F I E L D-B 0 K RAILROAI ENGINEERS I F I E L D-BOOK FOR RAILROAD ENGINEERS. CONTAINING FORMUL A FOR LAYING T IT CURVES, DETERMINING FROG ANGLES, LEVELLING, CALCULATING EARTH-WORK, ETC, ETC., TOGETHEFR WITS TABLES OF RADII, ORDINATES, DEFLECTIONS, LONG CHORDS, MAGNETIC VARIATION, LOGARITHMS, LOGARITHMIC AND NATURAL SINES, TANGENTS. ETC., ETC. BY JOHN B."IHENCK, A.M., CIVIL' ENGINEER. NEW YORK: D. APPLETON AND COMPANY, 443 AND 445 BROADWAY. LONDON; 16 LITTLE BRITAIN. 1866. Entered according to Act of Coneress, in the year 1864, by D APPi-rrON AND (OMPANT, tI the Clerk's Office of the its.rlct. Court for the Southern District of New Ub r PREFACE. THE object of the present work is to supply a want very generally felt by Assistant Engineers on Railroads. Books of convenient form for use in the field, containing the ordi nary logarithmic tables, are common enough; but a book combining with these tables others peculiar to railroad work, and especially the necessary formulae for laying out curves, turnouts, crossings, &c., is yet a desideratum. These formula, after long disuse perhaps, the engineer is often called upon to apply at a moment's notice in the field, and he is, therefore, obliged to carry with him in manuscript such methods as he has been able to invent or collect, or resort to what has received the very appropriate name of" fudging." This the intelligent engineer always considers a reproach; and he will, therefore, it is hoped, receive with favor any attempt to make a resort to it inexcusable. Besides supplying the want just alluded to, it was thought that some improvements upon former methods might be made, and some entirely new methods introduced. Among the processes believed to be original may be specified those in ~ 41-48, on Compound Curves, in Chapter II., on Parabolic Curves, in ~~ 106-109, on Vertical Curves, and in the article on Excavation and Embankment. It is Vi PREFACE. but just to add, that a great part of what is said on Reversed Curves, Turnouts, and Crossings, and most of the Miscellaneous Problems, are the result of original investigations. In the remaining portions, also, many simplifications have, been made. In all parts the object has been to reduce the operation necessary in the field to a single process, indicated by a formula standing on a line by itself, and distinguished by a 1z. This could not be done in all cases, as will be readily seen on examination. Certain preliminary steps were sometimes necessary, and these, whenever it was practicable, have been indicated by words in italics. Of the methods given for Compound Curves, that in ~ 46 will be found particularly useful, from the great variety of applications of which it is susceptible. Methods of laying out Parabolic Curves are here given, that those so disposed may test their reputed advantages. Two things are certainly in their favor; they are adapted to unequal as well as equal tangents, and their curvature generally decreases towards both extremities, thus making the transition to and from a straight line easier. Some labor has been given to devising convenient ways of laying out these curves. The method of determining the radius of curvature at certain points is believed to be entirely new. Better processes, however, may already exist, particularly in France, where these curves are said to be in general use. The mode of calculating Excavation and Embankment here presented, will, it is thought, be found at least as simple and expeditious as those commonly used, with the advantage over most of them in point of accuracy. The usual Tables of Excavation and Embankment have been omitted. To include all the varieties of slope, width of road-bed, and depth of cutting, they must be of great extent, and unfitted PREFACE. VII for a field-book. Even then they apply only to ground whose cross-section is level, though often used in a mannei shown to be erroneous in ~ 128. When the cross-section of the ground is level, the place of the tables is supplied by the formula of ~ 119, and when several sections are calculated together, as is usually the case, and the work is arranged in tabular form, as in ~ 120, the calculation is believed to be at least as short as by the most extended tables. The correction in excavation on curves (~ 129) is not known to have been introduced elsewhere. In a work of this kind, brevity is an essential feature. The form of "Problem" and "Solution" has, therefore, been adopted, as presenting most concisely the thing to be done and the manner of doing it. Every solution, however, carries with it a demonstration, which is deemed an equally essential feature. These demonstrations, with a few unavoidable exceptions, principally in Chapter II., presuppose a knowledge of nothing beyond Algebra, Geometry, and Trigonometry. The result is in general expressed by an algebraic formula, and not in words. Those familiar with algebraic symbols need not be told how much more intelligible and quickly apprehended a process becomes when thus expressed. Those not familiar with these symbols should lose no time in acquiring the ready use of a language so direct and expressive. It may be remarked that it was no part of the author's design to furnish a collection of mere " rules," professing to require only an ability to read for their successful application. Rules can seldom be safely applied without a thorough understanding of the principles on which they rest, and such an understanding, in the present case, implies a knowledge of algebraic formulae. The tables here presented will, it is hoped, prove relia V111 PREFACE. ble. Those specially prepared for this work have been computed with great care. The values have in some cases been carried out farther than ordinary practice requires, in order that interpolated values may be obtained from then more accurately. For the greater part of the material composing the Table of Magnetic Variation the author is indebted to Professor Bache, whose distinguished ability ir conducting the operations of the Coast Survey is equalled only by his desire to diffuse its results. The remaining tables have been carefully examined by comparing them with others of approved reputation for accuracy. Many errors have in this way been detected in some of the tables of corresponding extent in general use, particularly in the Table of Squares, Cubes, &c., and the Tables of Logarithmic and Natural Sines, Cosines, &c. The number of tables might have been greatly increased, but for an unwillingness to insert any thing not falling strictly within the plan of the work or not resting on sufficient authority. J. B. H. BOSTON, February, 1854. TABLE OF CONTENTS. CHAPTER I. CIRCULAR CURVES. ARTICLE I. - SIMPLE CURVES. 5nO. PAXG 2. Definitions. Propositions relating to the circle.. 4. Angle of intersection and radius given, to find the tangent 3 5. Angle of intersection and tangent given, to find the radius 3 6. Degree of a curve. 4 7. Deflection angle of a curve...... A. Method by Deflection Angles. 9. Radius given, to find the deflection angle.. 4 10. Deflection angle given, to find the radius.. 4 11. Angle of intersection and tangent given, to find the deflection angle 5 12. Angle of intersection and deflection angle given, to find the tangent..... 13. Angle of intersection and deflection angle given, to find the length of the curve..... 6 14. Deflection angle given, to lay out a curve.... 7 16. To find a tangent at any station...... 8 B. M/[ethod by Tangent and Chord Deflections. 17. Definitions....... 8 18. Radius given, to find the tangent deflection and chord deflection 9 19. Deflection angle given, to find the chord deflection.. 9 21. To find a tangent at any station..... 9 22. Chord deflection given, to lay out a curve.... 10 X TABLE OF CONTENTS. C. Ordinates. BECT. rPAG 24. Definition......... 11 25. Deflection angle or radius given, to find ordinates.. 11 26. Approximate value for middle ordinate. 13 27. Method of finding intermediate points on a curve approximately......14 D. Curving Rails. 29. Deflection angle or radius given, to find the ordinate for curving rails......14 ARTICLE II. -REVERSED AND COMPOUND CURVES. 30. Definitions.....15 31. Radii or deflection angles given, to lay out a reversed or compound curve.....16 A. Reversed Curves. 32. Reversing point when the tangents are parallel... 16 33. To find the common radius when the tangents are parallel 16 34. One radius given, to find the other when the tangents arc parallel...... 17 35. Chords given, to find the radii when the tangents are parallel 18 36. Radii given, to find the chords when the tangents are parallel 18 37. Common radius given, to run the curve when the tangents are not parallel....19 38. One radius given, to find the other when the tangents are not parallel......19 39. To find the common radius when the tangents are not parallel 21 40. Second method of finding the common radius when the tangents are not parallel..... 22 B. Compound Curves. 41. Common tangent point........23 42. To find a limit in one direction of each radius... 24 44. One radius given, to find the other.. 25 45. Second method of finding one radius when the other is given 26 46. To find the two radii........27 47. To find the tangents of the two branches.. 29 48. Second method of finding the tangents of the two branches. 30 TABLE OF CONTENTS. l ARTICLE III. —TURNOUTS AND CROSSINGS. sICT. pAGS 49. Definitions.....31 A. Turnout from Straight Lines. 50. Radius given, to find the frog angle and the position of the frog 32 51. Frog angle given, to find the radius and the position of the frog 33 52. To find mechanically the proper position of a given frog. 34 53. Turnouts that reverse and become parallel to the main track 34 54. To find the second radius of a turnout reversing opposite the frog....... 35 B. Crossings on Straight Lines. 55. References to proper problems.... 36 56. Radii given, to find the distance between switches...36 C. Turnout from Curves. 57. Frog angle given, to find the radius and the position of the frog 38 58 To find mechanically the proper position of a given frog. 41 59 Proper angle for frogs that they may come at the end of a rail 41 60 Radius given, to find the frog angle and the position of the frog 42 62 Turnout to reverse and become parallel to the main track. 44 D. Crossings on Curves. 63. References to proper problems...... 45 64. Common radius given, to find the central angles and chords 45 ARTICLE IV. MIaSCLLANEOUS PROBLEMS. 65. To find the radius of a curve to pass through a given point 46 b6. To find the tangent point of a curve to pass through a given point 47 67. To find the distance to the curve from any point on the tangent......47 68 Second method for passing a curve through a given point. 47 69. To find the proper chord for any angle of'deflection.. 48 70.- To find the radius when the distance from the intersection point to the curve is given...... 48 71. To find the distance from the intersection point to the curve when the radius is given....... 49 xIi TABLE OF CONTENTS. SECT. PA" 72. To find the tangent point of a curve that shall pass through a given point.....5C 73. To find the radius of a curve without measuring angles 51 74. To find the tangent points of a curve without measuring angles.......... 5? 75. To find the angle of intersection and the tangent points when the point of intersection is inaccessible. 52 76. To lay out a curve when obstructions occur.. 55 77. To change the tangent point of a curve, so that it may pass through a given point....56 78. To change the radius of a curve, so that it may terminate in a tangent parallel to its present tangent.... 57 79. To find the radius of a curve on a track already laid.. 58 80. To draw a tangent to a given curve fiom a given point.. 59 81. To flatten the extremities of a sharp curve... 59 82. To locate a curve without setting the instrument at the tangent point..... 60 83. To measure the distance across a river..63 CHAPTER II. PARABOLIC CURVES. ARTICLE I.-LOCATING PARABOLIC CURVES. 84. Propositions relating to the parabola..... 65 85. To lay out a parabola by tangent deflections... 66 86. To lay out a parabola by middle ordinates. 67 87. To draw a tangent to a parabola...... 67 89. To lay out a parabola by bisecting tangents... 68 90. To lay out a parabola by intersections.... 69 ARTICLE II.-RADIUS OF CURVATURE. 92. Definition.......... 71 93. To find the radius of curvature at certain stations... 71 95. Simplification when the tangents are equal... 76 TABLE OF CONTENTS. Xii CHAPTER III. LEVELLING. AIRTICLE I.-HEIGHTS AND SLOPE STAKES. BEOT. PAGE 96. Definitions......78 97. To find the difference of level of two points. 78 98. Datum plane.....79 99. To find the heights of the stations on a line.. 8C 100. Sights denominated plus and minus..... 81 101. Form of field notes....... 82 102. To set slope stakes...... 82 ARTICLE II.-CORRECTION FOR THE EARTH'S CURVATURE AND FOR REFRACTION. 103. Earth's curvature.....84 104. Refraction.......... 84 105. To find the correction for curvature and refraction.. 85 ARTICLE III. — VERTICAL CURVES. 106. Manner of designating grades..... 86 107. To find the grades for a vertical curve at whole stations. 86 109. To find the grades for a vertical curve at sub-stations.. 88 ARTICLE IV.- ELEVATION OF TIIE OUTER RAIL ON CURVES. H10. To find the proper elevation of the outer rail,.. 89 1l. Coning of the wheels........ 89 CHAPTER IV. EAR'TH-WORK. ARTICLE I. —PRISMOIDAL FORMULA. 112 Definition of a prismoid...... 92 113. To find the solidity of a prismoid...... 92 ARTICLE II.- BORROW-PITS. 114. Manner of dividing the' ground.... 93 Xiv TABLE OF CONTENTS. SET. PFAQ 115. To find the solidity of a vertical prism whose horizontal section is a triangle.........9 116. To find the solidity of a vertical prism whose horizontal section is a parallelogram....94 117. To find the solidity of a number of adjacent prisms having the same horizontal section.... 95 XRTICLE III.-EXCAVATION AND EMBANKMENT. A. Centre Heights alone given. 119. To find the solidity of one section..... 97 120. To find the solidity of any number of successive sections. 98 B. Cerdre and Side Heights givem 121. Mode of dividing the ground....... 9 122. To find the solidity of one section..... Io 123: To find the solidity of any number of successive sections. 104 125. To find the solidity when the section is partly in excavation and partly in embankment.... 105 126. Beginning and end of an excavation.... 107 C. Ground very Irregular. 127. To find the solidity when the ground is very irregular. 108 128. Usual modes of calculating excavation..... 109 D. Correction in Excavation on Curves. 129. Nature of the correction......110 130. To find the correction in excavation on curves... 112 132. To find the correction when the section is partly in excavation and partly in embankment.....113 TABLES. NO. PAGE I. Radii, Ordinates, Tangent and Chord Deflections, and Ordinates for Curving Rails......115 II. Long Chords.......... 119 TABLE OF CONTENTS. Xi WO. PAGE III. (correction for the Earth's Curvature and for Refraction. 120 IV. Elevation of the Outer Rail on Curves... 120 V. Frog Angles, Chords, and Ordinates for Turnouts.. 121 VI. Length of Circular Arcs in Parts of Radius... 121 VII. Expansion by Heat.......122 VIII. Properties of Materials.......123 IX. Magnetic Variation...... 126 X. Trigonometrical and Miscellaneous Fcrmula. 133 XI. Squares, Cubes, Square Roots, Cube Roots, and Reciprocals.......... 137 XII. Logarithms of Numbers.......155 XIII. Logarithmic Sines, Cosines, Tangents, and Cotangents 171 XIV. Natural Sines and Cosines.... 219 XV. Natural Tangents and Cotangents... 229 XVL Rise per Mile of Various Grades... 24 EXPLANATION OF SIGNS. THE sign + indicates that the quantities between which it is placed are to be added together. The sign - indicates that the quantity before which it is placed s 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 between which it is placed is to be divided by the second. The sign = indicates that the quantities between which it is placed are equal. The sign co indicates that the difference of the two quantities be. tween which it is placed is to be taken. The sign.. stands for the word " hence " or " therefore." Tie 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:6, and the ratio of c to d by c: d. A proportion ex presses the equality of two ratios. Hence, 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 Para. bolic. Circular curves are divided into Simple, Reversed, and Corn. pound Curves. We begin with Simple Curves. 2. Let the arc A D E FB (fig. 1) represent a railroad curve, unit. / C/ Fig. 1. G' / / 2 CIRCULAR CURVES. ing the straight lines GA and B H. The length of such a curve is measured by chords, each 100 feet long.* Thus, if the chords A D, DE, E F, and FB are each 100 feet in length, the whole curve is said to be 400 feet long. The straight lines GA and B H are always tangent to the curve at its extremities, which are called tangent points. If GA 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 a little beyond C to K, the angle K CB, formed by one tangent with the other produced, is called the angle of intersection, and shows the cbanqRc of direction in passing from one tangent to the other. The following propositions relating to the circle are derived from Geometry. I. A tangent to a circle is perpendicular to the radius drawn through the tangent point. Thus, A C is perpendicular to A 0, and B C to B 0. II. Two tangents drawn to a circle from any point are equal, and if a chord be drawn between the two tangent points, the angles between this chord and the tangents are equal. Thus A C - B C, and the angle BAC = ABC. IIl. An acute angle between a tangent and a chord is equal to half the central angle subtended by the same chord. Thus, C A BAOB. IV. An acute angle subtended by a chord, and having its vertex in the circumference of a circle, is equal to half the central angle subtended by the same chord. Thus, D A E = A D 0 E. V. Equal chords subtend equal angles at the centre of a circle, and also at the circumference, if the angles are inscribed in similar segments. Thus, A OD DOE, and D E = EA 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 CB = A O B. 3. In order to unite two straight lines, as GA and BH, by a curve. the angle of intersection is measured, and then a radius for the curve may be assumed, and the tangents calculated, or the tangents may be assumed of a certain length, and the radius calculated. * Some engineers prefer a chain 50 feet in length, and measure the length cf a urve 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 smt chords of any length. See also ~ 13. SIMPLE CURVES. 3 4 Problem. Given the angle of intersection K CB = I (fig. I), and the radius A 0 = R, to find the tangent A C = T. /"j C/ Fig. 1. 0S T Solution. Draw C 0. Then in the right triangle A 0 C we have AC (Tab. X. 3) A- = tan. A O C, or, since A 0 (= ^ I (~ 2, VI.), = tan. B I; t.'. T- =R tan.. Example. Given Z= 22~ 52', and R= 3000, to find T. Here R = 3000 3.477121 ^ 1= 11~ 26' tan. 9.305869 T= 606.72 2.782990 5. Problem. Given the angle of intersection K C B = I (fig. 1 ) and the tangent A C _ T, to find the radius A 0 = R. 4 CIRCULAR CURVES. Solution. In the right triangle A 0 C we have (Tab. X. 6) AO R = cot. A OC, or - = cot. A I; ^~c ~~T 1, ~.~. R== Tcot. i. Example. Given I = 31~ 16' and T = 950, to find R. Here T= 950 2.977724 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 0 D = 6 (fig. 1), ADEFB is a 6~ curve. 7. The deflection angle of a curve is the acute angle formed at any point between a tangent and a chord of 100 feet. The deflection angle is, therefore (~ 2, III.), half the degree of the curve. Thus, CA D or CBF is the deflection angle of the curve ADEFB, and is half A 0D or half FO B. A. Method by Deflection Angles. 8. The usual method of laying out a curve on the ground is by means of deflection angles. 9. Problem. Given the radius A 0 = R (fig. 1), to find the deflection angle C B F = D. Solution. Draw 0 L perpendicular to BF. Then the angle B OL -- B O F= D, and BL B F= 50. But in the right triangle OBL we have (Tab. X. 1) sin. B OL -= B BO 50..sin. D-. 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 CB F =. D (fig. 1), to find the radius A O = R. METHOD BY DEFLECTION ANGLES. 50 Solution. By the preceding section we have sin. D =, whence l sin. -= 50; 50 or.R. = 50e sin. Dl By this formula the radii in Table I. are calculated. Exramnp Given D = 1', to find R. Here 50 1.698970 D 1l sin. 8241855 R = 2864.93 3.457115 11. Problem. Given the angle of intersection K C B = I (fig. 1). and the tangent A C 7 T, to find the deflection angle CA D = D. Solution. From ~ 9 we have sin. D = and from ~ 5, R Tcot. I L Substituting this value of R i t.he first equation, we get sin. D 0 Tcot. j I' fr.. sin. D - 50 tan. T Example. Given I 210 and T = 424.8, to find D. Here 50 1.698970 I == 10~ 30t tan. 9.267967 0.9669bN T = 424 8 2.628185 D) - I~ 15' sin. 8.338752 12. Problem. Given the angle of intersection KCB = I (fig. 1) and the deflection angle CA D = D, to find the tangent A C = T. Soldion. From the preceding section we have sin. D = tan. - Hence, T sin. D = 50 tan. - I; l T.. T = 50 tan. & sin. D Example. Given I = 28~ and [) = 1~, to find T. Here T 0 tan. 14Tr - - - 714 31. sin. 1' 6 CIRCULAR CU IVES. 13. Problem. Given the anyle of intersection KCB = I (Yfiq. 1), and the deflection angle CA D = D, to find the length of the curve. Solution. By ~ 2 the length of a curve is measured by chords of 100 feet applied around the curve. Now the first chord A D makes with the tangent A C an angle CA D = D, and each succeeding chord D E, E F, &c. subtends at A an additional angle D A E, E A F, &c., each equal to D; since each of these angles (~ 2, IV.) is half of a central angle subtended by a chord of 100 feet. The angle CA B = A A O B = I1 is, therefore, made up of as many times D, as there are chords around the curve. Then if n represents the number of chords, we have nD = i I;.. n — -- D If D is not contained an even number of times in ^ I, the quotient above will still give the length of the curve. Thus, in fig. 2, suppose D is contained 41 times in ^ I. This shows that there will be four whole chords and ~ of a chord around the curve from A to B. The angle GA B, the fraction of D, 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 I = 16~ 52' and D = 1~ 20', to find the length of h I 80 26' 506' the curve. Here n = - == - = -- = 6.325, that is, the curve D 10209 80' is 632.5 feet long. To find the arc itself in this example, we take from Table VI. the length of an arc of 16o 52', since the central angle of the whole curve is equal to 1 (~ 2, VI ), and multiply this length by the radius of the.... Arc 10 -=.1745329 " 6~ -.1047198 " 50Q =.0145444 " 2' =.0005818' 16~ 52' -.2943789 * This method of finding the length of a sub-chord is not mathematically aceurate; 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 rchords of these arcs. In railroad curves, the error arising from this suppositior ir too small to be regarded. METHOD BY DEFLECTION ANGLES. 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 defletion angle D, to lay oat a curm from a fiven tangent point Fig. 2. Hl SodutioH. Let A (fig. 2) be the given tangent point in the tangent H C. Set the instrument at A, atd lay off the given deflection angle D from A C. This will give the direction A D, and 100 feet being measured from A in tdis direction, the point D will be determined Lay of in succession the additional angles D A E, E A F, &c., each eqhal to D, and make DE, EF, e&. each 100 feet, and the points E, F, &e. will be determined. The points D, E, F, &c., thus determined, are points oa the required curve (4 7, and { 2, III., IV.), and are called stations. If there is a sub-chord at the end, as G B, the sub-deflection angle G A B must be the same part of D that G B is of a whole chord (~ 13). 15. It is often impossible to lay out the whole of a curve, without removing the instrument from its first position, either on account 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 8 CIRCULAR CTRVES. 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 FL is the tangent required. Produce A F to K, and we have the angle KFL = A F C. But (~2, II.) A F C FA C. Therefore KFL = FA C. Now FA C is the sum of all the deflection angles laid off from the tangent at A, that is, in this case, FA C = 3 D, and the tangent FL is, therefore, obtained by laying off from A F produced an angle KFL equal to the total deflection from the preceding tangent. If the curve is afterwards continued beyond F, as, for instance, to B, a tangent B N at B is obtained by laying off from FB produced an ingle MBN= L BF= LFB, the total deflection from the pro ceding tangent FL. B. Ilethod 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 \- ~ ~- -X/ A_ 0 Produce the tangent E A, and from B draw B G perpendicular to A a. Produce also thle chords A B and B C. and make the produced METHOD BY TAN rENT AND CHORD DEFLECTIONS. 9 Darts B H and C K of the same length as the chords. Draw CI and D IK. B G is called the tangent deflection, and CH or D K the chord drflection. 18. Problemn. Given the radius A 0 = R (fig. 3), to find the tatn.ent deflection B G, and the chord deflection C HI. Solution. The triangle C 11 is similar to B O C; for the angle B 0 C = 180 - ( O B C + B C ), or, since B C0 = A B O, BO C = 180 - (0 B C + A B 0) = CB H, and, as both the triangles are isosceles, the remaining angles are equal. The homologous sides are, therefore, proportional, that is, B 0: B C = B C: C Il, or, representing the chord by c and tile chord deflection by d, R: c = c: d; c2... d —' To find the tangent deflection, draw B M to the middle of CII, bisecting the angle CB Ii, and making BM C a right angle. Then the right triangles B M1 C and A G B are equal; for B C- A B, and the angle CBM=- CBH= iBOC= AOB=BAG (~2, III.). Therefore B G = CMA= * CH = ^ 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 d(fflection d. Soltion. By the preceding section we have d = -, and- by 10, R -si — Substituting this value of R in the first equation, we find c2 sin. D d 50 This formula gives the chord deflecqtion for a chord c of any length, though D is the deflection angle for a chord of 100 feet (~ 7). When c = 100, the formula becomes d = 200 sin D, or for the tangent deflection d =- 100 sin. D. By these formulae tle tangent and chord deflections in Table I. may be easily obtained from the table of natural sines 20. The length of the curve may be found by first finding D (~ 9 or 11), and then proceeding as in ~ 13. 21. Problem. To dran a tangent to the curve at any station, as B (Jig. 3). Solution. Bisect tne chord deflection H C of the next station in M. 2 10 CIRCULAR CURVES. A line drawn through B and Ml will be the tangent required; for it has been proved (~ 18) that the angle CB M is in this case equal to k B 0 C, 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 tire chain then swung round, keeping the end at B fixed, until IIJ1 = 1 d, B Al will be the direction of the required tangent.* 22. Problem. Given the chord deflection d, to lay out a curve from a given tangent point. Solution. Let A (fig. 3) he the given tangent point, and suppose d has been calculated for a chord of 100 feet. Stretch a chainn of 100 feet from A to G on the tangent EA 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 fioml B to H on AB produced, and having marked this point, swing the chain round, uptil H C is equal to the chord deflection d. C is the second station on the curve. Continue to lay off the chord deflection from the preceding chord produced, until the curve is finished. Should a sub-chord DF occur at the end of the curve, find thle tangent D L at D (~ 21), lay off from it the proper tangent deflection L/ for the given sub-chord, making DF of the given len tth, and F will be a point on the curve. The proper tangent deflection for the slub chord may be found thus. Represent the sub-chord by c'. and tlhe corresponding chord deflection by d', and we have (~ 18) d' =-; lbut since i d -2' we have i d: d c'2: c2. Therefore d' =- d( Example. Given the intersection angle I betweei 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 - 1 = 1250 tan. 8~ 15- = 181.24; c2 1002 (~18) d= — 1250=8; * The distance B M is not exactly equal to the chord, but the error arising from taking it equal is too small to be regarded in any curves but those of very small radius. If necessary, the true length of B M may be calculated; for B M - / H - H af4 ORDINATES. 11 5i 50 ( 9) sin. D =- = 1250.04 = nat. sin. 2~ 17^t; R 1250 I 8D 15' 495' (~ 13) n..... = - 3.60. D 2) 17-I' 13.5' These results show, that the tangent point A (fig. 3) on the first tan gent is 181 24 feet from the point of intersection,- that the tangent deflection G B = ^ d = 4 feet, - that the chord deflection H C 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 tangent DL drawn, the tangent deflection for the sub-chord of 60 feet will be, as shown above, d' =. (1) = 4 X.6 = 4 X 36 = 1 44. LF= 1.44 feet being laid off fiom DL, the point F will, if the work is correct, fall upon the second tangent point. A tangent at F may be found (~ 21) by producing DF to P, making FP = DF= 60 feet, and laying off PNV = 1.44 feet. FN will be the direction of the required tangent, which should, of course, coincide with the given tangent. 23. Curves may be laid out with accuracy by tangent and chord deflections, if an instrument is used in producing the lines. But if an instrument is not at hand, and accuracy is not important, the lines may be produced by the eye alone. The radius of a curve to unite two given straight lines may also be found without an instrument by ~ 73, or, having assunied a radius, the tangent points may be found by ~ 74. C. Ordinates. 24. The preceding methods of laying out curves determine points 100 feet distant from each other. These points are usually sufficient for grading a road; but when the track is laid. it is desirable to have intermediate points on the curve accurately determined. For this purpose the chord of 100 feet is divided into a certain number of equal parts, and the perpendicular distances from the points of division to the curve are calculated. These distances are called ordinates. If the chord is divided into eight equal parts, we shall have points on the curve at every 12.5 feet, and this will be often enough, if the rails, which are seldom shorter than 15 feet, have been properly curved (~ 28). 25. Problen. Given the deflection angle D or the radius R of a curve, to find the ordinatesfor any chord. Solution. I. To find the middle ordinate. Let AEB (fig 4) be a portion of a curve, subtended by a chord A B, which may be de 12 CIRCULAR CURVES. noted by c. Draw the middle ordinate E D, and denote it by in. Produce E D to the centre F, and join A F and A E. Then (Tab. X 3\ A B Fig. 4. 1' G ED - = tan. E A D, or E = A D tan. E A D. But, since the angle A D EA D is measulicd by half the arc BE, or by half the equal arc A E. we have E A D = A FE. Therefore E D = A D tan. A FE, or E-P m c tan. AFE. Whenc = 100, AFE = D (~ 7), and in = 50 tan., D, whence,m may be obtained from the table of natural tangents, by dividing tan J D by 2, and removing the decimal point two places to the right. The value of m may be obtained in another form thus. In the triangle A D F we have D F-= /A F2" A D== V /R2 -; c2. Then m = EF- DF= R- DF, or n ==R~- /R2- c*. II. To find any other ordinate, as R N 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 =-= /R F - F G' /'RB - b, and R N R G - NG = R G -D F. Substituting thu value of R G and that of DF found above, we have F^ RN = R2 -= 2 __ — R2 - c2. ORDINATES. 13 By these formulae the ordinates in Table I are calculated. The other ordinates may also be found fiom the middle ordinate bv 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 point will divide the are E B into the same number of equal parts, and upon the further supposition, that the tangents of small angles are proportional to the angles themselves. These suppositions give rise to no material error in finding the ordinates of railroad curves for chords not exceeding 100 feet. Making, for example, four divisions of the chord on each side of the centre, and joining A R, AS, and AT, we have the angle RAN==, EAD, since RB is considered equal to a E B. But E A D = A FE. Therefore, R A N= I A FE. In the same way we should find SA 0 = 4 A FE, and TA P = f A FIE. We have then for the ordinates, R N= A Ntan. RA N = g c tan. I A FE, S 0 = A 0 tan. SAO = i c tan. A FE, and TP = AP tan. TAPP c tan. A FE. But, by the second supposition, tan. { A FE = i tan. A AFE, an AFEttan. A FE an A E,and tan. A FE = tan. A FE. Substituting these values, and recollecting that i c tan. I A FE = ti, we have r fl=T15 1 15 RN= 16 X 2 c tan. AFE = m, 8 3 S -- X c tan. A FE = m, 4 2.2A TP - X i- c tan. I A FE =- i n. In general, if the number of divisions of the chord on each side of the centre is represented by n, we should find for the respective ordi(n + 1) (n -1) m (n+2)(n-2)m nates, beginning nearest the centre,,,~i -' (n + 3) (Ln —8)m __, &e. rxample Find the ordinates of an 8~ curve to a chord of 100 feet. 15 8 Here m = 50 tan. 2~ = 1.746, R = N -- m=1.637, S 0- i m= 1.310, and P 7' —: 6 = 0.764. 26. An approximate value of m also may be obtained from the formula ma = R - ^/ - c2. This is done by adding to the quantity under the radical the very small fraction 4 R2, making it a perfect .'4 CIRCULAR CURVES. C2.quare, the root of which will be R - -. We have, then, m = R ~2 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 CHI = i ED. Intermediate points on a curve may, therefore, be very readily obtained, and generally with sufficient accuracy, in the following manner. Stretch a cord from A to B, and by means of the middle ordinate determine the point E. Then stretch the cord from A to E, and lay off the middle ordinate CH = A ED, thus determining the point C, and so continue to lay off from ihe successive half-chords one fourth the preceding ordinate, until a sufficient number of points is obtained. 1). Cnrvien lRails. 28. The rails of a curve arc usnally curved before theyv are lInid. Todo this propecrlv. it is necessary to know the middle orlinate of tihe curve for a (hord of the length of a rail. 29. Problem. Given the radius or deflection anyle of a curve, to find the middle ordinate for curving a rail of given lenlyth. Solution. Denote the length of the rail by 1, and we have (~ 25) the exact formula mn = R - V/R2 - I', and (~ 26) the approximato formula 2R This formula is always near enough for chords of the length of a rail If we substitute for R its value (~ 10) R = sinD we have, sin. D m= ~/~X ~. 100 Example. In a 10 curve find the ordinate for a rail of 18 feet in length. Here R is found by Table I. to be 5729 65, and therefore. REVERSED AND COMPOUND CURVES. I5 by the first formula, mr: - 11459.3 = 00707. By the second formula: i =.81 sin. 30' =.00707. The exact formula would give the same result even to the fifth decimal. By keeping in mind, that the ordinate for a rail of 18 feet in a 1~ curve is.007, the corresponding ordinate in a curve of any other degree may be found with sufficient accuracy, by multiplying this deci. mal by the number expressing the degree of the curve. Thus, for a curve of 50 36' or 5.60, the ordinate would be.t,7 X 5.6 =.039 ft. =.468 in. For a rail of 20 feet we have, ~ = 100, and, consequently, -- sin. D. This gives for a 1~ curve, i =.0087. The corresponding ordinate in a curve of any other degree may be found with sufficient accuracy, hy multiplying this decimal by the number expressing the degree of the curve. By the above formula for m, the ordinates for curving rails in Table I. are calculated. ARTICLE II.-REVERSED AND COMPOUND CURVES. 30 Two curves often succeed each other having a common tang en at the point of junction. If the curves lie on opposite sides of the corn mon tangent, they form a reversed curve, and their radii may be the same or different. If they lie on the same side of the common tangent, Fig. 5. they have different radii, and form a compound curve. Thus A / ('fig. 5) is a reversed curve, and A B D a comnound curve. 16 CIRCULAR CURVES. 31. Problem. To lay out a reversed or a compound curve, when the radii or dflection 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 ~ 21). Then B F will be the tangent also of the second portion B C of a reversed, or B D of a compound curve, and firom this tangent either of these portions may be laid off in the usual manner. A. Reversed Curves. 32 Theo-ren. Tie reversiyn point of a reversed curve 1etsween parallel tangents is in the line joining the tangent points. Fig. 6. F _-W ^~B K 13A Demonstration. Let A CB (fig. 6) be a reversed curve, uniting the parallel tangents HIA and B K, having its radii equal or unequal, and reversing at C. If now the chords A C and CB are drawn, we have to prove that these chords are in the same straight line. The radii E C and C F, being perpendicular to the common tangent at C (~ 2, 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 AE C= CFB, and, consequently, E CA. the half supplement of A E C, is equal to F C B, the half supplement of C I l; hut these angles cannot be equal, unless A C and CB are in the same straight line. 33. Problem. Giren the perpendicular distance between tu'o parallel tangents B D = b) (fig 6), and the distance between the hwo tangent points A B = a, to determine the reversing poin' C and the common radius E C = CF = R ofa reversed curve unitiin the tangents HA and B K. Solution. Let A C B be the required curve. Since the radii are REVERSED CURVES. 17 ctual, and the angle A E C B F C, the trianglcs A E C and B F C are equal, and A C = C B a. TAe reversing point C is, therefore, die middle point of A B. To find R, draw E G perpendicular to A C. Then the right triangles A E G and B A D are similar, since (~ 2, III.) the angle AB AD)=AEC= AE G. Therefore AE: A G = AB: BD, or R: a = a b; OF'~.. a. 4b Corollary. If R and b are given, to find a, the equation R =- givcs a2 = 4 Rb; li.'. a = 2.//1. b.'xanmples. Given b = 12, and a = 200, to determine R. Here 2002 10000! = X12 = — 12 = 833. Given R 675, and b = 12. to find a. Here a = 2,/675 X 12 = 2'/8100 = 2X 90 = 180. 34. Problem. Given the perpendicular!istance between two parrllel tangqents B D = bI (fig. 7), the distance betw.een the two tangent points,A == a, and the first radius E C = R of a: eversed curve uniting th( Frnaentfs HA and B K, to find the chords A C -- ca' and C B = a", and the second radius CF = Rt. n _____ A _____________ _______ __ /h_____.. A, Fig. 7. Sfution. Draw the perpendiculars E G and FL. Then the right triannles AB D and EA G are similar, since the angle BA D - i8 CIRCULAR CURVES. AE C A= AE G. Therefore AB: BD = E A: A G, or a: b R: ia'; 2 R b..a = —.. a Since a' and al' are (~ 32) parts of a. we have all a - a'. To find R' the similar trialngles A B D and FB L give A B: B 1 FB: B L, or a: b= I': (a"t; a at..~ Rl=,.... EIxample. Given b = 8, a = 160, and R =900, to find a', a"', and 2 X 900 X 8 /. tHere at'= 16 = 90, a" = 160- 90 = 70, and R = tG) x 70.X8 = 700. 35. Corollary 1. If b, a', and a" are given, to find a, R, and R', we have (~ 34) a ai R. a all L a = a' + al"; R = a- R I a 2b 2b Example. Given b = 8, a' = 90, and a"- = 70, to find a, R, and 1/ 160 X 90 160 X 70 Here a = 90 + 70 = 160. R = 2x =900, and R' 2- -x 700. 36. Corollary 2. If Ih, RI, and b are given, to find a, a', and a", a a' + a a" a (a' a (a + ) a2 we have (~ 35), R + R'= 2b 2b = -' Therctbre. = 2b (R + R'); ii-.. a = -/2 b (R +- R). Having found a, we have (~ 34) a't= 2 l b., _7 2 R' b a a Krample. Given R -- 900, R? =~ 700. and b = 8, to find a, a', anq a". lere a = ^/2 X 8(900 + 700) = - 16 X 1600 =X 160,' - 2 x900 x8 2 xx700x8 160 - ~ 90, and aL" = - - - 70. REVERSED CURVES. 19 37. Problem. Given the angle A K B = K, which shows the change of direction of two tangents HA and B K (fig. 8), to unite these tangents by a reversed curve of given common radius R, starting fromn a given tangent point A H /F - A -IN ^E Fig. 8. Soiution. With the yiven radius run the curve to the point D, where tIh;tsnent D N becomes parallel to B K. The point D is found thus. Since the angle N G K, which is double the angle H A D (~ 2, II.), is to he made equal to A KB K, lay off from HA the angle HA D = i K Measure in the direction thus found the chord A D = 2 R sin. ^ A This will be shown (~ 69) to be the length of the chord for a deflectiov angle I K. Having found the point D, measure the perpendicular distlance D M = b between the parallel tangents. The distance DB = 2 D C = a may then be obtained from the fr. mlla (~ 33, Cor.),Ka- - 2 2/ ib. The second tangent point B and the reversing point' are now (I. terniined. The direction of D B or the angle B D N may also be obD M tained; for sin. B D N = sin. l B MI= — D, or sin. B D)- b a a3. Probleln. G'ven the line A B = a (fig. 9) which joins tie fixed ta(ngent points A and B, the angles HA B = A and A B L = B, dnd the first radius A E = R, to find the second radius B F R' of a reversed curve to u., ite the tangents It' A and B K. P'irst Solution. With the given radius run the curve to the. point D, vhere the tangent D N becomes parallel to B K. The point D is found 2.0 CIRCULAR CURVES. thus. Since the angle H G N, which is double HA D (~ 2, II.), is equal to A cn B, lay off from l A the angle HA D -= (.4 c B), and measure in this direction the chord A D = 2 R sin. ~ (A c B) (~ 69)..F Fig. 9. / III, G/) /N Setting the instrlnument (t D, run the curve to the rreersinq7 point C in the line from D to B (~.32), and measure D C and C B. Then the similar triangles D E C and B F C give D C: DE = CB: B F or D C:'I CB: RI;.' R = C B X R.. C 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 BD. A B = a, A D = 2R sin. 2 (A - B), and the included angle DA B = HA 1 - HA D = A -- (A - B)= (A + B). Find in this triang,e (Tab. X. 14 and 12) BD and the angle A B D. Find also the angle D B L = B + A B D. Then the chord CB = 2 R' sin. ^ B F C = 2 R sin. D B L, and the chord D C == 21 sin. I DE C = 2R sin. DBL. (~ 69). But CB = BD- DC; whence 2R' sin. DBL = BD - R sin DBL; B D 2 sin. D B L When the point D falls on the other side of A, that is, when the angle B is- greater than A, the solution is the same, except that the angle D A B is then 18o0~- (A + B3). lanui the angle D B L = - A B D. REVERSED CURVES. 21 39. Problenl. Given the length of the commion tangent D G -- a, and the angles of intersection I and I (fig. 10), to determine the common radius CE = C F-'- ft of a reversed curve to unite the tangents HA an(t 1 L. Fig. 10. /' Soltion. By ~ 4 we nhve D C = R tan. 4 7, and( C G = tan. /'; whence R (tan. 1 + tan. - I') = 1) C +- C G =,. o0 tan. 1 1+ tan. ii This formula may be ndapted to calculation by logarithms; for we sin. (r+T) have (Tab. X. 35) tan.. I + tan.' - cos. Icos. Substituting this value, we get a cos. 1 cos. I' sin. - (1r+ P) The tangent points A and B are obtained by measuring from D a distance A D = R tan. I, and from G a distance B G = R tan. 7 1t. Example. Given a - o 600, 12~, and I' = 8, to find R. Here a = 600 2.778151 1 = 6~ cos. 9.997614' = 40 cos. 9.998941 2.774706 (c1+ 1') - 100 sin. 9.23967u R = 3427.96 3.535036 ) CCIRCULAR CURVES. 10 Problem. Given the line 13 = a (.fir 10). uhich joins the i.red tlangent points A and B, the an/le DA 1A = A. and the an(le A 1; C — =. to find the common radits E C -C' b= oJ a i,v rsl:~'rrle to rtnite the!anlents H A and B L. Fi. 10. // Solution. Find first the anx.vliltary angle A KE = B KF~, which i irtf be denoted by K. For this purpose the triangle A E K gives A E.: E.' = sin. K': sin. E A K. Therefore EK sin. K =_ A E sin. E~A K = R cos. A, since EA K=. 90o- A. In like manner, the triangle BFK gives FK sin K=- BF sin. FBK= R cos. B. Adding these equations, we have (E K q+ FK) sin. K = R (cos. A +- cos. B), or, since E K + F K = 2 R, 2 R sin. K = R (cos. A + cos.') Therefore, sin. K = ~ (cos. Al -+ cos. B). For calculation by logarithms, this becomes (Tab. X. 28) Pr sin K -- cos. I (A + B') cos.. (A - B). Having found K, we have tile angle A E = B E= 1803 -A'EA K - 180-3 - R (90 -.4) =! + A - K, an(d the angle B FK = F -= 1800 ~ —- FB = 180~ - K — (9(P -- ) = 90~ -+ B -. M oreoveri the triangle A E K gives A h J A K= sin. K: sin. E, or R sin. E = A K sin A and the triangle B F/K gives' F': BK = sin. K: sin F, or si.= B s in. B s.. Adding these equations, wc have /' (sin. E - sin. F) = (A K -' BA ) sin. K = 4 sin. K Sutibsituting for sir., E -+ sin. its alu 2 sin. (Es +- F COMPOUND CURVES. 23 cos. (- F) (Tab. X. 26), we have 2 II sin. 1 (E + F) cos.! a sin. K (E -F) = a sin. K. Therefore R sin. (E+Fo (EF)* Finally, substituting for E its value 90~ + A - K, and for F its value 900 + B -K. we get I (E +F)== 90- [K- (A + B)], and (E- F) - (A - B); whence rT" R= __ _ a sin. K cos. [K — (A + B)] cos. I (A - B) Exalmple. Given a =1500, A = 18~ and B = 6~, to find R1. Here (A + B) = 12~ cos. 9.990404 2 (A - B) = 6~ cos 9.997614 K = 76~ 36' 10/ sil. 9.988018 a = 750 2.8750(1 2.860379 A' - 2 (A + B) - 640 36' 10' cos. 9.632347 ( - B) = 60 cos. 9.997614 9.629961 R= 1710.48 3.233118 B.- Conmpound Curves. 41. Theorem. If one branch of a compound curve be produced, ttitil 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 paru!lel tangents. Demonstration. Let A CB (fig. 11) be a compound curve, uniting the tangents HA and B K. The radii CE and CF, being perpendicnlar to the common tangent at C (~ 2, T.), are in the same straight line. Continue the curve A C to D, where its tangent OD becomes parallel to BK, and consequently the radius DE parallel to BF. Then if the chords CD and CB be drawn, we have the angle CE D ='CFB; whence E C D, the half-supplement of C E D, is equal to F CB, the half-slipplement of CFB. But E CD cannot be equal to FC B, nnless C D coincides withl CB. Therefore the line BD pro luced passes through ihe commnon tangent point C 24 CIRCULAR CURVES. 42. PlroblenI. 7o find a limit in one direction of each radius of a ccflnfons(l curve. N // ii;Stllion. Let A I and BI (fig. 11) be the tangents of the curve. Through the intersection point I, draw I M bisecting the angle A 1 3. Draw A L and B. M perpendicular respectively to A I and B I, mccting I M in L and 1M. Then. tile radius of the branch commencing on the shorter tangent A Imust be less than A L, and the radius of the branch commencing on the longer tangent B I must be greater than B M. For suppose the shorter radius to be made equal to A L, and make IN =A 1, and join 1/ N. Then the equal triangles' A I 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 cominon tangent point would be the intersection of the straight line through B and N with the first curve; but in this case there can he 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 B1. In this case the extremity of the COMPOUND CURVES. 25 curve will fall outside the tangent B r in the line A Nproduced, and a straigllt line through B and this extremity will again fail to intersect the curve already drawn. As no common tangent point can be found when this radius is taken equal to 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. l. These values are' therefore, the limits of the radii in one direction. 43. If nothing were given l)ut the position of the tangents and the tangent points, it is evident that an indefinite number of different 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. Probletm. Given the line A B = a (fi. 11), which joins tlc fixed tangent points A and B, the anyle B A I = A, the angle A B I 13, and the first radius A E =- f. to find the second radius B F = R' of a compound curve to unite the tangents H A and B K. Solutionl Suppose the first curve to be run with the given radius from A to D, where its tanogent D 0 becomes parallel to B 1, and the angle IA D = 2 (A + B). Then (~ 41) the common tangent point C is in the line B L) produced, and the chord CB- = C)D + B D. Now in the triangle A B D we have A B =, A 1) = 2 R sin. I (A + B) ({ 69), and the included angle DA B = IA B - IA D == A (- (A - B) = (A - B). Find in this tritanle (Tab. X. 14 and 12) the angle A B D and the side B D. Find olso the angle CBI = B-A BD. Then (~ 69) the chord CB = 2 i' sin. CB 1, and the chord CD = 21t sin CDO = 2 R sin. CBI. Substituting these values of CB and CD in the equation found above, C B = CD + B D, we have 2 R sin. CBI= 2R sin. CBI+BD; BD,^.:...R'=R-^ ^+ = 2 sin. CBI When the angle B is greater than A, thit is, when the greater radius is given, the solution is the same, except that the angle D A B - 26 CIRCULAR -CURVES.. (BI -- A), and C B I is found by subtracting the suple)ment of A B D firom B. We shall also find CB = CD - B D, and consequently BD -'2sin. CB If more coni'enient, the point D may be determined in the field, by laying off the angle IA D = I (A + B), and measuring the distance A 1 --- 2 R.sit. _ iA + B). B D and CB I may then be measured, instead <,f leinl calculated as above. Example. Given a = 950, A -- 8, B = 7~, and R = 3000. to find II'. Iere A ) = 2 X 3000 sin. (8~ + 7~) = 783.16, and DA B i (80 -70) = 30. Tllen to find A B D we have.- B - D -- 166 84 2.222300 2 (At ) 3 + A4 B 1)) 890 45' tan. 2.36018C 4.582480. B + A D = 1733.1i:3.2388. (.i /J B A B D)) - 87 24' 17" tan. 1.343, 1 l.-.3 II =- 20' 4:3" Next, to find B 1),.1 =- 783.16 2.893849 ). 3 = —:30' sin. 7.940842 0.834691 AB ) - 2~ 20' 43" sin.'8.611948 B D 167.01 2.222743 L' A B D CBI - 4 391 17" sin. 8.909292 2 (R'- R) = 2058.03 3.313451.. R -R = 1029.01 I'. = 3000 + 1029.01 = 4029.01 To find the central angle of each branch, we have C'FB = 2 C B I = 9~ 18' 34", which is the central angle of the second branch; and AEC=A ED- CED — A - B- 2 CBI = 5~ 41' 26",which is the central angle of the first branch 45. Problen. Given (fig. 11) the tangents A I T, BI = T', the angle of intersection = 1, and the first radius A E = Il, to find the second radius B F = R'. Solution. Suppose the first curve to be run with- the given radn'u fiom A to D, where its tangent DO becomes parallel to B I. Throug.b COMPOUND CURVES. 27 D draw DP parallel to A I, and we have I P = DO = A O = R tan. 7 I (~ 4). Then in the triangle DPB we have D P = O = A I -AO = T -R tan. I, BP =BI - IP = T'-R tan. I 1, and the included angle DP B = A I B = 180'" -1. Find in this tliangle the angle CB, and the side B D. The remainder of the sol/t;on is the same as in ~ 44. The determination of the point D in the field is also the same, the angle IA D being here = I 1. When B is greater than A, that is, when the greater radius is given, the solution is the same, except that DP = R tan. I- T, and B P = R tan. 1 1 Tt. Example. Given T= 447 32, T7' = 510.84, 1 = 15, and R =.3Ota to find R'. Here R tan. I [ = 3000 tan. 7 ~ = 394.96, D P = 417.32 - 394.96 = 52.36, BP = 510.84 - 394.96 = 115.88; and DPB —= 1800 - 15~ = 165~. Then (Tab. X. 14 and 12) BP - DP 6352 1.802910 j (B D P + P B D) = 7 30' tan. 9.119429 0.922339 B P + D P -= 168 24 2.225929 (BDP- P B D) 2~ 50' 44" tan. 8 696410'.PBD = CBI- 4"39' 16" Next, to find B D, D P = 52.36 1.719000 DPB= 15~ sin 9.412996 1.131996 P B D = 40 39' 16t" sin. 8.909266 BD = 167.005 2.222730 The tangents in this example were calculated from the example in 4 44. The values of CB I and B D here found differ slightly fron those obtained before. In general, the triangle DBP is of better form for accurate calcuFation than the triangle A D B. 46. If no circumstance determines either of the radii, the condition may be introduced, that the common tangent shall be parallel to the iine joining the tangent points. Problem. Given the line A B = a (fig. 12), which unites the *fixrl tangent points A and B, the angle IA B = A, and the anle.4 B I = B, to fi'..d the radii A E -- R and B F = R' of a compolund rucre, hvinrg the comnion taInrent D) G parallel to A B. 28 CIRCULAR CURVES. Solaiion Let A C and B C be the two branches of the require.i curve a.:;d draw the chords A Cand BC. These chords bisect the I / Fig. 12. / / C G A1 - - F angles A and B; for the angle D A C = ID G = IA B, and ilic angle GBC= G = 1 A BI. Then in the triangle A CB we have A C: A B = sin. 4 B C: sin. ACB. But ACB= 180'(CA 1 + CB A) = 1800 - (A + B), and as the sine of the supplement of an angle is the same as the sine of the angle itself, sin. A CB sin. I (A + B). Therefore A C: a = sin. 2 B: sin. a sin t B - (A + B), or A C =. (A +s B). In a similar manner we should a sin. A B A C find B C = -in (A B)' Now we have (~ 68) - sinA and AC R' =s l.. - B, or, substituting the values of A C and B C just found. p R = 1 f sin. B. s a sin. A sin. lA sin. I (A + B)' sin. L B sin. (A B) Era:mple Given a = 950, A = 8~, and B = 70. to find R and R' Here COMPOUND CURVES. 29 A a = 475 2.676694 4 B = 3~ 30t sin 8 785675 1.462369. A = 43 sin. 8.843585 (A -+ B) = 7~ 30' sin. 9.115698 7.959283 R1 = 3184.83 3.503086 i'ransposing these same logarithms according to the formula for It e have 1 a = 475..676694 A = 40 sin. 8.843585 1.520279 2 B =3~ 30' sin. 8.785675 2 (A + B)= 7 30' sin. 9115698 7.901373 R' = 4158.21 3.6i8906 47. Problem. Given the line A B = a (fig. 12), whic'h unites the fixed tangent points A and B, and the tangents A I = T' and B I = Tl,.ofind the tangents A D = x and B G = y of the two branches of a conmpound curve, having its common tangent D G parallel to A B. Solution. Since D C = A D = x, and C G = BG == y, we have D G = x + y. Then the similar triangles ID G and IA B give ID IA = D G: A B, or T- x: T = x + y: a. Therefore T — ax = Tx + Ty (1). Also AD AI = BG: B I, or x: T =y. T. Therefore Ty = Tl x (2). Substituting in (1) the value of Ty in (2), we have a'- ax = Tx + T' x, or a x + Tx +'l'x = a T; _ T nd, since fiom (2) T' and, since from (2), y h' - a IT _ _ a a +l+ Tl' The intersection points D and G and the common tangent point C are now easily obtained on the ground, and the radii may be found by the usual methods. Or, if the angles IA B = A and A B I = B 30 CIRCULAR CURVES. have been measured or calculated, we have (~ 5) Ri = x cot.. A, and R -- y cot.. B. Substituting the values of x and y found above, we a Tcot. A a T'cot. A B have 1= - T, and R1 --,+ T +T x,'~am}le. Given a = 500, T -- 250, and 7' = 290, to find x and y. Ilere a +' + T' - 500 + 250 + 290 - 1040, whlence x = 500 X 250 - 1040 = 120.19, and y = 500 X 290 ~- 1040 - 139.42. 48. Probleml. Given the taclents A I - 7: 1 =- T', mid the angle of intersection 1, to uloite the tangont points A inl I).i:/. 13) by a compound curve, on condition that the two branches sh/:./l itc.e It/ lr angles of intersection 1 D G and I G D equal. Fig. 13. D 7 E / u.,rmi. Since I D G = I GD = 1, we have ID = I G. Represent the line I, = I G bq.r. Then if the perpendicular IH be let * The radii of an oval of given length and breadth, or of a three-centre arch of given span and rise, may also be found from these formulae In these cases A + B = 90', aT and the values of R and R' may be reduced to R = T and R' a + TI - T a- _ T'.These values admit of an easy construction, or they may be readily a culated- alculated TUlNOLT' S AND CROSSINGS. 31 Fall from 1, we have (T'al. X. I) D H = ID cos. ID = - (x cos. 1, and 1) G - 2.r cos. I I. But 1) G = /) C + CG-,4) + B (] T - x + T' — = T + 7''- 2.r'Tlerefore 2 cos. = T - T' 2'.r or 2 x + 2 x cos. I =-' + T'; whence x (T+ - T') t,, + oi, or (Tab. X. 25) _4 (Tq- T') cos. 4 I The tangents A D = T- and B G = T' - x are now readily found. With these and the known angles of intersection, the radii or deflection angles may be found (~ 5 or ~ 11). This method answers very well, when the given tangents are nearly equal; but in general the preceding method is preferable. Example. Given T= 480, T' = 500, and I= 18, to find x. Here (T + 7T')= 245 2.389166 iI = 4 30' 2 cos. 9.997318 x - 246.52 2391848 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 90, we find the radii A E == 233.48 cot. 43 30' =- 2966.65, and B F = 253.48 cot. 4~ 30' - 3220.77. ARTICLE III.- TURNOUTS AND CROSSINGS. 49. THE usual mode of turning off from a main track is by switching a pair of rails in the main track, and putting in a turnout curve tangent to the switched rails, with a frog placed where the outer rail of the turnout crosses the rail of the main track. A B (fig. 14) represents one of'the rails of the main track switched, B Frepresents the outer rail of the turnout curve, tangent to A B, and Fshows the position of the frog The switch angle, denoted by S, is the angle D A B, formed by the switched rail A B with A D, its former position in the main track. The frog angle, denoted by F, is the angle G FM made by the crossing rails, the direction of the turnout rail at F being the tangent FM at that point. In the problems of this article the glauge of the track D C. denoted by g, and the distance D B, denoted by d. are supposed to be known. The switch angle S is also supposed to be known, since its sine (Tab. X. 1) is equal to d divided by the length O)5. CIRCULAtA CURVES. of the switched rail. If, for example, the rail is 18 feet in length and d =.42, we have S -= 1 20'. A. Turnout fiomt Straight Lines. 50. I'roblc.ll Given fthe rddius' of the centre line of,t ft.rnoui (.,fr. 14), to find the frog angle G F l = F and the chord B K'. Fig. 14. // E -i R Solution. Through the centre E draw E K parallel to the mini; track. Draw B I and FK perpendicular to E K, and join EF:'Then, since E F is perpendicular to FM and F K is perpendicular to k'G, the angle E FK -- G FM1 = F; and since E B and B H are respectively perpendicular to A B. and A D, the angle E B H = D A B FK S. Now the triangle E FK gives (Tab. X. 2) cos. E F = K i F lBuit E the radius of the outer rail, is equal to R. + g, and' = -- C H = B H- B C= B E cos. E B H- B C=, + y) cos. S - (g - d). Substituting these values, we have cos. E FK - (R + ig) cos. S -(g - d) R l;-g, or cos. F = cos. S _ 9_-d R+bg From this formula F may be found by the table of natural cosines To adapt it to calculation by logarithms, we may consider g- d to be tqual to (a - d) cos. S, which will lead to no material error, since TURNOUT FROMI STRAIGHT LINES. 33 - d is very miatl~l and cos. S almost equal to unity The value of cos. F then becomes cos. F = (R - ~ + d) cos. S. R + 1y R+2y To find BF, the right triangle B CF gives (Tab. X. 9) B F in BF But BC = y - d and the angle BFC = BFECFE - (900 - BE F)-(90~ - F)= F- B E F. But BEF- BLF- EBL = F S. Therefore BFC = F2 (F -8) = 2 (F- 4 S). Substituting these values in the formula (or B F, we have B F = g- d sin.' (F+ S) By tle abtve formule the columns headed F and B Fiu Table V are calculated. Example. Given g = 4.7, d =.42, S== 1~ 20t, and R = 500, to find Fand B F. Here nat. cos. S =.999729, g - d = 4.28, R + q g -- 502.35, and 4.28.- 502 35 =.008520. Therefore nat. cos. F =.99'!729 -.008520 =.991209, which gives F = 7~ 36' 10". Next, to ilitl i) F, g - d 4.28 0.631444 j (F+ S) = 4~ 281 5" sin. 8.891555 BF== 54.94 1.739889 1 Problemn. Given the frog angle GFM = F (fig. 14), to find the radius R of the centre line of a turnout, and the chord B F. Solution. From the preceding solution we have cos. F cos. S- (g - -d), or cos. S - cos. F For calculation by logarithms this becomes (Tab. X. 29) _ Rq= ____ (g — d) R 2 sin. (F+ S) sin. (F- S) Having thus found R + 2 g, we find R by subtracting. g. B F is found, as in the preceding problem, by the formula si. Fi= q- -d 3 sin. (F -S) 32 34 CIRCULAR CURVES. Example. Given g = 4.7, d =.42, S = 10 20', and F = 7~, to find R. Here (g — d) 2.14 0.330414 (F - S) = 4 10' sin. 8.861283 (F- S) 20 50' sin. 8.693998 7 555281 R + g = 595 85 2.775133..R -, 593.5 52. Problem. To find mechanically the proper position of a given frog. Solution. Denote the length of the switch rail by 1, the length of the frog byf, and its width by w. From B as a centre with a radius B H = 21, describe on the ground an are G H K (fig. 15), and from the inside of the rail at G measure G H - 2 d, and from H measure IlK such that H K: B H =. w: f, or H K: 21 = w: f; that is, i I IIK-. Then a straight line through B and the point K will 4trike the inside of the other rail at F, the place for the point of the -A Fig. 15. rog. For the angle HB K has been made equal to F, and if B M be drawn parallel to the main track, the angle M BH is seen to be equal to I S. Therefore, MB K = B F C (F - S), and this was shown (~ 50) to be the true value of B F C. 53. If the turnout is to reverse, and become parallel to the main track, the problems on reversed curves already given will in general be sufficient. Thus, if the tangent points of the required curve are fixed, the common radius may be found by ~ 40. If the tangent point at the switch is fixed, and the common radius given, the reversing oint and the other tangent point may be found by ~ 37, the change f direction of the two tangents being here equal to S. But when the TURNOUT FROM STRAIGHT LINES. 35 fi'og angle is given, or determined from a given first radius, and the point of the frog is taken as the reversing point, the radius of the second portion may be found by the following method. 54. Problem. Given the frog angle F and the distance H B - b (Jig. 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 t)oint of the frog. Fig. 16. A / /!!l~ ~ ~ ~ K B Solution. Let the arc FB be the inner rail of the second branch, F G -= RI - g its radius, and B the tangent point where the turnout becomes parallel to the main track. Now since the tangent FK is one side of the frog produced, the angle HFK = F, and since the angle of intersection at K is also equal to F, BFK= K F (~ 2, II.); whence BFH = iF. Then (~ 68) FG -= in F or R- g R = jBF F HR Sub sin. F.But B F- s=in. BFH (Tab. X. 9), or B F sin. Su stituting this value of I B F, we have sin. F In measuring the distance H B = b, it is to be observed, that the widths of both rails must be included. 31 6CIRCULAR CURVES. Example. Given b 6 2 and F = 8, to find R'. Here. b 3.1 0.491362 F = 4 sil. 8.843585 B -F= 44.44 1.647777 F= 4' sin. 8.843585 RI- g = 63708 2.804192.'. - 639.43 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 onl both tracks a crossing on a straight line is a reversed curve between parallel tan gents. Let HD and N K (fig. 17) be the centre lines of two paralle, tracks, and HA and B K the direction of the switched rails. If now the tangent points A and B are fixed, the distance A B = a may be measured. and also the perpendicular distance B P = b between the tangents HP and BK. Then the common radius of the crossing A C 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:ro. 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 (fqi. 17) between the centre lines of two parallel tracks, and the radii E C = R and CF - RI of a crossing, to find the chords A C and B C. Solution. Draw E G perpendicular to the main track, and A L, C jM, and B L' parallel to it. Denote the an.yle A E C by E. Then, since the angle A EL = A HG = S, we have CEL = E + S, and in the right triangle CEM1 (Tab. X. 2), CE cos. CEM = R cos. (E + S) = E M = - E L - L. But E L = A E cos. A E L R cos. S, and L M: L' M = A C: B C. Now A C: B C EC: CF =R R:'. Therefore, L MI: L' M =- R: R', or LM 3: LM + L'M = R: R + RI; that is, L M: h - 2 d = R: R + R', whence R (b - 2 d) L M = R( + R'. Substituting these values of E L and L il in the equation for R cos. (E + S), we have R cos. (E + S) = IR cos. S'R (b 2 d) R+ RI' CROSSINGS ON STRAIGHT LINES. 37.. COS. (E + S) = cos. S- d. Having thus found E.+S, we have the angle E and also its equal CFB. Then (~ 69) i A C= 2 R sin.U E; BC= 2 Rt'sin. E. We have also A B = A C+ B C, since A C and B Care in the same straight line (~ 32), or A B = 2 (R + R') sin j E. F H C 1.._ ___ j_ D_ L_ A EB^ Fig. 17. When thle two radii are equal, the same formulae apply hv making' =- R. In this case, we have cos. (E + S) cos. S - d 2R B IBA C = B C= 2R sin. I E. Example. Given d..42, g = 4.7, S = 1 20', b = 11, and the angles of the two fiogs each 70, to find A C = B C = A B. The common.radius R, corresponding to F= 70, is fobnnd (~ 51) to be 593.5. Then 2R = 1187. b - 2 d = 10.16, and 10.16.- 1187.00856. Thereforec nat. cos. (E + S).=.99973 -.00856 =.99117; whence E + S = 7~ 37 15". Subtracting S, we have E = 6~17' 15" Next 2 1187 3.074451 E = 308' 37"1 sin. 8.739106 A C = 65.1.813557 38 CCIRCULAR CURVES. C. Turnout fiom Curves. 57. Problem. Given the radius R of the centre line of the marr track and the frog angle F, to determine the position of the frog by means of the chord BF (figs. 18 and 19), and to find the radius R' of the cen Ftra line of the turnout. D Fig. 18 Solution. I. When the turnout is from the inside of the curve (fig. 18). Let A G and CF be the rails of the main track, A B the switch rail, and the arc B F the outer rail of the turnout, crossing the inside rail of the main track at F. Then, since the angle E FK has its sides perpendicular to the tangents of the two curves at F, it is equal to the acute angle made by the crossing rails, that is, E FK -= F Also E B L = S. The first step is to find the angle B K F denoted by K. To find this angle, we have in the triangle B FK (Tab. X. 14), BK+KF: B K-KF=tan (BFK+ FBK): tan. (BFK —FBK). But B K = R + -g- d, and KF = R- g. Therefore, BK - KF — 2R-d, and B K- KF= g - d. Moreover, BFK= BFE + EFK=BFE+F, and FBK=EBF-EBK= B FE-S. Therefore, BFK- FB K= F+ S. Lastly, B F + FBK= 1800 - K. Substituting these values in the preceding roportion. we have 2 R — d: g -d - tan. (900 - K): tan. I (F+ S). 2 2/ ~ rhj TURNOUT FROM CURVES. 39 (2 R - d) tan. j (F + S) or tan. (900 K- h) = -t - )d But tan. (900 - K) = cot. K = ta. K; Io..tan.K= g-d (2R- d) tan. (F+S) Next, to find the chord B F, we have, in the triangle B FC BC sin. BC-F (Tab. X. 12), B F = in. B' ButBC=g-d, andBCF= 1800 -F CK = 1800 - (90~- K) = 900 + K, or sin. B CF =cos. K. Moreover, BFC= I(F+S); for BFK=KFC - BFC,andFBK=KCF -B C = KFC-BBFC. Therefore, BFK- FB K= 2 BFC. But, as shown above, BFKF B K = F+- S. Therefore, 2 BF C = F+ S,or B FC = (F+ S). Substituting these values in the expression for BF, we have B F= (9g — d) cos. 9 K. sin. j (F+- S) Lastly, to find R', we have (~ 68) R' + g = EF = sin BEF But BEF = BLF- EBL, and BLF = LFK+ LKF= F+ K. Therefore. BEF F F- K- S, and B F R q = sin. (F+ K- S) II. When the turnout is from the outside of the curve, the preceding solution requires a few modifications. In the present case, the angle EFK' = F (fig. 19) and EB = S. To find K, we have in tlle triangle B F K, K F + B K: KF- B K = tan. I (FBK * BFK):tan. (FBK —BFK). But KF = RR+ g,andBK =R- g+ d. Therefore, KF + BK = 2 R + d and KFBK = g - d. Moreover, FB K 1800 - FB = 1800(E B F -E B L) 1800 -(E B F-S), and B FK = 180 BFKI - 180' - (BFE + EFK') = 1800 - (EBF + F). Therefore, FB K-B FK = F + S. Lastly, FB K + B FK - 180 - K. Substituting these values in the preceding proportion, we have 2 R + d:. - d = tan. (900 - K): tan. i (F + S), or - (2 R + d) tan. (F + S) ) tan. (90 - - 2 =d Buttan.(9002 g-d Eot. = t M *r *. tan. KR= -- (2 R + d) tan. 2 (F+ S) 40 CIRCULAR CURVES. Next to find BF, we have, in the triangle B Ft' 3F; Csin. BCF sin. -FC But B C = g-d, and BCF= 905. h of B F may be obtained g- d by making cos. K = 1. This gives BF which is identical sin. j (F+ +S)' with the formula for BF in ~ 50. Table V. will, therefore, give a close approxination to the value of B F on curves also, for any value of F contained in the table. TURNOUT FROM CURVES. 41 But B F - BL F:- EBL, andBLF= LFK - LKF - F - K. Therefore, B E F =.F-K- S,and -- Rf+.g= g BF. sin. (F- K — S) Eranuple. Given y = 4.7, d ==.42, S -= 1 20', R = 4583.75, and F = 7', to find the chord L/B' and the radius I1' of a turnout from the outside of the curve. Here - d = 4.28 0.631444 0.631444 2 R + d = 9167.92 3.962271 (1'" + S) = 4 10' tan. 8.862433 sin. 8.861283 2.824704 1.770161 2 - = 22' 1.8" tan. 7.806740 cos. 9.999991 B' = 58.905 1.770152 2 0.301030 } (F — K- S) = 2` 271 58.2" sin. 8.633766 8.934796 RI + g = 684.47 2.835356..' = 682.12 58. Problemic.. -To find mechaclically the proper position of a gitve frog. Solution. The method here is similar to that already given. w-lcn tle turnout is fiom a straight line (2 52). Draw B 11 (is. 18 iand 19) parallel to F' C, and we have FB Al B F C = ( (F + S), as just shown (~ 57). This angle is to he laid off fiom B Al; but as F is the point to te found, the chord F C can be only estimated at first, and BM taken liarallel to it, froim which the angle i (F' + S) may lie laid off bh the method of~ 52. Ill this case, however, the first mneasure on the-arc is: d. and not 2 d. since we have here to start from B 1M, and not from the rail Having thus determined the point F approxinately, B 3I may be laid off more accurately, and F found anew. 59 When frogs are cast to be kept on hand, it is desirable to have them of such a pattern tlat they will fall at the beginning or end of a certain rail; that is, the chord B F is known, and the angle F is required. 42 CIRCULAR CURVES. Problem. Given theposition of a frog by means of the chont B P (Jfis. 14, 18, and 19), to determine the frog angle F. Solution. The formula BF g= - (+ which is exact on sin. {J (F- S))' straight lines (~ 50), and near enough on ordinary curves (~ 57, note), gives t;_ ^sin. i (F + S) - g —d B F By this formula I (F + S) may be found, and consequently F. 60 Problem. Given the radius R of the centre line of the main trck, and the radius R' of the centre line of a turnout, to find the frog angle F, and the chord B F (figs. 18 and 19). Solution. I. When the turnout is from the inside of the curve (fig. 18). In the triangle BE Kfind the angle B E K and the side E K. For this purpose we have B E = RI +- g, B K = ft + g- d, and the included angle E B K = S. Then in the triangle E FK we have E K, as just found, E F = R'I + g, and FK = R- g. The frog angle E FK = F ay, therefore, be found by formula 15, Tab X., which gives E^1t, tan. F 4(s -b) (s -c) s (s-a) where s is the half sum of the three sides, a the side E K, and b and e the remaining sides. Find also in the triangle EFK the angle FEK, and we have Mie angle B E F = B E K - FE K. Then in the triangle BE F we have (~ 69) 17 B F = 2 (R + 1 g) sin. l BE F.* II. When the turnout is from the outside of the curve (fig. 19). In the triangle B E K find the anqle B E K and the side EK. For this purpose we have B E = R' + gq, B K= - R-+ g + d, and the ineluded angle EB K = 180- S. Then in the triangle EFK wp have E K, as just found, E F = R' + g, and FK = R + - g. Tlic angle E FK may, therefore, be found by formula 15, Tab. X, which C sEK -- b) (s - e) gives tan. EFK = ( - s(s-a)'. But the angle EFK' = F * The value of B F may be more easily found by the approximate formula B F = g-d i. - (F +-) and generally with sufficient accuracy. See note to ~ 67. This remark applies also to B Fin the second pat of this solution. mark applies also to B F in the second part of this solution. TURNOUT FROM CURVES. 43 - 180 - E FK. Therefore 1 F = 90~ L E FK, and cot B;' I tan. j E FK; __..cot. F = (s-b) (s-c) s (s -a) where s is the half sum of the three sides, a the side E K, and b and c the remaining sides. Find also in the triangle E FK the angle FE K, and we have the angle BE F= FEK - BEK. Then in the triangle B E F we have (~ 69) 1 =B F = 2 (RI + 9g) sin.! BE F. Exani;le. Given g = 4.7, d =.42, S -= 1 20, 2 R == 4583.75, and IR' -- 682.12, to find and the chord B F of a turnout fiom the outsid6 of the curve. Here in the triangle B E K (fig. 19) we have BE = R1t + g = 684.47, B K = R - I g + d - 458182, and the angles EK+ BKE = S = 10 20'. Then B K - B E = 3897.35 3.590769 2 (BEK-+ BKE) - 40' tan 8.065806 1.656575 B K+ BE = 5266.29 3.72150F (B E K- B KE) = 29.6029' tan. 7.935070.. B E K == 1 9.6029' BK sin EBK EK is now found by the formnla E K = sin B EK' or log. E i 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 FK, s = (5266 12 - 684.47 + 4586.10) = 5268.34, s - a = 2.22, s - b - 4583.87, and s - c = 68224. s - b = 4583.87 3.661233 s - c = 682.24 2.833937 6.495170 s = 5268.34 3.721674 s - a = 2.22 0.346353 4.068027 2)2.427143 F= 3~ 3(' cot. 1.213571..F- 7 * This angle and the sine of lo 9 6029' below, are found by the method given in connection with Table XIII. If the ordinary interpolations had been used, we should have found F = 70 7T whereas it should be 73, since this example is the ec.wrerse-of tiat in ~ 57. 44 CIRCULAR CURVES. To find-FEK, we have s as before, but as a is here the side FK opposite the angle sought, we have s - a = 682.24, s -- b 4583 87, and s - c = 222. Then by means of the logarithms just used, we find J FE K = 3" 2' 45". Subtracting 2 B E K = 341 48", we have.3 E F = 2~ 27' 57".,Lastly, BF = 1368.94 sin. 2~ 27' 57" - 58.897. g-d The fornmula BF sin. (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 formulm, but to inaccuracies inci(ent to tle calculation. 61. If the turnout is to reverse, in order to join a track parallel to the main track, as A C/B (fig. 20), it will be necessary to determine the reversing points C and B. IThese poilnt- will be determined, if we find the angles A E Cand BFC, and the clhords l. C anld C I/ 62 Problem. Given tle radius D = R1 (fig 20) of thie centre line of the main track. the conmlon radius E C 1= C F = of the centre line of a turnout, and the distance B G == b lbetieen the centie lines of the parallel tracks, to find the central angles A E U Con 13 F C aid ile chords A CandB C......D / /C! Fig. 20. Slution. In the triangle A E K find the angle A EK and the side CROSSINGS ON CURVES. 45 E K. For this purposee ehave- A E -R', A K- R- d, and the included angle E A K- S. Or, if the frog angle has been )cpreviously calculated by ~ 60, the values of A E K and E K are already known.* fai'd in the triangle E FK the anyles E FK and FE K For this purpose we have E K, as just found, E F = 2 R', and 1 K = 1t + R - b. Then AE C = AEK - FEK, and B F -- EFK. Lastly, (~ 69)! AC==2R'sin AEC; CB=2R'sin. BFC. 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 the angles which A B makcs with the tangents at A and B The common ralius of the crossing, may tlen te found by ~ 40; or if one radius of the iossing is given, the other may be found by ~ 38. But if one tangent ihoint A is fixed, and the common radius of the crossing is given, it will be necessary to determine the reversing point C and the tangent ploint B. These points will be determined, if we find tlie angles A E C and B F C, and the chords A C and C B. 6 t. Problem. Given the radius D K = I (Jfq. 21) of the centre;ine?f the main track, the eommon rcdius' E C =: C F —'`k of the centre line of a. crossing, and the distance LD G = b between the..entre lines of the {(arlallel tracks, to find the centlrd angles A I C and B' C anld the clho, ds A. Cand CB. Solution..In the triangle A E K find the angle A E K and the side E K. For this purpose we have A E = R', A K:R -d, and the included angle E A K =S.'Find in.the. triangle B FK the angqle B FK and the side FK. For tlis purpose we have BF -., BK.= R - b.- + d, and the included almgle'B K = 180~-S. Find in the triangle E FK the angles FE K and EFK. For this -:* The triangle A E:K does not correspond precisely with BE K in:,60, A being on the centre line and B on the outer rail; but the difference is too slight to affect the calculations. i6 CIRCULAR CURVES. purpose we have E K and FK as just found, and E F =- 2 1?'. TiLr AEC= AEK-FEK, and BFC== EFK-BFK Lastly (~ 69,) L A C= 2R'sin. I AE C; CB= 2R' sin. B L('. D Fig. 21. ARTICLE IV.-MISCELLANEOUS PROBLEMS. 65. Problem. Given A B = a (fig. 22) and the perpendiculat B' C = b, to find the radius of a curve that shall pass through C and the tangent point A. Solution. Let 0 be the centre of the curve, and draw the radii A 0 and C O and the line CD parallel to A B. Then in the right triangle COD we have O C = CD2 + OD2. But O C= R, CD = a, and OD = A O AD R-b. Therefore, R = a2 + (R -b) = a2 + R2 - 2 Rb + b2, or 2 Rb = a2 + b2; t...R=_ + b. 2b Example. Given a = 204 and b = 24, to find R. Here R 204+ 2879. X-4'+ 2 867 - 12 = 879. MISCELLANEOUS PROBLEMS. 47 66. Corollary 1. If R and b are given to find A B =c a, that is, to determine the tangent point from which a curve of given radius A B D C Fig. 22. must start to pass through a given point, we have (~ 65) 2 Rb = a2 +', or a = 2 Rb b2; WQJ.-. a =,/b (2R —b). K'.raqmple. Given b =- 24 and R = 879, to find a. Here a -,/1I (1758 - 24) = 41616 = 204. 67. Corollary 2. If R and a are given, and b is required, we have (~ 65) 2 Rb = a2 + b2, orb2 - 2 R b =- a. Solving this equation, we find for the value of b here required, 5 b1 = R - JR2 - a. 68. Problem. Gicen the distance A C = c (fi/. 22) and the anq/e B A C = A, to find the radius R or deflection angle D of a curve, that (hall pass through C and the tangert point A. Solution. Draw 0 E perpendicular to A C. Then the angle A 0 E -- A 0 C = BA C = A (~ 2, III.), and the right triangle A OE gives AE (Tab. X. 9) A 0-si. A O in. A...= ~ sin. A To find D, we have (~ 9) sin. D =. Substituting for R its value just found, we have sin. D = 50 - sinA; t8 CIRCULAR CURVES... sin. D) 100 sin. A S. sin. ZD= c Example. Given c = 285.1 and A - 50, to find R anti /) Hfere 142.7 100 sin. 50 sin. 5-' in. 5 -- 1637.3; and sin. D _- 285.4 - 28r4 - or D = 1 45'. 69. Problenl. Given the radius R aol the de fiectioun auyle D o/ a crl'e, (cd thie l!(jle B A C A (fi/. 22), made by any chord uwith the tfanyent at A, to fJid lhe lenillh of the clord A C = c. Solution. If RI is given, we have (~ 68) R -= sin-A; 100.cs1s in. A If D is given, we have (~ 68) sin. D -100 si A. 100 sin. A sin. D This formula is useful for finding the length of chords, when a curve is laid out ht points two, three, or more stations apart. Thus, suppose that the c(nurve A Cis four stations long, and that we wish to find the length of the chord A C. In this case the angle A = 4 D and c 100 sin. 4 D. 13.Bv this method Table II. is calculated. sin. D Exuample. Given R = 2455.7 or D = 10 10' and A ==40 401 to tind c. Here, by the first formula, c = 4911.4 sin. 40 40' = 399 59. 100 sin. 40 40' Iy the second formula, c si n. 10' 399.59. 70. Problen. Given theangle of intersectioon KCB = (fi/. 23), and the distance CD = b from the ihtersection point to the curve in the direction of the centre, to find the tangent A C = T, and the radius A 0 = R. Solution. In the triangle A D C we have sin. CA D): sin. A D C - CD.AC. But CA D = A OD= I (~ 2 11. I and VI.), and as the sine of an angle is the same as the sine of its supplement, sin A D C = sin A DE =-cos. DA E cos. jL. Moreover, CD -- ) and. A C. T. Substituting these values in the preceding proportion, we have sin. 1I: cos. I - b: T, or T - b. s i whence (Tab. X. 33) MISCELLANEOUS PROBLEMS. 49 T=bcot. I. To find R, we have (~ 5) R = T cot. I. Substituting for'' its value just found, we have UtV R- =b cot. I cot. I /K Fig. 23. \ example. Given I = 30o, b- 130, to find 7' and R. Here b = 130 2.113943 I = 70 30' cot. 0.880571 7' = 987.45 2.994514 I= 150 coL. 0.571948 R = 3685.21 3.566462 71. Problem. Given the angle of intersection K CB = 1 (.fi. 23) znd the tangent A C = T, or the radius*A 0 = R, to find C D- b. Solution. If T is given, we have (~ 70) T = b cot. / I, or b = T eot''r 3!t:..*. bb==T tan. 1. If R is given, we have (~ 70) R = b cot. I cot. 1, orb =.R cot Icot. ~ I' CS^~ ~.'. b -= R tan. 7 Itan. I. 50 CIRCULAR. CU RVES. Exanple1 Given I= 270, T= 600 or R = 2499 18, to find i Here b = 600 tan. 60 45' = 71.01, or b = 2499.18 tan. 60 45 tan. 130 30' = 71.01. 72. Problem. Given the angle of intersection I of two tangents A C and B C (fig. 24), to find the tangent point A of a curve, that shall pass through a point E, given by CD = a, D E - b, and the angle CD E = I. C +F G Solution. Produce DE to the curve at G, and draw CO to the centre 0. Denote D Fby c. Then in the right triangle C D F we have (Tab. X. 11) DF= CD cos.CDF, or 1I c = a cos. I. Denote the distance A D from D to the tangent point by x. Then, by Geometry, x2 = DE X DG. But DG = D F+- FG = DF + EF= 2DF- DE = 2 c-b. Therefore, x = b(2 c -b),and Ut x = Vb (2 c - b). Having thus found A D, we have the tangent A C = A D + D C -x + a. Hence, Ror Dmaybe found (~5 or ~ 11). If the point E is given by E H and CH perpendicular to each other, a and b may be found from these lines. For a = CH + D i CH+ EHcot. (Tab. X. 9!. andb = DE = s^I sin. A I' MISCELLANEOUS PROBLEMS. 51 E.'xample. Given I = 200 16', a - 600, and b = 80, to find x and R. Here c = 600 cos. 100 8' = 590.64, 2c - b = 1101 28, and x = v80 X 1101.28 = 296.82. Then T= 600 + 296.82 = 896.82, and R - 896.82 cot. 100 8' = 5017.82. 73. Problem. (iven the tangent A C (fig. 25), and the chord A B, uniting the tangent points A and B, to find the radius A 0 — R. C Fig. 25. \ / Solution. Mensr'e or calculate the perpendicular CD. Then if CD be produced to the centre 0, the right triangles AD C and CA 0, having thin ngle at C common, are similar, and give CD: A D A C: A 0, or RADx AC C If it is inconvenient to measure the chord.1 B, a line E F, parallel to it, may be obtained by laying off from C equal distances CE and CF. Then measuring E G and G C, we have, from the similar triGExAC angles E G Cand CA 0, C G: GE = A C: A 0, or R- cG Example. Given A C 246 and A D = 240, to find R. Here 240 x 246!D = 54, and R = 4 - 1093.33. 64 3.3 52 CIRCULAR CURVES. 74. Probleln. Giaven the radius A 0 = -1- (fi, 25), to. find tlh tangent A C = Tof a, curve to unite two straight lines given on khe ground -Solution. Lay of' fronm the intersection- C of the yiven straiygt lines any equal distances CE and CF. Draw tlte perpendicular C G to the middie ff E F, (amel measure G E and C G. Then the right triangles E G C and C A 0, having the angle at C common,.arc similar, and give GE: C G A 0: A C, or _7 rT=_CGx AO GE By this problem and the preceding one, the radius or tangent points of a curve may be found without an instrument for measuring angles. E.xamplt.r Given ]1 - 1t)9:3, (E = 80, and C G = 18, to find T. 18 X 10931 Here 80 26. 75 Problem. To find th e uanfle of intersee'li( I of two straiight lines, when the point of inftrseclion is iarcessible, and to determiine the tangent points, wlhen the lenyth of the trn/enls is.qir.n Solution. I. To find the angle of intersection I Let A C and C V (fig. 26) be the given lines Sight f-oln- some point A on one line to a point B on the other, and measure the anyles C A B and T B V. These angles make up the change of direction in passing from one tangent to the other. But the angle of intersection (~ 2) shows thie change of direction between two tangents, and it must, therefore, be equal to the sum of CA B and TB V, that is, 0I II= CAB+ T V But if obstacles of any kind render it necessary to pass fiom A Cto B V by a broken line, as A D E F B, measure the angles CA D, ND E, P E F, R F B. and S B V, observing to note those angles as minus which are laid off contrary to the general direction of these angles. Thus the general direction of the angles in this case is to the righ.t;: but the angle PE F lies to the left of: DE produced, and is therefore to he marked lminus. The angles to be measured show the successive changes of direction in passing from one tangent to the other. Thus C A D shows the change of direction between the first tangent and A D, ND E shows the change between A D produced and D E, P E F the change between DE produced and E F, R FB tile change lbetween E F produced and FB, and, lastly, S B V the change between B,' pro IISCELLANEOUS PROBLEMS. 58 duced and the second tangent. But the angle of intersection (4 2) shows the change of. direction in:passing from one tangent to another, and it nust, therefore, he equal. to the sum -of the partial changes measured, that is,: I1 -CAD+ NDE-PEF+ FB + SBV. C Fig. 26. p T w b d II. To determine the tangent points. This will be done if we find the distances A C and B C; for then any other distances from C may be found. It is supposed that the distance A B, or the distances A D, DE, E F, and FB have been measured. If one line A B connects A and B,find A C and B C in the triangle A B C. For this purpose we have one side A B and all the angles. if a broken line A D E F B connects A and B, let. fall a perpendicultr B G from B3 upon A C, produced,if necessary, and find A G arid B (C b.q the usual method of working a tcraverse. Thus, if A C is takeh as a meridian line, and D K E L, and FM are drawn parallel to A C, and D H, E K, and FL arc dlrawn parallel to B G, the difference of latitude A G is equal to the sum of the partial differences of latitude A H, D K, EL, and F.M, and the departure B G is equal to the sum of the partial departures D H, E K, FL, and B M. To find these partial differences of latitude and departures, we have the distances A D, DE, EF, and FB, and the bearings may he obtained from the angles already measured Thus the bearing of A D is CA D, the hearing of DE is KDE = KDN+ NDE- = CAD + NDE, the bearing of EFis LEF= LEP-PEF= KDE-PEF, and the 64 CIRCULAR CURVES. bearing of FB is MFB = MFR + RFB - L E F + RFB: that is, the bearing of each line is equal to the algebraic sum of the preced ing bearing and its own change of direction. The differences of latitude and the departures may now be obtained from a traverse table, or more correctly by the formulae: Diff. of lat. = dist. X cos. of bearing; dep. = dist. X sin. of bearing Thus, A H = A D cos. CA D, and DH= AD sin. CA D. Having found A G and B G, we have, in the right triangle B G C, BG (Tab. X. 9) G C = B Gcot. B CG, and B C = sin. C O But B C G = 1800 -I. Therefore, cot. B C G =-cot. I, and sin. B C G BG = sin. I. Hence G C=- BG cot. 1, and B C = sin. Then, since A C = A G + G C, we have!F AC AG- BG cot. I; B C ==..BG sin. I When I is between 900 and 180~, as in the figure, cot. I is negative, and -B G cot. I is, therefore, positive. When I is less than 90~, G will fall on the other side of I; but the same formula for A C wil still apply; for cot. I is now positive, and consequently, -B G cot. I is negative, as it should be, since, in this case, A C would equal A G minus G C. Example. Given A D = 1200, DE = 350, E F = 300, FB310, CAD = 20~, NDE = 44~, PE F =- 25~, R FB = 31 and SB V'== 30~, to find the angle of intersection I, and the distances A C and B C. Here I =20~ + 44~- 25~ + 31~ + 300 = 100~. To find A G and B G, the work may be arranged as in the following table:Angles to the Right. Bearings. Distances. N. 0o 0 20 N. 20 E. 1200 1127.63 410.42 44 64 350 153.43 314.58 -25 39 300 233.14 188.80 31 70 310 106.03 291.30 1620.23 1205.10 The first column contains the observed angles. The second contains *ile bearings, which are found from tne angles of the first column, in MISCELLANEOUS PROBLEMS. 55 the manner already explained. A C is considered as running north fiom A, and the bearings are, therefore, marked N. E. The other columns require no explanation. We find A G = 1620.23, and B G 1205.10 Then G C - B G cot. I = - 1205.1 X cot. 1000~ 212.49. This value is positive, because it is the product of two negative factors, cot. 1000 being the same as -cot. 800, a negative quantity. Then A C= A G + G C = 1620.23 4- 212.49 = 1832.72, and 1205.1 B U-sin 1003 — 1223.69. Having thus found the distances of A and B from the point of intersection, we can easily fix the tangent points for tangents of any given length. 76. Problenm. To lay out a curve, when an obstruction of any kind prevents the use of the ordinary methods. Fig. 27. 0 Solution. First Method. Suppose the instrument to be placed at A (fig. 27), and that a house, for instance, covers the station at B, and also obstructs the view from A to the stations at D and E. Lay off fiom A C, the tangent at A, such a multiple of the deflection angle D, as will be sufficient to make the sight clear the obstruction. In the figure it is supposed that 4 D is the proper angle. The sight will then pass through F, the fourth station from A, and this station will be determined by measuring from A the length of the chord A F, found by .i5fi -:CIRCULAR CURVES. i 69 or by Table II. From the station at F the stations at D and / may afterwards be fixed, by laying off:the proper deflections fiom the tangent at F. Second Method. This consists in running an auxiliary curve paral lel to the true curve, either inside or outside of it. For this purpose lay offperpendicular to A C, the tangent at A, a line A A' of any con venient length, and from A' a line A' C' parallel to A C. Then A' C' is the tangent from which the auxiliary curve A'E' is to he laid off. The stations on this curve are made to correspond to stations of 100 feet on the true curve, that is, a radius through 3' passes through B. a radius through D' passes through D, &c. The chold A' l' is, therefore, parallel to A B, and the angle C' A'B' = CA B; that is, the deflection angle of the auxiliary curve is equal to that of the title curve It remains to find the length of the auxiliary clior(is Ai' 3'. /3' D', &c. Call the distance A A' = b. Then the similar triangles 4 13 0 and A' B' 0 give A 0: AO' = A B: A B', or t: R -I= 100: A B'. 100 (R - b) =10 100 b Therefore, A' B= 1' = 0 If he auxiliary curve were on the outside of the true curve, we should find in the same way 100 b A'B' =B 100 + R. It is well to make b an aliquot part of R; for the auxiliary chord is then more easily found. Tlius. if 71 is any R1 100 6 whole number, and1 we make 1 --, we lhave Al' B'= 100 ~: ~ 100 / = 100 ~ ~. If, for example, b — io, we have n =- 100 and A' B -- 100 1 = 101 or 99. When the auxiliary curve has been run, the corresponding stations on the true curve are found, by laying off in the proper direction the distances B B', D D', &c., each equal to b. 77. Problem. Having run a curve A B (fig. 28), to change the tangent point friom A to C, in such a way that a curve of the same radius may strike a given point D. Solution. Measure the distance B D fiom the curve to D in a direction paralldto the tangent CE. This direction may be sometimes judged of by the eye, or found by the compass. A still more accurate way is to make the angle D B E equal to the intersection 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 D B A may be made equal to B A E. MAeasure on the tanyent (backward or forward, as the case may be) a dis tance A C = B D, and C will be the new tangent point required. For, if H be drawn equal and parallel to A F, we have FH equal and par MISCELLANEOUS PROBLEMS. 57 allel to A C, and therefore equal and parallel to B D. Hence D H = B F = A F = CH, and D H being equal to CH, a curve of radius H from the tangent point C must pass through D. \\<^~\ ^^ Fig. 28. C A E 78. Problem. Having run a curve A B (fig. 29) of radius R or deflection angle D, terminating in a tangent B D, to find the radius R' or deflection angle D' of a curve A C, that shall terminate in a given parallel tangent CE. / Fig. 29. JL K Solution. Since the radii B F and C G are perpendicular to the par. allel tangents CE and B D, they are parallel, and the angle A G C.4 FB. Therefore, A C G, the half-supplement of A G C, is equal to 4 58 CIRCULAR CURVES. A B F, the half-supplement of A FB. Hence A B and B C are in the same straight line, and the new tangent point C is the intersection of A B produced with CE. Represent A B by c, and A C = c + B C by c. Measure B C, or, if more convenient, measure D C and find B C by calculation. To calculate DC B C from D C, we have B C = sin D B(Tab. X. 9), and the angle DB C = A BK = B A K, the total deflection from A to B. Then the triangles AFB and A GC give AB:AC = BF: C G, or c: c' =R IRI;..R'= c R. c 50 50 To find D', we have (~ 10) R' =in D, and R D. Suo. 50 stituting these values in the equation for Rt, we have -~-. D c 50 c ^sin.Ds c< 60 X sin. D ~,~.. ~. sin. D' = sin. D. 79. Problem. Given the length of two equal chords A Cand B ( (fig. 30), and the perpendicular CD, to find the radius R of the curve. G Fig. 30. 0 Solution. From 0, the centre of the curve, draw the perpendicular 0 E. Then the similar triangles OBE and BCD give BO: BE ~B C: CD. or R B C = BC: CD. Hence R2 BC RCD' MISCELLANEOUS PROBLEMS. 59 This problem serves to find the radius of a curve on a track already laid. For if from any point C on the curve we measure two equal chords A C and B C, and also the perpendicular CD from C upon the whole chord A B, we have the data of this problem. 80. Problem. To draw a tangent F G (fig. 30) to a given curve from a given point F. Solution. On any straight line F A, which cuts the curve in two points, measure F C and FA, the distances to the curve. Then, by Geometry,, F=,FG = FC xFA. This length being measured from F, will give the point G. When F G exceeds the length of the chain, the direction in whilc to measure it, so that it will just touch the curve, may be found by one or two trials. 81. Problem. Having found the radius A 0 = R of a curve (fig. 31), to substitute for it two radii A E = R1 and D F = R, the longer of which A E or B E' is to be used for a certain distance only at earh end of the curve. / I\ Fig. Assume the longer s f length whih ma be t oolution. Assume the longer radius of any length which may be thowaht 60 CIRCULAR CURVES. proper, and find (~ 9) the corresponding deflection angle DI. Suppose that each of the curves A D and B D' is 100 feet long. Then drawing C O, we have, in the triangle F O E, O E: FE = sin. OFE: sin. FOE. But the side OE =AE-AO =-R-R, FE= DE-DF== R -R, the angle FO E = 1800 - A O C = 1800 -- 1, and the angle OFE = A OF- OEF= I - 2 D1, since O E F= 2 D, (~ 7). Substituting these values, and recollecting that sin. (1800 -2 I) =sin. I, we have R - R: RI - R = sin. (11-2D ): sin. Hence R -R - (R1 - R) sin. l.' sin. (1 I- 2 DA) R2 is then easily found, and this will be the radius from ) to D', or until the central angle DFD' = I —4 Di. The object of this problem is to furnish a method of flattening the extremities of a sharp curve. It is not necessary that the first curve should be ju'st 100 feet long; in a long curve it may be longer, and in a short curve shorter. The value of the angle at E will of course change with the length of A D, and this angle must take the place of 2 D1 in the formula. The longer the first curve is made, the shorter the second radius will be. It must also be borne in mind, in choosing the first radius, that the longer the first radius is taken, the shorter will be the second radius. Example. Given R = 1146.28 and I= 450, to find R2, if Rl is assumed = 1910.08, and A D and B D' each 100. Here, by Table I., D = 10 30'. Then R, - R = 763.8 2.882980 l= 22~ 30' sin. 9.582840 2.465820 -- 2 DI = 190 30' sin. 9.523495 Ri - R2= 875.64 2.942325 =. R, = - 875.64 1034.44 82. Problem. To locate the second branch of a compound or reversed curve from a station on the first branch. Solution. Let A B (fig 32) be the first branch of a compound curve, and D its deflection angle, and let it be required to locate the second branch A B', whose deflection angle is D', from some station B on A B. MISCELLANEOUS PROBLEMS. 61 Let n be the number of stations from A to B, and nt the number of stations from A to any station B' on the second branch. Represent by V the angle A B B', which it is necessary to lay off from the chord B A to strike B*. Let the corresponding angle A B' B on the other curve be repreT A r Fig. 32. sented by V'. Then we have V + V= 1800 -- BAB'. But if T TT be the common tangent at A, we have TA B + TI A BI =_ n D 4 n'D' = 1800 - BAB'. Therefore, V+ V' = nD + n'D'. Next in the triangle ABB' we have sin. V: sin. V = AB:A B'. But A B: A B' = n: n', nearly, and sin. VI: sin. V= V': V, nearly. Therefore we have approximately V': V = n: na, or VI =,. Substituting this value of V' in the equation for V + V', we have V_+ - V = n D+ n' D'. Therefore, n' V+ n V= n' (nD + —' D'), or ~ V =_- n' (n1D + n' D') n + n' The same reasoning will apply to reversed curves, the only change being that in this case V+ Vt = n D - n' D', and consequently = n' (n D - n' D') n + nt When in this formula na D' becomes greater than n D, V becomes minus, which signifies that the angle Vis to be laid off above B A instead of below. This problem is particularly useful, when the tangent point of a curve is so situated, that the instrument cannot be set over it. The same method is applicable, when the curve A B' starts 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 62 CIRCULAR CURVES. V= n + na' When n and n' are each 1, the formula for Vis in all cases exact; for then the supposition that V': V= n: n is strictly true, since A B will equal A B' and Vand V', being angles at the base of an isosceles triangle, will also le equal. Making n and n' equal to 1, we have V== (D + D'). When the curve starts from a straight line, this formula becomes, by making D = 0, V = DI. We have seen that when n or n' is more than 1, the value of Vis only approximate. It is, however, so near the truth, that when neither, nor n' exceeds 3, the error in curves up to 50 or 60 varies fiom 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 AB BB, in which the sides A B and A B' may be found from Table II., and the included angle at A is known. The extent to which these formula may be safely used may be seen by the following table, whi(h gives the approximate values of Vfor several different values of n, nt, D. and D', anil also the error in each case. Compound Curves. Reversed Curves.n. D. D.. D'. V. Error. n. D. n'. /. V. Error. 0 0 03 if 0 0 O I It 1 0 5 1 4 10 0.9 1 3 4 3 7 12 27.2 1 0 5 3 12 30 253 2 3 4 3 4 0 23.5 2 0 3 3 5 24 22.1 3 3 4 3 142 8.3 3 0 3 3 4 30 29.7 3 3 3 3 45 24.0 1 1 5 3 13 20 18.6 2 1 4 0 40 0.1 2 1 3 1 20 0.7 2 1 4 2 4 0 11.0 2 3 3 748 15.0 1 2 6 4 0 23.5 2 4 3 10 40 24.7 1 5 3 5 7 30 51.8 3 3 3 4 10 30 54.0 2 3 5 3 6 257 52.8 As the given quantities are here arranged, the approximate values of Vare all too great; but if the columns n and n' and the columns D and D' were interchanged, and Vealculated, the approximate values of V would be just as much too small, the column of errors remaining the same. MISCELLANEOUS PROBLEMS. 63 83. Problem. To measure the distance across a river on a gzveni utraight line. Solution. First Method. Let A B (fig. 33) be the required distance Measure a line A C along the bank, and take the angles B A C and 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 A CB = D A C can be laid off to a point on the farther side, we have A B C = D A C -- A CB. Therefore, without calculation, A B = A C. B'iC 34 Fig. 34. Second Method. Lay off A C (fig. 34) perpendicular to A 1. Measure A C, and at Clay off CD perpendicular to the direction CB, and meeting the line of A B in D. Measure A D. Then the triangles A CD and A B C are similar, and give A D A C = A C: A B. A C2 Therefore, A B = A D -If from C. determined as before, the angle A CB' be laid off equal to A C B, we have, without calculation, A B = A B'. Third Method. Measure a line A D (fig. 35) in an oblique direction from the bank, and fix its middle point C. From any convenient point E in the line of A B, measure the distance E C, and produce 64 MISCELLANEOUS PROBLEMS. E C until CF = E C. Then, since the triangles A CE and D CF are similar by construction, we see that D F is parallel to E B. Find Fig. 35. now a point G, that shall be at the same time in the line of CB and of D F, and measure G D. Then the triangles A B C and D G C are equal, and G D is equal to the required distance A B. As the object of drawing EF is to obtain a line parallel to A B, this line may be dispensed with, if by any other means a line GF be drawn through D parallel to A B. A point G being found on this parallel in the line of CB, we have, as before, GD = A B. PARABOLIC CURVES. 60 CHAPTER II. -PARABOLIC CURVES. ARTICLE I. - LOCATING PARABOLIC CURVES. 84. LET AE B (fig. 36) be a parabola, A C and B C its tangents. tnd A B the chord uniting the tangent points. Bisect A B in D, and oin CD. Then, according to Analytical Geometry, - Fig. 36. M" A -D B I. CD is a diameter of the parabola, and the curve bisects CD in E II. If from any points T, T', T", &c., on a tangent A F lines lbe drawn to the curve parallel to the diameter, these lines TM, T' Ml T"AM'", &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 ED (fig. 37), drawn from the middle of a chord A B to the curve, and parallel to the diameter, may be called the middle ordi nate of that chord; and if the secondary chords A E and B E be drawn, the middle ordinates of these chords, K G and L H, are each equal to ED. In like manner, if the chords A K, KE, E L, and L B be drawn, their middle ordinates will be equal to i K G or i L H. IV. A tangent to the curve at the extremity of a middle ordinate, is parallel to the chord of that ordinate. Thus MF, tangent to the curve at E, is parallel to A B. ;6 6PARABOLIC CURVES. V. If any two tangents, as A C and B C, be bisected in M and P the line M F, joining the points of bisection, will be a new tangent, its middle point E being the point of tangency. 85. Problem. Given the tangents A C and B C, equal or unequal {fig. 36,) and the chord A B, to lay out a parabola by tangent deflections. Fig. 36. T M/ A D B Solution. Bisect A B in D, and measure C D and the angle A CD); or calculate CD* and A CD from the original data. Divide the tangent A C into any number n of equal parts, and call the deflection TM for the first point a. Then (~ 84, II.) the deflection for the second point will be T' Mt = 4 a, for the third point T" M" = 9 a, and so on to the nth point or C, where it will be n* a. But the deflection at this last point is CE C= CD (~ 84, I.). Therefore, n a = C E, and CE Having thus found a, we have also the succeeding deflections 4 a, 9 a, 16 a, &c.'Then laying off at T, Tt, &c. the angles A TM, A T' MA', &c. each equal to A CD, and measuring down the proper deflections, just found, the points M, M', &c. of the curve will be determined. The curve may be finished by laying off on A C produced n parts equal to those on A C, and the proper deflections will be, as before, a multiplied by the square of the number of parts from A. But an * Since C D is drawn to the middle of the base of the triangle A B C, we have, by Ioometry, C D2 = (A C1 + B C2)- A DM. LOCATING PARABOLIC CURVES. 61 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 U. If the number of parts on B C be made the same as on A C, it is obvious that the deflections from both tangents will be of the same length for corresponding points. The angles to be laid off from B C must, of course, be equal to B CD. The points or stations thus found, though corresponding to equal distances on the tangents, are not themselves equidistant. The length of the curve is obtained by actual measurement. 86. Problem. Given the tangents A C and B C, equal or unequal, (Jig. 37,) and the chord A B, to lay out a parabola by middle ordinates. C Fig. 37. A D B Solution. Bisect A B in D, draw CD, and its middle point E will be a point on the curve (~ 84, I.). D E is the first middle ordinate, and its length may be measured or calculated. To the point E drawn the chords A E and BE, lay off the second middle otdinatcs G AK ald II L, each equal to i D E (s 84, III ), and K and L are points on the curve. Draw the chords A K, K E, E L, and L B, and lay off third middle ordinates, each equal to one fourth the second middle ordinates, and four additional points on the curve will be determined. Continue this process, until a sufficient number of points is obtained 87. Problem. To draw a tangent to a parabola at any station. Solution. I. If the curve has been laid out by tangent deflections (~ 85), let M"' (fig. 36) be the station, at which the tangent is to lbe drawn. From the preceding or succeeding station, lay off, parallel to CD, a distance M" Noi EL equal to a, the first tangent deflection (~ 85), and MI" N or M"' L will be the required tangent. The same thing may be done by laying off from the second station a distance 31' T' 4a, or at the third station a distance GP = 9a; for the 68 PARABOLIC CURVES. required tangent will then pass through T' or G. It will be seen, also, that the tangent at M"' passes through a point on the tangent at A corresponding to half the number of stations from A to M"'; that is, M"' is four stations from A, and the tangent passes through T', the second point on the tangent A C. In like manner, M"' is six stations from B, and the tangent passes through G, the third point on the tangent B C. II. If the curve has been laid out by middle ordinates (~ 86), the 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. 37) were divided into four equal parts corresponding to the number of stations fiom A to E, the method of tangent deflections would give the same points on the curve, as were obtained by the method of ~ 86. In this case, the tangent deflection for one station would be a = -l- CE = I DE; but the last middle ordinate was made equal to d G K or 1E 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 (~ 84, IV.). 88. In laying out a parabola by the method in ~ 85, 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 ~ 87 to any station already found, as at M"' (fig. 36), and the tangent deflections a, 4 a, 9 a, &c. may be laid off from this tangent, precisely as from tie first tangent. These deflections must be parallel to CD, and the distances on the new tangent must be equal to TI'Nor NVMI, whiclh may be measured. 89. Problem. Given the tangents A C and B C, equal or unequal. (fig 38,) to lay out a parabola by bisecting tangents. Solution. Bisect A C and B C in D and F, join D F, and find E, the middle point of D F. E will be a point on the curve (~ 84, V.). We have now two pairs of what may be called second tangents, A D and ) E, and EF and FB. Bisect AD in G and DE in H, join G II, and its middle point Mwill be a point on the curve. Bisect E F and FB in Kand L, join KL, and its middle point N will be a point on the curve. We have now four pairs of third tangents, A G and G M, M H and H E, EK and K N, and NL and L B. Bisect each pair in turn, join the points of bisection, and the middle points of the joining LOCATING PARABOLIC CURVES. 69 lines will be four new points, M', M", N", and N'. The same method may be continued, until a sufficient number of points is obtained. C Fig. 38. D / 11 E M N L A B 90. Problem. Given the tangents A C and B C, equal or unequal, I fig. 39,) and the chord A B, to lay out a parabola by intersections. Fig. 39. A. G K B Solution. Bisect A B in D, draw CD, and bisect it in E. Divide the tangents. A C and B C, the half-chords A D and D B, and the line CE, into the. same number of equal parts; five, for example. Then the intersection Mof A a and F G will be a point on the curve. For FJM — I Ca, and Ca = 5 CE. Therefore, FM= i CE, which is the proper deflection from the tangent at Fto the curve (~ 85). In like manner, the intersection N of A b and HK may be shown to be a point on the curve, and the same is true of all the similar intersections indicated in the figure. If the line D E were also divided into five equal parts, the line A a would he intersected in Mon the curve by a line drawn from B through a'. the line A b would ce intersected in Non the curve by a line drawn 70 PARABOLIC CURVES. from B through b', and in general any two lines, drawn from A and B through two points on CD equally distant from the extremities C and D, will intersect on the curve. To show this for any point, as AI, it is sufficient to show, that Bat produced cuts F G on the curve; for it has already been proved, that Aa cuts FG on the curve. Now LDa': M G = B D: B G = 5: 9, or M G = ID a'. But Da' = CE. Therefore, MG = -9 CE. Again, FG: CD = A G: A D = 1:5. Therefore, FG = CD) = CE. We have then FM = FG - M G = 2 CE - 2 CE = CE. As this is the proper deflection fiom the tangent at F to the curve (~ 85), the intersection of B a' with 1,G is on the curve. This furnishes another method of laying out a parabola by intersections. 91. The following example is given in illustration of several of the preceding methods. Example. Given A C = B C= 832 (fig. 40), and A B = 1536, to lay out a parabola A E B. We here find CD = 320. To begin with the method by tangent deflections (~ 85), divide the tangent A C into CE 160 eight equal parts. Then a -- = -6 == 2.5. Lay off trom the divisions.on the tangent F = 2.5, G2 = 4 X 25 = 10, H1 = 9 X 2.5 = 22.5, and K4 = 16 X 2.5 = 40. Suppose now that it is inconvenient to continue this method beyond K. In this case we may C Fig. 40. I/ M N O E \rL A D1 B find a new tangent at E, by bisecting A C and B C (~ 89), and drawing KL through the points of bisection. Divide the new tangent KE ==, A D - 384 into four equal parts, and lay off from KE the RADIUS OF CURVATURE. 71 same tangent deflections as were laid off from A K, namely, M15 = 22.5, N6 = 10, and 07 = 2.5. To lay off the second half of the curve by middle ordinates (~ 86), measure E B- 784.49. Bisect E B in P. and lay off the middle ordinate P R'= D E = 40. Measure E R = 386.08, and BR = 402.31, and lay off the middle ordinates S T and V W, each equal to 4 PR = 10. By measuring the chords E T, TR, R W, and WB, and laying off an ordinate from each, equal to 2.5, four additional points might be found. ARTICLE II.-RADIUS OF CURVATURE. 92. THE curvature of circular arcs is always the same for the same are, and in different arcs varies inversely as the radii of the arcs. Thus. the curvature of an arc of 1,000 feet radius is double that of an arc of 2,000 feet radius. The curvature of a parabola is continually changing. In fig. 39, for example, it is least at the tangent point A, tie extremity of the longest tangent, and increases by a fixed law, until it becomes greatest at a point, called the vertex, where a tangent to the curve would he perpendicular to the diameter. From this point to B it decreases again by the same law. We may, therefore, consider a parabola to be made up of a succession of infinitely small circular arcs, the radii of which continually increase in going from the vertex to the extremities. The radius of the circular 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 necessary, in order that the rails may be properly curved (~ 28), to know hlow the radius of curvature may be found. It will, in general, he necessary to find the radius of curvature at a few points only. In short curves it may be found at the two tangent points and at the middle station, and in longer curves at two or more intermediate points besides. The rails curved according to the radius at any point should be sufficient in number to reach, on each side of that point, half-way to the next point. 93. Problem. To find the radius of curvature at certain stations on a parabola. Solution. Let A EB (fig. 41) be any parabola, and let it be required to find the radii of curvature at a certain number of stations 72 PARABOLIC CURVES. front A to E. These stations must be selected at regular intervals from those determined by any of the preceding methods. Let n denote the number of parts into which A E is divided, and divide CD into the same number of equal parts. Draw lines from A to the pointa Fig. 41. /^ A D B of division Thus, if n = 4, as in the figure, divide CD into four equal parts, and draw A F, A E, and A G. Let A D = c, A F = c1 A E = c,, A G = c3, and A C = T. Denote, moreover, CD by d, and the area of the triangle A CB by A. Then the respective radii for the points E, 1, 2, 3, and A will be C3 C13 co3 C33 T3 R, R, =', R — = R = -, R4 = -. The area A may be found by form. 18, Tab. X.; c and T are known; and cl, c2, c3 may be found approximately by measurement on a figure carefully constructed, or exactly by these general formulae: - T2 - c2 (n -1) d C12- + - n' T - c2 (n-3) d C2 = + -na - a~ T2 - ce (n-5) d2 3 C22 + n --' T2 - c2 (1- 7) dg C42 C32 + n -- n-' &c. &c. It will be seen, that each of these values is formed from the preceding, TI _e2 dd2 by adding the same quantity,2 and subtracting 2 multiplied in;-A^oC.F-f5? - 1,7n - - 5, &c Mak-inafv =, e,^,\ RADIUS OF CURVATURE. 73 C12 = + i(T -c2)- Ad', C2 = C12 + i ( T - c2) - r dM, C32 =22 + i (T2 - C2) +' ds. 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 of curvature at any given point on a parabola is, by the DifferCalculus. R-2 si= - E, in which p represents the parameter of,arabola for rectangular coordinates, and E the angle made with,ameter by a tangent to the curve at the given point. First, let the ddle station E (fig. 42) be the given point. Then the angle E is the C Fig. 42. J. B angle made with E D by a tangent at E, or since A B is parallel to the tangent at E (~ 84, IV.), sin. E = sin. A DE = sin. B DE. Let p' be the parameter for the diameter E D. Then, by Analytical Ge ometry, p = p' sin.2 E. Therefore, at this point R = 2- sin.3 ='p' sin.2 E p' AD2 C2 e2 2 3s =.-2 sin. E* But pl —D d Therefore, R d E 2,sin E- 2 sin.' E P- ED =d' dsdsin. cd in. E = A; since A = cd sin. E (Tab. X. 17). Next, to find R1, or the radius of curvature at H, the first station from E. Through H draw F G parallel to CD, and from F draw the tangent FK. Join A K, cutting CD in L. Then from what has just been proved for the radius of curvature at E, we have for the radius A G3 =: C Df curvature at H,R-, -RFI. Now AG' AL= AF:A C = "4 PARABOLIC CURVES. n-i n —:n,orA G = X AL. ButAL=c,. For,sinceAFz n- 1 (ni- 1)2 d n X A C the tangent deflection FH = 2. (~ 84, II.), and FG= 2F (n — ]p FG=2FHI== z- 1 d. Then, since CL:FG= -AC:AF = nz o n —1 n -- 1 n:n-1, CL= —1 X FG -= d. Hence LD= d- d = d, that is, A L = cl. Substituting this value in the expresn-1 sion for A G above, we have A G = c. Moreover, since n —I A F = X A C, aod because similar triangles are to each other as tlhe squares of their homologous sides, we have the triangle A F G = -- X ACL. But A CL:ACD= CL: CD=n-1:n, or a-1 ( - 1)3 A C L = X A CD. Therefore, A FG = X A CD, and A F 2 A FG= (1) X A CB = (n - 1) A Substituting A G3 these values of A G and A FKin the equation R, =A F K, and reducing, we find R1 =- -A By similar reasoning we should find R2r) ~ 33, 3 —, &C. A R3 AC&C It remains to find the values of c, c2, &c. Through A draw A M perpendicular to CD, produced if necessary. Then, by Geometry, we have AD2 = AL + LD2-2 LD X LM, and A C2 = L2 + CL2 +J 2 CL X L M. Finding from each of these equations the value of 2 L M, and putting these values equal to each other, we have AL2 + LD - AD2 AC2 -A L2 - CL2 1 LD -= CL ~. ButAL-, c, LD = nd, A D = c, A C = T, and CL = n d. Substituting these values in the last equation, and reducing, we find T2 (n -1 )2 (n - )ds 2 =-"n + ~n - n-' By similar reasoning we should find 2 T2 ( - 2)c2 2(n- 2)d', = n ~ n n' 3 T2 (n - 3)c" 3(n - 3)d' C? n +. n n2 &e. &c. RADIUS OF CURVATURE. 75 From these equations the values of c12, C22,, &c. given on page 72 are readily obtained. That given for c12 is obtained from the first of these equations by a simple reduction; that given for 22 is obtained by subtracting the first of these equations from the second, and reducing; that given for C32 is obtained by subtracting the second equation friom the third, and reducing; and so on. 94. Example. Given (fig. 41) A 0 = T= 600, B C = T = 520, and A D = c = 550, to find R, R1, R2, R3, and R4, the radii of curvature at E, 1, 2, 3, and A. To find CD = d, we have, by Geometry, d2 = 1 (T2 + Tt 2) -c2, which gives d2 = 12700. To find the area of A CB = A, we have (Tab. X. 18) A = s (s - a) (s - b) (s - c). s = 1110 3.045323 s-a = 590 2.770852 s- b = 510 2.707570 - c = 10 1.000000 2)9.523745 log. A 4.761872 ~~~~~~~1 ~~1150 X 50 Next n ( T2 - c) = ( T + c) (T - c)= - = 14375, and d2 12700 - - 1-6- - 793.75. Then C2 = 5502 = 302500 c12 = 302500 + 14375 - 3 X 793.75 = 314493.75 c22 = 314493.75 + 14375 - 793.75 = 328075 c32 = 328075 + 14375 + 793.75 = 343243.75 To find R, we have R -=, or log. R = 3 log. c- log. A. c -550 2.740363 c3 8.221089 A 4.761872 R = 2878.8 3.459217 C13 8 To find R1, we have R =, or log. R, = — log c2 - log. A. cl" = 314493.75 5.497612 c 3 8.246418 A 4.76 872 p, = 3051.7 3.484546 76 PARABOLIC CURVES. In the same way we should find R2 = 3251.5, R3 = 3479.6, R4 = 3737.5. To find the radii for the second part EB of the parabola, the same formulae apply, except that T' takes the place of T. We have then 1 1070 x - 80 (T - c2) = (T' + c) (TI - c) = 4- -8025 Hence c,2 = 302500 - 8025 - 2381.25 = 292093.75 c2 = 292093.75 - 8025 - 793.75 = 283275. c.2 = 283275 - 8025 + 793.75 == 276043.75 To find RI, we have RB =, or log. R1 = 2 log. c,2 - log. A c,2 = 292093.75 5.465523 C13 8.198284 A 4.761872 RI = 2731.6 3.436412 In the same way we should find R. = 2608.8, K3 = 2509.5, R4 = 2433. It will be seen, that the radii in this example decrease from one tangent point to the other, which shows that both tangent points lie on the same side of the vertex of the parabola (~ 92). This will be the case, whenever the angle B CD, adjacent to the shorter tangent, exceeds 900, that is, whenever c2 exceeds T'2 + d2. If B CD = 900. the tangent point B falls on the vertex. If B CD is less than 900, one tangent point falls on each side of the vertex, and the curvature will, therefore, decrease towards both extremities. 95. If the tangents T and T' are equal, the equations for c12, c22, &c. will be more simple; for in this case d is perpendicular to c, and T' - s = d2. Substituting this value, we get C12 = c+ +-2 3 d C22 = C12 + -- 5sd c32= C2 + -,, &c. &c. Example. Given, as in ~ 91, T= T' = 832, c = 768, and d. RADIUS OF CURVATURE. t7 320, to find the radii R, R,, and R2 at the points E, 4, and A (fig. 40), Here A = cd = 245760, n = 2, and c-2 = c2 + -d2 = 615424 03 c2 7682 C13 T3 Then R =d 1843.2, R1 - d' and R2 - d' c d't - 320 c843 cx2 = 615424 5.789174 cs3 8.683761 cd = 245760 5.390511 R1= 1964.5 3.293250 T = 832 2.920123 T3 8.760369 cd = 245760 5.390511 R2 - 2343.5 3.369858 R is the radius at the point R also, and Rs the radius at the point B 78 LEVELLING. CHAPTER IIL LEVELLING. ARTICLE I. HEIGHTS AND SLOPE STAKES. 96. THE Level is an instrument consisting essentially of a telescope, supported on a tripod of convenient height, and capable of being so adjusted, that its line of sight shall be horizontal, and that the telescope itself' may be turned in any direction on a vertical axis. The instrument when so adjusted is said to be set The line of sight, being a line of indefinite length, may be made to describe a horizontal plane of indefinite extent, called the plane of the lend. The levelling rod is used for measuring the vertical distance of anJ point, on which it may be placed, below the plane of the level. This distance is called the sight on that point. 97. Problem. To find the dfference of level of two points, as A and B (fi.q. 43). Solution. Set the level between the two points,* and take sights on both points. Subtract the less of these sights from the greater, and the difference will be the difference of level required. For if F P represent the plane of the level, and A G be drawn through A parallel to FP, A F will be the sight on A, and B P the sight on B. Then the required difference of level B G = BP - G = B P- A F. If the distance between the points, or tue nature of the ground, makes it necessary to set the level more than once, set down all the backward sights in one column and all the forward sights in another. Add up these columns, and take the less of the two sums from the greater, and the difference will be the difference of level required. Thus, to find the difference of level between A and D (fig. 43), the level is first set between A and B, and sights are taken on A and B; the level is then set between B and C, and sights are taken on B and * The level should be placed midway between the two points, whe. practicable, In order to neutralize the effect of inaccuracy in the adjustment of the instrument, and for the reason given in ~ 105. HEIGHTS AND SLOPE STAKES. 79 0; lastly, the level is set between C and D, and sights are taken on Cand D. Then the difference of level between A and D is ED - (BP+ KC+ OD)-(AF-+ BI+ NC). For ED = HC- LC = HM+iMC-LC. ButHM=I G = BP -AF, M C=KC-BI, and LC C= N C --- 0 D. Substituting these values, we have ED == BPAF+ KC-BI-NC+ OD= (BP + KC + OD) - (A F+ BI + NC). 98. 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 distance above or below, any permanent object, called a bench-mark, or simply a bench. It is most convenient, in order to avoid minus heights, to assume the datum plane at such a distance below the benchmark, that it will pass below all the points on the line to be levelled. Thus if A B (fig. 44) were part of the line to X H n -- f Z be levelled, and if A were the starting point, we should assume the datum plane CD at such a distance below some permanent object near A, as would make it pass below all the points on the line. If, for instance, we had reason to believe that no point on this 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 m i line to the datum plane would be positive. Lines before being levelled are asually divided into regular stations, the height of each of which above the datum plane is required. 80 LEVELLING. 99. Problem. Tofind the heights above a datum plane of the se eral stations on a given line..F. Solution. Let A B (fig. 44) represent a portion of the line, divided into regular stations, marked 0, 1, 2, 3, 4, 5, &c., and let CD represent the datum plane, assumed to be 25 feet below a bench-.^_ M __ lo _ a _mark near A. Suppose the level to be A^~~I~~ \ I set first between stations 2 and 3, and a sight upon the bench-mark to be taken, *H _ e and found to be 3.125. Now as this sight shows that the plane of the level EFis 3.125 feet above the bench-mark ~. ~ X ~~~and as the datum plane is 25 feet below this mark, we shall find the height of the plane of the level above the da~_~ stum 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 instru. _4 ment. If now a sight be taken on station 0, /. we shall obtain the height of this sta. tion above the datum plane, by subtracting this sight from the height of 0 ___o__ _ the instrument; for the height of this station is 0 C and 0 C = E C-E 0. Thus if E0 = 3.413, 0 C = 28.125 - e _______ ~ 3.413 = 24.712. In like manner, the heights of stations 1, 2, 3, 4, and 5 may be found, by taking sights on them in succession, and subtracting these sights from the height of the instrument. Suppose these sights to be respectiveF^ ~ An ~ly 3.102, 3.827, 4.816, 6.952, and 9.016, and we have height of station 0 = 28.125 - 3.413 = 24.712, " " " 1 = 28.125 - 3.102 = 25.023, tRIGiHTS AND SLOPE STAKES. 81 height of station 2 = 8.25 -- 3.82' = 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 G K = G 5 + 5 K. Suppose this sight to be G 5 = 2.740, and we have GK-= 19.109 + 2.740 = 21.849. A point like station 5, which is used to get the height of the instrument after resetting, is called a turniny point. The height of the instrument being found, sights are taken on stations 6, 7, 8, 9, and 10, and the heights of these stations found by subtracting these sights from the height of the instrument. Suppose these sights to be respectively 3.311, 4.027, 3.824, 2.516, and 0.314, and we have height of station 6 = 21.849 - 3.311 = 18.538, 7" " " 7 21.849 - 4.027 = 17.822, " " " 8 = 21.849 - 3.824 18.025, " " " 9 = 21.849 - 2.516 = 19.333, "~ 4" " 10 = 21.849 - 0.314 - 21.535. The instrument is now again carried forward and reset, station 10 is used as a turning point to find the height of the instrument, and every thing proceeds as before. At convenient distances along the line, permanent objects are selected, and their heights obtained and preserved, to be used as starting points in any further operations. These are also called benches. Let us suppose, that a bench has been thus selected near station 9, and that the sight upon it from the instrument, when set between stations 7 and 8, is 2.635. Then the height of this bench will be 21.849 - 2.635 = 19 214. 100. From what has been shown above, it appears that the first thing to be done, after setting the level, is to take a sight upon some point of known height, and that this sight is always to be added to the known height, in order to get the height of the instrument. This first sight may therefore be called a plus sight. The next thing to be done is to take sights on those points whose heights are required, and to subtract these sirhts fiom the height of the instrument, in order to get the required heights. These last sights may therefore be called minus eights. 5 82 LEVELLING. 101. The field notes are kept in the following form. The first col unn in the table contains the stations, and also the benches marked B., and the turning points marked t. p., except when coincident with a station. The second column contains the plus sights; the third column shows the height of the instrument; the fourth contains the minus sights; and thefifth contains the heights of the points in the first column. Station +.S. H. 1. -8. H. B. 3.125 25.0)0 0 28.125 3.413 24.712 1 3.102 25.023 2 3 827 24.298 3 4.816 23.309 4 6.952 21.173 5 2.740 9.016 19.109 6 21.849 3.311 18.538 7 4.027 17.822 8 3.824 1 18.025 9 2.516 19.333 B. 2.635 6 19.214 10 0.314 21.535 The height of the bench is set down as assumed above, namely, 25 feet; the first plus sight is set opposite B., on which point it was taken, and, being addea to the height in the same line, gives the height of the instrument, which is set opposite 0; the minus sights are set opposite the points on which they are taken, and, being subtracted from the height of the instrument, give the heights of these points, as set down in the fifth column. The minus sights are subtracted from the same height of the instrument, as far as the turning point at station 5, inclusive. The plus sight on station 5 is set opposite this station, and a new height obtained for the instrument by adding the plus sight to the height of the turning point. This new height of the instrument is set opposite station 6, where the minus sights to be subtracted fiom it commence. These sights are again set opposite the points on which they were taken, and, being subtracted from the new height of the instrument, give the heights in the last column. 102. Problem. To set slope stakes for excavat;ons and embank. ments. Solution. Let A B 1K C (fig. 45) be a cross-section of a proposed excavation, and let the centre cut A M = c, and the width of the road HEIGHTS AND SLOPE STAKES. R3 bed fK = b. The slope of the sides B H or C Kis usually given by the' ratio of the base KNto the height EN. Suppose, in the present case, that KN: EN = 3:2, and we have the slope = 2. Then if the ground were level, as D A E, it is evident that the distance from Fig. 45. ~''P~~~~~~~~:.a the centre A to the slope stakes at D and E would be A D = AE = MK+ KN== b -+ c. But as the ground rises from A to C through a height C G = g, the slope stake must be set farther out a distance E G = 2 g; and as the groudl falls from A to B through a height B F = g, the slope stake must be set farther in a distance DF To find B and C, set the level, if possible, in a convenient position for sighting on the points A, B, and C. From the known cut at the cntre find the value of A E-= b + 2 c. Estimate by the eye the rise from the centre to where the slope stake is to be set, and take this as the probable value of g. To A E add 2 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 corresponding distance out. The distance out will then be Lb + 3 c +. 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 DDF= b +Lc-9g. 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 smbankments corresponds to a ficll on excavations, and vice versd. This will be evident by inverting figure 45, which will then represent ^4ft~ sLEVELLING. an cmbankment. 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 embankment, the method above applies directly only to the side which is in excava tion at the same time that the centre of the road-bed is in excavation, or in embankment at the same time that the centre is in embankment On the opposite side, however, it is only necessary to make c in the expressions above minus, because its effect here is to diminish the distance out. The formula for this distance out will, therefore, become b - c + g. ARTICLE II. -CORRECTION FOR THE EARTH'S CURVATURE AND FOR REFRACTION. 103. LET AC (fig. 46) represent a portion of the earth's surface, Then, if a level be set at A, the line ofsiqht of the level will be the tangent A D, while the true level will be A C. The difference D C between the line of sight and the true level is the correction for the earth's curvature for the distance A D. 104. A correction in the opposite direction arises from refraction. Refraction is the change of direction which light undergoes in passing from one medium into another of different density. As the atmosphere increases in density the nearer it lies to the earth's surfatce, light. [passilg fiom a point B to a lower point A, enters continually air ol greater and greater density, and its path is in consequence a curve concave towards the earth. Near the earth's surface this path may be taken as the arc of a circle whose radius is seven times the radius of the earth." Now a level at A, having its line of sight in the direction A D, tangent to the curve A B, is in the proper position to receive the light from an object at B; so that this object appears to the observer to be at D. The effect of refraction, therefore, is to make an object appear higher than its true position. Then, since the correction for the earth's curvature D C and the correction for refraction D B are in opposite directions, the correction for both will be B C = DC - DB. * Peirce's Spherical Astrononmy, 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, aud sometimes the rays are refracted horizontally as well as vertically. EARTH'S CURVATURE AND REFRACTION. F5 This correction must be added to the height of any object as determined by the level. 105. Problem. Given the distance A D = D (fig. 46), the radiua of the earth A E = R, and the radius of the arc of refracted lyht - 7 R, Jo find the correction B C = dfor the earth's curvature andfor refraction. A D Fig. 46. Sdution. To find the correction for the earth's curvature D C, we have, by Geometry, D C(D C + 2E C) ==A D or D C (D C 2R) = D2. But as D Cis always very small compared with the diameter of the earth, it may be dropped from the, parenthesis, and we have D2 D C X 2 R = D2, or D C -= 2-'. The correction for refraction D B may be found by the method just used for finding D C, merely changD2 ing R into 7 R. Hence DB 14R. We have then d = B C= 1)2 D2 DC-DB -2 - 1R or d 3 3D By this formula Table III. is calculated, taking R = 20,911,790 ft., as given by Bowditch. The necessity for this correction may be avoided, whenever it is possible to set the level midway between the points whose height is required. In this case, as the distance on each side of the level is the same, the corrections will be equal, and will destroy each other. W9~~6 ~ LEVELLING. ARTICLE III.-VERTICAL CURVES. 106. VERTICAL curves are used to round off the angles formed bt the meeting of two grades. Let A C and CB (fig. 47) 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 respectively by g and g'; that is, g denotes what is added to the heigh t at every station on A C, and g' denotes what is added to the height at every station on CB; but since CB is a descending grade, the quantity added is a minus quantity, and g' will therefore be negative. The parabola furnishes a very simple method of putting in a vertical curve. 107. Problem. Given the grade g of A C (fig. 47), the grade g of C B, and the number of stations n on each side of C to the tangent points A and B, to unite these points by a parabolic vertical curve. Fig. 47. C G A p -p." Q-KR A r Pl Solution. Let A E B be the required parabola. Through B and C draw the vertical lines FK and C H, and produce A C to meet FK in F. Through A draw the horizontal line A K, and join A B, cutting C I in D. Then, since the distance from C to A and B is measured horizontally, we have A H = HK, and consequently I D = DB. The vertical line CD is, therefore, a diameter of the parabola (~ 84, I.), and the distances of the curve in a vertical direction from the stations on the tangent A Fare to each other as the squares of the number of stations from A (~ 84, II.). Thus, if a represent this distance at the first station 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 FR from A, it would be 4 n' a; that is, FB = 4 n2 a, or a == 2. To find a, it will then be necessary to find FB first. Through C draw the horizontal line C G, and we have, from the equal triangles CF G and VERTICAL CURVES. W1 A C I, F G = C H. But C H is the rise of the first grade g in the n stations from A to C; that is, CH = ng, or F G = ng. GB is also the rise of the second grade g' in n stations, but since y' is negative (~ 106), we must put G B = -ng'. Therefore, FB = F G + G B = ng- ng'. Substituting this value of FB in the equation for a ng —ngo we have a == 4,,or a = 9 - 9_. 4 n The value of a being thus determined, all the distances of the curve from the tangent AF, viz. a, 4 a, 9 a, 16, &c., are known. Now if T and T' be the first and second stations on the tangent, and vertical lines TP and T' P' be drawn to the horizontal line A K, the height TP of the first station above A will beg, the height T' P' of the second station above A will be 2g, and in like manner for sueceeding stations we should find the heights 3g, 4g, &c As we have already found TM = a. T' I' =- 4a, &c., we shall have for the heights of the curve above thle level of A, MP = TP - TM g - a, M P' = T' P - T' Mt = 2g - 4 a, and in like manner for the succeeding heights 3g - 9a, 4g - 16a, &c. Then to find the grades for the (urve 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)-O0 = g -a, (2y-4a) - (g -a) =g -3 a, (3g -9a) - (2g -4a) =g - 5a, (4 g - 16 a) - (3 g - 9 a) =g -7 a, &c. The successive grades for the vertical curve are, therefore, g —a, g-3 a, g-5a, g-7a, &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 fiom various combinations of ascending and descending grades. If proper figures were drawn to represent the remaining cases, the above solution, with due attention to the signs, would apply to them all; and lead to precisely the same formulae. 168. Ex.amp/les. Let the number of stations on each side of C be 3, and let A C ascend.9 per station, and CB descend.6 per station. Here g -- g' 9-(-.6) 1.5 - 3,.g =.9, and g' = —.6. Then a = gg = 4-3 -12 ~..125, and the grades fiont A to B will be 88 LEVELLING. g- a =.9 -.125 =.775, g- 3 a=.9 -.375 =.525, g- 5 a=.9 —.625 =.275, g- 7a =.9 -.875 =.025, g - 9a =.9- 1.125 - -.225, g-l a =.9 - 1.375 -=.475. As a second example, let the first of two grades descend.8 per sta tion, and the second ascend.4 per station, and assume two stations on each side of C as the extent of the curve. Here g —.8, g' =.4, -.8-.4 -1.2 and n 2. Then a = 4 2 =- 8 -.15, and the four grades required will be - a =-.8- (-.15) =-.8 +.15 —.65, g -.3 a -.8- (-.45) =-.8 +.45 =-.35, g-5a = —.8 - (-.75) =-.8 +.75 =-.05. g- 7a =-.8 - (- 1.05) =-.8 + 1.05 = +.25. It will be seen, that, after finding the first grade, the remaining grades may be found by the continual subtraction of 2 a. Thus, in the first example, each grade after the first is 25 less than the preceding grade, and in the second example, a being here negative, each grade after the first is 3 greater than the preceding grade. 109. The grades calculated for the whole stations, as in the foregoing examples, are sufficient for all purposes except for laying the track. The grade stakes being then usually only 20 feet apart, it will he necessary to ascertain the proper grades on a vertical curve for these sub-stations. To do this, nothing more is necessary than to let g and y' represent the given grades for a sub-station of 20 feet, and n the number of sub-stations on each side of the intersection, and to apply the preceding 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 ir 200 feet is 10. We have then g =.16, g =.08, and n = — 10 - 16 -.08 -.24 Hence a 4X10 = 40- =-o.00f. The first grade is, there fore, g - a = -.16 +.006 = -.154, and as each subsequent grade increases.012 (~ 108), the whole may be written down without farther trouble, thus: -.154, -.142, -.130, -.118, -.106, -.094, -.082, -.070, -.058, -.046, -.034, -.022, -.010, +.002, +.014, +.096, +.038 +'.050, + 062, +.074. ELEVATION OF THE OUTER RAIL ON CURVES. 89 ARTICLE IV.-ELEVATION OF THE OUTER RAIL ON CURVES. 110. 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 ele*)alion e of the outer rail of the curve. Solution. A.car moving on a curve of radius R, with a velocity per second = v, has, by Mechanics, a centrifugal force = R. To counteract this force, the outer rail on a curve is raised above the level of the inner rail, so that the car may rest on an inclined plane. This elevation must be such, that the action of gravity in forcing the car down, the inclined plane shall be just equal to the centrifugal force, which impels it in the opposite direction. Now the action of gravity on a body resting on an inclined plane is equal to 32.2 multiplied lv! the ratio of the height to the length of the plane. But the height of teil plane is the elevation e, and its length the gauge of the track y.'Ilhi action of gravity, which is to counteract the centrifugal force is, tlcrefore,= g. Putting this equal to the centrifugal force, we have 32.2 e v2 g R Hence g- V e= 9v 32.2 R If we substitute for R its value (~ 10) R = — si D, we have eg' v sin. D 0 X 822 -.00062112 g v2 sin. D. If the velocity is given in miles M X 5280 per hour, represent this velocity by M, and we have v 6o= x 60 Substituting this value of v, we find e =.0013361 g M2 sin. D. When = 4 7, this becomes e =.00627966 M2 sin. D By this formula Table IV. is calculated. In determining the proper elevation in any given case, the usual practice is to adopt the highest customary speed of passenger trains as the value of M. 11i. Still the outer rail of a curve, though elevated according to the preceding formula, is generally found to be much more worn than the inner rail On this account some are led to distrust the formula, and to give an increased elevation to the rail. So far, however. as the centrifugal force is concerned, the formula is undoubtedly correct, and the evil in question must arise from other causes, - causes which are not counteracted by an additional elevation of the outer rail. The principal of these causes is probably improper " coning" of the wheels. Two wheels, immovable on an axle, and of the same radius. must, if 90 I.EVELLING. 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 dif ference in length of the two rails of the curve. To remedy this, coning of the wheels was introduced, by means of which the radius of the outer wheel is in effect increased, the nearer its flange approaches the rail, and this wheel is thus enabled to traverse 4 greater distance than the inner wheel. To find the amount of coning for a play of the wheels of one inch, let r and rt represent the proper radii of the inner and outer wheels respectively, when the flange of the outer wheel touches the rail. Then 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, proportional to the radii of these rails. If R be taken as the radius of the inner rail, the radius of the outer rail will be R -pg, and we shall have r: r' = R: R + g. Therefore, r R + rg = r' R, or r -r - rg As an example, let R 600, r = 1.4, and g = 4.7. Then we have 1.4 x 4.7 r- ~ r- 60 = 011 ft. For a tire 3.5 in wide, the coning would be 3.5 X.011 =.Q385 ft., or nearly half an inch. Wheels coned to this amount would accommodate themselves to any curves of not less than 600 feet radius. On a straight line the flanges of the two wheels would be equally distant fiomn the rails, making both wheels of the same diameter. On a curve of say 2400 feet radius, the flange of the outer wheel would assume a position one fourth of an inch nearer to the rail than the flange of the inner wheel, which would increase the radius of the outer wheel just one fourth of the necessary increase on a curve of 600 feet. Should the flange of the outer wheel get too near the rail, the disproportionate increase of the radius of this wheel would make it get the start of the inner wheel, and cause the flange to recede from the rail again. If the shortest radius were taken s 900 feet, rand remaining the same we sould 4.7 as 900 feet, rand g remaining the same, we sltould have r?- r -- ELEVATION OF THE OUTER RAIL ON CURVES. 91 -=.0073, and for the coning of the whole tire 3.5 X.0073 =.0256 ft., or about three tenths of an inch. Wheels coned to this amount would accommodate themselves to any curve of not less than 900 feet radius. If the wheels are larger, the coning must be greater, or if the gauge of the track is wider, the coning must be greater. If the play of the wheels is greater, the coning may be diminished. Hence it might be advisable to increase the play of the wheels on short curves, by a slight increase of the gauge of the track. Two distinct things, therefore, claim attention in regard to the 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 curiilinear path, and which may be effectually counteracted for any given velocity by elevating the outer rail. The second is the unequal length of the two rails of a curve, in consequence of which two wheels fixed on an axle cannot traverse a curve properly, unless some provision is made for increasing the diameter of the outer wheel. Coning of the wheels seems to be the only thing yet devised for obtaining this increase of diameter. At present, however, there is little regularity either in the coning itself, or in the distance between the flanges of wheels for tracks of the same gauge. The tendency has been to diminish the coning,* without substituting any thing in its place. If the wheels could be made to turn independently of each other, the whole difficulty would vanish; but if this is thought to be impracticable. the present method ought at least to he reduced to some system. * Bush and Lobdell, extensive wheel-makers, say, in a note published in Appletons' Mechanic's Magazine for August, 1852, that wheels made by them for the New York and Erie road have a coning of but one sixteenth of an inch. This coning on a track of six feet gauge with the other data as given above, would suit no curve of less than a mile radius. 92 EARTH-WORK. CHAPTER IV. EART.H-WORK. ARTICLE I.-PRISMOIDAL FORMULA. 112. EATHI-WORK iiu'ludls the regular excavation and embank ment on the line of a road, borrow-pits, or such additional excavations as are made necessary when the embankment exceeds the regular ex cavation, and, in general, any tran.fers of e:lrth lhat 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. 113. 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 section midway between the parallel faces and parallel to them, and the altitude is the perpendicular distance between the parallel faces. If now b represents the base of any prism of altitude a, its solidity is a b. If b represents the base of a regular wedge or half-parallelopipedon of altitude a, its solidity is Jab. If b represents the base of a pyramid of altitude a, its solidity is i a b. The solidity of these three bodies admits of a common expression, which may be found thus. Let 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 = i b, and in the pyramid, mn == b. Moreover, the upper base of the prism = b, and the upper base of the wedge or pyramid = 0. Then the expressions a b, 1 a b, and ab may be thus transformed. Solidity of prism ab=a X 6b =a (b b 4 b) a (b + b + 4 m), 6 6 6 a a a wedge = ab = X 3b = (O + b + 2b)= ( + b+ 4 m), 6 6 6 pyramid ==ab= - X 26 =- (O +b+ b) = (O+6 +4 m). 6 6 6 BORROW-PITS. 93 Hence, the solidity of either of these bodies is found by adding together the area of the upper base, the area of the lower base, and four times the middle area, and multiplying the sum by one sixth of the altitude. Irregular wedges, or those not half-parallelopipedons, may be measured by the same rule, since they are the sum or differcnce of a regular wedge and a pyramid of common altitude, and as the rule applies to both these bodies, it applies to their sum or differcnce. Now a prismoid, being made up of prisms. wedges, and pyramids of common altitude with itself, will have for its solidity the sum of the solidities of the combined solids. But the sum of the areas of the upper and lower bases of the combined solids is equal to B + B', the sum of the areas of the parallel faces of the prismoid; and the sum of the middle areas of the combined solids is equal to 1l, the middle area of the prismoid. Therefore p S = a (B + B +4M). 6 ARTICLE II. - BORROW-PITS. 114. FOR the measurement of small excavations, such as borrowpits, &c., the usual method of preparing the ground is to divide the surface into parallelograms * or triangles, small enough to be considered planes, laid off from a base line, that will remain untouched by the excavation. A convenient bench-mark is then selected, and levels taken at all the angles of the subdivisions. After the excavation is made, the same subdivisions are laid off from the base line upon the bottom of the excavation, and levels referred to the same bench-mark are taken at all the angles. This method divides the excavation into a series of vertical prisms, generally truncated at top and bottom. The vertical edges of these prisms are known, since they are the differences of the levels ait 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 measured horizontally. 115. Problem. Given the edges h, hA, and h2, to find the solidity * If the ground is divided into rectangles, as is generally done, and one side be made 27 feet, or some multiple of 27 feet, the contents may be obtained at once in cubic yards, by merely omitting the factor 27 in the calculation. 94 EARTH-WORK. S of a vertical prism, whether truncated or not, whose horizontal section is a triangle of given area A. Fig. 48. \ C A ---- G Solution. When the prism is not truncated, we have h = b = h2. The ordinary rule for the solidity of a prism gives, therefore, S = A h = A X (h + h, + h2). When the prism is truncated, let A B CF G H (fig. 48) represent such a prism. truncated at the top. Through the lowest point A of the upper face draw a horizontal plane A D E cutting off a pyramid, of which the base is the trapezoid B D E C, and the altitude a perpendicular let fall from A on DE. Represent this perpendicular by p, and we have (Tab. X. 52) the solidity of the pyramid = pX BDEC= -p X DEX 1(BD CE)= -pX DE X A (BD + CE) = A X A (BD + CE), since Ip X DE = A DE = A. But ~ (B D + CE) is the mean height of the vertical edges of the truncated portion, the height at A being 0. Hence the formula already found for a prism not truncated, will apply to the portion above the plane A D E, as well as to that below. The same reasoning woild apply, if the lower end also were truncated. Hence. for the solidity of the whole prism, whether truncated or not, we have -a-l S x= A X (h + h, + h). 116. Problem. Given the edges h, h,, h2, and ha, tofind t'l solidity S of a vertical prism, whether truncated or not, whose horiotal ^~ction is a parallelogram of given area A. BORROW-PITS. 95 Solution. Let B H (fig. 49) represent such a prism, whether trun. rated or not, and let the plane B FHD divide it into two triangular Fig. 49. A I i prisms A FH and CFH. The horizontal section of each of these prisms will be i A, and if h, hl, h, and h3 represent the edges to which they are attached in the figure, we have for their solidity (~ 115), AFH= A X (h + + h3),and CFH = A X (h, + h + h3). Therefore, the whole prism will have for its solidity S = A X i (h + 2 h1 + h2 + 2 h3). Let the whole prism be again divided by the plane A E G C into two triangular prisms BE G and D E ( Then we have for these prisms, BE G -= A X J (h + hi + /,2), and I)E G -= A X (h + hA + -h), and for the whole prism, S = A X J (2 h + h +- 2 h,2 + hj). Adding the two-expressions found for S. we have 2 S = - A (h + h, + h2 + h3), or l S = A X (h + h + hl h2 +h3). It will be seen by the figure, that J (h + h3) = KL = - (kh + h3), or hI + h12 = hi + ha. The expression for S might, therefore, be reduced to S = A X i (h1- h2), or S= A X i (h, + h3). But as the ground surfaces A B CD and E F GH are seldom perfect planes, it is considered better to use the mean of the four heights, instead of the mean of two diagonally opposite. 117. Corollary. When all the prisms of an excavation have r}e same horizontal section A, the calculation of any number of thlem 96 EARTH-WORK. may be performed by one operation. Let figure 50 be a plan of such an excavation, the heights at the angles being denoted by a, 1,, a2, b a a, w a_____1_, _______ K ~ len C ________ &____ Fi_. _K Fig. 50. bi, &c. Then the solidity of the whole will be equal to A multiplied by the sum of the heights of the several prisms (~ 116). Into this sum the corner heights a, a2, b, b5, c5, d, and d4 will enter but once, each being found in but one prism; the heights a,, b4, c. d,, d2, and d3 will enter twice, each being common to two prisms; the heights b,, b3, and c4 will enter three times, each being common to three prisms, and the heights h.,, c,, c, and c3 will enter four times, each being common to four prisms. If, therefore, the sum of the first set of heights is represented by s,, the sum of the second by s2, of the third by s3, and of the fourth by s4, we shall have for the solidity of all the prisms S- == A (s,-+ 2s2 + 3s3- + 4s4). ARTICLE III. - EXCAVATION AND EIBANKMENT. 118. As embankments have the same general shape as excavations, it will be necessary to consider excavations only. The simplest case is when the ground is considered level on each side of the centre line. Figure 51 represents the mass of earth between two stations in an excavation of this kind. The trapezoid GBFH is a section of the mass at the first station, and G, B, Fi H, a section at the second station; A E is the centre height at the first station, and A, E, the centre height at the second station; H H, F, F is the road-bed, G GI B, B the CENTRE HEIGHTS ALONE GIVEN. 9 surface of the ground, and G G1 HI H and B B, F, F the planes forming the side slopes. This solid is a prismoid, and might be calculated by the prismoidal formula (~ 113). The following method gives the same result. A. Centre Heights alone given. 1 i9. Problem. Given the centre heights c and cl, the width of the road-bed b, the slope of the sides s, and the length of the section l, to find the solidity S of the excavation.'B ~ AtE ^~~~-1~Fig. 51. Solution. Let c be the centre heieht at A (fig. 51) and cl the height at A,. The slope s is the ratio of the base of the slope to its perpendicular height (~ 102). We have then the distance out AB -= b + sc, and the distance out Al B, = b +- s c- (~ 102). Divide the whole mass into two equal parts by a vertical plane A Al El E drawn through the centre line, and let us find first the solidity of the righthand half. Through B draw the planes B E E,, B A El, and BEl Fi, dividing the half-section into three quadrangular pyramids. having for their common vertex the point B, and for their bases the planes A A, El E, E El F, F, and Al B1 F, El. For the areas of these bases we have Area ofAA ElE - = -EEI X (AEE+ AEl) -= l(c cl), " EElF, F -EFX EE, =b l, A' A B1 F1 E1= A1E X (El,F,+A,Bi)= I (bcl +scl'); and for the perpendiculars from the vertex B on these bases, produced when necessary, 98 EARTH-WORK. Perpendicular on A A1 E1 E = A B = b +s c,. " EEEI FF =AE =c, " "AI BiF, El ==EEi=1. Then (Tab. X. 52) the solidities of the three pyramids are B-AAE,EE - J(-b+sc) X 2l (c+c1) = l(~bc+-bc -l sc2 + s.cc1) B-EEF,F ==c X bl = —lbc, B-AlB1 F E= 4 I X (b c + sc2) =I (bc + sc2). Their sum, or the solidity of the half-section, is S= 1 [b (c + cl) + s (C + c,2 + cci)]. Therefore the solidity of the whole section is S = l L[ b (C + C1) + s (c2 + c,2 + cc,)], or VP S = [b (c + cl) + s (c0 + co + cc,)]. When the slope is 11 to 1, s =, and the factor,s = may be dropped..120. Problem. To find the solidity S of any number n of successive sections of equal length. Solution. Let c, cl,, C3, &c. denote the centre heights at the snc-:essive stations. Then we have (~ 119) Solidity of first section = 1 [b (c +- c) + ~ s (c2 +- c, + c,)], " " second section = 1 [b (c, + c) +- Is (c,- + c22 + ci Ce)],' third section - I 1 [b (c2 + c3) + ~ s (C22 + C3- +c C c3)] &c. &c. For the solidity of any number n of sections, we should have I I mul tiplied by the sum of the quantities in n parentheses formed as thos( iust given. The last centre height, according to the notation adopted will be represented by Cn, and the next to the last by c, _. Collect ing the terms multiplied by b into one line, the squares multiplied b" U s into a second line, and the remaining terms into a third line, we have for the solidity of n sections S= S l b (c + 2c + 2 c + 2 3... - 2c,, + c,,) + ~ s (c2 + 2 cl2 + 2 c - 2 3....-, + + En + I S (C c, + C,2 + Cc C3.+ c 4 + Cn-I n). When s = -, the factor ~ s 1I may be dropped. CENTRE AND SIDE HEIGHTS GIVEN. 99 Exumple. Given I == 100, b == 28, s = 2, and the stations and centre heights as set down in the first and second columns of the annexed table. The calculation is thus performed. Square the heights, and set the squares in the third column. Form the successive products tc,, c, c2, &c., and place them in the fourth column. Add up the last three columns. To the sum of the second column add the sum itself, minus the first and the last height, and to the sum of the third column add the sum itself, minus the first and the last square. Then 86 is tlhi multiplier of b in the first line of the formula, 592 is the second line, since i s is here 1, and 274 is the third line. The product of 86 by b - 28 is 2408, and the sum of 274, 592, and 2408 is 3274. This multiplied by - 1 = 50 gives for the solidity 163,700 cubic feet. Station. c.. cc. 0 2 4 1 4 16 8 2 7 49 28 3 6 36 42 4 10 100 60 5 7 49 70 6 6 36 42 7 4 16 24 46 306 274 40 286 592 86 592 2408 28 2)3274 2408 163700. B. Centre and Side Heights given. 121 When greater accuracy. is required than can be attained by the preceding method, the side heights and the distances out (~ 102) are introduced. Let figure 52 represent the right-hand side of an excava cion between two stations. A A, B, B is the ground surface; A E =- and A, E =- cl are the centre heights; B G =.h and B, G, = hi, the side heights; and d and d,, the distances out, or the horizontal distances of' Band B, fiom 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; hut neither of these suppo. * It is easy in any given case to ascertain whether a surface like A Al B, B is a 100 EARTH-WORK. sitions is sufficiently accurate to serve as the basis of a general metlhod. 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 Al, from which the ground slopes downward on each side to A and B1. Instead of this, a depression might run from A to B1, and the ground rise each way to A, and B. If the ridge or depression is very marked, and does not cross the centre or side lines at the regular stations, intermediate stations must be introduced to make the triangular planes conform better to the nature of the ground. If the surface happens to be a plane, or nearly so, the diagonal may be taken in either direction. It will be seen, therefore, that the following method is applicable to all ordinary ground. When, however, the ground is very irregular, the method of ~ 127 is to be used. 122. Problem. Given the centre heights c and cl, the side heights on the right h and hi, on the left h' and htl, the distances out on the right d and d,, on the left d' and dtl, the width of the road-bed b, the length of the section 1, and the direction of the diagonals, to find the solidity S of the excavation. Solution. Let figure 52 represent the right-hand side of the excavation, and let us suppose first, that the diagonal runs, as shown in the figure, from B to Al. Through B draw the planes B E E, B Ai E1, and B E1 F, dividing the half-section into three quadrangular pyramids, having for their common vertex the point B, and for their bases the planes A Al El E, E El F1 F, and A, B, Fl E,. For the areas of these bases we have AreaofAAElE = EE1 X(AA +AlE) -=l (c -c,), " EEIFF = EFxEEi. bl, ( " lBl El =E AIEXdL+ VE, FXhA =d, c1+jbhA, and for the perpendiculars from the vertex B on these bases, produced when necessary, plane; for if it is a plane, the descent from A to B will be to the descent from A I to B,, as the distance out at the first station is to the distance out at the second station, that is, c -: cl — h = d: da. If we had c = 9, h = 6, cl = 12, HI = 8, d = 24, and di = 27, the formula would give 3 4 = 24: 27, which shows that the surface is not a plane. CENTRE AND SIDE HEIGHTS GIVEN, 101 Perpendicular on A A, E E EG = d,..." EEI FF ==BG =h, " " A B F1 E1-=E E = I. Fig. 52. AtI'G Then (Tab. X. 52) the solidities of the three pyramids are B-A A EIE = d X II(c+ c) = l(dc+de1), B-EE1F1F = h X bl -= lbh, B-AlB1 F,E1 =l X d (dl c1 -- b bh) = (di cl + bl-). Their sum, or the solidity of the half-section, is 11 (dc + d 1c + de, + bh + bhl). (1) Next, suppose that the diagonal runs from A to B1. In this case, hrough B1 draw the planes B1 E1 E, B, A E, and B, E F (not rep.'esented in the figure), dividing the half-section again into three [uadrangular pyramids, having for their common vertex the point 3l, and for their bases the planes A A1 E1 E, EEl F1 F, and A BFE.'or the areas of these bases we have rea of AA EE = EEI X (AE+A El) = I (c + lc), 4 " EEFF =EFX EEl =-bl,." ABFE =-AEXd+ EFxh =Idc+ibh; nd for the perpendiculars from B, on these bases, produced when ecessary, 102 EARTH-WORK. Perpendicular on A Al El E = E G 1 = di,.." "E E F F F B, G = hl, " ABFE = EE =l.1 Then (Tab. X. 52) the solidities of the three pyramids are B, - A A, E E = d, X (c c) 1 (dc + dlc,), B - EEI F F = ~h/ X bl =- bhl, B,-ABFE =-I X (dc+ -bh) =- l(dc + bh). Their sum, or the solidity of the half-section, is 1 (d c+ di cl + di c + bl + -bh). (2) We have thus found the solidity of the half-section.for both direc tions of the diagonal. Let us now compare the results (1) and (2), and. express them, if possible, by one formula. For this purpose let (1) be put under the form 1I [dc + d, c + dc, + 2 b (h + hl + h)l, and (2) under the form l [dc + di c + d, c + b)(h + h, + h,)]. The only difference in these two expressions is, that dct and the last h in the first, become d, c and h, in trhe second. But in the first case, c, and h are the heights at the extremities of the diagonal, and d is the distance out corresponding to h; and in the second case. c and h, are the heights at the extremities of the diagonal, and dl is the distance out corresponding to h,: Denote the centre height touched by the diagonal by C, the side height touched by the diagonal by H, and the distance out corresponding to the side height H by D. We may then express both de, and d, c by D C, 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 i [d c + d, c, + D C + b (h + h, + H)1. (3) To obtain the contents of the portion on the left of the centre line, we designate the quantities on the left by the same letters used for cor. responding quantities on the right, merely attaching a (') to them to distinguish them. Thus the side heights are h' and h',, and the distances out d' and dal, while D, C, and H become D', C', and Ht The solidity of the half-section on the left may therefore be taken di rectly from (3), which will become CENTRE AND SIDE HEIGHTS GIVEN. 103 td' c + d'1 ci + D' C' + b (h' + h' + H')]. (4) Finally, by uniting (3) and (4), we obtain the following formula for the solidity of the whole section between two stations - S 61[(d - +d')c + (d +- dt,) cl + D C+ DI C'+ - b (h + h, + H + h' + h + H)]. Example. Given 1 = 100, b= 18, and the remaining data, as ar ranged in the first six columns of the following table. The first collinn 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 cl = 8 and from c - 13.6 to h, = — 12. V1ta. d'. h'. c. h. d. d+d'. (d+d')c. D' C' D C. 0 21 8 13.6. 10 24 45 612 1 15 4 8.0 12 27 42 336 168 367.2 12 12 168 20 367.2 54 X 9 486 6)1969.20 32820. To apply the formula, the distances out at each station are added together, and their sum placed in the seventh column; these sums, multiplied by the respective centre heights, are placed in the eighth column; the product of d' = 21 (which is the distance out corresponding to the side height touched by the left-hand diagonal) by c, = 8 which is the centre height touched by the same diagonal) is placed the ninth column, and the similar product of d, = 27 by c = 13.6 s placed in the last column. The terms in the formula multiplied by b are all the side heights, and in addition all the side heights touched )y diagonals; or 8 + 4 + 10 + 12 + 8 + 12 = 54. Then by sub.;titution in the formula, we have S = } X 100 (612 + 336 + 168 - 367.2 + 9 X 54) = 32,820 cubic feet." The example here given is the same as that calculated in Mr. Borden's " Sys 104 EARTH-WORK. By applying the rule given in the note to ~ 121, we see that the surface on the left of the centre line in the preceding example is a plane; since 13.6 - 8: 8 - 4 - 21:15. The diagonal on that side might, therefore, be taken either way, and the same solidity would be obtained. This may be easily seen by reversing the diagonal in this example, and calculating the solidity anew. The only parts of the formula affected by the change are D' C' and b HI. In the one case the sum of these terms is 21 X 8 + 9 X 8, and in the other 15 X 13.6 + 9 X 4, both of which are equal to 240. 123 Problem. To find the solidity-S of any number n of successive sections of equal length. Solution. Let c, c,, C2, C3, &c. be the centre heights at the successive stations; h, Ah, h2, h,, &c. the right-hand side heights; h', h', h'2, h'3, &c. the left-hand side heights; d, dl,, d, d, &c. the distances out on the right; and d', d'l,, d', d, &e. the distances out on the left. Then the formula for the solidity of one section (~ 122) gives for the solidities of the successive sections 11[(d + l) c + (d, +d,)c, D C+ D'C' + b (h + + H+ h' + hA + H')], l [(d +d d,)c, + (d2 +d',) c2 + D, C, + D', C,+ ib (hi +A 2+ HI, + hA' h + H',)],' 1 [(d2 + d'^) c, + ((1 + dl3) c3 + D2 C2 + D'2 C'2 + 2 b (h2 + hA + H + h'+ h' +'H')1, and so on, for any number of sections. For the solidity of any number n of sections, we should have l 1 multiplied by the sum of n parentheses formed as those just given. Hence GS=l (d+ d') c + 2 (d,+ d',) c,+ 2 (d2 + d'2) c... + (dn + d',) Cn +DC+ D'C' + DIC, +- D'C'^C + DC2Q- D'C2 + &c. + blh + 2h, + 2h....+ h,,+ H+H + H + &c. + h'+ 2 h -+ 2 h....+ In + Hl'+IHI+H2 + &c. tern of Useful Formulae, &c," page 187. It will be seen, that his calculation makes the solidity 32,460 cubic feet, which is 360 cubic feet less than the result above. This difference is owing to the omission, by Mr. Borden's method, of a pyramid inclosed by the four pyramids, into which the upper portion of the right-hand half section is by that method divided. CENTRE AND SIDE HEIGHTS GIVEN. 105 Example. Given I = 100, b = 28, and the remaining data as given in the first six columns of the following table. Sta. d'. h'. c. h. d. d + d'. (d+d) c. D'C'. DC. 0 17 2, 2 2 17 34 68 1 18.5 3 >4'-5 21.5 40 160 681 43 2 20 4,-1 5_ -6 23 43 215 80 92 3 23 6 61 -8 26 49 294 115 130 4 21.5 5- f6 >7 24.5 46 276 129' 147 5 20 4 - 6 _l4 20 40 240 120' 147 6 15.5 11 4j 3 18.5 34 136 93' 80 25 35 1389 605 639 22 30 1185 22 37 605 69 102 639 102 2394 171 X 14 2394 6)6212 103533 cubic feet. The data in this table are arranged precisely as in the example for calculating one section (~ 122), and the remaining columns are calculated as there shown. Then, to obtain the first line of the formula, add all the numbers in the column headed (d + 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 D C, which give respectively 605 and 639. To obtain the first line of the quantities multiplied by b, add all the numbers in column h, making 35, next all the numbers except the first and the last, making 30, and lastly all the numbers touched by diagonals (doubling any one touched by two diagonals), making 37. The second line of the quantities multiplied by I b is obtained in the same way from the column marked h'. The sum of these numbers is 171, and this multiplied by lb = 14 gives 2394. We have now for the first line of the formula 1389 + 1185, for the second 605 + 639, and for the remainder 2394. By adding these together, and multiplying the sum by 1 = -, we get the contents of the six sections in feet. 124. When the section is partly in excavation and partly in embankment, the preceding formulae are still applicable; but as this application introduces minus quantities into the calculation, the following method, similar in principle, is preferable. 125. Problem. Given the widths of an excavation at the road-bed 6 106 EARTH-WORK. A F = w and A, F, = wi (Jig. 53), the side heights h and h,, the length of the section 1, and the direction of the diagonal, to find the solidity S of the excavation, when the section is partly in excavation and partly in em. bankment. Fig. 53. Al P Solution. Suppose, first, that the surface is divided into two triangles by the diagonal BA1. Through B draw the plane BAi F,, dividing that part of the section which is in excavation into two pyramids B - A A, F F and B - Al B, Fl, the solidities of which are B-A Al F F = h X 1 (w + w,) = lI(wh + uh), B- Al BI I =' X i lwlhl. Thie whole solidity is, therefore, S = 1 (w h + wl h, + w, h). Next, suppose the dividing diagonal to run from A to Bl. Through B1 draw a plane Bi A F (not represented in the figure), dividing the excavation again into two pyramids, of which the solidities are B -AA, FF= h, X lI(w+ -wi) = I (w h + wi ), BI-ABF = I X l wh = =lwh. The whole solidity is, therefore, S== l (wh + wi h + wh,). The only difference in these two expressions is, that w, h in the first becomes w Ah in the second. But in the first case the diagonal touches wi and h, and in the second case it touches w and h,. If, then, we designate the width touched by the diagonal by W, and the height touched by the diagonal by /, we may express both wo h and wl h, by WIT; so that the solidity in either case may he expressed by CENTRE' AND SIDE HEIGHTS GIVEN. 107 IP S==l (wh +wh, + WH). Corollary. When several sections of equal length succeed one another, the whole may be calculated together. For this purpose, the preceding formula gives for the solidities of the successive sections ll(wh +w, h+ WH), 6 (wl hl + two h. + WI Hj), I(wh, + w.h3 +3 W. H), and so on for any number of sections. Hence for the solidity of any number n of sections we should have S= 11(w h+ - 2 W h, + 2 w+ h2....+ wh+ WH+ WI + W2 2 + &c.) Example. Given I = 100, and the remaining data as given in the first three columns of the following table. Station. w. h. w. WH. 0 2.\ 2 1 8< 6 48 8 2 10 7 70 56 3 13 7 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 of those widths and heights touched by diagonals. The sum of the products in the fourth column is 247, the sum of all but the first and the last is 209, and the sum of the products in the fifth column is 186. These three sums are added together, multiplied by 100, and divided by 6, according to the formula. This gives the solidity of the four sections 10700 cubic feet. 126. When the excavation does not begin on a line at right angles to the centre line, intermediate stations are taken where the excavation begins on each side of the road-bed, and the section may be calcu. 1Ot EARTH-WORK. lated as a pyramid, having its vertex at the first of these points, and for its base the cross-section at the second. The preceding method gives the same result, since w and h in this case become 0, and reduce,he formula to S = I wl hA. The same remarks apply to the end of an excavation. C. Ground very Irregular. 127. Problem. To find the solidity of a section, when the ground is very irregular.'ig..54. ~~A~ ~~ ~~~- I EL G Solution. Let A HB FE - A CD B1 F, El (fig. 54) represent one side of a section, the surface of which is too irregular to be divided into two planes. Suppose, for instance, that the ground changes at H, C, and 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 vertically downward through the side line B B, to the plane of the road-bed, we may form as many vertical triangular prisms as there are triangles on the surface. This will be made evident by drawing vertical planes through the sides * Tt will often be necessary to introduce intermediate stations, in order to make the subdivision into triangles more conveniently and accurately. GROUND VERY IRREGULAR. 109 A C, H C, HD, and H BR. Then the solidity of the half-section will be equal to the sum of these prisms, minus the triangular mass B G - B, F, 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 ~ 115. To find the solidity of the portion B F G - B, F, G, which is to be deducted, represent the slope of the sides by s (~ 102), the heights at B and B, by h and h, and the length of the section by 1. Then we have F G = s h, and F1 G, = s h. Moreover, the area of B F G = s il, and that of B, FI (Gi = s h,2. Now as the triangles B F G and B, F Gi are similar, the mass required is the frustum of a pyramid, and the mean area is / sh2 X s 1i2 == shhl. Then (Tab. X 53) the solidity is BF Gi- B1 F1 G1 = ls (h -h12 hhl). Example. Given I = 50, b =18, s = 2, the heights at A, H, and B respectively 4, 7, and 6, the distances A H= 9 and HB = 9, the heights at Al, C, D, and Bi respectively 6, 7, 9, and 8, and the distances A, C= 4, CD = 5, and D B = 12 Then the horizontal section of the first prism adjoining the centre line is I I X A1 C, since the distance Al C is measured horizontally; and the mean of the three heights is A (4 + 6 + 7) = i X 17. The solidity of this prism is therefore I2 X Al C X X 17= - X 4 X 17, that is, equal to Ul 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 11(4 X 17+9 X 18+5 X 23+ 12 X 24+9 X 21) = l X 822. For the frustum to be deducted, we have U X 2(62 +82+6X8) =1l X 222. Hence the solidity of the half-section is (822 - 222) = - X 50 X 600 = 5000 cubic feet. 128. Let us now examine the usual method of calculating excavation, when the cross-section of the ground is not level. This method consists, first, in finding the area of a cross-section at each end of the mass; secondly, in finding the height of a section, level at the top, equivalent in area to each of these end sections; thirdly, in finding from the average of these two heights the middle area of the mass; 110 EARTH-WORK. and, lastly, in applying the prismoidal formula to find the contents The heights of the equivalent sections level at the top may be found approximately by Trautwine's Diagrams,* or exactly by the following method. Let A represent the area of an irregular cross-section, b the width of the road-bed, and s the slope of the sides. Let x be the required height of an equivalent section level at the top. The bottom of the equivalent section will be b, the top b + 2 s x, and the area will be the sum of the top and bottom lines multiplied by half the height or [x(2b + 2sx) = s'x + bx. But'this area is to be equal to A Therefore, s 2 +_ b x = A, and from this equation the value of x may be found in any given case. According to this method, the contents of the section already calculated in ~ 122 will be found thus. Calculating the end areas, we find the first end area to be 387 and the second to be 240. Then as s is here 3 and b = 18, the equations for finding the heights of the equivalent end sections will be 3 + 18x = 387, and x + 18x = 240. Solving these equations, we have for the height at the first station x = 11.146, and at the second, x = 8. The middle area will, therefore, have the height I (11.146 + 8) = 9.573, and from this height the middle area is found to be 309.78. Then by the prismoidal formula (~ 113) the solidity will be S = I X 100 (387 + 240 + 4 X 309.78) = 31102 cubic feet. But the true solidity of this section was found to be 32820 cubic feet, a difference of 1718 feet. The error, of course, is not in the prismoidal formula, but in assuming that, if the earth were levelled at the ends to the height of the equivalent end sections, the intervening earth might be so disposed as to form a plane between these level ends, thus reducing the mass to a prismoid. This supposition, however, may sometimes be very far from correct, as has just been shown. If the diagonal on the right-hand side in this example were reversed, that is, if the dividing line were formed by a depression, the true solidity found by ~ 122 would be 29600 feet; whereas the method by equivalent sections would give the same contents as before, or 1502 feet too much. D. Correction in Excavation on Curves 129. In excavations on curves the ends of a section are not parallel * A New Method of Calculating the Cubic Contents of Excavations and Embank ments by the aid of Diagrams. By John C. Trautwine. CORRECTION IN EXCAVATION ON CURVES. 111 to each other, but converge towards the centre of the curve. A section between two stations 100 feet apart on the centre line will, therefore, measure less than 100 feet on the side nearest to the centre of the curve, and more than 100 feet on the side farthest from that centre. Now in calculating the contents of an excavation, it is assumed that the ends of a section are parallel, both being perpendicular to the chord of the curve. Thus, let figure 55 represent the plan of two sections of iB, iB I Ai A L \ / Fig. 55. an excavation, E F G being the centre line, A L and CM the extreme side lines, and 0 the centre of the curve. Then the calculation of the frst section would include all between the lines A, C1 and B, D) while the true section lies between A C and B D. In like manner, the calculation of the second section would include all between HK and NP; while the true section lies between BD and L M. It is evident, therefore, that at each station on the curve, as at F, the calculation is too great by the wedge-shaped mass represented by KFD,, and too Fig. 56. M B Dm B small by the mass represented by B1 F'B. These masses balance 112 EARTH-WORK. each other, when the distances out on each side of the centre line are equal, that is. when the cross-section may be represented by A DFRiE (fig. 56). But if the excavation is on the side of a hill, so that the distances out differ very much, and the cross-section is of the shape A DFBE, the difference of the wedge-shaped masses may require consideration. 130. Problenl. 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 R or deflection angle D. Solution. The correction, from what has been said above, is a triangular prism of which B FR (fig. 56) is a cross-section. The height of this prism at B (fig. 55) is B, H, the height at R is RI S, and the height at F is 0. B1 H and R1 S, being very short, are here considered straight lines. Now we have the cross-section B FR FB E G - FREG = (I2cd + ibh) - (c11d + bh') = Ic(d- d') + b (i --- h'). To find the height B, H, we have the angle BFH = BFBI = D, and therefore B1 H = 2 HF sin. D = 2d sin. D. In like manner, R? S = KDi = 2KF sin. D = 2d1 sin. D. Then since the height at Fis 0, one third of the sum of the heights of the prism will be R (d + d') sin. D, and the correction, or the solidity of the prism, will be (~ 115) C = [l c (d - d') -+ b (h - h')] X. (d + d ) sin. D. When R is given, and not D, substitute for sin. D its value (~ 9) 50 sin. D = T. The correction then becomes C= c (d - d ) + b (h -h')] X ( + ). 3 R This correction is to be added, when the highest ground is on the convex side of the curve, and subtracted, when the highest ground is on the concave side. At a tangent point, it is evident, from figure 55, that the correction will be just half of that given above. Example. Given c = 28, h = 40, h' = 16, d = 74, d' = 38, b = 28, and R = 1400, to find C. Here the area of the cross-section B FR.= 28 28 2 (74 -38) + 4 (40 - 16) = 672, and one third of the sum of the heights of the prism is 00 ( + ) Hence C= 672 X 8 t792 cubic feet. CORRECTION IN EXCAVATION ON CURVES. 11; 131. When the section is partly in excavation and partly in em1,ankment, the cross-section of the excavation is a triangle lying wholly on one side of the centre line, or partly on one side and partly on the other. The surface of the ground, instead of extending from B to D (fig. 56), will extend fiom B to a point between G and E, or to a point between A and G. In the first case, the correction will be a triangular prism lying between the lines B F and HF (fig. 55), but not extending below the point F. In the second case, the excavation extends below F, and the correction, as in ~ 129, is the difference between the masses above and below F. This difference may be obtained in a very simple manner, by regarding the mass on both sides of F as one triangular prism the bases of which intersect on the line G F (fig. 56), 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. 132. Problem. Given the width of the excavation at the road-bed w, the width of the road-bed b, the distance out d, and the side height h, to find the correction in excavation C, at any station on a curve of radius R r' deflection angle D, when the section is partly in excavation and partly in embankmentt. 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. 56) is (~ 130) B H- 2 HF sin. D = 2 d sin. D. In asimilar manner, the height at E will be 2 G E sin. D = b sin. D, and at the point intermediate between G and E, the distance of which from the centre line is Ib - w, the height will be 2 (. b- w) sin. D (b - 2 w) sin. D. Hence, the correction, or the solidityof the prism, will be (~ 115) C- wh X (2d-b+b- 2w) sin. B -~wh X w ( +d- b- w) sin. D. When the excavation lies on both sides of the centre line, the correction, fromu what has been sai# above, is a triangular prism having also for its cross-section the c isection of the excavation. Its area will, therefore, be I w h. The height of this prism at B is also 2 d sin. D, and the height at E, b sin. D; but at the point intermediate between A and G, the distance of which from the centre line is w - ~ b, the height will be 2 (wo - b) sin. D = (2 w - b) sin D. As this height is to be considered minus, it must le subtracted from the others, and the correction required will be C =- w h X { (2 d + b-2 w + b) sin. P 114 EAriTH-WORK. - i w h X (d + b - w) sin. D. Hence, in all cases, when the see. tion is partly in excavation and partly in embankment, we have the formula Mt C _= w h X ~ (d + b - w) sin. D. When R is given, and not D, substitute for sin. D its value (~ 9) 50 sin. D =. The correction then becomes C= wh X 100 (d + b- w) 3R 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 is wh = 17 X 12 =!o0 (d-i- b —w) 204. and one third of the sum of the heights of the prism is 3 R 100 (51 +8 0 - 17) 4 4 = 1 1600 -) =. Hence C = 204 X 8 = 272 cubic feet. 133. The preceding corrections (~ 130 and ~ 132) suppose the length of the sections to be 100 feet. If the sections are shorter, the angle B FH (fig. 55) may be regarded as the same part of D that F G is ol 100 feet, and B1 FB as the same part of D that E Fis of 100 feet. The true correction may then be taken as the same part of C that the sum of the lengths of the two adjoining sections is of 200 feet. TABLE I. RADII, ORDINATES, DEFLECTIONS, AND ORDINATES FOR CURVING RAILS. Formula for Radii, ~ 10; for Ordinates, ~ 25; for Deflections, 19I for Curving Rails, ~ 29. 116 TABI.E I. RADII, ORDINATES, DEFLECTIONS, 7Ordinates. TnetIhod Ordinates for jOrdinates. Tangent Chord Rails. Degree. Radii. - Deflec- Deflec126. 25. 37A. 50. tion. tion. 18. 20. I 0 5 68754.94.08.014.017.01. 073.0 145.001.001 10 34377.48.016.027.034.036.145.291.001.001 15 2291833.024.405.01 051 5.218.436.002.002 23 17188.76.032.055.063.073.291.582.002.003 25 13751.02.040.068.085.091.364.727.003.004 30 11459.19.018.082.102.109.436.873.004.004 35 9322.18.056.095.119.127.509 1.018.004.005 40 8594.41.064.109.136.145.582 1.164.005.006 45 763949.072.123.153.164.654 1.309.005.007 50 6375.55.080.136.170. 182.727 1.454.006.007 55 6250.51.037.150.187.200.800 1.600.006.008 1 0 5729.65.095.164.205.218.873 1.745.007.009 5 5238.92. 103.177.222.236.945 1.891.008.009 10 4911 15.111.191.239.255 1.018 2.036.008.010 15 4583.75.119.205.256.273 1.091 2.182.009.011 20 4297.28.127.218.273.291 1.164 2.327.009.012 25 4044.51.135.232.290.309 1.236 2.472.010.012 30 3819.83.143.245.307.327 1.309 2.618.011.013 35 3618.80.151.259.324.3451 1382 2.763,011.014 4C 3437.87.159.273.341.364 1.454 2.909.012.015 45 3274.17.167.286.358.382 1.527 3.054.012.015 50 3125.61.175.300.375.400 1.600 3.200.013.016 55 2989.48.183.314.392.418 1.673 3.345.014 017 9 0 2864.93.191.327.409.436 1.745 3.490.014.017 5 2750.35.199.341.426.455 1.818 3.636.015.018 10 2644.58.207.355.443.473 1.891- 3.781.015.019 15 2546.64.215.368.460.491 1.963 3.927.016.020 20 2455.70.223.3-2.477.509 2.036 4.072.016.020 25 2371.04.231.395.494.527 2.109 4.218.017.021 30 2292.01.239.409.511.545 2.181 4.363.018.022 35 2218.09.247.423.528.564 2. 2.- 4.508.018.023 40 2148.79.255.436.545.582 2.327 4.6:4.019.023 45 2083.63 -.263.450.562.600 2.400 4.799.019.024 SG 2022.41.270.464.580.618 2.472 4.945.020.025 55 1964 64.278.477.597.636 2.545 5.090.02i.025 3 0 1910.08.286.491.614.655 2.618 5.235.021.026 5 1858.47 -294.505.631.673 2.690 5.381.022.027 10 1809.57.302.518.648.691 2.763 5.526.022.028 15 1763 18.310.532.665.709 2.836 5.672.023.023 20 1719.12.318.545.682.727 2.908 5.817.024.029 25 1677 20.326.559.699.745 2.981 5.962.024.030 30 1637.28.334.573.716.764 3.054 6.108.025.031 35 1599.21.342.586.733.782 3.127 6.253.025.031 40 1562.88.350.600.750.800 3.199 6.398.026.032 45 1528.16.358.614.767.818 3.272 6.544.027.033 50 1494.95.366.627.784.836 3.345 6.6S9.027.033 55 1463.16.374.6411.801.855 3417 6.835.028.034 4 0 1432.69.382.655.818.873 3.490 6.980.028.035 5 1403 46.390.663.835.891 3.563 7.125.029.036 10 1375.40.398.682.8.52.909 3.635 7.271.029i.036 15 1348.45.406.695.869.927 3.708 7.416.030.037 20 1322.53.414.709.886.945 3.781 7.561.031.038 25 1297.58.422.472.903.964 3.853 7.707.0311.039 30 1273.57.430.736.921.982 3.926 7.852.032i.039 35 125042.438.750.938 1.000 3.999 7.997.0321.040 40 1228.11.446.764.955 1.018 4.071 8.143.033'.041 45 1206.57.454.777.972 1.036 4.144 8.288.034.041 50 1185.78.462.791.989 1.055 4.217 8.433.034.042 55 1165.70.469.805 1.006 1.073 4.289 8.579.03.043 5 1146.28.477.818 1.023 1.091 4.362 8.724.035.044 (',, —-= —-— ~~~~~~~~~~~~~~~ AND ORDINATES FOR CURVING RAILS. 117 Ordinates. 1J.Fi ( IIir4* Ordinates for Ordinates. Tangentl Chord Rails. Degree. Radii. Dee- Deflec- Defle- 121. 25. 374. 50. tion. tion. 18. 20. 5 5 1127.50.485.832 1.040 1.109 4.435 8.869.036.044 10 1109.33.493.846 1.457 1.127 4.507 9.014.037.045 15 1091 73 501.859 1.074 1.146 4.580 9.160.037.046 20 1074.68.509.873 1.091 1.164 4.653 9.305.038.047 25 1058.16.517.887 1.108 1.182 4.725 9.450.038.047 30 1042.14.525.900 1.125 1 200 4.798 9.596.039.048 35 1026.60.533.914 1.142 1.218 4.870 9.741.039.049 40 1011.51.541.928 1.159 1.237 4.943 9.886.040.049 45 996.87.549.941 1.176 1.255 5.016 10.031.041.050 50 982.64.557.9.55 1.193 1.273 5.088 10.177.041.051 55 968.81.565.968 1.210 1.291 5.161 10.322.042.052 6 0 95537.573.9S2 1.228 1.309 5.234 10.467.042.052 5 942.29.581.996 1.245 1.327 5.3C6 10.612.043.053 10 929 57.589 1.009 1.262 1.346 5.379 10.758.044.054 15 917.19.597 1.023 1.279 1.364 5.451 10.903.044.055 20 905.13.605 1.037 1.296 1.382 5.524 11.048.045.055 2.5 893.39.613 1.050 1.3131 AO 5.597 1 1.193.045.056 30 881.95.621 1.064 1.330 1.418 5.669 11.339.046.057 35 870.79.629 1.078 1.347 1.437 5.742 11.44.047.057 40 859.92.637 1.091 1.364 1.455 5.814 11.629.047.058 45 849.32.645 1.105 1.381 1.473 5.887 11.774.048.059 50 838.97.653 1.118 1.398 1.491 5.960 11.919.048.060 55 828.88.661 1.132 1.415 1.510 6.032 12.u65.049.060 7 0 819.02.669 1.146 1.432 1.528 6.1U5 12.210.049 061 5 809.40.677 1.159 1.449 1.546 6.177 12.355.050.062 10 800.00.6851 1.173 1.466 1.564 6.250 12.500.i051.063 15 790.81.693 1.187 1.483 1.582 6.323 12.645.051.063 20 781.84.701 1.200 1.501 1.600 6.395 12.790.052.064 25 773.07.709 1.214 1.517 1.619 6.468 12.961.052.065 30 764.49.717 1.228 1.535 1.637 6.540 13.081.053.065 35 756.10.725 1.242 1.552 1.655 6.613 13.226.054.066 40 747.89.733 1.255 1.569 1.673 6.685 13.371.054.067 45 739.86.740 1.269 1.586 1.691 6.758 13.516.055.068 50 732.01 -.748 1.283 1.603 1.710 6.831 13.661.055.068 55 724.31.756 1.296 1.620 1.728 6.903 13.806.056.069 8 0 716.78.764 1.310 1.637 1.746 6.976 13.961.057 070 5 709.40.772 1.324 1.654 1.764 7.048 14.096.057.070 10 702.18.780 1.337 1.671 1.782 7.121 14.241.058.071 15 695.09.788 1.351 1.688 1.801 7.193 14.387.058.072 20 638.16.796 1.365 1.705 1.819 7.266 14.532.059.073 25 681 35.804 1.378 1.722 1.837 7.338 14.677.059.073 30 674.69.812 1.392 1.739 1.855 7.411 14.822.060.074 35 668.15.820 1.406 1.757 1.873 7.483 14.967.061.075 40 661.74.828 1.419 1.774 1.892 7.556 15.112.061.076 45 655.45.836 1.433 1.791 1.910 7.628 15.257.062.076 50'649.27.844 1.447 1,808 1.928 7.701 15.402.062,077 55 643.?2.852 1.460 1.825 1.1946 7.773 15.547.063.078 9 0 637.27.80 1.474 1.842 1.965 7.846 15.692.064'.078 5 631.44.868 1.488 1.859 1.983 7918 16.837.0641.079 10 625 1.876 1.501 1876 2.001 7.991 16.962.065.080 1 1 620.09.884 1.515 1.893 2.019 8.063 16.127.065.081 20 614.66.802 1.5291 1910 2.037 8.136 16.272.066.081 25 609.i14.900 1.542 1.927 2.056 8.208 16.417.066.082 30 603.80.908 1.556 1.944 2.074 8.281 16.562.0671.083 35 59857.916 1.570 1.961 2.092 8.353 16.707.068;.084 40 593.42.924 1.583 1.979 2.110 8.426 16.852.068;.084 45 588.36.932 1.597 1.996 2.128 8.498 16.996.069.085 50 583.38.940 1.611 2.013 2.147 8.571 17.141.069.086 55 578.49.948 1.624 2.030 2.165 8.643 17.286.070.086 10 0 573.69.956 1.638 2.047 2.183 8.7161 17.431.71 087 118 TABLE I. RADII, ORDINATES, DEFLECTIONS, &C. ~dnts Ordinates. f or ~Ordinates. Tangent Chord Rails. Degree Radii. Deflec- Deflec12. 25. 37-. 60. tion. tion. 18. 20. 10 1'J 564.31.972 1.665 2.081 2.219 8.860 17.721.072.089 20 555.23.988 1.693 2.115 2.256 9.005 18.011.073.090 30 546.44 1.004 1.720 2.149 2.292 9.150 18.300.074.092 40 537.9-2 1.020 1.748 2.184 2.329 9.295 18.590.075.093 50 529.67 1.036 1.775 2.218 2.365 9.440 18.880.076.094 11 0 521.67 1.052 1.802 2.252 2.402 9.585 19.169.078.096 10 513.91 1.003 1.830 2.286 2.438 9.729 19.459.079.097 20 506.38 1.084 1.857 2.320 2.475 9.874 19.748.080.099 30 499.06 1.100 1.834 2.354 2.511 10.019 20.038.081.100 40 491.96 1.116 1.912 2.389 2.547 10.164 20.327.082.102 50 485.05 1.132 1.938 2.423 2.5.84 10.308 20.616.084.103 12 0 478.34 1.148 1.967 2.457 2.620 10.453 20.906.085.105 10 471.81 1.164 1.994 2.491 2.657 10.597 21.195.086.106 20 465.46 1.180 2.021 2.525 2.693 10.742 21.484.087.107 30 459.28 1.196 2.049 2.560 2.730 10.887 21.773.088.109 40 453.26 1.212 2.076 2.594 2.766 11.031 22.063.089.110 50 447.40 1.228 2.104 2.628,2.803 11.176 22.352.091.112 13 0 441.68 1.244 2.131 2.662 2.839 11.320 22.641.092.113 10 436.12 1.260 2.159 2.697 2.876 11.465 22.930.093.115 20 430.69 1.277 2.186 2.731 2.912 11.609 23.219.094.116 30 425.40 1.293 2.213 2.765 2.949 11.754 23.507.095.118 40 420.23 1.309 2.241 2.799 2.985 11.898 23.796.096.119 50 415.19 1.325 2.268 2.833 3.022 12.043 24.085.098.120 14 0 410.28 1.341 2.296 2.868 3.058 12.187 24.374.099.122 10 405.47 1.357 2.323 2.902 3.095 12.331 24.663.100.123 20 400.78 1.373 2.351 2.936 3.131 12.476 24.951.101.125 30 396.20 1.389 2.378 2.970 3.168 12.620 25.240.102.126 40 391.72 1.405 2.406 3.005 3.204 12.764 25.528.103.12S 50 387.34 1.421 2.433 3.039 3.241 12.908 25.817.105.129 15 0 333.06 1.437 2.461 3.073 3.277 13.053 26.105.106.131 10 378.88 1.453 2.438 3.107 3.314 13.197 26.394.107.132 20 374.79 1.469 2.515 3.142 3.330 13.341 26.632.108.133 30 370.78 1.486 2.543 3.176 3.387 13.495 26.970.109.135 40 366.86 1.502 2.570 3.210 3.423 13.629 27.258.110.136 50 363.02 1.518 2.598 3.245 3.460 13.773 27.547.112.138 16 0 359.26 1.534 2.625 3.279 3.496 13.917 27.835.113.139 10 355.59 1.550 2.653 3.313 3.533 14.061 28.123.114.141 20 351.98 1.566 2.630 3.347 3.569 14.205 28.411.115.142 30 348.45 1.582 2.708 3.332 3.606 14.349 28.699.116.143 40 344.99 1.598 2.736 3.416 3.643 14.493 28.986.117.145 50 341.60 1.615 2.763 3.450 3.679 14.637 29.274.119.146 17 0 338.27 1.631 2.791 3.485 3.716 14.781 29.562.120.148 10 335.01 1.647 2.818 3.519 3.752 14.925 29.850.121.149 20 331.82 1.663 2.846 3.553 3.789 15.069 30.137.122.151 30 328.63 1.679 2.873 3.588 3.825 15.212 30425.123.152 40 325.60 1.695 2.901 3.622 3862 15.356 30.712.124.154 50 322.59 1.711 2.923 3.656 3.898 15.500 31.000.126.155 18 0 319.62 1.723 2.956 3.691 3.935 15.643 31.287.127.156 10 316.71 1.744 2.983 3.725 3.972 15.787'31.574.128.158 20 313.86 1.760 3.011 3.759 4.008 15.931 31.861.129.159 30 311.06 1.776 3.039 3.794 4.045 16.074 32.149.130.161 40 308.30 1.792 3.066 3.828 4.081 16.218 32.436.131.162 50 305.60 1.809 3.094 3.862 4.118 16.361 32.723.133.164 19 0 302.94 1.825 3.121 3.897 4.155 16.505 33.010.134.165 10 300.33 1.841 3.149 3.931 4.191 16.648 33.296.135.166 20 297.77 1.857 3.177 3.965 4.228 16.792 33.583.136}.168 30 295.25 1.873 3.204 4.000 4.265 16.935 33.870.137i.169 40 292.77 1.890 3.232 4.034 4.301 17.078 34.157.138.171 50 290.33 1.906 3.259 4.069 4.338 17.222 34.443.140.172 20 0 287.94 1.922, 3.287 4.103 4.374 17.365 34.730.141.174........ TABLE II. LONG CHORDS. 119 TABLE II. LONG CHORDS. ~ 69.. Degree of 2 Stations. Stations. 4 Stations. 5 Stations 6 Stations. Curve. 0 10 200.000 299.999 399.998 499.996 599.993 20 199.999.997.992.933.970 30.998.992.981.962.933 40.997.986.966.932.882 60.995.979.947.894.815 1 0 199.992 299.970 399.924 499.848 599.733 10.990.959.896.793.637 20.986.946.865.729.526 30.983.932.829.657.401 40.979.915.789.577.260 50.974.898.744.488.105 2 0 199.970 299.878 399.695 499.391 598.934 10.964.857.643.285.750 20.959.834.586.171.550 30.952.810.524.049.336 40.916.783.459 498.918.106 50.939.756.389.778 597.862 3 0 199.931 299.726 399.315 498.630 597.604 10.924.695.237.474.331 20.915.662.154.309.043 30.907.627.068.136 596.740 40.898.591 398.977 497.955.423 50.888.553.882.765.091 4 0 199.878 299.513 398.782 497.566 595.744 10.868.471.679.360.383 20.857.428.571.145.007 30.846.383.459 496.921 594.617 40.834.337.343.689.212 50.822.289.223.449 593.792 5 0 199.810 299.239 398.099 496.200 593.358 10.797.187 397.970 495.944 592.909 20.783.134.837.678 446 30.770.079.700.405 591.968 40.756.023.559.123.476 50.741 298.964.413 494.832 590.970 6 0 199.726 298.904 397.264 494.534 590.449 10.710.843.110.227 589.913 20.695.779 396.952 493.912.364 30.678.714.790.588 588.800 40.662.648.623 257.221 50.644.579 453 492.917 687.628 7 1) 199.627 298.509 396.278 492.568 587.021 10.609.438.099.212 686.400 20.591.364 395.916 491.847 585.765 30.572.289.729.474.115 40 *553.212.538.093 584.451 50.533.134.342 490.704 583.773 8 0.513 298.054 395.142 490.306 583.081 120 TABLE III. -TABLE IV. TABLE III. CORRECTION FOR THE EARTH'S CURVATURE AND FOR REFRACTION. ~ 105. D. d. D. d. D. d. D. d. 300.002 1800.066 3300.223 4800.472 400.003 1900.074 300.237 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 1 mile.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 4.500.415 8 36.566 1600 052 3100.197 4600.434 9 46.279 1.700.059 3200.210 4700.453 10" 57.135 TABLE IV. ELEVATION OF THE OUTER RAIL ON CURVES. ~ 110. Degree. M 15. M =20. M = 26. M 80. M= 40. M 60. 1.012.022 034.049.088.137 2.025.044.068.099.175.274 3.037.066.103.148.263.411 4.049.088.137.197.351.548 5.062.110.171.247.438.685 6.074.131.205.296.526.822 7.OS6.153.240.345.613.958 8.099.175.274.394.701 1.095 9.111.197.308.443.788 1.232 10.123.219.342.493.876 1.368 __,.... t~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE V. - TABLE VI. 121 TABLE V. FROG ANGLES, CHORDS, AND ORDINATES FOR TURNOUTS. This table is calculated for g = 4.7, d =.42, and S = 1~ 20'. For mula for frog angle F, and chord B F, ~ 50; for m, the middle or. dinate of B F, ~ 25; for m', the middle ordinate for curving an 18 ft rail, ~ 29. R.. BF. m. nm. R. F. BF. m. m'. 0 j I it 1000 527 44 72.22.651.041 9 6 57 48 59.17.727.068 975 5 31 39 71.53.655.042 575 7 6 26 58.16.733.070 950 5 35 44 70.83.659.043 550 7 15 40 57.12.739.074 925 5 39 59 70.1I.663.044 525 7 25 33 56.05.745.077 900 5 44 24 69.33.667.045 509 7 36 10 54.94.752.081 875 5 49 1 68.64.671.061 475 7 47 37 53.79.758.085 850 5 53 50 67.88.676.018 450 8 0 1 52.61.765.090 825 58 52 67.10.630.049 425 8 13 30 51.37.773.095 800 6 4 9 66.30.685.051 400 8 2S 14 50.09.780.101 775 6 9 41 65.49.690.052 375 8 44 26 48.75.788.108 750 6 15 30 64.65.695.054 350 9 2 29 47.35.796.116 725 6 21 37 63.80.700.056 325 9 22 16 45.88.805.125 700 6 28 4 62.92.705.05S 300 9 44 39 44.34.814.135 675 6 34 52 62.02.710.060 275 10 10 1 42.72.824.147 650 6 42 4 61.09.716.062 250 10 39 6 41.00.834.162 625 6 49 42 60.14.721.065 225 11 12 55 39.16.8451.180 TABLE VI. LENGTH OF CIRCULAR ARCS IN PARTS OF RADIUS. 0 II 1.01745 32925 19943 1.00029 08882 08666 1.00000 48481 36811 2.0.3490 65850 39887 2.00058 17764 17331 2.00000 96962 73622 3.05235 98775 59830 3.00087 26646 25997 3.00001 45444 10433 4.06981 31700 79773 4.00116 35528 34663 4.00001 93925 47244 5.08726 ~4625 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 7.00003 39369 576781 8.13962 63401 59546 8.00232 71056 69326 8.00003 87850 944891 9.15707 96326 79490 9.00261 79938 77991 9.00004 36332 313001 122 TABLE VII. EXPANSION BY HEAT. TABLE VII. EXPANSION BY HEAT. Bodies. 823 to 2120. 10. Authority. Platina,.0008842.000004912 Hassler. Gold,.001466.000008141 s Silver,.001909.000010605 I Mercury,.018018.0001001 " Brass,.00189163.000010509 " Iron,.00125344.000006964 " Water,.0466 not uniform. " Granite,.00036850.000004825 Prof. Bartlett. Marble,.00102024.000005668 "! uidrton,.00171576.000009532. _~~~~~~~~ TABLE VIII. PROPERTIES OF MATERIALS. 123 TABLE VIII. PROPERTIES OF MATERIALS. The authorities referred to by the capital letters in the table are:B Barlow, On the Strength of L. Larnm. Materials. M. Musschenbroek, Int. to Nat Be. Bevan. Phil. Br. Lieut. Brown. R. Rennie, Phil. Trans. C. Couch. Ro. Rondelet, L'Art de Batir. F. Franklin Institute, Report on T. Telford. Steam Boilers. Ta. Taylor, Statistics of Coal. G. Gordon, Eng. Translation of W. Weisbach, Mech. of MachinWeisbach. ery and Engineering. H. Hodgkinson, Reports to Brit. The numbers without letters are Association. taken from Prof. Moseley's EnHa. Hassler, Tables. gineering and Architecture In finding the weights, a cubic foot of water has, for convenience, been taken at 62.5 lbs. Tile numbers for compression taken from Hodgkinson were obtained by him from prisms high enough to allow the wedge of rupture to slide freely off. He shows that this is essential in experiments on compression. The modulus of rupture S is the breaking weight of a prism 1 in. broad, 1 in. deep, and 1 in. between the supports, the weight being applied in the middle. To find the corresponding breaking weight W'of a rectangular beam of any other size, let I = its length, b = its breadth, andd -its depth, all in inches. Then W = X S. The numbers in the last three columns express absolute strength For safety, a certain proportion only of these numbers is taken. The divisors for wood may be from 6 to 10, for metal from 3 to 6, for stone O0, and for ropes 3. When double numbers are used in the column headed "Crushing Force per Square Inch in lbs.," the first applies to specimens moderately dry, the second to specimens turned and kept dry in a warm place two months longer. In the case of American Birch, Elm, and Teak, the numbers apply to seasoned specimens. 124 TABLE VIII. PROPERTIES OF MATERIALS. Weight Tensile Crushing M1o Materials.. 1;umc ~q uar e Materials. Specific Cperi Strength Force perof up in lbs. in lbs. in lbs. Metals. I',r cst,.... 8.399 524.94 17963 R. pipe, ast,.... 8.607 537.94 19072 " relld,... 8.864 F. 554.00 32826 F. * rike-drawn,... 8.878 554.87 61228 Go14........ 19.258 Ia. 1203.62 IGon, est'. 19.361 Ha. 1210.06 Iron, cast Canlon'o. 2, cold blast, 7.066 H 441.62 16683 H. 106375 H. 38556 H. " ho lo " 7.046 H. 440.37 13505 H. 108540 H. 3750311. Devon No. e,. cold " 7.295 H. 455.94 36288 H. hot " 7.229 H. 451,1 21907 H. 145435H. 43497 H. Buffery Yo. 1, cc l' 7.079 II. 442.44 17466 H. 93335 H. 37503 H. " " clho. " 6.998 H. 437.37 13434 H. 86397 H. 35316 H. Iron, wrough', English bar,.. 7.700 481.25 57120 L. 56000!G. 54000 G. Welsh ". 64960 T. Swedish;'.... 64960 T. L, "s;t,... 7.478 F. 467.37 58184 F. Lancaster Co.,.:,%. 7.740 F. 483.75 58661 F. Tennessee ". 7.805 F. 487.81 52099 F. Missouri ". 7.722 F. 482.62 47909 F. Iron wire, English, diam..0' i. 80214 T. Phillipsb'g, Pa." 83 " 7 27 F. 482.94 841/ 6 F. 4" ".4s9 ( 738~8 F. "c ".1i6 " 89162 F. Lead, cast,...... 11 16. 715.37 1824 R. Lead wir%,...... 707.31 2531 M. Mercury,.......3.6 4. 849.87 pi 1.500 HA.'218.75 Platina,.......12.669 1a. 1116.81 Silver,....... 10.474 ILH. 654.62 40902 M. Steel, sft,...... 7.780 480.25 123000 " razor-te.-pered,.. 7.840 490.)0 10000 Tin, caot,.... 7.291 45' 63 3322 M. Zinc, fused,.. 7050 W. 440.32 " rolled,.... 7.540 W. 471 2i Wi,ds. 8 Ash, English,.... 760 B.5.' )M. 3 12156 B Birch, English,....792B. 49.50 lC 60297. 11. 10920 B. " Americn,....648 B. 40.50t 1663 H. 9624 B Box,....... 960 B. 60.060 20006 B. 9771 11. Cedar, Canadian,... 909 C. 56.81 11400 RI' I. Chestnut,......657 Ro. 41.06 13300. Deal, Christiania midR, |.693 B. 43.62 12400 9364 B. Memel.590 B. 36.87 10386 B. "* Norway Spruce,...340 21.25 17600 " English,.470 29.37 7000 Elm, seasoned,... 3 B. 34.55 13489 M. 10331 H 3078 B. Fir, New England,. 553..34.56 66 2. " Riga,....753B. 47.06 12000 B. { 651 I 57' 4. Lignum-vitae,. 1.220 76.25 11800M. Mahogany, Spanish,...800 50.00 16500 {1 TABLE VIII. PROPERTIES OF MATERIALS. 125 Weight MTensile Crushing Modulus _ —. e - f-p^ex^i Materials. Speci Cubic Square of up_ - u-I in lbs. Inch in lbs. Inc in lbs. Woods. Oak, English, 93. 58.37 10000 B. { } 10032 B. Canadian,.....872B. 54.50 10253 4 10596 B. Pine, pitch,...;..660 B. 41.25 7818 M. 6790 I 9792 B. " red,...657B. 41.06 395.} 8046 B. American, white,..455 Br. 23.44 7829 Br. " Southern,.872 Br. 54.50 13937 Br. Poplar,.333.. 23.94 7200 Be. 31071 5124 1H. Teak,..745 B. 46.56 15000B. 12101 14772B. Other Materials. Brick, red,...... 2.16 R. 135.50 280.803 R. 340 W. pale red,... 2.05 R. 130.31 300 562 R. 130 W. 2.784 174.03 Chalk,....... 1.869 116.81 501 R. Coal, Penn. anthracite,. 1.327 Ta. 82.94 1.700 Ta. 106.25 " " semi-bituminous, 1.453 Ta. 90.81 " Md. " 1.552 Ta. 97.00 " Penn. bituminous,. 1.312 Ta. 82.00 Ohio " ~ 1.270 Ta. 79.37 English. 1.29 Ta. 78.69 Earth, loamy hard-stamped, fresh, 2.060 W. 128.75 "(' " ~dry, 1.930 V. 12).62 garden, fresh,... 2.05) W. 123.12' dry,.... 1.630W. 101.87 dry, poor,...... 1.340 W. 83.75 Glass, plate,..... 2.453 153.31 9420 Gravel,....... 1.920 120.00 Granite, Aberdeen,... 2.625 R. 164.06 10914 R. Ivory...... 1.826 114.12 16626 Timestone...... 2.400 W. 150.00 1500 W. 700 W. Limestone,.... 2.860W. 178.75 6000 W. 1700 Marble, white Italian,.. 2.638 H. 164.87 9583 G. 1062 " black Galway,.. 2.695 H. 168.44 2664 Masonry, quarry stone, dry, 2.400 W. 150.00 " sandstone, " 2.050 W. 128.12 brick1.470W. 91.87 " " brick, ~dry, { 1.590 W. 99.37 Ropes, hemp, under 1 inch diam., 9280 W. " from 1 to 3 in. " 7218W. " over 3 inches " 5156 W. Sand, river,.... 1.836 117.87 Sandstone, 1.900W. 118.75 1400 W. 600 W. Santone,. 2.700 W. 168.75 13000 W. 800 W " Dundee,.. 2.530 R. 158.12.6630 R. " Derby, red and friable, 2.316 R. 144.75 3142 R. Slate, Welsh,.... 2.888 180.50 12800 " Scotch, 9600 126 TABLE IX. MAGNETIC VARIATION. TABLE IX. MAGNETIC VARIATION. THE following table has been made up from various sources, prin cipally, however, from the results of the United States Coast Survey kindly furnished in manuscript by the Superintendent, Prof. A. D Bache.'- These results," he remarks in an accompanying note, "ar( from preliminary computations, and may be somewhat changed by the final ones." Among the other sources may be mentioned the Smith sonian Contributions for 1852, Trans. Am. Phil. Soc. for 1846, Lond Phil. Trans. for 1849, Silliman's Journal for 1838, 1840, 1846, anc 1852, and the various American, British, and Russian Governmen Observations. The latitudes and longitudes here given are not always to be relied on, as minutely correct. Many of them, for. places in the Western States, were confessedly taken from maps and other uncer tain sources. Those of the Coast Survey Stations. however, as wel as those of American and foreign Government Observatories and Sta tions, are presumed to be accurate. It will be seen that the variation of the magnetic needle in th( United States is in some places west and in others east. The line of n, variation begins in the northwest part of Lake Huron, and runs througl the middle of Lake Erie, the southwest corner of Pennsylvania, the central parts of Virginia, and through North Carolina to the coast All places on the east of this line have the variation of the needle west, —all places on the west of this line have the variation of the needle east; and, as a general rule, the farther a place lies from thii line, the greater is the variation. The position of the line of no varia tion given above is the position assigned to it by Professor Loomis for the year 1840. But this line has for many years been moving slowly westward, and this motion still continues. Hence places whose varia tion is west are every year farther and farther from this line, so thi. the variation west is constantly increasing. On the contrary, places whose variation is east are evejy year nearer and nearer to this line so- that the variation east is constantly decreasing. The rate of this increase or decrease, as the case may be, is said to average about 2' fo the Southern States, 4' for the Middle and Western States, and 6' foi the New England States.* The increase in Washington in 1840 - was 3' 44.2; n oono in Toronto in 1841- 2 it was 4 46 2. The changes i * Prof Loomis in Silliman's Journal, Vol. XXXIX., 1840. TABLE IX. MAGNETIC VARIATION. 127 Cambridge, Mass. may be seen from the following determinations of the variation, taken from the Memoirs of the American Academy for 1846. Cambridge, 1708, 9 0 Cambridge, 1788, 6 38 " 1742, 8 0 Boston, 1793, 6 30 *" 1757, 7 20 Salem, 1805, 5 57 " 1761, 7 14 " 1808, 5 20 " 1763, 7 0 " 1810, 6 22 " 1780, 7 2 Cambridge, 1810, 7 30 C" 1782, 6 46 " 1835, 8 51 " 1783, 6 52 " 1840, 9 18 But besides this change in the variation, which may be called secular, there is an annual and a diurnal change, and very frequently there are irregular changes of considerable amount. With respect to. the annual change, the variation west in the Northern hemisphere is generally found to be somewhat greater, and the variation east somewhat less, in the summer than in the winter months. The amount of this change is different in different places, but it is ordinarily too small to be of any practical importance. The diurnal change is well determined. At Washington in 1840-2, the mean diurnal change in the variation was,* - it I II I Summer, 10 4.1 Autumn, 6 21.2 Winter, 5 9.1 Spring, 8 10.7 At Toronto the means were, t - 1841. 1848. 1845. 1847. 1849. 1850. 1851. Winter, 6.67 5.64 5.73 7.28 8.25 8.01 7 01 Spring and Autumn, 9.46 9.36 9.15 10.08 12.25 10.90 10.82 Summer, 12.38 1170 13.36 13.84 14.80 13.74 12.61 The diurnal change in the variation is such that the north end of the needle in the Northern hemisphere attains its extreme westerly position about 2 o'clock, P. M., and its extreme easterly position about 8 o'clock, A. M. In places, therefore, whose variation is west, the maximum variation occurs about 2 P. M., while in places whose variation is east, the maximum variation occurs about 8 A.M. In Washington, according to the report of Lieutenant Gilliss, the maximum variation, taking the mean of two years' observations, occurs at 1h' 331'" P. M., the minimum at 8h. 6m' A. M. The determinations of the Coast Survey are distinguished by the letters C. S. attached to the name of the observer. In some instances the name of the nearest town has been added to the name of the Coast Survey station. * Lieut. Gilliss's Report, Senate Document 172, 1845 London Philosophical Transactions. 1852 128 TABLE IX. MAGNETIC VARIATION. Place. Lti-Authority. ate. Vartion. tude. tude. A rity. Maine. o o Io Agamenticus, 43 13.4 70 41.2 T. J. Lee, C. S. Sept., 1847 10 0.0 W Bethel, 44 28.0 70 51.0 J. Locke, June, 1845 11 50.0 " Bowdoin Hill, Portland, 43 38.8 70 16.2 J. E. Hilgard, C S. Aug., 1851 11 41.1 " CapeNeddick,York 43 11.6 70 36.1 J. E. Hilgard C S. Aug., 1851 11 9.0 " Cape Small, 43 46.7 69 50.4 G. W. Dean, C. S. Oct., 1851 12 5.5 " Kennebunkport, 43 21.4 70 27.8 J. E. Hilgard, C. S. Aug., 1851 11 23.6 " Kittery Point, 43 4.8 70 43.3 J. E. Hilgard, C. S. Sept., 1850 10 30.2 " Mt. Pleasant, 44 1.6 70 49.0 0. W. Dean, C.. Aug., 1851 14 32.0 " Portland, 43 41.0 70 20.5 J. Locke, June, 1845 11 28.3 " Richmond Island, 43 32.4 70 14.0 J. E. Hilgard, C. S. Sept., 1850 12 17.9 " NTew Hampshire. Fabyan's Hotel, 44 16.0171 29.0 J. Locke, June, 1845 11 32.0 W. Hanover, 43 42.0 72 10.0 Prof Young, 1839 9 15.0 Isle of Shoals, 42 59.2 70 36.5 T. J. Lee, C. S. Aug 1847 10 3.4 " Patuccawa, 43 7.2 71 11.5 0. W. Dean, C. S. Aug., 1849 10 42.9 " Unkonoonuc, 42 59.0 71 35.0 J. S. Ruth, C. S. Oct, 1848 9 5.6 " Vermont. Burlington, 44 27.0 73 10.0 J. Locke, June, 1845 9 22.0 W. Massachusetts. Annis-squam, 42 39.4 70 40.3 0. W. Keely, C. S. Aug., 1849 11 36.7W. Baker's Island, 42 32.2 70 46.8 0. W. Keely, C. S. Sept., 1849 12 17.0 " Blue Hill, Milton, 42 12.7 71 6.5 T. J. Lee, 0. S. SeOt. d, 9 13.8 " Cambridge, 42 22.9 71 7.2 W. C. Bond, 185210 8.0' Chappaquidick,Edgartown, 41 22.7 70 28.7 T. J Lee, C. S. July, 1846 8 47.7 " Coddon's Hill, Marblehead, 42 31.0 70 50.9 G. W. Keely, C. S. Sept,1849 11 49.8 " Copecut Hill, 41 43.3 71 3.3 T. J. Lee, C. S. { Sept a 9 12.1 4" Dorchester, 42 19.0 71 4.0 W. C. Bond, 1839 9 2.0 " Fort Lee, Salem, 42 31.9 70 52.1 G. W. Keely, C. S. Aug., 1849 10 14.5 " Hyannis, 41 38.0 70 18.0 T.J Lee, C... Aug., 1846 9 22.0 " Indian Hill, 41 25.7 70 40.3 T. J. Lee, C. S. Aug., 1846 8 49.3 " Little Nahant, 42 26.2 70 55.5 G. W. Keely, C. S. Aug., 1849 9 40.9 " Nantasket, 42 18.2 70 54.0 T. J. Lee, 0. S. Sept., 1847 9 33.5 " Nantucket, 41 17.0 70 6.0 T J. Lee, C... July, 1846 9 14.0 " New Bedford, 41 38.0 70 54.0 T. J. Lee, C. S. Oct., 1845 8 54.6" Shootflying Hill, Barnstable, 41 41.1 70 20.5 T. J. Lee, C. S. Aug., 1846 9 40.1 " Tarpaulin Cove, 41 28.1 70 45.1 T. J. Lee, C. S. Aug., 1846 9 10.1 " Rhode Island. Beacon-pole Hill, 41 59.7 71 26.7 T. J Lee, C. S. Oct. and 9 29.8W. Nov.,l844 I McSparran Hills 41 29.7 71 27.1 T. J. Lee, C.. July, 1844 8 53.3 "4 Point Judith, 41 21.9 71 28.9 R.H. Fauntleroy,C.S. Sept, 1847 8 59.4 " Spencer Hill,. 41 40.7 71 29.3 T. J. Lee,. S. July aund 1 9 11.9 " Connecticut. Black Rock, Fairfield, 41' 8.6 73 12.6 J. Renwick, C. S. Sept., 1845 6 53.5W. Bridgeport, 41 10.0 73 11.0 J. Renwick, C. S. Sept.1845 6 19.3 " Fort Wooster, 41 16.9 72 53.2 J. S. Ruth, C. S. Aug., 1848 7 26.4 " Groton Point, New London, 41 18.0172 0.0 J. Renwick, C. S. Aug., 1845 7 29.5" L._. TABLE IX. MAGNETIC VARIATION. 129 Place. tude. Lndgi- Authority. Date. Variation. _lt 0. o I 0o Milford, 41 16.0 73 1.0 J. Renwick. C S. Sept., 1845 6 3.3 W. New Haven, Pavilion, 41 18.5 72 55.4 J. S. Ruth,. S Aug., 1848 6 37.5 " New Iaven, Yale College, 41 18.5 72 55.4 J. Renwick. C. S. Sept., 1845 6 17.3 " Norwalk, 41 71 73 24.2 J. Renwick, C. S. Sept., 1844 6 46.3 " Oyster Point, New Haven, 41 17.0 72 55.4 J. S. Ruth, C. S. Aug., 1848 6 32.3 " Sachem's Head, Guilford, 41 17.0 72 43.0 J. Renwick, C. S. Aug., 1845 6 15.2 " Sawpits, 40 59.5 73 39.4 J. Renwick, C. S. Sept., 1844 6 1.6 " Saybrook, 41 16.0 72 20.0 J. Renwick, C. S. Aug., 1845 6 49.9 " Stamford, 41 3.5 73 32.0 J. Renwick, C. S. Sept., 1844 6 40.4 " Stonington, 41 20 54.0 J. Renwick, C. S. Aug., 1845 7 38.2 " New York. Albany, 42 3.( 7;3 -.() Regents' Report, 1836 6 47.0 V. Bloomingdale Asylum. 40 488 73 57.4 J. Locke, C. S. April, 1846 5 109" Cole, Staten Island, 40 31.8 74 13.tJ J. Locke, C. S. April, 1846 5 33.8 " Drowned Meadow, L.., 40 56.1 73 3.5 J. Renwick, C. S. Sept., 1845 6 3.6" 4 Flatbush, L. I., 40 40.2 73 57.7 J. Locke, C. S. April, 1846 5 54.6 "4 Greenport, L. I., 41 6.0 72 21.0 J. Renwick, C. S. Aug., 1845 7 14.6 " Leggett, 40 48 973 53 0 R.H. Fauntleroy,C.S. Oct., 1847 5 40.6 " Lloyd's Harbor, L. I., 40 55.6 73 24.8 J. Renwick, C. S. Sept., 1844 6 12.5 " New Rochelle, 40 52.5 73 47.0 J. Renwick, C. S. Sept., 1844 5 31.5 " New York, 40 42.7 74 0 I J. Renwick, C. S. Sept., 1845 6 25.3 " Oyster Bay, L. I., 40 52.3 73 31 3 J. Renwick, C. S. Sept., 1844 6 53.6 " Rouse's Point, 45 0.t 73 21.0 Boundary Survey, Oct., 1845 11 23.0 " Sands Lighthouse, L.., 10 51.9173 43.5 R.H. Fauntleroy,C.S. Oct., 1847 6 9.7 " Sands Point, L. T., 10 52.0 7:? 43.0 J. Renwick. C. S. Sept., 1845 7 14.6 " Watchhill. Fire Isl-i and, -11 -1.72 5 9,R.H.E'auntleroy,C.S. Oct., 1847 7 33 5 " West Point. I11 2.0173 56.0 Prof. Davies, Sept., 1835 6 32.0" Il 2I 0~7: 56 0IProf.. New Jersey. I Cape May Light- house, 33 55.8 74 57 61J. Locke, C. S. June, 1846 3 3.2W. Chew, 39 48.217.- 9;J. Locke, C. S. July, 1846 3 20.4, Church Landing, 39 41) 9 75 3(3:J.1. Locke, C. S. June, 1846 *5 45.8 " Egg Island, 39 10.4 75 7 eJ. Locke, C. S. June, 1846 3 18.2 " Hawkins, 39 25.5 75 7 1!.1. Locke, C. S. June, 1846 2 58.7 " Mt.Rose,Princeton, 40 22.2 74 42.ll.. E. Ililgard, C. S. Aug.,'1852 5 31.8" Newark, 40 44.8 74 7. ilJ. Locke, C. S. April, 1846 5 32.7" Pine Mountain, 39 25.0175 19 1.1. Locke, C. S. June, 1846.2 52.0 " Port Norris. 39 14.5175 1. 1. Locke, C.S. June, 1846 3 6.5 Sandy Hook, 40 28.0 73 59., J. Renwick, C. S Aug., 1844 5 54.0 " Town Bank, Cape May, 3.3 58.6 74 57.4.1. Locke, C. S. June, 1846 3 3.2 " Tucker's Islaad, 39 30.8 74 16.9 T.'J. Lee, C. S. Nov., 1846 4 23.8 " White Hill, Bor- dentown, ^ 140 8.3174 43 81J.' Loeki CS. April, 1t46 4 22.5 Pennsylvania. Girard College, Philadelphia, 39 58 4 75 9.9J. Loa, C. S. May, 11 46 3 50 7 W. Pittsburg, 10 26.0179 5S.1t J. Lotie, May, 1 45. 0 33.1 " Vanuxem, Bristol, 4. 5.97 52.7 J. Loke, C. S. i July, 1346 4 20.5 * Local attraction exists here, accorlding to Prof. Loeke. 130 TABLE IX. MAGNETIC VARIATION. [i Place. Lati- LongiBoErn tude. t.ude. Authority. Date. Variation. Delaware. Bombay Hook o o 0 o Lighthouse, 39 21.8 75 30.3 J. Locke, C. S. June, 1846 3 17.9 W Fort Delaware, Delaware River, 39 35.3 75 33.8 J. Locke, C. 8. June, 1846 3 16.0 " Lewes Lauding, 33 48.8 75 11.5 J. Locke, C. S. July, 1846 2 47.7 " Pilot Town, 38 47.1 75 9.2 J. Locke, C. S. July, 1846 2 42.2 " Sawyer, 39 42.0 75 33.5 J. Locke, C. S. June, 1846 2 47.8 " Wilmington, 39 44.9 75 33.6 J. Locke, C. S. May, 1846 231.8 " Maryland. Annapolis, 38 56.0 76 35.0 T. J. Lee, C. S. June, 1845 2 14.0 W. Bodkin, 39 8.0 76 25.2 T. J. Lee, C. S. April, 1847 2 2.6 "c Finlay, 39 24.4 76 31.2 J. Locke, C. S. April,1846 219.5 " Fort McHenry, Baltimore, 39 15.7 76 34.5 T. J. Lee, C. 8. April, 1847 2 13.0 " Hill, 38 53.9 76 52.5 G. W. Dean, C. S. Sept., 1850 2 15.4 " Kent Island, 39 1.8 76 18.8 J. Heuston, C.S. July, 1849 2 30.5 " Marriott's, 33 52.4 76 36.3 T. J Lee, C. S. June, 1849 2 5.2 North Point, 39 11.7 76 26.3 T. J. Lee, C. S. July, 1846 1 42.1 " Osborne's Ruin, 39 27.9 76 16.6 T. J. Lee, C. S. June, 1845 2 32.4 " Poole's Island, 39 17.1 76 15.6 T. J. Lee, C. S. June, 1847 2 28.5 " Rosanne, 39 17.5 76 42.8 T. J. Lee, C. S. June, 1845 2 12.0 " Soper, 39 5.1 76 56.7 G. IV. Dean, C. S. July,.1850 2 7.0 " South Base, Kent Island, 38 53.8 76 21.7 T J. Lee, C. S. June, 1845 2 26.2 " BusquehannaLighthouse, Havre de Grace, 39 32.4 76 4.8 T. J. Lee, C.. July, 1817 2 51.1 " Taylor, 33 59.8 76 27.6 T. J. Lee, C. S. May, 1847 2 18.4 " Webb, 39 5.4 76 40.2 G. W. Dean, C. s. Nov., 1850 2 7.9" District of Columbia. Causten, Georgetown, 38 55.5 77 4.1 G. W. Dean, C. S. June, 1851 2 11.3W. Washington, 33 53.7 77 2.8 J. M. Gilliss, une, 1842 1 26.0" Virginia. Charlottesville, 38'2.0 78 31.0 Prof. Patterson, 1835 0 0.0 Roslyn, Peterslburg, 37 14.4 77 2.3.5 G. W. Dean, C. S. Aug., 1852 0 26.4 W. Wheeling, 40 8.0 80 47.0 J. Locke, April, 1845 2 4.0 E. North Carolina. Bodie's Island, 35 47.5 75 31.6 C. 0. Boutelle, C.. Dec., 1846 1 13.4 W. Shellbank, 36 3.3 75 44.1 C. 0. Boutelle, C. S. Mar., 1847 1 44.8 " Stevenson's Point, 36 6.3 76 11.0 C. 0. Boutelle, C. S. Feb., 1847 1 39.7" South Carolina. Breach. Inlet, 32 46.3 79 48.7 C. 0. Boutelle. C. S. April, 1849 2 16.5 E. Charlefon, 32 41.0 79 53.0 Capt. Barnett May, 1841 224.0 " Bast Base, Edisto, 32 33.3 80 10.0 G. Davidson, C. S.. April, 1850.2 53.6 " Georgia. Athens, 34 0.0 83 20.0 Prof. McCay. 1837 431.0E. Columbus, 32 28.0 85 10.0 Geol. Survey, 1839 5 30.0 " Milledgeville, 33 7.0 83 20.0 Geol.. Survey, 1838 5 51.0 " Savannah, 32 5.0 81 5.2 J. E. Iilgard, C.. April, 1852 3 45.0 " S.Apil TABLE IX. MAGNETIC VARIATION. 131 Place. Lati- Longi- Authority. Date. Variation. tude. tude. Florida. Cape Florida, 25 39.9 80 9.4 J. E. Hilgard, C S. Feb., 1850 4 25.2 B. Cedar Keys, 29 7.5 83 2.8J. B. Hilgard, C. S. Mar., 1852 5 20.5 " St. Marks Light, 30 4.5 84 12.5J. E. Hilgard, C. S. April, 1852 5 29.2 " Sand Key, 24 27.2 81 52.0J. E. Hilgard, C. S. Aug., 1849 5 29.0 " Alabama. Fort Morgan, MobileBay, 30 13.8 88 0.4 R.H. Fauntleroy,C.S. May, 1817 7 3.8 E. Tuscaloosa, 33 12.0 87 42.0 Prof. Barnard, 1839 7 28.0" East Pascagoula, 30 20.7 88 31.4 R.H. Fauntleroy,C.S. June, 1847 712.4 E. Texas. Dollar Point, Galveston, 29 26.0 94 53.0 R.II. Fauntleroy,C.S. April, 8IF48 8 57.2 E. Mouth of Sabine, 29 43.9 93 51.5 J. D. Graham, Feb., 1 840.2 Ohio. Carrolton, 39 38.0 84 9.0J Locke, Sept., 1845 4 45.4 E. Cincinnati, 39 6.0 84 22.0 J. Locke, April, 1845 4 4.0 " Columbus, 39 57.0 83 3.0 J. Locke, July, 1845 2 29.3 " IHudson, 4! 15.0 81 26.) E. Loomis, 1840 0 52.0 " Marietta, 39 26.0 l8 29.) J. Locke, April, 1845 2 2.5.0 Oxford, 39 30.0 84 38.0 J. Locke, Aug., 1845 4 50.0 St. Mary's, 40 32.0. l1i, J. Locke, Sept., 1845 3 4.0" Tennessee. Nashville,.36 10.0 (; I:. Prof. Iamilton, 1835 7.0 E. Michigan. Detroit, 42 24.0 82 58.0 Geol. Report, 1840 20.0 O. Indiana. Rlichmond, 39 V'I 84 47.01J Locke, Sept., 1845 452.0E. South Hanover, 38 45.0 85 23.0 Prof. Dunn, 1837 4 35.0 " Illinois. Alton, 38 52.0 90 12.0 H. Loomis, 1840 7 45.0 E. Missouri. St. Louis, 3:'f6.0 89 36.0 Col. Nicolls, 1835 849.0E. Wisconsin. Madison, 43 5.0 89 41.0 U. S. Surveyors, Nov., 1839 7 30.0E. trairiedu Chien, 43 1.0 91 8.0 U.S. Surveyors, Oct., 1839 9 5.0' owa. " Brown's Settlement 42 2.0 91 18.0 J. Locke, Sept., 1839 9 4.0 E. Davenport, 4130.0 90 34.0 U. S. Surveyors, Sept., 1839 750.0 11 Farnmer's Creek, 4213.0 90 39.0 J. Locke, Oct., 1839 9 11.0" SVapsipinnicon River, 41 44.0 90 39.0 J. Locke, Sept., 1839 8 25.0 " California. Point Conception. 13 26.9 120 26.0 G. Davidson, C. S. Sept., 1850 13 49.5 E. j ii -.-31 6.-10 6.1G.D dsn C.S. et 132 TABLE IX. MAGNETIC VARIATION. PlaCe. Isti- I - Authority. Date Variation. tude. tude. Point Pinos, o t o i 0 Monterey, 36 8.0 121 54.0. Davidson, C. S. Feb., 1851 14 58.0 E. Presidio, San Francisco, 37 47.8 122 27.0 G. Davidson, C. S. Feb., 1852 15 26.9 " San Diego, 32 42.0 117 14.0 G. Davidson, C.. May, 1851 12 29.0 " Oregon. Cape Disappointment, 46 16.6 124 2.0 G. Davidson, C S. July, 1851 20 45.0 E. Ewing Harbor, 42 44.4 124 21.0 G. Davidson, C S. Nov., 1851 18 29.2 " Washington Territory. Scarboro' Harbor, 48 21.8 124 37.2 G. Davidson, C.. Aug., 1852 21 30.2 E. BRiuns AMEaICA. Montreal, 45 30.0 73 35.0 Capt. Lefroy, 1842 8 58.0 W. Quebec, 46 49.0 71 16.0 Capt. Lefroy, 1842 14 12.0 " St. Johns, C. E. 45 19.0 73 18.0 Capt. Lefroy, 1842 11 22.0 " Stanstead, 45 0.0 72 13.0 Boundary Survey, Nov., 1845 11 33.0 " Toronto, 43 39.6 79 21.5 British Govern., Sept., 1844 1 27.2 NEw GBRIADA. Panama, 8 57.2 79 29.4 W H. Emory, Mar., 1849 6 54.6. EASTERN HBMISPHERE. Greenwich,England, 51 28.0 0 0.0 Prof. Airy, 1841 23 16.0 W. Makerstoun, Scotland, 55 35.0 2 31.0 W. J. A. Broun, 1842 25 23.6 " Paris, France, 48 50.0 2 20.0 E. Paris Observatory Nov., 1851 20 25.0 " Munich, Bavaria, 48 9.0 11 37.0 " 1842 16 48.0 " St. Petersburg, Russia, 59 56.0 30 19.0" Russian Govern., 1842 6 21.1 " Catherinenburg Siberia, 56 51.0 60 34.0 " Russian Govern., 1842 6 38.9 E. Nertchinsk, Siberia, 51 56.0 116 31.0" Russian Govern., 1842 3 46.9W. St. Helena, 15 56.7 S. 5 40.5 W. British Govern., Dec., 1845 23 36.6 " Cape of 000Od Hope, 33 56.0 " 18 28.7 E. British Govern, July, 1816 29 8.0 " Hobarton, Van Diemen's Ld.,42 52.5" 147 27.5 " British Govern., Dec., 1848 10 8.0. TABLE X. TRIGONOMETRICAL FORMULE. 133 TABLE X. TRIGONOMETRICAL AND MISCELLANEOUS FORMULAi LET A (fig. 57) be any acute angle, and let a perpendicular B Cbe 4rawn from any point in one side to the other side. Then, if the sides Fig. 57. A 7, C of the right triangle thus formed are denoted hy letters, as in the fig are, we shall have these six formula: - 1. sin. A =. 4. cosec. A = b c 2. cos. A = -. 5. sec. A a b 3. tan. A =. 6. cot A =. b a Solution of Right Triangles (fig. 57). e Given. Sought. Formula. 7 I.^ A, B, bsin. A = -, cos. B ==, b= ^(c +a) (c-a C C 8 a, b A,, c tan.A=, cot. B=, c /a + b 9 A,a B, b, c B= 900o-A, b = a cot. A, = sin A. 10, b B.a.c B= 90-A, a = btan.A, c - os. 11 A, c B, a, b B==900 - A, a==c sin.A, bc =ccos. A. 1 84 TABLE X. TRIGONOMETRICAL AND Solution of Oblique Triangles (fig. 58). Fig. 58. /A~ \ \Ca A/ \ Given. Sought. Formule. a sin. B 12A,B,a b b= in. A 13 A,a, b B sin. B bsin. (a - b) s i n. (A + B) 14 b, C A-B tan. I (A -B) (a-) tan ( + ab-be 15 a, b c JA cos.IA= tan. -- b 4 (s-b)s sin. A -_ 2 /J (s - a) (s - b) (s - ) bc a2 sin. B sin C 16 A. BC,,a a area ea= 17 A, b, c area area = bc sin. A. 18 a, b, c area s=- (a { b + c), area==-/s(s-a) (s-b) (s- c). General Triqonometrical Formulce. 19 sin.2 A + cos.2 A 1. 20 sin. (,4 13) = sin. A cos. B ~ sin. B cos. A. 21 cos. (A ~ B) = cos. A1 cos. B:: sin. A sin. B. 22 sin. 2 A = 2 sin. A cos. A. 23 cos. 2A ==cos.2 A-sin.2 A = 1-2 sin.2 A 2 cos. A- 1. 24 sin. A == - cos. 2 A. $ cos,' A =- + = cos. 2 A. 26 sin. A + sin. B 2 sin. I (A + B) cos. (A - B). 27 sin. A - sin. B 2 cos. (A + B) sin.. (A- B). 28 cos. A + cos. B = 2 cos. (A + B) cos. (A - B). 29cos. B - cos. A = 2 sin. (A + B) sin. (A - B). 30 sin.2 A - sin.2 B cos.2 B- os2 A = sin. (A + B) sin. (A -. 31'cos.' A- sin.2 B =cos. (A + B) cos. (A - B). MISCELLANEOUS FORMULRE. 135 sin. A 32 tan. A - cos. A cos. A 33 cot. A siA tan A_~ tanB 34 tan. (A ~ B) = 1-tan. A tan. B sin (A ~ B) 35 tan. A - tan. B -- osAcs B' sin (A - B) 36 cot. A cot. B = sin A in. B' s sin. A sin B tan I (A + B) sin. A - sin. B tan. (A - B).sin A -- sin. B 38cos. A + cs. tan- (A + B). cos + cos.B-, 2 sin. A + sin B ( 40sB-os.A - cota. (A - B). sin. A - sin. B A l + icos7 t a. -+ B). 41cos. B- cos.A - A+B). sin A 42 tan. A = 1+ cos. A' 43 cot. I - sin.A 2 1 - cos AMiscellaneous Formnule. i Sought. Given. Formnim i Area of 44 Circle Radius = r r r2. 45 Ellipse Semi-axes = a and b a b. 46 Parabola Chord = c, height = h c Ih.* 47 Regular Polygon Side -a, numberof i a cot. ~47 Reglr g sides a cot. Surface of 48; Sphere Radius - r 14 n. T2. 49 Zone Radius = r, height = h 12, rh. ( ldins ofsphere=r ) s (n-2)180 50 SphericalPolygon sum of angles =-S n r2X 18 number of sides = n ) SoUdityf of 51 Prism or Cylinder Base = b, height = h b h. 52 Pyramid or Cone Base = h, height = h Ijb h. 53 Frustumrn of Pvr- Bases b and h,'(4 +i 6 +, ~ 1 ) amid or Cone height I * The area of a circular segment on railroad curves, wvhre the chorl1 is very Iong in proportion to the height, may be found with great accuracy by the above formula. '6 TABLE X. MISCELLANEOUS FORMULME. Sought. i Given. Foormulve. Soliditi of i 54 Sphere Radius = 7 r E. h e Radii of bases = r 2 55I Sp;hericalSegment and r, height = h h (' 1 - F h liolate Sbpheoid Semi-transverse axis 4 b2 of ellipse = a 57 ()latc Spheroid PSemi.conjugate axis - b of ellipse = b 57,lrolate Spheroid of~ eip aese-b x 4 n ab2. 581 Paraboloid Radius of base = r, 1 r2 h. I height h-2 ~T =3.14159 26535 89793 23846 26433 83280. Log. r- 0.49714 98726 94133 85435 12682 88291 United States Standard Gallon 231 cub.in. = 0.133681 cub.ft " " " ~Bushel = 2150.42 " = 1.244456 British Imperial Gallon = 277.27384" = 0.160459 According to Hlassler. As usually given. French Metre, = 3.2817431 ft., = 3.280899 ft. " Litrc, 61.0741569 cub. in., = 61.02705 cub. in. " Kilogram, = 2.204737 lb. avoir., = 2.204597 lb. avoir Weight of Cubic Foot of Water,.Barom. 30 inches, Therm. Fahr. 39.83~, = 62.379 lb. avoir. " "'; 620, = 62.321 Length of Seconds Pendulum at New York = 39.10120 inches..". " " " London = 39.13908.." " " " Paris - 39.12843 Equatorl.l Radius of Earth according to Bessel = 20,923,597.017 feet Polar " " " = 20,853,654.177 " TABLE XI SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND RECIPROCALS OF NUMBERS fROM I TO 10S. 138 TABLE XI. SQUlARES, CUBES, SQUARE ROOTS, No. Squares. SquarSquare Roots. Cube Roots. Reciprocals. 1' I 1.00 1 0000 1.000000 1.0 0000 2 4 4! 8 1.4142136 1.2399210.50000000(1 i 3 9 27 1.7320508 1.4422486.333333333 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 1.9129312.142857141 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 2.2894286.03333333 13 169 2197 3.6055513 2.3513347.076923077 14 196 2744 3.7416574 2.4101422.071428571 15 22. 3375 3.8729833 2.4662121.066666667 16 256 4096 4.0000000 2.5198421.062500000 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 5529 12167 4.7958315 2.8438670.04347826 24 576 13824 4.8989795 2.8844991.041666667 25 625 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 6.3851648 3.0723168.034482759 30 900 27000 5.4772256 3.1072325.033333333 1 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 6.7032039 3.5568933.022222222 46 2116 97336 6.7823300 3.5830479.021739130 47 2209 103823 6.8556546 3.6088261.021276600 48 2304 110592 6.9282032 3.6342411.020833333 49 2401 117649 7.0000000 3.6593057.020403163 50 2500 125000 7.0710678 3.6940314.02000000 51 26()1 132651 7.1414284 3.7034298.019607843 52 2704 140608 7.2111026 3.7325111.019230769 53 2809 148877 7.2301099 3.7562958.018867925 51 2916 157464 7.3484692 3.7797631.018518519 55 3025 166375 7.4161985 3.8029525.018181818 56 3136 175616 7.4833148 3.8238624.017857143 57 3249 185193 7.5498344 3.8485011.017543860 58 3364 195112 7.6157731 3.8703766.017241379 59 3481 205379 ~7.6811457 3.892996.3.016949153 60 3690 216090 7.7459667 3.91-t8676.016666667 61 3721 226981 7.8102497 3.9364972.016393443 62 3344 238328 7.8740079 3.957891 3.016129032 = —----------- __, c — - CUBE ROOTS, AND RECIPROCALS. 139 No. Square. bes. Square Roots. Cube Roots. Reciprocals. 63 3969 250047 7.9372539 3.9790.71.015873016 64 40.f6 262144 8.0000000 4.0000U00.015625000 65 4225 274625 8.0622577 4.02;7.-6.015384615 66 4356 267496 8.1240384 4.041241)1.015151515 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 l5041 357911 8.4261498 4.1408178.014084507 72 5184 373248 8.4852814 4.1601676.013888889 73 53%9 389017 8.5440037 4.1793390.013698630 74 5476 405224 8.6023253 4.1983364.013513514 75 5625 421875 8.6602540 4.2171633.013333333 76 5776 433976 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 4.32674b7.012345679 82 6724 551368 9.0553s51 4.3444815.012195122 83 6889 571787 9.1104336 4.3620707.012048193 84 7056 592704 9.1651514 4.3795191.0119(4762 85 7225 614125 9.2195445 4.3968296.011764706 86 7396 636056 9.2736185 4.4140049.011627907 87 7569 658503 9.3273791 4.4310476.011494253 88 7744 681472 9.,:s315 4.4479602.011363636 89 7921 704969 9.4339811 4.4647451.011235955 90 8100 729,(30 9.4868330 4.4814047.011111111 91 8231 7535 71 9.5393920 4.4979414.010989011 92 8464 7786 38 9.5916630 4.5143574.010869565 93 8649 804357 9.6436508 4.5306549.010752688 94 8836 8353 4 9.6953597 4.5468359.010638298 95 9095 857375 9.7467943 4.5629026.010526316 96 9216 884736 9.7979590 4.578b570.010416667 97 9409 912673 9.8488578 4.5947009.010309278 98 960)4 941192 9.8994949 4.6104363.010204082 99 980i ~ 970299 9.949j744 4.6260650.010101010 100 10000 1000000 10.0000000 4.6415888.010000000 101 101 1030301 10.0498756 4.6570095.009900990 102 10404 1061208 10.0995049 4.6723287.009803922 103 10609 1092727 10.1488916 4.6875482.0097087'38 104 i 10816 1124864 10.198(1390 4.7026694.019 6153&5 105 11025 1157625 10.2469508 4.7176940.009523810 I106 11236 1191016 10.2956301 4.7326235.009433962 1()7 11449 1225043 10.3414004 4.7474594.009345794 103 11664 1259712 10.3923048 4.7622032.009259259 109 11881 1295029 10.4403065 4.7768562.009174312 110 12100 1331000 10.4880085 4.7914199.009090919 111 12321 1367631 10.5356538 4.8058955.009009009 112 12544 1404928 10.5830(052 4.8202845.00 928571 113 12769 1442897 10.6301458 4.8345881.008349558 114 12996 1481544 10.6770783 4.8486076.008771930 115 13225 1520375 10.7238053 4.8629442.008695652 41 13456 1560896 10.7703(1:6 4.8769990.008620690 117 13689 1601613 1.S'16638 4.8909732.008%4i7009 118 13924 1613(1:2 1 0.627i05 4.9048681.i00-474576 119 114161 16S515.9 10.90,7 121 4.9186847.00-403361 12') 11400 17280(0 1i09.^54)12 4.9324242.003:3333 121 14641 1771.'i 11.(0000(0 4.946(74.(00(64463 121z2 14884 1815,4, 11.045361!) 4.9596757.0((81;6721 123 15129 1860)867 11.0905365 4.97;31 898.008130081 12t, 15376 19066z4 I 11 1.352S7 1 4.9-66310.008064516 110 TABLE Xl. SQUARES, CUBES, SQUARE KOOTS, 1 — -. --...... —. -No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 12.3 15625 1953125 11.1803399 5.0000000.0030)0000 126 15876 2000376 11.2249722 50132979.007936503 127 16129 2048333 11.2694277 5.0265257.007874016 123 16334 2097152 11.3137035 5.0396342.007812500 I 12. 16611 2146639 11.3578167 5.0527743.007751938 13) 16900 2197000 11.4017543 5.0657970.007692308 1:31 17161 2243091 11.4455231 5.0787531.007633588 132 17424 2299963 11.4891253 5.0916134.007575758 133 17639 2352637 11.5325626 5.1044637.007513797 134 17956 2106104 11.5758369 5 1172299.007462687 135 18225 2460375 11.6189500 5 1299278.007407407 136 18496 2515456 11.6619033 5 1425632.007352941 137 13769 2571353 11.7046999 5.1551367.007299270 138 19044 262307? 11.7473444 5 1676493.007246377 139 19:121 2683619 11.7893261 5.1801015.007194245 140 196(00) 227-140)0 11.8321596 5.1924941.007142857 141 19331 22303221 11.8743421 5.2048279.007092199 142 20164 2363238 11.916:3753 52171034.007042254 143 20449 2921207 11.9532607 5.2293215.006993007 144 20736 2935934 12.0000000 5 2414328.006944441 145 21025 3048625 12.0415946 5.2535379.006396552 146 21316 3112136 12.0330460 5.2656374.006,349315 147 21609 3176523 12.1243557 5.2776321.006302721 148 21 904 3211792 12.1655251 5.2395725.006756757 149 22201 3307949 12.2065556 5.3014592.006711409 150 22500 3375000 12 2174437 5.3132923.006666667 151 22801 3442951 12 2332057 5.3250740.006622517 152 23104 3511803 12.3238230 5 3363033.006578947 153 23409 3531577 12 3693169 5.3434812.006535948 154 23716 3652264 12.4096736 5.3601084.006493506 155 24025 3723375 12.4493996 5.3716354.006451613 156 24336 3796416 12.4399960 5.3332126.006410256 157 24649 3369393 12.5299641 5.3946907.006369427 158 24961 3944312 12.5693051 5.4061202.006329114 159 25281 4019679 12.6095202 5.4175015.006239308 160 25600 4096000 12.6491106 5.4238352.006250000 161 25921 4173231 12.6335775 5;4401218.006211180 162 26244 4251523 12.7279221 5.4513618.006172340 163 26569 4330747 12.7671453 5.46255.56.006134969 164 26896 4410944 12.8062485 5.4737037.006097561 165 27225 4492125 12.8452326 5.4848066.006060606 166 27556 4574296 12.8340937 5.4958647.006924096 167 27889 4657463 12.9223480 5.5063784.005988024 163 28224 4741632 12.9614814 5.5178484.005952381 169 23561 4826309 13.0000000 5.5287748.005917160 170 23900 4913000 13.0334048 5.5396583.005382353 171 29241 5000211 13.0766968 5.5504991.005847953 172 29584 5038443 13.1148770 5.5612978.005813953 173 29929 5177717 13.1529464 5.5720546.005780347 174 30276 5263024 13.1909060 5.5827702.005747126 175 30625 5359375 13.2237566 5.5934447.005714236 176 30976 5451776 13.2664992 5.6040787.005631818 177 31329 5545233 13.3041347 5.6146724.005649718 178 31634 5639752 13.3416641 5.6252263.005617978 179 32041 5735339 13.3790382 5.6357408.005536592 180 32400 5832000 134164079 5.6462162.005555556 181 32761 592741 13.4536240 5.6566529.005524862 182 33124 6023563 13.4907376 5.6670511 005494505 183 33489 6123487 13.5277493 5.6774114.005464481 1 84 33356 6229504 13.5646600 5.6S77340.005434733 185 34225 6331625 13.6014705 5.6980192.0054054 0 186 34596 1 6434856 13.6331817 1 5.7032675.005376344 CUBE ROOTS, AND RECIPROCALS. 141 No. Squares. Cubes. Square Roots. Cube Roott. Reciprocala. 187 34,69 6539203 136747943 5.7184791.005347594 188 35:344 6644672 13.7113(92 5.7286543.005319149 189 35721 6751269 13.7477271 5.7387936 005291005 190 36100 6859000 13.7840488 5.7488971.005263158 191 jt64il 6967871 13.8202750 5.7589652.005235602 192 36864 7077888 13.8564065 5.7689982.005208333 193 37249 7189057 13.8924440 5.7789966.005181347 194 37636 7301384 13.9283883 5.7889604.005154639 195 38025 7414875 13.9642400 5.7988900.005128205 196 38416 7529536 14.0000000 58087857.005102041 197 38809 7645373 14.0356688 5.8186479.005076142 198 39204 7762392 14.0712473 5.8284767.005050505 199 39601 7880599 14.1067360 5.8382725.005025126 200 40000 8000000 14.1421356 5.8480355.005000000 201 40(401 8120601 14.1774469 5.8577660.004975124 202 40804 8242408 14.2126704 5.8674643.004950495 203 41209 8365427 14.2478068 5.8771307.004926108 204 41616 8489664 14.2828569 5.8867653.004901961 205 42025 8615125 14.3178211 5.8963685.004878049 206 42436 8741816 14.3527001 5 9059406.004854369 207 42849 8869743 14.3874946 5.9154817.004830918 208 43264 8998912 14.4222051 5.9249921.004807692 209 43681 9129329 14.4568323 5.9344721.004784689 210 44100 9261000 14.4913767 5.9439220.004761905 211 44521 9393931 14.5258390 5.9533418.004739336 212 44944 9528128 14.5602198 5.9627320.004716981 213 45369 9663597 14.5945195 5.9720926.004694836 214 45796 9800344 14.6287388 5.9814240.004672897 215 46225 9938375 14.6628783 5.9907264.004651163 216 46656 10077696 14.6969385 6.0000000.004629630 217 47089 10218313 14.7309199 6.0092450.004608295 218 47524 10360232 14.7648231 6.0184617.004587156 219 47961 10503459 14.7986486 6.0276502.004 66210 220 48400 10648000 14.8323970 6.0368107.004545455 221 48841 10793861 14.8660687 6.0459435.004524887 222 49284 10941048 14.8996644 60550489.0045!4505 223 49729 11089567 14.9331845 6.0641270.0044443(!5 224 50176 11239424 14.9666295 6.0731779.004464286 225 50625 1139062.5 15.0000000 6.0822020.004444444 226 51076 11543176 15.0332964 6.0911994.004424779 227 51529 11697083 15.0665192 6.1001702.0044(15286 228 51984 11852352 15.0996689 6.1091147.004385965 229 52441 12008989 15.1327460 6.1180332.004366812 2.30 52900 12167000 15.1657509 6.1269257.004347826 231 53361 12326391 15.1986842 6.1357924.004329004 232 53824 12487168 15.2315462 6.1446337.0(431(1345 233 54289 12649337 15.2643375 6.1534495.004291845 234 54756 12812904 15.2970.55 6.1622401.(0427:304 235 55225 12977875 15.3297097 6.1710058.0042r:5319 236 55696 13144256 15.3622915 6.1797466.004237288 237 56169 13312053 15.3948043 6.1884628.004219409 238 56644 13481272 15.4272486 6.1971544.0(04201681 239 57121 13651919 15.4596248 6.2058218.004184100 240 57600 13824000 15.4919334 6.2144650.00416t(667 241 58081 13997521 15.5241747 6.2230843.004149378 242 58564 14172488 15.5 63492 6.2316797.004132231 243 59049 14348907 15.5S884573 6.24 0S515.0041 15226 244 59536 14526784 15.6204994 6.2487998.004('98t61 245 -60025 14706125 15 6524758 6.2573248.004(!81633 246 60516 14886936 15.6843871 6.265i266.004065041 247 61009 15069223 15.7162336 6.2743054.0((404S583 248 61504 15252992 15.7480157 6.2527613.004032258 142 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 249 62001 15438219 15.7797338 6.2911946.0)1016064 250 62500 15625000 15.8113383 6.2996053.004000000 2.51 63001 15I13251 15.8429795 6.3079935.0039S4064 2;)- 6:3504 16003008 15.8745079 6.3163596.003963254 2.53 64009 16194277 15.9059737 6.3247035.003952569 254 64516 16387064 15.9373775 6.3330256.003937008 255 65025 16581375 15.9637194 6.3413257.003921569 256 65536 16777216 16.0000000 6.3196042.003906250 257 66049 16974593 16.0312195 6.3578611.003391051 258 66564 17173512 16.0623784 6.3660968.003875969 259 67031 17373979 16.0934769 6.3743111.003861004 260 67600 17576000 16.1245155 6.3825043.003846154 261 68121 17779581 16.1554944 6.3936765.003831413 262 68644 17981728 16.1864141 6.3938279.0)3316794 263 69169 18191447 16.2172747 6.4069555.003,02281 264 69696 18399744 16.2430768 6.4150637.003787879 265 7022. 18609625 16.2788206- 6.4231533.003773585 266 70756 18821096 16.3095064 6.4312276.003759398 267 71239 19134163 16.3401346 6.4392767.003745318 263 71824 1924:33 2 16.3707055 6.4473057.003731343 269 72361 19465109 16.4012195 6.4553148.003717472 270 72900 19633000 16.4316767 6.4633041.003703704 271 73441 19902511 16.4620776 6.4712736.003690037 272 73934 20123643 16.492422.5 6.4792236.003676471 273 743.29 20346417 16.5227116 6.4871541.003663004 274 75076 20570324 16.5529454 6.4950653.003649635 275 75625 20796375 16.5831240 6.5029572.003636.364 276 76176 21024576 16.6132477 6.5103300.003623188 277 76729 21253933 16.6433170 6.51 6839.003610108 278 77234 21484952 16.6733320 6.5265189.003597122 279 77841 217176:39 16.7032931 6.5343351.003534229 280 78400 21952000 16.7332005 6.5421326.003571429 281 78961 22188011 16.7630546 6.5499116.003553719 282 79524 22425763 16.7928556 6.5576722.003546099 233 80039 2266.317 16.8226038 6.5654144.003533569 234 80656 22906304 16.8522995 6.5731385.003521127 235 8122 23149125 16.8319430 6.5803443.003508772 236 81796 23393656 16.9115345 6.5885323.003496503 237 82369 23639903 16.94 10743 6.5962023 003484321 233 82944 23337872 16.9705627 6.603?545.00347222 239 83521 24137569 17.0000000 6.6114890.003460208 290 84100 24:39030 17.029:364 6.6191060.00344'276 291 84631 21642171 17.0537221 6.6267034.003436426 292 85264 24897038 17.0380075 6.6342374.003424658 293 83849 25153757 17.1172428 6.-6113322.003412969 294 86436 2.412184 17.1164282 6.6193998.003401361 295 87025 25672375 7 17755610 6.6569302.003339831 296 87616 25931336 17.20465305 6.6614,137.003378378 297 882)9 26190r)73 17.23:3G379 6 6719103.003367003 293 88804 26t6:3592 17.2626765 6.6794200.0(13355705 299 89101 26730399 17.2916165 6.6368831.003344482 300 90030 270 10090 17.3205031 6.6393295.003333333 301 9)691 27270901 17.3493316 6.7017593.003322259 332 91214 2751365)3 17.3781472 6.7(91729.0(13:311258 3 )3 91893 271 127 17.49,63952 6.7165700.003300330.3'1 92416 233)91161 17. 4355958 6.7239:508.00:3294 74 335 9:3123) 231726 25 17.464I2492 6.73:13155.003278369 306 9:)3636 23 652616 17.4923.557 6.73-64 iI1.003267974 307 91249 29311 11 3 1 7.521 4155 6.7459'67.003257329 313 91 61 29211 12 17.5499238 6.75:3134. 03246753 3!19 93 3 290:3629 17.578395.3 6.76'6143.003236246 310 9 I 0 ) 2979101)0 17.6: 163169 6.7678995.003225806 CUBE ROOTS, AND RECIPROCALS. 1}4 No. Squares. Cubes. Square Roots. Cube Roots. lReciprocals. 311 96721 30080231 17.6351921 6.7751690.003215434 312 97344 30371328 17.6635217 6.72:429..003205128 313 97969 30664297 17.6918060 6.7 5 66613.003194888 314 98596 30959144 17.7200451 6.7968844.003184713 315 99225 31255875 17.7432393 6.8040921.003174603 316 99%876 31554496 17.7763888 6.8112847.003164557 317 100489 31855013 17.80449:38 6.8184620.003154574 318 101124 32157432 17.8325545 6.8256242.0(3144654 319 101761 32461759 17.8605711 6.8327714.003134796 320 102400 32768000 17.8885438 6.8399037.003125000 321 103041 33076161 17.9164729 6.8470213.003115265 322 10363S 33386248 17.9443584 6.8541240.003105590 323 104329 33698267 17.9722008 6.8612120.003095975 324 104976 34012224 18.0000000 6.8682855.003086420 325 1056 25 34323125 18.0277564 6.8753443.003076923 326 106276 31615976 18.0554701 6.8323388.003067485 327 106929 34965783 18.0831413 6.8894188.003058104 32S 107534 352S7552 18.1107703 6.8964345.003048780 329 103241 35611239 18.1383571 6.9034359.003039514 330 1039)00 35937000 18. 1659021 6.9104232.003030303 331 109561 36264691 18.1934054 6.9173964.003021148 332 110(224 36594368 18.2208672 6.9243556.003012048 333 110(89 36926037 18.2432876 6.9313008.003003003 334 1 11556 37259704 18.2756669 6.9382321.002994012 335 112225 37595375 18.303(0052 6.9451496.002935075 336 112396 37933056 18.33(3028 6.9520533.002976190 337 113569 38272753 18.3575598 6.9589434.002967359 333 114214 336141472 18.3847763 6.9658198.002958580 339 114921 38958219 18.4119526 6.9726826.002949853 310 115600 3930400( 18.4390889 6.9795321.002941176 341 11621S 39651821 18.4661853 6.9863681.002932551 342 116964 40001688 18.4932420 6.9931906.002923977 343 117649 40353607 18.5202592 7.0000000.002915452 344 118:336 407(07584 18.5472370 7.0067962.002906977 345 119325 41063625 18.5741756 7.0135791.002898551 346 119716 41421736 18.6010752 7.0203490.002S90173 347 120409 41781923 18.6279360 7.0271058.002881844 348 121104 42144192 18.6547581 7.0338497.002873563 349 121801 42508549 18.6815417 7.0405806.002865330 350 122500 42875000 18.7082869 7.0472987.002857143 351 12:301 43243551 18.7349940 7.0540041.002849003 352 1 123904 43614208 18.7616630 7.0606967.002840909 353 121460 43986977 18.7882942 7.0673767.002832861 35t 125316 44361864 18.8148877 7.074(440.002824859 355R 126025 44738875 18.8414437 7.0806988.002816901 S.6 126736 45118016 18.8679623 7.0873411.00(2808989 3-7 1127449 45499293 18.8944436 7.0939709.002801120 353 128164 45832712 18.9203879 7.1005885.002793296 359 1233881 46258279 18.9472953 7.1071937.002735515 360 129600 466;6000 18.9736660 7.1137866.002777778 361 130321 47045881 19.0000(00 7.1203674.002770083.362 131044 47437928 19.0262976 7.1269360.002762431 363 131769 47832147 19.0525589 7.1334925.002754221 364 132496 48228544 19.0787840 7.1400370.002747253 365 133225 48627125 19.1049732 7.1465695.002739756 366 133956 49027896 19.1311265 7.1530901.002732240 367 134169 49430863 19.1572441 7.1595988.002724796 363 1335424 49836032 19. 1833261 7.166(1957.0()2717391 369 136161 50243409 19.2093727 7.1725809.00271(00(27 370 136900 50653000 19.2353841 7.179544.00271170()3 371 137641 51064811 19.2613603 7. 1855162.002695418 372 133384 51478848 19.2873015 7.1919663.002688172 144 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocabs. 373 139129 51895117 19.3132079 7.1934050.002630965 371 133376 52313624 19.3390796 7.20483'2'.002673797 375 140623 52734375 19.3649167 7.2112479.002666667 376 141376 53157376 19.3907194 7.2176.522.002659574 377 142123 535326:33 19.4161378 7.2240450.002652520 378 142334 54010152 19.442221 7.2304263.002645503 379 143611 54439939 19.4679223 7.2367972.002638522 330 144400 54872000 19.4935337 7.2431565.002631579 331 145161 55306:341 19.5192213 7.2495045.002624672 332 143924 55742963 19.5443203 7.2558415.002617801 333 146539 56181887 i9.5703:58 7.2621675.002610966 334 147456 56623104 19.59.39179 7.2684324.002604167 335 143225 57066625 19.6214169 7.2747864.002597403 336 143995 57512156 19.6463327 7.2810794.002590674 337 149769 57960693 19.6723156 7.2873617.002583979 333 150544 58411072 19.6977156 7.2936330.002577320 339 151321 58363369 19.7230329 7.2993936.002570694 390 152100 59319033 19.744 177 7.3061436.002564103 391 152331 59776171 19.7737199 7.3123823.002557545 332 153661 60236233 19.7939399 7.3186114.002551020 393 154 49 60633457 19. 242276 7.3248295.002544529 394 15.5236 61 162334 19.8494332 7.3310369.002533071 335 156025 61623375 19.8746969 7.3372339.002531646.395 156316 62099136 19.8997437 7.3434205.002525253 397 157609 62.570773 19.9243538 7.3195966.002518892 393 153401 63944792 19.9499373 7.3.557624.002512563 339 159201 63521199 19.9749344 7.3619178.002506266 433 160030 61009009 20.000000 7.3630630.002500000 491 160901 61481201 29.0249344 7.3741979.002493766 402 161601 61961303 20.0499377 7.3893227.002487562 493 162109 654.9327 20.0743599 7.3364373.002431390 401 163216 6.5939264 23.0997512 7.3925418.002475243 495 160125 663 1125 29.1246118 7.3986363.002469136 406 161336 66923116 2. 1494417 7.4047206.092463054 407 165649 6741 143 29.1742410 7.4107950.002457002 403 165464 67917312 20.1999039 7.4163595.002150980 403 167231 63417929 23.2237434 7.4229142.002444988 410 163103 63921003 21.2484.567 7.4289589.002439024 411 163921 69426531 20.2731349 7.4349933.002433090 412 169744 63934523 29.2977831 7.4410189.002427184 413 170569 70444997 2r.3224014 7.4470342.002421308 414 171396 70957914 20.3169399 7.4530:399.0(2415459 415 17222.5 71473375 20.3715488 7.4590359.0)2409639 416 173956 71991296 20.3960781 7.4650223.0124(1346 417 173389 72511713 20.4205779 7.4709991.0 2398082 418 174724 73316:32 20.4450133 7.4769661.002392314 419 175561 73569059 20.4694395 7.4829242.0023-6635 423 176400 7493303) 20.4939015 7.4888724.092330952 421 177241 746 18161 20.51 82 45 7.4943113.002375297 422 - 178334 75151443 29. 54263-6 7.5007406.002369663 423 173929 75636967 20.5669633 7.5066607.002364066 421 179776 76225124 29.5912693 7.5125715.03235.891 425 138625 7676562 5 20.61 55281 7.51 4730.002352941 426 181476 77393776 29.6397674 7.5243652.002347418 427 182329 77354433 29).6633783 7.5312482.092341(120 423 183134 78102752 20.68 1609 7.5361221.0,2336149 429 181911 789.3539 20.7123152 7.5419367.002331002 439 1 3909 79507000 20.7361414 7.5473423.092:325581 431 185761 80062991 29.7695395 7.5536388.00(2:3 o11 86 432 186624 8)621563 2).7846997'7.5595263.0(2314815 433 137489 81182737 20.8936520 7.5653548.092:39469 434 1-3356 81746304 20.8326667 7.5711743.002394147 CUBE ROOTS, AND RECIPROCALS. 145 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 435 189225 82312875 20.8566536 7.5769349.002298851 436 190096 82881856 20.8806130 7.5827865.002293578 437 190969 3-i 3453 20.9045450 7.5885793 002288330 438 191844 840z7672 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 8693S307 21.0475652 7.6231519.002257336 444 1971.36 87528.384 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 20430 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 S6702579 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 7.7306141.002164602 463 214369 99252847 21.5174348 7.7361877.002159827 464 215296 99897344 21.5406592 7.7417532.002155172 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 21.6561078 7.7694620.002132196 470 220900 103323000 21.6794834 7.7749801.002127660 471 221841 104487111 21.7025344 7.7804904.002123142 472 222784 105154048 21.7255610 7.7859928.002118644 473 223729 105823S17 21.748.5632 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 433 233289 112678587 21.9772610 7.8460134.002070393 484 234256 113379904 22.0000000 7.8514244.002066116 485 235225 114084125 22.0227155 7.856S281.002061856 486 236196 114791256 22.0454077 7.8622242.002057613 487 237169 115501303 22. 060765 7.8676130.002053388 4S8 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 119095488 22.1810730 7.8944468.002032520 493 243049 119823157 22.2036033 7.8997917.002028398 494 2440136 120553784.22.2261108 7.9051294.002024291 495 245025 121287375 22.2485955 7.9104599.002020202 496 246016 122023936 22.2710575 7.9157832.002016129 146 TABLE XI, SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 497 247099 122763473 22.2934968 7.9210994.002012072 498 218r9) 12.3505992 22.3159136 7.9264085.002008032 499 249001 124251499 22.3383079 7.9317104.002004008 500 250330 125000000 22.3606798 7.9370053.002000000 501 2-510)1 125751501 22.3830293 7.9422931.001996008 502 252004 126506008 22.4053565 7.9475739.001992032 503 253009 127263527 22.4276615 7.9528477.001988072 504 254016 123024064 22.4499443 7.9581144.001984127 505 25).)5025 128787625 22.4722051 7.9633743.0019S0198 506 256;)36 129554216 22.4944438 7.9686271.001976285 507 257049 130323843 22.5166605 7.9733731.001972387 508 258064 131096512 22.5338553 7.9791122.001968504 509 259031 131872229 22.5610283 7.9343444.001964637 510 260100 132651000 22.5831796 7.9395697.001960784 511 261121 133132831 22.6053091 7.9947883.001956947 512 262144 131217728 22.6274170 8.0000000.001953125 513 263169 135005697 22.6495033 8.0052049.001949318 514 264196 135796744 22.6715681 8.0104032.001945525 515 265225 136590375 22.69.36114 8.0155946.001941748 516 266256 137388096 22.7156334 8.0207794.001937984 517 267289 138183413 22.7376340 8.0259574.001934236 518 26S324 138991832 22.7596134 8.0311287.0019.30502 519 269361 139798359. 22.7815715 8.0362935.001926782 520 270400 140603000 22.8035085 8.0414515.001923077 521 271441 141420761 22.8254244 8.0466030.001919386 522 272484 142236648 22.8473193 8.0517479.001915709 523 273529 143055667 22.8691933 8.03,68862.001912046 524 274576 143S77824 22.8910463 8.0620180.001903397 525 275625 144703125 22.9123785 8.0671432.001904762 526 276676 145531576 22.9346899 8.0722620.001901141 527 277729 146363183 22.9564806 8.0773743.001897533 528 278784 147197952 22.9782506 8.0824800.001893939 529 279841 148035889 23.0000000 8.0875794.001890359 530 230900 148877000 23.0217289 8.0926723.0018S6792 531 281961 149721291 23.0434372 8.0977589.001883239 532 233)24 150563768 23.0651252 8.1028390.001879699 533 234(139 151419437 23.0867928 8.1079128.001576173 534 285156 152273304 23.1034400 8.1129803.001872659 535 286225 153130375 23.1300670 8.1180414.001869159 536 27296 153990656 23.1516738 8.1230962.001865672 537 238369 154854153 23.1732605 8.1281447.001862197 533 289444 155720972 23.1948270 8.1331870.001858736 539 290521 156590819 23.2163735 8.1382230.001855288 540 291600 157464000 23.2379001 8.1432529.001851852 541 292681 158310121 23.2594067 8.1482765.001848429 542 293764 159220038 23.2808935 8.1532939.001845018 543 294849 160103007 23.3023604 8.1583051.091841621 544 295936 160989184 23.3238076 8.1633102.001838235 545 297025 161878625 23.3452351 8.1683092.001834862 546 293116 162771336 23.3666129 8.1733020.001831502 547 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 23.4733S92 8.1931753.001814882 552 301704 163196608 23.4946802 8.2031319.001811594 553 305809 169112377 23.5159520 8.203n325.001808318 554 306916 170031461 23.5372046 8.2130271.001805054 555 303025 170953375 23.5584330 8.2179657.001801802 556 309136 171879616 23.5796522 8.2223985.001798561 557 310249 172303693 23.60)3474 8.2278254.001795332 558 311364 173741112 23.6220236 8.2327463.001792115 CUBE ROOTS, AND REClPROCALS. 147 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 559 312481 174676879 23.6431808 8.2376614.001783909 560 313600 175616000 23.6643191 8.2425706.001785714 561 314721 176558481 23.6854386 8.2747474.001782531 562 315844 177504328 23.7065392 8.2523715.001779359 563 316969 178453.547 23.7276210 8.2572633.001776199 564 318096 179406144 23.7486842 8.2621492.001773050 665 319225 180362125 23.7697286 8.2670294.001769912 566 320356 181321496 23.7907545 8.2719039.001766784 567 321489 182284263 23.8117618 8.2767726.001763668 568 3z2624 183250432 23.8327506 8.2316355.001760563 569 323761 1842200J,23.8537209 8.2864928.001757469 570 324900 185193000 23.8746728 8.2913444.001754386 571 326 ) 186169411 23.8956063 8.2961903.001751313 572 327184 187149248 23.9165215 8.3010304.001748252 573 323329 188132517 23.9374184 8.3058651.001745201 574 3291.6 189119224 2.3.9582971 8.3106941.001742160 575 330625 190109375 23.9791576 8.3155175.001739130 576 331776 191102976 24.0000000 8.3203353.001736111 577 332929 192100033 24.0208243 8.3251475.001733102 578 334084 193100552 24.0416306 8.3299542.001730104 579 335241 194104539 24.0624188 8.3347553.001727116 580 336400 195112000 24.0531891 8.3395509.001724138 581 337561 196122941 24. 1039416 8.3443410.001721170 582 338724 197137368 24.1246762 8.3491256.001718213 583 339889 1931552S7 24.1453929 8.3539047.001715266 584 341056 199176704 24.166(0919 8.3586784.001712329 585 342225 200201625 24.1867732 8.3634466.001709402 586 343396 201230056 24.2(174369 8.3682095.001706195 587 344569 202262003 24.2230829 8.3729668.001703578 588 345744 203297472 24.2487113 8.3777188.001700680 589 346921 204336469 24.2693222 8.3824653.001697793 590 348100 205379000 24.2899156 8.3872065.001694915 591 349281 206425071 24.3104916 8.3919423.001692047 592 350)464 2074746138 24.3310501 8.3966729.001689189 593 351649 203527857 24.3515913 8.4013981.001686341 594 352836 209584584 24.3721152 8.4061180.001683502 595 354025 210644875 24.39262f8 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 3.58801 214921799 24.4744765 8.42963S3.001669449 600 360000 216000000 24.4948974 8.4343267.001666667 601 36121)1 217031801 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 221445125 24.59f)7478 8.457'6906.001652893 606 367236 222545016 24.6170673 8.4623479.001650165 607 368449 22361S543 24.6373700 8.4670(11.001647446 608 369664 224755712 24.6576560 8.4716471.001644737 609 370381 225866529 24.6779254 8.4762892.001642036 610 3721(0 226931000 24.6981781 8.4809261.001639344 611 373321 228099131 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 37 325 232608375 24.7991935 8.5040350.001626016 616 379456 233744896 24.8193473 8.5086417.001623377 617 3S0639 234885113 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 8.5270189.001612903 _ _ - _ _- _ _ _ _ - _ _ _ _ - _ _ _ _ j 148 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 621 335611 239483061 24.9198716 8.5316909.001610306 622 336334 210611848 24.9399278 8.5361780.001607717 623 333129 241801367 24.9599679 8.5407501.001605136 624 339376 242970624 24.9799920 8.5453173.001602564 625 39)625 241140625 25.0000000 8.5493797.001600000 626 391876 245314376 25.0199920 8.5544372.001597444 627 393129 246491883 25.0399631 8.5589899.001594896 6; 334334 247673152 25.0599232 8.5635377.001592357 623 395611 248335189 25.0798724 8.5630807.001589325 630 396993 250017000 25.0993003 8.5726189.001537302 631 393161 251239591 25.1197134 8.5771523.001584786 632 399124 252435963 25.1396102 8.5816809.001582278 633 403639 253636137 25.1594913 8.5862047.001579779 631 401956 254340104 2.5.1793566 8.5907238.0)1577287 635 40322.5 256947875 25.1992063 8.5952330.001574803 636 404496 257259156 25.219041(4 8.5997476.001572327 637 405769 253474853 25.2338589 8.6042525.001569359 633 407044 259694072 25.2586619 8.6037526.001567393 639 403321 260917119 25.2784493 8.6132480.001564945 640 409690 262144000 25.2932213 8.6177383.001562500 611 410331 263374721 25.3179778 8.6222248.001560062 612 412161 264609238 25.3377189 8.6267063.001557632 613 413449 265347707 25.3574447 8.6311830.001555210 614 414736 267089934 25.3771551 8.6356551.001552795 615 416025 263336125 25.3963502 8.6401226.001550338 616 417316 269586136 25.4165301 8.6445855.001547988 647 418609 270340023 25.4361917 8.6490437.001545595 648 419904 272097792 25.4558441 8.65:34974.001543210 649 421201 273359449 2.5.4754784 8.6579165.001540832 650 422500 274625000 25:4950976 8.6623911.001533462 651 423301 275994451 25.5147016 8.6668310.001536098 6.52 425101 277167803 25..5342907 8.6712665.001533742 653 426109 278445077 23.5538647 8.6756974.001531394 654 427716 279726264 25.5731237 8.6301237.001529052 655 429025 231011375 25.5929678 8.6345456.001526718 6.56 43(336 232300416 25.6124969 8.6389630.001524390 657 431649 233593393 25.6320112 8.6933759.001522070 653 432961 234390312 25.6515107 8.6977843.001519757 659 431231 236191179 25.6709953 8.7021882.001517451 660 435600 287496000 25.6904652 8.7065377.001515152 661 436921 233304781 25.7099203 8.7109827.001512359 662 433244 291117528 25.7293607 8.7153734.001510574 663 439569 291434247 25.7487864 8.7197596.001508296 661 440396 292754944 25.7631975 8.7241414.001506024 665 442225 294079625 25.78759.39 8.7235187.001503759 666 443556 295103296 25.8069758 8.7323918.001501502 667 444399 296740963 25.8263131 8.7372604.001499250 663 446224 293077632 25.8456360 8.7416246.001497006 669 447561 299418309 25.8650343 8.7459846.001494763 670 44890) 300763900 25.8343532 8.7503401.001492537 671 450211 302111711 2?.9336577 8.7546913.001490313 672 451584 3)3161443 25.9229623 8.7590333.001488095 673 452929 301321217 25.9422435 8.7633309.001435884 674 454276 306182024 25.9615100 8.7677192.001483630 675 455625 307546375 25.9307621 8.7720532.001481481 676 456976 309915776 26.01)(0000 8.7763330.001479290 677 453329 310233733 26.0192237 8.7807034.001477105 678 459634 311665752 26.0334331 8.7850296.001474926 679 461011 313)46339 26.0576234 8.7893466.001472754 630 462403 314432003 26.0763096 8.7936593.001470583 631 463761 315321241 26.0959767 8.7979679.001463429 632 463121 317214563 26.1151297 8.8022721.001466276 CUBE ROOTS, AND RECIPROCALS. 149 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 683 466489 318611987 26.1342687 8.8065722.001464129 6S4 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 683 473344 325660672 26.2297541 8.8280099.001453488 689 474721 327082769 26.2488095 8.8322850.001451379 690 476100 328509000 26.2678511 8.8365559.001449275 691 477481 329939371 26.2868789 8.8408227.001447178 692 478864 331373888 26.3058929 8.8450854.001445087 693 480249 332812557 26.3248932 8.8493440.001443001 694 481636 334255384 26.3438797 3.8535985.001440922 695 483025 335702375 26.3628527 8.8578489.001438849 696 484416 337153536 26.3818119 8.8620952.001436782 697 485809 3.3S608873 26.4007576 8.8663375.001434720 698 487204 340068392 26.4196896 8.8705757.001432665 699 488601 341532099 26.4386081 8.8748099.001430615 700 490000 343000000 26.4575131 8.8790400.001428571 701 491401 344472101 26.4764046 8.8832661.001426534 702 492804 345948408 26,4952826 8.8874882.001424501 703 494209 347428927 26.5.141472 8.8917063.001422475 704 495616 348913664 26.5329983 8.8959204.001420455 705 497025 350402625 26.5518361 8.9001304.001418440 706 498436 351895816 26.57f16605 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 26.66458.33 8.9253078:001406470 712 506944 360944128 26.6833281 8.9294902.001404494 7-13 508369 362467097 26.7020598 8.9336687.001402625 714 509796 363994344 26.7207784 8.9378433.001400560 715 511225 365525875 26.7394839 8.9420140.001398601 716 5.12656 367061696 26.7581763 8.9461809.001396648 717 514089 368601813 26.7768557 8.9503438.001394700 718 515524 370146232 26.7955220 8.9545029 001392758 719 516961 371694959 26.8141754 8.9586581..001390821 720 518400 373248000 26.8328157 8.9628095.001388889 721 519341 374805361 26.8514432 8.9669570.001386963 722 521284 376367048 26.8700577 8.9711007.001385042 723 522729 3779.33067 26.8886593 8.9752406.001383126 724 524176 379503424 26.9072481 8.9793766.001381215 725 525625 381:078125 26.9258240 8.9835089.001379310 726 527076 382657176'26.9443872 8.9876373.001377410 727 528529 384240583 26.9629375 8.9917620.001375516 728 5299S4 385828352 26.9814751 8.9958829.001373626 729 531441 387420489 27.0000000 9.0000000.001371742 730 532900 389017000 27.0185122 9.0041134.001369863 731 534361 390617891 27.0370117 9.0082229.001367969 732 535824 392223168 27.0554985 9.0123288.001366120 733 537289 393832837 27.0739727 9.0164309.001364256 734 533756 3954 4604 27.0924344 9.0205293.001362398 735 540225 397065376 27.1108834 9.0246239.001360544 736 541696 398638256 27.1293199 9.0287149.001358696 737 543169 400315653 27.1477439 9.0328021.001356852 738 544644 401947272 27.1661554 9.0368857.001355014 739 546121 403583419 27.1845544 9.0409655.001353180 740 547600l 405224000~ 27.2029410 9.0450419.001351351 741 549081 40686902t 27.2213152 9.0491142.001349528 f42 550564 408518488 27.2396769 9.0531831.001347709 743 552049 410172407 27.2580263 9.0572482.001345895 744 553536 411830784 27.2763634 9.0613098.001344086 I,~~~~~~~ 150 TABLE X1 SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocala. 745 555025 41319362.5 27.2946831 9.0653677.001342282 746 556516 415160936 27.3130006 9.0694220.001340483 747 558009 416832723 27.3313007 9.0734726.001338688 748 559504 418508992 27.3495887 9.0775197.001336898 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 563516 428661064 27.4590604 9.1017265.001326260 755 570025 430363875 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 435976000 27.5680975 9,1258053.001315789 761 579121 440711081 27.5862284 9.1298061.001314060 762 580644 442450723 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 588239 451217663 27.6947648 9.1537375.001303781 768 589824 452934832 27.7123129 9.1577139.001302083 769 591361 454756609 27.7308492 9.1616369.001300390 770 592900 456533000 27.7488739 9.1656565.001298701 771 594441 453314011 27.7668868 9.1696225.001297017 772 595984 460099648 27.7848880 9.1735852.001295337 773 597529 461839917 27.8028775 9.1775445.001293661 774 599076 463634824 27.8208555 9.1815003.001291990 775 600625 465494375 27.8388218 9.1854527.001290323 776 602176 467288576 27.8567766 9.1894018.001288660 777 603729 469097433 27.8747197 9.1933474.001287001 778 605284 470910952 27.8926514 9.1972997.001285347 779 606841 472729139 27.9105715 9.2012286.001283697 780 603400 474552000 27.9284801 9.2051641.001282051 781 609961 476379541 27.9463772 9.2090962.001280410 782 611524 478211768 27.9642629 9.2130250.001278772 783 613089 480048687 27.9821372 9.2169505.001277139 784 614656 481890301 28.0000000 9.2203726.001275510 785 616225 483736625 2S.0178515 9.2247914.001273885 786 617796 485587656 28.0356915 9.2287068.001272265 787 619369 487443403 28.0535203 9.2326189.001270648 788 620944. 489393872 28.0713377. 9.2365277.001269036 789 622321 491169069 28.0891438 9.2404333.001267427 790 624100 493039000 28.1069386 9.2443355.001265823 791 625631 494913671 28.1247222 9.2482344.001264223 792 627264 496793038 28.1424946 9.2521300.001262626 793 628849 498677257 28.1602557 9.2560224.001261034 794 6:39136 500566184 28.1780056 9.2599114.001259446 795 6:32025 502459375 23.1957444 9.2637973.001257862 796 633616 504358336 28.2134720 9.2676798.001256281 797 635209 506261573 28.2311884 9.2715592.001254705 798 636804 508169592 28.2488938 9.2754352.001253133 799 633401 510082399 28.2665831 9.2793081.00125164 800 640000 512000000 28.2942712 9.2831777.001250000 801 641601 513922401 28.3019434 9.2870440.001248439 802 643204 -515849608 28.3196045 9.2909072.001246883 803 644309 517781627 28.3372546 9.2947671.001245330 804 646416 519718464 28.3548938 9.2986239.001243781 805 618025 521660125 28.3725219 9.3024775.001242236 806 649636 523606616 28.3901391 9.3063278.001240695.~~~~~~~~~~~~~~~~~. J.I CUBE ROOTS, AND RECIPROCALS. 151 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 807 651249 525557943 28.4077454 9.3101750.001239157 808 652364 527514112 28.4253408 9.3140190.001237624 809 654481 529475129 28.4429253 9.3178599.001236094 810 656100 531441000 28.4604989 9.3216975.001234568 811 657721 533411731 23.4780617 9.3255320.001233046 812 659344 535337328 28.4956137 9.3293634.001231527 813 660969 537367797 28.5131549 9.3331916.001230012 814 662596 539353144- 28.5306852 9.3370167.001223501 815 664225 541313375 23.5482048 9.3403386.001226994 816 665356 543338196 23..5657137 9.3146575.001225490 817 667489 5i5338513 28.5832119 9.3484731.001223990 818 669124 54734:3432 23.6006993 9.3522357.001222494 819 670761 549353259 23.6181760 9.3560952.001221001 820 672403 551363000 23.6356421 9.3599016.001219512 821 674041 553337661 28.6530976 9.3637049.001218027 S82 675684 555412248 28.6705424 9.3675051.001216545 823 677329 557441767 28.6379766 9.3713022.001215067 821.678976 559476224 28.7054002 9.3750963.001213592 825 630625 561515625 28.7228132 9.3783373.001212121 826 632276 563559976 28.7402157 9.3326752.001210654 827 633929 565609233 28.7576077 9.3361600.001209190 823 635584 567663552 28.7749891 9.3902419.001207729 829 637241 569722789 28.7923601 9.3940206.001206273 830 683900 571787000 28.8097206 9.3977964.001204819 831 690561 573856191. 23.8270706 9.4015691.001203369 832 692224 575930368 28.8444102 9.4053387.001201923 833 693389 578009537 28.8617394 9.4091054.001200480 834 695556 580093704 28.8790582 9.4123690.001199041 835 697225 582182375 28.8963666 9.4166297.001197605 836 693396 534277056 28.9136646 9.4203373.001196172 837 700569 5S6376253 28.9309523 9.4241420.001194743 833 702244 583430472 28.9482297 9.4278936.001193317 839 703921 590589719 28.9654967 9.4316423.001191895 840 705600 592704000 28.9327535 9.4353880.001190476 841 707291 594823321 29.0090000 9.4391307.001189061 842 708964 596947638 29.0172.363 9.4428704.001187648 843 710649 599077107 29.0344623 9.4466072.001186240 844 712336 601211534 29.0516781 9.4503410.001184834 845 714025 603351125 29.0638837 9.4510719.001183432 846 715716 605495736 29.0360791 9.4577999.001182033 847 717409 607645423 29. 1(32644 9.4615249.001180638 848 719104 609800192 29.1204396 9.4652470.001179245 849 720301 611960049 29.1376046 9.4689661.001177856 850 722500 6141250)0 29.1547595 9.4726824.001176471 851 724201 616295051 29.1719043 9.4763957.001175088 852 725904 618470203 29.1890390 9.4801061.001173709 853 727609 620650477 29.2061637 9.4833136.001172333 854 729316 622335364 29.2232784 9.4875182.001170960 855 731025 65026375 29.2403333 9.4912200.001169591 856 732736 627222016 29.2574777 9.4949188.001168224 857 734449 629422793 29.2745623 9.4986147.001166861 858 736164 631623712 29.2916370 9.6023078.001165501 859 737881 633839779. 29.3087018 9.5059980.001164144 860 739600 636056000 29.3257566 9.5096354.001162791 861 741321 638277331 29.3428015 9.5133699.001161440 862 743044 640503923 29.3598365 9.5170515.001160093 863 744769 642735647 29.3768616 9.5207303.001158749 864 746496 644972544 29.3933769 9.5244063.001157407 865 743225 647214625 29.4103823 9.5280794.001156069 866 749956 649461896 29.4278779 9.5317497.001154734 867 751689 651714363 29.4448637 9.5354172.001153103 863 753424 653972032 29.4618397 9.5390818.001152074 152 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 869 755161 656234909 29.4788059 9.5427437.001150748 870 756900 658503000 29.4957624 9.5464027.001149425 871 758641 660776311 29.5127091 9.5500589.001148166 872 760384 663054848 29.5296461 9.5537123.0011467,-9 873 762129 665338617 29.5465734 9.5573630.001145475 874 763876 667627624 29.5634910 9.5610108.001144165 875 765625 669921875 29.5803989 9.5646559.001142857 876 767376 672221376 29.5972972 9.5682982.001141553 877 769129 674526133 29.6141858 9.5719377.001140251 878 770884 676836152 29.6310648 9.5755745.001138952 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 -.QO 1132503 884 781456 690807104 29.7321375 9.5973373.001131222 885 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 799236 714516984 29.8998328 9.6333907.001118568 895 801025 716917375 29.9165506 9.6369812.001117318 896 802816 719323136 29.9332591 9.6405690.001116071 897 804609 721734273 29.9499583 9.6441542.001114827 898 806404 724150792 29.9666481 9.6477367.001113586 899 808201 726572699 29.9833287 9.6513166.001112347 900 810000 729000000 30.0000000 9.6548938.001111111 901 811801 731432701 30.0166620 9.6584684.001109878 902 813604 733870808 30.0333148 9.6620403.001108647 903 815409 736314327 30.0499584 9.6656096.001107420 904 817216 738763264 30.0665928 9.6691762.001106195 905 819025 741217625 30.0832179 9.6727403.001104972 906 820836 743677416 30.0998339 9.6763017.001103753 907 822649 746142643 30.1164407 9.6798604.001102536 908 824464 748613312 30.1330383 9.6834166.001101322 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 840339 771095213 30.2820079 9.7153051.001090513 918 842724 773620632 30.2985148 9.7188354.001089325 919 844561 776151559 30.3150128 9.7223631.001088139 920 846400 778688000 30.3315018 9.7258883.001086957 921 848241 781229961 30.3479818 9.7294109.001085776 922 850084 783777448 30.3644529 9.7329.309.001084599 92.3 851929 786330467 30.3809151 9.7364484.001083423 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 869329 796597983 30.4466747 9.7504980.001078749 928 861184 799178752 30.4630924 9.7539979.001077i86 929 863041 801765089 30,4795013 9.7675002.001076426 930 864900 804357000 30.4969014 9.7610001.001075269 CUBE ROOTS, AND RECIPROCALS. 153 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocala. 931 866761 806954491 30.5122926 9.7644974.001074114 932 86 624 809557568 30.5286750 9.7679922.001072961 933 870439 812166237 30.5450487 9.7714845.001071811 934 872356 814780501 30.5614136 9.7749743.001070664 935 874225 817400375 30.5777697 9.7784616.001069519 936 876!)96 820025356 30.5941171 9.762'3466.001063376 937 877969 8226.36953 30.6104557 9.7854238.001067236 938 879344 825293672 30.6267857 9.7889037.001066098 939 881721 827936019 30.6431069 9.7923861.001064963 940 883600 830534000 30.6594194 9.7953611..001063830 941 835481 833237621 30.6757233 9.7993336.001062699 942 8S7364 835396338 30.6920185 9.8023036.001061571 943 839249 838561807 30.7083051 9.8062711.001060445 944 891136 841232384 30.7215830 9.8097362.001059322 945 893025 843903625 30.7403523 9.8131989.001058201 946 894916 846590536 30.7571130 9.8166591.001057082 947 896309 849278123.30.7733651 9.8201169.001055966 948 398704 851971392 30.7896086 9.8235723.001054852 949 900601 854670349 30.8058436 9.8270252.001053741 950 902500 857375030 30.8220700 9.8304757.001052632 951 904401 860085351 30.8382879 9.8339238.001051525 952 9063 4 862801408 30.8544972 9.8373695.001050420 953 903209 865523177 30.8706931 9.8408127.001049318 934 910116 863250664 30.8863904 9.8442536.001048218 955 912025 870933875 30.9030743 9.8476920.001047120 956 913936 873722316 30.9192497 9.8511230.001046025 957 915849 876467493 30.9354166 9.8545617.001044932 958 917764 879217912 30.9515751 9.8579929.001043841 959 919631 831974079 30.9677251 9.8614218.001042753 960 921600 884736000 30.9838663 9.864848.3.001041667 961 923521 887503631 31.0000000 9.8632724.001040583 962 92.5444 890277128 31.0161248 9.8716941.001039501 963 927369 893056347 31.0322413 9.8751135.001038422 964 929296 895841314 31.048:3494 9.8785305.001037344 965 931225 898632125 31.0644491 9.8319451.001036269 966 933156 901428696 31.0895405 9.8853574.001035197 937 935039 904231063 31.0966236 9.8887673.001034126 963 937024 907039232 31.1126984 9.8921749.001033058 969 933961 909853209 31.1287648 9.8955801.001031992 970 940303 912673000 31.1448230 9.8939830.001030928 971 942841 915498611 31.1603729 9.9023835.001029366 972 944784 918330043 31.1769145 9.9057817.001028507 973 946729 921167317 31.1929479 9.9091776.001027749 974 194676 924010424 31.2039731 9.9125712.001026694 975 950625 926359375 31.2249900 9.9159624.001025641 976 952576 929714176 31.2409987 9.9193513.001024590 977 954529 932574833 31.2.569992 9.9227379.00102.3541 978 956484 935441352 31.2729915 9.9261222.001022495 979 958441 936313739 31.2889757 9.9295042.001021450 980 960400 941192000 31.3049517 9. 923839.001020408 931 962361 944076141 31.3209195 9.9362613.001019368 982 964324 946966168 31.3368792 9.9396363.001018330 983 966239 949862087 31.3528308 9.9430092.001017294 934 963256 952763904 31.3637743 9.9463797.001016260 985 970225 955671625 31.3847097 9.9497479.001015223 986 972196 958585256 31.4006369 9.9531138.001014199 987 974169 961504803 31.4165561 9.9564775.001013171 938 976144 964430272 31.4324673 9.9598389.001012146 989 978121 967361669 31.4433704 9.9631981.001011122 990 90100 970-299000 31.4642654 9.9665549.001010101 991 932081 973242271 31.4801525 9.9699095.001009082 992 934064 976191438 31.4960315 9.9732619.001008065 154 TABLE XI. SQUARES, CUBES, &C. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 993 986049 979146657 31.5119025 9.9766120.001007049 994 988036 982107784 31.5277655 9.9799599.001006036 995 990025 985074875 31.5436206 9.9833055.001005025 996 992016 988047936 31.5594677 9.9866488.001004016 997 994009 991026973 31.5753068 9.9899900.001003009 998 996004 994011992 31.5911380 9.9933289.001002004 999 998001 997002999 31.606J613 9.9966656.001001001 1000 1000000 1000000000 31.6227766 10.0000000.001000000 1001 1002001 1003003001 31.6385840 10.0033322.0009990010 1002 1004004 1006012008 31.65438.36 10.0066622.0009980040 1003 1006009 1009027027 31.6701752 10,0099899.0009970090 1004 1008016 1012048064 31.6859590 10.0133155.0009960159 1005 1010025 1015075125 31.7017349 10.0166389.0009950249 1006 1012036 1018108216 31.7175030 10.0199601.0009940358 1007 1014049 1021147343 31.7332633 10.0232791.0009930487 1008 1016064 1024192512 31.7490157 10.0265958.0009920635 1009 1018081 1027243729 31.7647603 10.0299104.0009910803 1010 1020100 1030301000 31.7804972 10.0332228.0009900990 1011 1022121 1033364331 31.7962262 10.0365330.0009891197 1012 1024144 1036433728 31.8119474 10.0398410.0009881423 1013 1026169 1039509197 31.8276609 10.0431469.0009871668 1014 1028196 1042590744 31.8433666 10.0464506.0009861933 1015 1030225 1045678375 31.8590646 10.0497521.0009852217 1016 10322.56 1048772096 31.8747549 10.0530514.0009842520 1017 1034289 1051871913 31.8904374 10.0563485.0009832842 1018 1036324 1054977832 31.9061123 10.0596435.0009823183 1019 1038361 1058089859 31.9217794 10.0629364.0009813543 1020 1040400 1061208000 31.9374388 10.0662271.0009803922 1021 1042441 1064332261 31.9530906 10.0695156.0009794319 1022 1044484 1067462648 31.9687347 10.0728020.0009784736 1023 1046529 1070599167 31.9843712 10.0760863.0009775171 1024 1048576 1073741824 32.0000000 10.0793684.0009765625 1025 1050625 1076890625 32.0156212 10.0826484.0009756098 1026 1052676 1080045576 32.0312348 10.0859262.0009746589 1027 1054729 1083206683 32.0468407 10.0892019.0009737098 1028 1056784 1086373952 32.0624391 10.0924755.0009727626 1029 1058841 1089547389 32.0780298 10.0957469.0009718173 1030 1060900 1092727000 32.0936131 10.0990163.0009708738 1031 1062961 1095912791 32.1091887 10.1022835.0009699321 1032 1065024 1099104768 32.1247568 10.1055487.0009689922 1033 1067039 1102302937 32.1403173 10.1088117.0009680542 1034 1069156 1105507304 32.1558704 10.1120726.0009671180 1035 1071225 1108717875 32.1714159 10.1153314.0009661836 1036 1073296 1111934656 32.1869539 10.1185882,0009652510 1037 1075369 1115157653 32.2024844 10.1218428.0009643202 1038 1077444 1118386872 32.2180074 10.1250953.0009633911 1039' 1079521 1121622319 32.2335229 10.1283457.0009624639 1040 1081600 1124864000 32.2490310 10.1315941.0009615385 1041 1083681 1128111921 32.2645316 10.1348403.0009606148 1042,085764 1131366088 32.2800248 10.1380845.0009596929 1043 7087849 1134626507 32.2955105 10.1413266.0009587738 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 1098304 1151022592 32.3728281 10.1575062.0009541985 1049 1100401 1154320649 32.3882695 10.1607359.0009532888 1050 1102500 1157625000 32.4037035 10.1639636.0009523810 1051 1104601 1160935651 32.4191301 10.1671893.0009514748 1052 1106704 1164252608 32.4345495 10.1704129.0009505703 1053 1108809 1167575877 32.4499615 10.1736344.0009496676 1054 1110916 1170905464: 32.4653662 10.1768539.0009487666 TABLE XII. LOGARITHMS OF NUMBERS FROM I TO 10,000 156 TABLE XII. LOGARITHMIS OF NUMBERS. No. 0 l1 _3 4 5 6 i7 1 8 9 Dff. 103 00X)U0 000434 00S6S 010J 3001 7:34 C02166 002598 003029 b03461 003891 432 1 4321 4751 5181 5609 603- 6466 6894 73211 7748 8174 428 2 8600 90261 9451 9876 010300 010724 011147'011570 0119931012415 424 3012337013259 01'3630 014100 4521 4940 5360 57791 6197 6616 420 4 70:33 74531 7868 8284 8700 9116 9532 9947 021361 020775 416G 5 021189 021603 022016 02242S 022S41 023252 023664 024075 44t 6 4896 412 6 5306 5715 6125 6.:33 6942 73)5 7757 8164 8571 8978 408 7 9384 9789 030195 030600 031(104 031408 031812 032216 032619 033021 404 8 033424 033326 4227 4628 5029 5430 5830 6230 6629 7028 400 9 7426 7825 8223 8620 9017 9414 9811 0402071040602 040998 397 110 041393 041787 042182 0425761042069 043362 043755 044148 044540 044932 393 1 6323 5714 6110 69 6885 7275 7664 80531 8442 8830 390 2 9218 9606 9993 050380 050766 051153 051538 051924 052309 052694 3S6 3 033078 053463 053346 4230 4613 4996 5378 5760 6142 6524 383 4 6905 7286 7666 8046 8426 8805 9185 9563 9942 060320 379 5 060698 061075 061452 061829 062206 (062582 0629.58 063333 063709 4083 376 6 4458 4832 5206 5580 5953 6356 6699 7071 7443 7815 373 7 8186 8557 892S 9298 9668 070038 070407 070776 071145 071514 370 80718S2 0722501072617 072985 073352 3718 4085 4451 4816 51821 366 9 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 120 079181 079543 079904 380266 080626 080987 081347 0817071082067 102426 360 10032785 083144 033503 3361j. 4219 145761 4931:. 5291 5647 6004 357 2 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 355 3 9905 090258 090611 090963 091315 091667 092018 092370 092721 093071 352 409.3422 3772 4122 4171 4820 5169 5518 5866 6215 6562 349 5 6910 7257 7604 7951 8298 8644 8990 9335 9681 100026 346 61100371 100715 101059 1014(13 101747 102091 102434 102777 103119 3462 343 7 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871.341 8 7210 75.49 7888 8227 8565 8903'9241 9579 9916 110(2.53 338 9 110590 110926 111263 111599 111934 112270 112605 112940 113275 3609 335 130 11-t33 11i4z7 1i4611 114944 115278 115611 115943 116276 116608 116940 333 1 7271 7603 7931 8265 8595, 8926 9256 9586 9915 120245 330 21120574 120903 121231 121560 121888 12221 5122544 122871 123198 3525 328 3 3352 4178 4504 4830 5156 54811 5806 6131 6!56 6781 325 4 7105 7429 7753 8076 4399 87221 9045 -9368 9690 130012 323 5 130334 130655 130977 131298 131619 131939 132260 132580 132900 3219 321 6 3539 3358 41771 4-1S61 4814 5133 5451 5769 6086 6403 318 7 6721 7037 7354 76 1 7987 83031 8618 8934 9249 9664 316 8 9879 140194 1405031140822 141136 141450 141763 142(076 142389 142702 314 9 143015 3327 3639 3951 4263 4574 4885 5196 5507 5818 311 14014612 16128 146438 146748 147058 147367 147676 147985 148294 148603 148911 309 1| 9219 9527 9835 150142 150449 150756 151063 15137(0 151676 151982 307 2152288 152594 152900 3205 3510 3815 4120 4424 4728 5032 305 3 5336 564(0 5943 6261 6549 6852 7154 7457 7759 8061 303 4 8362 8664 8965 9266 9567 9861 16016811604691160769 161C68 301 51613631161667 16196711622661162564 162863 3161 34601 3758 4055 299 6 4353 4650( 4947 5244 5541 56838 6134 6430 6726 7022 297 7 7317 7613 79(81 8203 8497 8792 9086 9380 9674 9c68[ 295 8 170262 170555 170848 1711411171434 171726 172019 172311 172603 172895 293 9 3186 3478 37691 4060 4351 46411 4932 5222 5512 5802 291 150 176091 176381 176670 176959 177248 177536 1"7825 178113 178401 178689 289 1 8977 9261 9552 9839 180126 180413 110699 180986 181272 181558 287 2181844 182129 1824151182700 2985 3270 3555 3839 4123 4407 285 3 4691 4975 52591 5542 582.5 6108 6391 6674 6956 7239 283 4 7521 7803 80841 8366 8647 8928 9209 9490 97711190051 281 5 190332 190612 190S921191171 191451 191730 192010 192289 192567 28461 279 6 31265 3403 3681 3959 4237 4514 4792 5069.3461 5623 278 7 5900 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 8 8657 8932 9206 9481 9755120002912o003(3200577 2o00850 201124 274 9201397 201670 201943 202216 202488 2761 3033 33051 35771 3848 272 No. 0 1 31 4 5 61 7 8 Diff. TABLE XII. LOGARITHMS OF NUMBERS. 157 No. O 1 a 3 j 4 1 5 6 8 9 Diff. 16O0 2011290 231391 234663 2.)1931 29~521.'223175 205746 206016 206236 206.356 271 I 6326 7096 7365 7631 7904 8173 8441 87101 8979 9247 269 2 9515 9783 210)51 210319 210536 210353 211121 211333 211654 211921 267 31212133,212.131 2729 2936 3252 3518 3783 4019 4314 4579 266 4 4314 5109 5373 563 3 5902 6166 6130 6694 6957 7221 264 5 7434 7747 8010 8273 8336 8793 9069 932:3 9585 9346 262 6 22)103 2-3:370, 22631 220332 221153 221414 22167.5 221936 222196 222456 261 7 2716 2976 3236 3196 3755 4015 4274 4533 4792 5051 2599 8 5399 5563 5326 6 34 6342 6600 6358 7115 7372 7630( 258 9 7837 8144 8403 8657 8913 9170 9426 9632 9933 230193 256 170 230119 239701 239969 231215 231470 231721 231979 232231 23248S 232742 255 1 2996 32 0 3591 3757 4011 4261 4.17 4770 5023 5276 253 2 553 5781 6:33 6235 6537 6789 7041 7292 7541 7795 252 3 83)16 8297 85-3 8799 9019 9299 9550 9300 240350 240309 250 4 21053 t21079. 24104 241297 211546 231795 122044 242293 2.541 2790 240 5 3033 3236:331 3732 403) 4277 4525 4772 5019 5266 248 6 5 513 5759 6316 6253) 6199 6745 6991 7237 7482 7728 246 7 7973 8219 8464 8709 8951 9193 9443 9637 99:12 250176 245 8 2.301 2 230661 250J03 251151 2313915 231633 252183 1 252125 252363 2610 243 9 235.3 3)96 3383 353) 3322 4061 4306 4548 4790 5031 242 189 235273 255514 255755 255996 256237 256177 256718 256953 257193 257439 241 1 7679 7918 8153 8393 8637 8377 9116 9355 9594 9-33 2.39 2 269371 26310 43 26 654S 2J787 261025 261263 261501 261739 261976 262214 23S 3 2451 2633 29-2. 3162 3399 3636 3373 4109 4346 4582 237 4 4318 5051 5290 552.5 5761 5996 6232 6167 6702 6937 235 5 7172 7406 7611 7875 8110 8314 8578 8812 9016 9279 234 61 9513 9746 9930 270213 2704461270679 270912 271144 271377 271609 233 7 271812 272974 272396 2533 2770 3901 3233 3-61 3696 3927 232 8 4153 4339 4620 48355 5031 5311 5542 5772 6002 6232 230 9 6162 6692 6921 7151 7330 76J9 7833 8067 8296 8525 229 193 278751 278932 279211 279139 279657 279395 230123 280351 230578 230806 228 1 231033 231261 231483 2317151231942 232169 2396 2622 2849 30751227 2 3301 3527 3753 3979 420.5 4431 46.56 4832 5107 5332 226 3 5557 5782 6007 6232 6156 6631 6905 7130 7354 7578 225 4 78f12 8026 8219' 8473 8636 8929 9143 9366 9589 9812 223 51290335 293257 2904890299702j299925 291147 291369 291591 291813 292034 222 6 2256 2178 2699 29209 3141 3363 3584 3304 4025 4246 221 7 4466 4637 4907 5127 5347 5567 5787 6007 6226 6446 220 8 6665 6834 71041 7323 7542 7761 7979 8198 8416 8635 219 9'8353 9071 92-9 9507 972.5 9943300161 300378 300595 300813 218 200 301030 391247 301464 391631 30189 302114 302331 302547 302764 302980 217 31 9 3196 3412 323 334 4059 4275 4491 4706 4921 5136 216 2 5355 571 6211 6425 6639 6354 7068 7232 215 3 7496 7710 7924 8137 8351 8561 8778 8991 9204 9417 213 4 9639) 9343 310310056 31026313131063 31096 31 311330311542 212 5 3117541311966 2177 2339 2601 2312 323 3234 3445 3656 211 6 3367 4078 4239 4499 4710 4920 5130 5:340 5551 5760 210 7 5970 6180 6390 6599 6399 7013S 7227 7436 7616 7854 209 8 86:3 8272 8431 86S9 8393 9106 9314 9522 973(1 9933 203 9 3201466 2 9035 3262 320769329977 321181 321391 321598 321805 322012 207 210 322219 322426 322633 322339 323016 3232.52 323458 323665 323371 324077 206l 1 4232 4433 4694 4399 5105 5310 5516 5721 5926 6131 205 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 3 8330 8533 8787 8991 9194 9393 9601 930 330008330211 203 41331143361733031913310221331225331427331630331832 2034 2236 202 5 2133 2610 2342 3941 3236 3447 3649 3350 4051 4253 202 6 4451 46.35 4356 5037 5257 5459 5653 5359 6059 6260 2011 7 6161 6661 6360 706) 7260 7459 7659 7853 80538 8257 200 8 8456 8656 8355 9054 9253 9451 9650 9349 340047340246 199 9131t911 31061 2 3103-1 331039 311237 131143 31632 311839 2023 222.5 19 No. 0 1 2 3 4 5 6 7 8 9 Of 158 TABLE XII. LOGARITHMS OF NUMBERS. No. 0 1 2 3 4 5 6 8 9 Di. 220 342423 32620 342817 343014 343212 343409 343606 3802 343999 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 8694 8889 9083 9278 9472 9666 9860350054 194 4 350248 350442 350636 3 30829351023 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 58.34 192 7 6026- 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 8 7935 8125 8316 8506'8696 8886 9076 9266 9456 9646 190 9 9835 360025 360215 360404 360593 360783 360972 361161 361350 361539 189 230 361728 361917 i62105 362294 362482 362671 362859 363048 363236 363424 188 1 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 188 2 5488 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 185 5 371068 371253 371437 371622 371806 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 9849380030 181 240 380211 380392 380573 380754 30934 381115 381296 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 5249 5428 179 3 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 4 7390' 7568 7746 7923 8101 8279 8456 6634 8811 8989 178 5 9166 9343 9520 9698 9875 390051 390228 390405 390582 390759 177 6390935 391112391288 391464391641 1817 1993 2169 2345 2521 176 7 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 176 8 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 175 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 397940 398114 398287 398461 398634 398808 39S981 399154 399328 399501 173 1 9674 9847 400020 400192 400365 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 4834 5005 5176 5346 5517 5688 585b 6029 6199 6370 171 5 6540 6710 6881 7051 7 722 7391 71 756 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 411283411451 169 8 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 168 -9 3300 3467 3635 3S03 3970 4137 4305 4472 4639 4806 167 260 414973 415140 415307 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 421275421439 165 4 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 164 5 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 164 6 4882 5045 5208 5371 5534 5697 5860 6023 6186 6349 163 7 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 430720 43081 431042431203 161 270 431364 431525 431685 431846 43007 432167 432328 432488 432649 432809 161 1 2969 3130 3290 3450 3610 3770 3930 4090 4249 4409 160 2 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 159 3 6163 6322 6481 6640 6799 6957 7116 7275 7433 7592 159 4 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 5 9333 9491 9648 9806 9964 440122 440279 440437 440594 440752 158 6 440909 441066 441224 441381 441538 1695 1852 2009 2166 2323 157 7 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 8 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 156 9 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 155 No. 0 1 2 3 4 5 6 7 8 9 Dif. TABLE XII. LOGARITHMS OF NUMBERS. 159 lNo 0 1 2 1 3 4: 5 6 7 8 9 Diff.l 280 447158 447313 447468 447623 447778 447933 4408S8 448242 448397 448.52 155 1 8706 8861 9015 9170 9324 9478 96331 97871 9941 450095 154 2 450219 450403 430557 450711 450865 451018 451172 451326 451479 1633 154 3 1786 1940 2093 2217 2400 2553 2706 2859 3012 3165 153 4 3318 3171 3624 37771 3930 4082 4235 4387 4540 4692 153 5 4815 4997 5159 5302j 5454 5606 5758 5910 6062 6214 152 6 6366 6318 6670 6821 6973 7125 7276 7428 7579 7731 152 7 7832 8933 8184 8336 8487 8638 8789 8940 9091 9242 151 8 9392 9543 9694 9845 999314601461460296 460447 4605971460748 151 9 460898 46104( 461198 461348 461499 1649 1799 1948 2098 2248 150 290 462398 462548 462697 462847 462997 463146 463296 463445 463594 463744 150 1 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 2 53S3 5532 5630 5829 5977 6126 6274 642.3 6571 6719 149 3 6863 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 4 8317 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 5 9822 9969 470116 470263 470410 470557 470704 470351 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 4653 4799 4944 5090 5235 5381 5526 146 9 5671 5816 5962 6107 6232 6397 6542 6687 6832 6976 145 300 477121 477266 477411 477555 477700 477844 4779S9 478133 478278 478422 145 1 8566 8711 8S55 8999 9143 9287 9431 9575 9719 9863 144 2 480007 480151 480294 480438 140582 480725 480869 481012 481156 481299 144 3 1443 1536 1729 1872 2016 2159 2302 2445 2588 2731 143 4 2374 3016 3159 3302 3445 3587 3730 3872 4015 4157 143 5 4300 4442 4585 4727 4869 5011 5153 5295 5437 5579 142 6 5721 5863 6005 6147 6289 6430 6572 6714 6355 6997 142 7 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 141 8 8551 8692 8833 8974 9114 9255 9396 9537 9677 9818 141 9 9958 490099 490239 490380 490520 490661 490801 490941 491081 491222 140 310 491362 491502 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 5128 5267 5406 139 3 5544 5633 5822 5960 6099 6238 6376 6515 6653 6791 139 4! 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 5 83111 8448 8586 8724 8362 8999 9137 9275 9412 9550 138 6, 9687 9824 9962 5000991500236 500374 500511 5006481500785 500922 137 715010591501196 501313 1470 1607 1744 1880 2017 2154 2291 137 8 2427 2564 2700 2837 2973 3109 3246 3332 3518 3655 136 9 3791 3927 4063 4199 4335 4471 4607 4743 4878 5014 136 320 5051050 5236 505421 505557 505693 505828 505964 506099 506234 506370 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 98741510009 510143;510277 510411 134 4]510545 510679 510813 510947 511081 511215 1349 14821 1616 1750 134 5 1833 2017 2151 2284 2418 2551 2684 2818 2951 3084 133 6 3218 3351 3484 3617 3750 3881 4016 4149 4282 4415 133 71 4548 4611 4513 4941 5079 5211 5344 5476 5609 5741 133 8 5874 6006 6i39 6271 6403 653.1 6668 6S00 6932 7064 132 9 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 132 330 518514 518646 518777 518909 519040 519171 519303 519434 519566 519697 131 1 93281 9959 520090 523221 523353 520484 520615 520745 520876 521007 131 21521138521269 1400 1530 1661 1792 192'2 2053 2183 2314 131 3 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 130 4 3746 3376 4006 4136 4266 4396 4526 4656 4785 4915 130 5 5045 51741 5304 54341 5563 56931 5822 5951] 6081 6210 129 6 6339 6469 6593 6727 6856 6985 7114 7243 7372 7501 129 7 7630 7759 7888 8016 8145 8274 8402 8531 8660 8788 129 8 8917 9045 9!74 9302 9130 9559 9637 9815 9943 530072 128 915:30200 530323 530456 530584 530712 530840 530963 531096,531223 1351 128 No.i 1 3 4 5 6 7 8 9 D.,Di1 160 TABLE XII. LOGARITHMS OF NUMBERS. No.1 0 1 M 3 4 5 6 7 _ 8 __ 9 Ditf. 310 531479 531607 531734 531662 531990 532117 532245 532372 53250J 532627 128 1 2754 2882 3009 3136 3264 3391 3518 3645 3772 3499 127 2 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 127 3 5294 5421 5547 5674 5800 5927 6053 6180 6306 6432 126 4 6558 6685 6811 6937 7063 7189 7315 7441 7567 7693 126 5 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 126 6 9076 9202'9327 9452 9578 9703 9829 9954 540079 540204 125 7 540329 540450 540580 540705 540830 540955 541080 541205 1330 1454 125 8 1579 1704 I1829 1953 2078 2203 2327 2452 2576 2701 125 9 2825 2950 3074 3199 3323 3447 3571 3696 3320 3944 124 350 544068 544192 544316 544440 544564 544688 544812 544936 545060 545183 124 1 5.307 5431 5555 5678 5802 5925 6049 6172 6296 6419 124 2 6543 6666 6789 6913 7036 7159 7282 7405 7529 7662 123 3 7775 7898 8021 8144 8267 8389 8512 8635 8758 ~881 123 4 9003 9126 9249 9371 9494 9616 9739 9861 9984 550106 123 5 550228 550351 550473 550595 550717 550840 550962 551084 5512C6 1328 122 6 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 7 2668 2790 2911 3033 3155 3276 3398 3519 3640 3762 121 8 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 9 5094 5215 5336 5457 5578 5699 5820 5940 6061 6182 121 360 556303 556423 556544 556664 5567S5 556905 557026 557146 557267 557387 120 1 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 120 2 8709 8S29 8948 9068 9188 9308 9428 9.48 9667 9787 120 3 9907 560026 560146 560265 560385 560504 560624 560743 560863 560982 119 4 561101 1221 1340 1459 1578 1698 1817 1936 2055 2174 119 5 2293 2412 2531 2650 2769 2887 3006 31 2 3244 3362 119 6 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 119 7 4666 4784 4903 5021 5139 5257 5376 5494 6612 6730 118 8 5848 5966 6084 6202 6320 6437 6555 6673 6791 6209 118 9 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 56S202 568319 568436 568554 568671 568788 568905 569023 569140 569257 117 1 9374 9491 9608 9725 9842 9959570076 570193 57 0309 570426 117 2 570543 570660 570776 570893571010 571126 1243 1359 1476 1592 117 3 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 116 4 2872 2988 3104 3220 3336 3432 3568 3684 38001 3915 116 5 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 6 5188 5303 5419 5534 5650 5765 588O 5996 6111 6261 115 7 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 115 8 7492 7607 7722 7836 7951 80 811 8295 84101 8525 115 9 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 380 579784 579898 580012 580126 580241 580355 580469 580583 580697 580811 114 1 580925 581039 1153 1267 1381 1495 1608 1722 1836 1950 114 2 2063 2177 2291 2404 2518 2631 2745 28.51:,.21 3085 114 3 3199 3312 3426 3539 3652 3765 3879 39b2 -' 105 4218 113 4 4331 4444 4557 4670 4783 4896 5009. 5122 5235 5348 113 5 5461 5574 5686 5799 5912 6024 6137 6250 6362 6475 113 6 6587 6700 6812 6925] 7037 7149 7262 7374 7486 7599 112 7 7711 7823 7935 80471 8160 8272 8384.8496 ~608 8720 112 8 8832 8944 9056 91671 9279 9391 9503 9615 9726 9838 1)2 9 9950 590061 590173 590284 590396 590507 590619 590730 590842 590953 112 390 591065 591176 591287 591399 591510 591621 591732 591843 591955 592066 III 1 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 111 2 3286 3397 3508 361.8 3729 3840 3950 4061I 4171 4282 111 3 4393 4503 4614 4724' 4834 4945 5055 51651 5276 5386 110 4 5496 5606 5717 5827i 5937 6047 6157 6267 6377 6487 110 51 6597 6707 6817 6927 7037.7146' 7256 7366 7476 7586 110 6 7695 7805:.7914 8024 8134 8243 8353 8462,8572 8681 110 71 8791 89001 9009 9119 9228 9337, 9446 95561 9665 9774 109 8 98831 9992 600101 600210 600319 600428 6005371600646 600755 600864 109 91 60097316010821 11911 1299 1408 15171 1623 1734 1843 1951 109 No. O 1 2 3 14 5 16 7 8 9 Diff. ~~.... TABLE XII. LOGARITHMS OF NUMBERS. 161 (No.l I 1 13 14 5 6 7 1 3 Diff. 140 602 060 692169 602277 602386 62);494 60 26':3 6:2711 6:)2SI 6929;23 60)3(036 103 1:14 35 361 3169 3577 3686 3794 39021 4010 4118 1(OS 2 42-26 4334 44 12 4550 4658 4766 487 49284 50S9 5197 108 3 5:3 513 54 5521 5623 5736 5844 5951 6',59 G6166.6274 103 4 3:J31 61 9 6596 671 4 6311 6919 7126 7133 7241 7348 107 5 7455 7562 766J3 7777 7884 7991 8098 8-205 8312 8419 107 6 8526 s633 874) 8847 8954 9061 9167 9274! 93i1 9488 107 71 9594 9701 93803 9914'61(1f021 61(123 610231 610341 610447 61(1551 107 8 61066 1610767 610373 G109791 1036 1192 129 14051 1511 1617 106 I 1723 1829 1936 2042 2143 2254 2360 2466 2572 2678 106 4103112784 612390 612996 613102 613207 613313 613419 613525 613631 613736 106 1 3342 3947 4(5331 4159 4264 4:3701 4475 451 466 4792 106 2 4897 50)03 510I 52131 5319 5121 5529 5631 5740 5845 105 3 5930 6)155' 6160 6-265 6370( 6176 65 1 66-6 6790) 6395 105 4 7059 7105 7210) 7315 7 7420 25 7629 7734 7839 7943 1(05 5 8143 8153 82537 8:363 8466 8571 8676 8780( 8834 8)989 105 6 9)9:3 9193 93)02 9406 9511 961 5 9719 9824 9923 620(032 104 7 620136 620241( 620:344 620418 620532 620636 62(0760 62(1864 62096> 1072 104 8 1176 12S(1 1331 1483 1592 1695 1799 19:31 20:17 211(1 104 9 2214 2318 2421 252.5 2623 2732 233.5 2939 3042 3146 104 423 623249 6233533 623156 623559 623663 623766 623869 623973 624076 624179 103 1 42i2 4335 44S3 4591 4695 4 793 4(1 )1 5(1004 51(07 52101 1()3 21- 5312 5415 5518 5621 5724 53827 59129 60:32 6135 62:383 103 3 6310 6443 65 16 664 6751 68531 6956 70(53 7161 726:1 1(13 4 7366 7468 7571 7673 7775 7878 793(0 8(032 8185 82^7 102 5 8339 8491 859:3 8695 8797 89.10 9002 9104 9206 9.303 1)02 6 9110 9512 96 131 97135 9817 19919163)(0021 630(123 63022-1 630326 1(2 7163) 12? 6305:31 6306t:163:733 6 6033'6:3:936 1(038 1139 1241 1342 102 8 1444 1515 16471 1748 1849 19151 22 2 1 53 2255 2:36 1(01 9 2157 2559 2663 2761 2S62 21963 3)61 3165 3266 3367 101 43163346 633563 63370633170 6:3771 633372 633)73 634074 634175 634276 634376 1(01 1 4477 4578 4679' 4779 48310 49'I 503I1 5183 52833 5383 101 2.5131 5534 5685 578.5 53t6 59t6 6137 6187 6287 638- 100 3 6183 6538 69331 6789 6839 693.9 7089 7189 729(0 7390 100( 4 7490 759) 76931 779(31 9(1 99 809(1 8191) 829)! 8:39 1(1)i 51 8589 8339 8691 8789 833; 893S 9101. 91833 1927 9:337 10()1 61 918f6 9536 9636 9)785' 9835 9)934 61(003164018:3 61(i23 61033821 99 7 6141311 610531 64(I06i3640(773!61(l379 640(1'77i 10177 1177 1276 1375 99 8 1474 1573 1672 1771 1871 1970 2069 216; 2267 2:366 99 9 2.3563 26 2662 2761 236(1 2959 3(158 3156 3255 33534 99 41,613 153 613551 64135in G1:3749 61317 613946 614014 644l143 64424212614:340 98 1 41:39 4537 46f36; 47:31 4832 49311 I 029 5127 5226 5324 98 2 5122. 5521 5619 5717 513 59L13 6911 6110 62(18 63116 93 3 61(11 6:12 6301 669) 6796 68314 69192 7109 7187 7283.'; 93 4 7:313 7431 7676 7774 7872 719169 8067 816.5 826-2 983 5 8:360) 815i 8555 8653 875() 8343 8915 9043k 9t14 9237 97 6 93:35 9132 995331 9627 9721 9321 9919 63;(116 6350113 6350210( 97 7 653363 0 6501)41 630502; 639599 65316t6 65179:3 650390 09i7 l(134 1181 97 8! 1273 1375i 1472 156 9 1636 1762 1851 1956 2(53 2.151' 97 9 2216 2313 2440 2336 26:33 2739 2326 2923 3019 3116 97 4533,65.3213 6A339 63105 332 65353 63593 6536953 653791 653S33 63539q4 6.540(1 96 11 4177 427:3 4369 44635 4562 46531 4754 48503 4946 50(42 96 2' 513 53 535 53:331 5l27 552:3 5619 57155 5810 5996 6312 96 13 6 139 6191 6329 6336 6182 65.77 667:1 67631 63>64 6961) 9; 4 731) 71.532 7247 7313 7433 7531 7629 772'~ 7823) 7916 96 5 8i1 1 8 1 07 823921 823 3 8:39 13 8318 8534 8679 8774 8870 95 6 89)6-: 90(63 9155 92 )31 9346 9111 9536 96311 97'26 98321 9. 7 19: 16 633 I 1'66 1(6 66 1231 66(12916 661391 66(136.66.531 66 )676 66(1771 95 86613631 0363( 105,) I I153} 1215. 1339 1434 135291 162: 1718 95 9! 18131 137 2002l 2)96' 2191 2236 2:3301 247' 2.69 2663: 95 ol0 1 2 314 5 6 7 1 8 9 DH.f. 162 TABLE XII. LOGARITHMS OF NUMBERS. No. 0 I 1 X 3 4 5 6 7 8 9 Diff. 469 662753 662852 662947 663(41 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 7 9317 9410 9503 9596 9689 9782 9875 9967 670060 670153 93 8 670246 670339 670431 670524 670617 670710 670802 670895 0988 1080 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 5228 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 7516 91 6 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 91 7 8518 8609.8700 8791 8882 8973 9064 9155 9246 9337 91 8 9428 9519 9610 9700 9791 9882 9973 680063 680154 680245 91 9 680336 680426 680517 680607 68069 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 5025 5114 5204 5294 5383 5473 5563 5652 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 9575 9664 9753 9841 9930 690019 690107 89 490 690196 6902 690373 69 690550 690639 690728 690816 690905 690993 89 1 10811 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 3815 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 8535 8622 8709 8796 8883 87 500 698970 699057 699144 699231 699317 699404 699491 699578 699664 699751 87 I 1 9833 9924 700011 700098 700184 700271 700358 700444 700531 700617 87 2700704 700790 0877 09631 1050 1136 1222 1309 1395 1482 86 3 1568 1654 1741 18271 1913 1999 2086 2172 2258 2344 86 { 4 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 86 1 5 3291 3377 3463 35491 3635 3721 3807 3893 3979 4065 86 { 6 4151 4236 4322 44081 4494 4579 4665 4751 4837 4922 86 1 7 5008 5094 5179 5265; 5350 5436 5522 5607 5693 5778 86 ~ 8 5S64 5949 6035 61201 6206 6291 6376 6462 6547 6632 85 9 6718 6803 6888 69741 7059 7144 7229 7315 7400 7485 85 1510 707570 707655 707740 707826 707911 707996 703081 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 1 4 0963 1048 1132 1217, 1301 1385 1470 1554 1639 1723 84 5 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 84 1 6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3-107 84 7 3491 3575 3659 3742 3S26 3910 3994 4078 4162 4246 84 8 4330 4414 4497 4581 4665 4749 4333 4916 5000 5084 84, 9 5167 5251 335 5418 5502 5586 5669 5753 5836 5920 84 No 0 1 2 3 1 4 5 7 8 9 Diff. TABLE XII. LOGARITHMS OF NUMBERS. 163 No.j0O r0 2 | 3 j 4 5 6 7 8 9 Diff. 5207 716087 716 70171654171633 7716421 716504 716588 17167 71674 83 1 6835 6921 70041 7088 7171 7254 7338 7421 7504 7587 83 2 7671 7754 7837 79209 8003 8086 8169 8253 8336 8419 83 3 8502 8586 8668 8751 8834 8917 9000 9083 9165 9248 83 4 9331 9414 9497j 9580S 9663 9745 9828 9911 9994 720077 83 5720159 720242 720325 720407 720490 720573 720655 720733 720821 OJ03 83 6 0986 1063 1151 1233 1316 1398 1481 1563 1646 1728 82 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 82 8 26:34 2716 2798 2881 2963 3045 3127 3209 3291 3374 82 9 3456 353S 362.1 3702 3784 3866 3948 4030 4112 4194 82 5:30 724276 72435S 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 6616 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 5 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 6 916.3 9246 9327 9403 9489 9570 9651 9732 9813 9893 81 7 9974 730055 730136 730217 730298 730378 730459 730540 730621 730702 81 81730782 0363 0944 1024 1105 1186 1266 1347 1428 1508 81 9 1539 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 r540 732394 732474 732555 732635 732715 732796 732376 732956 733037 733117 80 1 3197 3278 3353 31381 3518 3598 3679 3759 3839 3919 80 2 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 3 4800 4880 4900 5040 5120 5200 5 79 5359 5439 5519 80 4 5599 5679 5759 5833 5918 5998 6078 6157 6237 6317 80 5 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 6 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 79 7 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 79 8 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 9 9572 9651 9731 9310 9889 9968 740047 740126 740205 740284.79 550 740363 740442 710521 740600 740678 740757 740836 740915 740994 741073 79 1,1152 123 1309 1388 1467 1546 1624 1703 1782 1860 79 2 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 79 3 2725 2304 2 382 2961 3039 3118 3196 3275 3353 3431 78 4 3510 35, 3667 3745 3823 3902 3980 4058 4136 4215 78 5 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 6 5075 5153 5231 5309 5337 5465 5543 5621 5699 5777 78 7 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 78 8 6634 6712 6790 6863 6945 7(023 7101 7179 7256 7334 78 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 748188 7482661 74343 748421 748498 748576 748653 748731 748808 748885 77 1 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 2 9736 9814 9391 9968 750045 750123 750200 750277 750354 750431 77 375050317505861750663750740 0817 0394 0971 1048 1125 1202 77 4 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 77 5 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 77 6 2816 2393 2970 3047 3123 3200 3277 3353 3430 3506 77 7 3583 3660 3736 3813 3389 3966 4042 4119 4195 4272 77 8 4343 4425 4501 4578 4654 4730 4807 4883 4960 5036 76 9 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 570 755875 755951 756027 756103 756180 756256 756332 756408 75648 756560 76 1 6636 6712 6788 6361 6940 7016 7(92 7168 7244 7320 76 2 7396 7472 7548 7621 7700 7775 7851 7927 8003 8079 76 38155 81 230 8306 8332 8458 8533 8609'8685 8761 8836 76 4 8912 8938 9063 9139 9214 9290 9.366 9441 9517 9592 76 5 9663 9743 9819 9394 9970 760045 760121 760196 760272 760347 75 6 76')422 760493 763573 7606.19,760724 0799 08775 0950 1025 1101 75 7 1176 1'25 1326 1402 1477 1552 1627 1702 1778 1853 75 8 19')2 20)3 2078 2153 2228 2303 2378 2453'2529 2604 75 9 26791 2754 2329 2904 2978 3053 3128 3203 3278 3353 75 No. 0 1 2 3 4 5 6 7 8 9 Duff. 164. TABLE XII. LOGARITIIIS OF NUMBERS.'NO. 9 O 1 1 3 4 5 6. 7 1 8 9 Diff. 580 1 76.;28; 63503' 7;3578 763653 76327 02 76377 7 763952 764027 7 64101 75 11 416 4251 4326 4400 4475 4550 4624 4699 4774 4848 75 2 4923 4998 5072 5147 5221 5296 5370 5445 5520 5594 75 3 5669 5743 5818 5892 5 66 6041 6115 6190 6264.63381 74 4 6413 64S7 6562 6636 6710 6785 685 693:3 7007 7(082 74 5 7156 7230 7304 7379 7453 7527 7601 7675 7749 7823 74 6 7898 7972 8046 8120 8194 8268 8342 8416 8490 8564 74 7 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 74 8 9377 9451 9525 9599 9673' 97461 9820 9894 9968 770(42 74 9 770115 770189 770263 770336 770410 770464 770557 770631 770705 0778 74 590 770852 710926 770999 771073 771146 771220 771293 771367 771440 771514 74 1 1587 1661 1734 1808 1881 1955 2028 21(2 2175 2248 73 2 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 3360 3933 4006 4079 4152 4225 4298 4371 4444 73 5 4517 4590 4663 4736 4809 4882 4955 5028 5100 5173 73 6 5246 5319 5392 5465 5538 5610 5683 5756 5829 5902 73 7 5974 6047.6120 6193 6265 6338 6411 6483 6556 6629 73| 8 6701 6774 6846 6919 6992 7064 7137 7209.7282 7354 73 9 7427 749 7572 7644 7717 7789 7862 7934 8006 8079 72 600 778151 778224 778296 778368 778441 778513 778585 77,658 778730 778802 72 1 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 2 9596 9669 9741 9813 9885 9957 7800291780101 7801731780245 72 3178031778783 780461 780533780605780677 0749 0821 0893 0965 72 4 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 5 1755 1827 1899 1971 2042 2114, 2186 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 9 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 71 610 53 7 85472 785543 785615 785686 785757 785828 785899 785970 71 1 6041 6112 6183 62541 6325 6396 6467 6538 6609 6t80 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 8163 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 791)004 7900741790144 790215 70 7 7902857903;56 7904261790496 790567 790637 0707 0778 0848 0918 70 8 0938 1059 1129 1199! 1269 1340 1410 1480 1550 16M0 70 9 1691 1761 1831 19011 1971 2041 2111 2181 2252 2322 70 620 792392 792462 792532 792602 792672 792742 792812 792S82 792952 793022 70 1 3092 3162 3231 3301 3371 3441 3611 3581 3651 3721 70 2 3790 3860 3930 4(000 4070 4139 4209 4279 4349 4418 70 3 448S 4558 4627 4697 4767 4836 4906 4976 5045 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 706) 7129 7198 69 7 7268 7337 7406 7475 7545 7614 7643 7752 7821 7890 691 8 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 9 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 639 799341 799409 799478 799547 7996167996 79 99754 799823799892 799961 69 1 800029 800098 80(0167 800236 800305 800373 800442 800511 800580 800648 69 2 0717 0786 0854 0923 0992 1161 1129 1198 1266 1335 69( 3 1404 1472 1541. 1609 1678 1747 1815 188S4 1952 2021 69' 4 2089 2L58 2226 2295 2363 2432 2500 2568 2637 237) 68 5 2774 2842 2910 2979 3047 3116 3184 3252 3321 3379 C8 6 3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 68 7 4139 4208 4276 4344 4412 4480 4548 4616 4685 4453 68 3 4821 4889 4957 5025 5093 5161 5229 5297 565 5433 68 9 5501 5569 6637 5705 5773 5841 5908 5976 6044 6112 68'No. 0 1 12 f 3 4 I 5 I 6 7 8 9 Diff. J{ 6 6 6 TABLE XII. LOGARITHMS OF NUMBERS. 165 No.l O 0 I 2a 1 3 14 5 6 7 8 9 9 DiLf. 61018061801803218 8(0631 6183:34 806451 806.319 8U6587 80665. 806723 806780 681 I 6S581 69 6 6994 7061 7129 7197 7261 7332 7400 7467 68 2 7535 7603 7670 7738 7806 7873 794 8008 8076 8143 6S 3 8211! 82791 8316 8414 8451 8549 86 81 8751 8818 67 4 8 -6 8953 9)21 9033 9156 9223 9290 9358 9425 9492 67 5 9560 9S27 9694 971 929 92 9896 9964 810031 81009S 810165 67 6 810233 8103001810367 810434 810501 810569 810636 0703 0770 0837 67 7 09041 0971 1039 1106 1173 1241) 1307 1374 1441 1 58 67 8 1575 1612 1709 1776 1843 1910 1977 2044 2111 2178 67 9 2245 2312 2379 2 14.) 2512 2579 2646 2713 2780 2I47 67 650 812913 812930 813047 813114 813181 813217 813314 813381 813448 813514 67 1 3581 3648 3714 3781 3348 3914 3981 4048 4114 4181 67 i 2 4248 4314 4331 4447 4514 4581 4617 4714 4780 4'47 67 3 4913 4930 ()046 5113 5179 5261 5312 5378 5445 5511 66 4 5578.5,644 57111 57777 5843 59101 5976 6042 6109 6175 66 5 62411 6308 6374 6440 6506 657.3 6639 6705 6771 6338 66 6 6304 6970 7035 7102 7169 7235 7301 7367 7433 7499 66 7 7565 7631 7693 7764 78301 7896 7962 8028 8094 8160 66 8 8226 8 82'32 8358 2 82 8490 8556 8622 -868 8754 882) 66 9 8335 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819544 319610 819676 819741 181,07 819873 819939 820004 820070 3-20136 66 1 1202)1 8320267 320333 382')399 820464 820530 320595 0661 0727 0792 66 2 0i58 0924 0939 1055 1120 1186 1251 1317 1332 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 5 2822 2357 29.321 33 303 314 321: 33279 3:344 3409 65 6 3474 3539 3605 3670 3735 33(00 3865 3930 3996 4061 65 7 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 65 8 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 65 9 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 65 670 826075 8261 441 826204 826269 826334 826399 826464 326528 826593 826651 65 1 6723 6787 6352 6917 6931 7046 7111 7175 7240 7:305 65 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 651 3 8015 8030 8144 8209 8273 8333 8402 8467 8.331 8595 64 4 8650 8724 8789 8553 8918 8982 9046 9111 9175 9239 64 5 9304 9363 9132 9-197 9561 9625 9690 9754 94181 9382 64 6! 9947 3SS0'1 1 i30075 330139 830204 830263 830332 83)396 8304C0 330525 61 7i,830339 0)653 0717 0781 0845 0909 0973 1037 1102 1166 64 8 123l 1294 1338 1422 1486 1550 1614 1678 1742 1806 61 9 1870 1931 1933 2062 2126 2189 2253 2317 2381 244n 64 630 832509 332573 332637 832700 332761 q32328323292 832936 833020 833033 64 1 3147 3211 327.3 3333 3412 3466 3.533 3;93 36-37 3721 61 2 3784 3343 3912 3975 4039 4103 4166 4230 4294 4357 61 3 4121 4484 4543 4611 4675.47:39 4-02 4866 4929 4993 64 4 50'36 5120 5183 5247 31 5310 53:731 5437 5500 55641 5627 6 3 5 561 5754 5317 5331 5941 6307 6371 61:34 6197 6261 63 6 6321 6:387 6131 6.14 6577 6611 6704 6767 6830 6894 6:3 7 6957 7i>123 7(0331 7146 7210 7273 73:6 7:39. 74621 752.5 6:f 8 7.538 7632 77151 777,1 7841 7904 7967 80:30 80931 8156 63 9 8219 8232 8345 8403 8471 8534 8597 8660 8723 8786 63 690 833349 33 912 S3397.38393133 33101 -l39164 839227 -39239 839352 8391l5 63 1 9473 9.41 96'11 9667 9729 9792 93-5 9918 99 931 140; 43 63 2 840100 340163 810232 84029:1 88103.7 340420 340432 8405345 840603 0671 6:1 3 073:3 179616 03591 0921 09A4 10)46 1109 1172 1234 1297 6:3 4 13.59 1422 1435' 1547 1610 1672 173. 1797 1860 1922 63 5 1935 2147 2110 21721 223.1 2297 2360 2422 2184 2547 62 6 2619 2672 2734 2796 2359 2921 29-3 3'46 3103 3170 62 7 323:3 3295 3357 3120 3t12 3511 3696 3669 37:31 379: 62 8 3355 3913 3930 49012 41(1 4166 4221 4291 435:) 4-15 62 9 44771 45339 46)11 4661 4726 4783 4850 4912 4974 5036 62 No. 1 2 3 4 5 6 7 8 -9 iff. 166 TABLE XII. LOGARITHMS OF NUMBERS. lNo. O 1 2 3 1 4 5 6 71 8 9 Di.f 700 845098 845160 845222 845284845346 845408 845470 845532845594 84656 62 1 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 62 2 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 621 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 621] 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 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 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 4852 61 6 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 7 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 61 8 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 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 5 860338 860398 860458 860518 860578 0637 0697 0757 0817 0877 60 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 601 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 601 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 863323 863382 863442 863501 863561 863620 863680 863739 163799 863858 59 1 3917 39771 4036 4096 4155 4214 4274 4333 4392 4452 59 2 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 3 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 59 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 59 5' 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 7 7467 7526 7585 7644 7703 7762 7821 7860 7939 7998 59 8 8056 81151 8174 8233 8292 8350 8409 8468 8527 8586 59 9 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 869232 869290 869349 869408 869466 869525 869584 869642 869701 869760 59 1 9818 98771 9935 99941870053 870111 870170 870228870287 870345 59 2 870404 870462 8705211870579 0638 0696 0755 0813 0872 0930 58 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 58 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 6 2739 27971 2855 2913 2972 3030 3088 3146 3204 3262 58 7 3321 33791 3437 3495 3553 3611 3669 3727 3785 3844 58 8 3902 39601 4018 4076 4134 4192 4250 4308 4366 4424 58 9 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 581 750 875061 875119 875177 875235 875293 875351 875409 875466 875524 8755S 2 581 1 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 58 2 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 58 3 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 58 4 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58' 5 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 57 6 8.522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 7 90961 9153 9211 9268 9325 9383 9440 9497 9555 9612 57 8 9669 9726 9784 9841 9898 9956S880013 88007018801271880185 57 9 8802421880299 880356 880413 880471 880528 0585 0642 0699 0756 57 No. 0 1 i 3 4 5 6 7 8 9 Diff. L TABLE XII. LOGARITHMS OF NUMBERS. 167 jNo.l 0 1 3 4 5' 6 7 8 9 Diff. 760) 880814 880371 88092S 880985 881042 881099 881156 881213 881271 881328 57 1 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 57 2 1955 2012 2069 2126 2183 2240' 2297 2354 2411 2468 57 3 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 57 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 57 5 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 57 6 4229 4235 4342 4399 4455 4512 4569 4625 4682 4739 57 7 4795 4352 4909 4965 5022 5078 5135 5192 5248 5305 57 8 5361 5418 5474 5531 5587 5644 5700 5757 5813 5870 57 9 5926 5933 6039 6096 6152 6209 6265 6321 6378 6434 56 770 886491 886547 886604 886660 886716 886773 886829 886385 886942 886998 56 1 7054 7111 7167 7223 7230 7336 7392 7449 7505 7561 56 2 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 56 3 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 56 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 56 5. 9302 9358 9414 9470 9526 9582 9633 9694 9750 9806 56 6 9362 9918 9974 890030 893086 890141 890197 890253 890309 890365 56 7 8904211 90477 890533 0589 0645 0700 0756 0812 0363 0924 56 8 0930 1035 1091 1147 1203 1259 1314 1370 1426 1482 56 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 56 7380892095 892150 892206 892262 8 923173 892429 892484 892540 892595 56 1 2651 2707 2762 2S18 2373 2929 2985 3040 3096 3151 56 2 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 56 3 3762 3317 3373 3928 3984 4039 4094 4150 4205 4261 55 4 4316 4371 4427 4432 4533 4593 4648 4704 4759 4814 55 5 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 55 6 5423 5478 5533 5588 56.44 5699 5754 5809 5864 5920 55 7 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 55 8 6526 6581 6636 6692 6747 6302 6357 6912 6967 7022 55 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 55 790 897627 397632 897737 897792 897847 897902 897957 893012 898067 893122 55 1 8176 8231 8236 8:341 8396 8451 8506 8561 8615 8670 55 2 8725 8730 8835 8890 8944 8999 9054 9109 9164 9218 55 3 9273 9323 9333 9437 9492 9547 9602 9656 9711 9766 55 4 9821 9375 9930 9935 900039 900094 900149 900203 900258 900312 55 5 190367 930422 900476 900531 0586 0640 0695 0749 0804 0859 55 6 0913 0963 1022 1077 1131 1186 1240 1295 1349 1404 55 7 1455 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 8 2003 2057 2112 2166 2221 2275 2329 2334 2438 2492 54 9 2547 2631 2635 2710 2764 2318 2873 2927 2981 3036 54 800 Q03090 903144 903199 903253 903307 903361 903416 903470 903524 903578 54 1 3633 3687 3741 3795 3349 3904 3958 4012 4066 4120 54 2 4174 4229 4233 4337 4391 4445 4499 4553 4607 4661 54 3 4716 4770 4821 487, 4932 4936 5040 5094 5148 5202 54 4 5256 5310 5364 5418 5472 5526 5580 5634 5683 5742 54 51 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 61 6335 63S9 6443 6497 6551 6604 6658 6712 6766 6820 54 7 6374 6927 6931 7035 703g 7143 7196 7250 7304 7358 54 8 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 54 9 7919 8002 8056 8110 8163 8217 8270 8324 8378 8431 54'30 903485 903539 908592 908646 903699 90S753 9088071908860 908914 908967 54 1 9021 9074 9123 9181 9235 9239 9342 9396 9449 9503 54 2 9556 9610 9663 9716 9770 9323 9877 9930 9984 910037 53 3 910091 910144 910197 910251 910304 910358 910411 910464 910518 0571 53 4 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 53 5 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 53 6 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 53 7 2222 2275 2328 2331 2435 2488 2541 2594 2647 2700 53 8 2753 2306 28591 2913 2966 3019 3072 3125 3178 3231 53 9 3234 3337 3390 3143 3496 3549 3602 3655 3708 3761 53 No. 1 0 1 2 3 4 5 6 7 8 9 Di. 168 TABLE XII. LOGARITHMS OF NUMBERS. No. O 1 3 4 5 6 7 8 9 Diff. 82() 13-S814 91 367 9120913973914026 14079 914132 914184 914237 914290 53 1 4343 4396 4449 4502 4555 460s 4660 4713 47 C6 4819 5311 2 487 2 4925 4977 5030 5083 5136 5189 5241 5294 5347 53tt 3 5400 5453 5505 5558 5611 5C64 5716 5769 5822 5875 ~311 4 5927 5980 6033 60.5 6138 6191 6243 6296 6349 64011 53 5 6454 6.507 6559 6612 6664 6717 6770 6822 6875 6927 53 6 6980 7033 7085 7 3 7190 7243 729- 7348 7400 7453 53 7 7506 76558 7611 7663 7716 7765 782( 7873 7925 7978 52 8 8030180S3 8135 8188 8240 8293 8345 8397 8450 8502 52 9 8.555 86J7 8659 8712 8764 8816 8;69 8921 8973 ~2026 52 83)9 919078 919130 919183 919235 919287 919340 919392 919444 919496 919549 52 1 9601 9653 9706 9758 981(1 9862 9914 9967 920019 920071 52 2 92()123 920176 920228 920280 920332 920384 9204361 20489 1 O10593 52 l 0645 0697 (1749 0801..0853 09(6.095b- 1010 1062 1114 52 4 1166 1218 1271) ]322 1374 1426 1478 1530 1582 1i34 52 5 16-6 1738 179(1 1842 1894 1946 1998 2050 2102 2154 52 6 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 52 7 2725 2777 2829 28l1 2933 2985 3037 3089 3140 3192 52 8 3244 3296 3341 3399 3-151 3503 3555 3607 6658 371(1 52 9 3762 3S14 3865 3917 3969 4(021 4072 4124'4176 4228 52 840 92-1279 924331 q24331 924434 924486 924538 9245S9 924641 924693 924744 52 1 4796 448 4899 4951 5003 5054 51C06 5157 5209 5261 52 2 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 52 3 5828 5879 5931 5982 6034 6085 6137 6188 6240 6291 51 4 6312 6394 6445 6497 65481 6600 6651 67021 6754 6805 51 5 6857 6908 6959 7011 7062 7114 7165 7216 7268 73191 51 6 7370 7422 7473 7521 7576 7627 7678 7730 7781 7832 51 7 7883 793.3 79S6 8037 8088 81140 8191 8242 8293 8345 51 8 8396 8447 8498 8549 8601 8652 87031 8754 88o05 8857 51 9 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 51 850 929419199470929 521 929572 929629296329674 929725 929776 92982 1929879 51 1 9930 9981 930032 930083 930134 930185 93(0236 930287 932331 9()3891 51 219301440930491 0542 0592 0643 0694 0745 07961 0847 0898 51 3 0949 1000 1051 1102 11531, 1204 1214 13(5 1356 1407 51 4 1458 1509 1560 1610 1661 1712 1763 1814 18651 191 51 5 1966 2017 2063 2118 2169 222(0 2271 2322 2:372 2423 51 6 2474 2524 2575 2626 2677 2727 2778 2,29 28791 293(0 51 7 2911 3031 3082 3133 3183 3234 3285 3335 33-6 3437 51 8 3487 3538 35S9 3639 3691 3740 3791 3841 3822 943 51 3993 4044 4094 4143 4195 4246 4295 4347 4397 4448 51 6f i9:3449S 931549 934599 934650 9347001934751 93~4801 931552 934102 934953 50! 5003 5054 51(0404 5-1 5205 5255 5306 5356, 54(61 5457 50 2 5507 5558 56(18 5658 57(19 5759 5809 5S60 5910 5960 50 3 6011 6061 6111 61621 6212 6262 6313 6363 6413 6463 50 4 6514 6564 6614 6665 6715 6765 6815 6-651 6916 6i 6650 7016 70(66 7117 7167 7217 7267 7317 7.671 7418[ 7468 01 6 751S 7768^ 7618 7661 7718 7769 7819 7869 7919 7i69 50 7I 8019 8069 8119 8169 8219 8261, 8321! 8370' 8420( 8471 85-20 857( i860 8670 8720 8771 8820 8870, 8920 897(0 50) 9020 907. 0 9120 9170 9220 9220 93227 9(i 93419 9469 50 18d: 9319:.19 939.) 939 99619 9396697939999719 939i769 939019 9?9869 939918 93S96. r5 1!140 S 940016 94 > 11 940!)6 21 9402 67 941(317 940(1367 940417 940467 50 2 03516 (1566 0616 0666 06 716 0;76; 08. 0-6.5 (1915 (. 64 501 3 I14 106 1 14 116 1 1213 1263 1313 1362 1412 1462 50 4 1511 1561 1611 16601 171(1 176() 1809 18159 19(19 195S 50 51 2<1(:3 21058 21(17 2157 2-2117 2256 23(;6 23.5 24(105 241 5 50 6 25-14 25.;)4 2 60:31 26., 2701z 2752 2801 2851 2901 2950 50 71,3010 3,1491 3099 314.1 3191- 3247 3297 334-6 33.61 344 49 3195 3.441 3.-)3 3613 62 3742 3791 3841 3- 901 3'39 49 i 39 40 41038 4088 4313 1 46 46 36 4-285 4:135 4341 4433. 49 Nho. O 1 1 2 3 4 5 6 7 8 9 Diff. TABLE XII. LOGARITHMS OF NUMBERS, 169 No. 0 1 2 3 4 5 16 7 8 9 Dilf. 880 944483 J44-:53 —94415sI 9446:3 94468!)944729 944779 944S28 944(77 944927 49 1 4976 5025 5074 51241 5173 5222 5272 5321 5370 5419 49 2 i,69 551- 5567 5616 5665 5715 5764 5813 5862 5912 49 3 5961 6390 6059 ( 103 6157 6207 6256 6305 6351 6403 49 4 6452 61 61551 6600 C649 6698 6747 6796 6845 6394 49 5 6943 6992 7011 7090 7140 7189 723 72871 733i 73851 49 6 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 7 7924 7973'8022 8070 8119 8168 8217 8266 8315 8364 49 8 8413 -482 8511 8560 86091 8657 8706 8755 8804 8853 49 9 8902 8931 8999 9048 9097 9146 9195 9244 9292 9341 49 890 949390 ) 4C 94948 9 9435 995S5 949634 919683 949731 949780 99829 49 1 9378 9926 9975 950024 95(073 950121 950170 9.0219 950267 950316 49 2 9.50365 950414 950462 0511 0560 0631 ((657 0706 0754 0803 49 31 0351 0900 0949 0997 1046 1095 114 1192 1241) 1289 49 4 133S 13-6 14:35 1483 1532 1530 1629 1677 1726 1775 49 5 1823 1872 1920 1969 2017 20661 11-! 2163 2211 2260 48 6 2:338 2356 215 245 2.502 2550 2599 2647 2696 2744 48 7 2792 2841 2339 2933 2936 3034 3083 3131 3180 3223 48 8 3276 3325 3373 3421 3470 3518 3566 3615 3663 3711 43 9 3760 3303 3356 3905 3933 -001 4049 4098 4146 4194 48 900 9 95423942919339 95437 954433 95484 99532 954580 954628 954677 48 1 4725 1773 4321 4s69 4918 4966 5014 5062 511(0 5158 48 2 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 43 3 5683 5736 5784 5832 5380 5923 5976 6024 6072 6120 48 4 6163 (2161 6265 6313 6361 6109 67 6505 6553 6601 48 5 6649 6697 6745 6793 6340 6138 6936 6984 7032 7080 43 6 7123 7176 7224 7272 7320 7363 7416 7464 7512 7559 48 7 7607 76.55 7703 7751 7799 7847 7894 7942 7990 803S 48 8 8036 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 9 8561 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 910 959041 95903919591371959185 9 -9232 959230 959328 959375 959423 959471 48 1 95181 9566 9614 9661 9709 9757 9304 9352 9900 9947 48 2 9995 96032 960090 (96013 960189601233 960230 960328 960376 960423 48 3960471 0518 0566 0613 0661 0709 0756 0804 0351 0399 48 4 0946 0994 1041 1039 11:16 1184 1231 1279 1326 1374 48 5{ 1421 1469 1516 1563 1611 16531. 1706 1753 1801 1848 47 6 1895 194;3 1993 2033 2035 21 213 180 2227 2275 2322 47 7 2369 2417 2461 2511 2559 2606 2631 2701 2743 279)5 47 8 2343 2890 2937 2935 3032 3079 3126 3174 3221 3263 47 9 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 47 920 963783 963S35 963332 963929 963377 961024 964071 9641 18 964165 964212 47 1 426 ) 4307 43534 4401 444S 4495 4542 4590 4637 46314 47 2 4731 4778 4325 4872 4919 4966 5013 5061 5108 5155 47 3 5202 5249 5296 5343 5399 5437 5434 5531 5578 5625 47 4 5672 5719 5766 5313 5-69 5907 5954 6001 6048 6095 47 5 6142 6189 6236 62833 6329 6376 6423 6470 6517 6564 47 6 6611 6653 670.1 6752 6799 6S45 6892 6939 69S6 7033 47 7 7080 7127 7173 7220 7267 7314 7361 7403 7454 7501'47 8 7543 7595 7642 7633 77351 7782 7829 7873 7922 7969 47 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 47 930 963483 963530 963576 963823 963670 963716 9687631 96831 n963856 96S903 47 1 8950 8996 9043 9090 9136 - 9183 9229 9276 9323 9369 47 2 1 9 1631 96309 9556 9602 9649 9695 9742 9789 9835 47 3 9S32 9923 99751970021197006 970114 9701611970207 9702541970300 47 419703471970393 970440 04-6 0533 0579 06261 (672 0719 07653 46 5 0812 0(358 0901 0951 0997 1044 1090 1137 1183 1229 46 6 1276 1322 1369 1415 1461 1508 1554 16(01 1647 1693 46 7 1740 1786 18321 1879 1925 1971 2018 2064 211(0 2157 46 8 22(03 2249 2295 2342- 2333 2434 2431 25.27 2573 2619 46 9 2666 2712 2758 2 2 851 2397 2943 2989 3035 30832 46 No. 0 1 2 i 3 4 1 5 6 7 8 9 Diff. 170 TABLE XII. LOGARITHMS 3F NUMBERS. No. 0 1 2 3 4 5 1 6 7 1 8 1 9 IDiff. 94. 973128 973174 973220 973266 973313 973359 973405 973451 973497 9735431 6 1 3590 3636 3682 3728 3774 3,20 3E66 3913 3959 4(W.?l 46 2 4051 4097 4143 4189 4235 4281' 4327 4374 4420 4466 46 S 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 46 4 4972 5018 5064 5110 5156 5202 5248 5294 5340 53,61 16 t 5432 5478 5524 5570 5616 5662 5707 5753 5799 5845 46 6 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 46 6350 6396 6442 6488 6533 6579 6625 6671 6717 67;531; 6808 6854 6900 6946 6992 7037 7083 7129 7175 722u;1 46 9 7266 7312 7358 7403 7449 7495 7541 7686 7632 76781 46 950 977724 977769 977815 977861 977906 977952 977998 978043 978089 97815, 46 1 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 46 2 8637 8683 8728 8774 8819 865 8911 8956 9002 90417 6 3 9093 9138 9184 9230 9275 9321 9266 9412 9457 95C5i 46 4 9.548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 5 980003 980049 980094 980140 980185 980231 980276 980322 980367 95042} 15 6 0458 0503 0549 0594 0640 0685 0730 0776 0821 OSC' 45 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 8 1366 1411 1456 1501 1547 1592 1637. 1683 1728 1773 lo1 9 1819 1864 1909 1954 2000 2045 2090 2135 2181 222Cj 45 960 982271 982316 982362 982407 982452 982497 982543 982588 982633 982671 t5 1 2723 2769 2814 2859 2904 2949 2994 3040 3085 3' 45 2 3175 3220 3265 3310 3356 3401 3446 3491 3536 35811 45 3 3626 3671 3716 3762 3807 3852 3897 3942 3987 40321 15 4 4077 4122 4167 4212 4257 4302 4347 4392 4437 44 ^ 45 5 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 6 4977 5022 5067 5112 5157 5202 5247 5292 5337 53821.5 7 5426 5471 5516 5561 5606 6651 5696 5741 5786 5^! 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 956772 936817 986861 99869 51 986996 987040 987085 987130 987175 45 1 7219 7264 7309 7353 7398 7443 7488 7532 7577 76221 45 2 7666 7711 7756 7800 7845 7890 7934 7979 8024 80C;, 45 3 8113 8157 8202 8247 8291 8336 8381 8425 847(0 8514 45 4 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960! 45 5 9005 9049 9094 9138 9183 9227 9272 9316 9361 94 i J45 6 9450 9494 9539 9583 962S 9672 9717 9761 9806 90oo01 44 7 9895 9939 9983 990028 990072 990117 990161 990206 990250 9902941 44 8 990339 990383 990428 0472 0516 0561 0605 0650 0694 0735' 41 9 0783 0827 0871 0916 090960 1004 1049 1093 1137; i( 1 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 2 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 44 3 2554 2598 2642 2686 2730 2774 2819 2863 2907 29'-. 11 4 2995 3039 3083 3127 3172 3216 3260 3304 3348?7'i^ ~4 5 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 6 3877 3921 3965 4009 40.53 4097 4141 4185 4229 427^! 14 7 4317 4361 4405 4449 4493 4537 4581 4625 4669 4'' 4 8 4757 4801 4845 4889 4933 4977 5(121 5065. 5108 51521 44 9 5196 5240 5284 5328 5372 5416 5460 6504 5547 559 1! 44} 990 995635 995679 995723 995767 995811 995854 995998 995942 995986 9960301 44 1 6074 6117, 6161 6205 6249 6293 6337 6380 6424 64681'14 2 6512 6555 6599 6643 6687 6731 6774 6818 6862 6^,., 4 3 6949 6993 7037 7080 7124 7168 7-212 7255 7299':3q3 44 4 7386 7430 7474 7517 7561 7605 7648 7692 7736 77791 44 5 7823 78671 791(0 7954 7998 8041 8085 8129 8172 89:;'. 6 8259 8303 8347 8390 8434 8477 8521 8564 8608 bet;2\ 44 7 8695 8739 87821 8826 8869 8913 8956 9(000 9043 9087 44 8 9131 9174 9218 9261 9305 9348 9392 9435 9479 952-'3 4 9 9565 9609 9652 9696 9739 9783 9826 9870 9913 95' {1_-3 a.I 0 1 I a2 3 4 5 6 7 8 _ 9 iDiff. j~~~~~~~1 _ - _=.2.. TABLE XIII. LOGARITHMIC SINES, COSINES, TANGENTS, AND COT. kNGENTS. 172 TABLE XIII. LOGARITHMIC SINES, NOTE. THIE table here given extends to minutes only. The usual method of extending such a table to seconds, by proportional parts of the difference between two consecutive logarithms, is accurate enough for most purposes, especially if the angle is not very small. When the angle is very small, and great accuracy is required, the following method may be used for sines, tangents, and cotangents. I. Suppose it were required to find the logarithmic sine of 5' 24". By the ordinary metbhi we should have log. sin. 5' = 7.162696 diff. for 24" = 31673 log. sin. 5' 24" = 7.194369'iice more accurate method is founded on the proposition in Trigonometry, that the sines or tangents of very small angles are proportional to the angles themselves. In the present case, therefore, we have sin. 5': sin. 5' 24' - 5: 5' 24'" = 300"': 324'. Hence sin. 5' 24' 324 sin. 5i -= 300, or log. sin. 5' 24" = log. sin. 5' + log. 324 - log. 300. The difference for 24" will therefore, be the difference between the logarithm of 324 and the iogarithm of oG0.'he operation will stand thus: log. 324 2.510545 log. 300 = 2 477121 diff. for 24 = 33424 log. sin. 5' = 7.162696 log. sin. 5' 24" = 7.196120 Comparing this value with that given in tables that extend to seconds, we find it exact even to the last figure II. Given log. sin. A = 7.004438 to find A. The sine next less than this in the table is sin. 3' = 6.940847. Now we have sin. 3': sin. A 3 sin. A 3 A. Therefore, A = sin' or log. A = log. 3 + log. sin. A - log. sin. 3'. Hence it appears, that, to find the logarithm of A in COSINES, TANGENTS, AND COTANGENTS. 173 minutes, we must add to the logarithm of 3 the difference between log. sin. A and log. sin. 3t. log. sin. A = 7.004438 log. sin. 3' = 6.940847 63591 log. 3 = 0.477121 A - 3.473 0.540712 or A = 3' 28.38". By the common method we should have found A = 3' 30.54". The same method applies to tangents and cotangents, except that in the case of cotangents the differences are to be subtracted. ~' The radius of this table is unity, and the characteristics 9, 8,?, and 6 stand respectively for -1, -2, -3, and -4. 174 TABLE XIII. LOGARITHMIC SINES, 03 1790 M. Sine. D. I. Cosine. D. 1'. Tang. D. 1"1. Cotang. M. 0 Inf. neg. 0.000000 00 Inf. neg. Infinite. 60 1 6.463726 017 000000 6.46 3726 17 17 3.536274 59 2.764756 293485.000000 0.764756 2934.235244 58 3.940847 208231.000000 *.940847 208231.059153 57 8.366S16.999999 02.366817'.633183 52 9.417968 762:62.999999:1.417970 762.63 582030 51 10 7.463726 9.999998000 7.465727 2.536273 50 15.50511268 691..0.505120 698.494880 49 12.542906 9.629.999997.0.542909 6 1 457091 48 13.577668 0-3.999997.0.577672 579.3 422328 47 14.360985 6.53 *.9999996.609857 36.42 390143 46 165.63916 845.1 1999996.639820 46715.360180 45 16.667845 438.81.999995 01.667849 43882.332151 44 18.718997 39135.999994.719003 3913..280997 42 19.742478 371.27.999993.01.742484 371.28.257516 41 20 7.764754 351 9.9999993 7.764761 3. 2.235239 40 21.785943 33672.999992.01.785951 336.73.214049 39 22.806146 32.7.999991.01 806155 6 93845 249.81.0 1 629.8174540 23.825451 3083.99999990.842460 303.7 145460 37 25.861662 28388.999989 02 861674 295.49 138326 35 26.878695 27317.9999988.878708 273.8 121292 34 27.895085 26323.999987 02.89099 3..104901 33 28.910879 25399.999986 02.910894 2 089106 32 29.926119 245:38.999985.926134 24540.073866 31 30 7.9484 37..9 99983 02 7.940858 23 2.059142 30 31.955082 22980.999982.0.9551600 2.32 3 1 044900 29 32.963870 2223.999981..968889 2229. 031111 28 33.982233 21608.999980 02.982253 2161.017747 27 34.995198 20981 9999.995219 200 3.2004781 26 35 8.007787 20390.999977 02 8.007809 2032 1.992191 25 36.020021 198.3.9999976 0.020044 1833.979956 24 37.031919 193.999975 02.031945 19.968055 23 38.043501..999973.043527 9564713 22 39.054781 183 999972.054809 18327.945191 21 40 8.065776 17872 9.999971 02 8.065806 1.934194 20 41.076500 174.42.999969.076531 7.923469 19 42.086965 1703.999968.0.086997 174.913003 1 8 43.097183 166.39.999966.03.097217 166.42.902783 17 44.107167 295.9999964 03.107203 16268 892797 16 45.116926 1598 999963.3.116963 152..883037 15 46.126471 15566.999961 03.126510 155..873490 14 47.135810 152-38 0.9135851 1524.864149 13 48.144953 149-.999908 03.144996 127..855004 12 49.153907 14622.9999956 03.153952 14625.846048 1 50 8.162681 3.999954 8.162727 13 1.837273 10 51.171280 140.54.999952 03.171328 140.57.828672 9 52.179713 13786.999950 03.179703 130.820237 8 53.187985 1329.999948 0.188036 13.811964 7 5.196102 13280.999946 03.196156 124.803844 6 55.204070 13041 999944.204125 13.44.795874 5 56.21958 12810.999942 03.211953 5 1.788047 4 57.219587 125..9999940..219641 1251.780359 3 58.9227134 2.72.999938 0.227195.772805 2 216.. 02 216.1072 59.234557.999936.234621 12.765379 1 o161. 02.249 9.7945191 21 60 241855 9.9993497 1 8.065806.758079 M. Cosine. D.. Sine... Cotang..0. 174.44 Tang. M. 45 116926 0.999963.1169639.830 1....,......, COSINES, TANGENTS, AND COTANGENTS. 175 1~]LO~~~~~~~~~ ~178S i _o ___-___- ___ _____ A. Sine, D. 1". Cosine. D 1'l. Tang. D. 1". Cotang. M.! 8.241855 9.999931 8.241921 1967 1.758079 60.249033 119.6.999932.4.249102 11.7.750898 59 2.256094 115.8.999929 04.256165 1.-.743835 58 3.263042 113..999927.4.263115 1144.736885 57 *.269381 12.21'.999925.0.269956 1122.730044 56 11221.999923 11.23 55 5.276614 11050.999922.04.276691 11.723309 55 6.283243 83.999920 04.23323..716677.54.289773 107 2.999918 04.289856'.710144 53 8.296207 1072.999915.296292 0.703708 52 9.32.546.999913.302634 10'.697366 51 104.13 04 104.18:)' 8.303794 102 9.999910 04 8.303884 100 1.691116 50 11.314954 01.22.999907 04.313046 1026.684954 49 1.2.321027 9982.999905 04.321122 99..678878 48.3.327016 98..999902.05.327114 9851.672886 47 14.332924 97.999899 05.333025 9.666975 46!5.333753 97.4.999397 05.338856 9.9.661144 45 A..344504 946.999394.5.344610 5..655390 44 17.350181.999891.350289.649711 43 8) 93.38 05: 93.43 18.355783 93 3.999888 05.355895 94.644105 42 L I.361315.999885'.361430.638570 41 91.03.05 91.08 20 8.366777 89 0 9.999382 05 8.366995. 1.633105 40 21.372171 8880.999379.5.372292 89.5.627708 39 z2.377499 8772.999876.05.377622 8.78.622378 38 23.382762 667.999373.382889 867.617111 37 24.387962 8..999370 05.338092 8.7.611908 36.i.393101 8.64.999867. a.393234 85.7.606766 35 26.398179 8.64.999864 05.398315 4.69.601685 34 27 403199 82.1.999861 05.403338 82.76.596662 33 i.403161 8177.999858 O^.408304 812.591696 32:413068 1.77 O.3 81.82 29.413063 80.999854.413213.586787 31'05 80'91 30 S.417919 796 9.999851 06 8.418068 2 1.581932 30 31.422717 7909.999348 06.422869 7914.577131 29 32.427462 78.2.999844.6.427618 78..572332 28 33.432156 7740.999841 06.432315 7 4.567635 27.34.436800 765.999838 06.436962 763.563038 26 35.441394 7.7.999834 0.441560 75..558440 25 36.445941 74..999831..446110 75..553890 24 y.450440 74:2.999827.6.450613 742'.549387 23.8 45493 73.4.999824 06.455070 7'.544930 22 39.459301 72:73.999820 06.459481 7279.540519 21 40 8.463665 720 9.999816 06 8.463849 726 1.536151 20 41.467935 71.2.999813 06.468172 71.5.531828 19 42.472263 7..999909.472454.527546 18 ~13.476498 699.999805.0.476693 6..523307 17 44.480693 6924.999301 06.480892 9.'.519108 16 45.484848 689.999797..485050 69'.514950 15 t6.488963 67'.999794..489170.510830 14 47.493040.999790.4932.50..506750 13 67.31- 73 506750 13 48.497078 6.999786 07.497293.502707 12 I!9. 501080 16608.999782 07 5.5 3 498702 11 50 8.505045 6 48 9.999778 07 8.505267 655 1.494733 10 51.508974 999774.509200 9.490800 9.5.512867.,.999769.513098'.486902 8 53.516726 6375.999765.516961 6'2.483039 7 54.520551 63..999761 07.520790 63 2.479210 6 55.524343 6265.999757 07.524586 6272.475414 5 56.528102 62.999753 07.528349 6218.471651 4 57.531823 61 3.999748 07.532080 616.467920 3 81.535523 6106.999744 07.535779 6113.464221 2 59.539186 6055.999740 07.539447 062.460553 1 60.542319 _.99973.5.543084..456916 0.11. Cosine. D. 1'. Sine. -D. 1". Cotang. I D. 1". Tang. M. 910 880 176 TABLE XIII. LOGARITHMIC SINES, 177C M. Sine. D. 1". Cosine. D. 1". Tang. D 1. Cotang. 3I. 0 8.542819 9.999735 8.543084 1.456916 60 1.546422 60.04.999731 7.546691 6.62 453309 59 2 549995 9-6.999726 08.550268 5914.449732 58.553539..9 991)722 08.553817 r.6.446183 57 68 58' O3 ( 0 8'r. o8.66 4.557054 58.5.999717.557336 8.442664 56.560540.1.999713 08.569328 5773.439172 55 6.563999 57.o.999708 08.564291 57.435709 54 7.567431 57.9 99704..567727 56,.432273 53 8.57(0836.999699.571137 5.428863 52 9.574214 55.7.999694 8.574520 5.425480 51 10 8.577566 9.999639 8 8.577877 5 1.422123 50 11.580392 55..999685.'.581208'5..418792 49 12.584193 55..999680 08.584514 546..415486 48 13.587469 54..999675 8.57795.4122(15 47 54919 08'r, 54.27 14.591)721 5..999670 08.591(051 387.408949 46 15.593948 539.999665.08.594283..405717 45 16.597152 53.0.999660 08.597492 5308.402508 44 17.610332 52.6.999655 0.600677 52.0.399323 43 18.6034S9 52.1.999650 08.603839 523.396161 42 19:606623 52.23 ~.08 52.32 19.606623 518.999645 09.606978 5194.393022 41 51.86.09.606978.4 20 8.609734 9.999640 n9 8.610094 51 8 1.389906 40 21.612823 51.4.999635.613189 12.386311 39 22.615891 5..999629.616262 5 -..383738 38 00.77 c.09. 50.85 23.618937 50.7.999624 09.619313 5, r 380687 37 21.621962 50.01.999619 0.622343'0. 377657 36 50.06. 09 50.15 25.624965.999614.6253529 374648 35 49 72... 9 49.81 26.627948 4..999608..628340..371660 34 27.630911 49..999603 0.631308 49.36692 33 49.0O1 o 9 49 13' 28.633854.999597 09.634256 4880.365744 32 48 71' r.09 48.80 29.636776 4839.99592.637184 4.362816 31 48.39 09 48.48 30 8.639630 9.999586 0 8.640093 486 1.359907 30 48.06 0(9 48.16 31.642563 4.).999581 -9.6429,2 4784.357018 29 32.645428 47..999575.0.645853 473.354147 28 33.648274 4.999570 09.648704'-.351296 27 34.651102 4.2.999564 09.651537 4691.348463 26 35.653911 6.-.999558.'-.6:352 46..345648 25 36.656702 6.2.999553.657149 4631.342851 24 37.659475.-9.999547.0.659928 4602.340072 23 33.662230 5..999541.1.662689.-.337311 22 39.664968 45.999535.665433.334567 21.66 45:35.10 45.45 40 8.667639 07 9.999529 8.663160 45 1.331840 20 41.670393 45..999524.670870 44 8.329130 19 42.673080 44.7.999518.673563 441.326437 18 43.675751 4451 999512.10 676239 44'6 -323761 17 44.24.10 44.34 44.678405 43 9.999506'.678.67890321100 16 43.97.10 44.07 45.631043 437.999500.0.61544.318456 15 46.63366.5 4344.999493 10.684172 43'.315828 14 47.6'6272 43.999487.686784 8.313216 13 43 1 8 1 10' 3 43.28 48.638863..999481.639381 4.310619 12 49.691438.999.4691963 4.308037 11 50 8.693998 422 9.999469 8.694529 42o2 1.305471 10 51.696543 42-7.999463 10.697081 42.2.302919 9 52.699073.1.9994,56 I..699617 42 0.310383 8 53.701589 41..999450.11 702139 4.7.297861 7 54.704090 44.999443 1 704646 1.295354 6 41.44.11 70 40 55.706577 42.999437.707140 4132.292860 5 56.709049 4097.999431.1.709618 4108.290382 4 57.711.7 4.999424 2.712083 -.287917 3 58.713952 4074.999418.714534 402.285466 2 40.51.11 40.62 59.716383 429.999411 11.716972.283028 1 40.29:it 40.40 60.718800.999404.:.719396 o'.280604 0 M. Cosine. D. 1". Sine. -D. 1". Cotng. D. 1'. Tang. M. @9~3 rij3 COSINES, TANGENTS, AND COTANGENTS. 177 33 _____ 1760 _ M. Sine. D 1". Cosine. - D. 1". Tang. D. 1". Cotang. M. 0 8.718800 0 9.999404 8.719396 17 1.280604 60 1.721204 39..999398..721806 395.278194 59 2.723595 392.999391..724204 39..275796 58 3.725972 34.999334.726588 392.27:3412 57 4.728337 391.999378 II.723959 39.31.271041 56 5.730688 3898.999371.731317 39.26368:3 ) 6.7330(27 3377.999364.'.733663 3 89.266337 54 7.735354.999357 1.735996 33 6.264004 53 38.357.115 78 67 8.737667 3.999350 12.738317 334.261633 52 9.739969 38.999343 2.740626 3827.259374 51 38.16.12.I Io 10 8.742259 9 9.999336 12 8.742922 38 07 1.257078 50 11.744536 37.6.999329 12.745207 378.254793 49 12.746302 37'6.999322 1.747479'68.252521 48 13.749055.999315 12.749740 3..250260 47 14.751297 37.999308 12.751989 3729.248011 46 15.753528 36..999301 2.754227 371.245773 45 16.755747 6.8.999294 12.756453 6'92.243547 44 17.757955 3.999237 1.758668 3673.241332 43 18.760151 36.2.999279 12.760872,.239123 42 19.762337 364.999272:2.763065 36.236935 41 20 8.764511 9.999265 2 8.76.246 68 1.234754 40 21.766675 3588.999257 12.767417 36..2.32583 39 2635..777333 999220.778114 35.22186 34 27.779434 31.81 999212..780222 3197.219778 33 28.73 37.74999205 13.782320 3480.2176 0 32 29.783605 36.1.999197 1.784408 346.215592 31 30 8.785675 3 9.999189 13 8.786486 1.213514 30 31.787736 3418.999181 13.788554 34.1.211446 29 3235.1.78999174 13.790613 4.209337 23 331420.13 34.15 33.79182 33.8.999166 13.792662..207333 27 34.79359 33.70.999158 3.791701 3383.205299 26 35.795891..999150.13.796731.33.203269 25 36.797894 3.999142.793752.201248 24 37.7991397 3..999134,.800763 33. 199237 23 38.801892 33..999126 13.802765 3322.197235 22 39.803376.999118 1.804758 337.195242 21.93 101 I.3 33.07 40 8.805852 3278 9.999110 14 8.806742 3292 1.193253 20 41.897819 32-6.999102 14.803717 277.191283 19 42.809777 4.3999094 14.810683 3262.189317 18 43.811726 32.49.999036.812641.32.187359 17 44.813667 32.34.999077 14.814589 32.48.185411 16 45.815599 32.05.999069.14.816529 32.19.183471 15 46.817522.9996.818461.181539 14 47.819436 31.9.999053 1 82034 39.179616 13 48.82113.9.9044.822298 31.177702 12 31.63.14 31.63 49.823240 3.99 14.824205 3163.175795 11 50 8.82513') 9.9907 8.826103 1.173397 10 51.827011 31..9990191 14.827992 31.0.172003 9 52.82S9 31.22.14 31.36 52.828834 31.03.999?10 14 829874 313.170126 8 53.830749.9912 4.831748 31.16252 7'~a~~/ 32.~,!.13 31.09 54.832687 3. 2.993993 14.833613 9.166337 6 3L.34456 1 4 31.92 55.8.34456.99.993 14.I83-471 39.164529 5 56.836297 3.6.993976 l4.837321 3.8-3.162679 4 30 56.63.15 30.70 57.833130 3043.999967.839163 7.16937 3 58.839956 39303 999.8 3.9998.159002 2 38.30 I99,95 / 841 59.841774 30-.998950 5.842825 30.157175 1 93. 86 51, 279,., 120 9 /iP 178 TABLE XIII. LOGARITHMIC SINEs, t0 1755 M. Sine. D. 1". Cosine. D.1". Tang. D. 1". Cotang. IM. 0 8.843585.99 94l 8.844644 0 1.155356 60 1.84.37 29.92.989932.1~. 384.64055.15 5 59.2.847183 29.998923.45260 2.15 574 5.,, 7 29.940 5 7 -3 871 29.68.998914.5.850037 29.149943 4.871 29.9989015.851846.141 54 56 5.8 52-5.998896.853628 27.146372 6.85429.1 -29.998887 5.855403 29.8 144597 5 29.31.1' 29.46 7.856049 2919 998878.1.857171 29. 142829 53 8.857801 29.998869.858932'.141068 52 29 05.998.0'15 29.23 9.859546 2896.998860.860686 1.139314 51 28.96 998 17 29.11 10 8.861293 2884 9.998851 8.862433 29,0 1.137567 5( 11.863014'73.998841 1.864173',.135827 49 la S..15 28P.3~~73.,o 12.86473 -61.998832.865906 2..1 34094 48 ~- 8. -15 2A 77 13.866455 998823 16.867632 2.7.132.368 47 14.863165 2-.998813 16.869351 23.6.130649 46 ~2.39.16 28.50 15.8696 28.998804 16.871064 28.3 128936 16.87156) 2' 1.998795 16.872770 283 127230 44 17.873255- 2..998785 16.874469' 22.125531 4 18.87493 279.998776 6.876162 281.123838 12 27.95.16 2,. 11 19.87661[ 27:84.998766 16.877849 280.122151 41 20 8.878285 273 9.998757 16 8.8 529 27 1.120471 4') 21.879949 2763.998747 16.881202 27.9 118798 22.831607 2752.998738 16.833269 27.8.117131 23.883258 2742.998728..884330 27..115470 37 16'.~'~ 27.55 36 24.834903 273.998718 16.886185.113815 25.886542 27'1.998708 16.887833 27.7.112167 26.838174 2711.99-699 16.889476 7.7.110524 34 27.839801.998689..891112 27..10888 3 2700 16 27.l7 73 23.891421 2690.998679 16.892742 2707.107258 32 29.893335 26:80.998669 17.894366 267.105634 31 30 8.894643 26 9.9986959 17 8.895984 267 1.104016 30 31.896246 266.998649 17.897596 67.102404 29 32.897842 26..998639 17.899203 2667.100797 2S 33.899432 26.1.998629 17.900803 265.099197 27 34.901017 2631.998619 17.90239 268.097602 26 35.93296 262.99S609.1.903987 26..096013 25:36.901169 26 12.998599.1.905570 26. 94430 24 3.7 t17.9 6.29 04 37.905736 26.998589 17.907147 6 20.092F53 23 3.907297 2093.998578 17.908719 26.091291 22 39.903853 2.998563 17.910285 2i01.089715 21 40 8.910404 25.75 9.998558 17 8.911846 2592 1.098154 20 41.911949 2566.998.548.17.913401 25..086599 19 42.91348 2.998537 17.914951 2-3.085049 18 43.915022 25.56.998527. 1.916495 25.74.083505 17 41 916550 25.47 998516..918034 25.6.081966 16 45 918073 25.2.998506 8.919568 25..080432 15 Zi.919-)91 25.21.998495 18.921096 25.38.078904 14 47.921103 2512.998485.1.922619 25.29.077381 13 24.94.18 25.12 48.922610 2503.998474 18.924136 25.2.075864 12 49.921112.9 98464:18 4.925619 252.074351 11 50 8.925609 2486 9.998453 1 8.927156 2504 1.072844 10 51.927109 2'77.993442 8.928658 245.071342 9 52.923587 2.6.99431 1.930155 247.069845 8 53 9301 24.I8 53.9300263. 998421.931647 24.8.063353 7 51.93144 240.998410 8.933134 24.7.066866 6 55.933015 2i.998399 1.934616 24.0.065384 5 56.934431 24.998388.936093.063907 2433 18 24.53 06243 57.935 12 24.27.98377 8.937565 21 0624 3.18 24.45 58.937393 2.998366 1.939032 4.060968 2 59.933850 2 1.998355.940494 24.059506 1 60.91026.9 983104.949 52.058048 0 M. 0,0B. D.1,I Sine. D. 11". Cotang. D. Tang. M 940 850 COSINES, TANGENTS, AND COTANGENTS. 179 50 174o M. Sine. D. 1n. Cosine. D. 1". Tang. D. 1'. Cotang. M. 0 8.940296 24103 98.941952 2421 1.058048 6C 1.941733 2.9983333 19.94:3404 2413.056596 59 2.913174 23.998322.19.944852 240.055148 58 3.944606 2.998311 19.946235 239.053705 57 4.916034 23.7.993309 9.947734 23-9.052266 56 5.947456 1.998239.199168 -232.050832 55 6.948874 23.63 9927 19.95597.049403 54 7.950287 23.5.998266.9.952021 23-67.047979 53 8.951636.48.998255.9.953441 23..046559 52 9.953109 23.40.993243 19.954856.2'045144 51 23.32.19 23:51 10 8.954499 23 5 9.998232 9 8.956267 23 44 1.043733 50 11.955394 231.993220 19.957674 2336.042326 49 12.957284 231.998209 19.959075 2329.040925 48 13.958670 2302.993197 19.960473 2322.039527 47 14.960052 22 9.998186'19.961866 234.038134 46 15.961429 22..998174.19.963255 2307.036745 45 16.962 01 221.998163..964639 230.035361 44 19 23.00 17.964170 22..998151.966019 223.033981 43 18.965531 22..998139 20.967394 226.032606 4 2 19.96693 2259.998128.20.968766 2279.031234 41 20 8.968249 222 9.998116 20 8.970133 2.72 1.029867 40 21.969600 22.4.993104 20.971496 2265.02504 39 22.970947 223.993092.20.97255 2'-5.027145 38 23.972239..998080.974209 225.025791 37 24.97 3.9 8.9736275560 22..024440 36 25.974962 22.2.998056.2.976906 22I-.023094 35 26.976293 227.998044 0.978248.021752 34 27.977619 22. 0.998032 20.979586 22'.020414 3.3 23.978941 2.9.993820.2.980921 2217.019079 32 29.980259 2190.998008.20.982251 22:10.017749 31 30 8.981.573 283 9.997996 20 8.983577 2204 1.016423 30 31.932383 277.997984.2.984899 2197.015101 29 32.984189 2170.997972 20.986217 2191.013783 28 33.935491 2.997959.987532'24.012468 27 34.936789 21- 997947.9'8842 21'78.1158 26 35.938033 2 5.997935.2.990149 21 7.90851 25 36.989374 21.997922.991451 2 1.008549 24 37.990660 21.4.997910 21.992750 21 5.007250 23 3 9.38.21 21.5988 39.991943 211.997897.21.994045 2152.005955 22 39.993222 2125.937835 21.995337 2146.004663 21 40 8.991497 2119 9.997872 2, 8.996624 2140 1.003376 20 41.995763 21..997860 21.997908 21.002092 19 42.997036 21.0.997847.2.999188 21.7.000812 18 43.993299 210. 9973:15 21 9.000465 2121 0.999535 17 44.999560 20.9.997822 21.001738 2115.99262 16 45 9.03316 2088.997809.1.003007 210.996993 15 46.032069 2.997797.21.004272 21.03.995728 14 47.003313 207.9'774 21.005534 2097.994466 13 43.004563 20.7.997771 21.006792 201.993208 12 497.0058021 2 20 91 2 49.005 305' -!.99775s 2.008047 20:5.991953 11 50 9.007041t 20. 9.99774.5 22 9.009298 20.8 0.990702 10 51.003273 2 52.997732..010546 2074.99454 9 52.009510 2 4.997719.22.011790 988210 8 20.46 011790 21)63 988210 8 53.010737 2040.997706.22.013031 202.986969 7 54.011962 20.3.997693.2.014268.56.985732 6 55.013182 0.9763 0155..984498 5 56.014409 202.937667.2.016732 2045.8268 4 20.23'.22 20.45 57.015613 20.1.997654 22.017959 20.982041 3 53.016324.997641.2.019183.3.980817 2 59.018031 2002.997623.020403 20.979597 1 60.019235.997614..021620.978380 0. Oosin. | D. 11. Sine. D. 1". Cotang. D. 1". Tang. M. 950 84 180 TABLE XIII. LOGARITHMIC SINES, 60 173a M. Sine. D. 1". Cosine. D. 11. Tang. D. 1". Cotang. M. 0 9.019235. 9.997614 9.021620 202 0.978380 60 1.020435 19.95.997601 22.022834 20.9 7166 59 2.(21632 19.89.997588.22.024044 2.12.975956 58 3.022825 19.84.997574 22.025251 2) ).974749 57 4.024016 1978.997561 22.026455 2001.735'15 56 5.025203'19.7.997547 22.027655 1 9'.972345 55 6.026386 19.7.997534.'.028852 19'0.971148 54 7.027567 19.6.997520.23.030046 19..~69954 53 8.028744 19.6.997507.031237 19879.968763 52 9.029918 1951.997493 23.032425 1974.967575 51 10 9.0310.9 1 9.997480 o23 9.033609 19' f9 0.966391 50 11.032257 19.4.997466 23.034791 1964.965209 49 12.033421 1936.997452 23.685969 1958.964031 48 13.034582 -3.997439 23.337144 1953.962856 47 14.035741 19'25.997425.23.038316 19'48.9616P4 46 15.036896 1920.997411 23.039485 19 43.960515 45 16.038048 19.15.997397'23.040651 1938.959349 44 17.039197 19.10.997383.23.041813 19.3.958187 43 18.040342 1905.997369.2.042973 19'28.957027 42 19.041485 19 00.997355 23.044130 1923.955870 41 20 9.042625 1895 9.997341 23 9.045284 1918 0.954716 40 21.043762 18.90.997327 23.046434 19'3.953566 39 22.044895 886.997313 24..047582'08.952418 38 23.046026 18..997299.048727 1903.951273 37 24.047154 18..997285.24.049869 18.98.950131 16 25.048279 1'.997271.051008.948992 35 26.049400 18.997257 24.052144 18.93.947856 34 27.0150519 86.997242.24.053277 18.89.946723 33 28.051635 18.-'.997228 24.054407 18.945593 32 29.052749 180.997214 24.055535 1874.944465 3 30 9.053359 1846 9.997199 9.056659 1870 0.943341 30 31.054966 18.4.997185.24.057781 1865.942219 29 32.056071 18.36.997170.24.058900 18.60.941100 28 33.057172 1831.997156.24.060016 1856.939984 27 31.053 1.997141.24.061130 1851.938870 26 35.059367 18.997127.24.062240'.937760 25 i /.;'18.22 2.99727 /25 36.06040 18.17.997112 24.063348'42.936652 24 37.061551 18.13.997098.24.064453 i 8.37.935547 23 ~33 3 9.62639.997083.065556.934444 22 39.06.3724'8.997068 225.066655 182.933345 21 40 9.0649 -6 1 9.997053 9.067752 1824 0.932248 20 41.065885 17.99 997039.068846 18.931154 19 42.066962 17.95.997024.25.069938.1.930062 18 43.06036 1786.997009.2.071027 11.928973' 17 44.069107 17.86.996994.072113'. 92 887 16 45.070176 177.8.996979.073197 186.926803 15 46.071242 1772.996964.2.074278 1797.925722 14 47.072306 177.996949.075356 17'93.924644 13 48.073.366 1.996934.25.076432 1'78.923568 12 49.074424 7:59.996919 2.077505 1784.922495 11 50 9.07.5480 9.996904 2 9.078576 1 0.921424 10 51.076533 17.5.996889 25.079644 17.80.920356 9.995274 25.070710 1772.919290 8 53.078631 17.42.996858.25.081773 1767.918227 7 54.079676 /. 9968.43.2.082833 1763.917167 6 55.080719 17.34.996828.26.08389 1759.916109 5 56.081759 17.29.996812.26.084947 1755.915053 4 57.082797 17*2.996797.26.086000 7' 1.914000 3 3.083832 172 087050..912950 2 59.084864 17..996766.26.088098 1..911902 60 85894.085894.996751.26.089144 1~4.10S65 0 M. Cosine. D. 1". SIne. D. 1". Cotang D. 1". Tang. M 960 83" COSINES, TANGENTS, AND COTANGENTS. 181 70:_____ 17_ -' M. Sine. D. 1. Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.85394 1 1 9.996751 6 9.089144 0.910356 60 1.036922 17; 3.996735 *26.090187 17.39.90913 59 2.037947 1. i.996720 *26.091228 131.908772 58 3.088970 17 0.996704 26.092266 1727.907734 57 4.039990 16.9.99633'26.093302 17..906698 56 5.091003 1- 9.996673 26.094336 1'.905661 55 6.092024 1683.996657 *26.095367 171'.904633 54 7.093037 68.996641 26.096395 1711.903605 53 8.094047 1.996625 26.097422 17'.902578 52 9.095056 1660.996610 26.098446 17:03.901554 51 10 9.096)62,.3 9.996594.7 9.099468 16.99 0.900532 50 11.097065 I.99678 27.100487 169.899513 49 12.093066 16.6.996562 27 -101504 6.9.893496 48 13.099365 11.996546 27.102519 16..897481 47 14.100362 16 57.996530 27.103532 16.8.896463 46 15.101056 1'.3.996514 27.104542 6.80.895458 45 16.102048 16.49 96..105550 176.894450 44 17.103037 16.99t6182 27.106556 1672.893444 43 18.104025 16..996465 27.107559 166.892441 42 19.105010 163.99649 27. 15996449 1. 891440 41 20 9.105992 1631 9.996133 27 9.109559 161 0.890441 40 21.106973 16.30.996117 27.110556 16.5.889444 39 22.107951 6 27.996400 27.111551 1654.888449 38 2.3.103927 1623.996334 27.112543 16'.887457 37 24.109901 6..996363 27.113533 647.886467 36 25.110373 16..996351 7.114521 14.835479 35 16.16 199435.27 16.43 9 26.111842 2 1 996335 28.115507 16.3.884493 34 27.112309 1603.9963128.1 16491 1636.883509 33 28.113774 16 0.9963'2 2.117472 1-3.882523 32 29.114737 16:01.9962i5 28.118452 16 29.881548 31 30 9.115693 1598 9.996269 8 9.119429,- 0.880571 30 31.116656 15:94.996252.2'.120404 16.2.879596 29 32.117613 1'.9.996235 2.12377 1..878623 23 33.118567 15..996219.2.122.348 16..87 7652 27 31.119519 15.8.996202.2.123317 6.11.87663 26 3.5.120469 15..996185..124284 6..875716. 25 36.121417 15.7.996163.2.125249 6.08.874751 24 37.122362 1;5.996151 28.126211 16.0.873789 23 33.123306 1 5'7.996134 28.127172 16.0.872328 22 39.124242 156.996117 8.128130 15.9.871870 21 15.66.28 15.94 40 9.125187 1!6 9.996100 28 9.129087 191 0.870913 20 41.126125 a.996083 2.130041 5.869959 19 42.127060.996066 28.130994 15.8.869036 13 43.127993 1.996019.131944.863056 17 44.12392.5..996032 2.132893 15.8.867107 16 i 45.1293.5 1545.996015 29.133339 1.866161 1 46.130781 15'.2 995998 29.134784 5.71.865216 14 47.131706 15'33.995980..135726 15.7.861274 13 4-.132630 1.995963 2.136667 5'.863333 12 49.133551 13.995946 29.137605 15.6.862395 I 15.32.29 15.61 1 I 50 9.134470,- 9.995928. 9.138542 15 0.861458 10 51.135337 1526.993911 9.139476 -.8605214 9 52.136303 15.2'.995891'29.140409 15. 1.5 991 8 53.137216.9976.141310.866 7 543'i2.29 15.48 54.133123'11.995559 29.142269 15.4.857731 6 55.139037 15.1.995341 9.143196 154.856304 5 56.139914 1.995323 2'.144121.855879 4.5.09.29 15.39 57.140350 1 5.995806 9.145044 539.854956 3 58.141754 1.995788..145966 15.'.854034 2 59 142655' 995771 2.14685 15.853115 1 15. 01:.30 15.29 69.143555.995753.147803.85297 M. Cosine.. 1. I Sine. D. 1". Cotang. D. 1'. Tang. I M. O97 82 182 TABLE XIII. LOGARITHMIC SINES, 80 1710 M. Sine. D. 1". Cosine. D. I Tang. D. I". Cotang. M. 0 9.143555 14 9995753 0 9.147803 1526 0.852197 60 1.144453 143.995735.148718 1523.851282 59 2.145349 14[9;.995717..149632 120.85036S 58 ~,~'A', [.193 4 I 6v 57 3.146243' 487.995699.3.150544 15.8496 57 4.147136 14.8.995681 30.151454 5.14.848546 56 5.148026 148.995664 30.152.363 151.847637 55 6.148915 1 7.995646.153269 108.846731 54 7.149802 1475.995628 30.164174 105.845826 53 14.99 84215.02 51 8.1586S6 1472.995610.3.156077 1 02.844923 52 9.151569 146.995591.155978.44022 51 14.69.30 14.99 10 9.15451 1466 9 9.557 156877 14.96 0.843123 50 1.153330.63 995.99157775 8422 49 14.6' I 14.93 81 12.15420 14.995537.158671 14 841329 48 1315508 14 90 15.083 13.15508.3 14.5.995519..159565 8.840435 47 14.155957.995501.160457 148.839543 46 15.*15630 14'r1.995482..161347 1481.838653 45 16.157700 1.995464 -.3.162236.837764 44 14.48.31 14.78 837764 44 17.158569 144.995446.1.163123 1475.836877 43 18.159435 14.42.995427.31.164008 1473.835992 42 19.160301 4 439.995409 31.164892 1470.835108 41 14.39 20 9.161164 1, 9.995390 9.165774 0.834226 40 21.162025 14.3.995372 31.166654 14.6.833346 39 22.162885 1430.995353 -3.167532 1464.8.32468 38 23.163743 1427.995334 31.168409 14'5.831591 37 24.161600 142.995316' 1.169284 1456.830716 36 25.165454 14.2.995297 31.170157 145.829843 35 14.22.31 14.53 26.16630(7 141.995278 31.171029 1.5.828971 34 27.16f159.995260.171899.828101 33 28 16S008 113.995241.172767 14 827233 32 14.10:31 14.42 31 29.163856 14.995222.173634 4.826366 31 14.10.31 14.42... 30 9.169702 7 9.995203, 9.174499 9 0.825501 30 14.07.31 14.39 31.170547 14.0.995184 32.175362 1436.824638 29 32.171389 140.995165..176224 4'.823776 28 33.172230, 14.995146 32.177084 14.3.822916 27 13.99.32 14.31.822058 26 31.173070 136.995127.177942.822058 26 35.173903 1394.995108'2.178799 1425.821201 25 36.174744 11.995089 3.179655 1423.820345 24 37.175578 13.8.995070.3.180508 14-20.819492 23 33.176411 185.995051 32.181360 117.818640 22 39.177242 13.83.995032 32.182211 14:15.817789 21 40 9.178072 380 9.995013 3 9.183059 1412 0.816941 20 41.178900 17.994993.183907 1409.816093 19 42.179726 17.994974..184752 0.815248 18 43.180551 13.75.994955 32.18597 14-4.814403 17 13.72.32 14.04 44'.181374 13. 99.994935.186439 1402.813561 16 45.182196 1367.994916 332.187280 1399.812720 15 46.183016 1364.994896 -.188120 397.811880 14 47.183834 1361.994877.188958 1394.811042 13 48.184651 3'.994857.3.189794 1 3.810206 12 49.185466 13:56.994838 33.190629 13:89.809371 11 50 9.186280 14 9.994818 9.191462 386 0.808538 10 51.187092 15.994798.192294 1.807706 9 52.187903 1348.994779.33.193124 138.06876 8 53.188712 134.994759 33.193953 13'7.806047 7 54.189519 13.994739..194780 1376.805220 6 55.190325 3.43.994720 33.195606 1374.804394 5 13.41.33 13.74 56.191130 1338 994700.196430 371.803570 57.191933 1336.994680 33.197253 1369.802747 3 58.192734 133.994660..198074 1366.801926 2 i- 812'59.193534 1331.994640 33.198894 1364.801106 1 60.194332 994620 199713.800287 0 sic COSINES, TANGENTS, AND COTANGENTS. 183 fs~~~c ________________~~~~~170C X Sine. D. 1". Cosine. D. I". Tang. D. 1". Cotang. M. 0 9.14332 13.2 9.994620.33 9.199713 1362 0.800287 60 1.195129 13.991600.200529 13.799471 59 2.195925 132..994.80.201345 13.798635 58 3.196719 132.994560 4 202159 1354 797841 57 4.197511 131.994540 3.202971 13 -.797029 56 5.198302 1316.994519.3.203782 13.4.796218 55 6.199091 1313.994499.204592 1347.795403 54 7.199379.991479 34.205400 13-4.794600 53 8.200666 139.994459 3.206207 1342.793793 52 5 0 9.3 13.2 1 9.201451..994433 3.207013 130.792987 51 10 9.202231 9994418 9.207817 0.792183 50 13.9.202 13.38 11.203017 131.994398.3.203619 133.791381 49'12.203797 1I9.994377'3.209420 1333.790580 48 13.204577 12. 99437.210220 1331.789780 47 12.9.994336 3 138 14.205354 12.91 994336. 3.211018 13.788982 346I 15.206131 2. 994316 - 3.211815 13..788185 45.2613 1... 112.9 16.23606 12..994295 3..212611 1324.787389 44 17.27679 12.87.994274:34.213405 1321.786595 43 20 9.203992 12 9.994912.3 9.215780 15 0.784220 40 21.210760 12..994191..216568.783432 39 22.211526.994171.217356'.782644 38 23.212291 12:75.994150.218142.781858 37 12.73.35 13.08 24.213055 12.7 994129.. 218926 13.0.781074 36 25.213818 12..994103 *3.219710 13.07.780290 35 26.21459 12.994037 23.220492 1.77950 34 27.215338 12. 994066 3.221272 12..778723 33 23.216097 12..994045.222052 12.777943 32 29.216354 12..994024.3.22233.777170 31 24.5 216354 12.59.35. 12.95 30 9.217609 1257 9.994003.3 9.223607 12.92 0.776393 30 12.5 3 122 7618 31.218363 2.993932 7.224332 120.775618 29 32.219116 1253.993960.3.225156 128.774844 28 33.219339 1 2..993939.3.2225929 -.774071 27 34.220618 12..993918.226700.773300 12.48. 1284 35.221367.997.227471.772529 2.5 12.46.36 12.82 36.222115 12.4 99:1375.36.223239.771761 24 37.222361 12. 933354.3.229007 12.9.770993 23 33.223606 12.9.993332.36.229773 12.7.770227 22 39.221349 12.3.993311.230539 12-.769461 21 39.224349 12.37.36 12.73 40 9.223.5092 13 9.993789 3 9.231302 0.763698 20 41.225333.0.993763.232065.767935 19. 99' 2 12.33 1269 42.226573 1231.993746 36.232826 1267.767174 18 43.227311 1229.993725 36.2.33586!26.766414 17 44.223013 1.993703.234345.76.5655 16 45.223784 124.993631 36.235103 12.69.764397 15 46.229518 1222.993660 3.235359 12..764141 14 47.230252 {.993633 36.236614 12.5.763336 13 48.2.30931.993616' 1.253.27626329 12i 12.18.36 3 12.54 49.231715 12.16 1.36 12.52'.761880 11I 50 9.232444 1214 9.993572 9.238872 1250 0.761123 10 51.2.33172 1212.993550 3.239622 1248.760378 9 12.10.99 32..24037 52.2.33399.993528.240371 124.759629 8 53.231625 121.37 12.46 277 7 53.231625 1207.993506.241118 1244.758332 51.23 12.993184.241865 1242.757135 6 155.236073 -.0993162.242610 1.757390 5 56.236795.993440.243354 1233.756646 4 57.237515 11.99.993418.244097.755903 3 53.23326 11.97.993396.244839 1234.755161 2 59.233953 5.993374.245579.754421 1 63.2396-).993351'3.216319 /.753681 0 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1". Tang. 3M..222~~~~~~~~~~~~~~~~~115 184 TABLE XIII. LOGARITHMIC SINES, 103 1 69; M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M 0 9. 239670 1. 93 9.99 1 9.12439 130 0.753681 60 I1.24036 11.993329.7.217057 1 20 1 9 1.37 12.29.75294: 2 241101 1.9 99337)7.247794 226.7522()6 5 3.241814.8I.993284 7.24S53(1 1224.751470 57 4.242526 ]1.8.993262..249z64 2.22.750736 56 5.243237 11.83.993240'.249998 12.20.750002 55 6.243947 11..993217.250730.749270 54 7 244606 1:81 38 1218 i 7.244636 11.993195 7.251461 I..748539 53 1~.19'38 II 12.17' 8.245363 17.993172.252191 12.747809 52 9.246069 1.993149.252920 1213.747080 51 1011.75.38 12.1375 10 9.246775 11.7 9.993127 9.253648 2 0.746352 50!11.247478 11.993104.254374 120.745626 49 12.24911 11 69.993081.255100 107.744900 48 13.248883 116..993059..255824 12.744176 47.111.67.993036.38 1]'2.05 46 14.249583 11.6.993036.256547 123 7434 46 15.250282 11.6.993013..257269 1 1.742731 45 16.250980 1.992990.257990 1.742010 44 17.251677 11:9.992967.258710 1.741290 43 18.252373 1.5.992944.3.259429 198.740571 42 11.58 G.38 11.96 [ 19.253067 116.992921 38.260146.739854 41 20 9.253761 1. 9.992898 3 9.260863. 1192 0.739137 40 21.254453 11.5.992475..261578 1190.738422 39 22.255144 1..992852.262292 1189;737708 38 23.255834..992829. -.263005 7.736995 37 24.256523 48.992806..263717 *8.736283 36 2.25721 I'11. 4.992783.3.264428 11. 5.735572 35 26.25789 11.4.992759.3.265138 1.83.734862 34 27.258583.4.992736;.265847 119.734153 33 11.41.39 11.79 28.25926S.'39.992713 39.266555 11 7.733445 32 29.259951 11:37.992690 39.267261 11:76.732739 31 30 9. 26063.3 9.992666 9.267967 0.732033 30 31.261314 11.3 992643.268671 1. 731329 29 32.261994 113.992619.269375 73062 2 33 262673. 992596.3.270077 11.729923 27 3.26.3351 I.3'.992572 3.270779 1167.72921 26 35'.264027D 727822 35.264027 11.2.992 4.271479 116.728521 25 36.264703 1.2.992525..272178 1'.727822 24 37.265377,.o.992501.'.272876 1'.727124 23 11.22.39 11.62 22 38.266051 11.2.992478 273573 11.6 726427 2 39.266723 11.1.992454.0.274269 11.725731 2 40 9.267395 1117 9.992430 40 9.274964 11.57 0.72,5036 20 41.268065 11.17.992406 40.275658 1.724342 19 42.268734 11.1.992382 40.276331.53.723649 18 43.269402 11.1 992359.277043.5.7229-7 17 44.270069 11.1.992335..277734 11.50.722266 16 44 27009 11.10.9928 40 11.50' 721576 45.270735 10.992311.278424 11.4.721576 15 46.271400 11.06.992287 40.279113 11.6.720887 14 {'60.718142 10 47.272064 1.0.992263.40.279801 1.4.720199 13 48.272726 1103 992239.230488 43.719512 12 49.273388 1101.992214 40.281174 1141.718826 11 50 9.274049 9 9.992190 40 9.281858 14 0.718142 10 51.274708 10.9.992166.4.282542 138.717458 9 52.27537 10.992142 40.283225 11.3.716775 53.276025 109.992118.41.283907 11. 716093 7 54.276681 192.992093 41.284588 1.33.715412 6 10.92 i90.714~2. 55.277337 10.9.992069 A4.285268 11..714732.56.277991 109.992044..285947 11.3.714053 4 57.278645 108.992020 41.286624 1.28.713376 3 58.279297 10.8.991996.41.287301 11.26.712699 2 59.279948 10.84.991971 41.287977 11 712023 60.280599.991947.288652.711348 0 M. Cosine. D. 1". Sine. D. 11. Cotang. D. 1'. Tang. M. 5(.3 COSINES, TANGENTS, AND COTANGENTS. 185 11o 1650 M. Sine. TD. I". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.230599 102 9.991947 41 9. 8652 113 0.711348 60 1.281243 10..991922 A4.289326 1122.710674 59.281897 1O. 91 41 2 2.281897 1079.991897 41.239999 1120.710001 58 3.22544 77.991873 4.290671 18.709329 57 4.283190.991834.291342 111.708658 56 10.76. 41 11.17' 5.2336 0.991823 41.292013 1115.707987 55 6.284480.991799 41.29 2 1114.707318 54 7.285124 10.7.91774..29:3350 1112.706650 53 8.235766.991749..291.705983 52 9.236108 167.991724 42.294684 11'9.705316 51 10 9.287048 3 10 9.991699 42 9.295349 1107 0.704651 50 11.287638 10.6.991674..296 113 1',6.703987 49 12.288326 10.6.991619 42.296677 1104.703323 48 13.283964 061.991 -24'*.2973391 1'.702661 47 14.239600 1,.991599 42.298001 1101.701999 46 15.290236 o..991574 42.2986621.'.701338 45 16.290370 10..991549 42.299:'22 10'9.700678 44 17.291504 10.5.991524 42.299930 10'9.700020 43 18.29t137 1055.991498 42.300638 1097.699362 42 19.292768 53.991473.0.1295 698705 4 10.51.42 10:93 23 9.293,99 10 9 9.99144 4 9.30i51 102 0.69S049 40 21.294029.991422 42.3026)7 -.697393 39 22.294658 14.991397 42.303261.696739 38 23.295286 10.4.991372 42.303914 1089.696086 37 24.295913 10.45.991346 42.304 67.695433 36 25.296539 04.991321 4.305218 10'.694782:35 26.297164 104.991295 43.305869 108'3.694131 34 27.297788 1..991270.306519.693481 33 23.293412 10.3.991244 43.307168 10'8(.692832 32 29.299034 g.991218 4.307816 1078.692184 31 39 9.299655 104 9.991193 43 9.308463 1077 0.691537 30 31.300276 033.991167.309109 107.690891 29 10.3 3 10.76 32.300395 131.991141 43.309754 0.690246 28 33.331514 10-30.991115 43.310399 1073.689601 27 34.302132 210..991090 43.311042 10.71.688958 26 35.332748 10.2.991068 3.311685 10.70.638315 25 36.303.361 10.2.991038.312327 3 10..687673 24 37.303979 1023.991012 4.312968 10..68703-2 23 38.301593 10'2.990936 3.313608 10.65.686392 22 39.305207 020.990960.314247 0.685753 21 10.20 43 10.64 40 9.335819... 9.990934 9.314885 1 02 0.628.115 20 41.306133 101.990908 4.315523 106.684477 19 42.337041'1.9908.32.316159.683841 18 43.307650 1..990355'.316795 /.683205 17 44.3 18259 1013.829 44.317430 10.57.682570 16 47.310)80 13.9907.50 4.319330 10.680670 13 43.310635 09.990724 4.319961 1.6s0039 12 49.311239 10':.990697'4.320592 lo'.6794038 I 10.01 6 4433f. 50 9.311893 1001 9.990671 9.321222,OI 0.678778 10 51.312495 1003.99 0645.32151 lO.7.678119 9 542.313)97 O.01 990618 44.322179 ~10.46 677.521 S 4~ 13 10.01.44 10.46 53.31369 I..990591.323106 1044.6763i4 7 51.314297.3.990.565 44.3237.33 1043.67267 6 9l93 44 10543 55.314397 99.9905338 4.324353..67.;042 5 9.96 9901 45 10.40 57.316392.99085.325607 1.674393 3 54.316639.990458 a 32 231.673769 5J9.31193[ I 9.9 961 9.31245 102.5.31.31723951 ]:'.I999341 /.3I263353 1336.673147 I 65.18' 9..9 45 10.36 63.3178791 9..990434 _-.327475.672525 2 0 M. osine. D.I 1. Sine. D. 1. Cotang. D. 1". Tang. M. 1010- 78C 186 TABLE XII1, LO0ARITIMIC SINES, 120 167C M. Sine. D. 1". Cosine. D. 1". Tang. I D. I". Cotang. M. 0 9.317879 9.990404 9.327475 0.672525 60 1.318473 9.90.45 10.35 1.318473 9.990378.328095 33.671905 59 2.319066.990351.328715.671285 58 9.87.45 10.32 3.31968 96.99(1324.329334 131.67C666 57 4.320249 984.990297.329953 1029.670(47 56 5.320840 9.990270.330570 1.669430 55 6.321430 9.81.99(0243 5.331187 1027.668813 54 7 322019 9.322019 990215.331803.668197 53 8.322607.990188.332418 10 24.667582 52 9.323194.990161 45.333033 1023.666967 51 10 9.323780 76 9.990134 9.333646 10 21 0.666354 50 11 324.9901.32436634259 1020.665741 49 12.324950).990079 46.334871 1019.665129 48 13.325534 972.990052 46.335482 1017.664518 47 14.326117 970.990025 46.336093 1016.66397 46 15.3-26700 9.989997 46.336702 1015.663298 45 965`8.45 10.23 16.327281.989970 46.337311 1014.662689 44 17.327862 96.989942 46.337919 1012.662081 43 18.328442 96.989915 46.338527 1o1.661473 42 19.329021 9.989887 46.339133 1010.660867 41 20 9.329599 9 6 9.989860 9.339739 0.660261 40 9.62.46 10.08 21.330176.989832 6.340344 1007.696596 39 22.330753 9.60.989804 46.340948 1006.659052 38 23.331329 9.68.989777 46.341552 1005.658448 37 24.331903 9.5.989749 46.342155 1003.657845 36 25.332478 95.989721 46.342757 1002.657243 35 26.333051.989693 4.343358 1001.656642 34 28.334195 95.989637.344558 9.655442 32 29.334767 9:50.989610..345157 9:97.654843 31 30 9.355337 9 9.989582. 9.345755 96 0.654245 30 31.335906 948.989553 47.346353 9.653647 29 32.336475 946.989525.346949.653051 28 33.337043..989497.347545 92.652455 27 34.337610 94.989469.4.348141 991.651859 26 35.338176 9..989441 4.348735.651265 25 36.338742.989413.349329.650671 24 9.41.47 9.88 37.339307 940.989385 47.349922 9:87.650078 23 38.339871 9.989356.47.350514 986.649486 22 39.340434 9.989328.351106 985.648894 21 40 9.340996 9 9.989300. 9.351697 84 0.648303 20 41.341558.989271.352287 82.647713 19 42.342119 34.989243.352876 81.647124 18 43.342679 932.989214 48.353465 80.646535 17 44.343239 3.989186 4.354053 9.645947 16 45.343797 9.30.989157 48.354640 978.645360 15 46.344355 9.29.989128 48.355227 976.644773 14 47.344912 92.989100.355813.644187 13 47..48 9.'i 644176 1 1.1 48.345469 926.989071 *48.356398 9.643602 12 49.346024 9:25.989042 48.356982 9 7.643018 60 9.346579 9 2 9.989014 9.357566 9 72 0.642434 10 51.347134 22.988985 48.358149 970.641851 9 52.347687 9.22.988956.48.358731.69.641269 8 53.348240 9.988927 48.359313 96.640687 7 54.348792 19.988898 8.359893 67.640107 6 55.349343.988869 48.360474.6.639526 5 56.349893 9.988840 48.361053.638947 4 9.16.48 9.65.350443 915.988811 48.361632 9.638.368 3 58.350992 14.988782.362210 62.637790 3 59.351540 93.988753.362787 961.637213 3 60.32088 988724.363364.636636i 0 M. Cosine. D. 1". "Sina. D. 1. Cotang. D. I". Tang. 10o 770o COSINES, TANGENTS, AND COTANGENTS. 187 130 o 166 M. Sine. D. 1". Cosine. D. 1" Tang. D. 1". Cotang. M. 0 9.35-2:38 9 9.938724. 9.363364 9 0 0.636636 60 1.352635 9 10.983695 4.363940..636060 59 2.353181 909.933666 49.364515 9.5.63545 58 3.353726 0.9336:6 4.365090 9.58.634910 3.35649' ~.~ 56 4.354271..988607 9.3 5664..634336 56 5.3-54815 90.9i78 49.366237 9.54.633763 55 6.3.5358 904.93548 4 366810 3.633190 54 7.355901 903.933519 49.367:332.632618 53 8.356143 9-02.938489 4.367953 9.5.632047 52 9.356934 901.988460 49.363524 950.631476 51 10 9.357524 8 9.9334130 49 9.369094 49 0.630906 50 11.353064 8'9.93401 4.369663 9..630337 49 12.358603 897.938371 4'.370-232 97.629768 48 13.359141 8.96.933342'0.370799 9 4.629201 47 14.359678 8.938312.'.371.367 944.628633 46 15.360215 894.938282'.5.371933 9 3.628067 45 16.360752 892.938252.5.372499 942.627501 44 17.361237 91.993223.373064 941.626936 43 18.361822 9.988193.373629 94.626371 42 19.362356 8:9.983163 50.374193 9439.625807 41 20 9.362389 8 9.988133 I 9.374756 | 0.625244 40 21.363122 88.988103.5.375319.624681 39 22.363954 886.958073.5.375881 96.624119 38 23.364485 84.988043.50.376442 9 35.623558 37 24.365016 8.988013.377003.622997 36 25.365546 882.97987983..377563 932.622437 35 26.366075 881.937953'.378122 9 31.621878 34 27.3S6634 880.937922'..378681 9.621319 33 28.367131 89.987892.5.379239 9 2.620761 32 29.367659 878.987862 5.379797 92.620203 31 30 9.363185 876 9.937832 I1 9.336354 927 0.619646 30 31.363711 875.937801 51.380910 926.619090 29 32 369236 74.987771.331466 925.61534 23 33 369761 873.987740.5.332020 20.617980 27 31.370285 8.2.987710 5.382575 923.617425 26 35.370303 8.7.937679..383129 22.616371 25 36.371330 8.7.987649.5.383632 921.616318 24 37.371852 8 69.987618 1.384234 92.615766 23 33.3.937538 51.384786 9.615214 22 39.372894 866.987557 51.385337 9.614663 21 8.66'9875'7.51 3 9.18 40 9.373414 85 9.937526 9.335S88 917 @.614112 20 41.373933 864.987496.1.336438 916.613562 19 42.374452 863.987465.3869S7 95.613013 1. 43.374970 862.937434..387536 9.4.612464 17 44.375137 861.957403.5.383084 912.611916 16 4..376 )3.987372.5.388631 91.611369 1 46.376519 85.987341 52.389178 910.610822 14 47.3770:33 8.5.987310'5.389724 909.610276 13 4~.377519 5.937279 52.390270 08.609730 12 49.378063 8:56.987243 52.390815 9:07.6)09185 I1 50 9.37.8577 85 9.937217 2 9.391360 9 0.6')S640 10 51.379339.937186.52.391903 9.0.63097 52.379691 8.52.937155 52.392447 904.607553 8 53.339113 8.5.97124 5.3929'9 9.6J711 i 7 8 51' 52'979 9.03.51.33624 8.50.987092.52.393.,31.6)6469 6 55.331131.937061.394073.60927 5 5 ~6.3i 143 8.49'.52 9.01 | 5 6.331643 8.4.987030.52.394614..605336 4 57.332 152 847.936J98 5.3951.54 8.6)4346 3 53.332661.936967.395694.8 143)6 2 8.46 /'~; I.52 8-.93.9 333163..96936.. 362^33.7.603767 1 60.33575 8.4.9369014.396771.6!)3229 0 M. Cosine. D. 1". Sine. D. I". Coteng.. " Tang. M. 10^ ya~~~~~~~~~~~e 188 TABLE XIII. LOGARITHMIC SINES, 14~o 1650 M. Sine. D. ". Cosine. D. 1". Tang. D. 1". Cotang M. 0 9.33675 8.44 9.986904 9.396771 8.96 0.603229 60 1.334182 8.4.986873 53.397309 8'96.602691.59 2.384687 8.4.986341.5.397846 8.602.154 58 3.385192 8.41.986809 I53.398383 8.94.601617 I57 4.38.5697 8.40.986778 *53.398919 8 93.601081 56 5.336201 8..936746 93.399455 8-92.600345 55 6.386704 8.38.986714.5.399990 891.600010 54 7 387207 8.37.986683.53.400524 890.599476 53 8.387709 8.3 986651.53.401058 889.598942 52 9.388210 8:3.986619 53.401591 888.598409 51 1 0 9.333711. 9.986587. 9.402124 887 0.597876 50 11.339211 8.3.986555.53.402656 8 86.597344 49 12.389711 8.33.936523.53.403187 8.85.596813 48 1 3.390-210 8.32.936491.53.403718 8.84.596282 47 14.390708 8.31.986459.53.404249 8.83.595751 46 15 391206 8.30.986427.54.404778 8382.595222 45 16.391703 8.29.986395.54.405308 8.81.594692 44 17,392199 8.2.936.363.54.405836 8.80.594164 43 18.392695 8.27.986331 54.406364 8.79.593636 42 1 9 393191 8-.986299 5.406892 8.593108 41 20 9.393685 8o24 9.986266 54 9.407419 8 77 0.592581 40 21.394179 82-.986234'.407945'76.592055 39 22.394673 8.23.986202.54.408471 87.591529 38 23.395166 8.22.986169.54.408996 8 75.591004 37 24.395658 8.21.936137.409521 8.7.590479.36 25.396150 8.20.986104.54.410045 8.73.589955 35 26.39S641 8.1.986072 54.410569 8.72.589431 34 27.397132 8.18.986039.54.411092 8.71.588908 33 28.397621 8.17.936007.411615 8 70.588385 32 29.398111 1'6.985974.412137 86.587863 31 30 9.398600 o 1 9.935t42. 9.412658 6 0.587342 30 3 1.399088 8. 1.985909 54.413179 867.586821 29 32.399575 8.1.985876.55.413699 8 66.586301 28 33.400062 8.1.985843.55.414219 8-65.585781 27 34.400549 81.985811..414738 865.585262 26 35.401035 8.985778.415257 8 4.584743 25 36.401520 8 0'.985745 5. 415775:6.584225 24 37.402005.'.985712.416293.583707 23 38.402489 1 9805679..416810 ^ 8'62.583190 22 39.402972 805.985646 155.417326 8.60.582674 21 40 9.403455 o,8 9.95M613. 9.417842 8 9 0.582158 20 41.403938 8.4.985-580 55.418358 8'5.581642 19 42.404420 8.03.985.547.55.418873 8578.581127 18 43.404901 8.02.935514.55.419387 88.5.580613 17 44.405382 8.0.985480.55.419901 8.56.580099 16 45.405862 8.00.985447.55.420415 85.579585 15 46.406341 7.9.985414.55.420927 8 54.579073 14 47.406820 7.98.985331.5.421440 8 53.578560 13 48.407299 7.97.935347 421952.578048 12 49.407777 7796.985314.56.422463 8.1.577537 I 1 50 9.408254 7. 9.985280. 9.422974 8 50 0.577026 10 51.4098731 79.985247.56.423484 8-49.576516 9 52.409207 7.9.98.5213.56.423993 8.49.57607 8 53.409632 79.935180 56.424503 848.575497 7 54.410157 7.91.985146.56.425011 847.574989 6 55.410632 7.985113..425519 8.46.574481 5 56.411106 7.89.985079.56.426027 8-45.573973 4 57.411579 7.8.985045.56.426534 844.573466 3 58.412052 7.8.935011.427041.43.572959 2 59.412524 7.984978.56.427547 843.572453 1 60.412996'.oo.984944.428052..571948 0 Co. i.a Ds. 1ls I.'Cotang. D. 1. Tang. M 049~.,' COSINES, TANGENTS, AND COTANGENTS. 189 15C 1643 M. Sine. D. ". Cosine. D.1". Tang. D. 1". Cotang. M. 0 9.412996 7.85 9.984944.6 9.428052 82 0.571948 60 1.413467 78.984910..428558 8.571442 59 7.84.57 8.41 2.413938.984876.5.429062 8.570938 58 3.414408 77.83.948442.7.429566 8.3.570434 57 4.414878 7.984808 5.430070 838.569930 56 5.415347 7 1.984774.57.430573 88.569427 55 6.415815 7.984740.5.431075 8.3.568925 54 7.416283 77.984706 57.431577 83.568423 53 8.416751 77.984672 5.432079 8.3.567921 52 9.417217 777.984638 432580 83.567420 51 10 9.417684 y6 9.984603 9.433080 833 0.566920 50 11.418150 7..984569.5.43358 8.3.566420 49 12.418615 77.984535.7.434080 83.565920 48 832.565990 48 13.419079.984500'.434579 8.565421 47 14.419544 77.984466 57.435078 83.564922 46 15.420007 7'.984432 7.435576 8.3.564424 45 16.420470.7.984397..436073 8.2.563927 44 17.420933 77.984363.436570 128.5j3430 43 18.421395 7.984328 58.437067 8..56233 42 19.421857 769.984294 437563.562437 41 7.68.58 8.26 5623 20 9.422318 67 9.984259 9.438059 0.561941 40 21.422778 7.984224'58.438554.561446 39 22.423238 7.6.984190.58.439048 8.24.560952 38 23.423697 7.6.984155 58.439543 8.2.560457 37 24.424156 7.6.984120.58.440036 8.2.559964 36 25.424840.4 40529 8.1.559471 35 26.425073 7..984050.8.441022 820.558978 34 27.425530 7..984015.'.441514 82.558486 33 28.425987 7.61.99831 58.442006 8.2.557994 32 7.61.58 8.19 29.426443 761.983946.58 442497 81.557503 31 30 9.426899.9 9.983911. 9.442988 17 0.557012 30 31.427354.983875.443479 81.556521 29 32.427809..983840.5.443968 8.16.556032 28 33.428263 7..983805 I. 44458.555542 27 34.428717 7.5.983770.9.444947 8. 1.555053 26 35.429170.983735.445435 1.554565 25 36.429623 -'.983700 I.445923 13.554077 24 37.430075 7..983664 -..446411 8.13.553589 23 38.430527 752.983629.5.446898.1.553102 22 59.430978.983594.447384.1.552616 21 7.51.59 5610 40 9.431429 9.983558 9.447870 09 0.552130 20 41.431879 7.983523 59.448356 809.551644 19 42.43-2329 4.983487.9.448841 8.0.551159 18 43.432778 7.983452.5.449326 8.0.550674 17 44.433226.983416.449810.550190 16 7.47' 1 59'8.06 45.433675 7.983381 59.450294 86.549706 15 46.434122 7.4.983345 450777 80.549223 14 47.434569 4.983309.451260.48740 13 48.435016.983273..451743 8.548257 12 49.435462..983238 452225 8.547775 11 50 9.435908 7 9.983202 9.452706. 0.547294 10 51.436353 7..983166.60.453187 8.0.546813 9 52.436798 741.983130.0.453668.546332 8 53 v.437242 40.983094 60.454148 -00.545852 7 54.437686..983058.60.454628.545372 6 55.4381 29.983022.455107 7.544893 5 55.4819.60 7. 56.438572.982986.0.455586.544414 4 7.37.60 7.97 57.439014 36.982950.0.456064 79.543936 3 53.439456.982914.0.456542 79.543148 2 rO.439897 7.3.982878.60.457019 7.542981 1 60.440338 73.98842.457496 7.5.542504 0 M. Cosine. D 1. Sine. D.l'. Cotang. ID. l'. Tang. M. 10 yx,, D. 190 TABLE XIII. LOGARITHMIC SINES, 160 163C _ 1 i M Sine. D 1". Cosine. D. ". Tang. D. 1I. Cotang. M. 0 9.440333 73 9.932842 60 9.4'74'6 7 0.542504 60 1.440778 7..982805 0'.457973 74.542027 59 2.441218 3.982769'1.4.5449..541551 58 7.32.61 7.93 3.441658 31.9733 61.4538925 792.541075 57 4.442096 731.9826J6 61.459400 791.540600 56 5.442535 7.30.93866) 61.459375 71.5401-25 55 6.442973 7..9 9S26'61.460349 7.9.539651 54 7.413410 72.9'2537 61.460323 7 8.539177 53 8.4433847.982551'.461297.538703 52 9.44421 727.932314 6.461770 78.538230 51 10 9.444720 726 9.92477 61 9.462242 787 0.537758 50 11Z.44 5155 11.445155..982441 61.462715 7' 6.537285 49 12.445590,.982404 61.463186.536814 48 13.446025 7..982167.6.4636.58 7.536342 47 14.446459 72.982331 61.464128 7.53572 46 15.446393 7.2.982 1.464599 73.535401 45 16.447326 721.982257 I.465069 783.534931 44 17.447759 7.20 982220 62.465539 782.534461 43 18.443191 72.932183.6 466003.533992 42 19.448623.92146.466177 71.533523 41 7.19 3214 7.81 20 9.449054 9.-9210J.62. tG6:45- 7. ( 0. 533055 40 21.44943:5 7.932072 -6.17413 7.9.532587 39 22.449915 7.1.3 3 -.47380.532120 38 23.450345 7.17 2 16'317 73.531653 37 24.450775 7,1.981961 62.156814 7.7.531186 36 25.451204.!1t 4.6.46928 7.530720 35 26.451632. 313-6 62.9746 30254 34 27.452060 7.3.931349 62.470211 r.529739 33 23.452133 712.9)3112 I.470676 7.,.529324 32 29.452915 711.931774 6.471141 74.52335.39 31 30 9.453342 7 9.931737 62 9.471605 0.523395 30 31.453763 71.931700 2.472069.527931 29 32.451194 7.09.93166.63.472532 7.7.527463 28 33.451619.931625.472995.527005 27 31.455044 70.93157 6.473-157 70.526:4 1 35.45-469 7.07.91519 63.473919 769.52608 1 2 36.455393.981512.474381.525619 24 37.456316 7.05 931474.6.474842 7..52515 23 33.456739 7..931436 63.475303 7.524697 22 39.457162 4.981399 6.475763 7.67.524237 21 7.04 5.9313 7.67 40 9.4' 07534 91.931361: 9.4762-23 7' 6 0.523777 20 41.458006 7.981323.6.476633 7.6.523317 19 42.458427 70.981235.63.477142 7.65.522858 18 43.458348 7.01.91247.477601 7.522:399 17 41.459268 7..931209 63.478059.521941 16 45.459688..981171 63.478517 763.521483 15 46.460103 699.931133 6.478975 6.521025 14 47.460527 693.91095 63.479432 7.62 520563 13 48.460946 6.9 9.931057.4.479389 761.520111 12 49.461364 6.9.981019 64.480345 7.6.519655 11 50 9.461782 96 9.980981 64 9.480801 79 0.519199 10 51.462199 6.9.980942.481257.5.518743 9 52.462616 6.9.9830904 64.481712 7.5.518288 8 53.463032 6..980366'64.482167 7.5.517833 7 54.4348 6.93.98027.4.482621..517379 6 55.463864,.980789 6.483075 5.516925 5 56.464279 6..930750 64.483529 7.5 516471 4 6.91 7.55 516018 3 457 19. 946694.90712 64.483982 5.516018 3 53.4651038.990673.484435.515565 2 59 465522.9 063.462 84837..515113 6.89 6196 4' -05.465935.9596.485339.514661 M. Cosine. I D. 1'. Sine D. 1". Cotng. D. 1. Tang. M. 0L 6 _.5.08f'iIo COSINES, TANGENTS, AND COTANGENTS. 1^l lY3 163J M. Sine. D 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.465935 8 9.930596 64 9.485339 7 0.514661 60 1.46634 688.930555 64.485791 7.5.514209 59 2.466761 6.8.930519'.486242 7.5.513758 58 3.467173 686.930480 65.486693 7.513307 57 4.467585 685.980442 *65.487143 750.512857 56 5.467996 68.980403 65.487593..512407 55 6.463407 64.980364 65.488043 7.4.511957 54 7.463817 6.930325.6.488492 74.511503 53 8.469227 683.930286 65.488941.511059 52 9.469637 6 82.980247 65.489390.510610 51 10 9.470046 9.980208 9.489833 0.510162 50 11.470455 81.980169 65.490286 7.46 509714 49 12.470863 680.980130 65.490733 7.509267 48 13.471271 679.980091.6.491180.508820 47 14.471679 678.980052.65.491627 7.4.508373 46 15.472086.78.980012.65.492073 74.507927 45 16.472492 67 979973 65.492519.4.507481 44 17.47289 676 979934..492965.507035 43 18.473304 676.979895 6.493410 74.506590 42 19.473710.979855 66.493354 741.506146 41 6:75!.66' 7:41 20 9.474115 4 9.979816 9.494299 0 0.505701 40 21.474519 74.979776 6.494743..505257 39 22.474923 67.979737.6.495186.3.504814 33 23.475327.979697.495630.504370 37 24.475730 6.979658.66.496073 7.3 503927 36 25.476133 6.979618 6.496515.503485 35 26.476536.979579.6.49697.3 503043 34 27.476938.979539 66 497399 7.36.502601 33 28.477340 6.69.497949502159 32 29.477741 979459 498232.501718 31 6.68.66 7.34 30 9.478142 667 9.979420 66 9.498722 0.501278 30 31.478542 67.979330 66.499163 7.3.500837 29 32.478942 6..979340.6.49960.3 73.500397 28 33.479342 6.6.979300 67.500042 7..499958 27 3465 67.4799519 26 35.480140 64.979220 67.500920 7.31.499080 25 36.480539 6.979180'.501359 7..498641 24 37.480937 6.3.979140 67.501797 730.49203 23 38.481334 6..979100 6.502235 7..497765 22 39.481731 6.6.979059 67 502672 7 497328 21 6.61.67 7.28 40 9.482128 6 9.979019.67 9.503109 0.496391 20 41.482525 6..978979 7 503546 7.2.496454 19 42.482921 6.0.978939 67 503982 727.496018 18 43.483316 6.59.978898.6.504418 727.495582 17 44.483712 6.5.97858 67.504854 7.26.495146 16 45.484107 6.58.978817 67.505289 7..494711 15 46.484501 6.57.978777 67.505724 7.2.494276 14 47.484895 6.5.978737 68.506159..493341 13 48.485289 6.5.978696 68.506593 7.493407 12 49 485682 6.978655.5097027..492973 11 6.55.68 507027 23 50 9.486075 9.978615 6 9.507460 7 0.492540 10 51.486467 6.4.978574.6.507893.492107 9 52.486860 6.54.978533.508326 1.491674 8 53.487251 6.53.978493...508759 721.491241 7 54.487643 6.52.978452.68.509191 720.490809 6 55.488034 6.52.978411 68.509622 7.20.490378 5 56.488424 6.51.978.370 68.510054 7.489946 4 57.488814 650.978329.108.5 7.18 489515 3 650 ~ 68 4 830 58.489204 6.50.9782383.510916 7.1.489084 2 59.489593 6..978247 63.511346 717.4886.54 1 60.439932.4.978206'~.511776.48224 0 M1. Cosne. I D.1". Sine D. i. Cotang. D. 11". Tang. M. 1017 c 192 TABLE XIII. LOGARITHMIC SINES, 183 161C M. Sine. D. 1". Cosine. D. 1". Tang. 1". Cotang. M. 0 9.4399O 2 9.4978206 9.511776 I 0.488224 60 1.490.371 6.4.978165 *69.512206 7.16.487794 59 2.490759 6.978124..512635 7.1.487365 58 3.491147 646.978033 69.513064 1.456936 57 4.491535.978042 9 5134'93.486507 56 5.491922 6.4~.97S001..513921.486079 6.492303..977959.514349 485651 54 1 6..69 7.1364' 7.49'2695 6.43.977918 69.514777 7.1.485223 53 8.493081 6.977877.69.515204 7.12.44796 52 9.493466 4.977835 69.515631 7:.484369 51 6.42.69 7.11 10 9.493351 6 9.977794 9.516057 10 0.483943 50 11.494236 41.977752 69.516484 10.483516 49 12.494621 6.40.977711'69.516910 7.09.483090 48 13.495005 6-39.977669 6.517335 79.482665 47 14.495388 6.977628 6.517761 7.482239 46 15.495772 63.977586 69.518186 0.481814 45 6.35'69 7. 08 16.496154 63.977544..51~610 7..481390 44 17.496537 63.977503.519034 77.480966 43 18.496919 636.977461 70.519458 7.480542 42 19.497301 6.977419..519882.480118 41 6.36.70 7'05 20 9.497632 a 9.977377' 9.520305' 0.479695 40 21.493064.3.977335 7.2728 70.479272 39 22.498444.977293'.521151 4.478849 38 23.498325'.977251.521573..478427 37 24.4992,4 63.977209.0.521995.3.478005 36 6.33 70 7.03 25.499584 63.977167..522417.477583 35 6.32.70 7.02 26.499963 31.977125 -70.522833 02.477162 34 6.31.70 7 02 27.500342 6.977083'.523259.476741 33 23.5)0721 6.977041.523680 701.476320 32 29.109 76999 70.524100 475900 31 6.30.70 7.00 31 9.501476 6.29 9.976957 70 9.524520 9 0.475480 3( 31 50.5(8. 62.976914 71.524940 6.9.475060 29 32.5022;31 f6.2i.976372 7.525359 69.474641 23 33.502607 627.976830 71.525778 69.47422 27 313.98.47303 26 31.502984 27.976787 71.526197 69.473303 26 35.503360.976745.526615 69.47333 25 36.503735 6.25.976702.527033 6.9.472967 21 37.504110 976660.527451.472549 2 6.25 6.96 22 33.504485..976617 71 527868 6.472132 6.24.9765774:58 39.504860 624 976574.52825 95.471715 2 40 9.505234 9.976532 9.528702 0.471298 20 41.505608 62.976489'71.529119 6.9.470881 19 42.505981 622.976446 7.5295:35 694.470463 18 6.22 -.71 6.93 43.506354 621.9761 71.529951 6.3.470149 17 41.506727 621.9 6:3GI 1.530366 6..469634 16 45.507099 2.97631. 7.530781 6.469219 15 6.29.72 6.91 46.507471 6..976275. 2.531196 6..463804 14 47.507843.9.9 6232.5316191.46:3389 13 6.19.72 6.90 48.53214 61.976189.7.532012.5 6'.467975 12 6.18S 72 6.9n 49.505.5fi) 6.976146.532439 69.467561 11 50 9.5039.56 6~17 9.97610.3 9.53-2353 689 0.467147 10 51.509326.976060 72.533266 68.466734 9 52.509696 16.976017 72.533679 68.466:321 8 53.51(65 6.975974'..534(092 68.465908 7 51.5104.34 6'1.975930.531504 6.465496 6 6.15 i 7 6.87 55.510803.975887.534916.8 465084 5 6.14 1 72 6.86 56.511172 6.14 7844 2.535328..464672 4 57.511540.97800.535739.464261 3 53.511907 6.12.97757..5350.4630 2 59,. 225.5 122756 1. 973.714 / 7'_,I.536561.463439 60.512642 6.97670.536972 4.463028 0 M. Csine. Dl. 1 Sine. D. IP. Cotang. D.J'. Tang. M. 7tI COSINES, TANGENTS, AND COTANGENTS. 193 193 160~ M. Sine. D. 1'". Cosine. D. 1" Tang. D.'. Cotang. M. 0 9.512642 6 9.975670 7 9.5-36972 6 S 0.4630)2 6) 1.513009 6.1.97.27.73.537332 -.462618 59 2.513375 6.975533 73.537792 33.46220 58 3.513741 6.09.975539 73.53202 6.461793 57 4.514107 609.975496 7.533611 682.461339 56 5.514472 6.0.975152 *73.539020 1.460930 55 6.514837 03.975408 7.539429 61.460571 64 7.515202 0.975365.539837.460163 53 8.515566 6.7.975321'.540245 80.459755 52 9.515930 6.975277 73.540653 79.459347 51 10 9.516294 60 9.975233' 9.541061 69 0.458939 50 11.516657 05.975189.73.541463 6.7.458532 49 12.517020 60.975145 3.541875 6.78.458125 48 13.517382 60.975101..542281 67.457719 47 14.517745 6.0.975057.7.542688 677.457312 46 15.518107 03.975013 4.543094 67.456906 45 16.518468 60.974969.543499 76.456501 44 17.518829 602.974925 7.543905 6.456095 43 18.519190.974880.544310 5.455690 42 6 01' 74 6.75 19.519551 60.974836.544715 674.455285 41 20 9.519911 9.974792 4 9.545119 4 0.454881 40 21.520271 699.974748 74.545524 673.454476 39 22.520631..974703 74.545928 673.454072 38 23.520990.9.974659'74.546331'72.453669 37 24.521349 98.974614'74.546735 62.453265 36 25.5207 5.97.974570 74.547138 6.71.452862 35 26.522066.974525 7.547540 71.452460 34 27.522424 6.974481 7.547943.452057 33 28.522781.9.974436 7.548345 70.451655 32 29.523138 595.974391.548747.451253 31 5.95:.75 6.69 30 9.523495 94 9.974347 7 9.549149.9 0.450851 30 31.523852.974302.549550.450450 29 32.524208..974257.7.549951.68.450049 23 33.524564 5..974212 5.550352.67.449648 27 34.524920 5.9.974167.550752 67.449248 26 35.525275 92.974122..551153..67.448847 25 36.525630 5.974077 75.551552 6.6.448448 24 37.525984 5.91.974032'.551952 666.448018 23 38.526339 590.973987.552351 665.447649 22 39.526693 589.973942 75.552750 65.447250 21 40 9.527046 9.973897 9.553149 0.446351 20 1 520 5.89.75 6.61 41 |.527400.973852.553548 6.6.446152 19 42.527753 588 97307.553946.446)05 181 43.528105 58.973761.554314.4456>6 17 5.87 9.75' - 6.63 44.528458 587.973716 75.551741 66.445259 16 45.528810 86.97.3671 76.55513.444S61 15 46.529161 5'8.973625.5355536 66.444464 14 5.86 973627..76 6.61,47.529513 58.973580 7.555933 661.444067 13 48.529364 5.5.973.35.556329 6.413671 12 49.530215 5.84 34.556725 66.443275 11 5.84 97 66.76 50 9.530-565 r7 9. 2 1 9. 973444- 0.412i79 10 51.530915.97. 393.55.i-7.4243 9 5.83 973393.76 6.59.44243 9 52.531265 |.97335)2.7.5571i3 9.442037 8.8 6 -.76 6.59 53.531614 5.82.973307.76 5508.441692 7 54.531963 5.81 973261 6.55703 ~6.441297 6 55.532312 581.9732;15'.5976 65.440903 5 56.532661 5.81.973169 6.559191 57.440509 4 57.533009 80.973124 76.5593 6 44015 3 58 5337.337 973078..560;29.439721 2 59.533704 r 9.973032 7 56673 66.439327 1 60.534052 1~.972986.561066' 438934 0 M. Cosine. I DI". Sine. D.. Cotang. D. T. Tang. M. 1090 70 194 TABLE XIII. LOGARITHMIC SINES, 200 1509 M. Sine. D. 1". Coine. D. 1". Tang. D.1". Cotang. M. 0 9.534052 578 9.972986 9.561066 0.438934 60 1.534399 78.972940.561459 6.5.438541 59 2.534745 5..972894 77.561851 64.438149 58 5.77.77 6.54 3.535092 5.972848.562244 6.54.437756 57 4.53438 5.7.972802.77.562636 653.437364 56 5.535783 76.972755.563028 63.436972 55 6.536129 7.972709.7.563419 5.4361 54 7.536474 5.972663 7.563811 65.436189 53 8.536818 5.972617.564202 65.435798 52 9.537163 5.972570 7.564593.435407 51 ~5.73.77 6.50 10 9.537507 73 9.972524 9.564983 6 0.435017 50 11.537851 5.972478.77.565373 650.434627 49 12.538194 72.972431 78.565763 65.434237 48 13.5385.38 71.972385 78.566153 64.433847 47 14.538880 5.7.972338 8.566542 69.433458 46 15.539223 7.972291 7.566932 6.4.433068 45 16.539565 H.972245 78.567320.4.4326S 44 a5.70.78 6.48 17.53990(7 5..972198 78.567709 6.4.432291 43 18.540249 5.69.972151 78.568098.431902 42 19.540590 5.6.972105 8.568486 6 431514 41 5.68.78 6.46 20 9.540931 5 9.972058 9.568873 0.431127 40 21.541272 5.6.972011 78.569261 646.430739 39 22.541613 5.67.971964 78 569648 64.430352 38 23.74953 6.43 23.541953 5.6 971917 78 570035 6.45.429965 37 24.542293. 971870 78.570422 44.429578 36 25.5426-32 565.971823 78.570809 6.44.49191 35 26.542971 565.971776 78 571195 6.4.428805 34 27.543310.971729.571581 643.428419 33 28.543649 5.971682 571967 43.428033 32 29.543987 B563.971635 79 572352 42.427648 31.572352 6.42.427358 31 30 9.544325 63 9.971588 7 9.572738 642 0.427262 30 31.544663 5..971540..573123 6.426877 29 32.545000 62 971493..573507 6.41.426493 28 33 545338 6.971446.7.573892 40 426108 27 34.545674 51.971398.574276 40.425724 26 35.546011..971351.574660 6..425340 25 36.546347 5..971303.79.575044 6.424956 24 37.546683 5..971256.575427 639.424573 23 38.547019 5.971208 9 575810 38.424190 22 39.547354.971161.676193.423807 5.58.79.38 40 9.547689 9.971113 9.676576 63 0.423424 20 41.548024..971066.9.576959 37 423041 19 42.548359 5..971018.577341 37.422659 18 43.548693.970970 0.577723 36.422277 17 44.549027 5.970922 80.578104 636.421896 16 45.549360 5.970874 80.578486.421514 15 46.549693 5..970827..578867 635.421133 14 47.550026 55.970779 80.579248 6.4 420752 13 5.54.80 6.34 48.550359 5.4.970731 *8.579629 6.34 420371 12 49.550692 5..970683 80.580009 6:34.419991 11 50 9.551024' 9.970635 80 9.580389 6 0.419611 10 51.551356 5.5.970586.7.580769 33.419231 9 52.551687 5.5.970538 80.681149 6.3.418851 8 53.552018 552.970490 80.681528 32.418472 7 54 552349 5.5.970442 80.81907 32.418093 6 55.552680..5 970394.8.582286.417714 5 56.553010 55.970345 81.582665 31.417335 4 57.553341.970297.583044 6 -.416956 3 5.50.81 6630 58.553670 5.4.970249 81.583422'.416578 2 59.554000 54.970200..583800 63.416200 1.9 0. 80 6.30 60.554329 5.970152.584177 ~.415823 0 M. Coslne. D. -l. Sine. D. 1". Cotang. D.1. Tang. M. tl00 6Q9 COSINES, TANGENTS, AND COTANGENTS. 195 210 1580 -... M. Sine. D. 1I. Cosine. D. 1". Tang. D. 1". Cotang. M1. 0 9.554329 48 9.970152 81 9.584177 629 0.415823 60 1.554658..970103.584555.415445 59 2.554987.97055.84932 62.415068 58 3.555315.4.970006..585309 8 414691 57 5 5.47.81 6.28 4.55.r643 5.4.969957.585686 28.414314 56 5.555971 5.46.969909 81.586062..413938 55 6.556299 5.45.96960 81.586439 6.27.413561 54 7.256626 55.969811 8.5860 15 62.413185 53 8.556953 44 969762 81.587190 26.412810 52 9.557280.969714.587566.412434 51 5.44.81 6.26 10 9.557606 5 4 9.969665 82 9.587941 0.4I2059 50 11.55728;.29, 11.557932 4.969616 82.588316 6.25.411684 49 12.558258 *.969567 82.588691 62.411309 48 13.558583 5.969518 *.589066.4.410934 47 5.42.969468 82.589066 624.410934 46 14.558909.96949 82.589440.24.410560 15.559234 5.42.969420 82.589814 62.410186 45 16.559558 54.969370.590188.409812 44 5.411309 46 17.559883 5..969321 82.590562 62.409438 43 18.560207 5.40.969272 82.590935 6.22.409065 42 5.39.82 6.22 19.560531 519.969223 82.591308.408692 41 20 9.560855 r.969173 9.591681 21 0.408319 40 21.561178 5.969124 82.592054 21.407946 39 22..561501 5.38.969075.8.592426 6.20.407574 38 23.561824.38.969025.82.592799.20.407201 37 24.562146..968976 83.593171 620.406829 36 25.562468.968926 8.593542.406458 35 26.562790 968877 83 593914 6.406086 34 27.563112 536.968827 83.594285 6.405715 3 28.56.3433 5.36.968777.594656 6.18.405344 32 29.563755.968728.595027 61.404973 31 5.35.83 6.18 30 9.564075 9.968678 9.595398 6 0.404602 30 31.564396 5. 968628.3 59576 6..404232 29 32 Ei.34 I 83 6.17 32.564716 968578..596138 17.403862 28 33.565036 5.968528..596508 616.40342 27 34.565356.968479 83.596878 1.403122 26 35.565676 5.32.968429 83.597247 6.16.402753 25 36.565995 5.32.968379 83.597616 615.402384 24 37.566314 968329 83.597985 6. 15.402015 23 38 566632 5..968278.598354.401646 22 39.566951 5.968228.598722.401278 21 5.30.9.84 6.14 0 40 9.567269 9.968178 * 9.599091 6 0.400909 20 41.567587 5.0.968128 8.599459 6.400541 19 42.567904 52.968078.599827.400173 18 43.568222 5. 963027 8.600194 61.3990C6 17 44.568539 5.2 967977.4.600562 612.399438 16 5.28.84 6.12.399438 16 45.568856.967927.6(0929.391 1 46.569172 5.28 9.9601296 61.398704 14 47.569488 5.967826 601663.398337 1 48.569804 5..967775 4.602029 6.11.397971 12 ~49.570120 5.26.84.62 49.570120 5.26.967725 8.602395 610.397605 1 50 9.570435 9.967674 9.602761 10 0.397239 10 51.57051 5.25.967624 603127 6..39683 9 52 57106 5..5967573.8.603493 6.9 396507 8 53.571380 5..967522 85.603858 69.396142 7 54.571695 5.24.967471 85.604223 6.0.395777 6 55.572009 5.24.967421 85.604583 6.0.395412 5 56.572323.967370.604953 07.395047 4 57.572636 5.2.967319.85.605317 6..394683 3 8 572950..967268.605682.394318 2 60.573575 5.21.967166.85.606410.393590 0 M. Cosine, D. 1". Sine. D. I". Cotang. D. 1". Tang. M. Iinc s68 196 TABLE XIII. LOGARITHMIC SINES, l29 ________________________ 157' M. Sine. D. 1". Cosine. D. U'. Tang. D. ". Cotang. M. 0 9.573575 2 9.967166 8 9.606410 0.393590 60 1.573888 521.967115.5.606773 6.06.393227 59 2.574200 520.96764 85.607137 6.06.39263 58 3.574512 %20.967013.85.607500 6.392500 57 4.574824 2.966961'8.607863 6.0o.392137 56 5.575136 519.966910. 5.608225 6.05.391775 55 6.575447..966859.85.608588 6.04.391412 54 1575758.!8 I 86 6.04 3 7.575758 5.1..966808.86.608950 6.03.391050 53 8.576069 5.966756 6.609312 6.390688 52 9.576379 5.966705.609674.390326 51 10 9.576639 5. 9.966653 8 9.610036 60 0.389964 50 1.576999 5.16.966602.86.610397 6.02.389603 49 12.577309 5.16.966550.86.610759 6.02.389241 48 13.577618 5.16.966499 86..611120 6.388880 47 14.577927 5.15.966447 86.611480 601.388520 46 15.578236 5.966395.86.611841 6.388159 45 16.578545 5.1.966341.86.612201 600.387799 44 17.578853 5.14.966292 86.612561 600.387439 43 13.579162. 5.1.966240.86.612921 6.00.387079 42 19.579470 513.966188 86.613281 6900.386719 41 20 9.579777 2 9.966136 8 9.613641', 0.336359 40 21.580035 5.i.966085 87.614000..386000 39 2:2.580392 5..966033;87.614359 5.98.335641 38 23.580699 I11.965981 87.614718 5.98.385282 37 24.5s1005 5.11 965929.87.615077 5.9.384923 36 25..581312 5.1 965876'87.615435 5.97.384565 35 26.58161 S 5.1.965S24.615793 59.384207 34 27.581924 5.10.965772.87.616151 5.96.383849 33 23.532229 5.965720.66509.383491 32 29.582.535 509.965660 87.616867 5.96.383133J 31 30 9.582340 9.965615 8 9.617224 9r5 0.382776 30 31.533145.0.965563 87.617582 5.95.382418 29 32.534. 49.965511 -.617939 95.382061 23 33.58754.965453.618295,9.381705 27 34.58140.3 5.07.965406.87.618652 5.94.381348 26 35.534361 965353 619008.30992 25 36 665 5.06.965301 88.619364 5.94.380636 24 37.534963 5.0.9624 8.619720 5.93.330280 23 33.535272 5.05.965195 88.620076.379924 22 39.535574 5.04.965143 88.620432 92.379568 21 40 9.585377 9.965090 8 9.62073? 9 0.379213 20 41.586179 5.04.965037..621142.3788.58 19 42.536432 5.04.96494.88.62497 5.92.378503 18 43 586783 5.03.5964931.8.621852 5.91.378148 17 44.537035 5.964879 8.62227.377793 16 45.53736 5.02.964826-.83.622561 5.91.377439 15 46.568.96773 2.622915 5.90.377085 14 46 5876389 35,0.8 6 5 47.57939 5.01 964720.88 623269 5.90.376731 13 48.583239 5.01.964666.88.625823 5.90.376377 12 49.533599 5.0.964613.89 623976 5.89.376024 11 50 9.533399 e 9.96156) 3'.624330 0.375670 10 51 539193 5.00.964507.89.624633 5.89.375317 9 54.99.89 5/88 52. 539439.99 964454.625036 5.88.374964 8 53.539789.964400.89.625388 88.374612 7 54.59003 4.96347.89.625741 5.8.374259. 6 55.59337.961294.89.626093 5.87.373907 5 56.590636 4.9.964240.89.626445 5.87.373555 4 57.59934 4.97 964187.89.626797 5.373203 3 53.59122.961133.89.627149 5.86.372S51 2 69 -.591373 7 496.961026.'~.627852.372148 0 M. Cosine. 1D. 1". Sine. D. 1". Cotang. D. 1'. Tang. M..I'78.67 COSINES, TANGENTS, AND COTANGENTS. 197 230 156a M. Sine. D. 1". Cosine. D. 1". Tang. D.". Cotang. M. 0 9.591878 4.96 9.964026 89 9.627852 5.85 0.372148.60 1.592176.95.963972 89.628203 5.371797 59 2.592473 4.963919 90.628554 5.8.371446 58 3.592770 4..963865.0.628905.84.371095 57 4.593067 4..963811 90.629255 5.370745 56 5.593363 49.963757 90.629606 54.370394 55 6.593659 493.963704 9.629956 5.8.370044 54 7.593955 49.963650 90.630306 5.3.369694 53 8.594251 49.963596 0.6.30656 5..369314 52 9.594547.963542 90.631005 82.368995 51 10 9.594842 4 92 9.963488 9' 9.631355 2 0.368645 50 11.595137 49.963434'.6:31704 5.8.368296 49 12.595432 491..632053 5.2.367947 48 13.595727 9.963325 90.632 5.81.367598 47 4.91.96375 90 35 4.81 14.596021 4.963271 *90.632750 5.81.367250 46 15.596315 9.963217 90.633099 5. 8.366901 45 16.596609 489'.963163..633447 5..366553 44 17.596903 489.963108 91.633795..366205 43 18..597196 489.96:3054 9.34143.36857 42 19.597490.962999 9.634490 59.365510 41 20 9.597783 88 9.962945.91 9.634838 _ 0.365162 40 21.595075 48.962S90..635185 ".64-1' 39 22.59836 487.962836 91.6355:32 578.364468 38 5 78.. 23.598660 4,7.962781 1.635379 5.8.364121 37 24.598952 86.962727 91.636226 578.363774 36 /25 4j.5994'4..962672.636572.363428 35 26.59916 4.6.962617 91.6.36919.363081 34 27.599827 48.962562 91.637265.362735 33 23.600118 485.962.508.637611 76.362389 32 29.600409 48.962453 92.637956 5:6.362044 31 30 9.600700 4 4 9.962398 92 9.638302 576 0.361698 30 31.600990 4..962343..638647.7.361353 29 32.601280 483.962288 92.638992.361008 28 4.83.92 5.75 33.60 l570 483.962233 92.639337 5.360663 27 34.601860 483.962178 92.639682 5.7.360318 26 35.602150 482.962123 92.640027.359973 25 36.602439 4.962067'.640371.359629 24 37.602728 481.962012 92.640716.359284 23 4.81.92 5.73 I 33.603017 8.961957.641060.358940 22 4.81.90 5.73 39.603305 481.961902 92.641404 73.358596 21 40 9.603594 48 9.961846 92 9.641747 0.358253 20 41.603s82 40.961791 92.642091 5..357909 19 42.694170 4'7.961735 92.642434 57.357566 18 43.604457 47.961630 9.642777 572.357223 17 44.604745.7.961624.9.643120 5.7.356880 16 45.C05032.9619.6-4363 71.356537 15 46.605319 78.961513.64306.71.356194 14 47.605606.9614s 3..644148 5..355852 13 48.605892.961402 59.644490'7.355510 12 49.606179 4.961346 9.644832.355168 11 50 9.606465 4.76 9.961290 93 9.65174 50 0.3.54826 10 51.606751 7.961235.645516 569.354484 9 52.607036 476.961179.9.645857 569.354143 8 53.607322 4.961123.9.646199 569.35.:3801 7 54.607607 47.961067 3.646540 5.353460 6 55.607892'7.961011..646881 5.6.353119 5 i 6.608177 4.960955.3.647222.352778 4 4.74 98.3~ -2 4 57.60-461'74.960399'.647.562.352438 3 58.608745 4.960843 4.647903 5.6.35207 59.609029 43.960786.643243 67.351757 1 60.609313'.960730.648583.351417 0 M. Cosine. D. ". Sine. D. 1". Cotng. ang.. M. 1130 66c 198 TABLE XIII. LOGARITHMIC SINES, o 15_____________________ M. Sine. D. 1". Cosine. D. 1". Tang. D. 11. Cotang. M. 0 9.609313 7 9.960730 9.648583 67 0.351417 60 I.609597 4.72.960674.94 643 5.67.351077 59 2.609380 472.960618'.64926350737 5866 3.610164 4-72.969561.9.649602 5 6.350398 57 4.610147 71.960505.94.649942 566.35)058 56 5.610729 4-71.960448.94.650231 5.6;.349719 55 6.611012 471.960392.0 52.650620 54 7.611294 4'70.960335..650959 5.349041 53 8.611576 4 70.960279 94.651297 5.64.348703 52 40.94 5.61 34 6 9.611853 4:69.9602 2.6.51636.34364 51.94 5.61 10 9.612140 4.69 9.960165' 9.651974 0.34026 50 11.612421 4.69.960109.9'.6.52312 5.6.3476388 49 12.612702 4 68.969052.9.652650 5.63 347350 48 463 95.6)2650 563.3473.0 13.612983 4'63.959995. 5.652983 563.347012 47 14.613264 4 68.959933.95.653326 62.346674 46 15.613545 4-67.959882.95.653663 5.62.346337 45 16.613825 467.959825 I95.654000 5.6.346000 44 17.614105 4.6.959763S -.654337 5.62.345663 43 13.6143S5 4466.959711.9.654674 5.62 345326 42 19.614665 4:66.959634 95.655011 5.61 344989 41 20 9.614944 4.65 9959596 9 9.655348 561 0.344652 40 21.615223 4.65.959539.95.655634 561.344316 39 22.615502 465.95942.95.656920 56.343980 38 23.615781 464.9.59425.95.656356 5.60.343644 37 24.616060 4.6.959363.9.656692 5.60.313308 36 46.616 25.616333 44.959310.6.657028 5.9.342972 35 26.616616 463.959253.96.657364 559 ^12636 34 27.616391 463.959195.96.657699.5 42301 33 28.617172 63.959133.658034 5.58.341966 32 29.61740 4 62.959080.96.658369 55.5.341631 31 30 9.617727 6 9.959023. 9.658704 5 5. 0.341296 30 4.61 96 34 31.618001 4.61.959965.96.659039 5.58.340961 29 32.6182-31 4.61.958903.96.659373 5.5.340627 28 33.61S553 4.6.958850 96.659703 5.57.340292 27.61333 4 4.6.95792 96.660042 5.57.3:39953 26 35.619110 4.60.9587.34.96.660376 5.5.3.39624 25 36.619336 4.60 953677 96.660710.56.339290 24 37.619662 4.60.958619.661043.338957 23 38.619933 4.5.953561 97.661377 5.56.333623 22 39.620213 459.53.503 97.661710 556.333290 2.1 40 9.620188 48 9.958445.9 9.662043.5 0.337957 20 41.620763 4.53.953337.97.662376 5..337624 1 9 42.621033 4.58.958329.97.662709 5.337291 18 43.6-213l3 4.5.958271.'7.663042 4.336958 17 44.621537.958213.663375.336625 16 45.621861l.958154.663707 5.54.336293 15 4.57 9514 97.6307 9 46.6 2 21 35.958096 7.664039 3.335961 1 4 47.622109.56 958033 9.664371 5.53.335629 13 48.6226'2 4..957979.664703 335297 12 49.6-22956 45.957921.97.665035 553.33496) 1 1 50 9.623229 4 9.957863. 9.665366 5.52 0.334634 10 5J.623592 4.54 957804 98.665698 552.331302 9 i.623774 4-54.957746 98.666029 5*52.33:3971 8 53.621047 4.4.957687.9.666360 5..331640 7 54.624319.5.957628.9.666691 5.51.:lr?09 6 5.51 55.621591.957570 98.667021.332979 56.624863 4.53.957511 98.667352 5.51.33X48 4 135 ~~ 4.53.98 ~5.58 57.62513 4.52.957452.98.667682 5.51.33 18 3 58.625406 4.5.957393 93.663013 5.50.331987 2 59.625677 4.52.957335.9.668343 5.50.331657 1 60.62594S 4.52.957276'.9.668673 5.0.331327 0 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1". Tang. M. 1140 653 tl~~~~~~~o ~~~~66177. 382 COSINES, TANGENTS, AND COTANGENTS. 199 25O 1564 8r.......______D. - _ _ - _ _ 15: M. Sine. D. I1. Cosine. D 1". Tang. D. I". Cotang. M. 0 9.625948 4 51 9.957276 98 9.668673 50 0.331327 60 1.626219 4 957217 *9.669002.330998 59 2.626490 4.957168 98.669332.330668 58 3.626760 4.51 957099 98.669661 5.9.330339 57 4.627030 4.'.957040.98.669991.49.330009 56 5.627300 450 956981 9.670320 54.39680 55 6.627570 4.4.956921.99.670649.8.329351 54 7.627840 49.956862 99 670977 5.39023 53 8.628109 4'.956803 9.671306 5.7.328694 52 9.628378 448.956744 9.671635 5147.328365 51 10 9.628647 4.48 9.956684 99 9.671963 7 0.328037 50 I I.628916 448.956625.672291 54.327709 49 12.629185.956566.672619 54.327381 48 13.629453 4'4.956506.9.672947 5.6.327053 47 14.629721 4 7.956447 9.673274 546.326726 46 15.629989 46.956387 9.673602 5..326398 45 16.630257 446.956327..673929.4.326071 44 17.630524 46.966z68.674257.325743 43 18.630792..95628 00.674584 5.325416 42 19.631059 4.956148 1.674911.325089 41 4.45 1.00 5.45 20 9.631326 4.5 9.956089 100 9.675237.4 0.324763 40 21.631593.956029.675564.324436 39 22.631859.9o.5969 0.675890.2411 38 23.632125 4"44.955909 0.676217 5.4.323783 37 24.632392..955849..676543 5.4.323457 36 25.632658.95579 1.00.676869 5.3.323131 35 26.632923.955729 1.00 677194 5.322806 34 4'43 1'00 27.633189 442.955669 100.677520 5.43.322480 33 28.633454 42.955609 1..677846 5.42.322154 32 29.633719 442.955548 00.678171 321829 31 4.42 1.00 1 5.42 30 9.633984 41 9.955488. 9.678496 542 0.321504 30 31.634249 441.955428 101.678821 5.41.321179 29 32.634514 441.955368 10.679146 5.320854 28 33.634778 440.955307 o I.679471 5.1.320529 27 34.635042 44.955247 1..679795 5.1.320205 26 35.635306 440.955186.680120 o.41.319880 25 36i.635570.955126.680444 54.319556 24 37.6.35834 43.955065.680768.319232 23 3.6:36097..955005 1.681092 5.0.318903 22 4.955005 661416 5.40 39.636360.954944 0.681416.318584 21 40 9.636623 4.3 9.954883 101 9.681740 0.318260 20 41.63686 4-3.954823 10.682063 5.3.317937 19 42.637148.954762.682387.317613 18 43.637411.93.54701..682710 5.38.317290 17 44.637673 4.37.954640 1.0.683033 5.38.316967 16 45.63795 4.3 954579 2.683356 5.316644 15 46.638197 436.954518 10.683679 38.316321 14 47.638458 436.954457 1.634001 5..315999 13 48.638720 435.954396 02.684324 37.315676 12 49.63981.954335 02.6S4646 -37.315354 11 4.35 1.02 5:37 i 50 9.639242 435 9.954274 02 9.684968. 0.315032 10 51.639503 1.9 54213.685290 5.36.314710 9 4434 1302 5 3t 52.639764.954152 0.685612 56.314388 8 53.644 1.95400 02.685934 536.314066 7 4.64028.9542.9 10.686255 536.313745 6 55.64 4.3.953968 02.66577 5.3.313423 5 6.640804.953906 02.686898 5.3.313102 4.57.641064 4.953845 1.0.687219 5.3.312781 3 58.64132 4.32 953783 1.03 687540.312460 2 59.641583.953722 687861.312139 1.641842 3.953660.688182.311818 0 M. Cosine. D. ". Sine. D. 1. Cotang. D. 1 Ta 1M. 1150 604,~ Csie Dl 200 TABLE XII1. LOGARITHMIC SINES, 260 1534 M Sine 1". Cosine. D Tang. D. 1". Cotang. M. 0 9.641812 4 32 9.953660 1.03 9.688182 34 0.31 1813 60 1.642101.953599 103.688502 5-.311498 59 2.642360 431.953537 1'03.638823 5..311177 58 3.642618 43.95347 103.689143.310357 57 4.642877 430.953413 103.689463 533.310537 56 5.643135 4.953352 103.689783 33.310217 55 6.643393 4-3.953290 103.690103.3.309897 54 7.613650 4 2.953228 03.690423.309577 53 8.643908 429.953166 103.690742 5.32.309258 52 9.644165 29.953104 103.691062 532.3093 51 10 9.644 123 28 9.953042 103 9.691381 2 0.308619 50 I.61450 4..95290 1.691700 5.3.308300 49 12.64436 4.2.952918 10.692019 5.3.307981 18 13.645193 427.952855 1.692338 5.31.307662 47 14.645450 27.952793 04.692656 5.31 307344 46 15.645706 427.952731 104.692975.307025 45 16.645962 42.952669 104.693293.306707 44 17.646218 426.952606 104.693612 53.306388 43 18.646474 426.952544 04.693930 530.306070 42 19.646729 426.952481 14.694248 5.305752 41 21.647240 425.952356 104.694883 29.305117 39 22.647494 2.952294 1.695201 5.29 304799 3 23.647749 424.952231 04.695518..304482 37 24.648004 42.952168 *.695836 5.29.30416 36 26.648512.952043 1.696470 28.303530 34 27.643766.951980.696787 2..303213 33 23.649020 4..951917 10.697103 5..302897 32 29.649274 4.23.951854.697420.302580 31 4.22. 1:0 5.27 30 9.649527 4 22 9.951791 10 9.697736 527 0.302264 30 31.619781 422.951723 10'.698053 527.301947 29 82.650334 422.95166.5 i5.698369 52..301631 28 33.65037.951602.6968S5 5.301315 27 4.2i 1.05 5.26 31.6.5039 421.951539 105.699001 26.300999 26 35.650792 4.2.951476 o.699316 526.30064 25 36.651044 420.951412 105.699632.2.300363 24 37.651297..951349.699947.2.30053 23 38.6;1549 20.951286 1.700263.299737 22 39.651800 4 19.951222 06.700578 5.25.299422 21 40 9.652052 49 9.951159 i 9.700393 0.299107 20 41.652304 419.951096 106.701208 525.298792 19 42.6525 55 4..951032 06.701523 5.2.298477 18 43.652306 418.950963.7(01837 5.24.29 163 17 44.653057 4.1.950905..702152 5.24.29754 16 4918 106 524 45.653303 418.950841.70-2466.297534 15 46.6.4535 4 4.950778.2.702781 5.2 297219 14 4 6 4.17 1.06 5.24 47.63503 4.950714 16.703095 5.2.296905 13 48.654059 17.950650 1..703409 5.2.296591 12 49.654.309 4 16.950536 106.703722 5.23.296278 11 50 9.6545 53 9.950522 7 9.704036 0.295964 10 51.6.54083 4.16.950453 10.704350.295650 9 52.655058 46.950394 1 0.704663 5.22.295337 8 53.655307..950330 07.74976 5.22.295024 7 54.6.5556 4.15.950266 17.705290 5.22.294710 6 55.6805..8950202 10.705603 5.2.294397 5 56.6.36954 415.95013 10.705916.22.294084 4 57.656302 414.950074 17.706228 5.21 293772 3. 58.6-56551..950010.706541 293459 2 4.14 1.07.90211 59.656799 4.1 949945 1.7.706854 5.1.293146 1 60.657047.949381 1.707166.292834 0 M. Cosine. | D. l. Sine. D. 1". Cotang. D. 1". Tang. M. 1160 63 COSINES, TANGENTS, AND (:OTANGENTS. 201 270 1520 M. Sine. D. 11. Cosine. D. 1" Tang. D. 11. Cotang. M. 0 9.657047 4 3 9.949831 1 7 9.707166 5 0 0.292834 60 1.657295 413.919816 o7.707478 50.292522 59.657512 4'~ 1..') ~ t,.. 5.20 2.657542.12.949752 17.707790 520.292210 58 3.657790 412 9.949638 1.703102. 0.291898 57 4.65037 4'1.949623 1.70414.7044 291586 56 5.658234 1.949553.708726 20.291274 55 4.12 L.0i 5.10 6.655531 4.1 949494 10.709037 5.19.290963 54 7.658778 41.949429 L.709349 5.19.290651 53 8 659(025 41.949364 108.709660 5.19.290340 52 9.69271 1.949300.709971 18.290029 51 10 9.659517 41n 9.949235 8 9.710282 8 0.289718 50 I1.659763 410.949170.0.710593 5.8.239407 49 12.660009.4. 949105 1.08 710904 5.18.289096 48 13.660255 1.0 949040 103.711215 5.1.2i8785 47 14.6605m1 409.948975 I0.711525 5.1.238475 46 15.660746 09.948910 1.0.711836 s.238164 45 16.660991 40.948845 1.0.712146 5.7.2878354 44 17.661236 408.948780 1.09.712456 5.17.287544 43 18.661481 408.948715 1.0.712766 517.237234 42 19.661726 4.8.948650 109 713076..286924 41 20 9.661970 4 07 9.943584 1. 9.713386 0.286614 40 21.652214 407.948519 1 0.713696 56.236301 39 22.662459 407.948454 19.714005..285995 33 23.662703 406.948388 1.09.714314.6.285686 37 24.662946 40.948323 109.714624 5.'1,235376 36 25.663190 406.948257 09.714933 5.15.285067 3/ 26.663433 4.5.948192.9 715242 5..284758 34 27.663677 4.948126.715551,.234149 33 28.663920.948060.715860.284140 32 4.05 1.014 3 5.1431 29.664163.947995.716168 5.14.283832 31 30 9.664406 0 9.947929 10 9.716477 0.283523 30 31.66464891.947863 1.1'.716785 514.233215 29 32.664891 I4.947797 10.717093.14.282907 23 33.665133 40.3.947731 1.1.717401 513.282599 27 34.66375 403.947665.1.717709 513.282291 26 35.665617 403.947600'.718017 63.231983 25 36.665339.947533.0.718325 5.13.281675 24 37.666100 4.0947467 1I.718633.13..281367 23 4302 662 5.13 38.666342 4.2.917401 10.718940 5..281063 22 39.666583 0.947335 1.719248.280752 21 4.02 1. to 5.12 40 9.666824 01, 9.947269 9.719555 1 0.280445 20 41.667065 01.947203 1.1.719862 512.280133 19 42.667305 401.917136..720169..279831 43.667546 401.9i7070.720476.279524 17 44.667786.917004 111.720783.279217 16 45.663027 400.916937 1.721089.278911 15 46.663267 400.916371 11.721396 5.1.278604 14 47.663506 3.946304 1.1.721702.11.278298 13 43'.663746..946733 1.11.72209 5.o.277991 12 49.663986.96671 1'1.722315 10.277685 I1 50 9.669225: 9.916601. 9.722621 0.277379 10.:66: 19.61 39 1 0.279110 51.6691610 39.946533 1.11.722927.0.277073 9 52.669703 9.946471.723232,.276768 8 1.95 1.11 5.09'. 53.669912 39.946104.723.538.276462 7 1.11 5.09 54.670181 39.916337 1.1.723844 09.276156 6 55.670419 3.9{16270 1..724149.275851 5 56.670658 39.946203 1.12 724454 5.09.275.546 4 57.670396 97.946136 1.12.724760.09.275240 3 53.671134 396.946069 1.12 725065.274935 2 59.671372..946(102 1.12 725370..274630 1 g. 160.671609 1942.35.725674.274326 0 M. Cosine. DI. 1. Sine. D. 1". Cotang. D."it. Tang. M. 1170 10 202 TABLE XIII. LOGARITHMIC SINES, 8O. 151 M. Sine. D. ". Cosine. D. 1". Tang. D 1". Cotang. M 0 9.671609 9.94535 112 9.725674 508 0.274:6 60 1.671847 396.94536S 112.725979 5.274021 59 2.6720 31..945800..7262.2737J6 58 3.672321 95733.945733 412 4.672553.945666.2.726892.0.273103 O 6 3.95 1.12 5.07 5.672795 3.9.945598.727197 5.27203 55 6.673032 39.945531 1.1.727501 07.272199 54 7.67363.. 945464 1 727805.0.272195t 53 7.673-)63 3.94 1.13 5.06 8.673505 3.94.945396 1.13 728109.06.271891 t 52 9.673741 13.945328 13.728412 5..271588 51 3.93 1.13 5.06 10 9.673977 9.945261 9.728716 0.271284 50 1. 1.6743.945193 113.729020 6.270980 49 3.93 12.674143 33.945125 113.729323.270677 48 3.93 1.13 7.05 13.674634.945058 1.729626 5.270374 47 5.06 14.674919 3.92.944990 113.729929 50o.270071 46 15.675155 3.92.944922 13.730)233 5.269767 45 16.675390 3..941854.13.730535 05.26946.5 44 17.675624 391.944786 113.730538 505.269162 43 18.675353 31.944718 113.31141.26859 42 19.6760J4 3 91.944650.731444.26556 41 20 9.676328 390 9.944582 14 9.731746 5.04 0.268254 40 21.676562 30.944514.14.732043 5.267952.39 22.676796 390.944446.1.732351.04.267649 38 23.677030.90.944.377 14.732653.0.267347 37 24.677264 39.944309.732955.03.267045.36 25.677498 389.944241 1.7 33257 5.266743 35 26.677731 389.944172 1 4.73356 58.266442 34 27.677964 3 8.944104 1I.733860 5 3.266140 33.678197 3.944036 1.1.734162 02.265838 32 3.68 4 a 9 29.678430 38.943967.734463 5.265537 31.30 9.678663 8 9.943899 4 9.734764 r 02 0.2652.36 30 31.6789. 3.8.943330 11.735066.02.264934 29 32.679123 8.943761 1.735367 52.261633 28 33.679360 87.943693 1..735663 5 01.264332 27 34.679592 37.943624 1.73969 01.264031 26 336 11. 7 5.01:231 35.679324 3..943555 1I.736269 5.01.263731 25 36.60056 3.6.943486 15.736570 501.263430 24 37.630233 86.943417 1.1.736370 501.263130 23 33.630519.943318.737171.26229 22 39.630750 86.943279.737471.262529 21 3:85 1194 737 5.00 40 9.630932 9.943210 1 9.737771 5 00 0.262229 20 41.631213.943141 11.738071 00.261929 19 42.63144:3 3',.943072..738371.00.261629 18 43.631674 34.943003 11.733671.261329 17 3.84 1.15 738971 o.00 0 1 44.61905 3.942934 1.735971.261029 16 45.632135.94864 116.739271 49.260729 15 46.632365.942795 16.739570.260430 14 47.632595 383.942726 16.739870.260130 13 48.682323 3.83.942656.16 740169.2598331 12 49.683055 3:83..94287 16 40468 8.259532 I 50 9.633234 9.942517 116 9.740767 4598 0.259233 10 51.633514 382.942448 116.741066 98.2.58934 9 52.633743 82.942378 16.741,365 98.258635 8 3.82 1.516 4.98 53.633972 2.942.308 1..741664 498.258336 7 54.634201 381.942239 16.741962 498.258038 6 3.80 I. i 4.97.643(155 3.443.942169 16.742261.257739 5 53.611.941959 1..743156.256344 2 59,.63534 f3.941889: 117.7434 497 6 3.6-I 4.97 i' 60.635571.941819 1.743752 5624 0 M. Cosine. D.I. Sine. D. 1. Cotang. D. 1". Tang.. sO } r~~~~~~~~~~~~~~~~~~~~r >~~~~~~ COSINES, TANGENTS, AND COTANGENTS. 203 90 1500 M Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.685571 380 9.941819 117 9.743752 496 0.256243 60 1.635799 3.911749 17.741050 496.255950 59 2.66,27 3 7.911679 1.1.744348 496.255652 58 3.636254 37.941609 117.744645 96.255355 57 4.664S2 379.941539 1.7.744943 496.255057 56 5.6367) 3-78.941469 17.745240 496.254760 55 6.636936 378.941398 117.745533 4 9.254462 54 7.637163 378.9413283.7453 4.35 5.254165 53 8.637339.941258.746132..253863 52 9.637616 7.941187.746429 253571 51 6766 3.77 918 1.17 4.95 10 9.63734:3 77 9.941117 18 i.r46726 0.253274 50 11.6:3306J 37.911046.1.747023 4.5.252977 49 12.683295 7.940975 118.747319..252631 48 13.633521 376.910905 18.747616.252334 47 14.638747 376.940334 118.747913 494.252087 46 15.63972 376.940763 1.748209.251791 45 16.63919 376.940693 1'8.748505.251495 44 17.639423 3.940622 1.748801 3.251199 43 18.639618 5.940551 118.749097.93.250903 42 19.639873.940480 8.749393 93.250607 41 20 9.690098 3.5 9.940499 118 9.749689 4.3 0.250311 40 21.690323 37.940338 118.749985 3.250015 39 22.690548.940267 119.750281 3.249719 38 23.69)772 3 7.940196 119.750576 49.249424 37 24.693996 3.74.940125 1.1.750872.249128 36 25.691220 37.940054 1.19.751167 492.248833 35 26.691444 3-73.939982 119.751462 492.248533 34 27.691663 33.939911 11.751757 92.248243 33 28.691892 373.93940 1.19.752052 42.217948 32 29.692115.939768.752347 49.247653 31 3.72 1.19 4:91 30 9.692339 3 72 9.939697 19 9.752642 41 0.247358 30 31.692562 372.939625 119.752937 491.24763 29 32.692785 372.9395 4 19.753231 491.246769 23 33.693098 3 71.939482 119.753526 I.246474 27 31.693231 371 9:39410 1.9.753320 491.246180 26 35.693153..9.3.339 1'.754115 4 9.245885 25 36.69.3676 371.939267 12.754409 49.245591 24 37.633393 370.939195 1'20.754703 4:90.245297 23 38.691120 370.939123 12.754997.245003 22 39.694342 370.939052 120.755291 90.244709 21 40 9.694561 3 70 9.939830 120 9.755585 49 0.244415 20 41.6947S6 369.933903 120.755878 489.244122 19 42.695007 369.9338^36 120.756172 4 9.24.3829 18 43.6952269.933763 120.756465 89.243535 17 44.695450 369.933691 120.756759 4.8.243241 16 45.695671 363.933619 1 2.757052 489.242948 15 46.695892 36.933547 12.757345 4.9.242655 14 47.696113 3.63.9133475 21'.757638 4.8.242362 13 48.66331 363'.933402 121.757931 488.242069 12 49.69655.933330 1.758224.241776 11 50 9.636775 3 67 9.933258,1,2 9.758517 4. 0.241483 10'5.696995 367.93i185 121.758810 488.241190 9 52.697215 67.93i113 21.769102 4.8.240898 8 53.697435 366.933040,1.759395 487.240605 7 51.6 63; 3.6.937967 21.759687 4 7.240313 6 55.697874 3 6.937895. 1.759979 4.8.240021 5 56.693091 366.937822 1:21.760272 487.239728 4 57.698313 6.937749 21.760564 487.239436 3 5.3.6 1.91 93 58.698532 36.937676 12.760856..239144 2 93.65 1.21 4.86 59.698751 36.937604.1.761148 4.8.238852 1 60.698970 *~.937531'___.761439 _4~.238561 0 M. Cosine. D. 1". Sine. D. 1. Cotang. D. 1". Tang. M. i19= 600 204 TABLE XIII. LOGARITHMIC SINES, 303 14:9..65 1.22 4.86 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.698970 3.65 9.937531 1.22 9.761439 4.86 0.238561 60 1.699189 36.937458 122.761731 6.238269 ~9 2.699407 364.937385 1 22.762023 4.6.237977 58 3.699626 64.937312 22.762314 4'6.237686 57 4.699841 3.64.937238 22.762606 4.6.237394 66 5.700062 363.937165 22.762897 4..237103 55 6.700280 6.937092 122.763188'.236812 54 7.700498 3.937019 1 22.763479.236521 3.63 8.700716 363.936946 122.763770 4.5.236230 52 9.700933 362.936872 122.764061 485.23939 51 10 9.701151 62 9.936799 9.764352 0.23648 50 11.701368 3..936725'23.764643 4.8.235357 49 12.701585 362.936652 23 764933 4.235067 48 13.701802 361.936578 123.765224 4.84.234776 47 14.702019 361.936505 123.765514.8.234486 46 15.702236 361.936431 123.765805 484.234195 45 16.702452 361.936.357 12.766095 4.84.233905 44 17.702669 36.936284 23.766385.233615 43 3.60 1.23 ~~~~4.83 18.702885 360.936210 123.766675 4.83.23332 42 1 19.703101 360.936136 123.766965 4.83.233035 41 20 9.703317 9.936062 9.767255 0.232745 40 3.60 1.23 4.83 21.7035.33.935988 I.767545 83.232455 39 3.59 1/23 4.383 22.703749 3.935914 23.767834 4.83.232166 3 23.703964..935840 23.768124 4..231876 37 24.704179 3.935766 12.768414 4.2.231586 36 25.704395 3.935692 124.768703 42.231297 35 26.704610 58.935618 24.768992 82.231008 34 27.704825 8 935543 24.769281 42.2:30719 33 28.705040 38.9-35469.4.769571 42.230429 32 29.705254 3.935395 14.769860 1.230140 31 3.58 1.24 4.89 30 9.705469 3 9.935320 24 9.770148.81 0.229852 30 31.705683.57 -935246 124.770437 41.229663 29 32.705898 3.935171 124.770726 4.81.229274 28 37.706967 3.56.934798 1-25.772163 71.227832 23 38.707180 36.934723 712.772457 4.8.227543 22 39.707393.93504649.772745 9727297 21 i.67 4'96 1.25 772745 4.80 40.707606 3 9.9350574 12 9.773033 14 0.226967 20 41.707819 3.5.934499 125 773321 80.226679 19408 42.70802 3.56.934424 1.2.77368 4.81 226392 43.708245 3.54.934349 1.25 773896 4 O9.226104 17 3.708453..93274 3 25.77184 1.225816 16 45.708670 354.934199 1.774471 4.225529 1 36.708882.934123 12.774759 4.225241 14 7.709094 3.5 3448 775046.224954 1 43.709306.933973 12.775333 479.224667 12 49 1.0718.9343898 1 2.77621.224379 1 50 9.709730 9.933822 1/. 9.775908 4. 0.224092 10 51.709941 5.933747 126.776195 4.80.223805 952.710153.9343671 1.776482.223518 18 53.710364.933596.776768.223232 7 54.70575 3.9393520 12.777551 4.81.222945 1 55 710786 3.52 93445 1.26.777342 1.2226.3 15 56.710997 35 933369 2.777628 ^.22232 14 7 3.569 1.26 4. 2 57.711208.51 933293 16.777915 4..222085 58.711419 5 933217 126.778201 47.221799 1 53526. 9 22380 59.711629 35.933141 26.778488 1 7.221512 1 60.7118394 35.933066 1.2.778774.7 221226 0 I 1. Io2e.I D. 1d. Sim D. 1". Cotang. D. 11. Tang. M. _t900 35]. 22 3 88.1141 1.2 4:1. 7 5' 6.7787 4.4 60.719ri1839933898 1.21i.778774:22436 II. Cosine. " Sin. 1 tang. 1 1" Tang. COSINES, TANGENTS, AND COTANGENTS. 20~ 310 1480 M. Sine. D. 1". Cosine. D. 1". Tang. D. 11". Cotang M. 0 9.711839 0 9.933066 27 9.778774 4 0.221226 60 1.71200.932990 17.779060'.220940 59 2.712260 3.5 932914 1.27 779346 477.22(64 58 2260 3.50 1.93294 27 779346.77.2264 58 3.712469 3.50.932838.779632 47.220368 57 4.712679 3..932762 27.779918.22002 56 5.712889..932685 1.27 780203.219797 55 6.71309 3.49.932609 1 7 4.76.219511 54 7.713308 3.49.932533 127 780775 4.76.219225 53 8.713517.932533 1.27 4rB6'.21922.5'i3 8.713517 348 932457 127.781060 4i6.218940 52 9.713726 3.48.932380 1.27.781346 4.218654 51 10 9.713935 3 9932304 1.27 9.781631 475 0.218369 50 11.714144 48.932228'27.781916.218034 49 12.714352 3.43.932151 1'.782201..217799 48 13.714561..932075 1.782486 475.217514 47 14.714769 3.47.931998 1.2.782771 {.217229 46 15.714978 3.47 931921.783056.216944 45 16.715186 3.47 931845.783341.216659' 44 17.71539- 3.6 931763 128.783626..216374 43 18.715602 3.4.931691 12'.783910.216090 42 3.46 1.28 4 I74 19. 715809.931614 1.784195 44.215805 41 20 9.716017 9.931537 9.784479 0.215521 40 21.716224 3.46.931460.784764.215236 39 22.716132 3.45.931333 1.2.785048 474.214952 38 23.716639 3.4.931306 128.785332 4.4.21466 37 24.716846 345.931229 129.785616 47.214384 36 25.717053 3.931152.785900.214100 35 Z9 3.45 931075 I'29 4.73 S-2.717459.931075 729.786184 43.213816 34 3.44 1.29 4.73 27.717466.93099 29.76468 73.213532 33 28.717673 l.930921 19.786752.213248 32 29 71i7879 144.930843 129.787036 4'73.212964 31 30 9.718085 - 3 9.930766 9.787319 0.212681 30 31.718291.930638 19.787603'.212397 29 3.43 1.29 4.72 32.718497.930611 129.787886 472.212114 28 33 718703 33.930533 129.788170 472.211830 27 34.718909.930456 29.788453 1 7.211547 26 35.719114 32.930378 1.9.788736 472 211264 2. 36.719320 3.42.930300 130.789019 472.210981 24 37.719525 342.930223 130.789302 472.210698 23 38.719730 42.930145 13.789535.210415 22 II42 1.30 4.71 39.71993 3.41.930067 1 30.789868 4:71.210132 21 40 9.720140 341 9.929989 130 9.790151 4, 0.209849 20 41.720)345.4.929911 790434.209566 19 42.720549 341.929833 130.790716 ^1.209284 18 43.720754 3.929755 1.790999 4.71.209001 17 44.720958.929677 0.791281.20719 16 3.40 1.30 4.71 45.721162 3.40.929599 1.791563 4.7.203437 15 3.40.929521 46.721366.929521'.791846 4. 7 -.203154 14 3.40 1.30' 4.70 47.721.570 34.929442.3.792128 4.7.207872 13 48.721774 3..929364 11.792410 4.7.207590 12 49.721978 339.929286 131.792692 4 7.207308 11 3.39 91.71 4270 50 9.722181 3 9.929207 /, 9.792974' 0.207026 10 51.722385 3 929129' 1..793256 4.70.206744 / 52.722588 3 929050 131 793538 470.206462 8 53.722791 3.9.923972 131 793819 0 206181 7 54.722994 3.38.928893 1.31 794101 4.69.205399 6 55.723197.928815 13.794383 469.205617 5 56.723400 3.3.928736 31 794664 4.69 205336 4 57.723603.928657 794946 4.205054 3 3.37.723815 1.3 3 4.691 58.723305,.923578 13.795227 9.204773 2 59.724007 37.92.3499 132.795508 46.204492 1 60.724210.3 923420 1..795789!~-..204211 0 M. Cosine. D. I. Sine. D. 1~. Cotang. D. 1. Tang. M 9Vt -5 206 TABLE XIII. LOGARITHMIC SINES, 32g 1147~ M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.724210 9.928420 32 9.795789 48 0.204211 60 1.724412.928342 132.796070.6.203930 59 2.724614 3 928263. 796351 68.203649 58 3.724316 3.36 928183 796632 46.203368 5 4.725017 36.928104 1.796913 468.203087 56 5.725219.928025.797194.2028(6 55 6 725420 36 927946 3.797474.202526 54 7.725622 3..927867..797755 48.202245 53 8.725823.927787 1..798036 4.6.201964 52 9.726024 33.927708 132'798316 4 67.201684 51 10 9.726225 3.35 9.927629 1.32 9.798596 4.7 0.201404 50 11.726426 927549 798877 467.201123 49 12.726626 3.927470 1.33.799157 467.200843 48 13.726827 34 927390 33 799437 67.200563 47 14.727027 4.927310 133.799717 467.200283 46 15 -.727228 34.927231 3.799997 4.6.200003 45 16.727428.927151 13 800277 466.199723 44 3.33 1.33 17.727628.927071 33 800557 466.199443 43 18.727828.926991 33.800836 46.199164 42 19.728027 3.926911 133 801116 466.198884 41 20 9.728227 9.3 9926831 1 3 9.801396 4 0.198604 40 21.728427 3.926751.33.801675 4.66.19325 39 22.728626 2.926671 1..801955.66.198045 38 23.728825.32 926591.802234.197766 37 24.729024 3.32 926511 34 802513 65.197487 36 3.32 1.34 4.65 25.729223 331.926431.802792 4 5.197208 35 26.729422 31 926351 1..803072 465.196928 34 27.729621 331.926270 34.803351 65.196649 33 28.729820 3 31 926190 34 803630 465 96370 32 29.730018 331.926110 803909 4.196091 31 30 9.730217 3 30 9.926029 34 9.804187 65 0.195813 30 31.730415 3.92.949 1.3.804466 46.195534 29 32.730613..925868 134.804745 464.195255 28 33.730811 3.925788 34.805023.194977 27 34.731009 30.925707 13.805302 64.194698 26 35.731206 32.925626 13.805580 6.194420 25 36.731404 3.925545.805859 64.194141 24 37.731602 3.925465 1.5.806137 64.193863 23 38.731799 32.925384 135.806415 4.193585 22 39.731996 28.92503 35.806693 63.193307 21 40 9.732193 28 9.925222 9.806971 0.193029 20 41.732.390 28.925141 1.5.807249 463.192751 19 42.732587 328.925060 135.807527 463.192473 18 43.732784 8 924979 35.807805 463.192195 17 44.73290 3.27.924897 5.808083 46.191917 i 45.733177 327.924816 135.808361 463.191639 15 46.733373 27.924735 136.808638 463.191362 14 47.733569 327.924654.3.808916 42.191084 13 48.733765 327.924572 136.809193 462.190807 12 49.733961 26.924491 1:36.809471 4:62.190529 11 50 9.734157 326 9.924409 136 9.809748 62 0.190252 10 51.734353 326.924328 36.810025 462.189975 9 52.734549.924246.810302.189698 8 3.26 1.36 4.62 53.734744 32.924164 1.810580 62.189420 7 54 734939 3.25.924083 1.36.810857 4'62.189143 6 55.735135 35.924001 136.811134 4.61.188866 5 56.7353.30 325.923919 136.811410 4 6.188590 4 57.735525.5 923837 1.811687.61.188313 3 58.735719 3.923755'.811964 4.188036 2 59.735914.923673 1..812241 4 61.187759 1 60.736109.923591.812517.187483 0 M. CosIne. D. I". Sine. Cotang B. 1" Tang M. i220 5b7C COSINES, TANGENTS, AND COTANGENTS. 207 330 * M. Sine. D. 1". Cosine. D. 1. Tang. D. 1. Cotang. M. 0 9.736109 3.24 9.923591 1.37 9.812517 461 0.187483 6( 1.736303 24 923509.812794 46.187206 59 2.736498.923427.37.813070.61.106930 58 3.736692 33.9 23345 1.3.813347 46.186653 57 4.7:36386 23.923263.37.813623 4.6.186377 56 5.737080 3.923181..813899.186101 55 6.737274 323.923098 13.814176 4.60.185824 54 7.737467 23.923016 37.814452 46.185548 53 8.737661 22.922933 137.814723 4.185272 52 9.737855 22.922851 38.815004 40.184996 51 10 9.733048 22 9.922768 9.815280 0.184720 50 11.733241 3.22 922686 1.38 81.555 4.61 184445 *49 12.738434 3.22 922603 1.3.815-31.184169 48 13.738627..922520 1.38.816107 4.59.183893 47 14.738820 3.21.922438 1.3.816382 9.183618 46 15.739013 3.21 92235 1.3 816658.183342 45 3.21.9221 1.38 16.739206 321 922272 1.3.816933 4.59.183067 44 17.739398.922189 38.817209 9 18279 43 18.73990 3.21 922106 1..817484.6.182516 42 19.739783 20.922023 a3.817759 59.182241 41 20 9.739975 320 9.921940 1 9.818035 0.181965 40 21.740167 20.92187 1.39.818310 4.181690 39 22.740359 3.20.921774..818585 4.58.181415 33 2.3.740550 3.921691 1.39 818860.5.181140 37 24.740742 3.19 921607 39 819135 48.180865 36 25.740934 3.921524 1..819410..180590 35 3.61 1.39 4.58 26.741125.19.921441 1.39.819684 4.58.180316 34 27.741316.921357 1.39 819959.180041 33 3.19 1.39 4.58 7 28.741508 3.19.921274 1.39 820234 4.58.179766 32 29.741699 3.18.921190 1.39.820508 458.179492 31 3.18 1.39 4.58 30 9.741889 3.18 9.921107 139 9.820783 0.179217 30 31.74280 318.921023 9.821057 7.178943 29 32.742271.920939 1.3.821332 4.5.178668 28 33.742462 3.18.920856 40.821606 4.57.178394 27 34.742652 3.7.920772 1.40.821880. 178120 26 35.74242 317.920688 140.822154.177846 25 36.743033 3.17.920604 14.822429 4.5.177571 24 37.743233 3.920520 1..822703 4.5.177297 23 38.743413 3.1.920436 40 822977.177023 22 39.743602 3.920352 1.40.823251.176749 21 3.16 1.40 4.58 40 9.743792 3 16 9.920268 1 40 9.823524 56 0.176476 20 41.74392 16.920184 140.823798 4.56.176202 19 42.744171 16.920099 40 824072 4.56.175928 18.741508 3.16.920099 1.40.823456 43.744361.920015 1. 824345.17655 17'4 3.15 1.41 4.56 44.744550.15.919931 1.41.824619 456.175381 16 45.744739..919846..824893.175107 15 46.744928.15.919762 41 825166.174834 14 47.745117 3.15.919677 41 825439 4.174561 13 48.745306 3.1.9195923 1.825713.174287 12 49.745594 314.919508 1413.825986 4155.174014 11 50 9.74563 3 4 9.919424 4 9.826259 0.173741 1 0.742462 3.14 1.41 4.55 51.745871.919339.826532 4.173468 9 52.746060 3.1 919254.826805.173195 8 53.746248 3.14.919169 1.41 827078 4.55.172922 7 35.7428136 3. 91 141 54.746436 3.13.919035 14 827351 4.55 172649 6 55 3.13 1.42 4.55; 55.746624 3.13.919000 42 827624 4. 172376 5 56 3.1 1 1742:872 56.746812.3.918915 827897 5 172103 4 57.746999 3.13.918830 142 828170 4. 171830 3 58.747187 3.123.918745 142 828442 54.171558 2 59.747374.918659 828715 4.171285 1 918659 1.42 8271 60.747562 3.12.918574..828987 4..171013 0 M. Cosine. D. I". Sine D. 1". Cotang. D. 1". Tang. 31..7433 539 1.40 4.57.17023 2 208 TABLE XIII. LOGARITHMIIC SINES, 340 145 M. Sine. D. 1'. Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.747562 9.918574 i [ 9.8-3987 0.171013 60 1.747749 3.91H489 1.829260 1"5.170740 59 2.747936 3 918404 14.829532.17,)46i 58 3.12 1'42 4.893':.16 58 3.748123 3.1.918318. 14.829305 4.170195 657 4.748310 3.11.918233 [.830077 4.5.163923 56 5.748497.918147 [.830349 4'5.163651 55 6.748683.918062 147.062' 169379 54 7.743370 3.917976 1.830393.169107 53 8.749056 3.1.917891 4'.831163i 1'.16835 52 3~~~~~~~~~]~{.156{510 9.749243 3.917805 14.831437.168563 51 1 0 9.749129 31 9.917719 1.4 9.831709' 0.168291 50 II..749615 310.917634 1.43.831981 4:.168019 49 12.749801 31.917548 1'3.832253.167747 48 13.749987 310.917462 143.832525 4.167475 47 14.750172 309.917376 143.832796 4'5.167204 46 15.750358 30.917290 14.833068.166932 45 16.750543 309.917204 143.833339 452.166661 44 17.750729 309.917118 1'44.833611 4.166389 43 18.75094 3 09.917032 144.833882 /'.166118 42 19.751099 3.916946 1:4.834154 4:..165846 41 20 9.751284 3, 9.916359 {.4 9.834425 4e 52 0.165575 40 21.751409',.916773 144.834696 4'5.165304 39 22.7516.54 308.916687 1.834967 4'2.165033 38 23.751839 308.916603 144.835238 452.164762 37 24.752023 30^'.916511.835509 4.52 164491 36 25.752203 3.916427 8.35780 164220 35 26.752392 307.916341 144.836051 1'.163949 34 27 752576 3'7 916254 14.836322.163678 33 28 ~752760 3.07.916167 1~4.836593 4.51.163407 32 29.752944 \ 3.916031.836864 1 163136 31 3.06:1.45.394 4.51 3 30 9.753123 30 9.915994, 9.837134 0.1628 30 31.75:33121 9.837134 451 0.162866 30/ 31.753312 306.915907 |;5.837405 4.5.162595 29 32 {.753495 3.06^.915820 }.'.837675 451.16232.5 28 33 {.7536.',79 I.0 915733 1.45.837946 4.5.162054 27 34.753362 3.0 915646 ~1.4.838216 451.161784 26 35.754046 3.05.915559 1.4.838487 4-51.161513 25 36.75422 3.03.9135472 1.4.838757 450.161243 24 37.754412 3. 915335 1.4-.839027 4 5.160973 23 3 -,-, 075''1459'.'t;[38 3.704595 ~,',.915297 ~.4.839297 [.160703 22 I r. O.U5' I 3.~!1.' 4!.50 39.754778.915210 83956.160432 21 3.05 1.46 83956 4.50 40 9.754960 304 9.915123 / 9.839838.45 0.160162 20 41.755143.915035.84o1 08.159892 19 914948 1.463.840108.582 1 42.7553326 304.91 4948'.840378 450.159622 18 43.755503 304.914860 14.840648.50.1593,2 17 {44./75.3690 /'0.914773 6.840917 4.159083 I t; 44.753690 ~3.04 1.4645 45.[755872 [ 30].914635 1.841187.158813 15 46.756051 303:914598 1.46.841457 ].158543 14 47.756236 30.914510.46.841727 4'4.158273 I3 48.756118.914422 1.46.841996 7 158004 12 49.756600 934.842266.15778 12 50 9.756782.03 914334 1.46.84199 9.91'42'/46 4.15773.'1 1II 50 9.756782 302'9.914246 47 9.842535 44 0.15746.5 10 51.7.56963 0.914158, 842805.157195 9 52.757144 -^.914070 1.47 843074'.156926 8 53.757326 913982 1.47 843343.15667 7 3.02~~~ 147' 44 r.16657 54.757507 302 913894 ] 843612'9.156388 6 55.757638 02.913366 1.47 843882.1561 8 5 3 -0 -1.47'' 4.49''' 56.757869 3.913718 1 844151.155849 4 57.758050 ~ 913630 1.4 844420 44 8 58.758230 30 913541 1.47 844689 448.1558'?3 59.758411 301.913453 1.'.844958 4.48.., I 60.758591 v.913365 14.845227 {... 15477 ~ 1. Coine. D. 1". Sine. D. ". Cotong. )I. Tang. IM. ia4t 59 COSINES, TANGENTS, AND COTANGENTS. 2C9 350 14*4 M. Sine. D. 1" Cosine. D. 1. Tang. D. 1". Cotang. M. 0 9.758591 9.913365 1 47 9.845227 448 0.154773 60 1.758772 300 1.48 84596 448.154504 59 2.758952 3.913187 48.845764.154236 58 3.759132.91309.846033.153967 57 3.00 1.48 4.48 4.769312 30.913010 4.846302.153698 56 5.759492 30.912922 14.846570 48 153430 55 6.759672 29.912833 148.846839 448.153161 54 7.759852 299.912744 148.847108 47.152892 53 8.760J31 299.912655 148.847376.152624 52 9.760211 299.912566 1.48.847644 4.47.152356 51 10 9.760390 2.99 9.912477 148 9.847913 47 0.152087 50 ~11.760569 299.9123 8 148.848181.151819 49 12.760748 2.912299 149.848449.151551 48 13.760927 298.912210 49.848717.151283 47 14.761106 298.912121 149.848986.151014 46 15.761285 29.912031 149.849254.150746 45 16.761464 298.911942 4.849522.150478 44 17.761642 29.911853 149.849790 44.150210 43 18.761821 297.911763 149.850057 446.149943 4 19.761999 2.911674 1.350325.149675 41 2.97 1.49 4.46 20 9.7C2177 27 9.911584 149 9.850593 46 0.149407 40 21.762356 297.911495 149.850861 446.149139 39 22.762534 297.911405 149.851129 446.148871 38 23.762712 2:98.911315 1.851396 4:46.148604 37 24.762889 296.911226 150.851664 446.148336 36 25.763067.9.911136 150.851931 446.148069 35 26.763245 2.9.911046 I.852199 446.147801 34 27.763422 296.910956 1 6.852466 446.147534 33 28.763600 295.910866..852733 446.147267 32 29.763777 295.910776 1:50.853001 4:45.146999 31 30 9.763954 2. 9.910686 /10 9.853268 4 4 0.146732 30 31.764131 2.9 10596 5.853355 4.146465 29 32.764308.95 910506 1.0.853802 45.146198 28 33.764485 295.910415 1.'.854C69.145931 27 34.764662 2.910325.854336.145664 26 35.764838 94.910235 1.54603.145397 25 36.765015 2.910144..854870 4-4.14130 24 37.765191 9.910054.855137.144863 23 2.9:903 1.51 4.45 38.765267 24.999963 855404.144596 2 39..765544 2,93.90973 1.85671.144329 21 40 9.765720 2 9.3 9.909782 151 9.855938 4'4 0.144062 20 41.765896.909691 15.856204 4.143796 18 42.766072.93.909601 151.856471 44.143529 1S 43.766247 29.909510.856737.143263 17 2.93 1.51 4.44 44.766423 93.909419 1.5.857004 4.142996 16 45.766593 2..909328 1.857270..142730 15 2,92 - 1.52 4.44 46.766774 292.909237 152.857537 4.142463 1 47.766949 2.92.909146 15.857803.142197 13 48.767124 9.909055 2.858069.141931 12 292.76'80r.4[.0s 2 4.44 49.767300 2.903964 15.85833.141664'2.1I92 1.52' 4.44 50 9.767475,29. 9.908873 9.858602 4 0.141398 10 51.767649 9.908781 12.858868 443.141132 9 52.767824.908690.859134.466 1.52 r., p 4.43 53.767999 21.908599.859400.140600 7 54.76873 2.91 90507 152.8596.140334 6 55.762348 91.908416'.859932.14068 5 56.768522 2..90832.860198 443.139.802 4 57.7687 2.90.90.8233 5.860464.139536 3 53.76871 290.908141.86 0730 4.139270 2 2.90'1'53.1.5'1 4.43 59.769045: 903049 1.80995 139005 60.769219 2.90.90958 153.861261.13739 0 M' Cosine. D.l Sine.'D.'. otng. -. 1 t. TD. ang. M. I50 54 210 TABLE XIII. LOGARITHMIC SINES, 360 143' M. Sine. D. I". Cosine. D. I". Tang. D. 1". Cotang. I M. 0 9.769219 290 9.907958 153 9.861261 0.138739 60 1.769393.907866'.861527.138473 59 2.769566 289.907774 1.53 861792.8613920 58 3.769740 289.907682 153.862058 42.137942 57 4.769913 289.907590 53.862323 42.137677 56 5.770087 289.907498 53.862589 4.42.137411 55 6.770260 289.907406 154.862854.42.137146 54 7.770433 288.907314 1.863119 4.4.136881 53 8.770606 288.907222 154.863385 442.136615 52 9.770779 2:88.907129 1:54.863650 4:42.136350 51 10 9.770952 2'8 9.907037 i 54 9.863915 442 0.136085 50 11.771125 288.906945 154.864180 442.13,5820 49 12.771298 2:88.906852 1.864445 442.135555 48 13.771470 2 87.906760 I.864710,42.135290 47 14.771643 287.906667'.864975.135025 46 15.771815 287.906575 154.865240 441.134760 45 16.771987 287.906482 155.865505 441.134495 44 17.772159 287.906389 I'I.865770 441.134230 43 18.772331 87.906296 1.866035 441.13.3965 42 19.772503 286.906204 ].866300 41.133700 41 20 9.772675 286 9.906111 1. 5 9.866564 441 0.1.33436 40 21.772847 286.906018 1.866829 441.133171 39 22.773018 2.905925'.867094.132906 38 23.773190 286.905832 I's.867358 441.132642 37 24.773361 85 905739.867623 41.132377 36 25.773533 85.905645'.867887 41.132113 35 26.773704.905552 1'5.868152.131848 34 2.85 156.773875 68416 441.131584 33 28.774046 2.905366 1.865680 4.131320 32 29.774217 2.85.905272 1.868945 440.131055 31 30 9.774388 9.905179 9.869209 0.130791 30 31.774558 284.905085 1/56.869473 440.130527 29 32.774729 2:94.904992 156.869737'40.130263 23 33.774899 2.84.904898.56.870001 440.129999 27 34.775070 2.84.904804 1.870265 4'.129735 26 35 775240 2.84.904711 /:'.870529 440.129471 25 36.775410.904617.870793.129207 24 2M. 815640'292017/ 24![ 37.775530 83 904523 57.871057;40.128943 23 38.775750.904429 157.871321 4.128679 22 39.775920 2.83 1.57.871585.40.128415 21 40 9.776090 2.8 9.90421 1.57: 9.871849 4.40 0.128151 20 41.776259.904147 57.872112.127888 19 42.776429.904053 157.872376.127624 18 43.776598.903959 57.872640.127360 17 44.776768 2.82 903864 157.872903.127097 16 45.776937 2.82 903770 57.873167 4 126833 15 46.777106 2.82 903676 157.873430. 126570 14 47.777275.903581.873694 126306 1 2.82 1657 4.39.166 48.777444.903487 1~8.873957.126043 1I 2 49.777613 2.81 903392 158.874220 125780 11 50 9.777781 9.903298 /1 8 9.874484 / 0.125516 10 51.777950'903203 2.874747 /.122553 9 52.778119 2.81 903108 158.875010..124990 8 53.778287 2.81.903014 1/58.875273.124727 7 54.778455 2.81.902919 15.875537.124463 6 55.778624 2.80.902824 1.875800.124200 5 56.778792 I 902729;.876063 /,123937 4 II 57 /.778960 /28,0.902634 /.876326.12:3674 3 58.779128 20.90253o9 /16.876589 /.123411 2 59.779295 2.8 902444 /.876852 /.123148 60.779463 2.79 902349.877114.1286 M. Cosine. D. ". Sine".D. 1". Cotang. D. 1,". Tag. 1 "60 53. COSINES, TANGENTS, AND COTANGENTS. 211 373 1412D 37 ______________ - ~__ ____ _____ _ 14.___ M. Sine. D. I". Cosine. D. 1I. Tang. D. 1". Cotang. 3M. 0 9.779463 279 9.(902349 1r9 9.877114 4 (. 122886 6(0 1.779631 9.02253.877377.122623 59 2.79 1.59 4.38 2.779798 279.90 2158 1'..812640 34..12236 5S 3.779966.9 3.87793.122097 57 ~-~ 9lr03 1.' 9.8779131.1 4I38 4.780133 27.901967 1.9.87165 8.121835 56 5.7803C00 278.901872 1.9.878428 38.121572 55 6.780467 278.901776 19.87s691 438.121309 54 7.780634 2.9016S31 1.878953 48.121047 53 8.780801 27.90155..879216 4.3.120784 52 z.78. ~ 4.37 9.780968 278.901490 159.879478 43.12(1522 51 10 9.781134 2.78 9.901394 1.60 9.879741 4 0.120259 50 II.781301 277.901298 160. 88K13.119997 49 12 8;1468 [ 4.37 12.781468 277.901202 60.880265 4.37.119735 48 13.781634 277.901106 6.8b1528.119472 47 14.781800 277.901010 I-.889 4.119210 46 15.781966 2-77.900914 I.8,105(2 37.118948 45 16.782132 2-77.900818 160 881314 7.1186S6 44 17.782298 7.900722 160.881577.118423 43 8.782464 2 60 64.37 18.782464 2.76.900626 60.881839.118161 42 19.782630 7.900529 1.882101.117899 41 2.76 1.61 4.37 20 9.782796 276 9.900433 161 9.882363 0.117637 40 21.782961 76.900337 161.882625.117375 39 22.783127 276.900240.882887.117113 38' I 1.61 4.36 2.3.783292 2.7.900144 161.88314 4.36.116852 37 24.783458 25.900047 16.883410 4'.159 36 2.75 90.0v4 4.36 I I 25.783623 27.899951 16.883672 4.116328 35 26.783788 275.899854..883934 4.36.11666 27.783953 275.899767 61.884196 6.115804 33 2.75 1 61'81 4.36 28.784118 75.899660 1'1.884457.115543 32 29.784282 74.899564.884719.1121 31 2.74 1.62 4.36 30 9.784447 274 9.899467 62 9.884980 36 0.115020.30 31.784612 2.899370 1.62.885242 6.114758 29 32.784776 24.899273 162.885504 4.114496 28 33.784941 274 899176 162 885765 36.114235 27 34.785105'.899078 162 886026.113974 26 35.785269 273.898981.8862S8 4.113712 25I 36.785433 2.898884 16.88649 4.36.1451 24 37.785597 273 898787 62.886811.113189 1 273' ~1.62 4.35 38.785761.89869.887072 3.112928 22 39.785925 273.898592 6.887333.112667 21 2.73 1.62 -1'862 8 4.3 5 40 9.786089 23 9.898494':1 9.887594 0.112406f 20 41.786252 3.898397.887855.112145 12 42.8641 4.2713.11168 42.786416 27.898299 16.888116 435.1114 1 43.786579 72.898202 1.888.378..111622 17 2472 1763 4435 86 44.786742 7.898104 1.888639.111361 16 45.786906 22.898006 13.88900 4.3.1111 i 1' 46.787069 272.8979(8 163.889161 435.11 839 1 1 47.787232 272.897810 163 889421 4.110 1 49.787557 27.897614.889943.110057 11 1.63 4.35 50 9.787720 9.897516 9.890204 (.109796 1 51.787883 71.897418 164.90465.11533 4.35 52.788045 21.897320.64 890725 4 5.l2.53.788208.897222.8909S6. 0I0.tt 7.54.788370 2..897123 1.6.891247 4..IS(;753 6 55.788532 2.897. 891.07.' (103 2.70.891507' [,[.C 5 56.788694.896926 1.64 891768 4.12:2 4 2.70 1:I 6t' 902 4.34l 57.7888.56 27.896828 1.4.892028. 0.107972 3 5S.789018 2.70.896729 1.64.892289.10(7711 2 59.789180 2..896631 1.64.892549 1.107451 1 60.789342 2.70.896532 1.64.892810.10(7190 0 M. ICoine. D. inI.. - D. 1". Cotang. i D.I". Tatng. MI. L27L. 212 TABLE XIII. LOGARITHMIC SINES, 380 __________________________14 1K M1i Si. S D. oinsine. DD. Coin D.1. ". Tan D.'. Cotang. M 0 9.789342 269 9.896532 1 6 9.892810 34 0.107190 60 1.789504 2.69 896433.65.893070 4.106930 59 2.789665 6.8963.35..893331 4.106669 58 3.789627 2.69.896236 1 6.893591 4.106409 57 4.789988 69.896137 65.893851 4.106149 56 5.790149 269.896038 65.894111..105889 55 6.790310 68.895939 6.894372 4.105628 54 2.68 1.65' 4.34 7.790471 2.895840 15.894632.1015268 53 8.790632 68.895741 165.894892 4.105108 52 9.790793 2.68.895641 165.895152.104848 51 10 9.790954 2 68 9.895542 166 9.895412' 0.104588 50 11.791115 68.895443..895672.104328 49 12.791275 67.895343 66.895932.104068 48 13.791436 26.895244'6.896192.103808 47 2.67 1.66 4.33 14.791596 26.895145 1.896452.3.103548 46 15.791757 267.895045 166.896712 4..103288 45 16.791917 267.894945 166.896971.103029 44 17.792077 267.894846 166.897231 4.1.102769 43 18.792237 67.894746.897491.102509 42 2.67 1.66 4 33 19.792397 2.66.894646 1.66.897751 33.102249 41 20 9.792557 66 9.894546 167 9.898010 a 0.101990 40 21.792716 266.894446 67.898270.101730 39 22.792876 2.66.894346 1.67 898530 4.33.101470 38 23.793035 66.-894246 167.898789 33.101211 37 24.793195 26.894146 1 6.899049..100951 36 25.793354 2.6.894046 16.899308 43 100692 36 26.793514 26.893946 17.899568 43.100432 34 ~2.6 1.67 4.32 27.793673.65.893846 167.899827 4.32.100173 33 28.793832 265.893745 1 7.900087 432.099913 32 29.793991 65.893645 1 967.900346.099654 31 30 9.794150 265 9.893544 68 9.900605 32 0.099395 30 31.794308 264.893444 68.900864 2.099136 29 32.794467 264.893343 168.901124.098876 28 33.794626 2.893243.901383.098617 27 34.794784 4.893142.901642 32.098358 26 33.79462.64.893243.901383. 032 8617 27 35.794942 26.893041 1..901901 4.098099 25 36.795101.64.892940 168.902160 4.097840 24 37.795259 264.892839 1.902420 4..097580 23 38.795417 23.892739 16.902679.09732] 22 39.795575.892638 68'^.902938.097062 21 2.63.61.88' 4'$,32 221 40 9.795733 2 63 9.892536 1 69 9.903197 4 32 0.096803 20 41.795891 63.92435 6.903456.096544 19 2.63 1.69~ 4;31 42.796049 23.892334 169.903714 4.096286 18 43.796206.892233.903973.096027 17 44.796364 2.6.892132 19.904232 431.095768 16 45.796521 262.892030.69.904491 31.095509 15 46.796679 262.891929 169.904750 41.095250 14 47.796836 262.891827 19.905008 431.094992 13 48.796993 262.891726 169.905267 3.094733 12 49.797150 21.891624 1.905526 1.094474 11 2.61''8~2 1.69 4'31 50 9797307 2 619891523 1 9.905785 431 0.094215 10 51.797464..891421.906043.9397 9 2.61' 1.70'.90 4b310 52.797621 261.891319 1 906302 431.093698 8 53.797777 2 1.891217 1.7.906560 41'.093440 7 54.797934.891115.9069.093181 6 2.61 1.70.9019 4:31.093 6 56.79S091 261.891013 1..907077 431.092923 5 56.798247 6.890911 1..907336 431.092664 4 57.798403 2.890809,v7.907594 4.31.092406 3 2.60 1.70 4.31 58.798560 260.890707 17.907853 4.31.092147 2 59.798716 60.890605 0.908111 4.3.091889 1 60.798872..890503.70.908369 4..091631 0 IL Cosrue. D. 1". 8.i D.!. Cotug. -D.l1,. TaLg. M. 980& 91ii COSINES, TANGENTS, AND COTANGENTS. 21 390 140; M. Sine. D. I".' Cosine. D. 1". Tang. D.1". Cotang. M. 0 9.7938722 9.890503 9.90S369 4 0.091631 60 1.799028 60.890100 1.1.908628 30.091372 59 2.799184 2..890298'.903886.091114 58 3.799339.890195 1.9044.090856.57 4.799495 5.890093 71.909402.090598 56 5.799651 259.889990 17.909660.090:340 55 2.59 - 1.71 4.30 6.799806 25.889888 171.909918 4.090082 54 7.799962 2,.889785 11.910177.089823 53 8.800117 2.59.889682 17.910435 4.08956. 52 9.800272 2.59.889579 171.910693 43.089307 51 10 9.800427 2 5 9.839477 72 9.910951 0 0.089049 50 11.800582.839374 12.911209 40.088791 49 12.803737 2.58 889271 1.7.911467.088533 48 13.800892 2.5 889168 1..911725..088275 47 14.801047 2.8.889064 1.72.911982 4.3.038018 46 15.801201.28 888961 172.912240 430.087760 45 16.801356 8.888853 172.912498 430.037502 44 17.801511 2.57.88755'72.912756.087244 43 18.801665.888651 17.913014.086986 42 19.801819 27.888548 17.913271'.086729 41 20 9.801973 2 7 9.888444 1 7 9.913529 4 29 0.086471 40 21.802123 2.5.888341 173.913787 429 086213 39 22.802282 257.888237 173.914044 429.085956 38 23.802436 2.5.888134 1.3.914302 429.085698 37 24.802589 256.888030 1.73.914560 2.085440 36 25.802743 429 25.802743 2.56.887926 1.73.914817 4.9.085183 35 28.803234. 4 29 26.802897 2.56 887822 1.73.915075 429.084925 34 27.803050 2.56.887718 1.3.915332 42.084668 33 28.803204 2.56.837614 1.73.915590 4.9.084410 32 29.803357 2:55.887510 1'74.915847 429.084153 31 30 9.803511 2 9.887406 74 9.916104 4 29 0.083896 30 31.803664 2.5.887302..916362 429.033638 29 2.55 1.74 4.29 32.8073817..887198 174.916619.083381 28 33.803970.837093 14.916377.083123 27 34.804123 2..88989.917134 082666 26 35.804276.55.86885 }..917391 42.02609 25 36.804428..886780 1.74.917648 1.082352 24 37.804581 2.54.886676 4.917906.082094 23 38.804734 2.54 886571 1.74.98163.081837 22 39.804886 254 886466.74.91 420 49.081580 21 2.54 - 1.76 910 4.29 0815 1 40 9.805039 254 9.886362 1 9.918677 4 28 0.081323 20 41.805191 25.886257'.7.918934..081066 19 42.805343 2.54.886152 1.7.919191 428.080809 18'43.805495 253.886047 1.75 919448 4.28.080552 17 44.805.80647 2.885942.919705 48.080295 16 45.805799.5.885837 1.75.919962 428.080038 15 46.805951 253.885732 17.920219 28.079781 14 47.806103 2.53.886627 17.920476 4..079524 13 48.806254.53.885522 175.920733 28.079267 12 49.806406 2.885416 1'7.920990 28.079010 I 50 9.806557.5 9.885311 9,.921247 42 0.078753 10 51.806709 52..82 05 176.921603 42.078497 9 52.806860 25.685100 176.921760 428.078240 8 2.52 8046 4.28 ] 53.807011 2.884994 22017 2.077983 7 4.807163 52.884889 7.922274 4.28.077726 6 55.8073I4 2.52.884783 1.7 i2530 4..077470 5 56.807465 2.51.884677 1.76.922787 428.077213 4 7.807615..884572 16.923044 428.076956 3 53.807766 2.51.884466..92300 428.076700 2 59.807917 2.51.884360 923557 42.076443 1 60.803067 2.1.884254 1.7 2314 4.2.076186 0.9.'Cosine. D.Ill..Sine. D. l1 9 Cotang. D/.l'. Tang. M.._ _,.. I.o. -. 214 TABLE XII. LOGARITHMIC SINES, 4Io 1390 M ine. 1 Cosine. I. 1". CoTang. DD. 1. TCotang. M. 0 9.808067 9.8842 54 9.923814 0.076186 60 1.808218 251.884148 1'77.924070 28 75930 59 2.808368 251.884042..924327 427.075673 58 3.808519 50.883936 17.924583 27.075417 57 4.808669.883829 77.924840 427.075160 56 5.808819 50.88.3723 177.925096 47.074904 55 6.808969 25'.883617 177.925352 427.074648 54 7.809119 2.883510 1.925609 427.074391 53 8.809269 2 50.883401 8.925865 427.074135 52 9.809419 2.50.883297 78.926122 4'27.073878 51 10 9.809569 2 9 9.&83191 1.8 9.926378 4 27 0.073622 50 11.809718 24.883084 I.926634.073366 49 12.809868 2..882977.78.926890 42.073110 48 13.810017 2.49 882871 1..927147 427.072853 47 14.810167 2.882764.78.927403 27.072597 46 15.810316 2.49.882657 18.92659 427.072341 45 16.810465 248.882550 178.927915 427.072085 44 17.810614 2.882443 179.928171.27.071829 43 18.810763 24.882336 79.923427 42.071573 42 2.48 179 4.27 071316 41 19.810912 2:4.882229:79.92S634 4 27.071316 41 20 9.811061 988-2121 17 9.928940 4 0.071060 40 2 811 2'48 1'79 4.27 21.811210 248.882014 1.99196.2.07004 39 22.811358 2.881907.929452 42.070548 38 I 23.811507 24.881799 1.9.297 4.2.070292 37 2247.88179 427.070292 37 24.8116.o 247 88183692.9329964 4 27.070036 36 25.811804.881584..93122() 47.069780 35 26.811952 247.881477 1,.93(0475 26.069525 34 27.812100 247.831369..930731 26.069269 33 28.812248 7.881261 180.930987 {26.06()13 32 2.47 4.26 29 812396 2.81153.931243.(63757:1 30 9.812544 9.881046 1 9.931499 2 0.068501 30, 31.812692 2.46.880933 1 8.93175.0624 29 32.812840 2.880330.93210 46.067990 23 33.812988..880722 8.9322066 426.067734 27 34.813135 2..880613 1..932-2 46.0674734 35.8132 2.46 803 1.80 932522 4.2626 35.81343 2.46.83050.3 10.932778 26.067222 25 36.813430.4.880397 1 8.933033 6.066967 24 37.813578.880289'.933289'426.066711 23 38.813725 2.5.880180..933545 26.066455 22 39.813872.45 880072.933 066200 21 0 2.45 1.81 6.4.26 40 9.814019 2 5 9.879963 / 8i 9.934056 426 0.065944 20 41.814166 24.879855.934311 426.065689 19 42.814313 2..879746 1..934567 426.065433 18 43 8 0 2.45 1.01 4.26 0 43 814460 2.4.879637.934822.065178 17 44.814607 244.879529 8.935078.064922 16 45.814763 2.44 2.87940.935333 4.064667 15 46.814900 24.879341 I' 1.935589 426.064416 1 2.44 1.82 1 4.26.064411 14 47.815046.879202.935844.064166 13 48.815193 2.44.879093 1.2.936100 426.063900 12 49.815339 2.44.878984 8.936355 6.063645 2.44 1.82 4.26.063645 11 50 9.815485 244 9.878875 1/82 9.936611 46 0.063389 10 51.815632 2.878766.936866 4.063134 9 562.815778 34.216 I 966. 52.815778 2.43.878656 82.937121 -.062879 8 63.815924 243.878547 182.937377 4.6 062623 7 54.816069.878438.937632 4.06236 6 55 816215 2.43 182.93763 2 6 S5.816215.'.878328.-.937887 4.25.062113 56.816361.878219..938142.061858 4 57.816.507.878109 1.8.938398 425.061602 3 9 68.81668 242.877999.938653.-.061347 2 11 59.879.4.877890..938908 4.2.061092 1 i6o.816943 87.877780 4/.939163.060837 0 If.. Cosine. D. 1. Sine. D. 1". Cotang. D. D.n Tang. M. 1Sr 4 COSINES, TANGENTS, AND COTANGENTS. 215 ~10 1383 M, ilu. Sine. D. 1". Cosine. D.. Tang. D. 1. Cotag. M. 0 9.816943 9.877780 9.939163 0.060837 60 1.817088.42.877670 I83 939418 425.060582 59 2.817233 242.877560 183.939673 4255.060327 58 3.817379 242.877450 83 939928.060072 57 4.817524 242.877340 184.940183 425.059817 56 5.817668 241.877230 184.940439 42..059561 55 6.817813 24.877120 1.940694 4 2.0593(6 54 7.817958.41.877010 1.4 940949 *.059051 53 8.818103 2.4.876899 184.941204 4.25.058796 52 9.818247 2.8 9 4.78941459'.058541 51 2.41 9425 10 9.818392 9.876678 8 9.941713 4 0.058287 50 11.818536 2.41.876568 1.4.941968 4.5.058032 49 12.818681 2.4.876457 1.84.942223 15.057777 48 13.818825 240.876347.84.942478 25 057522 47 14.818969 240.876236 185.942733 4.25.057267 46 15.819113 24.876125 185.942988 45.057012 45 16.819257 2.0.876014 5.943243.056757 44 17.819401 2.40 875904 1.85 943498.5.056502 43 18.819545 2.40.875793 185.943752 2.056248 42 19.819689:24.875682 15.944007 -.055993 41 2.39 1.85 4.25 20 9.819832 239 9.875571 185 9.944262 425 0.05573 40 21.819976 239.875459 185.944517 425.055483 39 22.820120 239.875348 1.944771 4.055229 38 23.820263 2.8752.37 18.945026 44.054974 37 24.820406 39.875126 186.94528l 424.054719 36 25.820550 2..875014 86.945535 *.054465 35 26.820693 2.3.874903 186.945790 424.04210 34 27.820836 2.8.874791 8.946045.053955 33 28.820979 238.874680..946299 4-2.053701 32 29.821122 2.874568 186 94654 24.053446 31 2.38 1.86 4.24 30 9.821265 38 9.874456 18 9.946808 424 0.053192 30 31.821407 238.874344 186.947063 4.4.052937 29 32.821550 238.874232 1.8.947318 424.052682 28 33.821693 23.874121 187.947572.052428 27 34.821835.3874009'87.947827 44.052173 26 35.821977 2..873896 1 8.948081 424.051919 25 36.822120 2.873784..948335.051665 24 37.822262 237.873672 1..9485901.051410 23 38.822404 237.873660,..948844 4'.051156 22 39 822546.873448 1:8.949099 44.050901 21 2.37 1487 40 9.822688 o7 9.873335.7 9.949363 44 0.050647 20 41.822830 23.873223 i..949608 424.050392 19 42.822972 23.873110 1..949862.050138 18 43.823114 236.872998 1.88.950116 4.4.049884 17 44.823255 2.872885.950371.049629 16 2.36 1.88 4.24 45.823397 23.872772 18.950625 2.049375 15 46.82353872659..8 9.950879 1.049121 14 47.823680 2..872547 1.88.951133.048867 13 48.823821 2.36.872434 1.88.951388 424.048612 12 49.823963 2.35.872321 1.88.951642 24.04838 11 2.35 1.88.324 50 9.824104- 35 9.872208 9.961896 0.048104 10 51.824245.872095.89.952150.047850 9 52.824386 3.871981 19.962405.047595 8 53.824527 2.35.871868 1.89.95269 4.24.047341 7 64.824f668 ^2.35.87 1755..952913..047087 6 55.824808 2-35.871641 1.89.953167'24.046833 5 56.824949!.871528.953421.04679 4 57.825090 24.871414 1.89.953675 2.046325 3 2.34 19M 4.23:.046371 2 58.825230 24.871301 1.89.953929.06071 2 59.825371 2.34 871187 189.954183.045817 1 2..82aS 4L 1.90 4.23.045563 0 60.825511. 871073.954437045563 M. Cosine. D. I". Sine. D. 11". Cotang. D. 1". Tg- M. 1310 -43 13s~~~~~~~~~-~4O 216 TABLE XIII. LOGARITHMIC SINES, 42o 1370 M. Sine. D. 1"' Cosine. D'.l". Tang. D. 1". Cotang. M. 0 9.825511 9.871073 9.954437 0.045563 60 1.825651 2,.34.870960 190.954691 423.045309 59 2.825791 233.870346 19.954946 42.045054 5 3.825931 23.870732 190.955200 423.044800 57 4.826071 233.870618 19.955454 423.044546 56 5.826211 3.870504 190.95708..044292 55 6.826351 23.870390 190.955961 23.044039.54 7.826191 233.870276 190.956215 423.043785 53 8.826631 2.870161 1'.956469 4.043531 52 9.826770 23.870047 1 9.956723 3.043277 51 10 9.826910 o'2 9.869933,19 9.956977 4o3 0.043023 50 11.827049 232.869818 191.957231 4.23.042769 4 12.827189 232.869704 1 91.957485 423.042515 4S 13.827328 23.869539 9.957739.042261 47 14.827467 232.869474 91.957993 423.042007 46 15.827606.3.869360 191.958247 43.041753 45 16.827745 232.869245 1.958500 423.041500 44 17.827884 21.869130 - 9.955754 42.041246 43 18.823023 23.869015'2.9509008'423.040992 42 2.31 1 92 4.23 19.823162.863900.959262 43.040738 41 2.3i r.se 4.23 20 9.823301 2,, 9.863785 2 9.959516 423 0.040484 40 21.823439 231.863670 192.959769 4'23.040231 39 22.823578 231.868.55, 1.92.960023 423.039977 38 23.828716 21.863140 19.963277.039723 37 24.8283853.863324.'.960530 4.2.039470 36 25.823993..86209 192.960784,43.039216 35 26.829131 230.863093. 9.961(03 423.033962 34 27.829269 23.867978 i.9.96129 4 2.038703 33 23.829107 230.867862.93.961515 42.3.0331 32 29.829545 2:30.867747 1:93.961799 4 23.033201 31 33 9.829633 230 9.867631 193 9.962052 42 0.0:3791 30 31.829321 230.867515 193.962306 423.037691 29 32.829959 22.867399 93.962560 42.0374 0 2J 33.830)97..867233 1.962313 42.037187 27 31.830234 229.867167 193.963 )67 43.036933 26 35 830372 22.867051 L 9.963320 4 036630 25, 2.29'.94 4.23 36.8509 229.866935.963574 036126 24 37.839646 229.866319 194.963328 423 0:6172 23 38.830784 2.866703 I.964081 4.3919 22 39.830921 229.866586 1:.964335 43.035665 21 40 9.831053 2. 9.866470 94 9.964588 4 0.035412 20 41.831195 28.866353 194.964842 422.035158 19 42.831332 22.866237 194.96.5095'.034905 18 43.831469 22.866120.965349 4.0346.1 17 2.,28'866120 I 94 4.2 034 44.831606 2 2.866004 1.965602.2.034393 16 45.831742 2.865837 1'.965855 4..034145 15 46.831879 2.865770'.9.966109 4..033391 14 47.832015 223..865653 195.966362 4.22.033638 13 48.832152.2.865536 1 5.9666 6 4.22.033384 12 49.832288 7.865419.966369.033131 11 2.27 1.95 4.22 50 9.83242.5 227 9.865302 1.95 9.967123 4.22 0.032977 10 51.832.561 2.865185 1 9.967376 422.032624 9 52.832697.7.865063. 5..967629 42.032371 8 53.832333 2..864950 19.967883 4.22.02117 7 54.832969.864833.968136 422.031864 6 55.833105 2.864716 19.968389 42.031611 5 56.83.324 26.864598 19.968643 42.031357 4 57.833377 226.864431 196.963896 2 031104 3 58.83312 26.864363 196.969149 422.030351 2 59.8.6t3 6' 26.864-245 i196.969403' 22.030597 I 60.833783.864127.969656 "'..030344 0 M. Cosine. I. D.I 8ine. D. I. Cotang. D. 1I. Tan g. 1i3' 4~' COSINES, TANGENTS, AND COTANGENTS. 217 1.30 136 M. Sine. D. I". Cosine. D. 1". Tang. D. 1". Cbtang. M. 0 9.833783 2 9.864127 96 9.969656 2 0.030344 60 1.833919 26.8641010 1..969909 42.030091 59 2.834054 2.863S92 1'9.970162 422.029838 58 3.834189 2..863774 1..970416 422.029584 57 4.834325.86366 197.970669 42.029331 56 5.834460 2.2.863533 197.970922 422.029078 55 C.831595 225.863419 19.971175 4.028825 54 7.834730 225.863301 1.971429 422.028571 53 8.834865 I5.863183 197.971682 422.0283i8 52 9.834999.25.863064 197.971935 4.22.028065 51 10 9.835134 24 9.862946 1 8 9.972188 2 0.027812 50 11.835269 224.862827.972441 422.027559 49 12.835403 2..862709 198.972695 22.027305 48 13.83538 24.862590 I98.972948 22.027052 47 14.835672 224.862471 198.973201 22.026799 46 15.835807 224.862353.98.973454 22.026546 45 16.835941 24.862234 198 973707 22.026293 44 17.836075 223.862115 1.973960 22.026040 43 18.836209 223.861996.98.974213 422.025787 42 19.836343 23.861877 19.974466 422.025534 41 20 9.836477 23 9.861758 9.974720 4 2 0.025280 40 21.836611 223.861638 9.974973 4.22.025027 39 22.836745 223.861519.99.975226 422.024774 38 23.836878 2.23.861400.99.975479 22.024521 37 24.837012,'.861280 1.975732 42.024268 36 25.837146 -.861161.975985.024015 35 2.22 1.99 4.22 26.837279..861041 1.9.976238 422.023762 34 27.837412.22.860922.I.976491 422.023509 33 28.837546 22.860802 20.976744 2.023256 32 29.8 9. 837679 60682 2 00.976997 4.023003 31 2.22 2.00 4q22 30 9.837812 222 9.860562 20 9:977250 42o 0.022750 30 31.837945 222.860442 2.00.977503 422.022497 29 32.838078.860322 2.00.977756 422.022244 28 33.838211 2..860202' 0.978009 422.021991 27 34.838344 2.21.860082 2.00.978262 422.021738 26 35.838477 221.859962 2.0.978515 422.021485 25 36.838610 2.21.859842.00.978768 4.22.021232 24 37.838742 2.21.859721 2.01.979021 4.22.020979 23 2.21 2.01 4.22 38.838875..859601 2.979274 42.020726 22 40 9.839140 2 2 9.859360 01 9.979780 0.020220 20 41.839272 22.859239'.980033 42.019967 19 42.839404 2.859119 2.01.980286 422.019714 18 43.839536 20.858998.980538 422.019462 17 44.839668 2.858877 2.1.980791.019209 16 45.839800 2.20.858756 2.02.981044 4.22 018956 15 46.839932 220.858635 2.02.98!297 421.018703 14 47.840064 20.858514 2.02.931550.018450 13 48.840196 2'.858393 2.02.91803 421.018197 12 49.840328 19.858272 202.932056 21.017944 1 50 9.840459 219 9.858151 2 9.9S230 4 0.017691 10 51.840591 19.858029 20.9 6.9.017438 9 52.840722 219.857908 2.2.9S3214 421.017186 8 53.840854 219.857786 2.03.933067 421.016933 7 54.840985 9.85766.5.983320 1.016680 6 55.841116 9 8575.. 9853 93.33 4 016427 5 56.841247.1.857422.983826 21.011174 4 57.841378 218.857300 2.03.984079 21.015921 3 58.841509.857178.984332 421.015668 2 59.841640 28.857056 23 984584 4.01541 6 1 60.841771.856934 2.3.934837 4..015163 0 b. Cosine. D. 1". Sine. D. 1". Cotng. D. 1. Tang. M. 1330 46c 218 TABLE XIII. LOGARITHMBIC SINES, &C. 440 135t M. Sine. D 1". Cosine. D, 1". Tang. D. 1". Cotang. M. 0 9.841771 218 9.856934 03 9984837 21 0.015163 60 1.841902 218.856812 204.985090 421.014910 59 2.842033 218.856690 20.985343 421.014657 58 3.842163 2 18.856568 04.985596 41.014404 57 4.842294 217.856446 2..985848.21.014152 56 5.842424 217.856323 204.986101 4.2.013899 55 6.842555 217.856201 2.986354 421.013646 54 7.842685 17.856078 204.986607 421.013393 53 8.842815 17.855956 20.986860 421.013140 52 9.842946.855833.987112 4.012388 51 10 9.843076 2 17 9.855711 2 05 9.987365 42o 0.01263.-5 50 11.843206 2 7.855588 2.987618 4 2.0123'2 49 12.843336 21.855465 25.987871.21.012129 48 13.843466 2 1.855342 05.988123 4.21.011877 47 14.843595 16.855219 2.0.988376 4.21.011624 46 15.843725 216.855096 2-.988629 42.011371 45 16.8438.55 216.854973 20.988882 42.011118 44 17.843934 216.851850 2.989134 4.1.010866 43 18.844114 216.854727 2.989387 4.21.010613 42 19.844243 2:16.854603 2.989640 21.010360 41 20 9.844372 2, 9.854480 2.6 9.989893 4. 0.010107 40 21.844502 215.854356 26.990145 42.009855 39 22.844631 215.8542.33 06 99039 421.009602 38 23.844760 2 5 — 854109 206.9905i1 421.009349 37 24.844889 215.853986 06.990903 4 1.009097 36 25.845018 2.5.853862 20.991156 21.008844 35 26.845147 15.85.3738 206,991409 4.2.008591 34 27.845276 215.853614 207.991662 421.008338 33 28.845405 214.853490 207.991914 4 1.008086 32 29.845533.853366 07.992167 21.007833 31 30 9.845662 2 14 9.853242 2, 9.992420 4o, 0.007580 30 31.845790 2.8.53118 2.7.992672 421.007.32 29 32.845919.14.52994 2.7.992925 4.21.007075 28 33.846047 214.852869 207.993178 4.2.006S22 27 34.846175 214.852745 207.993431 42.006.569 26 35.846304 214.852620 208.993683 421.006317 25 36.846432 213.852496 28.993936.21.006064 24 37.846563 2.852371 20.994189 421.005811 23 33.846638 13.852247 2.0.994441 21 005559 22 39.86816 2:13.852122 08.994694 4.005306 21 40 9.846944 2.13 9.851997 28 9.991947 421 0.005053 20 41.847071.3.851872 2.0.995199 421.004801 19 42.847199 2.3.851747 2..995452 421.004548 18 43.847327 2..851622..995705 4.1.004295 17 44.847454 2..851497 2.09.995957 4.2.004043 16 45.84752 2.12.851372 2.09.996210.21.003790 15 46.847709 2.12.851246 9 996463 42.003537 14 47.847336 12.851121 209.996715 42.003285 13 48.847964 22.850996.09.996968.. 003032 12 49.841091 2 12.850870.997221..002779 11 2.12 2.09 4.21 50 9.843218 2,2.850745 209 997473 0002527 10 51.843:315 212.850619 20.997726 421.002274 9 52.843472 21.850493 20.997979 421.002021 8 53.848599 21.8.50368 2..998231 4.2.001769 7 54 848726 21.8.50242 21.998484 421.001516 6 55.848852.11.850116 210.99737 421.001263 5 56.848979.49990 10.998989 421.001011 4 57.849106 2.11.849364 210.999242 421.000758 3 58.849232 2.11.849738 2.999495 421.000505 59.849359 1.849611.999747 42.00253 60.849435 2.849185 2.1 0.0.40000 4.21.00000 0 M. Cosine. i D. 1. Si.'" Cotang. DI.... Tang. lM. 134o 4 ^ TABLE XIV. NATURAL SINES AND COSINES 220 TABLE XIV. NATURAL SINES AND COSINES. 0o i 1 20o 30 40 M. SinSine. Cosin. Sine. osin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. 0.00000 One..01745.99985.03490.99939.05234.99S63.06976.99756 60 1.00029 One..01774.99984 03519.99938.05263.99s61.07005.99754 59 2.0058 One..01803 99934.03543.99937.05292.99860.07031.99752 58 3.00387 One..01832.99983.03577.99936 05321.99858.07063.99750 57 4.00116 One..01862.99983.03606.99935.05350.99857.07092.99748 56 5.03145 One..0189t.99932.03635.99934.05379.99855.07121.99746 55 6.00175 One..01920.99932.03661..99933.05408.99854.07150.99744 54 7.03201 One..01919.99981.03693.99932.05437.99852.07179.99742 53 8.002:33 One..01978.999(30.0372:3.99931.05466.99851.07208.99740 52 9.03262 One..02007.99980.03752.99930.05495.99849.07237.99738 51 10.00291 One..02936.99979.03781.99929.05524.99847.07266.99736 50 11.0332 1.99999.02065.99979.03510.99927.05553.99846.07295.99734 49 12.00343.99999.0204-.99978.03839.99926.05582.99344.07324.99731 48 13.03.378.99999.02123.99977.03363.99925.05611.99842.07353.99729 47 14.00107.99999.02 152.99977.03397.99924.05640.99841.07382.99727 46 15.03136.99999.02181.99976.03926.99923.05669.99339.07411.99725 45 16.0016;.9999.0-2211.99976.03955.99922.05698.99333.07440.99723 44 17.03195.99J99.0224'.99975.03934.99921.05727.99336.07469.99721 43 1 3.0).24.99333.0 263.99974.04013.99919.05756.99834.07498.99719 42 19.005333.9999-3.02293.99974.04042.99918.05785.99833.07527.99716 41 20.0,).'32.99993.02327.99973.04071.99917.05814.99831.07556.99714 40 21.00611.99993.02356.99972.04100.99916.05844.99829.07585.99712 39 22.03i64).99993.02335.99972.04129.99915.0537;3.99827.07614.99710 33 23.03669.99933.02414.99971.04159.99913.05902.99326.07643.99708 37 21.01698 99393.02443.99970.04183.99912.05931.99324.07672.99705 36 25.09727.93397.02472.99969.04217.99911.05960.99322.07701.99703 35 26.0 )756.99997.02501.99969.04246.99910.05989.99821.0773).99701 34 27. )7831.99397.0 2530.99963.01275.99909.06018.99319. 07759.99699 33 23.00314.99397.02560.99967.04304.99907.06047.99817.07788.99696 32 23.3)3411.99:396.02589.99966.04333.99906.06076.99315.07817.99694!31 3 0337:3.99996.02618 99966.04362.99905.06105.99813.07846.9r9I 1 30 31:0302.99996.02647 99965.04391.99904.06134.99312.07875.9768'. 29 32.09931.99996.02676.99964.04420.99902.06163.99310.07904.99687 28 33. 0960..93935.02705.99963.04449.99901.06192.99808.07933.99635 27 3 31.0939.93 995.02734.99963.0-1478.93900.06221.99306.07962.99633 26:3..01018.99995.02763.99962.04507.99898.06250.99304.07991.99680 25 36.10107.99995.02792.99961.04536.99397.06279.99-03.03020.99678 24 37.01976.999911.02821.99960.04565.99396.06303.99301.08049.99676 23 33 110.939991.02353.99959.01594.99894.06337.997997.08078.99673 22 39.01131.99991.02379.99959.04623.99393.06366.99797.03107.99671 21 40.01164.99993.02903.99953.04653.99392.06395.99795.08136.99668 20 41.01193.9399:3.02938.99957.04632.99890.05424.99793.03165.99666 19 42.01222.99993.02967.99956.04711.99389.06453.99792.08194.99664 18 43.01251.99992.02996.99955.04740.99388.(618IS2.99790.03223.99661 17 41.012301.99992.03025.99954.04769.99336.06511.99788.08252.99659 16 45.01339.933g1.03954.9999536.0798.995.0610.99361.0821.99657 15 46.01:3:33.99991.03033.99952.04827.99883.06569.99784.08310.99654 14 47.01367.99991.03112.99952.04856.99382.0659W.99782.03339.99652 13 43.01396.99)90.03141.99951.04885.99831.06627.99780.08363.99649 12 49.01425.9990.03170.99950.04914.99379.i06656.99778.08397.99647 11 53.01454.99939.03199.99949.04943.9978 3.06685.99776.08426.99644 10 51.01483.99939.03223.99943.04972.99876.06714.99774.08155.99642 9 52.01513.999339.032.37.99917.0.3001.99375.(16743.99772.0.3484.99639 8 53.0154-2.9393'3 03236.99946.05030.99373.06773.99770.08513.99637 7 54.01571.99933.03316.99915.05059.99872.0630(2.99763.03542.99635 6 55.01600.99987 03345.99944.05088.99870.06331.99766.08571.99632 5 56.01623.99937.03374.99943.05117.99869.06360.99764.086n0.99630 4 57.016538.99936.0343.99942.05146.99S67. 06389.99762.08629.99627 3 5.01687.99936.03432.99941.051 75.99966.06918.99760.08658.99625 2 59.01716.9993..03161.99940.05205.99364.06947.99758.08637.99622 1 63.0174.5.99935.03490.99939.0.234.996 63.06976.99756.03716.99619 0 M. Cosin. ine. sin. Sine. Cosin. Sine. Cosin. Sne. Cosin. Sine. M. 1 890 88 870 860 - 850 9 _ TABLE XIV. NATURAL SINES AND COSINES. 221 50 60 70 89 90 M. Sine. Cosin Sine Cosin. Sine. SCosen Sine. CoiCosin. Sine. I Cosin. M. 0.08716.99619.10453.99452.12187.99255.13917.99027.15643'.98769 60 1.08745.99617.10482.99419.12216.99251.13946.99023.15672.98764 59 2.03774.99614.10511.99446.12245.99248.13975.99019.15701.93760 58 3.08303.99612.1040).99443.12274.99244.14004.99015.15730.93755 57 4.08831.99609.10569.99440.12302.99240.14033.99011.15758.98751 56 5.03860.99607.10597.99437.12331.99237.14061.99006.15787.93746 55 6.08839.99634.10626.99434. 12360.99233.1- 9'>.99002.15816.98741 54 7.08918.99602.10655.99431.12389.99230.1411b.98993.15845.98737 53 8.03947.99599.10684.99423.12418.99226. 41 14-.9-994.15873.98732 52 9.03976.99596.10713.93424.12447.99222.14177.9990(.15902.98728 51 10.09905.99594.10742.99421.12476.99219.1429.5.989S6.15931.93723 50 I.09034.99591.10771.99418.12504.99215.14234.98932.15959.98718 49 12.09063.9958.10809.99415.12333.99211.14263.9897o.1593S.98714 48 13.09092.995836.10329.9911 2..12562.99203.14292.98973.16017.98709 47 14.09121.995:3.10358.199'093.12591.99204.1 43201.93969.16046.98704 46 15.09150.99580.10387.99106.1 262'.932001.14349.93965.16074.98700 45 16.09179.99578.10916.99192.12619.99197.1 137,'.9896 1.16103.98695 44 17.092913.99575.10945.99 S99.12678.99193.14407.96957.16132.98690 43 18.09237.99572.10973.99396.12706.99189.14136.98953.16160.98686 42 19.09266.99570.11002.99:39.12735.99186.14464.98948.16189.98681 41 20.09295.99567.11031:99390.12764.99182.14493.93944.16218.98676 40 21.09324.99564,.11069.993-36.12793.99178.14522.98940.16246.98671 39 22.093533.99.562.11039.99333.1282.99175.145531.9-936. 16275.91667 33 23.09332.99559..11118.99330.12851.99171.14580.98931.16304.98662 37 24.09411.99556. 11147.99377.12880.99167.14605.95927.16333.98657 36 25.09440.99553.11176.99374.12903.99163.14637.98923.16361.98652 35 26.09469.99.551.11205.99370.12937.99160.14666.98919.16.390.98648 34 27.09498.99548.11234.99367.12966.99156.14695.98914.16419.98643 33 23.09527.99545. 12163.99364.12995.99152:.14723.98910.16447.98638 32 29.09556.99542.11291.99360.13024.99148.14752.98906.16476.98633 31 30.09585.99540.11320.99357.13053.99144.14781.98902.16505.98629 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 33.09671.9953l.11407.99347.13139.99133.14867.98889.16591.98614 27 34.09700.99523.11436.99344.13163.99129.14896.98884.16620.98609 26 35.09729.99526.11465.993 13197.91 7 9125:.14925.98880.16648.98604 25 36.0975S.99523.11494.99337.13226.99122.14954.98876.16677.98600 24 37.09787.99.520.11523.99334.13254.99118.14982.98871.16706.98595 23 33.09316.99517.11552.99331.13283.99114.15011.98867..16734.98590 22 39.09345.99514.11580.99327.13312.99110.15040.93863.16763.98585 21 40.09374.99511.11609.99324.13341.99106.15069.98858.16792.98580 20 41.09933.993503.11633.99320.13370.99102. 15097.98854.16820.98575 19 42.09932.99506.11667.99317.13399.99098.15126.98849.16849.98570 18 43.09961.99503.11696.993'14.13427.99094.15155.98845.16878.98565 17 44.0.9990.99509.11725.99310.134561.99091.15184.98841.16906.98561 16 45.10019.99497.11754.99307.13485.99087.15212.98836.16935.98556 15 46.10043.99494.11783.99303.13514.99083.15241.98832. 16964.98551 14 47.10077'.99491.11812.99300.13-543.99079.15270.98827.16992.98.546 13 48.10106.99488.11840.99297.13572.99075. 15299.98823. 17021.98541 12 49.10135.99485.11869.99293.13600.99071.15327.98818.170.50.98536 I 50.10164.994S2. 189.99299.13629.999 67.153-56.98814.17078'.98531 10 51.10192.99179..11927.99286.13658.99063.15385.98809.17107.98526 9 52. 10 42 1.99476.11956.99283.13687.99059.15414.98805.17136.98521 8 53.10250.99473.11935.99279.13716.99055.1.5442.98800.17164.98516 7 54.10279.99470.12014.99276.13744.99051.15471.98796.17193.98511 6 55.10398.99167.12043.99272.13773.99047 15500.98791.17222.98506 5 56.10337.99464'.12071.99269.13802.99043.15529.98787.172.50.98501 4 571.10366.99461.12100.99265.13331.99039.15557.98782.17279.98496 3 581.10395.99458.12129.99262.13860.99035.15586.98778.17308.98491 2 59.10424.99455.121.58.99258.13889 99031.15615.98773.173.36.93486 1 6010453.99152 I12187.S99255.13917.99027.15643.98769.17365.98481 0 M. Coal. Sine. C osin. Sine. Cosin. Sine. o. Sine. Cosin. Sine. 840o 830 so: 81 800 222 TABLE XIV. NATURAL SINES AND COSINES. o103 li_11_ 12o 133 140 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. 0.17365.93431.19081.93163.20791.97815.22495.97437.24192.97030 60 1.17393.98476.19109.98157.20820.97809.22523.97430.24220.97023 59 2.17422.93471.19138.98152.20848.97803.22552.97424.24249.97015 58 3.17451.93466.19167.98146.20877.97797.22580.97417.24277.97008 57 4.17479.93461.19195.98140.20905.97791.22608.97411.24305.97001 56 5.17503.93455.19224.98135.20933.97784.22637.97404.24333.96994 55 6.17537.93450. 19252.98129.20962.97778.22665.97398.24362.96987 54 7. 17565.93445. 19281.98124.20990.97772.22693.97391.24390.96980 53 8.17594.93440.19309.98118.21019.97766.22722 97384.24418.96973 52 9.17623.93435.19338.98112.21047.97760.22750.97378.24446.96966 51 10.176.51.93430,19366.93107.21076.97754.22778.97371.24474.96959 50 11.17630.93425.19395.98101.21104.97748.22807.97365.24503.96952 49 12.17703.9$420.19123.98096.21132.97742.22835.97358.24531.96945 48 13.17737.93414.19152.98090.21161.97735.22863.97351.24559.96937 47 14.17766.93409.19481.98034.21189.97729.22892 97345.24587.96930 46 15.17794.93414.19509.98079.21218.97723.22920.97338.24615.96923 45 16.17823.98399.119.533.98073.21246.97717.22948.97331,24644.96916 44 17.17852.9.9391.19566.98067.21275.97711.22977.97325.24672.96909 43 18.17830.933839.19595.98:'61.21303.97705.23005.97318.24700.96902 42 19.17909.93333.19623.98056.21331.97698.23033.97311.24728.96894 41 20.17937.93378.19652.98050.21360.97692.23062.97304.24756.96887 40 21.17966.93373.19630.983044.21388.97686.23090.97293.24784.96880 39 22.17995.93363.19709.98039.21417.97680.23118.97291.24813.96873.38 23.18023.93362 19737.98033.21445.97673.23146.97234.24341.96866 37 24.18052.983.57. 19766.93027.21474.97667.23175..97278.24869.96858 36 25.18031.93352.19794.98021.21502.97661.232)3'.97271.24897.96851 35 26.18109.93347. 19823.98016.21530.97655.23231.97264.24925.96844 34 27.18133.93341.19851.98010.21559.97648.23260.97257.249.54.968.37 33 23.18166.983:36.19830.98004.21587.97642.23283.97251.24932.96829 32 29.18195.93331.19903.97998.21616.97636.23316.97244.25010.96822 31 30.18224.93.325.19937.97992.21644.97630.23345.97237.25038.96815 30 31.18252. 98320.19965.97987.21672.97623.23373.97230.25066.96807 29 32.18231.93315.19994.97931.21701.97617.23401.97223.25094.96300 28 33.18309.93.310.20022.97975.21729.97611.23429.97217.25122.96793 27 34.183.33.93304.20051.97969.21758.976M4.2.3458.97210.25151.96786 26 35.18367.93299.20079.97963.21786.97598.23136.97233.25179.96778 25 36.18395.93294.23108.97958.21814.97592.23514.97196.25207.96771 24 37.18424.93233.21136.97952'.21843.97585.23542.97189.25235.96764 23 33.18452.98283.20165.97946.21871.97579,23571.971B2.2-5263.96756 22 39.18481.98277.20193.97940.21899.97573.23599.97176.25291.96749 21 40.18509.93272.20222.97934.21928.97566.23627.97169.25320.96742 20 41.18533.93267.20250.97923.21956.97560.23656.97162.25348.96734 19 42.18567.9,261.20279.97922.21985.97553.23684.97155.25376.96727 18 43.18595.93256.20307.97916.22013.97547.2.3712.97148.25404.96719 17 44. 1624.93250.20336 {97910.22041.97541.23740.97141.2.5432.96712 16 45.18652.93245.20364.97905.22070.97531.23769.97134.25460.96705 15 46.18691.93240.20393.97899.22093.97528.23797.97127.25488.96697 14 47.18710.93234.20421.97893.22126.97521.23325.97120.2.5516.96690 13 43.18733.98229.20450.97887.22155.97515.23353.97113.25545.96632 12 49.18767.93223:.20478.97881.22183.97503 23832.97106.25573.96675 1 1 50.18795.93218.20507.97875.22212.97502.23910.97100.25601.96667 10 51.188-24.93-212.20535.97869.22240).97496.23933.97093.25629.96660 9 52.18852.93207.20563.97863.22263.97489.23966.97086.25657.9i,653 8 53.18831.93201.20592.978.57 22297.97483.23995.97079.25635.96645 7 54.18910.93196.23620.973.51.22325.97476.24023.97072.25713.96638 6 55.18933.98190.20649.97345.22353.97470.24051.9706.5.25741.96630 5 56.18967.9818.5.20677.97839.22332.97463.24(79.97058.25769.96623 4 57.18995.93179.20706.97833.22410.97457..24103.9705 1.25798.96615 3 53.19024.93174.20734.97827.22438.974.50.24136.97044.25826.96608 2 59.19052.93168.20763.97821.22467.97444.24164.97037.25854.96600 1 60.19081.93163.20791.97815.22495.97437.24192.97030.25882.96593 0 M. IGo0. Sie. Coth. Sine. 1qomln. Sine. Coair. Sine.: Cosin. ine. M __..7 ygro 0o.-~~c P yo 11. 1 _8.o TABLE XIV. NATURAL SINES AND COSINES. 223 15 163 I 7 18 190. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. 0.25O 2.9619. 1.961..29237.95630 30902.95106.32557.94552 60 1.25911.96535.27592.96118.29265.95622.30929.95097.32584.9454 2 59 2.2.5933.96578.27620.96110.29293.95613.30957.95088.32612.94533 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 5.26022.96555.27704.96036.29376.95588.31040.96061.32694.94504 55 6.26050.96547.27731.96078.29404.95579.3106S.95052.32722.94495 54 7.26079.96540.27759.96070.29432.95571.31095.95043.32749.94485 53 8.26107.96332.27787.96062.29460.95562.31123.9.5033.32777.94476 52 9.26135.96524.27815.96054.29487.95554.31151.95024.32804.94466 51 10.26163.96517.27843.96046.29515.95545.31178.95015.32932.94457 50 I11.26191.96509.27871.96037.29543.95536.31206.95006.32859.94447 49 12.26219.96502.27899.96029.29571.9552'.31233.94997.32887.94438 48 13.26247.96494.27927.96021.29599.95519.31261.94988.32914.94428 47 14.26275.96486.27955.96013.29626.95511.31289.94979.32942.94418 46 15.26303.96479.27933.96005.29654.95502.31316.94970.32969.94409 45 16.26331.96471.23011.95997.29682.95493.31344.94961.32997.94399 44 17.26359.96463.23039.95989.29710.95485.31372.94952.33024.94390 43 13.26337.96456.23067.95981.29737.95476.31399.94943.33051.94380 42 19.26415.96448.28095.95972.29765.95467.31427.94933.33079.94370 41 20.26443.96440.28123.95964.29793.95459.31454.94924.33106.94361 40 21.26471.96433.23150.95956.29821.954.50.31482.94915.33134.94351 39 22.26500.96125.28178.95913.2984-9.95441.31510.94906.33161.94342 33 23.26523.96417.23206.95940.29876.95433.31537.94897.33189.94332 37 24.26556.96110.23234.95931.29904 95424.31'565.94888.33216.94322 36 25.26534.96402.28262.9592.3.29932.95415.31593.94878.33244.94313 35 26.26612.96394.23290.95915.29960.95407.31620.94869.33271.94303 34 27.26610.96336.23318.95907.29987.95398.31648.94860.33298.94293 33 23.26668.96379.28346.95898.30015.95389.31675.94851.33326.94284 32 29.26696.96371.28374.95890.30043.95380.31703.94842.3!33.53.94274 31 30.26724.96363.23402.95382.30071.95372.31730.94832.33381.94264 30 31.26752.96355.28429.95874.30098.95363.31758.9482.3.33408.94254 29 32.26780.96347.28457.95865.30126.95354.31786.94814.33436.94245 28 33.26308.96340.28485.95857.30154.95345.31813.94805.33463.94235 27 34.26336.96332.28513.95849.30182.95337.31841.94795.33490.94225 26 35..26364.96324.23541.95841.30209.95328.31868.94786.33518.94215 25 36.26392.96316.28569.95832.30237.95319.31896.94777.33545.94206 24 37.2692).963083.23597.95324.30265.95310.31923.94768.33573.94196 23 3 3.26943.96301.28625.95916.30292.95301.31951.94758.33600.94186 22 39.26976.96293.28652.9580T.30320.95293.31979.94749.33627.94176 21 40.27004.96235.23630.95799.30348.95284.32006.94740.33655.94167 20 41 l..27032.96277.28703.95791.30376.95275.32034.94730.33682.94157 19 42.27060.96269.28136.95782.30403.95266.32061.94721. 33710.94147 18 43.27033.96261.23764.95774.30431.95257.32089.94712.33737.94137 17 41.27116.96253.28792.95766.30459.95248.32116.94702.33764.94127 16 45.27144.96246.28820.95757.30486.95240.32144.94693.33792.94118 15 46.27172.96233.28847.95749.30514.95231.32171.94634.33819.94108 14 47.27200.92623.23375.95740.30542.95222.32199.94674.33846.94098 13 48.27228.96222.29903.95732.30570.95213.32227.94665.33874.94088 12l 49.272.56.96214.28931.95724.30597.95204 32254.94656.33901.94078 11 50.27234.96206.28959.95715.30625.95195.32282.94646.33929.94063 10 51.27312.96193.28987.95707.30653.95186.32309.94637.33956.94058 9 52.27340.96190.29015.95698.30630.95177.32337.94627.33983.94049 8 53.27363.96 182.29042.95690.30708.95163.32384.94618.34011.94039 7 54.27396.96174.29070.95631.30736.95159.32392. 94609.34038.94029 6 55.27421.96166.29093.95673..30763.95150.32419.94599.34065.94019 5 56.227452.96153.29126.95664.30791.95142.32447.94590.34093.91009 4 57.27430.9615).29154.95656.30819.95133.32474.94580.34120.93999 3 53.27503.96142.29182.95647.30846.95124.32502.94571.34147.939S9 2 59.27536.96134.29209.95639.30374.95115.32529.94561.34175.93979 1 60.27564.96126,.29237.95630.3 0902.95106.32557.94552 1.3420)2.93969 0 M. Cosin.l Sine. -Cosin. Sine. Cosin. Sine. Cosin. ine. Cosin. Sine. M. 1 743 73 ) 72 ( 710 70: 3 224 TABLE XIV. NATURAL SINES AND COSINES. -20 _ aI o 10 0 23 240 M. Sine. Cosin Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. 3M. 0.31202. 9363J.35337.93338.37461.92716.39-73.92050.40674.91355 60 1.31229.93959.35S64.9334S.37488.92707.39100.92039.407010.91343 59 2.34257.93949.35391.93337.37515.92697.39127.9202S.40727.91331 58 3.34'234.93939.35918.93327.37542.92656.39153.92016.40753.91319 57 4 31311.93929.35915.93316.37569.92675.39180.92005.4070[.91307 ~6 5.31339.93919.35973.93306.37595.92664.39207.91994.40S06.91295 55 6..34366.93909.36000.93295.37622.92653.39234.919S2.40333.91283 54 7.3133 93899.36927.93285.37649.92642.3926u.91971.40S60.91272 53 8.34421.938S9.36054.93274.37676.92631.39237.91959.40886.91260 52 9.3444S.93379. 36031.93264.37703.92620.39314.9194.40913.91248 51 10.34475.93 369.36108r.93253.3773).92609.39341.91936.40939.91236 50( 1.34 503.93559.36135.93243.37757.92593.39367.919235.40966.91224 49 1I.34530.93549.36162.93232.37784.92587.39394.91914.40992.91212 48 13.35577.93S339.36190.93222.37811.92576.39421.91902.41019.91200 17 14.345 4.93329.136217.93211.373 3.92565.39448.91891.41045.9118-s 46 15.31612.93319.36244.93201.37863.925.54.39474.91879.41072.91176 45 16.3469.93 09.36271.93190.37892.92543.39501.91863.41098.91164 44 17.34666.93799.36298.93180.37919.9253-2.3952-.91856.41125.9l1152 43 13.34691.93789.36325.93169.37946.92-521.39555.91845.41151.91140 42 19.31721.93779.36352.93159.37973.92510.39-81.91833.41178.91128 41 20 [.31748.93769.36379.93148.37999.92499.39608.91822.41204.91116 40 21.34775.93759.36106.93137.33026.92488.39635.91810.41231.91104 39 22.3183 3.9374.36434.93127.380053.92477.39661.91799.41257.91092 35 23.34330.9373.36461.93116.3i3030.92166.39638.91737.41234.91030 37 21 3437.93723. 36438.931f6.338107.92455.39715.917756.41310.91068 36 25.34431.93718.36515.93095.3i134.92444.39741.91764.41337.91056 35 26.31912.93703.36512.93034.38161.92432.39768.91762.41363.91044 34 27.3 99139.9 3.36569.93074.33188.92421.39795.91741.41390.91032 33 23.31966.93633.36596.93063.33215.92410.39322.91729.41416.91020 32 29.34993.93677.36623.93052.38241.92399.39848.91718.41443.91008 31 30.35921.93667.36650.93042.33263.92388.39875.91706.41469.90996 80 31.359043.93657.36677.93031.3 32951.92377.39902.91694.41496.90984 29 32.33075.93617.35704.93020.33322.92366.39923.91633.41522.90972 28 33.3510:3.93637.35731.9301)0.33349.92355.39955.91671.41549.90960 27 34.:35130.93626.36758.92999.33376.92.343.39982.91660.41575.90948 26 3..31 157'.93616.35785.92983.38403.92332.40003.91643.4 602.90936 25 36.351 4.93606.36812.92978.38430.92321.40035.91636.41623.90924 24 37.35211.93596.36339.92967..38156.92310.40062.91625.41655.90911 23 33..35239.93535.36367.92956.3 433.92299.40088.91613.41681.90899 22 39.35266.93575.36394.92945.33510.92287.40115.91601.41707.90837 21 40.35r293.933565.36921.92935.38537.92276.40141.91590.41734.90875 20 41.35320.93555.36948.929214.38564.92265.40163.91578.41760.90863 19 42.35347.93544.36975.92913.3591.92254.40195.91566.41787.90851 18 43.3537.5.93534.37002.92902.33617.92243.40221. 9I555.41813.90839 17 44.35402.93524.37029.92392.33644.92231.40218.9154.3.41840.90826 16 45.35429.93514.37056.92881.38671.92220.40275.91531.41866.90814 15 46.35416.93503.37083.92370.33698.92299.40301.91519.41892.90802 14 47.354,4 1.93493.37110.92859.33725.92198.40328.91508.41919.90790 13 48.35511.93483.37137.92849.33752.92186.40355.91496.419451.90778 12 49.35533.93472.37164.92339.33778.92175 40381.91484.41972.90766 11 50.3356.3.93462.37191.92327.33905.92164.40403.91472.41998.90753 10 51.35592.93452.37218.92816 33832.92152.40434.91461.42024.90741 9 52.3561..93441.37245.92305.38859.92141.40461.91449.42051.90729 8 53.364 1.93431.37272.92794.38886.92130.40488.91437.42077.90717 7 54.35674.93420.37299.92784.33912.92119.40514.91425.42104.90704 6 55.35701.9341).37326.92773.39939.92107.40541.91414.42130.90692 5 56.35728.93400.37353.92762.33966.92096.40667.91402.42156.90680 4 57.35755.93.339.37380.92751.33993.92035.40594.91390.42183.90668 3 58.35782.93379.37407.92740.39020.92073.40621.91378.42209.90655 2 59.35810.93368.37434.92729.39046.92062.40617.91366.42235.90643 1 60.35i37.93358.37461.92718 3C~073.92050.40674.91355.42262.90631 0 M. Cosin. Sine. Co in. Sine 9. CoSi. Sine. Coin Sine. Cos in. Sine. M. 690 66870 6 660 650 TABLE XIV. NATURAL SINES AND COSINES. 225 7 s-* aeB5_ i>3 28g 29?' M. Sine. Cosin. Sine. Cosin Si Cosin. Sine. Cosin. Sine. Cosin. M. 0.4-262.90631.43S.37. s979.45399.891 1.46947.88295.48481.87462 60 1.42238.9.)618.4336:3.S9367.45425 89087.46973.88281.4506.8744s 59 2.42315.90 1iJ.43389. i9354.45451,89074.46999.88267.48532.87434 58 3.42.341.90)94.43916.89541.45477.89061.47024.882.54.48557.87420 57 4 42:367.90)i2.43942.89328.45503.89043.470.50.882411.48583.87406 56 5.42394.93.490 3963.89316.45529 8035.4.85.4.88226.4608.87391 55 6'42420.90557.43991.89803.45554.89021.47101.88213.48634.87377 54 71 42446.93543.44020.89790.45580.89008.47127.88199.48659.87363 53 8.42473.935:32.44046.89777.45606.88995.47153.88185.48634.87349 52 9.42499.90520).44072.89764.45632.889 1.47178.88172.48710.87335 51 10.42.525.90507.44098.89752.45658.88968.47204.88158.48735.87321 50 11.42552.9049.5.44124.89739.45684.88955.47229.88144.48761.87306 49 12.42578.90433.44151.89726.45710.88942.47255.88130.48786.87292 48 13.42604.90170.44177.89713.45736.88928.47281'88117.48811.87278 47 14.42631.904,53.44203.89700.45762.88915.473)6.88103.48837.87264 46 15.42657.90446.44229.89687.45787.88902.47332.88089.4886i2.87250 45 16.42683.90433.44255.89674.45813.88888.47358.88075.488&).87235 44 1.42709.90421.44231.89662.45839.88875.47331.88062.48913 871221 43 18.42736.90405.44307.89649.45865.88862.47409.88048.48938.87207 42 19.42762.90396.443.33.89636.45891.88848.47434.83034.48964'87193 41 20.42738.90383.44359.89623.45917.88835.47460.88020.48989.87178 40 21.42915.90371.44335 89610.45942.88822.47486.83006.49014.87161 39 22.42341.90358.44411.89597.45963.88808.47511'.87993.49040.87150 38 23.42367.90346.44437.89584.45994.88795.47537.87979.49065.87136 37 24.42394.90334.44464.89571.46021.88782.47562'87965.49090.87121 36 25.42920.90:321.44490.89.558.46046,88768.47538.87951.49116.87107 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 28.42999.90234.44563.89519.46123.88728.47665.87909.49192.87064 32 29.43025.90271.44594.89506.46149.88715.47690.87896.49217.87050 31 30.43051.90259.446fi20.89493.46175.88701.47716.87882.49242:87036 -30 31.43077.90216.44646.89480.46201.88688.47741.87868.49268.87021 29 32.43104.90233.44672.89467.46226.88674.47767.87854.49293.87007 28 33.43130.90221.44693.89454.46252.88661.47793.87840.49318.86993 27 34.43156.90208.44724.89441.46278.88647.47818.87826.49344.86978 26 35.43182.90196.44750.89423.4634.88631.47844.87812.49369.86964 25 36.43209.90183.44776.89415.46330.83620.47869.87798.49394.86949 24 37.432:35.90171.44802.89402.46:355.88607.47895.87784.49419.86935 23 33.43-261.90158.44823.89339.46381.88.593.47920.87770.49445.86921 22 39.43237.90146.44854.89376.46407.88580.47946.87756.49470.86906 21 401.43313.90133.44880.89363.4643.3.88566.47971.87743.49495:86892 20 41.433401.90120.44906.893:50.46453.88553.47997.87729.49521.86878 19 42.43366.90(108.44932.89337.46484.88.539.48022.87715.49546.8686.3 18 43.43392.90095.44953.89324.46510.88526.48048.87701.49571.86849 17 44.43118.90082.449,34.89311.46536.88512.48073.87687.49596.868341 16 45.43445.90070.45010.89293.46561.884 99.48099.87673.49622.86320 15 461:43471.90057.45036.89235.46587.88485.48124.87659.49647.86805 14 471.43497.90045.45062.89272.46613.88472.48150.87645.49672.86791 13 48.4.3523.93032.45088.89259.46639.88458.48175.87631.49697.86777 12 491.43549.90019.45114.89245.48664.88445.48201.87617.49723.86762 11 50.4.3575.90007.45140.89232.46690.88431 43226.87603.49748.86748 10 511.436021.89994.451661.89219 467161.88417.48252.87589.49773.86733 9 52.4.36231-.89931.45192.89206.46742.88404.48277.87575.49798.86719 8 53.43654.89968.45218.89193.467671.88390.48303.87561.49824.86704 7 54.43630.89956.45243.89180.46793.88377.48328 87546.49849.86690 6 551.437061.89943.45269.89167.463191.88363.48354.87532.49874.86675 5 56.437.33.89930.45295.89153.46844.88349.43379.87518.49899.86661 4 57.43759.89918.45321.89140.46870.88336.48405.87504.49924.86646 3 53.43785.89905.45.347.89127.46896.88322.48430.87490.49950.86632 2 59.43811.89892.45373.89114.46921.88308.48456 ^87476.49975.86617 1 60.43337.89879.45399.89101.46947.88295.48481.87462.50000.86603 0. loia. Si6e. Cosin. e. (Cos1n. Se - Cosin. Sine. 1Cosin. Sine. M 64o 0630 680 610 60) 1 1 226 TABLE XIV. NATUURAL SINES AND COSINES. 300 310 32o 330 340 UI. Sine. Cosin.Sine. Coir. Sine. Ccsin. Sine. Cosin. Sine. Csin. M. 0.50000.86603 51504.85717.52992.84805.54464.83867.55919.I2904 60 1.50025.86588 51529.85702.53017.84789.544'8.83851.55943.82887 59 2.5250.86573.51554.86687.53041.84774.545 13.83835.55968.82871 58 3.50076.86559.51579.85672.53066.84759.54537.83819.55992.82855 57 4 50101.86544.51604.866r7.53091 S.4743.54561.83804.56016.82839 66 51.50126.86530.51 28.85642.53115.84728.r4536.83788.56040.82822 55 6.50151.86515.61663.85627.63140.84712.54610.83772.56064.82806 54 7.50176.86501.51678.86612.5314.94697.54635.83756.56088.82790 53 8.50201.86488.51703.35597. 3189.84681.54659.83740.66112.82773 52 9.50227.86471.5178.35582.3214.84666.54683.83724.56136.82757 51 10.50252.86457.51753.3.5.67.63238.4650.5708.8370-.56160.82741 50 11 502777.642.5177.86651.532 63. 46:35..54732.83692.56184.82724 49 12.50302.86427.51803.85536.53288.84619.54756.83676.56208.82708 48 13.50327.86413.51b28.85521.53312.81604.54781.8366i1.56232.82692 47 14.50352.86398.51852.85596.53337.84858.54305.83645.562.56.82675 46 15.50377.86384.51877.8541.53361.84573.54829.83629.56280.82659 45 16.5040.86369.51902,.85476.53386.84557.54854.83613.563 5.82643 44 17.50428.86354.51927.85461.53411.84542.54s78.83597.56329.82626 43 1 3.50453.86340.51952.85446.53435.84526.54902.83581.56353.82610 42 19.50478.86:325.51977.85341.53460.84511.54927.83565.56377.82593 41 20.50503.86310.52002.8541 6.53434.84495.54951.83549.56401.82677 40 21.50528.86295.52026.85401.53509.81480.54975.8.3533.56425.82561 39 22.50553.86281.52051.85385.53534.84161.54999.83517.56449.82544 38 23.50578.86266.52076.85370.53558.844 1ts.55024.83501.56473.82528 37 24.50 60.86251.52101.85355.535.8433.55048.834 5.56497.82511 36 25.50628.86237.621-6.8534 1.53607 84417.55072.83469.56521.82495 35 26.50654.86252.52151.85325.53632.844(s.2.55097.83453.5545.82478 34 27.50629.907.52175.85310.53656.843.6.55121.83437.56569.82462 33 28.50704.86192.52200.85294.5363i'84370.5514-1.83421.56593.82446 32 29.50729.86178.52225,85279.53705 84355.551 69.83405.56617.82429 31 30.50754.86163.52250'.85264.53730.84339.55194.83389.56641.82413 30 31.50779.86148.52275.85249.53754.84324.56218.83373.56665.82396 29 32.501.04.86133.526299.85234.53779,84308.55242.83356.566S9.82380 28 33.50829 861 19.52324.85218.5304.84'292.55266.83340.56713.82363 27 31.50854.86101.52349.85203.53832.84277.55291.83324.56736.82347 26 35.50879.86089.52374.85188.53853,84261.55315.83308.56760.82330 25 36.60904.86074.52399.85173.53877'84245.55339. 83292.56784.82314 24 37.. 2.86059.52423.85157.53902'.84230.55363.83276.56808.82297 23 33.50954.8605.52448'85142.53926 84214.55388.83260.56832.82281 22 39.5097.86010.52473,.85127.53951.84198.55412.83244.56565.82264 21 40.51004.86015.52498.85112.53975.84182.55436.83228.56880.82248 20 41.51029.86000.62522.85096.564 00.81167.56460.83212.56904.82231 19 42.51054.85985.52547.85081.54024.84151.55484.83195.56928.82214 18 43.51079.85970.52572 85066.54049.84135.55509.83179.56952.82198 17 44.51104.85956.52.597.85051.54073.84120.55533.83163.56976.82181 16 45.51129.85941.52621.85035.54097.84104.55557.83147.57000.82165 15 46.51154.85926.52646.85020.54122.84088.55581.83131.57024.82145 14 47.51179.85911.52671.85005.54146.84072.55605.83115.57047.82132 13 43.51204.85596.52696.84989'.51171.84057.55630.83098.57071.82115 12 49.51229.85381.52720.84974..54195.84041.55654.83082.57095.82098 11 50.51254.85866.52745.84959.54220.84025.55678.83066.57119.82082 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 53.51329.85821.52819.84913.54293.83978.55750.83017.57191.82032 7 54.51354.85806.52844,.84897.54317.83962.55775.83001.57215.82015 6 55.51379.85792.52869.84882.54342.83946.55799.82985.57238.81999 5 56 151404.85777.52393.84866.54366.8390.55823.82969.57262.81982 4 57.51429.85762.52918.84851.54391.83915.55847.82953'.57286.81965 3 53.51454.85747.52943.84836.54415.83899 55871.82936.57310.81949 2 59.51479.85732.52967.84820.54440.83883.55895.82920.57334.81932 1 60.51504.85717.52992.84805 54464.83867.55919.82904.57353.81915 0 M. Cosin,. Sine. Cosin. Sine. Cosln. Sine. Cosin. Sine. Cosin. Sie. M. 590 58 570 560 55 TABLE XIV. NATURAL SINES AND COSINES. 227 350 360 370 380 390 v. Sine. Cosn. Sine. Cosin. S.ine. Cosin. Sine. Cosin. Sine. Cosao. IL O.67358.81915.58779.80902.60182.79864 61566.78801.62932.77715 60 1.57381.81899.68802.80885.60205.79846.615891.78783.62955.77696 59 2.57405.81882.68826.80867.60228.79829.61612.78765.62977.77678 58 3.57429.81865.58849.80850.60251.79811.61635.78747.63000.77660 57 4.57453.81848.68873.80833.60274.79793.61958.78729.63022.77641 56 5.57477.81832.58896.80816.60298.79776.61681.78711.63045.77623 55 6.57501.81815.58920.80799.60321.79758.61704.78694.63068.77605 54 7.57524.81798.58943.80782.60344.79741.61726.78676.63090.77586 53 8.57543.81782.58967.80765.60367.79723.61749.78658.63113.77568 52 9.67572.81765.58990.80748.60390.79706.61772.78640.63135.77550 51 10.57596.81748.69014..80730.60414.79688.61795.78622.63158.77531 50 11.57619.81731.59037.80713.60437.79671.61818.78604.63180.77513 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 14.57691.81681.59108.80662.60506.79618.61887.78550.63248.77458 46 15.67715.81664.59131.80644.60529.79600.61909.78532.63271.77439 45 16.57738.81647.59154.80627.60553.79583.61932.78514.63293.77421 44 17.57762.81631.59178.80610.60576.79565.61955.78496.63316.77402 43 18.57786.81614.59201.80593.60599.79547.61978.78478.63338.77384 42 19.57810.81597.59225.80576.60622.79530.62001.78460.63361.77366 41 20.67833.81580.69248.80558.60645.79512.62024.78442.63383.77347 40 21.57857.81563.59272.80541.60668.79494.62046.78424.63406.77329 39 22.57881.81546.69295.80524.60691.79477.62069.78405.63428.77310 38 23.57904.81530.59318.80607.60714.79459.62092.78387.63451.77292 37 24.57928.81513.59342.80489.60738.79441.62115.78369.63473.77273 36 25.67952.81496.59365.80472.60761.79424.62138.78351.63496 77255 35..57976.81479.59389.80455.60784.79406.62160.78333.6318.77236 34 27.57999.81462.59412.80438.60807.79388.62183.78315.63540 77218 33 28.58023.81445.59436.80420.60830.79371.62206.78297.6353.77199 32 29. 68047.81428.59459.80403.60853.79353.62229.78279.63585.77181 31 30.58070.81412.59482.80386.6076.79335.62251.78261.63608.77162 30 31.58094.81395. 9506.80368.60899.79318.62274.78243.63630.77144 29 32.68118.81378.59529.80351.609'22.79300.62297.78225.63663.77125 28 33.58141.81361.59552.80334.60945.79282.62320.78206.63675.77107 27 34.58165.81344.59576.80316.60968.79264.62342.78188.63698.77088 26 35.58189.81327.59599.80299.60991.79247.62365.78170.63720.77070 25 36.68212.81310.59622.80282.61015.79229.62388.78162.63742.77051 24 37.58236.81293.59646.80264.61038.79211.62411.78134.63765.77033 23 38.58260.81276.59669.80247.61061.79193.62433.78116.63787.77014 22 39.58283.81259.59693.80230.61084.79176:62456.78098.63810.76996 21 40.58307.81242.59716.80212.61107.79158.62479.78079.63832.76977 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 44.58401.81174.69809.80143.61199.79087.62570.78007.63922.76903 16 45.58425.81167.59832.80125.61222.79069.62592.77988.63944.76884 15'46.58449.81140.69856.80108.61245.79051.62615.77970.63966.76866 14'47.58472.81123.69879.80091.61268.79033.62638.77952.63989.76847 13 48.58496.81106.69902.80073.61291.79016.62660.77934.64011.76828 12 49.58519.81089.59926.80056.61314.78998 62683.77916.64033.76810 II 50.58543.81072.59949.80038.61337.78980.62706.77897.64056.76791 10 51.58567.81055.59972.80021.61360.78962.62728.77879.64078.76772 9 52.58590.81038.69995.80003.61383.78944.62751.77861.64100.76754 8 53.58614.81021.60019.79986.61406.78926.62774.77843.64123.76735 7 54.586.37.81004.60042.79968.61429.78908.62796.77824.64145.76717 6 55.58661.80987.60065.79951.61451.78891.62819.77806.64167.76698 5 56.58634.80970.60089.79934.61474.78873.62842.77788.64190.76679 4 57.58708.80953.60112.79916.61497.78855.62864.77769.64212.76661 3 58.58731.80936.60135.79899.61520.78837.62887.77751.64234.76642 2 59.58755.80919.60158.79881.61543.78819.62909.77733.64256.76623 1 60.58779.80902:60182.79864.61566.78801.62932.77715.64279.76604 0 M Oosln. Sine. osodn. Sine. Cosin. Sr1n. Coein. Sine. FCosn. Sine. n. 540 ~530 520 510 6 500 228 TABIE XIV. NATJRAL SINES AND COSINES. 44)0 41~ 4_O_ 430 440 KL. Sine. (Coan. Sine. Cosin. Sine. Cofin. Sine. Coehn. Sine. Cosin. M. 0.64279 76604. 6.75471.66913.74314.62073135.69466.71934 60.64301 i7656.65628.75452.66935.74296.68221.73116.69487.71914 59 2.64323.76507.65650.75433.6695.74276.68242.73096.69508.7189458 3.6436.76548.65672.75414.66978.74256.68264,73076.69529.71873 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 6.64412,7642.65738.753566.67043.74198.6-327.73016.69591.71813 54 7 64435'76473.65759.75337.67064.74178.68349.72996.69612.71792 53 8.64457.76455.65781.75318.67086.74159.6837.72976.69633.71772 52 9.64479;76436 65808.75299.67107.74139.68391.72957.69654.71752 51 10.64501,76417.65825.75280.67129.741201.68412.72937.69675;71732 50 II.64524,76398.65847.75261.67151.74101.68434.72917 69696.71711 49 12.64546 76380.6586.75241.67172.74080.68455.72897.69717.71691 48 13.61563.76361.65891.75222.67194.74061.68476 72877.6737.71671 47 14.64590,761M2.65913.75203.67215&.7041.6897.72857.69758.71650 46 15.64612.76323.6935.75184.67237.74022.68518.72837.69779.71630 45 16.64635.76.304.65956.75165.67258.74002..68539.72817.69800.71610 44 17.64657.7686;.65978.75146.67280.73983.685f1.72797.69321.71590 43 18.64679.76267.66(000.751 267301.63 73963.68582.72777.69842.71569 42 19.64701.76248.66022.75107.67323.73944.68603'.72757.69862.71549 41 0.64723.76229.66044.7508.67344.7384.68624.72737.68883.71529 40 21.64746.76210.66066.7506.6736.73904.68645 72717.69904 7150 39 22.61768.76192.66088.76050.67387.73885.666 72607;60925.71488 38 23.64790.76173.66109.75030.67409.73863.68688.72677.69946.71468 37 24.6481.768154.66131.750 11.67430.73846.68 70 7267.69966.71447 36 25.64834.76135.66153.74992.67452.73826.68730.72637 -.9987.71427 35 26.64856.76116.66175.74973.67473.73806.68751..7217.70008;71407 34 27.64878.76097.66197;74953.67496.73787.68772.72597.70029.71386 33 28.64901.76078.66818.74934.67516 173767.66793.72577.7009.7136 32 29.64923.76059.66240.749t5.67538.73747.66814.72557.70070.71345 31 30.64945.76041.66269.74896;.67559.73728.6683 7:257T.70091.71325 3 31.64967.76022.68284.74876.6750.73708.68857 172517.70112.71306 29 32.698.7600.663..74857.67602.7368.68878.72197.7013.71284 28 33.65011.75984.66327.74833.67683.7369.6899.72477.70153.71264 27 34.6033.75965.6631.74818.67645.73649; 689 7257.70174.71243 26 35.65056.75946.66371.74799.6766.73629.68941:72437.7019.7223 25 36.65077.75927.66393.74780.67688.736101.3962.7217.70215.71203 24 37.65100.75908.66I14.74760.67709.73590,68983 72397.70236.71182 23 33.65122.75389.66436.74741:67730.735701.69004.72377.70257:.7I62 22 39.65144.75870.664I5.74722.67752.73551.69025.72367.70277.71141 21 40.65166.75854.66480.74703.67773.73531.69046.72337.0298.71121 20 41.65188.75832 66A01.74683.67795:73511.69867.72317.70019~.71100 19 42.65210.75813.66623.74664.67816.73*41.69088 72297.7033.71080 18 43.65232.75794.66545.74644.1837.73472.60109.72277.70360.71059 17 44.65254.75775.66666.74625.67859.73 2.69130.72257.70381.71039 16: 45.65276.77575.66588.74606.67880.73432.69151.72236.70401.71019 15 46.65293.75738.66610.74586.67901.73413.6B172.72216.70422.7099 14 47.65320.75719.66632.74567.67923.73393.69193.72196.70443.70978 13 48.65342...66653.74548.79.694.87373.60214..72176.70463.70957 12 49 65364.75630.66675.74528.67965.73363.9235.72156.70484:70937 11 50.65386.7."6'.66697.74509.67987.73333.69256.72138.70505.70916 10 51.65408.7.614.66718.74489.68008.73.314.69277.72116.70525.70896 52.65430.75623.66740.74470.68029.73294.60298.72095.70546.70875 8 53.6.5452.7.-604.66762.74451.68051.73274.69319.7275.70567.70855 7 54.65474.75585.66783.74431.68072.73254.69640.72055.70587.70834 6 55.65496.75566.66305.74412.63093.73234.69361.72035.70608.70813 5 56.65518.7.5547.66827.74392.681151.732.15.60932:72015.70628.70793 4 57.65541).75528.66348.74373.68136.73195.69403.71995.70649.70772 3 58.65.562.75509.66370.74363.68157.73175.69424.71974 -.70670.70752 2 59.655384.75490.66391.74334.68179.7315&.69445.71954.70690.70731 1 60.65606.75471.66913.74314.68200.73135.69466.71934.70711.70711 10 I. Cosin. Sine. Cosi. Sine. Oaln. If ine Coea. Sie. Cofn. S I. 4:93 "43 47 o ~4.60 450 TABLE XV. NATURAL TANGENTS AND COTANGENTS 230 TABLE XV: NATURAL TANGENTS AND COTANGE j O~ 10o go 30 M. Tang. Coanng. Tang. Cotang.ng. Tan g. Cotang. M. 0.00000 Infinite..01746 57.2900.03492 28.6363.05241 19.0811 60 1.00029 3437.75.01775 56.3506.03521 28.3994.05270 18.9755 59 2.00058 1718.87.01804 55.4415.03550 28.1664.05299 18.8711 58 3.00087 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 572.957.020 52.08 3667 27.2715.05416 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 18.1708 51 10.00291 343.774.02036 49.1039.03783 26.4316.05533 18.0750 50 11.00320 312.521.02066 48.4121.03812 26.2296.05562 17.9802 49 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 15.00436 229.182.02182 45.8294.03929 25.4517.05678 17.6106 45 16.00165 214.858.02211 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 1.00669 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 2'.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 30.00873 114.589.02619 38.1885.04366 22.9038.06116 16.3499 30 3 0.00902 110.892.02648 37.7686.04395 22.7519.06145 16.2722:Lk 32.00931 107.426.0-2677 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 35.4313.045,70 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 34.0273.04637 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 17 44.01280 78.1263.03026 33.0452.04774 20.9460.06525 15.3254 16 45.01309 76.39,00.03055 32.7303.04803 W 9188.06554 15 2571 15 46.01338 74.7292.03084 32.4213.04833 2.6932.06584 15.1893 14 A7.01367 73.1390.03114 32.1181.04862 20.5691.06613 15.1222 13 48 01396 71.6151.03143 31.8205.04891 20.4465.06642 15.0557 12 49.01425 70.1533.03172 31.5284.04920 20.3253.06671 14.9898 11 50.01455 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 61.3829.0.3376 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 28.6363.05241 19.0811.06993 14.3007 0 M. Co TCotang. Tang. Cotang. Tang ng ang. M. 1 893 883 870o 860 I. TABLE XV. NATURAL TANGENTS AND COTANGENTS. 231 4:0 50 60 7O 0 iM. Tang Cotang Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.06993 14.30307.08749 11.4301.10510 9.51436.12278 8.14435 60 1.0702(2 14.2411.08778 11.3919 1.10540 9.48781.12308 8.12481 59 2.07051 14.1821.08807 11.3-540.10569 9.46141.12338 8.10536 58 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 5.07139 14.0079.08895 11.2417.10657 9.38307.12426 8.04756 55 6.07168 13.9507.08925 11.2048.10687 9.35724.12456 8.02848 54 7.07197 13.8940.06954 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 11.0594.10805 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 13.5634.091.30 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 22.076:36 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 25.07724 12.9469.09482 10.5462.11246 8.89185.13017 7.68208 35 26.07753 12.8981.09511 10.5136.11276 8.86862.13047 7.66466 34 27.07782 12.8496.09541 10.4813.11305. 8.84551.13076 7.6473-2 33 28.07812 12.8014 09570 10.4491.11335 8.82252.13106 7.63005 32 29.07841 12.7536.0960 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 12.6591.09658 10.3538.11423 8.75425.13195 7.57872 29 32.07929 12.6124.09688 10.3224.11452 8.73172.13224 7.56176 28 33.07958 12.5660.09717 10.2913.11482 8.70931.13254 7.54487 27 34.07987 12.5199.09746 10.2602.11511 8.68701.13284 7.52806 26 35.08017 12.4742.09776 10.2294.11541 8.66482.133l3 7.51132 25 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 38.08104 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 42.08221 12.1632.09981 10.0187.11747 8.51259.13521 7.39616 18 43.08251 12.1201.10011 9.98931.11777 8.49128.13550 7.37999 17 44.08280 12.0772.10040 9.96007.11806 8.47007.13580 7.36389 16 45.08309 12.0346.10069 9.93101.11836 8.44896.13609 7.34786 15 46.08339 11.9923.10099 9.90211.11865 8.42795.13639 7.33190 14 47.08368 11.9504.10128 9.87338.11895 8.40705.13669 7.31600 13 48.08397 11,9087.10158 9.84482.11924 8.38625.13698 7.30018 12 49.08427 11.8673.10187 9.81641.11954 8.36555.13728 7.28442 11 50.08456 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 53.08544 11.7045.10305 9.70441.12072 8.28376.13846 7.22204 7 54.08573 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 7.16071 3 58.08690 11.5072.10452 9.56791.12219 8.18370.13995 7.14553 2 59 08720 11.4685.10481 9.54106.12249 8.16398.14024 7.13042 1 60 1.08749 11.4301.10510 9.51436.12278 8.14435.14054 7.11537 0 M. Cotang. Tang. CotangI Tang. Cotang. Tang. Cotang. Tang. M. i 50 840 830 80o t32 TABLE XV. NATURAL TANGENTS AND COTANGE. 80 93 100 110. Tang. Cotang. Tang. Cotang Tang. Cotang. Tang. Cotang. M. 0.14054 7.11537.15838 6.31375.17633 5.67128.194.38 5.14455 60 1.14084 7.10038 15868 6.30189.17663 5.6616.5.19468 5.13658 59 2.14113 7.08546.1.5898 6.29007.17693 5.65205.19498 5.12862 58 3.14143 7.07059.15928 6.27829.17723 5.64248.19529 5.12069 57 4.14173 7.05579.15958 6.26655.17753 5.63295.19559 5.11279 56 5.14202 7.04105.15988 6.25486.17783 5.62344.19589 5.10490 556.14232 7.02637.16017 6.24321.17813 5.61397.19619 5.09704 54 i 7.14262 6.91174.16047 6.23160.17843 5.60452.19649 5.08921 53 8.14291 6.99718.16077 6.22003.17873 5.59511.19630 5.08139 52 9.14321 6.98263.16107 6.20851.17903 5.58573.19710 5.07360 51 10.14351 6.96823.16137 6.19703.179:33 5.57638.19740 5.06584 50 1-.14381 6.95385.16167 6.18559.17963 5.56706.19770 5.05809 49 12.14410 6.93952.16196 6.17419.17993 5.55777.19801 5.05037 48 13.14440 6.92525 16226 6.16283.18023 5.54851.19831 5.04267 47 14.14470 6.91104.16256 6.15151.18053 5.53927..19861 5.03499 46 15.14499 6.89683.16286 6.14023.18083 5.53007.19891 5.02734 45 16.14529 6.88278.16316 6.12899.18113 5.52090.19921 5.01971 44 17.14559 6.86874.16346 6.11779.18143 5.51176.19952 5.01210 43 18.14588 6.85475.16376 6.10664.18173 5.50264.19982 5.00451 42 19.14618 6.84032.16405 6.09552.18203 5.49356.20012 4.99695 41 20J.14648 6.82694.16435 6.08444.18233 5.48451.20042 4.98940 40 21.14678 6.81312.16465.6.W7340.1826.3 5.47548.20073 4.98188 39 22.14707 6.79936.16495 6.06240.18293 5.46648.20103 4.97438 38 23.14737 6.78564.16525 6.05143.18323 5.45751.20133 4.96690 37 24.14767 6.77199:16555 6.04051.18353 5.44857.20164 4.95945 36 25.14796 6.75S38.16585 6.02962.18384 5.43966.20194 4.95201 35 26.14826 6.74483.16615 6.01878.18414 5.43077.20224 4.94460 34 27.14856 6.73133.16645 6.00797.18444 5.42192.20254 4.93721 33 28.14886 6.71789.16674 5.99720.18474 5.41309.20285 4.92984 32 29.14915 6.70450.16704 5.98646.18504 5.40429.20315 4.92249 31 30.14945 6.69116.16734 5.97576.18534'5.39552.20345'4.91516 30 31.14975 6.67787.16764 5.96510.18564 5.38677.20376 4.90785 29 32.15005 6.66463.16794 5.95448.18594 5.37805.20406 4.90056 28 33.15034 6.65144.16i24 5.94390.18624 5.36936.20436 4.89330 27 34.15064 6.63831.16854 I5.93335.18654 5.36070.20466 4.8W605 26 35.15094 6.62523.16884 5.92283.18684 5.35206.20497 4.87882 25 36.15124 6.61219 16914 5.91236.18714 5.34345.20527 4.87162 24 37.15153 6.599*21.16944 5.90191.18745 5.33487.20557 4.86444 23 38.15183 6.58627.16974 5.89151.18775 5.32631.20588 4.85727 22 39.15213 6.57339.17004 5.88114.18805 5.31778.20618 4.85013 21 40.15243 6.56055.17033 5,87030.18835 5.30928.20648 4.84300 20 41.15272 6.54777.17063 5.86051.18865 5.30080.20679 4.83590 19 42.15302 6.53503.17093 5.85024.18895 5.29235.20709 4.82882 18 43.15332 6.52234.17123 5.84001.18925 5.28393.20739 4.82175 17 44.15362 6.50970.17153 5.82982.18955 5.27553.20770 4.81471 16 45 j.15391 6.49710.17183 5.81966.18986 5.26715.20300 4.80769 1.5 461.15421 6.48456.17213 5.80953.19016 5.25880.20830 4.80068 14 47.154 51 6.47206.17243 5.79944.19046 5.25048.20861 4.79370 13 48.15481 6.45961.17273 5.789.38.19076 5.24218.20891 4.78673 12 49..15511 6.44720.17303 5.77936.19106 5.23391.20921 4.77978 11I 50.15540 6.43484.17333 5.76937.19136 5.22566.20952 4.77256 10 51.15570 6.42253.17363 5.75941.19166 5.21744.20982 4.76595 9 52.15600 6.41026.17393 5.74949.19197 5.20925.21013 4.75906 8 I 53.15630 6.39804.17423 5.73960.19227 5.20107.21043 4.75219 7 1 54.15660 6.38587.17453 5.72974.19257 5.19293.21073 4.74534 6 55.15689 6.37374.17483 5.71992.19287 5.18480.21104 4.73851 5 56.15719 6.36165.17513 5.71013.19317 5.17671.211.34 4.73170 4 57.15749 6.34961 17543 570037.19:347 5.16363.21164 4.72490 3 58.15779 6.33761.17573 5.69064.19378 5.16058.21195 4.71813 2 59.15,809 6.3566.17603 5.68094.19408 5.15256.21225 4.71137 1 60.15838 6.3137'.17633 5.67128.19438 5..14455.21256 4.70463 0 M. Cotang. Tang. Cotang. Tang Cotang. ITang.. CotBng. Tang. M. 810~ 803 o7 9 780 O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE XV. NATURAL TANGENTS AND COTANGENTS. 233 120 130 140 150 M. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0 21256 4.70463.23087 4.33148.24933 4.01078.26795 3.73205 60 I.21286 4.69791.23117 4.32573.24964 4.00582.26826 3.72771 59 2.21316 4.69121.23148 4.32001.24895 4.00086.26857 3.72338 58 3.21347 4.68452.23179 4.31430.25026 3.99592.26888 3.71907 57 4.21377 4.67786.23209 4.30860.25056 3.99099.26920 3.71476 56 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 54 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 51 10.21560 4.63825.23393 4.27471.25242 3.96165.27107 3.68909 50 11.21590 463171.2.3424 4.26911.25273 3.95680.27138 3.68485 49 12.21621 4.62518.23455 4.26352.25304 3.95196.27169 3.68061 48 13.216.51 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 45 16.21743 4.59927.23578 4.24132.25428 3.93271.27294 3.66376. 44 17.21773 4.59283.23608 4.23550.25459 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 3.61705 40 21.21895 4.56726.23731 4.21387.25583 3.90890.27451 3.64289 39 22.21925 4.56091.23762 4.20842.25614 3.90417.27482 3.63874 38 23.21956 4.55458.23793 4.20298.25645 3.89945.27513 3.63461 37 24.21986 4.54826.23823 4.19756.25676 3.89474.27545 3.63048 36 25.22017 4.54196.23864 4.19215.25707 3.89004.27576 3.62636 35 26.22047 4.53568.23885 4.18675.25738 3.88536.27607 3.62224 34 27.22078 4.52941.23916 4.18137.25769 3.88068.27638 3.61814 33 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 3.60588 30 31.22200 4.50451.24039 4.15997.25893 3.86208.27764 3.60181 29 32.22231 4.49832.24069 4.15465.25924 3.85745.27795 3.59775 28 33.22261 4.49215.24100 4.14934.2.5955 3.85284.27826 3.59370 27 34.22292 4.48600.24131 4.14405.25986 3.84824.27858 3.58966 26 353.22322 4.47986.24162 4.13877.26017 3.84364.27889 3.58562 25 36.22353 4.47374.24193' 4.13350.26048.3.83906.27921 3,58160 24 37.222333 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 4.4545548.24285 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 1.24347 4.10736.26203 3.81630.28077 3.56159 19 42.22536 4.43735.24377 4.10216.26235 3.81177.28109 3.55761 185 43.22.567 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 3.54573 15 46.22658 4.41340.24501 4.08152.26359 3.79378.282.34 3.54179 14 47.226S9 4.40745.24532 4.07639.26390 3.78931.28266 3.53785 13 48.22719 4.40152.21562 4.07127.26421 3.78485.28297 3.53393 12 49.22750 4.39560.24593 4.06616.26452 3.78040.23329 3.53001 11 50,.22781 4.38969.24624 4.06107.26483 3.77595.28360 3.52609 10 51 I22811 4.38331.24655 4.05599.26515 3.77152.28391 3.52219 9 52.22842' 4.37793.24686 4.05092.26546 3.76709.28423 3.51829 8 53.22872 4.37207.24717 4.04586.26577 3.76268.28454 3.51441 7 54.22903 4.36623.24747 4.04081.26608 3.75828.28486 3.51053 6 55.22934 4.36(40.24778 4.03578.266.39 3.75388.28517 3.50666 5 56!.22964 4*3854.59.24809 4.03076.26670 3.74950.28549 3.502ff79 4 57.22995 4.34879.24840 4.(02574.26701 3.74512.285-0 3.49S94 3 58.93026 4U43(g00.24871 402074.26733 3.74075.2i8612 3.49509 2 59.23056 4.33723,24902 4.601.576.26764 3.78540.28643 3.49125 1 60.2:3087 4.33148.24933 4.01078.26795.3:73205.2675 3.48741 I0 M)f Cotatg. Tang. Cotang. Tang. Ctng. Tang. Ootang. Tang. I. 7 i J y-6. " sao. 7i 9i.5.rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr.21.49 244125.50.90 25636663 '34 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 160 170 180 190 M.f Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.28675 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 3.26067.32588 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 55 6.25864 3.46458.30764 3.25055.32685 3.05950.34628 2.88783 54 7.23895 3.46080.30796 3.24719.32717 3.05649.34661 2.88511 53 8.23927 3.45703.30828 3.24383.32749 3.05349.34693 2.88240 52 9.28958 3.45327.30860 3.24049.32782 3.05049.34726 2.87970 51 10.28990 3.44951.30891 3.23714.32814 3.04749.34758 2.87700 50 11.290-21 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 3.43829.30937 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 3.02372.35020 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 3.40369.31242 3.20079.33169 3.01489.35118 2.84758 39 22.29363 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 3.00028.35291 2.83439 34 27.29526 3.38679.31434 3.18127.33363 2.99738.35314 2.83176 33 28.29558 3.38317.31466 3.17804.33395 2.99447.35346 2.82914 32 29.29590 3.37955.31498 3.17481.33427 2.99158.35379 2.82653 31 30.29621 3.37594.31530 3.17159.33460 2.98868.35412 2.82391 30 31.29653 3.37234.31562 3.16838.33492 2.98580.35445 2.82130 29 32.29685 3.36875.31594 3.16517.33524 2.98292.35477 2.81870 2S 33.29716 3.36516.31626 3.16197.33557 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 25 36.29811 3.3.443.31722 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 -314605 -.33718 2.96573 -.35674. 2.80316 22 39.29906, 3.34377.s1818 3.14288.33751 2.96288.35707 2.80059 21 40.29938 3.34023..318501 3.13972.33783 2.96004.35740 2.79802 20 41.29970 3.336Q.31882- 3.13656.338f6 2.95721 1.35772 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 3.11775.34010'2.94028.35969 2.78014 13 48.30192 3.31216.32106 3.11464.34043 2.93748.36002 2.77761 12 49.30-224 3.30868.32139 3.11153.34075 2.93468.36035 2.77507 1 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 5 56.30446 3.28452.32363 3.08991.34303 2.91523.36265 2.75746 4 57.30478 3.28109.32396 3.08685.34335 2.91246.36298 2.75496 3 58.30509 3.27767.32428 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 0. Cotang. Tang. Cotang. T. otng. Tang. Cotang. Tang. M. I yy"73') 720 710 70 TABLE XV. NATURAL TANGENTS AND COTA.NGENTS. 235 200 j 21 I 22- i 230 M.Tang. o Ta ng. Cot an. Tang. ang. Cotang. M. 0.36397 2.74748.; 38:36 2.60)509.4 7 2.4509.42447 2.35585 60 1.36430 2.74499.3242(0 2.60283.40136 2.47302.424s, 2.35395 59 2.36463 2.74251.3-453 2.60057.4(470 2.47095.42516 2.35205 58 3.36496 2.74004.38487 2.59831.40.5(4 2.46883.42551 2.35015 57 4.36529 2.73756.38520 2.59606.40538 2.46632.42585 2.3482.5 56 \ 5.36562 2.73509.38553 2.59391.4'572 2.46476 -.42619 2.34636 55 61.36595 2.73263.38587 2.59156.406)6 2.46270.426.54 2.34447 54 1 7.36625 2.73017.38620 2.58932.40640 2.46065.42688 2.34258 5311 8.36661 2.72771.38654 2.58708.40674 2.45860.42722 2.34069 52 9.36694 2.72526.38687 2.58484.40707 2.45655.42757 2.33881 51 10.36727 2.72281.38721 2.58261.40741 2.45451.42791 2.33693 50 11.36760 2.72036.38754 2.580:38.40775 2.45246.42826 2.33505 49 12.36793 2.71792.38787 2.57815.40809 2.45043.42560 2.33317 48 13.36326 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 2.71062.38888 2.57150.40911 2.44433.42963 2.32756 45 16.36925 2.70819.38921 2.5692S.40945 2.44230.42998 2.32570 44 17.36958 2.70577.33955 2.56707.40979 2.44027.43032 2.32383 43 18.36991 2.70:335.38988 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 2.4.3422.43136 2.31826 40 21.37090 2.69612.39089 2.55827.41116 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 2.553S9.41183 2.42819.43239 2.31271 37 24.37190 2.68892.39190 2.55170.41217 2.42618.43274 2.31086 36 25.37223 2.68653.39223 2.54952.41251. 2.42418.43308 2.30902 35 26.37256 2.63414.39257 2.54734.41285 2.42218.13343 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 2.67700.39357 2.b4082.41387 2.41620.43447 2.30167 31 30.37338 2.67462.39391. 2.53865.41421 2.41421.43481 2.29984 30 31.37422 2.67225.39425 2.53648.41455 2.41223.43516 2.29801 29 32.37455 2.66989.39458 2.53432.41490 2.41025.43550 8.96619 28 33.37488 2.66752.39492 2.53217.41524 2.40827.43585 2.29437 27 34.37521 2.66516.39526 2.53001.41558 2.40629.43620 2.29254 26 3i.37554 2.66281.39559 2.52786.41592 2.40432.43654 2.29073 25 36.37588 2.66046.39593 2.52571.41626 2.40235.436S9 2.28891 24 37.37621 2.65811.39626 2.52357.41660 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 2.283418 21 40.37720 2.65109.39727 2.51715.41763 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 2.51289.4183) 2.39058.43897 2.27806 18 4:3.37820 2.64410.39829 2.51076.41865 2.38863.43932 2.27626 17 44.37853 2.64177.39862 2.50864.41899 2.38668.43966 2.27447 16 45.37887 2.63945.39896 2.50652.41933 2.38473.44001 2.27267 15 46.37920 2.63714.39930 2.50440.41968 2.38279.44036 2.27088 14 47 37953 2.6.3483.39963 2.50229.42002 2.38084.44071 2.26909 13 48.37986 2.63252..39997 2 50018.42036 2.37891.44105 2.26730 12 49.3s020 2.63021.40031 2.49807.42070 2.37697.44140 2.26552 11 50.38053 2.62791.400,65 2.49597.42105 2.37504.44175 2.26374 10 51.33086 2.62.56.40098 2.49386.42139 2.37311.44210 2.26196 9 52.-1 20 2.62332.40132 2.49177.42173 2.37118.44244 2.26018 8 53.333153 2.62103.40166 2.43967.42207 2.36925.44279 2.25840 7 54.38186 2.61874.40200 2.48758.42242 2.36733.44314 2.25663 6 55.3.220 2.61646.40234 2.48549.42276 2.36541.44349 2.254S6 5 56.33253 2.61418.40267 2.48340.42310 2.36349.44384 2.25309 4 57..3286 2.61190.40301 2.48132.42345 2.36158.44418 2.25132 3 58.33320 2 6096.3.40335 2.47924.42379 2.35967.44453 2.24956 2 59.33.53 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. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. I.!1 69, 668 6 I T I —-- = = - -----— ~I 236 TABLE XV. NATURAL TANGENTS AND COTANGENTS. i. 240 _25o 260 270 M. Tang. Cotang. Tang. Cotang. Tang. i Cotang. Tang. Cotang. M. 0.44523 2.24604.46631 2.14451.48773 2.05030.50953 1.96261 60 1.44558 2.44128.46666 2.14288.48809 2.04879.50989 1.96120 59 2.4459.3 2.24'52.46702 2.14125.48845 2.04728.51026 1.95979 58 3.44627 2.24077.46737 2.13963.48881 2.04577.51(63 1.95838 57 4.44662 2.23902.46772 2.13^01.48917 2.04426.51099 1.95698 56 5.44697 2.23727.46-08 2.13639.4S953 2.04276.51136 1.9557 55i 6.44732 2.23553.46343 2.13477.48989 2.04125.51173 1.95417 5-1 7.44767 2.2:3378.46379 2.13316.49026 2.03975.51209 1.95277 53 8.44802 2.23204.46914 2.13154.49062 2.03825.51246 195137 52 9.44837 - 2.23030.46950 2.12993.49098 2.03675.51283 1.94997 51 10.44872 2.22857.46985 2.12832.49134 2.03526.51319 1.94858 50 I 1.44907 2.22683.47021 2.12671.49170 2.03376.51356 1.94718 49 12.44942 2.22510.47056 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 2.02929.51467 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 2.02631.51540 1.94023 44 17.45117 2.21647.47234 2.11711.49387 2.02483.51577 1.93885 43 18.45152 2.21475.47270 2.11552.49423 2.02335.51614 1.93746 42 19.45187 2.21304.47305 2.11392.49459 2.02187.51651 1.936(08 41 20.45222 2.21132.47341 2.11233.49495 2.02039.51688 1.93470 I40 21.45257 2:20961.47377 2.11075.49532 2.01891.51724 1.93332 39 22.45292 2.24790.47412 2.10916.49568 2.01743.51761 1.93195 38 23.45327 2.20619.47448 2.10758.49604 2.01596.51798 1.93057 137 24.45362 2.20449.47483 2.10600.49640 2.01449.51835 1.92510 36/ 25.45397 2.20278.47519 2.10442.49677 2.01302.51872 1.92782 35 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 2.00862.51983 1.92371 32 29.45538 2.19599.47662 2.09811.49822 2.00715.52020 1.92235 31 30.45573 2.19430.47698 2.09654.49858 2.00569.52057 1.92098 30 31.45608 2.19261.47733 2.09498.49894 2.00423.5205 1 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 2.00131.51168 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 1 36.46784.18419.47912 2.08716.50076 11.99695.52279 1.91282 24 37.45819 2.18251.47948 20,560.50113 1.99550.52316 1.91147 23 38.45854.{ 2.18034.47984 2.08405.50149 1.99406.523.53 1.91012 22 39.45889 2.17916.48019 2.08250.50185 1.99261.52390 1.90876 21 40.45924 2.17749.48055 2.08094.50222 1.99116.52427 1.9074.1 20 41.45960 217582;48091 2.07939:.50258 1.98972.52464 1.90607 19 42.45995 2.17416.48127 2.07785.50295 1.98828.52501 1.90472 18 43.4603,)' 2.17249. 48163 2.076 30.50331 1.98684.52538 1.90337 17 44.46065 2.17083.48198 2.07476.06368 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 206860.50514 1.97966.52724.1.89667 12 49.46242 2.16255.48378 2.06706.5o5550 1.97823.52761 1.89533 11 50.46277 2.16090.49414 2.06553.50587 1 1.97681.52798 1.89400 10 51 1 -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 4.83 2.15596.48521 2.06094.50696 1.97253.52910 1.89000 7 5'4 /46418 2.15432.4857 2.05942.50733 1 1.97111. 88867 6 15.46454 2.15268.48593 2.05790.50769 1.96969.529S5 1.887345'5 56.46489 2.15104.48629 2.05637.508061 1.96827.53022- 1.886(2 4 71.465i25- 2.14940.48665 2:0r485.50843 1.96685.53059 1.88469 3 58 i 46560| 2.14777.48701 2.05333.50879 1.96544.53096 1.88337 2; 9.46595 2.14614 48737 2.05182.50916 1:96402.53134 1.88205 1 6t0 1G.i46631 2.14451:48773 2.065030.50953-1 1.96261.53171 1.84(073 (0 1 [ ot —g. 1 a' ng./ Cotang. a - t Coting. Tang. otang. Tang. M. t fis ~ ~ — ef ~~64o 630 _- 6 ..!ABLE XV. NATURAL TANGENTS AND COTANGENTS.,~ 8o ~ 39 300_ 3 10 M Tang. Cotang. Tang. I Cotang. Tang. Cotang. Tang. Cotang. M. 0.63171 1.88073.55431 1.80405.57735 1.73205.60086 1.6642- 60 1.53208 1,87941.55469 1.80231.57774 1.73089.60126 1.66318 59 2.53246 1.87809.55507 1,80158.57813 1.72973.60165 1.66209 58 3.53283 1.87677.55545 1.80034.57851 1.72357.60205 1.66099 57 4.53320 1.87546.55583 1.79911.57890 1.72741.60245 1.65990 56 5.53358 1.87415.55621 1.79788.57929 1.72685.60284 1.65881 55 6.53395 1.87283.55659 1.79665.57968 1.72509.60324 1.65772 54 7.53432 1.87152.55697 1.79542.58007 1.72393.60364 1.65663 53 8.53470 1.87021.55736 1.79419.58046 1.72278.60403 1.65554 52 9.53507 1.86891.55774 1.79296.58085 1.72163.60443 1.65445 51 10.53545 1.86760.55812 1.79174.58124 1.72047.60483 1.65337 50 11.53582 1.86630.55850 1.79051.58162 1.71932.60522 1.65228 49 12.53620 1.86499 ]5588 1.78929.58201 1.71817.60562 1.65120 48 13.53657 1.86369.55926 1.78807.58240 1.71702.60602 1.65011 47 14.53694 1.86239.55964 1.78685.58279 1.71588.60642 1.64903 46 15.53732 1.86109.56003 1.78663.58318 1.71473.60681 1.64795 45 16.53769 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 43 18.53844 1.85720.56117 1.78198.58435 1.71129.60801 1.64471 42 19.53882 1.85591.56156 1.78077.58474 1.71015.60841 1.64363 41 20.53920 1.85462.56194 1.77955.58513 1.7'0901.60881 1.64256 40 21.53957 1.85333.56232 1.77834.58552 1.70787.60921 1.64148 39 22.53995 1.85204.56270 1.77713.58591 1.70673.60960 1.64041 38 23.54032 1.8.075.66309 1.77592.58631 1.70560.61000 1.63934 37 24.54070 1.84946.56347 1.77471.58670 1.70446.61040 1.63826 36 25.54107 1.84818.56385 1.77351.58709 1.70332.61080 1.63719 35 26.54145 1.84689.56424 1.77230.58748 1.70219.61120 1.63612 34 27.54183 1.84561.56462 1.77110.58787 1.70106.61160 1.63505 33 28.54220 1.84433.56501 1.76990.58826 1.69992.61200 1.63398 32 29.54258 1.84305.56539 1.76869.58865 1.69879.61240 1.63292 31 30.64296 1.84177.56577 1.76749.58905 1.69766.61280 1.63185 30 31.54333 1.84049.56616 1.76629.58944 1.69653.61320 1.63079 29 32.54371 1.83922.56654 1.76510.58983 1.69541.61360 1.62972 28 33.54409 1.83794.56693 1.76390.59022 1.69428.61400 1.62866 27 34.64446 1.83667.66731 1.76271.59061 1.69316.61440 1.62760 26 3.5.54484 1.83540.56769 1.76151.59101 1.69203.61480 1.62654 25 36.54522 1.83413.56808 1.76032.59140 1.69091.61520 1.62548 24 37.,54560 1.83286.56846 1.75913.59179 1.68979.61561 1.62442 23 38.54597 1.83159.56885 1.75794.59218 1.68866,61601 1.62336 22 39.54635 1.83033.66923 1.75675.59258 1.68754.61641 1.62230 21 40.54673 1.82906.56962 1.75556.59297 1.68643.61681 1.62125 20 41].54711 1.82780.57000 1.75437:.59336 1.68531.61721 1.62019 19 42.54748 1.82654.57039 1.75319.59376 1.68419.61761 1.61914 18 43.54786 1.82528.57078 1.75200.59415 1.68308.61801 1.61808 17 44.54824 1.82402.57116 1.75082.59454 1.68196.61842 1.61703 16 45.54862 1.82276.57155 1.74964.59494 1.68085.61882 1.61598 15 46.54900 182150.57193 1.74846.59533 1.67974.61922 1.61493 14 47.54938 1.82025.57232 1.74728.59573 1.67863;,61962 1:61388 13 48.54975 1.81899.57271 1.74610..59612 1.67752'.62003 1.61283 12 49.55013 1.81774.57309 1.74492.69651 1.67641.62043 1.61179 11 50.55051 1.81649.57348 1.74375.59691 1.67530.62083 1.61074 10 51-.55089 1..81524.57386 1.74257.59730 1.67419.62124 1.60970 9 52.55127 1.81399:57425 1.74140 5:9770 1.67309.62164 1.60865 8 53.55165 1.81274.57464 1.74022'.59809 1.67198.62204 1.60761 7 54.55203 1.81150.57503 1.73905.59849 1.67088.62245 1.60657 6 55.55241 1.81025.57541 1.73788.59888 1.66978.62285 1360553 5 56.55279 1.80901.67580 1.73671.59928 1.66867.62325 1.60449 4 57.55317 1.180777.67619 1.73555.59967 1.667567.62366 1.60345 3 58.65355 1.80653.57657 1.73438.60007 1.66647.62406 1.60241 2 59.55393 1.80529.57696 1.73321.60046 1.66538.62446 1.60137 1 {0.56431 1.80405.57735 1.73205 6.60086 1.6428.62487 1.60033 0 X. Ctaxg. Tng. Cotang. TlaNg. Cotng. TaWg. Cotang. Tang.. M Ij __ c. pt __, S), 238 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 320 330 340 350 M Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.62487 1.60033.64941 1.3986.67451 1.43256.70021 I.42615 60 1.62527 1.69930.64982 1.53888.67493 1.48163.70064 1.42726 59 2.62568 1.59826.65024 1.53791.67536 1.48070.70107 1.42638 58 3.62608 1.59723.65065 1.53693.67578 1.47977.70151 1.42550 57 4.62649 1.59620.65106 1.53595.67620 1.47885.70194 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 8.62811 1.69208.65272 1.53205.67790 1.47514.70368 1.42110 52 9.62852 1.69105.65314 1.53107.67832 1.47422.70412 1.42022 51 10.62892 1.69002.65355 1.53010.67875 1.47330.70455 1.41934 50 11.62933 1.68900.65397 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 1.58388.65604 1.52429.68130 1.46778.70717 1.41409 44 17.63177 1.58286.65646 1.52332..68173 1.466S6.70760 1.41322 43 18.63217 1.58184.65688 1.62235.68215 1.46595.70804 1.41235 42 19.63258 1.58083.65729 1.52139.69258 1.46503.70848 1.41148 41 20.63299 1.57981.65771 1.52043.68301 1.46411.70891 1.41061 40 21.63340 1.7879.65813 1.51946.68343 1.46320.70935 1.40974 39 22.63330 1.57778.65854 1.51850.63386 1.46229.70979 1.40887 38 23.63421 1.57676.65896 1.51754.68429 1.46137.71023 1.40800 37 24.63462 1.57576.65938 1.51658.68471 1.46046.71066 1.40714 36 25.63503 1.57474.65980 1.51562.68514 1.45955.71110 1.40627 36 26.63544 1.57372.66021 1.51466.68557 1.45864.71154 1.40540 34 27.63584 1.67271.66063 1.51370.68600 1.45773.71198 1.40454 33 28.63625 1.57170.66105 1.61275.68642 1.45682.71242 1.40367 32 29.63666 1.57069.66147 1.51179.68685 1.45592.71285 1.40281 31 30.63707 1.56969.66189 1.51084.68728 1.45501.71329 1.40195 30 31.63748 1.56r68.66230 1.50988.68771 1.45410.71373 1.40109 29 32.63789 1.56767.66272 1.50893.68814 1.45320.71417 1.40022 28 33.6.3830 1.56667.66314 1.50797.68857 1.45229.71461 1.39936 27 34.63371 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 25 36.63953 1.56366.66440 1,50512 s68985 1.4495&.71593. 1.39679 24 37.63994 1.56263..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 40.64117 1.55966.66608 1.50133.69157 1.44598 71769 1.39336 20 41.64158 1.55866.66650 1.50038.69200 1L44508 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 15 46.64363 1.55368.66860 1.49566.69416 1.44060.72034 1.38824 14 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 1.5.5071.66986 1.49284.69545 1.43792.72167 1.38568 11 50.64528 1.54972.67028 1.49190.69588 1.43703.72211 1.38484 10 51.64.569 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 53.64652 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 59.64899 1.54085.67409 1.48349.69977 1.42903.72610 1.37722 1 60.64941 1.53986.67451 1.48-256.7(021 1.42815.726.54 1.37638 0 OoCtang. Tang. Cotang. Tang Cotang. Tang. Cotang. Tang.. ~57 5a SC) 5 40 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 239 360 370 380 390 m. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.72654 1.37638.75355 1.32704.78129 1.27994.80978 1.23490 60 1.72699 1.37554.75401 1.32624.78175 1.27917.81027 1.23416 59 2.72743 1.37470.75447 1.32544.78222 1.27841.81075 1.23343 58 3.72788 1.37386.75492 1.32464.78269 1.27764.81123 1.23270 57 4.72832 1.37302.75538 1.32384.78316 1.27688.81171 1.23196 56 5.72877 1.37218.75584 1.32304.78363 1.27611.81220 1.23123 55 6.72921 1.37134.75629 1.32224.78410 1.27535.81268 1.23050 54 7. 72966 1.37050.75675 1.32144.78457 1.27458.81316 1.22977 53 8.73010 13696.75721 1.32064.78504 1.27382.81364 1.22904 52 9.73055 1.36883.75767 131984.78551 1.27306.81413 1.22831 51 10.73100 1.3600 1. 75812 1.31904.78598 1.27230.81461 1.22758 50 11.73144 1.36716.75858 1.31825.78645 1.27 53.81510 1.22685 49 12.73189 1.36633.75904 1.31745.78692 1.27077.81558 1.22612 48 13.73234 1.36549.75950 1.31666.78739 1.27001.81606 1.22539 47 14.73278 1.36466.75996 1.31586.78786 1.26925.81655 1.22467 46 15.73323 1.36383.76042 1.31507.78834 1.26849.81703 1.22394 45 16.73368 1.36300.76088 1.31427.78881 1.26774.81752 1.22321 44 17.73413 1.36217.76134 1.31348.78928 1.26698.81800 1.22249 43 18.73457 1.36134.76180 1.31269.78975 1.26622.81849 1.22176 42 19.73502 1.36051.76226 1.31190.79022 1.26546.81898 1.22104 41 20.73547 1.35968.76272 1.31110.79070 1.26471.81946 1.22031 40 21.73592 1.35885.76318 1.31031.79117 1.26395.81995 1.21959 39 22.73637 1.35802.76364 1.30952.79164 1.26319.82044 1.21886 38 23.73681 1.35719.76410 1.30873.79212 1.26244.82092 1.21814 37 24.73726 1.35637.76456 1.30795.79259 1.26169.82141 1.21742 36 25.73771 1.35554.76.502 1.30716.79306 1.26093.82190 1.21670 35 26.73816 1.35472.76548 1.30637.79354 1.26018.82238 1.21598 34 27.73861 1.35389.76594 1.30558.79401 1.25943.82287 1.21526 33 28.73906 1.35307.76640 1.30480.79449 1.25867.82336 1.21454 32 29.73951 1.3,5224.76686 1.30401.79496 1.25792.823385 1.21382 31 30.73996 1.35142.76733 1.30323.79544 1.25717.82434 1.21310 30 31.74041 1.35060.76779 1.30244.79591 1.25642.82483 1.21238 29 32.74086 1.34978.76825 1.30166.79639 1.25567.82531 1.21166 28 33.74131 1.34896.76871 1.30087.79686 1.25492.82580 1.21094 27 34.74176 1.34814.76918 1.30009.79734 1.25417.82629 1.21023 26 35.74221 1.34732.76964 1.29931.79781 1.25343 82678 1.20951 25 36.74267 1.34650.77010' 1.29853 7989 1.25268.82727 1.20879 24 37.74312 1.34568.77057 1.29775.:79877 1.25193..82776 1.20808 23 38.74357 1.34487.77103 1.29696.79924 1.25118.82825 1.20736 22 39.74402 1.34405.77149 1.29618.79972 1.25044.82874 1.20665 21 40.74447 1.34323.77196 1:29541.80020 1.24969..82923 1.20593 20 41.74492 1.34242.77242 1.29463.80067 1:24895.82972 1.20522 19 42.74538 1.34160.77289 1.29385.80115 1.24820.83022 1.20451 18 43.74593 1.34079.77335 1.29307.80163 1.24746.83071 1.20379 17 44.74628 1.33998.77382 1.29229.80211 1.24672.83120 1.20308 16 45.74674 1.33916.77428 1.29152.80258 1.24597.83169 1.20237 15 46.74719 1.33835.77475 1.29074.80306 1.24523.83218 1.20166 14 47.74764 1.33754.77521 1.28997.80354 1.24449.83268 1.2005 13 48.74810 1.33673.77568 1.28919.80402 1.24375.83317 1.20024 12 49.74855 1.33592.77615 1.28842.80450 1.24301.83366:.19953 11 50.74900 1.3.3511.77661 1.28764.80498 1.24227.83415 1.19882 10 51.74946 1.33430.77708 1.28687.80546 1.24153.83465 1.19811 9 52.74991 1.33349.77754 1.28610.80594 1.24079.83514 1.19740 8 53.75037 1.33268.77801 1.28533.80642 1.24005.83564 1.19669 7 54.75082 1.33187.77848 1.28456.80690 1.23931.83613 1.19599 6 55.75128 1.33107.77895 1.28379.80738 1.2.3858.83662 1.19528 5 56.75173 1.33026.77941 1.28302.80786 1.23784.83712 1.19457 4 57.75219 1.32946.77988 /1.28225.80(34 1.23710.83761 1.19387 3 58.75264 1.32965.78035 1.28148.80882 1.23637.83811 1.19316 2 59.75310 1.32785.78082 1.23071.80930 1.23563.83860 1.19246.1 60.75355 1.32704.78129 1.27994.80978 1.23490.83910 1.19175 0 M.iT. ITan. CIog.tan. ITn Cotang. Tang. (otang. Tag. M. 530 520 o 500 240 TABLE.XV. NATURAL TANGENTS AND CO IANGENTS. 400 410 420 430 ___ ang. T Cotang. Tang. Cotang. Tang. Cotang. Tag. otangM. 0.83910 1.19175.86929 1.15037.90040 1.11061.93252 1.07237 60 1.83960 1.19105.86980 1.14969.90093 1.10996.93306 1.07174 59 2.84009 1.19035.87031 1.14902.90146 1.10931.93360 1.07112 58 3.84059 1.18964.87082 1.14834.90199 1.10867.93415 1.07049 57 4.84108 1.18894.87133 1.14767.90251 1.10802.93469 1.06987 56 5..84158 1.18824.87184 1.14699.90304 1.10737.93524 1.06925 55 6.84208 1.18754.87236 1.14632.90357 1.10672.93578 1.06362 564 7.84258 1.18684.87287 1.14565.90410 1.10607.93633 1.06800 53 8.84307 1.18614.87338 1.14498.90463 1.10543.93688 1.06738 52 9.84357 1.18544.87389 1.14430.90516 1.10478.93742 1.06676 51 10.84407 1.18474.87441 1.14363.90569 1.10414.93797 1.06613 50 II.84457 1.18404.87492 1.14296.90621 1.10349.93852 1.06551 49 12.84507 1.18334.87543 1.14229.90674 1.10285.93906 1.06489 48 13.84556 1.18264.87595 1.14162.90727 1.10220.93961 1.06427 47 14.84606 1.18194.87646 1.14095.90781 1.10156.94016 1.06365 46 15.84656 1.18125.87693 1.14028.90334 1.10091.94071 1.06303 45 16.8476 1.18055.87749 1.13961.90887 1.10027.94125 1.06241 44 17.84756 1.17986.87801 1.13394.90940 1.09963.94180 1,06179 43 18 84806 1.17916.87852 1.13823.90993 1.09399.94235 1.06117 42 19.84856 1.17846.87904 1,13761.91046 1.09334.94290 1.06056 41 20.84906 1.17777.87955 1.13694.91099 1.09770.94315 1.05994 40 21.84956 1.17703.880027.91153 1.09706.944)0 1.05932 39 22.85006 1.17633.83059 1.13561.91206 1.09642.9445.3 1.05870 38 23.85057 1.17569.88110 1.13494.91259 1.09573.94510 1.05809 37 24.85107 1.17500.88162 1.13423.91:13 1.09514.94565 1.05747 36 25.85167 1.174.30.88214 1.13361.91366 1.09450.94620 1.05685 35 26.85207 1.17361.88265 1.13295.91419 1.093S6.94676 1.05624 34 27.85257 1.17292.88317 1.13223.91473 1.09322.94731 1.0.5562 33 28.85308 1.17223.88369 1.13162.91526 1.092538.94786 1.05501 32 29.85358 1.17154.88421 1.13096.91530 1.09195.94841 1.05439 31 30.85403 1.17085.88473 1.13029.91633 1.09131.94396 1.05378 30 31.8.5158 1.17016.88524 1.12963.91637 1.09067.949.;2 1.05317 29 32.85.509 1.16947.88576 1.12397.91740 1.09003.95007 1.05255 28 33.85559 1.16878.88623 1.12331.91794 1.08940.95062 1.05194 27 34.85699 1.16309.83630 1.12765.91847 1.03376.95118 1.05133 26 3..85660 1.16741.88732 1.12699.91901 1.03813.95173 1.05072 25'36.85710 1.16672.88784 1-.26:33.91955 1.087491.95229'1.0010 24 37.85761 1.16603 -88836 1.12567.920081 1.036.9524 1.4949 23 38.85811 1.16535.88888 1.12501.92062 1.03622'.95340L 1.04838 22 39.85862 1.16466.88940 1.12435.92116 1.03559.95395 1.04827 21 40.85912 1.16398.88992 1.12369.92170 1.03496.95451 1.04766 20 41.85963 1.16329.89045 1.12.303.92224 1.08432.95506 1.04705 19 42.86014 1.16261.89097 1.122.38.92277 1.03369.95562 1.04644 18 43.86064 1.16192.89149 1.12172.92.331 1.03306.95618 1.04583 17 44.86115 1.16124.89201 1.12106.92385 1.08243.95673 1.04522 16 45.86166 1.16056 53 111.9s2439 1.03179.95729 1.04461 15 46.86216 1.15987..89306 1.11975.92493 1.08116 95785 1.04401 14 47.86267 1.15919,89358 1.11909.92547 1.08053.95841 1.04340 13 48.86318 1.15851.89410 1.11844.92601 1.07990.95897 1.04279 12 49'86368 1.15783.89463 1.11778.92655 1.07927,95952 1.04218 1 50.86419 1.15715.89515 1.11713.92709 1.07864.96003 1.04158 10 51.86470 1.15647.89567 1.11648.92763 1.07801.96064 1.04097 9 52.86521 1.15.579.89620 1.11582.92817 1.07738,96120 1.04036 8 53.86572 1.15511.89672 1.11517.92872 1.0767I.96176 1.03976 7 54'..86623 1.15443.89725 1.11452.92926 1.07613.96232 1.03915 6 55.86674 1.15375.89777 1.11387.92980 1.07550.96288 1.03355 5 56.67-25 1.15309.89830 1.11321.93034 1.07487.96344 1.0.3794 4 57.86776 1.15240 8983 1. 11256.93088 1.07425.96400 1.03734 3:58.86327 1.15172.89935 1.11.191.93143 107362.96457 1.03674 2 59,.86878 1.151014 89988 1.11126.93i97 1.07299.96513 1.03613 1.60.86929. 15037.90010 1.11061.93252 1.07237.96569 1.03553 0. coog. Tang. Cotag.,T>mg. Cotang. /apg. Qtang. 10. ~ 11 4. ^9_~ -. 9 4. 4.. I~.~7~ TABLE XV. NATURAL TANGENTS AND COTANGENTS. 241 440 440 440 M. Tang. Cotang. M. M. Tang. Cotang. M. M. Tang. Cotang. M. 0.96569 1.03553 60 20.97700 1.02355 40 40.98843 1.01170 20 1.96625 1.03493 59 21.97756 1.02295 39 41.98901 1.01112 19 2.96681 1.03433 58 22.97813 1.02236 38 42.98958 1.01053 18 3.96733 1.03372 57 23.97870 1.02176 37 43.99016 1.00994 17 4.96794 1.03312 56 24.97927 1.02117 36 44.99073 1.00935 16 5.96850 1.03252 55 25.97984 1.02057 35 45.99131 1.00876 15 6.96907 1.03192 54 26.98041 1.01993 34 46.99189 1.00818 14 7.96963 1.03132 53 27.98098 1.01939 33 47.99247 1.00759 13 8.97020 1.03072 52 23.98155 1.01879 32 48.99304 1.00701 12 9.97076 1.03012 51 29.98213 1.01820 31 49.99362 1.00642 11 10.97133 1.02952 50 30.98270 1.01761 30 50.99420 1.00583 10 11.97189 1.02392 49 31.93327 1.01702 29 51.99478 1.00525 9 12.97246 1.02832 48 32.98334 1.01642 28 52.99536 1.00467 8 13 97302 1.02772 47 33.98441 1.01583 27 53.99594 1.00408 7 14.97359 1.02713 46 34.98499 1.01524 26 54.99652 1.003r50 6 15.97415 1.02653 45.35.98556 1.01465 25 55.99710 1.00291 5 16.974'2 1.02593 44 36.98613 1.01406 24 56.99763 1.00233 4: 17.97529 1.02533 43 37.93671 1.01347 23 57.99826 1.00175 3 18.97586 1.02474 42 33.98728 1.01288 22 58.99884 1.00116 2 19.97643 1.02414 41 39.98786 1.01229 21 59.99942 1.00058 1 20.97703 1.02355 40 40.98843 1.01170 20 60 1.0000 1.00000 0 M. Cotang. Tang. M - Cotang. Tang. M.. Cotang. Tang. M. 453 _ 450 453 - 242 TABLE XVI. RISE PER MILE OF VARIOUS GRADES. RISE PER MILE OF VARIOUS GRADES. Rise per Grade Rise per Rise per Rise per per per Mil er Mile. tatio sn. t tl n. t b Sen..01.528.41 21.648.81 42.768 1.21 63.388.02 1.056.42 22.176.82 43.296 1.22 64.416.03 1.584.43 22.704.83 43. 21 1.23 64.944.04 2.112.44 23.232.84 44.352 1.24 65.472.05 2.640.45 23.760.85 44.80 1.2.5 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 2.5.344.88 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 1..34 70.752.15 7.920.55 29.040.95 50.160 1.35 71.280.16 8.448.56 29.568.96 50.68s 1.36 71.808.17 8.976.57 30.096.97 51.216 1.37 72.336.18 9.504.58 30.624.93 51.744 1.38 72.864.19 10.032.59 31.152.99 52.272 1.39 73.392.20 10.560.60 31.680 1.00 52.800 1.40 73.920.21 11.088.61 32.208 1 0l 53.328 1.41 74.448.22 11.616.62 32.736 i.J.^ 53.8.36 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 1.11 58.608 1.51 79.728.32 1'6.896.72 38.016 1.12 59.136 1.52 80.256.33 17.424.73 38.544 1.13 59.664 1.53 80.784.34 17.952.74 39.072 1.14 60.192 1.54 81.312.35 18.480.75 39.600 1.15 60.720 1.55 81.840.36 19.008.76 40.128 1.16 61.248 1.56 82.368.37 19.536.77 40.656 1.17 61.776 1.57 82.896.38 20.064.78 41.184 1.18 62.304 1.58 83.424.39 20.592.79 41.712 1.19 62.832 1.59 83.952.40 21.120.80 42.240 1.20 63.360 1.60 84.480 _ - -. =I TABLE XVI. RISE PER MILE OF VARIOUS GRADES. 243 Grade Rise per Grade e pe Grade Rise per Grade Rise per Station. tatio. aion. tation. 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 1.63 88.704 1.88 99.264 2.80 147.840 4.80 253.440 1.69 89.232 1.89 99.792 2.90 153.120 4.90 258.720 1.70 89.760 1.90 100.320 3.00 158.400 5.00 264.000 1.71 90.288 1.91 100.848 3.10 163.680 5.10 269.280 1.72 90.816 1.92 101.376 3.20 168.960 5.20 274.560 1.73 91.344 1.93 101.904 3.30 174.240 5.30 279.840 1.74 91.872 1.94 102.432 3.40 179.520 5.40 285.120 1.75 92.400 1.95 102.960 3.50 184.800 5.50 290.400 1.76 92.928 1.96'103.488 3.60 190.080 5.60 295.680 1.77 93.456 1.97 104.016 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 FHE ED