T F UC-NRLF B 3 131 FIELD-BOOK FOR RAILROAD ENGINEERS CIRCULAR AND PARABOLIC CURVES, TURNOUTS, VERTICAL CURVES, LEVELLING, COMPUTING EARTH-WORK, TRANSITION CURVES ON NEW LINES AND APPLIED TO EXISTING LINES, TOGETHER WITH TABLES OF RADII, ORDINATES, LONG CHORDS, LOGARITHMS, LOGARITHMIC AND NATURAL SINES, TANGENTS, ETC., AND A METRIC CURVE TABLE BY JOHN B. HENCK, A. M. LATE PROFESSOR OF CIVIL ENGINEERING IN THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY NEW YORK AND LONDON D. APPLETON AND COMPANY 1912 COPYRIGHT, 1854, 1881, 1896, BY D. APPLETON AND COMPANY Copyright, 1909, by John B. Henck, Alice C. Henck, and Edward W. Henck Printed in the United States of America PREFACE. IN revising this work for the second time, the original purpose of making the volume compact, so as to be of con- venient size for use in the field, has been adhered to. It is designed to contain such formulae and tables as are mat- ters of constant reference in the field, to the exclusion of such as are rarely used. Subjects that, though important in themselves, require large space for satisfactory treat- ment, or are best learned, once for all, in the office or from competent superiors in the field, are also excluded. The size of the volume will therefore be found not materially increased by the changes and additions now made. Table I. has been enlarged. The first column contains the degrees of curves for evary two minutes up to 10, for every four minutes up to 20, and for every ten minutes afterward. The deflection angles will thus be always whole minutes. Ordinates for the quarter points, both for 100 feet chords and for 30 feet rails, are new features. The column of chord deflections has been omitted, being easily supplied by doubling the tangent deflections. All the data required in laying out a curve are found on one line. Some changes have been made in the other tables, and, in connection with the short metric curve table, a method is given of extending it by means of Tables I., II., III., and IV. The length of the arc of a curve is seldom required, since a curve is sufficiently described by giving the number and length of the chords and the deflection angle iii 387476 IV PREFACE. used. When the length of the arc is desired, it may be found by the method given in 13, which is exact for curves laid out with chords of any length. Matters formerly in an Appendix have been transferred to their proper places in the text. Some of them have been more fully developed, especially those relating to turnouts tangent to the main line. Transition curves have been more fully treated, and by methods entirely new. These curves have assumed great importance in view of the high speed of modern trains. The shock on entering and leaving a curve, and the dan- ger of derailment, may be greatly reduced by a transition curve, if carefully located and laid with rails that have been accurately curved. Both these essentials are secured by the methods here given. Certain portions of the dis- cussion involve the calculus, but the actual laying out of the curve merely requires the engineer to fix upon the length of curve he deems best, after which all the data for locating the curve, either by tangent offsets or by deflec- tion angles, are found on a single line of a short table. The method of applying a transition curve to an existing track is equally simple. The deflection angle of the exist- ing circular curve and its tangent point being known, and the length of the proposed transition curve chosen, a single line of a short table gives the data for locating the curve. In this table the ratio of the two radii concerned is taken as .9, but the general formulae are not confined to any par- ticular ratio. It will be seen that these methods do not require the central circular curve to be of some whole degree. The deflection angle D of the central curve may have any value we please a manifest advantage. For curving the rails accurately the ordinates at the centre and at the quarter points are required. These are readily found, especially when the curve is made to begin at a joint. The chapter on the common parabola is retained, be- cause, though this curve has met with but little acceptance on railroads, it is well adapted to vertical curves, and also PREFACE. V affords a simple means of laying out curves on common roads and pleasure drives, and such as are used in land- scape gardening. In the first preface to this work (1854) it was said: u Among the processes believed to be original may be speci- fied those in 41-48, on Compound Curves, in Chapter II., 011 Parabolic Curves, in 106-109 (now 149-151) on Vertical Curves, and in the article on Excavation and Embankment. It is 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." The claims here made have been properly recognized by some authors, while others have thought it sufficient to acknowledge the merits of the pro- cesses involved by simply adopting them. J. B. H. MOKTECITO, CAL., January, 1896. TABLE OF CONTENTS. CHAPTEK I. CIRCULAR CURVES. ARTICLE I. SIMPLE CURVES. SECT. PAGE 2. Definitions. Propositions relating to the circle .... 1 4. Angle of intersection and radius given, to lind 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 4 A. Method by Deflection Angles. 9. Kadius given, to find the deflection angle 5 10. Deflection angle given, to find the radius 5 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 6 13. Angle of intersection and deflection angle given, to find the length of the curve 6 14. Deflection angle given, to lay out a curve 7 16. To find a tangent at any station 9 B. Method by Tangent and Chord Deflections. 17. Definitions 9 18. Kadius given, to find the tangent deflection and chord deflection 9 19. Deflection angle given, to find the chord deflection . . .10 20. To find a tangent at any station 11 21. Chord deflection given, to lay out a curve . . - , .11 Viii TABLE OF CONTENTS. C. Method by Offsets from Tangent. SECT. PAGE 23. Deflection angle given, to find points on the curve by offsets from the tangent 13 D. Ordinates. 24. Definition 15 25. Deflection angle or radius given, to find ordinates . . .16 26. Approximate value for middle ordinate 18 27. Method of finding intermediate points on a curve approximately 18 E. Curving Hails. 29. Deflection angle or radius given, to find the ordinate for curv- ing rails .19 ARTICLE II. EEVERSED AND COMPOUND CURVES. 30. Definitions 19 31. Kadii or deflection angles given, to lay out a reversed, or com- pound curve 20 A. Reversed Curves. 32. Reversing point when the tangents are parallel . . . .20 33. To find the common radius when the tangents are parallel . 21 34. One radius given, to find the other when the tangents are par- allel 21 35. Chords given, to find the radii when the tangents are parallel . 22 36. Radii given, to find the chords when the tangents are parallel . 23 37. Common radius given, to run the curve when the tangents are not parallel 23 38. One radius given, to find the other when the tangents are not parallel ........... 24 39. To find the common radius when the tangents are not parallel. 25 40. Second method of finding the common radius when the tan- gents are not parallel .26 B. Compound Curves. 41. Common tangent point of the two arcs 27 42. To find a limit in one direction of each radius . . . .28 44. One radius given, to find the other 29 45. Second method of finding one radius when the other is given . 31 46. To find the two radii 32 47. To find the tangents of the two branches 34 43. Second method of finding the tangents of the two branches , 35 TABLE OF CONTENTS. ARTICLE III. TURNOUTS AND CROSSINGS. SECT. PAGE 49. Three cases of turnouts 36 First and Second Cases. 50. Definitions 37 A. Turnout from Straight Main Track. 51. Radius given, to find the frog angle and the position of the frog 37 52. Frog angle given, to find the radius and the position of the frog 38 53. To find mechanically the proper position of a given frog . . 39 54. To find the second radius of a turnout reversing opposite the frog 40 B. Crossings on Straight Lines. 55. Keferences to proper problems .42 56. Kadii given, to find the distance between switches . . .42 C. Turnout from Curves. 57. Frog angle given, to find the radius of the turnout and the posi- tion of the frog 43 58. To find mechanically the proper position of a given frog . . 47 59. Position of a frog given, to find the frog angle . . . .47 60. Radius of turnout given, to find the frog angle and the position of the frog 48 62. Turnout to reverse and become parallel to the main track . . 51 D. Crossings on Curves. 63. Eeferences to proper problems 52 64. Common radius given, to find the central angles and chords . 53 Third Case. Turnouts Tangent to Main Track. 65. Proper length of switch-rail 53 A. Turnout from Straight Lines. 66. Kadius given, to find the frog angle and the position of the frog 54 67. Frog angle given, to find the radius and the position of the frog 54 68. Locating a turnout curve , . , t , , t f 55 X TABLE OF CONTENTS. B. Crossings on Straight Lines. SECT. PAGE 70. References to proper problems ....... 51 C. Turnout from Curves. 71. Frog angle given, to find the radius of the turnout and the po- sition of the frog 56 72. Eadius of the turnout given, to find the frog angle and the po- sition of the frog . . . . . . . . . .59 74. Turnout to reverse and become parallel to the main track . . 62 75. Position of a frog given, to find the frog angle . . . .63 D. Crossings on Curves. 76. Eeferences to proper problems 63 E. Double Turnouts. 77. Those turning opposite ways and those turning the same way . 63 78. Finding certain chords, frog angles, and degrees of turnouts . 65 ARTICLE IV. MISCELLANEOUS PROBLEMS. 79. To find the radius of a curve to pass through a given point . 66 80. To find the tangent point of a curve to pass through a given point 67 81. To find the distance to the curve from any point on the tangent 67 82. Second method for passing a curve through a given point . . 67 83. To find the proper chord for any angle of deflection . . . 68 84. To find the radius when the distance from the intersection point to the curve is given 69 85. To find the external, that is, the distance from the intersection point to the curve when the radius is given . . . .70 86. To find the tangent point of a curve that shall pass through a given point 70 87. To find the radius of a curve without measuring angles . . 71 88. To find the tangent points of a curve without measuring angles 72 89. To find the angle of intersection and the tangent points when the point of intersection is inaccessible 73 90. To lay out a curve when obstructions occur . . . .76 91. To change the tangent point of a curve, so that it may pass through a given point 77 92. To change the radius of a curve, so that it may terminate in a tangent parallel to its present tangent 78 9, To find the radius of a curve on a/ track already laid , , .79 TABLE OF CONTENTS. XI SECT- 94. To draw a tangent to a given curve from a given point . 80 95. To flatten the extremities of a sharp curve ..... 80 96. To locate a curve without setting the instrument at the tangent point ............ 82 97. To measure the distance across a river ..... 84 98. To change a tangent point so that the tangent may pass through a given point .......... 86 99. To connect two curves by a common tangent . . . .87 CHAPTER II. PARABOLIC CURVES. ARTICLE I. LOCATING PARABOLIC CURVES. 100. Propositions relating to the parabola 89 101. To lay out a parabola by tangent deflections . . . .90 102. To lay out a parabola by middle ordinates 91 103. To draw a tangent to a parabola 92 105. To lay out a parabola by bisecting tangents 93 106. To lay out a parabola by intersections 93 107. Example illustrating preceding methods 94 ARTICLE II. RADIUS OF CURVATURE. 108. Definition 95 109. To find the radius of curvature at certain stations . . .96 110. Example in finding radius of curvature 99 111. Simplification when the tangents are equal 101 112. Length of parabolic arcs 102 CHAPTER III. TRANSITION CURVES. 113. Object of transition curves 104 ARTICLE I. THE CUBIC PARABOLA. 114. The equation of the cubic parabola 104 115. Two preliminary problems to be considered .... 106 116. Angle of intersection and radius of central curve given, to find the tangent 106 117. Angle of intersection and tangent given, to find the radius of the central curve , t ,,,,.,, 107 Xil TABLE OF CONTENTS. SECT. PAGE 118. Length of the abscissa ^ of the transition curve . . . 108 119. Formulae when the abscissa x l is expressed in rail lengths of 30 feet 108 120. Laying out the transition curve by offsets . . . . 110 121. Table A. Data for the method by offsets . . . .110 122. Example when R or D is given Ill 123. Example when Misgiven Ill 124. Laying out the transition curve by deflection angles . . .112 124. Table B. Data for the method by deflection angles . . .113 125. Example of the method by deflection angles .... 113 ARTICLE II. THE CUBIC PARABOLA APPLIED TO AN EXISTING CIRCULAR TRACK. 126. Necessary formulae deduced ........ 113 127. Table C. Data for applying the cubic parabola to an existing track 115 128. Example of cubic parabola applied to an existing track . . 116 129. Length of transition curve in terms of its chords . . . 116 ARTICLE III. CURVING THE KAILS. 131. Ordinates for curving the rails of a transition curve . . . 118 ARTICLE IV. COMPOUND TRANSITION CURVE. 132. Coordinates of stations on a compound transition curve. . . 119 133. Two preliminary problems to be considered .... 120 134. Angle of intersection and radius of central curve given, to find the tangent 120 135. Example when angle of intersection and radius of central curve are given 122 136. Angle of intersection and tangent given, to find the radius of the central curve 122 137. Advantage of beginning a transition curve at a joint . . . 123 CHAPTER IV. LEVELLING. ARTICLE I. HEIGHTS AND SLOPE STAKES. 138. Definitions 124 139, To find the difference of level of two points , t f .324 TABLE OF CONTENTS. PAGE SECT< 140. Datum plane ^ 141. To find the heights of the stations on a line . . 1< 142. Sights denominated plus and minus 12 ^ 143. Form of field notes ^ 144. To set slope stakes 15 ARTICLE II. CORRECTION FOR THE EARTH'S CURVATURE AND FOR KEFRACTION. -I 01 145. Earth's curvature 1 ^1 146. Refraction 147. To find the correction for curvature and refraction . . .1* ARTICLE III. VERTICAL CURVES. 148. Manner of designating grades . . v ^ 149. To find the grades for a vertical curve at whole stations . . le 151. To find the grades for a vertical curve at sub-stations . .135 ARTICLE IV. ELEVATION OF THE OUTER KAIL ON CURVES. 152. To find the proper elevation of the outer rail . . 1< 153. Coning of the wheels . 137 ARTICLE V. EASING GRADES ON CURVES. 154. Resistance on curves and grades compared 138 ARTICLE VI. EXPANSION OF RAILS. 155. Formula for the proper distance between rails .... 139 CHAPTER V. EARTH-WORK. ARTICLE I. PRISMOIDAL FORMULA. 156. Definition of a prismoid 140 157. To find the solidity of a prismoid 140 ARTICLE II. BORROW-PITS. 158. Manner of dividing the ground 141 159. To find the solidity of a vertical prism whose horizontal section is a triangle 142 Xiv TABLE OF CONTENTS. SECT. PAGE 160. To find the solidity of a vertical prism whose horizontal section is a parallelogram 143 161. To find the solidity of a number of adjacent prisms having the same horizontal section 144 ARTICLE III. EXCAVATION AND EMBANKMENT. A. Centre Heights alone given. 163. To find the solidity of one section 145 164. To find the solidity of any number of successive sections . . 146 B. Centre and Side Heights given. 165. Mode of dividing the ground 148 166. To find the solidity of one section 148 167. To find the solidity of any number of successive sections . . 152 169. To find the solidity when the section is partly in excavation and partly in embankment . . . . . .154 170. Beginning and end of an excavation 156 C. Ground very Irregular. 171. To find the solidity when the ground is very irregular . . 156 172. Usual modes of calculating excavation examined . . . 158 D. Correction in Excavation on Curves. 173. Nature of the correction 159 174. To find the correction in excavation on curves . . . .160 176. To find the correction when the section is partly in excavation ' and partly in embankment 161 178. Note on the computation of earthwork 163 TABLES. I. Eadii, Ordinates, Tangent Deflections, and Ordinates for Curving Rails 165 II. Long Chords 174 III. Tangents and Externals of a One-degree Curve . . . 176 IV. Corrections for Table III 183 V. Turnouts Tangent to a Straight Main Track. . . .183 VI. Length of Circular Arcs in Parts of Radius .... 184 TABLE OF CONTENTS. XV NO. PAGE VII. Elevation of the Outer Kail on Curves 184 VIII. Correction for the Earth's Curvature and for Refraction . 185 IX. Rise per Mile of Various Grades 186 X. Trigonometrical and Miscellaneous Formula. . . .188 XI. Heights by Aneroid Barometer 194 XII. Heights by Aneroid Barometer 201 XIII. Squares, Cubes, Square Eoots, Cube Roots, and Reciprocals 203 XIV. Logarithms of Numbers . 221 XV. Logarithmic Sines, Cosines, Tangents, and Cotangents . 237 XVI. Natural Sines and Cosines 286 XVII. Natural Tangents and Cotangents 295 XVIII. Comparison of French and English Weights and Measures. 308 XIX. Metric Curve Table 309 EXPLANATION OF SIGNS. THE sign + indicates that tfte 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 +- or : indicates that the first of two quantities be- tween which it is placed is to be divided by the second. The sign = indicates that the quantities between which it is placed are equal. The sign oo indicates that the difference of the two quantities between which it is placed is to be taken. The sign . *. stands for the word " hence " or " therefore." The ratio of one quantity to another may be regarded as the 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 pro- portion 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. FIELD-BOOK. 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 D E F B (fig. 1) represent a railroad curve, Fig. 1 2 CIRCULAR CURVES. uniting the straight Ime,^ G,A 'and JB II. The length of such a curve is measured by chords, each 1QO feet long.* Thus, if the chords AD, RE, ;.E*F. and F B jvre each 100 feet in length, the whole curve is said to be 400 feet long*. The straight lines Cr A and B H are always tangent to the curve at its extremities, which are called tangent points. If G A and B H are produced, until they meet in C, A C and B C are called the tangents of the curve. If A C is produced beyond G to K, the angle KGB, formed by one tangent with the other produced, is called the angle of intersec- tion, and shows the change of direction in passing from one tan- gent 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 G 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 <7, 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, CA B = $AOB. 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, DAE = %DOE. 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, A D = D E, and D A E = E A F. 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 G B = A B. 3. In order to unite two straight lines, as Cr A and B H, by a curve, the angle of intersection is measured, and then a radius for the curve may be assumed, and the tangents calculated, or the * 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 formulae deduced may be very readily modified to suit chords of any length. See also 13. SIMPLE CURVES. tangents may be assumed of a certain length, and the radius cal- culated. 4. Problem. Given the angle of intersection K C B = 1 (fig. 1) and the radius AO R^to find the tangent A C = T. Fig. 1. Solution. Draw C 0. Then in the right triangle A C we AC have (Tab. X. 3) = tan. A C, or, since A C = 1 1 ( 2, VI.) T - = tan. i 7; Example. Given /= 22 52', and R = 3000, to find T. Here R = 3000 i /= 11 26' ^=606.72 3.477121 tan. 9.305869 2.7829UO CIRCULAR CURVES. 5. Problem. Given the angle of intersection (fig. 1) and the tangent A C = T, to find the radius A = R. Solution. In the right triangle A C we have (Tab. X. 6) .-.R=T cot. i /. Example. Given /= 31 16' and T= 950, to find R. Here T-950 2.977724 i I =15 38' 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), ADEF B 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 deflec- tion angle is, therefore( 2, III.), half the degree of the curve. Thus, GAD or CBFis the deflection angle of the curve ADEFB, and is half A D or half FOB. Remark. The mode of designating curves by their degree, given above, is objected t^ by some, because when curves are laid out by chords shorter than 100 feet, as is usual on sharp curves, the degree of the curve is slightly increased, though its designation remains the same. If the arc of 100 feet is substituted for the chord of 100 feet in the definition, this difficulty vanishes ; but so many greater difficulties are introduced that the general adoption of this method is not probable. Moreover, when American engineers use the met- ric system, as possibly they are now doing on Mexican roads, both these methods are inapplicable. We might designate a curve by the length of its radius, for this fixes the curve, however laid out, and any units of length may be used ; but when the deflection angle D is even, R is generally fractional, which makes it inconvenient for ex- act definition. The length of the radius is also an indirect desig- nation, when curves are laid out by deflection angles. If the curve were designated by its deflection angle for a certain length of chord any length of chord and any units of length might be used, and th' curve be still definitely described. Thus we might say : " Curve tx the right, deflection angle for chords of 50 feet, 2 10'," or, " Curve to the left, deflection angle for chords of 20 metres, 1 35'." METHOD BY DEFLECTION ANGLES. 5 A. Method by Deflection Angles. 8. The usual method of laying out a curve on the ground is by means of deflection angles. 9. Problem. Given the radius AO = R (fig. 1), to find the deflection angle C B F = D. Solution. Draw OL perpendicular to B F. Then the angle BOL = ^BOF=D^ and B L = $ B F = 50. But in the right 71 T triangle B L we have (Tab. X. 1) sin. BOL= - ; B ' D- ~~ R' Example. Given R = 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 1 curve, and its deflection angle is 30'. 10. Problem. Given the deflection angle CBF= D (fig. 1), to find the radius AO = R. Solution. By the preceding section we have sin. D = , whence R R sin. D = 50 ; By this formula the radii in Table I. are calculated. Example. Given D 1, to find R. Here 50 1.698970 D=l sin. 8.241855 R = 2864.93 3.457115 11. Problem. Given the angle of intersection KCB 1 (fig. 1), and the tangent AC = T, to find the deflection angle CAD = D. 50 Solution. From 9 we have sin. D , and from 5, H 5 CIRCULAR CURVES. R =. Tcot. J /. Substituting this value of R in the first equa- _ 5Q . 50 tan. 1 1 . . sm. D = . Example. Given J= 21 and T- 424.8, to find D. Here 50 1.698970 ^ 1 = 10 30' tan. 9.267967 0.966937 T- 424.8 2.628185 D = 1 15' sin. 8.338752 12. Problem. Given the angle of intersection KGB 1 (fig. 1), and the deflection angle CAD=D, to find the tangent AC= T. Solution. From the preceding section we have sin. D =f Hence, T sin. D = 50 tan. * J; sin.D Example. Given /= 28 and Z) = 1, to find T. Here = 714.31. 13. Problem. Given the angle of intersection KCB = 1 (fig. 1), and the deflection angle CAD D, to find the length of the curve. Solution. By 2 the length of a curve is measured by chords of 100 feet applied around the curve. Now the first chord A D makes with the tangent A C an angle G A D = Z>, and each suc- ceeding chord D E, E F, &c. subtends at A an additional angle DAE,EAF, &c., each equal to D ; since each of these angles ( 2, IV.) is half of a central angle subtended by a chord o 100 feet. The angle CAB = \AOB = \I\$, 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 = /; If D is not contained an even number of times in J, the quo- tient above will still give the length of the curve. Thus, in METHOD BY DEFLECTION ANGLES. 7 figure 2, suppose D is contained 4| times in /. This shows that there will be four whole chords and f of a chord around the curve from A to B. The angle GAB, the fraction of Z>, is called a sub-deflection angle, and G B, the fraction of a chord, is called a sub-chord* The length of the curve thus found is not the actual length of the arc, but the length required in locating a curve. If the actual length of the arc is required, it may be found by means of Table VI. Example. Given/ 16 52' and D = 1 20', to find the length of the curve. Here n =^=r- = -^-577, = -^ TT = 6.325, that is, the curve JLf 1 &0 oU is 632.5 feet long. To find the arc itself in this example, we take from Table VI. the length to radius 1 of an arc of 16 52', since the central angle of the whole curve is equal to 1 ( 2, VI.), and multiply this length by the radius of the curve. Arc 10 = .1745329 " 6 = .1047198 " 50' =.0145444 " 2' = .0005818 " 16 52' = .2943789 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. Problem. Given the, deflection angle D, to lay out a curve from a given tangent point. 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 de- flection 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 DAE, EAF, &c., each equal to D, and make D E, E F, &c., each 100 feet, and the points E, F, &c., will be determined. The points * 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. CIRCULAR CURVES. Z>, E, F, &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 O B, the sub-deflection angle GAB must be the same part of D that G B is of a whole Fig. 2. chord ( 13). If there is a sub-chord at the beginning, the first stake on the curve will be at the end of the sub-chord, and the sub-deflection angle will be the same part of D that the sub-chord is of a whole chord. In laying out a curve there is an obvious advantage in having the several deflection angles whole minutes. When the deflection angle is assumed, whole minutes would naturally be chosen. But when D is found from / and I 7 by 11, it generally happens that D does not come out even minutes. In such cases, unless it is necessary that the curve should commence exactly at the assumed tangent point, it is better to take D to the nearest minute, and calculate T for / and this new value of D by 12. If, however, there is a sub-chord at the beginning of the curve, the sub-deflec- tion angle will generally contain seconds, although D contains none. In this case, set the vernier back the amount of the sub- deflection angle, so that, when this angle is turned off, the instru- ment will read zero. All the subsequent angles will then be whole minutes. METHOD BY DEFLECTION ANGLES. 9 15. It is often impossible to lay out the whole of a curve, with- out removing the instrument from its first position, either on ac- count of the great length of the curve, or because some obstruction to the sight may be met with. In this case, after determining as many stations as possible, and removing the instrument to the last of these stations, we ought to be able to find the tangent to the curve at this station ; for 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. Problem. 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 F L is the tangent required. Produce A F to K, and we have the angle KF L = A F C. But ( 2, II.) AFC -F AC. Therefore K F L = F A C. Now J'M G T is the sum of all the deflection angles laid off from the tangent at A, that is, in this case, F A C = 3 Z>, and the tangent F L is, there- fore, obtained by laying off from A ^produced an angle K F L 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 F B pro- duced an angle MBN = LBF=LFB, the total deflection from the preceding tangent F L. B. Method 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. Produce the tangent E A, and from B draw B G per- pendicular to A G. Produce also the chords A B and B C, and make the produced parts B II and C K of the same length as the chords. Draw C H and D K. B G is called the tangent deflec- tion, and C H or D K the chord deflection. 18. Problem. Given the radius AO = R (fig. 3\ to find the tangent deflection B G, and the chord deflection C H. Solution. The triangle C B H is similar to BOG', for the angle BOC = 180 - (0 B C + B C 0\ or, since BCO = ABO, BOC= 180 - (0 B C + A B 0) = CB H, and, as both the tri- angles are isosceles, the remaining angles are equal. The ho- 10 CIRCULAR CURVES. mologous sides are, therefore, proportional, that is, B : B C = B C : C ff, or, representing the chord by c and the chord deflection by d, R : c = c : d ; To find the tangent deflection, draw B M to the middle of C H, bisecting the angle C B H, and making BMC a right angle. Then the right triangles BMC and A G B are equal ; f or B C Fig. 3. AB, and the angle C B M = J OB H = % B C = i A B = .1 # ( 2, III.). Therefore B G = C M = % C H = % d, that is, the tangent deflection is half the chord deflection. 19. Problem. Given the deflection angle D of a curve, to find the chord deflection d. 2 Solution. By the preceding section we have d = -=, and by 50 8 10, R = - =:. Substituting this value of R in the first equa- sin. D tion, we find c 2 sin. D ~m-> 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. Z>, or for the tangent de- METHOD BY TANGENT AND CHORD DEFLECTIONS. H flection \ d = 100 sin. D. By this formula the tangent deflections in Table I. may be easily obtained from the table of natural sines. The length of the curve may be found by first finding D ( 9 or 11), and then proceeding as in 13. 20. Problem. To draw a tangent to the curve at any station, as B (fig. 3). Solution. Bisect the chord deflection H C of the next station in M. A line drawn through B and M will be the tangent re- quired; for it has been proved ( 18) that the angle C B M is in this case equal to i B (7, and B M is consequently ( 2, III.) a tangent at B. If B is at the end of the curve, the tangent 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 H M = d, B M will be the direction of the required tangent.* 21. Problem. Given the chord deflection d, to lay out a curve from a given tangent point. Solution. Let A (fig. 3) be the given tangent point, and sup- pose d has been calculated for a chord of 100 feet. Stretch a chain of 100 feet from A to G on the tangent E A produced, and mark the point G. Swing the chain round towards A B, keeping the end at A fixed, until B G is equal to the tangent deflection i d, and B will be the first station on the curve. Stretch the chain from B to H on A B produced, and having marked this point, swing the chain round, until H C is equal to the chord deflection d. C is the second station on the curve. Continue to lay off the chord deflec- tion from the preceding chord produced, until the curve is finished. Should the curve begin or end with a sub-chord, denote, as be- fore, the whole chord by c, the sub-chord by c', the tangent deflec- tion for c by d, and that for c' by d 1 . Then (18) i d = = aJ C and i d' = ^. Therefore d : -J- d' = c* : c' 2 , &J& (c' \ 2 ) * The distance B M is not exactly equal to the chord, but the error aris- ing from taking it equal is too small to be regarded in any curves but those of very small radius. If necessary, the true length of B M may be calcu- lated ; f or B M - V ITTT 2 - H Jkf 2 . 12 CIRCULAR CURVES. If the curve begins with a sub-chord, produce the tangenfc a distance c', and from its extremity lay off a distance * d' for a point on the curve. But as we need a whole chord in order to produce it for continuing the curve, measure back on the tangent a distance c c' = c" and lay off the deflection proper to c", but in an opposite direction to d'. This will give a point on the curve supposed to be run back to the preceding whole station. The line joining these two points on the curve will now be a whole chord, and can be produced in the usual way. If the curve ends in a sub-chord, as D F (fig. 3), find the tangent DL ( 20), and lay off from it the proper tangent deflection LF for the sub- chord, found as above. Fig. 3. Example. Given the intersection angle / between two tangents equal to 16 30', and R 1250, to find T, d, and the length of the curve in stations. Here (4) T= R tan. * 7 = 1250 tan. 8M5' = 181.24; ( 9) sin. D = = ^ = .04 = nat. sin. 2 17*' ; ; ' - 495 ' -S60 77 19 ry r-/ rf -bl). METHOD BY OFFSETS FROM TANGENT. 13 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 = i d = 4 feet, that the chord deflection HC or K D = 8 feet, and that the curve is 360 feet long. The three whole stations B, C, and D having been found, and the tan- gent D L drawn, the tangent deflection for the sub-chord of 60 feet will be, as shown above, $d' = 4 ( j = 4 x .6 2 = 4 x .36 = 1.44. L F = 1.44 feet being laid off from D L, the point F will, if the work is correct, fall upon the second tangent point. A tan- gent at F may be found ( 20) by producing D F to P, making FP = D F = 60 feet, and laying off PN = 1.44 feet. F N will be the direction of the required tangent, which should, of course, coincide with the given tangent. Curves may be laid out with accuracy by tangent and chord de- flections, 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. On sharp curves, such as sometimes occur on street railroads, where the chords may not exceed 10 feet, a fine cord may be used for producing the lines. The radius of a curve to unite two given straight lines may also be found without an instrument by 87, or, having assumed a ra- dius, the tangent points may be found by 88. C. Method ly Offsets from Tangent. 22. By this method points on a curve such as C (fig. 3a) are de- termined by measuring from the tangent point certain distances along the tangent, such as A B, and offsets at right angles to the tangent, such as B C. 23. Problem. Given D, the deflection angle of a curve for a chord c, to find A B = a (fig. 3d) and B C = b for a point C on the curve, distant from the tangent point a certain number of stations, whole or fractional, denoted ~by the letter n. Solution. The angle B A C = n D, and the central angle A C = 2 n D. Draw C D parallel to the tangent. Then, in the triangle CD 0, we have a = CD = C sin. DOC-R sin. 2 n D. 14 CIRCULAR CURVES. Substituting for R its value . , | c sin. 2 n D sin. D To find b, we have b = C = AO-DO = R-R cos. 2 n D, or (Tab. X., 23) b = R R (1-2 sin. 2 rc D} = 2R sin 2 n D. Substituting for R its value . , c sin. 2 71 D In computing these values for successive points, the logarithms ( remain constant, which facilitates the work. : and of OI ; f\ MIIJI^A v/j. ^. . sm. D sin. D The position of the stakes is best fixed by measuring the successive chords, instead of depending on the right angle at B. If the offsets from the original tangent become inconveniently long, a new tangent is readily found. Thus a tangent T C at C is determined by measuring from Fig. 3a. A a distance A T = R tan. n D = A T R i c tan. n D m , . , - : ^=r . T C should, of course, sm. D prove equal to A T. Since n may be a fraction or a mixed number, as well as a whole number, n c may represent any sub- chord, such as would generally oc- cur at the beginning of a curve. The points on the curve determined by the formulas for a and. b will therefore be the regular stations continued from the straight line. In laying out a whole curve AEB (fig. 3&) by this method a tangent D G at the middle point of the curve is found by computing the equal distances A Z), D E, E G, and G B by the formula AD-DE-EG-GB- R tan. J /. As a check, the distance C E may be found from the triangle C E D. For C E D E tan. /. Substituting for D E its value R tan. i /, we have C E = R tan. | /tan. /. The station of the tangent point A being known, and the length OBDINATES. 15 of the curve having been found ( 13), the stations of E and B are readily found. Then, by the process just explained, find the off- sets from the tangent A D to the regular stations on, say, one Fig. 36. quarter of the curve. By the same process, beginning at the known station at E, find offsets to the regular stations on the curve. In like manner, offsets from the tangents E G and B (r will complete the curve, the regular stations being kept through- out. Curves may be laid out with great accuracy by this method. D. Ordinates. 24. The preceding methods of laying out curves determine points 100 feet distant from each other. These points are usually sufficient for grading a road ; but when the track is laid, it is de- sirable to have intermediate points on the curve accurately deter- "hined. For this purpose the chord of 100 feet is divided into a 16 CIRCULAR CURVES. certain number of equal parts, and the perpendicular distances from the points of division to the curve are calculated. These distances are called ordinates. 25. Problem. Given the deflection angle D or the radius R of a curve, to find the ordinates for any chord. Solution. I. To find the middle ordinate. Let AE B (fig. 4) be a portion of a curve, subtended by a chord A B, which may be G denoted by c. Draw the middle ordinate E Z>, and denote it by m. Produce E D to the centre F, and join A F and A E. Then (Tab. X. 3) =p^ = tan. E A D, or E D = A D tan. E A D. But, A. JLJ since the angle E A D is measured by half the arc B E, or by half the equal arc A E, we have E AD = %AFE. Therefore E D = ADtsin.lt A FE, or When c = 100, A F E = D ( 7), and m = 50 tan. $ D, whence m may be obtained from the table of natural tangents, by divid- ing 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 ORDINATES. 17 triangle A D F we have D F = \/A F* - A D* - Then m E F DF=R D F, or m = R II. To find any other ordinate, as R jV, at a distance D N = b .from the centre of the chord. Produce R N until it meets the diameter parallel to A B in G, and join R F. Then R G = - t>\ and RN=RG-NG=RG- D F. Substituting the value of R G and that of D F found above, we have The other ordinates may also be found from the middle ordi- nate 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 arc 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 suppo- sitions 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 R, A , and A T, we have the angle JRAN = IE AD, since R B is considered equal to J E B. But E A D = \AFE. Therefore, R A N = f A F E. In the same way we should find SA = J A F E, and TAP=\A F E. We have then for the ordinates, RN=A N tan. R A N = | c tan. f A F E, I c tan. ^AFE. But, by the second supposition, tan. f A F E = f tan. i A FE, tan. J A FE = | tan. $A F E, and tan. | A FE = ta,u.$AFE. Substituting these values, and recollecting that i c tan. | A F E m, we have = x ictan. \ S = -j x i c tan. A FE = -^ m, 7 7 jT-P = ^-5 x ^ c tan. ^ ^4. F E = 77; w. lo lo In general, if the number of divisions of the chord on each side 3 }g CIRCULAR CURVES. of the centre is represented by n, we should find for the respect- (n + 1) (n 1) m ive ordinates, beginning nearest the centre, - -^ , (n + 2) (n - 2) m (n + 3)(n-3)m ^ M* tf These values of the ordinates are precisely what we should ob- tain if we regarded A E B as the arc of a parabola ; for in this v case, as we shall see later, the offsets from a tangent at E to R, S, and T would be ^ m, ^ m, and ^ m. Subtracting these dis- lo lo lo tances from m, we should get the results given above. Example. Find the ordinates of an 8 curve to a chord of 100 feet. Here m = 50 tan. 2 = 1.746, RN = ~.m = 1.637, S = | w = 1.310, and TP = ^ m = 0.764. 26. An approximate value of m also may be obtained from the formula m = R \/ R* i c . This is done by adding to the c 4 quantity under the radical the very small fraction . ^ , making c 2 it a perfect square, the root of which will be R ^-~. We have, then, m - R - R - ~ ; 27. From this value of m we see that the middle ordinates of any two chords in the same curve are to each other nearly as the squares of the chords. If, then, A E (fig. 4) be considered equal to | A B, its middle ordinate C H = J 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 J5, 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 C H = % 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 ob- tained. E. Curving Rails. 28. The rails of a curve are usually curved before they are laid. To do this properly, it is necessary to know the middle ordinate REVERSED AND COMPOUND CURVES. 19 of the curve for a chord of the length of a rail, and the ordinates at the quarter points. 29. Problem. Given the radius or deflection angle of a curve, to find the middle ordinate for curving a rail of given length. Solution. Denote the length of the rail by Z, and we have ( 25) the exact formula m = R \/jR* / 2 , and ( 26) the approxi- mate formula This formula is always near enough for chords of the length of a 50 rail. If we substitute for R its value ( 10) R = ^ ^ , we have, Example. In a 1 curve find the ordinate for a rail 30 feet in length. For a rail 30 feet in length Z 2 = 225, and, consequently, m = fl.25 sin. D. This gives for a 1 curve, m = .02. The corresponding ordinate for a curve of any other degree may be found approximately by multiplying the ordinate for a 1 curve by the number expressing the degree of the curve. The ordinates from the chord at the quarter points are ( 25) each f m. In Table I. are given the values of m and m for a rail of 30 feet. From these ordinates the ordinates for a rail of any other length are ob- tained by simply multiplying by the square of the ratio of its length to 30. Thus for a rail of 27 feet this ratio is .9, the square of which is .81, and the ordinates for, say, a 4 curve, are .079 x .81 = .064 and .059 x .81 = .048. ARTICLE II. REVERSED AND COMPOUND CURVES. 30. Two curves often succeed each other having a common tan- gent 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, they have different radii, and form a com- pound curve. Thus ABC (fig. 5) is a reversed curve, and ABB a compound curve. 20 CIRCULAR CURVES. 81. Problem. To lay out a reversed or a compound curve, when the radii or deflection angles and the tangent points are known. Solution. Lay out the first portion of the curve from A to B (fig. 5), by one of the usual methods. Find B F, the tangent to A B at the point B ( 16 or 20). Then B F will be the tangent also of the second portion B C of a reversed, or B D of a com- pound curve, and from this tangent either of these portions may be laid off in the usual manner. A. Reversed Curves. 32. Theorem. The reversing point of a reversed curve be- tween parallel tangents is in the line joining the tangent points. Demonstration. Let A C B (fig. 6) be a reversed curve, uniting the parallel tangents H A and B K, having its radii equal or un- equal, and reversing at C. If now the chords A C and C B are drawn, we have to prove that these chords are in the same straight line. The radii E C and G 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 H A and B K, are parallel. Therefore, the angle A E C = C F B, and, consequently, EGA, the half supplement of A E C\ is equal to F C B, the half supplement of CFB; but these angles cannot be equal, unless A C and C B are in the same straight line. REVERSED CURVES. 21 33. Problem. Given the perpendicular distance between two parallel tangents B D = b (fig. 6), and the distance between the two tangent points A B = a, to determine the reversing point C and the common radius EC=CF=R of a reversed curve uniting the tangents H A and B K. Solution. Let A C B be the required curve. Since the radii are equal, and the angle A E C = B F (7, the triangles AEG and B F C are equal, and AC=CB = $a. The reversing point C is, therefore, the middle point of A B. To find R, draw E G perpendicular to A C. Then the right triangles AEG and BAD are similar, since ( 2, III.) the angle BAD = \AEC = AEG. Therefore AE : AG = AB:BD, or R\a a:b; Corollary. If R and b are given, to find a, the equation R = a* j-j- gives a 2 = 4 R b ; 4 HT . . a = 2\/R~b. Examples. Given b = 12, and a = 200, to determine R. Here _ 200* _ 10000 _ 1 TH To OOO'j. Given R = 675, and b = 12, to find a. Here a 2^/675 x 12 = 2^8100 = 2 x 90 = 180. 34. Problem. Given the perpendicular distance between two parallel tangents B D = b (fig. 7), the distance between the two CIBCULAR CURVES. tangent points A B = a. and the first radius E C = R of a re- versed curve uniting the tangents H A and B , to find the chords A C = a' and C B = a", and the second radius OF R'. Solution. Draw the perpendiculars EG and F L. Then the right triangles A B D and E A G are similar, since the angle BAD = \AEG-AEG. Therefore AB\BD^EA\ AG, or a : b = R : % a' ; 2Rb Since a' and a" are ( 32) parts of a, we have ft^~ a" = a a'. To find R' the similar triangles ABD and FB L give AB-.BD = FB\ BL, or a: b = R' : | a"; a a" Example. Given b = 8, a = 160, and R = 900, to find a', a", n v QOO v ft - = 90, a" = 160 - 90 = 70, and and R. Here a' = 35. Corollary 1. If 6, a', and a" are given, to find a, . and R', we have ( 34) , ,, ^ aa' , a a" a = a + a -, t = ^-r ; H REVERSED CURVES. Example. Given 6 = 8, a' = 90, and a" = 70, to find 160 x 70 a, R, 1fiO v QO Here a = 90 + 70 = 160, R = " = 900, and . 2x8 2x8 . = 700. 36. Corollary 2. If ^, -R', and 6 are given, to find a, a', and a", we have ( 35), R + R' = a o ft a< = - o ^ ^ jfV Therefore a 2 = 2 6 (JK + J2') ; Jg^~ .-.= Having found a, we have ( 34) Example. Given /jJ = 900, R' = 700, and b = 8, to find and a". Here a = \/2 x 8(900 + 700)'= V16 x 1600 2 x 900 x 8 2 x 700 x 8 a' = -- = 90,anda = -- - -70. a, a', 160, 37. Problem. Given the angle AKB-=.K, which shows the change of direction of two tangents HA and B K (fig. 8\ to -N B Fig. 8. unite these tangents by a reversed curve of given common radius R, startina from a given tangent point A. Solution. With the given radius run the curve to the point D, where the tangent D jV becomes parallel to B K. The point D is found thus. Since the angle N B K, which is double the angle 24 CIRCULAR CURVES. H A D ( 2, II.), is to be made equal to A KB = K, lay off from H A the angle HA D = \K. Measure in the direction thus found the chord AD = 2R sin. | K. This will be shown ( 83) to be the length of the chord for a deflection angle i K. Having found the point D, measure the perpendicular distance D M b between the parallel tangents. The distance BD = 2DC=a may then be obtained from the formula ( 33, Cor.) 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 obtained ; for sin. B D N = sin. D B M = - , or sin.BDN=-. a 38. Problem. Given the line A B a (fig. 9\ which joins the fixed tangent points A and B, the angles ffAB = A and A B L = B, and the first radius A E R, to find the second radius B F = R' of a reversed curve to unite the tangents H' A and B K. Fig. 9. First Solution. With the given radius run the curve to the point D, where the tangent D N becomes parallel to B K. The point D is found thus. Since the angle U G N, which is double KEVERSED CURVES. 25 H A D ( 2, II.), is equal to Av*B, lay off from HA the angle H A D \(Acr> B], and measure in this direction the chord AD = 2 R sin. \(Av*B) ( 83). Setting the instrument at /), run the curve to the reversing point C in the line from D to B ( 32), and measure D C and C B. Then the similar triangles DEC and B F C give D C : D E = CB-.BF, or DC:R= CB:R'\ W .'.R'=^x R. Second Solution. By this method the second radius may be found by calculation alone. The figure being drawn as above, we have, in the triangle A B D, A B = a, A D = 2 R sin. %(A B}^ and the included angle D A B = HA B H A D = A - % (A B} = 4 (A + B}. Find in this triangle (Tab. X. 14 and 12) B D and the angle A B D. Find also the angle D B L = B + ABD. Then the chord CB = 2 R' sin. 4 BFC = 2 R' sin. D B L, and the chord DC = 2R sin. | D E C = 2 R sin. Z> B L ( 83). But CB = BD-DC; whence 2 .#' sin. D B L = B D - 2 R sin. When the point D falls on the other side of J., that is, when the angle B is greater than A, the solution is the same, except that the angle D A B is then 180 - i (A + 5), and the angle DBL = B- ABD. 39. Problem. Given the length of the common tangent D 6r = a, and the angles of intersection I and I' (fig. 10\ to deter- mine the common radius C E = C F = R of a reversed curve to unite the tangents HA and B L. Solution. By 4 we have D C = R tan. } 7, and C G = R tan. /', whence R (tan. 4 / + tan. J /') = D C + (7 tf = a, or This formula may be adapted to calculation by logarithms ; for we have (Tab. X. 35) tan. 4 / + tan. 4 I = sm - ^ J + ^) Substi- cos. 4 /cos. 4 /' tilting this value, we get _ a cos. 4 /cos. 4 /' sin. 4 (/ + /') 26 CIRCULAR CURVES. The tangent points A and B are obtained by measuring from D a distance A D = R tan. /, and from G a distance B O = R tan. | /'. Example. Given a = 600, 1 = 12, and /' = 8, to find R. Here a = 600 2.778151 7=6 cos. 9.997614 iJT =4 cos. 9.998941 2.774706 i (/+/') = 10 sin. 9.239670 R = 3427.96 3.535036 40. Problem. Given the line A B = a (fig. JO), which joins the fixed tangent points A and B, the angle D A B = A, and the angle A B Gr = B, to find the common radius E C ' = C JP R of a reversed curve to unite the tangents HA and B L. Fig. 10. Solution. Find first the auxiliary angle A KE = B K F, w/iich may be denoted by K. For this purpose the triangle A E K gives AE\EK = sin. K : sin. E A K. Therefore E JTsin. K = A E sin. E A K = R cos. A, since E A K = 90 A. In like manner, the triangle BF If gives FKsiu. K= HI 1 sin. FBK= R cos. B. Adding these equations, we have (E K + F K) sin. K = R (cos. A + cos. B\ or, since EK+FK=2R, 2Rsin.K = COMPOUND CURVES. 27 R (cos. A + cos. B). Therefore, sin. K = i (cos. A + cos. B). For calculation by logarithms, this becomes (Tab. X. 28) jgf" sin. K = cos. | (A + B) cos. i ( J. B}. Having found K, we have the angle AEK=E = 180 K E AK= 180 K (90 -4) = 90 + A K, and the angle B FK= F= 180 - K- FBK= 180 - K- (90 - B} = 90 + B K. Moreover, the triangle A E K gives A E : A K = sin. K : sin. E, or 72 sin. E = A TTsin. 7T, and the triangle B F K gives B F \BK sin. 7T : sin. F, or 72 sin. F = B K sin. 7T. Adding these equations, we have R (sin. 7? + sin. F) = (A K + B K) sin. K=a sin. 7T. Substituting for sin. E + sin. T^ 7 its value 2 sin. | (E + jP) cos. * (E - F) (Tab. X. 26), we have 2 72 sin. ^(E + F) cos. ^ (E F) a sin. 7T. Therefore 72 = _ * T _ s -= ^-. Finally, substituting for E its sin. i(E+ F) cos. 1 (E - F) value 90 + A K, and for F its value 90 4- B K, we get %(E + F) = 90 [K %(A + B)], and \ (E F} = | (A B)\ whence __ i a sin. K ~ cos. [JST- i (A + )] cos. $(A B)' Example. Given a = 1500, A = 18, and B = 6, to find 72. Here * (A + B) 12 cos. 9.990404 i U - 5) = 6 cos. 9.997614 7T = 76 36' 10' ' sin. 9.988018 i a = 750 2^875061 2.863079 - (A + B} = 64 36' 10" cos. 9.632347 i (A - B) = 6 cos. 9.997614 9.629961 ^ = 1710.48 3.233118 B. Compound Curves. 41. Theorem. If one branch of a compound curve be pro- duced, until the tangent at its extremity is parallel to the tangent at the extremity of the second branch, the common tangent point of the two arcs is in the straight line produced, which passes through the tangent points of these parallel tangents. 28 CIRCULAR CURVES. Demonstration. Let ACB (fig. 11) be a compound curve, uniting the tangents H A and B K. The radii C E and C F, be- ing perpendicular to the common tangent at C ( 2, I.), are in the Fig. 11. same straight line. Continue the curve A G to D, where its tan- gent D becomes parallel to B K, and consequently the radius D E parallel to B F. Then if the chords CD and G B be drawn, we have the angle CED=CFB\ whence E C D, the half- supplement of C E D, is equal to F C B, the half-supplement of C FB. But E C D cannot be equal to F C B, unless CD coin- cides with CB. Therefore the line BD produced passes through the common tangent point C. 42. Problem. To find a limit in one direction of each radius of a compound curve. Solution. Let A I and B I (fig. 11) be the tangents of the curve. Through the intersection point /, draw I M bisecting the COMPOUND CURVES. 29 angle A 1 B. Draw A L and B M perpendicular respectively to A /and B I, meeting I M in L and M. Then the radius of the branch commencing on the shorter tangent A 1 must be less than A *L, and the radius of the branch commencing on the longer tan- gent B I must be greater than B M. For suppose the shorter radius to be made equal to A L, and make I N ' = A 7, and join L N. Then the equal triangles AIL and NIL give A L = L N; so that the curve, if continued, will pass through N, where its tangent will coincide with IN. Then ( 41) the common tan- gent point would be the intersection of the straight line through B and N with 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 B I. In this case the ex- tremity of the curve will fall outside the tangent B I in 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 com- mon tangent point can be found when this radius is taken equal to A L or greater than A L, no compound curve is possible. This radius must, therefore, be less than A L. In a similar manner it might be shown, that the radius of the other branch of the curve must be greater than B M. If we suppose the tangents A I and B I and the intersection angle I to be known, we have ( 5) A L A I cot. -$ /, and B M = B I cot. | /. These values are, therefore, the limits of the radii in one direction. 43. If nothing were given but the position of the tangents and the tangent points, it is evident that an indefinite number of dif- ferent compound curves might connect the tangent points; for the shorter radius might be taken of any length less than the limit found above, and a corresponding value for the greater could be found. Some other condition must, therefore, be introduced, as is done in the following problems. 44. Problem, Given the line A B = a (fig. 11), which joins the fixed tangent points A and B, the angle B A 1= A, the angle A B 1= B, and the first radius A E = R, to find the second ra- dius B F = R' of a compound curve to unite the tangents HA and B K. Solution. Suppose the first curve to be run with the given radius from A to Z), where its tangent D becomes parallel to 30 CIRCULAR CURVES. B /, and the angle I A D = | ( A + B). Then ( 41) the common tangent point C is in the line B D produced, and the chord C B = C D + B D. Now in the triangle A B D we have A B =^ a, Fig. 11. > = 2 R sin. $(A + B) ( 83), and the included angle D A B = 7 J. jB / A D = A (A + .5) = \(A B). Find in this tri- angle (Tab. X. 14 and 12) the angle A B D and the side B D. Find also the angle CBI=B ABD. Then ( 83) the chord C B = 2R' sin. C B I, and the chord C D 2 R sin. CDO = 2R sin. C B I. Substituting these values otCB and CD in the equation found above, C B = CD + B D, we have 2 R' sin. CBI=2Rsin. C B I + BD "2 sin. When the angle B is greater than A, that is, when the greater radius is given, the solution is the same, except that the angle > A), and C B I is found by subtracting the sup- COMPOUND CURVES. 31 plement of ABD from B. We shall also find CB = CD 7? T) B Z>, and consequently R ' = R . & sin. If more convenient, the point D may be determined in the field, by laying off the angle I A D = % (A + B), and measuring the distance A D = 2 R sin. (A + B). BD and C B I may, then be measured, instead of being calculated as above. Example. Given a = 950, A =. 8, B = 7, and ft = 3000, to find R'. Here A D - 2 x 3000 sin. i (8 + 7) = 783.16, and D A B = i (8 7) = 30'. Then to find A B D we have A B - A D = 166.84 2.222300 i (A D B + A B D) = 89 45' tan. 2.360180 4.582480 A B + A D = 1733.16 3.238839 i (A D B - A B D) - 87 24' 17" tan. 1.343641 . . A B D = 2 20' 43" Next, to find B D, AD = 783.16 2.893849 D A B = 30' sin. 7.940842 0.834691 A B D = 2 20' 43" sin. 8.611948 B D = 167.01 2.222743 B-ABD=CBI=4W 17" sin. 8.909292 2 (R ' - R) = 2058.03 3.313451 R' -.#=: 1029.01 R 1 = 3000 + 1029.01 = 4029.01 To find the central angle of each branch, we have CFB = 2 CBI= 9 18' 34", which is the central angle of the second branch; and A EC = AED - CED = A + B - 2CBI = 5 41' 26", which is the central angle of the first branch. 45. Problem. Given (fig. 11) the tangents AI= T,BI = T', the angle of intersection /, and the first radius A E = R, to find the second radius B F = R' . Solution. Suppose the first curve to be run with the given ra- dius from A to D, where its tangent D becomes parallel to B I. Through D draw D P parallel to A J, and we have / P = D = 32 CIRCULAR CURVES. AO = Rtfm.iI (4). Then in the triangle DPB we have DP= 10=. AI-AO= T- 12 tan.*/, BP=B1-1P= T 1 - R tan. | /, and the included angle DPB=AIB= 180 - /. Find in this triangle the angle C B I, and the side B D. The remainder of the solution is the same as in g 44- The determina- tion of the point D in the field is also the same, the angle IAD being here = | /. When B is greater than A, that is, when the greater radius is given, the solution is the same, except that DP = / T, and BP= R tan. j- / T'. Example. Given T= 447.32, T' = 510.84, 7- 15, and R = 3000, to find ^'. Here ^ tan. | / = 3000 tan. 71 = 394.96, D P = 447.32 - 394.96 = 52.36, J5 P = 510.84 - 394.96 = 115.88, and DPD = 180 - 15 = 165. Then (Tab. X. 14 and 12) BP-DP- 63.52 1.802910 i (B D P + PB D) = 7 30' tan. 9.119429 0.922339 B P + D P = 168.24 2.225929 - PB D) = 2 50' 44" tan. 8.696410 .-.PBD= CB 7=4 39' 16" Next, to find 5 Z>, DP =52.36 1.71900C DPB- 165 sin. 9.412996 1.13199C = 4 39' 16" sin. 8.90926^ B D = 167.005 2.222730 The tangents in this example were calculated from the example in 44. The values ofCBl and B D here found differ slightly from those obtained before. In general, the triangle D B P is oi better form for accurate calculation than the triangle A D B. 46. If no circumstance determines either of the radii, the con- dition maybe introduced, that the common tangent shall be para> lei to the line joining the tangent points. Problem. Given the line AB a (fig. 10), which unites th^ fixed tangent points A and B, the angle I A B = A, and the angle ABIB,to find the radii A E = R and B F = R' of a com- pound curve, having the common tangent D G parallel to A B. COMPOUND CURVES. 33 Solution. Let A C and B C be the two branches of the required curve, and draw the chords A C and B C. These chords bisect Fig. 12. the angles A and B ; for the angle and the angle GBC = ^DGI=^ABL Then in the triangle A CB we have A C : A B = sin. ABC: sin. A C B. But A CB = 180 -(CAB+CBA) = 180 - 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 C B = sin. | (A + B). Therefore A C : a = a sin. $ B sin. B : sin. | (A + ^), or A C manner we should find B C sin. (A + ' In a similar ( 82) R = C -, , and R = sin. -J A i of A C and J? (7 just found. . sin. t B C . Now we have + 1?) , or, substituting the values -,R = $ a sin. | J. sin. ^ B sin. -J ( J. 34 CIRCULAR CURVES. Example. Given a = 950, A = 8, and B = 7, to find R and R'. Here i a = 475 2.676694 i^ = 3 30' sin. 8.785675 1.462369 i 4 = 4 sin. 8.843585 + -B) = 7 30' sin. 9.115698 7.959283 ^ = 3184.83 3.503086 Transposing these same logarithms according to the formula for R we have i a = 475 2.676694 M=4 sin. 8.843585 1.520279 1 5 = 3 30' sin. 8.785675 | (A + B) = 7 30' sin. 9.115698 7.901373 R = 4158.21 3.618906 47. Problem. Given the line AB a (fig. 12\ which unites the fixed tangent points A and B, and the tangents AI=T and B I = T', to find the tangents A D = x and B G = y of the two branches of a compound curve, having its common tangent I) Q parallel to A B. Solution. Since D C = A D = x, and C & = B G = y, we have D G = x + y. Then the similar triangles IDG and I A B give ID : I A = D G : A B, or T x : T = x + y : a. Therefore a T ax = Tx + Ty (1). Also A D : A I = B G : B 7, or x\Ty\T. Therefore T y = T 1 x (2). Substituting in (1) the value of Ty in (2), we have a T ax = Tx + T' x, or ax + Tx + T' x = a T-, aT . * .X = a+T +T" T' x and, since from (2), y ~- , - y ~ a + T + T' ' COMPOUND CURVES. 35 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 I A B = A and A B I = B have been measured or calculated, we have ( 5) R = x cot. -J- A, and R' = y cot. B. Substituting the values of a; an 1 y found above, we have R = a + , and R = a+ T + T > Example. Given a = 500, T = 250, and T = 290, to find x and y. Here a + T + T' 500 4- 250 4- 290 - 1040 ; whence x = 500 x 250 -*- 1040 = 120.19, and y = 500 x 290 -f- 1040 = 139.42. 48. Problem. Given the tangents A I = T< B I = T' , and the angle of intersection I, to unite, the tangent points A and B Fig. 13. * 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 formulae. In these cases A + B = 90, and the values of R and R' may be reduced to R = aT aT' - and R' = a + T'-T tion, or they may be readily calculated. These values admit of an easy construe- 36 CIRCULAR CURVES. (fig. 13) by a compound curve, on condition that the two branches shall have their angles of intersection IDG and I G D equal. Solution. Since IDG = IGD = $I, we have I D - I G. Represent the line ID = I G by x. Then if the perpendicular IH be let fall from J, we have (Tab. X. 11) DH=ID cos. IDG = x cos. i J, and D G = 2 x cos. i I. But D G = D C + C G = AD+BG=T x + T' x = T + T' 2 x. Therefore 2xcos.$I= T + T' 2x, or 2x + 2xcos.il= T + T \ whence or (T b x ^ ' _ 1 + cos. " cos. 2 */ ' The tangents AD = T x &ud 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 an- swers very well, when the given tangents are nearly equal ; but in general the preceding method is preferable. Example. Given ^=480, T'=500, and J=18, to find x. Here J(5T+ r') = 245 2.389166 i/=430' 2 cos. 9.997318 x = 246.52 2.391848 Then A D = 480 - 246.52 = 233.48, and B G = 500 - 246.52 = 253.48. The angle of intersection for both branches of the curve being 9, we find the radii A E = 233.48 cot. 4 30' = 2966.65, and B F = 253.48 cot. 4 30' = 3220.77. ARTICLE III. TURNOUTS AND CROSSINGS. 49. The turnouts here considered are of three kinds : Those in which a pair of rails in the main track are switched, and the turn- out curve is made tangent to the switched rails ; those in which a point switch, sometimes called a split switch, is employed, to one side of which, when thrown, the turnout curve is made tangent ; and those in which a pair of rails of the main track are switched in such a way that they become part of the turnout curve, which thus becomes tangent to the main track. The problems that im- mediately follow ( 50 to 64) are applicable to the first two cases. Problems relating to the third case will follow (g 65 to 76). TURNOUT FROM STRAIGHT MAIN TRACK. 37 First and Second Cases. 50. Let A B (fig. 14) represent either a switched rail, or the side of a point switch when thrown. To this line the outer rail B F of the turnout is tangent, and crosses the main track at F. The angle G F M, denoted by F, is called the frog angle, and the an- gle D A B, denoted by S, is called the switch angle. The gauge of the track D (7, denoted by g, and the distance D B, called the ^hrow, denoted by d, are supposed to be given. The distance A B = Us also given, whence we have sin. S = -75 = 7 If, for A. Jj I example, we had A B = I = 18, and d = .42, we should have sin. S = ~ = .02333, or S = 1 20'. lo A. Turnout from Straight Main Track. 51. Problem. Given the radius R of the centre line of a turnout (fig. 14), to find the frog angle G F M = F and the chord Solution. Through the centre E draw E K parallel to the main track. Draw B H and F K perpendicular to E K, and join B F. Then, since E F is perpendicular to F M and F K is per- pendicular to F G, the angle E FK = GFM=F-, and since E B and B H are respectively perpendicular to A B and A D, the angle EBH-DAB S. Now the triangle EFK gives 38 CIRCULAR CURVES, rr JT" (Tab. X. 2) cos. EFK ^--^ . But E F, the radius of the outer rail, is equal to R + \ g, and FK=CH = BH-BC- B E cos. E BH BC =(R + \g} cos. S (g - d). Substituting ^ T ^ (R 4- -J- g} cos. $ (g d) these values, we have co$.EFK = y/ -^ ^ z , or R + ig cos. .F = cos. S - : . -/t + kg From this formula F may be found by the table of natural cosines. To adapt it to calculation by logarithms, we may con- sider g d to be equal to (g d) cos. S, which will lead to no material error since g d is very small, and cos. S almost equal to unity. The value of cos. F then becomes To find BF, the right triangle B CF gives (Tab. X. 9) B F =* T> rt But B C-g-d and the angle BFC=BFE - siu.BFC' CFE = (90 \BEF) (90 -F) = F-$BEF. But BEF = BLF E B L = F S. Therefore B F C = F | (F S) = | (F 4- S). Substituting these values in the formula for B F, we have BF= . ?-* . Example. Given g = 4.7, d .42, = 1 20', and R = 500, to find F and .B.F. Here nat. cos. S = .999729, g - d = 4.28, .# + ^ - 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 (F + S) = 4 28' 5" sin. 8.891555 B F = 54.94 1.739889 52. Problem. Given the frog angle G F M = F (fig. 14), to find the radius R of the centre line of a turnout, and the chord BF. Solution. From the preceding solution we have cos. F = TURNOUT FROM STRAIGHT MAIN TRACK. 39 (g-d) ^ Theref ore ( + } g) cos . F=(R + R + ig t g) cos. S (ff d), or For calculation by logarithms this becomes (Tab. X. 29) ^ E + ^ g = sin.lt(F+S)sm.$(F-Sy Having thus found R + \g, we find R by subtracting \g. BF is found, as in the preceding problem, by the formula Example. Given # = 4.7, d = .42, S = 1 20', and F = 7, to find R. Here I (g - d) = 2.14 0.330414 + S) = 4 10' sin. 8.861283 _ #) - 2 50' sin. 8.693998 7.555281 R + ! = 595.85 2.775133 .-.^ = 593.5 Frogs on some roads are designated by numbers denoting the ratio of the length of the frog to its width, the width being a line drawn across the widest part of the frog, and the length a per- pendicular on this line from the point of the frog; so that if the number of the frog be denoted by w, we shall have cot. \ F = 2 n. Then to find -J- F we find the angle whose cotangent is double the number of the frog. Thus for frog number 7 we look for the angle whose cotangent is 14, and we find $ F = 4 5' 8". The frog angles in Tab. V. are so computed. 53. Problem. To find mechanically the proper position of a given frog. Solution. Denote the length of the switch rail by /, the length of the frog by /, and its width by w. From B as a centre with a radius BH=:2l, describe on the ground an arc tf -fiT/T (fig. 15), 40 CIRCULAR CURVES. and from the inside of the rail at # measure @H=2d, and from H measure HK such that HK\BH^w : /, or HK: 21 = %w :/; that is, H K = ^. 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 frog. For the angle HBKhas been made equal to i F, and if B M be drawn parallel to the main track, the angle MB H is seen to be equal to $S. Therefore, MBK = B F C = i (F + ), and this was shown ( 50) to be the true value of BFC. 54. If the turnout is to reverse, and become parallel to the main track, the problems on reversed curves already given will in gen- eral be sufficient. Thus, if the tangent points of the required curve are fixed, the common radius may be found by 40. If the tangent point at the switch is fixed, and the common radius given, the reversing point and the other tangent point may be found by 37, the change of direction of the two tangents being here equal to S. But when the frog angle is given, or determined from a given first radius, and the point of the frog is taken as the revers- ing point, the radius of the second portion may be found by the following method. Problem, (riven the frog angle F and the distance HB = b (fig. 16) between the main track and a turnout, to find the radius R' of the second branch of the turnout, the reversing point being taken opposite F, the point of the frog. Solution. Let the arc F B be the inner rail of the second branch, F & R ' \g its radius, and B the tangent point where the turnout becomes parallel to the main track. Now since the tangent F K is one side of the frog produced, the angle HF K TURNOUT FROM STRAIGHT MAIN TRACK. 41 F, and since the angle of intersection at K is also equal to F t BFK--=ltF( 2, II.) ; whence B FH=^F. Then ( 82) FG = Fig. 16. $BF sin.BFK X.9),orJ^=^ have . But BF=- . sin. Y JP 0111. Y^- sm.BFH . Substituting this value of | B F, we In measuring the distance HB = b,it is to be observed, that the widths of both rails must be included. Example. Given b 6.2 and F = 8, to find R '. Here i 6 = 3.1 0.491362 i-F=4 sin. 8.843585 = 44.44 1.647777 =4 sin. 8.843585 > -% g = 637.08 2.804192 .-.#'=639.43 42 CIRCULAR CURVES. B. Crossings on Straight Lines. 55. When 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 be- tween parallel tangents. Let HD and N K (fig. 17) be the centre lines of two parallel tracks, and HA and .5 JTthe 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= b between the tangents HP and B K. Then the common radius of the crossing AC B 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 frog 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. Problem. Given the perpendicular distance G N = b (fig. 17) between the centre lines of two parallel tracks, and the radii E G = R and C F = R ' of a crossing, to find the chords A C and B C. Solution. Draw E G perpendicular to the main track, and A L, CM, and B L ' parallel to it. Denote the angle A E C by E. Then, since the angle AEL = Aff6r = S,wQ have C E L = E + S, and in the right triangle GEM (Tab. X. 2), CEcos. CEM= Rcos.(E + S) = EM=EL-LM. But EL = AEcos.A EL = E cos. S, and L M : L' M= A C : B C. Now A C : B C = EC: CF=R:R'. Therefore, L M : L' M= R : R\ or L M : LM + L' M=R\R + R' ; that is, LM : b -2d = R: R + R' whence LM= j_ ~ /. . Substituting these values of E L and L H in the equation for R cos. (E + S), we have R cos. (E + S) = . ' . cos. (E + S) = cos. S - - j-t + M Having thus found E + S, we have the angle E and also its equal CFB. Then ( 83) TURNOUT FROM CURVES. 43 We have also AB = AC + B C, since A C and B C are in the same straight line ( 32), or A B = 2 (R + R ') sin. J E. Fig. 17. When the two radii are equal, the same formulae apply by mak- ing R' R. In this case, we have cos. (E + S) = cos. S - 2 R Example. Given d = .42, g - 4.7, S = 1 20', b = 11, and the angles of the two frogs each 7, to find AC=BC=%AB. The common radius R, corresponding to F = 7, is found ( 52) to be 593.5. Then 2 R = 1187, b-2d= 10.16. and 10.16 -t- 1187 = .00856. Therefore, nat. cos. (E 4- S) = .99973 - .00856 = .99117; whence E + S = 7 37' 15". Subtracting S, we have E = 6 17' 15". Next 212 = 1187 3.074451 i^^38'37^" sin. 8.739106 AC= 65.1 1.813557 C. Turnout from Curves. 57. Problem. #wm the radius R of the centre line of the main track and the frog angle F, to determine the position of the frog by means of the chord B F (figs. 18 and 19), and to find the radius R of the centre line of the turnout. Solution. I. When the turnout is from the inside of the curve 44 CIRCULAR CURVES. (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, Fig. 18. crossing the inside rail of the main track at F. Then, since the angle E F K has its sides perpendicular to the tangents of the two curves at F, it is equal to the acute angle made by the cross- ing rails, that \s>,EFKF. Also E B L = S. The first step is to find the angle B KF denoted by K. To find this angle, we have in the triangle BFK (Tab. X. 14) BK+KF -.BE- EF = tan. i (B FE + FB E} : tan. ( B F E - F B E). But B E = R + \g - d, and E F = R - $ g. Therefore, B E + KF-^R-d, and BE-EF = g-d. Moreover, BFE = BFE + EFE=BFE + F,w&FBE=EBF-EBE = B F E - S. Therefore, BFE-FBE=F+S. Lastly, BFE + FBE 180 K. Substituting these values in the preceding proportion, we have 2 R dig d = tan. (90 But tan. (90 - | K) = cot. Jf = - 9-< g-d TURNOUT FROM CURVES. 45 Next, to find the chord B F, we have, in the triangle B F G (Tab. X. 12), f = S ^ n air F - But B C = g - d, and al II. Jj J> \j B CF = 180 - F C K = 180 - (90 - K) = 90 + i K, or sin. 5 CF=cos.$K. Moreover, BFC = i(F + S) ; toiBFK ~ Therefore, B F K FBK=2BFC. But, as shown above, BFK-FBK=F+ S. Therefore, 2 B F C = F + , or BFC = $(F + S). Substituting these values in the expressior for B F, we have (g-d) G o S .$K ' Lastly, to find .#', we have ( %%) R + \g = E F = . . sin = BLF- EBL,an& BLF = LFK + LKF = F + K. Therefore, B E F = F + K - S, and II. When the turnout is from the outside of the curve, the pre- ceding solution requires a few modifications. In the present case, the angle E F K 1 = F (fig. 19) and E B L = S. To find K, we have in the triangle B F K, KF+BK: KF-BK- tan. 1 (F B K + B F K) : tan. \(FBK-BFK). But K F = R + g, and BK R $ g + d. Therefore, KF + BK = 2R + d, and KF-BK=g-d. Moreover, .F JT = 180 - FBL = 180- (EBF - E B L) = 180-^^^- S), and BFK=\W - BFK' = 180 - (B F E + E F K') = ISO - (EBF + F). Therefore, FB K - BFK = F + S. Lastly, FBK+ BFK=18Q-K. Substituting these values in the preceding proportion, we have 2 ft + d : g d = tan. (90 * JT) : tan. * (^ + S), or tan. (90 - * K) = P * + fl ta But tan. (90 - JT) = cot. } K = tan. Next, to find B F, we have, in the triangle B F C, B F = = 9~d, and BCF=^-^K, or 46 CIRCULAR CURVES. sin. B C F = cos. K. Moreover, B F C = $ (F + S) ; for BFK= RFC - B F C, and FB K= KG F + B FC = KFC+BFC. Therefore, FB K- BFK- 2 B F C. But, as shown above, FBK-BFK-F+S. Therefore, 2BFC- F + S, or ^ F C = | (^ + S). Substituting these values in the expression for B F, we have, as before, 7? F - (ff - ) 57. This remark applies also to B F in the second part of this solution. TURNOUT FROM CURVES. 49 the angle E F K' = F = 180 - E F K. Therefore J F - 90 - \EFK, and cot. \F- tan. | E FK\ JE^P" . . W u. ^ ^ -|/ , , o ^o ~~ U) where s is the half sum of the three sides, a the side E K, and I and c the remaining sides. ^\'n^ also in the triangle E F K the angle F E K, and we have the angle BEF=FEKBEK. Then in the triangle B E F we have ( 83) BF = 2(R' Example. Given g = 4.7, d = .43, S = 1 20', R = 4583.75, and jK' = 682.12, to find F and the chord B F of & turnout from the outside of the curve. Here in the triangle B E K (fig. 19) we have 5 50 CIRCULAR CURVES. W + ig = 684.47, B K = R-g + d = 4581.82, and the angles B E K + B K E = S =1 2V '. Then B K - B E = 3897.35 3.590769 i (B E K + B KE ) = 40' tan. 8.065806 1.656575 B K + B E = 5266.29 3.721505 %(BEK- BKE}* = 29.6029' tan. 7.935070 .-.BEK=\ 9.6029' EK is now found by the formula sm. B E K log. E K= log. 4581.82 + log. sin. 178 40' log. sin. 1 9.6029' = 3.721491, whence EK= 5266.12. Then to find F, we have in the triangle E F K, s = \ (5266.12 + 684.47 + 4586.10) = 5268.34, s - a = 2.22, s - b = 4583.87, and 8 - o = 682.24. a - ft = 4583.87 3.661233 s - c = 682.24 2.833937 6.495170 s = 5268.34 3.721674 5 - a = 2.22 0.346353 4.068027 2)2.427143 i^=330' cot. 1.213571 . . F = r To find F E K, we have s as before, but as a is here the side F K 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 | FEK- 3 2' 45". Subtracting | B EK- 34' 48", we have $BEF = 22T 57". Lastly, B F = 1368.94 sin. 2 27' 57" = 58.897. _ The formula B F = - -^ ^- ( 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 formulae, but to inaccuracies incident to the calculation. * This angle and the sine of 1 9.6029' below, are found by the method given in connection with Table XV. If the ordinary interpolations had been used, we should have found F=7 7', whereas it should be 7, since this example is the converse of that in 57. TURNOUT FROM CURVES. 51 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 deter- mine the reversing points C and B. These points will be deter- mined, if we find the angles A E C and B F C, and the chords A G and C B. 62. Problem. Given the radius DK=R (fig. SO) of the centre line of the main track, the common radius EC C F H ' of the centre line of a turnout, and the distance B G = b be- tween the centre lines of the parallel tracks, to find the central angles AEG and B F C and the chords A C and B C. Solution. In the triangle A E K find the angle A E K and the side E K. For this purpose we have AE=R',AK=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 F K the angles E F K and F E K. For this purpose we have E K, as just found, E F = 2 R', and F K = * The triangle A EK does not correspond precisely with B E Kin 60, A being on the centre line and B on the outer rail ; but the difference is too slight to affect the calculations. 52 CIRCULAR CURVES. R + R' l. Then AEC=AEK FEK, and EFK. Lastly (83), B@P" AC=2Rsm.iAEC, CB = 2R'si This solution, with a few obvious modifications, will apply, when the turnout is from the outside of a curve. D. Crossings on Curves. 63. When a turnout enters a parallel main track by a second switch, 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 Fig. 21. the angles which A B makes 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 re- versing point C and the tangent point B. These points will be determined, if we find the angles AEG and B F (7, and the chords^. C and C B. TURNOUTS TANGENT TO MAIN TRACK. 53 64. Problem. Given the radius DK=R (fig. 21) of the centre line of the main track, the common radius E C = C F = R ' of the centre line of a crossing, and the distance D 6r = b be- tween the centre lines of the parallel tracks, to find the central angles AEG and B F C and the chords A C and C B. Solution. In the triangle A E K find the angle A E K and the side E K. For this purpose we have AE = R',AK=R d, and the included angle E A K S. Find in the triangle B F K the angle B F K and the side F K. For this purpose we have BF=R',BK=R b + d, and the included angle FBK= 180 - S. Find in the triangle E F K the angles F E K and E F K. For this purpose we have E K and F K as just found, and E F = 2R'. ThenAEC = AEK-FEK, and B F C =E F K - B F K. Lastly ( 83), Third Case. Turnouts Tangent to Main Track. 65. In this case a pair of rails of the main track are switched in such a way that they become parts of the turnout curve. Their length in relation to R, the radius of the turnout, must be deter- mined. Denote their length by I and the " throw " by d. Then on the centre line d is the tangent offset of a curve of radius R. By 18 this offset or deflection is equal to the square of the chord I 9 divided by twice the radius, or d = -= ; .-.1= \/2~Rd. By this formula column I in Tab. V. is calculated. A switch-rail may be made to take the proper curve in the fol- lowing manner : Suppose the length of the switch-rail, as calcu- lated above, to be 20 feet. A rail 30 feet in length is, for 10 feet back from the tangent point, spiked down, or otherwise securely fastened on the main track, leaving 20 feet free for the switch-rail. The free end being thrown in the usual way, a curve is formed, which, however, is not a circular curve, but an elastic curve. The inclination at the free end, in the case supposed, would be about 54 CIRCULAR CURVES. three-fourths of that of the circular curve that meets it. If it be desired to make the two inclinations equal, so that the two curves shall be tangent to each other, the switch-rail should be only three-fourths of the calculated length of I. The switch-rail may, however, be made to take a circular form by suitable stops at- tached to the sleepers. The full length, as calculated above, will then, of course, remain free. The offsets from the tangent to the stops will be to d as the squares of the distances from the tan- gent point are to Z*. A. Turnout from Straight Lines. 66. Problem. Given the radius E of the centre line of a turnout, and the gauge B G == g (fig. 22\ to find the frog angle GFM=F,and the chord B F. Solution. The angle C E F, having its sides perpendicular to G F and F M, is equal to G F M = F. In the triangle CEFwe 1 1 have cos. CE F= Draw ED perpendicular to BF. Then, from the similar tri- angles B F C and B E D, we have the angle BFC=BED= | F. Therefore, B F sin. $F 67. Problem. Given the frog angle GF M=F (fig. 22), and the gauge BC = g,to find the radius R of the centre line of a turnout, and the chord B F. Solution. From the preceding problem we have v-Saa Z> 17 9 ^ ".^irrr In the triangle B E D we have B E sin. B E D = | B F, or (R + $g)sm.$F TURNOUT FROM STRAIGHT LINES. DO To put R in another form, substitute for B F its value above, and transfer %g to the second member. We then have R = Iff If now the frog angle F is expressed by means of the ratio n of the length to the breadth of the frog, as explained in 52, we have cot. J F = 2 n, and, substituting this value in the expression for R, we have By the formulae of this section the values of F, B F, and R in Table V. are calculated. 68. A ready way of locating the turnout curve is to locate the outer rail first by stretching a cord from B to F, and from it fix- ing the curve by ordinates at the centre and at the quarter points. The middle ordinate m may be taken in all cases = \g. For - , and putting in the value of R + | g above, (26),ro = and reducing, we have m = B Fsin. $ F = g. For g = 4.708, m = 1.177. At the quarter points the ordinates will be f m = 0.883. The inner rail is then located by the gauge. 69. If the turnout is to reverse and become parallel to the main track, the formulae of 53 apply here also. 56 CIRCULAR CURVES. B. Crossings on Straight Lines. 70. When a turnout enters a parallel main track by a second curve, it becomes a crossing, and the two curves form a reversed curve between parallel tangents. The problems that arise here have been solved already ( 33-36). C. Turnout from Curves. 71. Problem. Given the radius R of the centre line of the main track and the frog angle F, to determine the position of the frog ~by means of the chord B F (figs. 23 and ), and to find the radius R of the centre line of the turnout. Solution. I. Turnout from the inside of the curve of the main track. Let B G and C F (fig. 23) be the rails of the main track, and the arc B F the outer rail of the turnout, crossing the inner rail of Fig. 23. the main track at F. Then, since the angle EFK has its sides perpendicular to the tangents of the two curves at F, it is equal to the acute angle made by the crossing rails; that is, E F K = F. TURNOUT FROM CURVES. 57 The first step is to find the angle B K F denoted by K. To find this angle, we have in the triangle BFK (Tab. X., 14) tan. k mFK irnin (BK-KJ?)te*.i(BFK+FBK) (tJ< K- S K) - - BK+KF BK- KF=BK - CK = g, and BK + KF - 2 R. Also, tan. %(BFK + FBK} = tan. i (180 - K} = tan. (90 - i K) = cot.^K, and BFK-FBK=BFK-BFE = F. Substi- tuting these values, we have tan. F = ^-^ 2 R tan K 1 or 2 R tan. | F tan. %K=g\ tan lK - tan. $ JL 75 - T-> . if, by the notation of 52, we put cot. J F = 2 n. To find the chord B F, we have in the triangle B F C, B F But B C = g, and sin. B CF = sin. F CK = cos. i K. Moreover, B F C = i F. ~ForBFK=KFC + BFC,KH&FBK=KCF-BFC = KFC-BFC. There- f ore, by subtraction, BFK-FBK=2BFC. But, as shown above, BFK FBK=F. Therefore B F C i F. Sub- stituting these values in the expression for B F, we have -9 cos, j K* ~ Lastly, to find J2', we have in the triangle B E F, EFsm. %BEF=%BF. But EF=R' + \g, and the exterior angle II. Turnout from the outside of the curve of the main track. Let J5 Gr and C jF 7 (fig. 24) be the rails of the main track, and the arc B F the outer rail of the turnout, crossing the outer rail of the main track at F. The frog angle F is now represented by the angle E F K' . The first step is to find the angle B K F, denoted * Since i Kis generally very small, an approximate value of B F may be obtained by making cos. \K= 1, whence BF . g . CT , which ia identical sin. 5- _r with the formula for B F in 66. This remark applies also to B F in the second part of this solution. 58 CIRCULAR CURVES. by K. To find this angle, we have in the triangle B F K (Tab. X.. BFK } 14), t - BFK) = EutKF-BJR:=g,)(*-<*) where s (s a) s is the half sum of the three sides, a the side opposite the angle sought, here denoted by A, and ~b and c the remaining sides, Putting FJZKfor A, and FKfor a, we shall have an expression for t&r\.%FEK= tan. | (180 AEG) = coi.^AEC, and put- ting E F K for A and E K for a, we shall have an expression for DOUBLE TURNOUTS. 63 tan. \EFK- tan. \BFC. Making the proper substitutions in the formula for tan.^J., we shall have Having found A E C and B F C, we have the chords This solution, with a few obvious modifications, will apply when the turnout is from the outside of the curve. 75. Problem. Given the position of a frog by means of the vhord B F (figs. 22, 23, and 24), to find the frog angle F. Solution. The formula BF = . g , CT , which is exact on sin. F straight lines ( 66). and near enough on ordinary curves ( 71, note), gives D. Crossings on Curves. 76. 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. 25) are fixed, the distance A B must be measured, and also the angles made by AB 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, the reversing point C and the second tangent point B may be found by the problem of 74. E. Double Turnouts. 77. The cases that arise when two turnouts start from the same point on the main track fall under problems already solved. 64 CIRCULAR CURVES. Thus when the outer rails of two turnouts, as B C F and B' C F' (fig. 26), turn opposite ways, B' C F' may be treated as a turnout from the outside of the inner rail B ' D of B C F. Then if the frog angle at C is given, the radius of B' OF' may be found by Fig. 26. 57 or 71, or if the radius of B ' C F' is given, the frog angle at C may be found by 60 or 72, Or, the third frog may be placed with its point in the centre line of the main track, and its angle may be taken as made up of two angles, F\ and F<*, one on each side of said centre line, as in figure 26. On a straight main track the two turnouts would in general be symmetrical, and FI be equal to F^. On a curved main track these partial angles may be equal or unequal. All the relations between the radii and the frog angles concerned may be determined by previous problems, substituting \g for g as the distance of the line C H from either rail. Thus in the figure the radius of B C and the partial frog angle FI depend on each other, so also do the radius of B' C and the partial frog angle F* When one of the chords, as B C, is fixed in length, the length of the other, B' C, is also fixed, whether equal to B C on straight lines or different on curves. The partial frog angle F^ being de- DOUBLE TURNOUTS. 65 pendent on the length of B ' C, is found by 59 or 75, and from it the radius of the curve B' C is calculated. When either curve beyond (7, as C F, is not a continuation of the curve B C, the relation between its radius and the frog angle F is to be determined by considering F l to be a switch angle, and the curve C F to commence at the but-end of the frog ( 50 or 51), using %g instead of g for the gauge. If both turnouts turn the same way, as in figure 27, the third frog J^a is on a turnout A FI F* from the inside of the curve AF, and its angle and position may be determined by 60 or 72. Fig. 27. 78. Remarks. 1. If the two turnouts of figure 26 are symmetri- cal and tangent to the straight main track, the chord B C is to the chord B F as 1 to y2. For the offset from the tangent B F 1 to C is \g, and the offset to F is g, and these tangent offsets or deflections are to each other ( 18) as the squares of the chords B C and B F. Therefore B C* : B F 9 = %g : g = 1 : 2, or B C : I:f2; whence (7 = ^ = i +/2 B F = .707 5 JP, nearly. 2. We have ( 66) sin. $ ^ = ^=, and sin. } Jft as & ss ^-g-^. Jj Jj Jj G Jj O Denote the whole frog angle at (7 by F' = %Fi, and we have sin.i^P' = X-^-TY- Also, since, as shown above, BF=BCv% & -D L> we have sin. $ F = -=^-j= - . Therefore, sin. i F' : sin. $ F = -tf L> V ' & ~ S-^TY : T.^ rt = V2 : 2, or sin. F' = -^ sin. i J^= .707 sin. \ F, 2BC BC y% 2 nearly. t)6 CIRCULAR CURVES. 3. We have seen ( 66 and 71) that for a given frog angle the length of the chord B F in the three turnouts represented in figures 22, 23, and 24 is practically the same, since we may put in the three cases B F = . , . To find the degree of each of the sin. F three turnout curves, we have only to find the central angle sub- tended by a chord of 100 feet ( 6). Now, in the three cases in question, we know that the central angles B E F, subtended by the equal chords B F, are, respectively, F, F + K, and F K. The central angles for 100 feet chords will be obtained from these 100 100 very nearly by multiplying by JTJ,. Denoting the fraction --r, by m and the degrees of the three turnout curves by AI, A 2 , and A 3 , we have A! = m F, A 2 = m (F + K), A 3 = m (F - K). Now m K is approximately the degree of the curve of the main track (figs. 23 and 24) since K is the central angle of this curve for a chord approximately equal to B F. Therefore, denoting the degree of the main track by A, we have, approximately, for the same frog angle, A a = AI + A, A 3 AI A. Thus in the example of 71 (fig. 23), where n = 8, we have by Tab. V. the degree of a turnout from a straight line AI = 9 31'. The degree of the main track is here A = 3. Therefore A 2 = A! + A = 12 31', the degree of the turnout from the curve. The radius found for this turnout was 457.52 and the degree corre- sponding would be 12 32' 53". It appears, then, that if, for a given frog, we take from Tab. V. the degree AI of a turnout from a straight main track, we may obtain approximately the degree A 2 of a turnout from the inside of a curved track by adding to AI the degree of the main track, and the degree A 3 of a turnout from the outside of a curved track by subtracting from AI the degree of the main track. ARTICLE IV. MISCELLANEOUS PROBLEMS. 79. Problem. Given A JB = a (fig. 28) and the perpendicular B C = b, to find the radius of a curve that shall pass through G and the tangent point A. Solution. Let be the centre of the curve, and draw the radii A and C and the line CD parallel to A B. Then in the right MISCELLANEOUS PROBLEMS. 67 triangle COT) we have C 2 = CD' 2 + 07) 2 . But 0(7 = 72, CD = a, and D = A - A D = R - b. Therefore, 72* = + & 2 , or 2 72 b = a 2 + 6 2 ; *2 Example. Given a = 204 and & = 24, to find 72. Here R = 80. Corollary 1. If R and & are given to find AB = a, that is, to determine the tangent point from which a curve of Fig. 28. given radius must start to pass through a given point, we have ( 79) 2 Mb = a* + & 2 , or a 2 = 2 JS & - & 2 ; Example. Given & = 24 and 72 = 879, to find a. Here a = ^24(1758-24) = V41616 = 204. 81. Corollary 2. If 72 and a are given, and b is required, we have (79) 2 R b = a* + &*, or & 2 - 2 R b = - a*. Solving this equation, we find for the value of b here required, 82. Problem. Given the distance A C = c (fig. 28) and the angle B A C = A, to 'find the radius R or deflection angle 68 CIKCTJLAK CURVES. D of a curve, that shall pass through C and the tangent point A. Solution. Draw E perpendicular to A C. Then the angle AOE = iAOC = BAC=A(2, III.), and the right triangle ^ Ogives (Tab. X.9MO = .. . sin. A 50 To find D, we have ( 9) sin. D = -^ . Substituting for 72 its i c value just found, we have sin. Z> = 50 -- -tP -r ; Example. Given c = 285.4 and A 5, to find R and D. Here R = = 1637.3 ; and sin. D = 100 = = sin. 5 285.4 2.854 sin. 1 45' or D = 1 45'. 83. Problem. Given the radius R or the deflection angle D of a curve, and the angle B A C = A (fig. %8}, made by any chord with the tangent at A, to find the length of the chord AC = c. 1 x Solution. If R is given, we have ( 82) R = -7 ; sin. A. = 2R sin. A. 10 sin - -^ T* r> /o onx :r> If D is given, we have ( 82) sin. Z) = _ 100 sin. A sin. Z> 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 = 10 sm '^ D . By this method Table II. sm. D is calculated. Example. Given R = 2455.7, or D = 1 10', and A = 4 40', to MISCELLANEOUS PROBLEMS. 69 find c. Here, by the first formula, c - 4911.4 sin. 4 40' = 399.59. 100 sin. 4 40' By the second formula, c : T^JTV 84. Problem. Given the angle of intersection KGB I (fig. %9\ and the distance C D b from the intersection point to the curve in the direction of the centre, to find the tangent AC = T, and the radius A = R. Fig. 29. Solution. In the triangle A D C we have sin. C A D : sin. AD C- CD\ AC. But CAD = ^AOD = lI (2, III. and VI.), and as the sine of an angle is the same as the sine of its sup- plement, sin. ADC sin. ADE = cos. D A E = cos. /. More- over, C D b and A C = T. Substituting these values in the preceding proportion, we have sin. \ / : cos. I = b : T, or T = 6 ? S '* J ; whence (Tab. X. 33) T = b cot. i 7. To find R, we have ( 5) R = Tcoi. i I. Substituting for T its value just found, we have R = b cot. i /cot. i I. 70 CIKCULAK CURVES. Example. Given / = 30, b 130, to find 2* and R. Here b = 130 2.113943 i 7 = 7 30' cot. 0.880571 T= 987.45 2.994514 i/=15 cot. 0.571948 R = 3685.21 3.566462 85. Problem. Given the angle of intersection KG B I (fig. 29\ and the tangent A(JT, or the radius A = R, to find CD=b. Solution. If T is given, we have ( 84) T = I cot. /, or b = T cot.i/' B^~ .-.&= Ttan.iZ If .72 is given, we have ( 84) R = b cot. J 7 cot. /, or & = . . b = R tan. i /tan. -J Z Example. Given J= 27, T= 600 or j^ = 2499.18, to find b. Here & = 600 tan. 6 45' = 71.01, or b = 2499.18 tan. 6 45' tan. 13 30' = 71.01. The distance b from the intersection point to the curve in the direction of the centre is usually called the external, and this term is adopted in Table III. 86. Problem. Given the angle of intersection I of two tan- gents A C and B C (fig. 30\ to find the tangent point A of a curve that shall pass through a point E, given by C D = a, D E = b, and the angle C D E = -J /. Solution. Produce D E to the curve at G, and draw C to the centre 0. Denote D F by c. Then in the right triangle CDF we have (Tab. X. 11) D F = C D cos. CD F, or jgjp" c a cos. $ /. Denote the distance A D from D to the tangent point by x t Then, by Geometry, a 2 = D E x D O. But D - D F + F Q = D F + b), and MISCELLANEOUS PROBLEMS. 71 = 2 c b. Therefore, x 9 = 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 H and C H perpendicular to each other, a and b may be found from these lines. For a = C H + DH=CH+ EHcoi. \ /(Tab. X. 9), and b = D E = Example. Given /= 20 16', a = 600, and b = 80, to find x and R. Here c = 600 cos. 10 8' = 590.64, 2 c b = 1101.28, and x = \/80 x 1101.28 = 296.82. Then T 600 + 296.82 = 896.82, and R = 896.82 cot. 10 8' = 5017.82. 87. Problem. Given the tangent A C (fig. 31), and the chord A B, uniting the tangent points A and B, to find the radius AO = R. Solution. Measure or calculate the perpendicular C D. Then if CD be produced to the centre 0, the right triangles ADC and 72 CIRCULAR CURVES. C A 0, having the angle at C common, are similar, and give CD \ AD = AC:AO,OT _AD x A C CD ' If it is inconvenient to measure the chord AB, a line E F^ parallel to it, may be obtained by laying off from C equal dis- tances CE and C F. Then measuring EG and # (7, we have, from the similar triangles E G C and CAO, CG : GE - AC : GE xAC Example. Given A C 246 and A D = 240, to find R. Here CD = 54, and R = 24 *. 246 = 1093.33. 54 88. Problem. Given the radius AO = R (fig. 31\ to find the tangent AC T of a curve to unite two straight lines given on the ground. Pig. 31. Solution. Lay off from the intersection C of the given straight lines any equal distances C E and C F. Draw the perpendicular C G to the middle of E F, and measure GE and C G. Then the MISCELLANEOUS PROBLEMS. 73 right triangles E Or C and C A 0, having the angle at C common, are similar, and give G E : C G = A : A C, or CGxAO W GE By this problem and the preceding one, the radius or tangent points of a curve may be found without an instrument for measur- ing angles. Example. Given R = 1093, G E = 80, and C G - 18, to find K Here T =**** = 246. 89. Problem. To find the angle of intersection I of two straight lines, when the point of intersection is inaccessible, and to determine the tangent points, when the length of the tangents is given. Solution. I. To find the angle of intersection /. Let A C and C V (fig. 32) be the given lines. Sight from some point A on one line to a point B on the other, and measure the angles CAB and T B V. These angles make up the change of direction in passing from one tangent to the other. But the angle of intersection (g 2) shows the change of direction between two tangents, and it must, therefore, be equal to the sum of C A B and T B V, that is, I=CAB + TB V. But if obstacles of any kind render it necessary to pass from A C to B V by a broken line, as A D E F B, measure the angles CAD, ND E, PEF, RFB, and SB V, observing to note those angles as minus which are laid off contrary to the general direc- tion of these angles. Thus the general direction of the angles in this case is to the right ; but the angle PEF 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 CAD shows the change of direction between the first tangent and A D, ND E shows the change between AD produced and DE, PEF the change be- tween DE produced and E F, RFB the change between E F produced and F B, and, lastly, SB V the change between B F produced and the second tangent. But the angle of intersection ( 2) shows the change of direction in passing from one tangent to 74 CIRCULAR CURVES. another, and it must, therefore, be equal to the sum of the partial changes measured, that is, NDE-PEF+RFB + SBV. II. To determine the tangent points. This will be done if we find the distances A C and B C ; f or then any other distances from C may be found. It is supposed that the distance A B, or the distances A Z>, D E, E F, and F B have been measured. If one line A B connects A and B,find A C and B C in the tri- angle ABC. For this purpose we have one side A B and all the If a broken line ADEFB connects A and B, let fall a per- pendicular B G from B upon A C, produced if necessary, and find A G and B G by the usual method of working a traverse. Thus, if A C is taken as a meridian line, and D K, EL, and F M are drawn parallel to A C, and D H, E K, and F L are drawn parallel to B G, the difference of latitude A G is equal to the sum of the partial differences of latitude AH,DK,EL, and F M, and the departure B G is equal to the sum of the partial depart- ures D H, EK, FL, and B M. To find these partial differences of latitude and departures, we have the distances AD,DE,EF, and F B, and the bearings may be obtained from the angles al- ready measured. Thus the bearing of A D is CAD, the bearing of DE is KDE = KDN + NDE = CAD + NDE, the bear- ing of EFi$LEF=LEP PEF=KDE PEf, and MISCELLANEOUS PROBLEMS. 75 the bearing of F B is M F B = M F R + R F B = L E F + RFB; that is, the bearing of each line is equal to the algebraic sum of the preceding bearing and its own change of direction. The differences of latitude and the departures may now be ob- tained from a traverse table, or more correctly by the formulas : Diff . of lat. = dist. x cos. of bearing ; dep. = dist. x sin. of bearing. Thus, A H = A D cos. CA D, and DH = A D sin. CAD. Having found A G and B G, we have, in the right triangle 7? C 1 EGG (Tab. X. 9), G C = B G cot, BCG,an&BC = - ^7775 . sin. jj (j (j But BCG = 180 - 7. Therefore, cot. BCG=- cot. /, and sin. B C G = sin. I. Hence G C = - B G cot. /, and B C = Then, since AC= AG + G C, we have ~ T . sin. / When / is between 90 and 180, as in the figure, cot. / is nega- tive, and B G cot. / is, therefore, positive. When / is less than 90, G will fall on the other side of C ; but the same formula for A C will still apply ; for cot. / is now positive, and consequently, B G cot. / is negative, as it should be, since, in this case, A C would equal A G minus G C. Example. Given A D = 1200, D E = 350, E F = 300, FB = 310, CAD = 2Q, ND # = 44, PE F= - 25 ; R FB = Z\\ and SB V= 30, to find the angle of intersection /, and the dis- tances A C and B C. Here J= 20 + 44 - 25 + 31 + 30 = 100 C . To find A G and B G, the work may be arranged as in the following table : Angles to the Right. Bearings. Distances. N. E. 20 44 25 31 N. 20 E. 64 39 70 1200 350 300 310 1127.63 153.43 233.14 106.03 410.42 314.58 188.80 291.30 1620.23 1205.10 The first column contains the observed angles. The second con- tains the bearings, which are found from the angles of the first 76 CIRCULAR CURVES. column, in the manner already explained. A C is considered as running north from A, and the bearings are, therefore, marked N. E. The other columns require no explanation. We find A a = 1620.23 and B G = 1205.10. Then GC=-BG cot. 1 = 1205.1 x cot. 100 = 212.49. This value is positive, because it is the product of two negative factors, cot. 100 being the same as cot. 80, a negative quantity. Then A C = A G + G C 1620.23 + 212.49 = 1832.72, and B C = = 1223.69. Hav- sm. 100 ing thus found the distances of A and B from the point of inter- section, we can easily fix the tangent points for tangents of any given length. 90. Problem. To lay out a curve, when an obstruction of any kind prevents the use of the ordinary methods. Fig. 33, Solution. First Method. Suppose the instrument to be placed at A (fig. 33), and that a house, for instance, covers the station at B, and also obstructs the view from A to the stations at D and E. Lay off from A (7, the tangent at A, such a multiple of the deflec- tion angle D, as will be sufficient to make the sight clear the ob- struction. In the figure it is supposed that 4 D is the proper an- MISCELLANEOUS PROBLEMS. 77 gle. The sight will then pass through F, the fourth station from A, and this station will be determined by measuring from A the length of the chord A F, found by 83 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 from the tangent at F. Second Method. This consists in running an auxiliary curve parallel to the true curve, either inside or outside of it. For this purpose lay off perpendicular to A C, the tangent at A, a line A A of any convenient length, and from A a line A C' parallel to A C. Then AC' is the tangent from which the auxiliary curve A E' is to be laid off. The stations on this curve are made to correspond to stations of 100 feet on the true curve, that is, a ra- dius through B' passes through B, a radius through D' passes through D, &c. The chord A B' is, therefore, parallel to AB, and the angle C' A B' C A 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 B', B' D', &c. Call the distance A A' = b. Then the similar triangles ABO and A' B ' give A : A' = A B : A' B ', or R : R - b = 100 : A'B '. Therefore, A B' = ~ = ioo -~. If the auxiliary 1 jft curve were on the outside of the true curve, we should find in the same way A B ' = 100 + p . It is well to make b an aliquot part of R-, for the auxiliary chord is then more easily found. 7~> Thus, if n is any whole number, and we make b = , we have 71 A'B' = m ^ = 100 . If, for example, b = ^~, we jK fl lUu have n 100, and A B = 100 1 = 101 or 99. When the aux- iliary curve has been run, the corresponding stations on the true curve are found, by laying off in the proper direction the distances B B', D D', &c., each equal to b. 91. Problem. Having run a curve A B (fig. 34\ to change the tangent point from A to (7, in such a way that a curve of the same radius may strike a given point D. Solution. Measure the distance B D from the curve to D in a direction parallel to the tangent C E. This direction may be sometimes judged of by the eye, or found by the compass. A still more accurate way is to make the angle D B E equal to the inter- 78 CIRCULAR CURVES. section angle at E, or to twice B A E, the total deflection angle from A to B ; or if A can be seen from B, the angle DBA may be made equal to B A E. Measure on the tangent (backward or forward, as the case may be) a distance A C = B Z), and C will be the new tangent point re- quired. For, if CHbv drawn equal and parallel to A F, we have F H equal and parallel to A C, and therefore equal and parallel to B D. Hence DH=F = AF=CH, and D H being equal to Off, & curve of radius C H from the tangent point C must pass through D. 92. Problem. Having run a curve A B (fig. 55) of radius R or deflection angle D, terminating in a tangent B D, to find the radius R ' or deflection angle D' of a curve A C, that shall termi- nate in a given parallel tangent C E. Solution. Since the radii B F and C G are perpendicular to the parallel tangents C E and B D, they are parallel, and the an- gle A G C- A FB. Therefore, A C G, the half-supplement of A G C, is equal to A B F, the half -supplement of A F B. Hence A B and B C are in the same straight line, and the new tangent point C is the intersection of A B produced with C E. Represent A B by c, and A C = c + B C by c'. Measure B (7, or, if more convenient, measure D C and find B C by calculation. DC To calculate B C from D C, we have B C = 1 (Tab. X. 9) sin. DB C v and the angle DBG- ABKBAK, the total deflection from MISCELLANEOUS PKOBLEMS. 79 A to B. Then the triangles A FB and A G C give A B : A C BF : CG, or c : c' = R : R' ', To find D, we have ( 10) R ' = -rr, , and R = -= . Sub- sin. D ' sm.D stituting these values in the equation for R ', we have - = c' 50 Sm ' - x =~; c sin. D . . sin. D' = sin.D. c 93. Problem. Given the length of two equal chords A C and B C (fig. 36), and the perpendicular CD, to find the radius R of the curve. Solution. From 0, the centre of the curve, draw the perpen- dicular E. Then the similar triangles B E and BCD give BO:BE=BC: CD, or R : \ B C = B C : CD. Hence BC* R = 2CD' 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 80 CIRCULAR CURVES. from C upon the whole chord AB, we have the data of this problem. 94. Problem. To draw a tangent F Gr (fig. 36) to a given curve from a given point F. Fig. 36. Solution. On any straight line FA, which cuts the curve in two points, measure FG and FA, the distances to the curve. Then, by Geometry, J^~ FG = ^FC x FA. 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. 95. Problem. Having found the radius A R of a curve (fig. 31\ to substitute for it two radii A E = Ri and D F = R* , the longer of which A E or BE' is to be used for a certain dis- tance only at each end of the curve. Solution. Assum.e the longer radius of any length which may he thought proper, and find (g 9) the corresponding deflection angle DI . Suppose that each of the curves A D and B D ' is 100 feet long. Then drawing C 0, we have, in the triangle FOE, OE :FE=sm. F E : sin. FOE. But the side OE = AE R l - R*, the angle .F i 7 = MISCELLANEOUS PROBLEMS. 81 180 - A C = 180 - i /, and the angle OFE = AOF- = iI-2D l , since E F = 2 D l ( 7). Substituting Fig. 37. these values, and recollecting that sin. (180 -J- /) = sin. /, we have Ei R : Hi R* sin. (| / 2 DI) : sin. - /. Hence HP sin. (| J- 2 A) 7? 2 is then easily found, and this will be the radius from D to Z>', or until the central angle D F D ' I 4 Z>i. 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 just 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 Z>, and this angle must take the place of 2Di 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. 7 82 CIRCULAR CURVES. Example. Given R = 1146.28 and I 45, to find R*, if R l is assumed = 1910.08, and A D and B D ' each 100. Here, by Table I.,A = 130'. Then R^ - R = 763.8 i/=22 30' |/-. 2 A = 19 30' Ri-R* = 875.64 2.882980 sin. 9.582840 2.465820 sin. 9.523495 2.942325 .-.R^Rt- 875.64 = 1034.44 96. Problem. To locate the second branch of a compound or reversed curve from a station on the first branch. Solution. Let A B (fig. 38) be the first branch of a compound curve, and D its deflection angle, and let it be required to locate the second branch A B', whose deflection angle is D', from some station B on A B. Let n be the number of stations from A to B, and n' the number of stations from A to any station B ' on the second branch. Rep- _T Fig. 38. resent by V the angle ABB', which it is necessary to lay off from the chord B A to strike B'. Let the corresponding angle A B' B on the other curve be represented by V. Then we have V + V 180 BAB'. But if T T' be the common tangent at A, we have T A B + T' A B' = n D + n' D' = 180 BAB'. Therefore, V + V = n D + n' D' . Next in the triangle ABB' we have sin. V : sin. V = A B : A B '. But A B : A B ' n : n', nearly, and sin. V : sin. V= V : V, nearly. Therefore we have approximately V :V=n : n', or V , V. Substituting this value of V in the equation for V + V, we have V + ,V = MISCELLANEOUS PROBLEMS. 83 n D -f- n' D '. Therefore, n'V+ n V = ri (n D + ri D '), or n'(nD + n'D') n + ri The same reasoning will apply to reversed curves, the only change being that in this case V + V = nD n' D', and conse- quently ri(nD-riD') uHy? , n + n When in this last formula n' D' becomes greater than nD, V be- comes minus, which signifies that the angle V is 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 from 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 _ n + n' ' When n and n' are each 1, the formula for V is in all cases ex- act; for then the supposition that V : V= n : n' is strictly true, since A B will equal A B', and V and V, being angles at the base of an isosceles triangle, will also be equal. Making n and n' equal to 1, we have When the curve starts from a straight line, this formula becomes, by making D = 0, F=*Z>'. We have seen that when n or n' is more than 1, the value of V is only approximate. It is, however, so near the truth, that when neither n nor n' exceeds 3, the error in curves up to 5 or 6 varies from a fraction of a second to less than half a minute. The exact value of V might of course be obtained by solving the triangle A B B', in which the sides AB and AB' may be found from Table II., and the included angle at A is known. The extent to which these formulae may be safely used may be seen by the fol- lowing table, which gives the approximate values of Ffor several 84 CIRCULAR CURVES. different values of n, n\ D, and Z>', and also the error in each case: Compound Curves. Reversed Curves. n. D. n'. D'. F. Error. n. D. n'. D'. V. Error. / // o / 1 5 1 410 0.9 1 3 4 3 712 27.2 1 5 3 1230 25.3 2 3 4 3 4 23.5 2 8 3 524 22.1 3 3 4 3 1 42f 8.3 3 3 3 430 29.7 3 i 3 3 345 24.0 1 1 5 3 1320 18.6 2 1 1 4 040 0.1 2 f 1 3 1 20 0.7 2 1 4 2 4 11.0 2 2 3 3 748 15.0 1 6 2 6 4 23.5 2 2 4 3 1040 24.7 1 5 3 5 730 51.8 3 3 3 4 1030 54.0 2 3 5 3 625f 52.8 As the given quantities are here arranged, the approximate values of V are all too great ; but if the columns n and n' and the columns D and D' were interchanged^ and F calculated, the approximate values of V would be just as much too small, the column of errors remaining the same. 97. Problem. To measure the distance across a river on a given straight line. Fig. 39. Solution. First Method. Let A B (fig. 39) [be the required distance. Measure a line A C along the bank, and take the an- gles BAG and A C B. 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 AC B ^DAC can MISCELLANEOUS PROBLEMS. 85 be laid off to a point on the farther side, we have A S C = = ACB. Therefore, without calculation, A B = A C. Fig. 40. Second Method. Lay off A C (fig. 40) perpendicular to A B. Measure A C, and at C lay off C D perpendicular to the direction C B, and meeting the line of A B in D. Measure A D. Then the triangles A C D and A B G are similar, and give A D : A C = A r* AC'.AB. Therefore, AB = ^^. If from (7, determined as before, the angle A C B' be laid off equal to A C B, we have, without calculation, A B = A B'. Third Method. Measure a line A D (fig. 41) in an oblique di- rection from the bank, and fix its middle point C. From any Fig. 41. convenient point E in the line of A B, measure the distance E (7, and produce EC until C F = E C. Then, since the triangles 86 CIRCULAR CURVES. ACE and D C F are similar by construction, we see that D F is parallel to E B. Find now a point 6r, that shall be at the same time in the line of C B and of D I 1 , and measure O D. Then the triangles ABC and D & C are equal, and O D is equal to the re- quired distance A B. As the object of drawing E F is to obtain a line parallel to A B, this line may be dispensed with, if by any other means a line O F be drawn through D parallel to A B. A point G being found on this parallel in the line of C J5, we have, as before, G D = AB. 98. Problem, To change a tangent point so that the tan- gent may pass through a given point. Solution. If the given point is at a considerable distance but visible, let C (fig. 42) be the distant point and D the required tan- Fig. 42. gent point. Estimate the probable position of Z>, and at A, a station back of D but near to it, measure the angle B A C made by A C with the tangent at A. Then, as the angle at C is sup- posed to be very small, the chord A E will be nearly parallel to D (7, and D may be taken to be midway between A and E. The angle B A D, which fixes the position of Z), will therefore equal i B A i Tjy TT G H '^i : ; . Hence C A E' is determined. a R sin. A In both these cases E G has been supposed larger than .F 77. If E G is smaller than F H, the point E' will fall on the other side of E, and the angle C A E' = $ A % E'F'K. It is obvious that, in both cases, it E G is exactly equal to F H, the angle E'F'K vanishes, and C A E' = \ A. PARABOLIC CURVES. 89 CHAPTER II. PARABOLIC CURVES. ARTICLE I. LOCATING PARABOLIC CURVES. 100. LET A EB (fig. 47) be a parabola, A C and B C its tan- gents, and A B the chord uniting the tangent points. Bisect A B in D, and join C D. Then, according to Analytical Geometry, Fig. 47. B I. C D is a diameter of the parabola, and the curve bisects C D in E. II. If from any points T, T', T", &c., on a tangent AF, lines be drawn to the curve parallel to tlie diameter, these lines T M, T M', T" M", &c., called tangent deflections, will be to each other as the squares of the distances A T, A T', A T", &c., from the tangent point A. III. A line E D (fig. 48), drawn from the middle of a chord A B to the curve, and parallel to the diameter, may be called the middle ordinate of that chord ; and if the secondary chords A E and B E be drawn, the middle ordinates of these chords, KG and L H, are each equal to %E D. In like manner, if the chords A K, KE, EL, and L B be drawn, their middle ordinates will be equal IV. A tangent to the curve at the extremity of a middle ordi- 90 PARABOLIC CURVES. nate is parallel to the chord of that ordinate. Thus M F (fig. 48), tangent to the curve at E, is parallel to A B. V. If any two tangents, as A C and B C (fig. 48), be bisected in M and F, the line M F, joining the points of bisection, will be a new tangent, its middle point E being the point of tangency. 101. Problem. Given the tangents A C and B (7, equal or unequal (fig. 47). and the chord A B, to lay out a parabola by tangent deflections. Fig. 47. A D B Solution. Bisect A B in D, and measure C D and the angle A CD; or calculate CD* and ACD from the original data. Divide the tangent A C into any number n of equal parts, and call the deflection T M for the first point a. Then ( 100, II.) the deflection for the second point will be T' M' = 4 &, for the third point T" M " = 9 a, and so on to the nth point or (7, where it will be n 2 a. But the deflection at this last point is CE = $ CD ( 100, I.). Therefore, n*a=CE, and CE a = r- n 2 Having thus found a, we have also the succeeding deflections 4 a, 9 a, 16 a, &c. Then laying oif at T 7 , T 7 ', &c., the angles A T M, A T' M', &c., each equal to ACD, and measuring down the proper deflections, just found, the points M, M', &c., of the curve will be determined. * Since C D is drawn to the middle of the base of the triangle A B (7, we have, by Geometry, CD* = (AC* + B C*) - AD*. LOCATING PARABOLIC CURVES. 91, The direction in which to measure the deflections may be ob- tained by dividing A D into the same number of equal parts as A C and joining corresponding points. If more convenient the chord A E may be drawn, and, being similarly divided, may take the place of A D. The curve may be finished by laying off on A C produced n parts equal to those on A (7, and the proper deflections will be, as before, a multiplied by the square of the number of parts from A. But an easier way generally of finding points beyond E is to divide the second 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 obvious that the deflections from both tan- gents 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 around the stakes. See also 112. 102. Problem. Given the tangents A C and B C, equal or unequal (fig. 4$}, and the chord A B, to lay out a parabola by middle ordinates. Fig. 48. Solution. Bisect A B in D, draw C D, and its middle point E will be a point on the curve ( 100, 1.). 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 B E, lay off the second middle ordinates G K and HL, each equal to D E ( 100, III.), and K and L are points on the curve. Draw the chords A K, KE, EL, and L B, and lay off third middle ordinates, each equal to one fourth the second middle ordinates, and four additional points on 92 PARABOLIC CURVES. the curve will be determined. Continue this process, until a suf- ficient number of points is obtained. 103. Problem. To draw a tangent to a parabola at any station. Solution. I. If the curve has been laid out by tangent deflec- tions ( 101), let M'" (fig. 47) be the station, at which the tangent is to be drawn. From the preceding or succeeding station, la\ off, parallel to CD, a distance M" N or EL equal to a, the first tangent deflection ( 101), and M'" N or M' " L will be the re- quired tangent. The same thing may be done by laying off from the second station a distance M' T' = 4 a, or at the third station a distance & P = 9 a ; for the required tangent will then pass through T' or 6r. It will be seen, also, that the tangent at M'" passes through a point on the tangent at A corresponding to half the number of stations from A to M '" ; that is, M'" is four sta- tions from A, and the tangent passes through T', the second point on the tangent A C. In like manner, M'" is six stations from B, and the tangent passes through 6r, the third point on the tangent BC. II. If the curve has been laid out by middle ordinates ( 102), the tangent deflection for one station is equal to the last middle ordinate made use of in laying out the curve. For if the tangent A C (fig. 48) were divided into four equal parts corresponding to the number of stations from A to E, the method of tangent de- flections would give the same points on the curve, as were ob- tained by the method of 102. In this case the tangent deflec- tion for one station would be a = -^ C E ^ 6 D E\ but the last middle ordinate was made equal to Gr K or ^ D E. Therefore, a is equal to the last middle ordinate, and a tangent 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 ordinate, by drawing a line through this extremity, parallel to the chord of that ordinate ( 100, IV.). 104. In laying out a parabola by the method in 101, it may sometimes be impossible or inconvenient to lay off all the points from the original tangents. A new tangent may then be drawn by 103 to any station already found, as at M'" (fig. 47), and the tangent deflections a, 4 a, 9 a, &c., may be laid off from this tan- gent, precisely as from the first tangent. These deflections must LOCATING PARABOLIC CURVES. 93 be parallel to C D, and the distances on the new tangent must be equal to T'N or N M' ", which may be measured. 105. Problem. Given the tangents A C and B C, equal or unequal (fig. 49), 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 will be a point on the curve (J5 100, V.). We have now two pairs of what may be called second tangents, A D and D E, and EF and FB. Bisect A D in G and 7) E in H, join G H, and its middle point M will be a point on Fig. 49. the curve. Bisect E F and F B in K and L, join KL, and its middle point N will be a point on the curve. We have now four pairs of third tangents, A O and G M, ME and HE, E TTand K N, and N L and L B. Bisect each pair in turn, join the points of bisection, and the middle points of the joining lines will be four new points, M', M", N". and N'. The same method may be con- tinued, until a sufficient number of points is obtained. 106. Problem. Given the tangents A C and B C, equal or unequal (fig. 50), and the chord A B, to lay out a parabola by intersections. Solution. Bisect A B in Z>, 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 C E, into the same number of equal parts; five, for example. Then the intersection M of A a and F Gr will be a point on the curve. For FM^Ca, and Ca ^CE. Therefore, F M = -% C E, which is the proper deflection from the tangent at F to the curve ( 101). In like manner, the intersection N of A b and HKwsij be shown to be a point on the curve, and the same is true of all the similar intersections indicated in the figure. 94 PARABOLIC CURVES. If the line D E were also divided into five equal parts, the line A a would be intersected in M on the curve by a line drawn from B through a', the line A b would be intersected in N on the curve Fig. 50. by a line drawn from B through b', and in general any two lines, drawn from A and B through two points on C D equally distant from the extremities C and Z>, will intersect on the curve. To show this for any point, as Jl, it is sufficient to show, that B a' produced cuts F Gr on the curve ; for it has already been proved, that A a cuts F G on the curve. Now Da 1 : MG = B D : B Gr = 5 : 9, or M G = f D a'. But D a' = J- C E. Therefore, M G = G:CD = Aa:AD = l:5. Therefore, F Q = We have then FM=FG MG = $CE F C E = -fa C E. As this is the proper deflection from the tan- gent at F to the curve ( 101), the intersection of B a' with F G is on the curve. This furnishes another method of laying out a parabola by intersections. 107. The following example is given in illustration of several of the preceding methods. Example. Given A C = B C - 832 (fig. 51), and AB = 1536, to lay out a parabola A E B. We here find CD 320. To be- gin with the method by tangent deflections (g 101), divide the C 1 W 1 fif) tangent A C into eight equal parts. Then a -^r = 2.5. Lay off from the divisions on the tangent F 1 = 2.5, G 2 = 4 x 2.5 = 10, HZ = 9 x 2.5 = 22.5, and K = 16 x 2.5 = 40. Sup- RADIUS OF CURVATURE. 95 pose now that it is inconvenient to continue this method beyond K. In this case we may find a new tangent at JE, by bisecting A C and B C ( 105), and drawing KL through the points of bi- section. Divide the new tangent K E = -J- A D = 384 into four equal parts, and lay off from K E the same tangent deflections as were laid off from A K, namely, M 5 = 22.5, N6 = 10, and 01 Fig. 51. 2.5. To lay off the second half of the curve by middle ordinates ( 102), measure EB = 784.49. Bisect E B in P, and lay off the middle ordinate PR = D E = 40. Measure E R = 386.08, and B R s = 402.31, and lay off the middle ordinates S T and V W, each equal to %PR = 10. By measuring the chords E T, T R, RW, and W B, and laying off an ordinate from each, equal to 2.5, four additional points might be found. ARTICLE II. RADIUS OF CURVATURE. 108. THE curvature of circular arcs is always the same for the same arc, and in different arcs varies inversely as the radii of the arcs. Thus, the curvature of an arc of 1,000 feet radius is double that of an arc of 2,000 feet radius. The curvature of a parabola is continually changing. In fig. 50, for example, it is least at the tangent point A, the extremity of the longest tangent, and in- creases 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 96 PARABOLIC CURVES. 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 arc, corresponding to any part of a parabola, is called the radius of curvature at that point. If a parabola forms part of the line of a railroad, it will be ne- cessary, in order that the rails may be properly curved ( 28), to know how the radius of curvature may be found. It will, in gen- eral, 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. 109. Problem. To find the radius of curvature at certain stations on a parabola. Solution. Let A E B (fig. 52) be any parabola, and let it be re- quired to find the radii of curvature at a certain number of sta- tions from A to E. These stations must be selected at regular Fig. 52. 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 of division. Thus, if n = 4, as in the figure, divide C D into four equal parts, and draw A F, A E, and A G. LetAD = c, AF = c 1 ,AE=c*,A G = c 3 ,&ndAC = T. De- note, moreover, CD by d, and the area of the triangle A C B by RADIUS OF CURVATURE. 9? A. Then the respective radii for the points E, 1, 2, 3, and A will be 7? 3 7? ClB 7? C ' 3 7? - Ca * 7? ^ 3 ^ = z> ^Z' * 4' ^'-Z' ^ 4 -T- The area J. may be found by form. 18, Tab. X. ; c and T are known ; and Ci, c a , c 3 may be found approximately by measure- ment on a figure carefully constructed, or exactly by these gen- eral formulae : 2 = c2 , c t s + T*-c* _ (n - 3) d n n* c 2 _ c 2 + r * ~ c2 _ (n ~ 5 ) ^ 2 71 7l 2 ' 2 - 2 T* -c* _ (n-T)d* C * ~~ * n 7i 2 ' &c., &c. It will be seen, that each of these values is formed from the pre- ^2 C 2 ^2 ceding, by adding the same quantity , and subtracting -^ multiplied in succession by n 1, n 3, n 5, &c. Making n = 4, we have 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 formulae is as follows : The radius of curvature at any given point on a parabola is, by the Differential Calculus, R = . 8 , 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. First, let the middle station E (fig. 53) be the given point. Then the angle E is the angle made with E D by a tangent at E, or since A B is parallel to the tangent at E ( 100, IV.), sin. E = sin. A D E = sin. B D E. Let p' be the parameter for the diam- 98 PARABOLIC CURVES. eter ED. Then, by Analytical Geometry, p =p' sin. 2 ^. There- fore, at this point R = ' = ~ A 7)2 r ^ _. = 3 . Therefore, R ED id ~ =, . But p' = 2 sin. E /.2 ,,% % C/ . - =. = ; since A dsin.fi cdsm.fi A c dsin.fi (Tab. X. 17). Next, to find Ri , or the radius of curvature at H, the first sta- tion from E. Through IT draw F G parallel to CD, and from F Fig. 53. draw the tangent F K. Join A K, cutting CD in L. Then from what has just been proved for the radius of curvature at E, we A z have for the radius of curvature at H, RI . 77 . A. JJ J\. Now A O : For, since AF = n-l:n, ov 1 x AL. x AC, the tangent deflection FH= ( 100, II.), and F G = 2 FH = Then. -l, C L = d. Hence L D = d- _ 1 1 n I d=- d, that is, ^ ^ ~ / ^ n n Substituting this value in the expression for A Gr above, we I . A Gr Moreover, since AF= - x A C, and n n cause 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 C D= C L: CD = n- 1 : n, or A CL = RADIUS OF CURVATURE. 99 x A CD. Therefore, A F G = ~ x A CD, and A FK = 2AFG = ( n ~ 1)S x A C B = ^ ~ 1)3 A. Substituting these if 1 6 /no A G* values of A G and A F K in the equation R l = AFK' an( * re " ducing, we find R\ = -j~ . By similar reasoning we should find It remains to find the values of e,, c a , &c. Through A draw A M perpendicular to CD, produced if necessary. Then, by Ge- ometry, we have AD* = AL* + LD*-2LD x LM, and AC 9 A L* + C L* + 2 C L x L M. Finding from each of these equations the value of 2LM, and putting these values equal to AL* + LD*-AD* AW-AL*- C L* each other, we have - ^r - -^-=. - . But AL = d, LD=-d,AD = c, AC = T, and CL = ^^ d. n n Substituting these values in the last equation, and reducing, we find f = T^ (n-l)c* _ (n-l)d^ n n n? By similar reasoning we should find _2T* (n-2)c* 2(n-2)d* C% - - + - -- a - 5 n n n 1 a , n n* &c., &c. From these equations the values of Ci 2 , c 2 2 , c 3 2 , &c., given above, are readily obtained. That given for Ci 2 is obtained from the first of these equations by a simple reduction ; that given for c a 2 is ob- tained by subtracting the first of these equations from the second, and reducing ; that given for c s 2 is obtained by subtracting the second equation from the third, and reducing ; and so on. 110. Example. Given (fig. 52) A C = T = 600, B C = T' = 520, and A D = c 550, to find R, R^ , J? 2 , 7? 3 , and jK 4 . the radii of curvature at E, 1, 2, 3, and A. To find CD = d, we have, by Geometry, d 9 = i(T* + T' 2 ) - c 2 which gives d* = 12700. 100 PARABOLIC CURVES. To find the area of A C B = A, we have (Tab. X. 18) A = a)(s b) (s c). 8= 1110 3.045323 s a = 590 2.770852 s 6 = 510 2.707570 8 c = 10 1.000000 2)9.523745 log. A 4.761872 Next 1 (T 2 - c) = i (T + c) (T-c) = 115 4 X 5 = 14375, and Then n* lo c 2 = 550 2 = 302500 ci = 302500 + 14375 - 3 x 793.75 = 314493.75 c a 2 = 314493.75 + 14375 - 793.75 = 328075 c a 2 = 328075 + 14375 + 793.75 = 343243.75. c 3 To find R, we have R = -? , or log. R = 3 log. c log. A, A c = 550 2.740363 c 3 8.221089 A 4.761872 R = 2878.8 3.459217 To find Ri , we have J?i = -j- , or log. Ji = - log. Ci 2 log. A, Cl 2 = 314493.75 5.497612 d 8 8.246418 A 4.761872 J2i = 3051.7 3.484546 In the same way we should find R* = 3251.5, R 3 = 3479.6, R* - 3737.5. To find the radii for the second part E B of the parabola, the same formula? apply, except that T' takes the place of T. We RADIUS OF CURVATURE. 101 have then - (T' 2 - c 2 ) = (T 1 + e) (T 1 - c) = 107 * ~ 3Q = n 4 - 8025. Hence c 1 2 = 302500 - 8025 - 2381.25 = 292093.75. c 2 2 = 292093.75 - 8025 - 793.75 = 283275. c 3 2 = 283275 - 8025 + 793.75 = 276043.75. c 3 S To find ^j, we have R l = -|- , or log. ^1=7; log. d 2 log. 4, -A tit d 2 = 292093.75 5.465523 d 3 8.198284 A 4.761872 7?! = 2731.6 3.436412 In the same way we should find R* = 2608.8, R 3 = 2509.5, R 4 = 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 ( 108). This will be the case, whenever the angle BCD, adjacent to the shorter tangent, exceeds 90, that is, whenever c 2 exceeds T' 2 + d?. If B CD = 90, the tangent point B falls on the vertex. It BCD is less than 90, one tangent point falls on each side of the ver- tex, and the curvature will, therefore, decrease towards both ex- tremities. 111. If the tangents Tand T' are equal, the equations for d 2 , c 2 2 , &c., will be more simple ; for in this case d is perpendicular to c. and T 2 c 2 = c? 2 . Substituting this value, we get = d 2 + - &c., &c. Example. Given, as in 107, T T' = 832, c = 768, and d = 320, to find the radii 7?, R^ , and R* at the points E, 4, and A (fig. 102 PARABOLIC CURVES. 51). Here A = c d = 245760, n = 2, and cj = c* + Then R = ^ = ^= ^ = 1843.2, ^ = , , and . cd d 320 cd d 2 = 615424 c d = 245760 fit = 1964.5 cd = 245760 ^ 2 = 2343.5 * = 615424. 5.789174 8.683761 5.390511 3.293250 2.920123 8.760369 5.390511 3.369858 Ri is the radius at the point R also, and J? a the radius at the point B. 112. Length of parabolic arcs. B The length s of the parabolic arc A B (fig. 54) from the vertex A to a point B whose rectangular coordinates are x and y is, by the Calculus, or, introducing the angle i which the tangent at B makes with the axis of x, x 2 s = -- [tan. i sec. i + hyp. log. (tan. i + sec. *)] ; or, by series, LENGTH OF PARABOLIC ARCS. 103 When y is small relatively to x, two terms of this series are often sufficient. Whence 2y* s = x -H Q nearly, o x The length s of the parabolic arc A B (fig. 55) from the origin of oblique coordinates A to a point B whose oblique coordinates are x and y, is given by the following formula, in which i is the Fig. 55. angle made by the tangent at B with a line perpendicular to the axis of the parabola, and j is the angle made by y with a perpen- dicular to the axis A X. x* cos. 2 //, . . i i tan. i + sec. i\ s = - = ( tan. i sec. i tan. j sec. j + hyp. log. : . ) 4 y \ to tan.^ + sec.j/' In many cases a near approximation is 2 y 2 cos. 2 / s = x + y sm.j + ^ 3 x + ysiu.j" 104 TRANSITION CURVES. CHAPTER III. TRANSITION CURVES. 113. THE object of a transition curve is to make the change easy from a straight line to a circular curve. The proper super- elevation of the outer rail of the circular curve is also arrived at by a gradual rise from the straight line. To make this rise uni- form, the radius of curvature of the transition curve must be in- finite at its beginning on the straight line, must decrease in such a way that, at any point of the curve, it shall be inversely as the distance of that point from the beginning, and, finally, become equal to the radius of the circular curve, where it joins that curve tangentially. The cubic parabola fulfils all the essential requi- sites of such a transition curve. The compound circular curve ( 132) forms another method of easing the change from a straight line to a circular curve. ARTICLE I. THE CUBIC PARABOLA. 114. Let GDC' (fig. 56) be the central circular curve of radius C = R. Let ABC and A' B' C' be the transition curves, con- necting the circular curve with the tangents at A and A '. Let x and y be the rectangular coordinates of A B C, with origin at A, and let Xi and yi denote the coordinates of the point C. Let the rise of the outer rail be taken as uniform for distances from A along the axis of x, instead of along the curve, an immaterial change, and let -. denote the rate of rise. Then the rise at any i/ distance x from A will be - . This rise may be expressed in an- other way. For let p denote the radius of curvature of the curve at the point whose abscissa is x, and we have the rise e by the for- mula of 152, e 9 V . Equating the two values, o2.53 p q v* i '=&!* THE CUBIC PARABOLA. 105 When the velocity v has been fixed, and also the rate of rise -7 , the quantity ~7 becomes a constant. At (7, the radius of curva- ture p becomes R, and x becomes Xi , so that equation (1) becomes and we have 7^7 = R %i . By substitution (1) becomes Another expression for p is, by the Differential Calculus, 106 TRANSITION CURVES. where d s is the differential of the length of the curve. In the present case, the differential d x of the abscissa is so nearly equal to d s, that we may put dx* ^dtf P ~ dxd*y~~ d*y' Equating the two values of p, and inverting, we have d*y _ x dz?~ Rx^ Integrating once, we have f*y dx and, integrating again, 115. This is the equation of a cubic parabola that is, of a curve in which the ordinates are proportional to the cubes of the ab- scissas. The curves ABC and A' B ' C' are, therefore, to be treated as cubic parabolas. Before doing this, however, two prob- lems require consideration. For in order to connect two straight lines or tangents, as A 1 and A' I, by a central circular curve, with a transition curve at each end, we have either to find A I = T y when the radius C = R of the circular curve is given, or to find R, when T is given. In both cases the intersection angle I is supposed to be known, and the value of Xi = A E to be assumed. 116. Problem. Given the intersection angle 1=2 G 01 (fig. 56\ the abscissa x lt and the radius C = R of the central curve, to find the tangent AI=T. Solution. In the figure the circular curve is produced to and R tan - A = ia; - : . * . A H = Xi -J- Xi = #1 .* Next to find ZT/, we have #/=: OHtsiu.iI=(R + 0IT) tan.i/. Gr His the perpendicular distance between the tangent A E and a tangent to the circular curve at G. This is usually called the shift, and may be denoted by s. To find Cr H s we have 5 = CE-GK=y l -GK. By equation (3) fti 3 _x^_ yi ' 6 R x, 6 72 ' and G Kis the middle ordinate of the circular curve for a chord = x l . Therefore, ( 26), G K = ^4r 5 so that O -ft Substituting this value ofs=GH in the equation for H J, we have JJ /= CK + i yi) tan. i 7. Finally, substituting the values found for A H and HI in the equation for T, we have T=%x l + (R + i 117. Problem. ^iVew the intersection angle I 2 Gr 1 (fig. 56\ the abscissa x^ and the tangent AI= T, to find the ra- dius C = R of the circular curve. * When thought necessary, A Hm&y be calculated accurately by the for- mula AH z\ - Rsin. A. t The formula GK=R(\ cos. A) gives the exact value of G K, but the difference is generally unimportant. 108 TRANSITION CURVES. Solution. From the preceding section we have Compute this value of R + 3/1, and from it subtract an assumed probable value of 3/1 . This will give an approximate value of x 2 R, and with this compute 3/1 by the formula i^/i 5-. If the value so found agrees nearly enough with the assumed value of 2/1 , the approximate value of R may be taken as the true value. Otherwise, a new approximation is to be computed. Gen- erally, however, the value of R thus found would be used only to select a convenient deflection angle for the central curve. The corresponding value of R may then be used to find, by section 116, a new value of T. A change in the value of T would of course change the position of the tangent point, but seldom materially. 118. Length of the abscissa Xi . Let us now consider the value to be given to x\ . The rate of rise of the outer rail being - , the total rise at the end of the transition curve will be -^ . This total rise is also expressed by e = Q( f p ( 152). Equating these values, ' we have = e, or Xi = i e. The length of Xi is, therefore, depend- i ent on i and e. The value of i may be taken as varying from 300 to 600, corresponding to grades of 17.6 feet to 8.8 feet per mile. The value of e depends upon the velocity of trains and the radius of the curve. For high speeds e may vary from e = .3 to e .5. A value of e = .5 allows a speed of 67 miles per hour on a 2 curve, of 30 miles per hour on a 10 curve, and of 25 miles per hour on a 14 curve ; so that this value of e would rarely be ex- ceeded. With i = 300, Xi need not exceed 150 feet, and with i = 600, x l need not exceed 300 feet. These lengths might of course in exceptional cases be increased. 119. Let the length of Xi be expressed in rail lengths of 30 feet each, and let n denote the number of such rail lengths. We shall then have x l = 3Qn. x 3 To express yi , we have from equation (3) yi = n-^ o a x\ THE CUBIC PARABOLA. 109 ?00n = 1BO . substituting for R its value, R = -^ , D be- 6 It t sin. u ing the deflection angle of the circular curve for chords of 100 150 n 2 sin. D feet, we have y\ = r^- , or ou y l = 3 n 2 sin. D. Fig. 56. To fix the position of the common tangent C F, we require the distance F E. The triangle CF E gives FE= i ( 116) tan. A = -- = 30 n 30 7i sin. D ' , and by lUU . ^ , ... = .3 n sm. D. Substitut- ing this value and that of 7/1 , we have 3fi 2 sin. D . ,., ^ FE = ^. =. .3 n sm. D 110 TRANSITION CURVES. 120. Method by Offsets. With R or /), T, x l , and y, known, the curves can now be laid out. J., the point of beginning or origin, is a fixed point, from which x l = 30 n is measured to fix the point E ; y l =3n' 2 sin. D fixes the point C ; and F E = $ Xi = 10 n fixes the position of the common tangent C F. Intermediate points on the transition curve are fixed by offsets or ordinates from the tangent A E, thus : divide A E into n equal parts and denote the successive offsets at the points of division by di , d* , d* , d n . Then d n = yi , and, as the ordinates are as the cubes of 7 Vi 3 n 2 sin. D 3 sin. D m , the abscissas, di = a = = . The successive - , n 9 n z n offsets are then #1=^!, d a = 8 d, , dj = 27 di , d n = yi . The circular curve CDC' is now run in the usual way from the tangent C F produced, with D as the deflection angle for 100 feet chords. The central angle of this carve is COC' = I2&. At C', E'C 1 should prove equal to y it The distance D I is equal to the ordinary external D L, increased by L I = (r //sec. i I \yi sec. /. The second transition curve A'B'C' is the same as ABC reversed, and is laid out in the same way. 121. The annexed table gives the necessary data for curves from 60 to 300 feet in length. D is the deflection angle of the central curve for 100 feet chords. For any other chord c it is only neces- 100 sary to multiply the values given for yi and di by . Thus if c D were the deflection angle for 50 feet chords, we should have y^ 6 ft 2 sin. D and di = . In computing y-i and di use nat- ural sines. TABLE A. n Ort .. y^ = 3 n a sin. D 3 sin. D dl n 2 60 12 sin. D 1.5 sin. D 3 90 27 sin. D 1. sin. D 4 120 48 sin. D .75 sin. D 5 150 75 sin. D .6 sin. D 6 180 108 sin. D .5 sin. D 7 210 147 sin. D f sin. D 8 240 192 sin. D f sin. D 9 270 243 sin. D i sin. /) 10 300 300 sin. D .3 sin. /> THE CUBIC PARABOLA. Ill It will be seen that this method applies directly, whether the central curve is of an even degree or not, since sin. D may be taken from the table for any value of Z>. 122. Example, when R or D is given. Given / = 72 40', D = 3 20', and n = 8. Here x, = 240, y l = 192 sin. 3 20' = 192 x .05814 = 11.16288. From Table I., R = 859.92, and yi = 2.79. First find T. R + iy 1= = 862.71 2.935865 i/=:36 20' tan. 9.866564 T - a?i = 634.496 2.802429 T- 754.496 Table A gives, for n = 8, d, = f sin. D = f x .05814 = .021802, and d, , multiplied in succession by 8, 27, 64, 125, 216, and 343, gives d* = .174, d 3 = .589, d* = 1.395, d 5 = 2.725, d* = 4.709, and d, - 7.478. To find A we have ( 119) tan. A = .3 n sin. D. For small an- gles we may put A = .3 n D. In this example A = 2.4 D 8, and the central angle of the circular curve / 2 A = 56 40'. This divided by 2 D gives 8.5, as the number of 100 feet chords from C to C'. * 123. Example, when T is given. Given I - 68 20', T = 764.3, and n = 5. Here x l = 150, and T i x l = 689.3. 689.3 2.838408 34 10' cot. 0.168291 # + iyi = 1015.5 3.006699 Comparing this approximate value of R with values given in Table I., we see that D = 2 50' might be selected as a convenient deflection angle. We have then R 1011.51, sin. D = sin. 2 50' = .04943, y, = 75 x .04943 = 3.70725, and R + iy, = 1012.44, to find the new T. 1012.44 3.005369 i/=34 10' tan. 9.831709 T - i x l = 687.19 2.837078 T= 762.19 "We next find di = .6 sin. D, and proceed as in the preceding example. 112 TRANSITION CURVES. 124. Method by Deflection Angles. The transition curve can also be laid out by deflection angles. These angles (fig. 57) are C a' Fig. 57. aAE,lAE,cAE, etc. Denote them by ^ , 8 2 , 5 3 , 8 n . Now cd' the tangent of any one of these angles, as 8 3 , is tan. 8 3 = -r-r, = y x* -. If in equation (3), which is y = ^5 , we divide both sides x o K x\ y x by x we have - = r-^ . This shows that the tangents of the de- X O Jit X\ flection angles are to each other as the squares of the abscissas. Now if a tangent be drawn to the curve at any point, as c, the tangent of the angle it makes with A E is by equation (2) -~ = x* r-^ . This is exactly three times the tangent of the deflection u 1 X\ angle just found for the same point. This relation being a gen- eral one, we have at 6 V , tan. C A E = tan. CFE or tan. 8 n = % tan. A. All these angles are ordinarily so small that the angles themselves may be substituted for their tangents. It follows that the deflection angles are to each other as the squares of the ab- scissas, and that 8 n = $ A. Taking A = .3 n Z>, as found above, nD , _ 8 n D m , we have 5 n = A = , and 81 = -r = T-T . The successive an- il) fr LOn gles to be laid off from A E with the transit at A are therefore 81 = . 8 2 45i , 5 3 = 9 5i , 5 n n 2 81 . The annexed 10 n table gives the necessary data for curves from 60 to 300 feet in length. D is the deflection angle of the central curve for 100 feet chords. For any other chord c multiply the values given by . THE CUBIC PARABOLA. 113 Thus if D were the deflection amgle for 50 feet chords, we should nD D nave A = .to n D, o n = , and $1 = - . 5 5 n TABLE B. n *=*nD 6- 6l = lL 2 .6D .2D JL/J 3 .9D .3D To ^ 4 1.2 D .4D To-^ 5 1.5 D .5D -Q D 6 1.8 D .6Z) fa D 7 2.1 D .7D Vo D 8 2.4 D .8D ^-D 9 2.7 D .9D ^D 10 3.0 D 1.0 D ^D 125. Example. Taking the data of the example in 122, we have n = 8, D = 3 20' = 200'. Table B, for n = 8, gives A = 2.4 D = 8, 8 = .8D = 2 40', and a x = ^D = 2'.5. Multiplying by the successive squares, 4, 9, 16, etc., we have 5i = 2 '.5, 8 2 = 10', 5 3 = 22'.5, 5 4 = 40', 8 5 = 1 2 .5, 8 6 = 1 30', S 7 = 2 2'.5. To lay out the circular curve, set the transit at <7, reverse from A, and from the line A C thus produced turn off an angle, to the left or right as the case may require, equal to 2 3 n . The line of sight will now be tangent to the circular curve. ARTICLE II. THE CUBIC PARABOLA APPLIED TO AN EXISTING CIRCULAR TRACK. 126. Let A'PQ (fig. 58) be the existing track of radius A' P R, and tangent at A' to A'L. From a point P on this curve a circular curve G C P of radius 0'P= R', less than R, is drawn, and having the same central angle as A'PQ. It has, therefore, its tangent Gr M parallel to A'L. A B C is a cubic parabola, running from a point A on the tangent of the original curve to a point C on the new circular curve. Produce 0' Gr to //, and draw the chords J/Pand Q- P. These chords are on the same straight line, because the angle PGM is half the central angle at and the angle PAL is half the equal central angle at ($ 2, III.). Now from the properties of the cubic parabola, al- ready explained ( 116), we know that A E = Xi may be taken as 9 114 TKANSITION CUKYES. X 2 bisected at H, and that the shift G H = s * , or putting 50 Xi = 3Qn (8 119), and for 72' its value - , we have s = sm. Z) f n 2 sin.7)', and y = E C = 4s = 3w 2 sin.Z>'. To obtain D' we O Fig. 58. have sin. D' : sin. D = R : R'. If we put R' m R,m being any sin. D assumed proper fraction, sin. D = . Now J.' is a fixed point on the ground, and if we find the dis- tance AH to the centre of Xi , the points A and E can be found by simply measuring $%i = 15 n each way from //. To fix the point P, A'L and PL must be found. Consider P M and (7iVto be tangent offsets to the curve G C P from the tangent G M, and we have, very closely, G M : G N CN A G : a P = ' : ' P - R - m R : m R = 1 - m : m . q-^-. Also, CN=EC-EN=4s-s = 3s , 1 m -.PM = PM THE CUBIC PARABOLA. 115 - . Substituting this value of 77-^. in the expression for o (1 Til) C JM / m / m G M, we have GM= & N A/ - - = 15 n A/ - -. V 3 (1 - w) r 3 (1 - m) NowAII:GM=00':0'P=l-m:m. .'.AH= 15 n (1 m) / m 1 m AJ ;. Squaring , and putting it m r o (1 m) m . under the radical, we have, after reduction, AH = 15 n A/ . V 3m Next, A'L : AH '= P: 0' = 1:1 m. . .A'L = - - = A/ - . Squaring the denominator 1 m, and put- 1 m \ 3m ting it under the radical, and reducing, we have A'L = I5n . Lastly, PL = PM + ML = S + s = 3 m (1 - m) 1 - m s 1 m' In deciding upon a proper value for m, it is obvious that R ' should not differ much from R. If we make m = .9, the change would not be too great. This value also simplifies the formulae very much. Making m = .9, we have !ndPL = - iQs = M O For the central angle GrO'C = A' of the transition curve, we have, as before ( 119), sin. A' = .3wsin. Z>', and for A = A' OP, A'L 50n |/3 SOnsin.Df 3 n . wehave 8 m.A = = ~^~ -^ - = p m.Dv3 = .3 n sin. D' t/3. The central angle of C P, the new circular curve, is C O'P = A A'. In the expressions for sin. A' and sin. A sub- stitute the angles themselves for their sines, and we have A' = 3n D' and A = .3 n D ' V 3 and A- A' = .3 nD' ( V 3 - 1) = .22 nD', nearly. 127. Table C gives the values of these expressions, and also those of yi and di for values of n from 2 to 10. As already shown, sin. D' = Y sin. D, or, more simply, D' * D. D and D' are deflection angles for 100 feet chords, but it is easy to modify the expressions for other chords. 116 TRANSITION CURVES. TABLE C. n *i A'H A'L y\ di PL A' A- A' 2 60 5.77 57.74 12 sin. D' sin. D' 2.5y, .6D' .44D' 3 90 8.66 86.60 27 sin. D' sin. D' 2.5T/! .9D' .66 IX 4 120 11.55 115.47 48 sin. D' | sin. D' 2.5 T/J 1.2D' .88D' 5 150 14.43 144.34 75 sin. D' sin. D' 2.5 y, 1.5 D' 1.10D' 6 180 17.32 173.21 108 sin. D' sin. Z>' 2.50J 1.8D' 1.32D' 7 210 20.21 202.07 147 sin. D' ? sin. D' 2.50! 2.1Z>' 1.54D' 8 240 23.09 230.94 192 sin. D' | sin. D' 2.60, 2 AD' 1.76D' 9 270 25.98 259.80 243 sin. D' sin. D' 2.5s/! 2.7 D' 1.98D' 10 300 28.87 288.68 300 sin. D' $s sin. D' 2.5 2/1 3.0D' 2.20 D' 128. Example. Given the deflection angle D = 3 of an exist- ing circular track A'P Q (fig. 58). We have for the deflection an- gle of the curve G C P, D ' = ^ D = 3 20'. Take a* = 150 feet, and we have from Table C, for n = 5, A'JT= 14.43, A'L = 144.34, y l = 75 sin. 3 20' = 75 x .05814 = 4.36, d, = .6 x .05814 = .03488, and P = 10.90. From the known tangent point A' of the existing track A'P Q, we measure 14.43 feet to H, and from H 75 feet each way to A and E. Then the point P is fixed by A'L = 144.34 and PL = 10.90. The transition curve is then put in by offsets from the tangent A E. Th^se offsets are d^ .03488, d z = 8 d* = .279, d 3 = 27 di = .942, d 4 = 64 d, = 2.232, d 5 = y l = 4.36. The central angle of the short circular curve OP is A A' 1.1 D' 3 40'. As the central, angle of this curve for a chord of 100 feet is 2 D\ the chord (7Pwill be the same part of 100 feet that 1.1 Z)' is of 2 D' or 55 feet, and if the work is correct, this will be the distance on the ground. A further check would be to find the tangent at C, and compute the proper offset to P. In regard to this check, it should be observed that the value PL 2.5^ is not exact, as it depends upon the assumption that C N : P M = a N* : & M \ which is not strictly true. PL may be computed accurately by the formula PL = R- OK=R- *J R* - A'L*. The radical under the form \/(R 4- A'L) (R A'L) is easily computed by logarithms. In the present case we should find PL 10.966. 129. Length of Curve in Terms of its Chords. The length of a transition curve, as measured by the sum of the chords used in laying it out, is slightly in excess of the abscissa x l . This excess is generally so small that it may be neglected. When, however, the curve is long, and the deflection angle of the circular curve THE CUBIC PARABOLA. 117 large, a method of calculating the excess may be desirable. Each chord is the hypothenuse of a right-angled triangle, whose base is 30 feet, and perpendicular the difference between two successive tangent offsets. These offsets are di , 8 di , 27 di , 64 d l , etc., and the successive differences or perpendiculars are di , 7 di , 19 di , 37 d l , etc. Let p denote any one of these perpendiculars, and for the corresponding chord c we have c = -\/30 2 + p 12 . By developing this radical, and retaining the first two terms only of the root, we have c = 30 + |- , nearly. Substituting for p its successive values, the excess of the first chord will be ^r, of the second chord, 49 dS , ,. ,, . , 361 dS ._ , of the third, ^7 , etc. For a curve of n chords we oU oU , 2 should have for e, the total excess, e = ^ (I 2 + 7 2 + 19* + 37* + etc.), the parenthesis containing always n terras of the series. For di substitute its value already found di = ( 120), D being the deflection angle of the circular curve for 100 feet chords, and we have, after reducing, e = >l0 s '* D (I 2 + 7 2 + 19 2 + 37 2 + etc.). If e is computed by this formula for D = 1, and different values of n. the excess for any other deflec- tion angle Di , and given n will be obtained, very closely, by multiplying the value so found for D = 1 and the given n by the square of the number denoting Z), in de- grees. The values of e for Z> = 1, and values of n from 2 to 10 have been calcu- lated, and the results placed in the annexed table, where e 2 is the excess for n = 2, e 9 the excess for n = 3, etc. 130. Example. Given the deflection angle of the circular curve = 3 = J, and n 6, to find the excess of the length of the transition curve measured by its chords over x\ . Here we multi- ply e 6 in the table by (|) 2 = 4 a , and we have the excess e = .01749 x \ a = .21425. For n = 6, x l = 180, so that the length of the curve by chords is 180.214. 118 TRANSITION CURVES. ARTICLE III. CURVING THE RAILS. 131. To secure the greatest ease of motion on a transition curve, it is of importance that the rails be properly curved. To do this we must have, as on a circular curve ( 28), the middle ordinate and the ordinates at the quarter points. We there found that the ordinates at the quarter points were each f m, m being the middle ordinate. Here we shall find that the ordinate at the first quarter point is slightly less than f m and the ordinate at the sec- ond quarter point slightly greater than f m. This is what might be expected from the gradual increase of the curvature. Let A G B (fig. 59) be a rail length on any part of a transition curve, and CD its projection on the axis of x. Let C be distant from the origin r rail lengths, and D distant r + 1 rail lengths, r being a whole or fractional number. Let d^ , as above, denote the tangent offset at the end of the first rail length from the origin. Then the offset A C = r 3 d l , and the offset B D = (r + I) 3 d l . The middle ordinate for curving the rail will be m = GF=EF EG. Now EF=i(A C + B D) = (r 3 + r 3 + 3r 2 + 3r + 1) = (r 8 + f r 2 + f r + $) ^ and E G (r + l) 3 ^ = (r 3 + f r 2 -+- f r + }) cZ v Subtracting and reducing, we have m = f (2 r + 1) d l . In a similar way the ordinates HI and KL at the quarter points are found. They are H I = (-!% r + { ) di = f m ^ 4 - ^ , K L = (-- s r _|_ |i) ^ | m + ^. ^ ^ If the curve does not begin at a joint, that part of a rail that comes on the curve may be curved by finding the proper tangent COMPOUND TRANSITION CURVE. 119 offset for its length, and bending the end from the straight line a distance equal to the offset. As the tangent offset for a whole rail is di, the offset for a fraction will be di multiplied by the cube of the fraction. Thus, if the fraction is .8 the offset would be .512 di . Except in extreme cases, this offset is so small that the rail remains practically straight. If the curve begins at a joint the middle ordinates for the suc- cessive rails will be obtained by making r successively 0, 1, 2, 3, etc. Denoting these ordinates by Wi, m a , m s , etc., we have m-i = f di , m-9 = di , m s = J 8 5 di , etc., or m l = f di , w 2 = 3 m l , m s = 5 wii, m 4 =7wi, etc. Taking three fourths of these ordinates, and subtracting and adding -fcdi, we have the quarter point ordinates. ARTICLE IV. COMPOUND TRANSITION CURVE. 13.2. Transition curves of this kind consist of successive circu- lar arcs, the deflection angles of which are such that if D is the deflection angle of the first arc, that of the second is 2 />, that of the third 3 Z>, and so on. The chords are all of the same length. A curve of this kind A B CD (fig. 60) may be readily laid out by offsets from the tangent A /, measuring at the same time the successive chords. Let c represent the length of each chord, n their number, and let D be the deflection for the first chord, 2 D that for the second chord, 3 D that for the third chord, and so on to the deflection angle ot the last chord, which will be n D. Then it is easily seen that the angles Ti AB,T*B C, T z CD, etc., will Fig. 60. be successively D, 4 D, 9 D, 16 Z), etc., up to n 8 D. Calling the re- quired offsets from the tangent A 7, di, d 2 , d 3 , etc., and recollect- ing that, since these angles are all small, we may put sin. 4 D = 4 sin. D, sin. 9 D 9 sin. Z), etc., we have di = c sin. D, d^ = di + 4 c sin. D d l + 4 di = 5 d l , d s = d* + 9 c sin. D = 5 d l + 9 di = 120 TRANSITION CURVES. 14 d l , etc.. the successive offsets being formed by multiplying di by the terms of the series 1, 5, 14, 30, 55, 91, etc., formed by the successive additions of the squares of the natural numbers. More accurate values of the offsets may be obtained thus. From the table of natural sines, set down in a column sin. 7), sin. 4 Z>, sin. 9 D, etc., up to sin. n* D. Then for d l , d% , d a , etc., multiply successively by c the first number so set down, the sum of the first two numbers, the sum of the first three numbers, and so on, until for d n multiply by c the sum of the whole column. The projections of the chords A TI, B T 2 , C T 3 , etc., may be found thus. A 1\ - c cos. D,BT^-c cos. 4 Z), C T 3 = ccos. 9D, etc. From the table of natural cosines, set down in a column cos. Z), cos. 4 D, cos. 9 D, etc., up to cos. n* D. Denote by p^ , p? , pa, etc., respectively, the first projection, the sum of the first two projections, the sum of the first three projections. Then to obtain Pi,p-i,p3, etc., multiply successively by c the first number in the column, the sum of the first two numbers, the sum of the first three numbers, and so on, until for p n multiply by c the sum of the whole column. 133. We have now to find (fig. 61) A I = T, when R the radius of the central curve is given, or to find R, when Tis given. In both cases the intersection angle / is supposed to be known, and the number n of chords in the transition curve to be assumed. 134. Problem. Given the intersection angle I and the ra- dius C R or the deflection angle D' of C M, the main or cen- tral curve (fig. 61), to find the deflection angle D for the first arc COMPOUND TRANSITION CURVE. 121 of the transition curve A C, the coordinates A E = a and E C = b of the point C, and the tangent A I. Solution. Let the number of chords in A C be denoted by n, and the length of each chord by c. C M is half the central curve, so that the angle H 1 = | /. Run C M back to 6r, where its tangent becomes parallel to A /, and draw G H and C K. De- note the deflection angle of the central curve for a chord equal to c by D'. This deflection angle is either given directly, or found from that given for a different chord. Then as D is the deflection angle of the first chord on A C, the deflection angle for the last chord will be n Z>, and for the first on C M, (n + 1) D = D ' Having D, we have also ( 132) di , d* , d 9 , etc. From the pre- ceding section, we have a = A E = c (cos. D + cos. 4 D + cos. 9 D + cos. n*D) = n c, nearly. & = E C = c (sin. D 4- sin. 4 D + sin. 9 D + sin. n*D) = di (1 + 4 + 9 + ft 2 ), nearly To find T we have T= A H + HI. Now A H= A E - HE - a R sin. COG. The angle C & is the sum of the.central an- gles of the seyeral arcs of A C. The central angle of the first arc is twice its deflection angle, or 2 Z>, that of the second arc is 2 x 2 Z), of the third 2x37), etc. Denote the sum of these angles by o, and we have a = 2 Z> (1 + 2 + 3 -f n) = n (n + 1) D. Therefore AH = AE HE = a R sin. a. Next, HI = H tan. HOI=(EC + OK) tan. \ 7, or HI- (l) + R cos. o) tan. 1 1. Substituting these values of A H and HI, we have B^~ T= a R sin. a + (& + R cos. a) tan. | J. An approximate formula for T, generally accurate enough in practice, may be found thus. Consider HE to be equal in length to the arc (r C and find the length of & C in chords of length c by dividing half its central angle or 4- a by its deflection angle D 1 = (n + 1) D. Hence HE = _ = nc and A j/ = (n 4- 1) 7) Also, 7T/= J^tan. i/ = TRANSITION CURVES. (R + G II) tan. - /. Omit G II as small relatively to R, and we have P. I R tan. 4- Z Substituting these values of J. H and # I in the formula T= A H + HI, we have T = i 7i c + R tan. /, nearly. 135. Example. Given 1 = 42, the deflection angle of the cen- tral curve = 2 for 100 feet chords, n = 5, and c = 30, to find the deflection angle D of the first arc of the transition curve A C (fig. 61), the coordinates a and b of the point C, and the tangent A I = T. Here the deflection angle of the central curve for 30 feet chords ' = 7^ x 2 = 36' and D = n + 1 6 csin.Z) = 30 x .001745 =: .05235. Computing by the exact for- mulae we find a = 149.956, b = 2.879, and T = 625.24. By the approximate formulae, we find a = 150, b = 2.879, and T = 624.85. 136. Problem. Given the intersection angle 7, and the tan- gent A 1= T, to find the radius C = R of the central curve CM (fig. 61). Solution. From the preceding section we have T= R tan. % I, nearly. t. | J, nearly. This approximate value of R may now be substituted in the exact formula for Tin the preceding section, and if the value of Tthus found does not change the tangent point too much, this value of COMPOUND TRANSITION CURVE. 123 R may stand, and D', D, and the other requisite data be com- puted. The principal inaccuracy in the formula for R is due to drop- ping Gr H in the expression for H 7, above. If we retain O 77, we should find R- (r-inc)cot.i/- Q H. To get a more accurate value of R, subtract G H, which may be computed by the formula G U=E C K G = b R(l cos. a). Generally, however, the approximate value of R would be used only for finding a convenient deflection angle for the central curve that is, one not involving seconds. A new value of R would result, and a new value of T would have to be computed. 137. To run the central curve C M, we must be able to fix the common tangent C F. This may be readily done if we find the distance F E. Now in the triangle C F E the angle C F E has its sides perpendicular to those of the angle C G, and is, there- fore, = a = n (n + 1) D. gy . . F'E = b cot. a = b cot. n(n + l) D. The central angle of the central curve will be 2@OM2a = I 2 n (n 4- 1) 7), and the number of chords will be found in the usual way by dividing the central angle by twice the deflection angle used in laying out the curve. 137. Remark. There are certain advantages in beginning a tran- sition curve at a joint. The ends of each rail would then be defi- nitely fixed by the offsets, and the rails could be more satisfactorily curved. It would be easier to maintain the track in its proper position, if the trackmen knew that the tangent point was at a joint, and when the rails were renewed, the new rails would be more likely to be properly curved, and placed in their true position. 124 LEVELLING. CHAPTEE IV. LEVELLING. ARTICLE I. HEIGHTS AND SLOPE STAKES. 138. THE Level is an instrument consisting essentially of a tele- scope, 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 ver- tical axis. The instrument when so adjusted is said to be set. The line of sight, being a line of indefinite length, maybe 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 ol the level. This distance is called the sight on that point. 139. Problem. To find the difference of level of two points, as A and B (fig. 62). 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 F P, A F will be the sight on A, and B P the sight on B. Then the required difference of level B Q = BP-PG= BP-AF. 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. 62), the level is first set between A and B, and sights * The level should be placed midway between the .two points, when prac- ticable, in order to neutralize the effect of inaccuracy in the adjustment of the instrument, and for the reason given in 148. HEIGHTS AND SLOPE STAKES. 125 Ed O are taken on A and B ; the level is then set between B and C, and sights are taken on B and C ; lastly, the level is set between C and Z), and sights are taken on C and D. Then the difference of level between A and DisED = (BP+KC+ D) - (AF + B I + NC). For J D = H C - L C = II M + M C - L C. But H M = B G = B P- A F, M C = KC-BI, and LC=NC- D. Substituting these values, we have ED = B P- AF + KC - BI-NC + OD= (BP + KG + OD)-(AF + BI + NC). 140. It is often convenient to refer all heights to an imaginary level plane called the datum plane. This plane may be assumed at starting to pass through, or at some fixed dis- tance 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 as- sume the datum plane at such a dis- tance below the bench-mark, that it will pass below all the points on the line to be levelled. Thus if A B (fig. 63) were part of the line to be lev- elled, 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 line was more than 15 or 20 feet below A, we might safely assume CD to be 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 126 LEVELLING. usually divided into regular stations, the height of each of which above the datum plane is required. 141. Problem. To find the heights above a datum plane of the several stations on a given line. Solution. Let A B (fig. 63) repre- sent a portion of the line, divided into regular stations, marked 0, 1, 2, 3, 4, 5, &c., and let C D represent the datum plane, assumed to be 25 feet below a bench-mark near A. Sup- pose the level to be set first between stations 2 and 3, and a sight upon the bench-mark to be taken, and found to be 3.125. Now as this sight shows that the plane of the level E F is 3.125 feet above the bench-mark, and as the datum plane is 25 feet be- low 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 height of the instrument, meaning by this the height of the line of sight of the instrument. If now a sight be taken on station 0, we shall obtain the height of this station above the datum plane, by subtracting this sight from the height of the instrument ; for the height of this station is C and QG=EC E 0. Thus if E = 3.413, C = 28.125 - 3.413 = 24.712. In like manner, the heights of stations 1, 2, 3, 4, and 5 may be found, by taking sights on them in succession, and subtracting these sights from the HEIGHTS AND SLOPE STAKES. 127 height of the instrument. Suppose these sights to be respective- ly 3.102, 3.827, 4.816, 6.952, and 9.016, and we have height of station = 28.125 - 3.413 = 24.712, " " 1 = 28.125 - 3.102 = 25.023, " " 2 = 28.125-3.827 = 24.298, " " " 3 = 28.125-4.816 = 23.309, " " " 4 = 28.125-6.952 = 21.173, " " 5 = 28.125 - 9.016 = 19.109. Next, set 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 GIC=@5 + 5K. Suppose this sight to be #5 = 2.740, and we have GK= 19.109 + 2.740 = 21.849. A point like station 5, which is used to get the height of the instrument after resetting, is called a turning point. The height of the instrument being found, sights are taken on stations 6, 7, 8, 9, 10, and the heights of these stations found by subtracting these sights from the height of the instrument. Suppose these sights to be respectively 3.311, 4.027, 3.824, 2.516, and 0.314, and we have height of station 6 = 21.849 - 3.311 = 18.538, " " 7 = 21.849-4.027 = 17.822, " " 8 = 21.849 - 3.824 = 18.025, " " " 9 = 21.849 - 2.516 = 19.333, " " " 10 = 21.849-0.314 = 21.535. The instrument is now again carried forward and reset, station 10 is used as a turning point to find the height of the instrument, and everything 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. 142. 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 in- 128 LEVELLING. strument. 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 subtract these sights from the height of the instrument, in order to get the required heights. These last sights may therefore be called minus sights. 143. The field notes are kept in the following form : The first column in the table contains the stations, and also the benches marked B., and the turning points marked t. p., except when co- incident 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. The height of the bench Station. + s. H.I. -S. H. B. 3.125 25.000 28.125 3.413 24.712 1 3.102 25.023 2 3.827 24.298 3 4.816 23.309 4 6.952 21.173 5 2.740 9.016 19.109 6 21.849 3.311 18.538 7 4.027 17.822 8 3.824 18.025 9 2.516 19.333 B. 2.635 19.214 10 0.314 21.535 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 add- ing the plus sight to the height of the turning point. This new height of the instrument is set opposite station 6, where the minus sights to be subtracted from it commence. These sights are again set opposite the points on which they were taken, and, being sub- HEIGHTS AND SLOPE STAKES. 129 tracted from the new height of the instrument, give the heights in the last column. 144. Problem. To set slope stakes for excavations and em- bankments. Solution. Let A B HK C (fig. 64) be a cross-section of a pro- posed excavation, and let the centre cut A M = c, and the width of the road-bed HK= b. The slope of the sides B H or C K is usually given by the ratio of the base KN to the height EN. Fig. 64. Suppose, in the present case, that K N : EN 3 : 2, and we have the slope = f . Then if the ground were level, as D A E, it is evi- dent that the distance from the centre A to the slope stakes at D and J would be A D = A E = M K + KN=%b + f c. But as the ground rises from A to C through a height C O = g, the slope stake must be set farther out a distance E G = f g ; and as the ground falls from A to B through a height B F = g, the slope stake must be set farther in a distance D F = f g. To find B and (7, set the level, if possible, in a convenient posi- tion for sighting on the points A, B, and C. From the known cut at the centre find the value ofAE=$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. Tcr A E add f g, as thus estimated, and measure from the centre a distance out, equal to the sum. Obtain now by the level the rise from the centre to this point, and if it agrees with the estimated rise, the distance out is correct. But if the estimated rise prove too great or too small, assume a new value for g, measure a corresponding distance out, and test the accuracy of the estimate by the level, as before. These trials must be continued, until the estimated rise agrees sufficiently well with the rise found by the level at the correspond- ing distance out. The distance out will then be^fc + fc + f*?. 10 130 LEVELLING. The same course is to be pursued, when the ground falls from the centre, as at B ; but as g here becomes minus, the distance out, when the true value of g is found, will be A F = A D D F =. iZ + c-f<7. For embankment, the process of setting slope stakes is the same as for excavation, except that a rise in the ground from the centre on embankments corresponds to a fall on excavations, and vice versa. This will be evident by inverting figure 64, which will then represent an embankment. What was before a fall 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 embank- ment, 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 neces- sary to make c in the expressions above minus, because its effect here is to diminish the distance out. The formula for this dis- tance out will, therefore, become & f c + f , tangent to the curve A B, is in the proper position to receive the light from an object at B ; so * Peirce's Spherical Astronomy, Chap. X., 125. It should be observed, 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. 132 LEVELLING. that this object appears to the observer to be at D. The effect of refraction, therefore, is to make an object appear higher than its true position. Then, since the correction for the earth's curvature D C and the correction for refraction D B are in opposite direc- tions, the correction for both will be B C D C D B. This correction must be added to the height of any object as deter- mined by the level. 147. Problem. Given the distance A D = D (fig. 66), the radius of the earth A E R, and the radius of the arc of re- fracted light = 7 R, to find the correction B C = d for the earth's curvature and for refraction. Fig. 66. Solution. To find the correction for the earth's curvature D C, we have, by Geometry, D C(D C + 2EC) = A Z> 2 , or D C(D C +' 2 R) = Z) 2 . 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 D C x 2 R = D\ or D C = ~. The correction & i for refraction D B may be found by the method just used for finding D (7, merely changing R into 7 R. Hence D B = We have then d = BC = DC- d = Z> 2 2R' D* UR UR' , or 1R By this formula Tab. VIII. is calculated, taking R 20,911,790 ft., as given by Bowditch. The necessity for this correction may VERTICAL CURVES. 133 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. ARTICLE III. VERTICAL CURVES. 148. Vertical curves are used to round off the angles formed by the meeting of two grades. Let A C and CB (fig. 67) be two grades meeting at C. 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 C B may be denoted re- spectively by g and g' ; that is, g denotes what is added to the height at every station on A (7, and g' denotes what is added to the height at every station on C B ; but since C B is a descending grade, the quantity added is a minus quantity, and g 1 will there- fore be negative. The parabola furnishes a very simple method of putting in a vertical curve. 149. Problem. Given the grade g of A C (fig. 7), the grade g' of C B, and the number of stations n on each side of C to the tangent points A and B, to unite these points by a parabolic verti- cal curve. Fig. 67. Solution. Let A EB be the required parabola. Through B and C draw the vertical lines F K and C H, and produce A C to meet F K 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 H H K, and con- sequently A D D B. The vertical line C D is, therefore, a di- ameter of the parabola (g 100, I.), and the distances of the curve in a vertical direction from the stations on the tangent A F are 134 LEVELLING. to each other as the squares of the number of stations from A ( 100, II.). Thus, if a represent this distance at the first station from A, the distance at the second station would be 4 a, at the third station 9 a, and at B, which is 2 n stations from A, it would -im T> be 4n 2 a; that is, F B 4n' 2 a, or a = r~^ . To find a, it will 4 n then be necessary to find F B first. Through C draw the hori- zontal line C 6r, and we have, from the equal triangles C F O and A C II, FO CH. But C II is the rise of the first grade g in the n stations from A to (7; that is, C II = ng, or F O = ng. OB is also the rise of the second grade g' in n stations, but since g' is negative ( 148), we must put OB ng'. Therefore, F B = F O + O B = ng ng'. Substituting this value of F B in the n a n a' equation for a, we have a y g , or * *** The value of a being thus determined, all the distances of the curve from the tangent A F, viz. a, 4 a, 9 a, 16 a, &c., are known. Now if Tand T' be the first and second stations on the tangent, and vertical lines TPand T'P' be drawn to the horizontal line AK, the height T P of the first station above A will be g, the height T'P' of the second station above A will be 2g, and in like manner for succeeding stations we should find the heights 3g,4g, &c. As we have already found TM=a, T'M' = 4a, &c., we shall have for the heights of the curve above the level of A, M P = TP TM=g a, M'P' T'P' T'M' = 2g 4a, and in like manner for the succeeding heights Sg 9 a, 4g 16 a, &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) = g a, (2g 4 a) (g a) = g 3 a, (3g The successive grades for the vertical curve are, therefore, g a, g 3 a, g 5 a, g 7 a, &c. In finding these grades, strict regard must be paid to the algebraic signs. The results are then general ; though the figure represents but one of the six cases that may arise from various combinations VERTICAL CURVES. 135 of ascending and descending grades. If proper figures were drawn to represent the remaining cases, the above solution, with due at- cention to the signs, would apply to them all, and lead to precisely the same formulae. 150. Examples. Let the number of stations on each side of G be 3, and let AC ascend .9 per station,,and C B descend .6 per station. Here n = 3, g = .9, and g' = .6. Then, a 9 ^- = .9 (6) _ 1^? _ tl 25 } an d the grades from A to B will be 4 x o \& g a = .9 .125 = .775, g - 3 a = .9 - .375 = .525, g- 5a= .9- .625 = .275, g- 7a^.9- .875 = .025, g - 9 a = .9 - 1.125 = - .225, g 11 a .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 sta- tions on each side of C as the extent of the curve. Here g = .8, g' = .4, and n = 2. Then a = ^ -^- jr- .15, and the four grades required will be g-a = - .8 - (- .15) = - .8 + .15 = - .65, g _ 3 a = - .8 - (- .45) = - .8 + .45 = - .35, 0_5 a= _.8- (- .75) = _ .8 + .75 = - .05, g - la = - .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 nega- tive, each grade after the first is .3 greater than the preceding grade. 151. The grades calculated for the whole stations, as in the fore- going examples, are sufficient for all purposes except for laying the track. The grade stakes being then usually only 20 feet apart, it will be necessary to ascertain the proper grades on a vertical curve for these sub-stations. To do this, nothing more is neces- sary than to let g and g' represent the given grades for a sub-sta- tion of 20 feet, and n the number of sub-stations on each side of 136 LEVELLING. the intersection, and to apply the preceding formulae. In the last example, for instance, the first grade descends .8 per station, or .16 every 20 feet, the second grade ascends .4 per station, or .08 every 20 feet, and the number of sub-stations in 200 feet is 10. We have then g = .16, g' .08, and n 10. Hence a = L - -~ = 24 ' = .006. The first grade is, therefore, g a .16 -4- .006 = .154, and as each subsequent grade increases .012 ( 150), the whole may be written down without farther trouble, thus : - .154, - JL42, - .130, - .118, - .106,' - .094, - .082, - .070, - .058, - .046, - .034, - .022, - .010, + .002, + .014, + .026, + .038, + .050, + .062, + .074. ARTICLE IV. ELEVATION OF THE OUTER RAIL ON CURVES. 152. Problem, Given 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 of mass M moving on a curve of radius J?, with a velocity per second = v, has, by Mechanics, a centrifugal M v* force = 77- . To counteract this force, the outer rail on a curve H 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 ac- tion 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 in- clined plane is equal to 32.2 M multiplied by the ratio of the height to the length of the plane. But the height of the plane is the ele- vation e, and its length the gauge of the track g. This action of gravity, which is to counteract the centrifugal force, is, therefore, 32 2 Me . Putting this equal to the centrifugal force, we have 32.2 Me Mv* - = = . Hence g R eog- r- gv -32^ZT If we substitute for R its value ( 10) R j. , we have e = ~ -00062112 g v* sin. D. If the velocity is given in miles ELEVATION OF THE OUTER BAIL ON CURVES. 137 V x 5280 per hour, represent this velocity by V, and we have v -^ bl) x bO Substituting this value of v, we find e = .0013361 g F 2 sin. D. When g = 4.7, this becomes e .00627966 F 2 sin. D. By this for- mula Table VII. is calculated. In determining the proper eleva- tion in any given case, the usual practice is to adopt the highest customary speed of passenger trains as the value of V. 153. 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 prob- ably improper " coning " of the wheels. Two wheels, immovable on an axle, and of the same radius, must, if 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 trav- erse 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 r' represent the proper radii of the inner and outer wheels respectively, when the flange of the outer wheel touches the rail. Then r' r will be the coning for one inch in breadth of the tire. To 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, propor- tional 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 r : r' R : R + g. Therefore, rR + rg = r' R, or 138 LEVELLING. As an example, let R = 600, r = 1.4, and g = 4.7. Then we 14x47 have r' r = ' = .011 ft. For a tire 3.5 in. wide, the con- bUU ing would be 3.5 x .011 = .0385 ft., or nearly half an inch. 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 counter- acted for any given velocity by elevating the outer rail. The sec- ond is the unequal length of the two rails of a curve, in conse- quence 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 was devised for this purpose ; but as the coning, when at all considerable, was found to produce an irregular sidewise motion of the train, the tendency has been to diminish the coning. The standard wheel- tread adopted by the Master Car Builders' Association has a con- ing of but iV of an inch in 2| inches of the tread next to the flange. ARTICLE V. EASING GRADES ON CURVES. 154. When a curve occurs on a steep grade it is desirable to ease the grade on the curve, so as to make the joint resistance of the grade and curve equal to that of the grade alone on straight lines. The resistance on a grade is proportional to the rise of the grade per station and the resistance due to a curve can be repre- sented as equivalent to that of a grade having a certain rise per station. The rise per station of the eased grade will be simply the original rise diminished by the rise that represents the curve resistance. The resistance caused by curves varies greatly with the state of the track and the kind of rolling stock, and is vari- ously estimated as equivalent on a 1 curve to the resistance of a grade of .025 to .06 of a foot per station. For a curve of any other degree the resistance increases with the degree ; so that a 6 curve, for example, has six times the resistance of a 1 curve. As an example let a rise of .04 per station be taken as the resist- ance on a 1 curve and suppose a 6 curve to occur on a grade of 1.6 per station. Then the reduced grade will be 1.6 .24 1.36 per station. EXPANSION OF BAILS. 139 ARTICLE VI. EXPANSION OF RAILS. 155. The rails of a track exposed to a summer sun may rise to a temperature of 130 Fahrenheit. When, therefore, a track is laid at a much lower temperature, as is usual, provision for the expan- sion of the rails must be made by leaving a proper space between successive rails. The expansion of a bar of iron or steel may be taken as .000 007 of its length for every degree of rise in tempera- ture. The space to be left between the rails will vary with the length of the rails and with the number of degrees below 130 of the temperature when the track is laid. Suppose 30-feet rails are laid at a temperature of 50. Then the number of degrees of possible rise of temperature is 130 50 = 80, and the space to be left between the rails is .000 007 x 80 x 30 = .0168 of a foot. In general, let s be* the space to be left between the rails, n the number of degrees that the temperature is below 130, and I the length of the rails in feet, and we have s = .000 007 n I. A convenient rule for 30-feet rails may be obtained by putting in the formula I = 30 and n = 5, whence, nearly enough, 8 = .001. That is, the space to be left is one-thousandth of a foot for every five degrees that the temperature is below 130. 140 EARTH-WORK. CHAPTER V. EARTH-WORK. ARTICLE I. PRISMOIDAL FORMULA. 156. EARTH-WORK includes the regular excavation and embank- ment on the line of a road, borrow-pits, or such additional excava- tions 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 prismoid 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. 157. Problem. Given the areas of the parallel faces B and B ' , the middle area M, and the altitude a of a prismoid, to find its solidity S. Solution. The middle area of a prismoid is the area of a sec- tion 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. Ifb represents the base of a regular wedge or half- parallelopipedon of altitude a, its solidity is $ a b. If b represents the base of a pyramid of altitude , its solidity is a b. The so- lidity of these three bodies admits of a common expression, which may be found thus : Let m 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, m = b, in the regular wedge, m = ^b 9 and in the pyramid, m = $b. Moreover, the upper base of the prism = , and the upper base of the wedge or pyramid = 0. Then the expressions a b, -J- a b, and $ a b may be thus transformed. Solidity of prism = ab = x6b=(b + b + 4b} = (b+b + 4 m), BORROW-PITS. 141 wedge = pyramid = i ab = | x 26 = | (0 + b + b) = | (0 + 6 + 4 m). 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-paral- lelopipedons, may be measured by the same rule, since they are the sum or difference of a regular wedge and a pyramid of com- mon altitude, and as the rule applies to both these bodies, it ap- plies to their sum or difference. Now a prismoid, being made up of prisms, wedges, and pyra- mids 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 pris- moid; and the sum of the middle areas of the combined solids is equal to M, the middle area of the prismoid. Therefore ARTICLE II. BORROW-PITS. 158. FOR the measurement of small excavations, such as borrow- pits, &c., the usual method of preparing the ground is to divide the surface into parallelograms * or triangles, small enough to be considered planes, laid off from a base line, that will remain un- touched by the excavation. A convenient bench-mark is then se- lected, 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 re- ferred to the same bench-mark are taken at all the angles. This method divides the excavation into a series of vertical prisms, generally truncated at top and bottom. The vertical edges of these prisms are known, since they are the differences of the * 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 ob- tained at once in cubic yards, by merely omitting the factor 27 in the calcu- lation. 142 EARTH -WORK. levels at the top and bottom of the excavation. The horizontal section of the prisms is also known, because the parallelograms or triangles, into which the surface is divided, are always meas- ured horizontally. 159. Problem. Given the edges )i, hi , and h* , to find the solidity S of a vertical prism, whether truncated or not, whose horizontal section is a triangle of given area A. Fig. 68. Solution. When the prism is not truncated, we have h = hi=. hi . The ordinary rule for the solidity of a prism gives, therefore, S= Ah = A x $(h + hi + h?). When the prism is truncated, let ABCFGH (fig. 68) represent such a prism, truncated at the top. Through the lowest point A of the upper face draw a hori- zontal plane AD E cutting off a pyramid, of which the base is the trapezoid B D E C, and the altitude a perpendicular let fall from A on D E. Represent this perpendicular by^>, and we have (Tab. X. 52) the solidity of the pyramid = lp x BDEC Ipx DE x i(BD + CE) = $p xDEx%(BD + C E} = A x (BD + CE\ since $p xDE = ADE=A. Eut^(BD + C E) 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 ap- BORROW-PITS. 143 ply, if the lower end also were truncated. Hence, for the solidity of the whole prism, whether truncated or not, we have S=A *k(h + hi + hi). 160. Problem. Given the edges h, hi , h* , and h s , to find the solidity S of a vertical prism , whether truncated or not, whose horizontal section is a parallelogram of given area A. Solution. Let B H (fig. 69) represent such a prism, whether truncated or not, and let the plane B F HD divide it into two Fig. 69, triangular prisms AFH and C FH. The horizontal section of each of these prisms will be A, and if h, hi, h*, and A$ repre- sent the edges to which they are attached in the figure, we have for their solidity ( 159) AFH=$Ax$(h + h l + h) 9 and CFH=%A x $(hi + h? + h 9 ). Therefore, the whole prism will have for its solidity S = iAx^(h + 2h l + h y + 2A S ). Let the whole prism be again divided by the plane A E G C into two tri- angular prisms BEG and D E G. Then we have for these prisms, BEG^A x $(h + Ai + 7* 2 ), and DEG = \A x $(h + h* + A 3 ), and for the whole prism, S = $A x ^(2h + h^ +2^ 2 + h 3 ). Adding the two expressions found for S, we have 2 S = $ A (h + hi + hi + h 3 ), or S = A x J (h f hi + /la + hs). 144 EARTH-WOJRK. It will be seen by the figure, that i (h + /i, 2 ) = KL = $(hi + // 3 ), or Ji + h 2 = hi -+- 7*3. The expression for S might, therefore, be reduced to S = A x | (h + /i 2 ), or S = J. x |(/h + ^ 3 ). But as the ground surfaces ABCD and E F Gr H are seldom perfect planes, it is considered better to use the mean of the four heights, instead of the mean of two diagonally opposite. 161. Corollary. When all the prisms of an excavation have the same horizontal section A, the calculation of any number of them may be performed by one operation. Let figure 70 be a plan a* b+ fa a. Fig. 70. of such an excavation, the heights at the angles being denoted by a, i , a 2 , b, bi , &c. Then the solidity of the whole will be equal to I A multiplied by the sum of the heights of the several prisms (160). Into this sum the corner heights a, a*, b, b 5 , c 6 , d, and d 4 will enter but once, each being found in but one prism ; the heights i , & 4 , c, di, d^, and d s will enter twice, each being com- mon to two prisms ; the heights bi , b 3 , and c 4 will enter three times, each being common to three prisms ; and the heights & 2 , Ci , C 2 , and c 3 will enter four times, each being common to four prisms. If, therefore, the sum of the first set of heights is represented by Si , the sum of the second by s 2 , of the third by s a , and of the fourth by s 4 , we shall have for the solidity of all the prisms S = i A (i + 2 5 2 + 3 s 3 + 4 4 ). CENTRE HEIGHTS ALONE GIVEN. 145 ARTICLE III. EXCAVATION AND EMBANKMENT. 162. As embankments have the same general shape as excava- tions, it will be necessary to consider excavations only. The sim- plest case is when the ground is considered level on each side of the centre line. Figure 71 represents the mass of earth between two stations in-an excavation of this kind. The trapezoid O B F II is a section of the mass at the first station, and G l Si FI Hi a sec- tion at the second station ; A E is the centre height at the first station, and AI EI the centre height at the second station ; H Hi Fi F is the road-bed, G Gi BI B the surface of the ground, and G Gi HI H and B BI FI F the planes forming the side slopes. This solid is a prismoid, and might be calculated by the prismoid- al formula ( 157). The following method gives the same result. A. Centre Heights alone given. 163. Problem. Given the centre heights c and Ci , the width of the road-bed &, the slope of the sides s, and the length of the section Z, to find the solidity S of the excavation. Solution. Let c be the centre height at A (fig. 71) and Ci the height at AI . The slope s is the ratio of the base of the slope to its perpendicular height ( 144). We have then the distance out AB \~b + sc, and the distance out AI BI == | b + s Ci ( 144). Divide the whole mass into two equal parts by a vertical plane A AI EI E drawn through the centre line, and let us find first the 11 146 EARTH- WORK. solidity of the right-hand half. Through B draw the planes JBEEi, BAiEi, and BEi$\, dividing the half -sect ion into three quadrangular pyramids, having for their common vertex the point B, and for their bases the planes A A 1 E 1 E J EE 1 F 1 F, and AiBiFiEi. For the areas of these bases we have Area of A A 1 E^ E = i E E^ x (A E + A l EJ = %l(c + d), " E EI F^F EF x EEi = iH " " A l JB, F l Ei = iA l E l x (Ei F l + A : B^} = | (b c, + s d 2 ), and for the perpendiculars from the vertex B on these bases, pro- duced when necessary, Perpendicular on A AI EI E = Then (Tab. X. 52) the solidities of the three pyramids are sc) x ^?(c + c 1 ) = i?(i&c + S C 2 + S C Ci B-A 1 B 1 F 1 E 1 = $1 x i(6d + 5C! 2 ) = 1 (b d + s d Their sum, or the solidity of the half -section, is iS=-bl[$b(c + c,) + s(c 2 + d 2 + cci)]. Therefore the solidity of the whole section is When the slope is 1| to I, s = f, and the factor f s = 1 may be dropped. 164. Problem. To find the solidity S of any number n of successive sections of equal length. Solution. Let c, Ci , c a , c s , &c., denote the centre heights at the successive stations. Then we have ( 163) Solidity of first section Z [6 (c + d) + f 5 (c 2 + d 3 + c d)], " " second section = % I [b (d + c a ) + f s (d 2 + c a 2 + d e*)), " third section = i I [ b (c, + c.) + f s (c 2 2 + c 3 2 + c 2 c 3 )], &c. & 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 CENTRE HEIGHTS ALONE GIVEN. 147 as those just given. The last centre height, according to the nota- tion adopted, will be represented by c n , and the next to the last by c n i. Collecting the terms multiplied by b into one line, the squares multiplied by f s into a second line, and the remaining terms into a third line, we have for the solidity of n sections (c + 2 d + 2 c a + 2 c 3 ..... S = When s = f , the factor f s = 1 may be dropped. 2 c n _i + c n ) 2 C*! + c 2 n ) Example. Given I = 100, b = 28, s = f , 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 suc- cessive products c Ci, CxCa, &c., and place them in the fourth col- umn. 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 f 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 i I 50 gives for the solidity 163,700 cubic feet. Station. c. c 3 . cc l . 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 28 2408 592 2408 2)3274 163700. 148 EARTH-WORK. B. Centre and Side Heights given. 165. When greater accuracy is required than can be attained by the preceding method, the side heights and the distances out ( 144) are introduced. Let figure 72 represent the right-hand side of an excavation between two stations. AAiBiB is the ground surface ; A E = c and AI EI = GI are the centre heights ; B G = h and B l G^ h^ the side heights ; and d and d l , the dis- tances out, or the horizontal distances of B and BI from the centre line. The whole ground surface may sometimes be taken as a plane, and sometimes the part on each side of the centre line may be so taken ; * but neither of these suppositions is sufficiently ac- curate 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 B to A i , from which the ground slopes downward on each side to A and BI . Instead of this, a depression might run from A to BI , and the ground rise each way to AI and B. If the ridge or de- pression 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, there- fore, that the following method is applicable to all ordinary ground. When, however, the ground is very irregular, the method of 171 is to be used. 166. Problem. Given the centre heights c and Ci , the side heights on the right h and h\ , on the left h 1 and h'i , the distances out on the right d and di , on the left d' and d\ , the width of the * It is easy in any given case to ascertain whether a surface like A A l B l B is a plane ; for if it is a plane, the descent from A to B will be to the de- scent from A l to 1?,, as the distance out at the first station is to the distance out at the second station ; that is, c - h : c t h l = d : d l . If we had c = 9, h = 6, Ci = 12. hi = 8, d = 24, and c?j = 27, the formula would give 3:4 = 24 : 27, which shows that the surface is not a plane. CENTRE AND SIDE HEIGHTS GIVEN. 149 road-bed b, the length of the section Z, and the direction of the diagonals, to find the solidity S of the excavation. Solution. Let figure 72 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 AI. Through B draw the planes BEE^ BAiEt, and BE 1 F 1 , dividing the half-section into three quadrangular pyramids, having for their common vertex the point B, and for their bases the planes A AI EI E, E E\ FI F y and AiBiFiJZi. For the areas of these bases we have Areaof AA 1 E 1 E = %EEi x (AE + A^Ei) =iZ(c + Ci), ^F =EF x and for the perpendiculars from the vertex B on these bases, pro- duced when necessary, Perpendicular on A A l E l E = E 6r = d, =h, A . Fig. 72. V"~" : G Then (Tab. X. 52) the solidities of the three pyramids are Their sum, or the solidity of the half-section, is (1) 150 EARTH- WORK. Next, suppose that the diagonal runs from A to Bi . In this case, through BI draw the planes B^EiE, BiAE, and B t E F (not represented in the figure), dividing the half-section again into three quadrangular pyramids, having for their common ver- tex the point B\, and for their bases the planes A AI EI E, E EI F L jP, and A B F E. For the areas of these bases we have Area of AAiEiE=$EEi x (A E + A, E,} = $l(c + E E\FiF = EF x EE l = iK " " ABFE = %AE x d + \EF x h = $dc + and for the perpendiculars from BI on these bases, produced when necessary, Perpendicular on A AI E t E EI G 1 = d 1} " E E l FiF = Bi Gi = hi, " " ABFE = E Ei = I Then (Tab. X. 52) the solidities of the three pyramids are Bi A A! Ei E = di x il (c + d) = I (di c + d l d), Bi EE^Fi F=%li l x i ~bl lllh^ Bi- ABFE =$1 x 1 (d c + 1 b h) = J Z (d c + i 5 ft). Their sum, or the solidity of the half -section, is + di d + di c + b hi + i ~b h). (2) We have thus found the solidity of the half -section for both di- rections of the diagonal. Let us now compare the results (1) and (2), and express them, if possible, by one formula. For this pur- pose let (1) be put under the form I [d c + di d + d d + | & (h + Ai + h)], and (2) under the form Z [d c + di Ci + di c + i b (h + ^i + 7^)]. The only difference in these two expressions is, that d ^ and the last h in the first, become d l c and hi in the second. But in the first case c\ and h are the heights at the extremities of the diago- nal, and d is the distance out corresponding to h ; and in the sec- ond case c and hi are the heights at the extremities of the diago- nal, and di 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 CENTRE AND SIDE HEIGHTS GIVEN. 151 aide height H by D. We may then express both d c and di c by D (7, and both h and h^ by H\ so that the solidity of the half- section on the right of the centre line, whichever way the diago- nal runs, may be expressed by DC + (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 to them to distinguish them. Thus the side heights are h' and h\, and the distances out d' and d\ , while Z>, (7, and H be- come D', C', and //'. The solidity of the half -section on the left may therefore be taken directly from (3), which will become D'C' (4) Finally, by uniting (3) and (4), we obtain the following formula for the solidity of the whole section between two stations : S= d')e D C+D' C 1 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 c\ 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 h' = 8 to Ci = 8 and from c = 13.6 to h, = 12. Sta. d '. h'. c h. d. d + d'. (d + d')c. D' C'. D C. 1 21 8\ IS 15 4 ^ 8 .6^ 10 2 .0 ^12 2 4 45 7 42 612 336 1 168 367.2 12 12 168 20 367.2 54 x ( 1 = 486 6)1969.20 32820. 152 EARTH- WORK. 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 diag- onal) by d = 8 (which is the centre height touched by the same diagonal) is placed in the ninth column, and the similar product of d^ 27 by c = 13.6 is placed in the last column. The terms in the formula multiplied by b are all the side heights, and in ad- dition all the side heights touched by diagonals, or 8 4- 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, By applying the rule given in the note to 165, we see that the surface on the left of the centre line in the preceding ex- ample 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 revers- ing the diagonal in this example, and calculating the solidity anew. The only parts of the formula affected by the change are D' C" and \IH '. In the one case the sum of these terms is 21 x 8 + 9 x 8, and in the other 15 x 13.6 + 9x4, both of which are equal to 240. 167. Problem. To find the solidity S of any number n of successive sections of equal length. Solution. Let c, d , c 2 , c 3 , &c., be the centre heights at the suc- cessive stations; h, 7^, A a , h a ,&c. 9 the right-hand side heights; h', h'i , h'z , h' a , &c., the left-hand side heights ; d t d lt d 9l d ai &c., the distances out on the right; and d', d' L , d' 9 , d' 3 ,&c., the distances out on the left. Then the formula for the solidity of one section (g 166) gives for the solidities of the successive sections %l[(d + d') c + (d l + d\) d + D C + D' C' + i b (h + h, + H + h f + h', + H')l i I [(d, + d' 3 ) c* + (d, + d' 9 ) c, + DiC* + D\ C\ //a + h'i + h' 3 + /J' 2 )], and so on, for any number of sections. For the solidity of any CENTRE AND SIDE HEIGHTS GIVEN. 153 number n of sections, we should have I multiplied by the sum of n parentheses formed as those just given. Hence DC + DC 1 + DiCi-\-D' l ( ib h + 2 hj. + 2 ^ + "i + // 2 + &c. Example. Given I = 100, b = 28, and the remaining data as given in the first six columns of the following table : Sta. d'. fc'. c. h. d. d + d'. (d + d') c. D'C'. DC. 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 (K /6\ ^fl 26 49 294 115 130 4 21.5 5^ 2* >7 24.5 46 276 129 147 5 20 4-^ ^^ ^4 20 40 240 120 147 6 15.5 !>- 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 ( 166), 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 + d') 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 D'C' and DC, which give respectively 605 and 639. To obtain the first line of the quantities multiplied by &, add all the numbers in column ^, 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 diago- nals), making 37. The second line of the quantities multiplied by ^ & is obtained in the same way from the column marked h'. The sum of these numbers is 171, and this multiplied by 5 = 14 gives 154 EARTH- WORK. 2394. We have now for the first line of the formula 1389 + 1185, for the second 605 + 639, and for the remainder 2394. By adding 100 these together, and multiplying the sum by I = ~~ , we get the contents of the six sections in feet. 168. When the section is partly in excavation and partly in embankment, the preceding formulae are still applicable ; but as this application introduces minus quantities into the calculation, the following method, similar in principle, is preferable. 169. Problem. Given the widths of an excavation at the road-bed AF=w and AiFi=Wi (fig. 73), the side heights h and hi , the length of the section Z, and the direction of the diago- nal, to find the solidity S of the excavation, when the section is partly in excavation and partly in embankment. Solution. Suppose, first, that the surface is divided into two triangles by the diagonal B A \ . Through B draw the plane B AI Fi , dividing that part of the section which is in excavation into two pyramids B AAi F l F and B A l B^ F^ , the solidi- ties of which are B- The whole solidity is, therefore, S = i I (w h + w l 7h + Wi h). Next, suppose the dividing diagonal to run from A to BI . Through BI draw a plane BiAF (not represented in the figure), dividing the excavation again into two pyramids, of which the solidities are CENTRE AND SIDE HEIGHTS GIVEN. 155 B l A Ai F! F Bi-ABF = x i I (w + x + w l hi), The whole solidity is, therefore, S = I (w h + Wi Jii +w hi). The only difference in these two expressions is, that w\ h in the first becomes whi in the second. But in the first case the diago- nal touches Wi and A, and in the second case it touches w and ^i . If, then, we designate the width touched by the diagonal by TF, and the height touched by the diagonal by H, we may express both wih and wh t by WH\ so that the solidity in either case may be expressed by S = %l (w h + Wi ^ + WH). Corollary. When several sections of equal length succeed one another, the whole may be calculated together. For this pur- pose, the preceding formula gives for the solidities of the succes- sive sections 1 (Wi ^ + W* hi + TTi Hi), J I (wi h^ + w 9 h- 3 + Wi Hi), and so on for any number of sections. Hence for the solidity of any number n of sections we should have Wi Hi + &c.) Example. Given I = 100, and the remaining data as given in the first three columns of the following table : Station. w. fc. wh. WH. 2 /I 2 1 8< 6 48 8 2 10\ \7 70 56 3 13^ >s? 91 70 4 9 \4 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 156 EARTH-WOKK. 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. 170. When the excavation does not begin on a line at right an- gles to the centre line, intermediate stations are taken where the excavation begins on each side of the road-bed, and the section may be calculated 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 h in this case become 0, and reduce the formula to S = ^lwihi. The same remarks apply to the end of an excavation. C. Ground very Irregular. 171. Problem. To find the solidity of a section, when the ground is very irregular. Solution. Let A HB F E - A l CD B l F^ E l (fig. 74) 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 Fig. 74. GROUND VERY IRREGULAR. 157 changes at IT, C, and D, making it necessary to divide the surface into five triangles running from station to station.* Let heights be taken at H, C, and D, and let the distances out of these points be measured. If now we suppose the earth to be excavated verti- cally downward through the side line B BI 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 A C, II C, HD, and HBi . Then the solidity of the half-section will be equal to the sum of these prisms, minus the triangular mass B F G B^Fi G\. 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 159. To find the solidity of the portion BFG B 1 F 1 G 1 , which is to be deducted, represent the slope of the sides by s ( 144), the heights at B and BI by h and hi , and the length of the section by 1. Then we have F G = sh, and FI Gi = s h\ . Moreover, the area of BFG = $sh\ and that of B l F 1 O l = ish^. Now as the triangles B F G and B l FI G^ are similar, the mass required is the frustum of a pyramid, and the mean area is y\ sh* x - s h^ = Then (Tab. X. 53) the solidity is B F G - BI F l G i = Example. Given I 50, b = 18, s f , the heights at A, H, and B respectively 4, 7, and 6, the distances A H 9 and HB = 9, the heights at AI , (7, D, and BI respectively 6, 7, 9, and 8, and the distances A i C = 4, CD = 5, and DB t = 12. Then the horizon- tal section of the first prism adjoining the centre line is 1 1 x A i (?, since the distance A\ C is measured horizontally; and the mean of the three heights is (4 + 6 + 7) = i x 17. The solidity of this prism is therefore -J- 1 x AI C x x 17 = \ I x 4 x 17, that is, equal to I 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 \l (4 x 17 + 9 x 18 + 5 x 23 + 12 x 24 + 9 x 21) = \ I x 822. * It will often be necessary to introduce intermediate stations, in order to make the subdivision into triangles more conveniently and accurately. 158 EARTH- WORK. For the frustum to be deducted, we have i Z x f (6 2 + 8 2 4- 6 x 8) = $1 x 222. Hence the solidity of the half-section is I (822 - 222) = $ x 50 x 600 = 5000 cubic feet. 172. Let us now examine the usual method of calculating ex- cavation, 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 average of these two heights the mid- dle area of the mass ; and, lastly, in applying the prisraoidal for- mula to find the contents. The heights of the equivalent sections level at the top may be found approximately by Trautwine's Dia- grams,* 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 &, 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 x (2 b + 2 s x} = s x* + bx. But this area is to be equal to A. Therefore, sx* + bx = 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 166 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 f and b = 18, the equations for finding the heights of the equivalent end sections will be f x* + 18 x = 387, and f x* + 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 -(11.146 + 8) = 9.573, and from this height the middle area is found to be 309.78. Then by the prismoidal formula ( 157) the solidity will be S = fc 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 * A New Method of Calculating the Cubic Contents of Excavations and Embankments by the aid of Diagrams. By John C. Trautwine. CORRECTION IN EXCAVATION ON CURVES. 159 at the ends to the height of the equivalent end sections, the inter- vening earth might be so disposed as to form a plane between these level ends, thus reducing the mass to a prismoid. This sup- position, 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 166 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. * 173. In excavations on curves the vertical planes forming the ends of a section are not parallel 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 B, B Fig. 75. chord of the curve. Thus, let figure 75 represent the plan of two sections of an excavation, EF G being the centre line, A L and C M the extreme side lines, and the centre of the curve. 160 EARTH- WORK. Then the calculation of the first section would include all be- tween the lines AI Ci and BiDi', while the true section lies between A C and B D. In like manner, the calculation of the second section would include all between H K and N P, while the true section lies between B D and L M. It is evident, there- fore, that at each station on the curve, as at F, the calculation is too great by the wedge-shaped mass represented by KFDi, and too small by the mass represented by B^ F 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 D F R E (fig. 76). But if the excavation is on the side of a hill, so that the distances out differ very Fig. 76. much, and the cross-section is of the shape A D F B E, the difference of the wedge-shaped masses may require considera- tion. 174. Problem. Given the centre height c, the greatest side height h, the least side height h, the greatest distance out d, the least distance out d', and the width of the road-bed b, to find the correction in excavation C, at any station on a curve of radius Tir or deflection angle D. Solution. The correction, from what has been said above, is a triangular prism of which B F R (fig. 76) is a cross-section. The height of this prism at B (fig. 75) is B l H, the height at R is R, S, and the height at F is 0. BI H and RI S, being very short, are here considered straight lines. Now we have the cross-section h). To find the height B^ H, we have the angle BFH= BFB l = D, and therefore B^ H 2 HFsin. D = 2 d sin. D. In like manner, 7^ S = K D l = 2 K F sin. D = 2d' sin. D. Then since the height at F is 0, one third of the sum of the heights of the prism will be f (d + d') sin. D, and the cor- rection, or the solidity of the prism, will be ( 159) CORRECTION IN EXCAVATION ON CURVES. 161 h)] x f (d + d') sin. D. When R is given, and not D, substitute for sin. D its value ( 9) sin. D -^ . The correction then becomes . This correction is to be added, when the highest ground is on the convex side of the curve, and subtracted, when the highest ground is on the concave side. At a tangent point, it is evident, from figure 75, that the correction will be just half of that given above. Example. Given e = 28, h = 40, h' - 16, d = 74, d = 3S,b = 28, and R = 1400, to find G. Here the area of the cross-section OQ OQ B FR = Y (74 - 38) + ~ (40 - 16) = 672, and one third of the . 100 (74 + 38) 8 sum of the heights of the prism is ^ = - . Hence C _ o X 1.4UU o 672 x | = 1792 cubic feet. o 175. When the section is partly in excavation and partly in em- bankment, 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 ex- tending from B to D (fig. 76), will extend from B to a point be- tween G and E, or to a point between A and G. In the first case, the correction will be a triangular prism lying between the lines B! F and H F (fig. 75), but not extending below the point F. In the second case, the excavation extends below F, and the cor- rection, as in 173, is the difference between the masses above and below F. This difference may be obtained in a very simple man- ner, by regarding the mass on both sides of F as one triangular prism the bases of which intersect on the line G F (fig. 76), in which case the height of the prism, at the edge below F must be considered to be minus, since the direction of this edge, referred to either of the bases, is contrary to that of the two others. The solidity of this prism will then be the difference required. 176. Problem. Given the width of the excavation at the road-bed w, the width of the road-bed b, the distance out d, and 12 162 EARTH-WORK. the side height h, to find the correction in excavation (7, at any station on a curve of radius R or deflection angle D, when the section is partly in excavation and partly in embankment. 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, i w h. The height of this prism at B (fig. 76) is ( 174) Bi H 2 H F sin. D = 2 d sin. D. In a similar manner, the height at E will be 2 Q E sin. D = b sin. D, and at the point in- termediate between (? and E, the distance of which from the cen- tre line is \ 1} w, the height will be 2(%b w) sin. D= (b 2w) sin. D. Hence, the correction, or the solidity of the prism, will be ( 159) C = i w h x i (2 d + ~b + I - 2 w) sin. D = | w h x j (d + i _ w } s j n> 2). 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 excava- tion. Its area will, therefore, be \ w h. The height of this prism at B is also 2dsin.Z), and the height at E, bsin.D; but at the point intermediate between A and G, the distance of which from the centre line is w i b, the height will be 2(w $b) sin. D = (2 w b) sin. D. As this height is to be considered minus, it must be subtracted from the others, and the correction required will be C\wJi x $(2d + b 2w + b) sm.D = $wh x f (eZ + b - w) sin. D. Hence, in all cases, when the section is partly in excava- tion and partly in embankment, we have the formula C = i w h x f (d + b w) sin. D. When R is given, and not Z>, substitute for sin. D its value ( 9) 50 sin. D = . The correction then becomes . o _n, This correction is to be added, when the highest ground is on the convex side of the curve, and subtracted when the highest ground is on the concave side. At a tangent point the correction will be just half of that given above. Example. Given w = 17, b = 30, d = 51, h = 24, and R = 1600, to find C. Here the area of the cross-section NOTE ON THE COMPUTATION OF EARTH- WORK. 163 12 204, and one third of the sum of the heights of the prism is 100(d + 6~w)_100(51 + 30-17)_4 _ 4 _ 3# 3x1600 ~3' X 3~ 272 cubic feet. 177. The preceding corrections ( 174 and 176) suppose the length of the sections to be 100 feet. If the sections are shorter, the angle B FH(fi.g. 75) 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 E F is 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. NOTE ON THE COMPUTATION OF EARTH-WORK. 178. The mode of computing earth- work on railroads by first finding equivalent level- top sections has already been examined in 172, and the assumption made in applying the prismoidal formula is shown to lead to possibly serious errors. Another as- sumption that forms the basis of many formula?, tables, and dia- grams, is that the natural surface of the ground of such a section as that calculated in 166 is a warped surface or hyperbolic parab- oloid. The solidity is then computed by the prismoidal formula. Computing the section just referred to on this assumption, we find the solidity 31 210 feet. Now we have seen in 172 that, with the diagonal running in one direction, the solidity is 32 820 feet, and, with the diagonal running in the other direction, the solidity is 29 600 feet. The assumption of a warped surface gives, therefore, an exact mean between these two results, being 1,610 feet too much or too little, according to the direction of the diag- onal. Errors so great would not perhaps be common ; but they are at least possible. The objection to these methods is that they involve general as- sumptions as to the natural surface of the ground assumptions that the engineer cannot readily test in the field for each section, or allow for, if seen to be wrong. No method would seem to be reasonably correct that does not require all the data used in the computation to be obtained directly in the field. Now the division of the ground into triangular planes, whether four as in 166, or more as in 171, satisfies this condition. Since three points de- termine a plane, it is comparatively easy to decide on the ground 164 EARTH-WORK. what heights should be adopted at the vertices, so that a triangu- lar plane shall be a fair average of the ground. Suppose the ground cross-sectioned in the usual way, and the actual cats marked on the stakes and recorded. These cuts remain to guide the contractor in his work ; but the engineer is to examine each triangle, and see whether these cuts require any correction in order to obtain a fair average* of the surface. As he goes from section to section, two of the heights or cuts would in general be already fixed, and, standing at the third vertex, he readily deter- mines whether the actual cut there should stand, or have one, two, three, or more tenths added or subtracted. The correction, if any, may be noted in small figures over the actual cut, and applied when the heights are taken off for the computations. Some additional labor is doubtless involved in thus obtaining directly all the data required, and dispensing with all general as- sumptions ; but if justice to the contractor and to the company require such additional labor, the engineer will not hesitate on that account. The computations, as arranged in 167, will be found, after a little practice, to admit of very rapid work. Of course, only final estimates require so much care. In preliminary estimates, where centre heights alone are taken, the method of 164 will be found sufficiently accurate, and if the computations are arranged as there shown, the work will be found very expeditious. In many cases where only approximate results are aimed at, especially in making the usual " monthly estimates," the method of averaging end areas may be employed. This method consists in finding the areas of the two cross-sections which bound a section of an excavation, and multiplying the aver- age of these areas by the length of the section to obtain the con- tents of the section. TABLE I. RADII, ORDINATES, TANGENT DEFLECTIONS, AND ORDINATES FOR CURVING RAILS. This table applies directly only to curves laid out with 100 feet chords. With shorter chords, it may still be made useful. When 50 feet chords are used with a deflection angle half that for 100 feet chords, the radius of the curve is so slightly shortened, that, for the purpose of finding the new ordinates and tangent deflec- tions from Table I., the curve is practically the same as when laid out with 100 feet chords. The change in the radius is easily found. Let D be the deflection angle for 100 feet chords, and XA Kf\ we have ( 10 and Tab. X., 22) R = -^ = . , r-= = ^ sm.D 2 sin. | Z> cos. D , and for Ri , the radius for 50 feet chords, RI = sin. -I- .D cos. \D' 05 . = R cos. \D. In a 12 curve, where R = 478.34 and D 6, we have R l = Rcos.3 = 478.34 x .99863 = 477.68. Now in the same curve the ordinates ( 27) and the tangent deflections ( 19) are to each other as the squares of the chords; that is, for 50 feet chords these quantities are one-fourth of those given in Table I. for 100 feet chords. The ordinates for curving 30 feet rails will, of course, be unchanged. In the present example the , , , 2.620 , 1.965 ordinates would be . = .655 and - = .491, the tangent de- flection = 2.613, and the ordinates for curving 30 feet rails .235 and .176. With 25 feet chords and a deflection angle of 1-J- we should have the radius R* = R cos. 3 cos. 1, and the ordinates and tangent deflection one-sixteenth of those in Table I., while the ordinates for curving 30 feet rails would still be unchanged. This curve, strictly speaking, could no longer be called a 12 curve. The new degree, here about 12 1', might be found, or the curve might be designated by the radius ; but the most con- venient and definite designation would be : Deflection angle 3 for 50 feet chords, or deflection angle 1 for 25 feet chords. TABLE I. RADII, ORDINATES, TANGENT DEFLECTIONS, De- gree. Radius, 10. Ordinates, 25. Tangent Deflec- tion, 19. Curving 30-ft. rails, 29. De- gree. m. fl*. m. |m. Infinite. .000 .000 .000 .000 .000 o / 2 171887.35 .007 .005 .029 .001 .000 2 4 85943.67 .015 .011 .058 .001 .001 4 6 57295.79 .022 .016 .087 .002 .001 6 8 42971.84 .029 .022 .116 .003 .002 8 10 34377.48 .036 .027 .145 .003 .002 10 28647.91 .044 .033 .175 .004 .003 12 14 24555.35 .051 .038 .204 .005 .003 14 16 21485.94 .058 .044 .233 .005 .004 16 18 19098.62 .065 .049 .262 .006 .004 18 20 17188.76 .073 .055 .291 .007 .005 20 22 15626.15 .080 .060 .320 .007 .005 22 24 14323.97 .087 .065 .349 .008 .006 24 26 13222.13 .095 .071 .378 .009 .006 26 28 12277.70 .102 .076 .407 .009 .007 28 30 11459.19 .109 .082 .436 .010 .007 30 32 10743.00 .116 .087 .465 .010 .008 32 34 10111.06 .124 .093 .495 .011 .008 34 36 9549.34 .131 .098 .524 .012 .009 36 38 9046.75 .138 .104 .553 .012 .009 38 40 8594.41 .145 .109 .582 .013 .010 40 42 8185.16 .153 .115 .611 .014 .010 42 44 7813.11 .160 .120 .640 .014 .011 44 46 7473.42 .167 .125 .669 .015 .011 46 48 7162.03 .175 .131 .698 .016 .012 48 50 6875.55 .182 .136 727 .016 .012 50 52 6611.12 .189 .142 .'756 .017 .013 52 54 6366.26 .196 .147 .785 .018 .013 54 56 6138.90 .204 .153 .814 .018 .014 56 58 5927.22 .211 .158 .844 .019 .014 58 1 5729.65 .218 .164 .873 .020 .015 1 2 5544.83 .225 .169 .902 .020 .015 2 4 5371.56 .233 .175 .931 .021 .016 4 6 5208.79 .240 .180 .960 .022 .016 6 8 5055.59 .247 ,185 .989 .022 .017 8 10 4911.15 .255 .191 1.018 .023 .017 10 12 4774.74 .262 .196 1.047 .024 .018 12 14 4645.69 .269 .202 1.076 .024 .018 14 16 4523.44 .276 .207 1.105 .025 .019 16 18 4407.46 .284 .213 1.134 .026 .019 18 20 4297.28 .291 .218 1.164 .026 .020 20 22 4192.47 .298 .224 1.193 .027 .020 22 24 4092.66 .305 .229 1.222 .027 .021 24 26 3997.48 .313 .235 1.251 .028 .021 26 28 3906.64 .320 .240 1.280 .029 .022 28 30 3819.83 .327 .245 1.309 .029 .022 30 32 3736.79 .335 .251 1.338 .030 .023 32 34 3657.29 .342 .256 1.367 .031 .023 34 36 3581.10 .349 .262 1.396 .031 .024 36 38 3508.02 .356 .267 1.425 .032 .024 38 40 3437.87 .364 .273 1.454 .033 .025 40 42 3370.46 .371 .278 1.483 .033 .025 42 44 3305.65 .378 .284 1.513 .034 .026 44 46 3243.29 .385 .289 1.542 .035 .026 46 48 3183.23 .393 .295 1.571 .035 .026 48 50 3125.36 .400 .300 1.600 .036 .027 50 52 3069.55 .407 .305 1.629 .037 .027 52 54 3015.71 .415 .311 1.658 .037 .028 54 56 2963.72 .422 .316 1.687 . .038 .028 56 58 2913.49 .429 .322 1.716 .039 .029 58 AND OBDINATES FOB CURVING BAILS. De- gree. Radius, 10. Ordinates, 25. Tangent Deflec- tion, 19 Curving 30-ft. rails, 29. De- gree. m. m. m. f m. / 2 2864.93 .436 .327 1.745 .039 .029 / 2 2 2817.97 .444 .333 1.774 .040 .030 2 4 2772.53 .451 .338 1.803 .041 .030 4 6 2728.52 .458 .344 1.832 .041 .031 6 8 2685.90 .465 .349 1.862 .042 .031 8 10 2644.58 .473 .355 1.891 .043 .032 10 12 2604.51 .480 .360 1.920 .043 .032 12 14 2565.65 .487 .365 1.949 .044 .033 14 16 2527.92 .495 .371 1.978 .045 .033 16 18 2491.29 .502 .376 2.007 .045 .034 18 20 2455.70 .509 .382 2.036 .046 .034 20 22 2421.12 .516 .387 2.065 .046 .035 22 24 2387.50 .524 393 2.094 .047 .035 24 26 2354.80 .531 '.398 2.123 .048 .036 26 28 2322.98 .538 .404 2.152 .048 .036 28 30 2292.01 .545 .409 2.181 .049 .037 30 32 2261.86 .553 .415 2.211 .050 .037 32 34 2232.49 .560 .420 2.240 .050 .038 34 36 2203.87 .567 .425 2.269 .051 .038 36 38 2175.98 .575 .431 2.298 .052 .039 38 40 2148 79 .582 .436 2.327 .052 .039 40 42 2122.26 .589 .442 2.356 .053 .040 42 44 2096.39 .596 .447 2.385 .054 .040 44 46 2071.13 .604 .453 2.414 .054 .041 46 48 2046.48 .611 .458 2.443 .055 .041 48 50 2022.41 .618 .464 2.472 .056 .042 50 52 1998.90 .625 .469 2.501 .056 .042 52 54 1975.93 .633 .475 2.530 .057 .043 54 56 1953.48 .640 .480 2.560 .058 .043 56 58 1931.53 .647 .485 2.589 .058 .044 58 3 1910.08 .655 .491 2.618 .059 .044 3 2 1889.09 .662 .496 2.647 .060 .045 2 4 1868.56 .669 .502 2.676 .060 .045 4 6 1848.48 .676 .507 2.705 .061 .046 6 8 1828.82 .684 .513 2.734 .062 .046 8 10 1809.57 .691 .518 2.763 .062 .047 10 12 1790.73 .698 .524 2.792 .063 .047 12 14 1772.27 .705 .529 2.821 .063 .048 14 16 1754.19 .713 .535 2.850 .064 .048 16 18 1736.48 .720 .540 2.879 .065 .049 18 20 1719.12 .727 .545 2.908 .065 .049 20 22 1702.10 .735 .551 2.938 .066 .050 22 24 1685.42 .742 .556 2.967 .067 .050 24 26 1669.06 .749 .562- 2.996 .067 .051 26 28 1653.01 .756 .567 3.025 .068 .051 28 30 1637.28 .764 .573 3.054 .069 .052 30 32 1621.84 .771 .578 3.083 .069 .052 32 34 1606.68 .778 .584 3.112 .070 .053 34 36 1591.81 .785 .589 3.141 .071 .053 36 38 1577.21 .793 .595 3.170 .071 .053 38 40 1562.88 .800 .600 3.199 .072 .054 40 42 1548.80 .807 .605 3.228 .073 .054 42 44 1534.98 .815 .611 3.257 .073 .055 44 46 1521.40 .822 .616 3.286 .074 .055 46 48 1508.06 .829 .622 3.316 .075 .056 48 50 1494.95 .836 .627 3.345 .07'5 .056 50 52 1482.07 .844 .633 3.374 .076 .057 52 54 1469.41 .1851 .638 3.403 .077 .057 54 56 1456.96 .858 .644 3.432 .077 .058 56 58 1444.72 .865 .649 3.461 .078 .058 58 168 TABLE I. RADII, ORDINATES, TANGENT DEFLECTIONS, De- gree. Radius, 10. Ordinates, 25. Tangent Deflec- ti.n,19. Curving 30-ft. rails, 29. De- gree. m. Im m. fm. 4 1432.69 .873 .655 3.490 .079 .059 4 2 1420.85 .880 .660 3.519 .079 .059 2 4 1409.21 .887 .665 3.548 .080 .060 4 6 1397.76 .895 .671 3.577 .080 .060 6 8 1386.49 .902 .676 3.606 .081 .061 8 10 1375.40 .909 .682 3.635 .082 .061 10 12 1364.49 .916 .687 3.664 .082 .062 12 14 1353.75 .924 .693 3.693 .083 .062 14 16 1341118 .931 .698 3.723 .084 .063 16 18 1332.77 .938 .704 3.752 .084 .063 18 20 1322.53 .946 .709 3.781 .085 .064 20 22 1312.43 .953 .715 3.810 .086 .064 22 24 1302.50 .960 .720 3.839 .086 .065 24 26 1292.71 .967 .725 3.868 .087 .065 26 28 1283.07 .975 .731 3.897 .088 .066 28 30 1273.57 .982 .736 3.926 .088 .066 30 32 1264.21 .989 .742 3.955 .089 .067 32 34 1254.98 .996 .747 3.984 .090 .067 34 36 1245.89 1.004 .753 4.013 .090 .068 36 38 1236.94 1.011 .758 4.042 .091 .068 -38 40 1228.11 1.018 .764 4.071 .092 .069 40 42 1219.40 1.026 .769 4.100 .092 .069 42 44 1210.82 1.033 .775 4.129 .093 .070 44 46 1202.36 1.040 .780 4.159 .094 .070 46 48 1194.01 1.047 .786 4.188 .094 .071 48 50 1185.78 1.055 .791 4.217 .095 .071 50 52 1177.66 1.062 .796 4.246 .096 .072 52 54 1169.66 1.069 .802 4.275 .096 .072 64 56 1161.76 1.076 .807 4.304 .097 .073 56 58 1153.97 1.084 .813 4.333 .097 .073 58 5 1146.28 1.091 .818 4.362 .098 .074 5 2 1138.69 1.098 .824 4.391 .099 .074 2 4 1131.21 1.106 .829 4.420 .099 .075 4 6 1123.82 1.113 .835 4.449 .100 .075 6 8 1116.52 1.120 .840 4.478 .101 .076 8 10 1109.33 1.127 .846 4.507 .101 .076 10 12 1102.22 1.135 .851 4.536 .102 .077 12 14 1095.20 1.142 .856 4.565 .103 .077 14 16 1088.28 1.149 .862 4.594 .103 .078 16 18 1081.44 1.156 .867 4.623 .104 .078 18 20 1074.68 1.164 .873 4.653 .105 .079 20 22 1068.01 1.171 .878 4.682 .105 .079 22 24 1061.43 1.178 .884 4.711 .106 .079 24 26 1054.92 1.186 .889 4.740 .107 .080 26 28 1048.49 1.193 .895 4.769 .107 .080 28 30 1042.14 1.200 .900 4.798 .108 .081 30 32 1035.87 1.207 .906 4.827 .109 .081 32 34 1029.67 1.215 .911 4.856 .109 .082 34 36 1023.55 1.222 .916 4.885 .110 .082 36 38 1017.49 1.229 .922 4.914 .111 .083 38 40 1011.51 1.237 .927 4.943 .111 .083 40 42 1005.60 1.244 .933 4.972 .112 .084 42 44 999.76 1.251 .938 5.001 .113 .084 44 46 993.99 1.258 .944 5.030 .113 .085 46 48 988.28 1.266 .949 5.059 .114 .085 48 50 982.64 1.273 .955 5.088 .114 .086 50 52 977.06 1.280 .960 5.117 .115 .086 52 54 971.54 1.287 .966 5.146 .116 .087 54 56 966.09 1.295 .971 5.175 .116 .087 56 58 960.70 1.302 .977 5.205 .117 .088 58 AND OBDINATES FOB CUBVING BAILS. De- gree. Radius, 10. Ordinates, 25. Tangent Deflec- tion, 19. Curving 30-ft. rails, 29. De- gree. m. im. m. f m. 6 955.37 1.309 .982 5.234 .118 .088 / 6 2 950.09 1.317 .987 5.263 .118 .089 2 4 944.88 1.324 .993 5.292 .119 .089 4 6 939.72 1.331 .998 5.321 .120 .090 6 8 934.62 1.338 1.004 5.350 .120 .090 8 10 929.57 1.346 1.009 5.379 .121 .091 10 12 92458 1.353 1.015 5.408 .122 .091 12 14 919.64 1.360 1.020 5.437 .122 .092 14 16 914.75 1.368 1.026 5.466 .123 .092 16 18 909.92 1.375 1.031 5.495 .124 .093 18 20 905.13 1.382 1.037 5.524 .124 .093 20 22 900.40 1.389 1.042 5.553 .125 .094 22 24 895.71 1.397 1.047 5.582 .126 .094 24 26 891.08 1.404 1.053 5.611 .126 .095 26 28 886.49 1.411 1.058 5.640 .127 .095 28 30 881.95 1.418 1.064 5.669 .128 .096 30 32 877.45 1.426 1.069 5.698 .128 .096 32 34 873.00 1.433 1.075 5.727 .129 .097 34 36 868.60 1.440 1.080 5.756 .130 .097 36 38 864.24 1.448 1.086 5.785 .130 .098 38 40 859.92 1.455 1.091 5.814 .131 .098 40 42 855.65 1.462 1.097 5.844 .131 .099 42 44 851.42 1.469 1.102 5.873 .132 .099 44 46 847.23 1.477 1.108 5.902 .133 .100 46 48 843.08 1.484 1.113 5.931 .133 .100 48 50 838.97 1.491 1.118 5.960 .134 .101 50 52 834.90 1.499 1.124 5.989 .135 .101 52 54 830.88 1.506 1.129 6.018 .135 .102 54 56 826.89 1.513 1.135 6.047 .136 .102 56 58 822.93 1.520 1.140 6.076 .137 .103 58 7 819.02 1.528 1.146 6.105 .137 .103 7 2 815.14 1.535 .151 6.134 .138 .104 2 4 811.30 1.542 .157 6.163 .139 .104 4 6 807.50 1.549 .162 6.192 .139 .104 6 8 803.73 1.557 .168 6.221 .140 .105 8 10 800.00 1.564 .173 6.250 .141 .105 10 12 796.30 1.571 .178 6.279 .141 .106 12 14 792.63 1.579 .184 6.308 .142 .106 14 16 789.00 1.586 '1.189 6.337 .143 .107 16 18 785.40 1.593 1.195 6.366 .143 .107 18 20 781.84 1.600 1.200 6.395 .144 .108 20 22 778.31 1.608 1.206 6.424 .145 .108 22 24 774.81 1.615 1.211 6.453 .145 .109 24 26 771.34 1.622 1.217 6.482 .146 .109 26 28 767.90 1.630 1.222 6.511 .147 .110 28 30 764.49 1.637 1.228 6.540 .147 .110 30 32 761.11 1.644 1.233 6.569 .148 .111 32 34 757.76 1.651 1.239 6.598 .148 .111 34 36 754.44 1.659 1.244 6.627 .149 .112 36 38 751.16 1.666 1.249 6.656 .150 .112 38 40 747.89 1.673 1.255 6.685 .150 .113 40 42 744.66 1.681 1.260 6.714 .151 .113 42 44 741.46 1.688 1.266 6.743 .152 .114 44 46 738.28 1.695 1.271 6.773 .152 .114 46 48 735.13 1.702 1.277 6.802 .153 .115 48 50 732.01 1.710 1.282 6.831 .154 .115 50 52 728.91 1.717 1.288 6.860 .154 .116 52 54 725.84 1.724 1.293 6.889 .155 .116 54 56 722.79 1.731 1.299 6.918 .156 .117 56 58 719.77 1.739 1.304 6.947 .156 .117 58 170 TABLE I. BADII, ORDINATES, TANGENT DEFLECTIONS, De- gree. Radius, 10. Ordinates, 25. Tangent Deflec- tion, 19. Curving 30-ft. rails, 29. De- gree. m. 1- m. m. f m. / 8 71678 1.746 1.310 6.976 .157 .118 8 2 713.81 1.753 1.315 7.005 .158 .118 2 4 710.87 1.761 1.320 7.034 .158 .119 4 6 707.94 1.768 1.326 7.063 .159 .119 6 8 705.05 1.775 1.331 7.093 .160 .120 8 10 702.18 1.782 1.337 7.121 .160 .120 10 12 699.33 1.790 1.342 7.150 .161 .121 12 14 696.50 1.797 1.348 7.179 .162 .121 14 16 693.70 1.804 1.353 7.208 .162 .122 16 18 690.91 1.812 1.359 7.237 .163 .122 18 20 688.16 1.819 1.364 7.266 .163 .123 20 22 685.42 1.826 1.370 7.295 .164 .123 22 24 683.70 1.833 1.375 7.324 .165 .124 24 26 680.01 1.841 1.381 7.353 .165 .124 26 28 677.34 1.848 1.386 7.382 .166 .125 28 30 674.69 1.855 1.391 7.411 .167 .125 30 32 672.06 1.863 1.397 7.440 .167 .126 32 34 669.45 1.870 1.402 7.469 .168 .126 34 36 666.86 1.877 1.408 7.498 .169 .127 36 38 664.29 1.884 1.413 7.527 .169 .127 38 40 661.74 1.892 1.419 7.556 .170 .128 40 42 659.21 1.899 1.424 7.585 .171 .128 42 44 656.69 1.906 1.430 7.614 .171 .128 44 46 654.20 1.914 1.435 7.643 .172 .129 46 48 651.73 1.921 1.441 7.672 .173 .129 48 50 649.27 1.928 1.446 7.701 .173 .130 50 52 64684 1.935 1.452 7.730 .174 .130 52 54 644.42 1.943 1.457 7.759 .175 .131 54 56 642.02 1.950 1.462 7.788 .175 .131 56 58 639.64 1.957 1.463 7.817 .176 .132 58 9 637.27 1.965 1.473 7.846 .177 .132 9 2 634.93 1.972 1.479 7.875 .177 .133 2 4 632.60 1.979 1.484 7.904 .178 .133 4 6 630.29 1.986 1.490 7.933 .178 .134 6 8 627.99 1.994 1.495 7.962 .179 .134 8 10 625.71 2.001 1.501 7.991 .180 .135 10 12 623.45 2.008 1.506 8.020 .180 .135 12 14 621.20 2.015 1.512 8.049 .181 .136 14 16 618.97 2.023 1.517 8.078 .182 .136 16 18 616.76 2.030 1.523 8.107 .182 .137 18 20 614.56 2.037 1.528 8.136 .183 .137 20 22 612.38 2.045 1.533 8.165 .184 .138 22 24 610.21 2.052 1.539 8.194 .184 .138 24 26 608.06 2.059 1.544 8.223 .185 .139 26 28 605.93 2.066 1.550 8.252 .186 .139 28 30 603.80 2.074 1.555 8.281 .186 .140 30 32 601.70 2.081 1.561 8.310 .187 .140 32 34 599.61 2.088 1.566 8.339 .188 .141 34 36 597.53 2096 1.572 8.368 .188 .141 36 38 595.47 2.103 1.577 8.397 .189 .142 38 40 593.42 2.110 1.583 8.426 .190 .142 40 42 591.38 2.117 1.588 8.455 .190 .143 42 44 589.36 2.125 1.594 8.484 .191 .143 44 46 587.36 2.132 1.599 8.513 .192 .144 46 48 585.36 2.139 1.604 8.542 .192 .144 48 50 583.38 2.147 1.610 8.571 .193 .145 50 52 581.42 2.154 1.615 8.600 .193 .145 52 54 579.47 2.161 1.621 8.629 .194 .146 54 56 577.53 2.168 1.626 8.658 .195 .146 56 58 575.60 2.176 1.632 8.687 .195 .147 58 AND ORDINATE8 FOB CURVING RAILS. 171 De- gree. Radius, 10. Ordinates, 25. Tangent Deflec- tion,!^. Curving 30-f t. rails, 29. De- gree. m. fm. m. | m. o / 10 573.69 2.183 1.637 8.716 .196 .147 10 4 569.90 2.198 1.648 8.774 .197 .148 4 8 566.16 2.212 1.659 8.831 .199 .149 8 12 562.47 2.227 1.670 8.889 .200 .150 12 16 558.82 2.241 1.681 8.947 .201 .151 16 20 555.23 2.256 1.692 9.005 .203 .152 20 24 551.68 2.270 1.703 9.063 .204 .153 24 28 548.17 2.285 1.714 9.121 .205 .154 28 32 544.71 2.300 1.725 9.179 .207 .155 32 36 541.30 2.314 1.736 9.237 .208 .156 36 40 537.92 2.329 1.747 9.295 .209 .157 40 44 534.59 2.343 1.758 9.353 .210 .158 44 48 531.30 2.358 1.768 9.411 .212 .159 48 52 528.05 2.373 1.779 9.469 .213 .160 52 56 524.84 2.387 1.790 9.527 .214 .161 56 11 521.67 2.402 1.801 9.585 .216 .162 11 4 518.54 2.416 1.812 9.642 .217 .163 4 8 515.44 2.431 1.823 9.700 .218 .164 8 12 512.38 2.445 1.834 9.758 .220 .165 12 16 509.36 2.460 1.845 9.816 .221 .166 16 20 506.38 2.475 1.856 9.874 .222 .167 20 24 503.42 2.489 1.867 9.932 .223 .168 24 28 500.51 2.504 1.878 9.990 .225 .169 28 32 497.62 2.518 1.889 10.048 .226 .170 32 36 494.77 2.533 1.900 10.106 .227 .171 36 40 491.96 2.547 1.911 10.164 .229 .172 40 44 489.17 2.562 1.922 10.221 .230 .172 44 48 486.42 2.577 1.932 10.279 .231 .173 48 52 483.69 2.591 1.943 10.337 .233 .174 52 56 481.00 2.606 1.954 10.395 .234 .175 56 12 478.34 2.620 1.965 10.453 .235 .176 12 4 475.71 2.635 1.976 10.511 .236 .177 4 8 473.10 2.650 1.987 10.569 .238 .178 8 12 470.53 2.664 1.998 10.626 .239 .179 12 16 467.98 2.679 2.009 10.684 .240 .180 16 20 465.46 2.693 2.020 10.742 .242 .181 20 24 462.97 2.708 2.031 10.800 .243 .182 24 28 460.50 2.722 2.042 10.858 .244 .183 28 32 458.06 2.737 2.053 10.916 .246 .184 32 36 455.65 2.752 2.064 10.973 .247 .185 36 40 453.26 2.766 2.075 11.031 .248 .186 40 44 450.89 2.781 2.086 11.089 .250 .187 44 48 448.56 2.795 2.097 11.147 .251 .188 48 52 446.24 2.810 2.108 11.205 .252 .189 52 56 443.95 2.825 2.118 11.263 .253 .190 56 13 441.68 2.839 2.129 11.320 .255 .191 13 4 439.44 2.854 2.140 11.378 .256 .192 4 8 437.22 2.868 2.151 11.436 .257 .193 8 12 435.02 2.883 2.162 11.494 .259 .194 12 16 432.84 2.898 2.173 11.552 .260 .195 16 20 430.69 2.912 2.184 11.609 .261 .196 20 24 428.56 2.927 2.195 11.667 .263 .197 24 28 426.44 2.941 2.206 11.725 .264 .198 28 32 424.35 2.956 2.217 11.783 .265 .199 32 36 422.28 2.971 2.228 11.840 .266 .200 36 40 420.23 2.985 2.239 11.898 .268 .201 40 44 418.20 3.000 2.250 11.956 .269 .202 44 48 416.19 3.014 2.261 12.014 .270 .203 48 52 414.20 3.029 2.272 12.071 .272 .204 52 56 412.23 3.044 2.283 12.129 .273 .205 56 172 TABLE I. RADII, ORDINATES, TANGENT DEFLECTIONS, De- gree. Radius, 10. Ordinates, 25. Tangent Deflec- tion, 19. Curving 30-ft. rails, 29. De- gree. m. *m. m. lift. / 14 410.28 3.058 2.294 12.187 .274 .206 o / 14 4 408.34 3.073 2.305 12.245 .276 .207 4 8 406.42 3.087 2.316 12.302 .277 .208 8 12 404.53 3.102 2.326 12.360 .278 .209 12 16 402.65 3.117 2.337 12.418 .279 .210 16 20 400.78 3.131 2.348 12.476 .281 .211 20 24 398.94 3.146 2.359 12.533 .282 .211 24 28 397.11 3.160 2.370 12.591 .283 .212 28 32 395.30 3.175 2.381 12.649 .285 .213 32 36 393.50 3.190 2.392 12.706 .286 .214 36 40 391.72 3.204 2.403 12.764 .287 .215 40 44 389.96 3.219 2.414 12.822 .288 .216 44 48 388.21 3.233 2.425 12.880 .290 .217 48 52 386.48 3.248 2.436 12.937 .291 .218 52 56 384.77 3.263 2.447 12.995 .292 .219 56 15 383.06 3.277 2.458 13.053 .294 .220 15 4 381.38 3.292 2.469 13.110 .295 .221 4 8 379.71 3.306 2.480 13.168 .296 .222 8 12 378.05 3.321 2.491 13.226 .298 .223 12 16 376.41 3.336 2.502 13.283 .299 .224 16 20 374.79 3.350 2.513 13.341 .300 .225 20 24 373.17 3.365 2.524 13.399 .301 .226 24 28 371.57 3.379 2.535 13.456 .303 .227 28 32 369.99 3.394 2.546 13.514 .304 .228 32 36 368.42 3.409 2.556 13.572 .305 .229 36 40 366.86 3.423 2.567 13.629 .307 .230 40 44 365.31 3.438 2.578 13.687 .308 .231 44 48 363.78 3.452 2.589 13.744 .309 .232 48 52 362.26 3.467 2.600 13.802 .311 .233 52 56 360.76 3.482 2.611 13.860 .312 .234 56 16 359.26 3.496 2.622 13.917 .313 .235 16 4 357.78 3.511 2.633 13.975 .314 .236 4 8 356.32 3.526 2.644 14.033 .316 .237 8 12 354.86 3.540 2.655 14.090 .317 .238 12 16 353.41 3.555 2.666 14.148 .318 .239 16 20 351.98 3.569 2.677 14.205 .320 .240 20 24 350.06 3.584 2.688 14.263 .321 .241 24 28 349.15 3.599 2.699 14.320 .322 .242 28 32 347.75 3.613 2.710 14.378 .324 .243 32 36 346.37 3.628 2.721 14.436 .325 .244 36 40 344.99 3.643 2.732 14.493 .326 .245 40 44 343.62 3.657 2.743 14.551 .327 .246 44 48 342.27 3.672 2.754 14.608 .329 .247 48 52 340.93 3.686 2.765 14.666 .330 .247 52 56 339.60 3.701 2.776 14.723 .331 .248 56 17 338.27 3.716 2.787 14.781 .333 .249 17 4 336.96 3.730 2.798 14.838 .334 .250 4 8 335.66 3.745 2.809 14.896 .335 .251 8 12 334.37 3.760 2.820 14.954 .336 .252 12 16 333.09 3.774 2.831 15.011 .338 .253 16 20 331.82 3.789 2.842 15.069 .339 .254 20 24 330.55 3.803 2.853 15.126 .340 .255 24 28 329.30 3.818 2.864 15.184 .342 .256 28 32 328.06 3.833 2.875 15.241 .343 .257 32 36 326.83 3.847 2.885 15.299 .344 .258 36 40 325.60 3.862 2.896 15.356 .346 .259 40 44 324.39 3.877 2.907 15.414 .347 .260 44 48 323.18 3.891 2.918 15.471 .348 .261 48 52 321.99 3.906 2.929 15.529 .349 .262 52 56 320.80 3.920 2.940 15.586 .351 .263 56 AND ORDINATES FOR CURVING RAILS. De- gree. Radius, 10. Ordinates, 25. Tangent Deflec- tion, 19. Curving 30-ft. rails, 29. De- gree. m. f m. m. im. 18 319.62 3.935 2.951 15.643 .352 .264 18 4 318.45 3.950 2.962 15.701 .353 .265 4 8 317.29 3.964 2.973 15.758 .355 .266 8 12 316.14 3.979 2.984 15.816 .356 .267 12 16 315.00 3.994 2.995 15.873 .357 .268 16 20 313.86 4.008 3.006 15.931 .358 .269 20 24 312.73 4.023 3.017 15.988 .360 .270 24 28 311.61 4.038 3.028 16.046 .361 .271 28 32 310.50 4.052 3.039 16.103 .362 .272 32 36 309.40 4.067 3.050 16.160 .364 .273 36 40 308.30 4.081 3.061 16.218 .365 .274 40 44 307.22 4.096 3.072 16.275 .366 .275 44 48 306.14 4.111 3.083 16.333 .367 .276 48 52 305.06 4.125 3.094 16.390 .369 .277 52 56 304.00 4.140 3.105 16.447 .370 .278 56 19 302.94 4.155 3.116 16.505 .371 .279 19 4 301.89 4.169 3.127 16.562 .373 .279 4 8 300.85 4.184 3.138 16.620 .374 .280 8 12 299.82 4.199 3.149 16.677 .375 .281 12 16 298.79. 4.213 3.160 16.734 .377 .282 16 20 297.77 4.228 3.171 16.792 .378 .283 20 24 296.75 4.243 3.182 16.849 .379 .284 24 28 295.75 4.257 3.193 16.906 .380 .285 28 32 294.75 4.272 3.204 16.964 .382 .286 32 36 293.76 4.287 3.215 17.021 .383 .287 36 40 292.77 4.301 3.226 17.078 .384 .288 40 44 291.79 4.316 3.237 17.136 .386 .289 44 48 290.82 4.aso 3.248 17.193 .387 .290 48 52 289.85 4.345 3.259 17.250 .388 .291 52 56 288.89 4.360 3.270 17.308 .389 .292 56 20 287.94 4.374 3.281 17.365 .391 .293 20 10 285.58 4.411 3.308 17.508 .394 .295 10 20 288.27 4.448 3.336 17.651 .397 .298 20 30 280.99 4.484 3.363 17.794 .400 .300 30 40 278.75 4.521 3.391 17.937 .404 .303 40 50 276.54 4.558 3.418 18.081 .407 .305 50 21 274.37 4.594 3.446 18.224 .410 .308 21 10 272.23 4.631 3.473 18.367 .413 .310 10 20 270.13 4.668 3.501 18.509 .416 .312 20 30 268.06 4.704 3.528 18.652 .420 .315 30 40 266.02 4.741 3.556 18.795 .423 .317 40 50 264.02 4.778 3.583 18.938 .426 .320 50 22 262.04 4.814 3.611 19.081 .429 .322 22 10 260.10 4.851 3.638 19.224 .433 .324 10 20 258.18 4.888 3.666 19.366 .436 .327 20 30 256.29 4.925 3.693 19.509 .439 .329 30 ^feL J&4.43 4.961 3.721 19.652 .442 .332 40 >/# K2.60 4.998 3.749 19.794 .445 .334 50 23 OCTJP 250.79 5.035 3.776 19.937 .449 .336 23 10 249.01 5.071 3.804 20.079 .452 .339 10 20 247.26 5.108 3.831 20.222 .455 .341 20 30 245.53 5.145 3.859 20.364 .458 .344 30 40 243.82 5.182 3.886 20.507 .461 .346 40 50 242.14 5.218 3.914 20.649 .465 .348 50 24 240.49 5.255 3.941 20.791 .468 .351 24 10 238.85 5.292 3.969 20.933 .471 .353 10 20 237.24 5.329 3.997 21.076 .474 .356 20 30 235.65 5.366 4.024 21.218 .477 .358 30 40 234.08 5.402 4.052 21.360 .481 .360 40 50 232.54 5.439 4.079 21.502 .484 .363 50 174 TABLE II. LONG CHORDS. TABLE II. LONG CHORDS. 83. Degree of Curve. 2 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. 10 200.000 299.999 399.998 499.996 599.993 20 199.999 299.997 399.992 499.983 599.970 30 199.998 299.992 399.981 499.962 599.933 40 199.997 299.986 399.966 499.932 599.882 50 199.995 299.979 399.947 499.894 599.815 1 199.992 299.970 399.924 499.848 599.733 10 199.990 299.959 399.896 499.793 599.637 20 199.986 299.946 399.865 499.729 599.526 30 199.983 299.932 399.829 499.657 599.401 40 199.979 299.915 399.789 499.577 599.260 50 199.974 299.898 399.744 499.488 599.105 2 199.970 299.878 399.695 499.391 598.934 10 199.964 299.857 399.643 499.285 598.750 20 199.959 299.834 399.586 499.171 598.550 30 199.952 299.810 399.524 499.049 598.336 40 199.946 299.783 399.459 498.918 598.106 50 199.939 299.756 399.389 498.778 597.862 3 199.931 299.726 399.315 498.630 597.604 10 199.924 299.695 399.237 498.474 597.331 20 199.915 299.662 399.154 498.309 597.043 30 199.907 299.627 399.068 498.136 596.740 40 199.898 299.591 398.977 497.955 596.423 50 199.888 299.553 398.882 497.765 596.091 4 199.878 299.513 398.782 497.566 595.744 10 199.868 299.471 398.679 497.360 595.383 20 199.857 299.428 398.571 497.145 595.007 30 199.846 299.383 398.459 496.921 594.617 40 199.834 299.337 398.343 496.689 594.212 50 199.822 299.289 398.223 496.449 593.792 5 199.810 299.239 398.099 496.200 593.358 10 199.797 299.187 397.970 495.944 592.909 20 199.783 299.134 397.837 495.678 592.446 30 199.770 299.079 397.700 495.405 591.968 40 199.756 299.023 397.559 495.123 591.476 50 199.741 298.964 397.413 494.832 590.970 6 199.726 298.904 397.264 494.534 590.449 10 199.710 298.843 397.110 494.227 589.913 20 199.695 298.779 396.952 493.912 589.364 30 199.678 298.714 396.790 493.588 588.800 40 199.662 298.648 396.623 493.257 588.221 50 199.644 298.579 396.453 492.917 587.628 7 199.627 298.509 396.278 492.568 587.021 10 199.609 298.438 396.099 492.212 586.400 20 199.591 298.364 395.916 491.847 585.765 30 199.572 298.289 395.729 491.474 585.115 40 199.553 298.212 395.538 491.093 584.451 50 199.533 298.134 395.342 490.704 583.773 TABLE II. LONG CHORDS. LONG CHORDS. 83. 175 Degree of Curve. 2 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. 8 199.513 298.054 395.142 490.306 583.081 10 199.492 297.972 394.939 489.900 582.375 20 199.471 297.888 394.731 489.486 581.654 30 199.450 297.803 394.518 489.064 580.920 40 199.428 297.716 394.302 488.634 580.172 50 199.406 297.628 394.082 488.196 579.409 9 199.3a3 297.538 393.857 487.749 578.633 10 199.360 297.446 393.629 487.294 577.843 20 199.337 297.a52 393.396 486.832 577.039 30 199.313 297.257 393.159 486.361 576.222 40 199.289 297.160 392.918 485.882 575.390 50 199.264 297.062 392.673 485.395 574.545 10 199.239 296.962 392.424 484.900 573.686 10 199.213 296.860 392.171 484.397 572.813 20 199.187 296.756 391.914 483.886 571.926 30 199.161 296.651 391.652 483.367 571.027 40 199.134 296.544 391.387 482.840 570.113 50 199.107 296.436 391.117 482.305 569.186 11 199.079 296.325 390.843 481.762 568.245 10 199.051 296.214 390.565 481.211 567.291 20 199.023 296.100 390.284 480.653 566.324 30 198.994 295.985 389.998 480.086 565.343 40 198.964 295.868 389.708 479.511 564.349 50 198.935 295.750 389.414 478.929 563.341 12 198.904 295.630 389.116 478.339 562.321 10 198.874 295.508 388.814 477.740 561.287 20 198.843 295.384 388.508 477.135 560.240 30 198.811 295.259 388.197 476.521 559.180 40 198.779 295.132 387.883 475.899 558.107 50 198.747 295.004 387.565 475.270 557.020 13 198.714 294.874 387.243 474.633 555.921 176 TABLE III. TANGENTS AND EXTERNALS TABLE III. TANGENTS AND EXTERNALS OF A ONE-DEGREE CURVE. FOR chords of 100 feet the radius of a one-degree curve is 5729.65 feet. To find its tangent for any intersection angle /, we have ( 4) T R tan.|J, and to find the external ( 85) b = Ttan. /. By these formulae this table is computed. To find T and b for a curve of any other degree (chords 100 feet), divide the tabular values for the proper intersection angle by the number of degrees, whole or fractional, designating the curve. Thus, to find T and b for a 3 20' curve we divide the proper tabular values by 3. This process supposes the radii of curves to be inversely proportional to their degrees. , This is not strictly true, as may be seen by referring to Table I. Thus the radius of a 10 curve is greater than one-tenth the radius of a 1 curve. The values of T and b obtained as above will, therefore, be too small, and the corrections to be applied will always be ad- ditive. When thought to be necessary, these corrections may be obtained from Table IV. ; but, in the ordinary use of such a table, they may be disregarded. When the intersection angle of a proposed curve is known, and one of the three quantities R, T, and b is known or assumed, the other two may be obtained from the table. Thus, if we have / = 48 45' and the external b = 129 feet, we find from the table for this value of /, b 560.7. Then we have the degree of the pro- posed curve = 1 x ^~= = 4 C .346 = 4 20', nearly. Also for a 1 i/oy curve the table gives T 2596.1 ; so that for the proposed curve 2596 1 T = ' = 599.1. In a similar way, if the tangent of a pro- ^8" posed curve is known or assumed, the degree of the curve and its external can be found. OF A ONE DEGREE CURVE. 177 I. T. &. I. T. b. I. T. &. ! 50.0 .22 6' 300.3 7.86 11 551.7 26.50 5> 54.2 .26 5' 304.5 8.08 5' 555.9 26.90 10 58.3 .30 10 808.6 8.31 10 560.1 27.31 15 62.5 .34 15 312.8 8.53 15 564.3 27.72 20 66.7 .39 20 317.0 8.76 20 568.5 28.14 25 70.8 .44 25 321.2 8.99 25 572.7 28.55 30 75.0 .49 30 325.4 9.23 30 576.9 28.97 85 79.2 .55 35 329.5 9.47 35 581.2 29.40 40 83.3 .61 40 333.7 9.71 40 585.4 29.82 45 87.5 .67 45 337.9 9.95 45 589.6 30.25 50 91.7 .73 50 342.1 10.20 50 593.8 30.69 55 95.8 .80 55 346.3 10.45 55 598.0 31.12 2 100.0 .87 7 350.4 10.71 12 602.2 31.56 5 104.2 .95 5 354.6 10.98 5 606.4 32.00 10 108.3 1.02 10 358.8 11.22 10 610.6 32.45 15 112.5 1.10 15 363.0 11-49 15 614.9 32.90 20 116.7 1.19 20 367.2 11 -J5 20 619.1 33.35 25 120.9 1.27 25 371.4 12.02 25 623.3 33.80 30 125.0 1.36 30 375.5 12.29 30 627.5 34.26 35 129.2 1.46 35 379.7 12.57 35 631.7 34.72 40 ias.4 1.55 40 383.9 12-85 40 635.9 35.19 45 137.5 1.65 45 388.1 13.13 45 640.2 35.65 50 141.7 1.75 50 392.3 13.41 50 644.4 36.12 55 145.9 1.86 56 396.5 13-70 55 648.6 36.59 3 150-0 1.96 8 400.7 13.99 13 652.8 37.07 5 154.2 2.07 5 404.8 14.28 5 i 657.0 37.55 10 158.4 2.19 10 409.0 14.58 10 ! 661.3 38.03 15 162.5 2.31 15 413.2 14.88 15 665.5 38.52 20 166.7 2.42 20 417.4 15.18 20 S 669.7 39.01 25 170.9 2.55 25 421.6 15.49 25 673.9 39.50 30 175.1 2.67 30 425.8 15.80 30 678.1 39.99 35 179.2 2.80 35 430.0 16.11 35 682.4 40.49 40 183.4 2.93 40 434.2 16.43 40 686.6 40.99 45 187.6 3.07 45 438.4 16.74 45 690.8 41.50 60 191.7 3.21 50 442.5 17.07 50 695.1 42.00 55 195.9 3.35 55 446.7 17.39 55 699.3 42.51 4 200.1 3.49 9 450.9 17.72 14 703.5 43.03 5 204.3 3.64 5 455.1 i 18.05 5 707.7 43.55 10 208.4 3.79 10 459.3 18.38 10 712.0 44.07 15 212.6 3.94 15 463.5 18.72 15 716.2 1 44.59 20 216.8 4.10 20 467.7 19.06 20 720.4 45.12 25 220.9 4.26 25 471.9 19.40 25 724.7 45.65 30 225.1 4.42 30 476.1 19.75 30 728.9 46.18 35 229.3 4.59 35 480.3 20.10 35 733.1 46.71 40 233.5 4.75 40 484.5 20.45 40 737.4 i 47.25 45 237.6 4.93 45 488.7 20.80 45 741.6 47.80 50 241.8 5.10 50 492.9 ! 21.16 50 745.8 48.34 55 246.0 5.28 55 497.1 21.52 55 750.1 48.89 5 250.2 5.46 10 501.3 21.89 15 754.3 49.44 5 254.3 5.64 5 505.5 22.25 5 758.6 50.00 10 258.5 5.8S 10 509.7 22.62 10 762.8 50.55 15 262.7 6.02 15 513.9 23.00 15 767.0 51.12 20 266.9 6.21 20 518.1 23.37 20 771.3 i 51.68 25 271.0 6.41 25 522.3 23.75 25 775.5 52.25 30 275.2 6.61 30 526.5 24.14 30 779.8 i 52.82 35 279.4 6.81 35 530.7 24.52 35 784.0 53.39 40 283.6 7.01 40 534.9 24.91 40 788.3 53.97 45 287.7 7.22 45 539.1 25.30 45 792.5 54.55 50 291.9 7.43 50 543.3 25.70 50 796.8 55.13 55 296.1 7.65 55 547.5 26.10 55 801.0 56.72 13 178 TABLE ITT. TANGENTS AND EXTERNALS. I. T. &. I. T. &. I. T. ft. IB- 805.2 56.31 21 1061.9 97.58 26 1322.8 150.7 S' 809.5 56.90 5' 1066.2 98.36 5' 1327.2 151.7 10 813.7 57.50 10 1070.6 99.15 10 1331.6 | 152.7 15 818.0 58.10 15 1074.9 99.95 15 1336.0 153.7 20 822.3 58.70 20 1079.2 100.7 20 1340.4 154.7 25 826.5 59.31 25 1083.5 101.5 25 1344.8 155.7 30 830.8 59.91 30 1087.8 102.3 30 1349.2 156.7 35 835.0 60.53 35 1092.1 103.2 35 1353.6 157.7 40 839.3 61.14 40 1096.4 104.0 40 1358.0 158.7 45 843.5 61.76 45 1100.8 104.8 45 1362.4 159.7 50 847.8 62.38 50 1105.1 -105.6 50 1366.8 160.8 55 852.0 63.01 55 1109.4 106.4 55 1371.2 .161. 8 17 856.3 63.63 22 1113.7 107.2 27 1375.6 162.8 5 860.6 64.27 5 1118.1 108.1 5 1380.0 163.8 10 864.8 64.90 10 1122.4 108.9 10 1384.4 ! 164.9 15 869.1 65.54 15 1126.7 109.7 15 1388.8 i 165.9 20 873.3 66.18 20 1131.0 110.6 20 1393.2 167.0 25 877.6 66.82 25 1135.4 111.4 25 1397.6 ' 168.0 30 881.9 67.47 30 1139.7 112.3 30 1402.0 169.0 35 886.1 68.12 35 1144.0 113.1 35 1406.5 170.1 40 890.4 68.77 40 1148.4 113.9 40 1410.9 171.2 45 894.7 69.43 45 1152.7 114.8 45 1415.3 172.2 50 898.9 70.09 50 1157.0 115.7 50 1419.7 173.3 55 903.2 70.75 55 1161.4 116.5 55 1424.1 174.3 18 907.5 71.42 23 1165.7 117.4 28 1428.6 175.4 5 911.8 72.09 5 1170.1 118.2 5 1433.0 176.5 10 916.0 72.76 10 1174.4 119.1 10 1437.4 177.6 15 920.3 73.44 15 1178.7 120.0 15 1441.8 178.6 20 924.6 74.12 20 1183.1 120.9 20 1446.3 179.7 25 928.9 74.80 25 1187.4 121.7 25 1450.7 180.8 30 933.1 75.49 30 1191.8 122.6 30 1455.1 181.9 35 937.4 76.18 35 1196.1 123.5 35 1459.6 183.0 40 941.7 76.87 40 1200.5 124.4 40 1464.0 184.1 45 946.0 77.57 45 1204.8 125.3 45 1468.5 185.2 50 950.2 78.26 50 1209.2 126.2 50 1472.9 186.3 55 954.5 78.97 55 1213.5 127.1 55 1477.3 187.4 19 958.8 79.67 24 1217.9 128.0 29 1481.8 188.5 5 963.1 80.38 5 1222.2 128.9 5 1486.2 189.6 10 967.4 81.09 10 1226.6 129.8 10 1490.7 190.7 15 971.7 81.81 15 1230.9 130.7 15 1495.1 191.9 20 976.0 82.53 20 1235.3 131.7 20 1499.6 193.0 25 980.2 ! 83.25 25 1239.7 132.6 25 1504.0 194.1 30 984.5 83.97 30 1244.0 133.5 30 1508.5 195.2 35 988.8 84.70 35 1248.4 134.4 35 1512.9 196.4 40 993.1 85.43 40 1252.8 135.4 40 1517.4 197.5 45 997.4 86.17 45 1257.1 136.3 45 1521.9 198.7 50 1001.7 86.90 50 1261.5 137.2 50 1526.3 199.8 55 1006.0 87.64 55 1265.9 138.2 55 1530.8 201.0 20 1010.3 88.39 25 1270.2 139.1 30 1535.3 202.1 5 1014.6 89.14 5 1274.6 140.1 5 1539.7 203.3 10 1018.9 89.89 10 1279.0 141.0 10 1544.2 204.4 15 1023.2 90.64 15 1283.4 142.0 15 1548.7 205.6 20 1027.5 91.40 20 1287.7 142.9 20 1553.1 206.8 25 1031.8 92.16 25 1292.1 143.9 25 1557.6 207.9 30 1036.1 92.92 30 1296.5 144.9 30 1562.1 209.1 35 1040.4 93.69 35 1300.9 145.8 35 1566.6 210.3 40 1044.7 94.46 40 1305.3 146.8 40 1571.0 211.5 45 1049.0 95.24 45 1309.6 147.8 45 1575.5 212.7 50 1053.3 96.01 50 1314.0 148.7 50 1580.0 213.9 55 1057.6 96.79 55 1318.4 149.7 55 1584.5 215.1 OF A ONE DEGREE CURVE. 179 I T &. I. T. 6. I. T. b. 31 1589.0 216.2 36 1861.7 294.9 41 2142.2 387.4 5' 1593.5 217.5 5> 1866.3 296.3 5' 2147.0 389.0 10 1598.0 218.7 10 | 1870.9 297.7 10 2151.7 390.7 15 1602.4 219.9 15 i 1875.5 299.1 15 2156.5 392.4 20 1606.9 221.1 20 1880.1 300.6 20 2161.2 394.1 25 1611.4 222.3 25 1884.7 302.0 25 2166.0 395.7 30 1615.9 223.5 30 1889.4 303.5 30 2170.8 397.4 35 1620.4 224.7 35 1894.0 304.9 35 2175.6 399.1 40 1624.9 226.0 40 1898.6 306.4 40 2180.3 400.8 45 1629.4 227.2 45 1903.2 307.8 45 2185.1 402.5 50 1633.9 228.4 50 1907.9 309.3 50 2189.9 404.2 55 1638.4 229.7 55 1912.5 310.8 55 2194.6 405.9 32 1643.0 230.9 37 1917.1 312.2 42 2199.4 407.6 5 1647.5 232.1 5 1921.7 313.7 5 2204.2 409.4 10 1652.0 233.4 10 1926.4 315.2 10 2209.0 411.1 15 1656.5 234.6 15 1931.0 316.6 15 2213.8 412.8 20 1661.0 235.9 20 1935.7 I 318.1 20 2218-6 414.5 25 1665.5 237.2 25 1940.3 319.6 25 2223.3 416 3 30 1670.0 238.4 30 1945.0 ! 321.1 30 2228.1 418.0 35 1674.6 239.7 35 1949.6 322.6 35 2232.9 419.7 40 1679.1 241.0 40 1954.3 324.1 40 2237.7 421.5 45 1683.6 242.2 45 1958.9 325.6 45 2242.5 423.2 50 1688.1 243.5 50 1963.6 327.1 50 2247.3 425.0 55 1692.7 244.8 55 1968.2 328.6 55 2252.2 426.7 33 1697.2 246.1 38 1972.9 330.1 43 2257.0 428.5 5 1701.7 247.4 5 1977.5 331.7 5 2261.8 430.3 10 1706.3 248.7 10 1982.2 333.2 10 2266.6 432.0 15 1710.8 250.0 15 1986.9 334.7 15 2271.4 433.8 20 1715.3 251.3 20 1991.5 336.2 20 2276.2 435.6 25 1719.9 252.6 25 1996.2 337.8 25 2281.1 437.4 30 1724.4 253.9 30 2000.9 339.3 30 2285.9 439.2 35 1729.0 255.2 35 2005.6 340.9 35 2290.7 441.0 40 1733.5 256.5 40 2010.2 342.4 40 2295.6 442.7 45 1738.1 257.8 45 2014.9 344.0 45 2300.4 444.5 50 1742.6 259.1 50 2019.6 345.5 50 2305.2 446.4 55 1747.2 260.5 55 2024.3 347.1 55 2310.1 448.2 34 1751.7 261.8 39 2029.0 348.6 44 2314.9 450.0 5 1756.3 263.1 5 2033.7 350.2 5 2319.8 451.8 10 1760.8 264.5 10 2038.4 351.8 10 2324.6 453.6 15 1765.4 265.8 15 2043.1 353.4 15 2329.5 455.4 20 1770.0 267.2 20 2047.8 354.9 20 2334.3 457.3 25 1774.5 268.5 25 2052.5 356.5 25 2339.2 459.1 30 1779.1 269.9 30 2057.2 358.1 30 2344.1 460.9 35 1783.7 271.2 35 2061.9 359.7 35 2348.9 462.8 40 1788.2 272.6 40 2066.6 361.3 40 2353.8 464.6 45 1792.8 273.9 45 2071 362.9 45 2358.7 466.5 50 1797.4 275.3 50 2076.0 364.5 50 2363.5 468.4 55 1802.0 276.7 55 2080.7 366.1 55 2368.4 470.2 35 1806.6 278.1 40 2085.4 367.7 45 2373.3 472.1 5 1811.1 279.4 5 2090.1 369.3 5 2378.2 473.9 10 1815.7 280.8 10 2094.9 371.0 10 2383.1 475.8 15 1820.3 282.2 15 2099.6 372.6 15 2388.0 477.7 20 1824.9 283.6 20 2104.3 374.2 20 2392.8 479.6 25 1829.5 285.0 25 2109.0 375.8 25 2397.7 481.5 30 1834.1 286.4 30 2113.8 377.5 30 2402.6 483.4 35 1838.7 287.8 35 2118.5 i 379.1 35 2407.5 485.3 40 1843.3 289.2 40 2123.3 380.8 40 2412.4 487.2 45 1847.9 290.6 45 2128.0 382.4 45 i 2417.4 489.1 50 1852.5 292.0 50 2132.7 384.1 50 2422.3 491.0 55 1857.1 293.4 55 i 2137.5 385.7 55 1 2427.2 492.9 180 TABLE III. TANGENTS AND EXTERNALS I. T. &. I. T. &. I, T. ft. 46' 2432.1 494.8 51 2732.9 618.4 56 3046.5 759.6 5' 2437.0 496.7 5' 2738.0 620.6 5' 3051.9 762.1 10 2441.9 498.7 10 2743.1 622.8 10 3057.2 764.6 15 2446.9 500.6 15 2748.8 625.0 15 3062.6 767.1 20 2451.8 502.5 20 2753.4 627.2 20 3067.9 769.7 25 2456.7 504.5 25 2758.5 629.5 25 3073.3 772.2 30 2461.7 506.4 30 2763.7 631.7 30 3078.7 774.7 35 2466.6 508.4 35 2768.8 633.9 35 3084.0 777.3 40 2471.5 510.3 40 2773.9 636.2 40 3089.4 779.8 45 2476.5 512.3 45 2779.1 638.4 45 3094.8 782.4 50 2481.4 514.3 50 2784.2 640.7 50 3100.2 784.9 55 2486.4 516.2 55 2789.4 642.9 55 3105.6 787.5 47 2491.3 518.2 52 2794.5 645.2 57 3110.9 790.1 5 2496.3 520.2 5 2799.7 647.4 5 3116.3 792.7 10 2501.2 522.2 10 2804.9 649.7 10 3121.7 795.2 15 2506.2 524.1 15 2810.0 652.0 15 3127.2 797.8 20 2511.2 526.1 20 2815.2 654.3 20 3132.6 800.4 25 2516.1 528.1 25 2820.4 656.5 25 3138.0 803.0 30 2521.1 530.1 30 2825.6 658.8 30 3143.4 805.6 35 2526.1 532.1 35 2830.7 661.1 35 3148.8 808.2 40 2531.1 534.1 40 2835.9 663.4 40 3154.2 810.9 45 2536.0 536.2 45 2841.1 665.7 45 3159.7 813.5 50 2541.0 538.2 50 2846.3 668.0 50 3165.1 816.1 55 2546.0 540.2 55 2851.5 670.3 55 3170.6 818.7 48 2551.0 542.2 3 2856.7 672.7 58 3176.0 821.4 5 2556.0 544.3 5 2861.9 675.0 5 3181.4 824.0 10 2561.0 546.3 10 2867.1 677.3 10 3186.9 826.7 15 2566.0 548.3 15 2872.3 679.6 15 3192.4 829.3 20 2571.0 550.4 20 2877.5 682.0 20 3197.8 832.0 25 2576.0 552.4 25 2882.8 684.3 25 3203.3 834.6 30 2581.0 554.5 30 2888.0 686.7 30 3208.8 837.3 35 2586.0 556.6 35 2893.2 689.0 35 3214.2 840.0 40 2591.1 558.6 40 2898.4 691.4 40 3219.7 842.7 45 2596.1 560.7 45 2903.7 693.8 45 3225.2 845.4 50 2601.1 562.8 50 2908.9 696.1 50 3230.7 848.1 55 2606.1 564.9 55 2914.2 698.5 55 3236.2 850.8 49 2611.2 566.9 54 2919.4 700.9 59 3241.7 853.5 5 2616.2 569.0 5 2924.7 703.3 5 3247.2 856.2 10 2621.2 571.1 10 2929.9 705.7 10 3252.7 858.9 15 2626.3 673.2 15 2935.2 708.1 15 3258.2 861.6 20 2631.3 575.3 20 2940.4 710.5 20 3263.7 864.3 25 2636.3 577.4 25 2945.7 712.9 25 3269.2 867.1 30 2641.4 579.5 30 2951-0 715.3 30 3274.8 869.8 35 2646.5 581.7 35 2956.2 717.7 35 3280.3 872.6 40 2651.5 583.8 40 2961.5 720.1 40 3285.8 875.3 45 2656.6 585.9 45 2966.8 722.5 45 3291.4 878.1 50 2661.6 588.0 50 2972.1 725.0 50 3296.9 880.8 55 2666.7 590.2 55 2977.4 727.4 55 3302.5 883.6 50 2671.8 592.3 55 2982.7 729.9 60 3308.0 886.4 5 2676.9 594.5 5 2988.0 732.3 5 3313.6 889.2 10 2681.9 596.6 10 2993 3 734.8 10 3319.1 891.9 15 2687.0 598.8 15 2998.6 737.2 15 3324.7 894.7 20 2692.1 600.9 20 3003.9 739.7 20 3330.3 897.5 25 2697.2 603.1 25 3009.2 742.1 25 3335.8 900.3 30 2702.3 605.3 30 3014.5 744.6 30 3341.4 903.2 35 2707.4 607.4 35 3019.8 747.1 35 3347.0 906.0 40 2712.5 609.6 40 3025.2 749.6 40 3352.6 908.8 45 2717.6 611.8 45 3030.5 752.1 45 3358.2 911.6 50 2722.7 614.0 50 3035.8 754.6 50 3363.8 914.5 55 2727.8 616.2 55 3041.2 757.1 55 3369.4 917.3 OF A ONE DEGREE CURVE. 181 I. T. &. I. T. &. I. T. 6. el- 3375.0 920.1 66* 3720.9 1102.2 71 4086.9 1308.2 s' 3380.6 923.0 & 3726.8 1105.4 5' 4093.2 1311.9 10 3386.3 925.8 10 3732.7 1108.6 10 4099.5 1315.6 15 3391.9 928.7 15 3738.7 1111.9 15 4105.8 1319.2 20 3397.5 931.6 20 3744.6 1115.1 20 4112.1 1322.9 25 3403.1 934.5 25 3750.6 1118.4 25 4118.4 1326.6 80 3408.8 937.3 30 3756.5 1121.7 30 4124.8 1330.3 35 3414.4 940.2 35 3762.5 1124.9 35 4131.1 1334.0 40 3420.1 943.1 40 3768.5 1128.2 40 4137.4 1337.7 45 3425.7 946.0 45 3774.4 1131.5 45 4143.8 1341.4 50 3431.4 948.9 50 3780.4 1134.8 50 4150.1 1345.1 55 3437.1 951.8 55 3786.4 1138.1 55 4156.5 1348.8 62 3442.7 954.8 67 3792.4 1141.4 72 4162.8 1352.6 5 3448.4 957.7 5 3798.4 1144.7 5 4169.2 1356.3 10 3454.1 960.6 10 3804 4 1148.0 10 4175.6 1360.1 15 3459.8 963.5 15 3810.4 1151.3 15 4182.0 1363.8 20 3465.4 966.5 20 3816.4 1154.7 20 4188.4 1367.6 25 3471.1 969.4 25 3822.4 1158.0 25 4194.8 1371.4 30 3476.8 972.4 30 3828.4 1161.3 30 4201.2 1375.2 35 3482.5 975.3 85 3834.5 1164.7 35 4207.6 1379.0 40 3488.2 978.3 40 3840.5 1168.1 40 4214.0 1382.8 45 3494.0 981.3 45 1171.4 45 4220.4 1386.6 50 3499.7 984.3 50 3852 6 1174.8 50 4226.8 1390.4 55 3505.4 987.3 55 3858.6 1178.2 55 4233.3 1394.2 63 3511.1 990.2 68 3864.7 1181.6 73 4239.7 1398.0 5 3516.9 993.2 5 3870.8 1185.0 5 4246.2 1401.9 10 3522.6 996.2 10 3876.8 1188.4 10 4252.6 1405.7 15 3528.4 999.3 15 3882.9 1191.8 15 4259.1 1409.6 20 3534.1 1002.3 20 3889.0 1195.2 20 4265.6 1413.5 25 3539.9 1005.3 25 3895.1 1198.6 25 4272.0 1417.3 30 3545.6 1008.3 30 3901.2 1202.0 30 4278.5 1421.2 35 3551.4 1011.4 35 3907.3 1205.5 35 4285.0 1425.1 40 3557.2 1014.4 40 3913 4 1208.9 40 4291.5 1429.0 45 3562.9 1017.4 45 3919.5 1212.4 45 4298.0 1432.9 50 3568.7 1020.5 50 3925.6 1215.8 50 4304.5 1436.8 55 3574.5 1023.6 55 3931.7 1219.3 55 4311 1 1440.7 64 3580.3 1026.6 69 3937.9 1222.7 74 4317.6 1444.6 5 3586.1 1029.7 5 3944.0 1226.2 5 4324.1 1448.6 10 3591.9 1032.8 10 3950.2 1229.7 10 4330.7 1452.5 15 3597.7 1035.9 15 3956.3 1233.2 15 4337.2 1456.5 20 3603.5 1039 20 ::962.5 1236.7 20 4343.8 1460.4 25 3609.3 1042.1 25 3968.6 1240.2 25 4350.4 1464.4 30 3615.1 1045.2 80 3974.8 1243.7 30 4356.9 1468.4 35 3621.0 1048.3 35 3981.0 1247.2 35 4363 5 1472.4 40 3626.8 1051.4 40 3987.2 '1250.8 40 4370.1 1476.4 45 3632.6 1054.5 45 3993.3 1254.3 45 4376.7 1480.4 50 3638.5 1057.7 50 3999.5 1257.9 50 4383.3 1484.4 55 3644.3 1060.8 55 4005.7 1261.4 55 4889.9 1488.4 65 3650.2 1063.9 70 4011.9 1265.0 75 4396.5 1492.4 5 3656.1 1067.1 5 4018.2 1268.5 5 4403.1 1496.5 10 3661.9 1070.2 10 4024.4 1272 1 10 4409.8 1500.5 15 3667.8 1073.4 15 4030.6 1275.7 15 4416.4 1504.5 20 3673.7 ! 1076.6 20 4036.8 1279.3 20 4423.1 1508.6 25 3679.5 1079.7 25 4043.1 1282.9 25 4429.7 i 1512.7 30 3685.4 1082.9 30 4049.3 1286.5 30 4436.4 1 1516.7 35 3691.3 1086.1 35 4055.6 1290.1 35 4443.0 j 1520.8 40 3697.2 1089.3 40 4061.8 1293.6 40 4449.7 ! 1524.9 45 3703.1 1092.5 45 4068.1 1297.3 45 4456.4 ! 1529.0 50 3709.0 1095.7 50 4074.4 1300.9 50 4463.1 ! 1533.1 55 3715 1099.0 55 4080.6 1304.6 55 4469.8 1537.3 182 TABLE III. TANGENTS AND EXTERNALS I. T. b. I. T. b. I. T. 76 4476.5 1541.4 81 4893.6 1805.3 86* 5343.0 2104.7 5' 4483.2 1545.5 5' 4900.8 1810.0 5' 5350.8 2110.0 10 4,189.9 1549.7 10 4908.0 1814.7 10 5358.6 2115.3 15 4496.7 1553.8 15 4915 2 1819.4 15 5366.4 2120.6 20 4503.4 1558 20 4922.5 1824.1 20 5374.2 2126.0 25 4510.1 1562.1 25 4929.7 1828.9 25 5382.1 2131.4 30 4516.9 1566.3 30 4937.0 1833.6 30 5389.9 2136.7 35 4523.7 1570.5 35 4944.2 1838.3 35 5397.8 2142.1 40 4530.4 1574.7 40 4951 .5 1843.1 40 5405.6 2147.5 45 4537.2 1578.9 45 4958.8 1847.9 45 5413.5 2152.9 50 4544.0 1583.1 50 4966.1 1852.6 50 5421.4 2158.4 55 4550.8 1587 3 55 4973.4 1857.4 55 5429.3 2163.8 77 4557.6 1591.6 82 4980.7 1862.2 87 5437.2 2169.2 5 4564.4 1595.8 5 4988.0 1867.0 5 5445.2 2174.7 10 4571.2 1600.1 10 4995.4 1871.8 10 5453.1 2180.2 15 4578.0 1604.3 15 5C02.7 1876.7 15 5461.0 2185.6 20 4584.8 1608.6 20 5010.0 1881.5 20 5469.0 2191.1 25 4591.7 1612.9 25 5017.4 1886.3 25 5477.0 2196.6 30 4598.5 1617.1 SO 5024.8 1891.2 30 5484.9 2202.2 35 4605.4 1621.4 35 5032.1 1896.1 35 5492.9 2207.7 40 4612.2 1625.7 40 5039.5 1900.9 40 5500.9 2213.2 45 4619.1 1630.0 45 5046.9 1905.8 45 5509.0 2218.8 60 4626.0 1634.4 50 054.3 1910.7 60 5517.0 2224.3 65 4632.9 1638.7 55 5061.7 1915.6 55 5525.0 2229.9 78 4639 8 1643.0 83 5069.2 1920.5 88 5533.1 2235.5 5 4646.7 1647.4 5 5076.6 1925.5 5 5541.1 2241 1 10 4653.6 1651.7 10 5084.0 1930.4 10 5549.2 2246.7 15 4660 5 1656.1 15 5091.5 1985.3 15 5557.3 2252.3 20 4667.4 1660 5 20 5099.0 1940.3 20 5565.4 2258.0 25 4674 4 1664.9 25 5106.4 1945.3 25 5573.5 2263.6 30 4681.3 1669.2 30 5113.9 1950.3 30 5581.6 2269.3 35 4688.3 1673.6 35 5121.4 1955.2 35 5589.7 2275.0 40 4695.2 1678.1 40 5128.9 1960.2 40 5597.8 2280.6 46 4702.2 1682.5 45 5136.4 1965.3 45 5606.0 2286.3 60 4709.2 1686.9 50 5143.9 1970.3 50 5614.2 2292.0 55 4716.2 1691.3 55 5151.5 1975.3 55 5622.3 2297.8 79 4723 2 1695.8 84 5159.0 1980.4 89 5630.5 2303.5 5 4730.2 1700.2 5 5166.6 1985.4 5 5638.7 2309.3 10 4737.2 1704.7 10 5174.1 1990.5 10 5646.9 2315.0 15 4744.2 1709.2 15 5181.7 1995.5 15 5655.1 2320.8 20 4751.2 1713.7 20 5189.3 2000.6 20 5663.4 2326.6 25 4758.3 1718.2 25 5196.8 2005.7 25 5671.6 2332.4 30 4765.3 1722.7 30 5204.4 2010.8 30 5679.9 2338.2 35 4772.4 1727.2 35 5212.1 2016.0 35 5688.1 2844.0 40 4779.4 1731.7 40 5219.7 2021.1 40 5696.4 2349.8 45 4786.5 1736.2 45 5227.3 2026.2 45 5704.7 2355.7 50 4793.6 1740.8 50 5234.9 2031.4 50 5713.0 2361.5 55 4800.7 1745.3 55 5242.6 2036.5 55 5721.3 2367.4 80 4807.7 1749.9 85 5250.3 2041.7 90 5729.7 2373.3 5 4814.9 1754.4 5 5257.9 2046.9 5 5738.0 2379.2 10 4822.0 1759.0 10 5265.6 2052.1 10 5746.3 2385.1 15 4829.1 1763.6 15 5273.3 2057.3 15 5754.7 2391.0 20 4836.2 1768.2 20 5281.0 2062.5 20 5763.1 2397.0 25 4843.4 1772.8 25 5288.7 2067.7 25 5771.5 2402.9 30 4850.5 1777.4 30 5296.4 2073.0 30 5779.9 2408.9 35 4857.7 1782.1 35 5304.2 2078.2 35 5788.3 2414.9 40 4864.8 1786.7 40 5311.9 2083.5 40 5796.7 2420.9 45 4872.0 1791.3 45 5319.7 2088.8 45 {805.1 2426.9 50 4879.2 1796.0 50 5327.4 2094.1 50 5813.6 2432.9 55 4886.4 1800.7 55 5335.2 2099.4 55 5822.1 2438.9 TABLE IV. TABLE V. 183 TABLE IV. CORRECTIONS FOR TABLE III. FOR TANGENTS ADD FOR EXTERNALS ADD 5 10 15 20 25 30 5 10 15 20 25 30 1. Curve. Curve. Curve. Curve. Curve. Curve. f. Curve. Curve. Curve. Curve. Curve. Curve. 10 .03 .00 .10 .13 .16 .19 10 .001 .003 .004 .006 .007 .008 20 .06 .13 .19 .26 .32 .39 20 .005 .011 .017 .022 .028 .034 30 .10 .19 .29 .39 .49 .60 30 .013 .025 .038 .051 .064 .078 40 .13 .26 .40 .53 .67 .80 40 .023 .046 .070 .093 .117 .141 50 .17 .34 .51 .68 .85 1.02 50 .037 .075 .112 .151 .189 .227 j 60 .21 .42 .63 .84 1.05 1.27 60 .054 .111 .168 .225 .283 .340 70 .25 .51 .76 1.02 1.28 1.54 70 .077 .159 .240 .321 .403 .485 80 .30 .61 .91 1.22 1.53 1.84 80 .110 .220 .332 .445 .558 .671 90 .35 .72 1.09 1.45 1.83 2.20 90 .145 .298 .451 .603 .756 .910 TABLE V. TURNOUTS TANGENT TO STRAIGHT MAIN TRACK. Gauge, g = 4.708 ; throw of switch-rail, d .417. Ordinates ;o E F for all valnes of n, at the centre 1.177, at quarter points 0.883 ( 68). Frog No., 52. Frog Angle F, 52. Switch- rail /, 65. Chord B F, 66. Radius, 67. Degree. Curving 30-ft. rail, 29. m. \m. 4 o / 14 15 11.21 37.96 150.66 / 38 46 .747 .560 4 12 41 12.61 42.63 190.67 30 24 .590 .443 5 11 25 14.01 47.31 235.40 24 32 .478 .358 5^ 10 23 15.41 52.00 284.83 20 13 .395 .296 6 9 32 16.81 56.69 338.98 16 58 .332 .249 *t 8 48 18.22 61.38 397.83 14 26 .283 .212 7 8 10 19.62 66.08 461.38 12 27 .244 .183 74 7 38 21.02 70.78 529.65 10 50 .212 .159 8 7 9 22.42 75.47 602.62 9 31 .187 .140 B) 6 44 23.82 80.18 680.31 8 26 .165 .124 9 6 22 25.22 84.87 762.70 7 31 .148 .111 * 6 2 26.62 89.58 849.79 6 45 .132 .099 10 5 43 28.02 94.28 941.60 6 5 .119 .090 104 5 27 29.42 98.98 1038.11 5 31 .108 .081 11 5 12 30.83 103.68 1139.34 5 2 .099 .074 1H 4 59 32.23 108.39 1245.27 4 36 .090 .068 12 4 46 33.63 113.09 1355.90 4 14 .083 .062 184 TABLE VI. TABLE VII. 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 59&30 3 .00087 26646 25997 3 .00001 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 20867 7 .12217 30476 39603 7 .00203 62174 60660 r .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 31300 TABLE VII. ELEVATION OF THE OUTER RAIL ON CURVES. 152. De- gree. V = 15. V = 20. V = 25. V = 30. V = 35. y 40. V = 45. V = 50. V^ 60. V = 70. V = 80. .012 .022 .034 .049 .067 .088 .111 .137 .197 .269 .351 2 .025 .044 .068 .099 .134 .175 .222 .274 .395 .537 .701 3 .037 .066 .103 .148 .201 .263 .333 .411 .592 .805 1.052 4 .049 .088 .137 .197 .268 .351 .444 .548 .789 1.074 5 .062 .110 .171 .247 .336 .438 .555 .685 .986 6 .074 .131 .205 .296 .403 .526 .666 .822 7 .086 .153 .240 .345 .470 .613 .776 .958 8 .099 .175 .274 .394 .537 .701 .887 1.095 9 .111 .197 .308 .443 .604 .788 .998 10 .123 .219 .342 .493 .670 .876 12 .160 .263 .410 .591 .804 1.050 14 .172 .306 .478 .689 .938 16 .197 .350 .546 .787 1.071 TABLE VIII. CORRECTION FOR THE EARTH'S CURVATURE. 185 TABLE VIII. CORRECTION FOR THE EARTH'S CURVATURE AND FOR REFRACTION. 145. Z). d. D. d. D. d. D. d. 300 .002 1800 .066 3300 .223 4800 .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 .281 5200 .554 800 .013 2300 .108 3800 .296 Imile .571 900 .017 2400 .118 3900 .312 2 2.285 1000 .020 2500 .128 4000 .328 3 5.142 1100 .025 2600 .139 4100 .345 4 9.142 1200 .030 2700 .149 4200 .362 5 14.284 1300 .035 2800 .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 57.135 186 TABLE IX. RISE PER MILE OF VARIOUS GRADES. TABLE IX. EISE PER MILE OP VARIOUS GRADES, Grade per station. Kise per Mile. Grade per Station. Else per Mile. Grade per Station Kise per Mile. Grade per Station. Eise per Mile. .01 .528 .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 3.168 .46 24.288 .86 45.408 1.26 66.528 .07 3.696 .47 24.816 .87 45.936 1.27 67.056 .08 4.224 .48 25.344 .88 46.464 1.28 67.584 .09 4.752 .49 25.872 .89 46.992 1.29 68.112 .10 5.280 .50 26.400 .90 47.520 1.30 68.640 .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 .34 70.752 .15 7.920 .55 29.040 .95 50.160 .35 71.280 .16 8.448 .56 29.568 .96 50.688 .36 71.808 .17 8.976 .57 30.096 .97 51.216 .37 72.336 .18 9.504 .58 30.624 .98 51.744 .38 72.864 .19 10.032 .59 31.152 .99 52.272 .39 73.392 .20 10.E60 .60 31.680 1.00 52.800 1.40 73.920 .21 11.088 .61 32.208 1.01 53.328 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 75.504 .24 12.672 .64 33.792 1.04 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 37.488 .11 58.608 1.51 79.728 .32 16.896 .72 38.016 .12 59.136 1.52 80.256 .33 17.424 .73 38.544 .13 59.664 1.53 80.784 .34 17.952 .74 39.072 .14 60.192 1.54 81.312 .35 18.480 .75 39.600 .15 60.720 1.55 81.840 .36 19.008 .76 40.128 .16 61.248 1.56 82.368 .37 19.536 .77 40.656 .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 1.59 83.952 .40 21.120 .80 42.240 1.20 63.360 1.60 84.480 TABLE IX. RISE PER MILE OF VARIOUS GRADES. 187 Grade per Station. Rise per Mile. Grade per Station. Rise per Mile. Grade per Station. Rise per Mile. Grade per Station. Rise per Mile. 1.61 85.008 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 98.736 2.70 142.560 4.70 248.160 1.68 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 1.97 104.016 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 6.00 316.800 188 TABLE X. TRIGONOMETRICAL AND TABLE X. TRIGONOMETRICAL AND MISCELLANEOUS FORMULA. LET A (fig. 77) be any acute angle, and let a perpendicular B C be drawn from any point in one side to the other side. Then, if Fig. 77. the sides of the right triangle thus formed are denoted by letters, as in the figure, we shall have these six formulae : 1. . i* . sin. A = . c 2. cos. A = - . c 3. 0' 4. cosec. A = . a c 5. sec. A = T . 6. cot. A = - . a Solution of Right Angles (fig. 77). 7 8 9 10 11 Given. Sought. Formulae. a, c a, ~b A, a A,b A,c A,B,l A,B,c B, 6, c B,a,c B,a,l> a a j sin. A = - , cos. B-, ~b y(c + a)(c a), c c tan. A = -r , cot. B = j- , c = \/a* + 6 2 . 7? QO A 7. n ^^4- A n sm. A 7? Qfi A ft 7i fan \ f> cos. A B = 90 A, a= c sin. A, ~b c cos. A. MISCELLANEOUS FORMULA. 189 Solution of Oblique Triangles (fig. 78). Fig. 78. 12 13 14 15 16 17 18 Given. Sought. Formulae. A,,a A, a, b a, b, C a,b,c A,B,C,a A, b, c a, b, c b B A 7? , a sin. B sin. A -o b sin. A a tan l(A B\ (a ~ b) tolL * (A + B} A area area area a + b Tf * - !(. + 7, + C ) iuiA- l/ ( *~~ 6) (S ~ C) be J cos i 4 - a /(*-) tn 4 ^ - . /(*-b)(*-C) \ be V S (a-a) ' sin A - 2 Vs(s - a) (s - b)(s - c) I &c a 2 sin. 5 sin. (7 area . 2 sin. J. area = |Jcsin. J.. s = %(a 4- & + c), area = \/s (5 a) (sb) (sc). General Trigonometrical Formulae. 19 sin. 2 A + cos. 2 ^4 = 1. 20 sin. ( A B) = sin. A cos. B sin. B cos. Jl. 21 cos. ( A B) = cos. A cos. ^ T sin. A sin. i?. 22 sin. 2 J. = 2 sin. A cos. A. 23 cos. 2 J. = cos. 2 A sin. 2 A = 1 2 sin. 2 A = 2 cos. 2 A 24 sin. 2 ^i = i i cos. 2 A. -1. 190 TABLE X. TRIGONOMETRICAL AND General Trigonometrical JTormulce, (Continued). 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 cos. 2 A = sin. A + sin. A cos. A + cos. B sin. 2 A - cos. 2 A $ + 1 cos. 2 J.. sin. B = 2 sin. | (J. + .#) cos. i (^L B). sin. J5 2 cos. (A + B) sin. i (A .#). cos. B = 2 cos. | (A + 7?) cos. i (A .#). cos. J. = 2 sin. $(A + B) sin. | (A #). sin. 2 B = cos. 2 .# cos. 2 A = sin. (A + B) sin. (J. B). sin. 2 ^ = cos. (A + ^) cos. (^4 - B). sin. J. tan. A cot. A = tan. (A tan. A cot. A sin. A + cos. A ' cos. A sin. J. " ^ tan. Jl tan. i? 1:F 8ta!uA*f cos. ^4 cos. B ' , sin. (A B) cot B + """ sin. A sin.l?" sin. B tan. $(A + B) sin. A sin. A + sin. 5 tan. ^ (^4 B) ' sin. -D , \i\ _]_ p\ cos. A + sin. A + o5Ti ;- * t4 f^ Sin< ^ rot 4 M ^ cos. B sin. A cos. A sin. 5 . , A Dx cos. A + sin. A cos. 5 sin>jB rot 4M + Jft cos. B tan. $ A -- cot. i .A : cos. J. sin. ^4 ~ 1 + cos. ^4 * sin. A ~~ 1 cos. A * MISCELLANEOUS FORMULAE. Miscellaneous Formulae. 191 Sought. Given. Formulas. Area of 44 Circle Radius = r Trr 2 . 45 Ellipse Semi-axes = a and b vab. 40 Parabola Chord = c, height = h f ch* 47 Regular Polygon ( Side a, number of j ( sides = n j . 180 ^ a 2 n cot. . n Surface of 48 Sphere Radius = r 47rr 2 . 40 Zone Radius = r, height = h 2 IT r h. 50 Spherical Poly- j gon j j Radius of sphere = r j j sum of angles = S } o 7,^, o\i Qn k j O ^/fr lOyiOU . 180 [number of sides = n j Solidity of 51 Prism or Cylin- ) der j Base = b, height = h i)fc 52 Pyramid or Cone Base = , height = h i&^. 5:* Frustum ofl Pyramid o r j. Cone ( Bases = b and &i , ) j height = h j W + ,4V^). 54 Sphere Radius = r J IT ?' 3 . 55 Spherical Seg- ) ( Radii of bases = r\ T,/ 2 2 7,2 ment j ( and 7*1, height = h \ lir/j,(r + n 4t* ) 50 Prolate Spheroid Semi-transverse axisl of ellipse = a ^ 57 Oblate Spheroid Semi-conjugate axis | . of ellipse = b |*a% 58 Paraboloid ( Radius of base = r, | 4" IT r^fi. ] height = h \ TT = 3.14159 26535 89793 23846 26433 83280. Log. v = 0.49714 98726 94133 85435 12682 88291. * The area of a circular segment on railroad curves, where the chord is very long in proportion to the height, may be found with great accuracy by the above formula. 192 TABLE X. TRIGONOMETRICAL AND Miscellaneous Formulas, (Continued). United States Standard Gallon = 231 cub. in. = 0.133681 cub. ft. Bushel = 2150.42 " =1.244456 " British Imperial Gallon =277.27384 " =0.160459 " Length of Seconds Pendulum, at sea-level, at Equator, 39.0152 in. " " " " " " " N.York, 39.1017 " " London, 39,1393 " Weight of a Cubic Foot of Pure Water, according to Rankine : At 39.4 Fahrenheit, 62.425 Ibs. ; at 62, 62.355 Ibs. Figure of the Earth, Clarke, Ency. Brit. Art. Geodesy : Equatorial radius = 20 926 202 feet, Polar radius = 20 854 895 " Degrees in arc equal to radius 57.29578 Minutes" " " " 3437.74677 Seconds " " " " " 206264.80625 To change common logarithms into hyperbolic multiply by .434 294 48 ; the logarithm of which is 9.637 7843. x = tan. x -J- tan.% + tan. 6 # | tan. 7 o; + &c. Let a = length of a flat circular arc, c = its chord, R radius, D = deflection angle for 100 ft. chords. a 3 c 3 Then approximately a c = _ . p 2 = 2 = a sin. 2 J9= c sm. 2 I>. TABLES XI. XII. HEIGHTS BY ANEROID BAROMETER. 193 TABLES XI. AND XII. HEIGHTS BY ANEROID BAROMETER. THESE tables facilitate the use of the formula given below for obtaining the difference of height between two stations by means of the aneroid barometer. The formula and tables are taken from No. 12 of the Professional Papers of the Corps of Engineers, U. S. A. The aneroid barometers used are supposed to be adjusted to agree with a mercurial barometer at a temperature of 32 Fahrenheit, at the level of the sea, in latitude 45. Frequent comparisons with a mercurial barometer are highly desirable. Simultaneous observations of the barometers and of the temperature of the air are to be made at the two stations, or, if only one barometer is used, the observations should differ in time as little as possible- In both cases, repeated observations should be made when prac- ticable. Let Z the difference of height of the two stations in feet. " h = the reading in inches of the barometer at the lower station. " H= " upper " t and t' = the temperatures (Fahr.) of the air at the two stations. Then Z = (log. h - log. H) x 60384.3 x (l + * + *' ~ 64 ) . \ 900 / Table XI. contains the products of 60384.3 and the logarithms of any number of inches from 17 to 31, except that, as the charac- teristic of all these logarithms is one, this characteristic is omitted throughout, because the difference of any two products is not af- fected thereby. Table XII. contains the values of the fraction in the last parenthesis of the formula for all values of t + t' from 30 to 189. Example. Readings at lower station h = 29.63 in., t = 68 ; at higher station, //= 27.21 in., t' = 61. Table XI. gives for 29.63 28485.2 " " " " 27.21 26250.8 difference, 2234.4 Table XII. gives for 129 .0722 . Z = 2234.4 x 1.0722 = 2396 feet. 14 194 TABLE XI. FOR ANEROID FORMULA. M bb 5 X o CO saqoni a ui ppi-HCOt^ (NOit^t^OS O ^t 1 Ot O5 ^^^^^^ ^H^So t-rH IrlOlOOOC^ pGOp-^-i-i Sffl ^S^OiO ercco^cot- 1OOSCOO5CO QT-HG^IOO opoopiofN O O 10 O O t^COOSIr-CO C500-tH5CO rHC JOCDOSOW3 O Tt 1 Tf Tt< "^ rt "^f >O iO O S JO IO O D CD S COOt-JOlO ttJOSCOOSO COt-^OCCSCO COl 1 CO CD CO C oJcocoiH THTJJOOOCO OlOiOOCO CDCDCOCOCO COCOCCCD;D iOCDt^QOOS C5 r-J O PQ . CO JO CC -THO^CO 00 00 CO CO OS i-it-OOOr- ~ e5 e* G ^StTiCvJOtG ng ni CO TC TH Tf i-l lOCOt-t- COOTf Tf O llslll iiill lilii ^y CO O CO Ct CO CO CO lf5 T)< GO O5 O GO f 00 ^ CO lOTfOJrHO i . t ^ lf ? 1 ~i e ^ ^3JrHt^CC35 TtCDCOOOC* 1O 00 rH Tjl o ^ CC CO <7< (?5 i I TH O O O5 C5 GO t- t- CO 1C5 grTHCSeorHlO COt-GOGOO5 OT^C^WT? ^??? P??S 5t-gP? S? St^g 00510t-0 THT I CO 00 TH O <0 CO U !(N<7tC C* C? C: K3 I-H CD TH O O rP OOrHTtt-os szm* r^Wt^OOiO Oi O t^ O TH OOTH^COC oDeoo5ior< ^odw^-c'^ 55 8 Q * OOSTfCD-* O5OCO CO .- T< ODOOCOrHCO 0*CftC*COr-( ^coT- S O > I G< JOTt! COi-HOOOO COOCOJOO5 lOTHCOiHCO ?xl TT2Xf- ^i$5?^S ffiSS={3?2 OOOO:OOO O5Or-ieO(? ^OSOSOSOSCO t-CO^^O COlCCNOOlO THt-0?t-G<{ sCOg-^Ogi OOS^-COIO^ G^ (OOOt- CD Tt< CO T-H p ^&SS82S92 *2os o^iiig? iii?il QOW3O5OOO rf CO CO O^ CO b- lO rH CO CO COOSOOlOOO O5 ^iliSi iiiii TABLE XII. FOR ANEROID FORMULA. TABLE XII FOR ANEROID FORMULA. t + V t + '64 t + t' t + t' 64 t + t' t + t' 64 t + f t + > 64 900 900 900 900 30 -0.0378 70 +0.0067 110' + 0.0511 150 +0.0956 31 .0367 71 .0078 111 .0522 151 .0967 32 .0356 72 .0089 112 .0533 152 .0978 33 .0344 73 .0100 113 .0544 153 .0989 34 .0333 74 .0111 114 .0556 154 .1000 35 .0322 75 .0122 115 .0567 155 .1011 36 .0311 76 .0133 116 .0578 156 .1022 37 .0300 77 .0144 117 .0589 157 .1033 38 .0289 73 .0156 118 .0600 158 .1044 39 .0278 79 .0167 119 0611 159 .1056 40 .0267 80 .0178 120 .0622 160 .1067 41 .0256 81 .0189 121 .0633 161 .1078 43 .0244 82 .0200 122 .0644 162 .1089 43 .0233 83 .0211 123 0656 163 .1100 44 .0222 84 .0222 124 .0667 164 .1111 45 .0211 85 .0233 125 .0678 165 .1122 46 .0200 86 0244 126 .0689 166 .1133 47 .0189 87 .0256 127 .0700 167 .1144 48 .0178 88 .0267 128 .0711 16S .1156 49 .0167 89 .0278 129 .0722 169 .1167 50 .0156 90 .0289 130 .0733 170 .1178 51 .0144 91 .0300 131 .0744 171 .1189 52 .0133 92 .0311 132 .0756 172 .1200 53 .0122 93 .0322 133 .0767 173 .1211 51 .0111 94 .0333 134 .0778 174 .1222 55 .0100 95 .0344 135 .0789 175 .1233 56 .0089 96 .0356 136 .0800 176 .1244 57 .0078 97 .0367 137 .0811 177 .1256 58 .0067 98 .0378 138 .0822 178 .1267 59 .0056 99 .0389 139 .0833 179 .1278 60 .0044 100 .0400 140 .0844 180 .1289 61 .0033 101 .0411 141 .0856 181 .1300 62 .0022 102 .0422 142 .0867 182 .1311 63 -0.0011 103 .0433 143 .0878 183 .1322 64 .0030 104 .0444 144 .0889 184 .1333 65 + 0.0011 105 .0456 145 .0900 185 .1344 66 .0022 106 .0467 146 .0911 186 .1356 67 .0033 107 .0478 147 .0922 187 .1367 68 .0044 108 .0489 148 .0933 188 ,1378 69 + 0.0056 109 + 0.0500 149 + 0.0944 189 +0.1389 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND RECIPROCALS OF NUMBERS. FROM 1 TO 1054. 204 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Eoots. Cube Koots. Reciprocals. 1 1 1 1.0000000 1.0000000 1.000000000 2 4 8 1.4142136 1.2599210 .500000000 3 9 27 1.7320508 1.4422496 .333333333 4 16 64 2.0000000 1.5874011 .250000000 5 25 125 2.2360680 1.7099759 .200000000 6 36 216 2.4494897 1.8171206 .166666667 7 49 343 2.6457513 1.9129312 .142857143 8 64 512 2.8284271 2.0000000 .125000000 9 81 729 3.0000000 2.0800837 .111111111 10 100 1000 3.1622777 2.1544347 .100000000 11 121 1331 3.3166248 2.2239801 .090909091 12 144 1728 3.4641016 2.2894286 .083333333 13 169 2197 3.6055513 2.3513347 .076923077 14 196 2744 3.7416574 2.4101422 .0714^571 15 225 3375 3.8729833 2.4662121 .060060067 16 256 4096 4.0000000 2.5198421 .002500000 17 289 4913 4.1231056 2.5712816 .058823529 18 324 5832 4.2426407 2.6207414 .055555556 19 361 6859 4.3588989 2.6684016 .052631579 20 400 8000 4.4721360 2.7144177 .050000000 21 441 9261 4.5825757 2.7589243 .047619048 22 484 10648 4.6904158 2.8020393 .045454545 23 529 12167 4.7958315 2.8438670 .043478261 24 576 13824 4.8989795 2.8844991 .041060067 25 6 5 15625 5.0000000 2.9240177 .040000000 26 676 17576 5.0990195 2.9624960 .038461538 27 729 19683 5.1961524 3.0000000 .037037037 28 784 21952 5.2915026 3.0365889 .035714286 29 841 24389 5.3851648 3.0723168 .034482759 30 900 27000 5.4772256 3.1072325 .033333333 31 961 29791 5.5677644 3.1413806 .032258065 32 1024 32768 5.6568542 3.1748021 .031250000 33 1089 35937 5.7445626 3.2075343 .030303030 34 1156 39304 5.8309519 3.2396118 .029411765 35 1225 42875 5.9160798 3.2710663 .028571429 36 1296 46656 6.0000000 3.3019272 .027777778 37 1369 50653 6.0827625 3.3322218 .027027027 38 1444 54872 6.1644140 3.3619754 .026315789 39 1521 59319 6.2449980 3.3912114 .025641026 40 1600 64000 6.3245553 3.4199519 .025000000 41 1681 68921 6.4031242 3.4482172 .024390244 42 1764 74088 6.4807407 3.4760266 .023809524 43 1849 79507 6.5574385 3.5033981 .023255814 44 1936 85184 6.6332496 3.5303483 .022727273 45 2025 91125 6.7082039 3.5568933 .022222222 46 2116 97336 6.7823300 3.5830479 .021739130 47 2209 103823 6.8556546 3.60882G1 .021276600 48 2304 110592 6.9282032 3.6342411 .020833333 49 2401 117649 7.0000000 3.6593057 .020408163 50 2500 125000 7.0710678 3.6840314 .020000000 51 2601 132651 7.1414284 3.7084298 .019607843 52 2704 140608 7.2111026 3.7325111 .019230769 53 2809 148877 7.2801099 3.7562858 .018867925 54 2916 157464 7.3484692 3.7797631 .018518519 55 3025 166375 7.4161985 3.8029525 .018181818 56 3136 175616 7.4833148 3.8258624 .017857143 57 3249 185193 7.5498344 3.8485011 .017543860 58 3364 195112 7.6157731 3.8708766 .017241379 59 3481 205379 7.6811457 3.8929965 .016949153 60 3600 216000 7.7459667 3.9148676 .016666667 61 3721 226981 7.8102497 3.930 1972 .016393443 6-2 3844 238328 7.8740079 3.9578915 .016129032 CUBE BOOTS, AND RECIPKOCALS. 205 No. Square*. Cubes. Square Roots. Cube Roots Reciprocals. 63 3969 250047 7.9372639 3.9790571 .015873016 64 4096 262144 8.0000000 4.0000000 .015625000 65 4225 274625 8.0622577 4.0207256 .015384615 66 4356 287496 8.1240384 4.0412401 .015151515 67 4489 300763 8.1853528 4.0615480 .014925373 68 4624 314432 8.2462113 4.0816551 .014705882 69 4761 328509 8.3066239 4.1015661 .014492754 70 4900 343000 8.3666003 4.1212853 014285714 71 5041 357911 8.4261493 4.1408178 .014084507 72 5184 373248 8.4852814 4.1601676 .013888889 73 5329 389017 8.5440D37 4.1793390 .013698630 74 5476 405224 8.6023253 4.1983364 .013513514 76 5625 421875 8.6602540 4.2171633 .013333333 76 6776 438976 8.7177979 4.2358236 .013157895 77 5929 456533 8.7749644 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 4.3267487 .012345679 82 6724 551368 9.0553851 4.3444815 .012195122 83 6889 571787 9.1104336 4.3620707 .012048193 84 7056 592704 9.1651514 4.3795191 .011904762 85 7225 614125 9.2195445 4.3968296 .011764706 86 7396 636056 9.2738185 4.4140049 .011627907 87 7569 658503 9.3273791 4.4310476 .011494253 88 7744 681472 9.3808315 4.4479602 .011363636 89 7921 704969 9.4339811 4.4647451 .011235955 90 8100 729000 9.4868330 4.4814047 .011111111 91 8281 753571 9.5393920 4.4979414 .010989011 92 8464 778688 9.5916630 4.5143574 .010869565 93 8649 804357 9.6436508 4.5306549 .010762688 94 8836 830584 9.6953597 4.5468359 .010638298 95 9025 857375 9.7467943 4.5629026 .010526316 98 9216 884736 9.7979590 4.5788570 .010416667 97 9409 912673 9.8488578 4.5947009 .010309278 98 9604 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 102 103 10404 10609 1061208 1092727 10.0995049 10.1488916 4.6723287 4.6875482 .009803922 .009708738 104 10816 1124364 10.1980390 4.7026694 .009615385 105 11025 1157625 10.2469508 4.7176940 .009523810 106 11236 1191016 10.2956301 4.7326235 .009433962 107 108 11449 11664 1225043 1259712 10.3440804 10.3923048 4.7474594 4.7622032 .009345794 .009259259 109 11881 1295029 10.4403065 4.7768562 009174312 110 111 12100 12321 1331000 1367631 10.4880885 10.5356538 4.7914199 4.8058955 .009090909 .009009009 112 113 114 115 116 117 118 12544 12769 12996 13225 13456 13689 13924 1404928 1442897 1481544 1520875 1560896 1601613 1643032 10.5830052 10.6301458 10.6770783 10.7238053 10.7703296 10.8166538 10.8627805 4.8202845 4.8345881 4.8488076 4.8629442 4.8769990 4.8909732 4.9048681 .008928571 .008849558 .008771930 .008695652 .008620690 .008547009 .008474576 119 14161 1685159 10.9087121 4.9186847 .008403361 120 121 14400 14641 1728000 1771561 10.9544512 11.0000000 4.9324242 4.9460874 .008333333 .008264463 122 14884 1815848 11.0453610 4.9596757 .008196721 123 124 15129 15376 1860S67 1906624 11.0905365 11.1355287 4.9731898 4.9866310 .008130081 .008064516 206 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, Ho. Squares, Cubes. Square Roots. Cube Roota. Reciprocate. 126 15625 1953125 11.1803399 5.0000000 .008000000 126 15376 2000376 11.2249722 5.0132979 .007936508 127 16129 2048383 11.2694277 5.0265257 .007874016 123 16384 2097152 11.3137085 5.0396842 .007812500 129 16641 2146689 11.3578167 5.0527743 .007751938 130 16900 2197000 11.4017543 5.0657970 .007692308 131 17161 2248091 11.4455231 5.0787531 .007633588 132 17424 2299968 11.4891253 5.0916434 .007575758 133 17689 2352637 11.5325626 5.1044687 .007518797 134 17956 2406104 11.5758369 5.1172299 .007462687 135 18225 2460375 11.6189500 5.1299278 007407407 136 18496 2515456 11.6619038 5. 1425632 .007a52941 137 18769 2571353 11.7046999 5.1551367 .007299270 133 19044 2628072 11.7473401 5.1676493 007246377 139 19321 2685619 11.7898261 5.1801015 .007194245 140 19600 2744000 11.8321596 6.1924941 .007142857 141 19881 2803221 11.8743421 5.2048279 007092199 142 20164 2863288 11.9163753 5.2171034 .007042254 143 20449 2924207 11.9582607 5.2293215 .006993007 144 20736 2985984 12.0000000 5.2414828 .006944444 145 21025 3048625 12.0415946 5.2535879 .006896552 146 21316 3112136 12.0830460 5.2656374 .006849316 147 21609 3176523 12.1243557 5.2776321 .006802731 148 21904 3241792 12.1655251 5.2895725 .006756767 149 22201 3307949 12.2065556 5.3014592 .006711409 160 22500 3375000 12.2474487 6.3132928 .006666667 151 22801 3442951 12.2882057 5.3250740 .006622517 152 23104 3511808 12.3288280 6.3368033 .006578947 153 23409 3581577 12.3693169 5.3484812 .006535948 154 23716 3652264 12.4096736 5.3601084 .006493506 156 24025 3723875 12.4498996 6.3716854 .006451613 156 24336 3796416 12.4899960 5.3832126 .006410256 157 24649 3869893 12.5299641 6.3946907 .006369427 163 24964 3944312 12.5698051 5.4061202 .006329114 159 25281 4019679 12.6095202 6.4175015 .006289303 160 25600 4096000 12.6491106 6.4288352 .006250000 161 25921 4173281 12.6885775 6.4401218 .006211180 162 26244 4251528 12.7279221 5.4513618 .006172840 163 26569 4330747 12.7671453 5.4625556 .006134969 164 26396 4410944 12.8062485 5.4737037 .006097,561 165 27225 4492125 12.8452326 5.4848066 .006060606 166 27556 4574296 12.8840987 5.4958647 .006024096 IK' 27889 4657463 12.9228480 5.5068784 .005988024 163 28224 4741632 12.9614814 5.5178484 .005952331 169 28561 4826809 13.0000000 5.5287748 .005917160 170 28900 4913000 13.0384048 5.5396583 .005882353 171 29241 5000211 13.0766968 5.5504991 .005847953 172 29584 5088448 13.1148770 5.5612978 .005813953 173 29929 6177717 13.1529464 6.5720546 .005780347 174 30276 5268024 13.1909060 5.5827702 .005747126 175 30625 5359375 13.2287566 6.5934447 .005714286 17(5 30976 5451776 13.2664992 6.6040787 .005681818 177 31329 5545233 13.3041347 5.6146724 .005649718 178 31634 5639752 13.3416641 5.6252263 .005617978 179 32041 5735339 13.3790882 5.6357403 .005586592 180 32400 5832000 134164079 5.6462162 .005555556 181 32761 5929741 13.4536240 5.6566528 .005524862 182 331^ 6028568 13.4907376 66670511 005494505 183 33489 612&487 13.5277493 5.6774114 005464481 134 33856 6229504 13.5646600. 5.6877340 .005434783 185 34225 6331625 13.6014705 5.6980192 .005405405 186 34596 6434856 13.6381817 5.7082675 .005376344 CUBE ROOTS, AND RECIPROCALS. No. Squares. Cubes Square Roots. Cube Roots. Reciprocate. 187 34969 6539203 13.6747943 5.7184791 .005347594 188 35344 6644672 13.7113092 6.7286543 .005319149 189 35721 6751269 13.7477271 6.7387936 005291005 190 36100 6869000 13.7840488 6.7488971 .005263158 191 36481 6967871 13.8202750 6.7589652 - .005235602 192 36S64 7077888 13.8564065 6.7689982 .005208333 193 37249 7189057 13.8924440 6.7789966 .005181347 194 37636 7301384 13.9283883 6.7889604 .005154639 195 38025 7414875 13.9642400 6.7988900 .005128205 196 38416 7529536 14.0000000 6.8087857 .005102041 197 38809 7645373 14.0356688 6.8186479 .005076142 198 39204 7762392 14.0712473 6.8284767 .005050505 199 39601 7880599 14.1067360 6.8382725 005025126 200 40000 8000000 14.1421356 6.8480355 .005000000 201 40401 8120601 14.1774469 6.8577660 .004975124 202 40804 6242408 14.2126704 6.8674643 .004950496 203 41209 8365427 14.2478068 5.8771307 .004926108 204 41616 8489664 14.2828569 6.8867653 .004901961 206 42025 8615125 14.3178211 5.8963685 .004878049 206 42436 8741816 14.3527001 6.9059406 .004854369 207 42849 8869743 14.3874946 6.9164817 004830918 208 43264 8998912 14.4222051 6.9249921 004807692 209 43681 9129329 14.4568323 6.9344721 004784689 210 44100 9261000 14.4913767 6.9439220 .004761906 211 44521 9393931 14.5258390 6.9533418 .004739336 212 44944 9628128 14.5602198 6.9627320 .004716981 213 45369 9663697 14.6945195 6.9720926 .004694836 214 45796 9800344 14.6287388 6.9814240 .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 .004646465 221 48841 10793861 14.8660687 6.0459435 .004524887 222 49284 10941048 14.8996644 6.0550489 .004604606 223 49729 11089567 14.9331845 6.0641270 .004484306 224 50176 11239424 14.9666295 6.0731779 .004464286 225 50625 11390625 16.0000000 6.0822020 .004444444 226 61076 11543176 15.0332964 6.0911994 .004424779 227 61529 11697083 15.0665192 6.1001702 004406286 228 51984 11852352 15.0996689 6.1091147 .004385965 22S 52441 12008989 15.1327460 6.1180332 .004366812 230 52900 12167000 16.1657509 6.1269257 .004347826 231 53361 12326391 16.1986842 6.1357924 .004329004 232 63S24 12487168 15.2315462 6.1446337 .004310345 233 64289 12649337 15.2643375 6.1534495 .004291845 234 54756 12812904 15.2970586 6.1622401 .004273504 235 65225 12977875 15.3297097 6.1710058 .004255319 236 65696 13144256 15.3622916 6.1797466 .004237288 237 66169 13312053 15.3948043 6.1884628 .004219409 238 56644 13481272 15.4272486 6.1971544 .004201681 239 67121 13651919 15.4596248 6.2058218 .004184100 240 57600 13824000 15.4919334 6.2144650 004166667 241 58081 13997521 15.5241747 6.2230843 .004149378 242 53564 14172488 15.5563492 6.2316797 004132231 243 69049 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 61009 15069223 15.7162336 6.2743054 .004048583 248 61504 15252992 15.7480157 6.2827613 .004032258 208 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 249 62001 15438249 16.7797338 6.2911946 .004016064 250 62500 15625000 15.8113883 6.2996053 .004000000 251 63001 15813251 15.8429795 6.3079935 .003984064 252 63504 16003008 15.8745079 6.3163596 .003968254 253 64009 16194277 15.9059737 6.3247035 003952569 254 64516 16387064 15.9373775 6.3330256 .003937008 255 65025 16581375 15.9687194 6.3413257 .003921569 25,6 65536 16777216 16.0000000 6.3496042 .003906250 257 66049 16974593 16.0312195 6. 35786 li .003891051 258 66564 17173512 16.0623784 6.3660968 .003875969 259 67081 17373979 16.0934769 6.3743111 .003861004 260 67600 17576000 16.1245155 6.3825043 .003846154 261 68121 17779581 16.1554944 6.3906765 .003831418 262 68644 17984728 16.1864141 6.3988279 00381 6794 263 69169 18191447 16.2172747 6.4069585 .003802281 264 69696 183997-44 16.2480768 6.4150687 .003787879 265 70225 18609625 16.2788206 6.4231583 .003773585 266 70756 18821096 16.3095064 6.4312276 .003759398 267 71289 19034163 16.3401346 6.4392767 .003745318 263 71824 19248832 16.3707055 6.4473057 .003731343 269 72361 19465109 16.4012195 6.4553148 .003717472 270 72900 19683000 16.4316767 6.4633041 .003703704 271 73441 19902511 16.4620776 6.4712736 .003690037 272 73984 20123648 16.4924225 6.4792236 .003676471 273 74529 20346417 16.5227116 6.4871541 .003663004 274 75076 20570824 16.5529454 6.4950653 .003649635 275 75625 20796875 16.5831240 6.5029572 .003636364 276 76176 21024576 16.6132477 6.5108300 .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 280 78400 21952000 16.7332005 6.5421326 .003571429 281 78961 22188041 16.7630546 6.5499116 .003558719 282 79524 22425768 16.7928556 6.5576722 .003546099 283 80089 22665187 16.8226038 6.5654144 .003533569 284 80656 22906304 16.8522995 6.5731385 .003521 127 285 81225 23149125 16.8819430 6.5808443 .003508772 286 81796 23393656 16.9115345 6.5885323 .003496503 287 82369 23639903 16.9410743 6.596*023 .003484321 288 82944 23887872 16.9705627 6.6038545 .003472222 289 83521 24137569 17.0000000 6.6114890 .003460208 290 84100 24389000 17.0293864 6.6191060 .003448276 291 84681 24642171 17.0587221 6.6267054 .003436426 292 85264 24897038 17.0880075 6.6342874 .003424658 293 85849 25153757 17.1172428 6.6413522 .003412969 294 86436 25412184 17.1464282 6.6493998 003401361 295 87025 25672375 17.1755640 6.6569302 .003389831 296 87616 25934336 17.2046505 6.6644437 .003378378 297 88209 26198073 17.2330879 6.67194' K .003367003 298 88804 26463592 17.2626765 6.6794200 .003355705 299 89401 26730899 17.2916165 6.6868831 .003344482 300 90000 27000000 17.3205081 6.6943295 .003333333 301 90601 27270901 17.3493516 6.7017593 .003322259 302 91204 27543608 17.3781472 6.7091729 .003311258 303 91809 27818127 17.4068952 6.7165700 .003300330 304 92416 28094464 17,4355958 6.7239508 .003289474 305 93025 28372625 17.4642492 6.7313155 .003278689 306 93636 28652616 17.4928557 6.7386641 .003267974 307 94249 28934443 17.5214155 6.7459967 .003257329 308 94864 29218112 17.5499288. 6.7533134 .003246753 309 95481 29503629 17.5783958 6.7606143 .003236246 310 96100 29791000 17.6068169 6.7678995 .003225806 CUBE ROOTS, AND RECIPROCALS. 209 No. Sqoaice. Cubes. Square Root* . Cube Roots. Reciprocals. 311 96721 30080231 17.6351921 6.7751690 .003215434 312 97344 30371328 17.6635217 6.7824229 .003205128 313 97969 30664297 17.6918060 6.7S96613 .003194888 314 98596 30959144 17.7200451 6.7968844 .003184713 315 99225 31255875 17.7482393 6.8040921 .003174603 316 99356 31554496 17.7763888 6.8112847 .003164557 317 100489 31855013 17.8044933 6.8184620 .003154574 318 101124 32157432 17.8325545 6.8256242 .003144654 319 101761 32461769 17.86057!! 6.8327714 .003134796 ' 320 102400 32768000 17.8885438 6.8399037 .003125000 321 103041 33076161 17.9164729 6.8470213 .003115265 322 103684 33386248 17.9443584 6.8541240 .003105590 323 104329 33693267 17.9722008 6.8612120 .003095975 324 104976 34012224 18.0000000 6.8682855 .003086420 325 105625 34328125 18.0277564 6.8753443 .003076923 326 106276 34645976 18.0554701 6.8823888 .003067485 327 106929 34965783 18.0831413 6.8894188 .003058104 323 107584 35287552 18.1107703 6.8964345 .003048780 329 108241 35611289 18.1383571 6.9034359 .003039514 330 108900 35937000 18.1659021 $.9104232 .003030303 331 109561 36264691 18.1934054 6.9173964 .003021148 332 110224 36594368 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 .002985076 336 112896 37933056 18.3303028 6.9520533 .002976190 337 113569 38272753 18.3575598 6.9589434 .002967359 333 114244 38614472 18.3847763 6.9658198 .002958580 339 114921 38958219 18.4119526 6.9726826 .002949853 340 115600 39304000 18.4390889 6.9795321 .002941176 341 116281 39651821 18.4661853 6.9863681 .002932551 342 116964 40001688 18.4932420 6.9931906 .002923977 343 117649 40353607 18.5202592 7.0000000 .002915452 344 118336 40707584 18.5472370 7.0067962 .002906977 345 119025 41063625 18.5741756 7.0135791 .002898551 346 119716 41421736 18.6010752 7.0203490 .002890173 347 120409 41781923 18.6279360 7.0271058 .002881844 343 121104 42144192 18.6547581 7.0338497 .002873563 349 121801 42508549 18.6815417 7.0405806 .002865330 350 122500 42875000 18.7082869 7.0472987 .002857143 351 123201 43243551 18.7349940 7.0540041 .002849003 352 123904 43614208 18.7616630 7.0606967 .002840909 353 124609 43986977 18.7882942 7.0673767 .002832861 354 125316 44361864 18.8148877 7.0740440 .002824859 355 126025 44738875 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 .002770083 362 131044 47437928 19.0262976 7.1269360 .002762431 363 131769 47832147 19.0525589 7.1334925 .002754821 364 132496 48228544 19.0787840 7.1400370 .002747253 365 133225 48627125 19.1049732 7.1465695 .002739726 366 133956 49027896 19.1311265 7.1530901 .002732240 367 134689 49430S63 19.1572441 7.1595988 .002724796 368 135424 49836032 19.1833261 7.1660957 .002717391 369 136161 50243409 19.2093727 7.1725809 .002710027 370 136900 50653000 19.2353841 7.1790544 .002702703 371 137641 51064811 19.2613603 7.1855162 .002695418 372 133384 51478848 19.2873015 7.1919663 .002688172 15 210 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 373 139129 61895117 19.3132079 7.1984050 .002680986 374 139876 52313624 19.3390796 7.2048322 .002673797 375 140625 52734375 19.3649167 7.2112479 .002666667 376 141376 53157376 19.3907194 7.2176522 .002659574 377 142129 53582633 19.4164878 7.2240450 .002652520 378 142884 54010152 19.4422221 7.2304268 .002645503 379 143641 54439939 19.4679223 7.2367972 .002638522 380 144400 54872000 19.4935887 7.2431565 .002631570 381 145161 55306341 19.5192213 7.2495045 .002624672 332 145924 55742968 19.5448203 7.2558415 .002617801 ' 383 146689 56181887 19.5703858 7.2621675 .002610966 384 147456 56623104 19.5959179 7.2034824 .002604167 385 148225 57066625 19.6214169 7.2747864 .002597403 386 148996 57512456 19.6468827 7.2810794 .002590674 387 149769 57960603 19.6723156 7.2873617 .002583979 338 150544 58411072 19.6977156 7.2936330 .002577320 389 151321 58863869 19.7230829 7.2998936 .002570694 390 152100 59319000 19.7484177 7.3061436 .002564103 391 152881 69776471 19.7/37199 7.3123828 .002557545 392 153664 60236288 19.79S9899 7.3186114 .002551020 393 154449 60698457 19.8242276 7.3248295 .002544529 394 155236 61162934 19.8494332 7.3310369 .002538071 396 156025 61629875 19.8746069 7.3372339 .002531646 396 156816 62099136 19.8997487 7.3434205 .002525253 397 157609 62570773 19.9248588 7.3495966 .002518892 398 153404 63044792 19.9499373 7.3557624 .002512563 399 159201 63521199 19.9749844 7.3619178 .002506266 400 160000 64000000 20.0000000 7.3680630 .002500000 401 160801 64481201 20.0249844 7.3741979 .002493766 403 161604 64964808 20.0499377 7.3803227 .002487562 403 162409 65450827 20.0748699 7.3864373 .002481390 404 163216 65939264 20.0997512 7.3925418 .002475248 405 164025 66430125 20.1246118 7.3986363 .002469136 406 164836 66923416 20.1494417 7.4047206 .002463054 407 165649 67419143 20.1742410 7.4107950 .002457002 408 166464 67917312 20.1990099 7.4168595 .002450980 409 167281 68417929 20.2237484 7.4229142 .002444988 410 168100 68921000 20.2484567 7.4289589 .002439024 411 168921 6&426531 20.2731349 7.4349938 .002433090 412 169744 69934528 20.2977831 7.4410189 .002427184 413 170569 70444997 20.3224014 7.4470342 .002421308 414 171396 70957944 20.3469899 7.4530399 .002415459 415 172225 71473375 20.3715488 7.4590359 .002409639 416 173056 71991296 20.3960781 7.4650223 .002403846 417 173889 72511713 20.4205779 7.4709991 .002398082 418 174724 73034632 20.4450483 7.4769664 002392344 419 175561 73560059 20.4694895 7.4829242 .002386635 ' 420 176400 74088000 20.4939015 7.4888724 ,002380952 421 177241 74618461 20.5182845 7.4948113 .002375297 422 178084 75151448 20.5426386 7.5007406 .002369668 423 178929 75686967 20.5669638 7.5066607 .002364066 424 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 428 183184 78402752 20.6881609 7.5361221 .002336449 429 184041 78953589 20.7123152 7.5419867 .002331002 430 184900 79507000 20.7364414 7.5478423 .002325581 431 185761 80062991 20.7605395 7.5536888 .002320186 432 186624 80621568 20.7846097 7.5595263 .002314815 433 187489 81182737 20.8086520 7.5653548 .002309469 434 188356 81746504 20.8326667 7.5711743 .002304147 CUBE HOOTS, AND KECIPROCALS. No. Squares. Cubes Square Roots. Cube Roots. Reciprocals. 435 189225 82312875 20.8566536 7.5769849 .002298851 436 190096 82881856 20.8806130 7.5827865 .002293578 437 190969 83453453 20.9045450 7.5885793 .002288330 438 191844 84027672 20.9284495 7.5943633 .002283105 439 192721 84604519 20.9523268 7.6001385 .002277904 440 193600 85184000 20.9761770 7.6059049 .002272727 441 194481 85766121 21.0000000 7.6116626 .002267574 442 195364 86350888 21.0237960 7.6174116 .002262443 443 196249 86938307 21.0475652 7.6231519 .002257336 444 197136 87528384 21.0713075 7.6288837 .002252252 445 198025 88121125 21.0950231 7.6346067 .002247191 446 198916 88716536 21.1187121 7.6403213 .002242152 447 199809 89314623 21.1423745 7.6460272 .002237136 448 200704 89915392 21.1660105 7.6517247 .002232143 449 201601 90518849 21.1896201 7.6574138 .002227171 450 202500 91125000 21.2132034 7.6630943 .002222222 451 203401 91733851 21.2367606 7.6687665 .002217295 452 204304 92345408 21.2602916 7.6744303 .002212389 453 205209 92959677 21.2837967 7.6800857 .002207506 454 206116 93576664 21.3072758 7.6857328 .002202643 455 207025 94196375 21.3307290 7.6913717 .002197802 456 207936 94818816 21.3541565 7.6970023 .002192982 457 208849 95443993 21.3775583 7.7026246 .002188184 458 209764 96071912 21.4009346 7.7082388 .002183406 459 210681 96702579 21.4242853 7.7138448 .002178649 460 211600 97336000 21.4476106 7.7194426 .002173913 461 212521 97972181 21.4709106 7.7250325 .002169197 462 213444 98611128 21.4941853 7.7306141 .002164502 463 214369 99252847 21.5174348 7.7361877 .002159827 464 215296 99897344 21.5406592 7.7417532 .002165172 465 216225 100544625 21.5638587 7.7473109 .002150538 466 217156 101194696 21.5870331 7.7528606 .002145923 467 218089 101847563 21.6101828 7.7584023 .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 105154048 21.7255610 7.7859928 .002118644 473 223729 105823817 21.7485632 7.7914875 002114165 474 224676 106496424 21.7715411 7.7969745 002109705 475 225625 107171875 21.7944947 7.8024538 .002105263 476 226576 107850176 21.8174242 7.8079254 .002100840 477 227529 108531333 21.8403297 7.8133892 .002096436 478 228484 109215352 21.8632111 7.8188456 .002092050 479 229441 109902239 21.8860686 7.8242942 .002087683 480 230400 110592000 21.9089023 7.8297353 .002083333 481 231361 111284641 21.9317122 7.8351688 .002079002 482 232324 111980168 21.9544984 7.8405949 .002074689 483 233289 112678587 21.9772610 7.8460134 .002070393 484 234256 113379904 22.0000000 7.8514244 .002066116 485 235225 114084125 22.0227155 7.8568281 .002061856 486 236196 114791256 22.0454077 7.8622242 .002057613 487 237169 115501303 22.0680765 7.8676130 .002053388 488 238144 116214272 22.0907220 7.8729944 .002049180 489 239121 116930169 22.1133444 7.8783684 .002044990 490 240100 117649000 22.1359436 7.8837352 .002040816 491 241081 118370771 22.1585198 7.8890946 .002036660 492 242064 1190954&5 22.1810730 7.8944468 .002032520 493 243049 119823157 22.2036033 7.8997917 .002028398 494 244036 120553784 22.2261108 7.9051294 .002024291 495 245025 121287375 22.2485955 7.9104599 .002020202 496 246016 122023936 22.2710575 7.9157832 .002016129 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Boots. Reciprocate. 497 247009 122763473 22.2934968 7.9210994 .002012072 498 248004 123505992 22.3159136 7.9264085 .002008032 499 249001 124251499 22.3383079 7.9317104 ,002004008 500 250000 125000000 22.3606798 7.9370053 .002000000 601 251001 125751501 22.3830293 7.9422931 .001996008 602 252004 126506008 22.4053565 7.9475739 .001992032 503 253009 127263527 22.4276615 7.9528477 .001988072 604 254016 128024064 22.4499443 7.9581144 .001984127 505 255025 128787625 22.4722051 7.9633743 .001980198 606 256036 129554216 22.4944438 7.9686271 .001976285 607 257049 130323843 22.5166605 7.9738731 .001972387 603 258064 131096512 22.5388553 7.9791122 .001968504 609 259081 131872229 22.5610283 7.9843444 .001964637 510 260100 132651000 22.5831796 7.9895697 .001960784 611 261121 133432S31 22.6053091 7.9947883 .001956947 612 262144 134217728 22.6274170 8.0000000 .001953125 513 263169 135005697 22.6495033 8.0052049 .001949318 614 264196 135796744 22.6715681 8.0104032 .001945525 615 265225 136590875 22.6936114 8.0155946 .001941748 616 266256 137388096 22.7156334 8.0207794 .001937984 617 267289 138188413 22.7376340 8.0259574 .001934236 518 268324 138991832 22.7596134* 8.0311287 .001930502 619 269361 139798359 22.7815715 8.0362935 .001926782 520 270400 140608000 22.8035085 8.0414515 .001923077 621 271441 141420761 22.8254244 8.0466030 .001919386 622 272484 142236648 22.8473193 8.0517479 .001915709 623 273529 143055667 22.8691933 8.0568862 .001912046 524 274576 143877824 22.8910463 8.0620180 .001908397 625 275625 144703125 22.9128785 8.0671432 .001904762 526 276676 145531576 22.9346899 8.0722620 .001901141 627 277729 146363183 22.9564806 8.0773743 .001897533 528 278784 147197952 22.9782506 8.0824800 .001893939 629 279841 148035889 23.0000000 8.0875794 .001890359 630 280900 148877000 23.0217289 8.0926723 .001886792 631 281961 149721291 23.0434372 8.0977589 .001883239 532 283024 150568768 23.0651252 8.1028390 .001879699 533 284089 151419437 23.0867928 8.1079128 .001876173 534 285156 152273304 23.1084400 8.1129803 .001872659 635 286225 153130375 23.1300670 8.1180414 .001869159 536 287296 153990656 23.1516738 8.1230962 .001865672 637 288369 154854153 23.1732605 8.1281447 .001862197 538 289444 155720872 23.1948270 8.1331870 .001858736 539 290521 156590819 23.2163735 8.1382230 .001855288 640 291600 157464000 23.2379001 8.1432529 .001851852 641 292681 158340421 23.2594067 8.1482765 .001848429 542 293764 159220088 23.2808935 8.1532939 .001845018 543 294849 160103007 23.3023604 8.1583051 .001841621 544 295936 160989184 23.3238076 8.1633102 .001838235 645 297025 161878625 23.3452351 8.1683092 .001834862 646 298116 162771336 23.3666429 8.1733020 .001831502 647 299209 163667323 23.3880311 8.1782888 .001828154 548 300304 164566592 23.4093998 8.1832695 .001824818 549 301401 ' 165469149 23.4307490 8.1882441 .001821494 550 302500 166375000 23.4520788 8.1932127 .001818182 551 303601 167284151 23.4733892 8.1981753 .001814882 652 304704 168196608 23.4946802 8.2031319 .001811594 553 305809 1691 12377 23.5159520 8.2080825 .001808318 554 306916 170031464 23.5372046 8.2130271 .001805054 555 308025 170953875 23.5584380 8.2179657 .001801802 656 309136 171879616 23.5796522 8.2228985 .001798561 557 310249 172808693 23.6008474 8.2278254 .001795332 553 311864 173741112 23.6220236 8.2327463 .001792115 CUBE HOOTS, AND KECIPROCALS. 213 Ho. Squares. Cubes. Square Roots. Cube Roots. Reciprocate. 569 312481 ! 74676879 23.6431808 8.2376614 .001788909 660 313600 175616000 23.6643191 8.2425706 .001785714 661 314721 176558481 23.6854336 8.2474740 .001782531 662 315844 177504323 23.7065392 8.2523715 .001779369 663 316969 178453547 23.7276210 8.2572633 .001776199 664 318096 179406144 23.7486842 8.2621492 .001773050 665 319225 180362125 23.7697286 8.2670294 .001769912 666 320356 181321496 23.7907545 8.2719039 .001766784 567 321489 182284263 23.8117618 8.2767726 .001763668 568 322624 183250432 23.8327506 8.2816355 .001760563 669 323761 184220009 23.8537209 8.2864928 .001767469 670 324900 185193000 23.8746728 8.2913444 .001754386 671 326041 186169411 23.8956063 8.2961903 .001751313 572 327184 187149248 23.9165215 8.3010304 .001748252 673 328329 188132517 23.9374184 8.3058651 .001745201 674 329476 189119224 23.9582971 8.3106941 .001742160 575 330625 190109375 23.9791576 8.3155175 .001739130 676 331776 191102976 24.0000000 8.3203353 .001736111 677 332929 192100033 24.0208243 8.3251475 .001733102 578 334084 193100552 24.0416306 8.3299542 .001730104 679 335241 194104539 24.0624188 8.3347553 .001727116 680 336400 195112000 24.0831891 8.3395509 .001724138 681 337561 196122941 24.1039416 8.3443410 .001721170 682 338724 197137368 24.1246762 8.3491256 001718213 683 339889 198155287 24.1453929 8.3539047 .001715266 684 341056 199176704 24.1660919 8.3586784 .001712329 686 342225 200201625. 24.1867732 8.3634466 .001709402 686 343396 201230056 24.2074369 8.3682095 .001706485 687 344569 202262003 24.2280829 8.3729668 .001703578 683 345744 203297472 24.2487113 8.3777188 .001700680 689 346921 204336469 24.2693222 8.3824653 .001697793 690 348100 205379000 24.2899156 8.3872065 .001694916 691 349281 206425071 24.3104916 8.3919423 .001692047 592 350464 207474688 24.3310501 8.3966729 .001689189 593 351649 203527857 24.3515913 8.4013981 .001686341 594 352836 209584584 24.3721152 8.4061180 .001683502 595 354025 210644875 24.3926218 8.4108326 .001680672 596 355216 211708736 24.4131112 8.4155419 .001677852 597 356409 212776173 24.4335834 8.4202460 .001675042 598 357604 213847192 24.4540385 8.4249448 .001672241 599 358801 214921799 24.4744765 8.4296383 .001669449 600 360000 216000000 24.4948974 8.4343267 .001666667 601 361201 217081801 24.5153013 8.4390098 .001663894 602 362404 218167208 24.5356883 8.4436877 .001661130 603 363609 219256227 24.5560583 8.4483605 .001658375 604 364816 220348864 24.5764115 8.4530281 .001655629 605 366025 221-445125 24.5967478 8.4576906 .001652893 606 367236 222545016 24.6170673 8.4623479 .001650165 607 363449 223648543 24.6373700 8.4670001 .001647446 608 369664 224755712 24.6576560 8.4716471 .001644737 609 370881 225866529 24.6779254 8.4762892 .001642036 610 372100 226981000 24.6981781 8.4809261 .001639344 611 373321 22S099131 24.7184142 8.4855579 001636661 612 374544 229220928 24.7386338 8.4901848 .001633987 613 375769 230346397 24.7588368 8.4948065 .001631321 614 376996 231475544 24.7790234 8.4994233 .001628664 615 378225 232608375 24.7991935 8.5040350 .001626016 616 379456 233744396 24.8193473 8.5086417 .001623377 617 380639 2348S5I13 24.8394847 8.5132435 001620746 618 381924 236029032 24.8596058 8.5178403 001618123 619 383161 237176659 24.8797106 8.5224321 001615509 620 334400 238328000 24.8997992 8.5270139 .001612303 214 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, 1 * Squares. Cubes. Square Boots Cube Roots. Reciprocals. 621 385641 239483061 24.9198716 8.5316009 .001610306 622 386884 240641848 24.9399278 8.5361780 .001607717 623 388129 241804367 24.9599679 8.5407501 .001605136 624 389376 242970624 24.9799920 8.6453173 .001602504 625 390625 244140625 25.0000000 8.5498797 .001600000 626 391876 245314376 25.0199920 8.6544372 .001597444 627 393129 246491883 25.0399681 8.5589899 .001594896 628 394384 247673152 25.0599282 8.6635377 .001592357 629 395641 243858189 25.0798724 8.5680807 .001589825 630 396900 250047000 25.0998008 8.5726189 .001687302 631 398161 251239591 25.1197134 8.5771523 .001584780 632 399424 252435968 25.1396102 8.5816809 .001582278 633 400689 253636137 25.1594913 8.5862047 .001579779 634 401956 254840104 25.1793566 8.5907238 .001677287 635 403225 256047875 25.1992063 8.5952380 .001574803 636 404496 257259456 25.21904(^4 8.6997476 .001572327 637 405769 258474853 25.2388589 8 6042525 .001569859 633 407044 259694072 25.2586619 8.6087526 .001567398 639 408321 260917119 25.2784493 8.6132480 .001564946 640 409600 262144000 25.2932213 8.6177388 .001562500 641 410881 263374721 25.3179778 8.6222248 .001560062 642 412164 264609233 25.3377189 8.6267063 .001557632 643 413449 265847707 25.3574447 8.6311830 .001555210 644 414736 267089934 25.3771551 8.6356551 .001552796 645 416025 268336125 25.3968502 8.6401226 .001550388 646 417316 269586136 25.4165301 8.6445855 .001547988 647 418609 270840023 25.4361947 8.6490437 .001645595 643 419904 272097792 25.4558441 8.6534974 .001543210 649 421201 273359449 25.4754784 8.6579465 .001540832 650 422500 274625000 25.4950976 8.6623911 .001538462 651 423801 275894451 25.5147016 8.6668310 .001536098 652 425104 277167808 25.5342907 8.6712665 .001533742 653 426409 278445077 25.5538647 8.6756974 .001531394 654 427716 279726264 ' 25.5734237 8.6801237 .001529052 655 429025 281011375 25.5929678 8.6845456 .001526718 656 430336 282300416 25.6124969 8.6889630 .001524390 657 431649 283593393 25.6320112 8.6933759 .001522070 658 432964 284890312 25.6515107 8.6977843 .001519757 659 434231 286191179 25.6709953 8.7021882 .001517451 660 435600 287496000 25.6904652 8.7065877 .001515152 661 436921 288804781 25.7099203 8.7109827 .001512869 662 438244 290117528 25.7293607 8.7153734 .001510574 663 439569 291434247 25.7487864 8.7197590 .001508296 664 440896 292754941 25.7681975 8.7241414 .001506024 665 442225 294079625 25.7875939 8.7285187 .001503769 666 443556 295408296 25.8069758 8.7328918 .001501502 667 444889 296740963 25.8263431 8.7372604 .001499250 668 446224 298077632 25.8456960 8.7416246 .001497006 669 447561 299418309 25.8650343 8.7459846 .001494768 670 448900 300763000 25.8843582 8.7503401 .001492537 671 450241 302111711 25.9036677 8.7546913 .001490313 672 451584 303464448 25.9229628 8.7590383 .001488095 673 452929 304821217 25.9422435 8.7633809 .001485384 674 454276 306182024 25.9615100 8.7677192 .001483680 675 455625 307546875 25.9807621 8.7720532 .001481481 676 456976 308915776 26.0000000 8.7763830 .001479290 677 458329 310288733 26.0192237 8.7807084 .001477105 678 459684 311665752 26.0384331 8.7850296 .001474926 679 461041 313046839 26.0576284 8.7893466 .001472754 680 462400 314432000 26.0763096 8.7936593 .001470588 681 463761 315821241 26.0959767 8.7979679 .001468429 682 465124 317214568 26.1151297 8.8022721 .001466276 CUBE ROOTS, AND RECIPROCALS. 215 Wo. Squares. Cubes. Square Roots. Cube Roots. Reciprocate. 683 466489 318611987 26. 1342687 8.8065722 .001464129 684 467856 320013504 26.1533937 8.8108681 .001461988 685 469225 321419125 26.1725047 8.8151598 .001459854 686 470596 322828856 26.1916017 8.8194474 .001457726 687 471969 324242703 26.2106848 8.8237307 .001455604 688 473344 325660672 26.2297541 8.8280099 .001453488 689 474721 327082769 26.2488095 8.8322850 .001451379 690 476100 328509000 26.2678511 8.8365559 .001449276 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 338608873 26.4007576 8.8663375 .001434720 693 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 .001422476 704 495616 348913664 26.5329983 8.8959204 .001420455 705 497025 350402625 26.5518361 8.9001304 .001418440 706 498436 351895816 26.5706605 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 26.16645833 8.9253078 .001406470 712 506944 360944128 26.6833281 8.9294902 .001404494 713 508369 362467097 26.7020593 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 001392768 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 621284 376367048 26.8700577 8.9711007 .001386042 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 728 529984 385828352 26.9814751 8.9958829 .001373626 729 531441 387420489 27.0000000 9.0000000 .001371742 730 632900 389017000 27.0185122 9.0041134 .001369863 731 634361 390617891 27.0370117 9.0082229 .001367989 732 535824 392223168 27.0554985 9.0123288 .001366120 733 537289 393832837 27.0739727 9,0164309 .001364256 734 638756 395446904 27.0924344 9.0205293 .001362398 735 540225 397065375 27.1108834 9.0246239 .001360544 736 541696 39S68S256 27.1293199 9.0287149 .001358696 737 543169 400315553 27.1477439 9.0328021 .001356852 738 544644 401947272 27.1661554 9.0368857 .001355014 739 546121 403583419 27.1845544 9.0409655 .001353180 740 741 742 i 743 744 547600 549081 550564 552049 553536 405224000 406869021 408518488 410172407 411830784 27.2029410 27.2213152 27.2396769 27.25R0263 272763634 9.0450417 9.0491142 9,0531831 9.0572482 9.0613098 .001351351 .001349528 .001347709 .001345895 .001344086 216 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, No. Sqaun*. Cubes. Square Roots. Gale Roots. Reciprocals. 745 555025 413493625 27.2946881 9.0653677 .001342282 746 556516 415160936 27.3130006 9.0694220 .001340483 747 553009 416832723 27.3313007 9.0734726 .001338688 748 559504 418508992 27.3495887 9.0776197 .001336898 749 561001 420189749 27.3678644 9.0815631 .001335113 750 562500 421875000 27.3861279 9.0856030 .001333333 751 564001 423564751 27.4043792 9.0896392 .001331558 752 565504 425259008 27.4226184 9.0936719 .001329787 753 567009 426957777 27.4408455 9.0977010 .001328021 754 568516 428661064 27.4590604 9.1017265 .001326260 755 570025 430368875 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 576081 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 764 583696 445943744 27.6405499 9.1417874 .001308901 765 585225 447697125 27.6586334 9.1457742 .001307190 766 586756 449455096 27.6767050 9.1497576 .001305483 767 588289 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 .001298701 771 594441 458314011 27.7668868 9.1696225 .001297017 772 595984 460099648 27.7848880 9.1735852 .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 778 602176 467288576 27.8567766 9.1894018 .001288660 777 603729 469097433 27.8747197 9.1933474 .001287001 778 605284 470910952 27.8926514 9.1972897 .001285347 779 606841 472729139 27.9105715 9.2012286 .001283697 780 608400 474552000 27.9284801 9.2051641 .001282051 781 609961 476379541 27.9463772 9.2090962 .001280410 782 611524 478211768 27.3642629 9.2130250 .001278772 783 613039 480048687 27.9821372 9.2169505 .001277139 784 614656 481890304 28.0000000 9.2208726 .001275510 785 616225 483736625 28.0178515 9.2247914 .001273885 786 617796 485587656 28.0356915 9.2287068 .001272265 787 619369 487443403 28.0535203 9.2326189 .001270648 788 620944 489303872 28.0713377 9.2365277 .001269036 789 622521 491169069 28.0891438 9.2404333 .001267427 790 624100 493039000 28.1069386 9.2443355 .001265823 1 791 625681 494913671 28.1247222 9.2482344 .001264223 792 627264 496793088 28.1424946 9.25213QP .001262626 793 628849 498677257 28.1602557 9.2560224 .001261034 794 630436 500566184 28.1780056 9.2599114 .001259446 795 632025 502459875 28.1957444 9.2637973 .001257862 796 633616 504358a36 28.2134720 9.2676798 .001256281 797 635209 506261573 28.2311884 9.2715592 .001254705 798 636804 508169592 28.2488938 9.2754352 .001253133 799 638401 510082399 28.2665881 9.2793081 .001251564 800 640000 512000000 28.2842712 9.2831777 .001250000 801 641601 513922401 28.3019434 9.2870440 .001248439 802 643204 515849608 28.3196045 9.2909072 .001246883 803 644809 517781627 28.3372546 9.2947671 ,001245330 804 646416 519718464 28.3548938 9.2986239 .001243781 805 648025 521660125 23.3725219 9.3024775 .001242236 806 649636 523606616 28.3901391 9.3063278 .001240695 CUBE ROOTS, AND RECIPROCALS. 217 No. Squares. Cubes. Square Boots. Cube Roots. Reciprocals. 807 651249 525557943 28.4077454 9.3101750 .001239167 808 6528*4 5275141 12 28.4253408 9.3140190 .001237624 809 654481 529475129 28.4429253 9.3178599 .001236094 810 656100 531441000 28.4604989 9.3216975 .001234568 811 657721 533411731 28.4780617 9.3255320 .001233046 812 659344 535387328 28.4956137 9.3293634 .001231527 813 660969 537367797 28.5131549 9.3331916 .001230012 814 662596 539353144 28.5306852 9.3370167 .001228501 815 664225 541343375 28.5482048 9.3408386 .001226994 816 665856 543338496 28.5657137 9.3446575 .001225490 817 667489 545338513 28.5832119 9.3484731 .001223990 818 669124 547343432 28.6006993 9.3522857 .001222494 819 670761 649353259 286181760 9.3560952 .001221001 820 672400 65136800C 28.6356421 9.3599016 .001219512 821 674041 553387661 28.6530976 9.3637049 .001218027 822 675684 555412248 28.6705424 9.3675051 .001216545 823 677329 557441767 28.6379766 9.3713022 .001215067 824 678976 559476224 28.7054002 9.3750963 .001213592 825 6S0625 561515625 28.7228132 9.3788873 .001212121 826 682276 563559976 23.7402157 9.3826752 .001210654 827 683929 565609283 23.7576077 9.3864600 .001209190 828 685584 567663552 28.7749891 9.3902419 .001207729 829 687241 569722789 28.7923601 9.3940206 .001206273 830 688900 571787000 28.8097206 9.3977964 .001204819 831 690561 573856191 28.8270706 9.4015691 .001203369 832 692224 5759303S8 28.8444102 9.4053387 .001201923 833 693889 578009537 28.8617394 9.4091054 .001200480 834 695556 580093704 23.8790582 9.4128690 .001199041 835 697225 582182375 28.8963666 9.4166297 .001197605 836 698896 684277056 28.9136646 9.4203873 .001196172 837 700569 586376253 28.9309523 9.4241420 .001194743 838 702244 588480472 28.9482297 9.4278936 .001193317 839 703921 590589719 28.9654967 9.4316423 .001191895 840 705600 592704000 28.9827535 9.4353880 .001190476 841 707281 594823321 29.0000000 9.4391307 .001189061 842 708964 5969476.38 29.0172363 9.4428704 .001187648 843 710649 599077107 29.0344623 9.4466072 .001186240 844 712336 601211584 29.0516781 9.4503410 .001184834 845 714025 603351125 29.068-8837 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 849 720801 611960049 29.1376046 9.4639661 .001177856 850 722500 614125000 29.1547595 9.4726824 .001176471 851 724201 616295051 29.1719043 9.4763957 .001175088 852 725904 618470208 29. 1890390 9.4801061 .001173709 353 727609 ,620650477 29.2061637 9.4838136 .001172333 854 729316 622835364 29.2232784 9.4875182 .001170960 855 731025 625026375 29.2403330 9.4912200 .001169591 856 732736 627222016 29.2574777 9.4949188 .001168224 857 734449 629422793 29.2745623 9.4986147 .001166861 853 736164 631623712 29.2916370 9.5023078 .001165501 859 737881 633839779 29.3087018 9.5059980 .001164144 860 739600 636056000 29.3257566 9.5096354 .001 162791 861 741321 638277331 29.3423015 9.5133699 .001161440 862 743044 640503923 29.3593365 9.5170515 .001160093 863 744769 642735647 29.3763616 9.5207303 .001158749 864 746496 644972544 29.3933769 9.5244063 .001157407 865 748225 647214625 29.4103823 9.5230794 001156069 866 749956 649161896 29.4278779 9.5317497 .001154734 867 751689 631714363 29.4443637 9.5354172 .001153403 863 753424 653972032 29.4618397 9.5390818 .001152074 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, No. Square*. Cubes. Square Roots. Cube Roots. Reciprocate 869 755161 656234909 29.4788059 9.5427437 .001150748 870 766900 658503000 29.4957624 9.5464027 .001149426 871 758641 660776311 29.5127091 9.6500589 .001148106 872 760384 663054848 29.5296461 9.5537123 .001146789 873 762129 665338617 29.5465734 9.5573630 .001 145475 874 763876 667627624 29.5634910 9.5610108 .001144166 875 765625 669921875 29.5803989 9.5646559 .001142867 876 767376 672221376 29.5972972 9.5682982 .001141553 877 769129 674526133 29.6141858 9.5719377 .001140251 878 770884 676836152 29.6310648 9.6755746 .001138952 879 772641 679151439 29.6479342 9.5792085 .001137656 880 774400 681472000 29.6647939 9.5828397 .001136364 881 776161 683797841 29.6816442 9.5864682 .001135074 882 777924 686128968 29.6984848 9.5900939 .001133787 883 779689 688465387 29.7153159 9.5937169 .001132503 884 781456 690807104 29.7321375 9.5973373 .001131222 886 783225 693154125 29.7489496 9.6009548 .001129944 886 784996 695506456 29.7657521 9.6045696 .001128668 887 786769 697864103 29.7825452 9.6081817 .001127396 888 788544 700227072 29.7993289 9.6117911 .001126126 889 790321 702595369 29.8161030 9.6153977 .001124859 890 792100 704969000 29.8328678 9.6190017 .001123596 891 793881 707347971 29.8496231 9.6226030 .001122334 892 795664 709732288 29.8663690 9.6262016 .001121076 893 797449 712121957 29.8831056 9.6297975 .001119821 894 896 799236 801025 714516984 716917375 29.8998328 29.9165506 9.6333907 9.6369812 .001118668 .001117318 896 802816 719323136 29.9332591 9.6405690 .001116071 897 898 804609 806404 721734273 724150792 29.9499583 29.9666481 9.6441542 9.6477367 .001114827 .001113586 899 808201 726572699 29.9833287 9.6513166 .001112347 900 810000 729000000 30.0000000 9.6548938 .001111111 901 902 903 811801 813604 816409 731432701 733870808 736314327 30.0166620 30.0333148 30.0499584 9.6584684 9.6620403 9.6656096 .001109878 .001108647 .001107420 904 817216 738763264 30.0665928 9.6691762 .001106195 906 819025 741217625 30.0832179 9.6727403 .001104972 906 820836 743677416 30.0998339 9.6763017 .001103753 907 322649 746142643 30.1164407 9.6798604 .001102536 908 824464 748613312 30.1330383 9.6834166 .001101322 I 909 826281 751089429 30.1496269 9.6869701 .001100110 910 828100 753571000 30.1662063 9.6905211 .001098901 911 829921 756058031 30.1827765 9.6940694 .001097695 912 831744 758550528 30.1993377 9.6976151 .001096491 913 833569 761048497 30.2158899 9.7011583 .001095290 914 835396 763551944 30.2324329 9.7046989 .001094092 915 837225 766060875 30.2489669 9.7082369 .001092896 916 839056 768575296 30.2654919 9.7117723 .001091703 917 840889 771095213 30.2820079 9.71530C1 .001090513 918 842724 773620632 30.2985148 9.7188354 .001089325 919 844561 776151559 30.3150128 9.7223631 .001088139 920 921 846400 848241 778688000 781229961 30.3315018 30.3479818 9.7258883 9.7294109 .001086957 .001085776 922 923 850084 851929 783777448 786330467 30.3644529 30.3809151 9.7329309 9.7364484 .001084599 .001083424 924 853776 788889024 30.3973683 9.7399634 .001082251 925 855625 791453125 30.4138127 9.7434758 .001081081 926 857476 794022776 30.4302481 9.7469857 .001079914 927 928 859329 861184 796597983 799178752 30.4466747 30.4630924 9.7504930 9.7539979 .001078749 .001077586 929 863041 801765089 30.4795013 9.7575002 .001076426 930 864900 ' 804357000 30.4959014 9.7610001 .001075269 CUBE HOOTS, AND RECIPROCALS. 219 INo. Squares. Cubes. Squaro Roots. Cube Roots. Reciprocals. 931 866761 806954491 30.5122926 9.7644974 .001074114 932 86S624 809557568 30.5286750 9.7679922 001072961 933 870489 812166237 30.5450487 9.7714845 001071811 934 872356 814780504 30.5614136 9.7749743 .001070664 935 874225 817400375 30.5777697 9.7781616 .001069519 936 876096 820025856 30.5941171 9.781-J466 .001063376 937 877969 822656953 30.6104557 9.7854238 .001067236 933 879844 825293672 30.6267857 9.7389037 .001066098 939 881721 827936019 30.6431069 9.7923361 .001064963 940 883600 830584000 30.6594194 9.7958611 .001063830 941 885481 833237621 30.6757233 9.7993336 .001062699 942 837364 835896388 30.6920185 9.8023036 .001061571 943 839249 838561807 30.7083051 9.8062711 .001060445 944 891136 841232384 30.7245830 9.8097362 .001059322 945 893025 843903625 30.7408523 9.8131989 .001058201 946 894916 346590536 30.7571130 9.8166591 .001057082 947 896809 849278123 30.7733651 9 8201 169 .001055966 94S 898704 851971392 30.7896086 9.8235723 .001054852 949 900601 854670349 30.8058436 9.8270252 .001053741 950 902500 857375000 30.8220700 9.8304757 .001052632 951 904401 860085351 30.8332879 9.8339238 .001051525 952 906304 862801408 30.8544972 9.8373695 .001050420 953 908209 865523177 30.8706981 9.8408127 .001049318 954 910116 868250664 30.8868904 9.8442536 .001043218 955 912025 870983875 30.9030743 9.8476920 .001047120 956 913936 873722816 30.9192497 9.8511280 .001046025 957 915849 876467493 30.9354166 9.8545617 .001044932 958 917764 879217912 30.9515751 9.8579929 .001043341 959 919631 881974079 30.9677251 9.8614213 .001042753 960 921600 884736000 30.9838668 9.8643483 .001041667 961 923521 887503631 31.0000000 9.8682724 .001040583 962 925444 890277128 31.0161248 9.8716941 .001039501 963 927369 893056347 31 0322413 9.8751135 .001038422 964 929296 895841344 31.0483494 9.8785305 .001037344 965 931225 898632125 31.0644491 9.8819451 .001036269 966 933156 901423696 31.0805405 9.8853574 .001035197 967 935089 904231063 31.0966236 9.8887673 .001034126 968 937024 907039232 31.1126984 9.8921749 .001033058 969 938961 909853209 31.1237648 9.8955801 .001031992 970 940900 912673000 31.1443230 9.8989830 .001030928 971 942841 915498611 31.1608729 9.9023835 .001029366 972 944784 918330048 31.1769145 9.9057817 .001028807 973 946729 921167317 31.1929479 9.9091776 .001027749 974 948676 924010424 31.2089731 9.9125712 .001026694 975 950625 926359375 31.2249900 9.9159624 .001025641 976 952576 929714176 31.2409987 9.9193513 .001024590 977 954529 932574833 31.2569992 9.9227379 .001023541 978 956484 935441352 31.2729915 9.9261222 .001022495 979 958441 938313739 31.2889757 9.9295042 .001021450 980 960400 941192000 31.3049517 9.9323839 .001020408 931 962361 944076141 31.3209195 9.9362613 .001019368 982 964324 946%6168 31.3368792 9.9396363 .001018330 983 966239 949H62087 31.3528308 9.9430092 .001017294 934 963256 952763904 31.3637743 9.9463797 .001016260 935 970225 955671625 31.3847097 9.9497479 .001015228 988 972196 958585256 31.4006369 9.9531138 .001014199 987 974169 961504303 31.4165561 9.9564775 .001013171 938 976144 264430272 31.4324673 9.9598389 .001012146 989 978121 967361669 31.4433704 9.9631981 .001011122 990 980100 970299000 31.4&42654 9.9665549 .001010101 991 982081 973242271 31.4801525 9.9699095 .001009082 992 934064 976191488 31.4960315 9.9732619 .001008065 220 TABLE XIII. SQUARES, CUBES, SQUARE ROOTS, &C. No. Squaw*. Cubes. Square Boots. Cube Boots. "Reciprocals. 993 986049 979146657 31 5119025 9.9766120 .001007049 994 988036 982107784 31.5277655 9.9799599 .001006036 995 996 990025 992016 985074875 988047936 31.5436206 31.5594677 9.9833055 9.9866488 .001005025 .001004016 997 993 994009 996004 991026973 994011992 31.5753068 31.5911380 9.9899900 9.9933289 .001003009 .001002004 ; 999 99S001 997002999 31.6061,613 9.9966656 001001001 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1000000 1002001 1004004 1006009 1008016 1010025 1012036 1014049 1016064 1018081 1000000000 1003003001 1006012003 1009027027 1012048064 1015075125 1018108216 1021147343 1024192512 1027243729 31.6227766 31.6385.340 31.6543836 31.6701752 31.6859590 31.7017349 31.7175030 31.7332633 31.7490157 31.7647603 10.0000000 10.0033322 10.0066622 10.0099899 10.0133155 10.0166389 10.0199601 10.0232791 10.0265958 10.0299104 .001000000 .0009990010 ! .0009980040 i .0009970090 .0009960159 .0009950249 .0009940358 .0009930487 .0009920635 .0009910803 1010 1011 1012 1020100 1022121 1024144 1030301000 1033364331 1036433728 31.7804972 31.7962262 31.8119474 10.0332228 10.0365330 10.0398410 .0009900990 .0009891197 .0009881423 1013 1014 1015 1026169 1028196 1030225 1039509197 1042590744 1045678375 31.8276609 31.8433666 31.8590646 10.0431469 10.0-464506 10.0497521 .0009871668 .0009861933 .0009852217 1016 1017 1032256 1034239 1048772096 1051871913 31.8747549 31.8904374 10.0530514 10.0563485 .0009842520 .0009832842 1018 1019 1036324 1038361 1054977832 1058089859 31.9061123 31.9217794 10.0596435 10.0629364 .0009823183 .0009813543 1020 1031 1022 1023 1040400 J042441 1044484 1046529 1061208000 1064332261 1067462648 1070599167 31 9374388 31.9530906 31.9687347 31.9843712 10.0662271 10.0695156 10.0728020 10.0760863 .0009803922 .0009794319 .0009784736 .0009775171 1024 1025 1026 1027 1048576 1050625 105267G 1054729 1073741824 1076890625 1080045576 1083206683 32.0000000 32.0156212 32.0312348 32.0468407 10.0793684 10.0826484 10.0859262 10.0892019 .0009765625 .0009756098 .0009746589 .0009737098 1028 1056784 1086373952 32.0624391 10.0924755 .0009727626 1029 1058841 10^9547389 32.0780298 10.0957469 .0009718173 1030 1031 1060900 1062961 1092727000 1095912791 32.0936131 32.1091887 10.0990163 10.1022835 .0009708738 .0009699321 1032 1033 1034 1035 1036 1037 1038 1039 1065024 1067089 1069156 1071225 1073296 1075369 1077444 1079521 1099104768 1102302937 1105507304 1108717875 1111934656 1115157653 1118386872 1121622319 32. 1247568 32.1403173 32.1558704 32.1714159 32.1869539 32.2024844 32.2180074 32.2335229 10.1055487 10.1088117 10.1120726 10.1153314 10.1185882 10.1218428 10.1250953 10.1283457 .0009689922 .0009680542 1 .0009671180 .0009661836 .0009652510 .0009643202 .0009633911 1 .0009624639 1040 1081600 1124864000 32.2490310 10.1315941 .0009615385 1041 1042 1043 1083681 1085764 1087849 1128111921 1131366088 1134626507 32.2645316 32.2800248 32.2955105 10.1348403 10.1380845 10.1413266 .0009606148 .0009596929 .0009587728 1044 1089936 1137893184 32.3109888 10.1445667 .0009578544 1045 1092025 1141166125 32.3264598 10.1478047 .0009569378 1046 1094116 1144445336 32.3419233 10.1510406 .0009560229 1047 1096209 1147730823 32.3573794 10.1542744 .0009551098 1048 1049 1098304 1100401 1151022592 1154320649 32.3728281 32.3882695 10 15^5062 10.1607359 .0009541985 .0009532888 1050 1051 1052 1102500 1104601 1106704 1157625000 1160935651 1164252608 32.4037035 32.4191301 32.4345495 10.1639636 10.1671893 10.1704129 .0009523810 .0009514748 .0009505703 1053 1054 1108809 1110916 1167575877 1170905464 32.4499615 32.4653662 10.1736344 10.1768539 0009496676 .0009487666 TABLE XIV. LOGARITHMS OF NUMBERS. FROM 1 TO 10,000. 222 TABLE XIV. LOGARITHMS OF NUMBERS. No . 7 8 9 Dlff. 1U 1 S j 1 )00000( 4321 1 860C 1 012837 7032 >02118 I 5306 )000434 4751 902t 01325S 7451 021 60c 571 0008& 518 945 01368( 786^ 0220 H 612L 300130] 560i 987t )01410( * 8284 5 022426 > 653c 00173 > 6035 > '01 030 ) 452 870 02284 694 00216 646 010724 494 911 02325 735 00259 689 01114 536 953 02366 775 00302 732 01157 677 994 02407 8164 00346 774 01199 619 02036 448 857 00389" 817 01241 661 02077 489 897 ~432 428 424 420 416 412 7 i 9384 (033424 978 03382 0301 9. f 4227 03060C 4625 031004 502 03140 543 03181 583 03221 623C 03261 662 03302 702 404 1 400 ' ' 7426 782 8223 862f 901 941 981 04020 04060 04099 397 lie i ^ 5 6 041393 5323 9218 053078 6905 060693 4458 04178 571 9606 05346 728 06107 483 042182042576 6105 6495 9993 05038C 053846J 4230 7666 8046 061452'061829 5206 5580 04296 688 05076 461 842 06220 595 04336 727 06115 499 880 06258 632 04375 766 05153 537 918 06295 669 04414 805 051924 576C 956 06333 707 04454 844^ 05230 6142 994S 06370S 7443 04493 883C 052694 6524 060320 4083 781 393 390 386 383 379 376 ! 373 8 9 8186 071882 6547 855 07225 591 8928 072617 6276 9298 072985 6640 9668 073352 7004 07003 371 7368 07040 408 773 07077 445 8094 071145 4816 8457 07151 6182 881 370 366 363 120 1 2 4 5 6 r fc 9 079181 082785 6360 9905 093422 6910 100371 3804 7210 110590 079543 083144 6716 090258 3772 7257 100715 4146 7549 110926 079904 083503 7071 090611 4122 7604 101059 4487 7888 111263 080266 3861 7426 090963 4471 7951 101403 4828 8227 111599 080626 4219 778 091315 4820 8298 101747 5169 8565 111934 08098 457 8136 091667 5169 8644 102091 6510 8903 12270 08134 4934 8490 092018 5518 8990 102434 6851 9241 112605 08170 529 884 09237 686 933o 102777 619 9579 12940 082067 6647 9198 092721 6215 9681 103119 6631 9916 113276 082426 6004 9552 09307 6562 100026 3462 6871 110253 3609 360 367 365 362 349 346 343 341 338 336 130 2 3 4 5 6 7 8 9 113943 7271 120574 3852 7105 130334 3539 6721 9879 143015 114277 7603 120903 4178 7429 30655 3858 7037 40194 3327 14611 7934 21231 4504 7753 30977 4177 7354 40508 3639 114944 8265 121560 4830 8076 131298 4496 7671 140822 3951 115278 8595 121888 5156 8399 131619 4814 7987 141136 4263 16611 8926 22216 5481 8722 31939 5133 8303 41450 4574 15943 9256 22544 5806 9045 32260 5451 8618 41763 4885 16276 9586 22871 6131 9368 32580 5769 8934 42076 5196 116608 9915 23198 6456 9690 32900 6086 9249 42389 5507 116940 20245 3525 6781 30012 3219 6403 9564 42702 5818 333 330 328 326 323 321 318 316 314 311 140 1 146128 9219 152288 46438 9527 52594 46748 9835 52900 147058 150142 3205 147367 150449 3510 47676 50756 3815 47985 51063 4120 48294 51370 4424 48603 51676 4728 48911 51982 5032 309 307 305 3 4 5 5336 8362 161368 5640 8664 61667 5943 8965 61967 6246 9266 162266 6549 9567 162564 6852 9868 62863 7154 60168 3161 7457 60469 3460 7759 60769 3758 8061 61068 4055 303 301 299 6 7 8 9 4353 7317 170262 3186 4650 7613 170555 3478 4947 7908 70848 3769 5244 8203 171141 4060 5541 8497 171434 4351 5838 8792 71726 4641 6134 9086 72019 4932 6430 9380 72311 5222 6726 9674 72603 6512 7022 9968 72895 6802 297 295 293 291 150 176091 176381 76670 176959 77248 77536 77825 78113 78401 78689 289 2 3 4 6 181844 4C91 7521 190332 9264 182129 4975 7803 90612 95521 9839 82415! 182700 5259 6542 8084 8366 90892 191171 80126 2985 5825 8647 191451 80413 3270 6108 8928 91730 80699 3555 6391 9209 92010 80986 3839 6674 9490 92289 81272 4123 6956 9771 92567 81558 4407 7239 190051 2846 287 285 283 281 279 7 8 9 3125 5900 8657 Z01397 ' 3403 6176 8932 &1670 3681 6453 9206 01943 3959 4237 6729 7005 9481 9755 302216 202-188 4514 7281 00029 2761 4792 7556 00303 3033 5069 7832 00577 3305 5346 8107 00850 3577 5623 8382 201124 3848 278 276 274 i 272 No.| 1 3 I 3 4 5 6 7 8 9 Iff. ' i. TABLE XIV. LOGARITHMS OF NUMBERS. No. O 1 3 3 4 5 6 7 8 9 Biff. 160 1 204120 6326 204391 7096 204663204934205204 7365: 7634 1 7904 205475 8173 205746 8441 206016 8710 206286 8979 206556 9247 271 269 2 9515 9733 210051 210319.210586 210853 211121 211388 211654 211921 287 8 212183 212454 2720 2986| 3252 3518 3783 4049 4314 4579 266 4 4344 5109 5373 5633, 5902 6166 6430 6694 6957 7221 264 5 7484 7747 8010 8273 8536 8798 9060 9323 9585 9346 262 220103 220370 220631220892221153 221414 221675 221936 222196 222456 261 7 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 8 5309 5568 5326 6084 6342 6600 6858 7115 7372 7630 258 9 7887 8144 8400 8657 8913 9170 9426 9632 9933 230193 256 170 230449 230704 230960 231215 231470 231724 231979 232234 232488 232742 255 1 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 253 2 5523 5781 6033 6235 6537 6789 7041 7292 7544 7795 252 3 8046 8297 8548 8799 9049 9299 9550 9800 240050 240300 250 4 240549 240799 241048 241297 241546 241795 242044 242293 2541 2790 249 5 3033 3286 3534 3782 4030 4277 4525 4772 5019 5266 248 6 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 7 7973 8219 8464 8709 8954 9193 9443 9687 9932 250176 245 8 250420 250664 250903 251151 251395 251633 251881 252125 252363 2610 243 9 2853 3096 3383 3580 3822 4064 4306 4548 4790 5031 242 180 255273 255514 255765 255996 256237 256477 256713 256958 257198 257439 241 1 7679 7918 8153 8398 8637 8377 9116 9355 9594 9833 239 2 260071 260310 260548 260787 261025 261263 261501 261739 261976 262214 238 3 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 4 4318 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 5 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 6 9513 9746 9980 270213 270446 270679 270912 2Z1144 271377 271609 233 7 271842 272074 272306 2538 2770 3001 3233 3464 3696 3927 232 8 4158 4389 4620 4850 5081 5311 5542 6772 6002 6232 230 9 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 229 190 278754 278982 279211 279439 279667 279895 280123 280351 280578 280806 228 1 281033 281261 281488 281715 231942 282169 2396 2622 2349 3075 227 9 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 3 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 225 4 7802 8026 8249 8473 8696 8920 9143 9366 9589 9312 223 5 290035 290257 290480 290702 290925 291147 291369 291591 291813 292034 222 6 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 221 7 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 8 6665 6334 7104 7323 7542 7761 7979 8198 8416 8635 219 9 8353 9071 9289 9507 9725 ' 9943 300161 300378 300595 300813 218 200 301030 301247 301464 301681 301898 302114 302331 302547 302764 302980 217 1 3196 3412 3623 3844 4059 4275 4491 4706 4921 5136 216 2 5351 5566 5781 5996 6211 6425 6639 6854 7068 7282 215 3 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 213 4 9630 9843 310056 310268 310481 310693 310906 311118 311330 311542 212 S 311754 311966 2177 2389 2600 2812 3023 3234 3445 3656 211 8 3887 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 7 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 8 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 9 320146 320354 320562 320769 320977 321184 321391 321593 321805 322012 207 210 322219 322426 322633 322339 323046 323252 323458 323665 323871 324077 206 1 4232 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 3 4 8380 330414 8583 330617 8787 330819 8991 331022 9194 331225 9398 331427 9601 331630 9805 331832 330008 2034 330211 2236 203 202 6 2433 2640 2842 3044 3246 3447 3649 3850 4051 425? 202 6 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 7 6460 6660 6860 7060 7260 7459 7659 7853 8058 8257 200 8 8456 8656 8855 9054 9253 9451 9650 9349 340047 340246 199 9 340444 310642 340341 341039 341237 341435 341632 341830 2028 2225 198 No. 1 3 3 4 5 6 7 6 9 DH. 224, TABLE XiV. LOGARITHMS OF NUMUEI.'S. No. 1 3 343999 Diff. 220 342423 342620 342817 343014 343212 343409 343606 343802 344196 197 1 4392 4589 4785 4981 5178 5374 5570 5766 5962 6157 196 2 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 195 3 8305 8500 8691 8889 9083 927S 9472 9666 9860 350054 194 4 350248 350442 350636 1350829 351023 351216 351410 351603 351796 1989 193 5 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 193 6 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 192 7 6026 6217 64(,3 6599 6790 6981 7172 7363 7554 7744 191 8 7935 8125 83] 6 850C 8696 8886 9076 9266 9456 9646 190 9 9835 360025 360215 360404 360593 360783 360972 361161 361350 361539 189 230 361728 361917 J62105 362294 362482 362671 362859 363048 363236 363424 188 1 3612 3800 3988 4176 4363 4551 4739 4926 5113 6301 188 2 6488 5675 5862 6049 6236 6423 6610 6796 6983 7169 187 3 7356 7542 7729 7915! 8101 8287 8473 8659 8845 9030 186 4 9216 9401 9587 9772! 9958 370143 370328 370513 370698 370883 186 6 371068 371253 371437 371622371806 1991 2175 2360 2544 2728 184 6 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 7 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 380030 181 240 380211 380392 380573 380754 380934 381115 331296 381476 381656 381837 181 1 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 180 2 3815 3995 4174 4353 4533 4712 4891 5070 6249 5428 179 3 5606 5785 5964 6142| 6321 6499 6677 6856 7034 7212 178 4 7390 7568 7746 7923: 810 8279 8456 8634 8811 8989 178 5 9166 9343 9520 9698 9S75 390051 390223 390405 390582 390759 177 6 390935 391112 391288 3914641391641 1817 1993 2169 23451 2521 176 7 2697 2873 3048 3221 3400 3575 3751 3926 4101 4277 176 8 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 176 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 397940 398114 398287 398461 398634 398808 393981 399154 399328 399501 173 1 9674 9847 400020 400192400365 400538 400711 400883 401056 401228 173 2 401401 401573 1745 1917 2089 2261 2433 2605 2777 2949 172 3 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 171 4 483-1 5005 5176 5346 5517 5688 5858 6029 6199 6370 171 6 6540 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 8 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 168 9 3300 3467 3635 3303 3970 4137 430f 4472 4639 4806 167 260 414973 415140 415307 415474 415641 415808 415974 416141 416308 416474 167 1 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 166 2 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 165 3 9956 420121 420286 420451 420616 420781 420945 421110 421275 421439 165 4 421604 1768 1933 2097 2261 2426 2590 2764 2918 3082 164 5 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 164 e 4882 5045 5108 5371 5534 5697 5860 6023 6186 6349 163 6511 6674 6836 6999 7161 7324 7486 7648 7811 7973 162 8 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 162' 9 9752 9914 430075 430236 430398 430559 43072C 430881 431042 431203 161 WO 431364 2969 431525 3130 431685 3290 431846 3450 432007 3610 432167 3770 432323 393U 432488 4090 432649 4249 432809 4409 161 160 ( 4569 4729 4888 5018 5207 5367 5526 5685 5844 6004 159 j 6163 6322 6481 6640 6799 6957 7116 7275 7433 7592 159 4 7751 7909 8067 8226 1 8334 3542 870 li 8859, 9017 9175 158 5 6 9333 440909 9491 441066 9648 441224 9806 9964 441381 1441533 4401 22 [ 440279 1695 1852 1 440437 440594 2009: 2166 440752 2323 158 157 2480 2637 2793 2950 3106 3263 3419 3576; 3732 3889 157 6 4045 4201 4357 4513 46691 4825 i 4981 51371 5293 6449 156 5604 5760 5915 6071 6226 6382 - 6537 6692 6848 7003 155 Ho 1 1 2 3 * 5 6 7 8 9 jDitf. TABLE XIY. LOGARITHMS OF NtTMBERS. 225 Mo O 1 8 3 4 5 6 7 8 9 Diflf. 280 447168 447313 447468 447623 447778 447933 448088 448242 448397 448552 155 1 8706 8861 90151 9170 9324 9478 9633 9787 9941 450095 154 2 450249 450403 450557450711 450865 451018 451172 451326 451479 1633 154 3 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 153 4 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 6 4845 4997 6160 6302 6454 5606 6768 6910 6062 6214 152 6 6366 6518 6670 6821 6973 7125 7276 7428 7679 7731 152 7 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 151 8 9 9392 460898 9543 461048 9694 461198 9845 461348 9995 461499 460146 1649 460296 1799 460447 1948 460697 2098 460748 2248 151 150 290 462398 462648 462697 462847 462997 463146 463296 463445 463594 463744 150 1 3893 4042 4191 4340 4490 4639 4788 4936 6085 5234 149 2 6383 5532 6680 6829 5977 6126 6274 6423 6571 6719 149 3 6868 7016 7164 7312 7460 7608 7766 7904 8052 8200 148 4 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 5 9822 9969 470116 470263 470410 470557 470704 470851 470998 471145 147 6 471292 471438 1585 1732 1878 2025 2171 2318 2464 2610 146 7 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 8 4216 4362 4508 4663 4799 4944 5090 5235 5381 6526 146 9 5671 5816 5962 6107 6252 6397 6542 6687 6832 6976 146 300 477121 477266 477411 477566 477700 477844 477989 478133 478278 478422 145 1 8566 8711 8855 8999 9143 9287 9431 9575 9719 9863 144 2 480007 480151 480294 480438 480582 480725 480869 481012 481156 481299 144 3 1443 1586 1729 1872 2016 2159 2302 2445 2588 2731 143 4 2874 3016 3159 3302 3445 3587 3730 3872 4015 4157 143 5 4300 4442 4585 4727 4869 6011 6163 6295 6437 6579 142 6 5721 6863 6005 6147 6289 6430 6672 6714 6856 6997 142 7 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 141 8 8651 8692 8833 8974 9114 9255 9396 9537 9677 9818 141 9 9963 490099 490239 490330 490620 490661 490801 490941 491081 491222 140 310 491362 491602 491642 491782 491922 492062 492201 492341 492481 492621 140 1 2760 2900 3040 3179 3319 3458 3597 3737 3876 4015 139 2 4155 4294 4433 4572 4711 4850 4989 6128 6267 6406 139 3 5544 6683 6322 6960 6099 6233 6376 6515 6653 6791 139 4 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 5 8311 8448 8586 8724 8862 8999 9137 9275 9412 9660 138 6 9687 9824 9962 500099 500236 500374 600511 500648 500785 600922 137 7 601059 601196 501333 1470 1607 1744 1880 2017 2164 2291 137 8 2427 2564 2700 2837 2973 3109 3246 3382 3518 3655 136 9 3791 3927 4063 4199 4335 4471 4607 4743 4678 6014 136 320 505150 505286 505421 605557 605693 505828 605964 606099 506234 606370 136 1 6505 6640 6776 6911 7046 7181 7316 7451 7586 7721 135 2 7856 7991 8126 8260 8395 8530 8664 8799 8934 9068 135 3 9203 9337 9471 9606 9740 9874 510009 510143 510277 610411 134 4 610545 510679 510313 510947 511081 511215 1349 1482 1616 1750 134 6 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 133 6 3213 3351 3484 3617 3750 3883 4016 4149 4282 4415 133 7 4548 4631 4313 4946 5079 5211 5344 6476 5609 5741 133 8 5874 6006 6139 6271 6403 6535 6668 6800 6932 7064 132 9 7196 7328 7460 7592 7724 7855 7987 8119 8261 8382 132 330 818614 518646 518777 518909 519040 519171 519303 519434 519566 519697 131 1 9823 9959 520090 520221 520353 520484 520615 520745 520876 621007 131 2 621138 521269 1400 1530 1661 1792 1922 2053 2183 2314 131 3 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 130 4 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 130 6 5045 6174 5304 5434 5563 5693 5822 5951 6081 6210 129 6 6339 6469 6598 6727 6356 6935 7114 7243 7372 7501 129 7 7630 7759 7888 8016 8145 8274 8402 8531 8660 8788 129 8 8917 9045 9174 9302 9430 9559 9687 9815 9943 530072 123 9 530200 530323 530456 530584 530712 530840 530968 531096 531223 1351 128 No 1 3 3 4 5 6 7 8 9 IKff. 226 TABLE XIV. LOGARITHMS OF NUMBERS. No. O 1 2 9 Dlff. 340 531479 531607 31734 531862 531990 532117 532245 )32372 632500 532627 128 1 2754 2882 3009 3136 3264 3391 3518 3645 3772 3899 127 2 4026 4153 4280 4407 4534 4661 4787 4914 5041 6167 127 3 5294 5421 5547 5674 5800 6927 6053 6180 6306 6432 126 4 6558 6685 6811 6937 7063 7189 7315 7441 7567 7693 126 6 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 126 6 9076 9202 9327 9452 9578 9703 9829 9954 5400791540204 125 7 540329 540455 540580 40705 540830 540955 541080 641205 1330 1454 126 3 1679 1704 1829 1953 2078 2203 2327 2452 2576 2701 125 9 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 124 390 644068 544192 644316 544440 544564 544688 544812 644936 645060 645183 124 1 6307 5431 5555 5678 5802 6925 6049 6172 6296 6419 124 2 6643 6666 6789 6913 7036 7159 7282 7405 7529 7652 123 3 7775 7898 8021 8144 8267 8389 8512 S635 8758 8881 123 4 9003 9126 9249 9371 9494 9616 9739 9861 9984 660106 123 5 550228 650351 550473 650595 650717 650840 650962 551084 651206 1328 122 6 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 7 2663 2790 2911 3033 3155 3276 3398 3519 3640 3762 121 8 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 9 6094 5215 5336 5457 6578 6699 5820 6940 6061 6182 121 360 666303 666423 656544 556664 666785 666905 657026 667146 657267 657387 120 1 7607 7627 7748 7868 7988 8108 8228 8349 8469 8589 120 2 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 b 9907 660026 560146 560265 660385 560504 660624 660743 660863 660982 119 4 661101 1221 1340 1459 1578 1698 1817 1936 2055 2174 119 5 8293 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 6021 6139 6257 6376 6494 6612 6730 118 8 6848 6966 6034 6202 6320 6437 6555 6673 6791 6909 118 9 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 870 668202 668319 668436 668554 568671 668788 668905 669023 669140 669257 117 1 9374 9491 9608 9725 9842 9959 670076 670193 670309 670426 117 2 670543 670660 570776 670893 671010 571126 1243 1359 1476 1592 117 3 1709 1825 1942 2058 2174 2291 2407 2523 2639 2765 11(5 4 2872 2988 3104 3220 3336 3452 3568 3684 380u 3915 116 6 4031 4147 4263 4379 4494 4610 4726 4841 4967 6072 116 6 6188 6303 6419 6534 5650 6765 6880 6996 6111 6226 115 7 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 115 8 7492 7607 7722 7836 7951 8066 8181 8295 8410 8525 115 9 6639 8764 8868 8983 9097 9212 9326 9441 9565 9669 114 380 679784 679898 680012 580126 680241 680355 680469 580583 680697 580811 114 1 680925 581039 1153 1267 1381 1495 1608 722 , 1836 1950 114 2 2063 2177 2291 2404 2518 2631 2746 J58 #972 3085 114 3 3199 3312 3426 3539 3652 3765 3879 J992 4105 4218 4 4331 4444 4557 4670 4783 4896 6009 6122 6235 5348 12 6 6461 6574 5686 6799 5912 6024 6137 6250 6362 6475 13 6 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 .12 7 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 112 8 8832 8944 9056 9167 9279 9391 9503 9615 9726 9838 112 9 9950 690061 590173 690284 590396 690507 590619 690730 690842 690953 112 390 691065 F91176 691287 591399 661510 691621 691732 691843 691955 692066 111 1 217? 2288 2399 2510 2621 2732 2843 2954 3064 3175 111 3286 3397 3508 3618 3729 3840 3950 4061 4171 4282 111 J 4393 4503 4614 4724 4834 4945 5055 5165 5276 6386 110 4 6496 6606 6717 5827: 6937 6047 6157 6267 6377 6487 110 5 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 110 ( 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 110 j 8791 8900 9009 9119 9228 9337 9446 9556 9665 9774 109 8 9883 9992 60010 600210 600319 60042S 600537 600646 600755 600864 109 9 600973 601082 1191 1299 1408 1517 1625 1734 1843 1951 109 Mo 1 3 3 ! 4 5 6 7 8 | 9 Diff. TABLE XIV. LOGARITHMS OF NUMBERS. 221 |No. 1 3 3 ! 4 5 6 7 8 9 Dlff 400 602060 602169 602277.602336602494 602603 602711 602819 602928 603036 108 1 3144 3253 3361 ; 3469 3577 3636 3794 3902. 4010 4118 108 2 4226 4334 4442 4550 4658 4766 4874 4982i 5039 5197 108 3 5305 5413 5521 i 5628 5736 5844 5951 6059 6166 6274 108 4 6381 6489 6596: 6704 6811 6919 7026 7133 7241 7348 107 5 1 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 107 8 8526 8633 8740 ! 8847 8954 9061 9167 9274 9381 9488 107 7i 9594 9701 9808 9914 610021 610128 610234 610341 610447 610554 107 8610660 610767 610373 610979 1086 1192 1298 1405 1511 1617 106 9 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 612784 612890 612996 613102 613207 613313 613419 613525 613630 613736 106 1 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 106 9 4897 5003 51081 5213 5319 5424 5529 5634 5740 5845 105 3 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 105 4 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 105 5 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 105 6 9093 9198 9302| 9406 9511 9615 9719 9824 9928 620032 104 7 620136 620240 620344 620443 620552 620656 620760 620364 620963 1072 104 8 1176 1280 1334 14-38 1592 1695 1799 1903 2007 2110 104 9 2214 2318 242 lj 25251 2628 2732 2835 2939 3042 3146 104 420 623249 623353 623456 623559 623663 623766 623869 623973 624076 624179 103 1 4282 4385 4483 4591 4695 4798 4901 5004 5107 5210 103 2 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 103 3 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 103 4 7366 7463 7571 7673 7775 7878 7980 8082 8185 8287 102 5 8339 8491 8593 8695 8797 8900 9002 9104 9206 9308 102 6 9410 9512 9613 9715 9817 9919 630021 630123 630224 630326 102 7 630423 630530 630631 630733 630835 630936 1038 1139 1241 1342 102 8 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 101 9 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367 101 430 633468 633569 633670 633771 633872 633973 634074 634175 634276 634376 101 I 4477 4573 4679 4779 4830 4931 5081 5182 5283 5383 101 2 5484 5584 5685 5785 5886 5986 6087 6187 6287 6388 100 3 6438 6533 6688 6789 6889 6989 7089 7189 7290 7390 100 4 7490 7590 7690! 7790 7890 7990 8090 8190 8290 8389 100 5 8489 8539 8689 S 8789 8888 8988 9088 9188 9287 9387 100 6 9436 9586 9686 9785 9885 9984 640084 640183 640283 640382 99 7 640481 640531 640680 640779 640879 640978 1077 1177 1276 1375 99 8 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 99 9 2465 2563 2662 2761 2360 2959 3058 3156 3255 3354 99 440 643453 643551 643650 643749 643847 643946 644044 644143 644242 644340 98 1 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 98 2 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306, 93 3 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 98 4 7333 7481 7579! 7676 7774 7872 7969 8067 8165 8262 98 5 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 97 6 9335 9432 9530 9627 9724 9821 9919 650016 650113 650210 97 7 650308 650405 650502 650599 650696 650793 650890 0987 1084 1181 97 8 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 97 9 2246 2343 2440 2536 2633 2730 2826 2923 3019 3116 97 460 653213 653309 653405 653502 653598 653695 653791 653888 653984 654080 96 1 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042 96 2 5138 5235 5331 5427 5523 5619 5715 5810 5906 6002 96 3 6093 6194 ! 6290 6386 6482 6577 6673 6769 6864 6960 96 4 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 5 8011 8107 8202 8293 8393 8488 8584 8679 8774 8870 95 6 8965 9060 9155 9250! 9346 9441 9536 9631 9726 9821 95 7 99 16 66001 11660 106 660201 660296 660391 660486 660581 660676 660771 95 8660865 0960| 1055J 1150 1245 1339 1434 1529 1623 1718 95 9 1813; 1907 2002 2096 2191 1 2286 2380 2475 2569 2663 96 1 a 0.1 o j i 3 3 4 5 6 T 8 9 DUT 1 228 TABLE XIV. LOGARITHMS OP NUMBERS. No. O i | a 3 4 5 1 6 r 8 1 9 Diff. 460 662758 662852 662947 663041 663135 663230 663324 663418 663512 663607 94 1 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 94 2 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 94 3 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 94 4 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 94 5 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 93 6 8386 8479 8572 8665 8759 8852 8945 9038 9131 9224 93 7l 8 9317 670246 9410 670339 9503 670431 9f96 670524 9689 670617 9782 670710 9875 670802 9967 670895 670060 0988 670153 1080 93 93 9 1173 1265 1358 1451 1543 1636 1728 1821 1913 2005 93 470 672098 672190 672283 672375 672467 672560 672652 672744 672836 672929 92 1 3021 3113 3205 3297 3390 3482 3574 3666 3758 3850 92 2 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 92 3 4861 4953 5045 5137 6228 5320 5412 5503 5595 5687 92 4 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 92 5 6694 6785 6876 6968 7059 7151 7242 7333 7424 7616 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 9428 9519 9610 9700 9791 9882 9973 680063 680154 680245 91 9 680336 680426 680517 680607 680698 680789 680879 0970 1060 1151 91 480 681241 681332 681422 681513 681603 681693 681784 681874 681964 682055 90 1 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 2 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 90 3 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 4 4845 4935 6025 5114 6204 5294 5383 5473 6563 6652 90 5 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 89 6 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 89 7 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 89 8 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 89 9 9309 9398 9486 9576 9664 9753 9841 9930 690019 690107 89 490 690196 690285 690373 690462 690550 690639 690728 690816 690905 690993 89 1 1081 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 3199 3287 3375 3463 3551 3639 88 4 3727 3315 3903 3991 4078 4166 4254 4342 4430 4517 88 5 4605 4693 4781 4868 4956 5044 5131 5219 5307 5394 88 6 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 87 7 6356 6444 6531 6618 6706 6793 6880 6968 7055 7142 87 8 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 9 8101 8188 8275 8362 8449 8536 8622 8709 8796 8883 87 500 698970 699057 699144 699231 699317 699404 699491 699578 699664 699751 87 1 9838 9924 700011 700098700184 700271 700358 700444 700531 700617 87 2 700704 700790 0877 09631 1050 1130 1222 1309 1395 1482 86 3 1568 1654 i741 1827! 1913 1999 2086 2172 2258 2344 86 4 2431 2517 2603 2689! 2775 2861 2947 3033 3119 3205 86 5 3291 3377 3463 3549! 3635 3721 3807 3893 3979 4065 86 6 4151 4236 4322 4408! 4494 4579 4665 4751 4837 4922 86 7 5008 5094 6179 5265 6350 5436 6522 5607 5693 5778 86 8 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 85 9 6718 6803 6888 6974 7059 7144 7229 7316 7400 7485 85 510 707570 707655 707740 707826 707911 707996 708081 708166 708251 708336 85 1 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 85 2 9270 9355 9440 9524 9609 9694 9779 9863 9948 710033 85 3 710117 710202 710287 710371 710456 710540 710625 710710 710794 0879 85 4 0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 84 5 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 84 6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 84 7 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 84 8 4330 4414 4497 4581 4665 4749 4833 4916 6000 6084 84 9 5167 5251 5335 5418 5502 5586 5669 5753 5836 5920 84 Ko 1 3 3 4 5 6 7 8 e JHff. TABLE XIV. LOGARITHMS OF NUMBERS. 229 520 716003716087 716170 716254 716337 716421 716504 716588 8 716671 9 716754 DHL 83 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 82 2 7671 7754 7837; 7920 8003 8086 8169 8253 8336 8419 82 3 8502 8585 8668J 8751 8834 8917 9000 9083 9165 9248 82 4 5 9331 720159 9414 720242 9497 9580 720325720407 9663 720490 9745 720573 9828 720655 9911 720738 9994 720821 720077 0903 83 83 6 7 0986 1811 1068 1893 1151J 1233 1975 2058 1316 2140 1398 2222 1481 2305 1563 2387 1646 2469 1728 2552 82 82 8 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 82 9 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 82 530 724276 724358 724440 724522 724604 724685 724767 724849 724931 725013 82 1 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 82 2 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 82 3 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 81 4 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 81 6 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 6 9165 9246 9327 9408 9489 9570 9651 9732 981 J 9892 81 7 9974 730055 730136 730217 730298 730378 730459 730540 730621 730702 81 8 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 81 9 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 640 732394 3197 732474 3278 732555 3358 732635 3438 732715 3518 732796 3593 732876 3679 732966 3759 733037 3839 733117 3919 80 80 2 3999 4079 4160 4240 4320 4400 4480 4660 4640 4720 80 3 4800 4880 4960 5040 5120 5200 5279 6359 5439 5519 80 4 5599 6679 6759 5838 5918 6998 6078 6157 6237 6317 80 6 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 6 7193 7272 7352 7431 7511 7590 7670 7749 7829 7906 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 9389 9963 740047 740126 740205 740284 79 660 1 740363 1152 740442 1230 740521 1309 740600 1388 740678 1467 740757 1546 740836 1624 740915 1703 740994 1782 741073 1860 79 79 2 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 79 3 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 78 4 3510 35.SS 3667 3745 3823 3902 3930 4058 4136 4216 78 6 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 6 5075 6153 5231 5309 5387 6465 5543 5621 5699 5777 78 7 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 78 8 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 78 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 748188 748266 748343 748421 748498 748576 748653 748731 748808 748885 77 1 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 2 3 9736 750508 9814 750586 9891 750663 9968 750740 750045 0817 750123 0894 750200 0971 750277 1048 750354 1125 750431 1202 77 77 4 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 77 6 2048 2125 2202 2279 2356 2433 2509 2586 2663 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 8 434 S 4425 4501 4578 4654 4730 4807 4883 4960 6036 76 9 5112 5189 5265 5341 6417 6494 5570 5646 5722 6799 76 570 755875 755951 756027 756103 756180 756256 756332 756408 756484 756560 76 1 6636 6712 6788 6864 6940 7016 7092 7168 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 ; 9668 760422 9743 760498 9819 760573 9894 760649 9970 760724 760045 0799 760121 0375 760196 0950 760272 1025 760347 1101 76 76 7 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 76 8 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 76 9 2679 2754 2829 2904 2978 3053 3128 3203 3278 3353 76 No.) 1 3 1 3 4 5 6 7 8 9 Dlff. 230 TABLE XIV. LOGARITHMS OF NUMBERS. No. 1 2 3 4 5 6 7 8 9 Dltt'. j 1680 763428 "63503 763578 763653 763727 763802 763877 763952 764027 764101 76 1 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 76 2 4923 4998 5072 5147 6221 6296 6370 5445 5520 5594 76 3 5669 5743 6818 5892 5966 6041 6115 6190 6264 6338 74 4 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 74 5 7156 7230 7304 7379 7453 7527 7601 7675 7749 7823 74 6 7895 7972 8046 8120 8194 8268 8342 8416 8490| 8564 74 7 8638 8712 8786 8860 8934 9G08 9082 9156 9230 9303 74 8 9377 9451 9525 9599 9673 9746 9820 9894 9968 770042 74 9 770115 770189 770263 770336 770410 770484 770557 770631 770705 0778 74 690 770852 770926 770999 771073 771146 771220 771293 771367 771440 771514 74 1 1587 1661 1734 1808 1881 1955 2028 2102 2175 2248 73 3 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 73 3 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 73 4 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 6 4517 4590 4663 4736 4809 4882 4955 6028 5100 6173 73 6 6246 6319 5392 5465 5538 5610 6683 6756 6829 6902 73 7 5974 6047 6120 6193 6265 6338 6411 6483 6556 6629 73 6 6701 6774 6846 6919 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 9C19 9091 9163 9236 9308 9380 9452 9524 72 2 3 9596 780317 9669 780389 9741 780461 9813 780533 9885 780605 9957 780677 780029 0749 780101 0821 780173 0893 780245 0965 72 72 4 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 6 1756 1827 1899 1971 2042 2114 2186 2258 2329 2401 72 6 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 72 7 3189 3260 3332 3403 3475 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 6116 6187 6259 71 610 786330 785401 785472 785543 785615 785686 786757 785828 786899 785970 71 1 6041 6112 6183 6254 6325 6396 6467 6638 6609 6680 71 2 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 71 3 7460 7531 7602 7673 7744 7815 7885 7956 8027 8098 71 4 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 71 5 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 71 6 9581 9651 9722 9792 9863 9933 790004 790074 790144 790215 70 7 790285 790356 790426 790496 790567 790637 0707 0778 0848 0918 70 8 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 70 9 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 792392 792462 792532 792602 792672 792742 792812 792882 792952 793022 70 1 3092 3162 3231 330r : 3371 3441 3511 3581 3651 3721 70 2 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 70 3 4488 4558 4627 4697 4767 4836 4906 4976 6045 5115 70 4 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 70 5 5880 5949 6019 6088 6158 6227 6297 6366 6436 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 69 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 799616 799685 799754 799823 799892 799961 69 1 800029 800098 800167 800236 800305 800373 800442 800511 800580 800648 69 c 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 69 : 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 69 * 2089 2158 2226 2295 2363 2432 2500 2568 2637 2706 68 5 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 68 6 3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 68 ' 4139 4208 4276 4344 4412 4480 4548 4616 4685 4753 68 8 4821 4889 | 4957 5025 5093 5161 5229 5297 5365 6433 68 9 6501 5569 | 5637 5705 5773 5841 5908 6976 6044 6112 68 j No O 1 1 * 3 4 5 6 7 8 9 DMT. TABLE XIV. LOGARITHMS OF NUMBERS. 231 Ho. | 1 a 8 9 Diff. 640 306180 806248 306316 306334 06451 06519 06587 06655 06723 806790 68 6858 69^6 6994 7061 7129 7197 7264 7332 7400 7467 68 2 7535 7603 7670 7738 7806 7873 7941 8008 8076 8143 68 3 8211 8279 8346 8414 8481 8549 861G 8684 8751 8818 67 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 5 9560 9827 9694 9762 9829 9896 9964 810031 10098 810165 67 6 810233 810300 810367 810434 310501 810569 10636 0703 0770 0837 67 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 8 1576 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 9 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847 67 650 812913 812980 813047 813114 813181 813247 813314 813381 813448 813514 67 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 67 2 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 67 3 4913 4980 5046 5113 5179 5246 5312 6378 5445 5511 66 4 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 66 5 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 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 8292 8358 8424 8490 8556 8622 8688 8754 8820 66 9 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819544 819610 819676 819741 819807 819873 319939 820004 820070 820136 66 1 820201 820267 820333 820399 820464 820530 820595 0661 0727 0792 66 2 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 66 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 65 4 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 65 6 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 6 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 65 7 4126 4191 4256 4321 4336 4451 4516 4581 4646 4711 65 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 65 1 5426 5491 5556 5621 5686 5751 5815 5830 5945 6010 65 670 826075 826140 826204 826269 826334 826399 826464 826528 826593 826658 65 1 6723 6787 6852 6917 6931 7046 7111 7175 7240 7305 65 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 65 2 8015 8080 8144 8209 8273 8333 8402 8467 8531 8595 64 4 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 64 5 9304 9368 9432 9497 9561 9625 9690 9754 9818 9882 64 e 9947 830011 830075 830139 830204 830268 830332 830396 830460 830525 64 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 64 8 1230 1294 1353 1422 1486 1550 1614 1678 1742 1806 64 9 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 832509 832573 832637 832700 832764 832828 832892 832956 833020 833083 64 1 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 64 2 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 4 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 63 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 63 6 6324 6387 6451 6514 6577 6641 6704 6767 6830 6394 63 7 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 63 8 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 9 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 838849 838912 833975 839038 839101 839164 839227 839289 839352 839415 63 I 9478 9541 9604 9667 9729 9792 9855 9918 9981 840043 63 2 840106 840169 840232:840294 840357 840420 840482 840545 840608 0671 63 3 0733 0796 0859! 0921 0984 1046 1109 1172 1234 1297 63 4 1359 1422 1485J 1547 1610 1672 1735 1797 1860 1922 63 5 1935 2047 2110; 2172 2235 229 2360 2422 2484 2547 62 6 2609 267:. 27341 2796 2859 292 2983 3046 3108 3170 62 7 3232 329o 3357 3420 3482 354 3606 3669 373 3793 62 8 3855 39 IS 398C 4042 4104 416 4229 429 4353 4415 62 9 4477 453 4601 4664 472 478 4850 4912 4974 5036 62 No.] 1 3 3 * 5 6 7 8 e Dtff. TABLE XIV. LOGARITHMS OF NUMBERS. No O 1 2 3 4 5 6 7 8 9 Diff. 700 845098 845160 845222 845284 845346 845408 84547C 845532 845594 845656 62 1 5718 5780 5842 5904 5966 6028 609C 6151 6213! 6275 62 2 6337 639D 6461 6523 6585 6646 6708 6770 6832 6894 62 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 8l2fe 62 5 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 6 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 61 I 7 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 ' 8 850033 850095 850156 850217 850279 850340 850401 850462 850524 850585 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 3 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 61 4 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 61 5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4862 61 6 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 7 5519 5580 5640 5701 5761 5822 6882 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 60 720 857332 857393 857453 857513 857574 857634 857694 857755 857815 857875 60 1 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 2 8537 8597 8657 8718 8778 8838 8898 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 860218 860278 60 6 860338 860398 860458 860518 860578 0637 0697 0757 0817 0877 60 i 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1476 60 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 1 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 863323 863382 863442 863501 863561 863620 863680 863739 863799 863858 69 1 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 2 4511 4570 4630 4689 4748 4808 4867 4926 4985 6045 69 3 5104 5163 5222 5282 5341 5400 5459 6519 5578 5637 59 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 69 5 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 69 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 69 7 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 69 8 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 69 9 8644 8703 8762 8821 8879 8938 8997 9066 9114 9173 69 740 869232 869290 869349 869408 869466 869525 869584 869642 869701 869760 69 1 9818 9877 9935 9994 870053 870111 870170 870228 870287 870345 69 1 2 870404 870462 870521 870579 0638 0696 0755 0813 0872 0930 68 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 68 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 6 2739 2797 2855 2913 2972 3030 308S 3146 3204 3262 58 7 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 63 8 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 68 9 4482 4540 4598 4656 4714 4772 4830 4888 4945 6003 68 760 875061 875119 875177 875235 875293 875351 875409 875466 875524 875582 ts 1 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 68 2 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 68 3 6795 6853 1 6910 6968 7026 7083 7141 7199 7256 7314 68 4 7371 7429; 7487 7544 7602 7659 7717 7774 7832 7889 68 5 7947 8004' 8062 8119 8177 8234 8292 8349 8407 8464 67 6 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 7 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 67 8 9 9669 880242 9726 9784 880299 880356 9841 880413 9898 8S0471 9956 880528 880013 0585 880070 0642 880127 0699 880185 0756 67 67 No. O 1 2 3 4 5 6 7 8 9 Diff. 1 TABLE XIV. LOGARITHMS OF NUMBERS. 233 Mo. 1 8 3 4 5 6 y 8 9 Dlff. 760 880814 830871 830928 880985 881042 881099 881156 881213 881271 881328 67 1 1335 1442 1499 1556 1613 1670 1727 1784 1841 1898 57 2 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 67 3 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 67 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 67 5 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 57 6 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 57 7 4795 4852 4909 4965 5022 5078 5135J 5192 5248 5305 57 i 8 5361 5418 5474 5531 5587 6644 6700 5757 5813 5870 57 9 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 66 770 886491 886547 886604 886660 886716 886773 886829 886385 886942 886998 66 1 7054 7111 7167 7223 7280 7336 7392 7449 7505 7561 66 2 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 66 3 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 66 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 66 5 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 66 6 9862 9918 9974 890030 890086 890141 890197 890253 890309 890365 66 7 890421 890477 890533 0589 0645 0700 0756 0812 0863 0924 66 8 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 66 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 66 780 892095 892150 892206 892262 892317 892373 892429 892484 892540 892596 66 1 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 66 2 3207 3262 3318 3373 3429 3434 3540 3595 3651 3706 66 3 3762 3817 3873 3928 3934 4039 4094 4160 4205 4261 65 4 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 65 5 4870 4925 4980 6036 6091 5146 5201 6257 5312 6367 66 6 6423 6478 5533 5588 5644 6699 6764 6809 5864 5920 66 7 6975 6030 6085 6140 6195 6261 6306 6361 6416 6471 66 8 6526 6581 6636 6692 6747 6802 6867 6912 6967 7022 66 9 7077 7132 7187 7242 7297 7352 7407 7462 7617 7672 66 790 897627 897682 897737 897792 897847 897902 897967 898012 898067 898122 66 1 8176 8231 8286 8341 8396 S451 8606 8561 8615 8670 65 2 8725 8780 8835 8890 8944 8999 9064 9109 9164 9218 66 3 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 66 4 9821 9875 9930 9985 900039 900094 900149 900203 900258 900312 66 6 900367 900422 900476 900531 0586 0640 0695 0749 0804 0859 66 6 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 66 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 64 8 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 64 9 9547 2601 2655 2710 2764 2318 2873 2927 2981 3036 64 800 903090 903144 903199 903253 903307 903361 903416 903470 903524 903578 64 1 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 64 2 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 64 3 4716 4770 4824 4878 4932 4936 5040 5094 5148 5202 64 4 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 64 5 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 64 6 6335 6389 6443 6497 6551 6604 6653 6712 6766 6820 64 7 6874 697 6931 7035 7039 7143 7196 7250 7304 7358 54 8 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 64 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 3431 64 610 908485 908539 908592 90S646 903699 908753 908807 908860 908914 908967 64 1 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 64 2 9556 9610 9663 9716 9770 9323 9877 9930 9984 910037 63 3 910091 910144 910197 910251 910304 910358 910411 910464 910518 0571 63 4 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 63 5 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 53 6 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 63 7 2222 2275 2328 2331 2435 2438 2541 2594 2647 2700 63 8 2753 2806 2859 2913 2966 3019! 3072 3125 3178 3231 63 9 3234 3337 3390 3443 3496 3549 3602 3655 3708 3761 63 Nat O 1 3 3 4 5 o 7 8 9 Dttf. TABLE XIV. LOGARITHMS OF NUMBERS. Mo. O 1 3 4 5 6 r 8 9 jDiff. 820 913814 913867 913920 913973 914026 914079 914132 914184 914237 914290 63 1 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 53 2 4872 4925 4977 5030 6083 5136 5189 5241 5294 5347 63 3 5400 5453 6505 5558 5611 5664 6716 6769 5822 5875 63 4 5927 6980 6033 6085 6138 6191 6243 6296 6349 6401 53 5 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 53 6 6980 7033 7085 7133 7190 7243 7295 7348 7400 7453 53 7 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 62 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 52 9 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 52 830 919078 919130 919183 919235 919287 919340 919392 919444 919496 919549 62 1 9601 9653 9706 9758 9810 9862 9914 9967 920019 920071 52 2 920123 920176 920228 920280 920332 920384 920436 920489 0541 0593 62 3 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 52 4 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 62 5 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 62 6 2206 2258 2310 2362 2414 2468 2518 2570 2622 2674 62 7 2725 2777 2829 2881 2933 2986 3037 3089 3140 3192 62 8 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 62 9 3762 3814 3S65 3917 3969 4021 4072 4124 4176 4228 52 840 924279 924331 924383 924434 924486 924538 924589 924641 924693 924744 62 1 4796 4848 4399 4951 6003 6054 6106 6157 6209 5261 62 2 6312 5364 5415 5467 6518 6570 5621 6673 6725 5776 52 3 5828 6879 6931 6982 6034 6085 6137 6188 6240 6291 61 4 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 61 5 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 61 6 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 61 7 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 61 8 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 61 9 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 61 850 929419 929470 929521 929572 929623 929674 929725 929776 929827 929879 61 1 9930 9981 930032 930083 930134 930185 930236 930287 930338 930389 61 2 930440 930491 0542 0592 0643 0694 0745 0796 0847 0898 61 3 0949 1000 1061 1102 1153 1204 1254 1305 1356 1407 61 4 1458 1509 1660 1610 1661 1712 1763 1814 1865 1915 61 6 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 61 6 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 61 7 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 61 8 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 61 9 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 61 860 934498 934649 934599 934650 934700 934751 934801 934852 934902 934953 60 1 5003 5054 6104 6154 6205 6255 5306 6356 5406 5457 60 2 5507 5558 6608 6658 5709 6759 6809 6860 5910 6960 60 3 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 60 4 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 60 6 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 60 6 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 50 7 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 50 g 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 60 9 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 60 870 939519 939569 939619 939669 939719 939769 939819 939869 939918 939968 50 1 940018 940068 9401 18 940163 940218 940267 940317 940367940417 940467 60 '* 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 60 2 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 60 4 1511 1561 1611 1660 1710 1760 1809 1859 1909 1958 60 5 2008 2058 2107 2157 2207 2256 2306 2355; 2405 2455 60 6 2504 2554 2603 2653 2702 2752 2801 2851 i 2901 2950 60 j 3000 30*9 3099 3148! 319S 3217 3297 3346 3396 3445 49 8 3495 3544 3598; 3643 3692 3742 3791 3841 i 3890 3939 49 9 3989 4038 4083 4137 4186 4236 . 4285 4335 4384 4433 49 No 1 3 ~3~!^~ 5 6 7 8 9 IMS TABLE XIV. LOGARITHMS OF NUMBERS. 235 No.' 1 3 3 4 5 6 7 8 9 Diff. 880 941483 J44532 944531 944631 944680 944729 944779 944323 944877 944927 49 1 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 49 2 5469 6518 5567 5616 5665 5715; 5764 5813 5862 5912 49 3 5981 6010 6059 6108 6157 6207! 6256 6305 6354 6403 49 4 6452 6501 6551 6600 6649 6698 ! 6747 6796 6845 6894 49 5 6943 6992 7041 7090 7140 71891 7238 7287 7336 7385 49 6 7434 7483 7532 7581 7630 7679 7728 7777 7826 7876 49 7 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 8 8413 .8462 8511 8560 8609 8657 8706 8755 8804 8853 49 9 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 S90 949390 949439 949488 949536 949585 949634 949683 949731 949780 949829 49 I 9878 9926 9975 950024 950073 950121 950170 950219 950267 950316 49 2 950365 950414 950462 0511 0560 0608 0657 0706 0754 0803 49 3 0851 0900 0949 0997 1046 1095 114i 1192 1240 1289 49 4 1338 13S6 1435 1483 1532 1580| 1629 1677 1726 1775 49 5 1823 1872 1920 1969 2017 2066 '2114 2163 2211 2260 49 6 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 43 7 2792 2841 2889 2938 2986 3034 3083 3131 3180 3223 48 8 3276 3325 3373 3421 3470 3518 3566, 3615 3663 3711 48 9 3760 3808 3856 3905 3953 4001 4049 4098 4116 4194 48 900 954243 954291 954339 954387 954435 954484 954532 954580 951628 954677 48 i 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 48 2 5207 5255 5303 5351 5399 5447 5495 5543 5592 6640 I* 3| 5688 5736 5784 5832 5380 5928 5976 6024 6072 6120 48 4 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 5 0049 6697 6745 6793 6840 6888 6936 6984 7032 7080 48 6 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 48 7 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 48 8 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 9 8664 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 910 959041 959089 959137 959185 959232 959280 959328 959375 959423 959471 48 1 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 48 2 9995 960042 960090 960138 960185 960233 960230 960328 960376 960423 48 3 ! 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 48 4 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 48 5 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 47 6 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 47 7 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 47 8 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 47 9 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 47 920 963788 963835 963882 963929 963977 964024 964071 964118 964165 964212 47 1 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 47 2 4731 4778 4825 4872 4919 4966 5013 5061 6108 6155 47 3 5202 5249 5296 5343 5390 5437 5484 5531 6678 5625 47 4 5672 5719 5766 5313 5860 5907 5954 6001 6048 6095 47 5 6142 6189 6236 6233 6329 6376 6423 6470 6517 6564 47 6 6611 6658 6705 6752 6799 6345 6892 6939 6986 7033 47 7 7080 7127 7173 7220 7267 7314 7361 7408 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 963623 968670 963716 968763 968310 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 9882 9928 9975 970021 970063 970114 970161 970207 970254 970300 47 4 970347 970393 970440 04-36 05331 0579 0626 0672 0719 0765 46 5 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 46 6 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 46 7 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 46 8 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 46 9 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 46 No. 1 3 3 4 5 6 7 8 9 Diff. 236 TABLE XIV. LOGARITHMS OF NUMBERS. No. O 1 2 3 4 5 6 7 8 9 Diff. 940 973128 973174 973220 973266 973313 973359 973405 973451 973497 97'3543 46 1 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 46 2 4051 4097 4143 4189 4235 4281 4327 43?'4 4420 44G6 46 3 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 46 4 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 46 5 5432 5478 5524 5570 5616 5662 5707 5753 5799 5845 46 6 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 46 7 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 46 8 6808 6854 6900 6946 6992 703?' 7083 7129 7175 7220 46 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 950 977724 977769 977815 977861 977906 977952 977998 978043 978089 978135 46 1 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 46 2 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 46 3 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 46 4 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 5 980003 980049 980094 980140 980185 980231 980276 980322 980367 980412 45 6 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 45 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 8 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 45 9 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 45 900 982271 982316 982362 982407 982452 982497 982543 982588 982633 982678 45 1 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 45 2 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 45 3 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 45 4 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 5 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 6 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 45 7 5426 5471 5516 5561 5606 5651 5696 5741 5786 5830 45 8 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 45 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 45 970 986772 986817 986861 986906 986951 986996 987040 987085 987130 987175 45 1 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 45 2 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 45 3 8113 8157 8202 8247 8291 8336 8381 8435 8470 8514 45 4 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 5 9005 9049 9094 9138 9183 9227 9272 9316 9364 9405 45 6 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 7 9895 9939 9983 990028 990072 990117 990161 990206 990250 990294 44 8 990339 990383 990428 0472 0516 0561 0605 0650 0694 0738 44 9 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 44 980 991226 991270 991315 991359 991403 991448 991492 991536 991580 991625 44 1 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 <> 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 44 3 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 4 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 5 3436 3480 3524 3568 3613 3657 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 4801 4845 4889 4933 4977 5021 5065 5108 5152 44 9 5196 5240 5284 5328 5372 5416 5460 5504 5547 5591 44 990 995635 995679 995723 995767 995811 995854 995898 995942 995986 996030 44 1 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 44 2 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 3 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 44 4 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779 44 5 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 44 6 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 44 7 8695 8739 8782 8826 8869 8913 8956 9000 9043 9087 44 8 9131 9174 9218 9261 9305 9348 . 9392 9435 9479 9522 44 9 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 No. O 1 4 206131 206906 .207679 .203452 .209222 13.04 13.01 12.99 12.96 12.94 12.92 12.89 12.87 12.85 12.82 o 994418 J94398 .994377 .994357 994336 994316 994295 994274 .994254 994233 34 34 34 34 34 34 34 34 .35 36 9.207817 .203619 209420 210220 211018 211815 212611 213405 214193 214989 13.38 13.35 13.33 13.31 13.28 13.26 13.24 13.21 13.19 13.17 0.792183 .791381 .790580 .789780 .788982 .788185 .787389 .786595 .785802 .78501 1 60 49 48 47 46 45 44 43 42 41 ; 20 1 21 ! 22 23 24 25 26 27 23 29 9.209992 .210760 .211526 .212291 .213056 .213818 214579 215338 216097 216S54 12.80 12.78 12.75 12.73 12.71 12.68 12.66 12.64 12.62 12.59 9.994212 994191 .994171 994150 994129 .994108 .994037 .994066 994045 .994024 35 35 35 35 35 .35 35 35 35 35 9.215780 216568 217356 218142 218926 219710 220492 221272 222052 222830 13.15 13.12 13.10 13.08 13.06 13.03 13.01 12.99 12.97 12.95 0.784220 .783432 .782644 .781858 .781074 .780290 .779508 .778728 .777948 .777170 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.217609 .218363 .219116 .219368 .220618 .221367 .222115 .222.361 .223606 .224349 12.57 12.55 12.53 12.50 12.48 12.46 12.44 12.42 12.39 12.37 9,994003 .993932 .993960 .993939 .993913 .993397 .993875 .993854 .993832 .993811 35 35 .35 .35 .36 36 .36 .36 .36 .36 9.223607 .224382 .225156 .225929 .226700 .227471 .223239 .229007 .229773 .230539 12.92 12.90 12.88 12.86 12.84 12.62 12.79 12.77 12.75 12.73 0.776393 .775618 .774844 .774071 .773300 .772529 .771761 .770993 .770227 .769461 30 29 28 27 26 26 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.225092 .225833 .226573 .227311 .228043 .228784 .229518 .230252 .230984 .231715 12.35 12.33 12.31 12.29 12.26 12.24 12.22 12.20 12.18 12.16 9.993789 993768 993746 .993725 .993703 993681 .993660 .993633 .993616 993594 .36 .36 .36 .36 .36 36 36 .36 36 36 9.231302 232065 232826 .233586 .234345 235103 .235,359 .236614 .237368 .238120 12.71 12.69 12.67 12.65 12.63 12.60 12.58 12.56 12.54 12.52 0.768698 .767935 .767174 .766414 .765655 .764897 .764141 .763386 .762632 .761880 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 59 60 9.232444 .233172 ! .233399 .234625 .235349 .236073 .236795 .237515 .2382:5 .233953 .239671 12.14 12.12 12.10 12.07 12.05 12.03 12.01 11.99 11.97 11 95 9.993572 .993550 .993528 .993506 .993484 .993462 .993440 .993418 .993396 .993374 .993351 37 .37 .37 .37 .37 .37 37 .37 .37 .37 9.238872 .239622 .240371 .241118 .241865 .242610 .243354 .244097 .244839 .245579 .246319 12.50 12.48 12.46 12.44 12.42 12.40 12.33 12.36 12.34 12.32 0.761128 .760378 .759629 .758882 .758135 .757390 .756646 .755903 .755161 .754421 .753681 10 9 8 7 6 5 4 3 2 M. Cosine. 1 D. 1". Sloe. D. 1". Cotang. D. 1" Tang. M. 800 250 TABLE XV. LOGAKITIIMIC SINES, 160 1 M. Blue. D. 1" Cosine. D. 1". Twig. D. 1". Cotang M 1 8 4 6 6 7 8 9 .239670 .240386 .241101 .241814 .242526 .243237 .243947 .244656 .245363 .246069 11.93 11.91 11.89 11.87 11.85 11.83 11.81 11.79 11.77 11 75 9.993351 .993329 .993307 .993284 .993262 993240 .993217 .993195 .993172 .993149 .37 .37 .37 .37 .37 .37 .38 .38 .38 .38 9.246319 .247057 .247794 .243530 .249264 .249998 .250730 .251461 .252191 .252920 12.30 12.28 12.26 12.24 12.22 12.20 12.18 12.17 12.15 12.13 0.753681 .752943 .752206 .751470 .750736 .750002 .749270 .748539 .747809 .747080 60 59 58 57 56 55 54 53 52 51 10 11 9.246775 .247478 11.73 9.993127 .993104 .38 OQ 9.253648 .254374 12.11 12 09 0.740352 .745626 50 49 12 13 14 15 16 17 18 .248181 .248883 .249583 ^250282 .250980 .861677 .252373 11.71 11.69 11.67 11.66 11.63 11.61 11.59 .993081 .993059 .993036 .993013 .992990 .992967 .992944 .38 38 .38 .38 .38 .38 QQ .255100 .255824 .256547 .257269 .257990 .258710 .259429 12.07 12.05 12.03 12.01 12.00 11.98 Uq/> .744900 .744176 .743453 .742731 .742010 .741290 .740571 48 47 46 45 44 43 42 19 .253067 11.58 11 56 .992921 .38 .260146 11.94 .739854 41 20 21 22 23 24 26 26 27 28 29 9.253761 .254453 .255144 .265834 .256523 .257211 .257898 .258683 .259268 .259961 11.54 11.52 11.60 11.48 11.46 11.44 11.42 11.41 11.39 11 37 9.992898 .992875 .992852 .992829 992806 .992783 .992759 .992736 ,992713 ,992690 .38 .38 .39 39 .39 39 39 39 39 39 9.260863 .261578 .262292 .263005 .263717 .264428 .265138 .265347 .266555 .267261 11.92 11.90 11.89 11.87 11.85 11.83 11.81 11.79 11.78 11.76 0.739137 .738422 .737708 .736995 .736283 .735572 .734862 .734153 .733445 .732739 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 36 36 37 38 39 9.260633 .261314 .261994 .262673 .263351 .264027 .264703 .265377 .266051 .266723 11.35 11.33 11.31 11.30 11.28 11.26 11.24 11.22 11.20 11 19 9.992666 .992643 .992619 .992596 .992572 .992549 .992525 .992501 .992478 .992454 39 39 39 39. .39 .39 .39 .39 .40 .40 9.267967 .268671 .269375 .270077 .270779 .271479 .272178 .272876 .273573 .274269 11.74 11.72 11.70 11.69 11.67 11.65 11.64 11.62 11.60 11.58 0.732033 .731329 730625 .729923 .729221 .728521 .727822 .727124 .726427 .725731 30 29 28 27 1 26 I 25 1 24 1 23 22 21 40 41 42 43 44 45 46 47 48 49 9.267395 .268065 .268734 .269402 .270069 .270735 .271400 .272064 .272726 .273388 11.17 11.15 11.13 11.12 11.10 11.08 11.06 11.05 11.03 11 01 9.992430 .992406 .992382 .992359 .992335 .992311 .992237 ,992263 .992239 .992214 .40 .40 .40 40 .40 .40 .40 .40 .40 40 9.274964 .275658 .276351 .277043 .277734 .278424 .279113 .279801 .230488 .231174 11.57 11.55 11.53 11.51 11.50 11.48 11.46 11.45 11.43 11 41 0.725036 .724342 .723649 .722957 .722266 .721576 .720887 .720199 .719512 .718826 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 69 60 9.274049 .274708 .275367 .276025 .276631 .277337 .277991 .278645 .279297 .279948 .280599 10.99 10.98 10.96 10.94 10.92 10.91 10.89 10.87 10.86 10.84 9.992190 .992166 .992142 .992118 .992093 .992069 .992044 .992020 .991996 .991971 .991947 .40 .40 .40 .41 .41 .41 .41 .41 .41 .41 9.28ia r j8 .282542 .283225 .283907 .284588 .285268 .285947 .286624 .287301 .237977 .238652 11.40 11.38 11.36 11.35 11.33 11.31 11.30 11.28 11.26 11.25 0.718142 .717458 .716775 .716093 .715412 .714732 .714053 .713376 .712699 .712023 .711348 10 9 8 7 6 5 4 3 2 M. Cofiin. D. 1". Sir*. D. 1". Cotang. D. 1". Tang. M. 1000 COSINES, TANGENTS, AND COTANGENTS. M. Sine D.1". Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 3 9.280599 .281248 .281897 .282544 10.82 10.81 10.7S 9.991947 .991922 .991897 .991873 .41 .41 .41 A 1 9.238652 .289326 289999 290671 11.23 11.22 11.20 0.711348 .710674 710001 709329 60 59 58 57 4 .283190 .991843 .291342 .708658 56 5 .233836 10.76 .991823 292013 707987 55 6 .284480 10.74 .991799 .2926S2 .707318 54 7 .285124 10.72 .991774 41 293350 1 1.14 706650 53 8 .285766 10.71 .991749 .291DI7 705933 52 9 .286408 10.69 10.67 .991724 42 .294684 11.11 11.09 .705316 51 1C 11 9.287048 .287688 10.66 9.991699 .991674 .42 9.295:349 .296;)I3 11.07 0.704651 .703987 50 49 12 13 14 ( l5 16 17 18 19 .288326 .288964 .289600 .290236 .290370 .291504 .292137 .292768 10.64 10.63 10.61 10.59 10.58 10.56 10.55 10.53 10.51 .991619 .991624 .991599 .991574 .991549 .991524 .991498 .991473 .42 42 42 42 .42 .42 .42 .42 .296677 .297339 .298001 .293662 299322 299980 300638 .301295 11.06 11.04 11.03 11.01 11.00 10.98 10.97 10.95 10.93 .703323 .702661 .701999 .701338 .700678 700020 .699362 .698705 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.293399 .294029 .294658 .295236 .295913 .296539 297164 .297788 2934 12 .299034 10.50 10.48 10.47 10.45 10.43 10.42 10.40 10.39 10.37 10.36 9.991443 .991422 .991397 .991372 .991346 .991321 .991295 .991270 .991244 .991218 .42 .42 42 .42 .42 .43 .43 .43 .43 .43 9.301951 .3026:17 .303261 303914 304567 .305218 305869 .306519 307168 307816 10.92 10.90 10.89 10.87 10.86 10.84 10.83 10.81 10.80 10.78 a 698049 697393 696739 .696086 .695433 .694782 .694131 .693481 .692832 .692184 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 8. 299655 .300276 300395 .301514 .302132 .302743 303364 303979 .304593 .305207 10.34 10.33 10.31 10.30 10.28 10.26 10.25 10.23 10.22 10.20 9.991193 .991167 .991141 .991115 .991090 .991064 .991038 .991012 .990936 .990960 .43 .43 .43 .43 .43 .43 .43 .43 .43 .43 9.308463 .309109 .309754 .310399 .311042 .311685 .312327 .312968 .313608 .314247 10.77 10.76 10.74 10.73 10.71 10.70 10.68 10.67 10.65 10.64 0.691537 .690891 .690246 .689601 .688958 .688315 .687673 .687032 .686392 .685753 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.305819 .306430 .307041 .307650 .303259 .308867 .309474 .310080 .310685 .311289 10.19 10.17 10.16 10.14 10.13 10.12 10.10 10.09 10.07 10.06 9.990934 .990908 .990882 .990355 .990829 .990303 .990777 .990750 .990724 .990697 .44 .44 .44 .44 .44 .44 .44 .44 .44 .44 9.314885 .315523 .316159 .316795 .317430 .318064 .318697 .319330 .319961 .320592 10.62 10.61 10.60 10.58 10.57 10.55 10.54 10.53 10.51 10.50 0.685115 .684477 683841 .683205 .682570 .681936 .681303 .680670 .680039 .679408 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 59 60 9.311893 .312495 .313097 .313693 .314297 .314897 .315495 .316092 .316689 .317284 .317879 10.04 10.03 10.01 10.00 9.93 9.97 9.96 9.94 9.93 9.91 9.990671 .990645 .990618 .990591 .990565 .990538 .990511 .990485 .990453 .990431 .990404 .44 .44 .44 .44 .44 .44 .45 .45 .45 .45 9.321222 .321851 .322479 .323106 .323733 .324358 .324983 .325607 .326231 326853 .327475 10.48 10.47 10.46 10.44 10.43 10.41 10.40 10.39 10.37 10.36 0.678778 .678149 .677521 .676894 676267 .675642 .675017 .674393 .673769 673147 .672525 10 9 8 7 6 5 3 2 1 M. Octloe. D. 1. Sine, D.l. Co tang. D. 1". Taug. M. 1010 TABLE XV. LOGARITHMIC SINES, 16T< M. Sine D. 1". Cosine. D.l". Tang. D. 1". Cotang. M. 1 2 9.317879 .318473 .319066 9.90 9.88 9.990404 .990378 .990351 .45 .46 9.327475 .328095 .328715 10.35 10.33 0.672525 671905 .671285 60 59 58 3 .319658 9.87 .990324 45 .329334 10 31 .670666 57 4 6 6 7 8 9 .320249 .320840 .321430 .322019 .322607 .323194 9.86 9.84 9.83 9.81 9.80 9.79 9.77 .990297 .990270 .990243 .990215 .990188 .990161 '.45 .45 .45 .45 .45 .45 .329953 .330570 .331187 .331803 .332418 .333033 10^29 10.28 10.27 10.25 10.24 10.23 .670047 .669430 .668813 .668197 .667582 .666967 56 55 54 53 52 51 10 11 12 13 14 16 16 9.323780 .324366 .324960 .325534 .326117 .326700 .327281 9.76 9.76 9.73 9.72 9.70 9.69 n eo 9.990134 .990107 .990079 .990052 .990025 .989997 .989970 .45 .45 .46 .46 .46 .46 9.333646 .334259 .334871 .335482 .336093 .336702 .337311 10.21 10.20 10.19 10.17 10.16 10.15 10 14 0.666354 .665741 .665129 .664518 .663907 .663298 .662689 60 49 48 47 46 45 44 17 .327862 y.oo Q AA .989942 IA .337919 in 19 .662081 43 18 19 .328442 .329021 .OO 9.66 9.64 .989916 .989887 !46 .46 .338527 .339133 iu. l/o 10.11 10.10 .661473 .660867 42 41 20 21 22 9.329599 .330176 .330753 9.62 9.61 Q Art 9.989860 .46 .46 9.339739 .340344 .340948 10.08 10.07 10 06 0.660261 .659656 .659052 40 39 38 23 24 25 26 27 28 .331329 .331903 .332478 .333051 .333624 .334195 y.ou 9.58 9.57 9.66 9.54 9.53 ! 989777 .989749 .989721 .989693 .989665 .989637 .46 .46 .46 .46 .46 .47 .341552 .342155 .342757 .343358 .343958 .344558 lo'.os 10.03 10.02 10.01 10.00 Q Qft .658448 .657845 .657243 .656642 .656042 .655442 37 36 36 34 33 32 29 .334767 9.52 9.50 .989610 .47 .47 .345157 y.yo 9.97 .654843 31 30 31 32 33 34 36 36 37 38 39 9.335337 .335906 .336475 .337043 .337610 .338176 .338742 .339307 339871 .340434 9.49 9.48 9.46 9.45 9.44 9.43 9.41 9.40 9.39 9.37 9.989582 .989553 .989525 .989497 .989469 .989441 .989413 .989385 .989356 .989328 .47 .47 .47 .47 .47 .47 .47 .47 .47 .47 9.345755 .346353 .346949 .347545 .348141 .348735 .349329 .349922 .350514 .351106 9.96 9.95 9.93 9.92 9.91 9.90 9.88 9.87 9.86 9.85 0.654245 .653647 .653051 .652455 .651859 .651265 .650671 .650078 .649486 .648894 30 29 28 27 26 25 24 23 22 21 4C 41 42 43 44 45 46 47 9.340996 .341558 .342119 .342679 .343239 .343797 .344355 .344912 9.36 9.35 9.34 9.32 9.31 9.30 9.29 Q 97 9.989300 .989271 .989243 .989214 .989186 .989157 .989128 .989100 .47 .47 .47 .48 .48 .48 .48 9.351697 .352287 .352876 .353465 .354053 .354640 .355227 .355813 9.84 9.82 9.81 9.80 9.79 9.78 9.76 &7fi 0.648303 .647713 .647124 .646535 .645947 .645360 .644773 .644187 20 19 18 17 16 15 14 13 48 .345469 y.&l .989071 .48 .356398 ./O .643602 12 49 .346024 9.26 9.25 .989042 .48 .48 .356982 9.74 9.73 .643018 11 60 51 52 53 54 55 56 9.346579 .347134 .347687 .348240 .348792 .349343 .349893 9.24 9.22 9.21 9.20 9.19 9.17 9.989014 .988985 .988956 .988927 .988898 .988869 .988840 .48 .48 .48 .48 .48 .48 9.357566 .358149 .358731 .359313 .359893 .360474 .361053 9.72 9.70 9.69 9.68 9.67 9.66 0.642434 .641851 .641269 .640687 .640107 .639526 .638947 10 9 8 7 6 5 4 57 .350443 9.16 .988811 .48 .361632 9.65 .638368 3 68 59 6C .350992 .351540 .352088 9.15 9.14 9.13 .988782 988753 .988724 .48 .49 .49 .362210 .362787 .363364 9.63 9.62 9.61 .637790 .637213 .636636 2 I M. Coelue. D. 1". Sine. D. I". Cotung. D.I". Tang. M 103 770 COSINES, TANGENTS, AND COTANGENTS. 253 M. Sine. D. 1". Cosine. D.l. Tang. D. 1". Cotang. M. 1 2 3 4 9.352038 .352636 .353181 .35375$ .354271 9.11 9.10 9.09 9.03 9 07 9.988724 .938695 .988666 .988636 .988607 .49 .49 .49 .49 49 9.363364 .363940 .364515 .365090 .365664 9.60 9.59 9.58 9.57 0.636636 .636060 .635485 .634910 .634336 60 59 68 67 66 6 6 r 8 9 .354815 .355353 .355901 .356443 .356984 9.' 05 9.04 9.03 9.02 9.01 .93.3578 .988548 .988519 .938489 .938460 149 .49 .49 .49 .49 .366237 .366810 .367382 .367953 .368524 9.55 9.54 9.53 9.52 9.51 9.50 .633763 .633190 .632618 .632047 .631476 65 54 53 52 51 10 11 12 13 14 15 16 17 18 19 9.357524 .358064 .358603 .359141 .359678 .360215 .360762 .361287 .361822 .362356 8.99 8.98 8.97 8.96 8.95 8.94 8.92 8.91 8.90 8.89 9.988430 .983401 .988371 .988342 .988312 .938282 .988252 .988223 .988193 .933163 .49 .49 .49 .60 .60 .60 .60 .60 .60 .60 9.369094 .369663 370232 .370799 .371367 .371933 .372499 .373064 .373629 .374193 9.49 9.48 9.47 9.45 9.44 9.43 9.42 9.41 9.40 9.39 0.630906 .630337 .829768 .629201 .628633 .628067 .627601 .626936 .626371 .625807 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 9.362889 .363422 .363954 .364485 .365016 .365546 8.88 8.87 8.86 8.84 8.83 Q OO 9.988133 .988103 .938073 .988043 .988013 .937983 .60 .50 .60 .60 .50 fin 9.374756 .375319 .375881 .376442 .377003 .377563 9.38 9.37 9.36 9.35 9.33 0.625244 .624681 .624119 .623558 .622997 .622437 40 39 38 37 36 35 26 27 23 29 .366075 .366604 .367131 .367659 O.O& 8.81 8.80 8.79 8.78 937953 .987922 .987892 987862 .ou .50 .50 .60 .51 .378122 .378681 .379239 .379797 9.'31 9.30 9.29 9.23 621878 .621319 .620761 .620203 34 33 32 31 30 31 32 33 34 35 36 37 9.363185 .368711 369236 369761 .370285 .370808 .371330 .371852 8.76 8.75 8.74 8.73 8.72 8.71 8.70 8 69 9.987832 .987801 987771 .987740 .987710 .987679 .987649 .987618 .51 .51 .61 .51 .51 .51 61 9.380354 .380910 .381466 .382020 .382575 .383129 .383682 .384234 9.27 9.26 9.25 9.24 9.23 9.22 9.21 0.619646 .619090 .618534 .617980 .617425 .616871 .616318 .615766 30 29 28 27 26 25 24 23 38 .372373 .987538 K1 .384786 Q 1 Q .615214 22 39 .372894 a 66 .987557 ,ul .61 .385337 y.iy 9.18 .614663 21 40 41 42 43 9.373414 .373933 .374452 .374970 8.65 8.64 8.63 Q Q 9.987526 .987496 .987465 .987434 .61 .61 .51 ei 9.385888 .386438 .386987 .387536 9.17 9.16 9.15 0.614112 .613562 .613013 .612464 20 19 18 17 44 45 i 46 .375487 .376003 .376519 O.O4 8.61 8.60 o cq .937403 .987372 .987341 .Ol .51 .52 .388084 .383631 .389178 9J2 9.11 Q in .611916 .611369 .610822 16 15 14 47 .377035 o.oy Q KQ .987310 eo .389724 y. iu .610276 13 43 .377549 o.Oo o c7 .987279 Mi .390270 9.09 .609730 12 49 .378063 O.O/ 8.56 .987248 .52 .52 .390815 9.08 9.07 .609185 11 60 51 52 53 54 55 56 57 68 59 60 9.378577 .379089 .379601 .330113 .380624 .331134 .381643 .382152 .332661 .333163 .333676 8.55 8.53 8.52 8.51 8.50 8.49 8.43 8.47 8.46 8.45 9.987217 .937186 .987155 .987124 .987092 .987061 .987030 .936993 .936967 .986936 .986904 52 .52 .52 .52 .52 .52 .52 52 .52 .52 9.391360 .391903 .392447 .392989 .393531 .394073 .394614 .395154 .395694 .396233 .396771 9.06 9.05 9.04 9.03 9.02 9.01 9.00 8.99 8.98 8.97 0.608640 .608097 .607553 .607011 .606469 .605927 .605386 .604846 .604306 .603767 .603229 10 9 8 7 6 6 4 3 1 M. Cosine. D.I'. Sine. D. 1". Cotang. D. 1". Tang. MT 1030 TV* 254 TABLE XV. LOGATUTIIMIC SINES, 166* M Sine. D. 1". Coelne. D.I*. Tang. D. 1". Cotang M. I 1 2 3 4 5 6 7 8 9.383675 .384182 .384687 .385192 .385697 .386201 .386704 .387207 .387709 8.44 8.43 8.42 8.41 8.40 8.39 8.38 8.37 9.986904 .986873 .986841 .986809 .986778 .986746 .986714 .986683 .986651 .53 .63 .53 .53 .53 .63 .53 .53 9.396771 .397309 .397846 .398383 .398919 .399455 399990 .400524 .401058 8.96 8.96 8.95 8.94 8.93 8.92 8.91 8.90 Q OQ 0.603229 .602691 .602154 .601617 .601081 .600545 .600010 .599476 .598942 so ! 59 58 57 56 55 54 53 52 9 .388210 8.36 8.35 .986619 53 .401591 8.88 .598409 51 10 11 9.388711 .389211 8.34 9.986587 986555 .53 9.402124 .402656 8.87 0.597876 .597344 50 49 12 13 14 15 16 17 18 19 .389711 .390210 .390708 .391206 .391703 .392199 .392695 .393191 8.33 8.32 8.31 8.30 8.29 8.28 8.27 8.26 8.25 .986523 .986491 .986459 .986427 986395 .936363 .986331 .986299 .63 .53 .53 .63 .54 .54 .54 .54 .54 .403187 .403718 .404249 .404778 .405308 .405836 .406364 .406892 8.85 8.84 8.83 8.82 8.81 8.80 8.79 8.78 .296813 .696282 .595751 .595222 .594692 .594164 .593636 .593108 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.393685 .394179 .394673 .395166 .395658 .396150 .396641 .397132 .397621 .398111 8.24 8.23 8.22 8.21 8.20 8.19 8.18 8.17 8.18 8.15 9.986266 .936234 .986202 .986169 .936137 .986104 .986072 .986039 .986007 .985974 .54 .54 .64 .54 .64 .64 .54 .54 .54 .64 9.407419 .407945 .408471 .408996 .409521 .410045 .410569 .411092 .411615 .412137 8.77 8.76 8.75 8.75 8.74 8.73 8.72 8.71 8.70 8.69 0.692581 .592055 .591529 .591004 .690479 .589955 .589431 .588908 .588385 .587863 40 39 38 37 36 35 34 33 32 31 30 9.398600 9.985942 9.412658 a (*a 0.687342 30 31 32 33 34 35 36 37 38 .399088 .399575 .400062 .400549 .401035 .401520 .402005 .402489 8.14 8.13 8.12 8.11 8.10 8.09 8.08 8.07 .985909 .985876 .985843 .985811 .985778 .985745 .985712 .985679 .54 .55 .55 .65 .55 .55 .65 .55 .413179 .413699 .414219 .414738 .415257 .415775 .416293 .416810 8.67 8.66 8.65 8.65 8.64 8.63 8.62 Q C1 .586821 .586301 .585781 .585262 .684743 .584225 .583707 .583190 29 28 27 26 25 24 23 22 39 .402972 8.06 8.05 .985646 .55 .55 .417326 8.60 .582674 21 40 41 42 43 9.403455 .403938 .404420 .404901 8.04 8.03 8.02 9.985613 .985580 .985547 .985514 .55 .55 .55 9.417842 .418358 .418873 .419387 8.59 8.58 8.57 Q EC 0.582158 .581642 .581127 .580613 20 19 18 17 44 45 .405382 .405862 8.00 .985480 .985447 .55 .55 .419901 .420415 8.56 .580099 .579585 16 15 46 47 48 .406341 .406820 407299 7.99 7.98 7.97 .985414 .985381 .985347 .55 .56 .56 .420927 421440 421952 8.54 8.53 .579073 .578560 .578048 14 13 12 49 .407777 7.96 7.96 .985314 .56 56 .422463 8.51 .577537 11 50 51 9.408254 .408731 7.95 9.985280 .985247 .56 9.422974 .423484 8.60 0.577026 .576516 10 9 52 53 54 .409207 .409682 .410157 T.94 7.93 7.92 .985213 .985180 .985146 .56 .56 .56 .423993 .424503 .425011 8.49 8.48 .576007 .575497 .574989 8 7 6 55 56 .410632 .411106 7.91 7.90 .985113 .985079 .56 .66 .425519 .426027 8.46 Q AK .574481 .573973 5 4 57 58 .411579 .412052 7.88 .985045 .98501 1 .56 .56 .426534 .427041 8.44 ,573466 .572959 3 2 59 60 .412524 .412996 7.87 7.86 .984978 .984944 .56 .56 .427547 .428052 8.43 .572453 .571948 1 M. Cosine. D.I" Sine. D. 1". Cotang. D. 1". Taug M. !| 1040 COSINES, TANGENTS, AND COTANGENTS. 255 164 M. Bine. D.P. Cosine. D. 1". Tang. D. 1". Cotang. M 2 3 4 5 6 7 8 9 9.412996 .413467 .413938 .414408 414878 .415347 .415815 .416283 .416751 .417217 7.85 7.84 7.84 7.83 7.82 7.81 7.80 7.79 7.78 7.77 9.984944 984910 .984876 .984842 .984808 .984774 984740 984706 984672 .984638 56 .57 57 67 57 .57 .67 .57 .67 .67 9.428052 .428558 .429062 .429566 430070 .430573 .431075 431577 .432079 432580 8.42 8.41 8.40 8.39 8.38 8.38 8.37 8.36 8.35 8.34 0.571948 571442 570938 570434 569930 569427 -568925 .568423 567921 .567420 60 59 68 57 56 55 54 53 52 51 10 11 12 13 14 15 16 9.417684 .418150 .4)3615 419079 .419544 .420007 420470 7.76 7.75 7.75 7.74 7.73 7.72 7 71 9.984603 984569 984535 .984500 984466 .984432 .984397 67 67 .67 .67 67 67 CQ 9.433080 433580 434080 434579 435078 435576 436073 8.33 8.33 8.32 8.31 8.30 8.29 . O OQ. 0.566920 566420 .565920 .565421 .564922 .564424 663927 50 49 48 4? 46 45 44 17 18 420933 .421395 7.70 7 AQ .934363 984328 .68 CO 436570 437067 8.28 Q Q7 663430 .562933 43 42 19 421857 7.68 .984294 .68 437563 8.26 .662437 41 20 9.422318 7 R7 9.984259 CO 9.433059 U OR 0.661941 40 21 22 .422778 *23238 7.67 .934224 984190 .68 .438554 .439048 8.24 .661446 .560952 39 38 23 24 25 26 27 28 423697 .424156 424615 425073 425530 425987 7.65 7.64 7.63 7.62 7.61 984155 .934120 984085 .984050 .984015 983981 .68 .68 68 68 .68 439543 .440036 440529 441022 441514 442006 8.23 8.22 8.21 8.20 8.20 660457 659964 .659471 .658978 .658486 .557994 37 36 35 34 33 32 29 426443 7.61 7.60 .983946 .58 .68 442497 8.19 8.18 .557503 31 30 31 32 33 34 35 36 37 38 39 9.426899 .427354 427809 428263 428717 .429170 429623 .430075 430527 430978 7.59 7.58 7.67 7.66 7.65 7.55 7.53 7.52 7.62 7.51 9.983911 .983875 983840 .983805 983770 983735 983700 983664 933629 983594 .68 68 69 59 .69 69 69 59 69 69 9.442988 443479 443968 444458 444947 445435 445923 446411 446898 447384 8.17 8.16 8.16 8.16 8.14 8.13 8.13 8.12 8.11 8.10 0.657012 .556521 656032 .555542 655053 .654565 .654077 .553589 .553102 .652616 30 29 28 27 26 25 24 23 22 21 40 41 42 13 M 45 1 * 6 9.431429 .431879 .432329 .432778 .433226 .433675 434122 7.50 7.49 7.49 7.48 7.47 7.46 9.983558 933523 933487 983452 983416 983331 983345 59 59 69 59 59 59 9.447870 448356 448841 449326 449810 450294 450777 8.09 8.09 8.08 8.07 8.06 8.06 0.552130 651644 651159 .550674 550190 .549706 .549223 20 19 IS 17 16 15 14 47 .434569 933309 .59 451260 8.05 .548740 13 43 49 435016 435462 7.44 7.43 983273 983238 .60 .60 60 451743 452225 8.04 8.03 8.03 .548257 .547775 12 11 60 9.435908 9.983202 9.452706 0.? 47294 10 51 .436353 933166 .60 .453187 8.02 .546813 9 52 436793 933130 60 453668 546332 8 53 .437242 983094 .60 454148 .545852 7 54 55 437636 438129 7.39 983058 983022 .60 ,60 .454628 455107 8.00 7.99 .545372 .544893 6 5 56 57 68 59 60 438572 439014 .439456 .439897 .4403X3 737 7.36 736 7.35 932986 932950 982914 982878 932342 .60 .60 .60 60 60 455586 456064 456542 457019 457496 7.98 7.97 7.97 7.96 7.95 .544414 .543936 .543458 542981 .542504 4 3 2 1 M. Cosine. | D.l". Sine. D. 1". Cotang. D. 1". Tang. M. 256 TABLE XV. LOGARITHMIC SINES, 160 163 M. Sine. D 1". Cosine. D. i": Tang. D. 1". Cotang. M. 1 2 3 4 5 6 7 8 9 9.440338 .440778 .441218 .441658 .442096 .442535 .442973 .443410 .443847 .444234 7.34 7.33 7.32 7.31 7.31 7.30 7.29 7.28 7.27 727 9.982842 .982805 .982769 .932733 .932696 .932660 .932624 .932587 .982551 .932514 .60 .60 .61 .61 61 .61 .61 .61 .61 .61 9.457496 .457973 .458449 .458925 .459400 .459375 .460349 .460823 .461297 .461770 7.94 7.94 7.93 7.92 7.91 7.91 7.90 7.89 7.83 7.88 0.542504 .542027 .541551 .541075 .540600 .540125 .539651 .539177 .638703 .533230 60 59 58 57 56 55 54 53 52 51 10 H 12 13 14 15 16 17 18 19 9.444720 .445155 .445590 .446025 .446459 .446893 .447326 .447759 .448191 .448623 7.26 7.25 7.24 7.24 7.23 7.22 7.21 7.20 7.20 7 19 9.982477 .982441 .982404 .932367 .982331 .982294 .982257 .982220 .982183 .932146 .61 .61 .61 .61 .61 .61 .61 .62 .62 .62 9.462242 .462715 .463186 .463658 .464128 .464599 .465069 .465539 .466008 .466477 7.87 7.86 7.86 7.85 7.84 7.83 7.83 7. 82 7.81 7.81 0.537758 .537285 .536814 .536342 .535372 .535101 .534931 .534461 .533992 .533523 60 ! 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 23 29 9.449054 .449485 .449915 450345 .450775 .451201 .451632 .452060 .452488 .452915 7.18 7.17 7.17 7.16 7.15 7.14 7.13 7.13 7.12 7.11 9.932109 .982072 .932035 .981993 .981961 .931924 .931836 .931849 .931812 .981774 .62 .62 .62 62 .62 .62 .62 .62 .62 62 9.466945 .467413 .467880 .468347 .463814 .469280 .469746 .470211 .470676 .471141 7.80 7.79 7.78 7.78 7.77 7.76 7.76 7.76 7.74 7.74 0.533055 .532587 532120 .531653 .531186 .530720 .530254 .529789 .529324 .528859 40 , 39 38 37 36 36 34 33 ( 32 31 30 31 32 33 34 35 36 37 38 39 9.453342 .453768 .454194 .454619 .455044 .455469 .455893 .456316 .456739 .457162 7.10 7.10 7.09 7.08 7.07 7.07 7.06 7.05 7.04 704 9.981737 .981700 .931662 .981625 .981587 .931549 .981512 .981474 .931436 .931399 .62 .62 .63 .63 .63 .63 .63 .63 .63 .63 9.471605 .472069 .472532 .472995 .473457 .473919 .474381 .474842 .475303 .475763 7.73 7.72 7.71 7.71 7.70 7.69 7.69 7.68 7.67 7.67 0.528395 .627931 .527468 .527005 .526543 .526081 .525619 .525158 .524697 .624237 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.457584 .458006 .458427 .458848 .459268 .459688 .460108 .460527 .460946 .461364 7.03 7.02 7.01 7.01 7.00 6.99 6.98 6.98 6.97 696 9.981361 .981323 .981285 .981247 .981209 .981171 .981133 .981095 .981057 .981019 .63 .63 .63 .63 .63 .63 .63 .64 .64 .64 9.476223 .476683 .477142 .477601 .478059 .478517 .478975 .479432 .479889 .480345 7.66 7.65 7.65 7.64 7.63 7.63 7.62 7.61 7.61 7.60 0.523777 .523317 .522858 .522399 .521941 .521483 .521025 .520568 .520111 .519655 20 19 18 16 16 14 13 12 11 50 61 52 9.461782 .462199 .462616 6.96 6.95 9.980931 .980942 .980904 .64 .64 9.480801 .481257 .481712 7.59 7.59 0.519199 .518743 .518288 10 9 8 63 64 55 56 67 58 59 60 .463032 .463448 .463364 .464279 .464694 .465108 .465522 .465935 6.94 6.93 6.93 6.92 6.91 6.90 6.90 6.89 .980366 .980827 .980789 .980750 .980712 .980673 .980635 .980596 .64 .64 .64 .64 .64 .64 .64 .64 .482167 .482621 .483075 .483529 .483982 .484435 .484887 .485339 7.57 7.57 7.56 7.55 7.55 7.54 7.53 .517833 .617379 .516925 .516471 .516018 .515565 .515113 .514661 7 6 5 4 3 2 M. Cosine. D. 1". Shuv -ZZ5Z D.l". Cotang. D. 1". Pang. M. COSINES, TANGENTS, AND COTANGENTS. 25 "J 17 16JT M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 9.465935 fi QQ 9.930596 64 9.485339 f Q 0.514661 60 1 .466343 O.OO A ftQ .930558 .435791 / .Oo .514209 59 2 3 4 6 6 7 8 9 .466761 .467173 .467585 .467996 .468407 .468817 .469227 .469637 O.OO 6.87 6.86 6.85 6.85 6.84 6.83 6.83 6.82 980519 .980480 980442 .980403 930364 980325 930236 .980247 .65 .65 .65 .65 .65 .65 .65 .65 .436242 .486693 .487143 .487593 .488043 .488492 .488941 .489390 7.52 7.51 7.51 7.50 7.50 7.49 7.48 7.48 7.47 .513758 .513307 .512857 .512407 .511957 .511508 .511059 .510610 58 57 56 55 54 53 52 51 10 11 12 13 14 9.470046 .470455 .470863 .471271 .471679 6.81 6.81 6.80 6.79 9.980208 .980169 980130 .980091 .980052 .65 .65 .65 .65 9.489838 .490286 .490733 .491180 .491627 7.46 7.46 7.45 7.44 0.510162 .509714 .509267 .508820 .508373 50 49 48 47 46 15 16 17 18 .472086 .472492 .472898 .473304 6.78 6.78 6.77 6.76 .980012 979973 .979934 .979895 .65 .65 .65 .66 .492073 .492519 .492965 .493410 7.44 7.43 7.43 7.42 .507927 .507481 .507035 .506590 45 44 43 42 19 473710 6.76 6.75 .979855 .66 .66 .493354 7.41 7.41 .506146 41 20 21 9.474115 .474519 6.74 9.979816 .979776 .66 9.494299 .494743 7.40 0.505701 .505257 40 39 22 23 .474923 .475327 6.74 6.73 .979737 .979697 .66 .66 .495186 .495630 7.39 7.39 .604814 .604370 33 37 24 .475730 6.72 .979653 .66 .496073 7.38 .503927 36 25 26 .476133 .476536 6.72 6.71 .979618 .979579 .66 .66 .496515 .496957 7.38 7.37 .503485 .503043 35 34 27 .476938 6.70 .979539 .66 497399 7.36 .502601 33 28 .477340 6.69 .979499 .66 .497841 7.36 .502159 32 29 .477741 6.69 6.63 .979459 .66 .66 .498282 7.35 7.34 .501718 31 30 9.478142 9.979420 9.498722 0.501278 3(1 31 .478542 6.67 .979380 66 .499163 7.34 .500837 29 32 .478942 6.67 .979340 .66 .499603 7.33 .500397 28 33 34 35 .479342 .479741 .480140 6.66 6.65 6.65 .979300 .979260 .979220 .67 .67 .67 .500042 .500431 .500920 7.33 7.32 7.31 .499958 .499519 .499080 27 26 25 36 37 .480539 .480937 6.64 6.63 .979180 .979140 .67 .67 .501359 .501797 7.31 7.30 .498641 .498203 24 23 38 39 .481334 .481731 6.63 6.62 .979100 .979059 .67 .67 .502235 502672 7.30 7.29 .497765 .497328 22 21 6.61 .67 7.28 40 41 42 9.482128 .482525 .482921 6.61 6.60 9.979019 .978979 .978939 .67 .67 9.503109 503546 503932 7.28 7.27 0.496891 .496454 .496018 20 19 18 43 44 .483316 .483712 6.59 6.59 .978393 .978858 .67 .67 .504418 .504354 7.27 7.26 .495582 .495146 17 16 ' 45 46 .484107 .434501 6.58 6.57 .978817 .978777 .67 .67 .505239 .505724 7.25 7.25 .494711 .494276 15 14 47 .434895 6.57 .978737 .67 .506159 7.24 .493841 13 48 .485289 6.56 .978696 .68 .506593 7.24 .493407 12 49 .485682 6.55 .978655 .68 .507027 7.23 .492973 11 6.55 68 7.23 50 9.486075 9.978615 9.507460 0.492540 10 i 51 .486467 6.54 .978574 .68 .507893 7.22 .492107 9 52 53 54 .486360 .487251 .437643 6.54 6.53 6.52 .978533 .978493 .978452 .63 .68 68 .503326 .508759 .509191 7.21 7.21 7.20 .491674 .491241 .490809 8 7 6 55 .438034 6.52 .978411 .68 .509622 7.20 .490378 5 56 .483424 6.51 .978370 .68 .510054 7.19 .489946 4 57 58 .483814 .489204 6.50 6.50 .978329 .978238 .68 .68 .510435 .510916 MS .489515 .489084 3 2 59 .439593 1 .978247 .63 511346 7.17 .488654 1 60 .489932 ! 6 ' 48 .973206 .68 .511776 7.17 .438224 M. Cosine. D. 1". Sloe. D. 1". Cotang. D. 1". Tang. M. 18 73 258 TABLE XV. LOGARITHMIC SINES, 180 1610 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. ~T 9.489982 A 4ft 9.978206 AQ 9.511776 0.488224 60 2 3 .490371 .490759 .491147 O.4O ti.47 6.46 .978165 .978124 .978083 ,Oo .69 .69 .512206 .512635 .513064 7.16 7.16 7.15 .487794 .487365 486936 59 63 67 4 .491535 2*52 .978042 .69 513493 186507 66 5 .491922 6.45 .978001 .69 .513921 l-\\ , 486079 66 6 .492308 A 44 .977959 an .514349 10 .485651 64 7 8 9 .492695 .493081 .493466 BA3 6.43 6.42 .977918 .977877 .977835 .69 .69 .69 .69 .514777 .615204 .515631 7.13 7.12 7.12 7.11 485223 .484796 .484369 63 62 61 10 9.493851 9.977794 9.516057 0.483943 60 11 .494236 o.41 A 41 .977752 .69 AQ .616484 7.10 .483516 49 12 .494621 O.41 A 4H .977711 .oy .616910 7.10 .483090 48 13 .495005 O.4U .977669 *2J 517335 7.09 .482665 47 14 .495388 6.39 A QQ .977628 69 .617761 7.09 .482239 46 15 16 17 18 19 .495772 .496154 .496537 .496J19 .497301 o.oy 6.33 6.38 6.37 6.36 6.36 .977586 .977544 .977503 .977461 .977419 .69 .70 .70 .70 .70 .518186 .518610 .519034 .519458 .519882 7.08 7.08 7.07 7.07 7.06 7.05 .481814 .481390 .480966 .480542 .480118 45 44 43 42 41 20 21 22 23 24 25 26 27 28 9.497682 .498064 .498444 .498825 .499204 .499584 .499963 .600342 .500721 6.35 6.34 6.34 6.33 6.33 6.32 6.31 6.31 A on 9.977377 .977335 .977293 .977251 .977209 .977167 .977125 .977083 .977041 .70 .70 .70 .70 .70 .70 .70 .70 9.520305 .620728 .521151 .621673 .521995 .622417 .522838 .523259 .523680 7.05 7.04 7.04 7.03 7.03 7.02 7.02 7.01 0.479695 .479272 .478849 .478427 .478005 .477583 .477162 .476741 .476320 40 39 38 37 36 35 34 33 32 29 .501099 O.OU 6.30 976999 .70 .70 .524100 r.oi 7.00 .476900 31 30 31 32 33 34 35 36 37 38 39 9.501476 .501854 .502231 .502607 .502984 .503360 .503735 .604110 .504485 .604860 6.29 6.28 6.28 6.27 6.27 6.26 6.25 6.25 6.24 6.24 9.976957 .976914 .976872 .976830 .976787 .976745 .976702 .976660 976617 .976574 .70 .71 .71 .71 .71 .71 .71 .71 .71 .71 9.524520 .524940 .525359 .625778 626197 .626615 .627033 .627451 .627868 .528285 6.99 6.99 6.98 6.98 6.97 6.97 6.96 6.96 6.95 6.95 0.476480 .476060 .474641 .474222 .473803 .473385 472967 .472549 .472132 .471716 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.605234 .605608 .505981 .506354 .606727 .607099 .607471 .607843 .508214 .508585 6.23 6.22 6.22 6.21 6.21 6.20 6.19 6.19 6.18 6.18 9.976532 .976489 .976446 .976404 .976361 .976318 .976275 .976232 .976189 .976146 .71 .71 .71 .71 .71 .72 .72 .72 .72 .72 9.528702 .529119 .629535 .629951 .530366 .530781 .531196 .531611 .532025 .532439 6.94 6.94 6.93 6.93 6.92 6.91 6.91 6.90 6.90 6.89 0.471298 .470881 .470465 .470049 .469634 .469219 .468804 .468389 467975 .467561 20 19 18 17 16 15 14 13 12 11 60 51 52 53 64 55 66 57 53 9.503956 .509326 .50%96 .510065 .510434 .610803 .M1172 .511540 .511907 6.17 6.16 6.16 6.15 6.15 6.14 6.14 6.13 9.976103 .976060 .976017 .975974 .975930 .975887 .975844 .975800 .975757 .72 .72 .72 .72 .72 .72 .72 .72 9.532853 .533266 .533679 .534092 .534504 .534916 .535328 .535739 .536150 6.89 6.88 6.88 6.87 6.87 6.86 6.86 6.85 0.467147 .466734 .466321 465908 .465496 .465084 .464672 464261 .463850 10 9 8 7 6 5 4 3 2 69 .512275 6.12 .975714 .72 536561 6.85 .463439 1 60 .512642 6.12 .975670 .72 .536972 6.84 .463028 If. Co&iuo. D. 1". Sine. D. 1". Cotang. D. 1". Tang. M. 108" 190 COSINES, TANGENTS, AND COTANGENTS. M Sine. D.F. Cofilne. D. 1". Tang. D. 1". Cotaag. M. 1 9.512642 .513009 6.11 All 9.975670 .975627 .73 73 9.536972 .537382 6.84 0.463028 .462618 60 59 2 3 4 5 6 7 8 9 .513375 .513741 .514107 .614472 .514837 .515202 515566 515930 6.10 6.09 6.09 6.08 6.08 6.07 6.07 6.06 .975583 .975589 .975496 .975452 .975403 .975365 .975321 .975277 .73 .73 .73 .73 .73 .73 .73 .73 .537792 .538202 .533611 .539020 .539429 .539337 .540245 .540653 6.33 6.82 6.82 6.81 6.81 6.80 6.80 6.79 .462208 .461798 .461389 .460980 .460571 .460163 .459755 .459347 58 57 56 65 54 53 52 51 10 11 12 13 14 15 16 17 18 19 9.516294 .516657 .517020 .517382 .517745 .518107 .518468 .518829 .519190 .519551 6.05 6.05 6.04 6.04 6.03 6.03 6.02 6.02 6.01 6.00 9.975233 .975189 .975145 .975101 .975057 .975013 .974969 .974925 .974880 .974836 .73 .73 .73 .73 .73 .74 .74 .74 .74 .74 9.541061 .541463 .541875 .542281 .542688 .543094 .543499 .543905 .644310 .644715 6.79 6.78 6.78 6.77 6.77 6.76 6.76 6.75 6.75 6.74 0.458939 .458532 .458125 .457719 .457312 .456906 .456501 .456095 .455690 .455285 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.519911 .520271 .520631 .520990 .521349 .521707 .522066 .522424 .522781 .523133 6.00 5.99 5.99 5.98 5.93 6.97 5.97 5.96 5.95 5.95 9.974792 .974748 .974703 .974659 .974614 .974570 .974525 .974481 .974436 .974391 .74 .74 .74 .74 .74 .74 .74 .74 .74 .75 9.545119 .545524 .545928 .546331 .546735 .547138 .547540 .547943 .548345 .548747 6.74 6.73 6.73 6.72 6.72 6.71 6.71 6.70 6.70 6.69 0.454881 .454476 .454072 .453669 .453265 .452862 .452460 .452057 .451655 .451253 40 39 38 37 36 35 34 33 32 31 30 31 32 9.523495 .523852 .524208 5.94 5.94 K QQ 9.974347 .974302 .974257 .76 .75 9.549149 .549550 .549951 6.69 6.68 0.450351 .450450 .450049 30 29 23 33 34 35 36 37 .524564 .524920 .525275 .525630 .525984 593 6.92 5.92 5.91 C Qf) .974212 .974167 .974122 .974077 .974032 .75 .75 .75 .75 .550352 .550752 .551153 .551552 .551952 6.67 6.67 6.67 6.66 .449648 .449248 .448847 .448448 .448048 27 26 25 24 23 38 39 .526339 .526693 5.90 5.89 .973987 .973942 .75 .76 .552351 .552750 6.65 6.65 .447649 .447250 22 21 40 41 42 43 44 45 46 47 48 49 9.527046 .527400 .527753 .528105 .528458 .528810 .529161 .529513 .529364 .530215 5.89 5.88 5.88 6.87 5.87 5.86 5.86 5.85 5.85 5.84 9.973897 .973852 .973807 .973761 .973716 .973671 .973625 .973580 .973535 .973439 .75 .75 .75 .75 .76 .76 .76 .76 .76 .76 9.553149 .553548 .553946 .554344 .554741 .555139 .555536 .555933 .556329 .556725 6.64 6.64 6.63 6.63 6.62 6.62 6.61 6.61 6.60 660 0.446851 .446452 .446054 .445656 .445259 .444361 .444464 .444067 .443671 .443275 20 19 18 17 16 15 14 13 12 11 1 50 51 52 9.530565 .530915 .531265 5.83 5.53 C oo 9.973444 .973398 .973352 .76 .76 9.557121 .557517 .557913 6.59 6.59 0.442879 .442483 .442037 10 9 8 53 54 55 .531614 .531963 .532312 5.82 5.81 C Ol .973307 .973261 .973215 .76 .76 .558308 .558703 .559097 6.59 6.53 6.58 .441692 .441297 .440903 7 6 5 56 57 58 59 60 .532661 .533009 .533357 .533704 .534052 5.80 5.80 5.79 5.79 .973169 .973124 .973078 .973032 .972986 .76 .76 .76 .77 .77 .559491 .559385 .560279 .560673 .561066 6.57 6.57 6.56 6.56 6.55 .440509 .440115 .439721 .439327 .438934 4 3 2 1 M. Oorine. D. 1". Slue. D. 1". Cotuug. D. F. Tang. M. TABLE XV. LOGARITHMIC SINES, 169* M. Sine. D. 1". Coeine. D. 1". Tang. D. 1". Cotang. M, 1 2 3 4 9.534052 .534399 .534745 .535092 .535438 6.78 5.78 5.77 5.77 9.972986 .972940 .972894 .972848 .972802 .77 .77 .77 .77 77 9.561066 .561459 .561851 .562244 .562636 6.55 6.54 6.54 6.54 6 53 0.438934 .438541 .438149 .437756 .437364 60 59 58 57 56 5 6 7 8 9 .535783 .536129 .536474 .536818 .637163 5.76 5.75 6.75 5.74 5.74 .972755 .972709 .972663 .972617 .972570 .77 .77 .77 .77 .77 .563028 .563419 .563811 .564202 .564593 6.53 6.52 6.52 6.51 6.51 .436972 .436581 .436189 .435798 .435407 55 54 53 52 61 10 11 12 13 14 15 16 17 18 9.537507 .537851 .538194 .538538 .538880 .539223 .539565 539907 .540249 5.73 5.73 6.72 5.71 5.71 5.70 5.70 5.69 9.972524 .972478 .972431 .972385 .972338 .972291 .972245 .972198 .972151 .77 .77 .78 .78 .78 .78 .78 .78 fO 9.564983 .565373 .565763 .566153 .566542 .666932 .567320 .567709 .568098 6.50 6.50 6.50 6.49 6.49 6.48 6.48 6.47 fi 47 0.435017 .434627 .434237 .433847 .433458 .433068 .432680 .432291 .431902 50 49 48 47 46 45 44 43 42 19 .540590 6.68 .972105 .78 .568486 6.46 .431514 41 20 21 22 23 24 25 9.540931 .541272 .541613 .541953 .542293 .542632 5.68 5.67 5.67 5.66 5.66 9.972058 .972011 .971964 .971917 .971870 .971823 .78 .78 .78 .78 .78 7ft 9.668873 .569261 .569648 .670035 .570422 .670809 6.46 6.46 6.45 6.45 0.44 ft 44 0.431127 .430739 .430352 .429965 .429578 .429191 40 39 38 37 36 35 26 27 28 29 .542971 .543310 .543649 .643987 6.65 5.64 6.64 5.63 .971776 .971729 .971682 .971635 .78 .79 .79 .79 .671195 .671581 .571967 .572352 6.43 6.43 6.43 6.42 .428805 .428419 .428033 .427648 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.644325 .544663 .645000 645338 .545674 .646011 646347 .546683 .647019 .647354 5.63 6.62 6.62 5.61 6.61 6.60 6.60 6.59 6.69 6.58 9.971588 .971540 .971493 .971446 .971398 .971351 .971303 .971256 .971208 .971161 .79 .79 .79 .79 .79 .79 .79 .79 .79 .79 9.572738 .573123 .673507 .673892 .674276 .574660 .676044 .675427 .675810 .676193 6.42 6.41 6.41 6.40 6.40 6.40 6.39 6.39 6.38 6.38 0.427262 .426877 .426493 .426108 .425724 .425340 .424956 .424573 .424190 .423807 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.647689 .648024 .548359 .648693 .549027 .549360 .549693 .550026 .550359 .550692 5.58 5.67 6.57 5.56 6.56 6.55 5.55 6.55 6.64 664 9.971113 .971066 .971018 .970970 .970922 .970874 ,970827 .970779 .970731 .970683 .79 .80 .80 .80 .80 .80 .80 .80 .80 .80 9.576576 .576959 .577341 .577723 .578104 .578486 .578867 .579248 .679629 .580009 6.37 6.37 6.37 6.36 6.36 6.35 6.35 6.34 6.34 6.34 0.423424 .423041 .422659 .422277 .421896 .421614 .421133 .420762 .420371 .419991 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 68 59 60 9.551024 .651356 .551687 .552018 .552349 .552680 .553010 .653341 .653670 .654000 .554329 6.53 5.53 6.52 5.52 5.51 5.51 5.50 6.50 6.49 6.49 9.970635 .970586 .970538 .970490 .970442 .970394 .970345 .970297 .970249 .970200 .970152 .80 .80 .80 .80 .80 .81 .81 .81 .81 .81 9.580389 .580769 .681149 .581528 .681907 .582286 .582665 .683044 .683422 .583800 .584177 6.33 6.33 6.32 6.32 6.32 6.31 6.31 6.30 6.30 6.30 0.419611 .419231 .418851 .418472 .418093 .417714 .417335 .416956 .416578 .416200 .415823 10 9 8 7 6 5 4 3 2 M. Otelue. D. 1' . Biiio. D. 1". Ootaug. D. 1". Tang. M. 1100 C9 ate COSINES, TANGENTS, AND COTANGENTS. M. Sloe. D.I". Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 3 4 5 6 7 8 9 9.554329 .554658 .554987 .655315 .555643 .555971 .556299 .556626 556953 .557280 6.48 5.48 6.47 6.47 6.46 6.46 6.45 6.45 5.44 5.44 9.970152 .970103 .970055 .970006 .969957 .969909 .&69SGO .969811 .969762 .969714 .81 .81 .81 .81 81 .81 .81 .81 .81 .81 9.584177 .584555 .584932 .585309 .685686 .586062 .586439 .586815 .687190 .687566 6.29 6.29 6.28 6.28 6.28 6.27 6.27 6.26 6.26 6.26 0.415823 .415445 .415068 .414691 .414314 .413938 .413561 .413185 .412810 .412434 60 59 68 67 66 55 64 63 52 61 10 11 9 557606 .557932 6.44 9.969665 .969616 .82 9.587941 .588316 6.25 0.412059 .411684 60 49 12 .558258 5.43 .969567 .82 QQ 588691 6.25 .411309 48 13 .558583 5.43 .969518 .CM .589066 6.24 .410934 47 14 15 16 17 .558909 .559234 .559558 .559883 5.42 5.42 5.41 6.41 .969469 .969420 .969370 .969321 .82 62 .82 .82 QQ .589440 589814 590188 .690562 6.24 6.24 6.23 6.23 .410560 .410186 .409812 .409438 46 45 44 43 18 .560207 6.40 .969272 .Ko .590935 6.22 .409065 42 19 .560531 6.40 6.39 .969223 .82 .82 .591308 6.22 6.22 .408692 41 20 21 22 23 9.560855 .561178 .661501 .561824 5.39 6.38 6.38 9.969173 969124 .969075 .969025 .82 .82 ' .82 oo 9.591681 .592054 .592426 .692799 6.21 6.21 6.20 0.408319 .407946 .407574 .407201 40 39 38 37 24 25 26 27 28 .562146 .562468 .562790 .563112 .663433 6.37 6.37 6.37 5.36 6.36 c qc .968976 .968926 .968877 .968827 .968777 .0.6 .83 .83 .83 .83 oq .593171 .593542 .593914 .694285 .594656 6.20 6.20 6.19 6.19 6.18 .406829 .406458 .406086 .405715 .405344 36 35 34 33 32 29 .563755 o.oo .968728 .OO .695027 6.18 .404973 31 5.36 .83 6.18 30 9.564075 K 1A 9.968678 oq 9.595398 0.404602 30 31 32 33 .664396 .564716 .565036 O.H 5.34 5.33 .968628 .968678 .968528 .00 83 .83 .595768 .596138 .596508 6.17 6.17 6.16 .404232 .403862 .403492 29 28 27 34 35 .565366 .565676 5.33 5.32 .968479 .968429 .83 .83 .596878 .597247 6.16 6.16 .403122 .402753 26 25 36 37 38 39 .565995 .566314 .566632 .566951 5.32 5.32 6.31 5.31 6.30 .968379 .968329 .968278 .968228 .83 .83 .83 .84 .84 .597616 .597985 .598354 .598722 6.15 6.15 6.15 6.14 6.14 .402384 .402015 .401646 .401278 24 23 22 21 40 41 42 43 44 45 46 47 9.567269 .567587 .567904 .568222 .568539 .568856 .569172 .569488 6.30 6.29 5.29 6.28 6.28 6.28 6.27 9.968178 .968128 .968078 968027 .967977 .967927 .967876 .967826 84 .84 .84 .84 .84 .84 .84 9.599091 .599459 .599.827 .600194 .600562 .600929 .601296 .601663 6.13 6 13 6.13 6.12 6.12 6.12 6.11 0.400909 .400541 .400173 .399806 .399438 .399071 .398704 .398337 20 19 18 17 16 15 14 13 48 49 .569804 .670120 5.27 6.26 5.26 .967775 .967725 .84 .84 .84 .602029 .602395 6.11 6.10 6.10 .397971 .397605 12 11 50 51 52 53 54 55 66 9.570435 .570751 .571066 .571380 .671695 .572009 .572323 6.25 6.25 6.24 624 6.24 6.23 9.967674 .967624 .967573 .967522 .967471 .967421 .967370 .84 .84 .85 .85 .85 .85 9.602761 .603127 .603493 .603858 .604223 .604588 .604953 6.10 6.09 6.09 6.09 6.08 6.08 0.397239 .396873 .396507 .396142 .395777 .395412 .395047 10 9 8 7 6 5 4 67 53 .672636 672950 6.23 6.22 .967319 .967268 .85 .85 .605317 .605682 6.07 6.07 .394683 .394318 3 2 69 60 .673263 .573575 6.22 6.21 .967217 .967166 .85 .85 .606046 .606410 6.07 6.06 .393954 .393590 1 M. OOBIUO. D. 1". Sine. D. 1". Octai*. D. 1". Tang. M. Llio 68 262 TABLE XV. LOGARITHMIC SINES, M. Sine. D. 1". Oodne. D. 1". Tang. D. 1". Co tang. M. 1 9.573575 .673888 6.21 9.967166 .967115 .85 oe 9.606410 .606773 6.06 ft Oft 0.393590 .393227 60 59 2 3 4 .674200 .674512 .574824 6.2C 5.2C 6.20 .967064 .967013 .966961 .OO .85 .85 .607137 .607500 .607863 o.uo 6.05 6.05 .392863 .392500 .392137 58 57 56 5 6 .575136 .575447 5.19 5.19 .966910 .966859 .85 .85 .608225 .608588 e'.oi .391775 .391412 55 54 7 .575758 5.18 .966808 .86 .608950 6.04 .391050 53 8 .676069 6.18 .966756 .86 .609312 6.03 .390688 52 9 .576379 5.17 6.17 .966705 .86 .86 .609674 6.03 6.03 .390326 51 10 11 12 13 14 9.676689 .576999 .577309 .577618 .677927 6.17 5.16 5.16 5.15 9.966653 .966602 .966550 .966499 .966447 .86 .86 .86 .86 9.610036 .610397 .610759 .611120 .611480 6.02 6.02 6.02 6.01 ft n i 0.389964 .389603 .389241 .388880 .388520 60 49 48 47 46 15 .578236 5.15 .966395 .86 .611841 O.Ul .388159 45 16 .578545 5.14 .966344 .86 .612201 6.01 ft no .387799 44 17 .578853 5.14 .966292 .86 .612561 o.uu .387439 43 13 .679162 5.14 .966240 .86 .612921 6.00 .387079 42 19 .679470 5.13 5.13 .966188 .86 .86 .613281 6.00 5.99 .386719 41 20 21 22 23 24 9.579777 .580085 .580392 .580699 .681005 5.12 5.12 6.11 6.11 9.966136 .966085 .966033 .965981 .965929 .87 .87 .87 .87 9.613641 .614000 .614359 .614718 .615077 5.99 5.98 5.98 5.93 0.386359 .386000 .385641 .385282 .384923 40 39 38 37 36 25 26 27 23 29 .581312 .681618 .681924 .582229 .582535 5.11 5.10 5.10 5.09 5.09 5.09 .965876 .965824 .965772 .965720 .965668 .87 .87 .87 .87 .87 .87 .615435 .615793 .016151 .616509 .616867 5.97 5.97 6.97 5.96 5.96 5.96 .384565 .384207 .383849 .383491 .383133 35 34 33 32 31 30 31 32 33 34 9.582840 .633145 .683449 .683754 .584058 5.08 5.08 5.07 5.07 9.965615 .965563 .965511 .965458 .965406 .87 .87 .87 .87 U.617224 .617582 .617939 .618295 .618652 5.95 5.95 5.95 5.94 0.382776 .382418 .382061 .381705 .381348 30 29 28 27 26 35 .584361 5.06 .965353 .88 .619008 5.94 .380992 25 36 .584665 5.06 .965301 .88 .619364 5.94 .380636 24 37 .584968 5.06 .965248 .88 .619720 5.93 .380280 23 33 .585272 5.05 .965195 .88 .620076 5.93 .379924 22 39 .585574 5.05 5.04 .965143 .88 .88 .620432 5.93 5.92 .379568 21 40 9.585877 9.965090 9.620787 0.379213 20 41 .586179 5.04 .965037 .88 .621142 5.92 .378858 19 42 .586482 5.04 .964984 .88 .621497 5.92 .378503 18 43 .586783 5.03 .964931 .88 .621852 5.91 .378148 17 44 .587085 5.03 .964879 .88 .622207 5.91 .377793 16 45 46 .587386 .587688 5.02 5.02 .964826 .964773 .88 .83 .622561 .622915 6.91 5.90 .377439 .377085 15 14 47 .587939 5.01 .964720 .88 .623269 5.90 .376731 13 43 .588289 6.01 .964666 .88 623623 5.90 376377 12 49 .588590 5.01 5.00 .964613 .89 .89 .623976 5.89 5.89 .376024 11 50 9.588890 9.964560 9.624330 r on 0.375670 10 51 .589190 5.00 .964507 .89 .624683 o.oy .375317 9 52 .589489 4.99 .964454 .89 .625036 5.88 .374964 8 53 .589789 4.99 .964400 .89 .625388 5.88 e OQ .374612 7 64 .590088 4.99 .964347 .89 .625741 O.OO e Q .374259 6 65 .590387 4.98 .964294 .89 .626093 o.o/ .373907 5 66 .590686 4.98 .964240 .89 .626445 5.87 .373555 4 67 58 .590984 .591282 4.97 4.97 .964187 .964133 .89 .89 .626797 .627149 6.87 6.86 C Ofl .373203 .372851 3 2 59 .591580 Ant* .964080 .89 .627501 O.OO { 0< .372499 I 60 .591878 4.96 .964026 .89 .627852 u.OO .372148 M. Cotdae. D. 1". Sine. D. 1". Cotaug. D. 1". Taug. M. 67 COSINES, TANGENTS, AND COTANGENTS. 263 330 166Q ft Sine. D. 1". Cosine. D.F. Tang. D. F. Cotang. M. 2 3 4 6 7 8 9 9.591878 .692176 .592473 .692770 .593067 .693363 .693659 .693955 .594251 .694547 4.96 4.95 4.95 4.95 4.94 4.94 4.93 4.93 4.93 4.92 9.9640*0 .963972 .963919 .963865 .963811 .963757 .963704 .963650 .963596 .963542 .89 .89 .90 .90 .90 .90 .90 .90 .90 .90 9.627852 .628203 .628554 .628905 .629255 .629606 .629956 .630306 .630656 .631005 5.85 5.85 5.85 5.84 5.84 6.84 5.83 5.83 5.83 5.82 0.372148 .371797 .371446 .371095 .370745 .370394 .370044 .369694 .369344 .368995 60 59 68 67 56 55 54 63 52 51 10 11 12 13 14 15 16 17 9.594842 .595137 .595432 .595727 .596021 .596315 .696609 .596903 4.92 4.91 4.91 4.91 4.90 4.90 4.89 4 ftQ 9.963488 .963434 .963379 .963325 .963271 .963217 .963163 .963108 .90 .90 .90 .90 .90 .90 .91 Ql 9.631355 .631704 .632053 .632402 .632750 .633099 .633447 .633795 5.82 6.82 6.81 6.81 5.81 6.80 6.80 c on 0.368645 .368296 .367947 .367598 .367250 .366901 .366553 .366205 60 49 48 47 46 45 44 43 18 19 .697196 .697490 4.89 4.83 .963054 .962999 .91 .91 .634143 .634490 6.79 6.79 .365857 .365510 42 41 20 21 22 23 24 9.597783 .598075 .698368 .69866C .698952 4.88 4.88 4.87 4.87 A aa 9.962945 .962890 .962836 .962781 .962727 .91 .91 .91 .91 Ql 9.634838 .635185 .635532 .635879 .636226 6.79 6.78 5.78 6.78 c 70 0.365162 .364815 .364468 .364121 363774 40 39 38 37 36 26 20 1 27 29 .699244 .699536 .599827 .600118 .600409 4.86 4.86 4.85 4.85 4.84 .962672 .962617 .962562 .962508 .962463 .91 .91 .91 .91 .92 .636572 .636919 .637265 .637611 .637966 6.77 6.77 6.77 6.76 6.76 .363428 ,363081 .362735 .362389 .362044 36 34 33 32 31 30 31 9.600700 .600990 4.84 4 ft! 9.962398 .962343 .92 Q9 9.638302 .638647 6.76 C fK 0.361698 .361353 30 29 32 33 .601280 .601570 4.83 A CO .902288 .962233 .92 .638992 .639337 6.76 .361008 .360663 28 27 34 35 36 37 38 39 .601860 .602150 .602439 .602728 .603017 .603305 4.83 4.82 4.82 4.81 4.81 4.81 .962178 .962123 .962067 .962012 .961957 .961902 .92 .92 .92 .92 .92 .92 .639682 .640027 .640371 .640716 .641060 .641404 6.74 6.74 5.74 6.73 5.73 6.73 .360318 .359973 .359629 .359284 .358940 .358696 26 25 24 23 22 21 40 il 42 43 9.603594 .603882 .604170 .604457 4.80 4.80 4.79 4 7Q 9.961846 .961791 .961735 .961680 .92 .92 .92 QO 9.641747 .642091 .642434 .642777 6.73 5.72 5.72 K 79 0.358253 .357909 .357566 .357223 20 19 18 17 44 45 .604745 .605032 4.79 A 70 .961624 .961569 .93 .643120 .643463 6.71 c 71 .356880 .356537 16 15 46 47 .605319 .605606 4.78 .961513 .961458 .93 .643806 .644148 6.71 .356194 .355852 14 13 48 19 .605892 .606179 4.7? 4.77 .961402 .961346 .93 .93 .93 .644490 .644832 6.70 6.70 .355510 .355168 12 11 50 9.606465 9.961290 9.646174 e /o 0.354826 10 51 52 .606751 .607036 4.76 4 7fi .961235 .961179 .93 .645516 .645857 5.69 * CO .354484 .354143 9 8 53 .607322 961123 .646199 .353801 7 54 55 56 : 57 58 69 60 .607607 .607892 .608177 .603461 .608745 .609029 .609313 4.75 4.74 4.74 4.74 473 473 .961067 .961011 .960955 .960399 .960843 .960786 .960730 .93 .93 .93 .93 .94 .94 .94 .646540 646881 647222 .647562 .647903 .648243 .643583 5.68 5.68 6.68 5.67 5.67 5.67 .353460 .353119 .352778 .352438 .352097 .351757 .351417 6 1 6 4 3 2 1 M. Cosine. D.F. Sine D. 1". Co tang D. F. Tang. M. 6G 264 840 TABLE XV. LOGARITHMIC SINES, M. Sine. D. 1". Cosine. D. 1. Tang. D.I'. Cotang. M. 1 2 3 4 5 6 7 8 9 9.609313 .609597 .609880 .610164 .610447 .610729 .611012 .611294 .611576 .611858 4.73 4.72 4.72 4.72 4.71 4.71 4.71 4.70 4.70 4.69 9.960730 .960674 .960618 .960561 .960505 .960448 .960392 .960335 .960279 .960222 .94 .94 .94 .94 .94 .94 .94 .94 .94 .94 9.648583 .648923 .649263 .649602 .649942 .650281 .650620 .650959 .651297 .651636 5.67 5.66 5.66 5.66 5.65 5.65 5.65 5.64 5.64 5.64 0.351417 .351077 .350737 .350398 .350058 .349719 .349380 319041 .348703 .348364 60 59 58 57 56 55 54 53 52 51 10 11 9.612140 .612421 4.69 d RQ 9.960165 .960109 .95 9.651974 .652312 5.64 ff /q 0.348026 .347688 50 49 12 .612702 4.O3 A CO .960052 .95 .652650 O.DO .347350 48 13 14 15 16 17 18 19 .612983 .613264 .613545 .613825 .614105 .614385 .614665 "I. Do 4.68 4.68 4.67 4.67 4.67 4.66 4.66 .959995 .959938 .959882 .959825 .959768 .959711 .959654 .95 .95 .95 .95 .95 .95 .95 .95 .652988 .653326 .653663 .654000 .654337 .654674 .655011 5.63 5.63 5.62 5.62 5.62 5.62 5.61 5.61 .347012 .346674 .346337 .346000 .345663 .345326 .344989 47 46 45 44 43 42 41 20 21 22 23 24 25 26 9.614944 .615223 .615502 .615781 .616060 .616338 .616616 4.65 4.65 4.65 4.64 4.64 4.64 9 959596 .959539 .959482 .959425 .959368 .959310 .959253 .95 .95 .95 .95 .96 .96 9.655348 .655684 .656020 .656356 .656692 .657028 .657364 6.61 5.61 5.60 6.60 5.60 6.59 0.344652 .344316 .343980 .343644 .343308 .342972 .342636 40 39 38 37 36 35 34 27 28 29 .616894 .617172 .617450 4.63 4.63 4.63 4.62 .959195 .959138 .959080 .96 96 .96 .96 .657699 .658034 .658369 5.59 5.59 5.58 5.58 .342301 .341966 .341631 33 32 31 30 31 32 33 34 35 36 37 38 39 9.617727 .618004 .618281 .618553 .618834 .619110 .619386 .619662 .619938 .620213 4.62 4.61 4.61 4.61 4.60 4.60 4.60 4.59 4.59 4.59 9.959023 .958965 .958908 .958850 .958792 .958734 .958677 .958619 .958561 .958503 .96 .96 .96 .96 .96 .96 .96 .97 .97 .97 9.658704 .659039 .659373 .659708 .660042 .660376 660710 661043 661377 ,661710 5.58 5.58 5.57 6.67 5.57 5.56 5.56 5.56 5.56 5.55 0.341296 .340961 .340627 .340292 .339953 .339624 .339290 .338957 .338623 .338290 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 9.620488 .620763 .621038 .621313 .621587 .621861 4.58 4.53 4.58 4.57 4.57 9.958445 .953387 .958329 .958271 .958213 .958154 .97 .97 .97 .97 .97 9.662043 .662376 .662709 .663042 .663375 .663707 6.55 6.55 6.54 6.54 5.54 0.337957 .337624 .337291 .336958 .336625 .336293 20 19 18 17 16 16 46 47 48 49 .622135 .622409 .622682 .622956 4.57 4.56 4.56 4.56 4.55 .958096 .958038 .957979 .957921 .97 .97 .97 .97 .97 664039 .664371 .664703 .665035 5.54 5.53 5.53 5.53 5.53 335961 335629 .335297 .334965 14 13 12 11 50 51 9.623229 .623502 4.55 9.957863 .957804 .97 9.665366 .665698 5.52 0.334634 .334302 10 9 52 53 .623774 .624047 4.54 4.54 .957746 .957687 .98 .98 .666029 .666360 5.52 5.52 .333971 .33-3640 8 7 54 .624319 4.54 .957628 .98 .666691 5.61 .333309 6 55 .624591 4.53 .957570 .98 667021 5.51 332979 5 56 .624863 4.53 A. Q .957511 .98 QQ .667352 5.51 K ei .332648 4 57 .625135 1.OO .957452 .yo .667682 D.OJ .332318 3 68 59 .625406 625677 4.52 4.52 .957393 .957335 .98 .98 .668013 .668343 5.50 5.50 .331987 .331657 2 60 .625948 4.52 .957276 .98 .668673 5.50 .331327 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1". Tang. II. 1140 COSINES, TANGENTS, AND COTANGENTS. 265 M Blue. D.l". Cosine. D 1" Tung. D.1". Ootaiig. M. "T i 3 9.625948 .626219 .626490 .626760 4.61 4.61 4.51 A RA 9.957276 .957217 .957168 .957099 .98 .98 .98 QO 9.668673 .669002 .669332 .669661 6.50 6.49 6.49 0.331327 .330998 .330668 .330339 ~6JT 69 68 67 4 6 7 .627030 .627300 .627670 .627840 4.OU 4.60 4.50 4.49 .957040 .956981 .956921 .956862 vQ .99 .99 .99 99 .669991 .670320 .670649 .670977 6.49 5.49 6.48 6.48 K dfl .330009 .329680 .329351 .329023 66 65 54 63 8 9 .628109 .628378 149 4.48 .956803 .956744 '99 .671306 .671635 O.4O 6.47 6.47 .328694 .328365 52 61 1C 11 9J62S647 .628916 4.48 9.956684 .956625 .99 9.671963 .672291 5.47 0.328037 .327709 60 49 12 .629185 4.48 .956566 .99 QQ .672619 6.47 R AK. .327381 48 13 .629453 I'lr .956506 V .672947 O.3O .327053 47 14 .629721 J* .956447 .99 .673274 6.46 .326726 46 16 16 ,629989 .630257 4.47 4.46 .956387 .956327 .99 99 .673602 .673929 6.46 6.46 .326398 .326071 45 44 17 18 19 .630524 .630792 .631059 4.46 4.46 4.45 4.45 956268 .956208 .966148 .99 .99 1.00 1.00 .674257 .674584 .674911 6.45 5.46 6.45 6.46 .325743 .325416 .325089 43 42 41 20 21 22 23 24 26 26 27 28 29 9.631326 .631593 .631859 .632125 .632392 .632658 .632923 .633189 .633454 .633719 4.46 4.44 4.44 4.44 4.43 4.43 4.43 4.42 4.42 4.42 9.956089 .956029 .955969 .955909 .955849 .955789 .955729 .955669 .955609 .956548 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 9.676237 .675564 .675890 .676217 .676543 .676869 .677194 .677620 .677846 .678171 6.44 5.44 5.44 5.44 6.43 6.43 6.43 6.42 6.42 6.42 0.324763 .324436 .324110 .323783 .323457 .323131 .322806 .322480 .322154 321829 40 39 38 37 36 36 34 33 32 31 30 31 9.633984 .634249 4.41 9.955488 .955428 1.00 9.678496 .678821 6.42 0.321504 .321179 30 29 32 33 34 36 36 37 38 39 .634514 .634778 .635042 .635306 .635570 .635834 .636097 .636360 4.41 4.41 4.40 4.40 4.40 4.39 4.39 4.39 4.38 .955368 .955307 .955247 .955186 .955126 .955065 .955005 .954944 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 .679146 .679471 .679795 .680120 .680444 .680768 .681092 .681416 6.41 6.41 6.41 5.41 6.40 6.40 6.40 6.40 6.39 .320854 .320529 .320205 .319880 .319556 .319232 .318909 .318534 28 27 26 26 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.636623 .636886 .637148 .637411 .637673 .637935 .638197 .638458 .638720 .638981 4.38 4.38 4.37 4.37 4.37 4.36 4.36 4.36 4.35 4.36 9.954883 .954823 .954762 .954701 .954640 .954579 .954518 .954457 .954396 .954335 1.01 1.01 1.01 101 1.02 1.02 .02 1.02 .02 .02 9.681740 .682063 .682387 .682710 .683033 .683356 .683679 .684001 .684324 .684646 5.39 5.39 5.39 5.38 6.38 5.38 5.38 6.37 6.37 6.37 0.318260 .317937 .317613 .317290 .316967 .316644 .316321 315999 .315676 .315354 19 18 17 16 15 14 13 12 11 50 61 9.639242 .639503 4.35 9.954274 .954213 .02 9.684968 .685290 6.37 0.315032 .314710 10 9 52 53 54 55 56 57 58 59 60 .639764 .640024 .640284 .640544 .640804 .641064 .641324 .641583 .641842 4.34 4.34 4.34 4.33 4.33 4.33 4.32 4.32 4.32 .954152 .954090 .954029 .953968 .953906 .953845 .953783 .953722 .953660 .02 .02 .02 .02 .02 .02 .03 .03 1.03 .685612 .685934 .686255 .686577 .686898 .687219 687540 687861 5.36 5.36 5.36 5.36 5.35 6.35 5.35 5.35 5.35 .314388 .314066 .313745 .313423 .313102 .312781 .312460 .312139 .311818 8 7 6 6 4 3 2 t M Cosine D 1" Sine D. 1". Cotang* D.I'. ; JB Tug. t = M. 64 266 TABLE XV. LOGARITHMIC SINES, 460 15* H. Sine. 1 D.I". Cosine. D. 1". Tang. D. 1". Cotang. V. - 2 3 4 5 6 7 8 9 9.64184* .642101 .642360 .642618 .642877 .643135 .643393 .643650 .643908 .644165 4.32 4.31 4.31 4.31 4.30 4.30 4.30 4.29 4.29 4.29 9.953660 .953599 .953537 .953475 .953413 .953352 .953290 .953228 .953166 .953104 1.03 1.03 1.03 1.03 1.03 1.03 1.03 103 1,03 1.03 9.688182 .688502 .688823 .689143 .689463 .689783 .690103 .690423 .690742 .691062 5.34 5.34 5.34 5.34 5.33 5.33 5.33 5.33 6.32 6.32 0.311818 .31 1498 .311177 ,310857 .310537 .310217 .309897 .309577 309258 .308938 60 59 58 57 56 55 64 53 52 61 10 11 12 13 14 16 16 17 18 19 9.644423 .644680 .644936 .645193 .645450 .645706 .645962 .646218 .646474 .646729 4.28 4.28 4.28 4.27 4.27 4.27 4.26 4.26 4.26 4.26 9.953042 .952980 .952918 .952855 .952793 .952731 .952669 .952606 .952544 .952481 1.03 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 9.691381 .691700 .692019 .692338 .692656 .692975 .693293 .693612 .693930 .694248 5.32 6.32 5.31 5.31 5.31 6.31 5.30 5.30 6.30 6.30 0.308619 .308300 .307981 ,307662 3Cf7344 .307025 .306707 .306388 .306070 .305752 60 49 43 47 l 45 44 ' 43 42 1 41 20 21 22 9.646984 .647240 .647494 4.25 4.25 9.952419 .952356 .952294 1.04 1.04 1 ClA 9.694566 .694883 .695201 6.29 5.29 0.305434 .305117 .304799 40 ' 39 33 23 24 25 26 27 28 29 .647749 .648004 .648253 .648512 .648766 .649020 .649274 4.25 4.24 4.24 4.24 4.23 4.23 4.23 4.22 .952231 .952168 .952106 .952043 .951980 .951917 .951854 1.04 1.05 1.05 1.05 1.05 105 1.05 .695518 .695836 .696153 .696470 .696787 .697103 .697420 6.29 5.29 6.28 6.28 6.2S 5.28 6.27 .304482 .304164 .303847 303530 .303213 .302897 .302580 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.649527 .649781 .650034 .650287 .650539 .650792 .651044 .651297 .651549 .651800 4.22 4.22 4.22 4.21 4.21 4.21 4.20 4.20 4.20 4.19 9.951791 .951728 951665 .951602 .951539 .951476 .951412 .951349 .951286 .951222 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.06 1.06 1.06 9.697736 .698053 .698369 .698685 .699001 .699316 .699632 .699947 .700263 .700578 6.27 5.27 5.27 5.26 6.26 6.26 6.26 6.26 5.25 5.25 0.302264 .301947 .301631 .301315 .300999 .300684 .300363 .300053 .299737 .299422 30 29 28 27 26 25 24 23 22 21 40 41 9.652052 .652304 4.19 9.951159 .951096 1.06 1 HA 9.700893 .701208 5.26 0.299107 .298792 20 19 42 43 44 45 46 .652555 .652806 .653057 .653308 .653558 4.19 4.18 4.18 4.18 4.18 .951032 .950968 .950905 .950841 .950778 1.06 1.06 1.06 1.06 1 DA .701523 .701837 .702152 .702466 .702781 5.24 5.24 5.24 5.24 .298477 .298163 .297848 .297534 .297219 18 17 16 16 14 47 .653808 4.17 .950714 .703095 .296905 13 48 49 654059 654309 4.17 4.17 4.16 .950650 .950586 1.06 1.06 .7034-^9 .703722 5.23 5.23 .296591 .296278 12 11 50 51 52 53 54 55 56 57 53 59 60 9.654558 .654808 .655058 .655307 .655556 .655805 .656054 .656302 .656551 .656799 .657047 4.16 4.16 4.15 4.15 4.15 4.15 4.14 4.14 4.14 4.13 9.950522 .950458 .950394 .950330 .950266 .950202 .950138 .950074 .950010 .949945 .949881 1.07 1.07 1.07 1.07 1.07 1.07 1.07 1.07 1.07 1.07 9.704036 .704350 .704663 .704976 .705290 .705603 .705916 .706228 .706541 .706854 ,707166 5.23 5.22 5.22 5.22 6.22 6.22 6.21 521 5.21 6.21 0.295964 .295650 .295337 .295924 .294710 .294397 .294084 .293772 .293459 .293146 .292834 10 9 8 7 6 6 4 3 2 1 M. Cosine. D. 1". Sine. D. 1". Ootang. D. 1". Tang M. 1160 COSINES, TANGENTS, AND COTANGENTS. 26? y i5 M. Slue. D. 1". Cosine. D.I" Tang. D. 1". Cotang. M. 2 3 4 5 6 7 c 9.657047 .657295 .657542 .657790 .658037 .658284 .658531 .658778 659025 4.13 4.13 4.12 4.12 4.12 4.12 4.11 4.11 9.949881 .949816 .949752 .949688 .949623 .949558 .949494 .949429 .949364 1.07 1.07 1.07 1.08 1.03 1.08 1.08 1.08 9.707166 .707478 707790 .708102 .708414 .708726 .709037 .709349 .709660 5.20 5.20 5.20 5.20 5.20 5.19 5.19 5.19 0.292834 .292522 .292210 .291898 .291586 .291274 .290963 .290651 .290340 60 69 58 57 56 65 54 53 52 9 i .659271 4.10 .949300 1.08 709971 5.19 5.18 .290029 51 10 9.659517 4 10 9.949235 1QQ 9.710282 s ift 0.289718 50 a 12 13 14 15 16 17 18 19 .659763 .660009 .660255 .660501 .660746 .660991 .661236 .661481 .661726 4.10 4.10 4.09 4.09 4.09 4.08 4.08 4.08 4.08 .949170 .949105 .949040 .948975 .948910 .948845 .948780 .948715 .948650 1.08 1.08 1.08 1.08 1.08 1.09 1.09 1.09 1.09 .710593 .710904 .711215 711525 .711836 .712146 .712456 .712766 .713076 6.18 6.18 6.18 5.17 5.17 6.17 6.17 5.17 5.16 .289407 .289096 .288785 .288476 .288164 .287854 .287544 .287234 .286924 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 23 29 9.661970 .662214 .662459 .662703 .662946 .663190 .663433 .663677 .663920 .664163 4.07 4.07 4.07 4.06 4.06 4.06 4.05 4.05 4.05 4.05 9.948584 .948519 .948454 .948388 .948323 .948257 .948192 .948126 .943060 .947996 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.10 9.713386 .713696 .714005 .714314 .714624 .714933 .715242 .715551 .715860 .716168 5.16 6.16 5.16 6.15 5.15 5.15 5.15 6.15 6.14 6.14 0.286614 .286304 .285995 .285686 .285376 .285067 .284758 .284449 .284140 .283832 40 39 38 37 36 36 34 33 33 31 30 81 32 33 84 35 36 37 38 39 9.664406 .664648 .664891 .635133 .665375 .665617 .665859 .666100 .666342 .666533 4.04 4.04 4.04 4.03 4.03 4.03 4.03 4.02 4.02 4.02 9:947929 .947863 .947797 .947731 .947665 .947600 .947533 .947467 .947401 .947335 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 9.716477 .716785 .717093 .717401 .717709 .718017 .718325 .718633 .718940 .719248 5.14 6.14 5.14 5.13 5.13 5.13 5.13 5.13 6.12 5.12 0.283523 .283215 .282907 .282599 .282291 .281983 .281675 .281367 .281060 .280752 30 29 28 27 26 26 24 23 22 21 40 41 42 1 43 44 45 46 47 43 49 9.666824 .667065 .667305 .667546 .667786 .668027 .668267 .663506 .668746 .668936 4.01 4.01 4.01 4.01 4.00 4.00 4.00 3.99 3.99 3.99 9.947269 .947203 .947136 .947070 .947004 .946937 .946871 .946804 .946738 .946671 1.10 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 9.719555 .719862 .720169 720476 .720783 .721089 .721396 .721702 .722009 .722315 5.12 5.12 5.11 5.11 6.11 5.11 5.11 6.10 5.10 5.10 0.280445 .280138 .279831 .279524 .279217 .278911 .278604 .278298 .277991 .277685 20 ! 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 66 57 68 69 60 9.669225 .669464 669703 .669942 .670181 .670419 .670653 .670396 .671134 671372 .671609 3.99 3.93 3.98 3.93 3.98 3.97 3.97 3.97 3.96 3.96 9.946604 .946538 .946471 .946404 .946337 .946270 ,946203 .946136 .946069 .946002 .945935 1.11 1.11 1.11 1.11 1.12 1.12 1.12 1.12 1.12 1 12 9.722621 .722927 .723232 .723538 .723844 .724149 .724454 .724760 .725065 .725370 .725674 5.10 6.10 6.09 5.09 5.09 5.09 5.09 5.08 5.08 5.08 0.277379 .277073 .276768 .276462 .276156 .275851 .275546 .275240 .274935 .274630 .274326 10 9 8 7 6 6 4 3 2 1 I M Cosine. D. 1". Sine. D 1". Cotang. D.I'. T*ng. M. 69Q 268 TABLE XV. LOGARITHMIC SINES, 880 1BJ M. Bice. D. 1". Cosine. D. 1". Tang. D. 1". Cotacg. M. 1 2 3 4 5 6 7 8 9 9.671609 .671847 .672084 .672321 .672558 .672795 .673032 .673268 .673505 .673741 3.96 3.96 3.95 3.95 3.95 3.94 3.94 3.94 3.94 3 93 9.945935 .945868 .945800 .945733 .945666 .945598 .945531 .945464 .945396 .945328 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.13 1.13 1.13 9.725674 .725979 .726284 .726588 .726892 .727197 .727501 .727805 .728109 .728412 6.08 5.08 5.07 5.07 6.07 6.07 5.07 5.06 6.06 5.06 0.274326 .274021 .273716 .273412 .273108 .272803 .272499 .272195 .271891 .271588 60 59 58 57 56 5f 54 53 52 51 10 11 12 13 14 16 16 17 18 19 9.673977 .674213 .674448 .674684 .674919 .675155 .675390 .675624 .675859 .676094 3.93 3.93 3.93 3.92 3.92 3.92 3.91 3.91 3.91 3 91 9.945261 .945193 .945125 .945058 .944990 944922 .944854 944786 .944718 .944650 1.13 1.13 1.13 1.13 ,.13 1.13 1.13 1.13 1.13 1.13 9.728716 .729020 729323 729626 729929 730233 730535 730838 731141 731444 5.06 5.06 5.05 5.05 5.06 5.05 5.06 5.05 5.04 5.04 0.271284 .270980 .270677 .270374 .270071 .269767 .269465 .269162 .268859 .268556 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 26 26 27 28 29 9.676328 .676562 .676796 .677030 .677264 .677498 .677731 .677964 .678197 .678430 3.90 3.90 3.90 3.90 3.89 3.89 3.89 3.88 3.88 388 9.944582 .944514 .944446 .944377 .944309 .944241 .944172 .944104 .944036 .943967 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 9.731746 .732048 .732351 .732653 .732955 .733257 .733558 .733860 .734162 .734463 5.04 5.04 6.04 5.03 6.03 6.03 6.03 5.03 5.02 5.02 0.268254 .267952 .267649 .267347 .267045 .266743 .266442 .266140 .265838 .265537 40 39 38 37 36 36 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.678663 .678895 .679128 .679360 .679592 .679824 .680056 .680288 .680519 .680750 3.88 3.87 3.87 3.87 3.87 3.86 386 3.86 3.86 385 9.943899 .943830 .943761 .943693 .943624 .943555 .943486 .943417 .943348 .943279 1.14 1.14 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1 16 9.734764 .735066 .735367 .735668 .735969 .736269 .736570 .736870 .737171 .737471 6.02 6.02 5.02 5.01 6.01 6.01 5.01 6.01 6.01 6.00 0.265236 .264934 .264633 .264332 .264031 .263731 .263430 .263130 .262829 .262529 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 18 49 9.680982 .681213 .681443 .681674 .681905 682135 .682365 .682595 .682825 .683055 3.85 3.85 3.84 3.84 3.84 3.84 3.83 3.83 3.83 383 9.943210 .943141 .943072 .943003 .942934 .942864 .942795 .942726 .942656 .942587 1.15 1.16 1.15 1.15 1.15 1.16 1.16 1.16 1.16 1 16 9.737771 .738071 .738371 .738671 .738971 .739271 .739570 .739870 .740169 .740468 5.00 5.00 5.00 6.00 4.99 4.99 4.99 4.99 4.99 4.98 0.262229 .261929 .261629 .261329 .261029 .260729 .260430 .260130 .259831 .25953, 20 19 18 17 J6 15 14 13 12 11 60 61 52 53 64 65 66 67 63 59 60 9.683284 .683514 .633743 .683972 .684201 .684430 .684658 .684887 .685115 .685343 .685571 3.82 3.82 3.82 3.82 3.81 3.81 3.81 3.80 3.80 3.80 9.942517 .942448 .942378 .942308 .942239 .942169 .942099 .942029 .941959 .941889 .941819 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.17 1.17 1.17 9.740767 .741066 .741365 .741664 .741962 .742261 .742559 .742858 .743156 .743454 .743752 4.98 4.98 4.98 4.98 4.98 4.97 4.97 4.97 4.97 4.97 0.259233 .258934 .258635 .258336 .258038 .257739 .257441 .257142 .256844 .256546 .256248 a 8 7 6 5 4 3 2 1 M. Goedue. D 1". Slue. D. 1". Cotang. D.I". Tang. M. 118 COSINES, TANGENTS, AND COTANGENTS. 269 M Sine. D. 1". Cosine. D. 1". Tang. D. 1". Gotang. M. 1 2 3 4 5 6 7 8 9 9.685571 685799 .686027 686254 .686482 .686709 .686936 687163 .687389 .687616 3.80 3.79 3.79 3.79 3.79 3.78 3.78 3.78 3.78 3.77 9.941819 .941749 .941679 .941609 .941539 .941469 .941398 .941328 .941258 .941187 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 9.743752 .744050 .744348 .744645 .744943 .745240 .745538 .745835 .746132 .746429 4.96 4.96 4.96 4.96 4.96 4.96 4.95 4.95 4.95 4.95 0.256248 .255950 .255652 .255355 .255057 .254760 .254462 .254165 .253863 .253571 60 69 68 57 68 55 64 | 53 62 61 1C 11 12 13 14 15 16 17 18 19 9.687843 .688069 .688295 .688521 .688747 .688972 .689198 .689423 .689648 .689873 3.77 3.77 3.77 3.76 3.76 3.76 3.76 3.75 3.75 3.75 9.941117 .941046 .940975 .940905 .940834 .940763 .940693 .940622 .940551 .940480 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 9.746726 .747023 .747319 .747616 .747913 .748209 .748505 .748801 .749097 .749393 4.95 4.95 4.94 4.94 4.94 4.94 4.94 4.93 4.93 4.93 0.253274 .252977 .252681 .252384 .252087 .251791 .251495 .251199 .250903 .250607 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.690098 .690323 .690548 .690772 .690996 .691220 ,691444 .691668 .691892 .692115 3.75 3.74 3.74 3.74 3.74 3.73 3.73 3.73 3.73 372 9.940409 .940338 .940267 .940196 .940125 .940054 .939982 .939911 .939840 .939768 1.18 1.18 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 9.749689 .749985 .750281 .750576 .750872 .751167 .751462 .751757 .752052 .752347 4.93 4.93 4.93 4.92 4.92 4.92 4.92 4.92 4.92 4.91 0.250311 .250015 .249719 .249424 .249128 .248833 .248538 .248243 .247948 .247653 40 39 38 37 3d 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.692339 .692562 .692785 .693008 .693231 .693453 .693676 .693898 .694120 .694342 3.72 3.72 3.72 3.71 3.71 3.71 3.71 . 3.70 3.70 3.70 S.939697 .939625 .939554 .939482 .939410 .939339 .939267 .939195 .939123 .939052 1.19 1.19 1.19 1.19 1.19 1.20 1.20 1.20 1.20 1.20 9.752642 .752937 .753231 .753526 .753820 .754115 .754409 .754703 .754997 .755291 4.91 4.91 4.91 4.91 4.91 4.90 4.90 4.90 4.90 4.90 0.247368 .247063 .246769 .246474 .246180 .245885 .245591 .245297 .245003 .244709 30 29 28 27 26 25 24 23 22 21 40 41 9.694564 .694786 3.70 9.933980 .938908 1.20 9.755585 .755878 4.89 0.244415 .244122 20 19 42 43 44 45 46 47 48 49 .695007 .695229 .695450 .695671 .695892 .696113 .696334 .696554 3.69 3.69 3.69 3.69 3.68 3.68 3.68 3.68 3.67 .938836 .938763 .938691 .933619 .933547 .938475 ,938402 .938330 1.20 1.20 1.20 1.20 1.20 1.20 1.21 1.21 1.21 .756172 .756465 .756759 .757052 .757345 .757638 .757931 .758224 4.89 4.89 4.89 4.89 4.89 4.88 4.88 4.88 4.88 .243828 .243535 .243241 .242948 .242655 .242362 .242069 .241776 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 59 60 9.696775 .696995 .697215 .697435 .607654 .697874 .698094 .698313 .698532 .698751 .698970 3.67 3.67 3.67 3.66 3.66 3.66 3.66 3.65 3.65 3.65 9.938258 .938185 .938113 ,938040 .937967 .937895 .937822 .937749 .937676 .937604 .937531 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.22 9.758517 .758810 .759102 .759395 .759687 .759979 .760272 .760564 .760856 .761148 .761439 4.83 4.88 4.87 4.87 4.87 4.87 4.87 4.87 4.86 4.86 0.241483 .241190 .240893 .240605 .240313 .240021 .239728 .239436 .239144 .238852 .238561 10 9 8 7 6 5 4 3 2 1 M. Coelne. D. 1". Sine. D. 1". Cotang. D. l'. Tnng. M. 600 270 TABLE XV. LOGARITHMIC SINES, 800 149' M. Slae. D.I*. Cofdne. D. 1". Ttag. D.l. Cottmg. M 7 8 9 9.698970 .699189 .099407 .699626 .699844 .700062 .700280 .700498 .700716 .700933 3.65 3.64 3.64 3.64 3.64 3.63 3.63 3.63 3.63 3.62 9.937531 .937458 .937385 .937312 .937238 .937166 .937092 .937019 .936946 .936872 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 9.761439 .761731 .762023 .762314 .762606 .762897 .763188 .763479 .763770 .764061 4.86 4.86 4.86 4.86 4.86 4.85 4.85 4.85 4.85 4.86 0.238561 .238269 .237977 .237686 .237394 .237103 .236312 .236521 .236230 .235939 60 69 68 57 66 55 54 63 62 61 10 11 12 13 14 15 16 17 18 19 9.7C1151 .701368 .701585 .701802 .702019 .702236 .702452 .702669 .702885 .703101 3.62 3.62 3.62 3.61 3.61 3.61 3.61 3.60 3.60 3.60 9.936799 .936725 .936652 .936578 .936505 .936431 .936357 .936284 .936210 .936136 1.22 1.23 1.23 1.23 1.23 1.23 1.23 1.23 1.23 1.23 9.764352 .764643 .764933 .765224 .765514 .765805 .766095 .766385 .766675 766965 4.85 4.84 4.84 4.84 4.84 4.84 4.84 4.83 4.83 4.83 0.235648 .235357 .235067 .234776 .234486 .234195 .233905 .233615 .233325 .233035 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.703317 .703533 .703749 .703964 .704179 .704395 .704610 .704825 .705040 .705254 3.60 3.59 3.59 3.59 3.59 3.59 3.58 3.58 3.58 3.58 9.936062 .935988 .935914 .935840 .935766 .935692 .935618 .935543 .935469 .935395 1.23 1.23 1.23 1.23 1.24 1.24 1.24 1.24 1.24 1.24 9.767255 .767646 .767834 .768124 .768414 .768703 .768992 .769281 .769571 .769860 4.83 4.83 4.83 4.82 4.82 4.82 4.82 4.82 4.82 4.82 0.232745 .232455 .232166 .231876 .231586 .231297 .231008 .230719 .230429 .230140 40 39 38 37 36 36 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.705469 .705683 .705898 .706112 .706326 .706539 .706753 .706967 .707180 .707393 3.67 3.67 3.67 3.57 3.56 3.56 3.56 3.56 3.55 3.55 9.935320 .935246 .935171 .935097 .935022 .934948 .934873 .934798 .934723 .934649 1.24 1.24 1.24 1.24 1.24 1.24 1.25 1.25 1.25 1.26 9.770148 .770437 .770726 .771016 .771303 .771692 .771880 .772168 .772457 .772745 4.81 4.81 4.81 4.81 4.81 4.81 4.80 4.80 4.80 4.80 0.229852 .229563 .229274 .228985 .228697 .228408 .228120 .227832 .227543 .227255 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9707606 .707819 .708032 .708245 .708458 .708670 .708882 .709094 .709306 .709518 3.55 3.55 3.54 3.54 3.54 3.54 3.54 3.53 3.53 3.53 9.934574 .934499 .934424 .934349 .934274 .934199 .934123 .934048 .933973 .933898 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.26 1.26 9.773033 .773321 .773608 .773896 .774184 .774471 .774759 .775046 .775333 .775621 4.80 4.80 4.80 4.79 4.79 4.79 4.79 4.79 4.79 4.78 0.226967 .226679 .226392 .226104 .225816 .225529 .225241 .224954 .224667 .224379 20 19 18 17 16 15 14 13 12 11 50 51 52 9.709730 .709941 .710153 3.53 3.52 9.933822 .933747 .933671 1.26 1.26 1 OA 9.775908 .776195 .776482 4.78 4.78 A 7Q 0224092 .223805 .223518 10 9 8 53 54 55 56 57 58 59 60 .710364 .710575 .710786 .710997 .711208 .711419 .711629 .711839 3.52 3.62 3.51 3.51 3.51 3.61 3.51 .933596 933520 933445 933369 933293 933217 .933141 .933066 1.26 1.26 1.26 1.26 1.26 1.26 1.26 .776768 .777055 .777342 .777628 .777915 .778201 .778488 .778774 4.78 4.78 4.78 4.77 4.77 4.77 4.77 .223232 .222945 .222658 .222372 .222085 .221799 .221512 .221226 7 6 6 4 3 2 1 M. ,: rr Ooelne. j D.1*. Sh*. D.l Cotang. D.l" Ttof. M. LJKP COSINES, TANGENTS, AND COTANGENTS. 271 10 1480 M. Bine. D. 1 Coeine. D. 1". f*ng. D. 1". Gotang M. 1 3 4 6 6 7 8 9 9.711839 .712060 .712260 .712469 .712679 .712889 .713098 .713308 .713517 .713726 3.50 3.60 3.60 3.60 3.49 3.49 3.49 3.49 3.48 3.48 9.933066 932990 .932914 932838 .932762 932685 932609 .932533 .932457 932380 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 9.778774 .779060 .779346 .779632 .779918 .780203 .780489 .780775 .781060 .781346 4.77 4.77 4.77 4.76 4.76 4.76 4.76 4.76 4.76 4 76 0.221226 .220940 .220654 .220368 .220082 .219797 .219511 .219225 .218940 .218654 60 69 68 57 56 56 64 53 52 51 10 11 12 13 14 15 16 17 18 19 9.713935 .714144 .714352 .714561 .714769 .714978 .715186 .715394 .715602 .716809 3.48 3.43 3.48 3.47 3.47 3.47 3.47 3.46 3.46 3.46 9.932304 .932228 .932151 .932075 .931993 .931921 .931845 .931768 .931691 931614 1.27 1.27 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 9.781631 .781916 .782201 .782486 .782771 .783056 .783341 .783626 .783910 .784195 4.76 4.76 4.76 4.75 4.75 4.75 4.75 4.74 4.74 4 74 0.218369 218034 .217799 .217514 .217229 .216944 .216659 .216374 .216090 .216806 60 49 48 47 40 46 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.716017 .716224 .716432 .716639 716846 .717053 .717259 .717466 .717673 .717879 3.46 3.46 3.45 3.45 3.45 3.45 3.44 3.44 3.44 3.44 9.931537 .931460 931383 .931306 .931229 .931 152 931075 .930998 .930921 .930843 1.23 1.28 1.23 1.28 1.29 1.29 1.29 1.29 1.29 1.29 9784479 784764 785048 .785332 .785616 785900 .786184 .786463 .786762 .787036 4.74 4.74 4.74 4.74 4.73 4.73 4.73 4.73 4.73 4 73 0.215521 .215236 .214952 .214668 .214384 .214100 213816 .213532 .213248 .212964 40 39 38 37 36 36 34 33 32 31 30 31 32 33 34 36 36 37 38 39 9.718086 .718291 .718497 .718703 .718909 .719114 .719320 .719525 .719730 .719935 3.43 3.43 3.43 3.43 3.43 3.42 3.42 3.42 342 3.41 9.930766 .930688 930611 .930533 .930456 .930378 .930300 .930223 .930145 .930067 1.29 1.29 1.29 1.29 1.29 1.29 1.30 1.30 1.30 1 30 9.787319 .787603 .787880 .788170 .788453 .788736 .789019 .789302 .789585 .789868 4.73 4.72 4.72 4.72 4.72 4.72 4.72 4.72 4.71 4 71 0.212681 .212397 .212114 .211830 .211647 211264 .210981 .210698 .210416 ,210132 30 29 28 27 26 26 24 23 22 21 40 41 42 43 44 46 46 47 48 49 9.720140 .720345 .720549 .720754 .720958 .721162 .721366 .721570 .721774 .721978 3.41 3.41 3.41 3.41 3.40 3.40 3.40 3.40 3.39 3.39 9.929989 .929911 .929833 .929755 .929677 .929599 929521 .929442 .929364 .929286 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.31 1.31 1 31 9.790151 .790434 .790716 .790999 .791281 .791563 .791846 .792128 .792410 .792692 4.71 4.71 4.71 4.71 4.71 4.70 4.70 4.70 4.70 4 ffl 0.209849 .209566 .209284 .209001 .208719 .208437 .208154 .207872 .207590 .207308 20 19 18 17 16 16 14 13 12 11 6C 61 62 63 64 65 66 57 68 69 60 9.722181 .722385 .722588 .722791 .722994 .723197 .723400 723603 .723805 .724007 .724210 3.39 3.39 3.39 3.38 3.38 3.38 3.38 3.37 3.37 3.27 9.929207 .929129 .929050 .928972 .928893 .928815 .928736 928657 .928578 .928499 .928420 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.32 9.792974 .793256 .793538 .793819 .794101 794383 794664 794946 795227 .795508 .795789 4.70 4.70 4.70 4.69 4.69 4.69 4.69 4.69 4.69 469 0.207026 .206744 .206462 .206181 .205899 .205617 .205336 .205054 .204773 .204492 .204211 10 9 8 7 6 5 4 3 2 M. 1 Oorfue. | D. i". Oil* D 1". Gotaug. D.iw. Tang M 080 272 TABLE XV. LOGARITHMIC SINES, fto 14T M. Sine. D. I". Cosine. D. 1". Tang D. 1". Cotang. M. I 8 4 6 6 7 8 9 9.724210 .724412 .724614 .724816 .725017 .725219 .726420 .725622 .725823 .726024 3.37 3.37 3.36 3.36 3.36 3.36 3.36 3.35 3.35 3.35 9.928420 .928342 .928263 .928183 .928.04 .928025 .927946 .927867 .927787 .927708 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 9.795789 .796070 .796351 .796632 .796913 .797194 .797474 .797755 .798036 .798316 4.68 468 4.68 4.68 4.68 4.68 4.68 4.68 4.67 4.67 0.2042H .203930 .203649 .203368 .203087 .202806 .202526 .202245 .201964 .201684 60 59 58 57 56 55 54 53 52 61 10 11 12 13 14 16 16 17 18 19 9.726225 .726426 .726620 .726827 .727027 .727228 .727428 .727628 .727828 .728027 3.35 3.34 3.34 3.34 3.34 3.34 3.33 3.33 3.33 3.33 9.927629 .927549 .927470 .927390 .927310 .927231 .927151 .927071 .926991 .926911 1.32 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 9.798596 .798877 .799157 .799437 .799717 .799997 .800277 .800557 .800836 .801116 4.67 4.67 4.67 4.67 4.67 4.66 4.66 4.66 4.66 4.66 0.201404 .201123 .200843 .200563 .200283 .200003 .199723 .199443 .199164 .198884 60 49 48 47 46 45 44 43 42 41 20 9.728227 q qq 9.926831 1 !W 9.801396 4 RR 0.198604 40 21 22 .728427 .728626 3.32 q qg .926751 .926671 1.33 Iqq 801675 .801955 4.66 A RR .198325 .198045 39 38 23 24 26 26 27 28 29 728825 .729024 .729223 .729422 .729621 .729820 .730018 3.32 3.32 3.31 3.31 3.31 3.31 3.31 .926591 .926511 .926431 .926351 .926270 .926190 .926110 1.34 1.34 1.34 1.34 1.34 1.34 1.34 .802234 802513 .802792 .803072 .803351 .803630 .803909 4.65 4.65 4.65 4.65 4.65 4.65 466 .197766 .197487 .197208 .196928 .196649 .196370 .196091 37 36 36 34 33 32 31 80 31 82 83 84 36 86 37 88 89 9.730217 .730416 .730613 .730811 .731009 .731206 .731404 .731602 .731799 .731998 3.30 3.30 3.30 3.30 3.30 3.29 3.29 3.29 3.29 3.28 9.926029 .925949 925868 .925788 .925707 .925626 .925545 .925465 .925384 .925303 1.34 1.34 1.34 1.34 1.35 1.35 1.35 1.35 1.35 1.35 9.804187 804466 .804745 805023 805302 .805580 .805859 .806137 .806415 .a)6693 4.65 4.64 4.64 4.64 4.64 1.64 4.64 4.64 4.64 4.63 0.195813 .195534 .195255 .194977 .194698 .194420 .194141 .193863 .193586 .193307 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 46 46 47 48 49 9.732193 .732390 .732587 .732784 .732980 .733177 .733373 .733569 .733765 .733961 3.28 3.28 3.28 3.28 3.27 3.27 3.27 3.27 3.27 3.26 9.925222 .925141 .925060 .924979 .924897 .924816 .924735 .924654 .924572 .924491 1.36 1.35 1.35 1.35 1.35 1.35 1.36 1.36 1.36 1.36 y 806971 .807249 ,807527 807805 .808083 .808361 .808638 .808916 .809193 .809471 4.63 4.63 4.63 4.63 4.63 4.63 4.G3 4.G2 4.62 4.62 0.193029 .192761 .192473 .192195 .191917 .191639 .191363 .19IOW4 .190807 .190529 20 19 18 17 16 16 14 13 12 11 60 61 9.734157 .734353 3.26 q OR 9.924409 .924328 1.36 IqR 9.809748 .810025 4.62 A RO 0.190252 .189975 10 9 62 63 64 66 66 67 .734549 .734744 .734939 .735135 .735330 .735525 3.26 3.26 3.25 3.25 3.25 .924246 .924164 .924083 .924001 .923919 .923837 1.36 1.36 1.36 1.36 1.36 .810302 .810580 .810857 811134 .811410 .811687 4.C2 4.62 4.62 4.61 4.61 .189698 .189420 .189143 .188866 .188590 .188313 8 7 6 6 4 8 68 69 60 .735719 .735914 .736109 3.25 3.25 3.24 923755 .923673 .923591 1.37 1.37 1.37 .811964 .812241 ,812517 4.61 4.61 .188036 187769 .187483 2 1 M. Cosine D. 1". * D.I" Ootaug D.I" Ttag M. COSINES, TANGENTS, AND COTANGENTS. 33 273 14O M. Sine. D. 1". Cosine. D. 1". Tang. D. I". Cotang. M. 1 2 3- 4 5 6 7 8 9 9.736109 .736303 .736498 .736692 .736886 .737080 .737274 .737467 .737661 .737855 3.24 3.24 3.24 3.23 3.23 3.23 3.23 3.23 3.22 3.22 9.923591 .923509 .923427 .923345 .923263 .923181 .923098 .923016 .922933 .922851 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.38 9.812517 .812794 .813070 .813347 .813623 .813899 .814176 .814452 .814728 .815004 4.61 4.61 4.61 4.61 4.60 4.60 4.60 4.60 4.60 4.60 0.187483 .187206 .1869:30 .186653 .186377 .186101 .185824 .185548 .185272 .184996 60 59 58 57 56 55 54 53 52 51 10 11 12 13 9.738048 .738241 .738434 .738627 3.22 3.22 3.22 391 9.922768 .922686 .922603 .922520 1.38 1.38 1.38 9.815280 .815555 .815831 .816107 4.60 4.60 4.59 A tQ 0.184720 .184445 .184169 .183893 50 49 48 47 14 15 16 17 .738820 .739013 .739206 .739398 3.21 3.21 3.21 .922438 .922355 .9222/2 .922189 1.38 1.38 1.38 .816382 .816658 .816933 .817209 4.59 4.59 4.59 .183618 .183342 .183067 .182791 46 45 44 43 18 19 .739590 .7397&3 3.20 3.20 .922106 .922023 1.38 1.38 .817484 .817759 4.59 4.59 .182516 .182241 42 41 20 21 22 23 9.739975 .740167 .740359 .740550 3.20 3.20 3.20 9.921940 .921857 .921774 .921691 1.39 1.39 1.39 9.818035 .818310 .818585 .818880 4.59 4.58 4.58 0.181965 .181690 .181415 .181140 40 39 38 37 24 25 26 27 28 .740742 .740934 .741125 .741316 .741508 3.19 3.19 3.19 3.19 3.19 .921607 .921524 .921441 .921357 .921274 1.39 1.39 1.39 1.39 .8191&5 .819410 .819684 .819959 .820234 4.58 4.58 4.58 4.58 4.58 .180865 .180590 .180316 .180041 .179766 36 35 34 33 32 29 .741699 3.18 .921190 1.39 .820508 4.58 .179492 31 30 31 9.741889 .742080 3.18 9.921107 .921023 1.39 9.820783 .821057 4-57 0.179217 .178943 30 29 32 .742271 31 A .920939 1 40 .821332 A K7 .178668 28 33 34 35 36 37 38 39 .742402 .742652 .742842 .743033 .743223 .743413 .743602 3.17 3.17 3.17 3.17 3.17 3.16 3.16 .920856 .920772 .920688 .920604 .920520 .920436 .920352 1.40 1.40 1.40 1.40 1.40 1.40 1.40 .821606 .821880 .822154 .822429 .822703 .822977 .823251 4.57 4.57 4.57 4.57 4.57 4.57 4.56 .178394 .178120 .177846 .177571 .177297 .177023 .176749 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.743792 .743982 .744171 .744361 .744550 .744739 .744928 .745117 .745306 .745494 3.16 3.16 3.16 3.15 3.15 3.15 3.15 3.15 3.14 3.14 9.920268 .920184 .920099 .920015 .919931 .919846 .919762 .919677 .919593 .919508 1.40 1.40 1.40 1.41 1.41 1.41 1.41 1.41 1.41 1.41 9.823524 .823798 .824072 .824345 .824619 .824893 .825166 .825439 .825713 .825986 4.56 4.56 4.56 4.56 4.56 4.56 4.56 4.56 4.55 4.55 0.176476 .176202 .175928 .175655 .175381 .175107 .174834 .174561 .174287 .174014 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 59 9.745683 .745871 .746060 .746248 .746436 .746624 .746812 .746999 .747187 .747374 3.14 3.14 3.14 3.13 3.13 3.13 3.13 3.13 3.12 9.919424 .919339 .919254 .919169 .919085 .919000 .918915 .918830 .918745 .918659 1.41 1.41 1.41 1.41 1.42 1.42 1.42 1.42 1.42 9.826259 .826532 .826805 .827078 .827351 .827624 .827897 .828170 .828442 .828715 4.55 4.55 4.55 4.55 4.55 4.55 4.55 4.54 4.54 0.173741 .173468 .173195 .172922 .172649 .172376 .172103 .171830 .171558 .171285 10 9 8 7 6 5 4 3 2 1 60 .747562 .918574 .828987 .171013 M. Cosine. D. I". Sine. D. I". Cotang. D. I". Tang. M. 133 06' 274 TABLE XV. LOGARITHMIC SINES, 340 1 M. Sine D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 3 4 5 6 7 8 9 9.747562 .747749 .747936 .748123 .748310 .748497 .748683 .748870 .749058 .749243 3.12 3.12 3.12 3.11 3.11 3.11 3.11 3.11 3.10 3 10 9.918574 .918489 .918404 .918318 .918233 .918147 .918062 .917976 .917891 .917805 1.42 1.42 1.42 1.42 1.42 143 1.43 1.43 1.43 1.43 9.828987 .829260 .829532 .829805 .830077 .830349 .830621 .830893 ,831165 .831437 4.54 4.54 4.54 4.54 4.54 4.54 4.53 4.53 4.63 4.53 0.171013 .170740 .170468 .170195 .169923 .169651 .169379 .169107 .168835 .168563 60 59 58 57 56 65 54 53 52 51 10 11 12 13 14 15 16 17 18 19 9.749429 .749615 .749801 .749987 .760172 .760358 .750543 .750729 .750914 .751099 3.10 3.10 3.10 3.10 3.09 3.09 3.09 3.09 3.09 308 9.917719 .917634 917548 .917462 .917376 .917290 917204 .917118 917032 .916946 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.44 1.44 1 44 9.831709 .831981 .832253 .832525 .832796 .833068 .833339 .833611 .833882 .834154 4.53 4.63 4.53 4.53 4.53 4.53 4.52 4.52 4.52 4.52 0.168291 .168019 .167747 .167476 .167204 .166932 .166661 .166389 .166118 .166846 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 26 26 27 23 29 9.751284 .761469 .751654 .751839 .752023 .752208 .752392 .752576 752760 .752944 3.08 3.08 3.08 3.08 3.07 3.07 3.07 3.07 3.07 3 06 9.916859 .916773 .916687 .916600 .916514 .916427 .916341 916254 916167 .916081 1.44 1.44 1.44 1.44 ft 1.44 1.44 1.45 1 45 9.834425 .834696 .834967 .835238 .835509 .835780 .836051 .836322 .836593 .836864 4.62 4.52 4.62 4.52 4.52 4.52 4.61 4.51 4.61 4.51 0.165575 .165304 .165033 .164762 .164491 .164220 .163949 .163678 .163407 .163136 40 39 38 37 36 36 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.753128 .753312 .753495 .753679 .753862 .754046 .754229 .754412 .754595 .754778 3.06 3.06 3.06 3.06 3.05 3.05 3.05 3.05 3.05 305 9.915994 .915907 .915820 .915733 .915646 .915559 .915472 .915385 .915297 .915210 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1 46 9.837134 .837405 .837676 .837946 .838216 .838487 .838757 .839027 .839297 .839568 4.51 4.51 4.51 4.51 4.51 4.51 4.60 4.50 4.60 4.60 0.162860 .162595 .162325 .162054 .161784 .161513 .161243 .160973 .160703 .160432 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.754960 .755143 .755326 .755508 .755690 .755872 .756054 .756236 .756418 .756600 3.04 3.04 3.04 3.04 3.04 3.03 3.03 3.03 3.03 3 03 9.915123 .915035 .914948 914860 .914773 .914685 .914598 .914510 .914422 .914334 1.46 1.46 1.46 1.46 1.46 1.46 1.46 1.46 1.46 1 46 9.839838 .840108 .840378 .840648 .840917 .841187 .841457 .841727 .841996 .842266 4.50 4.60 4.60 4.60 4.50 4.49 4.49 4.49 4.49 4.49 0.160162 .159892 .159622 .159352 .159083 .158813 .158543 .158273 .158004 .157734 20 19 18 17 16 15 14 13 12 11 60 61 62 63 64 66 66 57 68 69 60 9.756782 .756963 .757144 .757326 .757507 .757688 .757869 .758050 .758230 .758411 .758591 3.02 3.02 3.02 3.02 3.02 3.02 3.01 3.01 3.01 3.01 9.914246 .914158 .914070 .913982 .913894 .913806 .913718 913630 .913541 .913453 .913365 1.47 1.47 1.47 1.47 1.47 1.47 1.47 1.47 1.47 1.47 9.842535 .842805 .843074 .843343 .843612 .843882 .844151 .844420 .844689 .844958 .845227 4.49 4.49 4.49 4.49 4.49 4.49 4.48 4.48 4.48 4.48 0.157465 .157195 .156920 .156657 .156*88 .156118 .155849 .155580 .155311 .155042 .154773 10 9 8 7 6 5 4 3 1 1 M. Coeirve. D. 1". Bice. D.1" Cotang D.I*. Tang. fit 55 850 COSINES, TANGENTS, AND COTANGENTS. M Sine. D.I* Cosine D.I' Tang. D. 1". Cotang M < 4 B e 8 9 9.758591 .758772 .768952 .759132 .769312 .759492 .769672 .759862 .760031 .760211 3.01 3.00 3.00 3.00 3.00 3.00 2.99 2.99 2.99 2.99 9.91336 .9132? .91318 .91309 .91301 .912922 .912833 .912744 .912655 .912566 1.47 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 9.84522 .84549 .845764 .84603 .846302 .84657 .84683 .847108 .847376 .847644 4.48 4.48 4.48 4.48 4.48 4.48 4.48 4.47 4.47 4 47 0.15477 .154504 .154236 .15396 .153698 153430 .15316 .152892 .152624 .152356 60 81 67 56 66 64 63 52 51 10 11 12 13 14 16 16 17 18 19 9.760390 .760569 .760748 .760927 .761106 .761285 .761464 .761642 .761821 .761999 2.99 2.99 2.98 2.98 2.98 2.98 2.98 2.97 2.97 2.97 9.912477 .912388 .912299 .912210 .912121 .912031 .911942 .911853 .911763 .911674 1.48 1.48 1.49 1.49 1.49 1.49 1.49 1.49 1.49 1.49 9.847913 .84818 .848449 .848717 848986 .849254 .849522 .849790 .850057 .850325 4.47 4.47 4.47 4.47 447 4.47 4.47 4.46 4.46 446 0.152087 .161819 .151651 .151283 .151014 .160746 .150478 .150210 .149943 .149676 50 49 48 47 46 45 44 43 42 41 40 21 22 23 24 26 26 27 28 29 9.7C2177 .762356 762534 .762712 .762889 .763067 .763245 .763422 .763600 .763777 2.97 2.97 2.97 2.96 2.96 2.96 2.96 2.96 2.95 2.96 9.911584 .911495 .911405 .911315 911226 911136 911046 910956 .910866 .910776 1.49 1.49 1.49 1.60 1.60 1.60 1.60 1.60 1.60 1.60 9.850593 .850861 851129 .851396 .851664 .851931 .862199 .852466 852733 .853001 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 446 0.149407 .149139 148871 148604 .148336 .148069 147801 147534 .1472*7 146999 40 39 38 37 36 36 34 33 32 31 30 31 32 33 84 36 36 37 38 3d 9.763964 .764131 764308 764485 764662 .764838 .765015 .765191 .765367 .765544 2.95 2.95 2.95 2.95 2.94 2.94 2.94 2.94 2.94 2.P3 9.910686 910596 910506 910416 910325 .910235 910144 .910054 909963 909873 1.60 1.60 1.60 1.51 1.61 1.61 1.61 1.51 1.61 1.51 9.853268 .853535 .853802 .854069 .854336 854603 .854870 .855137 .855404 .866671 4.46 4.46 4.45 4.45 4.45 4.45 4.45 4.45 4.45 444 0.146732 .146465 .146198 .145931 -.145664 .145397 .145130 144863 144596 .144329 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 46 46 47 48 49 9.765720 .765896 .766072 .766247 .766423 .766598 .766774 .766949 .767124 .767300 2.93 2.93 2.93 2.93 2.93 2.92 2.92 2.92 2.92 2.92 9.909782 .909691 .909601 .909510 909419 909328 .909237 .909146 .909055 .908964 1.51 1.61 1.61 1.61 1.62 1.52 1.62 1 62 1.62 1.62 9.855938 .856204 .856471 .856737 .857004 .857270 .857537 .857803 .858069 .868336 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4 44 0.144062 .143796 .143529 .143263 .142996 .142730 .142463 .142197 .141931 .141664 20 19 18 17 16 15 14 3 2 60 61 62 63 64 66 66 67 68 69 60 9.767476 .767649 .767824 .767999 .768173 .768348 .768522 .768897 .768871 .769045 .769219 2.91 2.91 2.91 2.91 2.91 2.91 2.90 2.90 2.90 2.90 9.908873 .908781 .908690 .908599 .908507 .908416 .908324 .908233 .908141 .908049 .907958 1.52 1.52 1.52 1.52 1.52 1.53 1.53 1.53 1.53 1.63 .868602 .858868 .859134 .859400 .859666 .859932 860198 860464 860730 860995 861261 4.44 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 0.141398 .141132 .140866 .140600 .140334 .140068 .139802 .139536 .139270 .139005 .138739 9 8 7 6 5 4 3 2 1 M. Ooeine. D.P. Bine. .1". Cofang. D. 1". Tang. 6*0 TABLE XV. LOGARITHMIC SINES, M. Bine. D. 1". Godne. D. 1". Tang. D. 1". Ootang. M 1 3 6 6 8 9 9.769219 .769393 .769566 .769740 .769913 .770087 .770260 .770433 .770606 .770779 2.90 2.90 2.89 2.89 2.89 2.89 2.89 2.88 2.88 288 .907958 .907866 .907774 .907682 .907590 .907498 .907406 .907314 .907222 .907129 1.53 1.53 1.53 1.53 1.53 1.53 1.54 1.54 1.54 1.54 9.861261 .861527 .861792 .862058 .862323 .862589 .862854 .863119 .863385 .863650 4.43 4.43 4.43 4.42 4.42 4.42 4.42 4.42 4.42 4.42 0.138739 .138473 .138208 .137942 .137677 .137411 .137146 .136881 .136615 .136350 60 59 58 57 56 56 64 63 5* 51 10 11 12 13 14 16 16 17 18 19 9.770952 .771125 .771298 .771470 .771643 .771815 .771987 .772159 .772331 .772503 2.88 2.88 2.88 2.87 2.87 2.87 2.87 2.87 2.87 2 86 9.907037 .906945 .906852 .906760 .906667 .906575 .906482 .906389 .906296 .906204 1.54 1.54 1.54 1.54 1.54 1.64 1.55 1.55 1.55 155 9.863915 .864180 .864445 .864710 .864975 .865240 .865505 .865770 .866035 .866300 4.42 4.42 4.42 4.42 4.42 4.41 4.41 4.41 4.41 4.41 0.136085 .135820 .135555 .135290 .135025 .134760 .134495 .134230 .133965 .133700 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.772675 .772847 .773018 .773190 .773361 .773533 .773704 .7738T5 .774046 .774217 2.86 2.86 2.86 2.86 2.85 2.85 2.85 2.85 2.85 285 9.906111 .906018 .905925 .905832 .905739 .905645 .905552 .905459 .905366 .905272 1.55 1.55 1.55 1.65 1.55 1.55 1.55 1.56 1.56 1.56 9.866564 .866829 .867094 .867358 .867623 .867887 .868152 .868416 .868680 .868945 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.40 4.40 0.133430 .133171 .132906 .132642 .132377 .132113 .131848 .131584 .131320 .131056 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.774388 .774558 .774729 .774899 .775070 775240 .775410 .775680 .775750 .775920 2.84 2.84 2.84 2.84 2.84 2.84 2.83 2.83 2.83 283 9.905179 .905085 .904992 .904898 ,904804 .904711 .904617 .904523 .904429 .904335 1.56 1.56 1.56 1.56 1.56 1.56 1.66 1.67 1.57 1 67 9.869209 .869473 .869737 .870001 .870265 .870529 .870793 .871057 .871321 .871685 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 0.130791 .130527 .130263 .129999 .129735 .129471 .129207 .128943 .128679 .128415 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.776090 .776259 .776429 .776598 .776768 .776937 .777106 .777275 .777444 .777613 2.83 2.83 2.82 2.82 2.82 2.82 2.82 2.82 2.81 281 9.904241 .904147 .904053 .903959 .903864 .903770 .903676 ,903581 .903487 .903392 1.57 1.57 1.67 1.57 157 1.57 1.57 1.67 1.58 1.58 9.871849 .872112 .872376 .872640 .872903 .873167 .873430 .873694 .873957 .874220 4.40 4.39 4.39 4.39 4.39 4.39 4.39 4.39 4.39 4.39 0.128151 .127888 .127624 .127360 .127097 .126833 .126570 .126306 .126043 .125780 20 19 18 17 16 15 14 13 12 11 50 61 52 53 54 65 66 67 68 69 9.777781 .777950 .778119 .778287 .778455 .778624 .778792 .778960 .779128 779295 2.81 2.81 2.81 2.81 2.80 2.80 2.80 2.80 2.80 9.903298 .903203 .903108 .903014 .902919 .902824 ,902729 .902634 .902539 ,902444 1.68 1.58 1.58 1.58 1.58 1.58 1.58 1.58 1.59 ICQ 9.874484 .874747 .875010 .875273 .875537 .875800 .876063 .876326 .876589 .876852 4.39 4.39 4.39 4.39 4.38 4.38 4.38 4.33 4.38 4 38 0.125516 .125253 .124990 .124727 .124463 .124200 .123937 .123674 .123411 .123148 10 9 8 7 6 6 4 3 1 I 60 .779463 2.79 .902349 .877114 .122886 M Ocxdna. D.I" Sine. D.I' Ootang. D. 1" Tang. M. I860 COSINES, TANGENTS, AND COTANGENTS. 277 M. Sine. D.1*. Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 9.779463 .779631 .779798 2.79 2.79 9.902349 .902253 .902158 1.59 1.59 9.877114 .877377 .877640 4.38 4.38 0.122886 .122623 .122360 60 69 68 3 4 .779966 .780133 2.79 2 79 .902063 .901967 1.59 .877903 .878165 4.38 4.33 .122097 .121835 57 66 6 6 7 8 9 .780300 .780487 .780634 .780801 .780968 2.78 2.78 2.78 2.78 2.78 .901872 .901776 .901681 .901585 .901490 1.59 1.59 1.59 1.59 1.60 .878428 .878691 .878953 .879216 .879478 4.38 4.38 4.38 4.37 4.37 .121572 .121309 .121047 .120784 .120522 66 64 63 52 61 10 9.781134 9 7ft 9.901394 fin 9.879741 0.120259 50 11 12 13 14 15 16 .781301 .781468 .781634 .781800 .781966 .782132 2.77 2.77 2.77 2.77 2.77 9 77 .901298 .901202 .901106 .901010 .900914 .900818 1.60 1.60 1.60 1.60 1.60 i An .880003 .880265 .880528 .880790 .881052 881314 4.37 4.37 4.37 4.37 4.37 .119997 .119735 .119472 .119210 .118948 .118686 49 48 47 46 45 44 17 18 .782298 .782464 2.76 9 7fi .900722 .900626 1.60 i fin .881577 .881839 4.37 .118423 .118161 43 42 19 .782630 2.76 .900529 1.61 .882101 4.37 .117899 41 20 21 22 23 24 25 26 27 28 29 9.782796 .782961 .783127 .783292 .783458 .783623 .783788 .783953 .784118 .784282 2.76 2.76 2.76 2.75 2.75 2.75 2.75 2.75 2.75 2.74 9.900433 .900337 .900240 .900144 .900047 .899951 .899854 .899767 .899660 .899564 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.62 9.882363 .882625 .882887 .883148 .883410 .883672 .883934 .884196 .884457 .884719 4.37 4.37 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 0.117637 .117376 .117113 .116852 .116590 .116328 .116066 .115804 .115543 .115281 40 39 38 37 36 36 34 33 32 31 30 31 32 33 34 35 9.784447 .784612 .784776 .784941 .785105 .785269 2.74 2.74 2.74 2.74 2.74 9 7Q 9.899467 .899370 .899273 .899176 .899078 .898981 1.62 1.62 1.62 1.62 1.62 9.884980 .885242 .885504 .885765 .886026 .886288 4.36 4.36 4.36 4.36 4.36 0.115020 .114758 .114496 .114235 .113974 .113712 30 29 28 27 20 25 36 37 .785433 .785597 2.73 9 74 .898884 .898787 1.62 .886549 .886811 4.36 4.36 .113451 .113189 24 23 38 39 .785761 .785925 2.73 2.73 .898689 .898592 1.62 1.62 .887072 .887333 4.35 4.35 4.35 .112928 .112667 22 21 40 41 42 43 44 45 43 47 48 49 9786089 .786252 .786416 .786579 .786742 .786906 .787069 .787232 .787395 .787557 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 9.898494 .898397 .898299 .898202 .898104 .898006 .897908 .897810 .897712 .897614 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 9.887594 .887855 .888116 .888378 .888639 .888900 .889161 .889421 .889682 .889943 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 0.112406 .112145 .111884 .111622 .111361 .111100 .110839 .110579 .110318 .110057 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 57 58 69 60 9.787720 .787883 .788045 .788208 .788370 .788532 .788694 .788856 789018 .789180 .789342 2.71 2.71 2.71 2.71 2.70 2.70 2.70 2.70 2.70 2.70 9.897516 .897418 .897320 .897222 .897123 .897025 .896926 .896828 .896729 .896631 .896532 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.64 9.890204 .890465 .890725 .890986 891247 .891507 891768 .892028 .892289 .892549 .892810 4.35 4.35 4.34 4.34 4.34 4.34 4.34 4.34 4.34 4.34 0.1097% .109535 .109275 .109014 .108753 .108493 .108232 .107972 .107711 .107451 .107190 10 9 8 7 6 5 4 3 2 M. Coelno. D.l. Sine. D. 1". Cotaiig D. 1". Tang. M. 278 TABLE XV. LOGARITHMIC SINES, 980 14 It M. Sine. D. I". Cosine. D.I". Tang. D. 1". Cotang. M. 1 2 3 4 6 6 r 8 9 9.789342 .789604 .789665 .789827 .789988 .790149 .790310 .790471 .790632 .790793 2.69 2.69 2.69 2.69 2.69 2.69 2.68 2.68 2.68 2.68 9.896532 .896433 .896a35 .896236 .896137 .896038 .895939 .895840 .895741 .895641 1.65 1.65 1.65 1.65 1.65 1.65 1.65 1.65 1.65 1.65 9.892810 .893070 .893331 .893591 .893851 .894111 .894372 .894632 .894892 .895152 4.34 4.34 4.34 4.34 4.34 4.34 4.34 4.34 4.33 4 33 0.107190 .106930 .106669 106409 106149 .105889 .105628 .105368 .105108 .104848 60 59 68 67 66 55 54 63 52 51 10 11 12 13 14 16 16 17 18 19 9790954 .791116 .791276 .791436 .791696 .791767 .791917 .792077 .792237 .792397 2.68 2.68 2.67 2.67 2.67 2.67 2.67 2.67 2.67 2.66 9.895542 .895443 .895343 .895244 .895145 .895045 .894945 .894846 .894746 .894646 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 9.895412 .895672 .895932 .896192 .896452 .896712 .896971 .897231 .897491 .897751 4.33 4.33 4.33 4.33 4.33 4.33 4.33 4.33 4.33 433 0.104588 .104328 .104068 .103808 .103541 .103288 .103029 .102769 .102509 .102249 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 26 26 27 28 29 9.792567 .792716 .792876 .793035 .793195 .793354 .793514 .793673 .793832 .793991 2.66 2.66 2.66 2.66 2.66 2.66 2.65 2.66 2.65 2.65 9.894546 .894446 .894346 .894246 .894146 .894046 .893946 .893846 .893745 .893645 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 9.898010 .898270 .898530 .898789 .899049 .899308 .899568 .899827 .900087 .900346 4.33 4.33 4.33 4.33 4.33 4.32 4.32 4.32 4.32 432 0.101990 .101730 .101470 .101211 .100951 .100692 .100432 .100173 .099913 .099664 40 39 38 37 36 36 34 33 32 31 30 31 82 33 84 36 36 37 38 39 9.794150 .794308 .794467 .794626 .794784 .794942 .795101 .795259 .795417 .795576 2.65 2.64 2.64 2.64 2.64 2.64 2.64 2.64 2.63 2.63 9.893544 .893444 .893343 .893243 .893142 .893041 .892940 .892839 .892739 .892638 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 9.900605 .900864 .901124 .901383 .901642 .901901 .902160 .902420 .902679 .902938 4.32 4.32 4.32 4.32 4.32 4.32 4.32 4.32 4.32 432 0.099395 .099136 .098876 .098617 .098358 .098099 .097840 .097580 .097321 .097062 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 46 46 47 48 49 9.795733 .795891 .796049 .796206 .796364 .796521 .796679 .796836 .796993 .797150 2.63 2.63 2.63 2.63 2.62 2.62 2.62 2.62 2.62 2.61 9.892536 .892435 .892334 .892233 .892132 .892030 .891929 .891827 .891726 .891624 1.69 1.69 1.69 1.69 1.69 1.69 1.69 1.69 1.69 1 69 9.903197 .903456 .903714 903973 .904232 .904491 .904750 .905008 .905267 .905526 4.32 4.32 4.31 4.31 4.31 4.31 4.31 4.31 4.31 4 31 0.096803 .096544 .096286 .096027 .095768 .095509 .095250 .094992 .094733 .094474 20 19 18 17 16 15 14 13 12 i. 60 61 62 63 64 66 66 67 68 69 60 9.797307 .797464 .797621 .797777 .797934 .798091 .798247 .798403 .798560 .798716 .798872 2.61 2.61 2.61 2.61 2.61 2.61 2.61 2.60 2.60 2.60 9.891523 .891421 .891319 .891217 .891115 .891013 .890911 .890809 .890707 .890605 .890503 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 9.905785 .906043 .906302 .906560 .906819 .907077 .907336 .907594 .907853 .908111 .908369 4.31 4.31 4.31 4.31 4.31 4.31 4.31 4.31 4.31 4.31 0.094215 .093957 .093698 .093440 .093181 .092923 .092664 .092406 .092147 .091889 .091631 10 9 8 7 6 6 4 3 2 M. Cosine. D.I Sine. D. 1". Cotang, D. 1". Tang. M. COSINES, TANGENTS, AND COTANGENTS. 390 If. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 3 4 6 6 7 8 9 9.798872 .799028 .799184 .799339 .799495 .799651 .799806 .799962 .800117 .800272 2.60 2.60 2.60 2.59 2.59 2.59 2.59 2.59 2.59 2.69 9.890503 .890400 .890298 .890195 .890093 .889990 .889888 .889785 .889682 .889579 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 9.908369 .908628 .908886 .909144 .909402 .909660 .909918 .910177 .910435 .910693 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 0.091631 .091372 .091114 .090856 .090598 .090340 .090082 .089823 .089565 .089307 60 59 58 57 56 55 54 53 52 51 10 ll 12 13 14 16 16 17 18 19 9.800427 .800582 .800737 .800892 .801047 .801201 .801356 .801511 .801665 .801819 2.63 2.63 2.53 2.53 2.58 2.53 2.57 2.57 2.57 2.57 9.889477 .889374 .889271 .889168 .889064 .888961 .888853 .888755 .888651 .888548 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 9.910951 .911209 .911467 .911725 .911982 .912240 .912493 .912756 .913014 .913271 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 0.089049 .088791 .088533 .088275 .088018 .087760 .087502 .087244 .086986 .036729 50 49 43 47 46 45 44 43 42 41 20 21 22 23 24 9.801973 .802128 .802282 .802436 .802589 2.57 2.57 2.57 2.56 9.888444 .888341 .888237 .888134 .888030 .73 .73 .73 .73 9.913529 .913787 .914044 .914302 .914560 4.29 4.29 4.29 4.29 0.086471 .086213 .085956 .085698 .085440 40 39 38 37 36 26 26 27 28 29 .802743 .802897 .803050 .803204 .803357 2.56 2.56 2.56 2.66 2.56 2.56 .887926 .887822 .887718 .887614 .887510 .73 .73 1.73 1.73 1.73 1.74 .914817 .915076 .915332 .915590 .915847 4.29 4.29 4.29 4.29 4.29 .085183 .084925 .084663 .084410 .084153 35 34 33 32 31 30 31 32 33 34 35 36 37 38 9.803511 .803664 .803817 .803970 .804123 .804276 .804428 .804581 .804734 2.65 2.65 2.55 2.55 2.55 2.55 2.64 2.64 9.837406 .837302 .887198 .887093 .886989 .886885 .886780 .886676 .886571 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 9.916104 .916362 .916619 .916877 .917134 .917391 .917648 .917906 .918163 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 0.083896 .083633 .083381 .083123 .082866 .032609 .082352 .082094 .081837 30 29 28 27 26 25 24 23 22 39 .804336 2.54 2.64 .886466 1.74 1.76 .918420 4.29 .081580 21 40 41 42 43 44 45 46 47 43 49 9.805039 .805191 .805343 .805495 .805647 .805799 .805951 .806103 .806254 .806406 2.54 2.64 2.54 2.53 2.63 2.53 2.53 2.53 2.53 2.52 9.886362 .886257 .886152 .886047 .885942 .885837 .885732 .885627 .885522 .885416 1.76 1.75 1.76 1.75 1.75 1.75 1.75 1.75 1.75 1.76 9.918677 .918934 .919191 .919448 .919705 .919962 .920219 .920476 .920733 .920990 4.23 4.28 4.23 4.28 4.28 4.28 4.23 4.28 4.28 4.23 0.081323 .081066 .080809 .080552 .080295 .080038 .079781 .079524 .079267 .079010 20 19 18 17 16 15 14 13 12 11 60 61 62 63 54 66 66 67 63 59 60 9.806557 .806709 .806860 .807011 .807163 .807314 .807465 .807615 .807766 .807917 808067 2.52 2.52 2.52 2.52 2.52 2.52 2.51 2.51 2.51 2.51 9.885311 .885205 .885100 .384994 .884889 .884783 .884677 .884572 .884466 .884360 .834254 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.77 1.77 9.921247 .921503 .921760 .922017 .922274 .922530 .922787 .923044 .923300 .923567 .923814 4.28 4.28 4.28 4.28 4.28 4.28 4.23 4.28 4.23 4.23 0.078753 .078497 .078240 .077983 .077726 .077470 .077213 .076956 .076700 .076443 .076186 10 9 8 7 6 5 4 3 2 1 M. Casino D. 1". Slno. D. 1". Cotang D. 1". Tang M. 380 TABLE XV. LOGARITHMIC SINES M. Sine. D. 1". Cosine. D. 1". Tang. D.I . Cotang. M. 1 2 3 4 5 6 7 8 9 9.808067 .808218 .808368 .808519 .808669 .808819 808969 .809119 .809269 .809419 2.61 2.51 2.51 2.50 2.50 2.50 2.50 2.50 2.50 2.50 9.884254 .884148 .884042 .883936 .883829 .883723 .883617 .883510 .883404 .883297 1.77 1.77 1.77 1.77 1.77 1.77 1.77 1.77 1.78 1.78 9.923814 .924070 .924327 .924583 .924840 .925096 .925352 .925609 .925865 .926122 4.28 4.28 4.27 4.27 4.27 4.27 4.27 4.27 4.27 427 0.076186 .075930 .075673 .075417 .075160 .074904 .074648 .074391 .074135 .073878 60 59 58 57 56 55 54 53 52 51 10 11 12 13 14 15 16 9.809569 .809718 .809868 .810017 .810167 .810316 .810465 2.49 2.49 2.49 2.49 2.49 2.49 9 dfi 9.883191 .883084 .882977 .882871 .882764 .882657 .882550 1.78 1.78 1.78 1.78 1.78 1.78 9.926378 .926634 .926890 .927147 .927403 .927659 .927915 4.27 4.27 4.27 4.27 4.27 4.27 0.073622 073366 .073110 .072853 .072597 .072341 .072085 50 49 48 47 46 45 44 17 18 19 .810614 .810763 .810912 2.48 2.48 2.48 .882443 .882336 .882229 1.79 1.79 1.79 .928171 .928427 .923684 4.27 4.27 4.27 4 27 .071829 .071573 .071316 43 42 41 20 21 22 23 24 25 26 27 28 29 9.811061 .811210 .811358 .811507 .811655 .811804 .811952 .812100 .812248 .812396 2.48 2.48 2.48 2.47 2.47 2.47 2.47 2.47 2.47 2.47 9.882121 .882014 .881907 .881799 .881692 .881584 .881477 .881369 .881261 .881153 1.79 1.79 1.79 1.79 1.79 1.79 1.79 1.80 1.80 1.80 9.928940 .929196 .929452 .929708 .929964 .930220 .930475 .930731 .930987 .931243 4.27 4.27 4.27 4.27 4.27 4.27 4.26 4.26 4.26 426 0.071060 .070804 .070548 .070292 .070036 .069780 .069525 .069269 .069013 .068757 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.812544 .812692 .812840 .812988 .813135 .813283 .813430 .813578 .813725 .813872 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.45 2.45 2.45 9.881046 .880938 .880830 .880722 .880613 .880505 .880397 .880289 .880180 .880072 1.80 1.80 1.80 1.80 1.80 1.80 1.81 1.81 1.81 1.81 9.931499 .931755 .932010 .932266 .932522 .932778 .933033 .933289 .933545 .933800 4.26 4.26 4.26 4.26 4.26 4.26 4.26 4.26 4.26 426 0.068501 .068245 .067990 .067734 .067478 .067222 .066967 066711 .066455 .066200 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.814019 .814166 .814313 .814460 .814607 .814753 .814900 .815046 .815193 .815339 2.45 2.45 2.45 2.45 2.44 2.44 2.44 2.44 2.44 2.44 9.879963 .879855 .879746 .879637 .879529 .879420 .879311 .879202 .879093 .878984 1.81 1.81 1.81 1.81 1.81 1.81 1.82 1.82 1.82 1.82 9.934056 .934311 .934567 .934822 .935078 .935333 .935589 .935844 .936100 .936355 4.26 4.26 4.26 4.26 4.26 4.26 4.26 4.26 4.26 426 0.065944 .065689 .065433 .065178 .064922 .064667 .06441 1 .064156 .063900 .063645 20 19 18 17 16 15 ' 14 13 12 11 50 51 52 53 54 55 66 57 68 59 60 9.815485 .815632 .815778 .815924 .816069 .816215 .816361 .816507 .816652 .816798 .816943 2.44 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.42 2.42 9.878875 .878766 .878656 .878547 .878438 .878328 .878219 .878109 .877999 .877890 .877780 1.82 1.82 1.82 1.82 1.82 1.83 1.83 1.83 1.83 1.83 9.936611 .936866 .937121 .937377 .937632 .937887 .938142 .938398 .938653 .938908 .939163 4.26 4.26 4.26 4.25 4.25 4.25 4.25 4.25 4.25 4.25 0.063389 .063134 .062879 .062623 .062368 .062113 .061858 .061602 .061347 .061092 .060837 10 9 8 7 6 6 4 3 2 1 M. Cosine D. 1". Sine D. 1". Cotang. D.I" Tang. M. 49- COSINES, TANGENTS, AND COTANGENTS. 281 M. Slue. D.1-. Cosine. D 1". Tang. D.I*. Ootang. M. 1 2 3 4 6 6 7 8 9 9.816943 .817088 .817233 .817379 .817524 .817668 .817813 .817958 .818103 .818247 2.42 2.42 2.42 2.42 2.42 2.41 2.41 2.41 2.41 2.41 9.877780 .877670 .877560 .877450 .877340 .877230 .877120 .877010 .876899 .876789 1.83 1.83 1.83 1.83 1.84 1.84 1.84 1.84 1.84 1.84 9.939163 .939418 .939673 .939928 .940183 .940439 .940694 .940949 .941204 .941459 4.26 4.26 4.26 4.25 4.25 4.25 4.26 4.25 4.26 4.26 0.060837 .060582 .060327 .060072 .059817 .059561 .059306 .059051 .058796 .058541 60 69 68 67 66 65 54 53 52 61 10 11 12 13 14 15 16 17 18 19 9.818392 .818536 .818681 .818825 .818969 .819113 .819257 .819401 819545 .819689 2.41 2.41 2.40 2.40 2.40 2.40 2.40 2.40 2.40 2.39 9.876678 .876568 .876457 .876347 .876236 .876125 .876014 .875904 .875793 .875682 1.84 1.84 1.84 1.84 1.85 1.85 1.85 1.85 1.85 1.85 9.941713 .941968 .942223 .942478 .942733 .942988 .943243 .943498 .943752 .944007 4.26 4.26 4.25 4.26 4.26 4.26 4.26 4.26 4.26 4.26 0.058287 .058032 .057777 .057522 .057267 .057012 .056757 .056502 .056248 .055993 60 49 48 47 46 46 44 43 42 41 20 9819832 9.875571 9.944262 0.055738 40 21 22 23 I 24 1 26 26 27 28 .819976 .820120 .820263 .820406 .820550 .820693 .820836 .820979 2^39 2.39 2.39 2.39 2.39 2.38 2.38 .875459 .875348 .875237 .875126 .875014 .874903 .874791 .874680 1.85 1.85 1.85 1.86 1.86 1.86 1.86 1.86 .944517 .944771 .945026 .945281 .945535 .945790 .946045 .946299 4.26 4.26 4.24 4.24 4.24 4.24 4.24 4.24 .055483 .055229 .054974 .054719 .054465 .054210 .053955 .053701 39 38 37 36 36 34 33 32 29 .821122 & 33 .874568 1.86 1.86 .946554 4.24 4.24 .053446 31 30 31 32 33 34 35 36 37 38 39 9.821265 .821407 .821550 .821693 .821835 .821977 .822120 .822262 .822404 .822546 2.38 2.38 2.38 2.37 2.37 2.37 2.37 2.37 2.37 2.37 9.874456 .874344 .874232 .874121 .874009 .873896 .873784 .873672 .873560 .873448 1.86 1.86 1.87 1.87 1.87 1.87 1.87 1.87 1.87 1.87 9.946808 .947063 .947318 .947572 .947827 .948081 .948335 .948590 .948844 .949099 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 0.053192 .052937 .052682 .052428 052173 .051919 .051665 .051410 .051156 .050901 30 29 28 27 26 26 24 23 22 21 40 41 42 43 44 45 46 47 9.822688 .822830 .822972 .823114 .823255 .823397 .823539 .823680 2.37 2.36 2.36 2.36 2.36 2.36 2.36 9 3A 9.873335 .873223 .873110 .872998 .872885 .872772 .872659 .872547 1.87 1.88 1.88 1.88 1.88 1.88 1.88 9.949353 .949608 .949862 .950116 .950371 .950625 .950879 .951133 4.24 4.24 4.24 4.24 4.24 4.24 4.24 0.060647 .050392 .050138 .049884 .049629 .049375 .049121 .048867 20 19 18 17 16 15 14 13 48 .823821 A.OO 9 Q .872434 1.88 .951388 4.24 .048612 12 49 .823963 X.0D 2.35 .872321 1.88 1.88 .951642 4.24 4.24 .048358 11 60 51 52 53 54 55 56 57 58 9.824101 .824245 .824386 .824527 .824668 .824808 .824949 .825090 .825230 2.35 2.35 2.35 2.35 2.35 2.34 2.34 2.34 9.872208 .872095 .871981 .871868 .871755 .871641 .871528 .871414 .871301 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 9.951896 .952150 .952405 .952659 .952913 .953167 .953421 .953675 953929 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.23 0.048104 .047850 .047595 .047341 .047087 .046833 .046579 .046325 .046071 10 9 8 7 6 6 4 3 2 59 .825371 2.34 .871187 1.89 954183 4.23 . 045817 1 60 .825511 2.34 .871073 1.90 .954437 4.23 .045563 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1". Tang. M. 48 282 TABLE XV. LOGARITHMIC SINES, M. Sine. D. 1''. Cosine. D. 1". Tang. D. 1". Cotang. M. 1 2 3 4 5 6 7 8 9 9.825511 .825651 .825791 .825931 .826071 .826211 .826351 .826491 .826631 .826770 2.34 2.34 2.33 2.33 2.33 2.33 2.33 2.33 2.33 2.33 9.871073 .870960 .870846 .870732 .870618 .870504 .870390 .870276 .870161 .870047 1.90 1.90 1.90 1.90 1.90 1.90 1.90 1.90 1.91 1.91 9.954437 .954691 .954946 .955200 .955454 .955708 .955961 .956215 .956469 .956723 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 0.045563 .045309 .045054 .044800 .044546 .044292 .044039 .043785 .043531 .043277 60 59 58 57 56 55 54 53 52 51 1C 11 12 13 14 15 16 17 18 19 9.826910 .827049 .827189 .827328 .827467 .827606 .827745 .827884 .828023 .828162 2.32 2.32 2.32 2.32 2.32 2.32 2.32 2.31 2.31 231 9.869933 .869818 .869704 .869589 .869474 .869360 .869245 .869130 .869015 .868900 1.91 1.91 1.91 1.91 1.91 1.91 1.91 1.92 1.92 1.92 9.956977 .957231 .957485 .957739 .957993 .958247 958500 .958754 .959008 .959262 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 0.043023 .042769 .042515 .042261 .042007 .041753 .041500 .041246 .040992 .040738 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.828301 .828439 .828578 .828716 .828855 .828993 .829131 .829269 .829407 .829545 2.31 2.31 2.31 2.31 2.31 2.30 2.30 2.30 2.30 2.30 9.868785 .868670 .868555 .868440 .868324 .868209 .808093 .867978 .867862 .867747 1.92 1.92 1.92 1.92 1.92 1.92 1.93 1.93 1.93 1.93 9.959516 .959769 .960023 .960277 .960530 .960784 .961038 .961292 .961545 .961799 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 0.040484 .040231 .039977 .039723 .039470 .039216 .038962 .038708 .038455 .038201 40 39 38 37 36 33 34 33 33 31 30 31 32 33 34 35 36 37 33 39 9.829633 .829821 .829959 .830097 ,830234 830372 .830509 .830646 .830784 .830921 2.30 2.30 2.29 2.29 2.29 2.29 2.29 2.29 2.29 229 9.867631 .867515 .867399 .867283 .867167 .867051 .866935 .866819 .866703 .866586 1.93 1.93 1.93 1.93 1.93 1.94 1.94 1.94 1 94 1.94 9.962052 .962306 .962560 .962813 .963067 .963320 .963574 .963828 .964081 .964335 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 0.037948 .037694 .037440 .037187 .036933 .036680 .036426 .036172 .035919 .035665 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.831058 .831195 .831332 .831469 .831606 .831742 .831879 .832015 .832152 .832288 2.23 2.28 2.28 2.28 2.28 2.28 2.28 2.27 2.27 227 9.86647u .866353 .866237 .866120 .866004 .865887 .865770 .865653 .865536 .865419 1.94 1.94 1.94 1 94 1.95 1.95 1.95 1.95 1.95 1.95 9.964588 .964842 .965095 .965349 .965602 .965855 .966109 .966362 .966616 .966869 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 0.035412 .035158 .034905 .034651 .034398 .034145 .033891 .033638 .033384 .033131 20 19 18 17 16 15 14 13 12 11 50 51 52 53 54 55 56 67 58 59 60 9.832425 .832561 .832697 .832833 .832969 .833105 .833241 .833377 .833512 .8a3648 .833783 2.27 2.27 2.27 2.27 2.27 2.26 2.26 2.26 2.26 2.26 9.865302 .865185 .865068 .864950 .864833 .864716 .864598 .864481 .864363 .864245 .864127 1.95 1.95 1.95 1.96 1.96 1.96 1.96 1.96 1.96 1.96 9.967123 .967376 .967629 .967883 .963136 .968389 .968643 .968896 .969149 .969403 .969656 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 0.032877 .032624 .032371 .032117 .031864 .031611 .031357 .031104 .030851 .030597 030344 10 9 S 7 6 5 4 3 2 1 M. Oosiue. D. 1". Sine. D. 1". Cotang. D.I'. Tang. M. 133 430 COSINES, TANGENTS, AND COTANGENTS. M. Sine. D. 1". Cosine. D.I" Tang. D. 1". Cotaug. M. 1 2 3 4 6 6 7 8 9 9.833783 .833919 .834054 .834189 .834325 .834460 .834595 .834730 .834865 .834999 2.26 2.26 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 9.864127 .864010 .863892 .863774 .863656 .863538 .863419 .863301 .863183 .863064 1.96 1.97 1.97 1.97 1.97 1.97 1.97 1.97 1.97 1.97 9.969656 .969909 .970162 .970416 .970669 .970922 .971175 .971429 .971682 .971935 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 422 0.030344 .030091 .029838 .029584 .029331 .029078 .028825 .028571 .028318 .028065 60 59 68 67 66 55 64 53 52 61 10 11 12 13 14 16 16 17 18 19 9.835134 .835269 .835403 .835538 .835672 .835807 .835941 .836075 .836209 .836343 2.24 2.24 2.24 2.24 2.24 2.24 2.24 2.23 2.23 2.23 9.862946 .862827 .862709 .862590 .862471 .862353 .862234 .862115 .861996 .861877 1.98 1.98 1.98 1.98 1.93 1.98 1.98 1.98 1.98 1.99 9.972188 .972441 .972695 .972948 .973201 .973454 .973707 .973960 .974213 .974466 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 422 0.027812 .027559 .027305 .027052 .026799 .026546 .026293 .026040 .025787 .025534 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 26 26 27 23 29 9.836477 .836611 .836745 .836878 .837012 .837146 .837279 .837412 .837546 .837679 2.23 2.23 2.23 2.23 2.23 2.22 2.22 2.22 2.22 2.22 9.861758 .861638 .861519 .861400 .861280 .861161 .861041 .860922 .860802 .860682 1.99 1.99 1.99 1.99 1.99 1.99 1.99 2.00 2.00 2.00 9.974720 .974973 .975226 .975479 .975732 .975985 .976238 .976491 .976744 .976997 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 422 0.026280 .025027 .024774 .024521 .024268 .024015 .023762 .023509 .023256 .023003 40 39 38 37 36 36 34 33 32 31 30 31 32 33 34 36 36 37 33 39 9.837812 .837945 .838078 .838211 .838344 .838477 .838610 .838742 .838875 .839007 2.22 2.22 2.22 2.21 2.21 2.21 2.21 2.21 2.21 2.21 9.860562 .860442 .860322 .860202 .860082 859962 .859842 .859721 .859601 .859480 2.00 2.00 2.00 2.00 2.00 2.00 2.01 2.01 2.01 201 9.977250 .977503 .977756 .978009 .978262 .978515 .978768 .979021 .979274 .979527 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 422 0.022760 .022497 .022244 .021991 .021738 .021485 .021232 .020979 .020726 .020473 30 29 28 27 26 26 24 23 22 21 40 41 42 43 44 46 46 47 43 49 9.839140 .839272 .839404 .839536 .839668 .839800 .839932 .840064 .840196 .84C328 2.21 2.20 2.20 2.20 2.20 2.20 2.20 2.20 2.19 2.19 9.859360 .859239 .859119 .858998 .858877 .858756 .858635 .858514 .858393 .858272 2.01 2.01 2.01 2.01 2.02 2.02 2.02 2.02 2.02 2.02 9.979780 .980033 .980286 .980538 .980791 .981044 .981297 .981550 .981803 .982056 4.22 4.22 4.22 4.22 4.22 4.21 4.21 4.21 4.21 4.21 0.020220 .019967 .019714 .019462 .019209 .018956 .018703 .018450 .018197 .017944 20 19 18 17 16 15 14 13 12 11 60 61 62 63 64 66 66 67 63 69 60 9.840469 .840591 .840722 .840854 .840985 .841116 .841247 .841378 .841509 .841640 .841771 2.19 2.19 2.19 '* 19 2.19 2.19 2.18 2.18 2.18 2.18 9.858151 .858029 .857908 .857786 .857665 .857543 .857422 .857300 .857178 .857056 .856934 2.02 2.02 2.02 2.03 2.03 2.03 2.03 2.03 2.03 2.03 9.982309 .982562 .982814 .983067 .983320 .933573 .983826 .984079 .984332 .984584 .984837 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.017691 .017438 .017186 .016933 .016680 .016427 .016174 .01592! .015668 .015416 .015163 10 9 8 7 6 6 4 3 2 1 M. Oodne. D. 1". Sloe. D. 1". Cotaug. D.I". Tang. M. 284 TABLE XV. LOGARITHMIC SINES, 440 138" M Slue. D. 1". Cosine D. 1". Tang. D. 1". Cotang. M. 2 3 6 6 7 8 9 9.841771 .841902 .842033 .842163 .842294 .842424 .842555 .842685 .842815 .842946 2.18 2.18 2.18 2.18 2.17 2.17 2.17 2.17 2.17 2.17 9.856934 .856312 .856690 .856568 .856446 .856323 .856201 .856078 .855956 .855833 2.03 2.04 2.04 2.04 2.04 2.04 2.04 2.04 2.04 2.04 9.984837 .985090 .985343 .985596 .985848 .986101 .986354 .986607. .986860 .987112 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.015163 .014910 .014657 .014404 .014152 .013899 .013646 .013393 .013140 .012888 60 69 68 67 56 55 54 53 52 51 10 11 12 13 14 16 16 17 18 19 9.843076 .843206 .843336 .843468 .843595 .843725 .843855 .843984 .844114 .844243 2.17 2.17 2.16 2.16 2.16 2.16 2.16 2.16 2.16 2.16 9.855711 .855588 .855465 .855342 .855219 .855096 .854973 .854850 .854727 .854603 2.05 2.05 2.05 2.05 2.05 2.05 2.05 2.05 2.06 2.06 9.987365 .987618 .987871 .988123 .988376 .988629 .988882 .989134 .989387 .989640 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.012635 .012382 .012129 .01 1877 .011624 .011371 .011118 .010866 .010613 .010360 50 19 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 23 29 9.844372 .844502 .844631 .844760 .844889 .845018 .845147 .845276 .845405 .845533 2.16 2.15 2.15 2.15 2.15 2.15 2.16 2.15 2.14 2.14 9.854480 .854356 .854233 .854109 .853986 .853862 .853738 .853614 .853490 .853366 2.06 2.06 2.06 2.06 2.06 2.06 2.06 2.07 2.07 2.07 9.989893 .990145 .990398 .990651 .990903 .991156 .991409 991662 .991914 .992167 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.010107 .009855 .009602 .009349 .009097 .008844 .008591 .008338 .008086 .007833 40 89 38 37 36 36 34 33 82 31 30 31 32 33 34 35 36 37 33 39 9.845662 .845790 .845919 .846047 .846175 .846304 .846432 .846560 .846688 .846816 2.14 2.14 2.14 2.14 2.14 2.14 2.13 2.13 2.13 2.13 9.853242 .853118 .852994 .852869 .852745 .852620 .852496 .852371 .852247 .852122 2.07 2.07 2.07 2.07 2.07 2.08 2.08 2.08 2.08 2.08 9.992420 .992672 .992925 .993178 .993431 .993683 .993936 .994189 .994441 .994694 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.007580 .007328 .007076 .006822 .006569 .006317 .006064 .005811 .005559 .005306 30 29 23 27 26 26 24 23 22 21 40 41 42 43 44 45 46 47 43 49 9.846944 .847071 .847199 .847327 .847454 .847582 .847709 .847836 .847964 .848091 2.13 2.13 2.13 2.13 2.12 2.12 2.12 2.12 2.12 2.12 9.851997 .851872 .851747 .851622 .851497 .851372 .851246 .851121 .850996 .850870 2.08 2.08 2.08 2.09 2.09 2.09 2.09 2.09 2.09 2.09 9.994947 .995199 .995452 .995705 .995957 .996210 .996463 .996715 .996968 .997221 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.006053 .004801 .004548 .004295 .004043 .003790 .003537 .003285 .003032 .002779 20 19 18 17 16 16 14 13 12 11 50 61 52 63 64 55 66 67 68 59 60 9.848218 .848345 .848472 .848599 .848726 .848852 .848979 .849106 .849232 .849359 .849485 2.12 2.12 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 9.850745 .850619 .850493 .850368 .850242 .850116 .849990 .849864 .849738 .849611 .849485 2.09 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.11 9.997473 .997726 .997979 .998231 .998484 .998737 .998989 .999242 .999495 .999747 0.000000 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 0.002527 .002274 .002021 .001769 .001516 .001253 .001011 .000758 .000505 .000253 .000000 10 9 8 6 6 4 3 2 1 M. Ooelne. D. 1". Sine. D.F. Cotang. D. 1". Tang. M. TABLE XVI. NATURAL SINES AND COSINES. 286 TABLE XYI. NATUKAL SINES AND COSINES. ~ 00 10 30 30 40 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Codn. M. .00000 One. .01745 .99985 .03490 .99939 .05234 .99863 .06976 .99756 1 .00029 One. .01774 .99984 .03519 .99938 .05263 .99861 .07005 .99754 53 2 .00058 One. .01803 .99984 .03548 .99937 .05292 .99860 .07034 .99752 58 3 .00087 One. .01832 .99983 .03577 .99936 05321 .99858 .07063 .99750 37 4 .00116 One. .01862 .99983 .03606 .99935 .05350 .99857 .07092 .99748 56 5 .00145 One. .01891 .99982 .03635 .99934 .05379 .99855 .07121 .99746 56 6 .00175 One. .01920 .99932 .C8664 .99933 .05408 .99854 .07150 .99744 54 7 .00204 One. .01949 .99981 .03693 .99932 .05437 .99852 .07179 .99742 53 8 .00233 One. .01978 .99980 .03723 .99931 .05466 .99851 .07208 .99740 52 9 .00262 One. .02007 .99980 .03752 .99930 .05495 .99849 .07237 .99738 51 10 .00291 One. .02036 .99979 .03781 .99929 .05524 .99847 .07266 .99736 50 11 .00320 .99999 .02065 .99979 .03810 .99927 .05553 .99846 .07295 .99734 49 12 .00349 .99999 .02094 .99978 .03839 .99926 .05582 .99844 .07324 .99731 48 13 .00378 .99999 .02123 .99977 .03368 .99925 .05611 .99842 .07353 .99729 47 14 .00407 .99999 .02152 .99977 .03897 .99924 .05640 .99841 .07382 .99727 46 15 .00436 .99999 .02181 .99976 .03926 .99923 .05669 .99839 .07411 .99725 45 16 .00465 .99999 .02211 .99976 .03955 .99922 .05698 .99838 .07440 .99723 44 17 .00495 .99999 .0224'> ,rJ975 .03984 .99921 .05727 .99836 .07469 .99721 43 18 .00524 .99999 .02269 .99974 .04013 .99919 .05756 .99834 .07498 .99719 42 19 .00553 .99998 .02298 .99974 .04042 .99918 .05785 .99833 .07527 .99716 41 20 .00582 .99998 .02327 .99973 .04071 .99917 .05814 .99831 .07556 .99714 40 21 .00611 .99998 .02356 .99972 .04100 .99916 .05844 .99829 .07585 .99712 39 22 .00640 .99998 .02385 .99972 .04129 .99915 .05873 .99827 .07614 .99710 38 23 00669 .99993 .02414 .99971 .04159 .99913 .05902 .99826 .07643 .99708 37 24 .00698 99998 .02443 .99970 .04188 .99912 .05931 .99824 .07672 .99705 36 25 .00727 .99997 .02472 .99969 .04217 .99911 .05960 .99822 .07701 .99703 35 26 .00756 .99997 .02501 .99969 .04246 .99910 .05989 .99821 .07730 .99701 34 27 .00785 .99997 .02530 .99968 .04275 .99909 .06018 .99819 .07759 .99699 33 28 .00814 .999 3 7 .02560 .99967 .04304 99907 .06047 .99817 .07788 .99696 32 29 .00844 ,9 ( jyy6 .02589 .99966 .04333 .99906 .06076 .99815 .07817 .99694 31 30 .00873 .90996 .02618 .99966 .04362 .99905 .06105 .99813 .07846 99692 30 31 .00902 .99996 .02647 .99965 .04391 .99904 .06134 .99812 .07875 .9968$ 29 32 .00931 .99996 .02676 .99964 .04420 .99902 .06163 .99810 .07904 .99687 28 33 .00960 .99995 .02705 .99963 .04449 .99901 .06192 .99808 .07933 .99685 27 34 .00989 .99995 .02734 .99963 .04478 .99900 .06221 .99806 .07962 .99683 26 35 .01018 .99995 .02763 .99962 .04507 .99898 .06250 .99804 .07991 .99680 25 36 .01047 .99995 .02792 .99961 .04536 .99897 .06279 .99803 .03020 .99678 24 37 .01076 .99994 .02821 .99960 .04565 .99896 .06308 .99801 .08049 .99676 23 38 .01105 .99994 .02350 .99959 .04594 .99894 .06337 .99799 .08078 .99673 22 39 .01134 .99994 .02879 .99959 .04623 .99893 .06366 .99797 .08107 .99671 21 40 .01164 .99993 .02908 .99958 .04653 .99892 .06395 .99795 .08136 .99668 20 41 .01193 .99993 .02938 .99957 .04682 .99890 .06424 .99793 .08165 .99666 19 42 .01222 .99993 .02967 .99956 .04711 .99889 .06453 .99792 .08194 .99664 18 43 .01251 .99992 .02996 .99955 .04740 .99388 .06482 .99790 .08223 .99661 17 44 .01280 .99992 .03025 .99954 .04769 .99886 06511 .99788 .08252 .99659 16 45 .01309 .99991 .03054 .99953 .04798 .99885 .06540 .99786 .08281 99657 15 46 .01338 .99991 03083 .99952 .04827 .99883 .06569 .99784 .08310 .99654 14 47 .01367 .99991 .03112 .99952 .04356 .99882 .06598 .99782 .08330 .99652 13 48 .01396 .99990 .03141 .99951 .04885 .99881 .06627 .99780 .03363 .99649 12 49 .01425 .99990 .03170 .99950 .04914 .99879 .06656 .99778 .08397 .99647 11 50 .01454 .99939 .03199 .99949 .04943 .99378 .06685 .99776 .08426 .99644 10 51 .01483 .99939 .03223 -99948 .04972 .99876 .06714 .99774 .08455 .99642 9 52 .01513 .99989 .03257 .99947 .05001 .99875 .06743 .99772 .08484 .99639 8 53 .01542 .99988 03236 .99946 .05030 .99873 .06773 .99770 .08513 .99637 7 54 .01571 .99938 .03316 .99945 .05059 .99872 .06802 .99768 .08542 .99635 6 55 .01600 .99987 03345 .99944 .05088 .99870 .06831 .99766 .03571 .99632 5 56 .01629 .99937 .03374 .99943 .05117 .99869 .06360 .99764 .08600 .99630 4 57 .01658 .99986 .03403 .99942 .05146 .99867 .06389 .99762 .08629 .99627 3 58 .01637 .99936 .03432 .99941 .05175 .99866 .06918 .99760 .08658 .99625 2 59 .01716 .99933 .03461 .99940 .05205 .99364 .06947 .99758 .08687 .99622 1 60 .01745 .99935 .03490 .99939 .05234 .99363 .06976 .99756 .08716 .99619 M. Cosin. Slue Cosin. Sine. Cosln. Sine. Cosin. Sine. Coflln. Sine. M. 890 880 87 860 850 TABLE XVI. NATURAL SINES AND COSINES. 287 50 GO 70 80 90 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. .08716 .99619 .10453 .99452 .12187 .99255 .13917 .99027 .15642 .98769 60 1 .08745 .99617 .10482 .99419 .12216 .99251 .13946 .99023 .15672 .98764 59 2 .08774 .99614 .10511 .99446 .12245 .99248 .13975 .99019 .15701 .93760 58 3 .08803 .99612 .10540 .99443 .12274 .99244 .14004 .99015 .15730 .98755 57 4 .08831 .99609 .10569 .99440 .12302 .99240 .14033 .99011 .15758 .98751 59 5 .08860 .99607 .10597 .99437 .12331 .99237 .14061 .99006 .15787 .98746 56 6 .08889 .99604 .10626 .99434 .12360 .99233 .14090 .99002 .15816 .98741 54 7 .08918 .99602 .10655 .99431 .12389 .99230 .14119 ,98993 .15845 .98737 53 8 .08947 .99599 .10684 .99428 .12418 .99226 .14148 .93994 .15873 .98732 52 9 .08976 .99596 .10713 .99424 .12447 '.99222 .14177 .98990 .15902 .98728 51 10 .09005 .99594 .10742 .99421 .12476 .99219 .14205 .98986 .15931 .98723 50 11 .09034 .99591 .10771 .99418 .12504 .99215 .14234 .98932 .15959 .98718 49 12 .09063 .99583 .10800 .99415 .12533 .99211 .14263 .98978 .15988 .98714 48 13 .09092 .99586 .10329 .99412 .12562 .99203 .14292 .98973 .16017 .98709 47 14 .09121 .99583 .10858 .99409 .12591 .99204 .14320 .98969 .16046 .98704 46 15 .09150 .99580 .10387 .99406 .12620 .99200 .14349 .98965 .16074 .98700 45 16 .09179 .99578 .10916 .99402 .12649 .99197 .14378 .98961 .16103 .98695 44 17 .09208 .99575 .10945 .99399 .12678 .99193 .14407 .98957 .16132 .98690 43 18 .09237 .99572 .10973 .99396 .12706 .99189 .14436 .98953 .16160 .98686 42 19 .09266 .99570 .11002 .99393 .12735 .99186 .14464 .98948 .16189 .98631 41 20 .09295 .99567 .11031 .99390 .12764 .99182 .14493 .98944 .16218 .98676 40 21 .09324 .99564 .11060 .993S6 .12793 .99178 .14522 .98940 .16246 .98671 39 22 .09353 .99562 .11089 .99383 .12822 .99175 .14551 .98936 .16275 .98667 38 23 .09382 99559 .11118 .99330 .12851 .99171 .14580 .98931 .16304 .98662 37 24 .09411 .99556 .11147 .99377 .12880 .99167 .14608 .98927 .16333 .98667 36 25 .09440 .99553 .11176 .99374 .12903 .99163 .14637 .98923 .16361 .98652 35 26 .09469 .99551 .11205 .99370 .12937 .99160 . 14666 .98919 .16390 .98648 34 27 .09498 .99548 .11234 .99367 .12966 .99156 .14695 .98914 .16419 .98643 33 28 .09527 .99545 .11263 .99364 .12995 .99152 . 14723 .98910 .16447 .98638 32 39 .09556 .99542 .11291 .99360 .13024 .99148 .14752 .98906 .16476 .93633 31 30 .09585 .99540 .11320 .99357 .13053 .99144 .14781 .98902 .16505 .98629 30 31 .09614 .99537 .11349 .99354 .13081 .99141 .14810 .98897 .16533 .98624 29 32 .09642 .99534 .11378 .99351 .13110 .99137 .14838 .98893 .16562 .98619 28 33 .09671 .99531 .11407 .99347 .13139 .99133 .14867 .98889 .16591 .98614 27 34 .09700 .99528 .11436 .99344 .13163 .99129 .14896 .98884 .16620 .98609 26 35 .09729 .99526 .11465 .99341 .13197 .99125 .14925 .98330 .16648 .98604 25 36 .09758 .99523 .11494 .99337 .13226 .99122 .14954 .98876 .16677 .98600 24 37 .09787 .99520 .11523 .99334 .13254 .99118 .14982 .98871 .16706 .98595 23 38 .09816 .99517 .11552 .99331 .13283 .99114 .15011 .98867 .16734 .98590 22 39 .09845 .99514 .11580 .99327 .13312 .99110 .15040 .98863 .16763 .98585 21 40 .09874 .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 .99317 .13399 .99098 .15126 .98849 16849 .98570 18 43 .09961 .99503 .11696 .99314 .13427 .99094 .15155 .98845 .16878 .98565 17 44 .09990 .99500 .11725 .99310 .13456 .99091 .15184 .98841 .16906 .98561 16 45 .10019 .99497 .11754 .99307 .13485 .99087 .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 48 .10106 .99438 .11840 .99297 .13572 .99075 .15299 .98823 .17021 .98541 12 49 .10135 .99485 .11869 .99293 .13600 .99071 .15327 .98818 .17050 .98536 11 50 .10164 .99482 .11893 .99290 .13629 .99067 .15356 .98814 .17078 .98531 10 51 .10192 .99479 .11927 .99286 .13658 .99063 .15385 .98809 .17107 .98526 9 52 .10221 .99476 .11956 .99283 .13687 .99059 .15414 .98805 .17136 .98521 8 53 .10250 .99473 .11985 .99279 .13716 .99055 .15442 .98800 .17164 .98516 7 54 .10279 .99470 .12014 .99276 .13744 .99051 .15471 .98796 .17193 .98511 6 55 .10308 .99467 .12043 .99272 .13773 .99047 15500 .98791 .17222 .98506 5 56 .10337 .99464 .12071 .99269 .13802 .99043 .15529 .93787 .17250 .98501 4 57 .10366 .99461 .12100 .99265 .13331 .99039 .15557 .98782 .17279 .98496 3 58 .10395 .99458 .12129 .99262 .13860 .99035 .15586 .98778 .17308 .98491 2 59 .10424 .99455 .12158 .99258 .13889 .99031 .15615 .98773 .17336 .98486 1 60 .10453 .99452 .12187 .99255 .13917 .99027 .15643 .98769 .17365 .98481 M7 Cosln. Sine. CoainJ Sine. CoHtn. Sine. Cosin. Sine. Cosln. Sine. M. 8*0 83 83 810 800 288 TABLE XVI. NATURAL SINES AND COSINES. 100 , 110 130 13 14 M. Sine. Cosin. Sine. Cosin. Sine, i Cosin. Sine. Cosin. Sine. Corfn. M. ! 17365 .98481 .19081 98163 .20791 .97815 "32495 .97437 .24192 .97030 60 17393 .98476 .19109 98157 .20820 .97809 .22523 .97430 .24220 97023 59 2 17422 .98471 .19138 98152 .20848 .97803 .22552 .97424 .24249 .97015 58 3 17451 .98466 .19167 .98146 .20877 .97797 .22580 .97417 .24277 .97008 57 4 17479 .98461 .19195 .98140 .20905 .97791 .22608 .97411 .24305 .97001 56 5 17508 .98455 .19224 .98135 .20933 .97784 .22637 .97404 .24333 .96994 55 6 17537 .98450 . 19252 93129 .20962 .97778 .22665 .97398 .24362 .96987 54 I 7 17565 .98445 .19281 .98124 .20990 .97772 .22693 .97391 .24390 .96980 53 8 17594 .98440 .19309 .98118 .21019 .97766 .22722 .97384 .24418 .96973 52 9 17623 .98435 .19338 .98112 .21047 .97760 .22750 .97378 .24446 .96966 51 10 17651 .98430 .19366 .98107 .21076 .97754 .22778 .97371 .24474 .96959 50 11 17680 .98425 .19395 .98101 .21104 .97748 .22807 .97365 .24503 .96952 49 12 .17708 .98420 .19423 .98096 .21132 .97742 .22835 .97358 .24531 .96945 48 1 13 17737 .98414 .19452 .98090 .21161 .97735 .22863 .97351 .24559 .96937 47 1 14 .17766 .98409 .19481 .98084 .21189 .97729 .22892 97345 .24587 .96930 46 1 15 .17794 .98404 .19509 .98079 .21218 .97723 .22920 .97338 .24615 .96923 45 l 16 .17823 .98399 .19538 .98073 .2i246 .97717 .22948 .97331 .24644 .96916 44 17 .17852 .98394 .19566 .98067 .21275 .97711 .22977 .97325 .24672 .96909 43 18 .17880 .98389 .19595 .98061 .21303 .97705 .23005 .97318 .24700 .96902 42 19 .17909 .98383 .19623 .98056 .21331 .97698 23033 .97311 .24728 .96894 41 20 .17937 .98378 19652 .98050 .21360 .97692 .23062 .97304 .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 .98362 .19737 .98033 .21445 .97673 .23146 .97284 .24841 .96866 37 24 .18052 .98357 .19766 .98027 .21474 .97667 .23176 .97278 .24869 .96858 36 25 .18081 .98352 .19794 .98021 .21502 .97661 .23203 .97271 .24897 .96851 35 26 .18109 .98347 .19823 .98016 .21530 .97655 .23231 .97264 .24925 .96844 34 27 .18133 .98341 .19851 .98010 .21559 .97648 .23260 .97257 .24954 .96837 33 23 .18166 .98336 .19880 .98004 .21587 .97642 .23288 .97251 .24982 .96829 32 29 .18195 .98331 .19908 .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 .18281 .98315 . 19994 .97981 .21701 .97617 .23401 .97223 .25094 .96800 28 33 .18309 .98310 .20022 .97975 .21729 .97611 .23429 .97217 .25122 .96793 27 34 .18338 .98304 .20051 .97969 .21738 .97604 .23458 .97210 .25151 .96786 26 35 .18367 .98299 .20079 .97963 .21786 .97598 .23486 .97203 .25179 .96778 25 36 .18395 .98294 .20108 .97958 .21814 .97592 .23514 .97196 .25207 .96771 24 37 .18424 .98288 .20136 .97952 .21843 .97585 .23542 .97189 .25235 .96764 23 38 .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 .98272 .20222 .97934 .21928 .97566 .23627 .97169 .25320 .96742 20 41 .18538 .98267 .20250 .97928 .21956 .97560 .23656 .97162 .25343 .96734 19 42 .18567 .98261 .20279 .97922 .21985 .97553 .23684 .97155 .25376 .96727 18 43 .18595 .98256 20307 .97916 .22013 .97547 .23712 .97148 .25404 .96719 17 44 .18624 .98250 .20336 .97910 .22041 .97541 .23740 .97141 .25432 .96712 16 45 .18652 .98245 20364 .97905 .22070 .97534 .23769 .97134 .25460 .96705 15 46 .18681 .98240 20393 .97899 .22098 .97528 .23797 .97127 .25488 .96697 14 47 .18710 .98234 .20421 .97893 .22126 .97521 .23825 .97120 .25516 .96690 13 48 .18738 .98229 .20450 .97887 .22155 .97515 .23853 .97113 .25545 .96682 12 49 .18767 .98223 .20478 .97881 .22183 .97508 23882 .97106 .25573 .96675 11 50 .18795 .93218 .20507 .97875 .22212 .97502 .23910 .97100 .25601 .96667 10 51 .18824 .98212 .20535 .97869 .22240 .97496 .23938 .97093 .25629 .96660 9 52 .18852 ,98207 .20563 .97863 .22268 .97489 .23966 .97086 .25657 .96653 8 53 .18881 .98201 .20592 .97857 .22297 .97483 .23995 .97079 .25685 .96645 7 54 .18910 .98196 .20620 .97851 .22325 .97476 .24023 .97072 .25713 .96638 6 55 18938 .98190 .20649 .97845 .22353 .97470 .24051 .97065 .25741 .96630 5 56 .18967 .98185 .20677 .97839 .22382 .97463 .24079 .97058 .25769 .96623 4 57 .18995 .98179 .20706 .97833 .22410 .97457 .24108 .97051 .25798 .96615 3 58 .19024 .98174 .20734 .97827 .22438 .97450 .24136 .97044 .25826 .96608 2 59 .19052 .98168 .20763 .97821 .22467 .97444 .24164 .97037 .25854 .96600 1 60 .19081 .98163 .20791 .97815 .22495 .97437 .24192 .97030 .25882 96593 M. Coein. Sine. Cosin. Sine. Cosin. Sine. Cosir . Sine. Cosin. Sine. M. 790 yso 77 o 760 7o .,.- i TABLE XVI. NATURAL SINES AND COSINES. 289 150 160 170 180 190 M Sine. Cosin Sine. Cosin Sine. Cosin Sine. Cosin. Sine. Ciwin. M. .25882 .96593 .27564 .9~6f26 .29237 .95630 .30902 .95106 .32557 .94552 60 1 .25910 .96535 .27592 .96118 .29265 .95622 .30929 .95097 .32584 .94542 59 4 .25933 .96573 .27620 .96110 .29293 .95613 .30957 .95088 .32612 .94533 58 3 .25966 .96570 .27643 .96102 .29321 95605 .30985 .95079 .32639 .94523 57 4 .25994 .96562 .27676 .96094 .29348 .95596 .31012 .95070 .32667 .94514 56 5 .26022 .96555 .27704 .96086 .29376 .95538 .31040 .95061 .32694 .94504 55 6 .26050 .96547 .27731 .96073 .29404 .95579 .31068 .95052 .32722 .94495 54 7 .26079 .96540 .27759 .96070 .29432 .95571 .31095 .95043 .32749 .94435 53 8 .26107 .96532 .27787 .96062 .29460 .95562 .31123 .95033 .32777 .94476 52 9 .26135 .96524 .27815 .96054 .29487 .95554 .31151 .95024 .32804 .94466 51 10 26163 .96517 .27843 .96046 .29515 .95545 .31178 .95015 .32332 .94457 50 11 26191 .96509 .27871 .96037 .29543 .95536 .31206 .95006 .32859 .94447 49 12 .26219 .96502 .27899 .96029 .29571 .9552S .31233 .94997 .32887 .94438 48 13 .26247 .96-194 .27927 .96021 .29599 .95519 .31261 .94988 .32914 .94428 47 14 .26275 .96436 .27955 .96013 .29626 .95511 .31289 .94979 .32942 .94418 46 15 .26303 .96479 .27983 .96005 .29654 .95502 .31316 .94970 .32969 .94409 45 16 .26331 .96471 .28011 .95997 .29682 .95493 .31344 .94961 .32997 .94399 44 17 .26359 .96463 .28039 .95939 .29710 .95485 .31372 .94952 .33024 .94390 43 13 .26337 .96456 .23067 .95981 .29737 .95476 .31399 .94943 .33051 .94380 42 19 .26415 96448 .23095 .95972 .29765 .95467 .31427 .94933 .33079 .94370 41 20 .26443 .96440 .23123 .95964 .29793 .95459 .31454 .94924 .33106 .94361 40 21 .26471 .96433 .23150 .95956 .29821 .95450 .31482 .94915 .33134 .94351 39 22 .26500 .96425 .28178 .95943 .29849 .95441 .31510 .94906 .33161 .94342 38 23 .26523 .96417 .28206 .95940 .29876 .95433 .31537 .94897 .33189 .94332 37 1 24 .26556 .96410 .28234 .95931 .29904 95424 .31565 .94888 .33216 .94322 36 25 26534 .96402 .28262 .95923 .29932 .95415 .31593 .94878 .33244 .94313 35 26 .26612 .96394 .28290 .95915 .29960 .95407 .31620 .94869 .33271 .94303 34 27 .26640 .963S6 .2-3318 .95907 .29987 .95398 .31648 .94860 .33298 .94293 33 28 .26663 .96379 .28346 .95898 .30015 .95aS9 .31675 .94851 .33326 .94284 32 29 .26696 .96371 .28374 .95890 .30043 .95380 .31703 .94842 .33353 .94274 31 30 .26724 .96363 .28402 .95882 .30071 .95372 .31730 .94832 .33381 .94264 30 31 26752 .96355 .28429 .95874 .30098 .95363 .31758 .94823 .33408 .94254 29 32 .26780 .96347 .28457 .95865 .30126 .95354 .31786 .94814 .33436 .94245 28 33 26808 .96340 .28485 .95357 .30154 .95345 .31813 .94805 .33463 .94235 27 34 26836 .96332 .28513 .95349 .30182 .95337 .31841 .94795 .33490 .94225 26 35 26864 1.96324 .28541 .95841 .30209 95323 .31868 .94786 .33518 .94215 25 36 26892 .96316 .28569 95332 .30237 95319 .31896 .94777 .33545 .94206 24 37 269201.96303 .23597 95324 .30265 95310 .31923 .94763 .33573 .94196 23 38 26948 .96301 .28625 95316 .30292 95301 .31951 .94758 .33600 .94186 22 39 26976 .96293 28652 95807 .30320 95293 .31979 94749 .33627 .94176 21 40 27004 .96235 .23680 95799 .30348 95234 .32006 .94740 .33655 .94167 20 41 27032 .96277 .28708 95791 30376 95275 .32034 94730 .33682 .94157 19 42 27060 .96269 28736 95782 30403 95266 .32061 94721 .33710 .94147 18 43 27088 .96261 28764 95774 30431 95257 .32089 94712 .33737 .94137 17 44 27116 .96253 28792 95766 30459 95248 32116 94702 .33764 .94127 16 45 27144 .96246 28820 95757 30486 95240 32144 94693 .33792 .94118 15 46 27172 .96238 28847 95749 30514 95231 32171 94684 .33819 .94108 14 47 27200 .96230 28875 95740 30542 95222 32199 94674 .33846 .94098 13 48 27228 .96222 23903 95732 30570 95213 32227 94665 .33874 .94088 12 49 27256 .96214 28931 957^ 30597 95204 32254 94656 .33901 .94078 11 50 27234 .96206 28959 95716 30625 95195 32232 94646 .33929 .94068 10 61 27312 .96193 28987 95707 30653 95136 32309 94637 .33956 .94058 9 52 27340 .96,90 29015 95693 30680 95177 32337 94627 33983 .94049 8 53 27363 .96182 29042 95690 30708 95163 32364 94618 .34011 .94039 7 54 27396 .96174 29070 95631 30736 95159 32392 94609 34038 .94029 6 55 .27424 .96166 29093 95673 30763 95150 32419 94599 34065 .94019 5 56 .27452 .96153 29126; 95664 30791 .95142 32447 94590 34093 .94009 4 57 .27480 .96150 29M4 95656 30319 .95133 32474 94530 .34120 .93999 3 58 .27503 .96142 29 1 32 .95647 30346 .95124 32502 94571 .34147 .93989 2 59 .27536 .96131 29200 .95639 30874 >.951 15 32529 94561 34175 .93979 60 .27564 96126 29237 95630 309021.95106 32557 94552 34202 .93969 M. Coda. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. : 740 730 730 | 710 700 20 290 TABLE XVI. NATURAL SINES AND COSINES. 30 310 33 330 940 1 M Sine. Cosin Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Corfn. M. .34202 .93969 .35337 .93358 .37461 .92718 .39073 .92050 .40674 .91355 60 .34229 .93959 .35864 .93348 .37488 .92707 .39100 .92039 .40700 .91343 59 2 .34257 .93949 .35391 .93337 .37515 .92697 .39127 .92028 .40727 .91331 58 3 .34284 .93939 .35913 .93327 .37542 .92686 .39153 .92016 .40753 .91319 67 4 34311 .93929 .35945 .93316 .37569 .92675 .39180 .92005 .40780 .91307 66 5 .34339 .93919 .35973 .93306 .37595 .92664 .39207 .91994 .40806 .91295 55 6 .34366 .93909 .36000 .93295 .37622 .92653 .39234 .91982 .40833 .91283 54 7 .34393 .93899, .36027 .93285 .37649 .92642 .39260 .91971 .40860 .91272 53 8 .34421 .93889 .36054 .93274 .37676 .92631 .39237 .91959 .40886 .91260 52 9 .34448 .93879 .36081 .93264 .37703 .92620 .39314 .91948 .40913 .91248 51 10 .34475 .93369 .36108 .93253 .37730 .92609 .39341 .91936 .40939 .91236 50 11 .34503 .93859 .36135 .93243 .37757 .92598 .39367 .91925 .40966 .91224 49 12 .34530 .93849 .36162 .93232 .37784 .92587 .39394 .91914 .40992 .91212 48 13 .34557 .93339 .36190 .93222 .37811 .92576 .39421 .91902 .41019 .91200 47 14 .34584 .93829 .36217 .93211 .37838 .92565 .39448 .91891 .41045 ,91188 46 15 .34612 .93819 .36244 .93201 .37865 .92554 .39474 .91879 .41072 .91176 45 16 .34639 .93309 .36271 .93190 .37892 .92543 .39501 .91868 .41098 .91164 44 17 .34666 .93799 .36298 .93180 .37919 .92532 .39528 .91856 .41125 .91152 43 18 .34694 .93789 .36325 .93169 .37946 .92521 .39555 .91845 .41151 .91140 42 ; 19 .34721 .93779 .36352 .93159 .37973 .92510 .39581 .91833 .41178 .91128 41 20 .34748 .93769 .36379 .93148 .37999 .92499 .39608 .91822 .41204 .91116 40 21 .34775 .93759 .36406 .93137 .38026 .92488 .39635 .91810 .41231 .91104 39 22 .34803 .93748 .36434 .93127 .38053 .92477 .39661 .91799 .41257 .91092 38 23 .34830 .93738 .36461 .93116 .33080 .92466 .39688 .91787 .41284 .91080 37 24 .34857 .93728 .36488 .93106 .38107 .92455 .39715 .91775 .41310 .91068 36 25 .34884 .93718 .36515 .93095 .38134 .92444 .39741 .91764 .41337 .91056 35 26 .34912 .93703 .36542 .93084 .38161 .92432 .39768 .91752 .41363 .91044 34 27 .34939 .93693 .36569 .93074 .33188 .92421 .39795 .91741 .41390 .91032 33 28 .34966 .93688 .36596 .93063 .38215 .92410 .39822 .91729 .41416 .91020 32 29 .34993 .93677 .36023 .93052 .38241 .92399 .39848 .91718 .41443 .91008 31 30 .36021 .93667 .36650 .93042 .38263 .92388 .39376 .91706 .41469 .90996 30 31 .35048 .93657 .36677 .93031 .33295 .92377 .39908 .91694 .41496 .90984 29 32 .35075 .93647 .36704 .93020 .38322 .92366 .39928 .91683 .41522 .90972 28 33 .35102 .93637 .36731 .93010 .33349 .92355 .39955 .91671 .41549 .90960 27 34 .35130 .93626 .36758 .92999 .38376 .92343 .39982 .91660 .41575 .90948 26 35 .85157 .93616 .36785 .92983 .33403 .92332 .40008 .91648 .41602 .90936 25 36 .35184 .93606 .36812 .92978 .38430 .92321 .40035 .91636 .41628 .90924 24 37 .35211 .93596 .36339 .92967 .38456 .92310 .40062 .91625 .41655 .90911 23 38 .35239 .93585 .36367 .92956 .33483 .92299 .40088 .91613 .41681 .90899 22 39 .35266 .93575 .36894 .92945 .33510 .92287 .40115 .91601 .41707 .90887 21 40 .35293 .93565 .36921 .92935 .38537 .92276 .40141 .91590 .41734 .90875 20 41 .35320 .93555 .36943 .92924 .38564 .92265 .40168 .91578 .41760 .90863 19 42 .35347 .93544 .36975 .92913 .38591 .92254 .40195 .91566 .41787 .90861 18 43 .35375 .93534 .37002 .92902 .38617 .92243 .40221 .91555 .41813 .90839 17 44 .35402 .93524 .37029 .92892 .38644 .92231 .40248 .911543 .41840 .90826 16 45 .35429 .93514 .37056 .92831 .38671 .92220 .40275 .91531 .41866 .90814 15 46 .35456 .93503 .37083 .92370 33698 .92209 .40301 .91519 .41892 .90802 14 47 .35484 .93493 .37110 .92859 .38725 .92198 .40328 .91508 .41919 .90790 13 48 .35511 .93483 .37137 .92849 38752 .92186 .40355 .91496 .41945 .90778 12 49 .35533 .93472 .37164 .92838 38778 .92175 40381 .91484 .41972 .90766 11 50 .35565 .93462 .37191 .92827 .38805 .92164 .40408 .91472 .41998 .90753 10 51 .a5592 .93452 .37218 .92816 .38832 .92152 .40434 .91461 .42024 .90741 9 52 .35619 .93441 .37245 .92805 .38859 .92141 .40461 .91449 .42051 .90729 8 53 .35647 .93431 .37272 .92794 .38886 .92130 .40488 .91437 .42077 .90717 7 54 .35674 .93420 .37299 .92784 .38912 .92119 .40514 .91425 .42104 .90704 6 55 .35701 .93410 .37326 .92773 .38939 .92107 .40541 .91414 .42130 .90692 5 56 .35728 .93400 .37353 .92762 .38966 .92096 .40567 .91402 .42156 .90680 4 57 .35755 .93339 .37380 .92751 .38993 .92085 .40594 .91390 .42183 .90668 3 53 .35782 .93379 .37407 .92740 .39020 .92073 .40621 .91378 .42209 .90655 2 59 .35810 .93368 .37434 .92729 .39046 .92062 .40647 .91366 .42235 .90643 1 60 .35837 .93358 .37461 .92718 .39073 .92050 .40674 .91355 .42262 .90631 M Cofdn. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Corfu". Sine. M. 690 680 670 660 65 TABLE XVI. NATURAL SINES AND COSINES. 291 350 360 aro 380 390 M. Sine. Costa Sine. Cosin. Sine. Costa Sine. Costa Sine Costa M. .42262 .90631 .43837 89379 .4539 .8910 .4694 .88295 .4848 .87462 60 ] .42288 .90618 .43863 .39567 .45425 .89087 ,46973 .8828 .43506 .87448 59 < .42315 .90506 .43389 .39854 .4545 .89074 .4699 .88267 .48532 .87434 58 c .42341 .90594 .43916 .89341 .45477 .89061 .47024 .88254 .48557 .87420 57 - .42367 .90532 .43942 .89823 .45503 .89048 .47050 .8824! .48583 .87406 56 5 .42394 90569 .43968 .89816 .45529 .89035 .47076 .88226 .48608 .87391 55 6 .42420 .90557 .43994 .89803 45554 .89021 .4710 .88213 .48634 S7377 54 r t 42446 .90545 .44020 .89790 45580 .89008 .47127 .88199 .48659 .87363 53 8 .42473 .90532 .44046 .89777 .45606 .88995 .47153 .88185 .48684 .87349 52 i .42499 .90520 .44072 .89764 .45632 .88981 .47178 .88172 .48710 .87335 51 10 .42525 .90507 .44098 .39752 .45658 .88968 .47204 .88158 .48735 .87321 50 11 .42552 .90495 .44124 .89739 .45684 .88955 .47229 .88144 .4876 .87306 49 12 .42578 .90483 .44151 .89726 .45710 .88942 .47255 .88130 .48786 .87292 48 13 .42604 .90470 .44177 .89713 .45736 .88928 .47281 .88117 .48811 .87278 47 14 .42631 .90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 15 .42657 .90446 .44229 .89687 .45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48888 .87235 44 17 .42709 .90421 .44231 .89662 .45839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .90408 .44307 .89649 .45865 .88862 .47409 .88048 .48938 .87207 42 19 .42762 .90396 .44333 .89636 .45891 .88848 .47434 .88034 .48964 .87193 41 20 .42788 .90383 .44359 .89623 .45917 .88835 .47460 .88020 .48989 .87178 40 21 .42815 .90371 .44335 .89610 .45942 .88822 .47486 .88006 .49014 .87164 39 , 22 .42841 .90358 .44411 .89597 .45968 .88808 .47511 .87993 .49040 .87150 38 I 23 .42867 90346 .44437 .89534 .45994 .88795 .47537 .87979 .49065 .87136 37 24 .42894 .90334 .44464 .89571 .46020 .88782 .47562 .87965 .49090 .87121 36 25 .42920 .90321 .44490 .89558 .46046 .88763 .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 .90284 .44568 .89519 .46123 88728 .47665 87909 .49192 .87064 32 1 29 .43025 .90271 .44594 .89506 .46149 88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 .44620 .89493 .46175 88701 .47716 87882 .49242 .87036 30 31 .43077 .90246 .44646 .89430 .46201 88688 .47741 87868 .49268 .87021 2V> 32 .43104 .90233 .44672 .89467 .46226 88674 47767 87854 .49293 87007 28 33 .43130 .90221 .44693 89454 .46252 88661 47793 87840 .49318 86993 27 1 34 43156 .90208 .44724 .89441 .46278 88647 47818 87826 .49344 86978 26 1 35 431821.90196 .44750 .89423 .46304 88634 47844 87812 .49369 86964 25 1 36 43209 .90183 .44776 .89415 .46330 88620 47869 87798 .49394 86949 24 < 37 43235 .90171 .44802 .89402 .46355 88607 47895 87784 .49419 86935 23 33 43261 .90158 .44828 89339 .46381 88593 47920 87770 49445 86921 22 39 43287 .90146 .44854 89376 .46407 88580 47946 87756 49470 86906 21 40 43313 .90133 .44880 89363 46433 88566 47971 87743 49495 86892 20 41 43340 .90120 .44906 89350 .46458 88553 47997 87729 49521 86878 19 42 4a366 .90108 .44932 .89337 .46484 88539 48022 87715 49546 86863 18 43 43392 .90095 .44958 89324 .46510 88526 48048 87701 49571 86849 17 44 43418 .90082 .44984 89311 .46536 88512 48073 87687 49596 86834 16 45 43445 .90070 45010 89298 .46561 88499 48099 87673 49622 86820 15 46 43471 .90057 45036 89285 .46587 88485 48124 87659 49647 86805 14 47 43497 .90045 45062 89272 .46613 88472 48150 87645 49672 86791 13 48 43523 .90032 45088 89259 46639 88458 48175 87631 49697 86777 12 49 43549 .90019 45114 89245 46664 88445 48201 87617 49723 86762 11 50 43575 .90007 45140 89232 46690 88431 48226 87603 49748 86748 Ki 51 43602 .89994 45166 89219 46716 88417 48252 87589 49773 86733 9 52 43628 89981 45192 89206 .46742 88404 48277 87575 49798 86719 8 53 43654 .89968 45218 89193 .46767 88390 48303 87561 49824 86704 7 54 43680 .89956 45243 89180 .46793 88377 48328 87546 49849 86690 6 65 43706 .89943 45269 89167 .46819 88363 48354 87532 49874 86675 5 56 43733 .89930 45295 89153 .46844 88349 48379 87518 49899 86661 4 57 43759 .89918 45321 89140 .46870 88336 48405 87504 49924 86646 3 68 43785 .89905 45347 89127 .46896 88322 48430 87490 49950 86632 2 59 43811 .89892 45373 89114 .46921 88308 48456 87476 49975 86617 I 60 43837 .89879 45399 89101 .46947 88295 48481 87462 50000 86603 sr Cofiin. Sine. Cosin. Sine. Costa. Sine. Cosin. Sine. Coeln. Sine. M 640 1 630 630 610 600 292 TABLE XVI. NATURAL SINES AND COSINES. 300 310 330 330 840 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Ootdn. M. 50000 86603 51504 .85717 .62992 .84805 .54464 .83867 .55919 .82904 60 1 .60025 .86588 51529 .85702 .63017 .84789 .54488 .83851 .55943 .82887 59 2 .50050 86573 51554 .85687 .53041 .84774 .54513 .83835 ,55968 .82871 58 3 .50076 .86559 .51679 .85672 .53066 .84759 .54537 .83319 55992 .82855 67 4 .50101 .86544 .51004 .85067 .63091 .84743 .54561 .83804 .56016 .82839 56 5 .50126 .86530 .51628 .85642 .63115 .84728 .545.S6 .83788 66040 2822 55 6 .50151 .86515 .61663 .85627 .63140 .84712 .54610 .83772 56064 .82806 64 7 .50176 .86501 .51678 .86012 .53164 .84697 .54635 .83756 ,56088 .82790 63 8 .50201 .86486 .51708 .85597 .63189 .84681 .54659 .83740 .66112 .82773 52 9 .50227 .86471 .517X8 .85682 53214 .84666 .54683 .83724 .56136 .82767 51 10 .50252 .86457 .51753 .85567 53238 .84650 .54708 .83708 .56160 .82741 50 11 .50277 .86442 .51778 .86551 53263 .84635 .54732 .83692 .56184 .82724 49 12 .50302 .86427 .51803 .85536 .53288 .84619 .54756 .83676 .56208 .82708 48 13 .50327 .86413 .51828 .85521 .53312 .84604 .54781 .83660 .56232 .82692 47 14 .50352 .86398 .51852 .86606 .53337 .84588 .54805 .83645 .66256 .82675 46 15 .60377 .86384 .51877 .85491 .53361 .84573 .54829 .83629 .66280 .82659 46 16 .50403 .86369 .51902 .85476 .53386 .8-1557 .54854 .83613 .56305 .82643 44 17 .60428 .86354 .51927 .85461 .53411 .84542 .54878 .83597 .56329 .82626 43 18 .50453 .86340 .51952 .85446 .53435 .84526 .54902 .83581 .56353 .82610 42 19 .504T8 .86325 .51977 .85431 .53460 .84511 .54927 .83565 .66377 .82593 41 20 .60503 .86310 .52002 .85416 .53484 .84495 .54951 .83549 .66401 .82677 40 21 .50528 .86295 .52026 .85401 .53509 .84480 .54975 .83533 .56425 .82561 39 22 .50553 .86281 .52051 .85385 .53534 .84464 .54999 .83517 .56449 .82544 38 23 .50578 .86266 .52076 .85370 .53558 .84418 .55024 .83501 .56473 .82528 37 24 .50603 .86251 .52101 .85355 .53583 .84433 .55048 .83485 .56497 .82511 36 25 .60628 .86237 .62126 .85340 53607 .84417 .55072 .83469 .56521 .82495 35 26 .5M54 .88222 .52151 .85325 53632 .84402 .55097 .83453 .56545 .82473 34 27 .50679 .86207 52175 .86310 .53656 .84386 .55121 .83437 .56569 .82462 S3 28 .60704 .86192 .52200 .85294 .53681 .84370 .55145 .83421 .56593 .82446 32 29 .50729 .86178 .52225 .85279 .53705 .84355 .55169 .83405 .56617 .82429 31 30 .60764 .86163 .62250 .85264 .53730 .84339 .55194 .83389 .56641 .82413 30 31 .50779 .86148 .52275 .85249 .53754 .84324 .55218 .83373 .66665 .82396 29 32 .50804 .86133 .52299 .8523-1 .63779 .84308 55242 .83356 .66689 .82380 28 33 .50829 .86119 .52324 .85218 .53804 .84292 .55266 .83340 .66713 .82363 37 34 .50854 .86101 .52349 .85203 .53828 .84277 .55291 .83324 .56736 .82347 26 35 .60879 .86089 .52374 .85188 .53853 .84261 .55315 .83303 .56760 .82330 26 36 .60904 .86074 .52399 .85173 .53877 .84245 55339 .83292 .56784 .82314 24 37 .60929 .86069 .52423 .85157 .53902 .84230 .55363 .83276 .56808 .82297 23 33 .50954 .86045 .52448 .85142 .53926 .84214 .55388 .83260 .56832 .82281 22 39 .60979 .88030 .52473 .85127 53951 .84193 .55412 .83244 .56856 .82264 21 40 .51004 .86015 .52498 .85112 .53975 .84182 .55436 .83228 .56880 .82248 20 41 .61029 .86000 .52522 .85096 .54000 .81167 .55460 .83212 .56904 .82231 19 42 .61054 .85985 .62547 .85081 .54024 .84151 .55484 .83195 .56928 .82214 18 43 .51079 .85970 .52572 .85066 .54049 .84135 .55509 .83179 .56952 .82198 17 44 .61104 .85956 .52597 .85051 .54073 .84120 .55533 .83163 .56976 .82181 16 45 .51129 .85941 .52621 .85035 .54097 .84104 .55557 .83147 .57000 .82165 .16 46 .51154 .85926 .52646 .85020 .54122 .84088 .55581 .83131 .67024 .82148 14 47 .51179 .85911 .52671 .85005 .54146 .84072 .55605 .83115 .67047 .82132 13 48 .51204 .85896 .52696 .84989 .64171 .84057 .55630 .83098 .67071 .82116 12 49 .51229 .85381 .52720 .84974 .54195 .84041 .55654 .83082 .67095 .82098 11 60 .51254 .85866 .52745 .84959 .54220 .84025 .65678 .83066 .57119 .82082 10 51 .51279 .85851 .52770 .84943 .54244 .84009 .55702 .83050 .57143 ,82065 9 52 .51304 .85836 .52794 .84928 .54269 .83994 .55726 .83034 .57167 .82048 8 53 .51329 .85821 .52819 .84913 .54293 .83978 .65750 .83017 .57191 .82032 54 .51354 .85806 .52844 .84897 .54317 .83962 .55775 .83001 .57215 .82015 t 55 .51379 .85792 .52869 .84882 .54342 .83946 .55799 .82985 .67233 .81999 5 56 .51404 .85777 .52893 .84866 .54366 .83930 .55823 .82969 .57262 .81989 4 67 .51429 .85762 .52918 .84851 .54391 .83915 .65847 .82953 67286 .81965 3 58 .51454 .85747 .52943 .84836 .54415 .83899 .55871 .82936 .67310 .81949 2 59 .51479 .85732 .52967 .84820 .54440 .83883 .65895 .82920 .5733-1 .81932 1 60 .51504 .85717 .52992 .84805 54464 .83367 .55919 .82904 .57358 .81915 M. Cosiii. 81no. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. M. 590 58 570 560 550 TABLE XVI. NATURAL SINES AND COSINES. 93 360 3TO 380 390 M. Sine, Coeln Sine. Cosin. Sine. Cosln Sine. Cosin Sine. Cceiii M. .67358 .81915 .58779 .80902 .60182 .79864 .61566 .78801 .62932 .7771 60 .57381 .81899 .58802 .80885 .60205 .79846 .61589 .78783 .62955 .77696 69 .67405 .81882 .58826 .80867 .60228 .79829 .61612 .78765 .62977 .7767 68 2 .67429 .81865 .58849 .80850 .60251 .79811 .61635 .78747 .63000 .77660 67 4 .57453 .81848 .58873 .80833 .60274 .79793 .61658 .78729 .63022 .7764 56 5 .67477 .81832 .58896 .80816 .60298 .79776 .61681 .78711 .63045 .77623 66 | .57501 .81815 .58920 .80799 .60321 .79768 .61704 .78694 .63068 .77605 64 .57524 ,81798 .58943 .80782 .60344 .79741 .61726 .78676 .63090 .77586 53 8 .57548 .81782 58967 .S0765 .60367 .79723 .61749 .78658 .63113 .7756 52 9 .57572 .81765 .58990 .80748 .60390 .79706 .61772 .78640 .63135 77560 61 10 .57596 .81748 .59014 .80730 .60414 .79688 .61795 .78622 .63158 .7753 50 11 .57619 .81731 .69037 .80713 .60437 .79671 .61818 .78604 .63180 .7751 49 12 .57643 81714 .59061 .80696 .60460 .79653 .61841 .78586 .63203 .77494 48 13 .57667 .81698 .59084 .80679 .60483 .79635 .61864 .78568 .63225 .77476 47 14 .67691 .81681 .59108 .80662 .60506 .79618 .61887 .78550 .63248 .77458 46 15 .67715 .81664 .59131 .80644 .60529 .79600 .61909 .78532 .63271 .77439 46 16 .57738 .81647 .59164 .80627 .60553 .79583 .61932 .78514 .63293 .7742 44 17 .67762 .81631 .59178 .80610 .60576 .79565 .61955 .78496 .63316 .77402 43 18 .57786 .81614 .59201 .80593 .60599 .79547 .61978 .78478 .63338 .77384 42 19 .67810 .81597 .59225 .80576 .60622 .79530 .62001 .78460 .63361 .7736C 41 20 .67833 .81580 .59248 .80558 .60645 .79512 .62024 .78442 .63383 .77347 40 21 .67857 .81563 .59272 .80541 .60668 .79494 .62046 .78424 .63406 .77329 39 22 .67881 .81546 .59295 .80524 .60691 .79477 .62069 .78405 .63428 .77310 38 23 .57904 .81530 .59318 .80507 .60714 .79459 .62092 .78387 .63451 .77292 37 24 .67928 .81513 .59342 .80489 .60738 .79441 .62115 .78369 .63473 .77273 36 26 67952 .81496 .59365 .80472 .60761 .79424 .62138 .78351 .63496 .77255 36 26 67976 .81479 .59389 .80455 .60784 .79406 .62160 .78333 .63518 .77236 34 27 67999 .81462 .59412 .80438 .60807 .79388 .62183 .78315 .63540 .77218 33 28 68023 .81445 .59436 80420 .60830 .79371 .62206 .78297 .63563 .77199 32 29 68047 81428 .59459 80403 .60853 .79353 .62229 .78279 .63585 .77181 31 30 68070 81412 .69482 80386 .60876 .79335 .62261 .78261 .63608 .77162 30 31 68094 81395 .59506 80368 .60899 79318 .62274 .78243 .63630 .77144 29 32 68118 81378 .59529 80351 .60922 .79300 .62297 .78225 .63653 .77126 28 33 58141 81361 .59552 80334 .60945 79282 .62320 .78206 .63675 .77107 27 34 68165 81344 .59576 80316 .60968 79264 .62342 .78188 .63698 .77088 26 35 681891.81327 .59599 80299 .60991 79247 .62365 .78170 .63720 .77070 25 36 582121.81310 .59622 80282 .61015 79229 .62388 .78152 .63742 .77061 24 37 582361.81293 .59646 80264 .61038 79211 .62411 78134 .63765 .77033 23 38 58260 .81276 .59669 80247 .61061 79193 .62433 78116 .63787 .77014 22 39 68283 .81259 .59693 80230 .61084 79176 .62456 78098 .63810 76996 21 40 68307 .81242 .59716 80212 .61107 79158 62479 78079 .63832 76977 20 41 58330 .81225 .59739 80195 .61130 79140 62502 78061 .63854 76959 19 42 58354 .81208 .59763 80178 .61153 79122 62524 78043 .63877 76940 18 43 58378 .81191 .59786 80160 .61176 79105 62547 78025 .63899 76921 17 44 58401 .81174 59809 80143 .61199 79087 62570 78007 .63922 76903 16 45 58425 .81157 59832 80125 .61222 79069 62592 77988 .63944 76884 15 46 68449 .81140 59856 80108 61245 79051 62615 77970 .63966 76866 14 47 58472 .81123 69879 80091 .61268 79033 62638 77952 .63989 76847 13 48 58496 .81106 59902 80073 61291 79016 62660 77934 .64011 76828 12 49 58519 .81089 59926 80056 61314 78998 62633 77916 .64033 76810 11 60 58543 .81072 59949 80033 61337 78980 62706 77897 64056 76791 10 61 58567 .81055 59972 80021 61360 78962 62728 77879 64078 76772 9 52 58590 .81038 59995 80003 61383 78944 62751 77861 64100 76754 8 53 58614 .81021 60019 799S6 61406 78926 62774 77843 64123 76735 7 54 58637 .81004 60042 79968 61429 78908 62796 77824 64145 76717 6 55 58661 .80987 60065 79951 61451 78891 62819 77806 64167 76698 6 56 58684 .80970 60089 79934 61474 78873 62842 77788 64190 76679 4 57 58708 .80953 60112 79916 61497 78855 62864 77769 64212 76661 3 58 58731 .80936 60135 79899 61520 78837 62887 77751 64234 76642 2 59 58755 .80919 60158 79881 61543 78819 62909 77733 64256 76623 1 60 58779 .80902 60182 79864 61566 78801 62932 77715 64279 .7GC04 M. Cotdn. Sine. Cosln. Sine. Coatii. Sine. Cosin. Sine. Costn. Slue. M. 54o 530 5o 51 50 294 TABLE XVI. NATURAL SINES AND COSINES. 400 410 4?o 430 440 BL Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. o .64279 .76604 .65606 .75471 .66913 .74314 .68200 .73135 T69466 .71934 60 1 .64301 .76586 .65628 .75452 .66935 .74295 .63221 .73116 .69487 .71914 59 2 .64323 .76567 .65650 .75433 .66956 .74276 .68242 .73096 .69508 .71894 58 3 .64346 .76548 .65672 .75414 .66978 .74256 .68264 .73076 .69529 .71873 57 4 64368 .76530 .65694 .75395 .66999 .74237 .68285 .73056 .69549 .71853 56 5 .64390 .76511 .65716 .75375 .67021 .74217 .68306 .73036 .69570 .71833 55 6 .64412 .76492 .65738 .75356 .67043 .74198 .68327 .73016 .69591 .71813 54 7 .64435 .76473 .65759 .75337 .67064 .74178 .68349 .72996 .69612 .71792 53 8 .64457 .76455 .65781 .75318 .67086 .74159 .63370 .72976 .69633 .71 ?72 52 9 .64479 .76436 .65803 .75299 .67107 .74139 .68391 .72957 .69654 .71752 51 10 .64501 .76417 .65825 .75280 .67129 .74120 .68412 .72937 .69675 .71732 50 11 .64524 .76398 .65847 .75261 .67151 .74100 .68434 .72917 .69696 .71711 49 12 .64546 .76380 .65869 .75241 .67172 .74080 .68455 .72897 .69717 .71691 48 13 .64568 .76361 .65891 .75222 .67194 .74061 .68476 .72877 .69737 .71671 47 14 .64590 .76342 .65913 .75203 .67215 .74041 .68497 .72857 .69758 .71650 46 15 .64612 .76323 .65935 .75184 .67237 .74022 .68518 .72837 .69779 .71630 45 16 64635 .76304 .65956 .75165 67258 .74002 .68539 .72817 .69800 .71610 44 17 .64657 .76286 .65978 .75146 .67280 .73983 .68561 .72797 .69321 .71590 43 13 .64679 .76267 .66000 .75126 .67301 .73963 .68582 .72777 .69842 .71569 42 19 .64701 .76248 .66022 .75107 .67323 .73944 .68603 .72757 .69862 .71549 41 20 .64723 .76229 .66044 .75088 .67344 .73924 .6862-1 .72737 .69883 .71529 40 21 .64746 .76210 .66066 .75069 .67366 .73904 .68645 .72717 .69904 .71508 39 22 .64768 .76192 .66088 .75050 .67387 .73885 .68666 .72697 .69925 .71488 33 23 .64790 .76173 .66109 .75030 .67409 .73865 .68688 .72677 .69946 .71468 37 24 .64812 .76154 .66131 .75011 .67430 .73846 .68709 .72657 .69966 .71447 36 25 .64834 .76135 .66153 .74992 .67452 .73826 .68730 .72637 .69987 .71427 35 26 .64856 .76116 .66175 .74973 .67473 .73806 .68751 .72617 .70008 .71407 34 27 .64878 .76097 .66197 .74953 .67495 .73787 .68772 .72597 .70029 .71386 33 28 .64901 .76078 .66218 .74934 .67516 73767 .68793 .72577 .70049 .71366 32 29 .64923 .76059 .66240 .74915 .67538 .73747 .68814 .72557 .70070 .71346 31 30 .64945 .76041 .66262 .74896 .67559 .73728 .68835 .72537 .70091 .71326 30 31 .64967 .76022 .66284 .74876 .67580 .73708 .68857 .72517 .70112 .71305 29 32 .64989 .76003 .66306 .74857 .67602 .73688 .68878 .72497 .70132 .71284 28 33 .65011 .75984 .66327 .74838 .67623 .73669 .68899 .72477 .70153 .71264 27 34 .65033 .75965 .66349 .74818 .67645 .73649 .68920 .72457 .70174 .71243 26 35 .65055 .75946 .66371 .74799 .67666 .73629 .68941 .72437 .70195 .71223 25 36 .65077 .75927 .66393 .74780 .67688 .73610 .68962 .72417 .70215 .71203 24 37 .65100 .75908 .66414 .74760 .67709 .73590 .68983 .72397 .70236 .71182 23 38 .65122 .75389 .66436 .74741 .67730 .73570 .69004 .72377 .70257 .71162 22 39 .65144 .75870 .66458 .74722 .67752 .73551 .69025 .72357 .70277 .71141 21 40 .65166 .75851 .66480 .74703 .67773 .73531 .69046 .72337 .70298 .71121 20 41 .65188 .75832 .66501 .74683 .67795 .73511 .69067 .72317 .70319 .71100 19 42 .65210 .75813 .66523 .74664 .67816 .73491 .69088 .72297 .70339 .71080 18 43 .65232 .75794 .66545 .74644 .67837 .73472 .69109 .72277 .70360 .71059 17 44 .65254 75775 .66566 .74625 .67859 .73452 .69130 .72257 .70381 .71039 16 45 .65276 ,75756 .66588 .74606 .67880 .73432 .69151 .72236 .70401 .71019 15 46 .65293 .75738 .66610 .74586 .67901 .73413 .69172 .72216 .70422 .70998 H 47 .65320 .75719 .66632 .74567 .67923 .73393 .69193 .72196 .70443 .70978 13 48 .65342 .75700 .66653 .74548 .67944 .73373 .69214 .72176 .70463 .70957 12 49 .65364 .75680 .66675 .74528 .67965 .73353 .69235 .72156 .70484 .70937 11 50 .65386 .75661 .66697 .74509 .67987 .73333 .69236 .72136 .70505 .70916 10 51 .65408 .75642 .66718 .74489 .68008 .73314 .69277 .72116 .70525 .70896 9 52 .65430 75623 .66740 .74470 68029 .73294 .69298 .72095 .70546 .70875 8 53 .65452 .75604 .66762 .74451 68051 .73274 .69319 .72075 .70567 .70855 7 54 .65474 .75585 .66783 .74431 68072 .73254 .69340 .72055 .70587 .70834 55 .65496 .75566 .66805 .74412 .68093 .73234 .69361 .72035 ,70608 .70813 5 56 .65518 .75547 .66827 .74392 .68115 .73215 .69332 .72015 .70628 70793 4 57 .65540 .75528 .66848 .74373 .68136 .73195 .69403 .71995 .70649 .70772 3 58 .65562 .75509 .66870 .74353 .68157 .73175 .69424 .71974 .70670 .70752 2 59 .65584 .75490 .66891 .74334 .68179 .73155 .69445 .71954 .70690 .70731 1 60 .65606 .75471 .66913 .74314 .68200 .73135 .69466 .71934 .70711 .70711 M: Costa. Sine. Cosin. SineT Cosin. Sine. Cosin. Sine. Cosin. Sine. M. 490 480 470 i 400 450 TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 296 TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 10 o 30 M. Ttuig. | Cotung. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. .00000 Infinite. .01746 57.2900 .03492 28.6363 .05241 19.0811 60 1 .00029 3437.75 .01775 56.3506 .03521 28.3994 .05270 18.9756 59 2 .00058 1718.87 .01804 55.4415 .03550 28.1664 .05299 18.8711 58 3 .00037 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7678 57 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.6656 56 5 .00145 687.549 ,01891 52.8821 03638 27.4899 05387 18.5645 55 6 .00175 572.957 .01920 52.0807 .03667 27.2715 95416 18.4645 54 7 .00204 491.106 .01949 51.3032 .03696 27.0566 .05445 18.3655 53 8 .00233 429.718 .01978 50.5485 .03725 26.8450 05474 18.2677 52 9 .00262 381.971 .02007 49.8157 .03754 26.6367 .05503 - '8.1708 51 1C .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 8.0750 50 11 .00320 312.521 .02066 48.4121 .03812 26.2296 05562 17.9802 49 1 12 .00349 286.478 .02095 47.7395 .03842 26.0307 05591 17.8863 48 [ 13 .00378 264.441 .02124 47.0853 .03871 25.8348 05620 17.7934 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 05649 17.7015 46 16 .00436 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 45 16 .00465 214.858 .0221 1 45.2261 .03958 25.2644 .05708 17.5205 44 17 .00495 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 .00524 190.984 .02269 44.0661 .04016 24.8978 .05766 17.3432 42 19 .00553 180.932 .02298 43.5081 .04046 24.7185 .05795 17.2558 41 20 .00582 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 40 21 .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 39 22 .00640 156.259 02386 41.9158 .04133 24.1957 .05883 16.9990 38 23 .00669 149.465 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 24 .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 25 .00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 35 26 .00756 132.219 .02502 39.9655 .04250 23.5321 .05999 16.6681 34 27 .00785 127.321 .02531 39.5059 .04279 23.3718 .06029 16.5874 33 28 .00815 122.774 .02560 39.0568 .04308 23.2137 .06058 16.5075 32 29 .00844 118.540 .02589 38.6177 .04337 23.0577 .06087 16.4283 31 30 .00673 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 SO 31 .00902 110.892 .02648 37.7686 .04395 22.7519 .06145 16.2722 29 32 .00931 107.426 .02677 37.3579 .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 .04483 22.3081 .06233 16.0435 26 35 .01018 98.2179 .02764 36.1776 .04512 22.1640 .06262 15.9687 25 36 .01047 95.4895 .02793 35.8006 .04541 22.0217 .06291 15.8945 24 37 .01076 92.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 23 38 .01105 90.4633 .02851 35.0695 .04599 21.7426 .06350 15.7483 22 39 .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 20 41 .01193 83.8435 .02939 34.0273 .04687 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 .04745 21.0747 .06496 15.3943 IT 44 .01280 78.1263 .03026 33.0452 04774 20.9460 .06525 15.3254 16 j, 45 .01309 76.3900 .03056 32.7303 .04803 20.8188 .06554 15 2571 If 46 .01338 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.1893 14 ! 47 .01367 73.1390 .03114 32.1131 .04862 20.5691 .06613 15.1222 13 48 01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.G557 12 49 .01425 70.1533 .03172 31.5284 .04920 20.3253 .06671 14.9898 1! 5C .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 9 52 .01513 66.1055 .03259 30.6833 .05007 19.9702 .06759 14.7954 8 53 .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 7 54 .01571 63.6567 .03317 30.1446 .05066 19.7403 .06817 14.6685 6 55 .01600 62.4992 .03346 29.8823 .05095 19.6273 .06847 14.6059 5 56 .01629 61.3829 .03376 29.6245 .05124 19.5156 .06876 14.5438 4 57 .01658 60.3058 .03405 29.3711 .05153 19.4051 .06905 14.4823 3 58 .01687 59.2659 .03434 29.1220 .05182 19.2959 .06934 14.4212 2 59 .01716 58.2612 .03463 28.8771 .05212 19.1879 .06963 14.3607 1 60 .01746 57.2900 .03492 23.6363 .05241 19.0811 .06993 14.3007 M: Cotang. Tang. Cotang. Tang. Do tang. Tang. otang. Taug. M. i 890 880 870 j 860 TABLE XVIT. NATURAL TANGENTS AND COTANGENTS. 297 *o 50 60 TO M. Ttuig Cotang Tang. Cotang. Tang. Cotaug. Tung. Cotanjr. M. ~0 .06993 14.31)07 .08749 11.4301 10510 9.51436 12278 8.14436 60 .07022 14.2411 .08778 11.3919 .10540 9.46781 .12308 8.12481 69 2 .07051 14.1821 .08807 11.3540 10569 9.46141 .12338 8.10536 68 3 .07080 14.1235 .08837 11.3163 10599 9.43515 12367 8.08600 57 4 .07110 14.0655 .08866 11.2789 10628 9.40904 12397 8.06674 56 6 .07139 14.0079 .08895 11.2417 10657 9.3-3307 12426 8.04756 55 6 .07168 13.9507 .08925 11.2048 10687 9.35724 12456 8.02848 54 7 .07197 13.8940 .08954 11.1681 10716 9.33155 .12485 8.00948 53 8 .07227 13.8378 .08983 11.1316 10746 9.30599 12515 7.99058 52 9 .07256 13.7821 .09013 11.0954 10775 9.28058 12544 7.97176 51 10 .07285 13.7267 .09042 1 1 .0594 10805 9.25530 .12574 7.95302 50 11 .07314 13.6719 .09071 11.0237 10834 9.23016 12603 7.93438 49 12 .07344 13.6174 .09101 10.9882 10863 9.20516 .12633 7.91582 48 13 .07373 13.5634 .09130 10.9529 10893 9.18028 12662 7.89734 47 14 .07402 13.5098 .09159 10.9178 10922 9.15554 12692 7.87895 46 15 .07431 13.4566 .09189 10.8829 10952 9.13093 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 .07548 13.2480 .09306 10.7457 11070 9.03379 12840 7.78825 41 20 .07578 13.1969 .09335 10.7119 11099 9.00983 .12869 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 38 23 .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 37 24 .07695 12.9962 .09453 10.5789 11217 8.91520 .12988 7.69957 36 26 .07724 12.9469 .09482 10.5462 11246 8.89185 13017 7.68208 36 26 .07753 12.8981 .09511 10.5136 11276 8.86862 .13047 7.66466 34 27 .07782 12.8496 .09541 10.4813 .11305 8.84551 .13078 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 30 .07870 12.7062 .09629 10.3854 11394 8.77689 13165 7.59575 30 31 .07899 12.8591 .09658 10.3538 11423 8.75425 13195 7.67872 29 32 .07929 12.6124 .09688 10.3224 .11452 8.73172 .13224 7 66176 28 33 .07858 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 7.61132 26 36 .08046 12.4288 .09805 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 .08163 12.2505 .09923 10.0780 11688 8.55555 13461 7.42871 20 41 .08192 12.2067 .09952 10.0483 11718 8.53402 .13491 7.41240 19 421 .08221 12.1632 .09981 10.0187 11747 8.51259 13521 7.39616 18 431 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 17 44 | .08280 12.0772 .10040 9.96007 11806 8.47007 .13580 7.36389 16 45! .08309 12.0346 .10069 9.93101 .11836 8.44896 13609 7.34786 15 46 .0,8339 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 .10187 9.81641 .11954 8.36555 .13728 7.28442 11 50 i .08456 11.8262 .10216 9.78817 11983 8.34496 .13758 7.26873 10 51 .08485 11.7853 .10246 9.76009 .12013 8.32446 .13787 7.25310 9 52 .08514 11.7448 .10275 9.73217 12042 8.30406 13817 7.23754 8 53 i .08544 11.7045 .10305 9.70441 12072 8.28376 .13846 7.22204 7 54 i .08573 11.6645 .10334 9.67680 .12101 8.26355 .13876 7.20661 6 55 .08602 11.6248 .10363 9.64935 12131 8.24345 .13906 7.19125 5 56 08632 11.5853 .10393 9.62205 12160 8.22344 .13935 7.17594 4 57 08661 11.5461 .10422 9.59490 .12190 8.20352 .13965 '.16071 3 58 Ort690 11.5072 .10452 9.56791 12219 8.18370 13995 * 14553 2 59 .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 7.13042 1 6U .08749 11.4301 .10510 9.51436 .12278 8.14435 .14054 7.11537 M. Getting. Tbug. Tang. Cotang. Tang. Cotang. Tang. M. ! 650 840 830 8o 98 TABLE XVII. NATUKAL TANGENTS AND COTANGENTS. 8 9 100 11 M Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. ^li ~0 14054 14084 7.11537 7 10038 15838 15868 6.31375 6.30189 17633 17663 5.67128 5.66165 .19438 .19468 5.14455 5.13658 60 59 2 .14113 14143 7.08546 7.07059 15898 15928 6.29007 6.27829 17693 17723 5.65205 5.64248 .19498 .19529 5.12862 5.12069 bb 57 14173 7 05579 15958 6.26655 17753 5.63295 .19559 5. 1 1279 56 g 14202 7.04105 15988 6.25486 17783 5.62344 .19589 5. 10490 bb v 14232 7.02637 16017 6.24321 17813 5.61397 .19619 5.09704 64 | 7 3 9 13 14 15 .14262 .14291 .14321 .14351 .14381 .14410 .14440 .14470 .14499 7.01174 6.99718 0. 98268 6.96823 6.95385 6.93952 6.92525 6.91104 6.89688 .16047 16077 .16107 .16137 16167 .16196 .16226 .16256 .16286 6.23160 6.22003 6.20851 6.19703 6.18559 6.17419 6.16283 6.15151 6.14023 .17843 .17873 .17903 .17933 .17963 .17993 .18023 .18053 .18083 5.60452 5.59511 5.58573 5.57638 5.56706 5.55777 5.54851 5.53927 5.53007 .19649 .19680 .19710 .19740 .19770 19801 .19831 .19861 .19891 5.08921 5.08139 5.07360 5.06584 5.05809 5.05037 5.04267 5.03499 6.02734 b3 52 51 50 49 48 47 46 45 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 .14529 .14559 .14588 .14618 .14648 .14678 .14707 .14737 .14767 .14796 .14326 .14856 .14886 .14916 .14945 6.88278 6.86874 6.85475 6.84082 6.82694 6.81312 6.79936 6.78564 6.77199 6.75838 6.74483 6.73133 6.71789 6.70450 6.69116 .16316 .16346 16376 .16405 .16435 .16465 16495 .16525 .16555 .16585 .16615 .16645 .16674 .16704 .16734 6.12899 6.11779 6.10664 6.09552 6.0S444 6.07340 6.06240 6.05143 6.04051 6.02962 6.01878 6.00797 5.99720 5.98646 5.97576 .18113 .18143 .18173 .18203 .18233 .18263 .18293 .18323 .18353 .18384 .18414 .18444 .18474 .18504 .18534 6.52090 5.51176 5.50264 5.49356 5.48451 5.47548 5.46648 6.45751 5.44857 5.43966 5.43077 5.42192 6.41309 6.40429 6.39552 .19921 .19952 .19982 .20012 .20042 .20073 .20103 .20133 .20164 .20194 .20224 .20254 .20285 .20315 .20345 6.01971 5.01210 6.00451 4.99695 4.98940 4.98188 4.97438 4.96690 4.95945 4.95201 4.94460 4.93721 4.92984 4.92249 4.91516 44 43 42 41 40 39 ! 38 37 i 36 35 34 I 33 32 31 30 - 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 .14976 .15005 .16034 .15064 .15094 .16124 .15153 .15183 .15213 .15243 .15272 .15302 .15332 .15362 .15391 6.67787 6.66463 6.65144 6.63831 6.62523 6.61219 6.59921 6.58627 6.57339 6.56055 6.54777 6.53503 6.52234 6.50970 6.49710 .16764 .16794 .16824 .16854 .16884 16914 .16944 .16974 .17004 .17033 .17063 .17093 .17123 .17153 .17183 6.96510 5.95448 5.94390 5.93335 5.92283 5.91236 6.90191 5.89151 6.88114 5.87080 5.86051 5.85024 6.84001 5.82982 5.81966 .18564 .18594 .18624 .18654 .18684 .18714 .18745 .18775 .18805 .18835 .18865 .18895 .18925 .18955 .18986 6.38677 5.37805 5.36936 5.36070 5.35206 5.34345 5.33487 5.32631 6.31778 5.30928 6.30080 5.29235 5.28393 5.27553 6.26715 .20376 .20406 .20436 .20466 .20497 .20527 .20557 .20588 .20618 .20648 .20679 .20709 .20739 .20770 .20800 4.90785 4.90056 4.89330 4.88605 4.87882 4.87162 4.86444 4.85727 4.85013 4.84300 4.83590 4.82882 4.82176 4.81471 4.80769 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 46 47 48 49 50 51 52 53 54 55 66 57 58 59 60 .15421 .15451 15481 .15511 .15540 .15570 .15600 .15630 .15660 .15689 .16719 .16749 .15779 .15809 .15838 6.48456 6.47206 6.45961 6.44720 6.43484 6.42253 6.41026 6.39804 6.38587 6.37374 6.36165 6.3496 6.3376 6.32566 6.31375 .17213 .17243 .17273 .17303 .17333 .17363 .17393 .17423 17453 .17483 .17513 .17543 .17573 .17603 .17633 5.80953 5.79944 5.78938 5.77936 5.76937 5.75941 5.74949 5.73960 5.72974 5.71992 6.71013 5.70037 5.69064 5.68094 5.67128 .19016 19046 .19076 .19106 .19136 .19166 .19197 .19227 .19257 .19287 .19317 .19347 .19378 .19408 .19438 5.25880 5.25048 5.24218 6.23391 5.22566 6.21744 6.20925 5.20107 6.19293 5.18480 6.17671 6.16863 5.16058 6.15256 6.14455 .20830 .20861 .20891 .20921 .20952 .20982 .21013 .21043 .21073 .21104 .21134 .21164 .21195 .21225 .21256 4.80068 4.79370 4.78673 4.77978 4.77286 4.76595 4.75908 4.76219 4.74534 4.73851 4.73170 4.72490 4.71813 4.71137 4.70463 14 13 12 11 10 9 8 7 6 5 3 2 K Cotang Tang. Cotang Tang. Cotang T*ng. Co twig Tung. ML 810 800 TOO w TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 299 ia 13 140 150 If Tang. Cotang Tung. Cotang Tang. Cotang Tang. Cotang M. 21256 4.70463 .23087 4.33148 .24933 4.01078 .26795 3.73205 60 1 .21286 4.69791 .23117 4.32573 .24964 4.00582 .26826 3.72771 69 2 .21316 4.69121 .23148 4.32001 .24395 4.00086 .26857 3.72338 68 3 .21347 4.68452 .23179 4.31430 .25026 3.99592 .26888 3.71907 57 4 .21377 4.67786 .23209 4.30860 .25056 3.99099 .26920 3.71476 66 5 .21408 4.67121 .23240 4.30291 .25087 3.98607 .26951 3.71046 55 6 .21438 4.66458 .23271 4.29724 .25118 3.98117 .26982 3.70616 64 7| -21469 4.65797 .23301 4.29159 25149 3.97627 .27013 3.70188 53 8 .21499 4.65138 .23332 4.28595 .25180 3.97139 .27044 3.69761 52 9 .21529 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.69335 5] 10 21560 4 63825 .23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 21590 4.63171 .23424 4.26911 .25273 3.95680 .27138 3.68485 49 12 21621 4.62518 .23455 4.26352 .25304 3.95196 27169 3.68061 48 13 21651 4.61868 .23485 4.25795 .25335 3.94713 .27201 3.67638 47 14 21682 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 46 16 21743 4.59927 .23578 4.24132 .25428 3.93271 .27294 3.66376 44 17 .21773 4.592H3 .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 3.91364 .27419 3.64706 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 .23854 4.19215 .25707 3.89004 .27576 3.62636 36 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 28 .22108 4.52316 .23946 4.17600 .25800 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 3.60688 30 31 .22200 4.50451 .24039 4.15997 25893 3.86208 .27764 3.60181 29 32 .22231 4.49832 .24069 4.15465 .25924 3.85745 .27795 3.59776 28 33 22261 4.49215 .24100 4.14934 .25955 3.85284 .27826 3.59370 27 34 22292 4.43600 .24131 4.14405 .25986 3.84824 .27858 3.58966 26 35 22322 4.47986 .24162 4.13877 .26017 3.84364 .27889 3.58562 26 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 .22628 4.41936 .24470 4.08666 .26328 3.79827 .28203 3.64573 16 46 .22658 4.41340 .24501 4.08152 .26359 3.79378 .28234 3.54179 14 47 .22689 4.40745 .24532 4.07639 .26390 3.78931 .28266 3.53785 13 43 .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 4.3S969 .24624 4.06107 .26483 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 .28391 3.52219 9 62 .22842 4.37793 .24686 4.05092 .26546 3.76709 28423 3.51829 8 53 .22872 4.37207 .24717 4 04586 26577 3.76268 .28454 3.51441 7 64 .22903 4.36623 .24747 4.04081 .26608 3.75828 28486 3.51053 65 .22934 4.36040 24778 4.03578 .26639 3.75388 28517 3.50666 66 .22964 4.35459 24809 4.03076 .26670 3.74950 28549 3.50279 57 .22995 4.34879 24840 4.02574 26701 3.74512 28580 3.49894 68 23026 4.34300 24871 4.02074 .26733 3.74075 28612 3.49509 69 .23056 4.33723 24902 4.01576 26764 3.78640 28643 3.49125 60 .23087 4.33148 24933 4.01078 26795 3.73205 28675 3.48741 M: Cotoug. Tang. otang. Tang. ctang. Tang. otang. Taiig. M. 770 760 750 740 300 TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 100 17 18 190 1 M. Taiig. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotaug. M. ~o .28675 3.48741 .30573 3.27085 .32492 3.07768 .34433 2.90421 60 1 .28706 3.48359 .30605 3.26745 32524 3.07464 .34465 2.90147 59 2 .28738 3.47977 30637 3.26406 32556 3.07160 .34498 2.89873 58 3 .28769 3.47596 .30669 3.2606? .32588 3.06857 .34530 2.89600 57 4 .28800 3.47216 .30700 3.25729 32621 3.06554 .34563 2.89327 56 5 ,28832 3.46837 .30732 3.25392 .32653 3.06252 .34596 2.89055 65 6 ,28864 3.46458 .30764 3.25055 .32685 3.05950 .34628 2.88783 54 7 .28895 3.46080 .30796 3.24719 ,32717 3.05649 34661 2.88511 53 8 .28927 3.45703 .30828 3.24383 .32749 3.05349 .34693 2.88240 52 g .28958 ?.45327 .30860 3.24049 .32782 3.05049 .34726 2.87970 51 10 .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2.87700 50 U .29021 3.44576 .30923 3.23381 32846 3.04450 .34791 2.87430 49 12 .29053 3.44202 .30955 3.23048 32878 3.04152 34824 2.87161 48 13 .29084 3.43829 .30987 3.22715 32911 3.03854 .34856 2.86892 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 2.86356 45 16 29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.86089 44 17 ! 29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.85822 43 18 .29242 3.41973 .31147 3.21063 .33072 3.B2372 .35020 2.85555 42 19 .29274 3.41604 .31178 3.20734 .33104 3.02077 .35052 2.85289 41 20 .29305 3.41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 40 : 21 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 2.84758 39 22 .29368 3.40502 .31274 3.19752 .33201 3.01196 .35150 2.84494 38 23 .29400 3.40136 .31306 3.19426 .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 .33298 3.00319 .35248 2.83702 35 26 .29495 3.39042 .31402 3.18451 .33330 3.00028 .35281 2.83439 34 27 .29526 3.38679 .31434 318127 .33363 2.99738 .35314 2.83176 33 28 .29558 3.38317 .31466 3.17804 .33396 2.99447 .35346 2.82914 32 i 29 .29590 3.37965 .31498 3.17481 .33427 2.99158 .35379 2.82653 31 30 .29621 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 30 31 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 29 32 .29685 3.36875 .31594 3.16617 .33524 2.98292 .36477 2.81870 28 33 .29716 3.36516 .31626 3.16197 .33657 2.98004 .35510 2.81610 27 34 .29748 3.36158 ..31658 3.15877 .33589 2.97717 .35543 2.81350 26 35 .29780 3.35800 .31690 3.15558 .33621 2.97430 .35576 2.81091 26 36 .2981 1 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 37 .29843 3.35087 .31754 3.14922 .33686 2.96858 .35641 2.80574 23 38 .29875 3.34732 .31786 3. 14605 .33718 2.96573 .35674 2.80316 22 39 .29906 3.34377 .31818 3.14288 .33751 2.96288 .35707 2.80059 21 40 .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 .33881 2.95155 .35838 2.79033 17 44 .30065 3.32614 .31978 3.12713 .33913 2.94872 .35871 2.78778 16 45 .30097 3.32264 .32010 3.12400 .33945 2.94591 .35904 2.78523 15 46 .30128 3.31914 .32042 3.12087 .33978 2.94309 .35937 2.78269 14 47 .30160 3.31565 .32074 3.11776 34010 2.94028 .35969 2.78014 13 48 .30192 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 49 .30224 3.30868 .32139 3.11153 .34075 2.93468 .36035 2.77507 11 ! 50 .30255 3.30521 .32171 3.10842 .34108 2.93189 .36068 2.77254 10 51 .30287 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 3.28795 .32331 3.0929S .34270 2.91799 .36232 2.75996 6 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 3.08379 .34368 2.90971 .36331 2.75246 2 59 .30541 3.27426 .32460 3.08073 .34400 2.90696 .36364 2.74997 1 60 .30573 3.27085 .32492 3.07768 .34433 2.90421 .36397 2.74748 M: Cotaug. Tang. Cotang. Tang Cotang. Tang. Cotaug. Ttuig. M T3o TO yio TOO TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 301 | 200 210 230 230 M. Tang. Cotang. Tang. Cotang Tang Cotang Tang. Cotang M. .36397 2.74748 .38386 2.60509 .40403 2.47509 .42447 2.35585 6( 1 .36430 2.74499 .38420 2.6028 .4043 2.47302 .42482 2.35395 69 2 .36463 2.74251 .38453 2.6005 .40470 2.47095 .42516 2.35205 5* 3 .36496 2.74004 .38487 2.5983 .40504 2.46838 .42551 2.35016 57 4 .36529 2.73756 .38520 2.5960 .40538 2.46682 .42585 2.34825 56 6 .36562 2.73509 .38553 2.5938 .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 5i 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 2.5826 .40741 2.45451 .42791 2 33693 50 11 .36760 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.33506 49 12 .36793 2.71792 .38787 2.57815 .40809 2.45043 .42860 2.33317 48 13 .36826 2.71548 .38821 2.57593 .40843 2.44839 .42894 2.33130 47 14 .36859 2.71305 .38854 2.57371 .40877 2.44636 .42929 2.32943 46 15 .36892 2.71062 .33888 2.57150 .4091 1 2.44433 .42963 2.32756 45 16 .36925 2.70819 .38921 2.56928 .40945 2.44230 .42998 2.32570 44 17 .36958 2.70577 .38955 2.56707 .40979 2.44027 .43032 2.32383 43 18 .36991 2.70335 .38988 2.56487 .41013 2.43825 .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 36 26 .37256 2.68414 .39257 2.54734 .41285 2.42218 .43343 2.30718 34 27 .37289 2.68175 .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 2.6770U 39357 2. 54082 .41387 2.41620 .43447 2.30167 31 30 .37388 2.67462 39391 2.53865 .41421 2.41421 .43481 2.29984 30 31 .37422 2.67225 ^9425 2.53648 .41455 2.41223 .43516 2.29801 29 32 .37455 2.66989 ^58 2.53432 .41490 2.41025 .43550 2.29619 28 33 .37488 2.66752 3*49* 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.66516 .3952H 2.53001 .41558 2.40629 .43620 2.29254 26 35 37554 2.66281 .39559 252786 .41592 2.40432 .43654 2.29073 26 36 .37588 2.66046 .39593 2.52571 .41626 2.40235 .43689 2.28891 24 37 .37621 2.65811 .39626 252357 .41660 2.40038 43724 2.28710 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 2.5J715 41763 2.39449 .43828 2.28167 20 41 .37754 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 14 ; 37353 2.64177 .39862 2.50864 41899 2.38668 43966 2.27447 16 15 .37887 2.63945 .39896 2.50652 .41933 2.38473 44001 2.27267 15 16 37920 2.63714 .39930 2.50440 41968 2.38279 44036 2.27088 4 17 37853 2.63433 39963 2.50229 .42002 2.38084 44071 2.26909 3 43 .37986 2.63252 39997 250018 .42036 2.37891 44105 2.26730 2 49 .38020 2.63021 40031 2.49807 42070 2.37697 44140 2.26552 50 .38053 2.62791 40065 2.49597 42105 2.37504 44175 2.26374 51 .380,86 262561 40098 2.49386 42139 2.37311 44210 2.26196 9 52 i 38120 262332 40132 2.49177 42173 2.371 18 44244 2.26018 8 53 | 38153 2.62103 .40166 2.48967 42207 2.36925 44279 2.25840 7 54 .38186 2.61874 .40200 2.48758 42242 2.36733 44314 2.25663 6 55 .3S220 2.61646 4023-1 2.48549 42276 2.36541 44349 2.25486 6 56 .3S253 2.61418 40267 2.48340 42310 2.36349 44384 2.25309 4 57 .38286 261190 40301 2.48132 42345 2.36158 44418 2.25132 3 58 .38320 2.60963 .40335 2.47924 42379 2.35967 44453 2.24956 2 59 .38353 2.60736 .40369 2.47716 42413 2.35776 44488 2.24780 1 60 .38386 2.60509 .40103 2.47509 42447 2.35585 44523 2.24604 _0 M. Co tang. Tang. Dotang. 1 Tang. otang. Tang. otang. Tang. ! 690 680 670 660 H 302 TABLE XVII. NATURAL TANGENTS AND COTANGENTS. ~34^ 350 360 3TO M. Tang. Cotang. Tang. Cotaug. Tang. Cotang. Toiig Cotang. M. IF .44523 2.24604 .46631 2.14451 .48773 2.05030 .50953 1.96261 60 1 .44558 2.24428 .46666 2.14288 .48809 2.04879 .50989 1.96120 59 2 .44593 2.24252 .467(12 2.14125 .48845 2.04728 .51026 1.95979 58 3 .44627 2.24077 .46737 2.13963 .48881 2.04577 .51063 1.95838 57 4 .44662 2.23902 .46772 2.13801 .48917 2.04426 .51099 1.95698 56 5 .44697 2.23727 46808 2.13639 .48953 2.04276 .51136 1.95557 55 Q .44732 2.23553 .46343 2.13477 48989 2.04125 .51173 1.95417 54 7 ! 44767 2^23378 .46879 2.13316 .49026 2.03975 51209 1.95277 53 Q .44802 2 23204 .46914 2.13154 .49062 2.03825 .51246 1.95137 52 g .44837 2.23030 .46950 2.12993 .49098 2.03675 .51283 1.94997 51 10 .44872 222857 .46985 2.12832 .49134 2.03526 .51319 1.94858 60 11 .44907 2.22683 .47021 2.12671 .49170 2.03376 51356 1.94718 49 12 44942 222510 .47056 2.12511 .49206 2.03227 .51393 1.94579 48 13 .44977 2.22337 .47092 2.12350 .49242 2.03078 .51430 1.94440 47 14 45012 2 22164 .47128 2.12190 .49278 2.02929 .61467 1.94301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 .51503 1.94162 45 16 .45082 2.21819 .47199 2.11871 .49351 2.02631 .61540 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 .61614 1.93746 42 19 .45187 2.21304 .47305 2.11392 .49459 2.02187 .61651 1.93608 41 20 .45222 2.21132 .47341 2.11233 .49495 2.02039 .61688 1.93470 40 21 .45257 2.20961 .47377 2.11075 .49532 2.0189J .51724 1.93332 39 22 45292 2.20790 .47412 2.10916 .49568 2.01743 .61761 1.93195 38 23 .45327 2.20619 .47448 2.10758 .49604 2.01596 .51798 1.930D7 37 24 .45362 2.20449 .47483 2.10600 .49640 2.01449 .51835 1.92920 36 25 45397 2.20278 .47519 2.10442 .49677 2.01302 .51872 1.92782 36 26 .45432 2.20108 .47555 2.10284 49713 2.01155 .51909 1 92645 34 27 .45467 2.19938 .47590 2.10126 .49749 2.01008 .51946 I. 92508 33 28 45502 2.19769 .47626 2.09969 .49786 2.00862 .61983 1.92371 32 29 .46538 2.19599 .47662 2.09811 .49822 2.00715 .52020 1.92235 31 30 .45573 2.19430 .47698 2.09654 .49858 2.00569 .62057 1.92098 30 31 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52054 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 2.00131 .52168 1.91690 27 34 .45713 2.18755 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 35 .45748 2.18587 .47876 2.08872 .50040 1.99841 .52242 1.91418 25 36 .45784 2.18419 .47912 2.08716 .50076 1.99695 .62279 1.91282 24 37 .45819 2.18251 .47948 2.08560 .50113 1.99550 .52316 1.91147 23 38 .45854 2.18084 .47984 2.08405 .50149 1.99406 .52353 1.91012 22 39 .45889 2.17916 .48019 2.08250 .50185 1.99261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .60222 1.99116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 1.98972 .52464 1.90607 19 42 .45995 2.17416 .48127 2.07785 .50295 1.98828 .52501 1.90472 18 43 .46030 2.17249 .48163 2.07630 .50331 1.98684 .52538 1.90337 17 44 .46065 2.17083 .48198 2.07476 .60368 1.98540 .-52575 1.90203 16 45 .46101 2.16917 .48234 2.07321 .50404 1.98396 .52613 1.90069 15 46 .46136 2.16751 .48270 2.07167 .50441 1.98253 .52650 1.89935 14 47 .46171 2.16585 .48306 2.07014 .50477 1.98110 .52687 1 .89801 13 48 .46206 2.16420 48342 2.06860 .50514 1.97966 .52724 1.89667 12 49 .46242 2 16255 48378 2.06706 .50550 1.97823 .52761 1.89533 11 50 .46277 2.16090 .48414 2.06553 .50587 1.97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 1.97538 .52836 1.89266 9 52 .46348 2.15760 .48486 2.06247 .50660 1.97395 52873 1.89133 8 53 .46383 2.15596 .48521 2.06094 50696 1.97253 .52910 1.89000 7 54 .46418 2.15432 .48557 2.05942 .50733 1.97111 .52947 1.88867 6 55 .46454 2.15268 .48593 2.05790 .50769 1.96969 .52985 1.88734 6 56 .46489 2.15104 .48629 2.05637 .50806 1.96827 .53022 1.88602 4 57 .46525 2 14940 48665 2.05485 .50843 1.96685 .53059 1.88469 3 58 .46560 2.14777 .48701 2.05333 .50879 1.96544 .53096 1.88337 2 59 .46595 2 14614 48737 2.05182 .50916 1.96402 .53134 1.88205 1 60 .46631 2.14451 .48773 2.05030 .50953 1.96261 .53171 1.88073 M Cotang. Tang. Cotang. Tang. Cofcang. Tang. Cotang. Tung. M 600 640 630 oao TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 303 HI 390 300 310 M. Tang Cotang Tang Cotang Tang Cotang Tang Cotang c .6317 1.8807 .5543 T.8040 .57735 1.7320 .6008 1.6642 6U ] .63208 1.8794 .55469 1.8028 .5777 1.7308 .6012 1.6631 5 2 .63246 1.8780 .55507 1.8015 .5781 1.7297 .6016 1.6620 5 3 .53283 1.8767 .55545 1.80034 .5785 1.7285 .6020 1.66099 5! 4 .63320 1.8754 55583 1.7991 .5789 1.7274 6024 l.6599i 5, 5 .63358 1.8741 .55621 1.79788 .5792 1.72620 60284 1.6588 5 6 .53395 1.87283 .55659 1.7966 .57968 1.7250 .60324 1.65772 & 7 .63432 1.8715 .55697 1.7954 .5800 1.7239 60364 1.65663 5J ! 8 .63470 1.8702 .55736 1.7941 .58046 1.7227 .60403 1.655& 5! 9 .63507 1.8689 .55774 1.79296 .58085 1.7216 .60443 1.65445 51 10 63545 1.86760 .55812 1.79174 .58124 1.7204 .60433 1.65337 5( 11 53582 1.86630 .55850 1.7905 .58162 1.71932 60522 1.65228 41 12 .53320 1.86499 .55888 1.78929 .58201 1.7181 .60562 1.65120 41 13 63657 1.86369 .55926 1.78807 .58241 1.71702 .60602 1.65011 47 14 .53694 1.86239 .55964 1.78685 .58279 1.71588 .60642 1.6490J 41 16 .53732 1.86109 .56003 1.78563 .58318 1.71473 .60681 1.64795 45 16 .63769 1.85979 .56041 1.78441 .58357 1.71358 .60721 1.64687 44 17 .53807 1.85850 .56079 1.78319 .58396 1.71244 .60761 1.64579 4' 18 .53844 1.85720 .56117 1.78198 .58435 1.71129 .60801 1.64471 4' 19 20 .53882 .53920 1.85591 1.85462 .56156 .56194 1.78077 1.77955 58474 .58513 1.71015 1.70901 .60841 .60881 1.64363 1.64256 41 40 21 .53957 1.85333 .56232 1.77834 .58552 1.70787 .60921 1.64148 39 22 .63995 1.85204 .56270 1.77713 .58591 1.70673 .60960 1.64041 38 23 .64032 1.85075 .56309 1.77592 .58631 1.70560 .61000 1.63934 37 24 .54070 1.84946 .66347 1.77471 .58670 1.70446 .61040 1.63826 36 26 .54107 1.84818 .66385 1.77351 .58709 1.70332 .61080 1.63719 36 26 .64145 1.84689 .56424 1.77230 .58748 1.70219 .61120 1.63612 34 27 .54183 1.84561 .56462 1.77110 .58787 1.70106 .61160 1 63505 33 28 .54220 1.84433 .66501 1.76990 58826 1.69992 .61200 1.63398 32 29 .64258 1.84305 56539 1.76869 58865 1.69879 .61240 1.63292 31 30 .64296 1.84177 66677 1.76749 .58905 1.69766 .61280 1.63185 30 31 54333 1.84049 56616 1.76629 58944 1.69653 61320 1.63079 29 32 .64371 1.83922 56654 1.76510 58983 1.69541 61360 1.62972 28 33 .64409 1.83794 66693 1.76390 59022 1.69428 61400 1.62866 27 34 36 .54446 .64484 1.83667 1.83540 56731 56769 1.76271 1.76151 59061 59101 1.69316 1.69203 61440 61480 1.62760 1.62654 26 25 36 .64522 1.83413 56808 1.76032 69140 1.69091 61520 1.62548 24 (7 .64560 1.83286 56846 1.75913 59179 1.68979 61561 1.62442 23 38 .64597 1.83159 66885 1.75794 59218 1.68866 61601 1.62336 22 39 .64635 1.83033 56923 1.75675 59258 1.68754 61641 1.62230 21 .64673 1.82906 56962 1.75556 59297 1.68643 61681 1.62125 20 \ I .64711 1.82780 57000 1.75437 59336 1.68531 61721 1.62019 19 \ 42 .64748 1.82654 57039 1.75319 59376 1.68419 61761 1.61914 8 1 3 .64786 1.82528 57078 1.75200 59415 1.68308 61801 1.61808 7 I 44 .54824 1.82402 57116 1.75082 59454 1.68196 61842 1.61703 6 46 .64862 1.82276 57155 1.74964 59494 1.68085 61882 1.61598 5 46 .54900 1.82150 57193 1.74846 59533 1.67974 61922 1.61493 4 7 .54938 1.82025 57232 1.74728 59573 1.67863 61962 1.61388 3 8 .64975 1.81899 57271 1.74610 59612 1.67752 62003 1.61283 2 1 .65013 1.81774 57309 1.74492 59651 1.67641 62043 1.61179 1 5C .65051 1.81649 57348 1.74375 59691 1.67530 62083 1.61074 51 .65089 1.81524 57386 1.74257 59730 1.67419 62124 1.60970 9 62 .65127 1.81399 57425 1.74140 59770 1.67309 62164 .60865 8 >3 .55165 81274 57464 1.74022 69809 1.67198 62204 1.60761 7 A .55203 1.81150 57503 1.73905 59849 1.67088 62245 1.60657 6 66 .55241 1.81025 57541 1.73788 59888 1.66978 62285 .60553 6 66 .6527$ .80901 57580 1 73671 59928 1.66867 62325 .60449 4 57 .55317 .80777 57619 1.73555 59967 .66757 62366 .60345 3 68 .55355 .80653 57657 1 73438 60007 .66647 62406 .60241 2 69 .65393 .80529 57696 1 73321 600-16 .66538 62446 .60137 1 CO .55431 .80405 57735 1 73205 60086 .66428 62487 .60033 M. Cotang. Taug. otang. Tang. otang. Tang. otang. Tang. 610 603 590 580 304 TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 8o 330 M Tang. Cotaug. Tang. Cotang. long. Cotang. Taug. Cotang. M. ~o 62487 1.60033 64941 1.53986 67451 1.48256 .70021 1.42815 60 ' ] .62527 1.59930 64982 1.53888 .67493 1.48163 .70064 1 .42726 59 2 62568 1 59826 65024 1.53791 .67536 1.48070 .70107 1.42638 63 ! 3 62608 1.59723 65065 1.53693 .67578 1.47977 .70151 1.42550 67 4 .62649 1.59620 65106 1.53595 .67620 1.47885 .70194 1.42462 56 ' 5 .62689 1.59517 65148 1.53497 .67663 1.47792 .70238 1.42374 55 , 6 .62730 1.59414 65189 1.53400 .67705 1.47699 .70281 1.42286 54 ! 7 .62770 1.59311 65231 1.53302 .67748 1.47607 70325 1.42198 53 g .62811 1.59208 65272 1.53206 .67790 1.47514 .70368 1.42110 52 Q 62852 1.59105 .65314 1.53107 .67832 1.47422 .70412 1.42022 51 10 .62892 1.59002 .65355 1.53010 .67875 1.47330 .70455 1.41934 50 |l .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 .68002 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 .68088 1.46870 .70673 1.41497 45 16 63136 1.58338 .65604 1.52429 .68130 1.46778 .70717 1.41409 44 17 .63177 1.58286 .65646 1.52332 .68173 1.46686 .70760 1.41322 43 18 63217 1.58184 .65688 1.52235 .68215 1.46595 .70804 1.41235 42 19 .63258 1.58083 .65729 1.52139 .68258 1.46503 .70848 1.41148 41 20 .63299 1.57981 .65771 1.52043 .68301 1.46411 .70891 1.41061 40 21 .63340 1.57879 .65813 1.61946 .68343 1.46320 .70935 1.40974 39 22 .63380 1.57778 .65854 1.51860 .68386 1.46229 .70979 1.40887 38 23 .63421 1.57676 .65896 1.51754 .68429 1.46137 71023 1.40800 37 24 .63462 1.67575 .65938 1.51658 .68471 1.46046 .71066 1.40714 36 25 .63503 1.57474 .65980 1.61562 .68514 1.45955 .71110 1.40627 36 I 26 63544 1.57372 .66021 1.51466 .68557 1.45864 71154 1.40540 34 27 .63584 1.57271 .66063 1.51370 .68600 1.45773 .71198 1.40454 33 ! 28 .63625 1.57170 .66105 1.51275 .68642 1.45682 .71242 1.40367 32 29 .63666 1.57069 .66147 1.51179 .68685 1.45592 .71285 1.40281 31 80 .63707 1.56969 .66189 1.51084 .68728 1.45501 .71329 1.40195 30 31 .63748 1.56868 .66230 1.50988 .68771 1.45410 .71373 1.40109 29 32 .63789 1.56767 .66272 1.50893 .68814 1.45320 .71417 1.40022 28 33 .63830 1.56667 .66314 1.50797 .68857 1.45229 .71461 1.39936 27 ! 84 .63871 1.56566 .66356 1.50702 .68900 1.45139 .71505 1.39850 26 35 .63912 1.56466 .66398 1.50607 .68942 1.45049 .71549 1.39764 26 36 .63953 1.56366 .66440 1.50512 .68985 1.44958 .71593 1.39679 24 37 .63994 1.56265 .66482 1.50417 .69028 1.44868 .71637 1.39593 23 38 .64035 1.56165 .66524 1.50322 .69071 1.44778 .71681 1.39507 22 39 .64076 1.56065 .66566 1.50228 .69114 1.44688 71725 1.39421 21 j 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 .69286 1.44329 .71901 1.39079 17 44 .64281 1.55567 .66776 1 49755 .69329 1.44239 .71946 1.38994 16 ! 45 .64322 1.55467 .66818 1.49661 69372 1.44149 .71990 1.38909 16 46 .64363 1.55368 .66860 1.49566 .69416 1.44060 .72034 1.38824 14 I 47 64404 1 55269 66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .64446 1.55170 .66944 1.49378 .69502 1.43881 .72122 1.38653 12 49 .64487 1.55071 .66986 1.49284 .69545 1.43792 .72167 1.38568 11 50 .64528 1.54972 .67028 1.49190 .69588 1.43703 72211 1.38484 10 51 .64569 1.54873 ,67071 1.49097 .69631 1.43614 72255 1.38399 9 52 .64610 1.54774 .67113 1.49003 .69675 1.43525 72299 1.38314 8 63 .64652 1.54675 .67155 1.48909 .69718 1.43436 .72344 1.38229 7 54 .64693 1.54576 .67197 1.48816 .69761 1 .43347 .72388 1.38145 6 55 .64734 1.54478 .67239 1.48722 .69804 1.43258 .72432 1.38060 5 56 .64775 1.54379 .67282 1.48629 .69847 1.43169 .72477 1.37976 4 57 .64817 1.54281 .67324 1.48536 .69891 1.43080 .72521 1.37891 3 58 64858 1 54183 67366 1.48442 .69934 1.42992 .72565 1.37807 2 69 .64899 1.54085 .67409 i.48349 .69977 1.42903 .72610 1.37722 1 60 .64941 1.53986 .67451 1.48256 .70021 1.42815 .72654 1.37638 M Ootaug Taug. Cotang Tang Cotauff Tang. Cotang Tang. M. 57 560 55 6*0 i TABLE XVII. NATURAL TANGENTS AND COTANGENTS, 3Q5 390 1 - Tang Cotang Tang Cotan Tang Cotan Tang. Cotang. M. 1 .72654 .7269 1.37638 1.37554 .7535 .7540 1.32704 1.32624 .7812 .7817 1.27994 1.2791 .80978 .81027 1.23490 1 2341 60 59 i .7274 .72788 .72832 .72877 1.3747 1.37386 1.37302 1.3721 .7544 .75492 .75538 .76584 1.3254 1.32464 1.32384 1.32304 .78222 .7826 .7831 .7836 1.2784 1.27764 1.27688 1.2761 .81075 .81123 .81171 .81220 1.2334 1.2327 1.23196 1 23123 58 57 66 55 I 6 * 9 .7292 .72966 .73010 .73055 1.37134 1.37050 1.3696 1.36883 .7562 .76676 .7572 .75767 1.3222 1.3214 1.32064 131984 .78410 .7846 .78504 .7855 1.27535 1.27458 1.27382 1.27306 .81268 .81316 .81364 .81413 1.2305 1.2297 1.22904 1 2283 64 53 52 51 10 11 12 13 .73100 .73144 .73189 .73234 1.36800 1.3671 1.3663 1.3654 .75812 .75858 .75904 .75950 1.31904 1.31825 1.3174 1.31666 .78598 .78645 .78692 .78739 1.27230 1.27163 1.2707 1.2700 .81461 .81510 .81558 .81606 1.22758 1.22685 1.22612 1 22539 60 49 48 47 14 16 .73278 .73323 1.36466 1.36383 .75996 .76042 1.31586 1.31507 .78786 .78834 1.26925 1.2684 .8165 .81703 1.22467 1.22394 46 46 ] 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 .73368 .73413 .73457 .73502 .73547 .73592 .73637 .73681 .73726 .73771 .73816 .73861 .73906 .73951 .73996 1.36300 1.3621~ 1.36134 1.3605 1.35968 1.35886 1.35802 1.35719 1.35637 1.35554 1.35472 1.35389 1.35307 1.35224 1.35142 .76088 .76134 .76180 .76226 .76272 .76318 .76364 .76410 .76456 .76502 .76548 .76594 .76640 .76686 ,76733 1.31427 1.3134 1.3126 1.3119 1.31110 1.3103 1.30952 1.30873 1.30795 1.30716 1.30637 1.30558 1.30480 1.30401 1.30323 .78881 .78928 .78976 .79022 .79070 .79117 .79164 .79212 .79269 .79306 .79364 .79401 .79449 .79496 .79644 1.2677 1.26693 1.2662 1.2654 1.2647 1.2639 1.2631 1.26244 1.2616 1.2609 1.26018 1.25943 1.25867 1.25792 1.25717 .81752 .81800 .81849 .81898 .81946 .81995 .82044 .82092 .82141 .82190 .82238 .82287 .62336 .82385 .82434 1.2232 1.22249 1.22176 1.22104 1.22031 1.21959 1.21886 1.21814 1.21742 1.21670 1.21598 1.21626 1.21454 1.21382 1.21310 44 43 4 4i 31 8! 31 34 K 11 30 31 32 33 34 36 36 37 38 39 40 42 43 44 jl 46 74041 .74086 .74131 74176 .74221 74267 74312 74357 74402 74447 74492 74533 74583 74628 74674 1.35060 1.34978 1.34896 1.34814 1.34732 1.34650 1.34568 1.34487 1.34405 1.34323 1.34242 1.34160 1.34079 1.33998 1.33916 .76779 76825 76871 76918 76964 77010 77057 77103 77149 77196 77242 77289 77335 77382 77428 1.30244 1.30166 1.30087 1.30009 1.29931 1.29853 1.29775 1.29696 1.29613 1.29541 1.29463 1.29385 1.29307 1.29229 1.29152 .79591 .79639 .79686 .79734 .79781 79829 79877 79924 79972 80020 80067 80115 80163 80211 80258 1.25642 1.25667 1.25492 1.26417 1.25343 1.26268 1.25193 1.25118 1.25044 1.24969 1.24895 1.24820 1.24746 1.24672 1.24597 .82483 .82531 .82580 82629 1.21238 1.21166 1.21094 1.21023 1.20961 1.20879 1.20808 1.20736 1.20665 1.20593 1.20522 1.20451 1.20379 1.20308 1.20237 g 27 26 26 24 23 22 21 20 19 8 17 6 6 .82678 82727 82776 82825 82874 82923 82972 83022 83071 83120 83169 46 47 48 49 60 61 1 KQ 74719 .74764 .74810 .74855 .74900 .74946 1.33835 1.33754 1.33673 1.33592 1.33511 1.33430 77475 77521 77568 77615 77661 77708 1.29074 1.28997 1.28919 1.28842 1.28764 1.28687 80306 80354 80402 80450 80498 80546 1.24523 1.24449 1.24375 1.24301 1.24227 1.24163 83218 83268 83317 83366 83416 83465 1.20166 1.20095 1.20024 1.19953 1.19882 1.19811 4 3 2 1 9 1 1 eo .74991 1.33349 77754 .28610 80594 1.24079 83614 1.19740 64 65 .75037 .75082 .75128 1.33263 1.33187 .33107 77801 77848 77895 .28533 .28456 .28379 80642 80690 80738 1.24006 1.23931 1.23858 83564 83613 83662 1.19669 1.19599 1 19528 7 6 5 66 67 68 69 60 "53T 7 .75173 .75219 .76264 .75310 .76355 . .33026 .32946 .32865 .32785 .32704 77941 77988 78035 78082 78129 .28302 .28225 .28148 .28071 .27994 80786 80834 80882 80930 80978 .23784 .23710 .23637 .23563 .23490 83712 83761 83811 83860 83910 1.19457 1.19387 1.19316 1.19246 1.19176 4 3 2 M . ( bteng. Tang. tang. Tang. tang. Tang. tang. Tan*. 530 | 510 - 500 306 TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 1 P 41 o 42 43 M. Tang. otang. ang. otang. ang. otang. ang. otang. M. I \ 83910 83960 84009 84059 84108 84158 19175 19105 .19035 .18964 .18894 .18824 86929 6980 7031 7082 87133 87184 15037 .14969 .14902 .14834 .14767 .14699 0040 0093 0146 0199 0251 90304 11061 10996 .10931 .10867 .10802 .10737 3252 3306 3360 3415 93469 93524 .07237 60 .07174 59 .07112 58 .07049 57 .06987 56 .06925 55 I i 14 15 84208 84258 84307 84357 84407 84457 84507 84556 84606 84656 .18754 .18684 .18614 .18544 .18474 .18404 .18334 .18264 .18194 .18125 87236 87287 87338 87389 87441 87492 87543 87595 87646 87698 .14632 .14565 .14498 .14430 14363 .14296 .14229 .14162 .14095 .14028 90357 90410 90463 90516 90569 90621 90674 90727 90781 90834 .10672 .10607 .10543 .10478 .10414 .10349 .10285 .10220 .10156 .10091 93578 93633 93688 93742 93797 93852 93906 93961 94016 94071 .06862 54 06800 53 .06738 52 .06676 51 .06613 50 .06551 49 .06489 48 .06427 47 .06365 46 .06303 45 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 84706 84756 84806 84856 84906 84956 85006 85057 85107 85157 85207 85257 85303 85358 85408 .18055 .17986 .17916 .17846 .17777 .17708 1.17638 1.17569 1.17500 1.17430 1.17361 1.17292 1.17223 1.17154 1.17085 87749 87801 87852 87904 87955 88007 88059 88110 88162 88214 88265 88317 88369 88421 88473 .13961 1.13894 1.13828 1.13761 1.13694 1.13627 1.13561 1.13494 1.13428 1.13361 1.13295 1.13228 1.13162 1.13096 1.13029 90887 90940 90993 91046 91099 91153 91206 91259 91313 91366 91419 91473 91526 91580 91633 .10027 .09963 .09899 .09834 1.09770 1.09706 1.09642 1.09578 1.09514 1.09450 1.09386 1.09322 1.09258 1.09195 1.09131 94125 94180 94235 94290 94345 94400 94455 94510 94565 94620 94676 94731 94786 94841 94896 .06241 44 1.06179 43 1.06117 42 1.06056 41 1.05994 40 1.05932 39 1.05870 38 1.05809 37 1.05747 36 1.05685 36 1.05624 34 1.05562 33 1.05501 35 1.05439 31 1.05378 3f 31 32 33 34 35 36 37 P 4 43 44 4 85458 85509 85559 85609 85660 85710 .85761 .85811 .85862 85912 .85963 .86014 .86064 .86115 .8616 1.17016 1.16947 1.16878 1.16809 1.16741 1.16672 1.16603 1.16535 1.16466 1.18398 1.1632 1.1626 1.1619 1.1612 1.1605 88524 88576 88628 88680 .88732 .88784 .88836 .88888 .88940 .88992 .89045 .89097 .89149 .89201 .89253 1.12963 1.12897 1.12831 1.12765 1.12699 1.12633 1.12567 1.1250 1.12435 1.1236 1.1230 1.12238 1.1217 1.12106 1.1204 91637 91740 .91794 .91847 .91901 .91955 .92008 .92062 .92116 .92170 .92224 .92277 .92331 .92385 .92439 1.09067 1.09003 1.08940 1.08876 1.08813 1.08749 1.08686 1.0862 1.0855 1.0849 1.0843 1.0836 1.0830 1.0824 1.0817 94952 95007 .95062 .95118 .95173 .95229 .95284 .95340 .95395 .95451 .95506 .95562 .95618 .95673 .9572 1.05317 2 1.05255 2E 1.05194 2" 1.05133 2( 1.05072 21 1.05010 % 1.04949 2! 1.04888 2 1 1.04827 2 1.04766 21 1.04705 1.04644 1.04583 1.04522 1.04461 4 4 4 4 50 5 ! 54 6 1 1 .8621 .8626 .8631 .86368 .8641 .8647 .8652 .8657 .86623 .8667 .8672 .8677 .8682 .8687 .8692 1.1598 1.1591 1.1585 1.15783 1.1571 1.1564 1,1557 1.1551 1.1544 1.1537 1.1530S 1.1524 1.1517 1.15104 1.1503 .89306 .89358 .89410 .8946 .8951 .8956 .8962 .8967 .8972 .8977 .8983 .8988 .8993 .8998 .9004 1.1197 1.1190 1.1184 1.1177 1.1171 1.1164 1.1158 1.1151 1.1145 1.1138 1.1132 1.1125 1.1119 1.111 1.1106 .92493 .92547 .9260 .9265 .9270 .9276 .9281 .9287 .9292 .9298 .93034 .93088 .9314 .93197 .9325 1.0811 1.0805 1.0799 1.0792 1.07864 1.0780 1.0773 1.0767 1.076 1.0755 1.0748 1.0742 1.073 1.072 1.072 95785 .9584 .9589 9595 .96008 .9606 .9612 .9617 .96232 .9628 .9634 .9640 .9645 .9651 .9656 1.04401 1.04340 1.04279 1.04218 1.04158 1 1.04097 1.04036 1.03976 1.03915 1.03855 1.03794 1.03734 1.03674 1.03613 1.03553 Gotang Tang Cotau Tang Cotan Tant Cotan Tang. 1 490 48 470 460 TABLE XVII. NATURAL TANGENTS AND COTANGENTS. 30 7 44 44 44 M. Tang. Cotang. 31. M. Tang. Cotang. M. M. Tang. Cotang. M. .96569 1.03553 60 20 .97700 1.02355 40 40 .98843 1.01170 20 1 .96625 1.03493 59 21 .97756 1.02295 89 41 .98901 1.01112 19 2 .96681 1.03433 58 22 .97813 1.02236 88 42 .98958 1.01053 18 3 .96738 1.03372 57 23 .97870 1.02176 37 43 .99016 1.00994 17 4 .96794 1.03312 56 24 .97927 1.02117 86 44 .99073 1.00935 16 5 .96850 1.03252 55 25 .97984 1.02057 86 45 .99131 1.00876 15 6 .96907 1.03192 54 20 .98041 1.01998 34 40 .99189 1.00818 14 7 .96963 1.03132 53 27 .98098 1.01939 83 47 .99247 1.00759 13 8 .97020 1.03072 93 28 .98155 1.01879 82 48 .99304 1.00701 12 9 .97076 1.03012 51 29 .98213 1.01820 31 4 I M i w i ^ w W O <1 525 , are for chords of 20 metres. The radii are, therefore, computed by the formula 72 = . . In Table I sin. D the radii are computed by the formula R = -^- . The radii in the metric table are, therefore, each one-fifth or .2 of the radii in Table I. for the same deflection angle. Moreover, since the ordi- nates given above and the tangent deflections vary only with the radii, these ordinates and the tangent deflections may also be ob- tained from Table I. by simply multiplying the corresponding quantities by .2, keeping in mind that corresponding quantities are those belonging to the same deflection angle. Table I., ex- cept in regard to ordinates for rails, may, therefore, be used for metric curves by simply multiplying corresponding quantities by .2. The metre will, of course, be the unit of the resulting quantities. Example. Given in a metric curve D = 3 10', to find R and the ordinates ra and m. In Table I., R = 905.13, m = 1.382, and f m 1.037. Multiplying these values by .2, we have for the metric curve R = 181.03, m .276, f m = .207, as in Table XIX. Since the Long Chords of Table II. for the same deflection an- gle vary directly with the radii, we may use this table for metric curves by multiplying the values there found by .2. We thus ob- tain in metres the length of corresponding long chords in metric curves. Example. Given in a metric curve D = 2 20', to find the long chord for five stations. From Table II. we have for an ordinary curve the long chord 496.689. Multiplying by .2, we have the required long chord in the metric curve = 99.338 metres. Tables III. and IV. may also be used for metric curves, as all the quantities vary only with the radii. Therefore, using the same 312 USE OF TABLES. deflection angle, we convert these tables into metric tables by multiplying corresponding quantities by .2, the ratio of the radii. First find T and b from the tables, as for an ordinary curve, and multiply the values so found by .2 to obtain T and b for the cor- responding metric curve. Example. Given in a metric curve 2 = 90 and D = 10, to find T and b. From the tables we should have for an ordinary curve T = + 1.45 = 287.935 and b = ' + .603 = 119.268. These values multiplied by .2 give for the metric curve T 57.587 metres and b = 23.854 metres. It is obvious that if chords of 10 metres were used in laying out a metric curve, the multiplier, as used above, would be .1, and that if chords of 30 metres were used, the multiplier would be .3. (46) THE END. RETURN CIRCULATION DEPARTMEN1 202 Main Library LOAN PERIOD 1 HOME USE 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 71 1 -month loans may be renewed by calling 6-month loans may be recharged by bringing books tol Renewals and recharges may be made 4 days priof DUE AS STAMPED BELOWJ JNTERLIBRAF Y LOAN 1 I/I/-U 8 ^70 1 UWIV r*tr j-. ii v. wr CALJ ^* BER?-r, 1 1 FORM NO. DD 6, 40m 10 '77 UNIVERSITY OF CALIFOF| . BFPKF1FY CA YA 0685; UNIVERSITY OF CALIFORNIA LIBRARY