.6101828 7.7564023.002141328 468 219024 102503232 21.6333077 7.7639361.002136752 469 219961 103161709 21.6564078 7.7694620.002132196 470 220900 103823000 21.6794834 7.7749801.002127660 471 221841 104487111 21.7025344 7.7804904.002123142 472 222784 1051 540-1 21.7255610 7.7859923.002118644 473 223729 105623>17 21.7485632 7.7914875.002114165 474 224676 106496G124 21.7713411 7.7969745.002109705 475 225625 107171875 21.7944947 7.8024538.002105263 476 226576 107650176 21.8174242 7.s079254.002100S10 477 227529 1 08531333 2t. 8 103297 7.8133892.002096436 473 2224>4 109215352 21.8632111 7.8188456.002092050 479 229441 109902239 21.8860G66 7.8242942.002087683 480 2304100 110592000 21.90s9023 7.8297353.002083333 431 231361 111284611 21.9317122 7.83516SS.002079002 482 232324 11198016s 21.9544984 7.8405949.002074689 483 233289 112678587 21.9772610 7.8460134.002070393 464 234256 1-13379904 22.0000000 7.8.514244.002066116 4S5 235225 114084125 22.0227155 7.8568281.002061856 4>G 236196 1147912.56 22.0454077 7. 622242.002057613 437 237169 115501303 22. n660765 7.(>676130.0020533S8 4SS 238144 116214272 22.0907220 7.8729944.002049180 489 239121 116930169 22.1133444 7.8783684.002044990 490 240100 117649000 22.1359436 7.8S37352.002040816 491 241081 118370771 22.15S519S 7.8S90946.002036660 492 242064 11909.5488 22. 1810730 7.8944468.002032520 493 243049 119>23157 22.2036033 7.8997917.002028398 494 244036 120553784 22.2261108 7.905 L294.002024291 495 245025 121287375 22.2485955 7.9104599.002020202 496 246016 122023936 22.2710575 7.9157832.002016129 13 146 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 497 247003 122763473 22.2934963 7.9210994.002012072 493 223004 123505992 22.3159136 7.92640385.002008032 499 249001 124251499 22.3383079 7.9317104.002004008 500 250000 125000000 22.3606793 7.9370053.002000000 50L 2510031 125751501 22.3S30293 7.9422931.001996003 502 252004 126506003 22.4053565 7.9475739.001992032 503 253009! 127263527 22.4276615 7.9523477.001988072 5s1 254016 123021064 22.4499443 7.9581144.001934127 505 255023 12S787625 22.4722051 7.9633743.001930198 506 256036 129554216 22.494443S 7.9686271.001976285 507 257049 13032338413 22.5166605 7.9733731.001972387 503 253064 131096512 22.5338553 7.9791122.001968504 509 259031 131372229 22.5610233 7.9343444.001964637 510 260100 132651000 22.5831796 7.9895697.001960784 511 26112L 133132831 22.6053091 7.9947333.001956947 512 262144 134217723 22.6274170 8.0000000.001953125 513 263169 13-005697 22.6495033 8.0052049.001949318 514 264196 135796744 22.6715631 8.0104032.001945525 515 265225 136590375 22.6936114 8.0155946.001941743 516 266256 137338096 22.7156334 8.0207794.001937984 517 267239 133183413 22.7376340 8.0259574.001934236 518 263324 133991832 22.7596134 8.0311287.001930502 519 269361 139798359 22.7S15715 8.0362935.0019267832 520 2704:00 140603000 22. 03.5085 8.0414515.001923077 521 271441 141420761 22.8254244 8.0466030.001919386 522 272434 142236643 22.3473193 3.0517479.001915709 523 273529 143055667 22.8691933 8.056S362.001912046 524 274576 143377824 22.3910463 8.06201S0.001903397 525 275625 144703125 22.9123785 3.0671432.001904762 526 276676 145531576 22.9346S99 8.0722620.001901141 527 277729 1116363183 22.9564806 8.0773743.001897533 523 278734 147197952 22.9732506 8.0324800.001893939 529 279341 143035859 23.()000000 8.0375794.001890359 530 230900 143877000 23.0217239 8.0926723.001386792 531 231961 149721291 23.0434372. 0977589.001883239 5.32 233024 1 5056763 23.0651252 8. 1023390.001879699 533 23-14039 151419:137 23.0S67923 8.1079123.001876173 534 2385156 152973301 23.10 4400 8.1129803.001872659 535 236225 15313037) 23.1300670 8.1180414.001869159 536 237296 153990656 23.1516733 8. 1230962.001865672 537 233369 154354153 23.1732605 8.1281447.001862197 533 239444 155720372 23. 191270 8.1331870.001858736 539 290521 156590319 23.2163735 8. 1382230.001855288 510 291600 1.57461000 23.2379091 8.1432529.001851852 541 292631 153)04421 23.2594067 8.14832765.001848429 5-12 293764 159220993 23.230393 8. 1532939.001845018 5-13 294849 160103007 23.3023604 8.1583051.0013841621 5 —1 295936 1609391S 4 23 3233076 8.1633102.001s3s235 545 29702. 161878625 23 34523 1 8.1633092.001834862 546 293116 162771336 23.3666429 8.1733020.001831502 5,47 299209 163667323 23.3880311 8. 1782838.001828154 543 300304 164 66592 23.4093998 8.1832695.001824818 549 301401 169563149 23.4307490 8.1882441.001821494 550 302500 166.375000 23,4520733 8.1932127.00131S182 551 303601 1672S4151 33.47339.1991753. 001814S32 52 30-1704 163196603 23 4946302 S.2031319.001811594 553 305809 169112:377 23. 5 1 5950 8. 20S825.001803318 5.4 306916 170031461 293. 5372016 8.2130)271.001805054 555. 303025 1709533375 23.55,430 8.21796.7.0O0180302 565 303136 171879616 23.5796522 8.32232935.001798561 557 31021!) 172309693 23.60)3471 8.22732 54 01 0)179053332 553 311361 173741112 23.6220236 S.232791G3.0()31791 15 CUBE ROOTS, AND RECIPROCALS. 147 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 559 312481 174676879 23.6431808 8.2376614.001783909 560 313600 175616000 23.6643191 8.2425706.001785714 561 314721 176558481 23.6854386 8.247474(0.001782531 562 315844 177504328 23.7065392 8.2523715.001779359 56:3 316969 178453547 23.7276210 8.2572633.001776199 564 318096 179406144 23.7486842 8.2621492.001773050 565 319225 180362125 23.7697286 8.2670-294.001769912 566 320356 181321496 23.7907545 8.2719039.001766784 567 321489 182284263 23.8117618 8.2767726.001763668 563 322624 183250432 23.8327506 8.2816355.001760563 569 323761 1842200:33 23.8537209 8.2864928.001757469 570 324900 185193000 23.8746728 8.2913444.001754386 571 3260431 186169411 23.8956063 8.2961903.001751313 572 327184 187149248 23.9165215 8.3010304.001748252 573 328329 188132517 23.9374184 8.3053651.001745201 574 329476 189119224 23.9532971 8.3106941.001742160 575 330625 190109375 23.9791576 8.3155175.001739130 576 331776 191102976 24.0000000 8.3203353.001736111 577 332929 192100033 24.0208243 8.3251475.001733102 5 78 334054 193100552 24.0416306 8.3299542.001730104 579 335241 194104539 24.0624188 8.3347553.001727116 580 336400 195112000 24.031S891 8.3395509.001724138 581 337561 1.96122941 24.1039116 8.3443410).001721170 582 338724 197137:368 24.1246762 8.3491256.001718213 583 339889 193155287 24.1453929 8.3539047.001715266 584 341056 199176704 24. 1660919 8.3586784.001712329 585 342225 200201625 24.1867732 8.3634466.001709402 586 3-13396 2012930056 24.2074369 8.3682095.0017064S5 587 3644569 2026200J3 24.2230S29 8.3729668.001703578 583 345744 203297472 24.2487113 8.377718S.001700680 589 346921 204336469 24.2693222 8.3324653.001697793 590 348100 205379000 24.2699156 8.3872065.001694915 591 349231 206425071 2:1.3104916 8.3919423.001692047 592 350464 207474688 24.3310501 8.3966729.001689189 593 351649 208527087 24.3515913 8.4013981.001636341 594 352836 209545844 24.3721152 8.4061180.001683502 595 354025 210644875 24.3926213 8.4108326.001680672 596 355216 211708736 24.4131112 8.4155419.001677852 597 356409 212776173 2.43358341 8.4202460.001675042 598 3576045 213S47192 24.4540385 8.4249448.001672241 599 338801 214921799 24.4744765 8.4296383.001669449 600 360000 216000000 24.4948974 8.4343267.001666667 601 361201 217031801 24.5153013 8.4390098.001663S94 602 362404 218167208 24.5356883 8.4436877.001661130 603 363609 219256227 24. 55605S3 8.4483605.001658375 6'04 36416 220348864 24. 5764113 8.4530281.001655629 605 366023 221 4t-125 24. 5967478 8.4576906.001652893 606 367236 222545016 24.6170673 8.4623479.001650165 607 36S449 223649 543 24.6373700 8.4670001.001647446 603 369664 224755712 24.6 576560 8.4716471.001644737 609 370381 225s66.29 24.6779254 8.4762892.001642036 610 372100 226981000 24.691781 8.4S09261.001639344'6lI 73321 22S099131 24.7184142 s.4S557i.001636661 612 374.D4 229220929 24.7386338 8.490141S.001633987 613 37,769 230346397 24.7s588363 s8494Sn065.001631321 614 376996 23147i5 44 24. 7790234 8.4991233.001628664 l 615 3732925 2326083757 2.7991935 8.504(0350.001626016 616 379-156 23374480s) 24. 1 93473 8.50os6417.001623377 617 3S0689 231885113 24.8394847 8.5132435.001620746 6!8 I 3 19249 2360293032 24.8 596058 8.51784013.001618123 61 ) 331G1 237176659 24.87971(16 8.5224321.001615509 620 324-100 23 328000 24.8997992 S. 5270189.001612903 148 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 621 3356411 239483061 24.9198716 8.5316009.001610306 622 2 3868S4 240641843 24.9399278 8.5361780.001607717 623 333129 2118)-1367 24.9599679 8.5407501.001605136 624 3S9376 2129706224 24.9799920 8.5453173 i.001602564 625 39)625 244140625 25.0000000 8.5498797.001600000 626 391876 245314376 25.0199920 8.5544372.001597444 627 393129 246491883 25.0399681 8.5589399.001594896 623 394354 j 247673152 25.0599282 8.5635377.001592357 62 395611 2481385189 25.0798724 8.5680307.001589325 630 396903 250047000 25.0998003 8.5726189.001537302 631 393161 251239591 25.1197134 8.5771523.001584786 6o32 399124 25243596 23.1396102 8 5i816S09.001582278 633 400689 253636137 2O.1594913 8.5862047.001579779 631 4019.56 254340104 25. 1793566 8. 5907233.0)1577287 63. 40322.5 256047875 25.1992063 8.5952330.001574803 6436 404496 2572594l56 25.21904(1)4 8.5997476.001572327 637 405769 25847453 233.238589 8.6042525.001569359 633 407044 259694072 25.2586619 8.6037526.001567393 639 403321 260917119 25.2784493 8.6132430.001564945 640 409600 262144000 25.2932213 8.6177388.001562500 641 410381 263374721 25.3179778 8.6222248.001560062 642 412161 264609298 25.3377189 8.6267063.001557632 613 413449 265347707 25.3574447 8.6311830.001555210 614 414736 267039934 25.3771551 8.6356551.001552795 645 416025 268336125 25.3963502 8.6401226.001550388 6416 417316 269586136 25.4 165301 8.6445855.001547988 647 418609 270340023 25.4361947 8.6490437.001545595 648 419904 272097792 25.4558441 8.6534974.001543210 649 421201 273359449 25.4754784 8.6579465.001540832 650 422500 274625000 25.4950976 8.6623911.001538462 651 423301 275894451 25.5147016 8.6668310.001536098 652 423104 277167803 25.5342907 8.67L2665.001533742 653 426409 278445077 29.5538647 8.6756974.001531394 654 427716 279726264 25.5734237 8.6301237.0015290(52 655 429025 231011375 25.5929673 8.6345456.001526718 656 430336 232300416 25.6124969 8.6889630.001524390 657 431649 283593393 25.6320112 8.6933759.001522070 638 432964 234890312 25.6315107 8.6977843.001519757 659 434121 236191179 25.6709953 8.7021882.001517451 660 435600 237496000 25.6904652 8.7065377.001515152 661 436921 238304781 25.7099203 8.7109827.001512859 662 433244 290117528 25.7293607 8.7153734.001510574 663 439569 291434247 25.7487864 8.7197596.00150s296 684 440396 292754941 25.7631975 8.7241414.001506024 665 442225 294079625 25.7875939 8.7285187.001503759 666 443556 295408296 25.8069758 8.7328918.001501502 667 4448399 296740963 25.8263431 8.7372601.001499250 663 4462-24 298077632 25.83456960 S.7416246.001497006 669 417561 299418309 25.8650343 8.7459846.001494763 670 448900 300763000 25.8843582 8.7503401.001492537 671 450241 302111711 25.9036677 8.7546913.001490313 672 451584 3303164 43 25.9229628 8.75903S3.001488095 673 452929 304S21217 25.9422435 8.7633309.001435884 674 454276 306182024 25.9615100 8.7677192.001433680 675 455625 307546375 25.9307621 8.7720532.001481481 676 456976 303915776 26.000000 8.7763330.001479290 677 458329 3 0238733 26.0192237 8.7807034.001477105 678 459684 311665752 26.03q4331 8.7850296.001474926 679 461041 313046339 26.0576234 8.7893466.001472754 630 4624C(D 314132000 26.076 096 8.7936593.001470583 631 463761 3 1 L53 21211 26.0959767 8.7979679 3.01468429 6 32 46-5121 i 317214563 26. 111297 8.8022721.001466276 CUBE ROOTS, AND RECIPROCALS. 149 No. Squares. Cubes. Square Roots.. Cube Roots. Reciprocals. 633 466489 3186119S7 26,1342687 8.8065722.001464129 684 467856 320013504 26.1533937 8.8108681.001461988 685 469225 321419125 26.1725047 8.815L598.001459854 686 470596 3228288056 26.1916017 8,8194474,001457726 687 471969 324242703 26.2106848 8,8237307.001455604 6S8 473344 325660672 26.2297541 8,8280099.001453488 689 474721 327082769 26.2488095 8,8322850.001451379 690 476100 328509000 26.2678511 8.8365559.001449275 691 477481 329939371 26.2868789 8.8408227.001447178 692 478864 331373888 26.3058929 8.8450854.001445087 693 480249 332812557 26.3248932 8.8493440.001443001 694 481636 334255384 26.3438797 8.8535985.001440922 695 483025 335702375 26.3628527 8.8578489.001438849 696 484416 337153536 26.3818119 8.8620952.001436782 697 485809 338608373 26,4007576 8.8663375.001434720 698 487204 340068392 26.4196896 8.8705757.001432665 699 488601 341532099 26.4386081 8.8748099.001430615 700 490000 343000000 26.4575131 8.8790400.001428571 701 491401 344472101 26.4764046 8.8832661.001426534 702 492804 345948408 26.4952826 8.8874882.001424501 703 494209 347428927 26.5141472 8.8917063.001422475 704 495616 348913664 26.5329983 8.8959204.001420455 705 497025 350402625 26.5518361 8.9001304.001418440 706 498436 351895816 26.57( 6605 8.9043366.001416431 707 499849 353393243 26.5894716 8.9085387.001414427 708 501264 354894912 26.6082694 8.9127369.001412429 709 502681 356400829 26.6270539 8.9169311.001410437 710 504100 357911000 26.6458252 8.9211214.001408451 711 505521 359425431 26,6645833 8.9253078.001406470 712 506944 360944128 26.6833281 8.9294902.001404494 713 508369 362467097 26.7020598 8.9336687.001402525 714 509796 363994344 26.7207784 8.9378433.001400560 715 511225 365525875 26.7394839 8.9420140.001398601 716 512656 367061696 26.7581763 8,9461809,001396648 717 514089 368601813 26.7768557 8.9503438.001394700 718 515524 370146232 26.7955220 8.9545029.001392758 719 516961 371694959 26.8141754 8.9586581.001390821 720 518400 373248000 26.8328157 8.9628095.001388889 721 519341 374805361 26.8514432 8.9669570.001386963 722 521284 376367048 26.8700577 8.9711007.001385042 723 522729 377933067 26.8886593 8.9752406.001383126 724 524176 379503424 26.9072481 8.9793766.001381215 725 525625 381078125 26.9258240 8.9835089.001379310 726 527076 382657176 26.9443872 8.9876373.001377410 727 528529 384240583 26.9629375 8.9917620.001375516 723 529984 385828352 26.9814751 8.9958829.001373626 729 531441 337420489 27.0000000 9.0000000.001371742 730 532900 389017000 27.0185122 9.0041134.001369863 731 534361 390617891 27.0370117 9.0082229.001367989 732 535824 392223163 27.0554985 9.0123288.001366120 733 537289 393832837 27.0739727 9.0164309.001364256 731- 533756 395446904 27.0924344 9.0205293.001362398 735 540225 397065375- 27.1108834 9.0246239.001360544 736 541696 398688256 27.1293199 9.0287149.001358696 737 543169 400315553 27.1477439 9.0328021.001356852 738 544644 401947272 27.166155 4 9.0368857.001355014 739 546121 403583419 27.1845544 9.0409655.001353180 740 547600 405224000 27.2029410 9.0450419.001351351 741 549031 406869021 27.2213152 9,0491142.001349528 742 550561 408518488 27.2396769 9.0531831.001347709 7413 - 55 49 410172407 1 27.2580263 9.0572482.001345895 744 553536 411830741 27.2763634 9. 0613098.001344086 1,> x. 150 TABLE X1 SQUARES) CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 745 555025 413493625 27.2946381 9.0653677.001342282 746 556516 415160936 27.3130006 9,.0694220.001340183 747 558009 416332723 27.3313007 9.0734726.00133868S 748 559504 418503992 27,3495887 9.0775197.001336S93 749 561001 420189749 27.3678644 9.0815631.001335113 750 562500 421875000 27.3861279 9.0856030.001333333 751 564001 423564751 27.4043792 9.0396392.001331558 752 565501 425259508 27.4226184 9.0936719.001329787 753 ]567009 426957777 27.4403455 9.0977010.001328021 754 563516 428661064 27.4590604 9.1017265.001326260 755 570025 4303638875 27.4772633 9.1057485.001324503 756 571536 432081216 27.4954542 9.1097669.001322751 757 573049 433798093 27.5136330 9.1137818.001321004 758 574564 435519512 27.5317998 9.1177931.001319261 759 576031 437245479 27.5499546 9.1218010.001317523 760 577600 438976000 27.5680975 9.1258053.001315789 761 579121 440711081 27.5862284 9.1298061.001314060 762 580644 442450728 27.6043475 9.1338034.001312336 763 582169 444194947 27.6224546 9.1377971.001310616 7641 583696 445943744 27.6405499 9.1417874.001308901 765 585225 447697125 27.6586334 9.1457742.001307190 766 586756 449455096 27.6767050 9.1497576.001305483 767 583289 451217663 27.6947648 9.1537375.001303781 768 589824 452984832 27.7128129 9,1577139.001302083 769 591361 454756609 27.7308492 9.1616869.001300390 770 592900 456533000 27.7488739 9.1656565.00129S701 771 594441 458314011 27.7668868 9.1696225.001297017 772 595984 460099618 27.7848830 9.17358.52.001295337 773 597529 461889917 27.8028775 9.1775445.001293661 774 599076 463684824 27.8208555 9.1815003.001291990 775 600625 465484375 27.8388218 9.1854527.001290323 776 602176 467288576 27.8567766 9.1894018.001288660 777 603729 469097433 27.8747197 9.1933474.001287001 778 605234 470910952 27.8926514 9.1972897.001285347 779 606841 472729139 27.9105715 9.2012286.001283697 780 603400 474552000 27.9284801 9.2051641.001282051 781 609961 476379541 27.9463772 9.2090962.001280410 782 611524 478211768 27.9642629 9.2130250.001278772 783 613089 430048637 27.9821372 9.2169505.001277139 784 614656 481890304 28.0010000 9.2203726.001275510 785 616225 483736625 23.0178515 9.2247914.001273385 786 617796 435587656 28.0356915 9.22870683.001272265 787 619369 487443403 28.0535203 9.2326189.001270648 78 620944 439303872 28.0713377 9.2365277.001269036 789 622521 491169069 28.0891433 9.2404333.001267427 790 624100 493039000 28.1069386 9.2443355.001265823 791 625681 494913671 28.1247222 9.2482344.001264223 792 627264 4967930883 23.1424946 9.2521300.001262626 793 628849 498677257 28.1602557 9.2560224.001261034 794 630436 500566184 23.1780056 9.2599114.001259446 795 632025 502459375 23. 1957444 9.2637973.001257862 796 633616 504358336 23.2134720 9.2676793.001256281 797 635209 506261573 28.23118834 9.2715592.001254705 793 636304 508169592 28.24188938 9.27543;-2.0012.53133 799 633401 510082399 23.2665331 9.2793081.001251564 809 640000 512000000 2..2342712 9.2831777.001250000 801 641601 513922401 23.30194-34 9.2S70440.001241439 802 643204- 515819609 23.3196014 9.2909072.001246383 803 644809 [517781627 2.33725146 9.29)47671.001245330 80-1 646416 5197184-G1 2. 351493 9.2936239.0012O 3781 805 61S9023 5216 LS 125 2 3. 375219 ). )24 775.001242236 806 619636 5236)6616 23 I.01()1 1I0695 CUBE ROOTS, AND RECIPROCALS. 151 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 807 631249 525557943 2..4077454 9.3101750.001239157 808 652364 527514112 28.4253403 9.3140190.001237624 809 634481 529475129 23.4429253 9.3178.99.001236094 810 656100 531441000 23.4601989 9.3216975.001234563 811 657721 533411731 23.4780617 9.3255320.001233046 812 659344 535337323 29.4956137 9.3293634.001231527 813 660969 537367797 23.5131549 9.3331916.001230012 814 662596 539353144 23.5306352 9.3370167.001223501 815 664225 511343375 23.5482048 9.3403386.001226994 816 665356 543338496 23.5657137 9.3446575.001225490 817 667489 545338513 23.5832119 9.3484731.001223990 818 669124 547343432 28.6006993 9. 3522357.001222494 819 670761 549353259 23.6181760 9.3560952.001221001 820 67240) 551363000 23.6356421 9.3599016.001219512 821 674041 55333766 28.6530976 9.3637049.001218027 822 675634 535412248 29.6705424 9. 3675051.001216545 823 677329 557441767 23.6379766 9.3713022.001215067 824 678976 559476224 23.7054002 9.3750963.001213592 825 630625 561515625 28.7223132 9.3783873.001212121 826 632276 563559976 23 7402157 9.3326752.001210654 827 633929 5656092S3 23.7576077 9.3864600.001209190 823 6855384 567663552 28.7749891 9.3902419.001207729 829 637241 ] 56972:2739 28.7923601 9.3940206.001206273 830 683900 571787000 2..8097206 9.3977964.001204819 831 690561 573356191 23. 8270706 9.4015691.001203369 832 692221 575930363 23.8444102 9.4053337.001201923 833 693389 578009537 28. 8617394 9.409 1054.001200480 834 695556 530093704 28.8790582 9.4123690.001199041 835 697225 532182375 23.8963666 9.4166297.001197605 836 6933896 534277056 2S.9136646 9.4203373.001196172 837 700569 536376253 28.9309523 9.4241420.001 194743 833 702244 533480472 23.9482297 9.4278936.001193317 839 703921 590389719 23.9654967 9.4316423.001191895 840 705600 592704000 23.9327535 9.4353S80.001190476 841 707231 594823321 29.0000000 9.4391307.001189061 842 708964 596947688 29.0172363 9.4428704.001187648 843 710649 599077107 29.0344623 9.4466072.001186240 844 712336 601211534 29.0516781 9.4503410.001184834 845 714025 603351125 29.0638837 9.4540719.001183432 846 715716 605495736 29.0860791 9.4577999.001182033 847 717409 607645423 29.1032644 9.4615249.001180638 848 719104 609800192 29.1204396 9.4652470.001179245 819 720301 611960049 29.1376046 9.4689661.001177856 850 722500 614125000 29.1547595 9.4726824.001176471 831 724201 616295051 29.1719043 9.4763957.001175033 8.52 725904 618470203 29.1890390 9.4801061.001173709 853 727609 620650477 29.2061637 9.4833136.001172333 854 729316 622335364 29.2232784 9.4875182.001170960 855 731025 625026375 29.2403839 9.4912200.001169591 8;6 732736 627222016 29.2574777 9.4949188.001163224 857 734449 629422793 29.2745623 9.4986147.001166861 833 736164 631623712 29.2916370 9.5023078.001165501 859 737881 633339779 29.3037018 9.5059980.001164144 860 739600 636056000 29.3257566 9.5096354.001162791 861 741321 633277331 29.3128015 9.5133699.001161440 862 743044 610503923 29.3598365 9.5170515.001160093 863 744769 642735647 29.3763616 9.5207303.001158749 864 746496 644972.544 29.3938769 9.5244063.001157407 865 743225 647214625 29.4103323 9. 520791.001156069 866 749956 6 19161896 29.4278779 9. 5317497.001154734 867 751639 651714363 29.4443637 9.5354172.001153403 863 753124 653972u)32 29.461 397 9.5390313.001152074 152 TABLE XI. SQUARES, CUBES, SQOJUAI-i ROOTS, No. Squares. Cubles. Squcare loots. Culbe Roots. Rleciprocals. ~6o1 75-5161.i 2j G (.47 SS 059 9.5427437.001 I' 074S S70 756900 6 55030C0 29.4957624 9. 5464027.0011494125 871 7;)>6 t i 660i763[ L 1 29.5127091 9 550(559.00(141C8 6 872' 035 4 6(3I)-)4842 29.296461 9.5537123.001114716,-9 873 7G 6 2 66533c61L7 29.5465734 9,5573630,0011415 1 874 s7639l( 7 667627624 29>,634910 9.5610108.001144165 875 765625 669921875 29.5 03989 9.5646 559.001142S57 876 767376 672221376 29.5972972 9.5682982.001141553 877 769129 6(74526133 29.6141858 9.5719377.001140251 878 7708881 676 36 e 152 29.63106148 9.5755745.001135952 879 772611 679151439 29.6479352 9.5792085.0011o7656 880 774400 651472000 29.6647939 9.5828397.00113t,3(' 3881 776161 63797841 29.6516412 9, 5064682 00 1350 74 882 779;24 6861289G6 29-.691>484 9.5900939.0011337S7 883 779689 68:4Gt 53 7 29.7,153159 9.59371i9 001132503 884 78 14-)6 69007104 29.75:2137.5 9.5973373,001131222 835 73 6931412 2.7 4 9.6009545.001129?4 4 86 74-)996 695506-56 2976 521 9. 6041I69G.0 01128(66S S-87 7'86769 69i>764103 02 9.78295452 9.6081317.00(1127,3';C 8S 788.544 700O2,1,27 29).,'3993299 9,.6011791.0011261'6 829 790321 702936309 29.8161030 9,6153977.001124859 890 792100 704969000 29.8328672 9.6190017.001123596 8'91 793~S1 707347971 29.31 9.62260O:,0.001122.33 892 79G664 7 0 9 732283 29.8663690 9.6262016.001121)076 893 797449 712121957 29.SS31056 9.6297975.001119921 894 799236 714516984 29.899232S 9.6333907.00111S568 895 S01025 716917375, 29.91655()6 9.6369812.001117318 896 802S16 7193231 36 29. 9332591 9.6405690.001116071 897 801609 721734273 29.9499583 9.6441542.001114827 898 806404 724150792 29.9666481 9.6477367.001113>86 899 802201 726572699 29.9833287 9.6513166.001112347 900 010000 729000000 30.0000000 9 6542938.001111111 901 811801 731432701 30.0166620 9.6584694.001109278 902 813604 7338708(8)> 30.0333148 9.6620403.0011086-17 903 815409 736314327 30.1(4995284 9.6,65096.001107420 904 817216 738763264 30.0665928 9.6691762.001106195 905 819025 741217625 30.0832179 9.6727403.001104972 906 820936 7436'77416 30.099>339 9.6563(317.0011037 53 907 822649 74614264-13 30.116440(7 9.67969604.001102536 908 824464 7486 1331 2 30. 1 330383 9. 6341 6G.001101322 909 826281 751089429 30.1496269 9.6269701.000010110 910 828100 753:571000 30.1662(363 9.6905211.00109,9901 911 829921 7560 5O03 L 30. 1827765 9.6940694.00109769)5 912 831744 7585;50.528-2 30 1993377 9.6976151.001096491 913 S33569 761043497 30.2158999 9.7011583.001095290 914 835396 763551944 30.2324329 9.70469S9.001094092 915 837225 76696()875 30.24,99669 9.70223f;9.001092s96 916 839096 763575296 20,2654919 9.7117723.001()91703 917 840)539 771092 13 30.2S20079 9.7153051.001090513 918 842721 773620632 30.2985148 9.7188354.001089325 919 844361 776151509 30.3150128 9,7223631.001088139 920 846400 778688000 30.3315018 9,7258883.001086957 921 841241 781229961 30.3479S18 9.7294109.001085776 922 8.50084 783777448 30.3644 1,529 9.329309.0010841.99 923 851929 786330467 30 3909151 9.73644>4.001083 423 924 8-53776 7983S9024 30.39736S3 9.7399634.001032251 9235 85562. 791453121 5 30.413,127 9.74347158.0010810(l 926 85741.76 794022776 30!.4302481 9.7 46987.00107'914 927 8;59329 796597!193 30.4466747 9.75049390.0010787439 92 861 14 79917875'. 2 30. 463019241 3', 33917.0(1 07 929 S63041 801765093 30.4795013 9.7571500(21.n0107 G6-26 930 864900 804357000 1 30.4959014 9.7610001.0010 6929 CUBE ROOTS, AND RECIPROCALS. 153 No. Squares. Cubes. SquareRoots. Cube Roots. Reciprocals. 931 866761 806954491 30.5122926 9.7644974.001074114 932 865624 809557563 30.5299750 9,767922.001072961 933 870439 812166237 30.5450487 9.771445,001071811 934 872356 814780501 30.5614136 9.7749743,001070664 935 874225 817400375 30.5777697 9.77~>4616.001069519 936 876)96 820025j56 30.5941171 9.7239466.001060376 937 877959 822656953 30.6104557 9.7854238.00 1067236 9383 8798:44 825293672 30.6267857 9.78890>7.001066098 939 881721 827936019 30.6431069 9.7923861.001061963 910 8.83630 830534000 30.6594194 9.7958611.001063830 911 835481 833237621 30.6757233 9,7993336.001062659 942 8736-4 835996388 30.6920185 9.8029036,0010t1571 943 899219 8338561807 30,7083051 9.8062711.001060445 914 891136 841232384 30.7245830 9.8097362.001059322 945 893025 843903625 30.7403.523 9.8131989.001059201 946 894916 846590536 30.7571130 9,8166591.001057082 917 896909 849278123 30.7733651 9.8201169.001055966 913 893704 851971392 30,7896086 9,8235723.0010514852 919 9)0601 854670349 30.8053436 9.8270252.011053741 950 902500 857375090 30.8220700 9.83041757.001052632 9i 901401 860085351 30.83279 9339233 O.1525 95 ] 9063 )4 862901403 30,$544 972 9.8373695.0010.50420 901)3209 865523177 30.8706951 9.94098127.001019318 931:1 910116 862506061 30.$98501 9,841125i36.031008218 955 912025 870933>75s 3 0.9630743 9,8476920,001097120 976 913936 879372216 )0919297 9.112>0 2 0010416025 957 915419 8761467493 30.9:54136 9.~545617,00101144932 9.58 917764 8792179j12 30. 95315751 9.8579929.001043841 959 9196>1 8>1974079 30.96772531 9.8614218.001042753 960 921600 834736000 30.933663 9.8618483.001011667 961 9235-21 8>7503691 31.0000000 9.86>2721.001040533 962 925444 - 890277123 31.0161243 9.8716941.001039501 963 927369 803056A17 31.0322413 9.8751135.0010438422 964 929296 895341314 31.0483-194 9.87)305:001037344 965 931225 895632125 31.0641491 9.8>19451.001036269 966 933156 90142.695 31.08395405 9.8853574.001035197 947 9350.39 9014231063 31.0966236 9.8887673.001034126 963. 93702-1 907039232 31.11269.34 9.8921749.001033058 969 933961 939853209 31.12376-13 9,8955801,001031992 970 910903 912673000 31.1443230 9.89>9830.001030928 971 942841 915493611 31.1603729 9.9023835.001029866 972 94784 918330043 31.1769145 9.9057817.001028307 973 946729 921167317 31.1929479 9.9091776.001027749 974 948676 92401042.1 31.2039731 9..9125712.001026694 -975 950625 926359375 31.2249900 9.9159624.001025641 976 952576 929714176 31.2409937 9.9193513.001024590 977 954529 932574833 31.2569992 9.9227379.001023541 973 956484 935441352 31.2729915 9.9261222.001022495 979 9538441 928313739 31.2389757 9.9295042.001021450 980 960400 941192000 31.3049517 9.9323839.001020408 931 962361 944076141 31.3209195 9.9362613.001019368 932 961324 946966168 31.3363792 9.9396363.001018330 933 966289 949862037 31.3528303 9.9430092.001017294 934 969256 952763904 31.3687743 9.9463797.001016260 985 970225 955671625 31.3847097 9.9497479.001015228 96 972196 958585256 31.4006369 9.9531133.001014199 987 974169 961504803 31.4165561 9.9564775.001013171 933 976144 961430272 31.4324673 9.9598389.001012146 939 978121 967361669 31.4433704 9.9631981.001011122 990 930100 970299000 31,4642634 9.9665549.001010101 991 932(13l 9732-12271 31.4801525 9.9699095.001O099082 9 / 1) 3-11064 9761914995 31.4960315 9.9732619.001008065 154 TABLE XI. SQUARES5 CUBES, &Co No. Squares Cubes. Square Roots. Cube Roots. Reciprocals. 9931 98S6049 97c9146607 31,11 19025 9.9766120.001007049 994 988036 983 1z07784 31.527760 G;5 9.9799599.0010060:36 995 99'3cS 99 5074,>75 31.5436206 9.92333055.001005025 996 992o16 988047936 31.5594677 9.9 8S66488.001004016 997 993409 991026973 31.5753065 9.9899900.001003009 993 996004 994011992 31.5911380 9.9933289.001002004 999 99o001 997002999 31.606,,613 9.9966656.001001001 1000 1000003 1000000000 31.6227766 10.0000000.001000000 1(01 100I00 00 003021I0301 31.6385840 10.0033322.0009990010 1002 1004 00t 100601200S 31.6543836 10.0066622.0009980040 100:3 101609) 10(19027027 31.6701752 10.(0099899.0009970090 1004 1003016 1012048(064 31. 659590 1(.0133155.0009960159 10,15 1010025 1015073)125 31.7017349 10.01663S9.0009950249 1006 10120O6 1018108216 31. 7175030 10.0199601.0009940358 1007 1014049 10121147343 31. 732633 10.0232791.0009930487 1003 1016064 1024192512 31.7490157 10.0265958.0009920635 1009 1018081 1027243729 31.7647603 10.0299101.0009910803 1010 1020100 1030301030 31.7804972 10.0332228.0009900990 1011 1022121 103336433L 31.7962262 10.036.5330.0009891197 1012 1024144 1036433728 31.S81 19474 10.0398410.0009881423 1013 1026169 10 39509197 31.8276609 10.0431469.0009871668 1014 1028196 1042590744 31.8433666 10.0464506.0009861933 1015 1030225 1045678375 31.8590646 10.0497521.0009852217 1016 1032256 1048772096 31.8747549 10.0530514.0009842520 1017 1034289 1051871913 31.8904374 10.0563485.0009832842 1013 1036324 1054977832 31.9061123 10.0596435.0009823183 1019 1038361 11)O8089859 31.9217794 10.0629364.0009813543 1020 1040400 1061208000 31.9374338 10.0662271.0009803922 1021 1042441 1064332261 31.95030906 10.06951t56.0009794319 10(12 1044484 106746264S 3L.968 347 10.0728020.0009784736 1023 1046529 1070.99167 31.9843'712 10.0760263.0009775171 1024 104,.576 1073741824 32.0000000 10.0793684.0009765625 1025 1050625 1076890625 2 32.(11 5612 10.0826484.0009706098 10(26 1052676 100045576 32.0312348 10.0859262.0009746589 1027 1054729 1033206633 32.0468407 1(.0S92019.0009737098 1028 1056784 10S6373952 32.0624391 10.0924755.0009727626 1029 1058841 1089547389 32.0780298 10.0957469.0009718173 1030 1060900 1092727000 32.0936131 10.0990163.0009708738 1031 1062961 1095912791 32. 10915S7 10.1022835.0009699321 1032 1065024 1099104763 32.1247563 10.1055487.0009629922 1033 10670>9 1102302937 32.1403173 10.1088117.0009680542 1034 1069156 1105507304 32. 1558704 10.1120726.0009671180 1(135 1071225 1108717S75 32.1714159 10.1153314.0009661836 1036 1073296 1111934656 32. 1~69539 10. 1185882.0009652510 1037 1075369 1115157653 32.2024844 10.1218428.0009643202 1033 1077444 1118396872 32,2180074 10.1250953.0009633911 1039 1079521 1121622319 32.2335229 10.1283457.0009624639 1040 1081600 1124864000 32.2490310 10.1315941.000961,53S5 i041 1083631 1128111921 32.2645316 10.1348403.0009606148 1042 1085764 1131366048 32.2800248 10.1380845.0009596929 1(043 1037849 113-1626507 32.2955105 10.1413266.0009587738 1044 1089936 11378931 4 32.3109388 10.1445667.0009578544 1045 1092025 1141166125 32.3264598 10.1478047.0009569378 1046 109-1116 1144445336 32.3 119233 10.1510406.0009-560229 1047 1096209 11477.302)3 32,.3573794 10. 1542744.000)95r51098 1 4S 1009;304 1151022592 32. 3728281 10. 1575062.0009541985 10(49 1100)1 11154320649 32.3832695 10.1607359.00095328S8 150 1102-300 1157625000 32.4037035 10.1639636.0009 123810 1051 1104601 116009:35631 32.4191301 10. 1671 893.00n0914748 1052 1106704 116G1252603S 3.4 34495 10.1704129.00095105)03 1053 110o809 1167575> 77 -3.4-19615 10.1736344.00(>94I6676 1 54 1110916 117090541(;4 12.146-62 10.1768539.0009487666 TABLE X1II LOGARITHMS OF NUMBERS FIROM L TO 90~@@,o 156 TABLE XII. LOGARITH3IS OF NUILIBERS. No., 0 a 9 9 3 0 7 - 9 Bu iffo 100 ooooo0 o 000434 l00068 0 10 1 001734 002166 00259S 003029 0'03461 003891i 432 1 4321 47511 5111 5609 6033 6466 6894 7321i 7748 8174 428 2 8600 90261 9451 9876 010300 010724/011147 011570 0lI193 012415 424 31012337 0132,99013680 0141001 4521 4940 5360} 5779 6197 6616 420 41 7033 74511 7869 82944 8700 9116 9532 9947 020361 020775 416 5 0211S91021603 022016 022423, 02284 1023252 023664 024075 4486 4& 96 412 6( 5306 5715 G21 65331 69-12 73501 77571 81G4 8571 897S 408 7 933-1 9739 0301 90) 030(6001031001 031403 0318121032216 032619 033021 404 8 033424 0333261 4227, 4628.502'3 5430 5830 6230 6629 7Q028 400 9 7-128 7825.9 832:3 8 62( 9017 9414 9811 040207 010602 040998 397 110 I041 393 177 04 218210425676 0429G9 0-133G2 043755 044143S 0445,10 044932 393 1 53231 5711 61()5i 6-19.5 G6S 5 7275 76642 80531 5442 8830 390 2 9218 96061 9993 0503So005o 0 506611 5,o516531S0051 921,4 0.3051o9 052694 3s6 3 053078( 034603 053461 42:30i 4613 4996 0:37S 5760 6142 6. 24 3S3 41 6093)2 72;61 7'(666i 80o1 8426 S8O30 98l,85 9;563 93942 0603209 379 5106069> 061075 0611452061829i062200 069;821062958 063'333 063709 4083I 3576 6i 4458 4 932 520G6 55601 595'J G326| G!91 7071 71 43 7815 373 71 8186 85571 892i3 9298, 966~070073076>0010,07710 711477 071145 151 370 807132072250077220072617 07295'073352'3718 4 44;511 4816 5182 366 l 9 5547 5912! 6276 6640 7004 7.3G6 7731I 80944 84571 8319 363 120 07i9181 795343 O-997)) 012669,OS0)62G 0 0S(17987 0)S347 031707 0S2067 0S2426I 360 1 02786.) 0831440S3503 361 41 4- 291 5647 6004.i9 21 360 6716 70711 7-126 7781 813G6 84901 S3451 9193, 9552 355." o 9905. 090;21 090611 090963 091315 ) 091 G67 092018 0923701092721 [093 711 352 7 40934221 3772 412 2 4471/ 4920 5169 56518 5iS 62151 65621 3i49/ 5 6910l 72357 760141 79351 829S3 sG-1 14 s990 9335 6 9681 100026 3146 61003711100715 101059l014(03 10174710201 1102091 1 143 1027771 103119 3462 343 7 39804 414(6 431 44821 5G9 56510 595 1 6191 6531 67}1 341 8 7210 75491 73SS 8227 3 S565 s903 92411 9579 9916 1102535 333S 91L105901110926 111263 111599 11193-1 112270 112605 112940 113275 9 3609 335 130 113943 114277 1114611 1 19114 1h52' 11> 61 1 i1159131 116276Gi116OS 116940( 333 1 7271 760:3 7931 S3.2G95 8(59 S3926 92356 9365SG 9915 120 245 30|| 2 120574 12090i3 121231 121560 1218889 1222161 12244 1221871 123198 35251 32S 3 3352 4178 4501 48301 5159 54831 58o G 6131 64156 6781 325i 4 7105/ 7429 7753 s0761 399 8722 90451 36S 9690 130012 323 5 130334 130655 130977 13129S 131619 /13193) 132260 132530 132900 3219 321 W6 3539 3358 4177 4196' 481l1 5133 2451 57691 60s6 6l40 31S 7 6721 7037 7354 76710 7987 8 3303 S8618 S8934 9249 9564 316 S 9379 140194 140503 140S22 1411136 14140o 141163 1120376 1423891 142702 314 9 113015 3327 3639 39511 4263 4t574 4351 5196 5507 5813 311 140 146123 146438 14674S 147038 141367 147676 1479149 14294 143i603 11911 309 1 9219 9527 93351150142'1504491(07 G6 1.51063 1.51370 1151676 15193S2 307 21152238 152594 1529001 30205 3510 3135 4120 4124 4 723 5(32 305 1 31 5336 56640 5943 62316 6549 6S52 71354 7457 77';59 8061 303 4 8362 8664 S965 9266 9567 986t 16016 160469 16076911610638 301 | 5,161368 161667 161967 162266 162564[162.S63 3161 346(3 37591 4055[ 299 6 43953 46-50 4947 5244 55341 5'3' 6134 6-130 6726 7022 297 7 7317 7613 79083 8203 8497 192 9090 93S0 9674 963 2995 | 8170262| 1709555 1708481171141 171434 17 7266 172019 172311 172603 1728951 293 9'3186 3478 3769 4060 4351 4611 4932 5222 5512 5802 291 90 17601 1631 176670 176959 177248 177536 177325 178113 178401 1736S9 29 1 89/7 9264 9.9521 9S39 180126 18013 11R0699 1809S6 181272 1315569 287 2181844 182129 124115!182700 29.59 3270 3555 3839'4123 4407i 285 4691 4975 5259 552 5825 6108 6391 6674 6956 7239 283 4 752 731 73O 0 8366 9617 89921 9209 9490 9771 190051 281 51100332 19 061 190 392 191171 191451 131730 192010 192289 192567 246G 279 6 3125/ 3403 1 3G11 3959 42:3 7 45141 4792 5069;3-16 5623 27S 7 5900 6176 61-3' 6729 7/009 7/231 75:96 7832 8107 833.2 276 8657 89321 920 92706 941 9 2000299200303 200(577 200850 201 19- 274 1 9 201397'201670()201-143 202216!2)024S 2'761 3033 3305 357'7 34s 272 iNo 0 0 2 3 6 | s 9 iDifl'.1 TABLE XII. LOGARITIIIS OF NUMIBERS. 157 No. 0 1 1 3 5 |_ T 6 9 Diff. 1690 20-T120 204 431.2046 163 2 3.3 J 2-)5 i7, 0,i)5 4 6 206031 6 20o26 20G656 271 1 63276 O 736 7G.34 7904 3173 8141 SIo101 8979 9247 269'2 9515 9783 210-511210319 21056 21032111 11211, 33 211641 211921 267 3 212183122-14 2720' 2936 3327 3;.13 3733 4091 4:314 4579 266 1 431i 5109 5373 539 5902 ) G6 613 1 6 69 1 6'957 7221 264 5 7431 7747 8010 82731.S 3'3 8 97093 9306)0 932431 9%55 936i 262 6 229010 29-:370 2M'0563[1 21030') /211)53 22114 t22 i67 /221936'22219M 2'22156 261 7 2716 29376 3235 3 96 37.5. 4015* 4274 415:33 4792 505 1 259 8 5303 5G3 532( 6:)4 6:312 660)9 G3.73 7115 7372 7630) 253 9 7837 8114 84103 86.7 89103 9170 9026 9G32 99353230193 256 170 230119 230701i 233960 231215 231-170 2317 72 231979 2322 3 23323.8] 232742 255 1 2996 32 30 50 3737'0 4301 4It I 4. 17 47701 502:3 5276 253 2 55233 05781 6033 63 637 6:37 9 704lI 7 2 92 754t 7795 252 3 80416 8297 S854s 8799 9049 9299 / 9).50 93001240050 240300 250 41240549 210799 241048 241297 241546 211 795 122044 242293 2541 2790 249 5 303:3 3236 3.331 3782 40:30 4277 45'.z 4772 5019 5266 2481 6 55130 5759 6006 625232 6199 674 6991 7237 74832 7723 246 7 7973 8219 846-1 8709 890-4 9198 9-11:3 96S7 9932 250176 245 8 250120 250664 250303 251151 251395 251633 25181 2512 25 2523628 2610 243 9 2383 3096 333 3 330 3322 4004 4306 4548 4790 5031 242 180 255273 255514 2.55755 255996 256237 256177 256718 256938 257198 257439 241 1 7679 7918 815S 8393 8637 8377 9116 93,5 9594 9833 239 2 260071 260310 2690548 2603787 2610251261263 261501 261739 261976 2622141 238 3 2451 263S 29253 31 62 3399 3636 3373 4109 43416 4582 237 4 4818 50534 5290 552.3 576 1 5996 6232/ 6167 6702 6937 235 5 7172 7406 7641 7875 8110 8341- 8578 8812 9046 9279 234 6 9513 9746 9980 270213 2704416 270679 270912 271114 271377 271609 233 7 271842 272074 272306 2533 2770 3001 32:3:3 34646 3696 3927 232 8 4159 4339 4620 48350 5031 5311 5542 5772 6002 6232 230 9 60t62 6692 6921 7151 7380 7609 7833 8067 8296 8525 229 190 278754 278932 279211 279439 279667 279395 28012:3 280351 230578 280906 228 1 201033 231261 231433 281715 281942 232169 2396 2622 28419 3075 227 2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 3 5557 5782 6007 6232 6156 6631 6905 7130 7354 7578 225 4 7802 8026 8249 8473 8696 8920 9143 9366 9589 9812 223 5 2900351 290257 290480 290702 290925 291147 291369 291591 291813 292034 222 6 2256 2478 2699 2920 3141 3363 3584 3304 4025 4246 221 7 4466 4637 4907 5127 5347 53567 5787 6007 6226 6446 220 8 6665 6334 7104l 7323 7542'776L 7979 8198 8416 8635 219 9 8353 9071 9239 9507 972.5 99431300161 300378 300595 300813 218 200 301030 301247 301461 301631 30189 302114 302331 302547 302764 302980 217 1 3196 3412 36281 3844 4059 4275 4491 4706 4921 5136 216 2 53.51 5566 5781 5996 6211 6425 6639 6354 70681 7282 215 3 7496 7710 79241 8137 83511 85644 8778 8991 92041 9417 213 4 9630 9343 310056 3102631311)-181 3106931310906 311118 13113301311542 212 51311754 311966 2177 2339 2600 2312 3023 3234 3445 3656 211 6 3367 4078 4239 4499] 4710 4920 5130 5340 5551 5760 210 71 5970 6180 63901 6099 63809 70181 7227 7436 7646| 7854 209 8 8063 8272 8331 86391 8393 9L06 9314 9522 9730] 9931 203 9 320146 32034 320.2:32C07) i320977 321131 321391 321598 321532S0 322012 207 210 3222191322 126 322033:3223391323016 302 32-32 023458 323665 323371 324077 206 1 4232 44338 4094 43)991 5103 5310 53016 5721 5926 6131 203 2 6330 6541 60175) 59) 7155 735093 7563 7767 70372 8176 204 3 83)0 $8o3 S7)7 8991 9194 9 39,9/ )601 03905 33000 330211 203 41330114 33i40 171i33) 1i 9 331232I 3312213311271331630 331832 203411 2236 202 5 21334 2010: 242 I:3;ti1 3141H 31)-19 3)350 40)51 42033 202 16 4'1-5145-7 1 6 - 7 1 5',.1 5 -S 5359 600 591 660 201 7 63;301 G6;bo) 0836I 70 (, 72 741 ) ) 7f9 7S8:3 80)58 S2.7 200 8 81G 8 3-,1 5-,- 91.,1 ) I G0000!)011) 93 4) )I7|:314)246 i99 931(1-1.3. I l.11')1 1:3 11(J-','J 3 112"37,3-1 I1: 1 3 2f) _ 2 ) 9 No. 0 1 o 01 1 I 14 158 TABLE XII. LOGARITHIBIS OF NUiTIBERS. No. O ) 1 1 2 3 5- 61 9 Diff. 220 342423 342620 342817 343014 343212 343409 343606 714:3602 342C99 -441'- 6 197 1 4392 4589 4785, 49811 5178 5374 5570 5,66 5962 6157 196 2 6353 6549 6744 6939' 7135 7330 7525 7720 7915 1 10 195 3 8305 8500 8694 8SS9: 9083 9278 9472 "666 9S"60 350U54 194 4 350248 350442 350636 350829 351023 351216 351410 351G603 351 716 1989 193 5 218S3 2375 256S 2761L 2954 3147 3 3532 374 3916 193 6 410s) 4301) 44193 46351 487G 0GS 5260 425032 6413 5S34 192 7 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 8 79350 81 250 S8316 8506l E'696 8S8G 9076 9266 9456 9646 190 9 9835 36002 360215 360404 36059336073 3 360972 361161 361350 361539 189 230 36172S 361917 36 105 362294 36s2 3G 1 362S590 36304S 363236 363424 188 1 3,612 38300 3988 4176, 43131 4 551 4739 4926 5113 5301 188 2 5488 5675 5862 6049,1 6236G 6(423 6610 67S6 6983 7169 187 3 7356 7542 7729 7915| 8101 S2S7 8173 s659 9845 9030 186 4 9216 9401 9587 97721 9958 370143 370328 370513, 370698 370883 185 5137106 371253 3371437 371622 37180G 1991 2175 2360 2544 2728 184 6 2912 3096 3280 3464[ 36417 3,331 4015 4198 4382 4565 184 7 4748 4932 5115 5298 54181 56641 546 60-29 6212 6394 183 8 6577 67359 69-2 7124 730iG |7-188 7 670 7852 8034 8216 182 9 8398 8580 9761 8943 9124 9306 9487 9668 9S4913S0030 181 240 380211 390392 390573 30754 3093-01 34 1 135 I 3'1296 1 3S91476 7 381656 | 381837 181 1 2017 2197 237 7 29057 2737 29317 3097 3277 3456 3636 1l0 2 3815 3995 41741 43531 45331 4712 4S91 5070/ 5249 5428 179 3 5606 5785 5964 6142 6322 1 (199 6677 6956 7034 7212 178 4 7390 7568 7746 7923 8101 8279 8456 9634 89911 8989 178 5 9166 9343 9520 98) 9 8/375 390051 390228 390405 390562 390759 177 6139093539 11123912881391464 3916-1 1817 1993 2169 2345 2521 176 7 2697 2873 3049 3224 30400 3375 3751 3926 4101 4277 176 8 4452 4627 4802 4977 5152 5326 5301 5676 5850 6025 175 9 6199 6374 67054 6722 696 707L 7245 7419 7592 7766 174 2501397940 398114 39S297839-161 3998634 398093 399591 399154 399328 399501 173 1 9674 9847 40002140 94001 92400365 400538 40071 1 400883 401056 401228 173 2401401 401573 1745 1917 2089 2261 2433 2605 2777 2949 1272 1 3121 3292 3464 36.35 3807 3978 4149 4321 44192 4663 171 4 4834 500,5 5176 5346 5517 5688 598 6029 6199 6370 171 5 6340 6710 6881 7051 7221 7391 7561 7731 7901 8070 170 6 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 7 9933 410102 410271 410440 410609 410777 410946 411114 411283 411451 169 81411620 1789 1956 2124 2293 2461 2629 2796 2964 3132 168 9 3300 3467 3635 3S03 3970 4137 4305 4472 4639 4806 167 260 414973 415140 415307 4154741 45641 41580 415974 416141 41630 416474 167 1 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 166 2 8301 8467 8633 8793 8964 9129 9295 9460 9625 9791 165 3 9956 420121 420286 140451 1420616 420781 420945 421110 421275 421439 165 4 421604 1768 1933 2097 2261 2426 2590 2754 2918 3(082 164 5 3246 3410 35574 3737 3901 4065 4228 4392 4555 4718 164 61 4882 5045 5209 5371 5534 5697 5860 6023 6186 6349 163 7 6511 6674 6936 6999 7161 7324 7486 7648 7811 7973 162 8 8135 8297 8459 8621 9 783 99442 9106 9263 9429 9591 162 9 9752 9914 430075 |430236 430398 430559 430720 430891. 431042 431203 161 270 431364 431525 43165 1431S46 432007 432167 43232 9 43249 9 432649 43209 161 1 2969 3130 3290 3450 36101 3701 3930 4()90 4249 4409 160 2 4569 4729 438 50138 5207i 5367 552 6 56385 5844 6004 159 3 6163 6322 64931 6640 6799 6393.7 7116 7275 7433 7592 1659 4 7751 7909 80671 S226 83941 S5421 9701 8Ss59 9017 9175 1598 5 9333 9491 9648 9906' 9Y64-140 122440279 440437 4401594 440752 199 6 440909 441066 441224 441331 4415338 1695 1852 2009 2166 2323 157 7 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 8 4045 4201 4337 4513 46691 4S25 4981 5137 5293 5449 15 G 91 5604 5760 591 6071 6226 63821 6537 G6692 684 7003 155 No., I 1 1 W 0 3 - - -11. TABLE XII. LOGARITHMIS OF NUMIBERS. 159 No. 0 1 2 3 3 5 6 9 Diff. 230 447153144731:3 4f4746 7 i4lt7623 4-177738 447933 44~0.8 448-242 448397 448352 1 5 1 37(6 861 90135 91701 9324 9-73 96331 97871 9941 430095 154 214502 19 4.5-1()3 450557 45071 1450G65 451018 451172 451326 431479 16 33 154 3 1786 1901 2093 2247 2400 253 2706 2S59 3012 31 6 153 4 3313 3171 3624 3777 3930 4032 4235 437 454() 4692 153 5 4345 4997 51503 5302 i5454 5606 5758 5910 6062 62141 152 6 6366 6513 6670 6821 6973 7125 7276 7428 7579 7731 152 7 7832 8033 8184 8336 8487 8633 8789 8940 9091 9242 151 8 9392 9543 9694 9845 9995 460146 460296 460447 460597 460748 151 91460393 461045146119814613483 461499 1649 1799 1943 209 22-18 150 290 462398 46254S 462697146247 462997 46346 463296 46314 463594 463741 150 11 3593 40412 4191 4340 4490 4639 4788 4936 505 523-1 1491 2 5333 5532 5681) 5829 5977 6126 6274 6423 6571 6719 149 3 636s 7016 716-1 7312 7460 760, 7756 7904 8052 8200 148 4 8317 8495 $613 8790 8938 9835 9233 93SO 9527 9675 148 5.99322 9969 4701 16 470263 470410 4170~57 470704 470351 4709981 71145 147 6 471292 47143-3 1585, 1732 1878! 2061 2171 2318 2464[ 2610 146 7 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 8 4216 4362 4503 4653 4799 49144 5090 5230 531 5526 146 9 5671 5816 5962 6107 62-52 6397 6542 6637 G832 6976 145 3001477121 477266 477411 477555 477700 477844 477989 478133 478278 478422 145 1 8566 8711 8355 8999 9143 9237 9431 9575 9719 9863 144 2,480007 480151 1480294 4804331 438082 480725 430869 481012 4S156 1481299 14-1 3 1143 15S6 1729 1872 2016 2159 2302 2445 2588 2731 143 4 2374 3016 3159 3302 3145[ 3587 3730 3872 4015 4157 143 5 4303 4-t421 4585 4727 4869 5011 5153 5295 5437 5579 142 6 572[ 5363 60031 6147 6291 6430 6572 6714 63551 6997 142 7 7133 7230 742L 7563 7704 7845 7986 8127 S269 S410 141 8i 8;35;1 8692 8~33 897-l 9114 9255 9396 9537 9677 9318 141 9 993 4900991 490239 490380 4905210 490661 49001 4909449101 491222 140 310 491362 49150 1491642 491782 491922 492062 492201 492341 492481 1492621 140 1 2760 2903 3040 31791 3319 3458 3597 3737 3376 4015 139 2 4155 429- 4433 45721 4711 4850 4939 5128 5267 5406 139 3 554-1 5633 5822 5960 6099 6235 6376 6515 66-,3 6791 139 4 69 31 706 6 7206 7344 7183 7621 7759 6 7897 8035 8173 138 5 8: 11 844S3 8-586 8724 8S62 8999 9137 9275 9112 9550 133 6, 9637 9324 9962 50009915002361 50()374 500311 500648 500785 500922 137 715010-9 501196 501:333 1470 1607 1744 1880 2017 21541 2291 137 8 21271 2564 27)0 2837 2973 3109 3246 3382 3518 3655 136 9 3791 3927 4063 4199 43351 4471 4607 4743 4878 5014 136 320505150/5 052~61505421 505557 505693 505828 505964 506099 506234 506370 136 11 6,051. 660! 6776 6911 7046 7181 7316 7451 7586 7721 135 2 73536 7991 8126 8260 83951 8530 8664 8799 8934 9068 135 3 92031 9337 9471 9606 9740 9374 510009 510143 510277 510411 134 4 151045~1510679 51 013 5109471511031 511215 1349 1432 1616 1750 134 5 18331 2017 2151 2241 2418 2551 2634 2318 2951 3034 133 6 3218 33511 3484 3617 3750 3883 4016 4149 4221 4415 133 7 4514 46~1 4813 4946 5079 5211 5344 5476 5609 5741 133 8 5874 60061 6i39 6271, 6403 6535 6668 6800 6932 7064 132 9 7196 7323 7460 7592 7724 7855 7937 8119 8251 8382 132 330 518514 5186-16518777 518909 519040 519171 519303 5194341519566 519697 131 1 9328 99 9 1520090 520221 520353 5204834 520615 520745 520376 521007 131 2/521138 1521269 1400 15301 1661 1792 1922 2053 2183| 2314 131 3 2t414 25751 2705 23351 2'966 3096 3226 3356 3486 3616 130 41 37-16 3376 4006 41361 4266 4396 4526 4656 4785 4915 130 5 50f40)3s 5174 5.3() 4 5431 5563 5693 5822 5951 6081 6210 129 6 6339 6469 6593 6727 6356 6985 7114 7243 73721 7501 129 7 7630 7759 7881 8016 8145 8274 8402 8531 8660 87883 129 8 8917 9045 91741 89302 93:30 9559 9687 9815 99131530072 128 9 330200_530325304-156 330 34 5:3071 530340 530968 531096 531223 1331 128 No.4 O i 1 1 3 14 5 4 6 8 9 Dif..~~~~~~~~~~ — I 160 TABLE XII. LOGARITHMIS OF NUMIBER1S. No.1 0 1 a 3 & 5 6 7 8 9;Diff. 310 531479 531607 531734 531862 1531990 532117 532245 532372!532O i 53227 128 1 2754 2382 3009 3136 3264 3391 3518 3615 1772 o3 99 127 2 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 127 3 5294 5421 5547 5674 5800 5927 6053 6180 68306 6432 ]1G 4 6558 6685 6811 6937 7063 7189 7315 7441 7567 76;3 126 5 7819 7945 8071 8197 8322 8448 8574 8699 S825} 2 51 1 261 6 9076 9202 9327 9452 9578 9703 9829 9954 540079 540204 12,5 7 540329 540455 540580 540705 540830 540955 541080 541205 1330 14154 125 8 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 125 9 2825 2950 3074 3199 3323 3447 3571 3696 3S20 3944 124-1 350 544068 544192 544316 544440 544564 544688 544812 544936 545060 545183 124 1 5307 5431 5555 5678 5802 5925 6049 6172 6296 6419 124 2 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 123 3 7775 7898 8021 8144 8267 8389 8512 8635 87 58 8881 123 4 9003 9126 9249 9371 9494 9616 9739 9861 9984 550106 123 5 550228 550351 550473 550595 550717 550840 550962 551084 551206 1328 122 6 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 7 2668 2790 2911 3033 3155 3276 3398 35195 364() 3762 121 8 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 9 5094 5215 5336 5457 5578 5699 5820 5940 6061 6182 121 360 556303 556423 556544 556664 556785 556905 557026 557146 557267 557387 120 1 7507 7627 7745 78681 7988SS 8108 8228 8349 8469 8589 120 2 8709 8S29 8948 9068 9188 9308 9428 9548 9667 9787 120 3 9907 56002 1560146 5602651 560385 5605041 56062 560743 560863 156982 119 4 561101 1221 1340/1 459 1578 1698 1817 1936 2055 2174 119 5 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 119 6 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 119 7 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 118 8 5348 5966 6034 6202 6320 6437 6555 6673 6791 6909 118 9 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 563202 568319 5684361568554 568671 568788 568905 569023 569140 569257 117 1 93741 9491 960S 9725 9842 99591 570076 570193 570309 570426 117 2 5705413 570660 570776 572093 571010 571126 1243 1359 1476 1]592 117 3 1709 1825 1942 2058t 2174 2291 2407 2523 2639 2755 116 4 2872 2988 3104 3220 3336 3452 3568 3684 3800 3915 116 5 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 6 5188 5303 5419 56534 5650 57651 5880 5996 6111 6226/ 115 7 6341 6457 6572 6637 6802 6917 7032 7147 7262 7377 115 8 7492 7607 7722 7836 7951 8066 8181 8295 8410 8525 115 9 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 380 579784 579898 580012 580126 580241 580355 580469 580583 580697 580811 114 1 580925 581039 1153 1267 1381 1495 1608 17221 1836 1950 114 2 2063 2177 2291 24041 2518 2631 2745 285,81 3912 30851 114 3 3199 3312 3426 3539 3652 3765 3879 3922 41105 4218 113 41 4331 4444 4557 46701 4783 48961 5009 5122 5235 5348 113 5 5461 5574 5686 5799 5912 6024 6137 6250 6362 6475 113 6 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 112 7 7711 7823 7935 80471 1601 8272 8384 8496 8608 8720 112 8 8832 8944 9056 9167 9279 9391 9503 9615 9726 9838 112 9 9950 590061 590173 590284590396 590507 590619 590730 590842 590953 112 390 591065 591176 591287 5913991591510 591621 591732 591843 591955592066 111 1 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 111 2 3286 3397 3508 3618 3729 3840 39501 4061 4171 4282 ll 3 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 310 4 5496 5606 5717 58271 5937 6047 6157 6267 6377 6487 110 51 6597 6707 6817 69271 7037 7146 7256 7366 71476[ 7586 110 6 7695 7S05 7914 80241 8134 8243 8353 81G62i 8572 8681 110 7 8791 8900 9009 9119! 9228 9337 9446 95561 9665 9774 109 8 9883 9992 600101 600210 600319 600428 600537 600646 600755 600864 109 91600973 601082 1191 1299 1408 1517 1625 1734 I 1843 1951 109 jrNo.0 1 213 -I 51 6 8 | 9 1Diff. TABLE XII. LOGARITHMIS OF NUMIBERS. 161'No 0 1 1 i 3 1 i 5 6 6 8 9 Diff. 410)9 9(N11 G602 69 G602277 G(926 6'3 962-194 60 2G9 i:3 6J 71 I6J2 29 62)G292S >G()'.' 1 S> I I 31 41 3 533 33361 -169 3577 36_>6 39794 3902, 4010 L - I 181l (9 21 122 46 4:134 4t 112 4 50 4653 4766 4871 498S2 35059 51971 ( 109 3 j5 I) 5 1.3 553211 362S 3736 5>44 5931 6', GI GG6 (3:'71 4 3l'1 15; I 9 6;53961 67013 G311 6919 712G 7133I 72-411 7 -1 1)' i G 71 7- 560 7'669 7777 73,64 79391 09, S79 -04 1 ~ 81)1 1 s 4L1 1(77 6 s,26 ~33 s74') 31 47 S934 9061 9167 92741 9:31 94)-1 1()7 7 9593 9-70 L 9>1)S 9914 61)002L 61()12S 1 10 23-1 4610341 6104417 6 1()5.1 1 1()7 8 10G6(1) 610767 61(173'610979 1(36. 1192 13293i 11105 11L1 lG17 1061 9 172:3 1>29 1936 20-12 2148 2204 2:360 24660 23572 2967 106 410 12734 6129n90 129'96 613102 613207 613313 613419 61352o 61:3633 613731 106 I 3>12 31947 40531 4159 4264 4370 4-475 4531 4656 4792 1)9 21 4,597 5)(,3 3 511) 5213 5319 54241 5529 563-1 5740 5>15 101 31 5930) 6133I 61969 6265 6370 676 651 666! 696 791) 6S39. 14 79() ) 7103- 7210 731.5 7420 752.5 7629 773-1| 7339 7943 (1)3j S 5) )S 815s33 257i s0367 2 8466 S571 SG7G s780 854 8989 151 6! 91'39:3 919 931 3021 1061 9311 96 L[) 9719 9>824 992> 6200932 1() 7162 1136 62024 1620344 6204-1 62)3552 620f:61)6 1 23 760 620)S64 62096> 1072 104 18 1176 12) 13 14 1592 1693 1799 1903 20'17 2110 104 9 2214 23193 2421 2.253 262S 2732 2>33 2939 30-12 3149 104 420 6232419 923333 69234561623.339 9623663 623766 623S69 623973' 621076 1624179 103 1 422'2/ 4133 44>i9 4791 46995 4793 491 1 5004 5107 521) 103 21 5312 5415 r518 56211 5724 58 27 59)9 6032 61351 6233 1(13 3 9:340) 64:3 / 6;G 6694 6751 68953 6956 709.8 7161 7263 103 41 736/ 74631 7571 76731 7775 7878> 7930 S0321 8185 827 102 5 8339 8491 891 593 8695 S797 8900 9002 9104 92061 9303 102 9 61 0 9312 9913 9713 9>171 99t9 1630021 63(123 63022416309326 102l 7 30142 63053016396:311639733 630>33516309:36 10383 1139 1241 1342 102 8/ 11444 1545 16471 17481 1949 1951 23032 21531 2255 23:56 101 9 2337 2559 2660 2761 2>62 2963 3061 3135 3266 3367 101 1430 6334693 633;69 6336701633771 633372 133973 341074 634175 634276 6:34376 101 1 4477 4578 4679 4779 439(1 49:31 50>1 51852| 52>31 53>3 101 21 5484 5984 / 56.55 571 586 599 59>36 6)71 6187 62>71 6358 100 3 6S1.S 6533 69939 6789 6s9>9 69>9 70,S9 7189 72901 7390 100 41 7490 7590 76909 7790 7890 7991 80901 81901 82901 8399 10 / 5 684 91 8391 86999 87391 SS 1 8938 03 913> 927 9387 100 61 91>6 9996 96961 9785 93SS.1 9934161400)31640183 6102S3 640389 99 7 -130481 G405S161 406S)016-1779 6-1U(79 1640(78 1077| 1177 1276 1375 99 8 1474 1573 16721 1771 1871 197() 2069 216s 2267 2366 99 9 2465 25639 2662| 2761 2>60 2959 3058 3156 32351 3354 99 440l613153 61451 6-3650 613719 61347 6 1394 630440l 1 l644 143: 644242 6-1340 9S 1 44:39 4:37 146:36 147:31 4332 4)31 5029 5127 5226 5324 98 2 543 5521 5619 5717 581, 59131 6011 61101 62081 63061 9 3 6404 G65021 6601 6699 6796 6941 6992 7039 7187 7285 93 4 7333 7431 7579 7676 777- 7>72 7991 S0971 SIG5 S262 99 5 s3601 s4513 555 8653 750 S-13 S945 9043i 9140 9237 97 6 93351 94132 9.5301 9627 97241 9S21t 9919165()01 63501131650210 97 7 65)303 653040610. 9.3092 9650;5 99 16509 1651()793 3 165890 0)937 10)84 1181 97 8S 12781 1375 1472 1569 1666 1762 1lS59 19;56 20533 2150 97 9 2219 2313 2110 2336 2933 27-3T1 226 2923 3919 3116 97 1599 693213 953339:6313 90. ) 33332 69 3.59339 3 693i 6;3379 L 6j3339 653939>4 6343>S 0 96 1 4177 427:3 4369 l 4 3 4362 4995S 4734 4S0j 4946 50'12 9 2 3,139 523.1 5:331 t 127 5323 5691 5 7135 5610I 5906 600(2 99 3 6)901 A91:9 61 20 9 G:361 6132 6.5771 6673 6769 6641 69601 96 4 730'3 7152! 72471 731 743> 7.5331 7691 7723 782!0 7916 96 5 5 11 8107 S202 S2931 339 3 818S 8541 S679 57741 SS7o 9; 9 s93-o 9.13i 9 L55 9231 9346 9-111 953 96 31.1 9726 9321 9l 7 9916 6302)1 II6(;)106 66:39201 6602961660391 66o04>;`62660;1 669)6761 660771 95 > 8;6S)99s5> 0360i 10)51 1150! 1243 1339 1434( 1529i 1623 171t 95 l 91 1:$13 19)07 20021 2096 2191 2296 2.3S0 2-17. 2569 2663 9 No. O 2 3 4 5 | 7 8 9 Dil1 14 162 TABLE XII. LOGARITHMIIS OF NUMIBERS. No. 0 I 1 2 3 I L 1 5 11II 460 6;6275S i6628j52 662947 663()41 ijti6:3135 8G3:360663;32466 3418 66o12 i6 0i, 41 3701 3795 33S9 3983 4078 41721 42G66 413GOI 6 40.)i 4i;l 2 4642 4736 4830Q 4924 5018 51121 5206t; 5299 5 3'9I3' }7 U1 3 5531 5675 5769 58 62 5956 6050 61413 6237 63:31 61>-1 i Y 4 6518 6612 6705 67'39 6392 69S63 69 71/3 7266 73601 i 5 745:3 7946 7640 7733 78261 7920 8013 8106 8199 8253 3 3 6 S386 8479 8572 S6665 S759 8852 8945 903S 91.31 ((2-4 9' 7 9317 9410 9503 9596 5699 9782 9875 99767 670060 670153 3 8 670246 670339 67043 1 670524 6706 17 670710 67(1)02 670895 0928 100 93 9 1173 1265 135S 1451 1543 1636 1728 1821 1913 2007 > 470 672098 672190 672283 672375 672467 672560 672652 672744 672S36 672929 52 1 3021 3113 3205 3297 3390 3482 3574 3666 3758 o30 2 2 3942 4034 4126 4218 4310 4402 4494 4586 4677 476'39 3 4861 4953 50-15 5137 5228 532(0 5412 5503 5595 56S7 92 4 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 92 5 6G94 6785 6876 6968 7059 7151 7242 7333 7424 7516 91 6 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 91 7 8518 8609 8700 8791 8882 8973 9064 9155 9246 9337 91 -8 94283 9519 9610 9700 9791 9882 9973 680063 680154 680245 91 9 680336 680-26 680517 680607 68068 698 680789 680879 0970 1060 1151 91 460 661241 681332 6S1422 681513 631603 681693 681784 681874 681964 6S2055 90 1 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 2 3047 3137 3227 3317 3407 3497 35871 3677 3767 3857 90 3 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 4 4845 4935 5025 5114 5204 5294 5383 5473 5563 5652 90 5 5742 58331 5921 6010 6100 6189 6279 63681 6458 6547 89 6 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 S'38 7 7529 7618 7707 7796 786 7975 8064 8153 8242 8331 89 8 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 89 9 9309 9398 9486 9575 9664 9753 9841 9930 690019 690107 89 490 690196 690285 690373 690462 690550 690639 690728 690816 690905 690993 89 1 1031 1170 1258 1347 1435 1524 1612 1700 1789 1877 88 2 1965 2053 2142 2230 2318 2406 2494 2583 2671 2759 88 3 2847 2935 3023 3111 3193 3287 3375 3463 3551 3639 88 4 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 8S 5 4605 4693 4781 4863 4956 5044 5131 5219 5307 5394 88 6 5482 5569 5657 57144 5832 5919 6007 6094 6182 6269 S7 7 6356 6-144 6531 6618 6706 6793 6880 69631 7055 7142 87 8 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 9 8101 818 8275 83621 8449 8535 8622 8709 8796 8883 87 500 69S9701 699057 699144 699231 699317 699404 699491 69957S6699664 699751 87 1 9838 9924 700011 700093 700184 700271 700358 700444'700531 700617 87 2 700704 700790 0377 09631 1050 1136 1222 1309 1395 1482 86 3 1568 1654 1741 1s827 1913 1999 2086 2172 2258. 2344 86 4 2431 2517 2603 26391 2775 2861 2947 3033 3119 3205 S6 5 3291 3377 3463 33491 3635 3721 3807 3893 3979 4065 86 6 4151 4236 4322 4403 4494 4579 4665 4751 4837 4922 86 7 5008 5094 5179 52651 5350 5436 5522 5607 5693 5778 86 8 5364 5949 6135- 6120 6206 6291 6376 6462 6547 6632 651 9 671S 6803 638 6974 7059 7144 7229 7315 7400 7485 85 510 707570 707655 707740 707826 707911 707996 7080811 70S166 70S251 703336 85 1 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 8 5 2 9270 9355 9440 95241 9609 9694 9779 9863 9948 710033 85 3 710117 710202 710287 710371 710456 710540 7106251710710 710794 0879 85i 4 0963 1048 1132 1217j 1301 1385 1470 1554 1639 1723 64 5 1807 1892 1976 2060 2L44 2229 2313 2397 24S1 2566 84 6 2650 2734 2318 2902 2986 3070 3154 3238 3323 3407 84 7 3491 3575 36.9 3742 3326 3910 3994 4078 4162 4246 84 8 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 84 9 5167 5251 5:33-5 5418 5302 5586 5669 5753 5336 5920 841 INo 0 | 1 2 3: 5 6 1 7 8| 9 lDiff.J TABLE XII. LOGARITHSIS OF NUIIBERS. 163 No. O _ 2 1 3 si ~ 5 6 9 Diff. 520 716003 716037 71617()0 716254i7163i 73 16421 71G0)l1 716.3),; 716,671 7167a4 83 1 68328 6921 70041 7038i 7171 7254 7333 7421. 750)4 7587 83 2 7671 7754 7837 7923i 8003 S036 8169 S825:3 833:3 8419 83 3 8502 8585 86681 8751 8S34 89 L7 9000 9083 9165 9248 83 4 9331 9414 94971 950 9663 9745 9823 99911 720077 83 5 720159 720242 720325 720407 720490 720573 72065.5 720:73 7209821 0773 83 6 098(6 1068 1151 1233 1316 1398 1481 1563 1646 1'728 82 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2)552 82 8 2634 2716 2793 2881 2963 3045 3127 3209 3291 3:374 82 9 3456 353S 3623 3702 3784 3866 3948 4030 4112 4194 82 530 724276 724358 724440 724522 724604 724685 724767 724349 724931 725013 82 1 5095 5176 5258 5340 5422 55.03 5585 5667 5748 5830 82 2 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 82 3 6727 6809 6890 6972 7053 71341 7216 7297 7379 7460 81 4 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 81 5 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 6 916.5 9246 9327 9408 9489 9570 9651 9732 9813 9893 81 7 9974 730055 730136 730217 98 1730298730378 730459 730540 730621 730702 81 8 730782 0863 0944 1024 1105 I'1186 1266 1347 1428 1508 81 9 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 540 732394 732474 732555 732635 732715 732796 732376 732956 733037 733117 80 1 3197 3278 335S 3438 3518 3598 3679 3759 3839 3919 80 2 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 3 4800 4830 4960 5010 5120 5200 5279 5359 5439 5519 80 4 5599 5679 5759 5838 5918 5998 6078 6157 6237 6317 80 5 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 6 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 79 7 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 79 8 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 9 9572 9651 9731 9810 9889 9968 740047 740126 7402057 74084 79 550 740363 740442 740521 740600 740678 740757 740336 740915 740994 741073 79 1 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 79 2 1939 2018 2096 2175 2254 2332 2411 2489 2.568 2647 79 3 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 78 4 3510 358S 3667 3745 3823 3902 3930 4058 4136 4215 78 5 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 6 5075 5153 5231 5309 5387 5465[ 5543 5621 5699 5777 78 7 5855 5933 6011 6039 6167 6245 6323 6401 6479 6556 78 8 6634 6712 6790 6868 6945 7()23 7101 7179 7256 7334 78 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 748188 748266 7483 8421 748498 748576 748653 748731 748808 748885 77 1 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 2 9736 9814 9891 9963 750045 750123 750200 750277 750354 750431 77 3 750508 750586 750663 750740 0817 0894 0971 1048 1125 1202 77 4 1279 1356 1433 15101 1587 1664 1741 1818 1895 1972 77 5 2048 2125 2202 2279 2356 2433 2509 2586 26631 2740 77 6 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 77 7 3583 3660 3736 3813 3889 3966 4042 4119 4195 4272 77 8 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 76 9 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 570 755875 755951 756027 756103 756180 756256 756332 756408 756484 756560 76 1 6636 6712 6788 6864 6940 7016 7092 7163 7244 7320 76 2 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 76 3 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 76 4 8912 8988 9063 9139 9214 9290 9366 9441 9517 9592 76 5 9663 9743 9819 9894 9970 760045 760121 760196 760272 760347 75 6 7604'22 760493 760573 760649 760724 0799 0875 0950 1025 1101 75 7 1176 1251 1326 1402 1477 1552 1627.1702 1778 1853 75 8 1923 2003 2078 2153 2229 2303 2378 2453 2529 2604 75 9 2679 2754 2329 29041 2978 3053 312s 3203 3278 3353 75 No. O I1 2 3 3 4 5 6 T 9 Diff. 164 TABLE X11. LOGARITHMS OF NUBLBERS, I No.,l O i 1 2. 3 i'L I 5 6 G i 81 i D iii 58(0~it763F23;63,03: 73:57 7,31763T 727 76/02 763-77 76 89527640271 6 C74101 75I I 1 4t176 4251 43 U -6 4400 -1 175 4501 4624 46991 4774 4Q48 751 2 4923 4998 5072 5147 5221 5296 5370 544- 5$20 5 92 - 75I 3 5669 574-3 513 5I8 9 5892 66 6041 6115 61901 6`(64 6:33 4 6413 6487 6562 66364 6710 6785 6>5 J9 6933) 7(!07 7(0S2 74 5 7156 72:30 730 4 737i9 74533 7527 7 601 767. 7749 7823 74! 6 798 7972 S8046 8120 81941 26 3-12 1 41 6 8490 8 6 4 7 4 7 8638 8712 s8786 8860 8934 9o00 90)o 2 1 -156 92:3() 9'03 74 8 9377 9451 9525 9 99 96973 9746 9820 I989 t 9 4 6 770(142 7,4 9 770115 770189 770263 770336 77/0410 770484 770557 770631 7707i) 0778 74 590 770852 770926 770099 771073 771146 771220 771293 771367 771440 771514 741 1 1587 1661 1734 1908 181 1955 2028 21(2 2175 2248 73 2 2322 2395 246S 2542 2615 2688 2762 2835 2908 2981 73 3 3055 312s 3201 3274 3348 3421 3494 36 71 3640 3713 73 4 3786 3260 3933 4006 40i79 4152 4225{ 42981 4371 4444 73 5 4517 4590 4663 47361 42091 4282/ 4955 5028 5100 5173 73 6 5246 5319 5392 54651 5538 5610 5683 57561 5829 5902 73 7 5974 6047 6120 6t93 6265 6338 6411i 6483 6556 6629 73' 8 6701 6774 6846 69191 6992 7064 7137 7209 7282 7354 73 9 7427 7499 7572 7644[ 7717 7789 7862 7934 8006/ 8079 72 600 778151 778224 778296'778368! 778441 778513 778585 778658 778730 778802 72 1 8874 8947 9(19 9091 9163 9236 9308S 9380 9452 9524 72 2 9596 9669 9741 98131 98855 9957 780029 780101 780173 780245 72 3 780317 780389 780461 170533i780605 780677 0749 021 0893 0965 72 4 1037 1109 11811 1253 1324 1396 1468 1540 1612 1684 72 5 1755 1827 18991 1971 2042 2114 2186 2258 2329 2401 72 6 2473 2544 26161 26S81 2759 2831 2902 2974 3046 3117 72 7 3189 3260 33321 3403 3,t75 3546 3618 3689 3761 3832 71 8 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 9 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 71 6101785330 785401 785472 785543 785615 7856S6 785757 78528 785899 785970 71 1| 6041| 6112 6183 6254| 632.5 6396 6467 653S8 6609 6680i 7 1 2 6751 6822 68931 6964 7035 7106 7177 7248 7319 7390 71| 31 7460 7531 7602 76731 7744 7815 78851 7956 8027 8098 7111 4 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 71 5 8875 8946 90161 90871 9157 9228 9299 93691 9440 9510 71 6 9581 9651 9722 9792, 9863 9933 790004 790074 790144 790215 70 7790285 790356 790426 790496 790567 790637 0707 07781 0848 0918 70 8 0988 1059 1129 1199i 1269 1340 1410 14801 1550 1620 70 9 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 792392 792462 792532 792602 7926721 792742792812 792821 792952 793022 70 1 30921 3162 3231 3301. 3371 3441 3511 3581 36651 3721 70 2 37901 3860 3930 40(10 4070 4139 4209[ 4279 4349 4418 70 3 4488 45532 4627 4697 4767 4836 4906 4976 5045 5115 70 4 5185i 5254 5324 5393 5463 5532 56(02 5672 5741 5811 70 5 5801 5949 6019 61)88 6158 6227 62971 63661 64361 6505 69 6 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 69 7 7268 7337 7406 7475 7545 7614 7683 7752 7821 7890 691 8 7960 8029 8098 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 79961 6 799685 799754 799823 799S92 799961 69 1 800029 800098 8001671800236 800)305 8003731 800442 800511 800580800648 69 1 2 0717 0786 08541 0923 0992 1061 1129 1198 1266 1335 691 31 14041 1472 1541 1609 1678 1747 1815 1884 1952 2021 69 4 20891 2158 2226 2295 2363 2432 2500 2563S 2637 2705 G8 5 2774 2342 2910 2979 30-71 3116 31841 32521 3321 3389 CS 6 3457 352;5 3594 3662 37301 3798 3867 3935 4003 4071 68 7 4139 4208 4276 4344 4412 4420 4548 4616 4685 4753 68 8 4821 4889 41957 5025 5093 5161 5229 5297 5365 5433 68 91 5501| 5.69 5696371 57(15 5773} 5s41 59081 5976 60441 6112 68 No. 0 2 3 5 6 7 9 Dif _ TABLE XII. LOGARITHMS OF NUBIBERS. 165 No. No. 0 1 1 3 I 0L 5 6 7 8 9 Diff. 64i0 8061>80 S0 32 6042 18 20416 3 i 0631 6065.19 3U65887 806653 806723 S0679(J 6>3 1 6581 69 6 69941 7061 7129 7197 7264 7332 7400 7167 G8 2 753 5l 7603 7670 77:33 7806 787:3 794 8003 8076 8143 6S 3 821 1! 8279 83J 16 8414 8411 8549 86 1 S684 8751 8818 6'7 4 891'6 8953 91)21 9038 91.56 9223 9290 9358 9425 9492 67 5 9560 9627 9691 9 91 9896 9964 S10031 81()098 S10165 67 6 S10233 110300 810367 S104341 10501 10569 810636 07>03 0770( 0337 67 7 0904 0971 1039 1106 1173 1240 13()07 1374 1441 1.081 67 8 1575 16142 1709 1776 1843 1910 1977 20414 2111 2178 67 9 2245 2312 2379 214i.) 2512 2579 2646 2713 2780 2A47 67 650 812913 8129S0 013047 813114 313181 213247 813314 8133S1 813448 >1314 67 1 35SI 3668 3714 3781 3342 3914 3931 4048 4114 418[ 6'7 2 4248 4314 4331 4447 4514 - 4581 4647 4714 4780 4847 67 3 4913 4930 5046 6113 5179 6246 5312 5378 5445 6511 66 4 5578 6644 6711 6777 5>43 5910 5976 6042 6109 6175 66 5 6241 6309 6374 6440 6506 6573 6639 6705 6771 6>38 66 6 6904 6970 7036 7102 7169/ 7235 7301 7367 7433 7499 66 7 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 66 8 8226 23 83 8 8-124 8490 8556 8622 86S8 8754 8820 66 9 8885 8951 9017 9033 9149 9215 9231 9346 9412 9478 66 660 819644 819610 S19676 819741 819SO07 819873 S19939 820004 820070 20136 66 1 820201 820267 20333820399 820464 8205301 820595 0661 0727 0792 66 2 0358 0924 0939 1055 1120 1186 1231 1317 1382 1448 66 3 1514 1579 16453 1710 1775 1841 1906 1972 2037 2103 65 4 216[S 2233 2299 2364 2430 2495 2560 2626 2691 2756 65 5 23221 237 29521 3018 3083 3148 3213 3279 3344 3409 63 6 34'74 3539 3605 3670 3735 3800[ 3865 3930 3996 4061 65 71 4126 4191 4256 4321 4396 4451 4516 4531 4646 4711 65 8 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 651 9 5426 5491 5556 56621 56S6 5751 5815 5880 5945 6010 651 670 826075 82614() 826204 826269 826334 826399 S26464 326528 826593 S2665S 65 1 6723 6787 6352 6917 69S1 70146 7111 7175 7240 7305 65 2 7369 7434 7499 7563 762S 7692 77-7 7221 7886 7901 65 3 80l15 80SO 8144 8209 8273 03323 8002 8467 s531 8595 64 4 8660 8724 8729 8853 8918 8952 9016 9111 9175 9239 64 5 9304 9363 9132 9497 9561 9625 96901 9754 9. 18 9882 64 6 9947 830)1 1 130075 830139 830204 830261 8303:32 830396 830460 S30525 64 71830589{ 0653 0717 07231 0845 0909 0973 1037 1102 1166 64 8 1230 1291 1358 1422 1486 1550 1614 1678 1742 1806 61 9 1870 1934 1992 2062 2126 2189 2253 2317 2381 2445 64 63018325091 32573 S32637 832700 8327641 32S26 832292 232956 833020 833n03 64 1 3147 32LI1 3275 33323 3402 3466 3530 3593 3657 3721 64L 2 3784 3848 39121 3975 40)39 4103 4166 4230 4294 4357 64 3 4421 44S84 4548 4611 4675 4739 4S02 4866 4929 4993 64 4 50356 5120 51831 52471 5310 5373[ 5437 5500 5564 5627 63 5 5691 5754 5171 5831 5944 6007 6071 6134 6197 6261 63 61 6324 6:337 6151 6514 6577 6611 6704 6767[ 6S30 6394 63 7 6957 7(1)2) 70:33 7146 7210 7273 7336 7399 7462 7525 6:3 8 75>3 7632 7715 777> 7841 7904 7967 8030 8093 8156 63 9 8219 8 2.2l 83451 8403 8471 8534 8597 8660 8723 8786 63 690 833249 3>5912 833.75. 8393> 23:39101 [39 164839227 839289 839352 839415 63 1i 9417 9-741 96n04 9667 9729 9792 92355 9918[ 9911 840043 63 2140106 240161) 31402321 402941 40357 8404120 40482 S40545 184060S 0671 631 3 0733 (17116 0359 0921 0934 10(46 1109 1172 1234 1297 63 4 13359 1422 1485 1547 1610 167-2 1735 1797 1860 1922 63 5 1935 2)017 2110 2172 22335 2297 2360 2422 2484 2547 62 6 2609 2672 27341 2796 2859 292[ 29>3 3046 310> 3170 62 7 3233 3295 33571 3420 3482 3.544 3606 3669 3731 3793 62 8 3255 3918 39s01 4042 4104 4166 4229 4291 4353 4415 62 9 4477 45 39 46411 46641 4726 47833 4850 4912 4974 5036 62 No. 1 2 3 1 4 5 6 7 8 9 1iff. 166 TABLE XIIo LOGARITHMS OF NUMBERS. No. 0 2 3 1 4 5 7 S Diff. 700 845093 845160 845222452 84 84845346 843540845470 845532 S45594 845656 62 1 5718 5780 5842 5904 5966 6028 6090 61511 62131 6275 62 2 6337 6399 6461 6523 6585 6646 6708 6770 68532 6394 621 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 621 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 621 5 8189 8251 8312 8374 8435 8497 8559 8620 68S2 8743 62' 6 8805 8866 892S 8989 9051 9112 9174 9235 9297 9358 61 7 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 8 850033 850095 850156 850217 850279 S50340 850401 850462 850524 S50585 61 9 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 61 710 851258 851320 851381 851442 851503 851564 851625 851686 851747 851809 61 1 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 61 2 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 31 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 61 4 3698 3759 3S20 3881 3941 4002 4063 4124 4185 4245 61 5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 6 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 7 5519 55O 5640 5701 5761 5822 5882 5943 6003 6064 61 8 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 601 720 857332 857393 857453 857513 857574 857634 857694 857755 857815 8 75 60 1 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 2 8537 8597 8657 871 8778 8838 8S98 8958 9018 9078 60 3 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 60 4 9739 9799 9859 9918 9978 860038 860098 860158 8621181 860278 60 5 86033S 860398 860458 860518 860578 0637 0697 0757 0817 0877 601 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60, 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 601 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 863323 863382 63442 863501 863561 863620 863680 863739 863799 863858 59 1 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 o9 2 451.1 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 3 r5104 5163 |o5222 52S2 3341 5400 64559 5519Y 5578 5637 59 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 59 5 6287 6316 6405 6465 6524 6583 6642 6701 6760 6819 59 6 6878 6937 6996 7055 7114 7173 7232 7291 77350] 7409 59 7 7467 7526 7585 7644 7703 7762 7821 7880) 7939 7998 591 8 8056 8115 8174 8233 8292 83350 8409 8468 8527 8586 59 9 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 869232 869290 869349 869408 869466 869525 86G9584 869642 869701 869760 59 1 9818 98771 9935 99941870053 870111 870170 870228 870287 870345 599 2 870404 870462 8705211870579 0638 0696 0755 0813 08,72 0930 581 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 4 1573 1631 1690 1748 1806 1865 1923 1981 204(1 2098 58 5 21 56 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 6 2739 27971 2855 2913 2972 3030 3088 3146 3204 3262 58 7 3321 33791 3437 3495 3553 3611 3669 3727 3785 3S44 58 8 3902 39601 4018 4076 4134 4192 4250 4308 4366 4424 5S 9 44S2 45401 4598 4656 4714 4772 4830 4888 4945 5003 581 750 875061 875119 875177 875235 875293 875351 875409 875466 1875524 8755582 58 1 5640 56983 5756 5813| 5871 5929 5987 6045 6102 6160 581 2 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 581 3 6795 6853i 6910 6968/ 7026 7083 7141 7199 7256 7314 58 4 7371 74291 7487 7544 7602 7659 7717 7774 7832 78899 581 5 7947 80041 8062 8119 8177 8234 8292 8349 8407 8464 57 6 8522| 8579 8637 8694[ 8752 8809 8866 8924 8981I 9039 67 7 9096 9153 9211 92681 9325 9383] 9440 9497 9555 9612 57 8 9669 9726 97841 9841 9898 9956 880013 880070 1880127 8801851 57 9 880242 880299 88035618S0413880471 18805281 0585 0642 0699 0756 57 No.l0 1 1 2 13 1 4L- 5 16 | 8 9 Diff. TABLE XII. LOGARITHMS OF NUMIBERS. 167 No. 0 1 2 3 d 5 6 I 8 I 9 1Diff. 760 88014 180871 s30928 88093385 10 10 L09o99 136i 168'1213 61271t1;1 132 57 1 1385 1442 1499 1556 1613 1670 1727 1784 1411 1898 57 2 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 57 3 2525 2581 2638 26935 2752 2. 099 2&66 2923 2980 3037 57 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 57 5 3661 3718 3775 3832 3383 3945 40021 4059 l1115 41L72 57 6 4229 4235 4:342 4399 4155 4512 4569 4 625 4662 47339 57 7 4795 4852 49U9 4963 5022 5078 513, 51 92 5246 5305 57 8 5361 5418 53474 531 55387 5614 5700.5737 5313 5870 57 9 5926 5933 6039 6096 6152 6209 6265 6321 6S6 G6131 56 770 856491 886547 886604 886660 686716 886773 886329, KS 6835 86942 666998 56 1 7054 7111 7167 7223 7230 7336 7392 7449 750. 75) 61 56 2 7617 7674 7730 77861 7842,7893 7955 8011 8067 8123 56 3 8179 8236 8292 8348 8404 8460 8516 857;3 8629 S685 56 4 8741 8797 8853 8909 8965 9021 9077 913-1 9190 9246 56 5 9302 9358 9114 9470 9526 9582 9633 9694 97350 9806 56 6 9362 9918 9974 890030 890036 890141 890197 890253 1890309 890365 56 7 890421 190477 890533 0.539 0645 0700) 0756 0312 0366 0924 56 8 0930 10353 1091 1147 1203 1209 1:314 13704 1126 1482 56 9 1537 1593 1649 1705 1760 1816 1872 19 2>, 1 93 2039 56 780 892095 892150 892206 8692262 S92317 92373 892129 692164 892540 39259. 56 1 265 1 2707 2762 2618S 2873 2929 2935 30(10 30936 3151 56 2 3207 32621 3:318 337:3 3429 3 16-1 33510 3593) 36'531 3706 56 3 3762 3317 3373 3928 3964 4039 40i394 4150) 4 205 41261 55 41 4316 4:371 4427 4482 4-533 4593 46(18 4704 4759 4814 55 5 4670 49253 4980) 5036 5091 5146 5201 5257 5312 5367 55 6. 5423 5478 5533 5586 56-14 5699'3 5754 5S09 5S641 59320 55 7 5975 6030 6085 6140 6195. 6251 6306 6361 6416 6471 55 8 6526 6531 6636 6692 6747 6302| 6357 6912 6967 7022 55 9 7077 7132 718 7242 7297 7321 7407 7462 73;17 7572 55 790 897627 8976S2 S97737 897792 897847 197902 897957 869012 698067 893122 55 1 8176 8231 8286 8:341 8396 841.1 8506 8561 8615 8670 55 21 87251 8780 8s35 8890 8944 8999 9054 9109 9164 9218 55 31 9273 9328 9333 9437 94921 9537 9602/ 9656 9711 9766 55 4 921 975 9930 9985 900039 9000911 900149 900203 900258 900312 55 5 900367 900422 900476 900531 0586 0640 0695 0749 080-1 0859 55 61 0913 0963 1022 1077 1131 1131 1186 1240 1295, 1349 1404 55 71 1458 151:3 1567 1622 1676 1731 1785 18401 189-1 194S 54 8 2003 2057 2112 2166 2221 2275 2329 2364 2438 2492 54 9 2547 2601 26.33 2710 2764 2:1S 2S73 2927 2931 3036 54 S001903090 903144 903199 903253 90323 307 903361 903416903470 903524 90357 51 1 3633k 3687 3741l 3793 3349 39041 39358 401 4066 4120 54 21 41741 4229 4233 4337 4391 4445 41991 45531 4607 4661 51 3 4716 4770 4321 4S7,> 4932 4936 5040 5094 5148 5202 54 4 52,6 5310( 5364 5418 5472 5526 5560 5634 56831 5742 54 5! 5796 58301 5904 5958 6012 6066 6119 6173 6227 6281 54 6 63:35 66339 6443 6497 6.551 6604[ 66536 6712| 6766 6320 54 7 6374 6927 69831 7035 7039 7143 71961 72501 7304 7358 54 8 7411 746.3 7519 7573 7626 76>01 7734 7787 7841 7895 54 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 54 110 90485 1903539 9035921908646 903699 908753 908307 903860 908914 908967 54 1 9031 9074 9128 9181 9235 92S89 9342 9396 9449 9503 54 2 9.56 9610 9663 9716 9770 9823 9877 9930 9984 910037 53 3 910091 910144 910197 910251 9103011910353 910411 910464 910518 0571 53 41 0624 0678 0731 0784 0(33S 0391 0944/ 0991 1051 110-1 53 35 1158 1211 12641 1317 1371 1-424 1477 1530 1584 1637 53 6 16 90 1743 1797 18350 1903 1956 2009 2063 21161 2169 53 7 2222 2275 2328 2331 21.3 248 254 941 2594 2647 2700 53 8 2753 2306 2;359 2913 2966 3019 3072 3125 3178 3231 53 91 3284 33371 3390! 3443 3496 35491 3602 36;551 3708 3761 53 No. IL 2 3 4 | 5 G T 8 I 9 Diff. 168 TABLE XII. LOGARITHMNS OF NUMBERS. No. 1 2 3 1 5 8 I 9 Diff. 890 913314 913>67 913920 913973 9140)26 9'14079 914132 914 84 9 14237 9142'30 53 1 4343 4396 4449 4502 4355 460> 4660 4713 4766 4819 53 2 4872 4925 4977 5030 5083 5136 5IS9 5241 924 5347 531 3 5400 5453 5505 553 8 5611 65t64 5716 6 >22 5.7) 7; 3 4 5927 5980 6033 60`5 6138 6191 6243 6 3J76 64 9 101;() I 3 5 6454 6507 6559 661 2 6664 6717 67 70( 6422 6' 5 6t27 73 6 6980 7033 7085 713> 7190 7243 72935 7341 740 -15. 53 7 7506 755S 7611 766:3 7716 7768 7597 3 t73 7925 3 97 52 8 81)30 8083 8135 S188 S2410 S293 83153 8397 S 4;3( 0 8502 52 9 8355 8607 8659 8712 S764 8816 8-69 9921 S973 9026 52 830 919078 919130 919183 919235 919287 919340 916392 919-144 919496 919549 52 1 9601 9633 9706 975S 9810 9>6;2 9914 1196)7 92001) 92(0071 52 2 920123 920176 920228 9202S0 920332 9203S4 920436 9209 9 07-41 0593 52 3 0645 0697 0749 0801 0S53 0916 09>50 1010 11162 1114 52 4 1166 1218 1270 1322 1374 1426 1147S 1530 15 1634 52 5 16>6 173> 1790 1842 1S94 1946 1998 2100 21 021 2154 52 6 2206 22.53 2310 2362 2414 2466 231> 257(1 2622 2674 52 7 2725 2777 2829 281l 2933 2985 3037 307>9 3140 3192 52 8 3244 3296 3348 3399 3151 5303 35553 3607 5658| 3710 52 9 3762 3314 3865 3917 3969 4021 40721 424 4176 422> 52 840 924279 924331 192438.3 924434 924486 924533 92459 92464 11 924693 92474 52 1 4796 4848 4899 4951 5003 5054 5106 5157 5209 5261 52 2 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 52 3 5828 5379 5931 5982 6034 6085 6137 6JSS 624 ) 6291 51 4 6342 6394 6445 6497 6548 6600 6651 67021 6754 68(5 51 5 6857 6903 6959 7011 7062 7114 7165 72161 796> 7319 51 6 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 51 7 7883 7933 79S6 8037 808 S814) 8191 8s242 893 834 5 51 s 8396 8447/ 849,8 8549 8601 8652 s8703 S754 0 SJ5 57 511 9 890> 8959 9010l 9061 9112 9163 9215 9266 9317 9368 51 850 929419 929470 929521 929572 929623 929674 929725 929827 929879 51 1 9930 9981 9300)32 9300 3 930134 930185 930236 930287 930331 930)39 51/ 2 930440 930491 0542 0592[ 0643 069-1 07451 0796 0847 0898 51 3 09-19 1000 1051 1102 1153 1204 124 1305 13561 14(17 51 1 4 1458 1509 1560 1610 1661 1712 1763 18141 1865 191. 51 5 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 51 6 2474 2524 25751 2626 2677 2727 2778 2829 2879 2930 51 7 29>1 3031 3082 3133 3183 3234 3285 5335 3386 3437 51 i 8 3487 3538 3539 3639 3690 37441 3791 3841 3692 3943 51 9 3993 4044 4094 4113 4195 4246 4296 4347 4397 44-18 51 S 66 93149s 934549 931599 934650 934700 1934751 93401 934652 934902 931953 50 1 5003 504 51)104 5151 5205 5255 5306 5376 5406 5457 50 2 55071 5.55! 560S 5658 5709 5759 5809 5>60 5910 5960 50 |3 60)11 6061| 6111 61621 6212 6262 6313 6363 6113 6463 50 4 6314 6564| 66144 6665 6715| 6765J 6815 663 6 916 61.6) 5(1 7016 7066/ 7117 7167/1 7217| 7267 7317| 7367 74-lS 7461G 5(),1 6 7518 7563 76 761S 7661 771>8 77'69 7819 7869 1 7919 7S691 501 7 8019 8069 s119 81G9 8219 s269{ 8:3282 837o) 942)o 947)o 51) S8 2) S8570) (6201 8670 87s207 8770 89820 8870' 10o S 970) 5So 9 9020 9070 9120 9170 9220 9270 9320 9369 719 9469 50) 9S'0 9393119939569 939619 939669 9397192 935 69939819 939869399198 91691 50 1 i)9400189410063 94011 iS94016l 94021 940()26 940)3171940367 94704171 90467 50'i 2 0316 7)0566 0616 0666 0716| 0'76(5 0Sl 5 0 -65 ()9135 G64 50i 3 1014 1061 111-1 1163 1213 126:3 313 132(; 1412 1462 50 4 111 561t 161 1660 1710 17111) IS)9 1S5I9 1979 19 ) 5 2)> 20831 2107) 21357 227)7 22;36 23(61 235 24(1)5 2455 5)5 7 30)0O 3049i 3099 3t14>5 319> 3217 3297 33446 3396 3145 49'1 8 3495 3544 33593 3643 3692 3742 3791 3S41 3,9() 3939 493 9 39>9 4038 4088 4137 4>16 4236 42>5 4:.33 43S4 4433 49 No.0 1 2 3 1 4 6 | I1 8 9 Diff-.~~~~~~ _' 8, 11. TABLE XII. LOGARITI-MIIS OF NUI13BE S. 1 69 No. 0 1 w 3 1 5 s I9 Di 880 944483 4;)41531 9411 94 4631 9446-160 )!94-729 944177-9 944323 94-i77 t44971) 4' 1 4976 502)- 5074/ 5124 517:3 5222 5272 5321 5:370 5419 19 2.:1:69 551t- 5567 5616 5665 5715 5764 5813 5362 59129 49 3 5961 691( 6059 t; 108 6157 6207 6256 6305 6354 6(403 49 4 (6452 653 655 1 6600 f,649 6698 6747 6796 6845 6894 49 5 6943 6992 7041 7090 7140 71S9 7233 7287 7336 7385 49 6 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 7 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 83 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 49 9 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 890 949390 31~432 949488 949536 949585 949634 919683 949731 949780 949829 49 1 9878 9926 9975 950024 950073950121 950170 950219 950267 950316 4 9' 2 950365 950414 950462 0511 0560 0608 06571 0706 0754 080:3 49 3 0351 0900 0949 0997 1046f 1095 114' 1t192 1240) 1289 49 1 4 133S 1336 1435 1483 1532 1550 1629 1677 1726 1775 49 5 1823 1872 1920 1969 2017 2066 M11I -! 2163 2211 2260 48 6 2303 23;56 20)5 2453 2502 2550 2599 2647 2696 2744 48S 7 2792 2841 2389 293S 2936 3034 30831 3131 31S0 322S 48 8 3276 3325 3373 3421 34701 351t8 3566 3615 3663 3711 48 9 3760 3303 3856 3905 3953 4001t 4049 4098 4146 4194 48 900 954243 954291 954339 954337 954435 954484 954532 954580 954628 954677 48 1 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 48 2 5207 5255 5303 5351 5399 5447 5495 5.543 5592 5640 43'3 568S 5736 5784 5832 5380 5923 5976 6024 6072 6120 48 4 616t 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 5 6619 6697 6745 6793 6840 6SS88 6936 6984 7032 7080 43 6 7123 7176 7224 7272 7320 7363 7416 7464 7512 7559 43 7 7607 7635 7703 7751 7799 7847 7894 79412 7990 80:38 48 8 80386 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 9 8564 8612 8659 8707 8755 8303/ 8850 8898 8946 8994 48 910 959041 959089 959137 959185 959232 959230 959328 959375 959423 959471 48 1 9518 9566 9614 9661 9709 9757 98041 9852 9900 9947 48 2 9995960042 96000 96013 960185 960233 960280 96032963768 960376 960423 43 3 960471 0518 0566 0613 0661 0709 0756 0804 0351 0899 48 4 0946 0994 1041 1091 1136 1184 1231 1279 1326 1374 48 5 1421 1469 1516 1563 1611 1658| 1706 1753 1801 1843 47 6 1895 1943 1990 2033 2085 2132 2180 2227 2275 2322 47 7 2369 2417 2461 2511 2559 2606 26.53 2701 2748 2795 47 8 2843 2890 2937 2985 3032 30791 3126 3174 3221 3268 47 9 3316 3363 34101 3457 3504 3552 3599. 3646 3693 3741 47 920 9637881963835 963882 963929 963977 964024 961071 964118 964165 964212 47 1 4260( 4307 4354 4401 4448 4495 45421 4590 4637 4684 47 2 4731 4778 4825 4372} 4919 4966 50131 5061 5108 5155 47 3 5202 5249 5296 5343 5390 54371 5484 5531 5578 5625 47 4 5672 5719 5766 5813 5360 5907 5954 6001 6048S 6095 47 5 6142 6189 6236 6283 6329 6376 64231 6470 6517 6564 47 61 6611 6658 67035 6752 6799 6345 6892 6939 6936 7033 47 7 7080 7127 7173 7220 7267 7314 7361 7403 7454 7501 47 8 7548 7595 7642 7638 7735 7782 7829 7875 7922 7969 47 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 47 930 968483 968530 968576 968623 968670 9637161968763 968810 968856 968903 47 1 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 47 2 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 47 3 98382 9928 9975 970021 970068 970114 970161 970207 970254 970300 47 4 9703471970393 970440 04861 0533 0579 0626 0672 0719 0765 46 51 0812 0858 0904 09511 0997 1044 10901 1137 1183 1229 46 61 1]276 1322| 1369 14153 1461 1508 1554 1601 1647 1693 46 71 1740 1786 1S32 1879 1925 1971 2018 2064 2110 2157 46 8 220)3 2249 2295 2342 2338 2434 2431 2527 2573 2619 46 91 2666 2712 2758 2o0-41 2351 2s97 2943 2939 3035 3032 46 No.l 1 j 3 a 5 C 7 8 9 Diff. _~~~~~~-is. 170 TABLE XII. LOGARITHMIS OF UIMIBEUIS. No.0To, 11 2 3 _ _ 8' sDiit. 941 973128 197317'4 97320'977 66 9713 973:3917 O40O 973.,1'. 7 97 9o43'6 ] 3590 3636 3632 3728 3774 32(0 3866,;213' 3754:9 - 4(;25 46[ 4051 4097 4143 4189 4235 4281 4327 43741 442()0 4-66 4 6 7 4512 4558 4604 4650 4696 4742 4788 4S341 45s0 4926 46 4 4972 5018 5064 5110 5156 5262 5248 52941 534(t0,361 46 5432 5478 5524 5570 561t 6 5662 5707 5733 73 45 46 6 5891 5937 5983 6029 6075 6121 6167 6212 6253S 6304 4 6 7 6350 6396 6442 64886 6533 6579 6625 6671 6717 67663 46 8 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 950 977724 977769 977815 977861 977906 977952 97799s 978043 978089 978135 46 1 8181 8226 8272 8317 8363 8409 84154 8500 8546 8591 46 2 8637 8683 8728 8774 8819 8865 8911 8936 9002 90047 46 3 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 46 4 9548 9594 9639 9685 9730 9776 9821 9S67 9912 9958 46 51980003 980049 980094 980140 9501S 5 980231 980276 980322 980367 980412 45 6 0453 0503 0549 0594 0640 0685 0730 0776 0821 0O67 43 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 8 1366 1411 14.56 1501 1547 1592 1637 16s3 172s 1773 45 9 1819 1SG64 1909 1954 2000 2045 2090 2135 21S1 2226 45 960 982271 982316 982362 982407 982452 9S2497 9S2543 9S2558 982633 982671 45 1 2723 2769 2814 2.509 2904 2949 29941 3010 301 3130 45 2 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 45 31 3o626 3671 3716 3762 3S07 3852 3897 3942 397[ 41032 245 4 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45,5 4527 4572 4617 4662 4707 4752 4797 4842 4s7/ 11932 4r, G 4977 5022 5067 5112 5157 5202 5247 52)92 5337 5332 45 5426] 5471 5516 5561 5606 561lI 5696 5741 57861,830 45 S 5875 5920 5965 6010 60505 6100| 6144 6189 6234 6279 45 9 6324 6369 6413 6458 6503 6548 6593 6637 6G82 6727 45 970 936772 936817 986861 986906 9S6951 986996 987040 9870 5 987130 9S7175 45 1 7219 7264 7309 7353 73983 7443 7438 753'2 7577 7622 45 2 7666 7711 7756 7800 78-15 7890 7934 7979 8024 8068 45 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 8$514 45 4 8559 8604 8648 8693 s7373 S782[ 8826 8871 8916 8~960 45 5 9005 9049 9094 9133 9183 9227/ 9272 9316 9361 9405 45 6 9450 9494 9539 9583 962S3 9672 9717 9761 9806 9850 44 7 9895 9939 9983 990029 9900721990117 990161 990206 990250 990294 44 8 990339 9903831990428 0472 0516 0561 0605 0650 0694 0738 44 9 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 44 980 9912261991270 9913151991359 9914031991418 991492 991536 9915801991625 44 1 1669 1713 1758 1802 1846 1890 1935 1979 20231 2067 44 2 2111 2156 2200 2244 22881 23331 23771 2421. 24651 2509 44 3 25541 2598 2642 2636 2730 2774 2319 2863 2907 2951 44 4 29951 30391 3083 3127 31721 3216 3260 3304 33481 3392 44 5 3436 3480 3524 3565 3613 36571 3701 3745 3789 3833 44 6 3877 3921 3965} 4009 4053 4097 4141 4185 4229 4273 44 7 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 44 8 4757 4S01 4845 4889 4933 4977 5021 5065 5108 51,52 441 9 5196 5240 5284 5328 5372 5416 5460 5504 5547 5591 44 990 9956351995679 995723 995767 995811 995854 995S98 1995942 995986 996030 44 1 6074 6117j 6161 6205 6249 6293 6337 6380 6424 6468 44 2 6512 65551 6599 6643 6637 6731 6774 6818 6862 6906 43 6949 6993 7037 708on 7124 7168/ 7212 7210 7299 7343 44 4 73861 7430 7474 7517 7561 76(51 76418 7692 7736 7779 44 5 7823 75677 79101 79544 79981 804.1 S0851 8129 S1721 82161 41 6 8259 8303 8347 8390 84341 8477 8521 8564 1 |6 0 s8672 21 44 7 8695 8739 8782 8826 8869 8913 S956 96000 90 97 44 8 9131 9174! 9218 9261 930.5 93481 9392 9435 94.79 952 441 9 9 565 9609 9652 9696 9739 97831 9826 98701 99313 99571 43 No. 0 L 3 4 1 5 6 j8 9 Diff. TABL]E X III LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 172 TABLE XIII. LOGARITHMIC SINES9 NOTE. THE table here given extends to minutes only. The usual method 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' 241t. By the ordinary methed, we should have log. sin. 5' 7.162696 diff,. for 24' 31673 log. sin. 51 24't 7.194369 Tne more accurate method is founded on the proposition in Trigonometry, that the sines or tangents of very small angles are proportional to the angles themselves. In the present case, therefore, we have sin. 5': sin, 5' 24' = 5':5 241t =- 3001t: 324't. Hence sin. 5' 24' 324 sin. 5' 2300, or log. sin. 5' 24" - log. sin. 5t + log. 324 - log. 300. The difference for 24"t will, therefore, be the difference between the logarithm of 324 and the logarithm of 300. The operation will stand thus: log. 324 = 2.510545 log. 300 = 2.477121 diff. for 24 = 33424 log. sin. 5' 7.162696 log. sin. 5' 24t 7.196120 Comparing this value with that given in tables that extend to seconds, we find it exact even to the last figure II. Given log. sin. A - 7.004438 to find A. The sine next less than this in the table is sin. 3' = 6.940847. Now we have sin. 3': sin. A 3 sin. A - 3 A. Therefore, A sin 3, or log. A = log. 3 + log. sin. A log. sin. 31. I-ence it appears, that, to find the logarithm of A in COSINES, TANGENTS, AND COTANGEIN''S. 173 minutes, we must add to the logarithm of 3 the difference between log. sin. A and log, sin. 3/. log. sin. A 7.004438 log. sin. 3' 6.940847 63591 log. 3 = 0.477121 A = 3.473 0.540712 or A = 31 28.38't. 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 9, 8, 7, and 6 stand respectively for -1, -2, -3, and -4. 15* 174 TABLE XIII. LOGARITHMtilC SINES, 00 1790. Sine. D. 1". Cosine. D. 11". Tang. D. 1". Cotang. I I. 0 Inf. neg. 0.000000 00 Inf. neg. Infinite. 60 1 6.463726 O.000000 0 6.463726 3.5274 59 2.764r o017.17.00 6 5017 1 4 2.764756 2934.85.000000.764756 23348 23244 5 3.940847 2082.31.000000' 3.940847 08231.059153 57 4 7.065786 1615.17.000000 7.065786 1 2.93214 6 5.162696.0000.162696 1 19.69 837304 5 6.241877.' 1 1319 69 3 4 6 241877 1115.78 99999.24187 1.75122 54 7 9308824 6653.999999.30SS25.691175 53 966.O53.00 966.54 8.366316.999999.366817 9 633183 52 9.417968 852.54 999999 ~o 417970 852..582030 51 762.62.01 762.63 10 7.463726 68 9.999998 7.463727 6S 85 2. 36273 50 11.505118.629981 99999'01.505120 62981.494880 49 12.542906 9.'999997.01.54299 457091 4 6 79.37 01 542909 13.577663 536.41.999997.01.577672 536.42 422323 47 14.609853 99.999996.609857.390143 46 15.639316 497.38.999996 o1.639820 415 360180 45 16.667845 438. 999995 01.667849 43.82.332151 44 17.694173 41372 999995 1.694179 413.73.305821 43 18.718997 39'3 999994.01.719003 136.280997 42 19.742478 37127 999993.742484 3 257516 41.01 371.28 20 7.764754 353 9.999993 01 764761 33 16 2.235239 40 21.785943 33672.999992 1.785951 33673.214049 39 22.806146 32175.999991 oi.806155 21.76 193845 38 23.825451 30805.999990.825460 0 I.174540 37 24.843934 29547.999989 01 843944 3 156056 36 295~47 /.02 298.49 25.861662.999989 02 861674 2890.138326 3 26.878695 273'17.999988 02.878708 273' 1.121292 34 27.895085 263.999987 02.895099 26325.104901 33 28.910879 25399.999986 02.910894 OS9106 32 253~99.02 254.01... 29.926119 24538.999985 2 926134 245. 07366 31 30 7.940842 23733 9.999983.02 7.940858 23735 2.059142 30 31.955082 2 999982.955100 2 044900 29 32.963870.22273 ~999981 02.96SS9 222.75.031111 28 33.982233'22..999980 02.982253 216 10.017747 27 34.995198.999979 02.995219 20983.004781 26 35 8.007787 20390 999977 02 8.007809 20392 1.992191 25 36.020021 19831.999976 02.020044 1983 979956 24 37.031919 999975 031945 195 968055 23 193.02.02 193.05 38.043501 18801 02.043527 18803.956473 22 39.054781 1832.999972 02 054809 183:27 945191 21 40 8.065776 178 72 9.999971 02 8.065806 875 1.934194 20 41.076500 174.2 999969 03.07653 17444.923469 19 42.086965.17 999968 03.08699 17034 913003 18 43.097183 1 0.999966 097217 16642.902783 17 44.107167 166.39.999964 03 107203 162. 892797 16 45.116926 1629.68.999963 03.116963 15.1 83037 15 46.126471 66.999961 03.126510 873490 14 47.135810 15 999959 3.135851.864149 13 48.144953 14924.999958.03.144996 1927 41 855004 12 149..02 149.27 00 49.153907 146.22.999956.03.153952 146.25.846048 11 50 8.162681 133 9.999954 8.162727 14336 1.37273 10 51.17120 144 999952.03.171328 140 7 828672 9 52.179713 17.86 1.999950 03.179763 3790.820237 8 53.187985 1. 2.999948'03.188036 1 3'.2 811964 7 54.196102 13.029.999946.196156 1.32 803844 6 55.204070 130 41 999944 03 204126 13.7974 5 56.211895 12810.999942 03 211953 1S14.788047 4 v57.2195S1 1 125:187.999940 04 219641 2.739 53.227134.999933 04.227195 123.1.772805 2 59.2 34557 |.999936.234621.76379 1 60.241~855 _.999934 1 241921.75S079 0 AT. 1 - D COSINES, TANGENTS, AND COTANGENTS. 17D5 12 1 9:)3'A. Sine..D. 111 Cosine. D 1t_ Tang. D. 1. Cotang. 1~~~~~~~~~~~~~~~~ 0 89.24185-5 5 9.99993-1 S. 241921 1.7.7- ti I/ 7 7 119.63 (4 119.67 1.249033'1169..999932'0 249102.67 I70,9 3 602 113209,4.26163 2.26094.999929 0.2 ~,.74~-! 115.v1. 04 11.74;J 2693042 1 2.999927 3 263115 1'136 /.26 [7 113.98 ~~ ~ ~~.01 114.02.269831.99992-5, 269956.730044 112.21 04 112.25.276614 11.0.50 999922.04 276691 105 723309 11.28303 999922 0.2.2..332.716677 ) ~2 la 23 99042 9 107/826 /03/ 7.239773 1 93 u' 83 04 -S Iu9,87.710144 3 107.22 99991:'.6 1 0. 71014 8.29K07 105.66.04.296292 07 7{03708 [ 2 9.3026 10.66.99991.. 30264 1008 69766 10.1302546 104.13 991 6/6 10 8.30794 9.999910 02 1.691116 il~~~B.G 6 0 {. 3088 10o'70[ 11.314934.999907.3115046 1 ~ 4.64904 19 101922 999 111.26 12.321027.9990'21122.6797 4 99.82.04 99.87 13.327016 999902 32'~ 7114.6' 47 98.47.05 9).!5 14.332924.999899 3' 0352.666975 ]0 97 19 /~ 94.60.05 94 65 46 2373 9927.1 (| 3 16 3:314530 9.26.9993050961799 17.350121 999391.350229 96 0-19711 1'93.3S05 93 3 ii lil.335783 I,~999'53 {55.6144t[ 452 92 19 992 9 1 92 24 6.1361315 91.03.9992 0 361430 02 63 04 {10.05 ]10 20 1.366777 9.9 99322 S 36695 1 623105 49 89.90.99 I- 9.9 6 2L.9.72171 880 9992/9 2.3'2299.20 6773: 3 8;.3578.u~'"558q Ud~a 23.377499 872 99976.0 377622 2 /.3 2776 8 99973 02.3S29 86.67111 ~~~~~~~~~~~~1 ~;,i, 9 4 [~440 21 2 9 02/7962 6364 999270 // 322092 8672 6119,92 2.39319 46.99967 05.393231 82 60666 26.39 679 8 999261 0.392315 297.6016 1 (717.40319.999261.4033. 59666 3 492161 8177.999253. 402304 8 591696 2': 77.05 81. 6 l [4 29.413063 8026 99931 0.413213 8091.5S6787 3 8 8.417919 79 9.999 6 8.418062 so 02 1-521932 30 321.42717 79 999 06.422269 7 577141 3 437 7223 9991.06 427618 7099 572322 22.427615,5723~~2 33.43215 7740 999241', 432315 745 62763 774300 7.4 26.436'.76.52 999S33.06 436962 7663.563032 999S76' 0 66 o,> 89,:).4...~~ 25 41394 75.77 99934.06 44169 7583.55449 36.445941 9 999831 0 446110.553290 $499 97.0 5 37.37.450440 742 999227'.450613 7.4.2549387 74. 22.06 74.2u - 3.45493 99924 0.455070 7353.544930 39.459301 999820.459481 540519 72_73.992 06 493 72.79 30 49 8463663 72.09 9.999216 8.463249 72.06 1. 36131 20.4672 /129 99981' 06.462312 7.13 3182 19 c12.067. ~ ] 2.4722619 7 06 99909 06 1.47245 2.527546 3 82~~~~~~~~~~~~~~~~~~~~~~~~ 71 3.476493 6991 99905.404669. 91 1307 4 6924 99901 06 092 699 1 5 -0 -0 901.L101 51L0 4~.4,~ ST 6 59 99979,.0.48W) 0.6.514900 1 6931 219102 1 62259.06 42019 15.43963 679 999794 07.4917 601.51023 7.493910 6731 7 435.99900 4930 67.506750 13 1.497072 6 999726 0 497293 6.502707 12 39.501020 6 999722 07.51292 6615.492/02 11 63.03.07 6,13) 1 19 8'0045 9.999772 0 3502267 1.494733 10 65.48g.07 65.559.53994 649 99774 07 509200 64 96 4903200 9 2.5122617 9993,69 07.51309 64.436902 89 516726 37 9976 07.516961 6 483039 7 92.52051 6319.99761 07 299 626 479210 6 5 5.524343 3 999757.524586 475414 5.6.52 999742.07.72 1 4 46.532156' 62-I;'-.07 62.18 57.53182 9997415007.532/79 6 467920 3 53.535523 60.999744.0.535779 611 464221 2 59.239186 99910.53947 460553 1') 52 60.50.07 60.62 6 I_ 3 942319 979976.543024.456916 0' P. Sne..Co tn~,. D.Tag TI. 11 3 Cosine-' D 91", S/ne D' Ca.n D 1" Tan 4.2 41q=2'' 90 176 TABLE XIII. LOGARITHtMIC SINES9 2o I. Sine. D. 1"I. Cosine. I. I.- Tang. D. I|. Cotang. M1. 0 8.542819 9.999735 0 7 0I 3084060 1.456916 60 1.546422.999731. 546691 9. 53,09 59 I2 5.55.07 5962 4 2.549995 59..999726 O0.55026 59 44.14 732 3.553539 59.06,999722.0.553817 59.4 446183 57 4.557054 558.999717.0 557336. 442664 56 5.560540 57.l5.999713.08.560528 57.19.439172 55 6.563999 57..99970.08.564291 435709 54 7.567431 5719.999704.09.567727 56..4132273 53 8.570()836 56.74.999699 571137 56.82.428863 52 9.574214 57.999694 0.574520 55 42540 51 10 8.577566 4 9.999689 8.57777 552 1.422123 50 11.580392 55'02.999685.0. 581208 55.10 418792 49 12.584193 55.0.999680.08.54514 5406.415496 48 13.587469 54.60.999675.08.587795.427 412205 47 14.590721 54.19.999670.0.591051 53 87 408949 46 15.593948 53,39.999665 03.594283 405717 45 00.39 ] 09 ] 16.597152 53,39.999660..597492 5.402508 44 17.640332 53.00.999655 528.600677.20 399323 43 19.603499 52.61,09.6006 52370 18.60349 52. 999650.603839 52.396161 42 52.23.08 52.32 9.66623 51.86.999645 09.606978 51 94 393022 4 20 8.609734 51.49 9.999640.09 8.610094 51 1.389906 40 21.612823.999635.613189 51.21 3861 39 22.615S91 51..999629. 9 6616262 5.383738 38 23.618937 50.7.999624.09.619313 5).30(687 37 24.621962 5,.999619 0.622343 50..377657 36'09 50.5 25.624965.999614 62532 0.374648 35 -279-18'49.72.01 371660 34 26. 79-3 9..999608..62S340 494.371660 3 2.63354 41.999597.9.63-1256 4880 365744 32 29 6.36776 4.999592 09.637184.362816 31 30 s.6360 9.9996 09 8.640093 48 16.359907 30 31.6563 4 9995.09 64292 47.84 357018 29 6 44432.9995/ 5 32.64542 0 47.43. 999775,ns.645853 473 354147 28 33. 1 2 9999570 9.68704 4722 351296 27 34.611 2.82 99956 09.65 1 537 91 348463 26 35,65-3911 62 999558 64352 345648 25 36.6602 42.999553 10.67149 461.342851 24 37.69175 3 9947 0 659923 46.30072 23 39.662 9'30 963 999541 6O.662619 45.73.337311 22 39,6696 435 999535 0 665433'.334567 21 40 8.6669 4.07 9.99959 8.663160 416 1.331840 20 41,670393 0999524.10.60370 4488 32913( 19 4,6143 43.70 999 00.10 691544 4380 318456 15 46.6366 434 999493 6 2.4172 43'54 31528 14 47.6 6272 3.. 999487.10 6a6784 43,28.313216 13 49 69 -63 42 92 999491 690.6393S1 43 20 310619 12 49 691433 4267 999475 10.691963 42773.30037 11 50 8.693998 1242 9 999469 10 8.694529 42 2 1.305471 10 51.696543 4217 999463.697081 42 302919 9 52.699073 41. 3 9994 /6.11.699617 423.31383 8 53.70159 1. 999450.11 702139 413.297861 7 54.704090 4144 999443.1.70466 41.29534 6 55.70677 4121 99947 11.707140 41 32 29260 5 56.799049 40.97 999431.11.709618 41 3O 290382 4 57.711507 40'.999424.11.71203 40.1 8.27917 3 59.713952 40n 4 999418.1.7 14534 40(62.285466 2 59.716393 402 999411.11.716972 400.23028 60.719830.999404.719396 28004 0 M. Cosine.. 1". Sine. D. 3"1. Cotang. D. 1. Tang. M.'2.42. 44.34] — ~I. ICosin. I,'".,~] 999500,'.,~ / 44an.0 /,T' s~g COSINES, TANGENTS, AND COTAI4GENTS. 177 Sine. D. 1. i Cosine. D. 1" Tang. D. 11 Cotang. M. 0 8.718800 9.999404 719396 40.17 1.28060 60 1.721204.394.999398 721806 3 273194 59 2.723595 3962 999391.724204 74 27579 5 3.725972 39.41 999334 11,726588 3]9'9,273412 57 4.728337 3919,999378.728959 9.31 271041 56 5.730633,98 999371 11.731317 39 10 2683 5) 6.733027 3877,999364 I1.733663 3389 266337 54 7.7353.54 3599937 735996.264004 53 8.737667 38.,7 999350 3. 73839171 3, / 261683 52 3..7 36 99 12 7331 2 9.739969.99:33.16 12 740626 27 259374 51 10 8,742259 9,999336 12 8,742922 07 257078 50 11.744536 37.999329 12 745207 3788 254793 49 12.74602 37:56.999322 12.747479 37'6 252521 48 13.749035 37..7 999315.12 749740.250260 47 14.751297 37.17 999308 12.751989 372,9 248011 46 15.753528 36.9.999301 12 754227 3710 24773 4 16.755747 999291,756453.243547 44 16 579 36.80 999291 12 6 3692 243547 44 17.75795 3661.999237 1o 758668 36 73.241332 4:3 18.760151 36.2.999279,12,760872,239123 42 19.762337 366:24.999272 12 6306 36 3 23935 41 20 8.764511 36I06 9.999265 12 8,765246 36 1 1.2.34754 40 21.766675 36.806 999257 12 767417 36,00 232583 39 22.768828 35.7088,999250 2.769578,383 230422 38 23.770970.999242 12.771727.228273 37 35.53 92.12 3'.5 21 4 773101 35. 99923 13.773866 3548. 226131 36 25.775223.35.1 999227 13.775995 3,48 224005 35 26.777333 35.18 999220 13.778114.221886 34 27,779434 344 999212 13.780222 3.219778 33 28,781524. 999205 13.72320.9 2176 32 34.67.13 34 80 29.783605 34.67 999197.784403 0.216592 31 31.51:13 34.64 30 8.785675 9,999189 13 8.786486 47 1.213514 30 31.787736 3418.999181 13.788554 3431.211446 29 32.789787 340,2 999174 1 3.790613 3415.209387 28 33.791828 33.86.999166 13.792662.207338 27 34.793859 33.999158 13.794701 33 205299 26 35.795881.999150 13,796731 33683.203269 25 36 797394.3.39 999142 513.793752 33.52 201248 24 37.799897 3323.999134 13.800763.199237 23 38.801892 33'0.999126,13.802765 33,. 197235 22 32.93.13 33.07 39.803876 332:9o83.999118:3!.804758 33'07.195242 21 40 8,805852 9.999110 14 8.806742 3292 1,193258 20 42.807819 32.78.999102.14.808717 32.77 191283 19 42.809777 3249.999094.4.810683 32.62 189317 18 43.811726 2.34.999086 14.812641 3248.187359 17 44.813667 322.3 999077'14.814589 32.3 185411 16 45,815599 32.20 999069 14.816529 32',183471 15 46,17522 32.05.999061 14.818461 325. 181539 14 47 819436 31.91.999053 14.820384.179616 13 48.821343 l.77.999044.822293 7'.177702 12 49.823240 31.63 9.14 8229 3177 17702 12 31.49.14.82205 31'63.75795 11 50 8.825130 9.999027.14 8.826103 31 50 1173897 10 51 18270141 31.2.999019 4.827992 31.3 172008 9 31.22.90 59412 52.8238384 31.0.999010.14 829874 31.3.170126 8 53.830749 31..999002.14 831748 31.23 16252 7 5-1.532607 3.82 998993 4 833613 309. 16637 6 55.834456 3.69 99384.14.835471 3096.164529 5 56.836297 309 56 99976.1 837321 30.83 162679 4 57.83130 998967.1.839163 3070.160337 3 53.8:9936 o.9939583.84099.159002 2 9.841774 3 3.998950 5.82825 45 157175 1 30.17.15 30.32 6.84353.993941.844644 32 15536 0 M. Cosidne. D. 1 9 8 Sine. D V.. 19 Cotang. ID. If. 002 T 1 9e30. t~~~~~~~~~~~l~~844 [ ~ 18 1TABLE XIII. LOGARITHIIIC SINES,!0. 11I M. Sine. D. 1". Cosine. D. 1"l. Tang. |D. 1". Cotang. I. O 8 3843585 3005 9,998941 8,844644 3020 1.1 553 60 1.845387 2992.998932 i 15 846455 30,07.153545 59 2.847183 29.0 998923 848260 299.151740 3.848971 2968.998914 1 85007 2 140943.4 850751 2 "'i,998905.851846 9S..148154 29,55,i 29.0 1 55 5.852525 29'.4 998896 5 853628 29.50 146372 6.854291 29.'3 998887 8855403 29.8 144597 54 7 856049 29.31 993 878 5 8771 71, 142829 53 8 i857801 2919 998869 1.858932.141068 52 9.859546 28 96 998860 1 8606S6 29.139314 51 ~285.96.15 29,11 10 8.861283 28.84 9.998851 8.862433 2900 1.137567 50 11.863014.998841 864173 2.88.135827 49 12.864738 2 8.3 998832 15 865906.8 77 134094 18 28.61.132368 47 13.86614505.998823 15.867632 2. 1326 3 S 14.868165 2850.998S13.869351 25 130649 46 28.39 a16 0 25',,, 15.869868 2823.998304 16.871064 2843.128936 4 16.871565 6 28 7.998795 1 6.872770 8,127230.8732752 20..7 998780. 874469 2,52 I125531 4 1S.874933 27.95.998776.16.876162 28.22.123838 42 19.876615 27.84'998766 16.877849 280 122151 41:7.84.16 28.00 20 8.878285 27 9.99757 16 8.87929 27. 9 1.120471 40 21.879949 2764.998747.16.881202 9.18796 39 22.831607 27 52.998733.16.882869 27. 117131 33 23.883258 242.998728 16.84530 27.58.115170 7 24.884903 2.998718. 8835 27, 3815 25.886542 27,21.998708.16 887833. 11267 3 26.888174 271 1.998699.1.889476 2.110524 27.889801 2700.99869 6.891112 27.17.108888 33 28.891421 26790.998679 16.892742 2707.107258 32 29.893035 26680.998669 17 894366 32697.105634 1 26.80.17 26.97 30 8.894643 2670 9.998659 17 895984 2 1.104016 30 31.896246 26.60.998649 17 897596 26.77.102404 29 32.897842 2651.998639.17 899203 26.67.100797 28 33.S99432 26.41.998629'17 900803 2658.099197 27 34.901017 26.1.998619.17.902393 2648.097602 26 35.902596 2622.998609 17.903987 26.39.096013 25 36.904169 26:12.998599'17 905570 26.29.094430 24 37.905736 26 0.998589 17.907147 2620.092853 23 3.907297 2'93.998578.17.908719 26 10.091281 22 39.90853.998568.910285 2601.09715 21 25.84.17 26.01 40 8.910404 - 5 9.998558 8.911846 1.088154 20 41.911949 25a5.998548 17.913401.06599 19 42.91348 2566 998537 7.914951 2.085049 1 0 25.56.17 25.74 18 43.915022 7.998527.17.916495 2 6.083505 17 44.916550 2 998516 918034.081966 16 45.918073 2,529.998506 8.919568 2547.080432 15 411.919591 2 2.99495 18.921096 2543.0789(4 14.18 6 25.3 47.921103 25 1'998485 8.922619 2.077381 13 43,922610 5.-.998,174 |.924136 2.(75864 12 4 9 25.03 20' 9 18 25.21 19.924112 2943.998464 18.925619 2.074351 11 50 8.925609 248 9.998453 8.92756 1.0724 10 51.927100 24.77.998442 18.928658 204 1.071342 9 52.9285S7 24.69 993431 18.930155.06945 8 53.930063 24o60 998421 18.931647 24.068353 7 5-.931544 24'52.998410'18.933134 24173.066866 6 55.933015 24.998399..934616 24 2.065384 5 56.9.3-1481 24.998388 18 936093 24 6.063907 4 87 9391 98 I 28.53 7.933942 24.998377 7 18'937565 24453.062435 3 58.937393 2.2 9.99366 1'.939032 4237,060968 2 24.19 -.1i 24.37 59.933550.998355 J.940494 22.059506 60.940296 4.99S.384.941952.058048 M. Cosine. D. 1". Sine. ID. 1". Cotang. D. 1" Tang. M I3.0 __3_ COSINES, TANGENTS, AND COTANGENTS.'79 M. Sine. D. 1". Cosine. D. Tang. D. 1. Cotall. M. 0 8.940 96 203 9.993344 3.9-1192 21.()1 1 L.911733 2''.998333.9i434 9)4 f. (309 ~2:3,95, I 2:1. 12 2.913 74.993 19.9414352. 1 0, 5' 5 I 4 58 3.9-1-606 2.993; 1. 63- 37) 3 714.19 2:397~.9 t6031 23.7.9930.9773 9.(')9'2t;6 5( I. 9 17456 236 99339.19.~91 3 2' 1 2 6.94,S74 23 63.993277 919.905973 2'3 74.049403 5 |4 23.49.19 23.67 01 7.99 00374 23340.993266 19.95202. 23.047199 9 53 4 ~ 9.5t69t61 2:',X.99S25 5 1 l.953I'.)46d559 52 8.99.931.993232.95346.045l44 5 1 0 8.95L4499 2325 9.993232.19 8.956267 23.44 1.043733 50 -.~ 993220 19 l 1.9 94 974 222 36.42326 49 1 2.90724.1 993209'19.959075 2329 40925 4 13.953670 2302.99197 19.960473 23 22.039527 47 14.9609.52 2299 99331 9..961366 23 14.03134 46 ].5.961429 2233.993174.9632255 2307.036745 45 16.96230 2281.993163 19.964639 20.035361 41 17.964170 2273.99311 2.966019 22.033981 43 1 96.331..993139..967394 2.6.03266 43 19.966393 22,59.993123 20.963766 22.0312:4 41 20 3.963249 222 9.993116 3.2 8.970133 22 72 1.029367 40 21.969600,)2 4 993104 / 20.971496 22,.023S504 3!) 22.970917 2233 993092 20.9723.5 228.027145 33 23.972239 2231 993030 2.974209 92 025791 37 24.973623.24 993068.975560,'4.024440 36 25.974962 2217.993056.20.976906 22'37.023094 35 26 1.97693 2210 99044.20 978248 22.30.021752 34 27.977619 2203.993032 120).979596 22.4.020114 33 23.978941 99(3).0.2.99019079 3 29.930259 290 9920.20.980921 22.1.017749 31 2 9 5 21'.90 /.20.9221 22.10 3 38.931573 21 83 9.997996 20 S.9S3577 22.03 1.016123 30 31.932333 2177 997984'20.98499 21.97.015101 29 32,93419 21 70.97972 2'.986217 21.91.013783 28 33.935491 2164.99795.9 |'.937532 21.4 012468 27 31.936739'p77' 1.99947 2.9S384-2 2G 3?.933033 2, 1 997935 2'.990149 21.7.00951 2 36.51.21.91 21.71 009 2 36.99374 21.44.997922.21.991451 21.6.008549 24 ||37 990660 21.33.997910 21.9927/30 21 59.007250 23 33.991943 1.31.937897 21.994045 52.005955 22 39 o99322,21 9978.995337 21 5.004663 21o 21.25.21 21.46 40 8.994497 2119 9.997872 21 8.9966241 2140 1.003376 20 41.995763 2112.997860 { 21 99790 2134.002092 19 42.997036 2106 99747. 999188 21.27.000812 18 43.993299 997835 9.000465 0.999535 17 44.999560 21'0.997822 1 9.001738 5 2..993262 16 4) 9.000316.997309'21.003007 21'15.996993 15 46.002069 203 83.997797 | 21.004272 21.03 995728 14 20.82.21 21.03 47.003318.99774 2.00.?534 2097.994466 13 -. ~U. tO,.21 20, 97 13.004563 2070.997771.006792 0 993208 1 2 ~49 1 ~.00~0..20.7 997753 j 2L.008017 209 1.991953 1 1 49.0030 20'64 21 9 20.85 50 9.007041 2023 9997745 1 22 9.009293 2080 0.990702 10 5 1.00327S 2052.997732 22.010546 2074.9944 9 5.009510 2046.997719'22 I.011790 2063.988210 8 53.010737 294 |.997706'22.013031 2062.986969 7 54.01962 20.3.997693 22.014263 20'56.985732 6 2.0.182 2 2.2 /.015502 2051.984498 5 56 0144 00 20.3.997667 J 22.016732 20 45.983268 4 57.0156L3.997654 22.017959 20.39.982041 3 53 -.016324 2 17 99761. 019183.90817 2 53.018031 20 12.997621'22.020403 20.34 979597 20.06.22 20'28 61j.0192305.997614.021620.9783830 1 1. Cosine.. 1 ine D. 1f. Cotang. D. 1". Tangt. 1..... 5..-, 10 TABLE XIII. LOGARITHIIIC SliSES, 80o 173g 1. Sille. 0D. 11. Cosine. D, 1'f. Tag. D. 1". Cotaev. I. O 9.019235 9.997614 2 9.021620 20.23 0. 97380 60 1.02035 20.00.997601.22.022834 20.17.977166 r)9 2.021632 19.95.997588 2.0241014 20. 9759; 5 3.022825 19 997574.22.025251 06.974749 57 S4 22 ~.02 4.024016 19. 997561.22.026455 20.01 135 5.972345 5.025203 1973.997547 22.027655 972345 55 6 026386'.997534 *23.028852.071148 7.67.26 1 9.90.027567 19.6.997520.23.030046 19..699 54 - 23 0319.54 15 8.0239744 19.997507.23.031237 19..68763 52 19.57 04 967575 9.029918 19.997493 23.032425.967575 5 10 9.031OS9 46 9.997480 3 9.033609 19.69 0,966391 50 11.032257 19.41.997466.23.034791 1.965209 49 12.033421 136.997452 2.035969.9643 48 13.034582 1930 23997439.037144 19.53 96256 47 14.035741 1925.997425 23.038316 19.48 9616S4 46 15.036896 19.20.997411 23.039485 19.43 960515 45 1 6.038018 19.15.997397 *23.040651 19.3.9593-9 44 17.039197 19.10 997383.. 813 958187 43 18.040342.997369.042973.957027 42 19 8 19.05.23o 19.28 19.04148 1900.997355 23 044130 1923.955870 41 20 9.042625 1,,95 9.997341 23 9.0452S4 1 0.954716 40 2 1.043762 18.9.997327.23.046434.953566 39 22.044895 18.90.997313.2 0.047582 19.952418 38 23.046026 18.8.997299.24.048727 1.951273 37 24.047154.18.0 997285.24.049869 8.95O131 37 25.048279 18.7 997271.24.051008.948992 35 8.L~' ~24 18,93 26.049400 18.70 997257 24.052144 1.947856 34 18.65.24 18.89 27.050519 18.65 997242.24.053277.96723 33 23.051635 18.60.99722s.24.054407 18,84.945593 32 2I7 50. 14,79 29:052749.997214.055535.944465 31 18'50 24 18.74 30 9 053859 46 9.997199 24 9.056659 18.70 0.943341 30 31 054966 18.4.997185.4.057781 1.942219 29 32.056071.997170.058900.941100 28 3 18.36.24 18.60 33.057172 1.99716. 24.060016 18.56 939984 27 31i.058271 18.31.997141.24.061130.5.938870 26 35.059367 18.27 997127.24 062240 18.51 937760 25 36.060460 18.122 997112.24.063348 18.936652 24,.:,- l~ l/' z 1..4 2 37.061551 8.997098.064453.935547 23 18.13 24 18.37 ] 38.062639.997083..065556.934444 22 39.063724 1 004.997068.25.066655 1828 933345 21 40 9.064806 17 9.997053 | 9.067752.24 0.932248 20 41.065885 1 i9.997039.068846 119 931154 19 42.066962 7.90.997024.25.069938 18.19 930062 18 43.068036 -.997009.071027.928973 17 44.069107 17.6.996994.25.072113 18.10.927887 16 45.070176 17.77.996979 8.2 073197 18.02 926803 15 46.071242 1772.996964.2o.074278 18.02 925722 14 47.072306 17.6.996949.25.075356 1793.924644 13 48.073366 17.68.996934. 25.076432 179.923568 12 49.074424 17:59.996919.25.07750 1784 922495 11 50 9.075480 17 9.996904,2 9,078576 O 0.921424 10 51.076533 17..996789.25 079644 1776.920356 9 52.077583 1 7.996874.25 080710 1.919290 8 53.078631 17] 7' 93653.25'73 918227 7 54 079676 1742.965 0 54.07'9676 1733. 0'396843 23 02 17 63.917167 6 55.080719 17.34 996828.6 083891 17.916109 5 56.031759 17.2 996812.26 084947 179 915053 4 57.082797 17.2.996797.26 086000 17.914000 3 58.083832 996782.087050 17.91290 2 59.0,4S61 17.1.996766..089098 1747 911902 1 60.085894.996751.26.0891441.910 56 O M. Cosine.. D1". Sille. 1D. 3'. Cotang.. 13 P Tau1. I. 130o t33B COSINES, I'ANGENTS, AND COTANGENTS. 18 1 172o M. Sine D. 1. Csie... Tang. D. 1. Cotang. M. 0 9.08394 1713 9.996751 26 9.039144 90.910S56 60 1.0369221.996735.6 090187 9093 59 2.037947 17'.996720 26.091223 17'31.908772 53 3.038970 1..00 996704 1 26.092266 17'27 907734 57 4.039990 196'99G66.093302 1723.906698 56.996673.09100 16.9 73.094336 17.19 90661 5) 6.092024.996657 2.095367.90633 54 16.88.26 17.15i, 7.093037 16.83.996641.26.096395 17..903609 63 8.094047 16.84 996625.26.097422 17.07 902573 2 9.09506 166 996610 26.09446 1703 901554 51 10 9.096062 1673 9.99694 7 9,099463 0.900532 50 11.097065 16769.996578.2.100437 169.S9;1 49 12.093066 16 69 996562 27 101504 1 691.S93496 4 13.099065.16.99646.27.102519 1633.189741 47 14.100062 16.,.996530 27 103532 1 68.896463 46 15.101056 16,53'996514 27.104542 1 895493 45 16.1020438 164 996198.1055.0 16.89440 44 17.103037 16. 996482..106556 167 8934 4 43 16.46.27 16.72 18.104025 1642 996465 27 10759 1669.892141 42 19.105010 1633.996449 27.103560 1665.891440 41 20 9.105992 16 34 9.996433 27 9.109559 1661 0,890441 40 21.106973 16.30.996417 *27.110556 16..8S9444 39 22.107951 16'27.996400 27 111551 164.888449 30 2:3.108927 1623 996334 27 112543 160 37457 37 24.109901 1619.996363 27.113533 1647 86467 36 1153077 80449 25. 110373 16.19.6996351.27.114521 ]16.4 835479 35 26.111842 16.12 996335.2 115507 16.39 493 34 27.112309 16.03.996318.23.116491 1636.3509 33 23. 11371 691 i 29.113774 16.996302 3 117472.S2523 32 29.114737 01 996235.1132 1629.1543 31 16.01.23 16'29 30 9.115693 1.98 9.996269.28 9.119429 16.2 0,830571 30 31.116656 1.94 996252 2 120404. 0404879596 29 32.117613.99623..12377 16 878623 2 33.118567 15.87.996219 3.122:34 1.1.877652 27 34,,1',1.876683 26 3. 119519 r 996202 123.3.. 76633 26.9 15.83'996202 23 124341 16.11 35.120469 15.0.996185.23.124284 1608.875716 25 36.121417 1.76 996163.23.126249 16 04 874751 24 37.122362 173 996151 28.126211 601.87379 23 33.123306.9.996134.127172 1593.872828 22 2 69 2 15.93 8 39.124243 6.99117 3.2130 15.871870 21 42.127060 15. 996066 28.130994 15.869006 1 43.127993 19.52 996019.2.131944 158.86056 17!5.52.2 30 19.81 44.128925 149 996032 29.13293 1577.867107 16 45.129354 14.996015.29 133339 15.866161 15 19.45 ~5 19,1y 46.130781 15.42.995998 29 134784 15.74.863216 14 47.131706 15..995980 29.135726 1563.8642741 13 48.132630 15,35.995963 29.136667 15.6.863333 12 49.133551 1532,995916 29.137605 15.64 862395 11 50 9.134470 1t 29 9,9959283 29 9.133542, 0.861453 10 51.135337 99911 139476 1.860524 9 52.136303 2 99539.9. 140409,.859591 53.137216 15.99576 29 141340 1.358660 7 54.133123 19 99559 29.142269 15.48.857731 6 55.139037 15.16.9981.29.143196 15.42.856304 5 57.140350 1.06'99306.29.14044 1536.83496 3 53.141754 1503.995738.29.145966 15.32.854034 2 59.142655 500.995771.30.14635 15.29.853115 1 60.143555.995753.14703.852197 0 1I. Cosinlle. D.1. Sine. | 1I." Cotang. I D.1. Tatng. M. 5}t9"' 16 9,4 182 TABLE XIII. LOGARITHMIC SINES, 80o 1'1 MI. Sine. D. f/. Cosine. D. 1" Tang. D. 1.. Cotang. iI. 0 9.143555 1497 9.995753 30 9. 14703 126 0.852197 60.14445o 14.03.995735.30.148718 1.23.50 22 2.145349 14.90 995717 3.149632 15,20.85(365 58 3.146243 7 99699 I. 1501.10 57.4.87 995699.30.150544 15,17 / 4 [.147136' 995681 / 14713 iS4 99561.30.1 5144 15.14.24515 5.143026 14.81 o95i664 30.152363 1. 14 8176G37 55.9956' B.3015 1 / 6.148915 147 995646.153269 I 846731 3154 7.149602 14.99562S.154174 15.05 S45826 3 8.1506S6.995610,.155077.844923 2 14 72'99610 30 15.02 i 9.15159 14'69.995591.15597.844022 51 14.69.30 14,99 10 9 152451 14 9. 99573 30 9. 156877 14.96 0843123 50 11.153330 1463.995555 30.157775 842225 49 12.154208 1460.995537 130.15S671 14 841329 48 13.155)83 14.57.995519.159565 1487.840435 47.839545 46 14 155957 14.54.995501.30.160457 1484 8393 46 15.1530.995482.3.161347 181 83863 45 16.157700 1 4.48 995464.31.162236 14,.78.837764 44 17.158569 14.4.995446.31.163123 14.75.836877 43 18.159135 14 995427.164008.835992 42 19.160301 14 32.995409 31.164892 83510 20 9.161164 1436 9.99390 3 9.165774 0.S34226 40 21.16202o 14.33.995372 3.166654 14 64.833346 39 22.1625.995353. 1.167532 14.832468 3 23.163743 14,27 995334.31.168409 1468 831591 37 l4,27 3. 14.8 3767 24.164600 142 995316.31.169284 1.830716 3 25.160 454.995297.170157 14.53.82943 35 26 G.16630)7 1.995278.1.171029 14.53 828971 34 14.19 2 31 14.50 828101 33 27.167159 14.16 995260 31.171899 14.47 2101 3 28.168008 14 13 99241.172767 14.44 827233 3 29.16n3a56 14.10.995222 31.173634 1442 ~826366 30 9.169702 14 07 9.995203 31 9.174499 14.39 0.825501 30 31.170a47 14.0 995184.32 175362 14.36.S24638 29 32.171389 1405 99165.176224 4.3 14 823776 2 33.172230 1.99.995146 32.17704.33 822916 27 35.173903 1.94 995108.32.178799 14.25 821201 2 36 174744 3.9.995039 32 796 55 1423.8203492 2 37.17578.995070 32.S0508 120.81949 31388 4.20:81640 22 33 17611 13.38 995051.32.181360 14.17.81640 2 39.177242 1383.995032 32.182211 1415.817789 2 40 9.178072 3 80 9.995013 2 9.183059 14 12 0.816941 20 41.18900 137 994993.183907 1409.816093 19 42.179726.99974 32.184752.815248 18 43.180551 13. / 99953.32.185597 1404.814403 17 4-.181374 13692 994935.186439 14.02.813561 16 45.182196 1367.994916'32.187280 14'99.812720 15 46.183016 1364 994396 3.1S88120 1397.811880 14 47. 13334 136 994877. 1 SS958 13 94.811042 13 13.61.33 13.94.8206712 48.184651 13.994857 33.19794 1391.8120 1 49.18466 994838.190629 9.809371 11 50 9.1862830 135 9.9418 1 3 9.191462 13S6 O. 080538 10 51.187092 1354.994798 3.192294 1384 807706 9 52.187903 13.48.994779 33.193124 1381.806876 8 53 1812 1346 994759 33.19953 139 0607 7 54.189519 13.43 994739 33.194780 13.76 805220 55.190325 134 994720 33.195606 4 804394 1 56.11130 1. 994700.196430 1371 803570 4 1338 934700196430 13.71 7.191933 133.994630 33 97253 1369 802747 3 58.19273 1.94660.19074 1366 801926 2 59,193534 131.99460 19 894 1 01106 1 13.31.33 13.64 60.191332.994620.199713 8002,7 2 M. Cosine., D.'f. Sine. i D. 1. Cotang. D'. Tang 1 365 81744:, 958 ~ 1795 8034 COSINES, TANGENTS, AND COTANGGENTS. 183 9D 170o - Sine. 1D 1F'. ( Cosia,. 1 Ta D. 1".. Cotallo. i. 0 9. 1.) 9,3394620 9. 19i13 I 0.0'37 63 1.195129 {.2333 52 1.3 7 9 2.195925 13.26.91'330 15 3.196719 32.9915630. 20)151 I7! 9T4L 57 13.21 3. 29t139 13.5 4 7 4.197511I.13.16 1) 31. 202971 13] r.797029 56 5.1 9 13, 18.. 207 13.1.79 6.199091 3 994-19112.7204592 13. 7 (). 13.13 923379 31.3339 13.4. 7.199879 13.13.991479.3-'20054001 34 7 791600 53 13.11.3- 7 173,4 8.200666 99459.3 206207 73793 5 2 9.201451 13.06 994433 31,207013 13 40.79937 51 10 9.202231. 9.99441.34 9,207317 13.3 0.7921S3 50 11.203017 1301.9943983 3.203619 13.'..7913 81 49 12.203797.994377.209420.790530 4,3 20 1212.99. 20220 13.33 13.204577 12.96,99437 2100 1. 1.7 0 47 14.20533i3 12.9 994336 31.21101; 13.23.788932 46 15.206131 12.92 994316 34.21 115 13 26.,78185 45 16.206906 1289 994295 34.212611 1324.78739 44 17.207679 12.7 994274.34,213400 13 21.786595 43 18.23452 1285 99254.21419 3 19 785302 42 19.239222 12.82 994233 35.214939 13 17:785011 41 20 9.209992 12.80 9.994212 9.215780 315 0.74220 40 21.210760 12,7.994191.216568 13.12.783432 39 22.211526 1275.994171.3, 217356 13.10, 782644 36 23.212291 2.73.994150 3.218142.1303 781858 37 22.213055 12.7.994129.3..218926 13.06.781074 36 25.213818 12.71.99410.219710 13.780290 3 26.214579 12.66 994037.220492 13.01 779503 34 27.215333 1 994066.221272 1299.78723 33 23.216097.994045.22202 1297 77794 32 29.216654 12592 994024 35.222833 1295,.777170 31 30 9.217609 9.994003 3 9.223607 12.92 0.776393 30 31.218363 1255,993982.3-.224332 12 90.775618 29 32.219116 993960 22156.774844 26 12.53 { ) 12.89 33.219368.993939.225929 126 774071 27 12.50 ] 35'o6700 12.86 34.22061 12.0 993918.36.226700 1284.773300 26 12.48 67 29 25 35.221367 12.46 99397 36.227471 72 2 36.222115 12.44 993875 36.22239 1279 771761 24 37.222961 1242.993354 36.229007 12 77 770993 23 33.223606 12.2 993332 36.229773 1275 770227 22 39.224349 1237 99311.230339 1 7690461 21 12.37.38127 40 9.225092 12359.993789 9.2313012 7 0.76698 20 41.22533 2.3.993763 232065 12. 767935 19 12.33.936 36.36 1269 42.226573 12.31.993746 36.23226 12.67.767174 1 43.227311 12.2.993725 36.233586 12'65.766414 17 44.223048 12.9 993703.234345 163,76 655 16 12.26.36 12.63 76487 15 45.223784 12.24.993631.36.235103 12.7064397 1 46.2295168 12, 22.993660.36 2:33559 8 64141 14 12.20.',,, 47.230252 122.993633 36.236614 12.56 76'3,6 13 48.230991 12.1.993616 6.237363 129.762632 12 49.231715 12:16. 993594 336.233120.761880 11 50 9.232444 12 4 9.993572.37 9.23372 12.50 0.761128 10 51.23:3172 1212 99350.239622 1243.76037 9 52.233399 12.10 993..210371 1246 759629 53.234625 207,99306.2-111.75382 7 12.07 / 12.44 51.2353-9 12.0 9.931 S A.37.241865 12942 759.758133 ( 5;.236073 12.03 993-162 7 24126 10 12.40 77390 5 56.236793.99340. 243331 12 756646 4 2 12.1 0.,7 12.33 57.237513 1199 993418. 241097 12.36 755903 3 53.23323 13 7.993396 37.2i43839 12 34 755161 2 55.233503 i.993~7 ~.2-~55r9.75442137 59.2: 19 9933 7 12.3 421 6' ).239370 1.99331'216319. 73691 0 1I. Cosille. i D1. 1". Sine D. 1" Cotalg. 1 I).1". T| nw. 31. D.99 O 184 TABLE XIII. LOGARITHMIC SINES, ILOD -.M. Sine. D. 1". Cosine. D. 1"t, Taig. D. 1", Cotang. N. 0 9.239670 9.993351 9.246319 i 3 0.753681 60 I,240386 11.993329.3,217057 1,23.752943 59 2.241101 11.9 3307.37.247794 12'26.752206 58 3.241814 1187.99328aI 37.2485:30 1221 7531470 57 4.242526.993262'37.249264 12,22,750736 56 5.243237.993240.249998.7 000 5 6.243947.83 993217'.250730 19' 2.749270 54 7.2656 11.81.993195.251461.18.743539 5 3 11.79.38 12.17 r. 8.245363 11.9.993172.252191 1.1,747809 52 9.26069 11.77 9.93149 2529 12.747080 5 11.75.38 12.13 10 9.24:6775 73 9.993127 9253648 12 1 0.746352 50 11.73.33 1;.11' 26 11.247478 1 1.993104 /.254374 1209., 626 49 11.71, - 12.09 12.248181 1.993081 38o.255100 1207.744900 48 13.248883 116..993059.2 5S24 120.744176 47 14.249583 1.65.993936 38.25647 120.743453 46 15.250282 1163.993013.3 257269 1203,742731 45 16.25o090 11.61.992990.3.257990 120,.742010 44 17.251677 11.59.992967.,258710 1 0.0 741290 43.18 252373 11.5.992944.3,2.259429 11. 740571 42 19.253067 11,5.992921.38.260146 1194.739854 41 11.56 3 4.3 11.96 7 20 9.253761 1 9.992898 38 9,260863 11 92 0,739137 40 21.254453 1 5.992,75.38.261578 11.738422 39 22.255144 11.52.992852.39.262292 11.9 737708 38 23.255834 11.5 992829.39 263005 11 7 736995 37 24,256523 11.468 992806.263717 11.85 736283 36 25.257211 11.46.992783..39 264428 11,83.735572 35 26.257898 11.44.992719, 3.265138 1.8 734862 34 27.258583 11.42.992736.39.265847 11.734153 33 28.259268 11.41,992713 39 11276.733415 32.992713.266555'.733445 32 29.259951 11.39 9 39 2 11.78 29.259951 11.37.992690.9.267261 1176 732739 31 11.37.39 11'6 3 30 9.260633 3 9,992666 9,267967 1, 0.732033 30 31.261314 1133.992643.39,268671 11. 731329 29 32.2614 11; 11. 762 28 32.261994 11.31.992619.39.269375 11.70.730625 23 33.262673 11,.992596 | 39.270077 69 729923 27 11'69 36.262351 1!0.9925772.270779.729221 26 35.264027 1 1'.992549 /.271479 11.728521 25 36.264703 11.24.992525 39.272178 11.727822 24 37.265377 11 2.992501.272876 1 1' 727124 23 11.22.39 11.62 38.266051 11..992478 40.273573. 726427 22 39.266723.992454 40.274269.725731 21 11.19.40 11.52 40 9.267395 1117 9,992430 40 9.274964 0.725036 20 41.268065 [..992406.40.275658 1 57.724342 19 42.268734 1 1,.992382 40.27635 [ 11.53.723649 18 43.269402 11.12.992359.40.277043 11 51.722957 17 44.270069 11.12.992335..277734.722266 16 45.270735 1.10.992311.40.278424 1 1'58.721576 15 46.271400 11.0.992237 40,279113 11 46,720887 14 47.272064 11.06.992263.40 279801 11.1,720199 13 48.272726 11.0,.992239 40.2S0483 I.43,719512 12 49.273338 11.013.992214 0 2S1174 41 718826 11 11.01'.40 11 41 50 9.274049 1099 9,992190 40 9.281858 1140 0.718142 10 51.274703 10 9.992166 40.282542 11 3.717458 9 52.275367 109.992142.40.28322 11.36 716775 8 53.276025 1094.992118.41.283907 1 135 716093 7 54.276631 1.92 992093..284588 11.715412 6 55.277337 1.992069.41.285268 11 31.714732 5 56.277991 1 1.992044 41.285947 11'30.714053 4 57.273645 10.67 992020. 286624'11.'2.713376 3 58.279297 10.8.991996.4.27301 11..712699 2 59.279948 10.991971.41.237977 125.712023 1 60.230399.991947.2S8652.711-348 0 r. Cosine. D. 1", Sine. D. 1". Cotan. I. 11'. Tang. I. l00G9:t>)3 COSINES, TANGENTS, AND COTANGENTS. 135 1t t 1 680 M1. Sine. D. 1". Cosine. D. 1 D..". Cotan1g. M. 0 9.2S0599 10 9991947 41 9.2652.23 0.71l343 60 1.231248.108 ~991922 41 2 9326 1122.710674 59 2.281897 10 991897'41'' 39999.710001 5 10.79.412 11.20 9 3.232544 79 1 991373'4 1 293)671 1 1' I 709329 57 4.233190.99184 41 4.2913412 1 701.6513 56 5.233336 10.76.991823.2920131.707937 5 5 6I.284480 10.7.991799.41.29 2 1.707318 54 7.2S5124 10.72.991774 *41.293350 12 7066 5 10871.36 10. 71 8 235766 111.69.991749 *41.231[-J17 1 1.75975S3 52 9 23640 10.67.991724 42 294634 11 09.705316 5 1 10 9.287043 10 66 9.991699 4 9,295319 0 0.70465.1 50 1.96 4.) 11.'6 3.498 11.237633 10.6.991674.4.29619 61 3 703987 411 12.288326 10.63 ~99161! I.296677] 1 0 703323 4s 13 233964 10.63 99162-1 42 297339 11.04 702661 47 14.239600 1059.991599 42 293001 11.03 701999 46 15.290236 1 05.991574 42 29662 11.0.701:333 4 5 16.290370 1 05! 991549 142 2993~2 1 700678 44 17.291504 10.5'6 9151.299930 10937.700020 43 1 91 10.55 9 4) 10.97 56 1 3 298137 1053 991493 * 42.300633 10.95 69936 42 1 9.292763 1051.991473 42.391295 1093.6903705 41 20 9.293399 9.99144 4 9.301931 0.698049 40 10.50.42'. 10.9"' 21.291029.1 991422 / 13096'17 1,092.697393 39) 22.29465 10.48 991397.42.3)3261 10.89 696739 33 23.295286 ( 991372.303914 0 696086 37 10.45.42 10 8.' 24.293913 10.43. 991346 42.303914( 67 1. 695433 36 10.42.2. 10.84 25.2963539 10{ 4 991321 4 3052tS 1.694782 35 26.297164 10.40 99129.3 308369 1 691131 34 27.297783 10.39 991270 3 30659 3 6934 33 23.293412 1..991244 43.307163..692832 32 29.299034 10.367 991218 43.307816 10 3.692184 31 30 9.299655 9.991193 9.303463 0.691537 30 31.300276 1 991167.309109 1 690391 29 32.300395 10.33.99114 43.30974 1076.690246 20 33.301514 10.31 991115 *3.310399 10.7 639601 27 34.302132 10. 991090.4.31102 69 2 35.302743 10.23 99106.3.311635 1071.63335 25 36.303364 10.2 991033.43 312327 0..637673 24 37.303979 9901.312963.637032 23 3.304593 1023.343 10..66392 22 39.305207 10 20 990960 43.314247 10 65.65753 21 40 9.305819 10.1 9 9.99093 9.3145 1062 0.6511 20 41.306130 10.1.99090.41.315523 10..684477 19 42.307041.990332.316159 10..6 1 1S 43.307650 10.14.9905.44.31679 6320 17 44.303259.9929.317430 07.62570 16 45.303367 10.12 99003.318064 5.681936 15 46.309474 10.1.990777.318697 1.5.631303 14 47.310030.990750.319330 1.60 0670 13 43.310658 10.09.990724.44.319961 10..670039 12 49.311239 1006.990697.320592 100.67940 11 50 9.311893 9.990671 9.321222 0.673778 10 51.312495 1 99064.44 321851 0.4.67149 9 52.313097 10.0 990618.44 322479 10.4.677521 8 53.31369 10.0.9909.44 323106 1044 676394 7 54.314297..99056.44.323733 143.676267 6 55.314S97.99033.32438 0.4.675642 56.31549 9.9.99051.32493 67017 4 57.316092 9.96.990435.4 325607.674393 3 57 102 9.94.4o 10.39 53.316689.99045 4.32231 107.673769 2 59.31723 9.990431 4.3633 106.673147 1 60.317879.990404.12747I.672525 0 M. Cooaine. | D. 1". Sine. D. 1". Cotang. D 1". Tang...o Cosi6. Tan. r1. 1010 1 6*,gc 186 TABLE XII1. LOGAR1TIIT IIC SINES, 120 1670 M. _Sine. D. 1_. Cosine.. ".I Tang. D. 1". Cotang. M. 0 9,317879 9.90 9. 9934'04 45 9.327475 10.3 0.6725 60 1.318473.9 990378 /.328095 103.1671 05. 9 2.319066 9.990351 4.328715 10.32.6712S5 58 3 31958 986.990324.3293341.67CG61 5 9996.99032431 4.320249 984.990297.329953 109.69 0047 / 5.320840 9, 8.990270'.330570 1,', 66430/28;7,o o (3 5, IU.40 6.321430 900213i.331187 1027 8.s3 54 9.81.4.318 10.27.6GS) 3~ 7.322019 98 990215.331803.6697 3 9'J o'.4' 10.25 O. 97.322607 9'0.990188.332418 1.667582 52 9 -123194 9.990161 4.333033 1023 666967 51 10 9 3230 9990134' 9.333646 102 0.666354 50 0L7o' 95 9 10 21 1.324366 97.990107 4 2.3342591. 665741 49 12.324950 0 97.990079 46.33471.66129 49 9.73.46 ~~~~~~~~~~~~~~~~~~~~~~10 19 13.325534 ".990052 46 3354821. 664518 47 14.326117 9 2.990025 1.336093106.663907 46 15.326700 970.989997.46.336702 ['.663298 45 9.J OP' qO' |I.4 10.5 16.327281 96 989970.337311 1.6626 4.989970 6 10.15 17 327892 68 6~~.3373~11 999999.~2,44 17.327862 66 6989942.337919.662081 43.989915 6 10.11 1.328442.65 98991.6.338527 1...661473 42 19.329021 964.99887 46.339133 ioio.660867 41 20 9.329599 9'62 9.989860 46 9.339739 100 0.66061 40 ~~~~~~9.33939 n'os 10.60 3 |4 21 330176 9.61.989332 4.340344 1007.659656 39 22.330753 960.989804 46.340948 1006.659052 38 23.331329 9.5v 989777 /' 341552 io.658448 37 24.331903 9..989749 46.342155 1003.657845 36 25.332478 96.989721 46.342757 o].657243 35 26.3333031 94.989693.343358.656642 34 5~j4.4 10.01/ /o 27.338624 90,.989665.343958.65642 33 2,.33419 9.989637.344558.655442 32.0~2.47 9.9s 3 29.334767.989610.345157 64 31 9.50.47 9.97895823 30 9.355337 94 9.989582 47 9.345755 996 0.654245 30 31.4 -479'96 31.335906 948 95933.346353 9 63647 29 32,4.33617 a a.959~ 32.33647 94.939525.346949 9.630o51 28 33.337043 4.989497 34.47545. 9.652455 27 34 376 9.45.47 9.92 34.33 94610.9,9469 47.348141.651859 26 35.338176 9.43 989441 47.348735 990.651265 25 36.338742 941.989413..349329 9'.650671 24 37.339307 040.93938 | 4.349922 87 650758 23 38.339371 93.989356.350514 986.649486 22 39.340434.989328.351106 9.85.648894 21 40 9.340996 9,36 9.989300 9.351697 9 84 0.648303 20 41.341558 93.989271 4' 35287 982.647713 19'3.47' r 9 ~~ 42.342119 934.989243.3528s6 9.647124 IS 43.342679 6462989214 353465 s.646535 17 443.343239 ).12 989214 A1'353464 5 9.SO 44.34323 9'. 969166 | 354053.645947 16.' 9579 1 45.343797 3.98917 9 354640 978.64560,;.30 /.397 6.34:355 929.9S9126.355227 9.644773 14 9.29.48~~~~q 9'76 7.344912 989100. 355813' -.644187 13 9.27 AS 1 48.345469' 926 907 41 3,G56398 7.643602 12'i.6 43 0 1 S 1 49.346024.969042.356992.43018 ii 9.2;5.43 9.73 50 9.346579 924 9.989014 9.357566 972 0.642434 10 51.347134 9'22 98985 8.358149'64-11851 9 v 22z' ~' 48' u7u 52.347687 -.98,956. 358731 9.0.691269 8 53 3163408' 2 99 53.3S240. 9885927 359313.6-10687 7 1 9.20.48 9.6 54t.3t8792.983898' 359393.64o lo7 6 9. 19 48 967'640107 55.3-19343'917.988869.'3604/4'.639526 5 56.349893 9.16.998840 4 361053 9.633947 4 57.350443.988811.48 361632 6368 3 58 338992 9 3612l0 5.938381 3 59.350992 9'14.988782 I4'./362210 9620.637790./.s9 2 59'351540 13.98s753 49..36787.637213 1 60.35208.988724 363364.636636 0 10. Cosine D U'. Sine 1) 1" Cotang D.1'. Tang. M1. 1020 COSINES, TANGENTS, AND COTANGENTS. 187 130 166P M5. Sine. D. 11". Cosine. D. 1"., Tang. D. 1". Cotang. i. 0 9.352(38 911 9.938724 49 9.363364 0.636636 60 1.352635 9.10.988695.49.363940 9.60.636060 59 2.3a3181 9.09.98S666.49.364515 9.59.635485 58 3.353726.933636.49.365090 9 5.634910 57 4 ~354271 9.07.933607.49.355664 9.55.634336 56 5.3548315 9.05.9S9578.49.366237 9.54.633763 55 6.355358.98854.366810 54.633190 54 7.355901 9.03.9833519.367382 9..632618 53 8.356443 9.02 489.367953 9.52.632047 52 9.35694 9.01.938460.49 363524 9.51.631476 5t -10 9.357524 9.988430 9.369094 0.630906 50 11:358064 8.9.988401 49.369663 9.49.630337 49 8.9~ -9.0.49 9.48 12.358603 8.97.938371 370232.4.629763 4 13.359141 8.9388342.50.370799.629201 47 14.359678 8.95.938312.50.371367 9.44.628633 46 15.360215 894.938282.50.37193.43.606 45 16.360752 8.92.938252.50.372499.627501 44 17.36127 8.91.938223.50.373064 9.42.626936 43 18.361822 8.90.988193.50.373629 9.41.626371 42 19.362356 8.89.93163.50.374193 9.40.62507 41 8.69.09.39 20 9.362839 8 83 9.933133 9.374756 0.625244 40 21.363422 8.87.933103.50.375319.624681 39 22.363954 886.958073.50.375881.624119 3S 23.3644835.988043.50 376442 9.36.62355 37 24.365016 83.933013 5.377003 9.3.622997 36 25.365546 8.82 933.377563 32.622437 3 26.366075 8.81 7953. 37122 9.32.621878 34 27.386644 8.80.937922.50.378631 9.31.621319 33 28.367131 8.937892 50.379239 9.30.620761 32 29.367659 873.93762.51.379797 9.29.620203 31 8.9 8862.51 9.23 30 9.363185 8.76 9.97332.51 9.330354 0.619646 30 3L.363711 8.75.937801.330910 9.2.619090 29 32,369236 874.937771.51.331466 9.2.6183534 23 33.369761 8.73.97740 51 2020 61790 27 34.3702385 872.937710 51.32575 9..617425 2G6 35.370303 8.7l.937679.5[.383129 9.23.61671 25 36.371330 8.70.987649.5.33682 9.22.61631 24 37.371852 8.69.97618 51.34231.615766 23 3S.372373 8.63.97538.51.334786 9.20.615214 22 39.372394 8.66.9S7557 1.335337 9 1.614663 21 40 9.373414 8.65 9.938587526 9.17 0.614112 20 41.373933 8.6.937496.51.36438 916.613562 19 42.374452 8.63.937465.51.386937 9.1.613013 1S 43.374970 8.62.987434.51.337536 9.1.612464 17 44.375487.6t.937403 51.33S08-4 9.12.611916 16'3586311 9.11.376003 860.937372 52.38631 9.1 61369 1 46.376519 859.937311 52.3917.1.6102 14 47.377035 8.58.937310.52.389724 9.09.610276 13 49.377549 8.57.937279.52.390270.609730 12 49.378063 56.97243 52.39015 07.60915 11 8.56.5;2 9.07 59 9.373577 8.55 9.97217.52 9.391360 9.06 0.608640 10 51.379039 8.53.937186.5.391903.608097 9 52.379601 8,52.937155.52.392447 9.05.607553 8 53.330113 i.9S7124.392989 904 607011 7 51.330624 8.51.937092.393331 9.03.60669 6 8.n 3313 9,02 55.391134 8.5 0.937061 52.394073 902.605927 5 56,.3S]1643 1843 937030 52.394614 00.6053S86 4 57.332152 8.47.93699S..395154 99.604846 3 53.332661 346.96967 52.395694 8.93.604306 2 59.333163 84.93636 2.396233.60.3767 1 60.33367o.96904.396771 8.603229 0 1. Cosine. D. 1Pt.' Sine D. | 1D. - Cotang. D. 1". Tang. MI. 103D 76) 188 TABLE XIII, LOGARIHIIT iIC SINES,3 1,1o_ 1653 M. Sine. ID. 1". Cosine, D. 1".~ Tang. D.1. Cotang. - M. 0 89.33 44 9.936904 9.39671 - 0603229 60 1 334182 843.986873.397309 8 9602691 59 0346537 4.986841 3,3978346 602154 58 ~.4 ~-o'.,53 o.uo.3o192 841.986809.3933.601617 87 4.33'697.986778'S..093919 31.601081 96 5.40 ~~.o.3 8 93 6935 0 o 386201.936746. 3994559 3 60545 55 6 336704.936714 8 3 6.399990 5 83 5.38o697 387207' 986683.400524. 59976 ~.3 53 89 3.387709.986631 5.401053 8 3 o598942 52 9.383210 o',,.936619.401591.598409 51 I. ooI.5o! / o8o 10 9.383711 / 9.986537 9.402124 8 0.597876 50 11.339211 8.986565 [.402656.597344 49 833.53 -.oo 12.339711 8.9"6o23'.403187.596813 48 13.390210.956491.403719.596282 47 14.390708.93649.40249 8.595751 46 15.391206.936427.404778 8.595222 45 16.391703 8.93639'.405308 882.592692 44 17.392199 8.966363. 400336.594164 43 18.392695.98 6 33 1 406364 0.593636 42 19.393191 8. 2 9699.406892.959310 20 9.393635 9.966266 / 9.407419 0.592581 40 ~821.4 0 879 21.394179.966234.407945.592055 39 22.394673 2.936202 [ 403471 8.591529 38 23.395166 82.966169.408996.591004 37 24.395658.936137.409521 8 0 590479 36 25.396150 8.096104.410045.539955 33 26.396641. 9862.410569 8 53.89431 34 o lo.5~t 8./z 27.397132 8 1 936039.4 411092.588908 33 0.11'. x.-4 O. 1 23.397691.936007 / 411615 533335 32 8 16 4 8.70 6 5.394111 98677 41257 8 53743 25 36.401520 8 03 93074) 4l5 8 63 26 9 4 37.394600 7 9.935942 9.412658 0.537372 3 3 99028'.980 909.413109 861 o31 22 8~~~ ~~~~~~~.5851. 8 29 39.399575 8 o!9o846.413699 8. 6 586301 41.400062 8 9053 43.414219.585731 27 42.40051 9,930811.41473.8 [ 585262 26 3.401031' 98577'.415297.58 4743 17 45.403362,8 055 9 5.4 4 46.4011 93o44 41577 8 5.594225 4 O.U3 - DO o.uo 47.4(16320 93)31 42140 8 6.513707 23 43.40255 9 2.8 7 489.407777 7 93679 6 416310 2.573190 22 8.06.55 8.06 51.402972.93646.4173264.52674 21 o.uo!.50 II /.uu 52 9.409237 9 985613 6 9.417842 0.52158 20 53~~~~~.00963 9613 o 4939 849 549 7. 55,-./.9,57, 0.4183103 /.s42661 19 57.411799547 7 930.4144 732 34.581127 13 3.41205255 7.7 2 59.412952 934978, -813.25o613 17 X.U. Coie55 1 ot. 44.40538 [.980430.419901.580099 16 45.4058,',~.9~o147 /',.4204151 ~,./ 9585 15 46 ~'... 4(61.13/L.55.~10... 79 ~01 2927.579073 14 7.97' - 56' 8 04'''6.578560 13,,7].9 954 29.5704 50 9~~~.,'~, ] 9.938520 / ro]9.422974 /', /0.577026 1 51,40373L ~. /9q,,'-17 | ~423:184 /.576516 52.409207.9%213 " 4~3~993' 576007 7 93' - 56', ~849 53.40963 "n.9>~310' n.424003 o, 559;'~- I5 25 ~l ].....'3 51.410157.y~.9>s146' r, 4.011.574989 6 7 91 0~6',-847.4L0632.~9:-5113' 420019.~'-~ 1 7 90' ~ 56'.zi0 56.4111'-6.9o5079 /.~]4260;?7 /.../.57:3973 57.411579.9550~.~,~.42a.3-:,.i ~.5734166 53.41205 /'~~~.9850ll {;b 427041 }~'i /.572959 59.4L25' __.994978 rI.427047 } "|.572453 10t4 l~S COSINES, TAiNGENTS, AND COTA1NGENTS. 189 F15o 164o M. Sine. D. 1". Cosine..1. Tang. P. 1". Coteng.. 0 9.412996 8 9.9849~44 6 9.426092 84 0.571948 go 1.4134_67 784.94910 I.429558 241.571442,9 2.41393' 78.984876.429062 840.570938 5/ 3.414408 73.984842.429566.570434 57 7'82.9888 57 I.4070 4.414878 782 964808 O'.430070.569930 /56.415347 7 984774.430573.569427 6.415815 70.984740.431075.568925 54 7. 7468 ~ 94706. 1 52.41671 7.9.467.431577 8.568423 53 Jo175,8' 7 I 4320791.672.417217.984638 s ~.432580 84.567420 5 7.77.57 5.35 10 9.417684 7 6 9.984603 9.433080 8 33 0.566920 50 11.418150' 984569'.4335,0.66420 49 12.7.4161 5 4350 833.640 49 12/.418615 / ~'~' /.984535 /.434080 832.565920 48 1.15 o/3'. 13.419079, 984500.434579.56042! 47 14.419544.984466.435078.564922 46 7.73.4307 8.31 552 47i 15.420007 772.984432 435576.564424 1 6.2 5 82 44 16.420470 7.984397.s.436 8273 9.563927 17.420933. 58 ~,.436570 828.5634309 4 7.70.463 58.436570 828.563430 43 18.421395.984328's.437067 827.562933 42 7.69 ~ ~.;~ 19.421857 7.8. 64294.437563 8:6.562437 41 20 9.422318 7 7 9.984259 58 9.436059 0.561941 40 21.422778 767.984224 o.438554 [.561446 39 2 2.6.4238238 v~ 22.423238.984190.439048 824.560952 38 7.66 85 524 23.423697' 984155. 439543.560457 37 24.424156 7. 984120. 440036.559964 36 7.64.58.406 8.22 25.424615.984085.440529.559471 n 26.425073 7. 984050.58.441022 020.558978 34 27.425530 7.984015..441514 82.05a4a a/ 28.425987 7.983981..442006.557994/ 3 29.426443 7/ 983946.8.442497.557909 3 7.60.58 8:18.553 3 30 9.426899 [ 59 9.983911 9.442988 8 17 0.557012 30 31.427354 758.983875 816.556521 29 32/.427809 7.5.58.443968.556032 28 32.42786 /.983805'.444458.555542 27 34.4877 7.56 9870.59 8,15 34.42877. 983770.444947 814.555053 26 35/.429170/',./.98373o /'~~ 445430/5,-.65456~/ 20/ 7.55 8123.565 25 36.4296.3.983700'.445923 813.554077 24 37/.430075' 983664'.446411 812.553589 23'7 02 ~ 5.22 38 {.430.527 983629.446898 811 r 752 /5 81 39.430978.983594[.447384.552 22 7.51.59.5561 2 40 9.431429 7 9.983558 9.447870 0.552130 20 r'50 80 41.431879.983523.448356 8.551644 19 42.432329.983487.448841.551159 18 43.432778.983452'.550674 17 7,48.4432 8.07.510 44 4.433226 /'..983416.449810 806.550190 16 45.433675 746.983381.450294 806.549706 15 46.434122.983345'~.450777 8.549223 14 47.434569.983309.451260 804.548740 13 48.435016.983273.6.451743 /.548257 12 49.435462 983238.452225 7.43.60 8.03' 50 9.435908 742 9.983202 9.452706 0.547294 10 51.436353 741.983166..453187.8 7~~ ~ ~ ~~.4581 /6 8. 54613 9 52.436798 7:40.983130 6.453668 8.546332 8 53.437242'7.98309/.60. 454148 8'00.545852 7 54.437666.983058..454628.545372 6 55.438129 738.983022..455107.544893 I 56.433572.982986.60 455586 7.98.544414 4 /.3.50' /9 t5u' 7 w 57.439014 76 982950.'456064 _.54393 3 r,'7 36' 60' 797 58.439456 8.92914. 456542.543458 2 7.36 ~~~8.60.642 7.96.543458 2 59 7.43997.98287.457019.542981 1 7635. 4546.542504 0 60.1 433.... —-o__ -7.!o ~ ~%7~ ~,~,% 51. Cosine. D 1". Sine. P. 1'. Cotang. D. 1 Tang. 1. 1056 40 19[0 TABLE XIII, LOGARITHI\IC SINES, 16o 163 I. Sine. |D. 1'. Cosine. D.!. Tang. D. l. Cotang. 1. 0 9.440331 7,34 9.932842 9.4774716 05-12504 60 1.410778 7-33 932-05 3 9 457973 79.542027.9 2.4412138 *.932769 6 458449 7 54155.9723 3.61 7.93 3.441653 7.31 927393.61.4o0925' 792.41075 57 4.442096 731 0982696 1.459400 7 9 540600 56 5 442535 731 932660 61 45975 7.91.510125 6.442973 79 982624 61 460349 790 539651 541 7 o443410 7 92587 61 460323.589 39177 3 6.443S-17 7.27 982o51 61 461297 788.538703 5 7.27,61 7.8. O9 494234L 7277.93214.1 461770 83 533230 51 10 9.444720 7 26 9 982477 61 9.462242 7.7 0.537756 50 11.445155 72 92441'61 46271.5 7 3537285 49 12.445590 24, 932404 61 46316 786 53614 4 13.46025 724 98367 61 463652 47 14.446459.92331.61 464128 7.8-1.535872 46 15 44693 72 932291 61 464599 7 53401 45 3(330 7813 16 447326 721 9227 465069 7.3 534931 44 17.447759 723 932220 6.465539 7 52 34461 43 18 448191 70 493213 62 466003 71.533992 42 9.49054 7.18.232 9 923.80 23 7 93.632 750 0..3303 4 21.4194348 7.17 932072 (69.467413 37 8532587 39 22.449915.17 9203, 6.6730 7.532120 3 7718', 3.37 23.45035 17.98199.46337 78.531653 37 2-1.40775 7.1.931961.62.463814 7.7.5311S86 36 23.451204 7.14.91924.62.469280 7.76.530720 35 26.41632 7..9 6.62.469746 7.76.530254 3 27.4-2060 3.931849 2.470211 529789 33 23.452148 71.'13931812 63.470676 7.529324 32 29.432915 71.931774 (3.471141.528359 31 30 9.495334237 9.471605 0.528395 30 31.4i373 9,81700 62 472069.527931 29 32.454194 7..9816363.'63.472532 7171.527463 28 33.454619 7..981623.63.472995..527005 27 34.45504 7..93157..473457.70.526543 26 35.455469 7.07.931I49.473919 7 56.526031 25 36.45393 0.9112.474381 9.525619 24 37.456316 7.03.9815174.63.474842 7.6' 525158 23 33.46739 70.931136.47303.524697 22 39.47162 7.0.91399 3.475763 7.67.24237 21 40 9.45754 703 9.931361 63 9,476223 0.523777 20 41.45006 02.981323 3 476633 7.66.523317 19 42.45427.9125.63 477142 7 52285 18 43.45S348 701.9127.63.477601.6.522399 17 4.45926 701.9129 63.478059 7.6 21941 16 5 1.459633 03.931171.63.478517 7.63.521483 15 46.463103 6,98,981133 6:3.478975 7'62.521025 14 47.46027.981095.479432. 520683 13 48.460946 6.97.93105 7.64.479389 7.61.520111 12 49.461364 698.981019 64.480345 761.519655 11 6.96.6 7.60. 50 9.461782 96 9.980931 61 9,48001 7 0.51999 10 51,462199 6. 930942 64.481237.59.518743 9 52.462616 6.94.930904 64.481712 759.528 83 9.45903(3 6 4 12 7.519823 ||53.463032 |6.93.980366.64 432167 737.517833 7 5 46348 93.90327 64.482621 7.7 517379 6 55 468(3.3 55.46364 6 92 930739.64.453()75 7.56.516925 5 |56.461279 |691.980750 6.433329 7.5.516471 4 57.461694.930712 (3.4:3932 75.516018 3 53.4610()3 6.90.930673 (3.44433.515565 2 59.46).522.9S.6.43137 4.515113 1 30 6339.453.45593768.9SO596.485339 514661 ] Cosine. / 1". Sine D., D.21". TangIL013D' ~~: I COSINES, TANGENTS, AND COTANGENTS. 191 T3. Sine Cosine. D..Tan. D. Cotan. D. 1'. n. 0 9.46593 6 9.930596 9.485339 7 0.514661 60 1.466343.93055S.45791.514209 59 6.88 2 64 7.52 2.466761.9307 019..4862412 7..513755 53 3.467173 666.930480.6-.486693 7 51.513307 57 4.46735.902 6-.47143.51257 56 5.467996.930403 - 437593.512407 55 6.463407 68 930364..4043.51197 54 7.463517 684.90325 6 492.511503 53 8.469227 63.930256 6.453941.511059 52 9.469637 6 83.90247.489390 7 4.510610 51 6.82.6 747 10 9.470046 6 9. 950205 9.439 33 0.510162 50 11.470455 681.930169.65.49026 74.509714 49 12.470363 6.80.980130'65.490733.509267 48 4918.502523 47 13.471271 6.79.980091.65.491180 7.4.508820 47 14.471679 6.7.980052.65.491627 744.50373 46 15.472056.90012 6.492073.507927 45 16.47242.979973. 492519 43.507481 44 17.472893.9799346 66.492965 742.507035 4 15.473301 676.979895 66.493410 7'41.506590 42 19.473710 675.97985 66.493354 741.506146 41 6.75 7.41 20 9.474115 9.979316 66 9.494299 0.505701 40 21.47519. 979776.66.494743 7.505257 39 22.474923 6.73 979737.66.49186 504814 3 23.475327 6.7.979697 66 495630 7 504370 37 24.475730 672.979658 66.496073 7.o3.503927 36 25.476133 6.71.979618.66.496515 7,37.503485 35 26.476536 6.70.979579.66.496957 7536.503043 34 27.476935 6.70.979539.66 7.4973 36.502601 33 28.477340 6.69 979499.66.497841 7.36.502159 32 29.477741 6.63 979459.66.493232 73.501718 3 30 9.478142 6.67 9.979420 66 9.493722 7 0.501278 30 31.478542 6.67.979380 66.499163 7.33.500837 29 32.478942 6.6.979340 67.499603.3 00397 28 33.479342.979300 67.500042.499958 27 34.479741 6. 65.979260.67.500481 7.3t.499519 26 35.480140 66.979220'67.5(0920 731.499030 25 33 6.66.67'1.31 36.480539 6.64 979180 67.501359 7.30.492641 24 37.480937 6.63 979140 67.501797 7.30.498203 23 3S.481334 6.979100.67.502235 7.2.497765 22 39.451731 6 61.979059'67.502672 7.23.497323 21 6.61 72 40 9.452128 9.979019 67 9.503109 7 2 0.496391 20 41.482525 6.60.97 979 0346 7.496454 9 43.483316 6.59.9783904418 727 49552 1 7 44.423712 6.59.978853.67.504854 7.25.495146 1 6 45 484107 6.57 97898.67 50239 7 2.494711 1 5 3.423316.497322 4fi 44501 6957.2978777.67.505724 7 26.494276 14 44.483712 9.979 019 / 43.454107 657 67 [.501509 7 494711 i 47.43489. 6.56.978737 63.506159.493841 13 48.455289.978696..50693.493407 12 49.435632 6.55.9765 63.507027 723.492973 11 50 9.436075 9.9786615 9.507460 722 0.492540 10 51.456467 6.54.978574.6.5075393 7.2.492t07 9 52.486860 6 5.97333.63.50332 7.21.491674 8 53.4872 1.973493.50379.49 241 7 54 437643 6.52.9783432..509191 7.20.490309 6 5.4034 6.52.978411.6.509622 7..4903738 1 56.455124 6.51 63 7.19.439946 4 56 3~44.978370.510054 7.17 57.4533514 6.50.978329.6.51049 7.15.489515 3 53.4924 6.9782 3 s.5109126 7. 1.489084 2 59.48499593 6.43 9 978247. 50.5113716 7.17.4326 34 60.49932 6..978206.511776.438224 0 6.Cosine D.1 1"_ Tang. 1. M. (osine. D. 5., |Sine. D. 1't. Cotang. D. 1M. Tang. M. 110 192 TABLE XIII. LOGARITHMIIC SINES, 18o 1601 M3. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M4. O 9.439932 6 48 9.978206 63 9.511776 0.483322-L 60 1.490371.978166..5122016 4 4 59 2.490759.978 124 9.51263.. 4i73 589 3.491147 6.46.903.69.513064 71 t4.4 63 57 4.491535 6.4.97S042.69.53493.1.41607 56 5.491922 6.4.978001.69.513921 7.1.4~6379 55 6.492303 644.977959 69.514349 713.48 5651 54 7.492695 643.97791.69.514777 72.45223 53 8.493081 6.43.977877 69.515204 712.484796 52 9.493466 b642.978315 69.515631 711.484369 51 10 9.493851 6 9.97794 69 9.516057 7 10 0.483943 50 11.494236 6. 1.977752 69.516484 7 10.483516 4 12.494621 640 977711 69.516910 709.483090 48 13.49500.97669 69.517335.4S2665 47 14.4953388. 9 97762S 69.517761.4.0 2239 46 15.495772 6.3.977556.9.51816.481814 45 16.496154 6,33.977544.70.518610 7.07.481390 44 17.496537 6.37.977503.70.519034.7.480966 43 13.496919 636.977461 70.519458 7.0.480542 42 19.497301 6 36.977419.519882 7.06.480118 41 20 9.497632 9.977377 70 9.520305 705 0.479695 40 21.493064..977335 70.520728 704.479272 39 22.498444 635.977293 70.521151 704.478849 38 23.498825 6'33.977251.70.521573 7.03.478427 37 24.499234 633.977209 7.521995 7.3.478005 36 1033 [70 7.03 [ 25.499584 6i 32.977167 70.522417 7'02.477593 35 26.499963 31.977125 0.52283 72.477162 34 27.500342 6.31.977083..523259.0.476741 33 28.500721 6.30.977041 70.523680 7.01.476320 32 29.501099 6:30.976999.701524100 7.47900 31 30 9.501476 629 9.976957 9.524520 699 0.475480 30 31.501854 62.976914.7.524940 9.475060 29 32.502231.97672 525359 6.99.474641 2 33.502607 6.27.976830.71.525778 6.98.474222 27 3.502984.97677 71.526197 6.9,473803 26 35.503360 6.27.976745 1.26615. 47 73385 25 36.503735.26.976702.71.527033 6.97.472967 21 37.504110 6.2.976660.71 527451 6.96.472549 23 6.25 4 71.4 6.96, 38.5(4485 6624.976617.527868.472132 22 6.24.71 6.05 - 39.504860.976574 71.528285 6 4717115 21 40 9.505234 6 23 9.976532 71 9.523702 6 9 0.471298 20 41.505608 6..976489'71.529119 694.470881 19 42.505981 622:976446.529535 6.9.470465 18 43.506354.2.976404.529951 6.93 470049 17 44.506727 621.976361 71.530366 692.469634 16 45.507099 6'20.976318 72.530781 691 469219 15 4(.9507171 6.19.976275 72.531196 6'91.468804 14 47.507843 19.976232'72.531611 690 468389 13 48.503214 6.18.97619 2.532025 467975 12 618. 3'.97601 49.508585 6146.532439 6.467561 11 6.18.72 6.5/ 50 9.50.8956 6.17 9.976103 72 9.532853 6 0. 467147 1 0 51.509326 6.16.976060.72 533266 6.88.466734 9 52.509696 16.976017.72.533679 688.46632 8 53.51065 5 97974 2.534092 6.87.465908 7 54.510434 6.15.975930 72.534504 6.87.46549 6 55.510803 6.14.975887 7.534916 6 S6.46504 5 56.511172 6.94 7544..535328 66.464672 4 57.511540 6. 975800.72.535709 6..464261 3 53.511907 6.13 97577.72 536150'.46350 2 593S225 6.12.72 6.4 464 1 59.512275 6.12.975714 72 53661 64.43439 60.512642.975670.536972.463028 0 M.A Cosine. D. 1". Sine. D. i". Cotanr. I D. 1. Tang. M. 1087 7s O COSINES, TANGENTS, AND COTANGENTS. 198 196 10~ M5. Sine. DI.. I. Cosine. D. Il. Tang.. Co ang. M. I 0 9.512642 9975670 9.553972 0.46302 r 6.11.73 6.841: 1.513009.97'627.3 53732 I.462618 59 2 it 9/o0~' t~ ~ 6,$3 2.513375.9755 3.537792 6.4605.3 513741 603~~9.5336202 K I391 6.09.73.,o.,.513741 6.0 ]90G3 7.53S202 6.823.461798S 57 4.514107 609 975496..46139 56 6.09.7 661 46 67.514337',,75408 7.53429.460971 60~ ".u i f' 7.~ G.82 7.515602 607.975365.73639S37. s 460.63 53 6.07.73 GSO~~~6.5 8.515566 6.07.597521 73 31245 45755 52 607 9706 6. 4616 9.515930 66.975277 06) 6.459347 5 10 9.516294 605 9.975233 9.541061 0 458939 50 11.516657.975189.5431466.45S53 49 12 517020 6.975145 3 541675.458125 48 13 5173' 04 6,78 1 3.51738: 0 975101.7 0421 6.457719 47 14 517745 6.04.975057. 562 6.77 457312 46 6.03.7.3 6.77 15.518107 6 975013.53094.4656906 415 16 5163 603.974969.54399 66.456501 44 ~51702 ~ 7 6 6.02.74 ~~~~~~6.78 17 518829 602 946 024'543905 67 4609) 1 8.519190 602.974880. 4310 453690 42 6 01.74 6.75 425 1 19.519351 60.974856 74 344715 674 91 9.519911 9974792 9.54119 0,45551 40'6.00.7 5421 6.74 21.520271 9 974748.5424.454476 38 5.99,74 6~Tn, 22.520631 5 974703 74 545928 63 454072 3 14.9. o46~ 6.'77 23.520990.97469.546331 6 A45369 37 2403 6 -72 24.521349.974614 74.546735 672.4532650 3 25.52107 97470.74 547133 671.4526 3 26.522066 7.9742 74.517340 6.71 452260 34 27 522424 56.9741 7 0 ~547943.4520 3 2.522781.974436 74.5 70 456 30 9.523495 9.974347 9.549149 0.450851 30 31.52352' 974302.75 49)30.450450 29 5.94.7'5 6.75 32 1.52420.97427 7 54991 6.6 450549 28 33.524561 93,974212. 55032 6.68 449648 27. 2.93.75 6.67 449248 26 31.524920 5.97467 7.50752 135 3 523275 592 3 3.6974122.55115347 25 36 1.525630 91.974077.551552 6.6 448448 2 37.525934 5.90.974032 7 551952 6.66.44048 23 38.526339.973987.552351 6 447649 22 39.526693.973912.552730 6.65 447250 21 5..S~~9.75 6.65 40 9.527046 8 9.97397 9 553149 0.446851 20 5.89.75 6.61 41.527400' 6.973852 3348 6.64 446452 19 4 2.5277;53.973807 7.553916 6.63 446056 17 4 3.59~2105.973761.554)3444 4 54 6 56 1 7 261 87 6.75 16.63 44 57)2B8.973716 6 554741 6 445259 16 4.52SS 10 o..973671 355139 62 44661 145 462 )2 6.62 444464 14.76 6.52961 5q96].973623 76 3)36 6.61 4446 1 ~~~~~~~~~.545928] 4 231.96529 5 73.76 3.1440671 1 3 48 2)61 8.973535 6.5563.1 6.443627 12 5.S5.76 6.60 49.53021.9739 76.552 6.60 3275 11 50 9.53 5' 9.9744 76 9.557121 639.4279 10 2~ 52066,~ i7 6.9742 2 I.5 583 97339''.472483 9 51.. 13 3 9/3398 76 5.2'3126. 9733 557913 6 442087 8 5.,, 93.76 6.59 53 531614 2 9733307.5756308.4641692 7 281 5. S2.76 6.4 3.551 31963.9761.55703 441297 6 5 532:3L2 5 973215 67.55997.440903 5 56.53661. 1 973169 76.55991 6 440509 4 5.80.76 6.57 57.533009 so.975124 76 5595 637 440115 3 58.533357.973078.560279.439721 2 59.533704.973332 7 560673 656.43932 1 5.79.77 ~~~~~~6.69 60.5310352 5'.972986 7!561066 6.55 436934 0 AI. Cosine. D 1" Sine. D. I". Cotsan. D. 1"1 Tang. MI. 03'[90 194 TABLE XIII. LOGARITHSBIC SINES$ 20o 1590 M. Sine. D.11". Cosine. D. 1". Tang. D. 1". CotaLU j J. O 9.5 34052 9.972986 9.561066G 0.438934 (i)0 1.534399.972940.561459 6'.43 541 59 2.534745 5,77.972894.7.561851 r 43149 58 3.535092 5.77.972848.5 7.562244, t4.4377516 57 4.535438 5.7.972802.77.562636 6 5.433641 53 5.535783 5.76.972755.77.563023 3.6972 55 6.536129 5..972709 77.5634 19 6.32.4 36S ] 54 7.536474 5.75.972663.563811 62.436189 53 8.536818.972617.77.564202 6.51.43579S 52 9.537163 574.972570.564593 651.43507 10 9.537507 73 9.972524 9.564983 650.435017 50 11.537851 5.73.972478 *77.565373 6 50.434627 49 12.538194 5.72.972431 *78.565763.50.434237 48 13.538538 5.71.972385..566153..433847 47 ]7 8'.78 6.49 14.538880 5.71.972333.8 566542 6.49.433458 46 15.539223.972291.78 566932.49.433068 45 16.539565 570.972245 567320.432680 44 12.78.3 6.48 4 17.539907.70.972198.78.567709 6.48.432291 43 18.540249 569.972151.8.568098 6.47.431902 42 19.540590 568.972105 78.568486 6.46.431514 41 5.68.78 6.46 20 9.540931 568 9.972058 78 9.568873 646 0.431127 40 21.541272 567.972011.78.569261 6.46.430739 39 22.541613 67.971964 78.569648 6.46.430352 38 23.541953 5.66 971917 78.570035 6.45.429965 37 24.542293 5.66.971870 78.570422 6.44.429578 36 25.542632 5.65.971823..570809 6..429191 35 26.542971 5.65.971776 78.571195 6.44.428805 34 28.543649.971682.571967 6.3.428033 32 29.543987 563.971635 79.572352 6:42.427648 31 30 9.544325 9.971588 9.572738 42 0.427262 30 31.544663 5.62.971540.79.573123 641.426877 29 32.545000 5.62.971493.573507 641.426493 28 33.545338 5.61.971446 9.573892 40.426108 27 34.545674 561.971398.79 574276 60.425724 26 35.546011 5.60.971351 79.574660 640.425340 25 36.546347 560.971303 79.575044 639,424956 24 37.546683 5..971256 79.575427 3.424573 23 38.547019 5.5.971208..575810 6.39 424190 22 39.547354.558.971161.576193 6.38.423807 21 40 9.547689 8 9.971113 9.576576 0.423424 20 41.548024 -.5..971066 80.576959.42304 19 42.548359 5.57.971018 *8(.577341 637.422659 18 43 548693 5.5.970970.577723 637.422277 17 5.56. 63 4226 7 1 44.549027 5.56.970922.80.578104 6.36.421896 16 45.549360 5.56.97074.80.578486 6.36.421514 15 46.549693 5.55 970827.80.578867 635.421133 14 47.550026 5.55 970779 80.579248 6634.420752 13 48.50359.970731.579629 634.420371 12 49.550692 4 970683 8.50009 634.419991 11 5.254'.80 6:34 50 9.551024 53 9.970635.80 9.580389 63 0.419611 30 51.551356 5.5.970586 0.5S0769.419231 9 52.551687 5.5.970538.8.581149.3.418851 8 5.52 80.58149 3 53.552018 5.52 970490.8.581528 6.32.418472 7 54.552349 5.52 970442.5S1907 6.32.418093 6 55.552680 5.51 970394 81.582286 6.31.417714 5. 56.553010 35.1.970345..582665 6.31.417335 4 7.55341 50 1 6.31 57.553341.970297.53044.416956 3 5.60.1 5 6.30 58.553670 5..970249.583422 6.30.416578 2 59.55430 5 9303.570 44O 6.30!.416200 1 60.534329.970152.584177.415823 BM. Cosine. D. 11f. Sine. D; Vr.o Cotarg. D. P'. Tang. M.ikq' ] GOO COSINES, TANGEINTS, AND COTANGE1NTS. 195 2k ~~~~1t'~~~~:a1580 tl. Sine. D.'1". Cosine..1. Tang. D. 1. Coteang. JM. 0 9.554329 5,48 9.970152.81 9.584177 629 0!415 3 - ( 1.55465s 5..970103'I.584555 6.'9.41544 59 J 2.554:87 5.4.970055.584932 628.45068 5 3.555315 5.47.970006.1.585309 6.28.4141691 57 4.55s',643 5.46.969957 81.585686 6.28.414314 56 5.555971 546 969909.586062.27.413938 55 6.556299 5.46.96960'81.586439 6'27.413561 54 7.556626 5.969811 81.5o6S15 68 6.413185 53 8.556953 5.44.969762 1.5S7190 6 26.412810 52 9.557280.969714.587566 6 26.412434 51 10 9.557606 9.969665 82 9.5S7941 0.412059 50 [ 11.557932 5.43.969616.8 2.5831 6.25.4116S4 49 12.558258 5.43.969567 82.588691.411309 40 13.558583 5.969518'82.589066 6. 4.410934 4 7 14.558909 5.42.969469 2.589440 6.24.410560 46 15.559234 5.42.969420.82.539814 6.23.410186 45 16.559558 5.41.969370 82.590188 623.409812 44 17.559883 5.41, 969321,82.590562 6,22.409438 43 18.560207 5.40.969272 82.590935 6,22.409065 42 19.560531 5. 9.969223 82.591308 6:22.408692 41 20 9.560355. 9.969173.82 9.591681 1 0,408319 40 21.561178 5.3.. 969124 82.592054 6.21.407946 39 22.561501 5.969075.82.592426 6.20.407574 38 23.561324 5.3.969025'82.592799 6.20.407201 37 24.562146 5.37.968976.83.593171 6.20.406829 36 25.562463 5.37.968926.83.593542 6...406458 35 26.562790 5'.968877.83.593914 6.19.406086 34 27.563112 5'36.968827.83.594285 6.18.405715 33 23.563433 5.35.96877.56 83 96 6.1.405344 32 29.563755 5.35.9672.83.595027 618.404973 31 30 9.564075 4 9.968678 9.595393 0,404602 30 31.564396 5.3.968628.83.59576 6.17.404232 29 32.564716 5. 3.96578.83.596138 6.1.403862 28 33.565036 5.33.96328.83. 9650 6.16.403492 27 34.565356 5,32.968479.3 96872 6.16.403122 26 35.565676 5.3.968429 83.597247 6.15.402753 25 36.565995 5.32.968379.3.597616 6.15.402384 241 37.566314 5.32.963329 8.3.597985 6.15.402015 23 38.566632 5.31.968278.84'598354 6.14.401646 22 5r,- 6314 39.566951 530 968228 84 598722 6614.401278 21 40 9.567269 30 9.968178 9.599091 13 0.400909 20 41.567587 529.968128 84.599459 6.13.400541 19 42.567904 5,29.968078.84.599827 6.3.400173 18 43.568222 5.29 963027.84 600194 6.13.399806 17 44.568539 524.967977.84.600562 6.12.399438 16 458 563856 28.967927 84.600929 612.399071 15 46.569172 5.27.967876.84.601296 6.11.398704 1 4 47.569488 5.27.967826.84.601663 6.1.398337 13 43.56904 527.967775.84 602029 6..397971 1 2 49.570120 5.26.967725.84.602395 6,10.397605 1 1 50 9.570435 2. 9.967674.4 9.602761 10 0.397239 10 5 1.570751 5.25.967624 8.4.603127 6.09.396873 9 52.571()66 5.2.967573.85.603493 6.09.396507 8 53.571330 5.24.967522.85,603858 6'09.396142 7 5'24 967 oo' o 5u 54.571695 5.24.967471.85.604223 608.395777 6 55.572009 5.23.967421.S5.60453S..395412 5 56.572323 5.3.96730/. 604953 6.0.395047 4 57.572636 5.22.967319 So8.605317 60 7.394683 3 53.572950 22.967268..605682 607.394318 2 59.573263 5.22.967217.85.606046 6.393954 1 60.57357a5.967166.606410.393590 0 M. l Cosine. /D. 11 Sine. BD. l1. Cotang. D. 1". Tang. M. 1110_. 196 TABLE XIII. LOGARITHnI1C SINES, 222 __ _ 157 2 MI. Sine. D. 11. Cosine. D. 1". Tang. D. 1. Cotang. i. 0 9.573575 2 9.967166 9. 60G110 0.39039:3-90 60 1.573333 5,21,967115., f606773 606 393227 59 2 574200 5.20.967064.8.6071'37.. _ 323 2 50s3 3 574512 o.967013.607500 60'( 0 ( 7 4.574S24 5r 1.966961.6)7863 60'.392137 f56 5.19 6.05 5,575136 966910.6083225,.391775 6 575447 60385 604.6 575447. 966S59..f0353 f.36)1413 2 1 2 7 575763,9660S3.603950,39() 5:3 ot8.86 - 6.03 8.576069. 96675612.391605 56 9.576379.966705.609674 0.390326 51 5.17 886 6.03 10 9.576639 517 9966653 86 9.610036 602 0.339964 G0 11.576999.966602.61 0397.39603 2 9 5.16.86 6.02 3"9241 -.b 12.577309 5.1 9660.86.10759 602 341 13.577618 5. 1.966499 6 611120 6'01.33230 47 14.577927 5.15.9664147.86.611480.01.3S520 46 15.57236 51.6 619663941 601.3359 4 16.5783545.. 14 966344.6.612201 6.0.3S7799 1 17.578853 5.14 966292.6 612561 6.33739 43 18.579162 5.13.966240.36.612921 (00.337079 4 19.57970 96618.6132.336719 4l 5.13.86 5 99 20 9.579777 2 9.966136 9.613641 0.3368)9 40 21.530085 2 966035.614000.36000 39 22.530392 5.1.966033..614359'.325641 33 24.55100O.9629 7.615077 9 3492 36 25 5831312 5.11.965876.615435.334565 35 26.531618 5.10.965324.615793.97.3-1207 3 27.51924 5.10.96772.616151 96.3339:3 23.5229 5.0 965720 87.6165209.33-191 32 29.582535 5.09 96663 87.616367 5,9.333133 31 30 9.532340 9.965615617224 0.332776 30 31.533145.96563.6175482.32-118 29 5.03 — ).87 5.95 32.58349 5.07.96511.617939 5..382061 23 33.583754 5..965453.87.618295 5..33170 27 34.58403 5.0s.965406.8.618652 5.94 381348 26 35.534361 5.06.965353 83.619003 5.94.330992 25 36.534665 5.06.965301 33.619364 5.9.330636 2 37.584968 5.0.69720 5 330280 23 33.5325272 5.05.965195.620076.379924 22 39.515574 5.965143.620432 2,793563 21 40 9.585377 9.965090 9.620787 0.379213 20 41.586179 5.04.965037 8.621142 5.92.3783;S 19 42.536432 5.04.96493.8.62197 5..378503 I 43.53678:3 5.03.964931.8.621852 5.91.378143 17 44.53703 503 9679 6222 591.377793 16 45.537336 5.02.9626.62261 5.9.377439 1 46.537683 o 2 942 5 9 3 46 |.5'S | 502.964773 8.622915 5.90.377035 -14 47.53799 5.01.964720.623269.376731 13 43.533239 5 01 96666, 632 5.90.376377 12 49 j533590 j.964613.623976.376024 11 3100.89 531) 50 9.258390 9.964560'9 9.624330 5 0.375670 10 59951 09[90 4 964207.1.89 624633.375317 9 52.5399.964454.89 62036 5.374964 53.539789.964400.89 62533 3 35..374612 7 51.59003S 4.964347 89.62)741 57 374259 6 55.590337.964294 626093.37.907 4.97.89 5.7 t f 56 j.590636 1.3'964240 626445 7 4 57.590934 |497.964187.89.6267)7 05/ S 7C)0:372 3 liH.591232 |49.964133.8 627149.372351 2 4.97 L, 05 r 59 591530 497.964080 9 62701 5.324)9 1 59137 960. 95LO6 S9.67852 5.36 2L43 0. Cosine. Sine. D. 11. Cotang. D. 1' Tang 1 40 9.585877 5.04 ---, ——, [{- 9.620787 —- [ ~.,_,,., ------ COSINES, TANGENTS, AND COTANGENTS. 197 233o 156c 51. Sine. D. 1". Cosine. D. 1I. Tang. ID. 1P'. Cotang. 1. 0 9,591878 4.96 9.96-126.89 9,627852 0. 3721483 10 1,592176.96372.628203.3711797 2.592473 4.963919 90.628554 5.8.371446 538 3,t92770 4.99.96386t).90.628905 5.85.371095 57 4.593067 4.95.963811.9.629255 5.S.370745 56 593363.963757.629606.370394 5 4.054.90 9.9 o 6.593659 4'93 963704 *90.629956 5..370044 54 7,593955 3.9636o0.630306 5 83.369694 53 8.59251 4.9 963596 90.63066.83.369344 52 9.594547 492.963542 90.63100 5.82.36995 51 10 9.594342 4 92 9.963488 90 9.631355 2 0.36S645 50 1.59.5137 4'91.963434 90,631704 5'2,368296 49 12.595432 4 91.963379.90.632053 5.81 367947 48 13.595727 4 91.96332 9.6322 5.81 367598 47 14 596021 4 90.963271 90.632750 5.8 1.367250 46 1 5.o96315 4'90.963217.9.633099 5..366901 45 16.596609 489.963163..633447 5.0.366553 44 17.596903.9.963108 91.633795 5.80 366205 43 18.597196 4 89.963054.91 634143 59.363857 42 19.597490 4 88.962999 91.634490 579.365510 41 20 9.597783 488 9.962945 9.634536 0.365162 40 21.59S07.962890 91 635185 5.36415 39 22.59836 487 962836 91 635532.364468 38 23 598660 87 962781 91.63579 5..364121 37 24.598952 4'86.962727 9.1.636226 5.78.363774 36 25.599244.86.962672 91.636572 5.77.363428 35 26.599536 4.86.962617.91.636919 5.77.363081 34 27.599827 4.962562.637265 5.77.362735 33 29.600118 8.962503 91.637611.362389 32 29.600409 4.84.962453:92.637956 5:76.362044 31 30 9.600700 4 4 9.962398 92 9.638302 5.76 0.361698 30 31.600990.484 962343'92.638647 5.7.361353 29 32.601280 4.83 92288.633992 57.361008 231 4.83.9.62288 33.601570.43 962233 92.639337.75 360663 27 34.601860 4.83.962178 92.639682.360318 26 35.602150 4.82 96223 2 640027.359973 25 36.602439 4.82.962067.9.640371 5.74.359629 24 37.602728 481.962012 92 640716.359284 23 38.603017 481.961957 92 641060 573.3589410 22 39.603305 4.81 961902 92.641404 73.358596 21 41.603832 480.961791 92 642091 572 357909 19 42.604170.479 961735 92 642434 5:72 357566 18 43.604457 4.79.961680 642777 72 357223 17 44.604745.4'.961624.93 643120 5.71.356380 16 45.605032 4.961569.93.643463 5.71.36537 15 06 1.79 46.605319.961513.643806 571 356194 14 4.78.93 5.71 4:78.91 64J4 47.605606 4.78 961458 3 644148 570 o 2 1 48.605892 7 961402.93.644490.3675510 12 49.606179 4.77 961346 93 64832 570 355168 11 50 9.6)64165 4.76 9.961290 9.645174 5.69 0.354826 10 51.606751 7 96123 93 645516 5 354484 9 2.607036 4 6 961179 64557 69 354143 8 4.76.9 61 569 5:3 G607322 | 961123 93 646199 69 353801 7 54 697607.470 961067.66540 5 353460 6 55.607892 4..961051.93 681. 6 35319 5 56 9608177.960955 6317222.3-2778 4 4.74.93 5.68 57 60461 74 96099 91 61762 067 35243S 3 55 60745 4 7.960843 | 9.6. 7903 67.352097 2 59.603029 4..960786 1 6243 5 35177 1 GO.609313.960730 | 9 64.3o83 56 351417 0G M.| Cosine. I D,1 Sie_ I D. 1-. Cotang. D. 1-". Tang. n1. 1130 17* qGc 198 TABLE XIII. LOGARITHIIIC SINES, 24~o 155~ M. Sine. D. 1". (Cosine. ID.111. Tag D.. s D 1 Cotang. 3I. 0 9.609313 9.960730 9.6435S3 5 0.31117 060 1.609597 4.72.960674.94.648923 5..3o10 77 59 2.609880 4'72.960618.9.649263 5.66.350737 3.610164 4.72.960561 94 649602..350393 57 4.610447 4. 1.960505..649942 r565.3500(8 56 5.610729 4.71.960448.9 6021.319719 55 6.611012 4.71 960392.91.650620 5.65.349380 54 7.611294 4.71.960335 650959.319041 53 4.70.94 5..63 6 8.611576 470.960279 9.651297 563.3438703 52 9.61158 469.960222 94.651636 348364 51 10 9.612140 4 69 9.960165 9 9.651974 0.343026 50 11.612421 469.960109.9.62312 53.3476S8 49 12.612702 468.9052 9.652650 5.6.3417350 48 13.12983 4 959995.652988 5.347012 47 14.613264 4.68.959933'.9 6,3326 5.62.346674 46 15.613345 4 67.959882. 653663 5.62.346337 45 16.6 13825 467.959825.95.654000 5.6.346000 44 17.61410 467.959768 654337 5.62.345663 43 18.614385.959711.654674 345326 42 19.614665 4.66.95964 95.655011 561 344989 41 20 9.614944 9.4 6 959o96 9. 65348 61 0.344652 40 21.615223 465 959539.655684..344316 39 4.65.95 5I61 22.615502 4.5.959482.95 656020 5.6.343980 3 23.6157831 464'959425.95 656356 5.60.343644 37 24 616060.959368.656692.343308 36 25.61633 464 959310.96.657028 5..342972 35 26.616616 4.63.959253.96.657364 5.59.342636 34 27.616394.6.959195.96 657699 5.5 342301 33 28.617172 463.959138'96.658034 5"8 341966 32 29 617450 462.959030.658369 5.341631 31 4.62.96 5.5' 30 9.6177274 9.959023.96 9.658704 53 0.341296 30 31.618004 4.62.958965 9.659039 3 340961 29 32.618281 4.61.9589038 96.659373 57.340627 28 33.618558 461.59850.96 659708 5.340292 27 34.618834 4.6.95792 96.660042 5.5 339953 26 35.619110 4.60.958734.96.660376 5.57.339624 25 36.619386 4.60.958677.96.660710 5.56.339290 24 37.619662 460.958619.661043 5.5 338957 23 38.619938.958561 97.661377.6.33623 22 39.620213 4.59.958503 97 661710 55 338290 2.1 49.93850 7.6.7 40 9; 620488 4.58 9.953445 9.662043 0. 33797 20 41.620763 9.5.953387.97, 662376 5.337624 19 42.621038 4.58.958329.97.662709 5.54.337291 19 43.621313 4.58.953271 7 663042 5. 5.33695 17 44.621587 4.57.958213 97.663375 5.54.336625 16 45.621861 4'57.958154..663707.5.336293 15 46.622135 456.958096.97.664039 5.335961 14 47.622409 4.6.958038 66371.335629 13 4856.622.68 48.62262 4.56.957979.97 6647)3.335297 12 49.622956 455.957921 97.665035 53.3396 11 50 9.623229 9.957863 9.665366 0.334634 10 4.55.97 5.52 51.623502 4.957804.97.665698 5.52.334302 9 52.623774.957746 98.666029 5.52.333971 8 53.62-047 454,957687 93. 666360.333640 7 54.624319.957628..666691 51.33309 6 55.62591 453.957570.93 667021 5.51.332979 5 56.624863.95751.9 7352 8 667 51 57.625135 3.957452.66762 55 33) 3 58.625406 4.52 95739.63 60.33 1~87 2 59.625677 2.95733 9.663 0.33167 1 60.62594.957276'.66673.331327 0 M. Cosine. D. 1". Sine. D. 11". Cotang. D. 11'. Tangr. 3I. 1 4 [6' COSINES, TANGENTS, AND COTANGENTS. 199 250 154o MI. Sine. D. 1'. Cosine. D). 1'. Tang. D. 1". Cotang.. 0 9.625948 451 9.957276.98 9.668673 0.331327 60 1.626219.1.957217.669002.33(it)8 4.51.98 5.49 2.626490 4.51.957158.98.669332 5.49.330668 58 451 98. 330339 3.626760 4.50.957099.98.669661 549.330339 57 4.627030 4r50.957040 99.669991.330009 6 5.627300 4.50.956981 99.670320 5.48.329680 55 6.6275670 449.956921 99.670649 5.48.329351 54 7.627840.956862.670977 54.329023 53 8.628109 4 49.956803.99 671306 5.4.328694 52 9.628378 4:48.956744 99.671635 547.328365 51 4.48.99 5.47 10 9.628647 448 9.956684 9.671963 0.328037 50 11.628916 448.956625.672291 5.47 327709 9 12.629185.956566.672619.327381 43 13.629453.4.956506.99.672947 5.46 327053 47 14.629721 4.47.956447.99.673274 5.46.36726 46 15.629989 4.46.956387.673602 5.46 326398 45 16.630257 446.956327.99.673929 5.4.326071 44 17.630524 4.46.956268 99.674257 5.45.325743 43 18.630792.956208 100.674584 4:325416 42 19.631059.956148.674911.325089 41 4.45 1.00 5.45 20 9.631326 44 9.956089 100 9.675237 0.324763 40 21.631593.956029.675564.44.324436 39 22.631859 4.44.955969 1.00.675890 5.44.324110 38 23.632125 4.44.955909 100.676217 5.323783 37 24.632392 4.43.955849 100.676543 5..323457 O 36 25.632658 4.4.955789.676869 5.323131 35 26.632923 4.43.955729 1.00.677194 5.4.322806 34 27.633189 442.955669 1.00.677520 542.322480 33 28.633454 442.955609 100.677846 5.42.322154 32 29.633719 4:42.955548 1:00.678171 542 321829 31 30 9.633984 4 41 9.955488 1 00 9.678496 5.42 0.321504 30 31.63429.955428 101.678821 5.41.321179 29 32.634514 4.955368 01.679146 5.41.320854 28 33.634778.4.955307 101.679471 5.1.320529 27. 9553 07 6799. 326079 47 3.63042 4.0.955247 1.679795 5.41.32020 4.40 i 699 35.635306 440.955186.01.680120 5.4.319880 25 36t.635570.955126.680444 40.319556 2 37.63834 4'.955065 1.68076 5..319232 23 37.67474.37 438076 39.6360 4.39.954944 101.681416.318584 21 4.36 1.01 5.39 40 9.636623 9.954883 101 9.681740 0.318260 20 41.636886.954823 1.01.682063.39.317937 19 42.637148.954762.682387 5.3.317613 18 413.637411.37.954701.0 I.682710 5.34.317290 17 44.637673 4..95440.02.63033.316967 16 45.637995 4.36.954579 1.02.683356 5.38.316644 15 46 638197 4.3.95418.02.683679 5.38.316321 14 47.638458 436.954457.02.684001 5.37.315999 13 48.638720 4.5.954396 102.684324 5.37.315676 12 49.63S981 4.35.954335 1:02.654646.315354 11 50 9.639242 43 9.954274 102 9.684968 0.315032 10 51.639503.954213 02.685290 5.36.314710 9 52.639764.954152 1.02.685612 5.36.31438 8 53.6404 4.34.954090 102.685934.36.314066 7 54.640284 4.33 954029 1.02'686255 5.36.313745 6 55.640544 433.953968 1 02.686577 5.5.313423 5 56.640804 4.953906 1.686898 |.313102 4 57.641064 3.953845.687219.312781 3 I ~.6 3 120 511 158.641324 4-32.93783 1103 687540 1535.312460 2 59.641583 4-32.953722 1.03 687861 5.35.312139 1 4.32 1.03 5.35 IV. Cosine. ne. D.. Sine Cotang. D. 1". Tang. M1. 15.3'0 200 TABLE XII. LOGARITIIMIIC SINES5 M1. Sine. D 1'. Cosine. D. Tann. D. I.. Cotang. 1. 0 9.641842 |4 3 9.95360 1.03 9.G8132 34 0.31813 60 1.642101 43. 9399 03.6502.3 149S 59 2.64260 31.953537 1.3 623.311177 38 3.642618 31.93475.639143 31.31iOS7 57 4.642877 4.30.953413 1.03.69463.310537 5 5.643135 4.30. 9733 172 1,03.639783.310217 5 4.30'953 352 1.03 5.33 6.643393 4 30 953290 1 03 690103. 33 309897 4 7.63650 429 1 9532263 1 4 03.690423 33.30977 3 S.643903 4 29.953166 1 03.690742.309258 52 9.6-4165 4 29.953104 1 03.691062 532.308933 51 4.29 1.03 5,3' 10 9.644423 4.23 9.953042 1.03 9.6913SI 32 0.308619 50 11.644680 4.28.95290 04.691700 32.308300 49 12.644936 4.28,952918 104.692019 5"31.307931 48 13.645193 4 27.952855 1.0.692338 3 1.307662 47 14.645450 4,27 952793 1. 0 692656 5 31.307344 16 15.645706 4.27.952731 1 04.69975.307025 45 16.634962 4.2.952669 1 04.693293 0 31.306717 44 17.646218 4. 26.9o2606 1 04 693612 50.306388 43 18.646474 426.92544 1.04.693930 50. 9306o7(1 42 19.64729 4 26.92481 1.G91 424.303752 41 20 9.8469S4 4 25 9.952419 1.04 9.6945G6 0.3043- 40 21.647240 4.2.952356 104 6940S3 9.305117 39 22.647494 4.2'.92294 1.04 691021 5..30-4799 38 23.47749 4 24.952231 104 69551 9.301442 37 21.64004 4 24.952168' I.69:336.29' 0.30 1 164 36 25.61S253..9o2106 1.0.396153 3..329 3047 35 26.618512.952043 1 696-70.3073530 31 27.6313766 4.23.951930 1..696787 5.28 D 3213 33 23.619020 4 23.951917 1. 697103.23 32)397 32 29.649274 4 22 91a4 1,0.697420 3,.302580 31 ~4,~~22 1 o 5.27 30 9.619527 22 9.951791 10 9.697736 07.3226-1 30 3 1 619781.2.95 17289. 0.698033 5.27.301917 29 32 60034 4.2.95 1 66 1..693369 27.301631 2> 33.)027 2.951602 1.0.69635 2 oO131 27 34.650539 21 91539 1 0.699001.26 300999 26 35.60792 4.21.91476 1.699316 3.26 3006S4 25 36.6)1044 4.20.91412 1.0.69!9632 5.26.300:363 237.6)1297 4,20.951349 1. 06.699947 5.26.300053 23 38.6154 4.91286 1 06.700263 2.299737 22 39.651800 19.91222 1 06 7(0573 5 2.2422 21 40 9.6)2052 4 9.951159 1 06 9.70393.2;- 0.299107 20 41.6)2334 919.1096 1 6.7)1238 0.298792 19 42.6)2550 4.19.951032 1 06. 01523 2.298477 18 43.65206.18 950963 106.7(1837.24.298163 17 44 -.65307 4.95090 1 06.702152 5.24.297848 16 45.653303 9.1 0904 106.702466.24.297534 15 4.10 1.06 5.2 14 4.653553 4.9077 70781.. 24 2379 14 47.6333 17.950714 1 06 730.29690)5 13 48.65059.90650.2963291 12 49.654309 6.95056 6 703722.2G278 11 4.16 3.06 5.23 50 9.6455 16 9 950522 1 07 9.70-1036 2 0. 29964 10 51.654808.950493 1 07.704350 5.23.290560 9 21.6),058.950391.704663 5 2.295337 8 53.655307 4 15.9o0330 1.709760.09717 7 54.655556. 4.950266 1 07.70590 22,294710 6 55.6058S05 4 5.950202 1 7.705G603 22.294397 5 56.65)6054 4 14 90133.705916.29104 4 57.66302 4 14.950074 1 07.706228.21.293772 3 53.66551 44 950010 107 70641 1 29359 2 59 656799 13.949945 107.70654.21 293146 1 60.657017 1 94931.707166.29283 0.i Cosine. ID. 1' Sine. D. 3 1. Cotang/ ID.. TaLzn. 1~~~~~~~~~~~~~~~''>) 13.5 COSINES1, TANGENTS, AND COTANGENTS. 20I 2~'o 15~o |I. Sine. D. 1" CDosine. DD ". Tang. D. 11. C otang. 0 96.67047 4 13 9.949331 1 07 9.707166 520 0.292834 60 1.65729,5..919316. 7.'707478 5.20.292522 59 2.657542 1.949752 107.707790 5.20.292210 58 3.657790 4.12 949638 1'.708102 520.291S93 57 4.653037 4.12.949623'.(3.708414 5. 20 i291536 56 5.68294 4.12 5.708726 5.1;291274 55 6.65331 949.949 1.709037 9 290963 54 7.6-)778.C949129.709319.19.290651 53 8 1 6A9025 4.11.949364 1. 43.709060 5.19.290340 52 9 1 69271 411.919300 1 08.709971 5.19.290029 51 4.10 1.903 5118 10 9. 659317 9,949235 108 9.710232 0.289718 50 11.65963 410.94170 103.710593.239407 49 1 2.660t)09 41.0.9-1910 1' 03.710904 5..839096 48 13.6602 j 4'09 i949040 1 0s.711215 5.18.28785 47 14.66501 409 948975 1 0.711525 5.18.28475 46 15.66746 409.948910..711336.288164 45 16.660991 *03.948345 1.09.712146 7.28784 44 481 511 097 17.661236 4..9483730 1..712456 1.27544 43 18.661481 4.03.94715 1.09.712766 5.17.287234 42 19.661726 403.943650 1 09.713076 5 16.286924 41 20 9.661970 4,907 94S3 58 9.713386.286614 40 21.662214 407.94819 1.09.713696 5.16.2S6404 39 4.07 1.09 51 2 3 22.662459 407 948454 1.09.714005 25.16 995 38 23.662703 406 948338 1.09.714314 5.16 285686 37 24.6629-16 406.948323 1.09.714624 5.15.235376 36 25.663190.406 948257 1 09.714933 5,15.285067 35 26.663433 |.948192 109.715242 5.15 i2475 34 4.03 948126 5.15.27 6613677 |40 2.948126 10o.715551 5.5 24449 33 23.663920 4.05.948060 9.715860 15 284140 32 29.664163 4.05 947995 1.09.716163 5.14.233832 31 30 9.664406 4 04 9.947929 110 9.716477 5.14 0.283523 30 3 1.664648 404.947863..716785 514.283215 29 32.664891 4.04.947797 1.10.717093 5.14.282907 28 33.665133 403.947731 1.10.717401 5.14.282599 27 34.665375 403 947665 1.10 9 13.282291 26 3).665617.947600 1.10 718017,3.21983 25 403.1 718017 ] 5 281983 53 36.66589.947533.718325.13.21675 24 37.666100 4.03.947467 1.10.718633 5.13.281367 23 33.666342 4.02.947401 1.10.718940 5.13.281060 22 39.666583 4.02 947335 1.10.719248 512.20752 21 4.06 1.10 512 40 9.666824 401 9.947269 110 9.719555 12 0.280445 20 41.667065 401.947203 71962.23013 19:21169 5 12.280138 42.667305 0.947136 1.11.720169 5.1.279831 18 43.667546 4.01.947070 1.11 720476 5.11.279524 17 44 1.667786 4.00.947004 I.720783 5.11.279217 16 45.668027 400.946937 1.721089 511.278911 15 46.663267 00.946371.721396.278604 14 47.663506.946304..721702 5.1.278298 13 4.66746 9.96733 1.11.722009 5.10.277991 12 49.663986 3.99.946671 1.722315 0.277685 11 53 9.669225 9.946604 9.722621 10 0.277379 10 51.669461 9.946533 1.11.722927 5.10.277073 9 3l.669703.946471 1.1.723232 5.09.276768 8 53.669912.946404 1.11.723038 5.09.276462 7 51.670181.946337 112 724344 5.09.276156 6 55,670419 37 96270 112,724149 5.09.275851 5 56.67065.96203 72454.275346 4 57.670396 |.996136 1.12 72-4760 5.09.275240 3 5S.671134 396. 946069 1.12,725065 5.01.274935 2 396 9,9 2 70 03 59 |.671372 1 3.96.9416(0 ].312,725370 5.08.274630 1 6I0 0.671609 949 7. 5674 274326 0. Cosine. D. 1. Sine.I D1 " Cotang. D. 1. Tang. o1. 1675 2 ~02 TABLE XIII. LOGARITIiMIIC SINES9,~8o ~~~~~~~~~~~~1510 M. Sine. D. 10. Cosine. D. 1". Tang. D. 1%. Cotang. M. 0 9.671609 396.94593 112 9.725674 508 0.274326 60 1.671847 39.945868 1:12.725979 S.274021 59 2.672084.945800 1.726284 5.273716 50 3 6 35 I.12 73 507 2i3112 3.672321,5.945733 112.726588 507.27312 57 4.672558 9.9456661'12.726892,07.273108 56 5.672795 3.945598.1.27197.,27203 55 6.673032.9455 3112.727501 272199 54 3.94.1. 2, 5.07'.. 7.673268.945464 727800.272199 53 5 3. 94 1.13 5 06 8.673505..94396,,.72109.06.271891 52 3.9 4 -j..72841 9.673741..9 94532.1.7281 5.270 5 3.93 1.13.7842' 7188 5 10 9.673977 9.945261 1.13 9.72716.0 0.271284 50 3.93 ~ ~ ~ ~ ~ ~~~~ [.2128 50O 11.674213.945193.729020.270980 49 1.1 1.1641 5.1 12.674448 39.945125 113.729323 50.20677 48 113.674684 3.9[3.945058 11.729626..270374 47 14 67491 1.3.72962 4.674919 3,92.944990 113.729929 [.0.270071 46 15.675155.944922 1.73023'.269767 45 16P675390 3.92 1.13 -.,323 16.6753902 3.9.944854 113.730535 5 0.269465 44 3,91 1,13 5-05 1 7.675624 31 944786 1',.730838 ]0,.269162 43 1. 675859 3.91. 19:675859 3.91.944718 113.731141 504.26859 42 1 9.676094 3.91.944650 1:13.731444 04.268556 41 20 9.676328. 9.944582 / 9.731746 5 04268254 40 21 6756.9.948 11r4 5 21.676562 3.90.944514 1 14.732048 5..267952 39 22.676796'.944446 114.732351 504 267649 38 23 "677030 944377.. 732653 503 267347 37 24.677264.944309 1 732955 03.267045 36 3 8 9 1 114 733207 25 677498 3.8.944241..733257.03.266743 35 26.677731 3.89.944172 1.1733558 5..266742 31 27.677964 3..944104 14.733860 5.03 266140 33.733860 503 6 28.678197 3.88 944036. 734162.265838 32 29.678430 3.88 943967 114 734463 02.265537 31 3.88 1.4 o.02 30 9.678663 3 9.9438 114 9.734764 5 02 0.265236 30 3.6788955,0 31.678915 3.71 943830 14.735066 502.264934 29 32.679128 8.9437618 ].735367.264633 28 3.87 94i 1115 5.02 33.679360.943693 115.735668',.264332 27 34.679592 38.943624 115.735969 /.01.264031 26 35.679824 3.86.943555.736269 /.0' 263731 25 36.680056 3.86 943486 115.736570.263430 2 37.680519 3.86 943417 1.736870 5.01.263130 23 39.630519 3.943348.737171 5 262829 22 38.650750..943279 115.737471 0 262529 21 3.85 1.15 5.00 40 9.680952 8 9.943210 1115 9.737771 5 0.262229 20 41.681213..943141 115.738071 so.261929 19 42.681443'8.943072',.738371',.261629 18 43.681674 3.84 943003.738671.00.261329 17,.8 o1i 15, ~ ~ o uU'00 44.631905 3.84 942934 1.1.738971.261029 16 45.682135 3.84 942864 1'.739271 4.260729 15 46.632365 3.942795 116.739570.260430 14 47.682595.83.942726 116.739870.260130 13 48.632825.9426 6.740169 259 12 3.83 1.16 4.9'3 49.683055 3 942.47 1. 40461.259532 12 983 416 4.9842 50 9.63284 3.82 9.942517 1.16 9.740767 498 0.259233 10 51 5'o 3.82 942448 1.16.741066 49'89.258934 I9 52 6.3743'~.93.63743 3.82 94238 1.16.741365 498.258635 8 53.63972 82.9230.6 741664 498.2583o6 7 54.684201 3.81.942239 1.16.741962',.258038 6 55.684430 3.81 942169 1.16 742261.257739 5.6468.81 1 16'.r_ 47'r 56.64658 3.81 942099. 742559 |' 257441 4 57 ~ ~~~~~'. 1.16' 4.97' 57.64887 0 942029 742858 257142 3 53 6.11 1.17 4.97 5.685115 3.80 941959 7 743156.256 2 59.685343.941889 1.17 743454.256546 1.7 _. 25646{[ 60.685571.941819 1.743752.2562458 l. Cosine,. D, I". Sine D. 1". Cotan. D. I'( Tang. 1180, COSINES, TANGENTS, AND COTANGENTS. 203 29~ 1500 7M. Sine. |D. 1". |Cosine. D. 1".'Tang. D. 11". Cotang.. 0 9.63571 9.941819 9.743752.2 60 3.O 7 9735 4.96 1.6S5799.9t1749.I.744050 496.25595 5 9 2.63G127.911679.744348.255652 58 3.63674.94 609 117.744645 4'96.255:3.55 57 3.79 1.17 4.96 4.683192.7.941539 117.744943 96.255057 56 5.6*&670)3 3.78.941469 1 17.745240 4 96.254760 55 6.66936 38 941393 1.17 745538 4 r.254-G12 54 7.6373 378.941328 117.745835 4.5.254165 53 8.6373S9 3.78.9412583 117.746132 4.253368 52 9.67616 77 941187 117 746429 95.253571 51 10 9.637343 377 9,941117 118 9.746726 4 0,253274 50 1.633069 7.941046..747023 4'95.252977 49 12.6i329-3 77.940975 118.747319 494.2o2631 48 13.6.33521.940905 118.747616.,252384 47 14.6SS33747 376.940834 118.74791 94 252087 46 15.633972 3.940763.748209..251791 45 16.63919 3.76.940693 1. 18.748505 4.94.251495 44 17.6 494'3 3.7.940622 I.748801 3.251199 43 18.63948 3..940551 1 18.749097 493.250903 42 19.639373.940480 1 18.749393 4 93.250607 41 20 9.69009 37 9.940409 118 9.749689 4 93 0.250311 40 21.690323 3.940338 1'18.749985.250015 39 22.690543 3.940267 1.19.750281 4.93.219719 38 23.690772 4.940196 1 19.750576 492.249424 37 25.691220.940054 1 19.751167 492.243833 33 26.691444 3.73 939982 1.19.751462 492.248538 34 27.69166S.939911 119.751757 92.248233 33 -28.691892.939340 119.752052 492.247948 32 29.692115 3:72.939768 1 19.752347 491.247653 31 3.72 1.19 4.91 30 9.692339 72 9.939697 9.752642 0.247358 30 3 1.6923562 372.939625 1.19 752937 91.247063 29 32.692785 3.72 93955 119 4.97523 1 246769 2 3.72 1.19 33.693003 9394822.753526.246474 27 3.7 9 119 491 26 3.693231 3.71.939410 1 19.753820 4 91.246180 26 35 693453 3.71 1939339 1.20 754115 4'9.245885 25 36.693676. 939267 120.754409 4 0.245591 24 37.693393 3.7.939195 1'20.754703 4 90.245297 23 3.70. 899 38 69-1120.939123 2 754997' 245003 22 39'694342 70 939052.20 755291 49 244709 21 40 9.694561 3 7 9.933980 20 9.755585 0.244415 20 41.694786.938908 1/20.755S78 4.89.244122 19 42.695007 69.933836 1 0.756172 4.9.243828 18 43.695229 I369.933763 1 0.756465 489.243535 17 44.695450 369.9386911 1 20 75675 9.243241 1 16 45.695671 368.933619 1 29.757052 489.242948 15 46.695892 363.933547 120.757345 4'88.242655 14 47.696113 3683.938475 121.757638 48 8.242362 13 48.696331 368.938402 121.757931 88.242069 12 49.696554 367.933:330 121.753224 4:88.241776 11 50 9.696775 67 9.938258 1 21 9.758517 4.88 0.241483 10 51.696995 367.938185 1 1.758810 4|88.241190 9 52.697215 367.938113 121i.759102 /487.240898 8 53.697435 3.938040 1[21.759395 487.240605 7 54.697654 366.937967 121.75967 47.240313 6 55.697874 3.66.937895 1 1.'759979.240021 5 56.693094 3.66.937822 1'21.760272 4'87.239728 4 57.693313 3.65.937749 121.760564 4'87.239436 3 53.698532.937676 121.760856 486.239144 2 3 6' 1.21 4.86 59.69375a1 6.937604 1.761148 4.233852 1 60.698970.937531 I... 761439'.._.238561 0 M11. Cosin. D. 1. Sine. D. 1". Cotang. 1D. 3/.. Tang. Al. 1t190 600 204 TABLE XIII. LOGARiTIiMIC SINES, 30o 14:9O M. Sine. D.1". Cosine. D. 1". Tang. D. 18. Cotang. M. D. lU. I ang. D. X 0 9.698970 365 9.937531 1.22 9.761439 0.238561 60.699189'.937458 1247613 8.23S269 59 2.699107 36.937385 1 2.762023.S6.237977 5S 3.64.9626 4.6 3.699626 364 937312 1'22.762314 48'6.2376,S6 57 4.699844 3.64 937233 1 22.762606 4 96 237394 56 5.700062 363.937165 1.762897.237103 55 3.63 1.~~~~~~"22 4s 6.700280 363.937092 1 22.7631388,.236812 54 o.6 oo2 i ~ ~85 7.700498.937019 1 22.763479 4,.236521 53 8.700716.936946 22.763770.23630 52 9.700933 36.936872 122.764061 4 235939 51 3.62 1.22 4.85.'83 10 9.701151 3,62 9.936/99 1.22 9.764352 4.85 0.235648 50 ii.701368 362.936725 12.764643 4',8.235357 49 12.701585 3.936652 i23.764933 4'84.235067 48 3.62 ~~~1.23 48 13.i01802 3 61.936578 1 23.765224 484.234776 47 14.702019 361.936505 12.765514 4 84.234486 46 15.702236 361.936431 12.765805 4.234195 45 3,,,6, 1.2:0' ~ ~ 16.702452 361.936357 123.766095 4'84.233905 44 17.702669 360.936284 123.766385.233615 43 3'60. 45 18.702885 3'60.936210 123.766675 /.233325 42 3.60 ~ ~ ~ ~ 666.23303 418 19.703101.936136 76695.233035 41 3.60 1.23 4.83 20 9.703317 3 60 9.936062 1 23 9.767255 4 83 0.232745 40 21.703533. 9398 123.767545 483.232455 39 3~~ ~ ~ ~~~~~~.2340 [,39,I8 22.703749,5.935914 123.767834'S.232166 38 23.5 7.2 7. 23.703964 359.935840 1.23.768124 4 82.231876 37 24.704179 359.935766 12.768414 482.231586 36 25.704395 359.935692 124.768703.8.231297 35 26.704610 3:58.935618 124.768992 482.231008 34 27.704825 |, 935543.769281 482.230719 331 28.5'.2 5. 29.705040 12.93 469 2.769571 48.230429 32 29.705254 38.935395 769860 4:82.230140 31 30 9.705469 5 9.935320 1.24 9.770148 4.81 0.229852 30 31.705683.933246.770437.229563 29 32.5 1.24 4 81.224 2 32.705898.935171 124.770726 4.81 229274 27 33.706112 357.935097 5 24j771015 4.706326 5.9350022 124 i 771303 48.1.22697 26 35.5 1.2 4.81'248 2 35.706539. 934948 124.771592.81 228408 2 a ~.5 2.448 36.706753 3.934873 1,25.771880.80.228120 2 37.706967 356.934798 125.77216 480.227832 2 38.707180 3.934723 1 772457 4 227543 22 39.70127393 / /.934649.772745.4 227255 21 0 3 55 1.25 4.80 40 9.707606. 9.934574 0 9,773033 0.226967 20 41.707819 ['5[.9344994'.730321.226679 19 ~ ~3:55 1:25 4.80:232 I 42.708032 3.5 934424 1.0 773608 4.0.226392 is 43.708245 5 934349 1.2.977389.80.226104 17 44.708458 3 934274 1.25.774184.9.225816 16 45.7089670. 934199 1.25.7724718 4..225529 15 ~~.5 382 ].7o~8 46.708882.934123 1.25.774759..225241 15.7088832 /,: 1.25 3. 22521 14 47.709094.934548.775046 4 7:224954 13 3.53 1.25 4.79 34.709306.933973.775333.224667 12 3.53 1226 4.79 49.709518 933898 1.26.775621 478.224379 1 50 9.709730 9.933822 1:26 9,775905 457 0.224092 10 51.709941 3.52.933747 1 3 5776195'.2230'5 9 1 52.710153 3.52 671.16 6482 ] I 223518 8 34.7103645'3.933596.776768.223232 7 3.52 1.26 4.78 5.710575 32.933520 126.777055 478.22945 6 5.710786 3.51.934' 1.26.777342 4'78.222658 lo5 56 1.710997 351.933369.777628.222372 4 5.7 1120 3.51.933293 1.26.777915..222085 31 53.711419 3 5.933217 1.26.201 221799 2 59.311629 31 933141.77488,.221512 1 60.711839 3 5.93366.778774 4.77.221226 0 120..2 ~ 2466 } 2 50 9~. 7073 },-,c -.-, 19332 [,,-. /~l. 9oa~ 775908 I,, -, I.249 I l Ife~~~~~0.. b' izo' ~i7 COSINES, TANGENTS, AND COTANGENTS. 20r 310 148~ 1. Sine. D. 1". Cosine. D.". Tang. D.11 Cotang,. O 9,711839.5 9.93306 197 91S774 0.221226 6) 3. 50 ] 9.97.3'~'.... 1.71200 50.932991.21 77060 1'7 221)0940 59 2.712260. 932914 1 7794.2206544 53 3.50 ~~~~~1.27 4.77.712469 O 9383 779.2036I 57 0.9 9383 1.27 4.76 4.712679.932762.779918 I.220S2 56 349 1.27 4.6 i 5.7123)9 3.93265,127 70203 7.219797 55 6.713098,932609 1'27 780149.2129511 54 3.419'.27,.7u 7.713303 932533 1.27 707 476.219225 923 3.49 I.747 S.713517 932457 7.781060 476.218940 52.713726 3 93230.78146 476 21 65 1 3.48 ].2 4'76! 10 9.713935 349 9,932304'27 9,71631 75 0. 218369 50'.~9 1 27''4 11.714144 ~ 1322)8 12 781916.21834 49 3.7435 1,'7~5 {3 12.714352...32151 123.792201 217799 48 1 7151 3.4' 1.23 4.75 47 15.714561'932075 1.782 4.2175i14 47 14.714769.931998 12.79771 217229 46 3.47 1. 2 4.75 1 5 147 34.931921[ 1'23 783056.216944 45 16.715186 931845 12 783341.216659 44 3.,t7 I,2.93185/,'2160::9 44/ 17.715394 46 93176 13.783626 21634 43 181.715602 346.931691 1 3910.2167090 42 3.46 I.2 4.93~91.2'2~ 19.715S09.931614.78419 21505 41 3.46 1:2' 20 9.716017 46 9.931537 9.73479 0.215521 40 21.716224 6 931460 12 76 21236 9 3~46 1, 23 ~ 784764 4'74.215236 39| 22.716132 4 9313 123 7850438,21492 33 23.716639 93150631.23 32 4.74.214663 37 24.716969 2 2 716 3.45.931229 1. 83616 4.73.214384 356 25.717053 3.45 I931152 129 785900..214100 35 26.7172 4 129 7614.73.21316 341 27.717466 950993 129.76465.21332 3 3.-14 1.29 ~~~~~4 73 23.717673 3 930921 129 7652.2132 32 29.717879 3.44.930843 129 706.212964 31 3.44 1129 4.73.212964 31 30 9.719035 34 9.930766 129 9,77319 0.212681 30 3.43. 47 31.718291 93068 129.787603 472.2297 29,02 718497/ 3.43 [.2 4.9'011}i' 32.719497.4 -930611 129.79796 472,212114 28 3.43' ~.~ 4.72 03~3.7 ~ 3403.930533 129 88170.211830 27 34.719909 9350456 1.29 78543 42.211547 26 3.43 1.293045672'2 35.719114 32 93037.29 78736.211264 25 3.42 1.29 4.72! 6.719320 349 9030.789019 472.210981 24 }.42 ].3 4.9330 {' 37.719525 42.930223 30.79302 72.210693 23 3.43 ~ ~'7']3 4.72/2 33.719730 342.930145'78955 4,71.210415 22 39.71993 3.41.930067 1.350 7963 471.210132 21 40 9.720140 3.41 9.929989 9.790151 4 71 0.209949 20 3.41'.3 -.71, 41.720345 34 t.929911 30 790434 471.209566 19 42.720549 41.92933 30 790716 471.209294 19 43.72054 341.929755.790999 471.209001 17 3.41. 47 44.720959 349.929677 iso.791291 471.208719 16 45.721162 340.929599 iso.791563 470.208437 1i 46.721366 340.929521..791846 470.20154 14 43.r207t!.929755/ i'~3~!.209001 / 7/ 47.721570 3.40.929442.792129 470.207972 1 3.71& 3.40 13.7916 /~; 48.721774.929364.792410 470.207590 152 3.72~a a.40 4.7934 49.721978 339.92926 1.792692 4'70.207309 11 50 9.722181 39 9.929207 1 9.792974 470 0,207029 10 51.722385 39 929129 13.793256 70 206744 9 52.722588 3. 92900 131.79353 470.206462' 3.39 1.20308 11tl 53.722791 33.92S972 1. 79319 4.206191 7 14.722994.928923.794101 4.69.205999 6 1..9 1931 151.469 55.723197 923915 1' 794383 46.205617 5 57.723603 3 923657 11 794946 6.205054 3 3.3 9'.3 I.t' 7 59 8 72331)5 3" 929579.795223.2041773 2 59:724007 3O7.929499 1.31 795509 4.204492 1 3.37 ~~~~~I5.'2 a 4/6 53.721279 3.39 4.9297 6.7294210.923420.795789 20121 1 0 M. jCosine. 1D. If. Sine. D. 1". 1Cotang. I. 1". T.8. M.1.723197 ~'.~[. 794383 I 206 TABLE XIII. LOGARITII\IIC SINES, 32o 14io!L. Sine. D. 1"f. Cosine. D. ill Tang. D. 111. Cotang. M. 0 9.724210 9.928420 1 32 9 795789. 6 0.204211 G/ 1.724412. 928;342 1.3.796070 4.63.203930 59 2.724614 3.36.928263 1 32.796:351 46'S.203649 58 3.724816 36.92 183 1'2.796632.203368 57 4.725017 3.36.928104 132.796913.68.2030S, 56 3.36 1.32 4.6 5.725219,36.92S025 [.2.797194 4.68.202806 55 o. 336 l.,2 4.68 6.725420 3.o6.927946 1.32,797474 4.68.202526 54 7.725622 3.927867 1.32.797755 4 68.202245 53 8.725823 35.927787 1.32.798036 4.67.201964 52 9.726024 335.927708 1.32.798316 4'67.201684 51 10 9.726225 9.927629 2 9.798596 4 0.201404 50 11.726426 3.34.927549 1:33.798877 467.201123 49 12.726626 3.927470 1 33.799157 467.200843 48 13.726827 3.927390 133.799437 467.200563 47 14.727027.927310 1*33.799717 4'67.200283 46 15.727228 3'34.927231 1'33.799997 466.200003 45 16.727428'.927151 1 833 800277 4'6.199723 44 17.7,27628.3. 927071 1.800557 4'66.199443 43 18.727828 3.33 926991 133.800836.199164 42 19.728027 3:33.926911 1:33.801116 4:66.19884 41 20 9.728227 3.926831 133 9.801396 466 0.198604 40 21.728427 33.926751 1.33.801675 466.198325 39 22.728626 3.32 926671 1.33.801955 4.66.198045 38 233.32. 926591 1..802234 4 65.197766 37 24.72902 32.926511 134.02513 4..197487 36 27.729024 31 926270 1.34.803351.196649 33 25.729223 3.926190.03630 6.197208 3 26.730018 331.92631101 34.8030729 4'..196921 31 127.729621 331 1.34 4.6.. 230.73021720.926190.8036304 S7. 0.196370 30 29.730018.926110 1:34 4.6e 30 9.730217 3.0 9. 926029 1 34 9.804187 4 6 0.195813 30 31.730415 30.925949 1.34.804466 4.64.195534 29 32.730613 3.30.925863 1.4.804745 464.195255 28 33 730811 3.30 925788 1.805023 464.194977 27 34.731009 3.30,925707 1.3.805302.64.194698 26 35.731206 3. 925626 1. 3.805580 4.64.194420 25 36.731404 3.29 925545.85859 4.64.194141 24 37.731602 3.29.925465 1.35.806137 4 64.193863 23 38.731799 3.29.92334.3.806415 464.193585 22 39.731996 3.28.925303 35 806693 463.193307 21 40 9.732193 28 9.925222 9806971 4 0.193029 20 41.732390 328.925141 1.35.807249 4.3.192751 9 42.732587 3.28.925060 1.35.807527 463.192473 18 43.732784 3'28.924979 1'35.807805 4'63.192195 17 44.732980 327.924897 135.808083 4 63.191917 16 45.733177 327.924816 1.35.808361 4'63.191639 1} 46.733373 327.924735 136.808638 63.191362 14 47.733569 3.27.924654 1.36.808916 4'62.191084 13 48.733765 327.924572 1.36.809193 4.62.190807 12 49.733961 3:26.924491 1:36.809471 4:62.190529 11 50 9.734157 326 9.924409 1.' 6 9.809748 4.62 0.190252 10 51.734353 3.26.9243281 36.810025 462.S19975 9 52.734549 3'.924246 136.810302 462.189698 8 53.734744 o26.924164 13.810580 62.189420 7 54.734939 3. 26.924083 1.36.810857 462.189143 6 3.25 1.36 4.62 55.735135 325.924001 1 36.S11134 461.188866 5 56.735330.923919.811410.188590 4 57.735525 | 2.923837 1 37.811687 4 61.1S8S313 3 58.735719 2.923755 1.7.811964 61.188036 2 59.7359141'.923673 1.812241 4.187759 1 60.736109 3.24.9235.91 1 7.812517.14.61 S7483 0 M. Cosine. |D. 1. Sine. D. 1". 1DCotang D. 1"'Tang M. COSINES, TANGENTS, AND COTANGENTS. 2C)07 330 liG M. Sine. D1. 1" Cosine..'. Tang. D. 1)'. Cotang. MI. 0 9.736109 24 9.923591 1 -- 9.812517 1 6 I O. 1' 83 G6 1.736303 24.923509.1294.187206 59 2,736498 3.24.923427 137.813070 4 (1i' / 166930[ 58 3.736692 323.9233457 4 6 1 3.16653 576 4.736386.923263 2.8 13621 3.3377 3.23 1.37 ] 46 I 5.737080 323.923181 13.81399, l2 61O 6.737274 323.923098 1.37.14176 4.1 5 241 7.737467 3 23.923016 1.37'814452 1..185541 5, 8.737661 22.922933 137.1423 2 9.737855 22.922851 1 8100. 184996 1 10 9.733048 ] 22 9.922768 9.815280. 0.1 184720 50 11.73241 22.922686.815553 G.S14414 49 12.738434 3,22.922603 1.33.81t531 1 60.184169 43 3.22 1.38 4.. 9 13.738627 3.2.922520 1.33 816107...1843893 47 14.738820 321.922438.33.816:332 459.183618 46 15.739013 21.922355 18.816653 459.18332 4 16.739206 3.21.922272 1*33.816933 4' 59.133067 44 17.739398 32 1.922189..817209 4,59.182791 43 18.739590 32.922106 1.3'.817484 4'59.182516 42 19.739783.922023.3 1779 9.182241 41 3:20 1:38 ~ 59 4. 182241 41 20 9.739975 320 9.921940 139 9,818035 4 0.181965 40 21.740167 3.20 921857 1.39.818310 4.58.181690 39 22.740359 3,.921774 139.818585 4.58.181415 38 23.740550 319.921691 1.3.818860 4. |.181140 37 24.740742.3.19 921607 1.39.819135 4.r.180865 36 25 740934 3.19.921524 1.39.819410 4.58.180590 35 26;741125 3.19.921441 139.819684 4.58.180316 34 27.741316 3.19.921357 13.819959 48 180041 33 /2 75 i'~9]3.19. 139 4 758 28.741508 o.18.921274 1.39.820234.5 79766 32 29.741699.921190 820508 458 179492 31 3.18 139 4.58 30 9.741889 3o18 9.9211071 39 9.820783 4 57 0.179217 30 31.742080 3.18.921023 139.821057 4' 7.178943 29 32.742271 318.920939 140.821332.178668 28 33.742462 3.17.920856 140.821606 4.57.178394 27 34.742652.920772 1.40.821880 4.57.178120 26 35.742842 3.17.920638 1'40.822154 4.57.177846 25 36.743033.1.920604.822429. 177571 24 37.743233 3.17.920520 1.822703.177297 23.2.1700 14 457 38.743413 3.17.920436 1.40.822977 4.5.177023 22 39.743602 3.16.920352 140.823251 456 176749 21 40 9.743792 3. 9.920268 1 40 9.823524 4 56 0.176476 20 41.743982 316.920184 1"40.823798 4.176202 19 42.744171.920099.824072 4.56.175928 18 43.744361 3.1.920015 1'40.824345.5.175655 17 3.15 141 4.56 44.744550 3.15.919931 141.824619 4.56.175381 16 45.744739 3.15.919846 1.41.824893 4.56.175107 15 46.744928 3.15.919762 1.41.825166 4.56.174834 14 47.745117 3.5.919677 1:41.825439 4.56.174861 13 48.745306 3.15.919593 1 41.825713 4.56.174287 12 9. 745594 35 4.56. 49.745594.919503.825986.174014 11 3.14 1.41 4.55 50 9.745683 3.14 9.919424 1.41 9.826259 0.173741 10 51.745871 3.14.919339 141.826532..173468 9 52.746060 3.14.919254 1.4.826805 4 5.173195 8 53.746248 3.13.919169 1.41.827078 4. 5.172922 7 541 746436 3.13.919035 1.42.827351 4. 5.172649 6 55.746624 3.13.919000.42.827624 4r55.172376 5 56.746812 3.13.9189150.827897.172103 4 57.746999 o.718.918830 1.42.828170 4.o:.171830 3 58.747187 o. 12.918745 1.42.828442 4.54.171558 2 59.747374 3.I'2.918659 1.42.828715 4.54.171285 1 60.747562.918574.823987.171013 O BI. Cosine. D. 1'.. Sine. D. 111. Cotang. D. 1". Tang. M. 1231 56~ 208 TABLE X11I. LOGARIITHIIC SINLES, 034.0 14 D in. Sine. Dl.. Cosine. D. 1". Tang. D. P'. Cotg.. ( 97478;) 32 9.1.-1,74 9.82''77 0.171013.60.7477-9 1. 49 1.4 S292603 4.170740 59 2.74 79:36 - 33 12 918404 14,S932.17,468[ 58 3.12~1 2I 1'"Y2 ~ 824.54:.748123'3. 11.91831S 1.4'.829' ( 4, 4.170195 57 -,.7483110.918:a43.83 5077 4.4.163923 56 5;.74%197 339.161 {91847 1.4L.830349 4. 1 169651 55 (. 7486S3 in.1062 14 1 4 169379 54 7 5.4S 3~ I h 1.43 4,5 3 52,.79b70 3.11.917976 1'43 83093 3.169107 53 8.749056 3 l0.917,91 1.43.S:311 II 453.163S35 52 9.749243 3 10.917805 1 4 83t437.16S563 51 3.10 1.43 4.53 10 9.7493429 9.917719 143 9.S1709 0.168291 50 11.749615.917634 43 S3198.168019 49 12.749301.917548 143.83253.167747 48 13.749937 310.917462 143.832`25 453.167475 47 3,10 1943 4,53 14.750172 3.09.917376 43 832796.53.167204 46 15.7035 3.09.917290 1.3.8336 4.53.166932 45 3.19 4.13 52 1 17.750729.309 917118 1.44 833611 4.52.166339 43 13.750914 309.917032 1.44.833S2 452 166118 42 19.751099 30.916946 144 834154 4.165846 4L 3.03 1.44 4.52 20 9.751284 9.91659 9. 83425 0. 165575 40 3.03 1.44. 4.52 21.751469 3.3.916773.414,S314696 4,52.165304 39 22.7516 54.9166S7 144.834967 4. 2.165033 38 23.751839 3.916603..83'23S.5.164762 37 24.752023 30.916514,.3509 4.52.16491 36 3. O''9151.1 4350 25.752203.916427 1.44.835780 4,52 164220 35 26.752392.0.916341 1.44.836051 4.52.163949 34 27.752576 3'07.916254 1.44.836322 4.5.163678 33 28.752760 307.916167 1.44.836593 4.51.163407 32 29.752944 3:06.91601 5.836864 4.5.163136 31 30 9.75312 3 06 9 915994.4 9.837134 4 0.162866 30 31.753312 06.915907.837405.151 62595 29 3.06 1.45 4.51 32.753495 3306'915820.637675 4 1 162325 28 33.753679 3.06.915733 1.4'837946.162054 27 34.753S62 3,915646 1.45.838216 451.161784 26 35.754046 3.05 915559 14.838487 4.51.1615l3 25 36.754229 3.0.915472 1.4.838757 51.161243 24 37.754412.915385.839027.160973 23 38.754595 3O.915297 1.45.839297 4.50.160703 22 39.754778 05 915210 45.839568 160432 21 3.05 1.46 4.50 40 9.754960 04 9.915123 46 9.839838 0.160162 20 41.755143 3.04.915035 146.840103 4.50.15992 19 42.755326 304.914948 146.840378 4.50.159622 18 43 -.755508 3.04.91460 1.46.840643 4.50.159352 17 44.755690 91773.40917 4.50.159083 16 3'0 146 4[50 45.755872 303.914685 1.46.841187 44.158813 15 46.756054 3'03.914598 1.46.841457 4.49.158543 14 47.756236.914510 46.841727.158273 13 48.756418 303.914422 146.841996.15S004 12 49.756600 3:03.914334 146.842266 449.157734 11 3.03 1.46 4.49 50 9.756782 3.02 9.914246 1.47 9.842535 0.157465 10 51.756963 3.02.914158 147.42805.157195 9 52.757144 302.914070.843074.156926 8 53.757326 302.913982 1.47.843343 4.49.156657 7 54.757507.91394 1.47.843612.16388 6 55.757688'.913366 1.47 843382.156118 5 56.757869 3.02 913718 1.47.844151 4.48.155849 4 57.75050 301 913630 1.47 84420 48.1555 3 301'.834421.155580 3 2 58.7582230 3.01 913541 1.47.84469 4.155311 2 59.753411 3.:.913453 147.844958 448.155042 1 60.758591..913365.845227.154773 0. Cosine. D. 1'. Sine. D. 1". Cotanlg. 1. 1". Tang. 5l. d~40E [ 9 COSINES, TANGENTS, AND COTAINGENTS. 209 350 1449 Al. Sine. ID. 1". Cosine. ID. 1"1. Tang. D. 33. Cotang. M. 0 9.75S591 301 9.913365 147 9.84527 448 0154773 60 1.758772 300.913276 845496 4.154504 59 2.758952 300.913187 148 845764 44.154236 58 3.759132 300.913099 148.846033 448.153967 57 4.759312 300.913010 148 846302 4.153698 4 5.759492 3 00.912922 148 8.46570 44.153430 55 6.7`672 2,9.912833. 8465839 448.153161 54 7.759852 299.912744 148 847108 4' 152892 53 8.760031 2'99.912655.48 S47376.12624 52 9.760211 299.912566 1.48 847644.152356 51 10 9.760390 299 9.912477 14 9.847913 0.15087 50 11.760569 299.912388.'848181 I.4s151819 49 [/ ~.9 128 4. 1064,; 12.760748 2.98.912299 149.848449 4.151551 48 13.760927'.912210 149.848717.11283 47 2~~ ~ ~ ~~~~~~.1528 147. 14.761106 298.912121 149.'4S986.151014 46 15.761285 298.912031 149.849254.150746 45 16.761464 2'98.911942 149.849522.150478 44 17.761642 297.911853 149.849790.150210 43 18.761821 297.911763 149.850057 446.149943 42 19.761999 297.911674 1'.850325 6.149675 41 /J97 1.49 4.46 20 9.762177 297 9.911584 149 9.850593 446 0.149407 40 21.762356 297.911495 149'850861 446.149139 39 22.762534 297.911405 149.851129 446.148871 38 23.762712 2.96.911315 /'.851396 4:46.148604 37 2. ~o 1.5 ou ~ ~ q.762889..911226. 851664 446.148336 36 25.763067 9.911136 s.851931.148069 35 26 I763245 2.96.911046.852199 446.147801 34 27.763422 2.96.910956 1.50.852466 446.147534 33 28.763600.910866 150 852733 4 147267 32 2.95 1. 46 29.763777 2:95.910776. 853001 4 146999 31 2.95 1~~.50 4.45 30 9.763954 2 9.910686 9.53268 0.146732 30 3097694 2.95 1.0 9.5326 31.764131 29 910/596 1.'.853535.146465 29 ~1472664 2 32'764308 910506 0 853802.146198 28 ~ 2.95 9145 1.50:509 4.45 415 271 29.910776 J 34.764662 29 910325 1.51.854336 5.1456649 26 2.94 uo 1 4.ou o445 2 35 I764838 2.94,910235 1.5..854603 445.145390 24 36.765015 2.94.910144 1.51.8534870 4.146130 24 37.765191 2.94.910054 1.51.8551379.14463 23 38.765367 2.9.909963 1.51.855404.144596 22 /1~~~r~i 35.7483 /~ 39.765544 2:93.909873 1.31.855671.144329 21 //3 / / /.903 /.8 0/'/1.541 40 9.765720 9.909782. 9.855938 0.144062 20 41.765896 293.909691 1.3.856204.143796 19 42.766072 293.909601 1.51 856471.143529 18 43.766247 293.909510 1.51 856737'.143263 17 2.9,~.~dI1' ~ci4 44.766423 293.909419 15.857004.142996 16 45.766598 292.909328 152 857270.12730 1 2~~ ~ ~ ~~~~~~.1473 155 44 46.766774 9.909237 1.52 857537 142463 14 2.92 1.52.857..142473 15 47.766949 2 92 146.. 857803.4.142197 13 48.767124 292.909055 152.858069..141931 2.92 ~1.52 4.401 49.767300.90o964.858336.141664 11 2.92 1.52 4.44 50 9.767475 2 9.9073 1.5 9.8558602 [ 0. 141398 10 2.1 1., t O ~ ~ t9 Si.767649 291.908781 1.52.858868. 14112 9 /~2 21.v i.5',- -,,- a 52.767824 291.908690 152.89134 443.140866 8 53.767999 291.908599 15.859400 I' 140600 7 54.768173 291.903507 15 89666 4 140334 6 55.768348 291.908416.53 859932 140068 5 56.768522 290 908324 1.3 860198 139802 4 57.768697.90o,33.860464.13936 3 2.9 1.9 4.43 53.76871 2 90 08141..860430 139270 3 2.11 1.5 4.43 59.769045.908049 1. 860995 4 13900 1 2.90 1.53 4.43~~~~[300 60:769219 2.90 907958 13 61261.138379 0 M. Cosine. D. 1". Sine. ID. 1". Cotan D Tang. 5. 1~5o 18 540 210 TABLE XIII. LOGARITHMIIC SINES, 360-3 1 M. Sine. D. Cosine. D. 11". Tang. D. 1". Cotang. i M. 0 9.769219 2 9.907958 9.861261 0.138739 G6) 1.769393 2'90.907866 1.3.861527.13473 59 2.769566 289.907774 1.53.861792 4.43.133208 58 3.769740 2.9.907682 153.862058 4 42.137942 57 4.769913 2.89.907590 1.53.862323 4'42.137677 56 5.770087 289.907498 1.53.86289.137411 55 6.770260 2-89.907406 1.53.862854 4.42. 37146 54 2.o9 1.54 4.42' 7.770433 2.907314 1.54.863119 442.136881 53 8.770606 288.907222 54 86338 4.42.136615 52 288.907222]154 6 442 13638 51 9.770779 2:88.907129 86 54 360.136350 51 10 9.770952 2 83 9.907037 54 9.863915 442 0.136085 50 11.771125 2 8.906945 1.5.86410 442.135820 49 12.771298. 8.906852 1.54.864445 4.42.135555 48 13.771470 2.87.906760 1. 86710 4.42. 1 35290 47 14.771643 2.87.906667 1.54.864975 4.42.135025 46 15.771815 2.87.906575 865240.134760 45 16.771937 2.87.9061382 1.55.86505 4.41.134495 44 17.772159 2.87.906339 1'.865770 4.1 134230 43 18.772331 87.906296.55.866035 441.133965 42 19.772503 6.906204 1 5.866300 4.133700 41 2.S6 1.5~5 4.41 20 9.772675 286 9.906111 1 985 966564 441 0 133436 40 21.772847 286.906018 1.5.866829 441.133171 39 22.773018..905925.5.8670943.132906 38 23.773190 26.90832 5.867358 441.132642 37 24.773361 2.85.905739 1.55.867623 4.4.132377 36 25.773533 2.5.905645 15 5.867887 4.41.132113 35 26.773704.905532.55.868152 441.131848 34 27. 737.905459.S68416 441.131584 33 25.774046 2 8.9053661.866880.131320 32 29. 774217 28.905272. 36.868945 440.13105 31 2.85 1.56 4.40 30 9.77438. 9.905179 1 6 9.869209 440 0.130791 30 31.774558 2.84 90508 56.869473 4.40 130527 29 32.774729 2.84.904992 1.56.869737 4.40.130263 28 33.774899 2.84.904398 1.56.870001 4.40.129999 27 34.775070 2.84.904804 1 56.870265 4.40.129735 26 35 775240 2.84.904711,56I.870529 4.40.129471 25 36.775410. 4.901617.5.870793 4.40.129207 24 37.775530 2.83.904523 1.56.871057 4.40.128943 23 38.775750.904429 1.57.871321 440.128679 22 39~7 2.83 1.57 74.40 39.775920 2.83.904335 157.871585 4:40.123415 21 40 9.776090 2.83 9.904241 7 9.871849 4 40 0.1281.51 20 41.776259 283.904147 157.872112.127888 19 42.776429 2.8.904053 1 57.872376 4.39.127624 18 43.776593 2.82.903959 1.57.872640 4.39.127360 17 44.776768 2.82.903864:1 7.872903.127097 1 6 45.776937 2'82.9037701.873167 4.39.126833 15 46.777106 2.32.903676 1.57.873430 4.39.126570 14 47.777275.903581 1.57.873694.126306 1 i 48.777444 281.903487 1.5.873957.126043 12 49.777613 2 81.903392 1.8 87422 439.125780 1 1 50 9.777781 2.81 9.903298 1] 9.874484 4 3 0.125516 1 0 51.777950 281.903203 /..8747471 439.125253 9 52.778119 281.9031083.58.875010.39.124990 8 53.77827 2.81.903014 1.5.8752737 4 39.124727 7 54.778455 2.8.902919 158 75537.124463 6 55.778624 2.80.902824 1.5.875800 433.124200 5 56.778792 2.80 902729.8760631 438 123937 4 57.778960 2.80.902634 58.8763261 4'3.123674 1 3 58.779128 2.80 902539.5.876589.123411 2 28$0 2 I.59 87658 4.38 59.779295 279.902444 1.59.7682 4.38 123148 1 60.779463 7 902349 877114 38 12286 0 MI. Cosine. D. 1". Sine. D. 11". Cotang.'. 1". Tang. M. 1t O(D 530 COSINES, TANGENTS, AND COTAMJENTS. 211 370 1423 M. Sine. D. 1. Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.779463 9. 902349 9.9877114 0. 12,8~6 60 1.779631 2 9022.3 1.59 877377 3 122623 59 2 779798 2.79 902158.877640...122360 58 2.79 1.69 4.3( 3.779966 2.79 902063 1.59.877903.122097 57 2,7'9.~f9 4 3o 4.78001 27.901967 1.879165 3 121835 5 2.79.9 1.59 0.8 7 125 5.780300 2.901872.0.87423.121 5 2.'78 1.59 ['7828 6.780467 2.78 901776 1.786971.121309 54 12.71468 277.901 1.59.8265194 7.780634 27.901161 1'87S952..1217 53 2.78 [.5i9 [' v~~ 8.78801 27.90100 1.887920.120784 52 2.73 ~~1.59143 1.781966 2.90014190.8810o2.1208522 5 1.60 79478 4.37 10 9.781134 I 9901394 1"!79741 0.120259 2.781,04.' 1.781301 2.77.9018 160.880003 119997 49 1~2 I 1.60 I.3,8003 12.781468 27.901202 160.880265. 119735 4 2.77 I ]1.60 ].3,806['; 13.781634 27.9011062 1 805281.119472 47 14.781800 27 901010 16 880790 43 119210 19.7826 2.77.9009 16.8810 4.437 S94 17.782296 276.90072 1.60.8815 4.37.118423 43 2.76 90] 1.60 4.37.118161 42 11.781966 902 813 2.77 I.60 I 83 4I 3.118686 a 19.782630.900529.882101.117 2.76 161 4.36 20 9.782796 9275.900433 161.882363 17637 40 17 I.r 1.60 [.7.135 3 21.7852961 27.900337 161.882625 4.3 115 3 25.7832 2.76.9995 1.61.2887 4.32.71613 3 26.783278 275.900984 161.883934 4.36.116066 34 27 1734 2.75 904 161.883419 4.36.116582 33 25.78323 2.75.8995 1' u.88361161 26.7378.7.9954 1.61 8394 4.36.116066 34 7.783953 275.899757 1 84196 4.36, 115543 3 28.784118 275.899660 161.884457 4.36.115281 31 29.784282 2:.899564 1.884719 4 2.7 1. 4.363 3 2.74 9.99467 1.6 4.36 0115020 30.784612 274.899370 162.885242 4.36 114758 29 32.784776.899273 1.885504 4.6.11496 28 33 784941 2.74 899176 1.612 885765 4.36 2 34.785105 2..899078 1 0.88626.113974 25 2.74 1.62 4.36 35.785269 7 88981 1.62 8862 4',.113712 39 785433 2.73 898884 1.62 [ 9 4.36 24 37.785597 273.898787.88611.3.113189 23 40 9.786 79 9.89494 1.6 12406 20 241.786252 27 898397.887855 11214 19 42.786416.898299 163.888116 4 1 18 2.75I 1.6 1.62 43.786579 272.89 163.88378.111622 17 34.9551 27.89807234 44 78674 2.72.898104 1.63 888639 111361 16 45.78,690 7 898 3 88820.3.11.382.787069 2.72 1623 4.3 39 14 47.787232.89.889421 435.110579 13 4.787395 7.897712 1.63 889682 110318 12 439.78557 71.897614 889943 435 110057 11 2.71 1.63 4.35! 50 9.787720 2.71 9.897516 1.64 9.890204 4.5 0.109796 10 [ ]3 1.61 [.52.787883 271.8967418.1890465 35.1 3 9 52.788045 271.897320 1.64 890725 10927 8 53.788208 271.897222 164.890986 4.109014 I7 54.788370 2.70 897123 1.64 91247 4.3 10753 6 55.788532 270.897025.891507 108493 52 56.788694.896926.891768 108 232 4 2.70 1.64 4.34 1079 57.7856 2.896828 64.892028 4.10792 5S.789018 2.70.896729 1.4.929 43 107711 2 270 1 64 ~~~.8928479 59.789180 270.896631 1.64.892549 4 10741 1 60.789342.8965324.892810 17190 0 2.74 [ S 1.62 C9a'89o Tang 451 Cosine. D 1. Sine. D. 1. Coang.1" Tn.110 31. 1572 1.63 7.3 o 1103 31.78406128709.811 lO 2.74 1.62. ~ 1 47.787232.897810 ~.885242[.114~7529 32.78435 87712.868 105 2.7 4 1.62.3 33.78457.964.8943 110 1 2.74 1.6 2.3 51.787883.897418 ~~.857065 I,~X.114235 27 52.805.9340,280*26?.1139274 2.785105,3' 53.78825.26 9.SS09628...113712 2.73 1.62 [.3 54~~~~~,787 872 88491 ]'._1073414 2.7 4 ] I'2 eo 55.788543.90.95 143 ~ ~2.7043! 1.623 [f 37,886811 I ~'F,~.81 1 3 1 9 2 fL7,4.7855972 2.7 3 1.6 2 [.3 38,7897018.72l.'98....$1192 22 2.7 3 1.6 2, i oo 59.7895 863 59254.04.51 2.7 3 1.6 2.3 90.78693.9652.9S0.(79 RI~ ~ ~ ~ ~ ~ ~~~~~~~.z I ~o,- 0 ~112406 20ie,i. oa=.i" ag 212 TABLE XIII. LOGARITHISIC SINES, 380 1410o M. Sine. D. 11". Cosine. D. 1".1 Tang. D. l%. Cortang. M. 0 9.789342 2.69 9.896532. 9892810 4.34 0.107190 60 1.789504 2.69.896433 1. 6.893070 4. 3.106930 59 2.789665 269.896335 1.65.893331 34.106669 58 3.789827 2'69.896236 1.6.893591 4.34.106409 57 4.789988 269. 896137 1.65.893851 4. 106149 56 5.790149 2.69.86038 1 65.894111.105889 55 6.790310 2,68.895939 6.894372.105628 54 7.790471 2 68.895840 1 65.894632 34.105368 53 8.790632 6.895741.894892.105108 52 9.790793 2e68.895641 1:65.895152 4:33.104848 51 10 9.790954 2.68 9.895542 1.6 9.895412 4 33 0.104588 50 11.791115 2 68.895443 1.66.895672.104328 49 12.791275 2 67.895343 1.66.895932 4,33.104068 48 13.791436 2 67.895244 1'66.896192 433.103808 47 14.791596 267.895145 166.896452 4'33.103548 46 15.791757 2.67.8950415 166.896712 433.103288 45 16.791917 267.894945 166.896971.103029 44 17.792077 267.894846 166.897231 33.102769 43 18.792237 67 894746.897491.102509 42 19.792397.2667.897751.102249 41 2.66 894646 166 4.33 20 9.792557 2 66 9.894546 1.67 9.898010 4.33 0.101990 40 21.792716 266.894446 1 67.898270 4 33.101730 39 22.792876 266.894346 1 67.898530 4 33.101470 38 23.793035 266.894246 167.898789 4:33.101211 37 24.793195..894146 167.899049 4.33.100951 36 25.793354 65.894046 1 67.899308 432.100692 35 26.793514 2.65.893946 1 67.899368 432.100432 34 27.793673 2.65. 93846.899827.100173 33 28 |793832 2.65.893745 1 67.900087 4[32.099913 32 29.793991 2.65 893645 167.900346 4 32.099654 31 2.65 1.67 900346 432 9 30 9.794150 2.65 9.893544 1 68 9.900605 4.32 0.099395 30 31.794308 264.893444 168.900864 4 32.099126 29 32.794467 2..893343 l..901124 4'2.098876 28 33.794626 2.64.893243 1 G..901383 4 32.098617 27 34.794784 2.64.893142 1.68.901642 432.098318 26 35.794942 2.64 893041..901901 432.098099 25 36.795101 2.64.892940 1.68.902160 32.097840 24 37.795259 2.64.892839 1.68.902420 4.32.097580 23 38.795417 2.63.892739 1.68.902679 4'32.097321 22 39.795575 2263 892638 1.68 902938 432.097062 21 40 9.795733 2.63 9.892536 1 69 9.903197 4.32 0.096803 20 41.795891 2.63.892435 1.69.903456 4.32.096544 ]9 42.796049 263.892334 169.903714 4.31.096286 18 43.796206 263.892233 169.903973 431.096027 17 44.796364 262.892132 1. 9.904232 431.095768 16 45.796521 2 62.892030 169.904491 431.095509 15 46.796679 262.8919291 569.904750 4'o1.095250 14 47.796836.891827 69.905008 43.094992 13 48.796993 262.891726 169.905267 431.094733 12 49.797150 2 61.891624 1'69.905526 4 31.094474 11 50 9.797307 2 91 891523 9.905785 31 0.094215 10 51.797464 2.61 1.70.906043 4.31.093957 9 52.797621 2 61.891319 1 70.906302 4.31.093698 8 53.797777 261.891217 70.906560 4.31.093440 7 54.797934 2'61.891115 1.70.906819 4.3.093181 6 55.79S091 261.891013 1.70 907077 4.31.092923 56.798247 2'61 890911 1.70.907336 4.31.092664 4 57.798403 260.89n809, 7.907594 4.092406 3 58.79860 260.S890707 1.70.907853 4. 092147 2 59.798716.89065 9011 4.1.091889 1 60.798872 2.60 9.8905034.900369 4.91631 H. Cosine. D. 1. Sine. D. 1''. Cotang. D. 1". Tang. M. lf8 5Q t ( COSINES, TANGENTS, AND COTANGENTS. 213 390 1400 M. Sine. D. 111. " Cosine. DD 1" Tang. D. 111. Cotanw. M1. 0 9.793372 2.60 9890503 171 9.90369 43 0.091631 60 1.799023 8 9001.890400 1.71 903628 4 30 091372 80 2 7914 2.60.8023 1.71.90-,1 - 4.30.914 5 -;g918.509.9j~.....~4 5. 3.799339 2.5].890195 7 90J144 43.090356 57 4.79995 2.-9 890093 171 90J402 4 30 090593 116 5.799651 29 88 9900. 909660.090340 50'""0459 1' I71''du r 6.799306 2.59 893 1. 909918 430.090032 54 2.59 1.71 4.30 7.799962, 2'59 889785 1.910177 43I9823 53 9.800117 2.59.8962 171 910435.089565 52 9.800272.889579.910693.039307 51 2.59 1.71 4.30 10 9.800427 9.869477 9.910951 o0.09049 50 2.53 172 4.30 11.800582 7379374.911209 430.0>3791 49 12 2.58 172.91146 12 800737 2.5.839271 1 911467.038533 48 2.58 896 17 4.9172 13.800892 5 9168 4911725 30.088275 47 14.801047'2.5.889064 1.72.911932 430.03801 46 2.58 1."72 43 15 801201 /43 5.801201.58.883961 172 9122407760 45 16.801356.88388.72 430.037760} 45 1 62.57'.88888 1 72.912498 43. 0 37502 44[.a.801[11.838755 1 72.912756 430.087244 43 18 8165 2.57. 43 1.801665 2.57.883651 I72.913014 4.3.036936 42 9.801 219..8548.913271.06729 41 2.57 1.72 4.30 20 9.801973 257 9.838444 1.73 9.913529 4 0.086471 40 802 712313 4.29 21.802123.5.883341.7.913787.04 6213 39 22 ~'07 ].7 4.834 22.8392823 2.57.88237.73 914044.29.0085956 39 O'Z 7' I167 q 29 23.80436 2.56.81 173 914302 29.085695 07 2,56 ~~~~~~~4'29 24.802589.56.880301 173.914560 429 0835440 36 25.80274 2.56.88792 173.914817 429 0o8183 35 26.80297.56.837822 17.915075 429.04925i 34 27.803050.887718 173.915332 429.084668 33 28.803204 2.56.837614.1 915590 4.084410 32 29 3337 2.56 83751 4,29 29 335 2.55.887510 174.915847 4 29 0413 31 30 9.803511 25 9.874061. 9.916104 429 0033396 30! 31.803664.55.887302 1..916362.29.033630 29 32.80317 2.55.887198 1.74 916619.083381 28 3.803970 2.55.837090 I.74 916877 4 29.083123 27 34.804123 2.55 89 1.74.917134 429.082066 26 35 804276 2.55.868385 1.74.917391 429.02609 26 36.804428 2.54 836780 1.74 917648 4 29 0>2302 24;~ 54 1 t, z 37.80 2.54.86676 1.74.9179106 429.082094 23 2.54 86 1.74 4.29 0894 2 33 804734 2' 88657I 1.918163 2.081837 22 2 0 666 1.7 4 29 0387 2 39.804836:486466 1 918420 429 31580 21 2o54 /07 4,29 0350 2 40 9.805039 2.4 9.836362 17 9.918677 423 0.081323 20 41.805191 254.886257..918934 4 28.081066 19 2.5 o1tI 5'i 40 42.805343.836152.919191 428.08009 18 43 805495 2.836047 1'.919448 4'9 ~0>0>52 17 44.805647.82 5942 i'7.919705 23.0>0295 16 45 805799 25 3.83337.7.919962 4.030033 15 46.05951 2 835732...920219 2.079761 14 2.801 3 ].7,V [ ~47.806103. ~85627 ]920476.079524 13 2- 53' 8- 6 75 4281 48. 06 %), 2 3'552 920733.2 079267 12 2:53 1.75'90' 4.28 49.806406 2.52 3 816 1 920990.079010 1'807 079601 50 9.8)6557 9.885311 48921247 0.07>753 10 51.806709 252 85205 1.76 9.921403.078497 9 52.806360 22 83109 1.76.921760 423.078240 8 53,5 1 76.270 4.23 3.807011 252 84994.922017 423.077983 7 54.8071663 22.889 176 922274 428.077726 6 55.82734 >2.83573 176 92530. 077470 5 56 0 807465 2 614677 1.76 922737 4.23 077213 4 57.8076 5.864572.923044.076956 3 2.51 1.76 4.2.9 59.807766 21 834466 177 92300 23.076700 2 59.807917 25.834360 17 2357 423.076443 1 60.803067 2 ol 884254 1 977.076186 0 4. Cosine. 1" Sine. D. 1". I Ctang. I D. 1. Tang. SI. 129>) 50>) 214 TABLE XIIIo LOGARITIISIIC SINES, 400 1393 M. Sine. DI.. Cosine. DB.. Tang. D. 1"l. Cotang. M. n 9.808067 2.51 9.884254 7 9.923814.2S 0.076186 60 1.808218 21 r2 1.884148 1.77.924070.7530 59 2.808368 2.51.884012 77.92327 427.075673 58s 3.808519.50.83936 177.92483 427.075417 57 4.808669 2.50.883829 1.77.924840 4..075160 5 5.808819 2.50.883723 1.77.92096.074904 55 6.808969 2.50.83617 0 177.925352 4.27.074648 54 7.809119.83010 177.925609 427.074391 53 8.809269.883404.925865.074135 52 9.809419 2.500.883297' 1 78.926122 4'27.073878 51 2.50 1.78 4.27 10 9.809569 2.49 9.83191 18 9.926378 4.27 0.073622 50 11.809718.9.883084 1.78.926634 4.27.073366 49 12.809868 2.4.832977 1..92690 4.2.073110 48 13.810017 2.49.S82871 1.78.927147 4.27.072853 47 14.810167 249.882764.1.78 927403 4.27.072597 46 15.810316 2.49.832657 1.i8.927659 4.27.072341 45 1[49 1.8.5927659.072341 4 5 16.810465.882550 8 927915 4.27.072085 44 17.8lO0614 2'48.882443 1.7.928171 4.27.071329 43 1 8.8106 2.48 1.79.92427 4.27 0713 4 18.810763 248.8s2336.73 42 19.810912.882229 1.79.923634 4;27.071316 41 2:48 1.79 4.27 20 9.811061 2 48 9.882121 1.79 9.928940 427 0.071060 40 21.811210 2.3.882014 1 79.929196 4. 070804 39 22.811358 2.48.881907 1.9.929452 4.27.070548 38 23.811507 2.47.881799 1 79.929708 4.27.070292 37 24.811655 2:47.881692 1.'.929964 4.27.070036 36 25.811804 247.881584'79.930220 4.27.069780 35 26.811952 2.47.881477 1.9.930475 4.27.069525 34 27.812100 47.881369.9 930731 4.2.069269 33 2.47 1.80':: 4.26 069013 32 28.812248 2.47.ss1261 /.80'.930937,26.(69013 32 29.812396 2.47 881153 1.931243 4.26.068757 31.8 153 1.804.26 30 9.812544 9.881046 1 9.931499 0.068501 30 31.812692 2.830933 1 8.9317)5 4.26.068245 29 32.812840 2.46.80330 1. 0.932010 4.26.067990 28 33.812988 2.46.830722 1. O.932266 4.26.067734 27 34.813135 2.46.80613 1.80 932522 4.26.067478 26 35.813283 2.46.830050 1. 80 932773 4.26.067222 25 36.813430.880397 1..933033 4.26.066967 24 37.813578 2.46.80289 1.81.933289 4.26.066711 23 33.813725 2.45.80180 181.933545 4.26.066455 22 39.813872 2.45.80072 1.933800 4.26.066200 21 2.45 4.26 40 9.814019 2.45 9.879963 81 9.934056 426 0.065944 20 41.814166 2.45.879355 1.81.934311 4.26.065689 19 2445 8131 SA[h 42.814313 2.4 o.879746 1.81.934567 4.26.065433 18 43.814460 2.45.879637 81.934822 4.26.065178 17 44.814607 2.4.87929 1.1.935078 6.06-1922 16 2.44 [ 809922 16 ~/ 45.814753 2.44.879420 1.81.935333 4[26 064667 15 46.814900 2.44.879311 4...935589 4. 6.064411 14 47.815046 2.44.879202 1.8.93,344 4.26.064156 13 48.815193 2.44.879093 _.82.936100 4.26.063900 12 49.815339.S78934 1.96355.063645 11 2.44./ 2 4.26 50 9.813485 2.44 9.S71875 1.,2 9.936611 4 26 0.063389 10 51.815632 2.878766 2.936466 26.063134 9 52.815778.43.878656 I.8.9371 21 4.062879 S 53.815924 2.43 S78547 1.S2.937377 4.2.062623 7 54.816069 2.43.878438 1.82.9:37632 4.2.)62368 6 55 816215 2.43.87832.937387 4.2.062113 5 56.816361 243.889219.93142.0)61'58 4 57.81607 243.878109 13.93839S 061602 3 2.43 1.3 o 4.25.816652 2.877999.93 3 6)3 4.061347 2.4'2 1 3' 4.2'' 9.679 2.82 77890.9339803 4 061092 1 6 8). 169343 2.87780.939163 2).06037 0. Cosine. D. 1". Sine D. 1. Cotng D. 1" Tang. I. 130) 49 COSINES, TANGENTS, AND COTANCGENTS. 215 410 13~8 1. Sine. D. 1'. Cosine. D.1.. Tang. D. Ill Cotamr. M 0 9.816943 2.42 9.877780 1 83 9.939163 25 0.060337 60 1.817088 2.42.877670 i.3 939418 4.2.060582 59 2.817233 2'42.877560 1.3.939673 4.060327 58 3.817379 242.877450 1.3.939928 4.2.060072 57 4.817524 2.42.77340 84.94013.059817 6 5.817663 241.877230 1.84 940439 4 25.059561 55 6.817813 241.877120 1.84.940694 4.)5.0593('6 54 7.817958 2.41.877010 1.84.940949 4.25.059051 53 8.81,103 241.876S99 1.84 941204 4 2.058796 52 l.41' - ~ I8~', 4.25 9.818247 2:41.876789 184.941459 2.058541 51 10 9,818392 2'41 9.876678 1 84 9.941713 25 0.058287 50 11111 1,84 4.25 11.818'36 2.07656 1 84.941968 4.5.05(132 49 12.818631.81 876457 1 84 942223 4 2 057777 48 13.8182 240.876347 184.942478 ).057522 47 2.40 1.84 4.25 14.818969 2.40 876236 1 942733 4.057267 46 15.819113 2240.876125 1.8. 942988 42.057)12 45 16.81927.876014.943243 4 (06757 44 2,40 1.85 17.819401 240.875904.943498. 0 6 02 43 18.819545 2 1.875793 1 S'.943752.u 6248 42 19.819689.875682.944007.05,9 3 41 i.39 1.85 4,25 20 9.819332 239 9.875571 1S 9.944262 0.0.5573 40 21.8119976 2.9.875A49 1.'.944517 4.2 055483 39 22.820)120 239.875348 1.-.944771 424 053229 3S 23.820263 2.39.8371 915026 4.054974 37 24.820406 239.8126 34581 4.2.04719 36 25.820550 23 875014 1.86 9413 34. 4 65 35 26.820693.38.874903 1..945.79) 4.241 0-4210 34 27.820836 2.33.874791 1 8 916045.24 9.0.)395 33 28.820979 2.3.874680 6.9;99 424 0. O.3S) 01 32 30 9.S2126 238 9.874436 9.901603.14 0.0U3192 30 31.821407 2 74344 16.17063.0'5237 29 32.821t550 2.874232 1.6.4 S 44.052682 28 33.821693 2.3.874121 1.87.9147o72 4.94.)223 27 2.37 91/72 1 > 4 4.24 34.82183 2.37 8.74009 1.87.947827 424.052173 26 35.821 377 873" 96. 9408 31.05213919 2 36.822120 2. 37.873784 187.94335 4 24 (11 665 24 37.822262 2.37.873672 187.9459U 24.051410 23 3S.822104 237.873560 1.87.948844 424.01156 22 39 1.87 4824 5 6 39.822546 2 37.734 7.949099 424.050901 21 237 8 1.3 7 4.24 40 9.82268 237 9.873335 1. 9.94 9353 24 0.050647 20 41.822,939 236.873223 I.8.949608 4.24.050392 19 42.822972 2.36 873110 1.8.949362 4.24.0;50138 18 43.823114 236.872998 1.88.950116 424.049884 17 44.823235 236.872885 1.88.950371 4.24.49629 16 45.823397 2 36 872772 1.8.950625 4.049375 15 4-6.823539 2.872659 1.8.9879 4.24.049121 14 47.S23630 26.872547 1.88.951133.048867 13 48.823>21 2.|.872434 1.8.951388 4..048612 12 49.82363 23.872321 1.88.951642 4.24.048358 11 "2.35 I 1.88 4.24 50 9.824104 23 9.872208 9.951896 24 0.0-8104 10 51.82445 2.35.872095 1.8.952150 424.047850 9 52.824386 2.3 871981 1.89.952405 494.047595 8 53.824527 2.3 871863 1.952659 494.047341 7 54.824668 2 3 871755 1.89.952913 424.047087 6 871641 1,54.24 55.824808 2 34.871641 1.89.953167 24.046833 5 56.824949 34.87152.953421.046579 4 57.82'5090 871414 LS9.953675 4.046325 3 5.s825230 2.34.871301 189 953929 4.23.046071 2 2.34..92 4.23 4 59.825371 23 71187 19.94183 423.0417 1 I _. 60.821 871073.954437.045563 0 M. Cosine. I D. 1. Sine. D. 1. Cotang. D. 1". Tang. 1. 1310 216 TABLE XIII. LOGARITIHMI IC SINES, 4fGo BL37Q M. Sine. D. 1"1. Cosine. D. 1". Tang. D. 1". Cotalg. M. 0 9.825511 2.34 9871073 1 9.9544317 423 0.04)5563 60 1 8256 231.870960 190 95691 42 01531)9 59 2.825791 2.33 870 46 1 90.954916 4. 2.045054 58 3.825931.870732 1 90.95 200.4 044S00 57 4 826071 2"33.870618 1.9 54541 4.23.04t454 56 5.826211 2 33.870304 1.955708 4.2.044292 5 6.826351 2.3 870390 1 90 9.55961 2 01410!39 54 7.826491 2.33 670276 1 90 9.%62195 4'.043705 53 8 S826631 2 33.87016t 1 91.96169 4 93 043531 52 9 826770 3 7007 1 91 956723 4 23.043277 51 10 9.826910 2 32 9.869933 1 91 9.956977 4 23 0.043023 50 11 827049 232.869318 1.91.957231 423.042769 49 12.827189 2 ).869704 1'91.957485 4 3,042515 48 13.827329.86959 1 91.957739 23 042261 47 14.827467 2'32.8694741 191.95/993 4 23 042007 46 15 827606 2'32.869360 1,9.958247 423 9041753 45 16 827745 232.69245 1 91 958300 423.041500 44 17 827834 2 31.869130 1 92 958754 4 23.041246 43 19 829023 2 31.869015 1 92.959003 423.040992 42 19.823162 2 31.863900 1 92 959262 4 23 040739 41 20 9.829301 2 31 9.86785 1 92 9.959516 4 23 0.040484 40 21 9828439 231.863670 1 2.959769 4.23.00231 39 22 273 2.3 6855 65 1.92 960023 4'23.039977 39 23.823716 2.31.869140 1 92.960277 4'23.039723 37 24 S55 2 B31 9. L6324 192.960530 23.039470 36 265 9289913 2 3.0 86203 1 92.96(784 4.23 0392 16 35 26.82931 2'30.86S093 1 93.96103 43 2 038962 34 27.829269 3.30 867978 1.961292..038703 33 293.829407 2' 30 -.s67862 1.93.961554 4 23.038345 32 29.829545 2230.867747 1 3 961799 4 23 033201 31 30 9.S29693 2 9. 67631 1 9.962052 4 2 0.037948 30 31.S29321 2.867515.962306.037694 29 2.30 1.913'2 4'23 32.829959 2 29.867399 1.9.962O/60 23.03740 23 33.830097 2 2.S6723 193.962913 42.03'187 27 34.830231. 67167.963067 4. 0 6933 26 33 830372 2.29.867051 1.93.963320 4. 3.036630 25 36.830909 229.866935.963574 423.036426 24 37.830636 2.99.S6619 1.9.963323 4 2.036172 23 3.830784 2 29.66703 1'94.964081 4 23.035919 22 39 30921 2 9.8 6 1 94 964335 423.035663 21 40 9.83105)3 2 9.6670 19 9.96 3 0.035412 20 4 1.3119) 2 [. 866353 1.94.9643-12 4.2.03 198 1 9 42.831332 23.866237 1 9.9609 422.031905 1 4:3.831469 2.93.866120 1 94.965349 [42.0346; 1 17 4.831606 2.2 S6600- 19.96602 422.034393 16 4.S 33742 28.869387.96953 55.031145 15 46.831879., 2.865770 1.9.966109 4.033391 14 47.83)201 97.86 1,653 1.966362.033638 13 41.93152 2. 8636 1 9.96661 6.0334 12 49 98339.263119.966G69.03 31 31 11 50 9.832123.9 9.965302 9.967123 422 0.032377 10 51.S3261 7.S618 1.9.967376 4.22.032624 9 52. 3 36 2 7.6 )6 1.9.9 4.29 02 37 1 8 53.8332933 22 7.86495,0 1'9.967333 4 0.032117 7.326 2.27.964833 1.96.963136 4.22.(31861 6.8313) 2.26.864716 1.96.96339 4.22.031611 5 6.833241 226.86459 196.96643 42.031337 4 57 33377 2.26 S61 96 963896 4.22 031104 3 5 93A2 2 26. 16363 196.969149 4 22.0)3031 2 59.83368 26 8641246 1 6.969403 4 22.030597 1 60 833793 6.864127.969656..03034 0 Cosine. D. 1". I Sine. D. 1. Cotg. D. 1. Tangm. 1 32' 47A COSINES, TANGENTS, AND COTANGENTS. 217 4.30.136> Mr Sine. D. Ill. Cosine. D. 1". Tang. D. 1k'. Cotang. M. 9 9,833783 9.864127 9 969G 66 4 0.030344 60.8:99 2. 2.6 4.22.833919 842 969909 030091'519 2.834054 9. 26 ~1.97 42.8340[10 X'922.4863,92'. 97 0162 42.02938 58 8.834189 2'.863774 1" 970416 42',.029054 7 4 934325. 63656.970669. 0293o>4 5.83440 62.26' s3 1.97 90922 4.22.029o7 6.53'9_' "5 z o iu..... 5 9 22,25.863419 1,97 9117/5 ~02862..5 8>.0 127 91/ 422 9 4 7.834,30 9.63301 9.971429 422 028571 o3 8.834 60 2.25~.863183 1.97.9162 422.02318 52 9.834999 22 63064 97 97193 42.02065 51 10 9.835134 4 2 1 9 1o9 9.972188 42 0.027812 50 1.~30569 2'24.862S27 1.',..~72441 4.22 ]027559 /49 83:>403 22i', I'3, e, - 1 2'83 -43 22 862709 1',9, ].972(/95 4.2'~, I027305 /4a 1 3.8:35 5 3.'S 2 90 794.02775052 4 7.24 1.92 9 4.22 1 85G72.292171.1.98.9 320[.026799 > 6 13 2 3>~ 21. 2 1 3 198 14.029)16.. 1(;.1 11' 9. 9.2 2' 333.973707.06223 4 17.,83610)75 8211.9739360 42.026314() 4:3 1 s.836209 2 2 t.$6t [.9.9.9741213/ 422.025787 4 2 22 ~~~~~ 9~.718~3 (39 1 ( 2.2 161( 4.22 19.36313.8 691877.974466.02535 24 41 2.23 1, (9 4.22 -8 20 9.836477 9.86175 | 9.974720 0. 022>0 / 40 2.23 1.94.22 21 836611 i' 61638 13.974973.020027 39 2,23 1.04.22 22.836745 861519.975226. 024774 38 23.836979 2 S61400.97549 422 024521 37 2.23 1.~ 4.22 ( 2 4~ 2, 6,,-) qt 6 24.837012 >93 8612>0 1 /.532 4.22 0 25.837146 611.7595.0201 2.22 86119 1.2 26 837279 939 861041 1 976233 422.023762 34 27.837412 2.22 860922 1.09 976491 422.023509 0 2.222 200 4.22 0 23.93746 (200.976744.0232 32 3 307 2.22.860322 2.00 97776 4.22 0 2 33.837211 860202 97.2021091 7 34 83O14 1 82.21 6008 200 978262 422 021 2 35.838477 2'2 t.859962 2.00'976?4'45 4.22.0214S5 25 30 9.837812 I,.,,_ 9.860,562 |,' [9.97'7250 /,,,,/ 0.022750 /30 / 2.21 2.00 4.22 36.837910 22 604,2' 977603 022497 29 2.21,2 2.01 4.22 37.838742 22 21 7' o 21 0 ~ 979021 4.22 02(979 2' 3.83807 22 8936032 201 97 4726 0202264 282 39 99007 898 2.0 997 4/22 020473 21 40 9.839140 2.21 98960 99 4.22 002020 20 4 1.8399272 22 859239 2.01.990(33 42 019367 1 9 4 3.8396336 2.20.85899S 2.01.9S803S 4.22 0 1 94O62 17 4844 /81969 2 [2 86002 1970 262 49 42 019293 26 43 839 221 2 0 4.21 01 6 1 46 839932 22.98593 1 -0.912.7 / 2.021803 14 4 7.840061 2.20 514 2.02 981330 4.21 01840 13.840196 220 /59342 20 9 3 41 01819776 02 1232 2 2~~~~~~ 19 4 9.8403298.1 582872 2.02.9S20o6 4.21.017944 Il1 50 9.840459 29 1 202 992309 421 0.01691 10 i 1.84059 219.859029.9262 421.02017943 9 52.840722 219 937908 20. 982814 4'21.01o716 8 53.840854 219 8577S6 2.02 9>307 421.016933 7 54.8340985 866 2 993320.016198 2.2l ~"" 1 9i9027/.2T~i2 2.19 203 4.21 55.841116 2 S537543 7.9S3.016427 5 56.82952. 8749 2.03 936.2 2.218 203 4.01.2 403.8473971.856934.9437.01163 01 Cse839/. ~:.9 39in D. 9.003._D.1 T M 22 ~19 || 45.S3~~x'u.95'5 ~ 2 Ol'0-. OI!'S~' 42 8:3940:t / [ 8~ ~91190 ~ 0803 1 117~ ~~~~o ~8 0 I S98o2~6..9SC ~0. /o19,]4 / s/ 2.20 ~~~2 O 1 4 22 4s3 839~86 ] SS9:)350.197 ~.405 /.535 I]9820/9l /'o 017920 / ] l 2xo'.~ 2o02'- 4 ~:452 838ool~ -.s~-~ 8o,.it9s8oi4 /..~_/ o5;6 / s/ ~~2 2'h.- 2109 (~ ~4~[?~ 46.839932 / S7~ ~__/0 6 ~0 3 7 ~.1 so~-63 [ 29/..> 1 5,3/]/ 47 aooo/7 _.~_/,,34/2 I I 336 x o ~.1 2.0' 4 ~1 4s.8410196' 877 943.0156 59.84l ~ ~ ~5s33/ ~98IS5 3/.-.01s197| 1 ~,9 2 19' ~)~.~ 2 0 2 4,41 50. 3405 |' a.5151 |,,',. 218 TABLE XIII. LOGARITHBIIC SINES, &C. 440o l1 50o M. Sine. D 1'!. Cosine. D. 1'. Tang. D. 111. Cotang. _. 0 9.841771 2.18 9.856934 9.932 437 121 0 o011 60) 1.811902 2.1.85o12 2.04.80.()14910!) 2.84203 I.8590 2.04.933 4.1.01467 58 3.8042163 2.13.85 04.5 4.014404 57 949.84229 2.1.86446 2.04 4.1 01412 56 6.812 2 1.86323 2.0.96101 0199 2, 17 2.0-1 421. 8 421jj3.856201 2 7.84296 06. 16.856078. 0 319S607 1.21.0 13 99 5: 3 34.83 1 2 17 859.j6 2.04.93G860 4.21.013140, 9.842946 217.855633 204.97112 421.012 533 1 2.17 2.045 4.21 10 9.843076 217 9.85711 2.97365 421 00126395 50 11.43206 2.17.85 2.0 9376 4.21.0123 49 12.S43336 2.16.85.546 2.05.937371 4. 1.012129 48 13.843466.8342 2.0 4.21.01S77 1 5.843725 2.16.5 096 2 9..9383629 4..011 37I 4 5 16.8 2 16 209493913142.9 82 421.011113 44 17.843934 216 54350 0.989134 1.018366 43 138.844114 2167.8547/27 206 9 4.2.01013 42 19.844243 216.854603 2 06.99610 421.010360 41 20 9.84372 2.1 9.85440 2.06 9.9393 421 0.10107 40 21.4502 2. 1.854356 2.06.99014 4.21.00955 39 22.844631 2,1.8042633 2.990393 4.21.009602 33 23.844760 2 15 84109 2.06 99061 42.009349 37 24.44939 25.8-396..99903 491.009097 3 2.23 45018 2..853362 2.06 991156 4.21.00844 3 2.845147 215.85973lS 2.06.9914()9 4.21.003591 329.S1)9533 211.8593366 2.07 992167 4.21.007833 31 30 9.345662 214 9.8532- 2 9 992920.070 30 2.14 2.07 4.2I1 31.84790 214.531 I 207.99262 421.00732 29 32 3845919 2.14 89 9929. r9 2.075 4.00705 2 33.336047 2 14.352969 2077.9937S 4 21.006132 27 3 1 846175 2 )14 852745 2(07.993431 421. 006569 26 35.630 2.14.262 2.03 99633 4.21.06317 2 36.846432 2.13.2496 2. 3.993936 4 21 006064 24 37.846563 2.16.852373 2..9 9 421 00 1 23 33.84663 213.85227 2.0 994441..005559 22 39 343316 2 1 2.24994.852122 203 99.0053096 21 40 9.84694 13 931997 03 999 7.21 0.00 3 20 34 3 -7071 2.1.3.51872 2.0 9999.21 0041 42.847199.853747.99492.00448l 148 2.13. 2.05 4.21 43.8437327 2.13.85162 2.09.995705 42 00429 7 44 741 2.1 1497 29.995957 421 00043 1 45.81475 2 l.81372.996210 4 21 003790 14 46 347709 2.12.851246 2.09.996463 4.21.003537 1 4 4 37 )47336 2 812.851121 9.996715 4 t j0032853 13 347 2. 21 2.0 4.21 4.847904 2 12 S0996 2.09.996963 42 00332 1 2 49 809 2 070 209 997221 4.2.00279.850 9.8745 9.997473 421 0.00227 1 0 1.84450] 2.1 2.0 997726 421.002274 9 52.8472 2 2.10.997971 421 002021 3 5 813 2 12 99061 2.0 9 4.21 43 387299 2.11.839036S 2.10 99'3231 4.21 001769 7 54 8393726 1.502 2.993431 421.001516 6.55.81SS52 2.11.8.0116 2.10.99S737 4.21.001263 5 56.818979 49990 2.0 99 |. 001011 4 57 3.19106 2.11.849364 2.0.999242 4.21.000758 3 53.849232 211.849733 2.10 4 99995 2 [.000335 32 2.11 2.110 4.2 003L 59 939 2.1 2. 9997 47 4.21 00023 1 86)4.81949 1.1 849439 0 0000.000000 0. Cosine. D. 1" Sine. D. 1". tan99. 1 Tan- M.s#/2.15 2I.8 i —_ ~ ~~2.[ ~'5070 2.9 99221 4:~ oo~=9 —---- 9.1219[ 9998 TABLE XIV. NATURAL SINES AND COSINES. 220 TABLE XIV. NATURAL SINES ANTD COSINES. 00 10 f 30 0 M. Sine. oCosin. S ~. Cosin. i Sine. Cosin. Sine. Cosin. Sine. Cosin. 31. (.00000 Onle.. 01 4'.9993.5. 030 999-.39 i.0 234.9936:3.(6796 9 7 56 6L0 1.00029 One..01774.99934.03519.99938.0)5263 99161 0.:7005.99754 59 2.00058 One..0103().99934.0354].99937.05292.996Q60.070311 997/5 03:J.00037 One..01332.999'3.0377 1.99936.05321.99 05708.07063.99750 57 4.00116 One..01862.999 3.03606.99935.05350.99857i.07092.99743 56 5.00145 One..01391 9.99932.03635.99934.05379.993;55.0)121.99746 55 6.00175 One. 9.019 0/ 99992.0366l.99933.05409.99354.07150 1.99744 54 7.00201 One..01919 99931.03693.99932.05437.99352.07179.99742 53 8.00233 One..01978 999'30.03723.99931.05466.99531 1.0720.99740 52 9.00262 One..02007.99930.03752999.30.05495.99349.07237.99738 51 10.00291 One..02036.99979.03781.99929.0552-1.99847.07266.997i06 5C0 11.00320.99999.02065..99979.033-10.99927.05553.9946.0729.9973 49 12.00319.99999.020941.99978 03S39.99926.-05532. 99314.07324.99731 48 13.00378.9999.09123.99977.03631.99925 0.05611.99342.07353.99729 47 14.00407 1.99999.02152.99977.03697.99924. 05640.99841.07392.99727 46 15.00436.99999.02181.99976.03926 -.99923.05669.99339.07411.99725 45 16.001465.99999.02211.99976.03955.99922.05693.99333.07440.99723 44 17.00495.99999.0940 99975.02394.99975 92L.05727.99336.07469.99721 43 18.00524.9999.02269.99974.0-1013.99919.005756.99834.074983.99719 42 19.00553.99993.0229.99974.04042.99918.05785.99833.07527.99716 4 1 20.00532.99993.02327.99973.04071.99917.0.5314.99331.07556.99714 4() 2 l.00611.99993.02356.99972.04100.99916 6.05344.99329.075853.99712 39 22.006-0!).99993.02335.99972.04129.99915 0.05373.99827.07614.99710 3. 23.00669.99993.0214.99971.04159.99913.05902.99326.07643.99703 37 24.00698 99999.02443.99970.0-1833.99912.05931.9924 1.07672.99705 36 25.00727.99997.02472.99969.0-1217.99911.05960.99322.07701.99703 35 26.007056.99997.02501.99969.04246.99910.0O999.99321.07730.99701 34 27. 0073.5.99997.02530.99963.04275.99909.06018.99319.07759.99699 33 23.00814.99997.02560.99967.0)4304.99907.06047.99817.077838.99696 32 29.00344.99996.02589.99966.04333.99906.06076.99315.07817.996941.sl 30.00373.99996.02618.99966.04362.99905.06105.99313.07846.99692 30 31 00902.99996.02647.99965.04391.9990.06134.99312.07375.99689 29 32.00931.99996.02676,99964.04420.99902.06163.993310.079041 99687 28 33.00960.99995.02705.99963.04449.99901.06192.998083.07933.99635 27 34.00939.99995.02734.99963.04478.99900.06221.99806.07962.99633 26 35.01018.99995.02763.99962.04507.99398.06250.99304.07991.99601 25 36.01047.99995.02792.99961.04536.99397.06279.99303.0020.99673 24 37.01076.99994.02321.99960.04565.99396.06303.99801[.08049.99676 23 39.01105.99994.02350.99959.04594.99394.06337.99799.06078.99673 22 39.01134.99994.02379.999359.04623.99393.06366.99797 |.03107.99671 21 40.01164.99993.02903.99953.04653.99392.06395.99795.0136.99663S 20 41.01193.99993.02938.99957.04632.99390.00424.99793.0316 1.99666 1 9 42.01222.99993.02967.99956.04711.99339.064 53.99792.03194.99664 1s 43.01251.99992.02996.99955.04740)9983.99.o6452.99790.0223.99661 17 44.01230.99992.03025.99954.04769,99336 1.065 1 0.99738.082252.99659 16 45.01309.99991'.03054.99953.04798,99335.06540.997,936.08231.99657 15 46.01.'339.99991.03033.99952.04827.99383.06569.99791.03910.996 54 114 47.01367.99991.03112.99952. 04356. 9962. 065Y91. 997982.09339. 99652 13 43.01396.99990.03141.999.1.04S35.99331.06627.99780.03363.99649 12 49.01425.99990.03170.99950.04914.99379.06656.997 0 7.093971.906417 11 50.01454.99939.03199.99949.04943.993783.06685.99776.0`9426.996;144 10 51.01433.99939.03223.99943.04972.99876.06,14|.99774. 0s455. 9642 9 52.015131.99939.03257.99947.05001 1.993875. 6743 199772.0 1 4.99639 8 53.01542. 9993 03236.99946.05030,99373.06773.99770.0351 3 |.99637 7 54.01571.99933.03316.99915.05059.99372.06302.99763.00542.9963. 6 55.01600.99937 03345.999-14.05083.99370.06331 1.99766.03571.99632 5 56.01629.99937.03374.99943,05117,99869.06 60.99764.0361)0 930 4 57.016;5S. 99996.03403.99942 1.05146 1.9967..06 9 (.99762.(36)9.9969 7 3 53.01637. 99936.03432.99941..0)175.99.66 0.)691S.9.(760 0. 3 6|.096903 2 59.01716.99935.0316 1.99940.0)205.996i. 06947.99758 O 637.996 I22 1 6n.01743.9993 5.03 -90.99939.05234.99.:63 0.976G I.99756 037161.9619 0 AI. Cosin. Sine. Cosin. Sine. Cosin. Sine. gCosin. Sine. Coein. Sine. M. F389D9 _o. - Sin -# s in -e.6 B5; _] TABLE XIV. NATURAL SINES AND COSINES. 221 53 10 i9 90o. Sine. osin.. osin. Sine. Coin. Sie. Cosin. Sine. Cooin. TM. (1.0U3716. 99Y619-.10453.99452.1;187.992o5. 1331 7.9927.1564-13.93769 60' 1.03745.99617. 10IS2.991 14-).12216.9921.13916.991)23.15672.98764 59 2.03774.99614.10.511.99446.12245.99)48.13975.99019.15701.93760 58 3.0()S03.99612.10 -4)).9944:. 1274 1..924-.1-410)4 1099015.159730.98755 57 4.08331.99609. 105I69.99440.123025.9940.140:33.19.,75.987591 56 5.0O86S.99607.10597.99437. 12331.99237.14061 [.99006.15787.93746 55 6.03339.99604.10626.99434.12360.99233 1.1 )9). 99002.15816.98741 54 7.03918.99602 I.10 655.9431.12389.99230.14119.98'98.15345.98737 53 8.03947.99599. 106-31.99428. 12418.99226. 1414S.93994.15873.98732 52 9.03976.99596.10713.93124.12447.992;22.14177.98990.19902.93728 51 10.09905.99594.10742.99421.12476.99219.14205.98396.15931.9S723 50 11.09034.99591.10771.99418.12504 9921.14234.9392.1599.971 49 12.09063.99;53 B.10300.99415. 12533.99211. 14263.93978. 15938.98714 4T 13.09392.99596. 1029. 99- 1142.12;562.99203.14292.98973.16017.9S709 47 14.09121.99.5833.1053.99409.12591.99204.14320.93969.16046.98704 46 15.09150.99590.1 0387.99406.12620.99200.14349.9S965.16074.93700 45 16.09179.99578.10916.99102.12649.99197.1437,S.98961.16103.98695 44 17.09203.99575.109-15.99:399.12678.99193.14407.93957.16132.98690 43 18.09237.99572.10973.99396.12706.99189 -1.14436.9S953.16160.98686 42 19.09266.99570.11002.9939:3.12735.99186.14464.93948.16189.98631 41 20.09295.99567.11031.99390.12764.99182.14493.93944.16218.98676 40 21.09324.9994.11060. 99336.12793.99178.14522.98940.16246.98671 39 22.09353.99562.11039.99383.12322.99175.14551.9S936.16275.9S667 33 23.09382.99559.11113.993S0.12351.99171.14580.98931.16304.98662 37 24.09411.99556.11147.99377.12880.99167. 1460.983927.16333.98657 36 25.09440.99553.11176.99374.1290S.99163.14637.98923.16361.98652 35 26.09469.99551.1 12051.99370.12937.99160.14666.98919.16390.98648 34 27.09498.99548. 1 1234.99367.12966.99156.14695.98914.186419.98643 33 23.09527.99545.11263.99364.12995.99152.14723.98910.16447.98638 32 29.09556.99542.11291.99360.13024.99148.14752.98906].16476.98633 31 30.09585.99540.11320.99357.13053.99144.14781.93902.16505.98629 30 31.09614.99537.11349.99354.13081.99141.14810 1.98897.16533.98624 29 32.09642.99534.11378.99351.13110.99137.14838.98893.16562.98619 23 33.09671.99531.11407.99347.13139.99133.14867.98889.16591.98614 27 34.09700.9952S.1 1436.99344.13163.99129.14896.98884.16620.98609 26 35.09729.99526.11465.99341.13197.99125.14925.98880. 16648.98604 25 36.09758.99523.11494 1.99337. 13226.99122.14954.98876.16677.98600 24 37.09787.99520.11523.99334. 13254.99118.14982.98871.16706.98595 23 33.09316.99517.11552.99331.13283.99114.15011.98867.1.6734.98590 22 39.09345.99514.11580.99327.13312.99110.15040.98863.16763.98585 21 40.09374.99511.11609.99324.13341.99106.15069.98858.16792.98580 20 41.09903.99508.11638.99320.13370.99102.15097.98854.16820.98575 19 42.09932.99506.11667 1.99317.13399.99098.15126.98849.16849.98570 18 43.09961.99503.11696.99314.13427.99094. 15155.98845. 16378.98565 17 44.09990.99500.11725.99310.13456.99091.15184.93841.16906.98561 16 45.10019.99497.11754/.99307.13485.99037.15212.98836.16935.98556 15 46.10048.99494.11783.99303. 13514.99083.15241-.98832.16964.98551 14 47.10077.99491. 11812.99300.13543.99079.15270.98827.16992.98546 13 4S. 10106.99438 1.11,40.99297.13572.99075.15299.98323.17021.98541 12 49.10135.99435.11869.99293.13600.99071.15327.98818.17050.98536 1 1 50.10164.99482.118931.99290.13629.99067.153.56.93814.17078.98531 10 5 1.10192,.99179. 11927 1. 992S6.1 36538.99063.15385.98809 17107. 98526 9 52.10221.99476.11956.99283.13637.99059.15414.98305.17136.93521 8 53. 10250 994173.1 1935 1.99-279.13716 1.99055.15442.93800.17164.98516 7 54.10279.99470.12014.99276.13744.2 99051.15471.98796.17193.98511 6 55.1033.99167. 120i43.99272.13773.99047 1 5500.93791.17222.93506 5 56.10:337.99164.12071.99269.13302.99043.15529.93787.172s5(.98501 4 57.10366.99461.12100.9926.5,13331.99039.155.57.98782.17279.98496 3 53.10395.99145.12129 1.99262. 1360.99035.15586.93778.17308.98491 2 59.102-4.99415. 12158. 992-581.13S39 1. 99031. 15615 1.98773.17336.98436 1 60.10531 991452.12187.99255.13917.99027.15643.98769.17365.98481 0. Cosil Sine. Cosin. Sine. Cosin. ISine. Cosinl. Sine. Cosin. Sine. M. I 80 83) 9 0 8-10 80I 1981 222 TABLE XI-V. NATURAL SINES AND COSINES. 1_> 11 10 L 110 103 IL ]LO~]L3D 4 - 0 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin'I 0.17365.98481..1901.93163.20791.971 O.22495.97137.t9 )2,97)30 60) 1.17393.98476.19109.98157.20320.97809.22523.97430.24242().97023) 59 2.17422.93471.1913'3.98152.20848.9730()3 22552.974241.21249 9701D 5 5 3.17451.93466.19167.93146.20377.97797,2 42 S 0.97417.24277,97008 57 4.17479.98461.19195.9S140.20905.97791.22608.97411.24305.97001 56 5.17503.98455.19224.9S135.20933.97784.22637.97404.24:333. 9S699.4 55 6.17537.93450.19252.98129.20962.97778.22665.97393.24362.96987 54 7.17565.98445.19281.98124.20990.97772.22693.97391.24390.96930 53 8.17594.98440.19309.981181.21019.97766.22722.97334.24418.96973. 52 9.17623.93435.19338.98112.21047.97760.22750.97378.24446.96966 51 10.17651.98430.19366.98107.21076.97754.227781.97371.24474.96959 50 11.17630.99425.19395.98101.21104.97748.22807 1.97365.24503 1.96952 49 12.17703.98420.19423.93096.21132.97742.22835.97358.24531.96945 48 13.17737.98414. 19452.98090.21161.97735.22863.97351.24559.96937 47 14.17766.98409. 19481.98084.21189.97729.22892 97345.24587.96930 46 15.17794.98404.19509.98079.21218.97723.22920.97338.24615.96923 45 16.17823.98399.19533.98073.2i246.97717.22948.97331.24644.96916 44 17.17852.98394.19566.98067.21275.97711.22977.97325.24672.96909 43 18.17880.98389.19595.980'61.21303.97705.23005.97318.24700.96902 42 19.17909.98383.19623.98056.21331.97698.23033.97311 1.24728.96394 41 20.17937.98378.19652.98050.21360.97692.23062.97304 1.24756.96887 40 21.17966.98373.19680.98044.21388.97686.23090.97298.24784.96880 39 22.17995.98368.19709.98039.21417.97680.23118.97291.24813.96873 38 23.18023.93362.19737 1.98033.21445.97673.23146.97284.24841.96866 37 24.18052.98357.19766.9S027.21474.97667.23175.97278 1.24869.96858 36 25.18031.98352.19794 1.98021.21502.97661.2324)3.97271.24897.96851 35 26.18109.98347.19823.98016.21530.97655.23231.97264.24925.96844 34 27.18138.93341.19851.98010.21559.97648.23260.97257.24954.96837 33 28.18166.98336.19830.98004.21587.97642.23283.97251.24992.96829 32 29.18195.98331.19903.97998.21616.97636.23316.97244.25010.96822 31 30.18224.98325.19937.97992.21644.97630.23345.97237.25038.96815 30 31.18252.98320. 19965.97987.21672.97623 23373.97230.25066.96807 29 32.18231.98315.19994.979891.21701.97617.23401.97223.25094.96001 28 33.18309.98310.20022.97975.21729.97611.23429.97217.25122.96793 27 34.18338.93304.20051.97969.21758.97604.23458.97210.25151.96786 26 35.18367.98299.20079.97963.21786.97598.23486.97203.25179.96778 25 36.18395.98294.210 1.97958.21814.97592.23514.97196.25207.96771 24 37.18424.93288.20136.97952. 21843.97585.23542.97189.25235.96764 23 33.18452.98283.20165.97946.21871.97579.23571.97182.25263.96756{ 22 39.18481.98277.20193.97940.21899.97573.23599.97176.25291.96749 21 40.18509.93272.20222 [.97934.219289.97566.23627,.97169.25320.96742 20 41.18.538.9267.20250.9792S.219-56.97560.2:366.97162.25I.3 14.96734 19 42.1 567.98261.20279,.97922.21985.97553.23684.97155.25376.96727 18S 43.18095.98256.20307.97916.22013.97547.23712.97148.2354041.96719 17 44.18624.98250.20336.97910.22041.97541.23740.97141.254.32.96712 16 45.18652.98245.20364.97905.22070.97534.23769.97134.25460.96705 15 46.1861I.98240.20393.97899.22093.97528,23797.97127.25488.96697 14 47.18710.93234.20421.97893.22126.97521.23325.97120.25516 1.96690 13 48.18733.98229.20450.97887.22155.97515.23953.97113.25545.96632 12 49.18767.98223.20478.97881.22183.9750 23SS2.97106.25573.96675 1l 50.18795.93218.20507.97875.22212.9750(2.23910.97100o.25601.96667 10 51.18824.93212,20535.97869.22240.97496.2:3938 97093 l25629 196660 9 52.18852.9820(7.20563.97863.,22268.97489.23966.97096 2 25657'.9'653 8 53.18831.93201. 20592.973857 22297.974. 3 23995.97079.2.6835.96645 7 54.18910.98196.20620.97851.I 223 25.97476.240233.97072.257I13.966381 6 55.18938.98190.20649.97345.22353.97470.21051 1.97(6..25741.96630 5 56.18967.98185 1.20677.97839.22382.97463.24079.92705.25769.966233 4 57.18995.93179.20706.9783:3.22t10.97457.2-108.97051.2.579.96615 3 58.19024.93174-.20734.97827.22433.97450.21136.97014 2.2-58296.96608S 2 59.19052.93168.20763.97321 D.22467.97444 IG164.97037.25854.96600 1 60.19081.93163.20791.97815.22495.97437.24192.97030.25832.96593 0 M. Cosin. Si - Sine. C o.S inn. l Sine.i Cosin. Sine. Cosin. Sine. M. _ 793 s780 i 773 s 750 TABLE XIV. NATURAL SINES AND COSINES. 223 _153 9 1 186 17 190 MI. Sine. Cosin. Sine. I Cosin. Sine. Cosin. Sine. Cosin. ine. Cosin. M. 0.2.969.7564.96126.29237.956.30.3090 9-19 06.32557 9- 531 3.25910.96).55.27592.96118.29265.956)2;2.30929.95097.32.S4.94.j42 589 2.252935.96)578, 27620.961 10.29293.95613.30957.8.32612.94533 5 3.25966.96570.27648.96102.29321.95605.30985.935079.32639 4.2)3.571) 4.25994.96562.27676.96094.29:3483.9596 -.31012.95070.34667. 9414 6 5.26022.96555.27704.96036.29376.90558.310-410.9061.39694-.94504I5 6 1.26050.96547.27731.96078.29404..9 95 79. 3106.9502 32722 944 I 7.26079.965410 27759.96070.29432.90571.31095.95043.32749.94453 8.26107.96532,27787.96062.29460.935562.31123.95033.32777 594476 2 9.26135.96524,27315.9604.29487.9554.31151.95024.32804 1.94466 5 10.261G3.96517.27843.96046.29315.95545.31178.95015.32932.9-14,7 5()0 11 1.26191.96509.27871.96037.29543.95536.31206.95006,32859.9-417 49 12.26219.96502.27899.96029.29571.955289.31233.94997.32887.94438 41 13.26247.96494.27927.9602t 29599.9559.31261.949S8.32914.94423 47 14.2627 ).9643G 6 27955, 96013,29626.935511.31289.94979.32942.94418 46. 15.26303.96479.27933.96005 29654.95502.31316.94970.32969.94409 45 16.26331'.96471.23011.95997.29632.95493.31344.94961.32997.94399 44 17.26:359.9 646'- 1.251031.9599 9.29710.953485.31372.94952.33024.94390 443 18.26337.96456.23067.95981.29737.95476.31399.94943.33051.94380 42 1 9.26415.964463.2809.95972'29765).95467.31427.94933.33079.9-1370 41 20.26443.96440.28123.95964.29793.95459.31454.94924.33106.94361 40 21.26471 { 96433.2150.93956.29621.95430.31482.94915.33134.94351 39 22.26500.96425.28178.95943.29349.95441.31510.94906.33161.943-42 33 23.265923.96417.28206.95940 1.29376.95433.3 1 537.94897.33189.9433;2 37 24.26956,96410.28-234.95931.29904 95424.316565.94883.33216.94322 36 25.26594.96402.26262 95923.29932.95415.31593.9487.33244.94313 35 26.26612.96394.26290.9)915.29960.9.5407.31620.94869.33271.94303 34 27.26640.96336.2'3 t 8.95907.29987.95398. 3164 8.94860.33298.94293 33 2S.26668.96379.2316.95893!.300151.95339.31675.94851.33326.94284 32 29.26696.96471 23374'9915 30043 1 95380.31703.94842.33353.94274 31 30.26724.96363.2S402.95582.30071.95372.31730.94832.33381.94264 30 31.26752.96355.28429.95874.3009S.95363 3175.94823.33408 9.9254 29 32.26780.96347.23457.95865.30126.95354.31 736.94814.33436.94245 26 33.26306.96310.231835.9557.30154.9 345.31813.980.33463.94235 27 34.26336.96332.28313.95S49.30182.9.0337.313941.94795.33490.9422.5 26 35.26364.96324.28541.95841.30209.95323. 316s.9476.33518.94215 2 36.26S92.96316.28669.95332.30237.95319.31896.94777.33545.94206 24 37.26920.9630,8.2~)597.95324.30265.995310 t.31923.9476.3373.9496 23 38.269431.96301.28625 1 958 16.30292.95301.31951.94758.33600.9466 22 39. 26976.96-293.286)62 5.953807.30320.95293.31979.94749.33627.94176 21 40.270041.962835.26680.95799.30343.95284.32006.94740.33655 94167 20 41.27032.96277.23703.95791.30376. 95275.32034.94730.3362.9417 19 42.2706r3.961269.23736.95782.30-103.95266.32061.94721.33710.94147 16 43.27(138.96261.28764.95774.30431.9.5257.32089.94712.33737.94137 17 44.27116.96253.28792. 95766.30459.95248.32116.94702.33764.9127 16 45.27144.96246.28320.95757.30486.95240.32144.94693.33792.94116 13 46.27172.96233.28847.95749.30514.95231.32171.94634 M.33819.94108 14 47.27200.96230 1.23875).95740.331512.9.5222.32199.94674.33o16.9409 13 48.27228.96222.23903.95732.30570.95213.32227. 94665.33874.94088 12 49. 27256 |*9)62 14.28931.95724.30597.95204 ~ 32254.94656.33901.94078 11 50.2723-1.96206.28959.95715.30625 95195 3222 1.94646.33929.9406 10 5[.27.312.96198.28987.95707.30653.918G 6.32309.94637.3396.94058 9 52.27340.96190.2901 5 /.956993..30)60.95177.32337.94627.3393.94049 8 5 3.27363.96 182.290)-12.95691).30708.953163 1.32364.9661S6.34)1 1 1.94039 7 54.27396.96174.29070.95691 9.30736.951959.32392.9609.34038 94029 6 /55.274211.96166.29093.9 967:3.30 63.93-150.32419.94599.3406.5 i.9019 5 56.27452I.961533.29126.95661 3.791 95142.32447.94.9).3093.91009 4 571 27430 1 9615O.29154.95656.3319.951333,32417.1 94: i.34120 1.9399 3 58 *2;5!)3 S.25 6t4 29182 1.956-17 334( 95114.325)02.9-471.341471.93991 2 59i.27536.96134 29209{.95639..':-3 4 I 951 1 5 9 I 39.23 193-6 61329 3 031.95639 Oi4 "6)115.3>5)29I.94361 3-1175 939731 1 0.2W7561.)GI26.292'3.971 (563:-) 9021' 9;l6 1:. {:;);-) l5 I.'3.. 2 *3 1 2 9. 9. 60.2 9 9)3656.37>1.3!.(3969 611 ~M. IC~osil. Sie. Cosiu. 8ine. i (osil. Sine. Cosin. I Sine Cosin.l Sine. i.. jCoosin Si3 >. Sine Ci SneiCo. n'C. _7. _.I 224 TABLE XIV. NATURAL SINES AND COSINES. 203 21 " 212 ~ 33 2 30 MI. Sine. Cosin Sine. Cosin Sine. Cosin. Sine. Cosin. Sine. Cosin. i1. 0O. 31)80. 932S61 353's:7. 933; ~ 3 — 461. 9 7 1,1.:39)73.92050. 4)674 0.'1 1.34229.93959 1.3-6 L. 4 334s.37438.92707.39100.92039.40700 19343 r 9 2. 1.3457 0.93949.35S91.93337.37515.92697.39127.920'23.40727.91331 51 3.3434.93939 35913.93327 1.37542.92656.39153.92016.40753,91319 57 4 34311.93929.3 5915.93316.37569.92675.3910).92005.407S0 91307 56 5 1.34339.93919.35973.9.3306.37595.92664 1.39207.91994.40906.912995 55 6.34366.93909.36000.93295.37622.926-53.39234.919>62.40333.912S3 541 7.321393.93.99.36027.932S5'. 37649.92642.39 260.91971.4060.912721 50.3 8.34421.9389.36054.93274.37676.92631.39237.91959.4036.91260 52 9.34442.93379.360311.93264.37703.92620.39314.91943.40913.91248 51 10.34475.93363.36103.93253.37730).92609.39341.91936.40939.91236 50) 11.4.34503.93859.36135.93243.37757.92593.39367.9192:5.40966.91224 49 12.345304.9:349.:36162.93232.37784.9257.3939 4.91914.40992.91212 41 13.3455.7.93339.36190.93222.378 1 1.92.076.39421.91902.41019 91200 447 14.34534.93i329.36217.93211 I.37>3.92.56 5.39448.91891.41045.911 21 4G6 15.3i6121.93319.36244.93201.37865.92.5)54.39474.91279.41072.91176 45 16.34639 1.93 09.36271.93190..37892.925,43.39501.91863S.41098.91161 4.1 17 1.34666.93799.36298.93130.37919.92.532.39;2S.91856.4112)5.9 11 52 4 [ 13.34694.93789.363251.93169.37946.92521.39555.91845 9 4.411511.91 [14(! - 19.3-1721.93779.36352.93159.37973.92510.3958 1.91833.41178.911 2, 4 1 20.3i748.93769.36379.93148.37999 92. 396 0.91822.41204.91 16 -1 ( 21.34775.93759.36406.93137.33026.9243ss.39635.91810.41231.91104/ 39) 22.3403/..93743,9.364:34.93127.330053.92177.39661 1.91799.41257.91092 3s 23.34S30].93733.36461 9.93116.30303(.92466.396S8.91787.412s4.91030 37' 21.3435.7. 9372 63.'36t48.93 6lr.383107.92455..39715.91775.41310.9106,S 36 25.344 31.93710.36515.93095.3>13>1.92444.39741.91764.41337.91056I 363 26.34921).9 370.365'42.0930 41.3316i.92432. *.39768 6.91752.41363.91(044 34 27.31939.93693.36569.93074.3312 1.92421..39795.91741.41390.91032 33 2>|.3-1966.936s1.36596.93063.3>215.92410.39S22.91729.41416.91020 32 29.34993.93677.36623.93052.382-11.92399.39348.9171.41443.91008 31 30.35021.93667.36650.93042.3326>.92338.39375.91706.41469.90996 330 31 4.435045.93657.36677.93031.3 295.92377.39902.91694.41496.909>4 29 32.3(50)71. 93647 |.3704.93020.3S322.92366.3992>.916S3.41522.90972 21>; 33.35102.93637.33731.931)10.33349.92355.39955.91671.41o49.90960 27 34/.:5130.93626.36758.92999.333 6 1 32343 3992 91660 1.4157 rz 909451 26 35.35157.93616.371351.-929s3.3>403.92332.4000s.9164,.41602. 90936 25 36.331824.93606.3681 1. 9297S.38430.92321.400351.91636.4162 1.90924 24 37.3-5211.93596.36S39.92967.38456.92310.40062.91625,.41655.90911 23 3l.35239.93535.36367.9 2956.3 4333.92299.400>2.91613.416S l.90399 22 39.35~266.93>57>5. 36S94.929-45.35101.922>7.40115.91601.41707.9093S7 21 40.35293.93>565.36921.92935.33537.92276.40141.91590.41734].90875 20 41.35320 /.93555.33194S.92924.33564.92265.40163.91 5078.41760.90S63 19 42.35347.93544.36975.'9:913.33591.92254.40195.91566.41787.90S51 1 I 43.35375.93534.37002.92902. 3S617.92243.40221 [.9195505.41813.90839 1 7 44.35402.93524.37029.9'929|.3S641.92231.40218.91 543.41840.9026 16( 45 35429.93514 9.37056.92381.3S671.92220.40275.91531.4 866.90314 15 46.354561.93503.37083.92370.33693.92209.40301.91519.41892.90S02 14 47.334 1.93493.37 110.92 59.33725.9219>.4032>.91 5.41919.90790 l 3 43.35511 934s3.371371.2849 7.3 752.992106.40355.91496.41945.90773 12 49.3.33.93172.371 L.92333.33773.92175 40331.9144-1.41972.90766 11 50 /.3-56),.93462.37191.92027.33305.-.92164.40403. 91472.41998 >.9075>3 1 0 5l.3-5092.93152.3721s.92316.33S32.92152.40434.91461.420124 90741 9 521.356 9.93441.3724i5.92S05.30859.92141.40461.91449.420511.90729 8 53.3 56747 93431.37272.92794.333SS6.92130.40488.91437.42077.90717[ 7 54.3567-1.93420!.37299.92704.33912.92119.40514>.914259.42104.9070-1 6 55.35701.9341 (.37326.92773.3s939.92107.40541.91414.42130.9069 56.3,7231.93400.37353.92762.3s966.92096.40567.91402.42156.9060 4 57|.35755.93339.373S0.92751 >.3~993.92035.40594.91390.42 s3 1.9)66 3 53.35782.93379.37407.92740.3902q.92073.40621.91378.42209 1.906>5 2 r 91.3 510.93368.37434 192729 309046.92062.40647.91366.4 -223.) 06(; t I1 60 I.35x371 93358 4.7461.927138.3>073.92050.4067- 1.91 355/.'2;G2 /.942 G:1; 3I Coslin Sine Cosin Sine. Cosinl. Sine Cosin Sine cosill. Sine. I.T _ 69> _ > _ 6> 7 663 > _ __ T.:: TAiLL xSiV. NATUIR AL SINiES ANiD COSINES. ~25. Sine. Cosin. Sine. Cosin. Sine. C osin. Sine. Cosin. Sine. Cosin. 53 o0.42262 90631 -.43337 39.8- 4 91U1.46947 6.3295.48431.874626 60 1.42233.90618.43363.,9 367.45425.89087.46973.88s21.43506.87443 i59 2.42315.90N6 1.43389 /.93354.45451.89074.46999.83207.43532.87434 58 3.42341.90.59.43916.S9841.45477.89061 4702-1.8S321.48557.87420 57 -1.42367.90 32 1.43942.89323.,.45503.89043.47050.8324,,.48353.874506 56 5.42394.90569.43963.89316.4o5529.89035.4 076.88226.43608.87391l 55 6.424420.90557,43991.89803 1.45554.89021.47101.83213.48634.87377 54 7.42446.90545.440420.89790.45580.89008.47127.88199.48639.87363 53 8.42473.90532.44046.89777.45606.88995.47153.88185.48634.87349 52 9.42499.905929 0.44072.89764.45632.839s1.47178.88172.48710 i87335 5il 10.42525.905>07.44093.89752.456583.83968.47204.8158 j.48735,87321 50 11.42552.90495.44t24.89739.45684.848955.47229.88144 48761.87306 49 12.42578.90143.44151.89726.45710,88942.472;55.88130.48786.87292 48 13.42604.90470.44177.89713.45736.83923.47231.88117.4831 1.87278[' 47 14.42631.90453.44203.89700.45762.88915.47306.88103.48837.87264 46 15.42657.90446.44229.89637.45737.88902.47332.88089.48862.87250 45I 16.42633.90433.442551.89674.45813.83383.47358.88075.48883.87235) 44 17.42709.90421.44231.89662.45839.88875.47333.88062.48913.87221 43 1S.42736.90403.44307.89619.45865.88362,.47409.88048.48933.87207 42 19.42762.90396.44333.89636.45891.88848.47434.83034.48964.87193 41 20.42733.90333.44309.89623.45917.88835.47460.88020.48989.87178[ 40 21.4281.90371.44395,89610.45942.883322.47486.88006.49014.871641 39 22.42341.903583.44411.89597.45963.88803.47511.87993.49040.87150 33 }23.42S67.90346.44437.89534.45994.88795.47537.87979.49065.87136 37 24.42394.90334.44464.89571.460209.88782.47562.87965.49090.87121 36 25/.42920.90321 1.44490.95.58.46046.88768.47588.87951.49116.87107 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 23.42999.90234.44563.89519.46123.88728.47665.87909.49192.87064 32 29.43025.90271.44594.89506.46149.88715.47690.87896.49217.87050 31 30.43051.90259.44620.89493.46175.8S701.47716.87882.49242.87036 30 31.43077.90246.44646.89480.46201.s8688.47741.8786S.49263.87021 29 32.43104.90233.44672.89467.46226.88674.47767.87854.49293.87007 98 33.43130.90221.44693.89454.46252.38661.47793.87840.493 18.86993 27 34.43156.90208.44724.89441.46278.38647.479 81.87826.49344.86978 26 35.43182.90196.44750.89423,46304.8634.47844.87812.49369.86964 25 36.43209.90183.44776.89415.46330.88620.47869.87798.49394.86949 24 37.43235.90171.44802.89402.46355.83607.47395.87784.49419.86935 23 33.43261.90158.443S8.89339.46331.88593.47920.87770.49445.86921 22 39.43271.90146.44854.89376.46407.88580.47946.87756.49470.86906 21 40 1.43313.90133.44880.89363.46433.88566.479711.87743.49495.86892 20 4 1.43340.90120.44906.89350.46458.88553.47997.87729.49521.86878 /19 42.43366.90108.44932.89337.46484.88539.48022.87715.49546.86363 18 43 1.433921.90095.44958.89324.46510.88526.48048.87701.49571.86849 1 7 44.43418.90082.44984.89311.46536.88512.48073.87687.49596.86834 16 45.43445.90070.45010.89298.46061.88499.48099.87673.49622.86820 15 46 1.43471.90057.45036.89285.46587.88485.48124.87659.49647.86S05 14 471.434971.90045.45062.89272.46613.88472.48150.87645.49672.86791 13 483.43523.90032.45038.89259.46639.88458.48175.87631.49697.86777 12 49 1.43549.90019.45114.89245 1.46664.88445.48201.87617.49723.86762 11 50.43575.90007.45140.89232.46690.88431 48226.87603.49748.86748 10 51 1.43602 1.89994.45166.89219.46716.88417.48252.87589.49773.86733 9 52.43628 1.89931.451922.206.46742 1.88404.4277.8757.5.49798.86719 8 53.43654.89963.45218.89193.46767.88390.48303.87561.49824.86704 7 54.43680.89956.45243.89180.46793.88377.48328.87546.49849.86690 6 55.43706.899143.45269.89167.46819 1.8363.483354.87532.49874.86675 5 56 1.43733 89930.45295.89153.46844.88349.48379.87518.49899.86661 4 57.43759.89918.45321.89140.46870.88336.48405.87504.49924.86646 3 531.43785.89905.45347.89127.46896.88322.484301.87490.49950.86632 2 59.43811 1.8992.45373.89114.46921.88303.484561.87476.499751.86617 1 60.43337.89879.45399.89101.46947.88295.48481 1.87462.50000.86603 0 1. Cdsin. Sine. Cosin. Sine. Cosin. Sine.C Cosin. Sine. I. 1 f- ~ 630 620 1 Dj _ T1 iAB]LE XiV. NA.ATURAL: SINES AND COSIN'7E3. 30> 312o 33> 31:5I Sine.'oem. Si'l Cos 1. Sine csin. SCox ine. Cosin. sie. oin.. 0.500031.86603 WM504.8-5717.52992.84505 1181 ~r;-)4464;S7 r.5 5n9319,29)1 60 I. 5 35.86528 51529.85702.53017.84789.54411 >8: 3>1 >.5-943.>2>87 59 2.50,50.86573.515541.85687.53041.84774 4.:i51: 88335.55968.82871 58 3.50376.86559.51579.85' 72 1.53066.4/459.54537..35.55992.82855 57 50 01;.86544 1,51604.866571.53091.>174:3 6.546 >t.S3>04.45016.82>39 56.50126.86530.51628 1.*>56i42.53115.1472s.54 56/.83788.56010.828222 55 6.50151.865151.51653.85627.55140.>4712 ].54610.83772.560641,2>06 54 7.50176.86501.51678.85612.53154.81697.546:3)5.83756.56030.82790 53 l.50201t.86486.51703.85597.53189.84681.54659.83740.56112,>2773 52 9.502 27.86471 1,517281.5582.532141.4666.546>3 /.83724.561536,82757 I51 101.50-52.86457.51753/.$2567.53238.84650 i.54708>1I.37 0'.56160,82741 50( 1] 502777.86442].51778.85551.53263.84635.54732.83692.56184.62724 49 12.50302.86127.51803.85536.53288.84619.54756.83676 56208.82708 4> 1 8.50-327.85113.51828.85521.53312 81604.54781.8366(1.6238.82692 47 14.50352,86398.518521.85506.53337 8451881.5405.83615,56256.82675 46 15.50377.86384 1.51877.85131 J.533611,84573.54829.853629.556280.82659 45 16.50403.86369.51902.8;1) 4 76.53386.84557.54854.83615.563)5.8 2 42645_ 44 17 1.5042>[.863541.51927.85461.53411.84542.54578.83597.56321.82626 43 1I8.50453.86340.519532.85446.53435 1,84 26.54902.351.56353.82610 42 19.50478.86325 51977 8 1341 1.53460.84511.54927.83363.5.56377 S >2393 41 20.510503.86310,52002 1,85:1 6 ).534>4.84495.54951 3.835:-19.56401.82577 40 21.50528.86325.520216.854101.53309.84410.54975.835335.56425.82.561 39 22.50553,86281.52051.85335; 5:5314;84 t16.54999.83517.56141,. 82544 3> 23.50578.86266.52076 85370 53558. >41118.535024 /.83501.5647:.8252 37 2-1.50603.86251.52101.85355.53583.>4313.55048.634.5..56497.82511 56 25.50628.862>7.:521 8.8534,.5I07.84417.55072 316. 46).56521.>2495 3 5 26.50654,86252.52151.85325.53632.>-111)2.550937.8345:3.56535.8247> 31I 27.506791;862071.52175;85310..:3 j56 0 34:'I G 55121.>31-5 7.56569.82462 33 2 /.50704.86192.5220.8552941.536>1.84370.5514 -:83121.56593.824146 32 29 1.50729.86178.522251.85279.531'05 84345.551695. >3405 1.56617.82-129 31 30 1.50754.86163.52250.85264.53730.8-1339.55194.83:39.56641/ 82413 30 311.50779.86148.522751.85249.53754.84324.5521 1. >3373.566613:823996 29 32.50804.86133.52299.85234.53779.8431>.552-12.83356 1.56689.823>0 2>/{ 33.50829.861i9.52324.85218,53)41, 84292..55266, 83310.56713,3 363 27 31 l.50854.86101.52349.85203.53>2>t.84277.55291.833241 567367,82347 26 35.50879.860>9.52374.85 s18 >.53.133.8426,1.55315.8330>.56760.82330 25 36. 50904.86074.52399/.851~73. 535377]. 84 24 5.55339.83292.56784.82314 24 37.50929.86059.52423,85157 C.53902. 8-20 ~.55363.83276.56808 >.022971 23 38/.50954.86045.5244S.835142.539261.61214. 5533~.832601.56>32.8228 1 22 39 150979 1.86030.52473.85127.53951.8 119>.5541t2.d32-14 13.656 86>1.82264 2l 40 1.51004.66015.5249> 1.85112.53975 >9,$182.554360.832281.56801.82248> 20 4 t.51029 1860001.52522.85096 1i 54000.8 l(67 C 455460. 3 2 59.ri604.8223 1 19 42.51054.85985.52547.85081.54024.81151 t.5354>4.83195). 5692>.82214 18 143 151079.85970.52572.85066.54049.84135.55509.83179.56952.8219> 17 44.51104.895956. 5297 1.85051.54073.84120 ~.5533.831634.6)1976. 2181 16 45.51129.55941.52621.85035.54097.841041.55557 83147.57000.821l65 15 46.51152115926.52646.50201.54122/.840ss iS'>1' 83131.57024 s.8214>s I4 47.51179.8591.52671.8500.5 54146.84072.55605.83115.570471.82132 13 431.51204.85896.52696.84989.51171.84057.5563514 >30)>8.57071.82115 12 49!.51229.85881.527201.84974.54195)3.84041.55654.830828.57095.820981 1l 50 1.512.531.3s66/.527s45;.8 495;9.54220.84025.5567S 1.>306s.>57119.82082 1]0 51 1..5127919. 35 5 1.52770.84943. 854244.839009.55702 1,83050 3.571431.82055 9 5821.51:31I.s5S36.527941.8192>.54269.83994.537261.5 3034.57167.8204I 8 8431.5 1 3;z9.528:19 1.8:19)3.51293.83978.,55750.83017.57i191 /.20332 7 54>.51315.1.1>5 1) 6.52>4-1.84897.543171.839621;55775 >).83001.57215.220171 s 6 /5 3.5137!)1.85792.52>69..8482.74342.83946 I.55799.8295.57235>.81909 5 56 515)4)r >.35777.52>93.>4>66.54366 1.83930.558231.8269.57262.8619>2 4 57 1.51429 1.85762.52918.84S51.54391.83915.55847 1.82953.57286.81965: 3 / 5.51 /.85747 j.52943J:/.4836.54415.83899.55871/.829361.57810[.81949/ 2 5 1.51479.85732. 52967.84820.54440.53883.55895.82920.'57334.81932 1 60.51,5054.85717.52992.84805 45461.83>67.55i919.82901.57358.81915 0 cii. co~ll. Sine. Cosic. Sine. Cosin. Sine. co011. 8ine. C osin. Sine. I 1. 593 39> 5 50> 83 T'AY, LE XIV. NATURATL S:INES AND COSI-NES. 622 350 3oI 33s o. Sine. Cosin Sine. Cosin Sine. Cosin. S osin. Sine. Co0.siln. 0 5733,58.1913.55779.S0902.f01S2| 1 oI 566 /0 2 2.75 I 573Sl.81573 899.5S().80. 05.6020..796846 61559?. 6 5 7,.13 796 59 2.574 0518S2.182 6.588i.0.i67.60228.79829.61612.7876.,;2 77. 7 678 58 3.5742-.1865.S8I49.8 085( -.60257.79811 1.6 l635.78747 t:30(i(.7 7i6fi() 57 4/ 4 745 8l,8.58873.80833 6024.79793 5 189 J.;J. i6, 4 5.57477.81S32.5S8961.80816.S0298.79776.. 6 167S71,7871 G, 4j-5 1.77f623 5I 6.579-1.181(15.58920().80799.603321.79758.617(4.78694 6!ti.; 1.7 605 56 1 i 7 8,57 i. s1798.589453 807S2.60344 1.79741,61726.7>676.300)90 1.,77586,3 8.57a [ >. S1l7S2.58967,80765.60367.7972.61749.78658[.63i 1;3 8 7 7568 59,, 5.15 i':;'1 765 1.58993,80748.60390.797(6.61772.78640.63135,.77 550 51 l1 0).55).81748.591)14.80730. 60414.796o5.617'95.78622.631 58i.77531 5,0 11.57619.81731.59037.80713.60437.79671.61818. 7S6(4 I63 180).77513l 49 12 9.57613.S1714.59061.80696.60460 /79653.61841.7S586.63203.77494 4S 13 13.57667 8.1693.59084.80679.60483.79635.61864.78568.63225.77476 47 14,57691.S816s1,.59108.80662.60506.79618.61887 178550.63248.771458 46 15.57715.81664.59131,80644 60529.79600 | 61909.78532.63271 1.77439 45j 16.57733.81647.59154.S0627.60553,.79583.61932. 785614,6:32 93.7-421 442 17.57762.81631.59178.8061(.60576.79565.619-55.78496.633 I7G,402 43 1S.57786].81614..59201.80593.60599.79547.61978.78478.6'33381. 77381 42 19.57810.81.597.59225/.80576.60622.7953o0 g.62001.789460.633o61.77366 41 20.5783:31.81 50 59248.80558.60615.79512.62024 1.78442.6(3383 1.773471 40 2l 1.57857.81563.59272.805 1.6066 1.7949-4.62046 |.78424 ].673406.773291 39 22 1.57881 81546.59295 6.0524.60691.79-177.62069.78405.63-128.7 31I.380 9 23.579()1..81530.59318. 8(,507.60714.79459 ].62092.78357.63l4 51.77292 37 }21.0,57928.81513.59342.80489.60738.79441 |. 62115.78369.63473' 177273 36 0| 25.57952 |,81496 0,59365 (80(472.60761.79424 6, 621 38 78 351 ].634296/.772551 3-5 26.57976.81479.59359.80455.60784.79406.62160.78333.635118.77236| 34 27.57999.81462 59'12.80433.60S07.793883.62183.78315.16:03540.77218 331t 28f 58023.381445.59436/.8010 1.60S30.79371.62206.78297.63563.77199 32 2 29.589471.81:428.5:,L9459.801403 7.6(8S53).79353 62229.78279.635825.77181 31 1 30.58070.81412.59482. 80386'6.60876. 79335 62251i.78261.63608. 77162 30 31.58094.81395.59506.803G68.60899 1.7931.62274.78243.63630.77144 29 32.58118.81378.59529.80351 i.60922.79300).62297.7S225.636 5a3.77125 28 33.58141 t.81361.59552 1.S03354.609745.79292 1.62320.78206.63675.77107 27 34 l.9165 /. 34 4.557s6.03l 6.60968.79264.62342/.7818.6363S/.770SS| 26 35..58189.81327.59599.80299.60991.79247 L.62365,78170.63720.77070( 25 36.5212 l.81310.59622.S0282.61015.79229.623388.78152.63742.77051 24 37.58236.81293.59646.80264.61038.79211.62411.78134.63765.770331 23!3S.58260 1.81276.59669.S0247.61061.7919.3.62433 1.78116.63787.77014 22:39.58283 1.81.259.59693.80230.61084.79176 ~.624561.78098.63101.769961 21 40.53071.81242.59716.80212.61107.79158.62179.78079.63332.76977 20 41.53301.81225.59739.80195.61130 1.79140.62502 1.7S061.63854.76959 19 1 42.58354. 81208.59763.80178.61153.79122 t.62524 L780413.63877.76940 18 4 143.583378.81191.59786.80160.61176.79105.62547 78025.63899.76921 17 44.5S401.81 174 ].59809.80143.61199.79087.625-70.78007.63922.769039 16 45.58425 *.81157. 59832.80125.61222.79069.62 92].77988.639441.76884 15 46.58449.811(40.59856.80s108.61245.79051.62615.77970.63966.76866 14 47.58472.81123.59879.80091 1.61268.79033.6263 1 77952,.63989.76847/ 13 483.59496. 81106.59902.80073.61291.79016.62660.77934.64011.76828 12 49.585191.81089.59926. 80056 [.61314.7899S 62633 77916.64033.76810 11 0 435843.81072.59949.80038 61337.7980.62706.77897.64056.76791 10;51 58567.81055. 59972.80021.61360.78962.62728.77s79.64078.76772 9 591.585901.81038.59995.80003.61331.789- 4.62751.77861.64100.76754 8 I53.58614.81021.60919.79986.614(6 1.789261.62774.77843.64123.76735 7 5l.58637.81004.61042.79968.61479968 29.790 62796.77824.6445.76717 6 55 1.5S661 l.0387 160065.79951.G1451.78891.62819.77806.64167.76698} 5 56.5686 4 4.80970.60089.79934 [.61474.78873.62842 1.77788.64190.76679 4 57.58708(.80953.60112.79916.61497.78855.62864 77769.64212.76661 3 58 /.58731.80936, 60135 79,99.'61.20.78837.62887.77751.64234.76642 2 59.58755 1.80919.60158.79831.61543.78819.62909.77733.64256.76623 1 60.58779.80902 M 60182.79864 L61566.78801.629321.77715.64279.76604 0 /!,. lCosin. e. cosn.' Sine. Cososii. Sine. eosil I sine. Cosin. Sine. i.U ( 510 530 5Q0 9 Ji 500 228 TABLE XIV. NATURIAL SINES AND COSiINES. 03 O 410 42 4330 o M Sine Cosin Sine. fosin. Sine.'Cosin. Sine. i sin. Sine,1 I Cosn. MI 0.64273.76604 6;6)606.75471.66913.74314.6>200 31 13691 16;71:1 a3 l.64301.765S6.65628. 75-152.66935.74295.62211 73116 69417.71914 59 2.61323.76567.65650.75433.669356.74276.6X242.73096.69508.71894 58 3.64316.76548.65672.75-114.66978.74256.63264.73076.69529.71873 57 4 61363.76530.65694.75395.66999.74237.28.6251.73056.69549.718353 56 5.61390.76511 1.65716.75375.67021.74217 1.68306/.73036.69570.71833 55 6.644121.76492 2 65733 1.75356.67043.74198 g.68327.73016.69591.713813 51 7.6 44.35.76473.65759.75337.67064.74178.63349.72996.69612.71792 53 8.64457.76455.65781.753 18.670S6.74159,6370.72976.69633.71772 52 9.64479.76436.65303.75299.67107.74139.63391.72957.696.)4.71752 51 10.64501.76417.65825.75280 D.67129.74120.63412.72937 69675.71732 50) 11 }.64524.76398.65847.75261 167151.74100 X.68434.72917.69696.71711 49 12.64546.76330.65369.75241.67172.74080.68455.728971.69717.71691 48 13.64563.76361.65891.75222.67194.74061.63476.72877 69737.71671 47 14 1.64590).76342.63913.75203.672165.74041.638497.72857.69753.71650 46 15.64612.76323.65935.75184.67237.74022 ].68518.72837 1.69779.71630] 45 16.64635.76304.65956.75165!.67258.74002.68539.72817.69800.71610 44 171.64657.76236.65978.75146.672S0.73983 ].68561.72797 1.69S21.71590 43 1S 1.64679.76267.66(000.75126.6730(1.73963 1.68532.72777.69842.71569 42 19.64701.762408.66022.75107.67323.73944 ].63603.72757 69862.71549 41 20.64723.76229 1.66044.75038.67344.73924 ].63624.72737 1.69883.71529 4 0 2 1.64746.76210.66066.75069.67366.73904.68645.72717 1.699041.71508 39 22.647683.76192.66083.75050.67337,73835 g.68666.72697 1.69925 ].714883 38 23.64790.76173.66109.75030.67409.73365.6368t.72677 [.69946 1.71468 37 24.64812.76154.66131.75011.67430.73346.68709.72657.69966.71447] 36 25.64934.7613.5.66153.74992.67452.73826.68730.72637 | 69987.71427 35 26.64856.76116.66175.74973.67473.73606.63751.72617.70008.71471407 34 27.64878.76097.66197.74953.67495.73787.63772.72597!.70029.713861 33 28 1.64901.76078.66218.74934.67516'73767.637931.72577.70049.71366 32 29.61923.76059.66240.74915.67538.73747 ].68814.72557 ].70070.713451 31 30 1.64194.5.76041.66262.74396.67559.73728.63335.72537 1.70091.713251 30 31.64967.76022.66284.74876.67580.73703.68857.72517 1.70112.713051 29 32.64989.76003.66306.74857.67602.73688 A.68837.72497.70132.712841 28 33.65011.75984.66327.74833.67623.73669.63899.72477.70153 1.712641 27 34.65033.75965.66349.74810.67645.73649.689201.72457.70174.712431 26 /35.65055.75946.66371.74799.67666.73629.63941 172437 1.70195.712231 25 36.65077.75927.66393.74780.67683.73610.63962.72417 1.70215.71203 24 37.65101).75908.66414.74760.67709.73590.68983.72397.70236.71182 23 381.65122.75889.66436.74741.67730.73570.69004.72377.70257.71162 22 39.65144.75870.66453.74722;67752. 73551 e.69025.72357.70277 |.71141 21 t 40.65166.75851.66480.74703.67773.73531.69046.72337.70298.71121 20 411.651831.75832.66501.74683.67795.73511.69067.723171.70319.71100 19 42.65210.75813.66523.74664.67816.73491.69088;72297.70339.71080 18 43.65232 1.75794.66545.74644.67837.73472.69109.72277.70360 1.710591 17 44 1.65254.75775.66566.74625.67859.73452.69130.72257.70381.71039 16 45.65276.75756.66588.74606.67880.73432.69151.72236.70401.71019 15 46.6529 1.757308.66610.74586.67901.73-113..691721.72216.70422.70998 14 47.65320.75719.66632.74.567.67923.73393.69193.72196 1.70443.70978 13 13 1.65342.75700.66653.74'543.679-14.73373.69214.72176.704163.70957 12 49/ 65364.75630.66675.74523.67965.73353.69235.72156.70484.70937 11 50 1.65386.7566 1.66697.7450)9.67937 73333 1.6256 1.72136.70505 1.709161 10 51 [.654003.756-12.66718.74-1439 (.6;008/.733134.69277.72116.70525.70096 9 52[.43()0.7562_3.66740.7-1470.,69029.73294 69298.72095.70546.708751 8 5 1.651iT52.75604.66762.741-151 G.6051 1.73274.69311. 72,'75.70567.7S55 5 7 54.G6f74[.75585.66783.74431.68()72/.73254 1.693340.,72055.70587.70S334l 6 55.654-196.75566.66305.744112 s09'3.3234.69361.72035.70608 170813 5 56.635518.75547.66327.743992.6 15.72'3215.693032 172015.706283. 70793 4 571.65540.755205.6648.74373.63136.731935.694031.71995.70649.70772 3 53 [.6 6|2.75509.66370.74:3;3.63157.73175 4.6942-1471974.70670.70752 2 59.65534.75490.66091.74334.681791.73155.69445.71954.70690.70731 1 60.65606.75171.66913.74314.6200.73135.69166| 71934 70711.70711 0 M/ ICosin. Sine. Cosinj. Sine. 1Cosin.l Sine. Cosin Sine. Cosinl Sine. M. t 191 480 83 Z I 80 TABLE XV, NATURAL rANGENTS AND COTANGENTS, 20 280 TABLE XV,, NATUR BAL'TANGENT IS AND CO'i' GEN'iT 1 _03 10 30. BI., Ttang. Coattng. Tang. Cotang, Tang. Cotang. Tang. Cotarg.,I. 0.00000 Infinite..0174f 57.2'00).0 42 2.j:636.0.21 19.08(S11 0t 1.00029 3437.75.01775 56,330(J6.(03521 28.379'J4.03270 18.-)755 59 i 2.00058 1718.87.01804 55.4415.03550 2S.1664 (.052 9 18.8711 58 3.00087 1145.92.01833 54.5613,03579 27.9372.o05328 18. 7 678 571 4.00116 859.436.01862] 53,7086,03609 27.7117.03357 180.,G6 6 5.00145 687.549.01891 52,8821,03638 27,4899.05357 18.5645 55 6.00175 572,957.01920 52.0807.03667 27.271.5. (5416 18.4645 54 7.00204 491,106.019491 51,3032,03696 27.0566.05445 1,3655 53 8.00233 429.718,01978 50.5480,03725 26.8450,05474 18.,677 52 9.00262 381,971,02007 490,8157.03754 26X6367,05503 18.1708 51 10.00291 343.774,02036 4901039,037083 26.4316.05533 18.0750 50 11.00320 312,521.02066 48.4121 03312 26.2296.05562 17.9802 49 12.00349 286,478.02095 47.7395,03S42 26.0307.05591 17,863 48 13.0037 S 264.441.02124 47,0853,03871 25.8348.05620 17.7934 47 14.00-107 245.552,02153 46.4489,03900 25,6418.05649 17.7015 46 15.300136 229.182.02182 45.s294.03929 25.4537.05678 17.6106 45 1 6.0046 5 214, 8538.02211 45.2261.03958 25.2644.05708 17.5205 44 17.0(0'495 202.219.02240 44.63S6.03987 25,0798,05737 17.4314 43 1S.003521 190.984.02269 44.0661.04016 24.8978,05766 17.3432 42 19.0()0553 180.932.02298 43.5081.04046 24.7185,05795 17,2558 41 2(0.00532 171.885.02328 42.9641.04075 24.5418.05824 17.1693 40 21.00611 163.700.02357 42.4335.04104 24.3675.05854 17,0O337 39 22.0064OG0L 156.259.023S6 41.9153.04133 24.1957.05S83 16.9990 3[ 23.00669 149.465.02415 41.4106.04162 24.0263.05912 16.9150 37 24.00698 143.237.012414 40.9174.04191 23. S593.05941 16,8319) 36 25.00727 137.507.02473 40.4358.04220 23.6945. 0970 16.7496 35 26.00756 132.219.02502 39.9655.04250 23.5321.05999 16.6681 34 27.007s85!27.321.02531 39.5059.04279 23.3718.06029 16.5874 33 28.00315 122.774.02560 39.0568.04308 23.2137.06053 16.5075 32 29.00844 118.510.02!:.89 33.61 77.04337 23.0577.060O7 16.4283 31 30.0073 114.589.0261 9 38.1885.04366 22.9038.06116 16.3199 30 31.009502 110,092,026418 37,76S6.04395 22.7519.06145 16.2722 29 32.00931 107.426.02677 37.3379.04424 22.6020.06175 16.1952 28 33.00960 104.171.02706 36.9560.04454 22.4541.06204 16.1190 27 34.00989 101,107. 02735 36,5627,01483 22,3081,06C233 16.0435 26 35.01018 93.2179.02764 36.1776.04512 22.1640.06262 15.9687 25 36.01047 95.4895.02793 35.8006.0454 1 22.0217.06291 15.8945 24 37.01076 92.9085.02822 35.4313.04570 21.8813.06321 15.S211 23 38.01105 90.4633.02851 35.0695.0-599 21,7426.(16350 15.7483 22 39.01135 85.1436.02381 34,7151.04628 21.6()56.06379 15.6762 21 40.011641 85.9398.02910 34.3678.0465,8 21.4704.06408 15.6048 20 41.01193 83.8435.02939 34.0273.0-1637 21.3369.06437 15.5340 19 42.01222 81.8470.02968 33.6935.04716 21.2049.06467 15.4638 18 43.01251 79.9434.02997 33.3662.01745 21.0747.06496 15.3943 17 44.01280 78.1263.03(126 33.03i2.04774 20.9460.06525 15.3254 16 45.01309 76.3900.03 055 32.7303.04103.- 188.06554 15.2571 15 461.01338 74.7292.03034 32.4213.04833 2l0.6932.06584 151tS93 14 47.01367 73,1390).03114 32.111.04862 20.,691.06613 15.1222 13 48.01396 71.6151.03143 31.8205.04S91 20.4465.06642 15,0557 12 49.01425 70.1533.03172 31.5284.04920 20.3253.06671 14.998 1 1 50.01455 68.7501.03201 31.2416.01949 20.2056.06700 14.9244 10 51.01484 67.4019.032301 30.9599.04978 20.0872.06730 14.8596 9 52.01513 66.1055.0:3259 30.6833.05007 19.9702.06759 14.7954 8 53.015,12 |64.85O0.03289 30.4116.05037 19.8546.06788 14.7317 7 54.01571 63. 6567.003317 30.1446.05066 19.740.3.06517 14.66c5 6 55.01600 62.4992.03346 29.8823.05095 19.6273.06847. 141.6059 5 I56.01629 61.3S29.03376 29.6245.05124 19.5156.06876 14.54383 4 57.0165I8 60.3058.0C3415 29.3711.05153 19.4051.06905 14.4823 3 58.01GS7 59.2659.03431 29.1220.05 82 19.2959.06934 14.4212 2 359.01716 r58.2612.(13463 28.8771.05212 19.1279.06963 14.3607 1 601.01746 57.29001.03192 2'.6363.0-3211 19.011.06903 14,.3007 0 M.' Cotang. Tang. Cotrg.'lang. Cotag. Tang.,Cotang. Tang. M5. TABLE XVai NATUIl'AL TANIGE1NTS` AND COTANGlENTG. 231T oI 4o t I 60 o7 1 0. Tas.. T. Cotang. Tan Co. tan. Tang. Cot Tag. Cotang. M. 0.06993 14.3 07.0874,9 11.4.301 t.1 1.5. 14.1Z>5,28.1441:25 6C 1.070(22 14.2-111,0778 11. 319.10;40 9.4-711.1 230S 8.12481 59 2.07051 14.1821.0,o07 11.3;,40o.10,6363 9.46141.12338 S. 10;-:36 58 3.07080 14,1233. ().037 11.:3163.10599, 9,4:3315.12367 S,o8>oo 57 4.07110 14,0655,.0 8;66 11.2789.1062s 9,409041.123977 8.06674 56 5.07139 14.0079,08895 11,2417.10637] 9,3>307.12426 8,04756 53 6.0716S 13.9507.0892.5 11.22048.10687 9,357244.124156 8.024 547.07197 13,8940.09-54 11.1681.10716 - 9.33155.12485 8.00948 53 8.07227 13,8373,o09S3 11.1316.10746 9,303599.12515 7.99058 52 9.07256 13,7821.09013 11,0954.10775 9,28058.12544 7.97176 51 10.07285 13,7267.09042 11.0594.10 SC5 9.25530.12574 7.95302 50 11.07314 13.6719.09071 11.0237.10 S34 9.23016.12603 7.9343;q 49 12.07344 13,6174.09101 10, 9822.10 63 9,.20516.12633 7.91592 48 13.07373 13,5634.09130 10.9529.10>93 9, 102[.12662 7.89734 47 14.07402 13,5093.09159 10.9178.10922 9,15554.12692' 7.7S95 46 15.07431 13,4566.09189 10.8829,10952 9,113093.12722 7.86064 45 16.07461 13,4039.09218 10.8483.10981 9,10646.12751 7.84242 44 17.07490 13,3515.09247 10.8139.11011 9.08211.12781 7.82428 43 18.07519 13,2996.09277 10.7797.11040 9,05789. 12810 7.80622 42 19.07541 13,2480.09306 10,7457.11070 9.03379.12840 7.78825 41 20.07578 13.1969.09335 10.7119.11099 9,00983.12269 7.77035 40 21.07607 13.1461.09365 10.6783.11128 8,98598.12899 7.75254 39 22.07636 13.0958.09394 10,6450.11158 8,96227.12929 7.73480 3S 23.07665 13,0458.09423 10.6118,11187 8,93~67.12958 7.71715 37 24.07695 12.9962.09453 10.5789.11217 8,91520.129SS 7.69957 36 25.07724 12,9469.09482 10.5462.11246 8.89185.13017 7.68208 35 26.07753 12.8981.09511 10.5136.11276 8.86862.13047 7.66466 34 27.07782 12.8496.09541 10.4813,11305 8.84551.13076 7.64732 33 28,07812 12.8014'09570 10.4491.11335 8.82252.13106 7.63005 32 29,07841 12.7536.09600 10.4172.11364 8.79964.13136 7.61287 31 o0 07870 12.7062.09629 10.3354.11394 8.77689.13165 7.59575 30 31.07899 12,6591.09658 10.3538.11423 8.75425.13195 7.57872 29 32.07929 12;6124,09688 10,3224.11452 8.73172.13224 7.56176 28 33.07958 12.5660,09717 10.2913.11482 8.70931.13254 7.54487 27 34.07987 12,5199.09746 10.2602.11511 8.68701.13284 7.52806 26 35.08017 12,4742.09776 10.2294.11541 8.66482.13313 7.51132 25 36.08046 12.4288.09305 10.1988.11570 8.64275.13343 7.49465 24 37.08075 12.3838.09834 10.1683.11600 8.62078.13372 7.47806 23 38.03104 12.3390.09864 10.1381.11629 8.59893.13402 7.46154 22 39.08134 12.2946.09893 10.1080.11659 8.57718.13432 7.44509 21 40 -.03163 12.2505.09923 10.0780.1163s 8.55555.13461 7.42871 20 41.08192 12.2067.09952 10.0483.11718 8.53402.13491 7.41240 19 42.08221 12.1632.09981 10.0187.11747 8.51259.13521 7.39616 18 43.08251 12.1201.10011 9.98931.11777 8.49128.13550 7.37999 17 44.08280 12.0772.10040 9.96007.11806 8.47007.135SO 7.36389 16 45.08309 12.0346.10069 9.93101.11836 8.44896.13609 7.34786 15 46.08339 11.9923.10099 9.90211.11865 8.42795.13639 7.33190 14 47.08368 11.9504.10128 9,87338.11895 8.40705.13669 7.31600 13 48.08397 11,9087.10158 9;84482.11924 8.38625.13698 7.30018 12 49.08427 11.8673.101S7 9.81641.11954 8.36555.13728 7.28442 11 50.08456 11.8262.10216 9.78817.11983 Si34496.13758 7.26873 10 51.0S485 11.7853.10246 9.76009.12013 8.32416.13787 7.25310 9 52.03514 11.7448.10275 9.73217.12042 8.30406.13817 7,23754 8 53.083514 11.7045.10305 9.704,11.12072 S8.29376.138416 7.22204 7 54.08573 11.6645.10:331 9.67620.12101 8.26355.13876 7.20661 6 55.08602 11.6248.10363 9.64935.12131 8.24345.139()6 7.19125 5 56.08632 11.5853.10393 9.62'205.12160 8.22344.13935] 7,17594 4 57.08661 11.5461.10422 9.59190.12190 8,20352.13965 7.16()71 58.OG69o0 11.5072.10452 9.56791.12219 8.18370.13995 7.14553 2 59).03720 11.4(635.10481O 9.54106.12249 S. 16398S.14024 7.13042 1 60l.02719t 11 4301.10511) 9.i14136.12278 8.1443.5.1l1054 7.11537 O).MI. sotarg.i Tang. fCottr. Talng. Cotang.g TTang. iCotang. Tang. Ml. 835 8 83 820 f..,._ = 2 TABLE XV, ]NATtRAlL TANGESrNT'S AIND COT.ANGENTS, | 0 9D Oo 100 1 10o 1. Tang. Cotan. Tang. Cotang, aTa ng Cotang. Tang. Cotang. M. 0.14054 7.11537.15838 6.31375.17 633 5.67 128.19:43S 5. 1443; 60 1.140834 7, 10038 15868 6,30139,17663 5.661 1 19168 5.13658 59 2.14113 7,08546.15898 G,29007 17G693 5,65205,19493 5.128262 158l 3.14143 7,07059.15928 6.278291.17723 5, 64248,J9529 5.1206'3 57 4.14173 7,05579,15953 6.26655;17753 5,63952 1 95559 5.11279 5 0 5.14202 7.04105.15938 6,254866 11773 5.6234-1 19589 5.1049)0 5 6.14232 7.02637,16017 6,24321 i 17813 5, 61397 0'G 1) 5. 0904 54 7.14262 6,91174.16047 6.23160,1784-13 5,60452.1096-19 5)08921 53 8.14291 6.99718.16077 6.22003,17873 5,59511 1, 6o0 5,08139 52 9.14321 6,93826.16107 6,20851,17903 5,5o573,19710 5.07360 51 10.14351 6,96823.16137 6.19703,179):)3 5,57 38 2 19740 5,.065S4 50 11.14381 6.95385.16167 6, 18559.17i'63 5,56706, 19770 5,05809 49 12.14410 6.93952.16196 6,17419 I,17993 5,55777.1 9801 5,035037 48 13.14440 6; 925235,16226 6,10283,1]023 5 54851.19831 5,04267 47 14.14170 6,91104 16256 6,15151,18 )53 5. 53927 19861 5,03499 46 15.14499 6.S9685.16236 6.14023,13083 5,53007.19891 5,02734 45 16.14529 6.88278.16316 6.12S99.18113, 52090.19921.I 01971 44 17.14559 6.86374.16346 6.11779.18143 5;51176.19952 5,01210 43 18.14588 6.85475.16376 6,10664.18173 5,50264.19982 5,00451 42 19..14618 6.84032,.16-305 6, 09552,18203 5,49.3f6,.20012 4,99695 41 20,14648[ 6.82694.16435 6.0(444 18233 5843451.20042 4.98940 40 21.14678 6.81312.16465 6.017340.182603 5.47543.20)73 4,98188 39 22.14707 6.79936.16495 6.06-240,1829:3 5S46418.2010:1 4,97438 38 23.14737 6.78564.16525 6.05143,18323 5,45751.20133 4,96690 37 24.14767 6.77199.16555 6.01i)51,18353 5.44057,20164 4,95945 36 25.14796 6,75338,165S5 6,02962,18384 5,43266.20194 4.95201 35 26.14826 6.74483,16615 6.01878.18414 5.43077,20224 4.94460 34 27.14856 6.73133.16645 6.007977.18444[ 5,42192,20254 4,93721 33 28; 14886 6.71789.16674 5i,99720.18474 5.41309,202S5 4,92984 32 29.14915 6.70450.16704 5.98646L,18504 5,40429,20315 4,92249 31 30.14945 6.69116.16734 5,97576.18534 5,39552,20345 4.91516 30 31.14975 6.67787.16764 5.96510,18564 5,33677.20376 4,90785 29 32.15005 6.06463,16794 5.95448.183594 5.37805.20406 4.90056 28 33.15034 6.65144.16824 5.94390.18624 5i36936,20436 4.89330 27 34.15061 6.63831.16834 5.93335.18654 5,36070.20466 4.SS605 26 35.15094 6.62523.16884 5.92283.1 8684 5.35206.20497 4,87882 25 36.15124 6.61219 16914 5,91236.18714 5.34345.20527 4.87162 24 37.15153 6.59921.16944 5,90191.18745 5,33487,20557. 4,86444 23 38.15183 6.58627.16974 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 5,87080.18835 5.30928.20648 4,84300 20 41.15272 6,54777.17063 5,86051.18805 5.30080.20679 4,83590 19 42.15302 6,53503.17093 5.85024.18895 5.29235.20709 4.82882 18 43).15332 6.52234.17123 5,84001 18925 5,28393.20739 4.82175 17 4-1.1.15362 650970.17153 5,82982.18955 5.27553.20770 4,81471 16 45.15391 | 6,49710.17183 5.81966.18986 5.26715.0.o300 4.80769 15 /146.15421 648456|.17213 5.80953.19016 5.258S0.20830 4.80068 14 47.15451 6,47206.17243 5.79944.190-46 5.25048.20861 4.79370 113 1 48.15481 6.45961.17273 5.783938.19076 5.24218,20891 4.78673 12 49.15511 6.447201.17303 5.77936.19106 5.23391.20921 4.77978 11 01.15540 6.43484.17333 5.76937.19136 5.22566,20952 4.77286 10 51.15570 6.422531.17363 5.75941.19160 5,21744,20982 4.76595 9 521.15600 6.41020.17393 5.749-19.19197 5.20925.21013 4.75906 8 5:3.15630 6.39804.17423 5.73960(1.19227 5.20107,21043 4.75219 7,541.15I60 | 6.33587.174530 5.72974.19257 5.19293.210)73 4.74534 6.55 1569 6.37374.174833 5,71992.19287 5.18480.211041 4.73851 5 561.15719 6.36165.17513 5.71013.19317 5.17671.21134 4.73170 4 57.15749 6.34961.17543 5.70037.19347 5.16863.21164 4.72490 3 58.15779 6.33761.17573 5.69064.19378 5.16058,21195 4.71813 2 59.15809 6.32566.17603 5.68094.19403 5.15256.21225 4.71137 1 60.15838 6.31375.17633 5.67123.19433 5.14455.21256 4.70463 0 39.Cotaulg. Tang. Cotang. Tang. Cotanl.g. ng. Cotang. Tang. M. 810 I soD 90' 0 TABLE XV. N6ATURAL TANGENTS AND COTANGENTS, 233 120 130 1 015 IM. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotaug. I. 0-.21256 4.70463.2:30s7 4.3314.249333 4.01078.2t795 3.73205 60 1.21286 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 4.31430.25026 3.99592.26888 3.71907 57 4.21377 4.67786.23209 4.30360.25(.56 3.99099.26920 3.71476 56 5.21408 4.67121.23240 4.30291.2.5037 3.9S607.26951 3.71046 5I 6.21438 4.66458.23271 4.29724.25118 3.98117.26982 3.70616 54 7.21469 4.65797.23301 4.29159.25149 3.97627.27013 3.70188 53 8.21499 4.65 138.23332 4.28595.25180 3.97139.27044 3.69761 52 9.21529 4.64430.23363 4.28032.25211 3.96651.27076 3.69335 5i 10.21560 4.63S25.23393 4.27471.25242 3.96165.27107 3.68909 50 11.21590 4.63171.23424 4.26911.25273 3.95680.27138 3.684S5 49 -12.21621 4.62518.23455 4.26352.25304 3.95196.27169 3.68061 48 13.2165 1 4,61868.23485 4.25795.25335 3.94713.27201 3.67638 47 14.21632 4.61219.23516 4.25239.25366 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 4.59283.23608 4,23580.25459 3.92793.27326 3.65957 43 18,21804 4.58641.23639 4.23030.25490 3.92316.27357 3.65538 42 19.21834 4.58001.23670 4.22481.25521 3.91839.27388 3.65121 41 20.21864 4.57363.23700 4.21933.25552 3.913641.27419 3.61705 40 21.21895 4.56726.23731 4.21387.25583 3.90890.27451 3.64289 39 22.21925 4.56091.23762 4.20842.25614 3.90417.27482 3.63874 38 23.21956 4.55458.23793 4.20298.25645 3.89945.27513 3.63461 37 24.21986 4.54826.23823 4.19756.25676 3.89474.27545 3.63048 36 25.22017 4.54196.23354 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 4.52941.23916 4.18137.25769 3.88068.27638 3.61814 33 23.22103 4.52316.23946 4.17600.25S00 3.87601.27670 3.61405 32 29.22139 4.51693.23977 4.17064.25831 3.87136.27701 3.60996 31 30.22169 4.51071.24008 4.16530.25862 3.86671.27732 3.60588 30 311.22200 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 4.14934.25955 3.85284.27826 3.59370 27 34.22292 4.48600.24131 4.14405.25986 3.84824.27858 3.58966 26 35.22322 4.47986.24162 4.13877.26017 3.84364.27889 3.58562 25 36.22353 4.47374.24193 4.13350.26048 3.83906.27921 3.58160 24 37.22383 4.46764.24223 4.12825.26079 3.83449.27952 3.57758 23 38.22414 4.46155.24254 4.12301.26110 3.82992.27983 3.57357 22 39.22444 4.45548.24285 4.11778.26141 3.82537.28015 3.56957 21 40.22475 4.44942.24316 4.11256.26172 3.82083.28046 3.56557 20 41.22505 4.44338.24347 4.10736.26203 3.81630.28077 3.56159 19 42.22536 4.43735.24377 4.10216.26235 3,81177.28109 3.55761 18 43.22567 4.43134.24408 4.09699.26266 3.80726.28140 3.55364 17 44.22597 4.42534.24439 4.09182.26297 3.80276.28172 3.54968 16 45.222628 4.41936.24470 4.08666.26328 3.79827.28203 3.54573 15 46.22658 4.41340.24501 4.08152.26359 3.79378.28234 3.54179 14 47.22659 4.40745.24532 4.07639.26390 3.78931.28266 3.53785 13 48.22719 4.40152.24562 4.07127.26421 3.78485.28297 3.53393 12 49.22750 4.39560.24593 4.06616.26452 3,78040.28329 3.53001 11 50.22781 4.38969 1.24624 4.06107.26483 3.77595.28360 3.52609 10 5 1.22811 4.383S1.24655 4.05599.26515 3.77152.28391 3.52219 9 52.22842 4.37793.24686 4.05092.26546 3.76709.28423 3.51829 8 53.229872 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 4.36040.21778/ 4.03578.26639 3. 75388.28517 3.50666 56.22964 4.354 59.24809 4.03076.26670 3.74950.28549 3.50279 4 57.22995 4.3:4879.24S40 ) 4.0)2574.26701 3.74512.285fi0 3.49894 3 51.930C26 { 4.34300.24871 4.020(74.26733 3.74075.28612 3.49509 2 591.23056 4.33723.24902 4.01576.26764 3.73640.286413 3.49125 1 60.2387 4.331483.24933 4.0107 8.26795 3.73205.28675 3.48741 0 B.Cotang. Tang. tang.T Tang. Cotalg.| Tng. Cotang. Tang. M. _ 77D | 760 1 o |50 0 20* 234 TABLE XV. NATURAL TANGENTS AND COTAiNGENT'. 16o ] j3 18 J 19 MI. Tang. Cotang. TanI'g. Cotang. Tang. I Cotang.'rTang. t Cotang. LIT. 0.28675 3.48741.30573 3.270O5. 324i22:o.2776.4133 2.9011 60 1 1.28706 3.48359.30605 3.26745.:32.2; 3.07464.34465 2.90147 50) 2.28738 3.47977.30637 3.26406.325;56 3.071601.34498 2.89873 578 3.28769 3.47596.30669 3.26067.32oS 3.06537.3-1530 2.89600 57 4.28800 3.47216.30700 3.25729.32621 3.06554.34 563 2.89327 56 5.28832 3.46337.30732 3.25392.32653 3.06252.34596 2.,'9055 55 6.28864 3.464538.30764 3.25055.32636 3.05950.34628 2.63733 51 7.28895 3.46080.30796 3.24719.32717 3.05649.34661 2.S3511 53I 8.28927 3.45703.30828 3.24383.32749 3.05349.34693 2.88240 52 9.28958 3.45327.30860 3.24049.32782 3.05049.34726 2.87970 51 1 0.28990 3.44951.30391 3.23714.32814 3.04749.34758 2.87700 50 11.29021 3.44576.30923 3.23331.32646 3.04450.34791 2.87430 49 1 2.29053 3.44202.30955 3.23048.32S78 3.0-152.34624 2.87161 4S 13.29084 3.43829.30957 3.22715.32911 3.03854.34856 2.86S92 47 14.29116 3.43456.31019 3.22384.32943 3.03556.34889 2.86624 46 15.29147 3.43084.31051 3.22053.32975 3.03260.34922 2.86336 45 16.29179 3.42713.31033 3.21722.33007 3.02963.34954 2.86089 44 17.29210 3.42343.31115 3.21392.33010 3.02667.34987 2.85822 43 18.29242 3.41973.31147 3.21063.33072 3.02372.35020 2.85555 42 19.29274 3.41604.31178 3.20734.33104 3.02077.35032 2.85289 41 20.29305 3.41236.31210 3.20406.33136 3.01783.330S5 2.85023 40 21.29337 3.40369.31242 3.20079.33169 3.01489.35118 2.84758 39 22.29363 3.40502.31274 3.19752.33201 3.01196.35150 2.844941 3S 23.29400 3.40136.31306 3.19126.33233 3.00903.35183 2.84229 37 24,29432 3.39771.31338 3.19100.33266 3.00611.35216 2.83965 36 25.29463 3.39406.31370 3.18775.33293 3.00319.35248 2.83702 35 26.29495 3.39042.31402 3.18451.33330 3.00028.35231 2.83439 34 27.29526 3.38679.31434 3.18127.33363 2.99738.35314 2.83176 33 28.29558 3.38317.31466 3.17804.33395 2.99447.35346 2.82914 32 29.29590 3.37955.31498 3.17481.33427 2.99158.35379 2.82653 31 30.29621 3.37594.31530 3.17159.33460 2.98868.35412 2.82391 30 3 1.29653 3.37234.31562 3.16838.33492 2.98580.35445 2.82130 29 32.29685 3.36875.31594 3.16517.33524 2.98292.35477 2.81870 23 33.29716 3.36516.31626 3.16197.33557 2.98004.35510 2.81610 27 34.29748 3.36158.31658 3.15877.33599 2.97717.35543 2.8135 0 26 3.3.29780 3.35800.31690 3.15558.33621 2.97430.35576 2.81091 25 36.29811 3.35443.31722 3.15240.33654 2.97144.35603 2.80833 24 37.29843 3.35087.31754 3.14922.336S6 2.96858.35641 2.80574 23 38.29875 3.34732.31786 3.14605.33718 2.96-73.35674 2.80316 22 39.29906 3.34377.31818 3.14288.33751 2.96288.35707 2.80059 21 10.29938 3.34023.31850 3.13972.33783 2.96004.35740 2.79802 20 41.29970 3.33670.31882 3.13656.33816 2.95721.35772 2.79545 19 42.30001 3.33317.31914 3.13341.33848 2.95437.35805 2.79289 18 43.30033 3.32965.31946 3.13027.338 1 2.95155.35838 2.7903.3 17 4-4.30065 3.32614.31978 3.12713.33913 2.94872.35871 2.78778 1 6 415.30097 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 L4 47.30160 3.31565.32074 3.11775.34010 2.94028.35969 2.78014 13 48.30192 3.31216.32106 3.11464.34043 2.93748.36002 2.77761 12 49.30224 3.30863.32139 3.11153.34075 2.93468.36035 2.77507 11 50.30255 3.30521.32171 3.10842.34108 2.93189.36068 2.772541 10 51.302S7 3.30174.32203 3.10532.34140 2.92910.36101 2.77002 9 52.30319 3.29829.32235 3.10223.34173 2.92632.36134 2.76750 8 53.30351 3.29483.32267 3.09914.34205 2.92354.36167 2.76498 7 54.30382 3.29139.32299 3.09606.34238 2.92076.36199 2.76247 6 55.30414 3.2S795.32331 3.09293.34270 2.91799.36232 2.75996 5 56.30446 3.28452.32363 3.08991.34303 2 91523.36265 2.75746 4 57.30478 3.28109.32396 3.08685.34335 2.91246.36298 2.75496 3 58.30509 3.27767.32428 308379.34363 2.90971.36331 2.75246 2 59.30541 3.27426.32460 3.08073.34400 2.90696.36361 2.74997 1 60.30573 3.270,85.32492 3.07763.34433 2.90421.36397 2.74748 0 1.t Cotang. Tang. Cotang. T Tang. Cotan-g. Tang. Cotang. Tang.. t 73D 72D3 r02 I 0 I -------- ------- ---- --------— 00 j TABLE XV. NATURAL TANGENTS AND COTANGENTS. 283 _____ ____R> ___R f230 M. Tang. Cotang.'Tang. Cotang. Tanlng. Cotang.'T'ang, Cotang. DM. ().36397 2.747483.38386 2.60509.4(403 2.47509.42147 2.35585 60 1.36430 2.74499.33420 2.60S83.40436 2.47302.42482 2.35395 59 2.36463 2.74251,38453 2.60057.40470 2.47095.42516 2.35205 58 3.36496 2.74004.38487 2.59831.40504 2.468388.42551 2.35015 57 I 4.36529 2.73756.38520 2.59606.40538 2.46632.42585 2.34825 56 5.36562 2.73509.38553 2.59331.40572 2.46476.42619 2.34636 55 6.36595 2.73263.38587 2.59156.40606 2.46270.42654 2.34447 54 7.36628 2.73017.38620 2,58932.40640 2.46065.42688 2.34258 53 8.36661 2.72771.38654 2.58708.40674 2.45860.42722 2.34069 52 9.36694 2.72526.38687 2.58484.40707 2.45655.42757 2,33881 51 10.36727 2.72281.38721 2.58261.40741 2.45451.42791 2.33693 50 11.36760 2.72036.38754 2.58038.40775 2.45246.42826 2.33505 49 1 2.36793 2.71792.38787 2.57815.40809 2.45043.42860 2.33317 48 1 3.36826 2,71548.38821 2.57593.40843 2.44839.42894 2.33130 47 14.36859 2.71305.33854 2.57371.4(1877 2.44636.42929 2.32943 46 15.36892 2.71062.38888 2.57150.40911 2.44433.42963 2.32756 45 16.36925 2.70819.38921 2,56928.40945 2.44230.42998 2.32570 44 17.36958 2.70577.33955 2.56707.40979 2.44027.43032 2.32383 43 18.36991 2.70335.38988 2.56487.41013 2.43S25.43067 2.32197 42 19.37024 2.70094.39022 2,56266.41047 2.43623.43101 2.32012 41 20.37057 2.69853.39055 2.56046.41081 2,43422.43136 2.31826 40 21.37090 2.69612.39089 2.55827.41115 2,43220.43170 2.31641 39 22.37123 2.69371.39122 2.55608.41149 2.43019.43205. 2.31456 38 23.37157 2.69131.39156 2.55389.41183 2,42819.43239 2.31271 37 24.37190 2.68892.39190 2.55170.41217 2.42618.43274 2.31086 36 25.37223 2.68653,39223 2.54952.41251 2.42418.43308 2,30902 35 26.37256 2.68414.39257 2.54734.41285 2.42218.43343 2,30718 34 27.37289 2.63175.39290 2,54516.41319 2.42019.43378 2.30534 33 28.37322 2.67937.39324 2.54299.41353 2.41819.43412 2,30351 32 29.37355 2.67700.39357 2.b4082.41387 2.41620.43447 2.30167 31 30.37388 2.67462.39391 2.53265.41421 2.41421.43481 2.29984 30 31.37422 2.67225.39425 2.536-18.41455 2.41223.43516 2.29801 29 32.37455 2.66989.39458 2.53432.41490 2.41025.43550 2.29619 28 33.37488 2.66752.39492 2.53217.41524 2.40827.43585 2.29437 27 34.37521 2.66516.39526 2.53001.41558 2.40629.43620 2.29254 26 35.37554 2.66281.39559 2.52786.41592 2.40432.43654 2.29073 25 36.37588 2.66046.39593 2.52571.41626 2.40235.43689 2.28891 24 37.37621 2.65811.39626 2.52357.41660 2.40033.43724 2.2S8710 23 38.37654 2.65576.39660 2.52142.41694 2.39841.43758 2.28528 22 39.37687 2.65342.39694 2.51929.41728 2.39645.43793 2.28348 21 40.37720 2.65109,39727 2.51715.41763 2.39449.43828 2.28167 20 41.377.54 2,64875.39761 2.51502.41797 2.39253.43862 2.27987 19 42.37787 2.64642.39795 2 51289.41831 2.39058.43897 2.27806 18 43.37820 2.64410.39829 2.51076.41865 2.38863.43932 2.27626 17 4 1.37853 2.64177.39862 2.50864.41899 2.38668.43966 2.27447 16 45.37887 2.63945.39896 2.50652.41933 2.38473.44001 2.27267 15 46.37920 2.63714.39930 2.50440.4196S 2.38279.44036 2.27088 14 I 7.37953 2.63483.39963 2.50229.42002 2.38084.44071 2.26909 13 148.37936 2.63252.39997 2 50018.42036 2.37891.44105 2.26730 12 491).31020 2.63021.40031 2.49807.42070 2.37697.44140 2.26552 11 5)0 }.:38053 2.62791.40(t65 2.49597.42105 2.37504.44175 2.26374 10 5l.3> SG 2.62561,40093 2.49386.42139 2.37311.44210 2.26196 9 52.'3 L20 2.62332.40132 2.49177.42173 2.37118.44244 2.26018 8 ~1, ~1 *5:.:3 1 2.6)103.40166 2.43967.42207 2.36925.44279 2.25840 7 5. 3S 1 86 2.61874.40200 2.48758.42242 2.367533.44314 2.25663 6 ~, 5I.-3-2l2,) 2.616.40234 2.48>49,42276 2.36541.44349 2.254816 5,5,.:., 2.61418.40267 2.48340.42310 2.36349.44384 2.25309 4 57. >'.~G 2.61190.4030l 2.43132.42345 2.36158.44418 2.25132 3 i*:3.3-2!r) 2 60963.40-325 2.47924 49379 2.35967.44453 2.24956 2 3 2-.6036.40(3669 2.47,16.42413 2.35776.,44188 2.247,80 1 i6) i86.3 I 6 2.6()809.0to 1:3 2.47 509.42447 2.35585.44523 2.24604 0 I.l Cotng II'>'a>. Cot>lg. Tang. Cotang. Tang. Cotang. Tg... _ _ -,63 663. 236 TABLE XV. NATURAL TANGENTS AND COTAINGENTS. 1 24~ 25> 260 27~ I 11., Tang. Cotang. Tanlg. Cotang.'T'ng. Cotang. Tang. Cotang. M. 0.44o523 2.24604.46i631 2.14451.48773 2.0513(.509.53 1.966261 60 1.44558 2.24423.46666 2.142881.4SS09 2.()4879.509S9 1.,6t120 59 2.44593 2.24252.46702 2.14125.48845 2.04728.51026 1.95979 58 3.44627 2.24077.46737 2.13963,4~81 2, (4577.51063 1. 95838 57 4.44662 2.23902.46772 2.1:301,48917 2,04426.51099 1.95698 56 5.44697 2.23727.46808 2.13639.42953 2.04276.51136 1.95;557 55 6.44732 2.23553.468;43 2.13477 A49S9 2.04125.51173 1.95417 54 7.44767 2.23378.46879 2.13316.49026 2.03975.51209 1.95277 53 8.44802 2,23204.46914 2.13154.49062 2.03825.51246 1.95137 52 9.44837 2.23030.46950 2.12993.49098 2.03675.51283 1.94997 51 10.44872 2.22857.469'85 2.12832.49134 2.03526.51319 1.94858 50 11 -.44907 2.22683.47021 2.12671.49170 2.03376.51356 1.94718 49 112.44942 2.225 10.47056 2.12511.49206 2.03227.51393 1.94579 48' 13.44977 2.22337.47092 2.12350.49242 2.03078.51430 1.94440 1 47 14.45012 2.22164.47128 2.12190.49278 2.02929.51467 1.94301 46 15.45047 2.21992.47163 2.12030.49315 2.02780.51503 1.9-162 45 16.45082 2.21819.47199 2.11871.49351 2.02631.51540 1.94023 44 17.45117 2.21647.47234 2.11711.49387 2,02483.51577 1.93885 43 18.45152 2.21475.47270 2.11552.49423 2.02335.51614 1.93746 42 19.45187 2.21304.47305 2.11392.49459 2.02187.51651 1.93608 41 20.45222 2.21132.47341 2.11233.49495 2.02039.51688 1.93470 40 21.4.5257 2.20961.47377 2.11075.49532 2.01891.51724 1.93332 39 22.45292 2.20790.47412 2.10916.49568 2.01743.51761 1.93195 38 23.45327 2.20619.474483 2.10758.49604 2.01596.51793 1.93057 37 24.45362 2.20449.47483 2.10600.496-140 2.01449.51835 1.92920 36 25.45397 2.202783.47519 2.10442.49677 2.01302.518S72 1.92782 35 26.45432 2.20108.47555 2.10284.49713 2.01155.51909 1.92615 34 27.45467 2.19938.47590 2.10126.49749 2.01008.519i)6 1.92508 33 28.45502 2.19769.47626 2.09969.49786 2.00862.51983 1.92371 32 29.45538 2.19599.47662 2.09311.49822 2.00715.52020 1.92235 31 30.45573 2.19430.47698 2.09654.49S58 2.00569.52057 1.92098 30 31.45608 2.19261.47733 2.09498.49894 2.00423.52094 1.91962 29 32.45643 2.19092.47769 2.09341.49931 2.00277.52131 1.91826 28 33.45678 2.18923,47805) 2.09184.49967 2.001.31.52168 1.91690 27 34.45713 2.18755.47840 2.09023.50004 1.999S6.52205 1.91554 26 35.45748 2.18587.47876 2.03872.50040 1.99841.52242 1.91418 25 36.45784 2.18419.47912 2.08716.500)76 1.99695.52279 1.91282 24 37.45819 2.18251.47948 2.08560.50113 1.99550).52316 1.91147 23 38.45854 2 10S4.479S4 2.08405.50149 1.99466.52353 1.91(112 22 39.45889 2.17916,48019 2.08250.50185 1.99261.52390 1.90876 21 40.45924 2.17749.498055 2.0S094.50222 1.99116.52427 1.90741 20 41.45960 2.175S2.48091 2,07939.50258 1.98972.52464 1.9()607 19 42.45995 2.17416.48127 2.07785.50295 1.9882S.5250(11 1.90472 1S 43.46030 2.72419.48163 2.0763(.50331 1.98684.52538 1.90337 17 44.46065 2,17083.48198 2.07476,5036S3 1.95-40.52575 1.902(03 16 45.46101 2.16917.4823- 2.07321 150404 1.98396.52613 1.90069 15 46.46136 2.16751.4,270 2.07167.5011 1.98253.52650 1.9935 14 47.46171 2.16585.48306 2.07014.504t77 1,98 10.52687 1.89801 13 48.46206 2.16420.48342 2 06860.50514 1 97966.52724 1.89667 12 49.4624 2.16255.48378 2.06706 5(1550 1.97823.52761 1.89533 11 5o.462/77 2.16090.4414 2,06553.505s7 1.97681.5279s 1.89-100 10 o51.463121 2.15925.484>50 2.06400.50623 1.97533.52836 1.89266 9 52.46358 2.15760.48486 2.06247.50660 1,97395 52873 1.89133 S 53.46:383 2.1 5596.4S521 2.06094.50696 1.97253.52910 1.S90(0 7 54 |46418 2.15433.4S3,r7 2.n0942.50733 1.97111.52947 1.88867 6 55.46454 2. 1268.4893 2.0790n.50769 1.96969.52985 1.88734 5 56.46489 2.15104,48629 2. 0,637.50SO6 1.96827.53022 1.8S6(12 4 57.46-525 2.14940.4S66>5 2.n5)485.50S43| 1.966S5.530.59 1.SS469 3 58.46-)60 2.1477/7.4,701 2.0.,333.50ST79 1,96544.53096 1.8337 2 59 1. 4 6;595| 2.14614.4>737 2.0518S2.50916 1L96402.53134 1.8S82O5 1 60 1.6631 2.14 -1)1.4773 2.01).30 5093 1.96261 53171 1.Ss(073 ( 1. CotangI Tan). 0>Cot)g Tm CotRn- I Tang. 1Cotang. Tang. 51. i- - 65- * 6|3>.1 6 - - TABLE XV. NATURAL TANGENTS AND COTANGENTSo 237 20 - 29 1 300 0 310 DI Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.53171 1.88073.54731 1.60405.57735 1.73205.60086 1.66 42 60 1.53208 1.87941.55469 1.802S1.57774 1.73089.60126 1.66318 59 2.53246 1.87809.55507 1.80158.57813 1.72973.60165 1.66209 58 3.53283 1.87677.55545 1.80034.57851 1.72857.60205 1.66099 57 4.53320 1.87546.55583 1.79911.57890 1.72741.60245 1.65990 56 5.53358 1.87415.55621 1.79788.57929 1.72625.60284 1.65881 55 6.53395 1.87283 [55659 1.79665.57963 1.725;09.60324 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 1.86891.55774 1.79296.5803.5 1.72163.60443 1.65445 51 10.53545 1.86760.55812 1.79174.5124 1.72047.60483 1.65337 50 1 1.53582 1.86630.55850 1.79051.58162 1.71932.60522 1.65228 49 12.53620 1.86499'.558S8 1.78929.58201 1.71817.60562 1.65120 48 13.53657 1.86369.55926 1.78S07.58240 1.71702.60602 1.65011 47 14.53694 1.86239.55964[ 1.786S5.58279 1.71588.60642 1.64903 46 15.53732 1.86109.56003 1.78563.58318 1.71473.60681 1.64795 45 16.53769 1,85979.,56041 1.78441.58357 1.71358.65721 1.64687 44 17.53807 1.85850.56079 1.78319.58396 1.71244.60761 1.64579 43 18.53844 1.85720.56117 1.78198.58435 1.71129.60801 1.64471 42 19.53382 1.85591.56156 1.78077.58474 1.71015.60841 1.64363 41 20.53920 1.85462.56194 1.77955.58513 1.70901.60881 1.64256 40 21.53957 1.85333.56232 1.77834.58552 1.70787.60921 1.64148 39 22.53995 1.85204.56270 1.77713.58591 1.70673.60960 1.64041 38 23.54032 1.85075.56309 1.77592.58631 1.70560.61000 1.63934 37 24.54070 1.84946.56347 1.77471.58670 1.70446.61040 1.6:3826 36 25.54107 1.84818 [.56385 1.77351[.58709 1.70332.61080 1.63719 35 26.54145 1.84689.56424 1.77230.58748 1.70219.61120 1.63612 34 27.54183 1.S4561.56462 1.77110.538787 1.70106.61160 1.63505 33 23.54220 1.84433.56501 1.76990.58826 1.69992.61200 1.63398 32 29.542583 1.84305.56539 1.76869.58S65 1.69S79.61240 1.63292 31 30.54296 1.84177.56577 1.76749.5S905 1.69766.61280 1.63185 30 31.54333 1.84049.56616 1.76629.58944 1.69653.61320 1.63079 29 32.54371 1.83922.56654 1} 76510.58983 1.693541.61360 1.62972 28 33.54409 1.83794.56693 1.76390.59022 1.69428.61400 1.62S66 27 34.54446 1.83667.56731 1.76271 1.59061 1.69316.61440 1.62760 26 35.54484 1.83540.56769 1.76151.59101 1.69203.61480 1.62654 25 36.54522 1.83413.56308 1.76032.59140 I 1.69091.61520 1.62548 24 37.54560 1.83286.56346 1.75913.59179 1.6S979.61561 1.62442 23 38.54597 1.83159.56S85 1.757941.59218 1.68866.61601 1.62336 22 39.54635 1.83033.56923 1.75675.59258 1.6S754.61641 1.62230 21 40.54673 1.82906.56962 1.75556.59297 1.68643.61681 1.62125 20 41.54711 1.82780.57000 1.75437.59336 1.68531.61721 1.62019 19 42.54748 1.82654.57039 1.75319.59376 1.6S419.61761 1.61914 18 43.54786 1.82528.57078 1.75200.59415 1.68308.61801O 1.61808 17 44.54824 1.82402.57116 1.75032.59454 1.6S196.61842 1.61703 16 45.54862 1.82276.57155 1.74964.59194/ 1.68085.61SS2 1.6159S 15 46.54900 1.82150.57193 1.74846.59533 1.67974.61922 1.61493 14 47.54933 1.82025..57232 1.74728.59573 1.67863.61962 1.61388 13 48.54975 1.81899.57271 1.74610.59612 1.67752.62003 1.61283 12 49.55013 1.81774.57309 1.74492.59651 1.67641.62043 1.61179 11 50.55051 1.81649.57348 1.74375.59691 1.67530.62083 1.61074 10 51.55089 1.81524.57336 1.74257.59730 1.67419.62124 1.60970 9 52.55127 1.81399.57425 1.74140.59770 1.67309.62164 1.60865 8 53.55165 1.81274.57464 1.74022.59809 1.67198.62204 1.60761 7 54.55203 1.81150.57503 1.73905.59849 1.67088.62245 1.60657 f6 55.55241 1.81025.57541 1,73788.59888 1.66978.62285 1.60553 5 56.55279 1.80901.57580 1.73671.59928 1.66867.62325 1.60449 4 57.55317 1.80777.57619 1.73555].59967 1.667t:57.62366 1.60345 3 3 58.55355 1.80653.57657 1.73438.60007 1.66647.62406 1.60241 2 59.55393 1.80529.57696 1.73321.60046 1.66538.62446 1.60137 1 60.55431 1.80405.57735 1.73205.600S6 1.66428.62487 1.60033 0 M. Cotanu.| Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. 11. c o610 (60 59 — o o ti5S. _-:4~ —.g. -~ 238 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 320 330 340 350 M._ Tang. Cotag ag. Cotaang.TaCo Ctagg. Tang. Cotang. M. 0.624S77 1.60033.64941 1.539S6.67451 1.4S2536,.70021 1.42815 60 1.62527 1.59930.64982 1.53888.67493 1.48163.7006 1.42726 59 2.62563 1.59326.65024 1.53791.67536 1.48070.70107 1.42638 58 3.62608 1.59723.65065 1.53693.67578 1.47977.70151 1.42550 57 4.62649 1.59620.65106 1.53595.67620 1.47385.70194 1.42462 56 5.62689 1.59517.65148 1.53497.67663 1.47792.70238 1.42374 55 6.62730 ]1.59414.65189 1.534C00.67705 1.47699.70231 1.422S6 54 7.62770 1.59311.65231 1.533(02.67743 1.47607.70325 1.42198 53 8.62811 1.59208.65272 1.53205.67790 1.47514.70368 1.42110 52 9.62852 1.59105.65314 1.53107.67832 1.47422.70412 1.42022 51 10.62892 1.59002.65355 1.53010.67875 1.47330.7(0455 1.41934 50 11.62933 1.58900.65397 1.52913.67917 1.47238.70499 1.41847 49 12.62973 1.58797.65438 1.52816.67960 1.47146.70542 1.41759 48 13.63014 1.58695.65480 1.52719.63002 1.47053.70586 1.41672 47 14.63055 1.58593.65521 1.52622.68045 1.46962.70629 1.41584 46 15.63095 1.58490.65563 1.52525.6S03S 1.46S70.70673 1.41497 45 16.63136 1.5833S.65604 1.52429.68130 1.46778.70717 1.41409 44 17.63177 1.55286.65646 1.52332.68173 1.466S6.70760 1.41322 43 18.63217 1.58184.656S8 1.52235.68215 1.46595.70S04 1.41235 42 19.63258 1.58033.65729 1.52139.63258 1.46503.70848 1.41148 41 20.63299 1.57981.65771 1.52043.68301 1.46411.7089 1.41061 40 21.63340 1.57879.65813 1.51946.68343 1.46320.70935 1.40974 39 22.633S0 1.57778.65854 1.51850.63386 1.46229.70979 1.40887 33 23.63421 1.57676.65S96 1.51754.68429 1.46137.71023 1.40800 37 24.63462 1.57575.65938 1.51658.68471 1.46046.71066 1.40714 36 25.63503 1.57474.65980 1.51562.63514 1.45955.71110 1.40627 35 26.63544 1.57372.66021 1.51466.6X557 1.45864.71154 1.40540 34 27.63584 1.57271.660fi3 1.51370.68600 1.45773.7119S 1.40454 33 28.63625 1.57170.66105 1.51275.6$642 1.45652.71242 1.403(67 32 29.63666 1.57069.66147 1.51179.66835 1.45592.71285 1.40281 31 30.63707 1.56969.66189 1.51084.68723 1.45501.71329 1.40195 30 31.63748 1.56368.66230 1.509SS8 6S771 1.45410.71373 1.10109 29 32.63789 1.56767.66272 1.50893.63814" 1.45320'.71417 1.40022 28 33.63830 1.56667.66314 1.50797.63557 1.45229.71461 1.39936 27 34.63971 1.56566.66356 1.507(02.68900 1.45139.71505 1.39850 26 35.63912 1.56466.66398 1.50607 68942 1.4o049.71549 1.39764 25 36.63953 1.56366.66440 1.50512.68985 1.4495S.71593 1.39679 24 37.63994 1.56265.66-432 1.50417.69028 1 44868.71637 1.39593 23 38.64035 1.56165.665324 1.50322.69071 1.44773.716 1 1.39507 22 39.64076 1.5606.5.66566 1.50228.69114 1.44638.71725 1.39421 21 40.64117 1.55966.66608 1.50133.69157 1.44598.71769 1 39336 20 41.64158 1.55866.66650 1.50038.69200 1.44508.71813 1.39250 19 42.64199 1.55766.66692 1.49944.69243 1.44418.71857 1.39165 18 43.64240 1.55666.66734 1.49849.692S6 1.44329.71901 1.39079 17 44.64281 1.55567.66776 1.49755.69329 1.44239.71946 1.3S994 16 45.64322 1.55467.66818 1.49661.69372 1.44149.71990 1.38909 15 46.64363 1.55363.66860 1.49566.69416 1.44060.72034 1.38824 14 47.64,404 1.55269.66902 1.49472.69459 1.43970.72(78 1.38738 1 3 48.64446 1.55170.66944 1.49378.69502 1.4381.72122 1.3 653 12 49.64487 1.55071.66986 1.49234.69545 1.43792.72167 1.39568 11 50.64-28 1.54972.67028 1.49190.695S8 1.43703.72211 1.3S484 10 51.64569 1.54873.67071 1.49097.69631 1.43614.72255 1.38399 9 52.6,610 1.54774.67113 1.49003.69675 1.43525.72299 1.38314 8 53.64652 1.54675.67155 1.43909.69718 1.434936.723-44 1.38229 7 54.64693 1.54576.67197 1.48816.6,76 L 1.433 47.72338 1.3S145 6 55.64734 1.54478.67239 1.48722.69304 1.4325S.72432 1. 3060 5 56.64775 1.54379.67282 1.48629.6947.724 77 1.37976 4 57.64817 1.54281.67324 1.48536.69891 1.43080.72521 1.37S91 3 58.64858 1.54183.67366 I.4S4-12.69934 1.42992.72 65: 1.37807 2 59.64899 1.54085.674109 1.4,3319.69977 1.42903.72610 1.37722 1 60.64941 1.53986.67451 1.48456.7C(021 1.42S 15.72654 1.37638 0 M. Cotan..Tang. Cotang. Tang. Cotar. Taing. -Cotang. Tang. IM. 5 T 7 560 5503 50to TABLE XV. NATURAL TANGENTS AND COTANGENTS. 239 36o 370 j 380 390 IM.' Tang. Cotang Tang. Cotang. Tal. t Cotang. Tang. Cotang. 0.72654 11.37633 75:355 1.32704.76129 1.27994 780978 1.23490 60 1.72699 1.37554.75401 1.32624.78175 1.27917.81027 1.23416 59 2.72743 1.37470.75447 1 232544.7S222 1.27841.8107 1.23:343 58 3.72788 1.37386.75492 1..364694.782G9 1.27761.81123 1,23270 57 4.72332 1.37302.75538 1.23S84.78316 1.27 6S8.8117 1.23196 56 5.72877 1.37218.755384 1.32304.78363 1.27611.81220 1.23123 51 6.72921 1.37134.7a629 1. 3222.78410 1.27535 1.81268 1.23050 54 7.72966 1.37050).75675 1.32144.78457 1.27458.81316 1.22977 53 8.73010 1.36967.75721 1.32064.78604 1.27382.81364 1.22904 52 9.73055 1.36G83.75767 1.31984.78551 1..27306.81413 1.22831 5 51 10.73100 1.36800.75512 1.31904.78o598 1.27230.81461 1.2258 50 11.73144 1.36716.758.31825 7S645 1.2713.81510 1.22685 9 12.73189 1.36633.75904 1.31745.792 1.27077.81558 1.22612 48 13.73234 1.36549.75950 1.31666.78739 1.27001.81606 1.22539 47 14.73278 1.36466.73996 1.31586.78786 1.26925.81655 1.22467 46 15.73323 1.36383.76042 1.31507.78834 1.26849.S1703 1.22394 45 1 6.7'3363 1.36300.76088 1.31427.78881 1.26774.S1752 1.22321 44 17.73-413 1.36217.761341.31348.7928 1.31348.7892 2669. 2281800 1.22249 43 18.734537 1.36134.76180 1.31269.78975 1.26622.81849 1.22176 42 19.73502 1.36051.76226 1.31190.79022 1.26546.81898 1.22104 41 20.73547 1.3596S 76272 1.31110.79070 1.26471.81916 1.22031 40 21.73.J92 1.35885.76318 1.31031.79117 1.26395.8199-5 1.21959 39 22.73637 1.35s02 76364 1.30952.79164 1.26319.S2044 1.21 SS6 3S 23.736S1 1.35719.76410 1.30873.79212 1.26244.82092 1.21814 37 24.73726 1.35637.76456 1.30795.79259 1.26169.8211 1.21742 36 25.73771 1.35554.76.502 1.30716.79306 1.26093.82190 1.21670 35 26.733 1 6 1.35472.76548 1.30637.79354 1.26018.82238 121598 34 27,73S61 1.3539.76594 1.30558.79401 1.25943.82287 1.21526 33 28.73906 1.35307.76640 1. 3048/.79449 1. 2a67.82336 1.21454 32 29.73951 1.35224.766S6 1.3040 1.79496 1.25792.82385 1.21382 31 30.73996 1.35142.76733 1.30323.79544 1.25717.82434 1.21310 30 31.74041 1.35060.76779 1.30244.79591 1 25642.82483 1.21238 29 32.74086 1.34978.76825 1.30166.79639 1. 25567.82531 1.21166 28 33.74131 1.34896.76871 1.30087,796S6 1,25492 82580 1.21094 27 34.74176 1.34814.76918s 1.30009.79734 1.25417.82629 1.21023 26 35.74221 1.34732.76964 1.29931.79781 1.2343 82678 1.20951 25 36.74267 1.34650.77010 1.29853.7929 1.25268.82727 1.20879 24 37.74312 1.3456S.77057 1,29775.79877 1.25193.82776 1.20808 23 38.74357 1. 34487.77103 1.29696.79924 1.25118.82825 1.2(1736 22 39.744029.34429618.79972 1.205.77149 1.296.79972 1.25044.2874 1.2(665 21 40.74447 1.34323.77196 1.29541.80020 1.24969.82923 1.20593 20 41.74492 1.3472 42 1.29463.S0067 1.24S95.82972 1.20522 19 42.74538 1.34160.77289 1.29385.80115 1.24820.83022 1.20451 18 43.745S3 1.34079.77335 1.29307.80163 1.24746.83071 1.20379 17 44.74623 1.33998.77382 1.29229.80211 1.24672.83120 1.20308 16 45.74674 1.33916.77428 1.29152.80258 1.24597.83169 1.20237 15 46.74719 1.33835.77475 1.29074.80306 1.24523.8321 1.20166 14 47.74764 1.33754'. 77521 1.28997.80354 1.24449.83268 1.20095 13 48.74851 0 1.33673.77563 1.28919.80402 1.24375.83317 1.20024 12 49.74855/ 1.33592.77615 1. 2842.S0450 1.24301.83366 1.19953 11 50.74900 1.335 1.77661 1.23764.80498 1.24227.83415 1.198S2 10 51.74946. 33430.77703 1.28687, 0546.24l53.83465 1.19811 9 52.74991 1.33349.77754 1.28610.80594 1.24079.83514 1.19740 8 53.75037 1.33268.770 1.28533.8064 1.2-005.83564 1.19669 7 54.75032 1.33187.77848 1.28456.80690 123931.83613 1.19599 6 55.75128 1.33107.7795 1.29379.8073s 1.2358.83662 1.19528 5 56.75173 1.33026 77941 1.28302.80786 1.23784.83712 1.19457 4 57.7 5219 1.32946 77988 1.28225.80834 1.23710.83761 1.19387 3 58.75264 1. 32S65.78035 1.2814S.8 0882 1.23637.83811 1.19316 2 59.75310 1 32785.7082 1. 2071.80930 1.23563.83860 1.19246 1 60.753355 1.32704.78129 1.27994.80978 1.23490.83910 1.19175 0 M. Cotang. Tang. Cotng. Tag. Cotang. Tang. Cotang. Tang.. 530 552, 51 500 240 TABLE XV. NATURAL TANGENTS AND COTA'NGENTSo 1 403 _ _L __ 1_ 630 M.! Tang. Cotang. Tang.ang. Cotg. Tang. Cotanu l Tang. Cotang. j. V.83910 1.19175.869'29 1.153.9U0 1.11OG6.79362 02I. Z 7 it 1.83960 1.19105.86930 1.14969.90093 1.10996.93306 1.07174 59 2.84009 1.19035.87031 1.1490()2.90146 1.10931.93360 1.07112 58 3.84059 1.18964.87082 1.14S34.90199 1. 10367 93415 1.07049 57 4.84108 1.18894.87133 1.14767.90251 1.10302;.93469 1.06937 56 5.84158 1.18824.87184 1.14699.90304 1.1.0737.93524 1.06925 55 6.84203 1.18754.87236 1.14632.903537 1.10672.93573 1.06362 54 7.84258 1.18684.87287 1.14565.90410 1.10607.93633 1.0(600 53 8.84307 1.18614.8733S 1.1449s.90463 1.10433.936S8 1.06733 52 9.84357 1.18544.87389 1.14430.90516 1.10478.93742 1.()6676 51 10.84407 1.18474.87441 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 1.10295.93906 1.06489 48 13.84 556 1.182641.87595 1.14162.90727 1.10220.93961 1.0427 47 14.84606 1.18194.87646 1.14095.90781 1.101556.94016 1.06:365.- 46 15.84656 1.18125.87698 1.14028.90834 1.10091.94071 1.06303 45 16.84706 1.18055.87749 1.13961.90887 1.10027.94125 1.06241 44 17.84756 1.17936.87801 1.13S94.90940 1.09963.94180 1,06179 43 18.84806 1.17916.87852 1.13828.90993 1.09399.94235 1 ()6117 42 19.84856 1.17846.87904 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 1.17708.88007 1.13627.91153 1.09706.944100 1.05932 39 22.85006 1.17633.83059 1.13561.91206 1.09642.4435 1.05 S70 38 23.85057 1.17569.88110 1.13194.91259 1.09578.94510 11)5809 37 24.85107 1.17500.81 62 1.13428.91313 1.09514.945645 1.05747 36 25.85157 1.17430.82 214 1.13361.91366 1.09450.94620 1.05685 35 26.85207 1.17361.8326.5 1.13295.91419 1.09386.94676 1.05624 34 27.85257 1.17292.889317 1.13223.91473 1.09322.94731 1.05562 33 28.83530S 1.17223.88369 1.13162.91526 1.09258.94786 1 05501 32 29.8 353 1.17154.8421 1.13096.915S0{ 1.09195.91>441 1.05439 31 30.85403 1.17035.88473 1.13029.916:33 1.09131.94896 1.05378 30 31.85458 1.17016 86524 1.12963.91637 1.09067.94952 1,05317 29 32.85509 1.16947,83576 1.12397.91740 1.09003.95007 1.052355 28 331.85659 1 16378,86823 1 12>31.91794 1.089-10.93062; 1 05194 27 34.85609 1.16309,8360 1.127635.91847 1.08S76.95118 1.05133 26 35.85660 1.16741.88732 1.12699.91901 1.03313.95173 1. 0072 25 36.85710 1.16672.8784 1.12633.91955; 1.08749.95229 1 05010 24 37.85761 1.16603 83836 1.12567.92003 1.08636.935241 1.04949 23 38.85S>11 1.165337 8333 1.12501.92062 1.08622.95340 1. 0-83s 22 39.85862 1.16466,83940 1.12435.92116 1. 08359.95395 1. 0-127 21 40.85912 1.16393.88992 1.12369.92170 1.03496.95451.04766 20 41.85963 1.16329.89045 1.12303.92224 1.08432.9-506 1. 01705 19 42.86014 1.162261.89097 1. 12233.92277 1.0>369.95562 1.04644 18 43.86064 1.16192.89149 1.12172.92331 1.0306.95618 1.04583 17 1 44.86115 1.1612-1,89201 1.12106.923583 1.03243.9673 73 1.04522 16 45.86166 1.16056.89253 1.12041. 9139 1.0179.95729 1] 01461 15 46.86216 1.15987.89306 1.11975.92493 1.03116 95785 1:01401 14 47.86267 1.15919.8935)3 1.11909.92547 1.0803.95841 1.04340 1 3 48.86318 1.15851.8S9410 1.11844 92601 1.07990.95397 1.04279 12 49.86368 1.15783.89463 1.11778'92655 1.079'27.935952.04218 1 1 50.86-119 1.15715.89515 1.11713.92709 1.07864.96003 1.0415 5 1 0 51.86470 1.15647.89.567 1.11648.92763 1.07801.96064 1.04097 9 52.86521 1.15579.99620 1.11582.92817 1.07733.96120 1.01036 8 53.86572 1.15511.89672 1.1 1517.92872 1.07676.96176 1.03976 7' 54.86623 1.15443.89725 1.11452.92926 1.07613.96232 1.0391-5 6 55.86674 1.15375.89777 1.11337.92930 1.07/550.962S3 1.03355 5 56.86725 1.15303.89330 1.11321.930331 1.074>87.963144 1.03/794 4 57.86776 1.15240.899S3 1.11256.93083 1.07425.964010 1.03734 3 58.86627 1.15172.8993.5 1.11191.93143 1.07362.964)57 1.0374 2 59.86878 1.15104.89988 1.11126.93197 1.07299.96513 1.0363 1 60.86929 1.15037.900:10 1.1 106 93252 1.07237.96569 1.03553 0 M. Cotan. Tlang. Cotarg.I Tang. Cotang. Tang. CotairD, Tang. M. 4= = _ _> _ _ 0 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 241 44: 0 40 M. Tang-. Cotang. Ml. M. Tang. Cotang. M. I. Tang. Cotang. I. 0.96569 1.03553 60 20.97700 1.02355 40 40.98843 1.01170 20 1.96625 1.03493 59 2t.97756 1.02295 39 41.9S901 1.01112 19 2.96631 1.03433 58 22.97813 1.02236 38 42.9895S 1.01053 18 3.96733 1.03372 57 23.97870 1.02176 37 43.99016 1.00994 17 4.96794 1.03312 56 24.97927 1.02117 36 44.99073 1.00935 16 5.96850 1.03252 55 25.97984 1.02057 35 45.99131 1.00876 15 6.96907 1.03192 54 26.98041 1.01993 34 46.99189 1.00818 14 7.96963 1.03132 53 27.98098 1.01939 33 47.99247 1.00759 13 8.97020 1.03072 52 28.98155 1.01879 32 48.99304 1.00701 12 9.97076 1.03012 51 29.98213 1.01820 31 49.99362 1,00642 11 10.97133 1.02952 50 30.9S270 1.01761 30 50.99420 1.00583 10 11.97189 1.02S92 49 31.98327 1.01702 29 51.99478 1.00525 9 12.97246 1.02832 48 32.98384 1.01642 28 52.99536 1.00467 8 13.97302 1.02772 47 33.98441 1.01583 27 53.99594 1.00408 7 14.97359 1.02713 46 34.98499 1.01524 26 54.99652 1.00350 6 15.97416 1.02653 45 35.98556 1.01465 25 55.99710 1.00291 5 16.97472 1.02593 44 36.98613 1.01406 24 56.99768 1.00233 4 17.97529 1.02533 43 37.98671 1.01347 23 57.99826 1.00175 3 18.97586 1.02474 42 33.98728 1.01288 22 58.99884 1.00116 2 19.97643 1.02414 41 39.98786 1.01229 21 59.99942 1.00058 1 20.97700 1.02355 40 40.98843 1.01170 20 60 1.00000 1.00000 0 M. SCotanng. tagTang. M. S-LI. Cotang. Tang.. ot z450 50. = _0 5 = 1 =I 21 242 TABLE XVI. RISE PER MILE OF VARIOUS GRADES. TABLE XVIo RISE PER MILE OF VAR IOUJS GRADES. Grade Rise per Grade Rise per Grade Rise pe Grade eise per per Mile. Stper M Sile. per Mile. per Mile. Station. Station. Station. Station..01.523.41 21.648.81 42.768 1.21 63.888.02 1.056.42 22.176.82 43.296 1.22 64.416.03 1.584.43 22.704.83 43.824 1.23 64.944.04 2.112.44 23.232.84 44.352 1.24 65.472.05 2.640.45 23.760.85 44.880 1.25 66.000.06 3.163.46 24.288.86 45.408 1.26 66.528.07 3.696.47 24.816 o.87 45.936 1.27 67.056.08 4.224.48 25.344.8 46.464 1.28 67.584.09 4.752.49 25.872.89 46.992 1.29 63. 112.10 5.230.50 26.400.90 47.520 1.30 69.640.11 5.808.51 26.928.91 48.048 1.31 69.168.12 6.336.52 27.456.92 48.576 1.32 69.696.13 6.864.53 27.984.93 49.104 1.33 70.224.14 7.392.54 28.512.94 49.632 1.34 70.752.15 7.920 So 29.040.95 60 160 1.35 71.280.16 8.448.56 29.568.96 50.6883 1.36 71.808.17 8.976.57 30.096.97 51.216 1.37 72.336.18 9.504.58 30.624.98 51.744 1.33 72.864.19 10.032.59 31.152.99 52.272 1.39 73.392.20 10.560.60 31.680 1.00 52.800 1.40 73.920.21 11. 088.61 32.208 1.01 53.323 1.41 74.448.22 11.616.62 32.736.1.02 -53.856 1.42 74.976.23 12.144.63 33.264 1.03 54.384 1.43 750504.24 12.672.64 33.792 1.094 54.912 1.44 76.032.25 13.200.65 34.320 1.05 55.440 1.45 76.560.26 13.728.66 34.848 1.06 55.968 1.46 77.088.27 14.256.67 35.376 1.07 56.496 1.47 77.616.28 14.784.68 35.904 1.08 57.024 1.48 78.144.29 15.312.69 36.432 1.09 57.552 1.49 78.672.30 15.840.70 36.960 1.10 58.080 1.50 79.200.31 16.368.71 37.48 1.11 58.608 1.51 79.728.32 16.896.72 38.016 1.12 59.136 1.52 80.256.33 17.424.73 38.544 1.13 59.664 1.53 80.784.34 17.952.74 39.072 1.14 60.192 1.54 81.312.35 18.2480.75 39.600 1.15 60.720 1.55 81.840.36 19.008.76 40.128 1.16 61.248 1.56 82.368.37 19.536.77 40.656 1.17 61.776 1.57 82.896.38 20.064.78 41.184 1.18 62.304 1.58 83.424.39 20.592.79 41.712 1.19 62.832 1.69 83.952.40 21.120.80 42.240 1.20 63.360 1.60 84.480 TABLE XVI. RISE PER MILE OF VARIOUS GRADES. 243 Grade Grade Grade R p ade Grade Rise per Rise per e ise per Grade ise per per Mile per pe ile. pe ile Station. SMile. tation. tation. Station. le 1.61 85.003 1.81 95.568 2.10 110.880 4.10 216.480 1.62 85.536 1.82 96.096'2.20 116.160 4.20 221.760 1.63 86.064 1.83 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 1.65 87.120 1.85 97.680 2.50 132.000 4.50 237.600 1.66 87.648 1.86 98.208 2.60 137.280 4.60 242.880 1.67 88.176 1.87 93.736 2.70 142.560 4.70 248.160 1.63 88.704 1.88 99.264 2.80 147.840 4.80 253.440 1.69 89.232 1.89 99.792 2.90 153.120 4.90 258.720 1.70 89.760 1.90 100.320 3.00 158.400 5.00 264.000 1.71 90.288 1.91 100.848 3.10 163.680 5.10 269.280 1.72 90.816 1.92 101.376 3.20 168.960 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 5.40 285.120 1.75 92.400 1.95 102.960 3.50 184.800 5.50 290.400 1.76 92.928 1.96 103.488 3.60 190.080 5.60 295.680 1.77 93.456 1.97 104016 3.70 195.360 5.70 300.960 1.78 93.984 1.98 104.544 3.80 200.640 5.80 306.240 1.79 94.512 1.99 105.072 3.90 205.920 5.90 311.520 1.80 95.040 2.00 105.600 4.00 211.200 6.00 316.800 THE END.