FI ELD-BOOK FOR RAILROAD ENGINEERS FIE LD-BOOK FOR RAILROAP ENGINEERS. CONTAINING FO RM UL. fOR LAYING OUT CURVES, DETERMINING FROG ANGLES, LEVELLING, CALCULATING EARTH-WORK, ETC., ETC., TOGETHER WITH TA B L E S OF RADII, ORDINATES, DEFLECTIONS, LONG CIORDS, MAGNETIC VARIA TION, LOGARITfHMS, LOGARITIMIC AND NATURAL SINES, TANGENTS. ETC., ETC. BY JOHN B. HENCK, A.M., CIVIL ENGINEE R. NEW YORK: D. APPLETON & COMPANY, 549 & 551 BROADWAY. LONDON: 16 LITTLE BRITAIN. 1871. Entered according to Act o' "'-igress, in the year 1854, by D. APimON AND COMPANY, I the Clerk's Office of the District Courc for the Southern District of New YTeo 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 formulae, after long disuse perhaps, the engineer is often called upon to apply at a moment's notice in the field, and he is, therefore, obliged to carry with him in manuscript such methods as he has been able to invent or collect, or resort to what has received the very appropriate name of " 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, m 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, in(icated by a formula standing on a line by itself, and distinguished by a H'P. 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. V for a field-book. Even then they apply only to ground whose cross-section is level, though often used in a manner shown to be erroneous in ~ 128. When the cross-section of the ground is level, the place of the tables is supplied by the formula of ~ 119, and when several sections are calculated together, as is usually the case, and the work is 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 sel. dom 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 vii PREFACE. ble. Thpse specially prepared for this work have been computed with great care. The values have in some cases been carried out farther than ordinary practice requires, in order that interpolated values may be obtained from them more accurately. For the greater part of the material composing the Table of Magnetic Variation the author ix 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 Logarith. mic and Natural Sines, Cosines, &c. The number of tables might have been greatly increased, but for an unwillingness to insert any thing not falling strictly within the plan of the work or.not resting on sufficient authority. J. B. H. BOsTON, Februar, 1854. TABLE OF CONTENTS. CHAPTER I. CIRCULAR CURVES. ARTICLE I.-SIMPLE CURVES. XCT. IPAG 2. Definitions. Propositions relating to the circle. 1 4. Angle of intersection and radius given, to find the tangent 3 5. Angle of intersection and tangent given, to find the radius 3 6. Degree of a curve.... 4 7. Deflection angle of a curve....... A. Method by Deflection Angles. 9. Radius given, to find the deflection angle... 4 10. Deflection angle given, to find the radius... 1I. Angle of intersection and tangent given, to find the deflection angle...... 12. Angle of intersection and deflection angle given, to find the tangent.. 5 13. Angle of intersection and deflection angle given, to find the length of the curve..... 14. Deflection angle given, to lay out a curve.. 16. To find a tangent at any station...... 8 B. Method 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. SECT. PAW 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 are 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 wihen 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. m ARTICLE III.-TURNOUTS AND CROSSINGS. ICT. PAGe 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.-MISCELLANEOUS PROBLEMS. 65. To find the radius of a curve to pass through a given point 46 66. To find the tangent point of a curve to pass through a given point.....47 67. To find the distance to the curve from any point on the 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 1U TABLE OF CONTENTS. saar. PAU 72. To find the tangent point of a curve that shall pass through a given point..... 50 73. To find the radius of a curve without measuring angles. 51 74. To find the tangent points of a curve without measuring angles........ 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.. 5 80. To draw a tangent to a given curve from a given point. 59 81. To flatten the extremities of a sharp curve.... 59 82. To locate a curve without setting the instrument at the tangent point......... 60 83. To measure the distance across a river.. 63 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. ARTICLL I.- HEIGHTS AND SLOPE STAKES. SMT. PAGX 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 THE OUTER RAIL ON CURVES. 110. To find the proper elevation of the outer rail 89 11. Coning of the wheels........ 89 CHAPTER IV. EARTH-WORK. ARTICLE I. PRISMOIDAL FORMULA. 112, Definition of a prismoid...... 92 113.4o find the solidity of a prismoid......92 ARTICLE II. — BORROW-PITS. 114. Manner of dividing the ground.... 93 xiV TABLE OF CONTENTS. SECT. PAGQ 115. To find the solidity of a vertical prism whose horizontal section is a triangle.........93 116. To find the solidity of a vertical prism whose horizontal section is a parallelogram. 94 117. To find the solidity of a number of adjacent prisms having the same horizontal section..... e ARTICLE 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. Certre and Side Heights given. 121. Mode of dividing the ground....... 9 122. To find the solidity of one section.... 100 123. To find the solidity of any number of successive sections. 104 125. To find the solidity when the section is partly in excavation and partly in embankment.. 105 126. Beginning and end of an excavation... 107 C. 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. PAGS I. Radii, Ordinates, Tangent and Chord Deflections, and Ordinates for Curving Rails. 115 II. Long Chords......119 TABLE OF CONTENTS. Xi no. 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 V II. Properties of Materials....... 123 IX. Magnetic Variation........126 X. Trigonometrical and Miscellaneous Fcnmul.. 133 XI Squares, Cubes, Square Roots, Cube Roots, and Reciprocals..... 137 XII. Logarithms of Numbers.....155 XIII. Logarithmic Sines, Cosinee Tangents, and Cotangents 171 XIV. Natural Sines and Cosines... 219 XV. Natural Tangents and Cotangents.. 229 XVL Rise per Mile of Various Grades... 4 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 j 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." The ratio of one quantity to another may be regarded as the quo. tient of the first divided by the second. Hence, the ratio of a to b is expressed by a: b, and the ratio of c to d by c: d. A proportion ex presses the equality of two ratios. Hence, t. 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. THF railroad curves here considered are either Circular or Para, bolic. Circular curves are divided into Simple, Reversed, and Compound Curves. We begin with Simple Curves. 2. Let the arc ADEFB (fig. 1) represent a railroad curve, unitC/ Fig. 1. / \ A\\ B o 2 CIRCULAR CURVES. ing the straight lines GA and B I. The length of such a curve is measured by chords, each 100 feet long.* Thus, if the chords A D, D E, E F, and FB are each 100 feet in length, the whole curve is said to be 400 feet long. The straight lines G A and B H are always tangent to the curve at its extremities, which are called tangent points. If GA and BH 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 pro(luced, is called the angle of intersection, and shows the chanrqe of direction in passing from one tangent to the other. The following propositions relating to the circle are derived from Geometry. 1. 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 BA C =A B C. III. An acute angle between a tangent and a chord is equal to half the central angle subtended by the same chord. Thus, CA B AOB. IV. An acute angle subtended by a chord, and having its vertex in the circumference of a circle, is equal to half the central angle subtended by the same chord. Thus, D AE = i D 0E. 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 = DO E, and ) A E = EA F. VI. The angle of intersection of two tangents is equal to the central angle subtended by the clord which unites the tangent points. Thus, KCB = AO B. 3. In order to unite two straight lines, as GA and BH, by a curve. the arigle of intersection is measured, and then a radius for the curve may be assumed, and the tangents calculated, or the tangents maybe assumed of a certain length, and the radius calculated. * Some engineers prefer a chain 50 feet in length, and measure the length cf a turve by chords of 50 instead of 100 feet. The chord of 100 feet has been adopted throughout this article; but the formulae deduced may be very readily modified to suit chords of any length. See also ~ 13. SIMPLE CURVES. 3 4. Problem. Given the angle of intersection K R =B I (Jig!) and the radius A 0 =- 1, to find the tangent A C = T. K Fig. 1. A' * B 0 Solution. Draw CO. Then in the right triangle A 0 C we bave AC (Tab. X. 3) - tan. A 0 C, or, since A 0 C= I (~ 2, VI.) = tan. I I; 837'.'. T=Rtan.al. Example. Given I = 22~ 52', and R = 3000, to find T. Here R = 3000 3.477121 I -= 11~ 26t tan. 9.30586SC T= 60672 2.782990 5. Problem. Given the angle of intersection K C B = I (fig. I), ifd the tangent A C = T, to find thp ra(dilS A O = R. 4 CIRCULAR CURVES. Solution. In the right triangle A 0 C we have (Tab. X. 6) AO R -T = cot. A C, or - =cot. 1I; B^1.'. R =T cot. 1. Example. Given I= 31~ 16' and T= 950, to find R. Here T= 950 2.977724 ^ -- 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 60 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 AOD or half FOB. 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 CB F = D. Solution. Draw 0 L perpendicular to BF. Then the angle B OL = B O F= D, and B L = BF= 50. ]ut in the right triangle BL OBL we have (Tab. X. 1) sin. B OL = ~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 CBF = D (fig. 1), to find the radius A 0 = R. METHOD BY DEFLECTION ANGLES. 5 Solution. By the preceding section we have sin. D-= =, whence R sin. D = 50; 50 sin. D By this formula the radii in Table I. are calculated. Example. Given D = 10, to find R. Here 50 1.698970 D - 1~ sin. 8.241855 R = 2864.93 3.457115 11. Problem. Given the angle of intersection K CB = I (fig. 1), and the tangent A C T, to fild the deflection angle CA D = D. Solution. From ~ 9 we have sin. D =-, and from ~ 5, R= T cot. A I. Substituting this value of R in the first equation, we get 50 sin. D cot. I; sin. D = 50 tan. i I T Example. Given I = 21~ and T= 424.8, to find D. Here 50 1.698970 1 -= 10~ 30 tan. 9.267967 0.966937 T = 424.8 2.628185 D 1 15' sin. 8.338752 12. Problem. Given the angle of intersection K CB = I (fig. 1): and the deflection angle CA D = D, to find the tangent A C = T.,. Solution. From the preceding section we have sin. D = tan. T Hence, Tsin. D = 50 tan. ^ I;.. T =50 tan. ^7 sin. D Example. Given 1 = 28~ and D = 1~, to find T. Here T 50 tan. 14 71431. sin 714.31. sin 1~ 6 CIRCULAR CURVES. 13. Probleill. Gicen the angle of intrsetion K CB = I (.fi. 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, EF, &c. subtends at A an additional angle D A E, EA 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 CAB = A 0 B = ^ [ is, therefore, made up of as many times D, as there are chords around the curve. Then if n represents the number of chords, we have n D = i I; I. D If D is not contained an even number of times in I, thi quotient above will still give the length of the curve. Thus, in fig. 2, suppose ) is contained 41 times in 1 I. This shows that there will be four whole chords and j of a chord around the curve from A to B. The angle G A B, the fiaction of ID, is called a sub deflection angle, and G B. the fraction of a chord, is called a sub-chord. The length of the curve tlhus 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 = 10 20', to find the length of 80 26' 506' the curve. Here n- - = 20 - 8 -6.325, that is, the curve D 10 20' - - is 632.5 feet long. To find the arc itself in this example, we take from Table VI. the length of an are of 16~ 52', since the central angle of the whole curve is equal to I (~ 2, VI), and multiply this length by the radius of the curve. Arc 1(P -.1745329 " 6~.1047198 " 50' =.0145444 " 2t =.0005818 16~ 52' =.2943789 * This method of finding the length of a sub-chord is not mathematically accurate; for, by geometry, angles inscribed in a circle are proportional to the arcs on which they stand; whereas this method supposes them to be proportional to the chords of these arcs. In railroad curves, the error arising from this suppositinp iw too small to be regarded. METHOD BY DEFLECTION ANGLES. I The radius of the curve is found from Table I. to be 2148.79, and this multiplied by.2943789 gives 632.558 feet for the length of the arc. 14. Problem. Given the deflection angle D, to lay out a curve from a given tangent point. L Fig. 2. M Solution. Let A (fig. 2) be the given tangent point in the tangent H C. Set the instrument at A, and lay off the given deflection angle D from A C. This will give the direction A D, and 100 feet being measured from A in this direction, the point D will be determined Lay off in succession the additional angles D A E, E A F, &c., each equal to D, and make DE, EF, &c. each 100 feet, and the points E, F, &c. will be determined. The points D, E, F, &c., thus determined, are points on the required curve (~ 7, and ~ 2, III., IV.), and are called stations. If there is a sub-chord at the end, as G B, the sub-deflection angle GAB 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 tlie tangent to the curve at this station; for 8 CIRCULAR CURVES. then the curve could be continued by deflections from the new tangent. in precisely the same way as it was begun from the first tangent. 16. 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. Therefogre 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 angle MBN=LBF=L FB, the total deflection from the pro ceding tangent FL. B. Method by Tangent and Chord Deflections. 17. Let A B CD (fig. 3) be a curve between the two tangents E A and DL, having the chords A B, B C, and CD of the same length K'I Fig. 3. A Produce the tangent EA, and from B draw B G perpendicular to A G. Produce also the chords A B and B C, and make the produced METHOD BY TAN xENT AND CHORD DEFLECTIONS. & parts B H and CK of the same length as the chords. Draw CII and D K. B G is called the tangent deflection, and CH or D K the chord deflection. 18. Problem. Given the radius A 0 = R (fig. 3), to find the tangent deflection B G, and the chord deflection CH. Solution. The triangle CB H is similar to B 0 C; for the angle BOC = 1803 -(OB C+ B CO), or, since B CO = A B 0, BO C 180 - (0 B C - AB 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 H, or, representing the chord by c and the chord deflection by d, R: c = c: d;.. d=. To find the tangent deflection, draw B 2 to the middle of Cll, bisecting the angle CBH, and making B MC a right angle. Then the right triangles B M C and A G B are equal; for B C = A B, and the angle CBM=- CBH=i-BOC=^AOB=BAG (~2, IlI.). Therefore B G = CM = i 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 deflection d. Solution. By the preceding section we have d = -, and by ~ 10, i -tsin. D Substituting this value of R in the first equation, we find c2 sin. D ~~ d-.... 50 This formula gives the chord deflection for a chord c of any length, though D is the deflection angle for a chord of 100 feet (~ 7). When c = 100, the formula becomes d = 200 sin D, or for the tangent deflection E d= 100 sin. D. By these formulae the tangent and chord deflections in Table I. may be easily obtained from the table of natural sines. 20. The length of the curve may be found by first finding D (~ 9 or 5 11), and then proceeding as in ~ 13. 21. Problem. To draw a tangent to the curve at any station, as B (fig. 3). Solution. Bisect tne chord deflection H C of the next station in M. 2 10 CIRCULAR CURVES. A line drawn through B and M will be the tangent requirel; for it has been proved (~ 18) that the angle C B _M is in this case equll t, i 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 th! chord is extended to HI on A B produced, the point II liarkik,l, and iet chain then swung round, keeping the end att I fixed, until I I1 = (1, B i11 will he the direction of the required tangent.' 22. Problem. Given the chord defflction d, to laty out a curve froml 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 chain of 100 feet firom A to G on the tangent E A produced, and mark the point G. Swing the chain round towards A B. keeping the end at A fixed, until B G is equal to the tangent deflection 1 d, and B will be the first station on the curve. Stretch the chain from B to [I on A B pro duced, and having marked this point, swing the chain round, until IIC 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 pro duced, until the curve is finished Should a sub-cholrd D F occur at the end of the curve, find the tan gent D L at D (~ 21), lay off from it the proper tangent deflection L f for the given sub-chord, making DF of the given length, and F will be a point on the curve. The proper tangent deflection for the subchord may be found thus. Represent the sub-chord by c', and the corresponding chord deflection by d', and we have (~ 18) d' = R; but since i d = we have i d': 2 d = c2' c:. Therefore d' - d () Example. Given the intersection angle I between two tangents equal to 16~ 30', and R = 1250, to find T, d, and the length of the curve in stations. Here ( 4) T== R tan. I =1250 tan. 8~ 15t = 181.24; e2 1002 ( 18) d = - = 8; R 1250 * 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 1I may be calculated; for B M =,/ -2~- HA-i ORDINATES. 1) 50 50 ( 9) sin. D = -= 2 =.04 = nat. sin. 2~ 17'; 12-50 31 80 15' 495' ~13) nn 2= 17 --- -= 3.60. D 20 17~' 137.5' 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 KD - 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' = 4 ( )== 4 X.62 =4 X 36 = 144. LF= 1.44 feet being laid off from DL, the point F will, if the work is correct, fall upon the second tangent point. A tangent at F may be found (~ 21) by producing DF to P, making FP = DF= 60 feet, and laying off PN = 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 assumed a radius, the tangent points may be found by ~ 74. C. Ordinates. 24. The preceding methods of laying out curves determine points 100 feet distant from each other. These points are usually sufficient for grading a road; but when the track is laid, it is desirable to have intermediate points on the curve accurately determined. For this 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. Problem. Given the deflection angle D or the radius R of a curve, to fJid the ordinatesfor any chord. Solution. I. To find the middle ordinate. Let A EB (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 ED, and denote it by m. Produce ED to the centre F, and join A F and A E. Then (Tab. X. 31 C A A\ D N 0 P Fig. 4. F G RD - = tan. A D, or ED = A D tan. E A D. But, since the angle AD EA D is measured by half the arc BE, or by half the equal arc A E, we have E A D = A FE. Therefore E D = AD tan. J A FE, or IIP m= c tan. AFE. When c - 100, AFE = D (~ 7), and m = 50 tan. X D, whence m may be obtained from the table of natural tangents, by dividing tan. 4 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= JA F2 - A D2 = ^/R2 -- c. Then m =-EF-DF = R —DF, or O7 m = R- ^/R2 - - c. II. To find any other ordinate, as R N, at a distance DN= 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 = JR F* - F G =/R2 - b~, and R N= R G - NG = R G- D F. Substituting the value of R G and that of DF found above, we have M R N = /R2 - ~ RR" 2 c2. ORDINATES. 13 By these formulae the ordinates in Table I. are calculated. The other ordinates may also be found from the middle ordinate by the following shorter, but not strictly exact method. It is founded on the supposition, that, if the half-chord B D be divided into any number of equal parts, the ordinates at these points will divide the arc E B into the same number of equal parts, and upon the further supposition, that the tangents of small angles are proportional to the angles themselves. These 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 AR, AS, and AT, we have the angle RAN= E EAD, since RB is considered equal to E B. But E A D = AFE. Therefore, R A N= i A FE. In the same way we should find SA 0 -= A FE, and TA P =- A FE. We have then for the ordinates, R N= A Ntan. RAN = c tan. I AFE, SO=A 0 tan. SAO = i c tan. i A FE, and TP = A P tan. TA P = c tan. A FE. But, by the second supposition, tan. j AFE = i tan. i AFE, tan. A A FE = tan. I A FE, and tan. A FE = tan. A FE. Substituting these values, and recollecting that i c tan. I A FE = m, we have 15 15 RN= =6 X I c tan. I AFE = m, 3 3 Br - SO = X. c tan. 2 AFE= m, TP == X ~c tan. 2 AFE = m. In general, if the number of divisions of the chord on each side of the centre is represented by n, we should find for the respective ordi-.. (n+l)(n-1)m (a+2)(-~2)m nates, beginning nearest the centre, (n 1) (n-1) m 2)(-2) (n + 3) (n-3)m 102 X &C. Example Find the ordinates of an 8~ curve to a chord of 100 feet. 158 Here m = 50 tan. 2~ -- 1.746, RN= 6 m= 1.637, S O0 m = 1.310, and TP = 7 m = 0.764. 26. An approximate value of m also may be obtained from the formula m = R - ^/R2 - c2. This is done by adding to the quantity under the radical the very small fraction 64 R1 making it a perfect 14 CIRCULAR CURVES. square, the root of which will be R- R. We have, then, m R (R C2 ); -( 8)C2 8M= 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 A B, its middle ordinate C t = 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 fiom A to B, and by means of the middle ordinate determine the point E. Then stretch the cord fiom A to E, and lay off the middle ordinate CI = 1 ED, thus determining the point C, and so continue to lay off from the successive half-chords one fourth the preceding ordinate, until a sufficient number of points is obtained. D. Curvirng Rails. 28. The rails of a curve are usually curved before they are laid. To do this properly, it is necessary to know the middle ordinate of the curve for a chord of the length of a rail. 29. Problem. Given the radius or deflection angle of a "urve, to find the middle ordinate for curving a rail of given length. Solution. Denote the length of the rail by 1, and we have (~ 25) the exact formula m = RI - J/2 - 4 1, and (~ 26) the approximate formula 2 R This formula is always near enough for chords of the length of a rail If we'substitute for R its value (~ 10) R = — in D, we have, i2 ^m=4 - T X sin. D 100 Example. In a 1~ 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. 15 92 by the first formula, m -- iin459.3.00707. By the second formula, m =.81 sin. 30' =.00707. The exact formula would give the same result even to the fifth decimal. By keeping in mind, that the ordinate for a rail of 18 feet in a 1~ curve is.007, the corresponding ordinate in a curve of any other degree may be found with sufficient accuracy, by multiplying this decimal by the number expressing the degree of the curve. Thus, for a curve of 5~ 36' or 5.60, the ordinate would be.(t7 X 5.6 =.039 ft..468 in. For a rail of 20 feet we have J 12 = 100, and, consequently, mn sin. D. This gives for a 1~ curve, in =.0087. The corresponding ordinate in a curve of any other degree may be found with sufficient accuracy, by multiplying this decimal by the number expressing the degree of the curve. By the above formula for 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 tangent at the point of junction. If the curves lie on opposite sides of the common tangent, they form a reversed curve, and their radii may be the same or different. If they lie on the same side of the common tangent, Fig. 5. thev have different radii, and form a oompound curve. Thus A B C'fig. 5) is a reversed onrve, and A BD a compound curve. 16 CIRCULAR CURVES. 31. Problem. To lay out a reversed or a compound curve, whet the radii or deflection angles and the tangent points are known. Solution. Lay out the first portion of the curve from A to B (fig. 5), by one of the usual methods. Find B F, the tangent to A B, at the point B (~ 16 or ~ 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 from this tangent either of these portions may be laid off in the usual man ner A. Reversed Curves. 32 Theorem. The reversing point of a reversed curve between,arallel tangents is in the line joining the tangent points. Fig. 6. F H A Demonstration. Let A CB (fig. 6) be a reversed curve, uniting the parallel tangents HA and B K, having its radii equal or unequal, and reversing at C. If now the 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 HA and B K, are parallel. Therefore, the angle A E C= CFB, and, consequently, E CA, the half supplement of A E C, is equal to F CB, the half supplement of CFB; but these angles cannot be equal, unless A C and CB are in the same straight line. 33. Problem. Given the perpendicular distance between two par. allel tangents BD = b (Jig. 6), and the distance between the two tangent points A B = a, to determine the reversing point C and the common radizu E C = C F = R of a reversed curve uniting the tangents HA and B K. Solution. Let AC B be the required curve. Since the radii are REVERSED CURVES. 17 equal, and the angle A E C = B F C, the triangles A E C and B F C are equal, and A C = CB = -a. The reversing point C is, therefore, the middle point of A B. To find 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 BAD== AEC= AEG. Therefore AE: A G =AB: BD, or R: a = a: b; Pre.'.R= a". 4b Corollary. If R and b are given, to find a, the equation R = gives a2 = 4 R b; B... a = a= 2^b. Examples. Given b = 12, and a = 200, to determine R. Here 2002 10000 R = 4 x12 ~- 12- 833. Given R = 675, and b = 12, to find a. Here a = 2V675 X 12 2V8100 = 2 X 90 = 180. 34. Problem. Given the perpendicular distance between two parallel tangents B D = b (fig. 7), the distance between the two tangent points A B = a, and the first radius E C = R of a reversed curve uniting the tangents HA and B K, to find the chords A C = at and CB = a", and the second radius CF = Rt. -A Fig. 7. E Solution. Draw the perpendiculars E G and FL. Then the right triangles A B D and E A G are similar, since the angle BAD D 18 CIRCULAR CURVES. jAEC = E G. Therefore AB:BD = EA:AG, ora:b - R: ia'; t 2Rb...-.o^-_Since a' and art are (~ 32) parts of a, we have i7W all = aa-a'. To find RI the similar triangles A B D and FB L give A B: B D = FB: B L, or a:b = R': a'l; a aft Example. Given b = 8, a = 160, and R = 900, to find at, al", and 2 x 900 x 8 R'. Here a'= 60 90, al - 160- 90 = 70, and R' = 160 x 70 2X8= 700. 35. Corollary 1. If b, a', and at are given, to find a, R, and R', we have (~ 34) ar +a = at2 -+a"; Rt =a a;a a al. 2b 26 Example. Given b = 8, a' = 90, and a"' = 70, to find a, R, and R 160 X.90 160 X 70'Here a = 90 + 70 = 160, R 160 X - 900 and R' = 12 2X8 2X8 700. 36. Corollary 2. If R, R', and b are given, to find a, a', and a", we have (~ 35), + = aaaa a (a a) a2 Therefore 2b 2b b' Therefore 2 _ = 2b (R + R); F1 ^.'. a =^/2 b (R+ R). Having found a, we have (~ 34) a' _- 2 R b. a 2R' b a a J'lrample. Given Z? 900, R' = 700, and b = 8, to find a, a', ana al'. iere a = /2 X 8(900 + 700) =./16 X 1600 =: 160, a' 2X 900 X 8 2 X 700 8 -60 90, and " - 70. -- 160 -- 70 REVERSED CURVES. 19 37. Problem. Given the angle A KB = K, which shows the change of direction of two tangeins HA and B K (fig. 8), to unite, these tangents by a reversed curve of given common radius R, startingfrom a giv. en tangent point A. H F D __ _N BE Fig. 8. Solution. With the given radius run the curve to the point D, where the tangent D N becomes parallel to B K. The point D is found thus. Since the angle N G K, which is double the angle HA D (~ 2, II.), is to be 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. ^ K This will be shown (~ 69) to be the length of the chord for a deflectiop angle I K. Having found the point D, measure the perpendicular distance D M -= b between the parallel tangents. The distance DB -= 2 D C = a may then be obtained from the formula (~ 33, Cor.) VW a=-2.R/b. The second tangent point B and the reversing point C are now determnined. The direction of D B or the angle BDNmay also be obD M tained; for sin. B DN = sin. D B M = D, or UV" s.sin. BDN= N. a 38. Problesi. G:ven the line A B -= a (fig. 9) which joins the fixed tangent points A and B, the angles HA B = A and A B L = B, and the first radius A E = R, to find the second radius B F = I' of a reversed culve to a ite the tangqents IH A and B K. First Solutiozn TfVith the qy;cenl radiits run the curve to the point D, chere the tanjqent D) NT ber,;(-s parallel to B K. The point D is found 20 CIRCULAR CURVES. thus. Since the angle H GN, which is double HA D (~ 2, II.), is equal to A ce B, lay off from HIA the.angle HA D -= (A cM B), and measure in this direction the chord A D = 2 R sin. i (Aoo B) (~ 69) Fig. 9. ~ D / A Setting the instrument at D, run the curve to the reversing point C tn the linefrom D to B (~ 32), and measure D C and CB. Then the similar triangles DE C and BF C give D C: DE = CB: BF. or D C: R - CB:.Rt;...R' CB X R. DC Seconid 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 = 2 R sin. } (A - B), and the included angle DA B = HA 1I - HA D - A - (A -B) = i (A + B). Find in this triangle (Tab. X. 14 and 12) BD and tlie 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 DC = 2 R sin. DE C= 2R sin. D B L (69). But CB = BD - DC; whence 2R' sin. DBL = BD - 2 R sin DBL;.RI = BD _ R. 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 180~ -- (A + B), and the angle DBL = B - A B D. REVERSED CURVES. 21 39. Problem. Given the length of the common tangent D G - a, and the angles of intersection land PI (fig. 10), to determine the common radius CE = C F = R of a reversed curve to unite the tangents HA and B L. Fig. 10. Solution. By ~ 4 we have D C == R tan. j L, and C G = R tan. P; whence R (tan. I +- tan. I') = D C + C G = a, or 11? R a tan. I7 + tan. ~ IV This formula may be adapted to calculation by logarithms; for we sin. (I+ P) have (Tab. X. 35) tan. I I+ tan. P == -c.. t Substituting this value, we get R _- a cos. I cos. 1 P. sin. (I + P) The tangent points A and B are obtained by measuring from D a distance A D = R tan. I I, and from G a distance B G = R tan. I P. Example. Given a = 600, 1 = 12~, and P = 8~, to find R. Here a = 600 2.778151 r - 60 cos. 9.997614 1 = 40 cos. 9.998941 2.774706 i (I+ P) - 10~ sin. 9.239670 R = 3427.96 3.535036 22 CIRCULAR CURVES. 40. Problem. Given the line A B = a (fig. 10), which joins the fixed tangent points A and B, the angle DA B = A, and the angle A B G = B, to find the common radius E C= CF = R of a revered curve to unite the tangents HA and B L. Fig. 10. / D Solution. Find first the auxiliary angle A KE = B K F, which mi4 -5 21.5 40 160 68 43 2 20 4^i 5 -6 23 43 215 80 92 3 23 6-j, 6 — 8 26 49 294 115 130 4 21.5 5- 6 >7 24.5 46 276 129 147 5 20 4" -6/ 4 20 40 240 120' 147 6 15.5 1 4-f 3 18.5 34 136 93 80 25 35 1389 605 639 22 30 1185 22 37 605 69 102 639 102 2S94 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 2 b is obtained in the same way from the column marked h'. The sum of these numbers is 171, and this multiplied by jb = 14 gives 2394. We have now for the first line of the formula 1389 + 1185, for the second 605 + 639, and for the remainder 2394. 100 By adding these together, and multiplying the sum by 1 = —, we get the contents of the six sections in feet. 124. When the section is partly in excavation and partly in embankment, the preceding formula are still applicable; but as this application introduces minus quantities into the calculation, the following method, similar in principle, is preferable. 125. Problem. Given the widths of an excavation at the road-bed 6 106 EARTH-WORK. A F = w and A1 F, = wt (fig. 53), the side heights h and h,, the length of the section 1, and the direction of the diagonal, to find the solidity S of the excavation, when the section is partly in excavation and partly in embankment. Fig. 53. B. Al A w Solution. Suppose, first, that the surface is divided into two trian gles by the diagonal B A. Through B draw the plane BA, F,, dividing that part of the section which is in excavation into two pyramids B - A AI F1 F and B - AI B1 F1, the solidities of which are B-A A.F F=- F h X I (w+ w,) = I (wh + Wh), B-Ai B F1 = I X iw h = lw h,. The whole solidity is, therefore, S = I (w h + wl hi + wi h). Next, suppose the dividing diagonal to run from A to B.. Through B1 draw a plane B, A F (not represented in the figure), dividing the excavation again into two pyramids, of which the solidities are B,-AAlFF= - h, X Il(w + WI) = l(wh, + wh,), B,-ABF = l X wh == lwh. The whole solidity is, therefore, S = I(wh + w, h, + w hi). The only difference in these two expressions is, that w, h in the first becomes w h, 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 H, we may express both w, h and w h, by WIT; so that the solidity in either case may be expressed by CENTRE AND SIDE HEIGHTS GIVEN. 107 S= 11 (wh + w,h k+ WI). 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 L (wh + wA h + WH), I (w h, + wh2 + WI H), l (w2 h+ W3h+ W H2), and so on for any number of sections. Hence for the solidity of any number n of sections we should have tI3 S = 1 (w h +2 wl hi + 2 w2 h2.... - wn kn + WH+ Wi HI + W2 H2 + &c.) Example. Given 1 = 100, and the remaining data as given in the first three columns of the following table. Station. w. h. wh. WH. 0 2 1 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 calcn 1O0 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 the formula to S c= I w1 hi. 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 grmnd is very irregular. A, ~~A~~~~~ i ig. 54 ideoution. Let A Hs FE- Ac CDo Bi Fc Ei (fig. 54) represent one side of a section, the surface of which is too irregular to be divided into two planes. Suppose, for instance, that the ground changes at H, C, and D, making it necessary to divide the surface into five triangles running from station to station.* Let heights be taken at II, C, and D, and let the distances out of these points be measured. If now we suppose the earth to be excavated vertically downward through the side line B B to the plane of the road-bed, we may form as many vertical triangular prisms as there are triangles on the surface. This will be made evident by drawing vertical planes through the sides * It will often be necessary to introduce intermediate stations, in order to make tbe subdivision into triangles more conveniently and accurately. GROUND VERY IRREGULAR. 109 A C, H (, HD. and HB,. Then the solidity of the alf-section will be equal to the sum of these prisms, minus the triangular mass B F GB, 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 - BI F1 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 1' G, =s h. Moreover, the area of B F G - s 12, and that of Bi Fi G = I s h2. 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./- S /2 X s h12 -= 2 S h h. Then (Tab. X 53) the solidity is BF G- B1 F, GI = Is (h + hl2 + hhl). Example. Given I = 50, b =18, s =, the heights at A, H, and B respectively 4, 7, and 6, the distances A I = 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 Bi = 12 Then the horizontal section of the first prism adjoining the centre line is I 1 X At C, since the distance At 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 I1 X A1 C X'X 17 X 4 X 17, that is, equal to l multiplied by the base of the triangle and by the sum of the heights. In this way we should find for the solidity of the five prisms i1(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 l X (62 + 82 + 6x8) = X 222. Hence the solidity of the half-section is 1l (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 Jx(2b + 2sx) = sx'2 + bx. But this area is to be equal to A Therefore, s r2 + 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'lhen as s is here 3 and b = 18, the equations for finding the heights of the equivalent end sections will he x2 + 18 x = 387, and x2 + 18x = 240. Solving these equations, we have for the height at the first station z == 11.146, and at the second, x = 8. The middle area will, therefore, have the height g (11.146 + 8) = 9.573, and from this height the. miiddle area is found to be 309.78. Then by the prismioidal formula ( 113) the solidity will be S = X 100 (387 + 240 + 4 X 309.78) 31102 cubic feet. But the true solidity of this section was found to be 32820 cubic feet, a difference of 1718 feet. The error, of course, is not in the prisrnoidal 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 B, B'i An excavation/ 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 Al C, and B, DI; 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. B A B small by the mass represented by B F H 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 DFRE (fig. 56). But if the excavation is on the side of a hill, so that the distances out differ very much, and the cross-section is of the shape AD FBE, the difference of the wedge-shaped masses may require consideration. 130. Problem. Given the centre height c, the greatest side height h, the least side height t', 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, fiom 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 R, S, and the height at F is 0. B, iH and R, S, being very short, are here considered straight lines. Now we have the cross-secticn B FR = FB E G - FREG = l c d -+, 4hh) - (lcd' + I bh') =. c(d- d') + b ( --- h'). To find the height B, H. we have the angle B FH BFBI = D, and therefore Bi H = 2 IIF sin. D = 2d sin. D. In like manner, R, S = KD. = 2KF sin. D = 2(1' sin. D. Then since the height at Fis 0, one third of the sum of the heights of the prism will be I (d + d') sin. D, and the correction, or the solidity ot the prism, will be (~ 115) I' C = I c (d-d') + b (h - h')] X i (d + d) sin. D. When R is given, and not D, substitute for sin. D its value (~ 9) 50 sin. D =. The correction then becomes V- C=[Ic(d-d')+Ib(h — h')l X 100 (d + dl) V - 2 4 \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, 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 100 (74 + 38) 8 8 heights of the prism is 8 400. Hence C= 672 X = i792 cubic feet. CORRECTION IN EXCAVATION ON CURVES. 113 131. When the section is partly in excavation and partly in embankment, the cross-section of the excavation is a triangle lying wholly on one side of the centre line, or partly on one side and partly on the other. The surface of the ground, instead of extending from B to D (fig. 56), will extend from B to a point between G and E, or to a point between A and 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 osiers. 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 or deflection angle D, when the section is partly in excavation and partly in embankment. Solztion. 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 a similar manner, the height at E will be 2 G E sin. D = b sin. D, and at the point intermediate between G and E, the distance of which 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 solidity of the prism, will be (~ 115) C=2 wh X i (2d-+b- b-b 2w) sin. D ~ wh X ~ (d +b- w) sin. D. When the excavation lies on both sides of the centre line, the correction, from what has been said above, is a triangular prism having also for its cross-section the cross-section of the excavation. Its area will, therefore, be I 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 - I b, the height will be 2 (w - b) sin. D = (2 w-b) sin. D. As this height is to be considered minus, it must be subtracted from the others, and the correction required will be C = w h X i (2 d b - 2 w + b) sin. D 114 EAB TH-WORK. i w h X i (d - b - w) sin. D. Hence, in all cases, when the sees tion is partly in excavation and partly in embankment, we have the formula C = wh X i (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 = 100(d + b-w) 204, and one third of the sum of the heights of the prism is 3R 100 (51+ 30 -17) 4 4 8 X 1600 = 8 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 BFH (fig. 55) may be regarded as the same part of D that FG is ol 100 feet, and B, FB as the same part of D that EFis of 100 feet. The true correction may then be taken as the same part of C that the sum of the lengths of the two adjoining sections is of 200 feet. TABLE I. RADII, ORDINATES, DEFLECTIONS, AND ORDINATES. FOR CURVING RAILS. Farmula for Radii, 4 10; for Ordinates, ~ 25; for DeflectIou, 4 19 for Curving Rails, 4 29. 116 TABLE I. RADII, ORDINATES, DEFLECTIONS, Ordinates. Tangent Chord Rails. Degree. Radii. Deflec- Deflec12. 25. 370. 50. tion. tion. 18. 20. 0 _ 0 5 68754.94.008.014.017.018.073.145.001.001 10 34377.48.016.027.034.036.145.291.001.001 151 2291833.024.041 1. 055.218:436.002.002 20 17188,.76.032.055.063.073.291.582.002.003 25 13751.02.040.068.085.091.364.727.003.004 301 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 7639.49.072.123.153.164.654 1.309.005.007 50 6875.55.080.13.170.182.727 1.454.006.007 55 6250.5.07.150.187.0200.800 1.600.006.008 1 0 5729.65.095.164.205.218.873 1.745.007.009 5 52i8.92 103.177.222.236.945 1.891.008.009 10 4911.15.111.191.239.255 1.018 2.036.008.010 15 4583.75.119.205.256.273 1.091 2.182.009.011 20 4297.28.127.218.273.291 1.164 2.327.009.012 25 4044.51.135.232.290.309 1.236 2.472.010.012 30 3819.83.143.245.307.327 1.309 2.618.011.013 35 3618.80.151.259.324.345 1 382 2.763.011.014 40 3437.87.159.273.341.364 1.454 2.909.012.015 45 3274.17.167.286.358.382 1.527 3.054.012.015 50 3125.36.175.300.375.400 1.600 3.200.013.016 55 2989.48.183.314.392.418 1.673 3.345.014 017 1 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 2455.70.223.382.477.509 2.036 4.072.016.020 25 2371.04.231.395.494.527 2.109 4.218.017.021 30 2292.01.239.409.511.545 2.181 4.363.018.022 35 2218.09.247.423.528.564 2.254 4.508.018.023 40 2148.79.255.436.545.582 2.327 4.654.019.023 45 2083.68.263.450.562.600 2.400 4.799.019.024 50 2022.41.270.464.580.618 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 5.672.023.028 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 3.0.54 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 6.689.027.033 55 1463.16.374.641.801.855 3.417 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 1375.40.398.682.852.909 3.635 7.271.029.036 15 1343.45.40(6.695.869.927 3.708 7.416.0301.037 20 1322.53.414.709.886.945 3.781 7.561.0311.038 25 1297.58.422.723.903.964 3.853 7.707.0311.039 30 1273.57.430.736.921.982 3.926 7.852.032!.039 35 12.30 42.438.750.938 1.000 3.999 7.997.032'.040 40 1228.11.446.764.955 1.018 4.071 8.143.033.041 45 1206.57.454.777.9721 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 8.579.035.043 5 0 1146.28.477.818 1.023 1.091 4.3621 8.724.035.044.030. AND ORDTNATES FOR CURVING RAILS. 117 Ordinates for Ordinates. Tangent! Chord Rails. Degree. Radii. Deflec- Deflec12i. 25. 37i. 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.(157 1.127 4.507 9.014.037.045 15 1091 73 501.859 1.074 1.146 4.580 9.160.037.046 20j 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 50 982.64.557.955 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 0 955 37.573.982 1.228 1.309 5.234 10.467.042.052 5 942.29.581.996 1.245 1.327 5.3(C6 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 25 893.39.613 1.050 1.313 1.4001 5.597 11.193.045.056 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 11.484.047.057 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 12.065.049.060 7 0 819.02.669 1.146 1.432 1.528 6.lu5 12.210.049 061 5 809.40.677 1.159 1.449 1.546 6.177 12.355.050.062 10 800.00.685 1.173 1.466 1.564 6.250 12.500.051.063 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 12.936.052.065 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 13.951.057 070 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 20 688.16.796 1.365 1.705 1.819 7.266 14.53-2.059.073 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.f74 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 55 643.22.852 1.460 1.825 1.94r 7.773 15.547.063.078 9 0 637.27.860 1.474 1.842 1.965 7.846 15.692.064.078 5 631.44.868 1.488 1.859 1.983 7918 15.837.064.079 10 625 71.876 1.501 1.876 2.001 7.991 15.982.065.080 15 620.09.884 1.515 1.893 2.019 8.063 16.127.065,081 20 614.56.892 1.529 1.910 2.037 8.136 16.272.066.08i 25 609.14.900 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 35 598 57.916 1.570 1.961 2.092 8.353 16.707.0681.084 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 5 583.38.940 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 01 573.69.956 1.6381 2.047 2.183 8.716 17.431.071..087 ~~~~~~~~~~~~1.9961 118 TABLE I. RADII, ORDINATES, DEFLECTIONS, &C. Ordinates for Ordinates. Tangent Chord Rail Degree. RadiI Deflec- Deflec12J. 26. 87i. 60. tion. tion. 18. 20. 10 10 564.31.972 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 40 537.92 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 1.063 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 1.884 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 2.584 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 1.2.331 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 25.528.103.128 50 387.34 1.421 2.433 3.039 3.241 12.908 25.817.105.129 15 0 383.06 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 3.350 13.341 26.682.108.133 30 370.78 1.486 2.543 3.176 3.387 13.485 26.970.109.135 40 366.86 1.502 2.570 3.210 3.423 13.629 27.258.110.136 50 363.02 1.518 2.598 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 3.382 3.606 14.349 28.699.116.143 40 344.99 1.598 2.736 3.416 3.643 14.493 28.986.117.145 50 341.60 1.615 2.763 3.450 3.679 14.637 29.274.119.146 17 0 333.27 1.631 2.791 3.485 3.716 14.781 29.562.120.148 10 335.01 1.647 2.818 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 30 328.68 1.679 2.873 3.588 3.825 15.212 30425.123.152 40 325.60 1.695 2.901 3.622 3 862 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 1.728 2.956 3.691 3.935 15.613 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 33.870.1371.169 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 0 287.91 1.922 3.287 4.103 4.374 17.365 34.730.141.174......... TABLE II. LONG CHORDS. 119 TABLE II. LONG CHORDS. ~ 69. Degreeof 2 Stations. 8tations. 4 Stations. Stations. 6 Stations. Curve. 8 lb 200.000 299.999 399.998 499.996 599.993 20 199.999.997.992.983.970 30.998.992.981.962.933 40.997.986.966.932.882 50.995.979.947.894.815 1 0 199.992 299.970 399.924 499.848 599.733 10.990.959.896.793.637 20.986.946.865.729.526 30.983.932.829.657.401 40.979.915.789.577.260 50.974.898.744.488.105 2 0 199.970 299.878 399.695 499.391 598.934 10.964.857.643.6285.750 20.959.834.586.171.550 30.952.810.524'049.336 40.946.783.459 498.918.106 50.939.756.389.778 697.862 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 50.644.579 453 492.917 587.628 7 0 199.627 298.509 396.278 492.568 587.021 10.609.438.099.212 586.400 20.591.364 395.916 491.847 585.765 30.572.289.729.474.115 40.553.212.538.093 584.451 50.533.134.342 490.704 583.773 8 0.513 298.054 395.142 490.306 583.051 120 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 3400.237 4900.492 500.005 2000.082 3500.251 5000.512 600.007 2100.090 3600.266 5100.533 700.010 2200.099 3700.281 5200.554 800.013 2300.108 3800.296 1 mile.571 900.017 2400.118 3900.312 2 " 2.285 1000.020 2500.128 4(00.328 3 " 5.142 1100.025 2600.139 4100.345 4 " 9.142 1200.030 2700.149 4200.362 5 " 14.284 1300.035 2800.161 4300.379 6 " 20.568 1400.040 2900.172 4400.397 7 " 27.996 1500.046 3000.184 4500.415 8 " 36.566 1600 052 3100.197 4600.434 9 " 46.279 1700.059 3200.210 4700.453 10 " 57.135 TABLE IV. ELEVATION OF THE OUTER RAIL ON CURVES. ~ 110. Degree. M = 15. M = 20. M =25. M = 80. M = 40. M 6 0. 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 7.086.153.240.345.613.958 8.099.175.274.394.701 1.095 9.111.197.308.443.788 1.232 10.123.219.342.493.876 1.368 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 = 10 20'. For mula for frog angle F, and chord B F, ~50; for m, the middle ordinate of B F, ~ 25; for m', the middle ordinate for curving an 18 ft rail, ~ 29. R. F. B. m. m'. R. F. BF. m. mI 1000 5 27 44] 72.22.651.041 600 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 70.83.659.04:3 550 7 15 40 57.12.739.074 925 5 39 59 70).11.663.044 525 7 25 33 56.05.745.077 900 5 44 24 69.38.667.045 500 7 36 10 54.94.752.081 875 5 49 1 68.64.671.036 475 7 47 37 53.79.758.085 850 5 53 50 67.8&.676.018 450 8 0 1 52.61.765.090 825 5 58 52 67.10.680.019 425 8 13 30 51.37.773.095 800 6 4 9 66.30.685.051 400 8 28 14 50.09.780.101 775 6 9 41 65.49.690.052 375 8 44 26 48.75.788.108 750 6 15 30 64.65.695.054 350 9 2 20 47.35.796.116 725 6 21 37 63.80.700.056 325 9 22 16 45.88.805.125 700 6 28 4 62.92.705.058 300 9 44 39 44.34.814.135 675 6 34 52 62.02.710.060 275 1 10 1 42.72.824.147 650 6 42 4 61.09.716.062 250 1 39 6 41.00.834.162 625 6 49 42 60.14.721.065 225 11 12 55 39.16.845.180 TABLE VI. LENGTH OF CIRCULAR ARCS IN PARTS OF RADIUS. 1.01745 32925 19943 1.00029 08882 08666 1.00000 48481 36811 2.03490 65850 39887 2.00058 17764 17331 2.00000 96962 73622 3.05235 98775 59830 3.00087 26646 25997 3.00001 45444 10433.06981 31700 79773 4.00116 35528 34663 4.00001 93925 47244 5.08726 64625 99716 5.00145 44410 43329 5.00002 42406 84055i 6.10471 97551 19660 6.00174 53292 51994 6.00002 90888 20867 7.12217 30476 39603 7.00203 62174 60660 7.00003 39369 57678 8.13962 63401 59546 8.00232 71056 69326 8.00003 87850 94489 9.15707 96326 79490 9.00261 79938 77991 9.00004 36332 31300 122 TABLE VII. EXPANSION BY HEAT. TABLE VII. EXPANSION BY HEAT. Bodies. 820 to 2120. 1. Authority. Platina,.0008842.000004912 Hassler. Gold,.001466.000008141 A Silver,.001909.000010605 ~' Mercury,.018018.0001001 " Brass,.00189163.000010509 " Iron,.00125344.000006964 " Water,.0466 not uniform. " Granite,.00036850.000004825 Prof. Bartlett. Marble,.00102024.000005668 " andstone,.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. Lame. Materials. M. Musschenbroek, Int. to Nat Be. Bevan. Phil. Br Lieut. Brown. R. Rennie, Phil. Trans. C. Couch. Ro. Rondelet, L'Art de Battr. F. Franklin Institute, Report on T. Telford. Steam Boilers. Ta. Taylor, Statistics of Coal. G. Gordon, Eng. Translation of W. Weisbach, Mech. of MAachitWeisbach. 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. The 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 I 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, and d = its depth, all in inches. Then W = 3- X S. The numbers in the last three columns express absolute strength For safety, a certain proportion only of these numbers is taken. The divisors for wood may be from 6 to 10, for metal from 3 to 6, for stone 10, and for ropes 3. When double numbers are used in the column headed "Crushing Force per Square Inch in lbs.," the first applies to specimens 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 Modulus pi C Strength Force per of Rup Gravity. ubi per Square Squre ture S Foo Inchin l nc in lbs. in lbs. s. in lbs. Metals. -rst cst.,l..... 8.399 524.94 17963 R. toppee, cast,...-.. 8.607 537.94 19072 " rlled,... 8.864 F. 554.00 32826 F. " wie-drawn,... 8.878 554.87 61228 GII~olAr, l2 819.258 Ha. 1203.62 Gol.l.-1... 19.361 IIa. 1210.06 Tron, ca-st, Carnon co. 2, cold blast, 7.066 II 441.62 16683 H. 106375 II. 38556 H. "' ot " 7.046 H. 440.37 13505 H. 108540 I1. 37503 H. Devon No. 3, cold " 7.295 II. 455.94 36288 H. hot " 7.229 H. 451,81 21907 H. 14543511. 43497 I. Buffery Yo. l, c l " 7.079 H. 442.44 17466 H. 93335 I. 37503 H. Io" t oi " 6.998 I. 437.37 134341II. 86397 H. 35316 H. Iron, wrough', English ba-,.. 7.700 481.25 57120 L. 56000 G. 5400(0G. Welsh 6-1960 T. Swedish'.... 64960 T. 7.478 F. 467.37 58184 F. Lancaste."o,.' rrt,. 7.740 F. 483.75 58661 F. Tennessee 7.805 F. 487.81 52099 F. Missouri. 7.722 F. 42 62 47909 F. Iron wire, English, a'ai.. 0'.. 80214 T. Phillipsb'g,'a. 33"1 7.727 F. 482.94 84186 F.' 1i9' " 73888 F. " ".l 1 "t 89162 F. Lead, cast,..... 11 446 M. 715.37 1824 R. Lead wire,... 117 707.31 2581 M. Mercury,........ 3.598 W. 349.87.Platina,1 506 HA.'218.75 2la.669 Ia 1.116.81 Silver,....... IC 474 IL% 654.62 40902 M, Stel,s,..... 7.780 486.25 123000 " razor-tenmpermi,...840 190.O1 1I0000 Tin, cat,.......91 45. 63 0322 M. Zinc, fused,.... 7 050 W..14 " rolled,..'540 W 47i 2, Ash, English,......760 B 47.5) 1vO {I1 8683H 12156 B 9363 H.' 12156 B Birch, English,. 792B. 4n.50 {64021 10920 B. Americe.n,...648B. 40.501 11663 1. 9624 B. Box,.... 960 B 60.00' 2CX4l 7= 7771 HI. Cedar, Canadian,.. 909 C 5b.8l 11400 B. 5 H. Chestnut,...657 Ro. 41.06 13300Ro. Deal, Christiania midt ^.699 B. 43.62 12400 9864 B. Memel ".590 B 36.87 10386 B. * Norway Spruce,..340 21.25 17600 " English,.... 470 29.37 7000 Elm. seasoned,...553 B 34.5. 13489 M. 1033i bt f78 B. Fir, New England,..553 B. 34.56 fi 2 B. "Riga,....753 B. 47.06 12000 B. {676.I V ^ Lignum-vitse,.. 1.220 76.25 11800M. Mahogany, Spanish,...800 50.00 16500 | 8193 1. _ _ _ _. fABLE VIII. PROPERTIES OF MATERIALS. 125 Wleight Tensile Crushing Modulus S pecific pe r Strength[FOrce per of RupMateriallrs.;. Cusbic e SqSuare Gravity. Foot per SquareI ture oRu. Inch in lbs. in lbs. _ n lbs. in lbs. Woods. Oak, English,....934 B. 58.37 10000 B., {641SI} 10032 B. " Canadian,...872 B. 54.50 10253 4231 II1} 10596 B. Pine, pitch,.....660 B. 41.25 7818 M. }6790 9792 B. " red,.......657B. 41.06 {535H 8046 B. " American, white,. 455 Br. 28.44 7829 Br. 6" " Southern,.872 Br. 54.50 13987 Br. Poplar,........333 M. 23.94 7200 Be. { 3514 H. Teak,.........745 B. 46.56 15000 B. 12101 H. 14772 B. Other Materials. Brick, red,.. 2.168 R. 135.50 280 808 R. 340 W. " pale red... 2.095 R. 130.31 300 562 R. 180 W. Chalk,..... 2.784 174.00 501 R. 1.869 116.81 1.327 Ta. 82.94 Coal, Penn. anthracite,. 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 " 1.2.59 Ta. 78.69 Earth, loamy hard-stamped, fresh, 2.060 W. 128.75 C " " dry, 1.9-0 WV. 12).62 garden, fresh,.. 2.05 ) W. 128.12 " dry,.. 1.630 W. 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 Limestone.... 2.400 W. 150.00 1500 W. 700 W. 2.860ene,.. 178.75 6000 W. 1700 Marble, white Italian,.. 2.638 II. 164.87 9583 Q. 1062 " black Galway,.. 2.695 I. 168.44 2664 Masonry, quarry stone, dry, 2.400 W. 150.00 " sandstone, " 2.050 W. 128.12 " brick, dry, 1.470 W. 9187 1.590 W. 99.37 Ropes, hemp, under 1 inch diam., 9280 W. " from 1 to 3 in. " 7218 W. " over 3 inches " 5156 W. Sand, river,..... 1.886 117.87 Sandstone 1.900 W. 118.75 1400 W. 600 W..an stone,..... 2.700 W. 168.75 13000 V. 800 W. " Dundee, 2.530I R. 158.12 6630 R. " Derby, red and friable, 2.316 R. 144.75 3142 R. Slate, Wesh..... 2.888 180.50 12800 " Scotch, 9600 - 144.75 126 TABLE IX. MAGNETIC VARIATION. TABLE IX. MAGNETIC VARIATION. THE following table has been made up from variol.s sources, principally, however, from the results of the United States Coast Survey, kindly furnished in manuscript by the Superintendent, Prof. A. D. Bache.' These results," he remarks in an accompanying note, "are from preliminary computations, and may be somewhat changed by the final ones." Among the other sources may be mentioned the Smithsonian Contributions for 1852, Trans. Am. Phil. Soc. for 1846, Lond. Phil. Trans. for 1849, Silliman's Journal for 1838, 1840. 1846, and 1852, and the various American, British, and Russian Government Observations. The latitudes and longitudes here given are not always to be relied on as minutely correct. Many of them, for places in the Western States, were confessedly taken from maps and other uncer. tain sources. Those of the Coast Survey Stations, however, as well as those of American and foreign Government Observatories and Stations, are presumed to be accurate. It will be seen that the variation of the magnetic needle in the United States is in some places west and in others east. The line of no variation begins in the northwest part of Lake Huron, and runs through the middle of Lake Erie, the southwest corner of Pennsylvania, the central parts of Virginia, and through North Carolina to the coast. All places on the east of this line have the variation of the needle west,-all places on the west of this line have the variation of the needle east; and, as a general rule, the farther a place lies from this line, the greater is the variation. The position of the line of no variation 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 variation is west are every year farther and farther from this line, so th.t the variation west is constantly increasing. On the contrary, places whose variation is east are every year nearer and nearer to this line, so that the variation east is constantly decreasing. The rate of this increase or decrease, as the case may be, is said to average about 2' for the Southern States, 4' for the Middle and Western States, and 6' for the New England States.* The increase in Washington in 1840 - 2 was 3' 44.2'1; in Toronto in 1841- 2 it was 4t 46 2". The changes in * 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 ".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,* - Summer, 10 4.1 Autumn, 6 21.2 Winter, 5 9.1 Spring, 8 10.7 At Toronto the means were, t - 1841. 1843. 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, 12.88 11.70 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- 33m' 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. Lati- Longi- Authority. Date Variation. tude. tude. A ra i Maine. o I o I o I Agamenticus, 43 13.4 70 41.2 T. J. Lee, C. S. Sept., 1817 1 ) 1(.O0 Bethel, 44 28.0 70 51.0 J. Locke, June, 1845 11 50.0 " Bowdoin Iill, Portland, 43 38.8 70 16.2 J. E. IIilgard, 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. V. Dean, C. S. Oct., 1851 12 5.5 " Kennebunkport, 43 21.4 70 27.8 J. E. IIilard, C. S. Aug., 1851 II1 23.6 " Kittery Point, 43 4.8 70 43.3 J. E. Ililgard, C. S. Sept., 1850 10 30.2 " Mt. Pleasant, 44 1.6 70 49.(0 G.. Dean, C. S. 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. IIilgard, C. S Sept., 1850 12 17.9 " NewI Hampshire. Fabyan's Hotel, 44 16.0 71 29.0 J. Locke, June, 1845 11 32.0W. 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 G. V. 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 22.0W. Massachusetts. Annis-squam, 42 39.4 70 40.3 G. W. Keely, C. S. Aug., 1849 11 36.7 W. Baker's island, 42 32.2 70 46.8 0. IV. Keely, C. S. Sept., 1849 12 17.0 " Blue II, Milt, Mio 42 12.7 71 6.5 T. J. Lee, C.S. { Oct. 9 13.8 " Cambridge, 42 22.9 71 7.2 W. C. Bond, 1852 10 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.. { ct. 184 9 12.1 " Dorchester, 42 19.0 71 4.0 W. C. Bond, 1839 9 2.0 " Fort Lee, dalem, 42 31.9 70 52.1 G. W. Keely, C. S. Aug., 1849 10 14.5 " IIyannis, 41 38.0 70 18.0 T. J Lee, C. S. 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 4099 " Nantasket, 42 18.2 70 54.0 T. J. Lee, C. S. Sept., 1847 9 33.5 " Nantucket, 41 17.0 70 6.0 T J. Lee, C.'S. 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. { v. }a 9 29.8W. MeSparran Hill, 41 29.7 71 27.1 T. J. Lee, C. S. July, 1844 8 53.3 " Point Judith, 41 21.9 71 28.9R.I. Fauntleroy,C.S. Sept, 1847 8 59.4 i Spencer Hill, 41 40.7 71 29.3 T. J. Lee, C. S {July and 9 11.9 " Connecticut. Black Rock, Fairfield, 41 8.6 73 12.6!J. Renwick, C. S. Sept., 1845 6 53.5 W. Bridgeport, 4i 10.0 73 11'.OJ. 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 0 London, 41 18.0 72 0.0OJ. Renwick, C. S. Aug., 1845 7 29.5 " TABLE IX. MAGNETIC VARIATION. 129 Place. Lati- Longi- Authority. Date. Variatlon. 0o I 0 o Milford, 41 16.0 73 1.0 J. Renwick, C.S. Sept, 1845 6 33.3 W. New Haven, Pavilion, 41 18.5 72 55.4 J. S. Ruth, C. S Aug., 1848 6 37.5 " New Haven, Yale College, 41 18.5 72 55.4 J. Renwick, C. S. Sept., 1845 6 17.3" Norwalk, 41 7.1 73 24.2J. 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 IIead, 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 20.0 71 54.0 J. Renwick. C. S. Aug., 1845 7 38.2 " New York. Albany, 42 39.0 73 44.0 Regents' Report, 1836 6 47.0 W. Bloomingdale Asylum, 40 48.8 73 57.4 J. Locke, C. S. April, 1846 5 10.9" Cole, Staten Island, 40 31.8 74 13.8J. Locke, C. S. April, 1846 5 33.8 " Drowned Meadow, L. I., 40 56.1 73 3.5 J. Renwick, C. S. Sept., 1845 6 3.6 " Flatbush, L. I., 40 40.2 73 57.7 J. Locke, C. S. April, 1846 5 54.6 " Greenport, L. I., 41 6.0 72 21.0 J. Renwick, C. S. Aug., 1845 7 14.6 " Leggett, 40 48.9 73 530 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 0.0 73 21.0 Boundary Survey, Oct., 1845 11 28.0 " Sands Lighthouse, L. I., 40 51.9 73 43.5 R.H. Fauntleroy,C.S. Oct., 1847 6 9.7 " Sands Point, L. I., 40 52.0 73 43.0 J. Renwick. C. S. Sept., 1845 7 14.6 " Watchhill. Fire Island,!40 41.4 72 58.9 R.H. Fauntleroy,C.S. Oct., 1847 7 33 5 West Point, 141 25 0 73 56 0 Prof. Davies, Sept., 1835 6 32.0" New Jersey. Cape May Lighthouse, 3S. 55 8 74 57.6 J. Locke, C. S. June, 1846 3 3.2W. Chew,:39 4.2 75 97 J. Locke, C. S. July, 1846 3 20.4 " Church Landing, 39 4) 9 75 30.3 J. Locke, C. S. June, 1846 *5 45.8 " Egg Island, 39 10.4 75 7 8 J. Locke, C. S. June, 1846 3 18.2 " Hawkins, 39 25.5 75 17.1 J. Locke, C. S. June, 1846 2 58.7 " Mt.Rose,Princeton, 40 22.2 74 42.9 J. E. HIilgard, C. S. Aug., 1852 5 31.8 " Newark, 40 41.8 71 7.0 J. Locke, C. S. April, 1846 5 32.7 " Pine Mountain, 39 25.0 75 19 9 J. Locke, C. S. June, 1846 2 52.0 " Port Norris. 39 14.5 75 1.0 J. Locke, C. S. June, 1846 3 6.5 " Sandy Hook, 40 28.0 73 59.8 J. Renwick, C. S Aug., 1844 5 54.0 " Town Bank, Cape May, 38 58.6 74 57.4 J. Locke, C. S. June, 1846 3 3.2 " Tucker's Island, 39 30.8 74 16,9 T. J. Lee, C. S. Nov., 1846 4 23.8 " White Hill, Bordentown, 40 8.3 74 43 8 J. Locke, C S. April, 1846 4 22.5 Pennsylvania. Girard College, Philadelphia, 39 58.4 75 9.9 J. Locke, C. S. May, 1846 3 50.7W. Pittsburg, 40 26.0 79 58.0 J. Locke, May, 1845 0 33.1 " Vanuxem, Bristol, 40 5.9 74 52.7 J. Locke, C. S. July, 1846 4 20.5 " Local attraction exists here, according to Prof. Locke. 7 130 TABLE IX. MAGNETIC VARIATION. Place. Lati- Loi- Authority. Date. Variation. Place. tude. tude. t. Delaware. Bombay Hook o 0 o Lighthouse, 39 21.8 75 36.3 J. Locke, C. 8 June, 1846 3 1.9 W Fort Delaware, Delaware River, 39 35.3 75 33.8 J. Locke, C.. June, 1846 316.0" Lewes Landing, 38 48.8 75 11.5 J. Locke, C. S. July,1846 247.7 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 " Wilmingtn, 39 44.9 75 33.6 J. Locke, C. S. May, 1846 2 31.8 " 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.. April, 1847 2 2.6" 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,. S. April, 1847 2 13.0 " Hill, 33 53.9 76 52.5 G. W. Dean, C. S. Sept., 1850 215.4 Kent Island,:39 1.8 76 18. 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.. July, 1846 1 42.1" Osborne's Ruin, 39 27.9 76 16.6 T J. Lee, C. S. June, 1845 232.4" Poole's Island, 39 17.1 76 15.5 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. W. Dean, C.. 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 SusquehannaLighthouse, Havre de Grace, 39 32.4 76 4.8 T J. Lee, C.. July, 1847 2 51.1 " Taylor, 33 59.8 76 27.6 T J. Lee, C. S. May, 1847 2 18.4 " Webb, 39 5.4 76 40.2 G W. Dean, C. S. Nov., 1850 2 7.9 " District of Columbia. Causten, Georgetown, 38 55.5 77 4.1 G. W. Dean, C.. June, 1851 2 11.3 W. Washington, 33 53.7 77 2.8 J. M. Gilliss, June, 1842 1 26.0 " Virginia. Charlottesville, 38 2.0 78 31.0 Prof. Patterson, 1835 0 0.0 Roslyn, Petersburg, 37 14.4 77 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. S. 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 O. Boutelle, C. S. Feb., 1847 139.7 " South Carolina. Breach Inlet, 32 46.3 79 48.7 C. 0. Boutelle. C. S. April, 1849 2 16.5 E. Charleston, 32 41.0 79 53.0 Capt. Barnett May, 1841 2 24.0 " Bast Base, Edisto, 32 33.3 80 10.0 G. Davidson, 0d. S. April, 1850 2 53.6 " Georgia. Athens, 34 0.0 i3 20.0 Prof. McCa. 1837 431.0 E. Columbus, 32 28.0 85 10.O0Geol. Survey, 1839 530.0 Nll'ed feville, 33 7.0 33 20.0Geol. Survey, 1i33 5 51'.0 IlSavvlnah, 3-2 5.01-1 5i.2iJ. E..Ililgard, C. S. IApril, 1852 3 45.0" t!: -- TABLE IX. MAGNETIC VARIATION. 131 Lati- Longi. Place. tude. Lo- Authority. Date. Variation. Florida. ~25 399 80 Cape Florida, 25 39.9 80 9.4J. E. Iilgard, C S. Feb., 1850 4 25.2 E. Cedar Keys, 29 7.5 83 2.8J. E. Hilgard,.. 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 52.0 J. E. Hilgard, C. S. Aug., 1849 5 29.0 " Alabama. Fort Morgan, Mobile Bay, 30 13.8 88 0.4R.H. Fauntleroy,C.S. May, 1847 7 3.8 E. Tuscaloosa, 33 12.0 87 42.0 Prof. Barnard, 1839 7 28.0 " Mississippi. East Pascagoula, 30 20.7 88 31.4 R.H. Fauntleroy,C.S. June, 1847 7 12.4 E. Texas. Dollar Point, Galveston, 29 26.0 94 53.0 R.H. Fauntleroy,C.S. April, 1848 8 57.2 E. Mouth of Sabine, 29 43.9 93 51.5 J. D. Graham, Feb., 1840 8 40.2 Ohio. Carrolton, 39 38.0 84 9.0 J. 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 " Hudson, 41 15.0 81 26.0 E. Loomis, 184' 0 52.0 " Marietta, 39 26.0 81 29.0 J. Locke, April, 1845 2 25.0' Oxford, 39 30.0 84 38.0 J. Locke, Aug., 1845 4 50.0 " St. Mary's, 40 32.0 84 19.( J. Locke, Sept., 1845 3 4.0 " Tennessee. Nashville, 36 10.0 86 49.0 Prof. Hamilton, 1835 7 7.0E. Michigan. Detroit, 42 24.0 82 58.0 Geol. Report, 1840 2 0.0 B. Indiana. Richmond, 39 49.0 84 47.0 J Locke, Sept., 1845 4 52.0 E. South IIanover, 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, 33 36.0 89 36.0 Col. NicoUl, 1835 8 49.0 E. Wisconsin. Madison, 43 6.0 89 41.0 U. S. Surveyors, Nov., 1839 7 30.0 E. Prairie du Chien, 43 1.0 91 8.0 U. S.Surveyors, Oct., 1839 9 5.0" Iowa. Brown's Settlement 42 2.0 91 18.0 J. Locke, Sept., 1839 9 4.0 E. Davenport, 41 30.0 90 34.0 U. S. Surveyors, Sept., 1839 7 50.0 " Farmer's Creek, 42 13.0 90 39.0 J. Locke, Oct., 1839 9 11.0 " Wapsipinnicon River, 41 44.0 90 39.0 J. Locke, Sept., 1839 8 25.0 " California. Point Conception, 34 26.9 120 26.01G. Davidson, C. S. Sept., 1850 13 49.b E.~~...w.,_.. 132 TABLE IX. MAGNETIC VARIATION. Place. ati- Longi. Authority. Date. Variation. tude. tude. Point Pinos, o 1 o Monterey 36 38.0 121 54.0. Davidson, C. S. Feb., 185114 58.0E. Presidio, San Francisco, 37 47.8 122 27.0 G. Davidson, C. S. Feb., 185215 26.9" San Diego, 32 42.0 117 14.0 G. Davidson, C. S. 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.S. Aug., 1852 21 30.2 E. BRITISH AMERICA. 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 127.2' NEW GRENADA Panama, 8 57.2 79 29.4 W H. Emory, Mar., 1849 6 54.6 B. BASTERN HEMISPHERE. 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 28.6" 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 Good Hope, 33 56.0 18 28.7 E. British Govern, July, 1846 29 8.0 " Hobarton, Van Diemen'sLd., 42 52.5 147 27.5 British Govern., Dec., 1848 10 8.0. TABLE X. TRIGONOMETRICAL FORMULAE. 133 TABLE X. TRIGONOMETRICAL AND MISCELLANEOUS FORMULAE LET A (fig. 57) be any acute angle, and let a perpendicular B Cbe drawn from any point in one side to the other side. Then, if the sides Fig. 57. of the right triangle thus formed are denoted by letters, as in the fig nre, we shall have these six formulae: - 1. sin. A = -. 4. cosec. A = a. 2. cos. A =. 5. sec. A = 3. tan. A =. 6. cot. A Solution of Right Triangles (fig. 57).Given. Sought. Formulae. 7 a, c A,B, bsin., cos. = cosb. B, (c +a) (c-a) 8 a, b A, B,c tan. = b, cot. B =A, c = /as +'. 9 A, a Bb,c B == 90 A, b == a cot. A, c = -J. 10lA, B a, c B=900-A, a = b tan. A, c co. B. 11 A, c B,a, B= 90-A, a = c sin. A, b = c cos. A. 134 TABLE X. TRIGONOMETRICAL AND Solution of Oblique Triangles (fig. 58). B Fig. 58. b Given. Sought. Formule. 12 A, B, a b a sin. B bAasin. A 1 sin. A 13 A, a, b B sin. B si A (a - b) tan. (A - B) 14 a, b, C A-B tan. (A - B) a=- b Ifs= (a+b+c), sin.A= (s- b) ( — 15 a, b, c A. c A= s —a) tan. A- (s sin. A _ 2 ^/s (s a) (s-b) (s-c) bc a2 sin. B sin C 16 A,, C, area ara ea 2i. A 17 A, b, c area area = b c sin. A. 181a, 6, c area s= — (a + b + -,, area==/s(s-a) (s-b) (J- e}. General Trigonometr;ic' Formule. 19 sin.2 A + cos.2 A =. 20 sin. (A ~ B) = sin. A cos. B ~ sir is cos. A. 21 cos. (A ~ B) = cos. A cos. B: sin. 4.'ip. B. 22 sin. 2 A = 2 sin. A cos. A. 23 cos. 2A =cos. A- sin.2 A 1-2 sin 4 = 2cos A-1. 24 sin.2 A -= O cos. 2 A. 25 cos.2A = + I cos. 2 A. 26 sin. A + sin. B 2 sin. ( (A + B) cos. ( B). 27 sin. A - sin. B = 2 cos. (A + B) sin. (A B). 28 cos. A + cos. B = 2 cos. (A + B) cos. A (A- P). 29 cos. B - cos. A = 2 sin. (A + B) sin. (A-P) 30 Isin.2 A -sin.2 B = cos.2 B -cos. A = sin. (A + B) s ( s 31 cos." A - sin.2 B = cos. (A + B) cos. (A - B). MISCELLANEOUS FORMUL2E. 185 sin. A 32 tan. A = s. cos. A 33 cot. A cot. A -= sin. A' 34 tan. (A B) = 1 tan. A tan B sin (A ~ B) 35 tan. A ~ tan. B = cos. A cos. sin (A ~ B) 36cot. A: cot. B = i sin. A in. B' sin. A +- sin.B tan. ~ (A + B) sin. A - sin. B = tan. ~ (A + B) 37' sin. A - sin. B - tan. (A - B) sin A + sin. B 38 tan. L (A +- B) cos. A + cos. B - I sin. A + sin. B (A cos. - cos. =. (A - B) sin. A - sin. B 40 cos. cos. tan. ( B ). sin. A - sin. B 41 sin. A-sin. cot. 1 (A + B). cos. B - cos.A 2 sin A 42 tan. A = l osi' 2 1 + cos. A sin. A 43 cot. 1 A cos. A 2 - 1 - cos. A Mliscellaneous Formule, Sought. Given. Formula. Area of 44 Circle Radius = r r2. 45 Ellipse Semi-axes = a and b r ab. 46 Parabola Chord = c, height = h c h.* 47 Regular Polygon { Side n as n cot., Surface of 48 Sphere Radius r 4 7 r2. 49 Zone Radius = r, leight - h 2 7 r h. (Radius of sphere-r ) 2S-(n —2)1800 50 SphericalPolygon sum of angles = S, 1 X 180 number of sides = n Solidity of 51 Prism or Cylinder Base = b, height = h bh. 52 Pyramid or Cone Base = b, height = h k b h. 53 Frustum of Pyr- Bases b and b, } (b + b + b bl ) amid or Cone ] height A - h * The area of a circular segment on railroad curves, where the chord is very long In proportion to the height, may be found with great accuracy by the above formula. T~6 TABLE X. MISCELLANEOUS FOBMULAE. Sough>. Given. Formulae. Solidity of 54 Sphere Radius = r 4.r r3. 55 SpherlSement Radii of bases -= r 1_h (,.2_ ] q2 Al) SphricalSegmentand rl, height = h 56 Prolate Spheroid Semi-transverseaxis 4 ofellipse = a 3 ab. I57 Oblate Spheroid Semiconjugate axis 2 57 Oblate Splheroid of ellipse = b 3 7c a b. 58 Paraboloid hei t ", h 58 Paraboloid ~~~b~t";"E{Radius of base = r, } 2 Z r2 h. T - = 3.14159 26535 89793 23846 26433 83280. Log. T = 0.49714 98726 94133 85435 12682 88291 United States Standard Gallon -231 cub. in. = 0.133681 cub. ft C" " " Bushel = 2150.42 " 1.244456 " British Imperial Gallon = 277.27384" = 0.160459 " According to Hassler. As usually given. French Metre, = 3.2817431 ft., = 3.280899 ft. " Litre, = 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. 39.830, = 62.379 lb. avoir. " " "'; 62, = 62.321 " Length of Seconds Pendulum at New York = 39.10120 inches. " " " " " London = 39.13908 " I" " 4" " "Paris = 39.12843 " Equatorial Radius of Earth according to Bessel = 20,923,597.017 feet Polar " " " " 20,853,654.177 " TABLE XI. SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND RECIPROCALS OF NUMBERS lOM 1I TO 1054. 138 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 1 1 1 1.0000000 1.0000000 1.000000000 2 4 8 1.4142136 1.2599210.50000000( 3 9 27 1.7320508 1.44224S6.333333333 4 16 64 2.0000000 1.5874011.250000000 5 25 125 2.2360680 1.7099759.200000000 6 36 216 2.4494897 1.8171206.166666667 7 49 343 2.6457513 1.9129312.142857143 8 64 512 2.8284271 2.0000000.125000000' 9 81 729 3.0000000 2.0800837.111111111 10 100 1000 31622777 2.1544.347.100000000 11 121 1331 3.3166248 2.2239801.090909091 12 144 1728 3.4641016 2.2894286.083333333 13 169 2197 3.6055513 2.3513347.076923077 14 196 2744 3.7416574 2.4101422.071428571 15 225 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 23 629 12167 4.7958315 2.8438670.04347826 24 576 13824 4.8989795 2.3844991.041666f67 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 5.3851648 3.0723168.034432759 30 900 27000 5.4772256 3.1072325.033333333 31 961 29791 5.5677644 3.1413806.032258065 32 1024 32768 5.6568542 3.1748021.031250000( 33 1089 35937 5.7445626 3.2075343.03030303'0 34 1156 39304 5.8309519 3.2396118.029411765 35 1225 42875 5.9160798 3.2710663.02;571429 36 1296 46656 6.0000000 3.3019272.027777778 37 1369 50653 6.0827625 3.3322218.027027027 8 1444 54872 6.1644140 3.3619754.026315789 39 1521 59319 6.2449980 3.3912114.025641026 40 1600 64000 6.3245553 3.4199519.025000000 41 1631 68921 6.4031242 3.4482172.024390244 42 1 764 74088 6.4807407 3.4760266.023809524 43 1849 79507 6.5574335 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 103323 6.8556546 3.6088261.021276600 48 2304 110592 6.9282032 3.6342411.020833333 49 2401 117649 7.0000000 3 fi593057.020403163 50 2500 125000 7.0710678 3.6840314.0200G0000 51 2601 132651 7.14 14284 3.7084298.019607843 52 2704 140608,7.2111026 3.7325111.019230769 53 2809 148877 7.2301099 3.7562358.018867925 51 2916 157464 7.3434692 3.7797631.018518519 55 3025 166375 7.4161985 3.3029525.018181818 56 3136 175616 7.4;33148 3.8258624.017857143 57 3249 185193 7.5498344 3.8485011.017543860 58 3364 195112 7.6157731 3.8703766.017241379 59 3481 20.5379 7.6811457 3.8929965.0169-191 53 60 3600 216000 7.7459667 3.9148676.016666667 61 3721 226931 7.8102497 3.9364972.016393443, 62 3S44 238328 7.8740079 3.9573915 I.016129032 ~,~ ~ ~ _ __ _ =~ ~ CUBE ROOTS, AND RECIPROCALS. 139 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 63 3969 250047 7.9372539 3.9790571.015873016 64 40b6 262144 8.0000000 4.0o00ooU0.015625000 65 4225 274625 8.0622577 4.02 72'6.015384615 66 4356 287496 8.1240384 4.01124)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 5041 357911 8.4261498 4.1408178.014084507 72 5184 373248 8.4852814 4.1601676.013888889 73 5329 389017 8.5440037 4.1793390.013698630 74 5476 405224 8.6023253 4.1983364.013513514 75 5625 421875 8.6602540 4.2171633.013333333 76 5776 438976 1 8,7177979 4,2358236.013157895 77 5929 456533 8.7749644 4.2543210.012987013 78 6084 474552 8.8317609 4.2726586.012820513 79 6241 493039 8.8881944 4.2908404.012658228 80 6400 512000 8.9442719 4.3088695.012500000 81 6561 531441 9,0000000 4.3267487.012345679 82 6724 551368 9.0553851 4.3444815.012195122 83 6889 571787 9.1104336 4.3620707.012048193 84 7056 592704 9.1651514 4.3795191.011904762 85 7225 614125 9.2195445 4.3968296.011764706 86 7396 636056 9.2733185 4.4140049.011627907 87 7569 658503 9.3273791 4.4310476.011494253 88 7744 681472 9.3808315 4.4479602.011363636 89 7921 704969 9.4339811 4.4647451.011235955 90 8100 729000 9.4868330 4.4814047.011111111 91 8281 753571 9.5393920 4.4979414.010989011 92 8464 778688 9.5916630 4.5143574.010869565 93 8649 804357 9.6436508 4.5306549.010752688 94 8836 830584 9.6953597 4.5468359.010638298 95 9025 857375 9.7467943 4.5629026.010526316 96 9216 884736 9.7979590 4.5788570.0104i66t7 97 9409 912673 9.8488578 4.594700(9.010309278 98 9604 941192 9.8994949 4.610)463.01024(s82 99 9801 970299 9.9498744 4.6260650.010101010 100 10000 1000000 10.000000o 4.6415888.010000000 101 10201 1030301 10.0498756 4.63570095.009900990 102 10404 1061208 10.0995049 4.6723287.009203922 103 10609 1092727 10.1488916 4.6875482.(l9708;:8 104 10816 1124864 10.1980390 4.7026694.0,;6153 5 105 11025 1157625 10.2469508 4.7176940.009523810 106 11236 1191016 10.2956301 4.7326235.(100433962 107 11449 1225043 10.3440804 4.7474594.()09345754 108 11664 1259712 10.3923048 4.7622032.009259259 109 11881 1295029 10.4403065 4.7768562.009174312 110 12100 1331000 10.4880085 4.7914199.0090909( 9 111 12321 1367631 10.5356538 4.8058955.009009009 112 12544 1404928 10.5830052 4.8202845.008928571 113 12769 1442897 10.6301458 4.8345881.008849558 114 12996 1481544 10.6770783 4.8488076.008771930 115 13225 1520875 10.7238053 4.8629442.008695652 116 13456 1560896 10.7703296 4.8769990.008620690 117 13689 1601613 10.8166538 4.8909732.008547009 118 13924 1643032 10.8627805 4.9048681.00A474576 119 14161 1685159 10.9087121 4.9186847.008403361 120 14400 1728000 10.9544512 4.9324242.008333333 121 14641 1771561 11.0000000 4.9460S74.008264463 122 14884 1815848 11.0453610 4.9596757.008196721 123 15129 1860867 11.0905365 4.9731898'.008130081 124 15376 1906624 11.1355287 I 4.9866310.008064516 140 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 125 15625 1953125 11.1803399 5.0000000.008030000 126 15876 2000376 11.2249722 50132979.007936508 127 16129 2048383 11.2694277 5.0265257.007874016 123 16334 2097152 11.3137085 5.0396842.007812500 129 16611 2146639 11.3578167 5.0527743.007751938 130 16900 2197000 11.4017543 5.0657970.007692308 131 17161 224091 11.4455231 5.0787531 007633588 132 17424 2299968 11.4891253 5.0916134.007575758 13:3 176S9 2352637 11.5325626 5.1044687.007518797 134 17956 2106104 11.5758369 5 1172299.007462687 135 18225 2460375 11.6189500 5.1299278.007407407 136 18496 2515456 11.6619038 5 1425632.007352941 137 18769 2571353 11.7046999 5.1551367.007299270 133 19!),44 2623072 11.7473444 5 1676493.007246377 139 1 9321 2685619 11.7893261 5.1801015.007194245 140 19600 2744000 11.8321596 5.1924941.007142857 141 19331 28('3221 11.874,3421 5.2048279.007092199 142 20164 2363288 11.9163753 5.2171034.007042254 143 20449 2924207 11.9582607 5.2293215.006993007 144 20736 2985984 12.000000 5.2414828.006944444 145 21025 3048625 12.0415946 5.2535879.006896552 146 21316 3112136 12.0330460 5.2656374.006849315 147 21609 3176523 12.1243557 5.2776321.00680272_ 148 21904 32-11792 12.1655251 5.2895725.006756757 149 22201 3307949 12.2065556 5.3014592.006711409 150 22500 3375000 122174487 5.3132928.006666667 151 22801 3442951 12.2382057 5.3250740.006622517 152 23104 3511803 12.3288280 53363033.006578947 153 23409 3531577 12 3693169 5.3434812.006535948 154 23716 3652264 12.4096736 5.3601084.006493506 155 24025 3723375 12.4498996 5.3716354.006451613 156 24336 3796416 12.4399960 5.3832126.00(410256 157 24649 3369393 12.5299641 5.3946907 -.006369427 158 24964 3944312 12.5698051 5.406120'2.006329114 159 25281 4019679 12.6095202 5.4175015.006239308 160 25600 4096000 12.6491106 5.4238352.00625000 161 25921 4173281 12.6335775 5.4401218.006211180 162 26244 4251528 12.7279221 5.4513618.006172340 163 26569 4330747 12.7671453 5.4625556.006134969 164 26896 4410944 12.8062485 5.4737037.006097561 165 27225 4492125 12.8452326 5.4848066.006060606 166 27556 4574296 12.8840987 5.4958647.006024096 167 27889 4657463 12.9228480 5.5063784.005988024 168 28224 4741632 12.9614814 5.5174848.005952331 169 28561 4826309 13.0000000 5.5287748.005917160 170 23900 4913000 13.0334048 5.5396583.005882353 171 29241 5000211 13.0766968 5.5504991.005847953 172 29584 5088443 13.1143770 5.5612978.0058 13953 173 29929 5177717 13.1529464 5.5720546.0')5780347 174 30276 5263024 13.1909060 5.5827702.005747126 175 30625 5359375 13.2237566 5.5934447.005714286 176 30976 5451776 13.2664992 5.6040787.005631818 177 31329 5545233 13.3011347 5.6146724.005649718 178 31634 5639752 13.3416641 5.6252263.005617978 179 3:241 5735339 13.3790382 5.6357408.005536592 180 32100 53.32000 134164079 5.6462162.005555556 131 1 32761 5929711 13.4536240 5.6566528.005524862 132 33124 6238363 13.4907376 5.6670511 0054.94505 133 33139 6123487 13.5277,93 5.6774114.005464481 134 33:33356 6229504 13.5646600 5.6377340.005434733 1.3. 34225 6331625 13.6014705 5.6980192.005405407 186 34596 64:34356 13.6331817 5.7032675.005376344 CUBE ROOTS, AND RECIPROCALS. 141 No. Squares. Cubes. Square Roots. CubeRoots. Reciprocals. 187.34969 6539203 13.6747943 5.7184791.005347594 188 35344 6644672 13.7113092 5.7286543.005319149 189 35721 6751269 13.7477271 5.7387936 005291005 190 36100 6859000 13.7840488 5.7488971.005263158 191 36481 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 5.8087857.005102041 197 38809 7645373 14.0356688 5.8186479 -.005076142 198 39204 7762392 14.0712473 5.8284767.005050505 199 39601 7880599 14.1067360 5.8382725 005025126 200 40000 8000000 14.1421356 5.8480355.005000000 201 40401 8120601 14.1774469 5.8577660.004975124 202 40804 8242408 14.2126704 5.8674643.004950495 203 41209 8365427 14.2478068 5.8771307.004926108 204 41616 8489664'14.2828569 5.8867653.004901961 205 42025 8615125 14.3175211 5.8963685.004878049 206 42436 8741816 14.3527001 5 9059406.004854369 207 42849 8869743 14.3874946 5.9154817.004830918 208 43264 8998912 14.4222051 5.9249921.004807692 209 43681 9129329 14.4568323 5.9344721.004784689 210 44100 9261000 14.4913767 5.9439220.004761905 211 44521 9393931 14.5258390 5.9533418.004739336 212 44944 9528128 14.5602198 5.9627320.004716981 213 45369 9663597 14.5945195 5.9720926.004694836 214 45796 9800344 14.6287388 5.9814240.004672897 215 46225 9938375 14.6628783 5.9907264.004651163 216 46656 10077696 14.6969385 6.0000000.004629630 217 47089 10218313 14.7309199 6.0092450.004608295 218 47524 10360232 14.7648231 6.0184617.004587156 219 47961 10503459 14.7986486 6.0276502.004566210 220 48400 10648000 14.8323970 6.0368107.004545455 221 48841 10793861 14.8660687 6.0459435.004524887 222 49284 10941048 14.8996644 6 0550489.004504505 223 49729 11089567 14.9331845 6.0641270.004484305 224 50176 11239424 14.9666295 6.0731779.004464286 225 50625 11390625 15.0000000 6.0822020.004444444 226 51076 11543176 15.0332964 6.0911994.004424779 227 51529 11697083 15.0665192 6.1001702.004405286 228 51984 11852352 15.0996689 6.1091147.004385965 229 52441 12008989 15.1327460 6.1180332.004366812 230 52900 12167000 15.1657509 6.1269257.004347826 231 53361 12326391 15.1986842 6.1357924.004329004 232 53824 12487168 15.2315462 6.1446337.004310345 233 54289 12649337 15.2643375 6.1534495.004291845 234 54756 12812904 15.2970585 6.1622401.004273504 235 55225 12977875 15.3297097 6.1710058.004255319 236 55696 13144256 15.3622915 6.1797466.004237288 237 56169 13312053 15.3948043 6.1884628.004219409 238 56644 13481272 15.4272486 6.1971544.004201681 239 57121 13651919 15.4596248 6.2058218.004184100 240 57600 13824000 15.4919334 6.2144650.004166667 241 58081 13997521 15.5241747 6.2230843.004149378 242 58564 14172488 15.556.3492 6.2316797.004132231 243 59049 14348907 15.5884573 6.2402515.004115226 244 59536 14526784 15.6204994 6.2487998.004098361 245 6(0025 147(6125 15 6524758 6.2573248.004081633 246 60516 148S6936 15.6843871 6.2658266.004065041 247 61009 15069223 15.7162336 6.2743054.004048583 248 61504 15252992 15.74S0157 6.2827613.004032258 142 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciproeba. 249 62001 15438219 15.7797338 6.2911946.004016064 250 62500 15625000 15.8113383 6.2996053.004000000 251 63001 15813251 15.8429795 6.3079935.003984064 252 63504 16)93005 15.8745079 6.3163596.003963254 253 64009 16194277 15.9059737 6.3247035.003952569 254 64516 16357064 15.9373775 6.3330256.003937008 255 65025 16581375 15.9637194 6.3413257.003921569 256 65536 16777216 16.0090000 6.3496042.003906250 257 66049 16974593 16.0312195 6.3578611.003-91051 253 66564 17173512 16.0623784 6.3660968.003875969 259 67081 17373979 16.0934769 6.3743111.003861004 260 67600 17576000 16.1245155 6.3825043.003346154 261 68121 17779531 16.1554944 6.3906765.003831418 262 68644 17984728 16.1864141 6.3988279.003316,94 263 69169 18191447 16.2172747 6.4069585.003802281 264 69696 18399744 16.2430763 6.4150687.003787879 265 70225 18609625 16.2738206 6.4231583.003773585 266 70756 18821096 16.3)95064 6.4312276.003759398 267 71289 19034163 16.3401346 6.4392767.003745318 268 71824 192433:32 16.3707055 6.4473057.003731343 269 72361 19465109 16.4012195 6.4553148.003717472 270 72900 19633000 16.4316767 6.4633011.003703704 271 73441 19902511 16.4620776 6.4712736.003690037 272 73984 20123643 16.4924225 6.4792236.003676471 273 74529 20346417 1-6.5227116 6.4871541.003663004 274 75076 20570824 16.5529454 6.4950653.003649635 275 75625 20796375 16. 531240 6.5029572.003636364 276 76176 21021576 16.6132477 6.5103300.003623188 277 76729 21253933 16.6433170 6.5186839.003610108 278 77284 21484952 16.6733320 6.5265189.003597122 279 77841 21717639 16.70332931 6.5343351.003584229 280 78400 21952000 16.7332005 6.5421326.003571429 281 78961 22188041 16.7630546 6.5499116.003558719 282 79524 22425763 16.7928556 6.5576722.003546099 283 80039 22665187 16.8226038 6.5654144.003533569 284 80656 22906304 16.8522995 6.5731385.003521127 285 81225 23149125 16.8919430 6.5803443.003508772 286 81796 23393656 16.9115345 6.5885323.003496503 287 82369 23639903 16.9410743 6.5962023 903484321 288 82944 23887872 16.9705627 6.6038545.003472222 289 83521 24137569 17.0000900 6.6114890.0034613208 290 84100 24339000 17.0293361 6.6191060.003448276 291 84631 24642171 17.0587221 6.6267054.003436426 292 85264 21897038 17.0380075 6.6.342874.003424658 293 85849 25153757 17.1172428 6.6419522.003112969 294 86436 25412184 17.1464282 6.6493998.003401361 295 87025 25672375 17.1755610 6.6569302.003339831 296 87616 25934336. 17.2046.505 6.6614437.003378378 297 88209 26193073 17.2336979 6.6719403.003367003 298 88804 26463592 17.2626765 6.6794200.(03355705 299 89401 26730399 17.2916165 6.6363831.003344482 300 90000 2700090 17.3205081 6.6943295.003333333 301 90601 27270901 17.3493516 6.7017593.003322259 302 91204 275-13603 17.3781472 6.7091729.003311258 30,3 91809 27818127 17.4068952 6.7165700.003300330 304 92416 28094464 17.4355958 6.7239508.003289474 305 93025 28372625 17.4642492 6.7313155.003278639 306 93636 28652616 17.4928557 6.7336641.003267974 307 94249 28934443 17.5214155 6.7459967.003257329 303 94864 29218112 17.5499288 6.7533134.003246753 309 95481 29503629 17.57839.58 6.7606143.003236246 310 96100 29791000 17.6063169 6.7678995.003225806 CUBE ROOTS, AND RECIPROCALS. 143 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 311 96721 30080231 17.6351921 6.7751690.003215434 312 97344 30371328 17.6635217 6.76241J9.003205128 313 97969 30664297 17.6918060 6.7S86613.003194888 314 98596 30959144 17.7200451 6.79688i44.003184713 315 99225 31255875 17.7482393 6.8040921.003174603 316 99856 31554496 17.7763888 6.8112847.003164557 317 100489 31855013 17.8044938 6.8184620.003154574 318 101124 32157432 17.8325545 6.8256242.003144654 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 103684 33386248 17.9443584 6.8541240 *.003105590 323 104329 33698267 17.9722008 6.8612120.003095975 324 104976 34012224 18.0000000 6.8682855.003086420 325 105625 34328125 18.0277564 6.8753443.003076923 326 106276 34645976 18.0554701 6.8823888.003067485 327 106929 34965783 18.0831413 6.8894188.003058104 328 107584 35287552 18.1107703 6.8964345.003048780 329 108241 35611289 18.1383571 6.9034359.003039514 330 108900 35937000 18.1659021 6.9104232.003030303 331 109561 36264691 18.1934054 6.9173964.003021148 332 110224 36594368 18.2208672 6.9243556.003012048 333 110889 36926037 18.2482876 6.9313008.003003003 334 111556 37259704 18.2756669 6.9382321.002994012 335 112225 37595375 18.3030052 6.9451496.002985075 336 112896 37933056 18.3303028 6.9520533.002976190 337 113569 38272753 18.3575598 6.9589434.002967359 338 114244 38614472 18.3847763 6.9658198.002958580 339 114921 38958219 18.4119526 6.9726826.002949853 310 115600 39304000 18.4390889 6.9795321.002941176 341 116281 39651821 18.4661853 6.9863681.002932551 342 116964 40001688 18.4932420 6.9931906.002923977 343 117649 40353607 18.5202592 7.0000000.002915452 344 118336 40707584 18.5472370 7.0067962.002906977 345 119025 41063625 18.5741756 7.0135791.002898551 346 119716 41421736 18.6010752 7.0203490.002890173 347 120409 41781923 18.6279360 7.0271058.002881844 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 123201 43243551 18.7349940 7.0540041.0(2849003 352 123904 43614208 18.7616630 7.0606967.002840909 353 124609 43986977 18.7882942 7.0673767.002832861 351 125316 44361864 18.8148877 7.0740440.002824859 355 126025 44738875 18.8414437 7.0806988.002816901 356 126736 45118016 18.8679623 7.0873411.002808989 357 127449 45499293 18.8944436 7.0939709.002801120 358 128164 45882712 18.9203879 7.1005885.002793296 359 128881 46268279 18.9472953 7.1071937.002785515 360 129600 46656000 18.9736660 7.1137866.002777778 361 130321 47045881 19.0000000 7.1203674.002770083 362 131044 47437928 19.0262976 7.1269360.002762431 363 131769 47832147 19.0525589 7.1334925.002754821 364 132496 48228544 19.0787840 7.1400370.002747253 365 133225 48627125 19.1049732 7.1465695.002739726 366 1:3:3956 49027896 19.1311265 7.1530901.0(02732240 367 1316S9 49430863 19.1572441 7.1595988.002724796 36S 13.5424 49836032 19.1833261 7.1660957.002717391 369 136161 50243409 19.2093727 7.1725809.002710027 370 136900 5(1653000 19.2353841 7.1790544.002702703 371 137641 51064811 19.2613603 7 1855162.00269'541 8 372 138384 51478848 19.2873015 7.1919663.002688172 1361 5041 92163 71512.0651 144 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. ReciprocaSb. 373 139129 51895117 19.3132079 7.1984050.002680965 374 139876 52313624 19.3390796 7.2048322.002673797 375 140625 52734375 19.3649167 7.2112479.002666667 376 141376 53157376 19.3907194 7.2176522.002659574 377 142129 53532633 19.4164S78 7.2240450.002652520 378 142884 54010152 19.4422221 7.2304268.002645503 379 143641 54439939 19.4679223 7.2367972.00-2638522 380 144400 54872000 19.4935887 7.2431565.002631579 381 145161 55306341 19.5192213 7.2495045.002624672 382 145924 55742968 19.5448203 7.2558415.002617801 333 1 16639 56181887 19.5703:58 7.2621675.002610966 384 147456 56623104 19.5959179 7.2684824.002604167 385 148225 57066625 19.6214169 7.2747864.002597403 386 148996 57512456 19.6468827 7.2810794.002590674 337 149769 57960603 19.6723156 7.2873617.002583979 338 150544 58411072 19.6977'I56 7.2936330.002577320 339 151321 58363369 19.7230829 7.2998936.002570694 390 152100 59319000 19.7484177 7.3061436.002564103 391 152881 59776171 19.7737199 7.3123828.002557545 392 153664 60236288 19.7939899 7.3186114.002551020 393 154449 60693457' 19.8242276 7.3248295.002544529 394 155236 61162934 19.8494332 7.3310369.002538071 395 156025 61629375 19.8746069 7.3372339.002531646 396 156816 62099136 19.8997437 7.3434205.002525253 397 157609 62570773 19.9248538 7.3495966.002518892 398 158404 63044792 19.9499373 7.3557624.002512563 399 159201 63521199 19.9749844 7.3619178.002506266 400 160000 64000000 20.0000000 7.3680630.002500000 401 160301 64481201 20.0249344 7.3741979.002493766 402 161604 64964808 20.0499377 7.3803227.002487562 403 162409 65450827 20.0748599 7.3864373.002481390 404 163216 65939264 20.0997512 7.3925418.002475248 405 164025 66430125 20.1246118 7.3986363.002469136 406 164836 66923416 20.1494417 7.4047206.002463054 407 165649 6741J)143 20.1742410 7.4107950.002457002 408 166464 67917312 20.1990099 7.4168595.002450980 409 167281.68417929 20.2237484 7.4229142.002444988 410 168100 68921000 20.2484567 7.4289589.002439024 411 168921 69426531 20.2731349 7.4349938.002433090 412 169744 69934528 20.2977831 7.4410189.002427184 413 170569 70444997 20.3224014 7.4470342.002421308 414 171396 70957944 20.3469899 7.4530399.002415459 415 172225 71473375 20.3715488 7.4590359.002409639 416 173056 71991296 20.3960781 7.4650223.002403846 417 173889 72511713 20.4205779 7.4709991.002398082 418 174724 73034632 20.4450483 7.4769664.002392344 419 175561 73560059 20.4694895 7.4829242.002386635 420 176400 74088000 20.4939015 7.4888724.002380952 421 177241 74618461 20.5182345 7.4948113.002375297 422 178084 75151448 20.5426386 7.5007406.002369668 423 178929 75686967 20.5669638 7.5066607.002364066 424 179776 76225024 20.5912603 7.5125715.002358491 425 180625 76765625 20.6155281 7.5184730.002352941 426 181476 77308776 20.6397674 7.5243652.002347418 427 182329 778.54483 20.6639783 7.5302482.002341920 423 183184 78402752 20.6381609 7.5361221.002336449 429 184041 78953589 20.7123152 7.5419867.002331002 430 184900 79507000 20.7364414 7.5478423.002325581 431 185761 80062991 20.7605395 7.5536888.002320186 432 186624 80621568 20.7846097 7.5595263.002314815 433 187489 81182737 20.8086520 7.5653548.002309469 4.34 188356 81746504 20.8326667 7.5711743.002304147 CUBE ROOTS, AND RECIPROCALS. 145 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 435 189225 82312875 20.8566536 7.5769849.002298851 436 190096 82881856 20.8806130 7.5827865.002293578 437 190969 83453453 20.9045450 7.5885793.002288330 438 191844 84027672 20.9284495 7.59436.33.002283105 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 869.38307 21.0475652 7.6231519.002257336 444 197136 87528384 21.0713075 7.6288837.002252252 445 198025 88121125 21.0950231 7.6346067.002247191 446 198916 88716536 21.1187121 7.6403213.002242152 447 199809 89314623 21.1423745 7.6460272.002237136 448 200704 89915392 21.1660105 7.6517247.002232143 449 201601 90518849 21.1896201 7.6574138.002227171 450 202500 91125000 21.2132034 7.6630943.002222222 451 203401 91733851 21.2367606 7.6687665.002217295 452 204304 92345408 21.2602916 7.6744303.002212389 453 205209 92959677 21.2837967 7.6800857.002207506 454 206116 93576664 21.3072758 7.6857328.002202643 455 207025 94196375 21.3307290 7.6913717.002197802 456 207936 94818816 21.3541565 7.6970023.002192982 457 208849 95443993 21.3775583 7.7026246.002188184 458 209764 96071912 21.4009346 7.7082388.002183406 459 210681 96702579 21.4242853 7.7138448.002178649 460 211600 97336000 21.4476106 7.7194426.002173913 461 212521 97972181 21.4709106 7.7250325.002169197 462 213444 98611128 21.4941853 7.7306141.002164502 463 214369.99252847 21.5174348 7.7361877.002159827 464 215296 99897344 21.5406592 7.7417532.002155172 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 21.6564078 7.7694620.002132196 470 220900 103823000 21.6794834 7.7749801.002127660 471 221841 104487111 21.7025344 7.7804904'.002123142 472 222784 105154048 21.7255610 7.7859928.002118644 473 223729 105823817 21.7485632 7.7914875.002114165 474 224676 106496424 21.7715411 7.7969745.002109705 475 225625 107171875 21.7944947 7.8024538.002105263 476 226576 107850176 21.8174242 7.8079254.002100840 477 227529 108531333 21.8403297 7.8133892.002096436 478 228484 109215352 21.8632111 7.8188456.002092050 479 229441 109902239 21.8860686 7.8242942.002087683 480 230400 110592000 21.9089023 7.8297353.002083333 481 231361 111284641 21.9317122 7.8351688.002079002 482 232324 111980168 21.9544984 7.8405949.002074689 483 233289 112678587 21.9772610 7.8460134.002070393 484 234256 113379904 22.0000000 7.8514244.002066116 485 235225 114084125 22.0227155 7.8568281.002061856 486 236196 114791256 22.0454077 7.8622242.002057613 487 237169 115501303 22.0680765 7.8676130.002053388 488 238144 116214272 22.0907220 7.8729944.002049180 489 239121 116930169 22.1133444 7.8783684.002044990 490 240100 117649000 22.1359436 7.8837352.002040816 491 241081 118370771 22.1585198 7.8890946.002036660 492 242064 119095488 22.1810730 7.8944468.002032520 493 243049 119823157 22.2036033 7.8997917.002028398 494 244036 120553784 22.2261108 7.9051294 *.002024291 495 245025 121287375 22.2485955 7.9104599.002020202 496 246016 122023936 22.2710575 7.9157832.002016129 146 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 497 247009 122763473 22.2934968 7.9210994.002012072 498 248004 123505992 22.3159136 7.9264085.002008032 499 249001 124251499 22.3383079 7.9317104.002004008 500 250000 125000000 22.3606798 7.9370053.002000000 501 251001 125751501 22.3830293 7.9422931.001996008 502 252004 126506008 22.4053565 7.9475739.001992032 503 253009 127263527 22.4276615 7.9528477.0019,88072 504 254016 128024064 22.4499443 7.9581144.001984127 505 255025 128787625 22:4722051 7.9633743.001980198 506 256036 129554216 22.4944438 7.9686271.001976285 507 257049 130323843 22.5166605 7.9738731.001972387 508 258064 131096512 22.5388553 7.9791122.001968504 509 259081 131872229 22.5610283 7.9843444.001964637 510 260100 132651000 22.5831796 7.9895697.001960784 511 261121 133432831 22.6053091 7.9947883.001956947 512 262144 134217728 22.6274170 8.0000000.001953125 513 263169 135005697 22.6495033 8.0052049.001949318 514 264196 135796744 22.6715681 8.0104032.001945525 515 265225 136590875 22.6936114 8.0155946.001941748 516 266256 137388096 22.7156334 8.0207794.001937984 517 267289 133188413 22.7376340 8.0259574.001934236 518 263324 138991832 22.7596134 8.0311287.001930502 519 269361 139798359 22.7815715 8.0362935.001926782 520 270400 140608000 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 8.0568862.001912046 524 274576 143377824 22.8910463 8.0620180.001908397 525 275625 144703125 22.9128785 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 280900 148877000 23.0217289 8.0926723.001886792 531 281961 149721291 23.0434372 8.0977589.001883239 532 233024 150568768 23.0651252 8.1028390.001879699 533 234089 151419437 23.0867928 8.1079128.001876173 534 285156 152273304 23. 1084400 8.1129803.001872659 535 286225 153130375 23.1300670 8.1180414.001869159 536 287296 153990656 23.1516738 8.1230962.001865672 537 288369 154854153 23.1732605 8.1281447.001862197 538 289444 155720872 23. 1948270 8.1331870.001858736 539 290521 156590819 23.2163735 8.1382230.001855288 540 291600 157464000 23,2379001 8.1432529.001851852 541 292681 158340421 23.2594067 8.1482765.001848429 542 293764 159220088 23.2808935 8.1532939.001845018 543 294849 160103007 23.3023604 8.1583051.001841621 544 295936 160989184 23.3238076 8.1633102.001838235 545 297025 161878625 23.3452351 8.1683092.001834862 546 293116 162771336 23.3666429 8.1733020.001831502 547 299209 163667323 23.3880311 8.1782888.001828154 548 300304 164566592 23.4093998 8.1832695.001824818 549 301401 165469149 23.43074-90 8.1882441.001821494 550 302500 166375000 23.4520788 8.1932127.001818182 551 303601 167284151 23.4733892 8.1931753.001814882 552 304704 168196608 23.4946802 8.2031319.001811594 553 305809 169112377 23.5159520 8.2080825.001808318 554 306916 170031464 23.5372046 8.2130271.001805054 555 303025 170953875 23.5584330 8.2179657.001801802 556 309136 171879616 23.5796522 8.2228985.001798561 557 310249 172808693 23.60)3474 8.22782.54.001795332 558 311364 173741112 23.6220236 8.2327463.001792115 CUBE ROOTS, AND RECIPROCALS. 147 go. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 559 312481 174676879 23.6431808 8.2376614.001788909 560 313600 175616000 23.6643191 8.2425706.001785714 561 314721 176558481 23.6854386 8.2474740.001782531 562 315844 177504328 23.7065392 8.2523715.001779359 563 316969 178453547 23.7276210 8.2572633.001776199 564 318096 179406144 23.7486842 8.2621492.001773050 565 319225 180362125 23.7697286 8.2670294.001769912 566 320356 181321496 23.7907545 8.27190.39.001766784 567 321489 182284263 23.8117618 8.2767726.001763668 568 322624 183250432 23.8327506 8.2816355.001760563 569 323761 184220033 23.8537209 8.2864928.001757469 570 324900 185193000 23.8746728 8.2913444.001754386 571 326011 186169411 23.8956063 8.2961903.001751313 572 327184 187149248 23.9165215 8.3010304.001748252 573 328329 188132517 23.9374184 8.3058651.001745201 574 329476 189119224 23.9582971 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 24.0831891 8.3395509.001724138 581 337561 196122941 24.1039416 8.3443410.001721170 582 338724 197137368 24.1246762 8.3491256.001718213 583 339889 198155287 24.1453929 8.3539047.001715266 584 341056 199176704 24.1660919 8.3586784.001712329 585 34222.5 200201625 24.1867732 8.3634466.001709402 586 343396 201230056 24.2074369 8.3682095.001706185 587 344569 202262003 24.2280829 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 350464 207474688 24.3310501 8.3966729.001689189 593 351649 208527857 24.3515913 8.4013981.001686341 594 352836 209584584 24.3721152 8.4061180.001683502 595 354025 210644875 24.3926218 8.4108326.001680672 596 355216 211708736 24.4131112 8.4155419.001677852 597 356409 212776173 24.4335834 8.4202460.001675042 598 357604 213847192 24.4540385 8.4249448.001672241 599 358801 214921799 24.4744765 8.4296383.001669449 600 360000 216000000 24.4948974 8.4343267.001666667 601 361201 217081801 24.5153013 8.4390098.001663894 602 362404 218167208 24.5356883 8.4436877.001661130 603 363609 219256227 24.5560583 8.4483605.001658375 604 361816 220348864 24.5764115 8.4530281.0(1655629 605 366025 221445125 24.5967478 8.4576906.001652893 606 367236 222545016 24.6170673 8.4623479.001650165 607 368449 223648543 24.6373700 8.4670001.001647446 608 369664 224755712 24.6576560 8.4716471..001644737 609 370381 225866529 24.6779254 8.4762892.001642036 610 372100 226981000 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 37S225 232608375 24.7991935 8.5040350.001626016 616 379456 233744896 24.8193473 8.5086417.001623377 617 3806l9 234885113 24.8394847 8.5132435.001620746 618 331924 236029032 24.8596058 8.5178403.001618123 619 333161 237176659 24.8797106 8.5224321.001615509 620 334400 238328000 24.8997992 8.5270189.001612903. [ 148 TABLE XI. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocae. 621 385641 239483061 24.9198716 8.5316009.001610306 622 386884 240641848 24.9399278 8.5361780.001607717 623 388129 211804367 24.9599679 8.5407501.001605136 624 339376 242970624 24.9799920 8.5453173.001602564 625 390625 244140625 25.0000000 8.5498797.001600000 626 391876 245314376 25.0199920 8.5544372.001597444 627 393129 246491883 25.0399681 8.5589899.001594896 628 394334 247673152 25.0599282 8.5635377.001592357 629 395641 248858189 25.0798724 8.5680807.001589825 630 396900 250047000 25.0998008 8.5726189.001537302 631 398161 251239591 25.1197134 8.5771523.001584786 632 399424 252435963 25.1396102 8.5816809.001582278 633 400689 253636137 25.1594913 8.5862047.001579779 634 401956 254840104 25.1793566 8.5907238.001577287 635 403225 256047875 25.1992063 8.5952380.001574803 636 404496 257259456 25.2190404 8.5997476.001572327 637 405769 258474853 25.2388589 8.6042525.001569859 633 407044 259694072 25.2586619 8.6087526.001567398 639 408321 260917119 25.2784493 8.6132480.001564945 640 409600 262144000 25.2932213 8.6177388.001562500 641 410381 263374721 25.3179778 8.6222248.001560062 642 412164 264609238 25.3377189 8.6267063.001557632 643 413449 265847707 25.3574447 8.6311830.001555210 6t4 414736 267089934 25.3771551 8.6356551.001552795 645 416025 268336125 25.3968502 8.6401226.001550388 646 417316 269586136 25.4165301 8.6445855.001547988 647 418609 270840023 25.4361947 8.6490437.001545595 648 419904 272097792 25.4558441 8.6534974.001543210 649 421201 273359449 25.4754784 8.6579465.001540832 650 422500 274625000 25.4950976 8.6623911.001538462 651 423801 275894451 25.5147016 8.6668310.001536098 652 425104 277167808 25.5342907 8.6712665.001533742 653 426409 278445077 25.5538647 8.6756974.001531394 654 427716 279726264 25.5734237 8.6801237.001529052 655 429025 281011375 25.5929678 8.6845456.001526718 656 430336 232300416 25.6124969 8.6889630.001524390 657 431649 283593393 25.6320112 8.6933759.001522070 658 432964 234890312 25.6515107 8.6977843.001519757 659 434281 286191179 25.6709953 8.7021882.001517451 660 435600 287496000 25.6904652 8.7065877.001515152 661 436921 288804781 25.7099203 8.7109827.001512859 662 438244 290117528 25.7293607 8.7153734.001510574 663 439569 291434247 25.7487864 8.7197596.001508296 664 440396 292754944 25.7631975 8.7241414.001506024 665 442225 294079625 25.7875939 8.7285187.001503759 666 443.556 295408296 25.8069758 8.7328918.001501502 667 444899 296740963 25.8263431 8.7372604.001499250 668 446224 298077632 25.8456960 8.7416246.001497006 669 447561 299418309 25.8650343 8.7459846.001494768 670 448900 300763000 25.8843582 8.7503401.001492537 671 450241 302111711 25.9036677 8.7546913.001490313 672 451584 303164448 25.9229628 8.7590383.001488095 673 452929 304821217 25.9422435 8.7633809.001485884 674 454276 306182024 25.9615100 8.7677192.001483680 675 455625 307546875 25.9807621 8.7720532.001481481 676 456976 303915776 26.0000000 8.7763830.001479290 677 458329 310283733 26.0192237 8.7807084.001477105 678 459684 311665752 26.03843.31 8.7850296.001474926 679 461041 313046839 26.0576284 8.7893466.001472754 680 462400 314432000 26.0763096 8.7936593.001470588 681 463761 315821241 26.0959767 8.7979679.001468429 682 465124 317214563 26.1151297 8.8022721.001466276 CUBE ROOTS, AND RECIPROCALS. 149 Noo. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 683 466489 318611987 26.1342687 8.8065722.001464129 684 467856 320013504 26.1533937 8.8108681.001461988 685 469225 321419125 26.1725047 8.8151598.001459854 686 470596 322828856 26.1916017 8.8194474.001457726 687 471969 324242703 26.2106848 8.8237307.001455604 688 473344 325660672 26.2297541 8.8280099.001453488 689 474721 327082769 26.2488095 8.8322850.001451379 690 476100 328509000 26.2678511 8.8365559.001449275 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 338608873 26.4007576 8.8663375.001434720 698 487204 340068392 26.4196896 8.8705757.001432665 699 488601 341532099 26.4386081 8.8748099.001430615 700 490000 343000000 26.4575131 8.8790400.001428571 701 491401 344472101 26.4764046 8.8832661.001426534 702 492804 345948408 26.4952826 8.8874882.001424501 703 494209 347428927 26.5141472 8.8917063.001422475 704 495616 348913664 26.5329983 8.8959204.001420455 705 497025 350402625 26.5518361 8.9001304.001418440 706 498436 351895816 26.57 6605 8.9043366.001416431 707 499849 353393243 26.5894716 8.9085387.001414427 708 501264 354894912 26.6082694 8.9127369.001412429 709 502681 356400829 26.6270539 8.9169311.001410437 710 504100 357911000 26.6458252 8.9211214.001408451 711 505521 359425431 26.6645833 8.9253078.001406470 712 506944 360944128 26.6833281 8.9294902.001404494 713 508369 362467097 26.7020598 8.9336687.001402525 714 509796 363994344 26.7207784 8.9378433.001400560 715 511225 365525875 26.7394839 8.9420140.001398601 716 512656 367061696 26.7581763 8.9461809.001396648 717 514089 368601813 26.7768557 8.9503438.001394700 718 515524 370146232 26.7955220 8.9545029 001392758 719 516961 371694959 26.8141754 8.9586581.001390821 720 518400 373248000 26.8328157 8.9628095.001388889 721 519841 374805361 26.8514432 8.9669570.001386963 722 521284 376367048 26.8700577 8.9711007.001385042 723 522729 377933067 26.8886593 8.9752406.001383126 724 524176 ~ 379503424 26.9072481 8.9793766.001381215 725 525625 381078125 26.9258240 8.9835089.001379310 726 527076 382657176 26.9443872 8.9876373.001377410 727 528529 384240583 26.9629375 8.9917620.001375516 728 529984 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 9.0082229.001367989 732 535824 392223168 27.0554985 9.0123288.001366120 733 537289 393832837 27.0739727 9.0164309.001364256 734 538756 395446904 27.0924344 9.0205293.001362398 735 54022.5 397065375 27.1108834 9.0246239.001360544 736 541696 398688256 27.1293199 9.0287149.001358696 737 543169 400315553 27.1477439 9.0328021.001356852 738 544614 401947272 27.16615.54 9.0368857.001355014 739 546121 403583419 27.1845544 9.0409655.001353180 740 547600 405224000 27.2029410 9.0450419.001351351 741 549081 406869021 27.2213152 9.0491142.001349528 742 550564 408518488 27.2396769 9.0531831.001347709 743 552049 410172407 27.2580263 9.0572482.001345895 744 553536 411830784 27.2763634 9.0613098.001344086 ___ ___ _ 150 TABLE Xl SQUARES, CUBES, SQUARE ROOTS, No quae. be. SquareRoots. Cubes. Suare Roots. C Rootsoot. Reciprocals. 745 555025 413493625 27,2946881 9.0653677.001342282 746 556516 415160936 27.3130006 9.0694220.001340483 747 558009 416832723 27.3313007 9.0731726.001338688 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 27.5317993 9.1177931.001319261 759 576031 437245479 27.5499546 9.1218010.001317523 760 577600 438976000 27.5680975 9.1258053.001315789 761 579121 440711081 27.5862284 9.1298061.001314060 762 580644 442450728 27.6043475 9.1338034.001312336 763 582169 444194947 27.6224546 9.1377971.001310616 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 583289 451217663 27.6947648 9.1537375.001303781 768 589824 452934832 27.7128129 9.1577139.001302083 769 591361 454756609 27.7308492 9.1616869.001300390 770 592900 456533000 27.7488739 9.1656565.001298701 771 594441 458314011 27.7668868 9.1696225.001297017 772 595984 460099648 27.7848880 9.1735852.001295337 773 597529 461839917 27.8028775 9.1775445.001293661 774 599076 463684824 27.8208555 9.1815003.001291990 775 600625 465484375 27.8388218 9.1854527.001290323 776 602176 467288576 27.8567766 9.1894018.001288660 777 603729 469097433 27.8747197 9.1933474.001287001 778 605284 470910952 27.8926514 9.1972897.001285347 779 606841 472729139 27.9105715 9.2012286.001283697 780 603400 474552000 27.9284801 9.2051641.001282051 781 609961 476379541 27.9463772 9.2090962.001280410 782 611524 478211768 27.3542629 9.2130250.001278772 783 613039 480048637 27.9821372 9.2169505.001277139 784 614656 481890304 28.0000000 9.2208726.001275510 785 616225 483736625 28.0178515 9.2247914.001273885 786 617796 485587656 28.0356915 9.2287068.001272265 787 619369 487443403 28.0535203 9.2326189.001270648 788 620944 489303872 28.0713377 9.2365277.001269036 789 622521 491169069 28.0891438 9.2404333.001267427 790 624100 493039000 28.1069386 9.2443355.001265823 791 625681 494913671 28.1247222 9.2482344.001264223 792 627264 496793038 28.1424946 9.2521300.001262626 793 628849 493677257 28.1602557 9.2560224.001261034 794 639436 500566184 28.1780056 9.2599114.001259446 795 6.32025 502459875 28.1957444 9.2637973.001257862 796 633616 504358336 28.2134720 9.2676798.001256231 797 6:35209 506261573 28.2.311884 9.2715592.001254705 798 636304 508169592 28.2488933 9.2754352.001253133 799 633401 510082399 28.2665881 9.2793081.001251564L 800 640000 512000000 28.2942712 9.2831777.001250000 801 641601 51392'2401 28.3019434 9.2870440.001214439 802 643204 515849608 28.3196045 9.2909072.001246383 803 644809 517781627 28.3372.546 9.2947671.001245330 804 646416 519718464 28.3548933 9.2986239.001213781 805 648025 521660125, 28.3725219 9.3024775.001242236 806 649636 523606616 28.3901391 9.3063278.001240695 CUBE ROOTS, AND RECIPROCALS. 151 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 807 651249 525557943 28.4077454 9.3101750.001239157 808 652864 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 28.4780617 9.3255320.001233046 812 659314 535337328 28.4956137 9.32936.34.001231527 813 660969 537367797 28.5131549 9.33.31916.001230012 814 662596 539353144 28.5306852 9.3370167.001223501 815 664225 541313375 28.54S2048 9.3408386.001226994 816 665856 543338496 28.5657137 9.3446575.001225490 817 667489 545338513 28.5832119 9.3484731.001223990 818 669124 547343432 28.6006993 9.3522357.001222494 819 670761 549353259 23.6181760 9.3560952.001221001 820 672400 551368000 28.6356421 9.3599016.001219512 821 674041 553387661 28.6530976 9.3637049.001218027 822 675634 555412248 28.6705424 9.3675051.001216545 823 677329 557441767 28.6379766 9.3713022.001215067 824 678976 559476224 28.7054002 9.3750963.001213592 825 630625 561515625 28.7228132 9.3788873.001212121 826 682276 563559976 2. 7402157 9.3326752.001210654 827 683929 565609233 28.7576077 9.3864600.001209190 828 685584 567663552 28.7749891 9.3902419.001207729 829 637241 569722789 28.7923601 9.3940206.001206273 830 688900 571787000 28.8097206 9.3977964.001204819 831 690561 573856191 23.8270706 9.4015691.001203369 832 692224 575930368 28.8444102 9.4053387.001201923 833 693889 578009537 28.8617394 9.4091054.001200480 834 695556 580093704 28.8790582 9.4128690.001199041 835 697225 582182375 28.8963666 9.4166297.001197605 836 698896 534277056 28.9136646 9.4203873.001196172 837 700569 586376253 28.9309523 9.4241420.001194743 838 702-244 533480472 28.9482297 9.4278936.001193317 839 703921 590389719 28.9654967 9.4316423.001191895 840 705600 592704000 28.9827535 9.4353880.001190476 841 707281 594823321 29.0000000 9.4391307.001189061 842 708964 596947638 29.0172363 9.4128704.001187648 843 710649 599077107 29.0314623 9.4466072.001186240 844 712336 601211534 29.0516781 9.4503410.001184834 845 71402.5 603351125 29.0638837 9.4540719.001183432 846 715716 605495736 29.0860791 9.4577999.001182033 847 717409 607645423 29.1032644 9.4615249.001180638 848 719104 609800192 29.1204396 9.4652470.001179245 849 720801 611960049 29.1376046 9.4689661.001177856 850 722500 614125000 29.1547595 9.4726824.001176471 851 724201 616295051 29.1719043 9.4763957.001175038 852 725904 61847020 29. 1890390 9.4801061.001173709 853 727609 620650477 29.2061637 9.4838136.001172333 854 729316 622335864 29.22.32784 9.4875182.001170960 855 731025 625026375 29.2403330 9.4912200.001169591 856 732736 627222016 29.2574777 9.4949188.001168224 857 734449 629422793 29.2745623 9.4986147.001166861 858 736164 631623712 29.2916370 9.502.3078.00116.5501 859 737881 633839779 29.3087018 9.5059980.001164144 860 739600 636056000 29.3257566 9.5096354.001162791 861 741321 633277331 29.3428015 9.5133699.001161440 862 743044 640503923 29.3598365 9.5170515.001160093 863 744769 642735647 29.3768616 9.5207303.001158749 864 746496 644972544 29.3938769 9.5244063.001157407 865 748225 647214625 29.4103823 9.5230794.0011.56069 866 749956 649461896 29.4278779 9.5317497.001154734 867 751639 651714363 29.4443637 9.5354172.001153403 863 753424 653972032 29.4618397 9.5390818.001152074 85 381~63792.3071 9.509.00144 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 9.5937169.001132503 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 81.5409 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 840889 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 9.7329309.001084599 923 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 859329 796597983 30.4466747 9.7504930.001078749 928 861184 799178752 30.4630924 9.7539979.001077586 929 863041 801765089 30.4795013 9.7575002.001076426 930 864900 804357000 30.4959014 9.7610001.001075269 CUBE ROOTS, AND RECIPROCALS. 153 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 931 866761 806954491 30.5122926 9.7644974.001074114 932 863624 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 876096 820025356 30.5941171 9.7829466.001068376 937 877969 822656953 30.6104557 9.7854288.001067236 938 879844 825293672 30.6267857 9.7889037.001066098 939 881721 827936019 30.6431069 9.7923861.031064963 940 883600 830584000 30.6594194 9.7958611.001063830 941 885481 833237621 30.6757233 9.7993336.001062699 942 8S7364 835896888 30.6920185 9.8023036.001061571 943 889249 838561807 30.7083051 9.8062711.001060445 944 891136 841232384 30.7245830 9.8097362.001059322 945 893025 843903625 30.7408523 9.8131989.001058201 946 894916 846590536 30.7571130 9.8166591.001057082 947 896309 849278123 30.7733651 9.8201169.001055966 948 898704 851971392 30.7896086 9.8235723.001054852 949 900601 854670349 30.8058436 9.8270252.001053741 950 902500 857375000 30.8220700 9.8304757.001052632 951 904401 860085351 30.8382879 9.8339238.00)1051525 952 906304 862801408 30.8544972 9.8373695.001050420 9.3 903209 865523177 30.8706931 9.8408127.001049318 9. 910116 868250664 30.8868904 9.8442536.001048218 955 912025 870933875 30.9030743 9.8476920.001047120 956 913936 873722816 30.9192497 9.8511280.001046025 9 7 915849 876467493 30.9354166 9.8545617.001044932 958 917764 879217912 30.9515751 9.8579929.001043841 959 919681 881974079 30.9677251 9.8614218.001042753 960 921600 884736000 30.9838668 9.8648483.001041667 961 923521 887503631 31.0000000 9.8682724.001040583 962 925444 890277128 31.0161248 9.8716941.001039501 963 927369 893056347 31,0322413 9.8751135.001038422 964 929296 895841344 31.0483494 9.8785305.001037344 965 931225 898632125 31.0644491 9.8819451.001036269 966 933156 901428696 31.0895405 9.8853574.001035197 937 935089 904231063 31.0966236 9.8887673.001034126 963 937024 907039232 31.1126984 9.8921749.001033058 969 938961 909853209 31.1287648 9.8955801.001031992 970 940900 912673000 31.1448230 9.8989830.001030928 971 942841 915498611 31.1608729 9.9023835.001029866 972 944784 918330048 31.1769145 9.9057817.001028307 973 946729 921167317 31.1929479 9.9091776.001027749 974 948676 924010424 31.2089731 9.9125712.001026694 975 950625 926859375 31.2249900 9.9159624.001025641 976 952576 929714176 31.2409987 9.9193513.001024590 977 954529 932574833 31.2569992 9.9227379.001023541 978 956484 935441352 31.2729915 9.9261222.001022495 979 958441 938313739 31.2889757 9.9295042.001021450 980 960400 941192000 31.3049517 9.9328839.001020408 981 962361 944076141 31.3209195 9.9362613.001019368 982 964324 946966168 31.3368792 9.9396363.001018330 983 966289 949862087 31.3528308 9.9430092.001017294 994 968256 952763904 31.3687743 9.9463797.001016260 985 970225 955671625 31.3847097 9.9497479.001015228 986 972196 958585256 31.4006369 9.9531138.001014199 987 974169 961504803 31.4165561 9.9564775.001013171 988 976144 964430272 31.4324673 9.9598389.001012146 989 978121 967361669 31.4483704 9.9631981.001011122 990 980100 970299000 31.4642654 9.9665549.001010101 991 982081 973242271 31.4801525 9.9699095.001009082 992 984064 976191488 31.4960315 9.9732619.001008065 S. 154 TABLE XI. SQUARES, CUBES, &C. No. Squares. Cubes. Square Roots. Cubs 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 31.606.613 9.9966656.001001001 1000 1000000 1000000000 31.6227766 10.0000000.001000000 1001 1002001 1003003001 31.6385840 10.0033322.0009990010 1002 1004004 1006012008 31.6543836 10.0066622.0009980040 1003 100(6'09 1009027027 31.6701752 10.0099899.0009970090 1004 1008016 1012048064 31.659590 10.0133155.0009960159 10o5 1010025 1015075125 31.7017349 10.0166389.0009950249 1006 1012036 1018103216 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 10.0299101.0009910803 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 1032256 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 1 M34o00 P361208000 31 9374388 10.0662271.0009803922 1021 3 i42, 41 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 10n6 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 10.1022835.00(9699321 1032 1065024 1099104768 32.1247568 10.1055487.0009689922 1033 1067089 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 1118386372 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 IC42 1085761 1131366088 32.2800248 10.1380845.0009596929 i043 1087849 1134626507 32.2955105 10.1413266.0009587788 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 1(48A 1098304 1151022592 32.3728281 10.1575062.0009541985 1049 1100401 1154320649 32.3882695 10.1607359.0009532888 D105. 1102500 1157625000 32.4037035 10.1639636.0009523810 1051 1104601 1160935651 32.4191301 10.1671893.0009514748 1052 1106704 1164252608 32.4345495 10.1704129.0009505703 1053 11(08809 1167575877 32.4499615 10.1736344.0009496676 1:4. 1110916 1170905464 32.4653662 10.1769539.(;009487666,~-~-~ — L - I —-----— ~- - TABLE XII. LOGARITHMS OF NUMBERS FROM I TO 10,000 156 TABLE XII. LOGARITHMS OF NUMBERS. No. O I 3 4 5.6 7 8 9 Diff. 100 0000(0 000434 00868 001301 001734 002166 002598 003029 003461 003891 432 1 4321 4751 5181 5609 6038 6466 6894 7321 7748 8174 428 2 8600 9026 9451 9876 010300010724 011147 011570011993012415 424 3 012837 013259 013680 014100 4521 4940 5360 5779 6197 6616 420 4 7033 7451 7868 8284 8700 9116 9532 9947020361 020775 416 5 021189 021603 022016 022428 022841 023252 023664 024075 4486 4896 412 6 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 7 9384 9789 030195 0306003104 031408 031812 032216 032619 033021 404 8 033424 03.3826 4227 46281 5029 5430 5830 6230 6629 7028 400 9 7426 7825 8223 8620 9017 9414 9811 040207 040602 040998 397 110 041393 041787 042182 042576 042969 043362 043755 044148 044540 044932 393 11 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 390 2! 9218 9606 99931050380 050766 051153 051538 051924 052309052694 386 3:053078 053463 053846 4230 4613 4996 5378 5760 6142 6524 383 4! 6905 7286 7666 8046 8426 8805 9185 9563 9942 060320 379 51060698 061075 061452 061829 062206 062582 062958 063333 063709 4083 376 61 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 373 7 8186 8557 89281 9298 9668 0700381070407070776071145071514 370 81071882 072250 072617 072985 073352 3718 4085 4451 4816 5182 366 9 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 120 079181 079543 079904 080266 080626 080987 081347 081707 082067 082426 360 1032785 083144 083503 3861 4219 4576 4934 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 41093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 349 5 6910 7257 7604 7951 8298 8644 8990 9335 9681 100026 346 6 100371 100715 101059 101403 101747 102091 102434 102777 103119 3462 343 7 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 341 81 7210 7549 7888 8227 8565 8903 9241 9579 9916 110253 338 9 110590 110926 111263 111599 111934 112270 112605 112940 113275 3609 335 130 11393 1146 14277 14611 11944 115278115611 115943 113 276 116608 116940 333 1 7271 7603 7934 8265 8595 8926 9256 9586 9915 120245 330 2 120574 120903 1212311121560 121888 122216 122544 122871 123198 3525 328 3 3852 4178 4504 4830 5156 5481 5806 6131 6,56 6781 325 4 7105 7429 7753 8076 8399 8722 9045 9368 96901130012 323 5 130334 130655 130977 131298 13161911319391132260 132580 132900 3219 321 6 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 318 7 6721 7 0377354 7671 7987 8303 8618 8934 9249 9564 316 8 98791140194 140508 140822 1411361141450 141763 142076 142389 142702 314 9 143015 3327 3639 3951 4263 4574 4885 5196 5507 5818 311 14011461281146438 146748 147058 147367 147676 147985 148294 148603 148911 309 1 9219 9527 98351501421150449115075611510631151370 151676 151982 307 21522881152594 152900 3205 3510 3815 4120 4424 4728 5032 305 3 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 303 4 8362 8664 8965 9266 9567 98681160168 160469 1607691161068 301 51161368 161667 161967 162266 162564 162863 3161 3460 3758 4055 299 6 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 7 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 8 170262 170555 17084 1711411171434 171726 172019 172311 172603 172895 293 9 3186 3478 3769 4060 4351 4641 4932 5222 5512 5802 291 150 176091 176381 176670 176959 177248 177536 177825 178113 178401 178689 2891 1 8977 9264 9552 9839 180126 180413 180699 180986 181272 181558 287 21181844 182129 182415 182700 2985 3270 3555 3839 4123 4407 285 3 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 283 4 7521 7803 8084 8366 8647 8928 9209 9490 97711190051 281 5 1903321190612 190892 191171 191451 191730 1920101192289 192567 2846 279 6 3125 3403 3681 3959 4237 4514 4792 5069 53461 5623 278 7 5900 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 8 8657 8932 9206 9481 97551200029 200303 200577 2008501201124 274 9201397 201670 201943 202216 202488 2761 3033 3305 3577 3848 272 No.1 0 1 2 3 1 4 5 6 7 8 9 Diff. TABLE XII. LOGARITHMS OF NUMBERS. 157 No.I 0 3 i 4: 5 6 18 9 Diff. 160 2041201204391 204663 201934 205204 205-175 205746 206016 206286 206556 271 1 6826 7096 7365 7631 7904 8173 8441 87101 8979 9247 269 21 9515 9783 210351 210319 210586 210353 211121 211388 211654 211921 267 3 212183t212454 2720 2986 3252 3518, 3783 40191 4314 4579 266 4 48411 5109 5373 5633 5902 6166 6430 6694 6957 7221 264 5 7434 7747 8010, 8273 8536 8793 9060 9323 9585 9346 262 6 220103 220370 220631 220392 221153 221414 221675 221936 222196 222456 261 7 2716 2976 32361 3496 3755 4015 4274 4533 4792 5051 259 8 5309 5563 5826 6034 6342 6600[ 6858 7115 7372 7630 258 9 7337 8144 8400 8657 8913 9170 9426 9632 9938 230193 256 170 230149 230704 230960 231215 231470 231724 231979 2322.34 232488 232742 255 11 2996 3250 3504 3757 4011 426- 4517 4770 5023 5276 253 2 55-23 5781 6033 6235 6537 6789 7041 7292 7544 7795 252 31 8046 8297 8548 8799 9049 9299 9550 9800 240050 240300 250 4240549 210799 211048 241297 241546 241795 242044 242293 2541 2790 249 5 3(033 3236 3534 3782 4030 4277 4525 4772 5019 5266 248 6 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 7 7973 8219 8464 8709 8954 9193 9443 9687 99321250176 245 8 250120 250664 250903 251151 251395 251638 251881 252125 252363S 2610 243 9 2353 3096 3333 3530 3822 4064 4306 4548 4790 5031 242 180 255273 255514 255755 255996 256237 256477 256718 256958 257198 257439 241 1 7679 7918 8158 8398 8637 8377 9116 9355 9594 9833 239 2 260071 260310 260548 260787 261025 261263 261501 261739 261976 262214 233 3 2451 2633 2925 3162 3399 3636 3873 4109 4346 4582 237 4 4818 5051 5290 5525 5761 5996 6232 6167 6702 6937 235 5l 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 6i 9513 9746 9980 270213 270446 270679 2709121271144 271377 271609 233 7271842 272074 272306 2533 2770 3001 3233 3461 3696 3927 232 8 4153 4339 4620 4850 5031 5311 5542 5772 6002 6232 230 9 6462 6692 6921 7151 7380 7609 783S 8067 8296 8525 229 190i278754 278932 279211 279439 279667 279395 2301231'320351 280578 230806 228 1 231033 231261 2314883231715 231942 282169 2396 2622 2849 3075 227 2 3301 3527 3753 3979 4205 4431 4656 4382 5107 5332 226 3 5557 5782 6007 6232 6156 6681 6905 7130 7354 7578 225 4 7802 8026 8249 8473 8696 8920 9143 9366 9539 9812 223 5 290035 290257 2904801290702 290925 291147 291369 291591 291813 292034 222 6 2256 2478 2699' 2920 3141 3363 3581 3301 4025 4216 221 7 4466 4637 4907 5127 5347 5567 5787 60,07 6226 6446 220 8 6665 6334 7104 7323 7542 7761 7979 8198 8416 8635 219 9 8353 9071 9239 9507 97275 9943 300161 300378 300595 300813 218 200 301030 301247 301464 301631 301898 302114 302331 302547 302764 302980 217 1 3196 3112 3628 3314 4059 4275 4491 4706 4921 5136 216 2 5351 5566 5781 59956 6211 6425 6639 6854 7068 7232 215 3 7496 7710 79241 8137 8351 8564 8778 8991 9204 9417 213 4 9630 9343 310056!310263 310481 310693 3109061311118 3113301311542 212 51311754/311966 2177 2339 2600) 2312 3023 3234 3445 3656 211 I 6 3367 4078 4239 4499 4710 4920 5130 5340 5551 5760 210 7 5970 6180 6390 6599 6809 7018 7227 7436 7616 7854 209 8 8063 8272 8431 8639 8893 9106 9314 9522 97301 9938 203 9320146 320354 320562 320769 320977 321184 321391 321598 321805 322012 207 210 322219 322426 322633 322839 323046 323252 323458 323665 323871 324077 206 1 4282 4488 46941 4899 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 9805 3300083330211 203 4330114330617 330319 331022 331225 331427 3316301331832 2034 2236 202 5 2433 2640 2842 3044 3216 3447 3649 3350 4051 4253 202 6 4451 4655 4856 50.57 5257 5458 5658 5859 6059 6260 201 7 6460 6669 6360 7069 7260 7459 7659 7853 8058 8257 200 8 8456 8656 8855J 9054 9253 9451 9650 9349 340047 340246 199 9 340144 310612 340341 341039 341237 341435 311632 311830 2023 2225 190 1o. 0 1 - j 1'3 4 5 6 7 8 9 i1f. 158 TABLE XII. LOGARITHMS OF NUMBERS. No. 0 1 2 3 4 5 ~ 7 8 9 D. 220 342423 342620 342817 343014 343212 43409 343606 343802 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 9860 350054 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 5834 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 62105 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 53710681371253 371437371622371806 1991 2175 2360 2544 2728 184 6 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 7 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 380030 181 240 380211 380392 380573 380754 380934 381115 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 8634 8811 8989 178 5 9166 9343 9520 9698 9875 390051 390228 390405 390582 390759 177 6390935 391112 391288 391464 391641 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 398981 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 432) 4492 4663 171 4 4834 5005 5176 5346 5517 5688 5858 6029 6199 6370 171 5 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 170 6 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 7 9933 410102 410271 410440 410609 410777 410946 411114 411283 411451 169 8411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 168 9 3300 3467 3635 3803 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 421275 421439 165 4421604 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 430881 431042 431203 161 270 431364 431525 431685 431846 432007 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 44(594 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 IDiff. TABLE XII. LOGARITHMS OF NUMBERS. 159 No 0 1 1 3 4 5 6 7 i_8 9 jl)iff. 280 447158 447313 447468447 476 47778 447933 44-038 448242 448397 4438552 1 I5 1 8706 8861 90156 9170 9324 9478 9633 97871 99414500951 154 2 450219 450403 450557 450711 450865 451018 451172 451326 451479 16.33 154 3 1786 1940 2093 22171 2400 2553 2706 2859 3 165 153 4 3318 3471 3624 37771 3930 4082 4235 4387 4540 4692 1,3 5 4815 4997 5150 5302 5454 5606 5758 5910 6062 62141 152 6 6366 6518 6670 6321 6973 7125 7276 7428 7579 7731 1.52 7 7882 8033 8184 8336 8487 8638 8789 8940 9091 92412 151 8 9392 9543 9694 9845 99951460146 460296 460447 460597 460748 151 91460898 461048 461198 461348 461499 1649 1799 1948 2098 224" 1 5) 290463946 8 462548 462697 462847 462997 463146 463296 463445 463594 463744 150 1 3393 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 2 5333 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 3 6863 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 4 8317 8495 8643 8790 8938 9085 9233 9330 9527 9675 148 5 9322 9969 470116 470263 470410 470557 470704 4 470993 471145 147 614712921471433 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 6252 6397 6542 6687 6832 6976 145 300 477121 477266 477411 477555 477700 477844 477989 478133 478278 478422 145 1 8566 8711 8855 8999 9143 9287 9431 9575 9719 9863 144 2 4800071480151 480294 480438 480582 480725 480869 481012 481156 481299 144 3 1443 1586 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 5S63 6005 6147 6289 6430 6572 6714 6355 6997 142 7 7133 7280 7421 7563 7704 7845 7986 8127 8269 8410 141 8 85511 8692 8833 8974 9114 9255 9396 9537 9677.9818 141 -9 99583490099 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 51541 5683 5822 5960 6099 6233 6376 6515 6653 6791 139 4 6913 7063 7206 7344 7483 7621 7759 7897 8035 8173 138 51 S:ll1 8448 8586 8724 8362 8999 9137 9275 9412 9550 13s 6 9687 9824 99621500099 500236 500374 500511 500648 500785 500922 137 75010595501196 501333 1470 1607 1744 1880 2017 215-1 2291 137 8 2127 2564 2700 2837 2973 3109 3246 3321 3518 365?1 136 9 3791 3927 4063 4199 4335 4471 4607 4743 4873 5014 136 320 505150 505286 505421 505557 505693 505828 505964 506099 506234 506370 136 1 6505 6610 6776 6911 7046 7181 7316 7451 7586 7721 135 2 7856 7991 8126 8260 8395 8530 8664 8799 8934 9063 135 31 9203 9337 9471 96061 9740 98741510009 510143510277510411 13-1 514 5516795113510 51094715110811511215 1349 14821 1616 17501 134 5 1833 2017 2151 2284 2418 2551 2684 2818 2951 3084 133 6 3218 3351 3484 3617 3750 3883 4016 4149 4282 4415 1331 7 4518 46*1| 4813 4946 5079 5211 5344 5476 5609 5741 133 8 5874 6006 6i39 6271 6403 6535 6668 6800 6932 7064 1321 9 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 132 330 518514 518646 518777 518909 519040 519171 519303 5194341519566 519697 131 1 9828 9959 5200901520221 5203531520484 520615 520745 520876 521007 131 2521138521269 1400 1530 1661 1792 1922 2053 2183 2314 131 3 2144 2575 2705 2835 2966 3096 3226 3356 3486 3616 130 4 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 130 5 5045 5174 5304 5434 5563 5693 5822 5951 6081 6210 129 6 6339 6469 6593 6727 6356 6985 7114 72431 7372 7501 129 7 7630 7759 7888 8016 8145 8274 8402 8531 8660 8788 129 8 8917 9045 9174 9302 9430 9559 9687 98151 99431530072 128 9 530200 530328 530456 530584 530712 530840 530968 531096 531223 1351 128 No. O 1 K 3 4 5 6 71 8 9 Diff. 160 TABLE XII. LOGARITHMS OF NUMBERS. No. 0 I 2 13 4 5 6 7 8 9 iDiff. 340 534 9 531607153173453186253199 532117 532245 532372 532500 532627 128 1 2754 2882 3009 3136 3264 3391 3518 3645 3772 3S99 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 540455540580540705 540830 540955541080541205 1330 1454 125 8 1579 1704 1829 1953 2078 2203 2327 24.32 2576 2701 125 9 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 124 350 544068 544192 544316 544440 544564 544688 544812 544936 545060 545183 124 1 5307 5431 5555 5678 5802 5925 6049 6172 6296 6419 124 2 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 123 3 7775 7898 8021 8144 8267 8389 8512 8635 8758 8881 123 4 9003 9126 9249 9371 9494 9616 9739 9861 9984 550106 123 5 5150228 550351 550473 550595 550717 550840 550962 551084 551206 1328 122 6 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 7 2668 2790 2911 3033 3155 3276 3398 3519 3640 3762 121 8 3S83 4004 4126 4247 4368 4489 461( 4731 4852 4973 121 9 5094 5215 5336 5457 5578 5699 5820 5940 6061 6182 121 360 556303 556423 556544 5566 5 567S8 556905 557026 557146 557267 557387 120 1 7507 7627 774 7868 79688 8108 8228 8349 8469 8589 120 2 8709 8S29 8946 90683 9188 9308 9428 9548 9667 9787 120 3 9907 560026 560146 560263 560385 560504 560624 560743 560863 560982 119 41561101 1221 1340 145 9 1578 1698 1817 1936 2055 2174 119 5 2293 2412 2531 2650 2769) -887 30C6 3125 3244 3362 119 6 3481 3600 3718 3837 3955 4074 41921 4311 4429 4E48 119 7 4666 4784 4903 5021 5139 5257 53,6 5494 5612 5730 118 8 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 118 9 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 568202 56S319 568436 568554 568671 568788 568905 569023 5691.40 569257 117 1 9374 9491 9608 9725 9842 9959570076570193570309570426 117 2 57054315706601570776 570893'571010 571126 1243 1359 1476 1592 117 3 1709 1825 1942 20581 2174 2291 2407 2523 2639 2755 116 4 2872 29681 3104 32201 3336 3452 3568 3684 3800 3915 116 5 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 6 5188 5303 5419 5534 5650 5765 5880 5996 6111 6226 115 7 6341 6457 6572 66871 6802 6917 7032 7147 7262 7377 115 8 7492 7607 7722 78361 7951 8066 8181 8295 8410 8525 115 9 8639 8754 8868 89831 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 2825 - 2; 2 3085 114 3 3199 3312 3426 3539 3652 3765 3879 3992 11 1) 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 69251 7037 7149 7262 7374 7486 7599 112 7 7711 7823 7935 8047[ 8160 8272 8384 8496 8608 8720 112 8 8832 8944 9056 9167 9279 9391 9503 9615 9726 9838 112 9 9950 590061 590173 590284 590396 590507 590619 590730 590842 590953 112 390 591065 591176 591287 591399 591510 591621 591732 591843 591955 592066 111 1 2177 2288 2399 2510 2621 27321 2843 2954 3064 3175 111 2 3286 3397 3508 36181 3729 38401 3950 4061 4171 4282 111 3 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 110 4 5496 5606 5717 5827 5937 6047 6157 6267 6377 6487 110 51 6597 6707 6817 6927 7037 7146 7256 73661 7476 7586 110 6 769.3 7805 7914 8024 8134 8243 8353 8462 8572 8681 110 7 8791 8900 9009 9119 9228 9337 9446 9556 9665 9774 109 81 9883 99921600101 600210 600319 600428 600537 600646 600755i60064 109 91600973 6010821 1191 1299 1408 15171 1625 1734 1843| 1951 109 No. 1 21 3 5 6 7 8 9 Diff,...~~~~~~~~~~~~~~~1 TABLE XII. LOGARITHMS OF NUMIBERS. 1I3 No. 0 1I 2' 3 1 4 5 6 t 7 8 9 Diff. i 400 602060 602169 602277 602386 602494 602603 602711 602S19 602928 G030361 1'6 1 3144 3-253 3361 3469 3577 3686 3794 3902j 4010 4118 101, 2 4226 4334 4412 4550 4658 4766 4874 4982 5039 5197 108 3 5305 5413 5521 5628 5736 5844 5951 6059 6166 6274 103! 4 5381 6489 6596 6704 6811 6919 7026 7133 7241 7348 1071 5' 7455 7562 7669 7777 7884 7991 8093 8205 8312 8419 107 61 8526 8633 8740 8847 8954 9061 9167 9274 9381 9488 107 7 9594 9701 9808 9914 610021 610128 610234 610341 610447 610554 107 861066016107671610873 610979 1086 1192 1298 1405 1511 1617 106 9 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 612784 612890 612996 613102 613207 613313 613419 613525 613630 613736 106 1 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 106 2 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 105 3 5950 6055 6160 6265 6370 6476 6581 6686 6790 689.5 105 4 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 105 5 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 105 6 9093 9198 9302 9406 9511 9615 9719 9824 99281620032 104 7 620136 620240 620344 620448 620552 620656 620760 620864 620968 1072 104 8 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 104 9 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 104 420 623249 623353 623456 623559 623663 623766 623869 623973 624076 624179 103 1 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 103 2 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 103 3 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 103 4 7366 7468 7571 7673 7775 7878 7980 8082 8185 8237 102 5 8339 8491 8593 8695 8797 8900 9002 9104 9206 9303 102 6 9410 9512 9613 9715 9817 9919 630021 630123 630224 630326 102 71630428 630530 630631 630733 6308351630936 1038 1139 1241 1342 102 8 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 101 9 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367 101 430 633468 633569 633570 633771 633872 633973 634074 634175 634276 634376 101 1 4477 4578 4679 4779 4830 4981 5081 5182 5233 5333 101 2 5484 5584 5635 57853 5886 5986 6087 6187 6287 63838 100 3 6488 6533 6638 6739 6889 6989 7089 7189 7290 7390 100 4[ 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 1001 5 8489 8589 8639j 8789 88833 89338 903S 9183 9237 9387 100 6 9436 9586 9686 9785 9885 9984 640034 640183 640283640382 99 716404811640581 6406301640779 6403791640978 1077 1177 1276 1375 99 81 1474 1573 1672 1771 1871 1970 2069 2163 2267 2366 99 9 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 99 440 643453 643551 643650 643749 643847 643946 644044 644143 644242 644340 98 1 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 91 2 5422 5521 5619 5717 5815 5913 6011 6110 62)08 6306 98 3 6404 6502 6600 6693 6796 6394 6992 7089 71S7 728.5 9 4 7333 7481 7579 7676 7774 7872 7969 8067 8165' 8262 98 5 8360 8458 8555 8653 8750 8848 8945 9043i 9140 9237 97 6 9335 9432 9530 9627 9724 9821 9919165001616501131650210 97 71650308 1650405 650302 650599 650696 650793 650890 0937 1084 1181 97 8 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 97 9 2246 2343 2440 2.536 2633 2730 2826 2923 3019 3116 97 450 653213 653.30i9 653105 653502 653598 653695 653791 653888:653984 654080 96 1 4177 4273 4369 4465 4562 4658 4754 48501 4946 5042 961 2 5133 5235 5331 5427 5523 5619 5715 5810 5906 601'2 9611 3 6093 6194 6290 6386 6482 6577 6673 6769 6864 6960 96! 4 7056 7152 7247 7343 7433 7534 7629 772.3 7820( 7916 96 I 1 5 8011 8107 8202 8298 8393 8488 8534 8679 8774 8870 95 F 6 8965 9060 9155 9250 9346 9441 9536 96311 9726 9821, 91 7 9916 660011 660106 660201 660296 660391 660486!660581,660676 660771 951 81660365 0950 1055 1150 1245 1339 1434 15291 1623 1718i 95 9/ 1813| 1907 2002 2096! 2191 2286 23801 24751 2569 2663 95 No. 0 1 2 3 1 4 5 6 7 8 9 Diff. 3I4~~ 5.. 1 162 TABLE XII. LOGARITHMS OF NUMBERS. No. 0 1 2 3 4 5 6 7 8 9 Diff. 460 662758 662852 662947 663041 663135 663230 663324 663418 6635121 663607 94 1 3701 3795 38893983 4078 172 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 680698680789 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 690285 690373 690462,690550 690639 690728 690816 690905 690993 89 1 1031 1170 1258 1347 1435 1524 1612 1700 1789 1877 88 2 1965 2053 2142 2230 2318 2406 2494 2583 2671 2759 88 3 2847 2935 3023 3111 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 8s 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 1 9833 9924 700011 700098 700184 700271 700358 700444 700531 700617 87 217007041700790 0877 0963' 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 5 3291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 6 4151 4236 4322 4408! 4494 4579 4665 4751 4837 4922 86 7 5008 5094 5179 5265 5350 5436 5522 5607 5693 5778 86 8 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 85 9 6718 6303 6388 6974 7059 7144 7229 7315 7400 7485 85 510 707570 707655 707740 707826 707911 707996 708081 708166 708251 708336 85 1 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 85 2 9270 9355 9440 9524 9609 9694 9779 9863 9948 710033 85 31710117 1710021 87 710371 710456 710540 710625 710710 710794 0879 85 4 0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 84 5 180)7 1892 1976 2060 2144 2229 2313 2397 2481 2566 84 6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3107 84 7 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 84 8 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 84 9 — 5167 5251 5335 5418 5502 5586 5669 5753 5836 5920 84 No o 1 I3 4 5 7 8 9 Dif. I No ~ ~ I O 1 2 3 - / QL I 6- 7 1 s | 9:'Diff. l| TABLE XIL LOGARITHMS OF NUMBERS. 163 No.] O i __ 1 3 4 5 6 -7 1 8 9 Diff. 520 716003 716087 716170i716254 71633 716421 715504 716583 716671 716754 83 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 83 2 7671 7754 7837 792;) 8003 8086 8169 8253 8336 8419 83 3 8502 8585 8668 8751 8834 8917 9000 9083 9165 9248 83 4 9331 9414 9497 9580 9663 9745 9828 9911 9994 720077 83 5 720159 720242 720325 720407 720490 720573 720655 720733 72o021 0303 83 6 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728/ 82 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 82 8 2634 2716 2798 2881 2963 3045 3127 3209 3291 3371 82 9 3456 3538 362) 3702 3784 3866 3948 4030 4112 4194 82 530 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 6646 82 3 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 81 4 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 81 5 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 6 9165 9246 9327 9403 9489 9570 9651 9732 9813 9893 81 7 9974 730055 730136 730217 730298 730378 730459 730540 730621 730702 81 8 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 81 9 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 540 732394 732474 732555 732635 732715 732796 732876 732956 733037 733117 80 1 3197 3278 3358 3438 3518 3598 3679 3759 3839 3919 80 2 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 3 4800 4880 4960 5040 5120 5200 5279 5359 5439 5519 80 4 5599 5679 5759 5838 5918 5998 6078 6157 6237 6317 80 5 6397 6476 6556 6635 6715 6795 6S74 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. 12301' 1309 1388 1467 1546 1624 1703 1782 1860 79 2 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 79 3 2725 2804 2382 2961 3039 3118 3196 3275 3353 3431 78 4 3510 3538 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 6868 6945 7(23 7101 7179 7256 7334 78 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 748188 748266 748343 748421 748498 748576 748653 748731 748808 74888) 77 1 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 2 9736 9814 9891 9968 750045 750123 750200 750277 750354 750431 77 31750508 75058617506631750740 0817 0894 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 2893 2970 3047 3123 3200 3277 3353 3430 3506 77 7 3583 3660 3736 3813 3889 3966 4042 4119 4195 4272 77 8 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 76 9 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 5701755875 755951 7,56027 56175 0 03 756180 756256 756332 756408 756484 756560 76 1 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 2 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 76 3 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 76 4 8912 8988 9063 9139 9214 9290 9366 9441 9517 9592 76 5 9668 9743 9819 9894 9970 760045 760121 760196 760272 760347 75 6769422 760476498760573760649760724 0799 0375 0950 1025 1101 75 i 7 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 75 8 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 75 9 2679 2754 2329 2904 2978 3053 3128 3203 3278 3353 75 No.0 1 2 3 4 5 6 7 1 8 9 Dif. 164 TABLE XII. LOGARITHMS OF NUMBERS. No. 0 2 3 4 5 6 7 1 8 9 Dff. 580 763428 76 303 7563578 76 63653 763727 76802 7682 61387 7-63952 764027 764 101 75 1 4176 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 5966 6041 6115 6190 6264 6338 74 4 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 74 5 71-6 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 9746 9820 9894 9968 770042 74 9 770115 770189 770263 770336 770410 770484 770557 770631 770705 0778 74 590 770852 770926 770999 771073 771146 771220 771293 771367 771440 771514 74 1 1587 1661 1734 1808 1881 1955 2028 2102 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 3560 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 7499 7572 7644 7717 7789 7862 7934 8006 8079 72 600 778151 778224 778296 778368 778441 778513 778585 778658 778730 778802 72 1 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 2 9596 9669 9741 9813 9885 9957 780029 780101780173 780245 72 3 780317 780389 780461 780533 780605 780677 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 721 6 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 72 7 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 71 8 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 9 4617 4689 4760 4831 4902 4974 5045 5116 6187 5259 71 610 785330 785401 785472 785543 785615 78568 785757 785828 785899 785970 71 1 6041 6112 6183 6254 6325 6396 6467 6538 6 6 680 71 2 6751 6822 6893 6964 7035 7106 7177 248 7319 7390 71 3 7460 7531 7602 7673 7744 7815 7885 7956 8027 8098 71 4 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 71 5 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 71 6 9581 9651 9722 9792 9863 9933 790004 790074 790144 790215 70 71790285 790356 790426 790496 790567 790637 0707 0778 0848 0918 70 8 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 70 9 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 792392 792462 792532 792602 792672 7927421792812 792882 792952 793022 70. 1 3092 3162 3231 3301, 3371 3441 3511 3581 3651 3721 70 2 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 70 3 4488 4558 4627 4697 4767 4836 4906 4976 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 7060 7129 7198 69 7 7268 7337 7406 7475 7545 7614 7683 7752 7821 7S90 69 8 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 9 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 630 799341 799409 799478 799547 799616 799685 799754 799823 799892 799961 69 1 800029 800098 800167 800236 800305 800373 800442 800511 800580 800648 69 2 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 691 3 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 69 4 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 68 5 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389'C8 6 3457 3525 3594 3662 3730 3798 3867 3935 4003 4(71 68 7 4139 4208 4276 4344 4412 4480 4548 4616 4685 4753 68 8 4821 4889 4957 5025 5093 6161 5229 5297 5365 5433 681 9 5501 5569 5637 5705 5773 6841 5908 5976 6044 6112 68 l No.! 0 1 2 1 3 4 5 6 7 i 8 9 Diff. TABLE XII. LOGARITHMS OF NUMBERS. 165 No. O 11 2 3 4 5 6 7 8 9 Diff. 6401806180! 8052 8 806316 83)5394 806451 9806519 806587 806655) 806723 806790 68 1 6358 69 6' 6994 7061 7129 7197 7264 7332 7400 7467 68 2 7535 7603 7670 7738 7806 7873 794 8008 8076 8143 68 3 8211 8279 8346 8414 8481 8549 861?; 8684 8751 8818 67 4 8S36 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 5 9560 9527 9694 9762 9829 9896 9964 810031 810098 810165 67 6 810233 810300 810367 810434 810501 810569 810636 0703 0770 0837 67 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 8, 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 91 2245 2312 2379 2i4:) 2512 2579 2646 2713 2780 2847 67 650812913 812980 813047 813114 813181 813247 813314 813381 813448 813514 67 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 67 2 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 67 3 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 66 4 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 66 5 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 66 6 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 66 7 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 66 8 8226 8232 8358 8424 8490 8556 8622 8688 8754 8820 66 9 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819544 819610 819676 819741 819807 819873 819939 820004 820070 80136 66 1 820201 820267 820333 820399 820464 820530 820595 0661 0727 0792 66 2 0358 0924 0989 1055 1120 1186 1251 1317 1382 1448 66 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 65 4 2168 22.33 2299 2364 2430 2495 2560 2626 2691 2756 65 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 6 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 65 7 4126 4191 4256 4321 4336 4451 4516 4581 4646 4711 65 8 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 65 91 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 65 670 826075 826149 826204 826269 826334 826399 826464 826528 826593 826658 65 1 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 65 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 65 3 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 64 4 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 64 5 9304 9368 9432 9497 9561 9625 9690 9754 9818 9882 64 6 9947 830011 830075 830139 830204 830261 830332 830396 830460 830525 64 71830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 64 81 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 61 9 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 1 6SO 183250 3832573 832637 832700 832764 832828 832892 832956 8.33020 833083 64 11 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 64 21 3784 3548 3912 3975 4039 4103 4166 4230 4294 4357 64 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 41 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 63 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 63 6 6324 6337 6451 6514 6577 6641 6704 6767 6830 6894 63 7 6957 7020 7033 7146 7210 7273 7336 7399 7462 7525 63 8 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 9 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690i833849 838912 833975!83903 839101 839164 839227 839289 839352 839415 63 1 9178 9541 96041 9667 9729 9792 9855 9918 99811840043 63 2 840106 840169 840232 840294 840357 840420 840482 840545 840608 0671 63 3 0733 0796 0359 0921 0984 1046 1109 1172 1234 1297 63 4 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 63 5 19351 2017 2110 2172 2235 2297 2360 2422 2484 2547 62 6 26091 2672 2734 2796 2859 2921 2983 3046 3108 3170 62 71 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 62 8 3355 3918 3930 4042 4104 4166 4229 4291 4353 4415 62 91 44771 4.A39 461 4664 4726 4788 4850 4912 4974 5036 62 No. 2 l 3 4 5 6 7 8 9 Diff. r~~~~~~~~~~~ 166 TABLE XII. LOGARITHMS OF NUMBERS. No. 0 1 2 3 1 4 5 6 7 8_ 9 Di 700 845098 845160 84522 845284 845346 845408 845470 845532 8455948-45656 62 1 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 62 2 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 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 60 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 863323 863382 863442 863501 863561 863620 863680 863739 863799 863858 59 1 3917 3977 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 7880 7939 7998 59 8 8056 8115 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 9877 9935 9994 870053 870111 870170 870228 870287 870345 59 2 870404 870462 870521 870579 0638 0696 0755 0813 0872 0930 68 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 58 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 6 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 58 7 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 8 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 9 4482 4540 4598 4656 4714 4772 4830 4888 4945.5003 58 750 875061 875119 875177 875235 875293 875351 875409 875466 875524 875582 58 1 5640 56981 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 571 6 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 7 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 57 8 9669 9726 9784 9841 9898 9956 88001318800701880127 880185 57 9 880242 880299 880356 880413 880471 880528 0585 0642 0699 0756 57 No. 0 11 2 3 4 5 6 7 8 9 DIff. TABLE XII. LOGARITHMS OF NUMBERS. 16' No.0 1 2I 3 4 5 6 7 8 9 DDiff. 760 880814 880871 80928 880985 881042 881099 881156 881213 881271 1881328 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 4285 4342 4399 4455 4512 4569 4625 4682 4739 57 7 4795 4852 4909 4965 5022 5078 5135 5192 5248 5305 57 8 5361 5418 5474 5531 5587 5644 5700 5757 5813 5870 57 9 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 56 770 886491 886547 886604 886660 886716 886773 886829 886885 886942 886998 56 1 7054 7111 7167 7223 7280 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 9638 9694 9750 9806 56 6 9862 9918 9974 890030 890086 890141 890197 890253 890309 890365 56 7 890421 890477 890533 0589 0645 0700 0756 0812 0868 0924 56 8 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 56 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 56 780 892095 892150 892206 892262 892317 892373 892429 892484 892540 892595 56 1 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 56 2 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 56 3 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 55 4 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 55 5 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 55 6 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 55 7 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 55 8 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 55 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 55 790 897627 897682 8977371897792 897847 897902 897957 8801218980671 898122 55 1 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 55 2 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 55 3 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 55 4 9821 9875 9930 9985 900039 900094 900149 900203 900258 900312 55 5 900367 900422 900476 900531 0586 0640 0695 0749 0804 085'9 55 6 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 55 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 8 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 54 9 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 54 800 0 903449031 3199 903253 903307 903361 903416 903470 903524 903578 54 1 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 21 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 54 3 4716 4770 4821 4878 4932 4986 5040 5094 5148 5202 54 4 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 54 5 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 6 6335 6339 6443 6497 6551 6604 6658 6712 6766 6820 54 7 6874 6927 6981 7035 7039 7143 7196 7250 7304 7358 54 8 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 54 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 54 8\0 908485 908539 908592 908646 908699 908753 908807 908860 908914 908967 54 1 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 54 2 9556 9610 9663 9716 9770 9823 9877 9930 9984 910037 53 3 910091 9101441 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 2859 2913 2966 3019 3072 3125 3178 3231 53 9 3234 3337 3390 3443 3496 3549 3602 3655 3708 3761 53 No. 0 1 2 3 4 5 6 79 Di f. 168 TABLE XII. LOGARITHMS OF NUMBERS. No. 0 1 2 3 4 5 6 7 8 9 Diff. 820 913814 913867 913920 913973 14029164079 914132 914184 914237 914290 53 1 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 53 2 4872 492' 4977 5030 5083 5136 5189 5241 5294 5347 53 3 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 53 4 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 53 5 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 3 6 6980 7033 7085 713S 7190 7243 7295 7348 7400 7453 31 7 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 521 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 521 9 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 52 830 919078 919130 91918391 919287 919340919392919444 919496 919549 52 1 9601 9653 9706 9758 9810 9862 9914 9967 920019 920071 52 2 920123 920176192022819202801920332 920920436 920489 0541 0593 52 3 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 52 4 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 52 5 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 52 6 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 52 7 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 52 8 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 52 9 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 52 840 924279 924331 924383 924434 924486 924538 924589 924641 924693 924744 52 1 4796 4848 4899 4951 5003 5054 5106 5157 5209 5261 62 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 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 51 5 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 51 6 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 51 7 783 7935 7986 8037 8088 8140 8191 8242 8293 8345 51 8 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 51 9 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 51 850 929419 929470 929521 929572 929623 9S9674 725 929776929827 929879 51 1 9930 9981 930032 930083 930134 930185 930236 930287 930338 930389 51 2 930440 930491 0542 0592 0643 0694 0745 0796 0847 0898 51 3 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 51 4 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 51 5. 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 51 6 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 51 7 2931 3031 3082 3133 3183 3234 3285 3335 3386 3437 51 8 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 51 9 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 51!860 934498 934549 934599 934650 934700 934751 934801 934852934902 934953 50 1 5003 5054 5104 5154 5205 5255 5306 5356 5406 5457 50 2 5507 5558 5608 5658 5709 5759 5809 5860 5910 5960 50 3 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 501 4 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 50 5 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 50 6 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 50 7 8019 8069 8119 8169 8219 8269 8320 83701 8420 8470 50 8 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 50 9 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 50 870 9395 19 939569 939619 939669 939719 939769 939819 939869 939918 939968 50 1 9400181940068 940118 9401683940218 940267 9403171940367 940417 940467 50 2 0516 0566 0616 0666 0716 0765 0815 0865; 0915 0964 50 3 1014 1064 1114 1163 1213 1263 1313 1362' 1412 1462 50 4 1511 1561 1611 1660 1710 1760 1809 1859 1909 1958 50 5 2008 2058 2107 2157 2207 2256 2306 23551 2405 2455 50 6 2504 2554 2603 2653 2702 2752 2801 28511 2901 2950 50 7 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 49 8 34951 3544 3593 3643 3692 3742 3791 38411 3890 -3939 49 93989 40381 4088 4137 4186 4236 4285 4335 4384 4433 49 No. O 1 2 3 4 5 6 7 8 19 Diff. TABLE XII. LOGARITHMS OF NUMBERS. 169 No. 0 1 2 3 4 5 6 7 s 9 Diff. 880 944483 944532 944581 944631 944680 944729 944779 944828 944877 944927 49 1 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 49 2.5469 551 5567 5616 5665 5715 5764 5813 5862 5912 49 3 5961 6010( 6059 6108 6157 6207 6256 6305 6354 6403 49 4 6452 6501 6551 6600 6649 6698 6747 6796 6845 6894 49 5 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 49 6 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 7 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 8 8413 t462 8511 8560 8609 8657 8706 8755 8804 8853 49 9 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 890 949390 ) 1 431 949488 949536 949585 949634 949683 949731 949780 949829 49 1 9878 9926 9975 950024 950073 950121 950170 950219 950267 950316 49 2 950365 950414 950462 0511 0560 0608 0657 0706 0754 0803 49 3 0851 0900 0949 0997 1046 1095 114 1192 1240 1289 49 4 1338 1336 1435 1-183 1532 1530 1629 1677 1726 1775 49 5 1823 1372 1920 1969 2017 2066 1111 2163 2211 2260 48 6 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 48 7 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 48 8 3276 3325 3373 31 3470 3518 3566 3615 3663 3711 48 9 3760 3803 3856 3905 3953 4001 4049 4098 4146 4194 48 900 954243 954291 951339 954387 9 54454484 954532 954580 954628 954677 48 1 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 48 2 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 48 3 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 48 4 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 51 6649 6697 6745 6793 6840 688S 6936 6984 7032 7080 48 61 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 48 71 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 48 8 8036 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 9 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 910 959041 959089 959137 959185 959232 959280 959328 959375 959423 959471 48 1 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 48 2 9995 960042 9600901 96013960185 960233 960280 960328 960376 960423 48 3 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 48 4 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 48 5 1421 1469 1516 1563 1611 1658 1706 1753 1801 184S 47 6 1895 1943 1990 2038 2035 2132 2180 2227 2275 2322 47 7 2369 2417 2461 2511 2559 2606 2653 2701 2748 2795 47 8 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 47 9 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 47 920 963788 963835 963882 963929 963977 964024 964071 964118 964165 964212 47 1 4260 4307 4354 4401 4448 4495 4542' 4590 4637 4684 47 2 4731 4778 4825 4872 4919 4966 5013 5061 5108 5155 47 3 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 471 4 5672 5719 5766 5313 5860 5907 5954 6001 6048 6095 471 5 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 47 6 6611 6658 6705 6752 6799 6345 6892 6939 6986 7033 47 7 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 47 8 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 47 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 47 930 96 3 9688 30 9688535576 968623 96670 968716 968763 96810 96856 96903 47 1 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 47 2 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 47 3 982 9928 9975 970021 97006 970114 970161 7 97 9 70207 970490300 47 4970347 970393970440 0486 0533 0579 0626 0672 0719 0765 46 5 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 46 6 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 46 7 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 46 8 22(03 2249 2295 2342 2338 2434 2481 2527 2573 2619 46 9 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 46 No. 0 1 2 3 4 5 6 7 8 9 Diff. 170 TABLE XII. LOGARITHMS OF NUMBERS. o. 0 1 2 3 4 5 6 1 7 s8 9 iDif 94(1973129773138 97 0 7363174 973320 979 93405973451 973497 9735431 46 1 3590 3636 3682 3728 3774 3'20 3866 39131 3959! 46(,15 46 2 4051 4097 4143 4189 4235 4281 4327 4374! 44201 4466 46 3 4512 4558 4604 4650 4696 4742 4788 48341 48801 4926, 46 4 4972 5018 5064 5110 5156 5202 5248 52941 5340( 538i' t1 5 5432 5478 5524 5570 5616 5662 5707 57531 57599 5845 46 6 5891 5937 5983 6029 6075 6121 6167 6212! 6258! 6304 146 7 6350 6396 6442 6488 6533 6579 6625 6671 6717 67631 46 6808 6854 6900 6946 6992 7037 7083 7129 7175 722u 46 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 950 977724 977769 977815 977861 977906 977952 977998 978043 978089 9781351 46 1 8181 8226 8272 8317 8363 8409 8454 8500 8546 85911 46 2 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 1 6 3 9093 9138 9184 9230 9275 9321 9366 9412 9457 95C3 46 4 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 5 980003 98004980094 980140 980185 1 923180276 980322 980367 980412 15 6 0458 0503 0549 0594 0640 0685 0730 0776 0821 S067 45 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 8 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 15 9 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226i 45 960 982271 982316 982362 982407 982452 982497 982543 982588 982633 982678! 1 1 2723 2769 2814 2859 2904.2949 2994 3040 3085 3';f 45 2 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 45 3 3626 3671 3716 3762 3807 3852 3897 3942 3987 40(32 45 4 4077 4122 4167 4212 4257 4302 4347 4392 4437 44,Y 15 5 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 6 4377 5022 5067 5112 5157 5202 5247 5292 5337 5382 1.5 7 5426 5471 5516 5561 5606 5651 5696 5741 5786 5'I 45 8 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 45 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 45 970 986772 986817 986861 986906 986951 986996 987040 987085 987130 987175 45 1 7219 7264 7309 7353 7398 7443 7488 7532 7577 76221 45 2 7666 7711 7756 7800 7845 7890 7934 7979 8024 806i 415 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 85141 45 4 8559 8604 8648 8693 8737 8782 8826 8871 8916 89601 45 5 9005 9049 9094 9138 9183 9227 9272 9316 9361 940:! 45 6.9450 9494 9539 9583 9628 9672 9717 9761 9806 98ot1, 44 7 9895 9939i 9983 990028 990072 990117 990161 990206 990250 9902941 44 8 990339 990383 990428 0472 0516 0561 0605 0650 0694 07.S' Il 9 0783 0827 0871 0916 0960 1004 1049 1093 1137 1,jz 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 291: 14 4 2995 3039 3083 3127 3172 3216 3260 3304 3348 73':i i4 5 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 6 3877 3921 3965 4009 4053 4097 4141 4185 4229 42^I1 14 7 4317 4361 4405 4449 4493 4537 4581 4625 4669 4"' 14 8 4757 4801 4845 4889 4933 4977 5021 5065 5108 5152 44 9 5196 5240 5284 5328 5372 5416 5460 5504 5547 5591 44 990 995635 995679 995723 995767 995811 995854 995898 995942 995986 996030 44 1 6074 61171 6161 6205 6249 6293 6337 6380 6424 6468! 44 2 6512 6555 6599 6643 6687 6731 6774 6818 6862 69i),^.4 3 6949 69931 7037 7080 7124 7168 7212 7255 7299 73431 44 4 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779 44 5 7823 78671 7910 7954 7998 8041 8085 8129 8172 839.:' i 6 8259 8303 8347 8390 8434 8477 8521 8564 8608 b6o2 44 7 8695 8739, 87821 8826 8869 8913 8956 9000 9043 9087 44 8 9131 9174 9218 9261 9305 9348 9392 9435 9479 95?2 4 9 9565 9609 9652 9696 9739 9783 9826 9870 9913 99__ _ 3 KN. 0 1 2 13 4 5 6 7 8 9 IDiff. TABLE XI1I. LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 172 TABLE XIII. LOGARITHMIC SINES, NOTE. THE table here given extends to minutes only. The usual method of extending such a table to seconds, by proportional parts of the difference between two consecutive logarithms, is accurate enough for most purposes, especially if the angle is not very small. When the angle is very small, and great accuracy is required, the following method may be used for sines, tangents, and cotangents. I. Suppose it were required to find the logarithmic sine of 5' 24" By the ordinary methb" we should have log. sin. 5' = 7.162696 diff. for 24" = 31673 log. sin. 5' 24"t 7.194369''ti 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' = 51'5 24" = 300"t: 324'. Hence sin. 5' 24/ 324 sin. 5' - 30, or log. sin. 5' 24"- = log. sin. 5' + log. 324 - log. 300. The difference for 24"f will therefore, be the difference between the logarithm of 324 and the logarithm of 300. The operation will stand thus:log. 324 = 2.510545 log. 300 = 2.477121 diff. for 24 33424 log. sin. 5' = 7.162696 log. sin. 5' 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 $ sin. A 3; A. Therefore, A = in 7, 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. 3'. log. sin. A 7.004438 log. sin. 3't 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 characteristica 9, 8, 7, and 6 stand respectively for -1, -2, -3, and -4. 174 TABLE XIII. LOGARITHMIC SINES, 0o 179" M. Sine. D. I. Cosine. D. lV. Tang. D. 1u. Cotang. I M. 0 Inf. neg. 0.000000 00 Inf. neg. Infinite. 60 1 6.463726 5017.17.000000'0 6.463726 5017.17 3.536274 59) 2.764756 293485.000000 ~.764756 293485.235244 58 3.940847 208231.000000 00.940847 208231.059153 57 4 7.065786 161517.000000 0 7.065786 161517 2.934214 56 5.162696 131969 00000.162696 837304 55 6.241877 7 9.999999.241878 11.758122 54 7.308824.999999 oo.308825 96654.691175 53 8.366-816 85254.999999.366817 8-.55.633183 52 9.417968:.999999.01.417970.582030 51 762.62 999.01 762.63 10 7.463726 68988 9.999998 01 7.463727 68988 2.536273 5Q 11.505118 62981.999998 i.505120 6281.494880 4 12.542906.999997.01.542909.457091 48 13.577668 53641.999997.577672 53642.422328 47 14.609853 4.999996 i.609857.390143 46 15.639816 46714.999996 o.639820 46715.360180 45 16.667845 4381.999995.667849 438.82.332151 44 17.694173 41372.999995.694179 41373.305821 43 18.718997 13.999991.719003 39136.280997 42 391.35.99999 01 391.36 19.742478 3.12017.742484.257516 41.01 371.28 20 7.764754 353 9.999993 01 7.764761 353.16 2.235239 40 21.785943 33672.999992 0.785951 336.73.214049 39 22.806146 32175.999991.806155 321.76 19345 38 23.825451.999990.825460 308.07 174540 37 24.843934.999989.843944.156056 36 25.861662 999989.861674283.90.138326 35 26.878695 23..999988 02.878708 23.9 121292 34 273.17.02 273.18 27.895085 263 23.999987 02.895099 263.25 104901 33 28.910879 25399.999986 02.910894 254.01 089106 32 29.926119.999985 0.926134 24.073866 31 245.38 5.40.99998 30 7.940842 23733 9.99998.02 7.940858 23735 2.059142 30 31.955082 22980.999982.955100 22982.044900 29 32.968870 22273.999981 02.968889 22275.031111 28 33.982233 21608.999980.982253 21610.01747 27 34.995198 20 999979 02.995219 20983.004781 26 35 8.007787 203 999977.02 8.007809 20392 1.992191 25 36.020021 19831.999976 0.020044 19833.979956 24 37.031919.999975.031945 19305.968055 23 38.043501 188.01.999973.02.043527 18803.956473 22 39.054781 183.25.999972 02.054809 18327.945191 21 40 8.065776 178.72 9.999971 02 8.06580617875 1.934194 20 1.076500 17442.999969 03.07653 7444.923469 19 42.086965.999968 03.086997.913003 18 43.097183 166.39 999966 03.097217 6642.902783 17 44.1071697.\.6 999964 -0.107203 16268.892797 16 45.116926 1 999963 03.116963 15911.883037 15 46.126471 55.66.999961.03.126510 15569.873490 14 47 135810 999959.135851 5241.864149 13 48.144953 149.24.999958 03.144996 14927.855004 12 49.153907 146.22.999956.153952 14625.846048 11 50 8.162681 14333 9.999954 03 8.162727 14336 1.837273 10 51.171280 14054.999952 03.171328 14057 828672 9 52.179713 13786.999950 03.179763 13790.820237 8 53.15390 53.187985 135.2.999948 03.188036 13532.811964 7 54.196102 132.80.999946 03.196156 132 80384 803844 6 55.204070 130.41.999944 03.204126 130.795874 5 56.211895.999942.211953 1 788047 4 57 21958128.1 12591 57.219581 128.999940 0.219641 1.780359 3 58.227134 12.999938.227195 12376 772805 123.72. 09 133.06 59.234557.64.999936.234621.765379 1 60.241855.999934.241921.758079 0 M. Cosine. D. I". Sine. D. I". Cotang. D. I". Tang. iM. 900 89 COSINES, TANGENTS, AND COTANGENTS. 175 1o ____178c M.j Sine D. 1'-. Cosine. D 1'. Tang. D. 1". Cotang. M. 0 8.241855 1163 9.999934 8.241921 11967 1.758079 60 1.249033 1176.999932..249102 772.750898 59 2.256094 1158.999929.256165 1..743835 53 3.263042 113.8.999927 04.263115 1142.736885 57 4.269881 1122.999925.269956 1-.730044 56 5.276614.999922..276691 1125.723309 55 110 50' 04 6110.54 5 6.283243 110.999920 04.283323.716677.5 108.83.04 108.87 7.289773 9.99918.289856 0.710144 5 107.22' 04 107.26 8.296207 1.66.999915 04.296292 00.703703 52 9.302546 104:13.999913 04.302634 lo4..697366 51'04.13.04 104.18 LU 8.308794 9.999910 8.308884 7 1.691116 50 102.66.04' 102.70 11.314954 1122.999907 04.315046 1126.684954 49 12.321027 982.999905 4.321122 987.678878 48 t3.327016 9.999902 0.327114 98.672886 47 14.332924 974.999899.,.333025 971.666975 46 15.338753 95..999897..338856'90.661144 45 9.86 3405 9599099 i6.344504 94.6.999894.0.344610 96.655390 44 17.350181 999891.350289 94.649711 43 18.355783 9.999888.05.355895 924.644105 42 i9.361315 91.3.999885 0.361430 90.638570 41 91.03'05 91'08 20 8.366777 90 9.999882 5 8.366895 89^ 1.633105 40 21.372171 8.0.999879 05.372292 88.5.627708 39 22.377499 872.999876.0.377622 877.622378 33 23.382762 867.999873..382889 8672.617111 37 24.387962.6.999870 05.388092 870.611908 36.'25.393101.999867.5.393234 84.606766 35 26.398179 8.6.999864 05.398315 8371.60165 34 27.403199 8-..999861..403338 276.596662 33 82.71.05 82 76 28.408161.7.999858.408304 182.591696 32 29.413068 886.999854 0.413213 809.586787 31 30 8.417919 7 9.999851 06 8.418068 800 1.581932 30 79.96'080 0202 31.422717 7909.99948 06.422869 7914.577131 29 32.427462 7823.999844 06.427618 7829.572332 2 33.432156 7740.999841 06.432315 7.56765 27 34.436800.999838.436962.56303 26 76.58.06 76.63 26 35.441394 7..999834.0.441560.6-.558440 25 36.445941 7'4.999831 06.446110 75.'.553890 24 37.450440 7422.999827.450613 7..54937 2 38.454893 7347.999824 06.455(70 7'.544930 22 39 999820.459301.999820481 540519 21 72'73 06 72'79 40 8.463665 0 9.999816 0 8.463349 72 1.536151 20 41.467935 71..999813 06.468172 713.531828 19 42.472263 70..999809.472454.527546 18 43.476493 691.999805.476693 6.23307 17 44.480693 6924.99901 06.480892 691.519108 16 45.484848 689 999797 06.485050 686.514950 15 16.488963 94 999794 07.48970 6801.51030 14 47.493040 63.999790 07.493250 67.8.506750 13 48.497078 66..999786 07 497293 66.7.502707 12 49.501080 6608.999782 07.501298 6615.498702 11 50 8.505045 9.999778 07 8.505267 6 1.494733 10 51.508974 89 999774 07.509200 6496.49080 9 52.512867 64'.999769'7.513098.486902 8 53.516726.07 64.39 53.516726 37.999765.0.516961 382.483039 7 54.520551 63.999761 07.520790 6.2.479210 6 55.524343 6.999757.524586 6272.475414 5 56.528102 6211 999753 07.528349 6218.41651 4 57.531828 1.999748 7.532080 1.467920 3 61 58' 07 61.65 8.535523 6.999744 0.535779 6.464221 2 59.539186.999740.539447.460553 1 60.55 9 07 60.62 60.542819 6..~..999735.543084.456916 0 M. Cosine. D. 1". Sine. D. 1". Cotang. I D.1". Tang. M. gln ssa i76 TABLE XIII. LOGARITHMIC SINES, 20 177' M. Sine. D. 1". Cosine. D. 1". Tang. D. 1I. Cotang. M. 0 8.542819 6004 9.999735. 8.543084 60.12 1.456916 60 1.546422 9..999731 07.546691 62.453309 5,9 2.549995 59.6.999726.8.550268 5914.449732 58 3.553539 5..999722.08.553817 586.446183 57 4.557054 58.'.999717.8.557336 58..442664 56 5.560540 57.65.999713 08.560828 57.439172 55 6.563999 57.19.999708 08.564291 5727.435709 54 7.567431 5.74.999704 08.567727 6.2.432273 53 8.570836 56-.999699..571137 5'.428863 52 9.574214 53.999694 0.574520 56 3:425480 51 10 8.577566 9.999689 8.577877 5552 1.422123 50 55.44 08 55.52 11.580892 55..999685 0.581208 5510.418792 49 12.584193.999680.584514 6.415486 48 54.60.08 54.68 792 13.587469 54..999675 08.587795 54 7.412205 47 14.590721 5.9.999670 08.591051 5387.408949 46 15.593948 53.999665 08.594283.405717 45 16.597152 53.0.999660.08.597492 53'0.402508 44 17.600332 52.61.999655.0.600677 5270.399323 43 18.603489 52..999650..603839 -32.396161 42 19 52.23 08 52.32 19.606623.999645.606978 54.393022 41 20 8.609734 9.999640 8.610094 1.389906 40 21.612823 51.49.999635 09.613189 51..386811 39 2 6181 51.12.9691.09 3 51.21 386811 39 22.615891 50.7.999629.0.616262 50.5.383738 38 23.618937 50.1.999624.619313 50. 380687 37 24.621962 50.6.999619 09.622343 50.15 377657 36 25.624965.999614.625352.374648 35 2 49.72 09 49.81 26.627948..999608.628340 49.'.371660 34 27 609 49.38. 09.6234 27.630911 49.8.999603 0. 631308 49.368692 33 28 49.04 9 09'74913 28.633854..999597 09.634256 8..365744 32 48.71.09 48.80 29.636776 483.999592 09.637184 4848.362816 31 308.6396 80 4.9995,86 30 8.639663 9.999586 8.640093 1.359907 30 31.642563 47.7.999581 09.642982 4784.357018 29 32.645428 4.999575'.645853 3.354147 28 33.648274 47. 2.999570 0.648704 47-2.351296 27 34.651102.1.999564..651537 491.348463 26 35.653911 46.82.999558.09.654352 46..345648 25 36.656702 46.52.999553.657149 46.342851 24 37.659475 4-62.999547 10.659928 46.3.340072 23 33.662230..999541.0.662689 46..337311 22 39.664968 455.999535 10.665433 4545.334567 21 40 8.6676S9 4 7 9.999529 8.668160 4 1.331840 20 41.670393 5..999524 0.670870 448.329130 19 42.673080 4.7.999518 10.673563 44..326437 18 43.675751 44-24.999512.10.676239 44.1 323761 17 44.678405 44.4.999506 10.678900 44.0.321100 16 45.631043 437.999500'1.681544 4380.318456 15 46.683665 44.999493.10.684172 43'5.315828 14 47.636272 4318 999487 10.686784 4328.313216 13 47:94998.10 3 48.688863 42 92.999481 10.69381 4303.310619 12 49.691438 26.999475.691963 4277.308037 11 50 8.693998 422 9.999469 0 8.694529 422 1.305471 10 51.696543'.999463.697081.302919 9 52.699073 42.7.999456 11.699617 42.28.3(033 8 53.701589 4 1.9.999450.1.702139.297861 7 54.704090 418.999443 11.704646 417.295354 6 55.706577 41.4.999437 11.707140 41.5.292860 5 56.709049 41.21 999431.11.709618 41 3.290382 4 67.711507 40.7.999424.1.712083 408.287917 3 58.713952 40.999418.1.714534 408.285466 2 59.716383 40...999411 1 716972 40.283028 1 60.718800 0.9.999404 11.719396.280604 0 M. Cosine. D.1. SIin. D. ". Cotang. D. 1.. Tang. M. ~~3 ^r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ COSINES, TANGENTS, AND COTANGENTS. 177 M. Sine D ". Cosine. D. 1'". Tang. D. 1". Cotang. M. 0 8.718800 40(06 9.999404 a 8.719396 4017 1.28060 60 1.721234 3984.999398 ~.721806 3'.278194 59 2.723595 3962.999391 ~'].724204 397.275796 58 3.725972 3941.999384 [.726588 3952.27:3412 57 4.728337 3919.999378 [.728959 3931.271041 56 5..730688 3898.999371 [.731317 3910.26683 55 6.733027 3877.999364 ].733663 3889.266337 54 7.735354 857.999357.735996 38'.264004 53 8.737667 3836.999350.738317'8.261683 52 9.739969 38:16.999343 12.740626 38:.259374 51 3'6] /.12 38.27. 10 8.742259 3796 9.999336 12 8.742922 38 1.257078 50 I1.744536 3776.999329 12.745207 3788.254793 49 12.746302'.999322.747479 252521 48 1.8 37:56.999322 12 37.68 2521 48 13.749055 377.999315 12.749740 3'.250260 47 14.751297 3717.999308 12.751989 3729.248011 46 15.753528 3698.999301 12.754227 3710.245773 45 16.755747 3680.999294 12.756453 92.243547 44 17.757955 3661.999287 12.758668 3673.241332 43 18.760151 3642.999279 12.760872 36|5.239128 42 19.762337 3624.999272 12.763065 3636.236935 41 20 8.764511 3606 9.999265 12 8.765246 3618 1.234754 40 21.766675 82.999257 812.767417 3600.232583 39 22.768828 3570.999250 12.769578 3...230422 38 23.770970 3.5.999242 2.771727 3..228273 37 24 30.53.12 35'65 24.773101 353.999235 1.773866 35..226134 36 25.775223 35.1.999227 /.775995 353.224005 35 26.777333 35.999220 13.778114 35.1.221886.34 27.779434 35.1.999212 13.780222 3.4.219778 33 28.781524 34..999205.782320.21760 32 29.783605 34.999197.78440 34.8.215592 31 34.51 9.13 34.64 30 8.785675 3 9.999189 3 8.786486 I44 1.213514 30 31.787736 3 ".999181.788554 343.211446 29 32.789787 34.12.999174 13.790613 34.1.209337 28 33/.13 9415 33.7918283 34.0.999166 13.792662.207338 27 34.793359 33.86.999158 13.794701 3383.205299 26 35.795881 33.70.999150 13.796731 3368.203269 25 36.797894..9914 37.797894 999142 13.793752 33.201248 24 7.7999897 |..999134 13.800763..1992.37 23. 3h 8.801392337.3..0 999126 1 39.80182 30/.999126.802765 /.197235 22 3.803876 3.999118:.804758.195212 21 I 32.93.13 33.07 40 8.305852 ", 9.999110 14 8.806742 329 119325- 20 41'.807819 32..7 41.807819 I.63.999102 14.803717 32.77.191283 19 42.1109777 32.4.999094 14.8106.3 3.62.1893117 18 1.8161726 32.34.999086 14.812641 32.4.187359 17 44.813667.999077 1.814589.185411 16 45.8155999069.1.816529 3.183471 15 46.8175232. 999061.14.81461 0 181539 14 47.819436.9 99053 4.820384..179616 13 48.8213 31.77 93904.14 82229 31.177702 1 49.823240 31.63.14 1.-.177702 1 31.49 903.1~4 82425 3163.175795 1 50 8.82.513 336 9.999027 I 8.826103 1.173397 10 51.827011 31 99909.827992 1.. 172003 9 52.828884 31.22.99900.829874 31.36.1:70126 8 53.830749 3095 999002.14.831748 31.09 I 168252 7 54.832607 30.5.998993 4.833613 3.96.166337 6 55.834456 30.82 99994.14 835471 30.96 164529 5 56.836297 30.6 998976.14 I837321 30.83.162679 4 57.833130 306 998967.15.839163 30.7.160837 3 58.83 30. 99956.15 840998 30.5.159002 2 59 *.841774 303.998950.15.842825 30..157175 1 8_ _.998941..844644 3.155356 0 M.9 Cosine...1 Coang. D.' ang. M 930 9 86 178 TABLE XIII. LOGARITHMIC SINES, to 1755 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 8.843585 9.998941 8.844444 3020 1.155356 60 1.845387 292.998932.15.846455 300 153545 59 2.847183 2980.998923 15.848260 299.151740 58 3.848971 296.998914.5.850057 29.8.149943 57 4.850751 29.5.998905 15.851846 29.7.148154 56 5.852525 29..998896.5.853628 29..146372 55 6.854291.998887.15.855403 29..144597 54 7.856049 29..998878..857171 293.142829 53 2919 5'3 8.857801 2908.998869.1.858932 29.2.141068 52 9.859546 296.998860.15.860686 291.139314 51 28.998860 29..1 10 8.861283 2o4 9.998851 8.862433 2900 1.137567 50 11.863014 2.99841.15.864173 28.8.135827 49 12.864738 2861.998832.15.865906 28.134094 48 13.866455 280.998823.1.867632 286.132368 47 14.868165 2..998813.16.869351 285.130649 46 15.869868..99804. 6.871064 2.12936 5 2525 16.12593 45 16.871565 21..998795 16.872770 28..127230 44 17.873255 2.6.998785 16.874469 2.2.125531 43 18.874938 2795 977.9876162 251.123838 42 19.876615 27.84.998766 16.877849 28'.122151 41 20 8.878285 27 7 9.998757 8.879529 1.120471 4 ) 21.879949 263.998747 16.881202 27'9.118798 39 22.881607 272.998738 16.882869 276.117131 38 23.883258 2742.99728 16.884530 278.115470 37 24.884903 27.3.998718 16.886185 2747.113815 36 25'.886542 27.998708 16.887833 2737.112167 35 26.888174 71.998699 6.889476 7..110524 34 30 8.894643 26 7 9.998659 17 8.895984 287 1.104016 30 31.896246 26.6.998649 17.897596 2677.102404 29 32.897842 26.998639'7.899203 2 7.100797 28 33.899432 26.51.998629.1.900803 6.6.099197 27 3. 2901019.17 26 8.097602 26 35.902596 262.998609 17.903987 2639.096013 25 36.904169 261.998599 17.9. 94437 0 24 37.905736.2998589 17.907147 2620.09253 23 38.907297 293.998578 7.908719 2-0.091281 22 39.908853 258.998568.910285 26.089715 21 40 8.910404 2575 9.998558 17 8.911846 25.92 1.08154 20 41.911949 25..998548 7.913401.08699 19 42.913488 25.6.998537.17.914951 25.085049 18 43.915022 25.47.998527 17.916495 265.083505 17 41 916550 5..998516.17.918034 25.56.081966 16 4.918073 252.998506 18.919568 25.47.08432 15 46.91991 2.99495.1.921096 25.38.078904 14 47.921103 25.21 998485.18.922619 25.29.077381 13 48.922610 25.99174 18.924136 2.075864 12 49.92112 24 94.998464:18.925649 2512.074351 11 50 8.925609 24 9.998453 18 8.92716 04 1.072844 10 51.927100 247..998442 8.928658 24.95 0713412 9 52.923587. 7.69 998431 1.930155 4.8.06984 8 53.9:30063 24.6.998421 18.931647 24.7.068353 7 54 931 4 11.914.98410.933134 2470.066866 6 55.933015 2443.998399 18.934616 24-62.065384 5 56.9.34481.998388 18.936093 24.063907 4 57.935942 243.998377 1.937565 24.53.062435 3 58.937398 24-1.998366 1.06939032 2437 6 2 3. 9 459.93850 21-.9985C355.9 9.059506 1 60.940296.998344.9041952'2.058048 0 38. Cosine. D.I'. S9e.: D. ". Cotang. D. 1". I Tang. M QQ^ ~~~= ~ ~ 1 COSINES, TANGENTS, AND COTANGENTS. 179 50 1740 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1'. Cotang. M. 0 8.940296 2403 9.993344 18 8.941952 2421 1.058048 6C 1.941738 23-.998333 19.943404 24'13.056596 59 2.943174 I23.998322.19.944852 24.0-.055148 58 3.944606 23./.998311 19.946295 2397.053705 57 4.916034 231.998300.947734 2.390.052266 56 5.947456 23.71.998289 19.949168 2382.050832 55 6.948874 2-6.998277 19.950597 2374.049403 54 7.950287 23.54.998266 19.952021 23674.047979 53 8.951696 23.40.998255 *19.953441 2;359.046559 52 9.953100 23.32.998243:19.954856 23:51.045144 51 10 8.954499 23 9.998232 19 8.956267 23.44 1.043733 50 11.9558594 23.27.993220.19.957674 23'36.042326 49 12.957284 23 1.993209.19.959075 2329.040925 48 13.958670 2310.993197.19.960473 2322.039527 47 1 4.960052 2.I0.998186' 1.961866 23.14.038134 46 15.961429 22.9.998174.19.963255 23.07.036745 45 16.962301 22.81.993163.19.961639 23.00.035361 44 17.964170 22,7I.998151.20.966019 2293.033981 43 18.965534 22'66.998139 20.967394 2286.032606 42 19.966393 2259.9 98128 20.963766 22:79.031234 41 20 8.963249 2252 9.998116 ~20 8.970133 22'72 1.029867 40 21.969600 22.45.998104 20.971496 226'.02504:39 22.970917 22.998092 20.972855 22o.027145 38 23.972239 2231.993080 20.974209 2251.025791 37 24.973623 22.24.998068 20.975560 2244.024440 36 25.974962 2217.998056 20.976906 2237.023)94 35 26.976293 2210.999044 *20.978248 2230.021752 34 27.977619 2203.99.3032 20.979586 2'24.020114 33 23.978941 21.97.998020..980921 217.019079 32 29.3025 2'/.930259 2190.998008.2.92251 2210.017749 31 30 8.91573 21.83 9.997996.0 8.983577 2204 1.016423 30 31.93283 21.77.997984.20.984899 21'97.015101 29 32.934189 2170.997972 20.96217 2191.013783 28 33.93.5491 2164.997959 20.987532 21'84.012168 27 34.936789.21.997947.2.938842 21'78.011158 26 33.938083 21.51.997935 21.990149 21.71.009851 25 36.99374 21.44.997922 21.991451 21'.008549 24 37.990660 218.997910 21.992750 21:007250 2.3 33.991943 2131.997897 21.994042 1 2152.00595 22 39.993222 21.25.997885 21 2146. 004663 21 40 8.994497 2119 9.997872 21 8.996621 2140 1.003376 20 41.995763 2.9997860 21 2134.002092 19 42.997036 210. 97847 21.999188 2127.000812 18 43.,993299 210.99735 21 9.000465 2121 0.999535 17 44.0099956 20.9.997822 21.00173 2..993262 16 45 9.000316 2.99709 21.003007 21.09.996993 15 46.002069 20.8 9977297.21.004272 1 203.99572 14 47.003318 20.82 7734 ~^,'.00"5534 210 3 43 9.0606792 20'91.993208 12 49.005305 20:46.997758:2.008017 20'5.991953 1t 50 9.007041 20.3a 9.997745.22 9.009293 20 0.990702 10 51.003278 205.997732 22.010546 20.9454 52.099510 20.24.997719 *22.011790 20-4.988210 8 53.010737 20.4.997706 22.013031 2063.936969 7 54.011962 20/3.997693 22.014263 205.985732 6 55.013182 202 9.997630 22.015502.6.9449 5 56.014400 2023.997667 22.016732 2045.9812368 4 57.015613 2.'.997654.22.017959 20'9.98932041 3 53.016324 20.17.997641.019183 234.980317 2 59.018031 20.12.99762S.22.020403 2028.979597 60.019235 0.0.997614..021620.978380 _0 M. Cosine. D. I". Sine. D. 1". Cotang. D. 1". Tang. M. 95 8"c 180 TABLE XIII. LOGARITHMIC SINES, 60 1731 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. I. 0 9.019235 2000 9.997614.22 9.021620 20.23 0.978380 60'2.021 26 20 23 1.020435 199.997601 22.022834 2017.977166 59 2.021632 1.9.997588 22.024044 2012.975956 58.022825 19.8 997574.025251 6.974749 57 19 84 ~ 22 20.06 4.024016 19..997561 22.026455 200.97355 56 5.025203 1973 997547 22.027655 19.972345 55 6.026386 997534 23.028852'.971148 54 7.027567 19.997520 23.030046 19.69954 53 8.028744 19..997507.031237 19.968763 52 19.57.23 19.79 9.029918 1 997493.032425 17.967575 51 19.51.23 19.74 10 9.031089 19.46 9.997480 23 9.033609 19.69 0.966391 50 1.032257 1941.997466 23.034791 1964.965209 49 12.033421 3.997452 23.35969 1958.964031 45 13.034582 19 997439 23.037144 1953.962,56 47 14.035741 19'.997425 3.038316 1948.961684 46 15.036896 19.2.997411 23.039485 1943.960515 45 16.038048 19.5.997397 23.040651 938.959349 14 17.039197 19..997383 23.041813 1933.958187 43 18.040342 19.0.997369 23.042973 1928.957027 42 19.041485 19 997.997044130.955870 41 19.00.23 19.23 20 9.042625 189 9.997341 2 9.045284 1918 0.954716 40 21.043762 18.-.997327 3.046434 1913.953566 39 22.044895 18.8.997313.047582 1908.952418 38 23.046026 18.8.997299.048727 1903.951273 37 24.047154 18..997285 24.049869 1898.950131 36 25.048279 1870.997271 24.051008 1893.948992 35 26.049400 86.997257..052144 1889.947856 34 27.050519 1860.997242 24.053277 1884.946723 73 28.051635 18.5.997228.054407 1879.945593 32 29.052749 1850.997214 24.055535 1874.944465 31 30 9.053359 1 9.997199 9.056659 1870 0.943341 30 31.054966 18.997185 24.057781 1865.942219 29 32.056071 1836.997170.24.058900 1860.941100 28 33.057172 18-3.997156.060016 1856.939984 27 34.058271 185.997141.061130 185.938870 26 35.059367 182.997127.2.062240 1-4.937760 25 18.22.997127 24 1846 36..060460 18.2.997112 24.063348 1842.936652 234 37.061551 18.17.997098 24.064453 1837.935547 23 40 9.064-06 799 9.997053 r 9.067752 1824 0.932248 20 41.065885 17.9.997039.25.068846 1819.931154 19 42.066962 17.997024.25.069938 18 930(2 18 43.06S036.997009.25.071027.928973 17 44.060107 1..996994.25.072113 186.927887 16 45.070176 17.8.996979.25.073197 186.926803 15 46.071242 1.996964 5.074278 1-8.925722 14 17.72,',,. 25 17.97 47.072306 17. 6.996949.25.075356 1793.924644 13 48.073366 176.996934.25.076432 179.923568 12 49.074424 17.9.996919 25.077505 1784.922495 1I 50 9.075480 175 9.996904 9.078576 17.80 0.921424 10 51.076533 1751.99688.25.079644 1776.920356 9 52.077583.99674 080710.919290 8 53.078631 1742.996858.25.081773 1767.918227 7 54.079676 38.996843.082833 1763.917167 6 55.080719 1734.996828.26.083891 179.916109 5 56.081759 1729.996812.26.084947 1755.915053 4 57.082797 1725.996797.26.086000 1751.914000 3 53.083832.996782.6.087050.912950 2 59.084864.996766.088098.911902 1 0.085894.996751 2.089144.1036 0 41.0658,,,',,17.17' 086914 17.43 931 6 0 I M. Cosine. D. 1". Sine. D. 1". Cotang D. 1". Tang. IM _9 ~83 COSINES, TANGENTS, AND COTANGENTS. 181 70 172' M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.085894 1713 9.996751 26 9.089144 9 0.910356 6) 1.086922 1709.996735 *26.090187 173.9098 13 59 2.037947 70.996720 2.091228 7.3.908772 58 17.05.26 17.31 3.08970 17.00.996701 26.092266 17.27.907734 57 4.039990 16.96.996683 *26.093302 7.2 906693 56 5.091008 16..996673 26.094336 17.905664 55 6.092024 16.88.996657 26.095367 1715.904633 54 7.093037 1684.996641 26.096395 17.1.90360. 53 8.094047 16.80.996625 26.097422 17.902578 52 9.095056 16.76.996610 26.098446 7 90154 51 10 9.019962 1fi73 9.996594 27 9.099468 1699 0.900532 50 11.097065 6.996578 27.100487.899513 49 16.69'996578.27 16.95 12.093066 665.996562 27 101504.893496 48 13.099065 16.6.996546 27.102519 16..897481 47 14.100062 16.61.996530 27.103532 168.896463 46 15.101056 16.5.996514 2.104542 168.895458 45 16.102048 16.49.996198 27.105550 1676.894450 44 17.103037 1646.996482 27.106556 16 893444 43 16.46'996482.278944 4 18.104025 1642.996465 27.107559.892441 42 19.105010 163.996449.108560 16.891440 41 16:.38.27 16.67 20 9.105992 34 9.996433 27 9.109559 6 0.890441 40 21.106973 16..996417 27.110556 16.5.889444 39 22.107951 1627.996400 27 111551 165.888449 38 23.108927 1623.996334 27.112543 16.5.887457 37 24.109901 16.19.996363 27.113533 647.886467 36 25.110373 16.996351 27.114521 4.8479 35 26.111842 162 996335 28.115507.884493 34 27.112309 1.996318 2.116191 16.3.883509 33 28.113774.996302.117472 1..88252 32 16.05.28 16.32 29.114737 15.996235 28.118452 162.881548 31 16:01.925.28 16.29 30 9.115698 158 9.996269 28 9.119429 162 0.880571 30 31.116656 15.94.996252 28.120404 1622.879596 29 32.117613 1590.996235 28.121377.878623 23 33.281187 16.18 33.118567.87.996219 28.122348 11.877652 27 34.119519 5.83.996202 28.123317 161.876683 26 35..120469 15.996185 28.124284 160.875716 25 36.121417 15.996168.2.125249 16.874751 24 37.122362 15.7.996151 28.126211 6.01.873789 23 33.123306 156.996134 28.127172.872328 22 39.124248 15.66.996117 28.128130 94.871870 21 40 9.125187 5.62 9.996100 28 9.129087 151 0.870913 20 41.126125 1559.996083 28.130041 15..869959 19 42.127060 15..996066..130994.869006 18 43.127993 15.5.996019.2.131944 158.868056 17 44.123925 15.54.996032 29.132893 15.8.867107 16 15.49. 39 15.77 45.129354 15.4.996015 29.133339 5..866161 15 46.130781 15.42.995998 29.134784 15.7.865216 14 47.131706 139.995980.135726 15.861274 13 48.132630 5..995963 29.136667.863333 12 49.133551 1.99596 29.137605 16.862395 11 15.32 995946.29 16 50 9.134470 1529 9.995928 29 9.138542 158 0.861458 10 51.135337 1526.995911 29.139476 1-s.860524 9 52.136303 1522.995894 29.140409 15.5 859591 8 53.137216 1519.995876 29.141340 1548.858660 7 54.138128 151.995859..142269 15..857731 6 55.139037 15.1.995941 29.143196 15.4.856804 5 56.139944.995823.144121 1.855879 4 15.09.29 15.39 57.140850 15.06.995806.145044 1536.854956 3 58.141754 5.03.995788 29.145966 132.854034 2 59 142655 15..995771 30.146885 12.853115 1 60.45.995753.147803555.99573.14703 852197 9 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1'. Tang. M. 970o 8 182 TABLE XIII. LOGARITHMIC SINES, 80 171C M. Sine. D. 1". Cosine.'D. 1 Tang. D. 1". Cotang. M. 0 9.143555 1497 9.995753 9.147803 1526 0.852197 60 1.144453 1493.995735.30.148718 15.2.8512,82 59 2.145349 14.9.995717 30.149632 15.850368 58 3.146243 1487.995699 30.150544 1520.84946 57 4.147136 1484.995681 30.151454.1.848546 56 5.148026 1481.995664..152363 5.847637 55 6.148915 478.995646 0.153269 1.846731 54 14'78.13 2 6 0 846 53 7.149302 14I. 995628.3.154174.0.84526 53 8.150686 142.995610.155077 1.844923 52 9.151569 14.995591.155978 1..844022 51 10 9.152451 1466 9.995573 9.156877.19 0.843123 50 11.153330 1463'.995555.0.157775 149.842225 49 12.154208 146.995537 3.158671 4.9.841329 48 13.155083 14-7.995519.^.159565 148.840435 47 14.155957 1454.995501 3.160457 14..839543 46 15.156830 15.995482 3.161347 14..838653 45 16.157700.995464..162236 14.8.837764 44 17.158569 14.995446 31.163123 147.836877 43 18.159435 1445.995427 31.164008 14.7.835992 42 19.160301 14.3.995409 43.164892 1470.835108 41 20 9.161164 1436 9.995390 3, 9.165774 147 0.834226 40 21.162025 1433.995372.1.166654 14.6.833346 39 22.162885 40.995353 1.167532 14.832468 38 23.163743 127 99334 31.168409 145.831591 37 24.164600 1424.995316 31.169284.830716 36 25.165454 14.2.995297.3.170157 1.5.829843 3 26.166397 1419.995278.3.171029 1..828971 34 27.167159 1416.995260.171899 14..82810 3 28 168008 3.995241 1.172767..827233 3 14.13 31 173634 14.44 29.168856 1410.995222 31.173634 1442.826366 31 30 9.169702 1407 9.995203 3 9.174499 1439 0.825501 30 31.170547 1405.995184.2.175362 14..824638 29 32.171389 1-02.995165 32.176224 143.823776 28 33.172230 1399.995146.2.177084 14.3.822916 27 31.173070 1396.995127 2.177942 14 2.822058 26 35.173903 13-.995108 32.178799 14..821201 25 36.174744 13.9.995089 32.179655 142.820345 24 37.175578 13 8.995070..180508 1.819492 23 33.176411 13-8.995051 I3.181360 4.818640 22 13.85 21417 17 39.177242 1383.995032 32.182211 1415.817789 21 40 9.178072 1380 9.995013'. 9.183059 2 0.816941 20 41.178900 1..994993.32.183907 14.9.816093 19 42.179726 13 75 9.994974 184752 7.815248 18 43.180551 1372.994955 32.185597 14..814403 17 44.181374 1369.994935.2.186439 14.0.813561 16 45.182196 1367.994916 32.187280 139.812720 15 46.183016 1'64.994896.3.188120 1397.811880 14 47.183834'6.994877.188958 4.811042 13 48.184651 1359.994857..189794 1..810206 12 49.185466 13 56.994838.3.190629 1389.809371 11 50 9.186280 1354 9.994818 33 9.191462 1386 0.808538 1 0 51.187092 1351.994798..192294 138.807706 9 52.187903 13 48'.9193124 131.806876 8 53.188712 3.4.994759 33.193953 13.806047 7 51.189519 1343 99.9947780 137.805220 6 55.190325 1341.994720.195606 1374.804394 5 56.191130 138 994700.196430.1.803570 4 57.191933 13'.994680.197253 13.802747 3 13.36'994680 33 13.69 58.192734 13..994660 3.198074 136.801926 2 59.193534 13..994640.3.198894 13.6.801106 1 60.194332'..994620.3 199713.800287 0 M. Cosine. D. 1". Sine. D. 1". Cotang. D.I". Tang. M. go8 810 COSINES, TANGENTS, AND COTANGENTS. 183 170C _. Sine D 1". Cosine. D. I". Tang. D. 1". Cotang. M. 0 9.1943332 132 9.994620 9.199713 1362 0.803237 60 1.195129 13.2.991600..200529 13..799471 59 2.195925 13.23.994580 0 34.201315 13'.798655 58 3.196719 3.2.991560 3.202159 13'.797841 57 1321 39 13754 9 5 4.197511 13.99449..202971 13 2.797029 5.198302 1316.994519 34.203782 13..796218 5. 6.199091 131.994499 34.204592 1.49.795403 54 7],.19979 [ 9.9.7 1347 7.19979 13..99479.205400 135.794600 53 8.200666 13. 0.994459 34.206207 1342.793793 52 9.201451 16.994438 134.207013 13:40.792937 51 10 9.202234 9.994418 3 9.207817 0.792183 50 11.203017 13.0.994398 3.203619 13.3.791381 49 12.203797 132.994377 34.209420 1333.790580 48 13.204577 1296.99437.210220.789780 47 14.205354 1294.994336 3.211018 13.2.788982 46 15.2!36131 12.9.994316 34.211815 1326.788185 45 16.206906 12.9.994295 *.212611 13.787339 44 17.2(7679 128.994274 4.213405 132.786595 43 18.238452 12..994254 35.214193 1319.785802 42 19.239222 1282.994233 36.214939 1317.785011 41 20 9.209992 12.80 9.994212 35 9.215780 13. 0.784220 40.21 10760 12.7.994191 *3.216568 1312.783432 39 22.211526 12.7.994171 35.217356 1.10.782644 33 23.212291 127.994150 3.218142 13.781858 37 24.213355 12.1.994129 3-.218926 1306.781074 36 23..21381 25.213818 12.'.994103'..219710 13..780290 35 26.214579 12.66.994037 35.220492 3.01.779503 34 27.215338.994066.221272 1.778723 33 28.216097..994045 3.222052 2.777943 32 29.216354 1265.994024 35.222331), 29.777170 31 30 9.217609. 9.994003 9.223607 2.92 0.776393 30 31.218363 12.5.993932'5.224332 12.9.775618 29 32.219116 12.5.993960.3.225155 12 8.774844 28 33.219363 12.50.993939 3.225929 1286.774071 27 34.220618 1248.993918 36.226700 1284.773300 26 35.221367 1246.993897 36.227471 12.772529 25 36.222115 1244.993875 38.223239 1279.771761 24 37.222361..9933.54 36.229097 127.770993 23 33.223606 12.2.993332.6.229773 127.770227 22 39.224349 1237.993311 36.230539 1273.769461 21 40 9.225092 12 9.993789 36 9.2313(2 127 0.768698 20 41.225333 99376.92323)65 129.767935 19 42.226573 12..993746.232826 -6.767174 18 43.227311 1.1.993725 6.233586 12.67.766414 17 44.2230 12.2.993703.234345.765655 16 45.223784 12.26. 399 30.3 45.228784 122.99:3631 36.235103 12.6.764897 15 46.229I318 12.22.993660:36.235359 12.8.764141 14 47.230252 2.2.99363.3.236614 125.763336 13 48.230934 12..993616 6.237363 I12..762632 12 49.231715 12.16.993594:36.233120 12.52.761880 11 50 9.232444 1214 9.993572 7 9.238372 125 0.761128 10 51.233172 12-2.993550 37.239622 2.4.760378 9 52.233399 2.1.993523.7.240371 12..759629 8 53.231625 12.0.993506 37.241118 1244.758332 7 54.233319 12.0.993184 37.211865 12..758135 6 55.236073 12.0.93162 7,242610 12.^.757390 5 12:03. -2.9931 56.236795 12..993440 37.243354 12.0.756646 4 57.237515 1.0.993418 37.244097..755903 3 53.2332~.5 11.99 12.36'.755903 3 53.2332.- o 5.97.993396.244839 2.755161 2 59.233953'1.9.993374 37.245579 12'..754421 1 63.2396- 1'.993351.2 6319 __~"".753631 0 M. Cosine. I D. 1". Sine. D. 1I". Cotang. D. 1". Tang. M. _.>.8.. 184 TABLE XIII. LOGARITHMIC SINES, 10 169 M. Sine. D. I". Cosine. D. 1". Tang. D. I". Cotang. M. 90 9.239670, 9.993351 9.246319 90.753681 60 192386 11.93.37 12.30 0.753681 60 1.240386 11.9.993329.7.247057 12.2.752943 59 2.241101 11.9.993307..247794 12.2.752206 58 3.241814 1.8.993284 *.248530 122.751470 57.242 52611.87.37 12.24 24252.24226 993262..249264 1222.750736 56 5 243237 11.86.51 - l~ 5.243237 183.993240.249998 2.750002 55 6.243947.993217.3.250730 12.0.749270 54 7.244656 11.1.993195'.251461 121.748539 53 8.245363 11.7.993172 38.252191 12-1.747809 52 9.246069 1177.993149 38.252920 12.3.747080 51 10 9.246775,, 9.993127 9.253648 0.746352 50 11.73.38 12.11 1.247478,,.993104'3.254374 1'0.745626 49 12.248181 11.6.993081.3.255100 1207.744900 48 13.248883 11.67.993059 38.255824 120.744176 47 14.249583'11.6.993036 38.2.56547 12.0.743453 46 15.250282 11.63.993013 38.257269 12.0.742731 45 16.250980 11.61.992990.257990 2'00.742010 44 17.251677..992967 8.258710 11 9.741290 43 18.252373 11.5.992944.3.259429 11.'.740571 42 19.253067 11.56.992921 3.260146 119.739854 41 20 9.253761 1 9.992898 38 9.260863 1192 0.739137 40 21.254453 115.992875 38.261578 1.9.738422 39 22.255144 1.50.992852.9.262292 11.9.737708 38 23.255834 11.4.992829..263005 1187.736995 37 24.256523.46.992806.263717 /'8.736283 36 25.25721 11.44.992783.264428 1. 8.735572 35 26.257898 11.4.992759 3.265138 1.81.734862 34 27.258583 i.41.992736 39.265847 11.'.734153 33 28.2592631 11.3.992713 9.266555 11.7.733445 32 29.259951 11.37.992690.267261 1176.732739 31 30 9.260633 9.992666 39 9.267967 11. 0.732033 30 31.261314 11.33.992643 9.268671.72.731329 29 32.261994 1131.992619 3.269375 1170.730625 28 33.262673 1 -0.992596 3.270077 1169.729923 27 34.263351 2.992572 3.270779 11.6.729221 26 35.264027 11.2.992549..271479 11.6.728521 25 36.264703 11.2.992525 39.272178 11.6.727822 24 37.265377 -.992501.272876 11.6.727124 23 38.266051 11.2.992478.^.273573.726427 22 39.266723 11 20.992454 40.274269 11 58.725731 21 40 9.267395 117 9.992430 40 9.274964 0.725036 20 41.268065 1115.992406.4.275658 11.5.724342 19 42.263734 1.992382 40.276351 11.5.723649 18 43.269402 11.1.992359.40.277043 11.5.722957 17 44.270069 11.12.992335 40.277734 115.722266 16 45.270735 11.10 992311 40.278424 11.5.721576 15 46.271400 11.08.992237 40.279113 11.8.720887 14 47 11.06.40 11.46 47.272064'1.0.992263 40.279801 1..720199 13 42 11.05'.40 11.45 48.272726 11 1.992239 40.230488 14.719512 12 49.273388 110.992214 40.281174 1141.718826 11 50 9.274049 1099 9.992190 40 9.281858 0.718142 10 51.274708 10.992166 40.282542 1138.717458 9 52.275367 109.992142 40.283225 113.716775 8 53.276025 10..992118 41.283907 11.3.716093 7 54.276681 10.9.992093.41.284588 133.715412 6 10.92.4113 55.277337 10.1.992069 4.285268.714732 5 56.277991.992044.285947 1.714053 4 57.278645 0.9.992020.4.286624 11.3.713376 3 10.87.275 /41 11.28 I 58.279297 10.87.991996 41.287301 11.2.712699 2 59.279948 10.86.991971 41.287977 11.2.712023 1 60.280599 1.991947.288652.711348 0 M. Cosine. D. 1". Sine. D.11". Cotag. D. 1. Tang. M. 00oo 79G COSINES, TANGENTS, AND COTANGENTS. 185 Iio 1 68S M. Sine. D. 1". Cosine. D. 1". Tang. D. I". Cotang. M. 0 9.280599 1082 9.991947 1 9.2S8652 11,23 0.711348 60 1.281248 1081.991922 41.289326 1122.710674 59 2.281897 1079.991897 41.289999 11.2.710001 58 3.282544 1077.991873 1.290671 1118.709329 57 4.283190 1076.99184: 41.291342 1 1.708658 56 5.283836 074.991823 41.292013 -.707987 55 6.284480 1072.991799 41.292682 1114.707318 54 7.285124 1071.991774.293350 112.706650 53 8.285766.991749 41.291017 1.705983 52 9.286408 16.991724 42.294684 119.705316 51 10 9.287048 1 9.991699. 9.295349 1 0.704651 50 11.287688 1064.991674 42.296013 1 60.703987 49 10.64.42 11.06 12.288326 1.991649 -42.26677 11(.703323 48 13.288964 061.991624' 297339 |.702661 47 14.289600 1,.991599 2.298001 I.701999 46 15.290236 10.59.991574 2.298662!.7013:38 45 16.290870 10.58.991549 /.299322.00.700678 44 17.291504 10.56 991524 29990 98.700020 43.292137 10.55.991498 2.300638.699362 42 19.292768 10.991473 2.301295 0.698705 41 10.51.9 2 10.93 20 9.293399 1050 9.99144 42 9.301951 1 0.698049 40 21.294029 1048.991422 42.302607 10.69739 39 22.294658 1.991397 42.303261 1.696739 38 23.295286 10.45.991372.303914 87.696086 37 24.295913 104.991346 42.304567 18.695433 36 25.296539 10.991321..305218 i.694782 35 10.42.43 104 61 34 26.297164 104.991295.3805869 1'.694131 27.297788 [103.991270 ~4.306519 ].693481 33 28.298412 10.991244.307168 108.692832 32 29.299034 10.991218 43.307816 108.692184 31 30 9.299655 1034 9.991193 9.308463 1077 0.691537 30 31.300276 1033.991167.309109 076.690891 29 32.300895 1031.991141.309754 7 690246 28 33.301514 1030.991115 4.310399 1073.689601 27 34.302132 10.991090 ].311042 107.688958 26 35.302748 1.991064' 4.3116S5 107.688315 25 36.303364 I.9910386.312327 17 687673 24 10.25'9 312 10.68 37.303979.991012 4.3! 968 10.6.687032 23 38 ~.304593.23.990936.33608.3136 0.686392 22 10.22.43 10.65 39.305207 102.990960.314247 685753 21 10.20.43 10.64 40 9.305819 10 9.990934 9.314885 10 0.65115 20 41.306130 17.990908.315523.684477 19 10.17.44 10.61,' 42.307041 1016.990882 -11.316159..6.683841 18 43.307650.990355,316795 1.682570 17 44.308259 101.990829.317430.682570 16 45 308867 1012.990803.318064 10.55.681936 15 46.309474.990777.318697.681303 14 47.310080 100.990750.31933.680670 13 10.09.9W724, 44 319330 10.53 12 48.310685 1007 0724.319961.680039 12 49.311289 1006.990697 44.320592 10.50.679408 11 50 9.311893 1004 9.990671. 9.321222 1048 0.678778 10 51.312495 1.990645.321851 j'4.678149 9 52.313097.990618.322479 0.677521 8 53.313698.990591.323106.676894 7 54.314297 98.990565 -4.323733 10.43.676267 6 55.314897.990538.324358 4.675642 5 56.315495 996.990511.324983'40.675017 4 57.316092.990485.325607'.674393 3 58.316689 9.990458.5.326231 103.673769 2 59.317284 991.990431 {.326853 1037.673147 1 60.317879.990404.327475.672525 0 M. Cosine. D.. Sine. D. ". Cotang. D. Tang M 1010o 78 186 TABLE XIII. LOSARIT13MIC SINES, 120 167. M. Sine. D.. Cosine. D.1 Tang. D. 1". Cotang. M. 0 9.317879 990 9.990404 9.327475 10.35 0.672525 60 1.318473 9..990378.4.328095 033.671905 59 2.319066.990351.328715 032.671285 58 3.31968.990324 45.329334 031.676666 57 4.320249 984.990297 4.329953 029.670047 56 5.320840 983.990270.5.330570 10 2.669430 55 6.321430 9.990(243 5.331187 10.2.668813 54 7.322019 98.990215.331803 025.668197 53 8.32Z607 979.990188.332418 1024.667582 52 9.323194 77.990161 45.333033 1023.666967 51 10 9.32370 9 76 9.990134 9.333646 1 0.666354 50 11.324366 9.990107.334259 1'.665741 49 12.324950 93.90079 46.334871 19.665129 48 13.325534 972.990052 46.335482 1017.664518 47 14.326117 70.990025 46.336093 101.663907 46 15.326700 69.989997 46.336702 loi..663298 45 16.327281 968.989970 46.337311 101.662689 44 17.327862 966.989942 46.337919.142.662081 43 -'42 9.66.46 10.123 18.32842 965.989915 46.338527 o1.661473 42 19.329021 964.989887 46.339133 110.660867 41 20 9.329599 962 9.989860 46 9.339739 0.660261 40 21.330176 961.989832 46.340344 107 659656 39 22.33075.3 9.989804 6.340948 1006.659052 38 23.331329 58.989777 46.341552 1oos.658448 37 9.524.331903.989749 46.342155 1003.657845 36 25.332478 956.989721 46.342757 1002.657243 35 26.333051.989693 4.343358 i'oo.656642 34 27.333624 3.989665 343958 ioo.656042 33 29.334767 5.989610 7.345157 9.654843.3 30 9.355337 99 9.989582 9.345755 996 0.654245 30 31.335906 948.989553.4.346353 99.653647 29 32.336475 946.939525 47 346949 9.93.665351 28 33.337043.989497 347545.652455 27 34.337610. 989469.348141.651859 26 35.338176.989441.7.348735 9.65126 25 36.338742 941.989413 47.349329 9.650671 24 37.339307 9.989385..349922 87.650078 23 38.339871.989356..350514 9.649486 22 39.340434 9.989328:47.351106 9.85.648894 21 40 9.340996 936 9.989300' 9.351697 9 0.648303 20 18 339 9.31.8 9.794 41.341558.989271 4.352287 9.647713 19 42.342119.989243 9.352876 98.647124 18 9.24.48 9.71 43.342679 92.989214.353465 9..646535 17 44.343239 93.989186.8.354053.645947 16 45.343797.989157.48.354640.645360 15 46.344355 99 9891248.355227 9..644773 14 47.344912 97.989100.48 355813 97.644187 13 48.345469 6 989071 4.356398 643602 12 49.346024.989042.48.356982.6430186 II.50 9.3466579 24 9.989014 9.357566 9 72 0.642431 10 51.34715349.988985 48.358149 70.64181 9 52.34517687.988956.358731.641269 8 53.348240.988927.48.359313 96.640687 7 54.34792 919.988898 5.359893 9.640107 6 9.35 9.14.49 9.6 55.349343 917.988869 48.360474 96.639526 5 9 351540 1 988753.362787 9.6.637213 56.349483 99888410.3363 99.63847 43 60.3520.88 13.988724 49.363364.636636 9 M. Cosine. D. I". Sine. D. ". Cotang. D. 1". Tang. M. 09.80 TO:11-1~~~~~~9455 0oi. 6 5 4 2 4 530. D yrCoaa.Tag,~ 47 —------.339069.9 COSINES, TANGENTS, AND COTANGENTS. 18] 130 166C M. Sine. D. 1". Cosine. D. 1" Tang. D. 1". Cotang. M. 0 9.3528 9.11 9.988724 9.363364 960 0.636636 60? ^1 ^ ^.49 9.6 0.4 9.60.352635.988695.49.36390.636060 59 2.353181 909.988666.3641515 958.63545 58 3.353726.988636.365090.634910 57 4.354271 907 988607 4.335664 95.634336 56.3545 9.05 578.366237 9.5.633763 55.355358 904.988548 4.366810.6319 54 7.3559019.367382 52.632618 53 8.356443 9.0.98889.49.37953 95.632047 52 16.35611907.988459.3799. 9.356934 901.9836524 50.631476 51 10 9.3524 899 9.988430 5 9.369094 9 0.630906 50 8.9 98401.3696639 9.48.63337 49 12.38603 897 938371 370232 47.629768 48 13.359141 8.933342.370799.629201 47 14.359678 91832.371367.628638 46 16 3675 8.94.9 0 943.62750 44 17.361237 8.991.9382373064626936 43 18.361822 890.988193.373629 940.626371 42 19.3612356.37193 9:39.625807 41 20 9.362889 8 9.988133 9.374756 0.625244 40 21.363422 887.988103.o.375319 9..62461 39 22.363954 86.98073.375881.6241119 3 33'369761.50 9.4. 23.364485 84.988043.376442.623558 37 24.365016 83.9803 50.377003.622997 36 125.365546 882.9837953 s.377563.622437 35 26.366075 881.937953.378122 931.621878 34 27.38664 880.987922 o.3786881.621319 33 23.367131 79 937892.379239 929.620761 32 29.367659 87.987862.3979797.620203 31 30 9.363185 876 9.987832 51 9.330354 927 0.619646 30 31.36711 937801.380910 926.619090 29 32.369236 874.937771.331466 9.618534 2 33 369761 873.987740.332020 924.617980 27 31.370235 872.987710 i.392575 9.617425 26 35.370308 871.987679 ~.383129.616371 2 36.371330.987649.383632 9.616318 24 37.371852 869 7618 1.384231 9.20.615766 23 39.372394.987557.385337.614663 2 5833.:66.108.35. 6597 40 9.373414 86 9.987526 51.33588 0.614112 20 41.373933 8.65.351 9.17.613562 19 42.374452 63.987465.3697 15.63013 43.374970 862 s7434.387536 14.61241 17.2 6.3781 2 2 34 44.37587 81.957403 5.383034 12.611916 16 45.376003 60.97372.36231 9.611369 1 46.376519.987341 52.389178.610822 14 8.59.52 9.10 47.377035.93731 5.389724.610276 13 4S.377549.987279 52.390270 908 609730 12 49.378063.5.937248 52.390815 90.609185 50 9.378577 855 9.937217 52 9.391360 0.608640 10 51.379089 53.97186 52.391903 608097 9 52.379601 52.937155 52.392447.:607553 53.380113 8.51.987124 52.392939 903.607011 7 5.334 850.987092 52.393531 902.606469 6 55.33113 849.98701 52.394073 901.605927 5 56.331643.937030.394614:605386 3 57.342152 7.96998 52.395154 899.604846 2 63.332661 846.936967.395694.604306 2 59.33163.96936.396233 7 1 8.45.52 8.97 0 60.383675.936904.396771.603229 0 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1". Tang. M. 103V 7a: 188 TABLE XIII. LOGARITHMIC SINES, 4I-o 1650 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang M. 0 9.383675 4 9.986904 9.396771 96 0.603229 60 1.34182 843.986873 3.397309 96.602691 59 2.384687 8.4.986841 53.397846 95.602154 58 3.385192 841.986809 3.398383 894.601617 57 4.385697 840.986778.398919 893.601081 56 5.386201 839.986746..399455 92.60045 55 6.386704 8. 986714.39990 1.600010 54 8.33.53 8.91 7.387207 83.986683.400524 89.599476 53 8.337709 8.3.986651 3.401058 8.89.598942 52 9.388210.986619.401591.598409 51 8.3;5.53 8.88 10 9.383711 9.986587 9.402124 7 0.597876 50 11.39211 8..986555.402656.597344 49 12.389711 8.3.986523 5.403187 886.596813 48 13.390210 831.986491 53.403718 88.596282 47 14.390708 8.3.986459.5.404249 8.8.595751 46 15.391206 8..986427.5.404778 88.595222 45 16.391703 8.9.986395.405308 88.594692 44 17.392199 8.28.936363 54.405836 8.594164 43 18.392695 8.2.986331 5.406364 8..593636 42 19.393191 825.986299.406892 87.593108 41 8.25.54 8.78 20 9.393685 824 9.986266 9.407419 8 77 0.592581 40 21.394179 82.986234 4.407945 76.592055 39 22.394673 8.2.986202.408471 87.591529 38 23.39,516 8.22 8.75 23.395166..986169.408996 875.591004 37 24.39565 8.2.96137.54.409521 874.590479 36 25.396150 8.2.986104 5.410045 7.589955 35 26.396641 8.19.986072.410569 87.589431 34 8.18.54 8.72 27.397132 8.8.986039 5.411092 8.588908 33 28.397621 8.98607 54.411615 87.588385 32 29.398111 8.16.98.412137.587863 3 8.15.54 8.69 30 9.398600 9.985942 9.412658 8 0.587342 3 31.399088 8.1.98909.5.413179 67.586821 29 32.399575 8.1.985876.413699 66.586301 28 33.400062 8.1.985843'5.414219.585781 27 34.400549 8.11.985811.55.414738 865.585262 26 35.401035.985778.55.415257 86.584743 25 36.401520.9.98745.55.415775 864.584225 24 37.402005 8.08.985712 55.416293 8.63.583707 23 38.402489 8.07.985679 55.416810 8.62 583190 22 39.402972 80.985646 55.417326 8.61 582674 21 8.05.55 8.60 40 9.403455 8.4 9.985613 9.417842 859 0.582158 20 41.403938 8.3.985580.418358 858.581642 19 42.404420 8.985547.418873 87.581127 18 43.404901 8.0.985514..419387 5.580613 17 46.406341 7'.985414.5.420927 8.54.579073 14 47.4032.01.98'8054 47.406820 7.97.985381.56.421440 853.578560 13 48.40(7299.985347 56 421952.578048 12 49.407777.985314.422463.577537 1 7.96.951.56 8.51.V), 9.40832.54 9.985280 9.422974 50 0.577026 10 5.40731.985247.56.42484 849.576516 9 2.409207 7.9 98523 56.423993 84.576007 8 i:1.49632 7.985180.424503 8.5749 7 54.410157.9.985146 56.425011 84.574989 6 7.9 6.56 8.4 5 1 ).410632 790.985113.56.42519 46.574481 5 55.411106 7 985079.6.426027.573973 4 I 57.411579 88.985045.6.426534 44.573466 3 53.412052 787.985011.56.427041 43.572959 2 59.41-2524 86.984978.6.427547.572453 1 60.412996:.984944.56.428052..571948 0 AM. Cosine. ID. " Sine. D. Ilf. Cotang. D. 1". Tang. M. 4O4-0,y0 COSINES, TANGENTS, AND COTANGENTS. 189 15c 1640 M. Sine. D. I". Cosine. D.I". Tang. D.I". Cotang. M~ 0 9.412996 785 9.984944 56 9.428052 842 0.571948 60 1.413467 784.984910.7.428558 841.571442 59 2.413938.984876!..429062 840.570938 58 3.414408 783.984842.429566 8.570434 57 4.414878 I.984808 I..430070 83.569930 56 5.415347 78.984774.430573 8.569427 55 6.415815 7.984740 I..431075 83.568925 54 7.416283 1-8.984706.431577'.568423 53 7.79 7'4. 7 8.36o 8.416751 778.984672.432079 835.567921 52 9.417217.984638.432580 8.567420 51 10 9.417684 9.984603 9.433080 33 0.566920 50 11.418150.984569 1.433580 833.566420 49 12.418615 7.984535..434080 832I.565920 48 13.419079 /.984500 57.434579 831.565421 47 14.419544 7.984466 /.435078 830.564922 46 15.420007 772.984432 5.435576 829.564424 45 16.420470.984397.436073 8.563927 44 17.420933.984363 ~.436970.3430 43 18.421395 76.984328.437067 8.2.8633 42 19.421857 7:68.984294.437563 8.26.562437 41 7.68.58 8126 20 9.422318 7. 9.984259 8 9.438059 8 25 0.561941 40 21.422778 7.984224.438554.561446 39 22.423238 7.984190 58.439048 824.560952 38 23.423697 984155 439543.560457 37 24.4 2.1.658 82 25.424156.984120.440036.559964 36 25.424615 7.63.984085 58.440529 8.22.559471 35 26.425073 7.62.98400.441022 8.21.558978 34 27.425530 7.6.984015.441514 820.958486 33 28.425987.983981.442006.557994 32 29.426443 7.61 58.442497 8.19.557503 31 7.60.58 8:18 30 9.426899 75 9.983911 / 9.442988 81,7 0.557012 30 31.427354 /.983875 5.443479 816.556521 29 3-2 /.427809.5 983840.5.443968 816.556032 28 33.428263.983805.444458 8.555542 27 34.428717..983770'9.444947 81.555053 26 35.429170.983735.445435 8.13.554565 25 36.429623.983700.445923.554077 24 78.49623.59 8.13.553889 23 37.430075 752.983664.9.446411..553589 23 38.430527 752.983629.446898 811.653102 22 39.430978; 8.983594 9 -44738 10.552616 21 40 9.431429 9.983558 9.447870 809 0.552130 20 41.431879.983523.448356 809.551644 19 42.432329.983487 ~.448841 8.551159 18 43.432778.983452.449326.55067k 17 44.433226.983416 -.449810 8.0.550190 16 7.45.59 45.433675.983381.450294 8.549706 15 46.434122. 983345.450777 80.549223 14 47.434569.983309.451260 8I0.548740 13 48.435016 44.983273 60.451743 8.03.548257 12 49.435462 7.43 23 60.452225 803.5475 50 9.435908 74 9.983202' 6 9.452706 0.547294 10 51.4.36353 7.42.93166 60.453187 8.02.546813 9 52.436798 7.983130 60.453668 800.546332 8 53.437242 740.983094 60.454148 800.545852 7 54.437686.983058 60.454628 7.545372 6 55.438129.983022.455107 7.544893 5 56.438572.982986 60.455586 79.544414 4 57.439014..982950 60.456064 1.543936 3 7.36.982914".60 7.97.543458 2 58.439456..982914 0.456542 ".543458 2 fO.439897.982878 60.457019'.52981 1. 60.440338..982842.457496.542.504 0 CosIne. C D. 1". Sine. I D. 1". Cotang. D. 1". I Tang. M. t050 7&c 190 TABLE XIII. LOGARITHMIC SINES, 160 1163 M. Sine. D 1". Cosine. D. 1'. Tang. D. 1". Cotang. M. 0 9.440338 9.932842 9.4074 i 0.542504 60 7.34.60 7.99 1.440778.3.98-2805.6.457973 7.94.541)27 59 2.441218 732 9 9.982769 4449 7.9.541551 58 3.441653 73.9a2733'6.4589 5 7.92.541075 57 4.442096 731.982636 61.459400 791.540600 56 5.4425.35 7.. 9266'.6.459375 7.9.540125 5. 6.442973.26' 5 6.442973 729.9326s21 61.460349 7.5:39651 54 7.44340 7.23.92587 61.460323.8.539177 53 8.4447..982551 61.461297 7.538703 52 9.44424 7..7 92314 6.461770.533230 51 10 9.444720 7'26 9.932477 61 9.462242 7. 0.537758 50 11.445155.25.982441 61.462715 7.,.5372-5 49 12.445590 7.2.982404 61.4631S6 7.6.536814 48 13.446025 724.932 i67 61.46:36.8 7.8.536:344 47 14.446459 723.982331.61.464128 7.8.535372 46 15.446893 72.98291 6.464599 78.535401 45 16.447326 72.98-2257 61.465069 783.534931 44 17.447759 7.2.982220 *62.465539 7.8.534461 43 18.448191. 982183 6.466008 7.8.533992 42 19.448623 7..932146:62.466477 7.81.533523 41 7.191 20 9.4490.54 7 1 9.992109 62 9.466945 0.533055 40 21.449485 7.7.982072 62.467413.532587 39 22.449915.1.9s32035 62.4678-0 7.7.532120 38 23.450345 716.931993 62.468317 778.531653 37 24.4.50775.981961 2.463814.531186 36 25.451204 7.14.91924.6.469280 7.7.530720 35 26.451632 713.9318S6 62.469746.76.530254 34 27.452060 713.981849 2.470211 7..529789 33 28.452188 7.13.931312.2.470676 7..529324 32 29.452915 711.981774 62.471141 7.528859 31 30 9.453.342;10 9.931737 62 9.471605. 0.523395 30 31.453768.9700 31.453763 7.0.981700 62.472069 772.527931 29 32.45-1194 7.0.931662 63.472532 7.71.527468 28 33.454619 70.931625 63.472995 7.527005 27 34.455014 7.'.981587.63.473457 70.526543 26 35.455469 7.07.931549 63.473919 7.6.526081 25 36.455393 706.981512.6.474331 76.525619 24 37.456316.0.981474.3.474842 7.6.525158 23 33.456739 70.9314.36 6.475303 76.524697 22 39.457162 4.91399..475763 7..524237 21 7.04.63 7.67 40 9.457584'A 9.981361 3 9.476223 0.523777 20 41 7.03.43 7.66 41.453006 7..981323 *63.476633 7 66.523.317 19 42.458427 7.02.981235..477142 765.522858 18 43 7.01.63.472 43.458348 7.01.981247 63.477601 7.6.522399 17 44.459268 7. 16 44.45968 7.0.981209 63.478059 7.6 521941 16 45.459683 6.9.981171.6.478517 7..521483 15 46.460108 6.9 981133 63.478975 7.6.521025 14 47.460527 6.8 981095.6.479432 7.6.520563 13 43.460946..931057 -.47989 7.61 520111 12 4 6.97.64 761 3 12 49.461364 696.981019:64.480345 76.519655 11 50 9.461782 9.980981 64 9.480801 0.519199 10 51 6.96.64 7.59 51.462199 6.9.980942 64.481257 7.5.518743 9 5`2 6.91..8 09.04 52.462616.930904.481712 75.518288 8 53.463032.980366.482167.517833 7 5-1 6.93.64 7.57 54.463443 6.93.980327.6.482621.77 517379 6 55.463364 6.92.90789 6.48:3075 7.5.516925 5 6.464279.9.980750..483529 7..516471 4 57.461694 6.9.980712 64.483982 7 55.5164018 3 55 6.90 9.6 64..516018 3 5.46.510.980673.484435 7.515565 2 59.465522.980635.424837 7..515113 1 60.465935 _9_.980.96.485339.53.514661 0 M. Cosine. I D. 1'1. Sine. D. 1. Cotang. D. 11. Tang. M. 806 o COSINES, TANGENTS, AND COTANGENTS. 191 I7'o __ ______ 1622 M. Sine. D 1l". Cosine. D. 1'. Tang. D. 1". Cotang. M. 0 9.465935 6 8 9.930596 9.485339 3 0.514661 60 1.46634S 68.980553 64.485791. 2.514209 59 2.466761 8.930519 6.486242 7..513753 58 3.467173 68.930480..486693 7.5.513307 57 6 86..65 4 7.51 56 4.467585 68.980442 65.487143.512357 5 5.467996 65.980403. 4.487593.512407 6.463407 6.8.930364.6.488043 7..511957 54 7.463817 6.93035.6.488492 7.511503 53 8.469-227 6.8.980236.6.488941 74.511059 52 9.469637 6.8.980247 65.489390 7.4.510610 51 10 9.470046 6 9.980208 9.439833 7. 0.510162 50 11.470455 81.980169.6.490286 7.46.509714 49 12.470863 68.980130 6.490733 7..509267 43 13.471271 6..980091.5.491180..508820 47 6.79.65 7.44 14.471679 6..98002 6.491627 4.503373 46 15.47206 980012 492073.507927 45 16.472492 979973 65.492519 7.507481 44 17.472898.979934 66.492965.507035 43 18.473.304 76.9795.493110.506590 42 19.473710 75.979855 66 49334 41.506146 41 20 9.474115 6 9.979816 66 9.494299 40 0.505701 40 21.474519 74.979776.494743.505257 39 22.474923 67.979737 66.495186.504814 38 23.475327 6..979697 6.495630.504370 37 6.72.66 7.,8 [ 24.475730 7.979658.496073 7..503927 36 25.476133 6.71.979618 66.496515 7.503485 35 26.476536 6.7.979579.496957.36.503043 34 27.476938.979539 66 497399 36.502601 33 28.477340 69.979499.49784'.502159 32 23.477741 68.97959 66.498282 7 34.501718 31 30 9.478142 667 9.979420 66 9.498722 7,, 0.501278 30 31.478542 667.979380..499163.500837 29 32.478942 66.979340.6.499603 7.500397 28 33.479342 66.979300 67.500042 73.499958 27 34 479741 6.65.979260 67.50041 7.3.499519 26 35.480140 66.979220 67 500920 73.499080 25 36.480539 66.979180 67.501359 73.498641 24 37.480937 63.979140 67.501797 7..498203 23 38.481334 66.979100 6.502235 7..497765 22 39.481731 6.61.979059.67 502672 7.28.497328 21 40 9.482128.6 9.979019 67 3.503109 7 8 0.496891 20 41.432525 66.978979 67 503546 27.496454 19 42.482921 69.978939 67.503982 77.496018 18 43.483316 59.978898.6.504418 72.495582. 17 44.483712 6.5.978858 67.504854 726.495146 16 45.44107 6.5.97881 7.50529 7.25.494711 15.434501 6.978777 67.505724 72.494276 14 47.44395 6.56.978737.6.506159 24.49334 13 43.485289 6.5.978696'6.506593 72.493407 12 49.485632 6.5.978655 68.507027 723.492973 1 50 9.46075 54 9.978615 6 9.507460 7 Ci492540 10 51.486467 6.4.978.574'6.607893 72.492107 9 52.486860.978533'6.5033-26.491674 8 53.487251 653.978493 68.508759 720.491241 7 54 437613 52.978452.6.509191 7.490309 6 55.483034 652.978411 68.509622 7..490378 5 56 6.51.6' 7.19 9 56.483424 650.978370.6.510054 71.489946 4 57.48814 6.50.978329 6.51045 71.489515 3 58.489204 6.978283 68.510916 71.489034 2 9.489593 649.978247 -6.511346 717.488634 1 60.49932.978206.511776 ~.488224 0 M. Cosine. D.1". Sine D. ". Cotang. D. 1. Tang. M. 107M3 72C 192 TABLE XIII. LOGARITHMIC SINES, 180 161l M. Sine. D. IC. Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.489982 48 9.978206 68 9.511776 1 0.488224 60 1.490371 6..978165.512206 716.487794 59 2.490759 4.978124.69 512635 71.487365 58 ~41 6.46.69 7.15 14 3.491147 646.978083 *69.513064 714.486936 57 4.491535 645.978042 69 513493 714.486507 56 5.491922 645.978001'..513921.486079 55 6.492308 644.977959.69.514349 13.485651 54 7.492695 64.977918.69.54777 1.485223 53 8.493081 6.97877 69.515204 712.484796 52 9.493466 42.977835 515631.484369 51 6 42.69 7.11 10 9.493851 9.977794 9.516067 0.483943 50 11.494236 641.977752 69.516484 7.10.483516 49 12.494621 6.4.977711.69.516910 7..483090 48 13.495005 6.39 977669 69.517335 7.482665 47 14.495388 639.977628 69.517761.482239 46 15.495772 638.977586 69.518186 7.0.481814 45 16.496154 63 97.977544 70 518610 70.481390 44 17.496537 37.977503.19034.07.480966 43 18.496919 6.977461 70.519458 7.0 48052 42 19.497301 6.36.977419.519882.480118 41 20 9.497682 9.977377 9.520305 0.479695 40 21.498064 6.34.977335 70.520728 7.0.479272 39 22.498444 6.977293 70.521151.478849 38 23,498825 33.977251 70 521573.478427 37 24.499204 63.977209.21995.478005 36 6.33.70 7.03 25.499584 632.977167.822417 03.477583 35 26.499963 31.977125.70.22838 7.02.477162 34 27.500342.977083 70.53259.47741 3 6.31.70 r 701 33 28.500721 630.977041 70.52360 701.476320 32 29.501099 6.30 976999.524100.475900 31 30 9.501476 29 9.976957 9. 24520 0.475480 30 31.501854 628.976914 71.524940.9.475060 29 32.502231 628.976872.71.525359 6.9.474641 28 33.502607 62 976830 71.525778.474222 34.502984.976787 71.526197 69.473803 26 35.503.360 6.27.71 6.97 35.503360 626.976745 71.526615 67.473385 25 36.503735 625.976702 71.527033 6.9.472967 24 37.504110.976660.527451.472549 23 3 6.25.71 6.96 38.504485 62.976617 1.527868 6..472132 22 ii a. 7 0 4 4 65..47175 21 39.504860 624.976574 2825.471715 21 40 9.505234 623 9.976532 71 9.528702 6 0.471298 20 41.505608 622 976489 71.529119 94.470881 19 42.505981 22.976446 71.529535 93.470465 18 43-.506354 6 976404.529951.470049 17 6.21.7i 6.93 44.506727 21.976361 71.530366.469634 16 6.21.71 6.92 469219 15 45.507099 620 76318 72.530781.469219 15 46.507471 9.976275 72.531196 91.468804 14 47.507843 61.976232 72.531611.468389 13 4.19 6232.72 6.90 48.508214 619 976189 72 532025.467975 12 49.508585 618 6.90 49.5085.85 1.976146 72.532439 6..467561 i11 6.18.72 6.89 50 9.508956 6 17 9.976103 72 9532853 689 0.467147 10 51.509326 6.976060 2.533266..466734 9 52.509696 6..976017 72.533679 6.88.466321 8 6.16.72 6.88 53.510065.975974.534092 68.465908 7 54.510434 15.975930 72.534504.465496 6 55.610803 61.975387 7.534916 68.465084 5 6.14 6.86 56;511172 6.775844'72,35328.464672 4 57.511540.975800 72 535739 66.464261 3 58.511907 6.13 75757 72.536150 6.85.463850 2 59.512275 6.12.975714 72.536561 68.463439 1 60.512642 612.975670 ~.536972 6.4.463028 0 M. Cosine. D. 1". Sine. D. 1". Cotang.. Tang.. M. 1080 y COSINES, TANGENTS, AND COTANGENTS. 193 190 1600 M. Sine. D.1". Cosine. D.1". Tang. D. 1". Cotang. M. 0 9.512642 611 9.975670.73 9.536972 684 0.463028 60 6.11 6.84III 1.513009.975627.537382 83.462618 59 6.1 1.7 3 61-.459347 51 2.513375.975583.537792 63.462208 58 3.513741 97.38202 6.461798 57 6.09.73 6.82 6. 3 4.514107 609.97496.538611 682.461389 56 5.514472 608.975452.539020 681.46098 5 6.514837 608.975408 9.539429 681.460571 7.515202 607.975365.539837 680.4601635 53 8.5155665321 540245.459755 52 9.5159-30 6:06.975277:73.540653 6:79.459347 51 10 9.5162941 6.0 9.975423 3 9.541061 679 0.458939 50 1.516657 05.975189 73.54146 678.458532 49 12.517020 604.975145.541875 678.458125 48 13.517382 04 975101.542281 677.457719 47 14.517745.975057.54688 677.457312 46 6.0 7.73 1.7 3 ^ ^ ^ 15.518107 603.975013.543094 676.456906 5 16.518468 02.974969.5499.456501 44 17.518829 6.02.974925.543905 675.46095 43 18.519190 601.974880 7.544310 675.455690 42 20 9.519911 600 9.974792 7 9.545119 674 0.454881 40 21.520271.974748.545524.454476 39 22.520631.974703.545928 673.454072 38 23.520990 974659 74.546331 6.45369 37 24.52139 59.974614 7.546735 6 72.453265 36 25.521707. 974570 7.547138 6.7.452862 35 5.97'974570.74 6.71 26.522066 9.974525 7.547540 671.452460 34 27.522424 596.974481 4 6.554794319 70.452057 33 28.522781.59 974436:7.54835 670.451655 32 29.52138.974391 75.548747 669.45123 31 30 9.523495 9.974347 9.549149 669 0.450851 30 31.523852.974302.4955 668.450450 29 32.524208 5.9742857.549951 6.45049 28 33.524564.974212.550352 667.449648 27 34.524920 592.974167.550752 667.449248 26 35.52275 92 974122.551153 67.448847 25 36.525630 591.974077 7.551552 666.448448 24 37.525984 5:90.974032 75.551952 666.448048 23 38.56339 5 90.973987 75.552351 6.665.447649 22 39.526693 973942 7.5527509.447250 21 6.89 475 6.65 40 9.527046 9.973897.553149 0.446851 20 41.527400 5.89.75 6.64.446452 19 5.88 9752.75 6.64 42.57753 88 973807.553.5.446054 18 43.528105 587.973761.554344 663.445656 17 44.528458 87 973716 76.554741 662.4450259 416 45.528810 586.973671 76.555139 662.444861 15 46.529161 586 97362 76.555536 661.44444 14 47.529513 585.973580 76.55933 661.444067 13 48.529864 585.973535 76.556329 660.443671 12 49.530215 584.973489:76.556725 660.443275 1 50 9.530565 583 973444 76 9.557121 659 0.442879 10 51.530915 973398 76.557517 69.4424831 52.531265 82.93352 76.557913 59.44207 53.531614 582.973307 76.558308 658.441692 7 M. G,5e. D. I". Sine. D. I".C552g. D. 6.73 54.531963 81 973261 76.558703 58.441297 6 55.532312 5.973215 76 559097 57.440903 5 56.532661 580.973169.559491 57.440609 4 57.533009.973124 76.559885 656.440115 3 58.533357.973078.560279 6.56.439721 2 59.533704.973032.560673 655.439327 1 60.534052.972986.561066.438934 0. Coine. Sine. D.D1. Cotng. D. Tang. M. 1090 TOG 194 TABLE XIII. LOGARITHMIC SINES, 200 1590 M. Sine. D. 1". Cosine. D. 1". Tang. D. ". Cotang. M. 0 9.534052 78 9.972986 9.561066 6.55 0.4:38934 60 1.534399 5.78.972940.561459 54.43,541 59 2.534745 77.972894.561851 64.438149 58 3.535092 5..972848.562244.4.437756 57 4.535438 76.972802.7.562636 6.437364 56 5.535783 576.972755.77 56302 53.4:6972 55 6.536129 5.972709 77.56:319.5.436581 54 7.536474 75.972663 77.563811 652.436189 53 6.52 8.536818 r.972617.564202.435798 52 9.537163.972570.564593.435407 51 5.74.77.64593 6.51 10 9.537507 3 9.972524 9.564983 0 0.435017 50 11..537851 5..972478.7.565373.5.434627 49 12.538194 57.972431.7.565763 6.50 434237 48 13.538538 571.972385 78.566153 6.49 43387 47 14.538880 571.972338 78.566.542 6.9.433458 46 15.539223 70.972291 78.566932 6.4.433068 45 16.539565 570.972245 8.567320 6.48.432680 44 17.539907 5..972198 78.567709 6.4.432291 43 18.540249 569.972151 78.568098 6.47.431902 42 1 9.540590 568.9726.47 19.540590 5.68.972105 78.568486 646.431514 41 20 9.540931 56 9.972058 78 9.568873 6 0.431127 40 21.541272 567.972011 78.569261 6.46.430739 39 22.541613 56.971964 78.569648 64.430352 38 23.541953 56.971917 78.570035.5.429965 37 24.542293 566.971870 78.570422.44.429578 36 25.542632 565.971823 78.570809 644.429191 35 26.5427.971971776 78.571195 6.4.428805 34 27.543310 5.971729..571581 6.4.428419 33 28.543649 56.971682.7.571967 6.43.428033 32 29.543987 5663.971635.79.572352 6.42.427648 31 30 9.544325 5 63 9.971588 79 9.572738 6 42 0.427262 30 32.545000 62.971493.573507 641.426493 28 33 545338 561.971446.573892 640.426108 27 34.54.5674 5.1.971398.7.574276 ^.425724 26 35.546011 5.60.971351..574660 40.425340 25 5.60 3 6.43 424956 24 36.546347.971303.575044.424956 24 37.546683..971256 79.575427 639.424573 23 38.547019 9.971208 79.575810 6.38.424190 22 39.547354 5.58.971161 79.576193 6:38.423807 21 40 9.547689 9.971113 9.576576 3 0.423424 20 5.68.79 6.37.42.824 20 41.548024.971066 80.76959 6.423041 19 42.548359 5.971018 80.577.341 6.3.422659 18 43.548693 56.970970.577723.422277 17 44.549027 55.970922 80.578104 36.421896 16 45.549360 5.970874 80.578486 6.35.421514 15 46.549693.970827 80.578867 6.35.421133 14 47.550026 r.970779 80.579248 6.34.420752 13 48.5550359 5.970731.579629 6..420371 1 49.550692.970683.58(009 6.419991 11 5.54.80 6.34 50 9.551024 9.970635. 9.580389 6 0.419611 10 51.551356. 970586 80.580769 6.3.419231 9 52.551687 552.970538 80.581149 632.418851 8 53.552018 552.970490.80.581528 6..418472 7 54.552349 5.5.970442.5S1 907 6.3.4 18(93 6 55.552680.970394 81.58266 63.417714 5 56.553010.970315.52665.417335 4 7.553341.970297 58.583(44 63.416956 3 58.553670.5.970249..583422 30.416578 2 59.55 5.49.970200 81.58.380 630.416200 1 60.554329.97152..584177..415823 0 5. Costne. D. I.. 8ine. D. 1". Cotang. D. P. Tang. M. 110o 090 53. 521 COSINES, TANGENTS,, AND COTANGENTS. 195 2 1o 1580 M. Sine. D. 1". Cosine. D. 1". Tang. D. I". Cotang. M. 0 9.554329 548 9.970152 81 9.584177 629 0.415S23 60. ~ 5418::81 6:.29.S5 1.554658 5..970103 81.534555 29.415445 59 2.55497.4.970055.81.584932.28.415068 58 3.555315 7.970006 81.585309 6..414691 57 4.55.643 5.46 969957 81.585686 6.28.414314 56 5.555971 5.6.969909 1.586062 6..2.413938 55 6.556299 5..969S60 8.586439 27.413561 54.556626.8 9698611 8.426105 7. 566-26 5.45.969811 81.586815 6.2.413185 53 8.556953 5..969762.587190 6..412810 52 9.557280.969714 81.587566 26.412434 51 5.44:81 6.26 10 9.557606.44 9.969665.82 9.587941 625 0.412059 50 11.557932 5.969616 82.588316 6.25.411684 49 12.558258 543.969567 82.588691 6.24.411309 48 13.55858:3 542.969518 82.589066 6.24.410934 47 14.558909.558909 969469 89440 24.410560 46 15.559234 5.41.969420 82.589814 6.23.410186 45 16.559558 5.4.969370 82.590188 6.3.409812 44 17.559883 5.4.969321 82.590562 6.22.409438 43 18.560207 540.969272 82 590935 6.22.409065 42 19.560531 539.969223 82.591308 6:22.408692 41 20 9.560855 r. 9969173 82 9.591631 2 0.408319 40 21..561178.969124.592054 6.407946 39 22.561501.3. 969075 8.592426 620.407574 38 23.561824.3.969025 2.592799 20.407201 37 24.562146 537.968976 83.593171 6.20.406829 36 25.562468 5..968926 83.593542 6.406458 35 26.562790..968877 83.593914 61.406086 34 27.563112 5.36 968827.594285 61.405715 3 28.563433 5.3 968777 83.594656 61.405344 32 29.563755 968728.96.595027.404973 31 30 9.564075 9.968678 83 9.595398 6 0.404602 30 31.564396 5.968628 8.595768 17.404232 29 32.564716'34.968578.3.596138 61.403862 28 33.565036.33.968528.83.596508 616.403492 27 34.565356.33 968479 83.596878 616.403122 26 35.565676 5.3 968429 83.597247 616.402753 25 36.565995 5..968379.3.597616 615.402384 24 37.566314 3.96329 83.597985 6..402015 23 38.566632 5.31.968278..598354 614.401646 22 39.566951 30.968228 884.598722 6:14.401278 21 40 9.567269. 9.968178 84 9.599091 6 3 0.400909 20 41.567587 5.2.968128 8.599459 1.400541 19 42.567904 52.968078 84.599827.400173 18 43.568222 52 968027 84.600194 6.1.399806 1 44.56539 528.967977.600562 61.399438 16 45.568856 28.967927.60929 612.399071 15 46.569172 5.2.967876..601296 61.398704 14 47.569488 527.967826 4.601663 6.1.398337 13 43.569804 5.2.967775.602029 61.397971 12 49.570120 5.26.967725 84.602395 6:10.397605 1 50 9.570435 25 9.967674. 9.602761 10 0.397239 10 51.570751 2 967624.603127 6..396873 9 52.57166 5.2.96773.8.603493 609.396507 8 53.571380 524.967522 85.603858 69 396142 7 54.571695 524.967471.85.604223 6.0.395777 55.572009 5.2 967421.604588 08.395412 5 56.572323 3.967370.60953.395047 4 57 572636 5.2.967319.605317 6.0.394683 3 58 5729.50 522.967268.85 605682 607.394318 2 59.573263 52.967217 85.606046..393954 1 60.573575.967166.8.606410 6.06 393590 M: Cosine. D. 1". Sine. D. 1. Cotang. D. 1". Tang. M. 111o 68 196 TABLE XIII. LOGARITHMIC SINES, ~x ao ___________________________ _ 157' M. Sine. D. 1". Cosine. D. ". Tang. D. I". Cotang. M. 0 9.573575, 9.967166 8 9.606410 606 0.393590 60 1.573888 5.2.967115.5.606773 606.393227 59 2.574200 5.0.967064.8.607137 60.392863 58 3.574512 520.967013 85.607500 6.5.392500 57 4.574824 5..966961..607863 6.392137 56 5.575136 5.1.966910.8.608225 604.391775 55 6.575447 5..966859..608588 604.391412 54 7.575758 5 8.966808.8.608950 6..391050 53 8.576069 517.966756 86.609312 603.390688 52 9.576379 7.966705.609674.390326 51 5.17.~'86' 7 6.03 10 9.576689 9.966653 9.610036 6 o 0 0.389964 50 11.576999.966602 6.610397 6.389603 49 12.577309 1.966550.610759 02.389241 48 5.16.86 6.02 13.577618 5.1.966499 86.611120 6.0.388880 47 14.577927.55 966447 86.611480 601.388520 46 15.578236 5.966395..611841 60.388159 45 16.57545 5.14.966344 86.612201 60.387799 44 17.578853 5.14.966292 86.612561 600.387439 43 18.579162 54.966240.612921 60.387079 42 5.13.86 6 6.00 19.579470 13.966188.613281.386719 41 20 9.579777 9.966136' 9.613641 0.336359 40 21.580035 5..966085 7.614000 5.98.386000 39 22.580392 5.1.96603.3 87.614359 5.385641 38 23.580699 5.1.965981.87.614718 5.98.385282 37 24.581005 1 1.965929.7.615077 59.384923 36 25.581312 5.1.965876 87.615435.384565 35 26.581618 5..965824 7.615793.7.384207 34 27.581924 5.0.965772.8.616151..383849 33 25.582229 5.09.965720 87.616509 56.383491 32 5.87.15 5.96 3813 29.582535 59.965668.6686.68383133 31 5.09 87 5.96 30 9.582840 8 9.965615.7 9.617224 5 0.382776 30 31.583145 5.08.965563 87.617582..382418 29 32.583449'07.965511 87.617939 5.5.382061 28 33.583754 5.965458 87.618295 5.381705 27 34.584058 5.0.965406.618652.94.381348 26 5.06 9, 88 5.94 35.584361',.9653. 3 ~ 88.619008 5.9.380992 25 36.584665..95301.619364.380636 24 37.584968 5..965248.8.619720 5.9.380280 23 38.585272..965195 88.620076 5.379924 22 39.585574 55.965143 88.620432 5.379568 21 40 9.585877.4 9.965090 9.620787 0.379213 20 0.88 5.92 2378858 19 41.586179..965037.88.621142 5.2.378858 19 42.586482.4.964984..621497.9.378503 18 43.586783 5.03.964931 88.621852 5.91.38148 17 44.587085.3.964879 8.622207 5..377793 16 45.587386 5.02.964826.88.622561 5.91.377439 15 46.587688 5.02.964773.8.622915 5.90.377085 14 47.5799.01.964720.88.623269 5.90.376731 13 48.588289 5.0.964666.88.623623 90.376377 12 5.00.89 5.89 49.588590 5.0.964613 -89.623976 5899.376024 11 50 9.588890 9.964560 9 9.624330 89 0.375670 10 51.589190.0.964.507.8.624683 5..375317 9 52.589489 4..964454.89.625036 5.8.374964 8 53.589789.964400.8.625388 58.374612 7 54.590088 4 9.964347 9.625741 588.374259 6 55.590387.98.964294.626093 587.373907 56.590686 4. 9.964240.89.626445 57.373555 4 67.590984 4.97.964187.89.626797 5.87.373203 3 58.591282 4..964133.89.627149 58.372851 2 59.591580 4-.964080.9.627501 /'.372499 1 60.591878.96.964026 89.627852 5..372148 0 M. CosneD.Se. D.. Cot!ng. D. 1. -Tang. 6M. I 1~o 67C COSINES, TANGENTS, AND COTANGENTS. 197 230.156S M. Sine. TD.I ". Cosine. D. Tag. D D.I". Cotang. M. 0 9.591878 4 96 9.964026 89 9.627852 5 o 0.372148 60 1.592176 4.963972 89.628203 585.371797 59 2.592473 49.963919 90.628554 585.371446 58 3.592770.4 963865 9.628905 54.371095 57 4.593067 4.963811.629255.370745 56 5.593363 49.963757 90.629606 5.8.370394 55 6.593659 4'9.963704 90.629956 5.8.370044 54 7.593955 49.963650 90.630306 5..369694 53 8.594251 4'.963596 90.630656 5.83.369344 52 9.594547 49.963542.9.631005 582.3&6995 51 10 9.594842 o92 9.963488 90 9.631355 582 0.368645 50 11.595137 491.963434 90.631704 582.368296 49 12.595432 91.963379 90.632053 5:81.367947 48 13.595727 4 91.963325 90.632402 5..367598 47 14.596021 490.963271 90.632750 5.81.367250 46 15.596315 4'9.963217..633099 5.8.366901 45 16.5966099 9631 9.9631633447 5.80.366553 44 17.596903 489.963108 9.633795 5.80.366205 43 18.597196 89.963054 91.634143 5..365857 42 19.597490.962999.634490 5.365510 41 4:88.91 5.79 365510 20 9.597783 488 9.962945 91 9.634538. 0.365162 40 21.598075 4.962890 ^.635185 5..364815 39 22.598368 8.962836 1.63552 78 364468 38 4.87'962836.91 5.78 23.598660 4.8.962781 91.635879 578.364121 37 25.599244 4.8.962672.1.636572.363428 35 95 6E S 4.R | XM.91 5 78' 26.599536 48.962617.63691 9.363081 34 4.86.91 5.77 27.599827 485.962562 9'.637265 5.362735 33 28.600118 4 8.962508 9'8.637611 5 7.362389 32 29.600409 0.962453.9 2.637956.362044 31 30 9.600700 484 9.962398'9 9.638302 r.- 0.361698 3 l31.600990 484.962343 92.638647.i.361353 29 32.601280 483.962288.63992.361008 28 33.601570 483.962233 2.639337..360663 27 34.601860 483.962178 92.639682 57.360318 26 35.602150 4.962123.9.640027 5.359973 25 4.9 69.7 592 5 74 36.602439 482.962067 92.640371.359629 24 37.602728 481.962012 92.640716 5.359284 23 38.603017 481.961957 93.641060 73.358940 22 39.603305 481.961902:92.641404 5573.358596 21 40 9.603594 4 8 9.961846 9 9.641747. 0.358253 20 41.603882 4'0.961791 92.642091 5,2.357909 19 42.604170 4.961735 92.642434 572.357566 1I 43.604457 4.961680'.642777 5 7.357223 17 44.604745 7.961624 93.643120 5.7.356880 16 45.6050.32 47.961569.643463 57.356537 15 46.605319 4.7.961513.9.643806 5.7.356194 14 47.605606.961458.644148.355852 13 48.605892 4.961402 9.644490 70.355510 12 49.606179 77.961346 93.644832 70.355168 11 4.77.93 5.70 50 9.606465 4'76 9.961290 * 1 9.645174 0.354826 10 51.606751 76.961235.645516.354484 52.607036 76.961179.645857.354143 8 II 53 /.93 5269 53.607322.961123:.646199 69.353801 7 54.6067 4.75.96106607 16540 56.353460 6 55.607892 4.7.961011'.646881 56.353119 5 56.608177'.960955.647222 568.352778 4 57.60461 47.960399.647562 5.352438 3 58.608745.960843.647903 67.352097 2 59.609029 /.960786./ 443..351757 1 60.609313 4_'73.960730'9.648583 5.67.351417 0 M. Cosine. D.1I". Sine. I D. 1". Cotang. D. 111. Tang. M. 1130 66G 198 TABLE XIII. LOGARITHMIC SINES,. 240 _155c M. Sine. D. 1". Cosine. D. 1". Tang. D. 1'. Cotang. M. 0 9.609313 3 9.960730 9.648583 67 0.351417 60 1.609597 4.7.960674.4.648923 566.351077 59 2.609880 472.960618.9.649263 566.350737 58 3.610164 472.960561.649602 66.350398 57 4.610447 471.960505 4.649942 5..350()58 56 5.610729 471.960448.9.650231 5 5.349719 55 6.611012 4 7.960392 4.650620 5.5.349380 54 7.611294 71.960335.650959 56.319041 53 8.611576 470.960279.4.651297 564.348703 52 9.611858 469.960222.651636 6.348364 51 10 9.612140 4.69 9.960165 9- 9.651974 564 0.348026 50 I.612421 469.960109 9.652312 5.6.347688 49 12.6 2702 46.96!052.9.652650 563.347350 48 13.612983..959995.9.652988.63.347012 47 14.613264 8.959933..653326 52.346674 46 15.613.345 4.6.959382.9.65366:3 562.346337 45 16.613825 467.959325 9..654000 562.346000 44 17.614105 67.95976S.9.654337 5.6.345663 43 1.614335 4.6.959711..654674 5.6.345326 42 19.614665 466.9596.54 95.655011 5:61.344989 41 20 9.61-944 4 65 9959596' 9.655348 561 0.344652 40 21.6152 99539.6.6556S4.344316 39 22.615502 * 6.959482..656020 5..343980 38 23.615781 64.959425.656356 5.6.343644 37 24.616060 461.959363.9.656692 5.6.343308 36 25.616:333 4-4.959310..657028 5.5.342972 35 26.616616.6.959253 96.657364 9.3126:36 34 27.616391 463.959195.9.657699 59.342301 33 28.617172 463.959133 96.658034 558.341966 32 29.6174.) 462.959080 96.658369 558.341631 31 30 9.617727 2 9.959023 9.658704 0.341296 30 1.618001,4.953965..659039 55.340961 29 32.618231 4..958908 96.659373 5.340627 28 33.613553 61.958850.6.59708.340292 27 4.61.96 5.57.6134314 460.958792 6.66.66042 55.33995 26 35.619110 46.958734 96.660376 5.56.339624 25 36.619336..953677.96.660710 5 6.339290 24 37.619662 4'.958619.9.661043 5.5.338957 23 38.61993 4 5.953561.7.661377 556.333623 22 39.620213.958503.661710.338290 21 4.o9.97 5.55 40 9.620188 4 9.958445. 9.662043 / 0.337957 20 41.620763 4.5.953337.7.662376 5..337624 19 42.62103 4-5.958329 97. 662709 5:.337291 18 43.621313 7.958271.97.663042 55.3-36958 17 44.621587.958213.663375 5.336625 16 45.621861 4.57.958154.9.663707..4.336293 15 46.622135 4 6.958096.9.661039.5..33.)961 14 437.622409 6.95,033 /.664371 /.3356.9 13 48.6226 2 4'.6.957979 97.66473 5 3'.335297 12 49.6296.957921;3 49.6956 4.957921 9.665035.33496- 11 50 9.6232-29 4 9.957863. 9.665366.52 0.334634 10 5 51.623592.54.957804.9.665698. 2.3313)2 9 52.623774 4.)4.957746.8.666929 5 2.,:71 8 53.6210-17 1-.957687 9.66636f0 ]...o3.40 7 4.54.98 5.51:,.. 5.62.1319 7 628.66691 63::9 G 55.6)1531..957570.9.6677021..332979 5 56.621863 4..957511 98.667352.5.3;'43 4 57.6)135.. 97452.97. 667682 5.33i 3 58.625 16 4.2.957393 9.663013 0.33'87 2 59.625677 5.9733.5.66313 5.331657 1 4.52 98 60.625918.957276.663673 0.331327 0 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1". I Tang. M. 1140 651 COSINES, TANGENTS, AND, COTANGENTS. 199 ag0o 1540 M. Sine. D. I". Cosine. D 1".'rang. D. I". Cotang. M. 0 9.625948 4, 9.957276 98 9.668673 6. 0.331327 60 1.626219.957217 98.669002 49.330998 59 2.626490 51.957158.669332 5.330668 58 3.626760 1' 957099.669661.330339 57'~ 669661.303 6.627030 460.957040 9.369991.330009 56 4.627030 4650.970.39609 95 6.627300.956981.670320.329680 55 6.627570.956921.670649 58 1 4.49.329351 I,~ -7.627840.956862.670977 48.329023 53 4.49 99.676 5.48 8.628109.49 956803.671306 5.4.328694 52 9.628378 4:48.956744 5.67163 47.328365 51 10 9.628647 4 9.956684 9.671963 0.328037 50 11.628916 4.48 956625 9.672291 5.47.327709 49 12.629185 7.956566.672619 5.4.327381 48 13.629453.956506.672947.46.327053 47 14.629721.956447'.673274 5.46.32626726 46 15.629989 446.956387.673602 5.46.326398 45 16.630257.956327.673929.326071 44 17.630524 46.956268.674257 6.45.325743 43 18.630792 4.956208.674584.325416 42 1 00 ~ 5.45 19.631059 4145.956148.674911.325089 41 20 9.631326 4'45 9.956089 oo 9.675237 I44 0.324763 40 21.631593 44 956(029'.675564 5.44.324436 39 22.631859.955969.675890 5.44 324110 38 23.632125 44 955909.676217 5.44.323783 37 24.632392 4.955849 io.676543.323457 36 25.632658 1.955789 100.676869.323131 35 26.632923 43 955729 o.677194 5.43.322806 34 27.633189 42 955669 1.00 677520 5.4.322480 33 28.633454 442.955609 1.677846 5.42.322154 32 29.633719.955548.678171 5.42 321829 31 4:42 9.678496 5.42 3229 3 30 9.633984 9.955488 9.678496 2 0.321504 30 31.634249 41.955428.678821 54.321179 29 32 634514 41.955368.679146 5.41.320854 28 33.634778 440.955307!.679471.41.320529 2 34.635042 440.955247.679795.1.320205 26 4.40 1.01 5.41 35.63.5306 440.955186 I.680120 5.41.319880 25 36.635570 4.955126 1'0.680444 5.40.319556 21 37.635834 4.955065 1.680768 540.319232 23 ~ 1'. I.681092 5640 38.636097.955(,05.681092 40.318909 22 39.636360 954944 1.681416 39 318584 21 40 9.636623 4'38 9.954883, 9.681740 [ 0.318260 20 41.636886.954823 1(,1.682063.39.317937 1) 42.637148.954762.682.387 5.39.317613 18 43.637411.9.54701 io.682710 5.317290 17 44.637673.3 954640.68301667 16 45{.637935{ 1.02'6 5.38 3 45.637935 36.954579 02.683356 38.316644 15 46.638197 436.954518 102.683679 38.316321 14 47.638458 36.9457 102.64001.315!99 13 48.638720 4:35.954396 1'02.684324 5.315676 12 49.63981 5.954.335 1.02.64646 37.315354 11'50 9.639242' 9.954274 102 9.6849658 0.315032 10 51.639503.954213 102.685290 536.314710 9 52.639764.954152 102.6S5612 36.314388 8 53.640)24 43.9.54090 02.6S5934 36.314066 7 54.640284 4.954029 1h -.686255 5.36.313745 6 55.6-10.544 I.953968'2.686577.313423 5 56.640804.953906 102 686898 5.3131(2 4 57.641064 4.953845 1.687219.312781 3 57 953845 1.03 *6~75.312460 25.35 58.641324 4.3.953783 687540.312460 2 59.641583..95372*2.03 687861.312139 1 60.641842.953660.685182.311818 0 M. Cosine D. ". Sine. D.l". Cotang. D.1 Tang. -M. 1150 64o 200 TABLE Xllf LOGARITHMIC SINES, 260 153E M. Sine. D.- 1". Cosine. 1 Tang. D. 1. Cotang. M. 0 9.641842 32 9.953660 03 9.688182 0.311818.60 1.642101..953599 03.688502.311498 59 2.642360 431'.93537 688823 34.311177 58 3.642618 4 31 9.53475 689143.31057 57 4.31 1.0315.34 4.642877 430.953413 1.69463 3.310537 56 5.643135 430.953352 103.689783 33.310217 55 6.643393.953290 03.690103.309897 54 7.643650 29.953228 103.690423.309577 53 8.643908 4.953166 03.690742 -309258 52 9.644-165 429.953104 03.691062 532.308938 51 4.29 1.03 5.32 10 9.644123 428 9.953042 103 9.691381 32 0.308619 50 11.644680 48.952980 04.691700 5.2.308300 49 12.644936 4.28.952918 104.692019 5.31.307981 48 13.645193 4.2.952855 14.692338 31.307662 47 14.645450 27.952793 14.692656 53.307344 46 15.645706 427.952731 04.692975 31.307025 45 16.645962 46.952669 1.693293.306707 44 4.26 1.05.30 17.646218 426.952606 104.693612 5.3.306388 43 18.646474 426.952544 1.693930 5..306070 42 19.616729 426.952481 1 04.694248 5.305752 41 20 9.646984 5 9.952419 104 9.694566 59 0.305434 40 21.647240 4.25.952356.694883 29.305117 39 22.647494 4.25.952294 104.695201 529.304799 38 23.647749 424.952231 10.695518 29.304482 37 24.64004 424 952168 0.695836 529.304164 36 25.648258 24.952106 1 05.696153 5..303847 35 26.648512 4..952043.I.696470 5.28 303530 34 27.648766 423.951980 1..696787 523.303213 33 28.649020 423.951917 105.697103 5.28.302897 32 29.649274 4:22.951854 105.697420 5 27.302580 31 30 9.649527 4 22 9.951791 05 9.697736 5.27 0.302264 30 31.649781 22.951728.o.698053 27.301947 29 32.650034 4..951665 105.698369 527.301631 28 33.650287 4.21.951602 1.05.698685 5.26.301315 27 34.650539 4 1.951539.699001 526.300999 26 35.650792 421.951476 1.0.699316 5..300684 25 36.651044 42.951412 1..699632 526.300368 24 37.651297 420.951349 1.699947 526.300053 23 38.651549.64.1286 0.700263.299737 22 39.651800 419.951222 1 06.7)0578- 5.299422 21 40 9.652052 419 9.951159 106 9.700893 5'8 0.299107 20 41.652304 1'.951096 06.701208 25.298792 19 42.652555 41.951032 1.701523.298477 18 43.652806 4.18.950968 106.701837 5.24 298163 1 7 44.65.3057 418.950905 16.702152 2.297848 16 45.653308 418.950841 106.702466.2.297534 15 46.653558 4.1.950778 1..702781 5.24.297219 14 47.653808..950714 1.703095 5..296905 13 48.654059 4.17.950650 1.06.703409 5.2.296591 12 49.654309 417.950586 1 0.703722 23.296278 1 4.16 1.06 5.23 50 9.654558 9.950522 9.704036 0.295964 10 51.654808 4.16.9504583.704350.295650 9 52.655058 4..950394.704663 522.295337 8 53.655307.15.950330 1.704976 5..295024 7 54.655556 4.15.950266.705290 5.2.294710 6 55.655805 4.15.950202.07.70560 5.2.294397 5 56.656054 4.15.950138 1.07.705916 5.2.294084 4 57.656302 4.14.950074 1.07.706228 5.21.293772 3 58.656551 41.950010 1'0.706541 5.293459 2 59.656799 14 949945.70654 21.293146 1 60.657047 41.949881 1.707166 531.292834.657047!292_.394916 0 j M. Cosine. D. -". Sine. D. ". Cotang. D. ". Tang. M. 1160 63* COSINES, TANGENTS, AND (COTANGENTS. 201 270 152C M. Sine. D. 1". Cosine. D. 1". Tang. D. I'. Cotang. M. ) 9.657047 13 9.949881 1 07 9.707166 5 20 0.292834 60 1.657295 4.3.919816 107.707478 520.292522 59 2.657542 4.1.949752 1'7.707790 520.292210 58 3.657790 412.949638 08.708102 20.291898 57 4.653037 4.2.949623 o1..708414 20.291586 56 5.658234 412.949o5S 10.708726 51.291274 55 6.658531.949494.709037 19.290963 54 7.658778 11.949429 s.709349 19.290651 53 8 659025 11.949364 0.709660 59.290340 52 9 1.659271 4..949300 1.8.709971 5 18.290029 51 10 9.659517,40 9.949235 1 08 9.710282 5.18 0.289718 50 11.659763 410.949170 1..710593 5 18.289407 49 12.660009 410.949105 108.710904 518.239096 48 13.660255 4.1' 949040 1.0.711215 58.288785 47 14.660501 4 09 948975 1.08.711525 517.288475 46 15.660746 409.948910 1.08.711836 517.288164 45 16.660991 408.948845 1 09.712146 517.287854 44 17.661236 4..948780 1.9.712456 517.287544 43 18.661481.948715.712766 5.17.27234 42 19.661726 40 3.948650 109.713076 5.286924 41 20 9.661970 407 9.948584, 9.713386 5 1 0.286614 40 21.662214 4'7.948519 109.713696 516.286304 39 22.662459 407.948454 109.71400 5 16.285995 38 23.662703 4 06.948388 109.714314 5 15.285686 37 24.662946 4.94832.3 1.714624 5.285376 36. 25.663190 4.6.948257 109.714933.I.285067 35 26.663433 405.948192 109.715242 515.284758 34 27.663677 4..948126.09.715551 515.284449 33 23.663920 40.948060.715860 14.284140 32 4.05'979 IO 7 61. 5.141 29.664163 0.947995:.716163 1.283832 31.' 30 9.664406 4 9.947929 9.716477 4 0.283523 30 31.664648 0.947863 1.716785 514.283215 29 32.664391 404.947797'1.717093 5..282907 23 33.665133 403.947731 1i0.717401 I.282599 27 34.665375 403 947665..717709 13.282291 26 4.03' 1.10 5 13 35.665617 403.947600 10.718017 13.281983 25 36.665859 403.947533 10.718325 5 13.281675 24 37.666100 42.947467 110.718633 5. 3.281367 23 38.666342 402.947401 10.718940 512.281060 22 39.666533 402.947335 11.719248 512.280752 21 40 9.666824 4 01 9.947269 110 9.719555 5 12 0.280445 20 41.667065.1.947203..719862 12.280138 19 42.667305 4..947136.720169'5..279831 18 43.667546 401.947070 1.1.720476 5.1.279524 17 44.667786 400.947004 I.720783 5.1.279217 16 45,668027.00.946937 {1..721089 5.1.278911 15 46.663267 40.946371 I.721396 5.278604 14 47.663506.946304.721702 5 1.278298 13 43.663746 99.946733..722009 I.277991 12 49.668986 3.946671 111.722315 10.277685 11 50 9.669225.9 9.946604 9.722621 10 0.277379 10 51.669464 3..946538.722927 I.277073 9 52.669703 39.946471 1.1.723232 5.0.276768 8 53.669942 398.946404 11.723538 59.276462 7 54.670181.946337 11.723844 59.276156 6 55.670419.946270 1'1.724149 5.275851 5 56.67 06358 97.946203 112.724454..275546 4 57.670396 397.946136 1'.724760 *08.275240 3, 58.671134 396.946069 112.725065 5.8.274935 2 59.671372 396.946(02 1 I.725370 5..274630 1 60.671609.9459.35.725674.274326 0 M. Cosine. D. I1". Sine. D. 11'. Cotang. D. 1I. Ta-g. M. 1 170 6B2C 202 TABLE XIII. LOGARITHMIC SINES, o80 151 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.671609 - 9.945935 6 9.725674 0.274326 60 1.671847 96.945868 12 725979 08.274021 59 2.672084 3.945800 12.726284.273716 58 3.95 1.12 5.07 9 3.672321.945733 112.726588 507.273412 57 4.672558.945666 112.726892 07.273108 56 5.672795.945598 112.727197 507.272803 55 6.673032.945531 12.727501 5.272499 54 7.673268.945464 13.727805 06.272195 53 8.673505.945396 13.728109 506.271891 52 9.673741 3.93.945328 13.728412 06.271588 51 10 9.673977 9.945261 113 9.728716 506 0.271284 50 11.674213. 945193 113.729020 506.270980 49 12.674448.9451215 113 729323 505.270677 48 13.674684 392.94508 113.729626 1o1.270374 47 14.674919 392.944990 113.729929 505.270071 46 15.675155 3.92.944922 3.730233.9767 16.675390 3 944S54.730535.269465 17.675624 391.944786 113.730838.269162 43 18.675859 391.944718 113.731141 50.268859 42 19.676094 391.944650 113.731444 504.268556 41 3.91 1.13 5.04 20 9.676328 3 9.944582 14 9.731746 0.268254 40 21.676562 390.944514 14.732048 04.267952 39 22.676796 5.944446 14.732351.267649 38 23.677030 39.944377.732653 03.267347 37.67726 944309 732955.267045 36 25.677498 038.2743 25.677493.944241.733257.266743 35 3.89 1.4 5.03 26.677731 3.944172 14.733558 03.266442 34 27 677964 8.944104 14.733860 03.266140 33 28.678197 388.944036 114.734162 02.265 32 29.678430 38.943967.734463.0.265537 31 3.88 1.13 5.0. 30 9.678663 3'88 9.943899 114 9.734764 02 0.2652.36 30 31.678895 37.943830 114.735066 502.264934 29 32.67912 387.943761.735367.264633 28 33.679360 3.87.943693 1.735668 501.264332 27 34.679592 387.9436 11.735969 S01.264031 26 35.679824.943555 1.736269 01.263731 25 63861.71312 5 20 4 24 36.60056.943486 1.736570263430 37.630283.943417 5.736870.263130 23 33.680519.943348.737171.01.262829 22 39.680750 386.943279 1 15.737471.262529 21 3.85.915 5.00 40 9.680932 385 9.943210 1.15 9.737771 500 0.262229 20 41.681213..943141 115.738071 0.261929 19 42.681443 3.8.943072 1.738371 o.261629 18 43.681674 3.84.943003.738671.261329 17 44.681905 3.84.942934 1. 738971 49.261029 16 45.682135 3.942864 16.739271.260729 15 46.682365.94279.739570.260430 14 3.83 1.12 4 030 47.682595 3.942726 16.739870.260130 13 48.682825 353.942656 16.740169.259531 12 49.683055.942587.740468.259532 4 50 9.683284 382 9.942517 116 9.740767 498 0.259233 10 51.63514 82.942448 16.741066 98.258934 9 52.63743 3.82.942378.741365 98.258635 8 53.683972.942308.741664.258336 7 634201 3.82 1.16 4.98 3 4.634201 1.942239 116.741962 498.258038 6 55.634430 3.81.942169 1.16.742261 4.257739 5 56.684658 381.942099 116.742559 4.257441 4 57.684887 38.942029 117.742858.257142 3 53.685115 30.941959 17.743156.256844 2 3.80 1.1 4 4.97 59.65343 80.941889 1.743454 4.256546 1.6365571.941819.743752.256248 0 IM.J Cosine. D. I". Sine. D. I". Cotang. PD. 1". I Tang. M. 118- ~^~~~9437 COSINES, TANGENTS, AND COTANGENTS. 203 o _9 0 1 500 M Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.685571 380 9.941819 117 9.743752 496 0.256248 60 1.685799.941749 17.744050 496.255950 59 2.686027 9.941679 17.744348 96 255652 58 3.686254.941609 117.744645.255355 57 4.686482 1.941539.74493 96.255057 56 5.686709 3.79 1.17 4.96 55 5.686709 378.941469 117.745240 496 254760 6.686936 378.941398 117.745538 495 254462 5 7.687163 378.941328 117.745835 495.254165 53 8.687389 3.941258.746132 495 253868 5 9.637616.941187 117.746429 4.95 203571 1 10 9.687843 9.941117 118 t46726 45 0.253274 50 11.688069 377.941046 1:18.747023 4.252977 49 12.6295.910975 747319.252681 48 13.633521 376.940905 118 747616.252384 47 14.688747 376.940334 118.747913.252087 46 15.68972 376.940763 1.748209.251791 45 16.689198 376.940693 11.748505.251495 44 17.69423.940622.748801.251199 3.75 1.18 4.93 25 18.69648.940551 118.749097.250903 42 19.689873.940480 1 749393 493.250607 41 3.75 1.18 1 4.93 20 9.690098 3 9.940409 18 9.749689 0.250311 40 24.690996.940125 750872.249128 36 3.74 1.19 4.92 3.7 940054 1.1 167 4.92.248833 35 26.691444.939982 119 751462 492.248538 34 27.691663.939911 119.751757.93 33 2.691892.939840 119.7505.247948 32 29.692115 72.939768 19.752347 491.247653 31 30 9.692339 9.939697 19 9.75264 491.24 3.72 939625 1.19 752937 4.91.247063 29 33.693008 371.939482 119.753526 491.246474 27 34.693231 371.939410 119.75320 491.246180 26 35.693453 371 120. 754115 49.245885 25 36.693676 371.939267 20 754409 4.90.245591 24 37.693898.939195.75403 4.90 3 3 692 3.73 1.20 75245297 23 3.694 3.70'939123 120'9 4.90 245003 22 42.695007.938836.75612 4.9. 3.76 1. 17 43.695229 69.938763.756465 4.24535 1 3.72 1.17 4.96945 45 695671 3..93619 20.757052 4.89.242948 46.695892.93547 20 757345 88 242655 14 3.68 938475 121.757638 488.242362 13 4.6964 368 93 2.9102 79 48 242069 12 49.696554 367.938330 121.758224 4.88.241776 11 50 9.696775 9.938258 21 9.758517 488 0.241483 10 51.696995 367.938185 121.758810 488 241190 9 52.697215 67.938113 21.759102 87 240898 8 56.698094 937822 12.760272 8.239728 4 57.698313 3.66.7564.248 5 07 3.65 1.21 40.87 2 58.698532 3.6.937676 121.757.239144 2 59.69897 13.6 937604 1.22 4.86 18 60.698970 3.65:937531 _.7161439.2348561 0 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1". Tang. M. 1190.6 204 TABLE XIII. LOGARITHMIC SINES, 303 14:9o M. Sine. D. I*. 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 64.937458 122.761731 486.238269 59 2.699407 64.937385 122.762023 486.237977 58 3.699626 36.937312 122.762314 486.237686 57 4.699844 36.937238 122.762606 486.237394 56 5.700062 363.937165'22.762897 485.237103 55 6.700280 36.937092 122.763188 485.236812 54 7.700498 363.937019'22.763479 4 8.236521 53 8.700716 3.936946 22.763770 85.236230 52 9.700933.936872.764061 5.235939 51 10 9.701151 3 62 9.936799 1 22 9.764352 4 85 0.235648 50 11.701368 362.936725 123.764643 484.235357 49 12.701585.6.936652'2.764933 44.235067 48 13.701802 361.936578.23.765224 4..234776 47 14.702019 361.936505 123.765514 484.234486 46 15.702236 361.936431 123.765805 4.4.234195 45 16.702452 361.936357 123.766095 484.233905 44 17.702669 360.936284 23.766385 83.233615 43 1IS.702885 360.936210 123.766675 483.233325 42 19.703101 360.936136 123.766965 4:83.233035 41 20 9.703317 360 9.936062,3 9.767255 4 83 0.232745 40 21.7035.33 5.935988 123.767545 483.232455 39 2.703749 35.935914 123.767834 483.232166 38 2:3 703964 3'59.935840 123.768124 4'82.231876 37 24.704179 3 9.935766 124.78414 482.231586 36 25.704395 3'.935692 124.768703 4.82.231297 35 26.704610..935618 124.768992 482.231008 34 27.704825.935543'.769281 4 S.230719 33 28.705040.935469.24.769571 4.2.230429 32 29.705254 3.939.576960 48.230140 31 358 1'24 4.82 30 9.705469 9.935320 2 9.770148 481 0.229852 30 31.705683 3.935246 1..770437 1.229563 29 32.705898 3..935171 124.770726 48.229274 28 33.706112 357.935097 124.771015'.228985 27 34.706326 3.56.935022 2.771303 /.228697 26 35.706539 56.934948 24 771592 81.22408 2 3 56' 1 24' 4.81 36.706753 356.934873 12.771880 80.228120 24 37.706967 356.934798 125.772168 4.80.227832 23 38.707180.934723 1.772457 4.227543 22 39.707393 3.5.934649 12.772745.227255 21 40 9.707606 32 9.934574 1 9.773033 4 80 0.226967 20 41.707819 32.934499 25.773321 4'8.226679 19 42.708032 3.5.934424 125.773608 4.80.226392 18 3'54.904'49 43.708245.934349 15.773896.226104 17 44.708458 54.934274 125 774184 4.79.25816 16 45.708670 3..934199 125.774471 4 7.225529 15 46.708882 3..934123 25 774759.225241 14 47.709094 353.934048 1.25.775046 i479.224954 13 43.709306 35.933973 1.775333 49.224667 12 49.709518 33.933898 16.775621 478.224379 11 50 9.709730 3 9.933822 1 26 9.775908 4.'8 0.224092 10 51.709941 52.933747.7761951 4'.223805 9 52.710153 3'2.933671 126.776482 48.223518 8 53.710364 352.933596 126.776768 4.78.223232 7 54.710575 3.52 933520 1'26.777055 478.222945 6 55.710786.5 931445'6.777342 4.7.222658 5 56.710997 31 933369 126.777628.222372 4 57.711208 933293 26.777915 77.222085 3 58.711419 351 933217 2.778201 77.221799 2 59.711629 35l.933141 1 6.7784881 77.221512 1 3.51 1.26 7 4 77 2212 60.711839.933066.778774.221226 0 iM. Cosine. D. 1". Sine. D. 11. Cotang. D.l". Tang. M. 190o bOo COSINES, TANGENTS, AND COTANGENTS. 20E 310 14.80. Se. D.. sine. D. 1". Tang. D. 1". Cotang M 0 9.711839 3.50 9.933066 127 9.778774 77 0.221226 60 3.50 3 127 4 77 1.712(50 3.932990 1 *.779060..220940 59 2.712260 3..932914 27.779346 4.7.220654 58 3.71169 3.932838 27.779632 476.220368 57 4.712679 3.5 932762 12'7.779918 4.2200S2 56 5.712889.932685 27.780203.219797 55 6.713098..932609 1 27.780489 76.219511 54 7.713308 3.4 932533 27.780775.219225 53 8.713517 3.4 932457 127.781060 4.76.218940 52 3.48 4.76 9.713726 48 932380 27 781346 476.2154 51 10 9.713935 9.932304 1.27 9.781631 0.218369 50 1.714144 3.48.932228 127.781916 5 218034 49 12.714352 3.932151 ~28.782201.217799 48 13.714561.4 932075 128.782486 4.7.217514 47 14.714769.47 931998.782771.217229 46 15.71497 3.931921 128.783056 4.216944 45 16.715186 3.4.931845 2.783341.216659 44 3.47 1.28'73 4 ~216659 4 4 i 17.715394 3.4.931768 1..783626 47.216374 43 18.715602 3.46.931691 128.783910.216090 42 19.715809 36.931614 28.784195 4.215805 41 20 9.716017 346 9.931537 1'28 9784479 0.215521 40 21.716224 346.931460 1. 784764.215236 39 22.71632 3.4.931333 128 785048.214952 38 23.716639.931306 28.785332.214663 37 24.716846 45.931229 1 29.785616 473.214384 36 25.717053 3.931152 129 785900.214100 35 26.717259 3.4.931075 129.786184.213816 34 27.717466 3.4.930993 129.786468.213532 33 28.717673.44 9309'21 29.786752.213248 32 29.717879 44 93043 129.787036 73.212964 31 30 9.718085 343 9.930766 1 29 9.787319 73 0.212681 30 31.718291 43.9306S8'29.787603 472.212397 29 32.718497 33.930611 12.787886 4.7.212114 28 33.718703 4.930533 129.788170 472.211830 27 34.718909 3.43.930456 29.788453 472.211547 26 35.719114 42.930378 129.788736 472 211264 25 36.719320 342.930300'30.789019 472.210981 24 37.719525 342.930223 1.0.789302 472.210698 23 38.719730 342.930145 130.789535 4.7.210415 22 39.719935 3.41.930067 1:30.7898C,8 4.71.210132 2 1 40 9.720140 3 41 9.929989 1 30 9.790151 4 7 0.209849 20 41.720345 341.929911 3.790434.1.209566 19 42.720549 341.929833 1.790716 71.209284 18 43.720754 4.929755 130.790999 471.209001 17 44.720958 3.4.929677 1.3.791281 471.208719 16 45.721162 340.929599 130.791563 4.7.208437 15 46.721366 3-4.929521 130.791846 47.208154 14 47.721570 340.929442 -31.792128 470.207872 13 48.721774 33.929364 1 3.792410 47.207590 12 49.721978 39.929286 31.792692 47.207308 11 50 9.722181 [ 9.929207 1. 9.792974 4 70 0.207026 10 51.722385 3..929129 131.793256 4.0.206744 9 52.722588.929050 1.31 793538 47.206462 8 53.722791 338.928972 1.3.793819 4.6.206181 7 54.722994 33.928893 1.3.794101 69.205899 6 55.723197 3.8.928815 1.1.794383.69.205617 5 56.72.3400 3..928736 1..794664 4..205336 4 57.723603.923657 13.794946.69.205054 3 58.72.3305 37.923578 31.795227 4.69.204773 2 59.724007 3..928499 131.795508..204492 1 60.72421 0 9 2 795789 46.204211 0 M. Cosine. D.1 Si. D. 1D.. Cotang. D. ". Tang. M 21 V 589 206 TABLE XIII. LOGARITHMIC SINES, 320 147~ M. Sine. D.1". Cosine. D. 1" Tang. D. 1". Cotang. M. 0 9.724210 9.928420 1. 9.795789 468 0.204211 60 1.724412.928342 132.796070.203930 59 2.724614.928263 2.796351 4..203649 58 6 725420 3.36.927946 132.79747 4.68.202526 54 7.725622 335.927867 132 797755 4.68.202245 53 8.726823 35.927787 132.798036 4.67.201964 52 9.7260248.927708 32.798316.201684 51 10 9.726225 3 9.927629 13 9798596 4.6 0.201404 50.1 7264267.9217549 133 798877 46.201123 49 12.726626 3.927470 1.799157 4.6.200843 48 13.726827.9279 1.33 793 4.6.200563 47 14.727027 33.927310 133'799717 4.67.200283 46 15.727228.927231 33 799997 466.200003 45 16.727428 3.3927151 800277 466.199723 44 7.727628.927071. 800557 46.199443 43 8.72820.9299 1.3 80036 4.6.199164 42 19.728027.926911.801116.198884 41 3.33 1.33 4.66 20 9.728227 3 9.926831 1 3 9.801396 466 0.198604 40 21.728427 33.926751 33.801675 4.198325 39 22.728626 3.926671.801955 46.198045 38 3.32 23.728825 33.926591 1.34 802234 4.65.197766 37 24 729024 3.3 926511.02513 46.197487 36.7 3 204 1.34.803 2 4.6.1948 25.729223 1.926431 13.80-2792 46.197208 35 26.729422 3*.926351 1.9 3.803072.1 96928 34 27.729621 3.926270 1.803357 4.6.196649 33 3.31 1.33 4.67 28.729820 3.3.926190 1.803630 146.196370 32 29.730018 3.92611.803909 4.65.196091 31 30 9.730217 3 30 9.926029 1 34 9.804187.65 0.195813 30 31.733041 3.3.925949 1.80446 4.6.19534 29 32.732761 3.3.925868 1.804745 4.6.195255 28 33.730811 3.925788 13.480503.194977 27 34.731009 3.925707 135.805302 4.6.194698 26 35.731206 329.925626 135.805680 2 64.194420 25 36.731404 32.925545 1.35.80859 46.194141 24 37.731602 3.2.92465 1.8013.8.193863 23 38.731799 32.926384 1 3.806415 46.193585 22 39.731996 3.926303.806693.193307 21 3.33 1.35 4.63 40 9.732193 3 28 9.925222 13 9.806971 463 0.193029 20 41.732390 3.2.92514 1.3.807249 46.192751 19 42.732587 328.925060 135.807527 463.192473 18 43.732784 32.9247 1.3.078105 4.6.192195 17 44.732980 3.2.924897 13.808083 46.19917 i1 46.73373 327.924735 1.3.808638 4.6.191362 14 47.733569 32.924654 1.3.08916 4.191084 13 48 733765 3.3 924572 13.809193 4.6.190807 12 49.733961 3.924491.809471.190529 3 3.26'924491 1.34 4.62 50 9.73417 326 9.924409 369.809748 462 0.190252 10 51.734353 326.924328 13.810025 462.189975 9 2.734549 2.924246 13.810302 4.2 189698 8 3.29 1.35 4.65 53 734744 26.924164.810580 462.189420 7 54.734939.924083.810857 62.189143 6 5.735135 3.2 924001 811134 4.6.1888668 5 56.735330 3.25 923919.811410 61.188590 4 57.735525 32.923837 1.37.811687 4.6.188313 3 58 735719 25 923755 1.3.811964 4.61.188036 2 59.73591 3..923673 1.3.812241.6.187759 1 60.736109.923591.812517.187483 0 M. Cosine. D.I". Sine. D.I". Cotang D. 1" Tang M..73.80120 5 52 Csn.7345 4 9 Sn. D~'~ o~n DlrITagIM If~~Ro br3.2 COSINES, TANGENTS, AND COTANGENTS. 207 330 1l*6 M. Sine. D. 1". Cosine. D. 1". Tang. D. I". Cotang. M. 0 9.736109 34 9.923591 9.812517 461 0.187483 60 1.736303 24.923509 17.812794 46.187206 59' 2.73649 324.923427 1.37.81070 46.186930 58 3.736692 3.2.923345 1.37.813347 4.6.186653 57 4.736886 3.23.923263 37.813623 460.186377 56 5.737080 23.923181 137.813899 60.186101 55 6.737274 3.2.923098 37.814176 46.18524 54 7.737467 3..923016 1.7.814452 460.18548 53 8.737661 3.2.922933 13.81472S 4.185272 52 9.737855 322.922851 1.3.815004 46.184996 51 10 9.738048 22 9.922768 38 9.815280 460 0.184720 50 11.738241 32.922686 138.815555 46.1844-5 49 12.738434 322.92603.38.815-31.184169 48 13.738627 3.2.922520 138.816107.183893 47 14.738820 3.2.922438 38.816382 59.183618 46 15.739013 21.922355 138.816658.183342 45 16.739206 3.21.922272 38.816933 41.183067 44 17.739398 321.922189 1..817209.182791 43 18.739590 321.922106 1.38 817484.18216 42 19.739783 3.20.922023 1.38.817759 59.182241 41 20 9.739975 20 9.921940.9 9.818035 0.181965. 40 21.740167 320.921857 1.39.818310 58.181690 39 22.740359 320.921774 1.39.818585 58.181415 38 23.740550 3.2.921691 i.818860 45.181140 37 24.740742 19.921607 1.39.819135 48.180865 36 25.740934 19.921524 139.819410 58.180590 35 26.741125 319.921441 39.819684 45.180316 34 27.741316 3.19.921357 1.39.819959 4.8.180041 33 28.741508.1.921274 139.820234 458.179766 32 29.741699 318.921190 139.820508 458.179492 31 38 9.741889 8 9.921107 9.820783 7 0.179217 30 31.742080 3.18.921023.821057.178943 29 32.74-2271 3.1.920939 1.3.821332 4.5.178668 28 33.742462 3.1.920856.821606 4.178394 27 317 140 178124.57 34.742652 317 920772 140.821880 4.178120 26 35.742842 3.7.920688 140.822154 7.177846 25 36.743033 3.1.920604 1.822429.177571 24 37.743233 3..920520 1 822703.177297 23 3:17:822977 4.57 22 38.743413 31.920436 1.40.822977 4.5.177023 22 39.743602 316.920352 14.823251 4.56.176749 21 40 9.743'92.6 9.920268.40 9.823524 4.6 0.176476 20 41.743982 3.16.920184 140.823798 4.176202 19 42.744171 316.920099 140.824072.56.175928 18 43.744361 3.1.920015 1..824345.5.175655 17 44.744550 15.919931 1.41.824619.17531 1 45.744739 3.15 919846 1.41.824893 4.56.175107 1 46.744928 3.15.919762.825166.174834 14 47.745117 3.15.919677 141 825439 56.174561 13 3.15 9 1.41.4.56 48.745306 3.1.919593 141.825713.174287 12 49.745594 3.14.919508 41.825986 455.174014 11 3.14 1.41 4.55 50 9.745683 9.919424 9.826259 0.173741 10 51.745871 3.14.919339 1.1.826532 55.173468 9 52.746060 314.919254 1.41.826805 4.55 173195 8 53.746248 3.1.919169 141.827078 4.55 172922 7 54.746436 3.13.919085 142.827351.172649 6 55.746624 3.13.919000 14.827624.172376 5 56.746812 3.13.918915 142.827897 4.55.172103 4 57.746999 3.13.918830 14.828170.171830 3 58.747187 3.1.918745 42.828442.171558 2 59.747374 12.918659 42.828715 -.171285 1 3.12 1. 42 4.54 60.747562 3.1.918574..828987.5_.171013 0 M. Cosine. D. 1". Sine D. 1". Cotang. D. 1". Tang. M. 1233 56a TABLE X1V. NATURAL SINES AND COSINES. 225 - 250 26o 270 28 290o M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. 0.42262.90631.43837 8979.45399.89101.46947.88295.48481.87462 60 1.4228.90618.43863.89367.45425.89087.46973.88281.48506.87448 59 2.42315.90306.43389.89354.45451.89074.46999.88267.48532.87434 58 3.42341.90394.43916.89841.45477.89061.47024.88254.48557.87420 57 4.42367.90582.43942.89328.45503.89048.47050.88240.48583.87406 56 5.42394.90569.43968.89316.45529.89035.47076.88226.48608.87391 55 6.42420.90557.43994.89803.45554.89021.47101.88213.48634.87377 54 7 42446.90545.44020.89790.45580.89008.47127.88199.48659.87363 53 8.42473.90532.44046.89777.45606.88995.47153.88185.48634.87349 52 9.42499.90520.44072.89764.45632.88981.47178.88172.48710.87335 51 10.42525.90507.44098.89752.45658.88968.47204.88158.48735.87321 50 11.42552.90495.44124.89739.45684.88955.47229.88144.48761.87306 49 12.42578.90483.44151.89726.45710.88942.47255.88130.48786.87292 48 13.42604.90470.44177.89713.45736.88928.47281.88117.48811.87278 47 14.42631.90453.44203.89700.45762.88915.47306.88103.48837.87264 46 15.42657.90446.44229.89687.45787.88902.47332.88089.4886.2.87250 45 16.42683.90433.44255.89674.45813.88888.47358.88075.48883.87235 44 17.42709.90421.44281.89662.45839.88875.47383.88062.48913.87221 43 18.42736.90408.44307.89649.45865.88862.47409.88048.48938.87207 42 19.42762.90396.44333.89636.45891.88848.47434.88034.48964.87193 41 20.42788.90383.44359.89623.45917.88835.47460.88020.48989.87178 40 21.42815.90371.44385.89610.45942.88822.47486.88006.49014.87164 39 22.42841.90358.44411.89597.45968.88808.47511.87993.49040.87150.38 23.42867 90346.44437.89584.45994.88795.47537.87979.49065.87136 37 24.42894.90334.44464.89571.46020.88782.47562.87965.49090.87121 36 25.42920.90321.44490.89558.46046.88768.47588.87951.49116.87107 35 26.42946.90309.44516.89545.46072.88755.47614.87937.49141.87093 34 27.42972.90296.44542.89532.46097.88741.47639.87923.49166.87079 33 23.42999.90284.44568.89519.46123.88728.47665.87909.49192.87064 32 29.43025.90271.44594.89506.46149.88715.47690.87896.49217.87050 31 30.43051.90259.44620.89493.46175.88701.47716.87882.49242.87036 30 31.43077.90246.44646.89480.46201.88688.47741.87868.49268.87021 29 32.43104.90233.44672.89467.46226.88674.47767.87854.49293.87007 28 33.43130.90221.44698.89454.46252.88661.47793.87840.49318.86993 27 34 43156.90208.44724.89441.46278.88647.47818.87826.49344.86978 26; 35.43182.90196.44750.89423.46304.88634.47844.87812.49369.86964 25 36.4.3209.90183.44776.89415.46330.88620.47869.87798.49394.86949 24 37.43235.90171.44802.89402.46355.88607.47895.87784.49419.86935 23 38.43261.90158.44828.89389.46381.88593.47920.87770.49445.86921 22 39.43287.90146.44854.89376.46407.88580.47946.87756.49470.86906 21 40.43313.90133.44880.89363.46433.88566.47971.87743.49495.86892 20 41.43340.90120.44906.89350.46458.88553.47997.87729.49521.86878 19 42.43.366.90108.44932.89337.46484.88539.48022.87715.49546.86863 18 43.43392.90095.44958.89324.46510.88526.48048.87701.49571.86849 17 44.43418.90082.44984.89311.46536.88512.48073.87687.49596.86834 16 45.43445.90070.45010.89298.46561.88499.48099.87673.49622.86820 15 46.43471.90057.45036.89285.46587.88485.48124.87659.49647.86805 14 47.43497.90045.45062.89272.46613.88472.48150.87645.49672.86791 13 48.43523.90)32.45088.89259.46639.88458.48175.87631.49697.86777 12 49.43549.90019.45114.89245.46664.88445.48201.87617.49723.86762 11 50.43575.90007.45140.89232.46690.88431 48226.87603.49748.86748 10 51.43602.89994.45166.89219.46716.88417.48252.87589.49773.867.33 9 52.43628.89981.45192.89206.46742.88404.48277.87575.49798.86719 8 53.43654.89968.45218.89193.46767.88390.48303.87561.49824.86704 7 54.43680.89956.45243.89180.46793.88377.48328.87546.49849.86690 6 55.43706.89943.45269.89167.46819.88363.48354.87.532.49874.86675 5 56.43733.89930.45295.89153.46844.88349.48379.87518.49899.86661 4 57.43759.89918.45321.89140.46870.88336.48405.87504.49924.86646 3 68.43785.89905.45347.89127.46896.88322.48430.87490.49950.86632 2 59.43811.89892.45373.89114.46921.88308.48456.87476.49975.86617 1 60.43837.89879.45399.89101.46947.88295.48481.87462.50000.86603 0. Coain Sine C. ie CinSine. Cosin. ine. Cooin. Snine. M. 640 630 620 610 600 11 COSINES, TANGENTS, AND COTANGENTS. 209 350 1440 M. Sine. D.1" Cosine. D. ". Tang. D. I". Cotang. M. 0 9.758591 301 9.913365 147 9.845227 448 0.154773 60 1.758772 300.913276 148.845496 448.154504 59 2.758952 300.913187 148.845764 448.154236 58 3.759132 3.9130,9 148.846033 1 48.153967 57 4.769312 300.913010 148.8463(02 448.153698 56 5.759492 300.912922 148.846570 448 153430 55 6.7672 299.912833 148.846839 448.153161 54 7.759852 99.912744 48.847108.152892 53 8.760031 299.9126155 1.847376'.152624 2 9.760211 2 99.912566 1:48.847644 4:47.152356 51 10 9.760390 299 9.912477 148 9.847913 4 0.152087 50 1.760569 299 91238 148.8481 81 4.47.151819 49 11.760927 298.912210 149.848717.151283 47 14.761106 298.912121 149.848986 4.151014 46 15.761285 298.912031 149.849254 47.150746 45 16.761464 298.911942 1.849522 47.150478 44 17.761642 297.911853 149.849790 446.15021043 18.761821 297.911763 1.850057 446.14975 42 19.761999 2:97.911674 1:49.350325 4:46 1 20 9.7G2177 297 9.911584 1.49 9.850593 446 0.149407 40 21.762356 97I.911495 49.850861 I6I.149139 22.762534 297.911405 149.851129 446.148871 38 23.762712 296.911315 150.851396148604 io' ]~]~ I ~ 148I6 04l: 24.762889 296.911226 1.851664 446 148336 25.763067 296.911136 150.851931 46.148069 35 2 7693 ] ^ 4'46. 1427309152 26.763245 2.911046 150.852199 46.147801 27.763422 296.910956.852466.147534 33 28.763600.910866 150.852733 446.147267 32 29.763777 2:95.910776 1:50.853001 4:45.146999 31 30 9.763954 95 9.910686 9.853268 0.146732 30 31.764131 2.9'910596.853535 44.146465 29 32.764308'95.910506 1'I.853802 5.146198 33.764485.910415.854069 145931 27 2.95 1'901 1.51 4.45 593 34.764662 94.910325.854336 45.145664 26 35.764838 294.910235.854603.145397 25 36.765015 2.910144.854870 3.14130 24 37.765191 294.910054. 855137 45.144863 23 38.765367 294.909963.855404.14496 22 39.765544 2:.9.909873 8671 44..144329 21 40 9.765720 2.93 9.909782 1.51 9.855938 0.144062 20 41.765896 293.909691,.856204.143796 19 42.766072 293.909601 1.86471.143529 18 43.766247.93 909S10 1.856137 4..143263 17 44.766423 293.909419 152.857004.142996 16 45.766598 292.909328.857270 4.44.142730 15 46.766774.909237 15.83.142463 143 47.766949 292.909146 12 857803 4.44.142197 12 48.767124 292.909055 152.858069 44.141931 1 49.767300.908964.858336 444 2.92 1.52'858336'141664 1/ 50 9.767475 2.91 9.908873 152 9.858602 44 0.141398 10 51.767649 291.908781 152.858868 4.43.141132 52.767824 291.908690 12.859134 43.140866 8 53.767999 291.908599 2.859400 140600 7 54.76173 2.9087 859666 6 55.768348 291.908416 1.3.859932 443 140068 5 56.768522 2.90.908324 1.53 860198 4.43 13902 4 57.768697 2.90 908233 1.53 860464 43 /139536 3 58.768871.908141..8607301 443.139270 2 59.769045 290.90l049 1..860995 43.139005 1 2.90 1.53 4.43... 60.769219.907958 3.8F61261.138739 0 M. Cosine. D. 1". Sine. D. 1i". Cotang D. 1 Tang. M. 1~~~~~~~~~330~ ~ ~~~~~I0] 210 TABLE XIII. LOGARITHMIC SINES, 360 _143 M. Sine. D. 1. Cosine. D.. Tang. D. 1". Cotang. M. 0 9.769219 290 9.907958 53 9.861261 0.138739 60 1.769393 290.907866 53.861527.138473 59 2.769566 28 907774 1.861792.138208 58 3.769740.907682 53.862058 42.137942 57 4.769913.9 907590 153.862323 442.137677 56 5.770087 289 907498 1.53.862589 2.137411 55 6.770260 89 907406 54.862854 42.137146 54 7.770433 8 907314.863119 42.136881 53 1.64'863119 442.136615 52 8.770606 28 907222 1..863385 44.136615 62 9.770779 88 907129 54.863650.136350 51 10 9.770952 2.88 9.907037 154 9.863915 442 0.136085 50 11.771125 2..906945 54.864180 42.135820 49 12.771298 88 906852 54.864445. 135555 48 13 771470 87 906760.864710 42 35290 47 14.771643 87.906667 54.864975 442 135025 46 15.771815 287 906575 14.865240 41.134760 45 16.771987 2.87.906482 1.55.865505 441.134495 44 17.772159 287.906389 1.55.865770.134230 43 18.772331 287.906296 i'5.866035 41 13395 42 19.772503 2.86 906204 1:55.866300 41.133700 41 20 9.772675 6 9.906111 1. 9.866564 4 41 0.133436 40 21.772847 286.906018.5.866829 41.133171 39 22.773018 286.905925 1.55.867094 4.132906 38 23.773190 286 905832 1.867358 41.132642 37 24.773361.905739 155.867623 44.132377 36 25.773533 85.905645 5.867887 41.132113 35 26.773704 85.905552.868152 4.131848 34 27.7738r 5.905459 16.868416.131584 33 28.774046 85.905366 156.868680 44.131320 32 29.774217 285.905272 156.868945 40.131055 31 2.85 1.56'6 868 4.40 1 30 9.774388 9.905179 9.869209 440 0.130791 30 31.774558 284.905085 156.869473 4.130527 29 32.774729 284.904992 156.869737 44.130263 28 33.774899 284.904898 156.870001 40 129999 27 34.775070 2.904804.56.870265 40.129735 26 35 775240 284.904711.870529 44.129471 25 36.775410 2.8.904617 1.56.870793.40.129207 24 37.775530 2.83.904523 1.57.871057 4.40 128943 23 38 775750 2.83.904429 57.871321 40.128679 22 39.775920 2.83.904335 157.871585 40.128415 21 40 9.776090 283 9.904241 57 9.871849 440 0.128151 20 41.776259 283.904147.872112.127888 19 42.776429 282.904053 57.872376 4.39.127624 18 43.776598 2.903959 1.5.872640 9 127360 17 44.776768 2.8.903864 57.872903.127097 16 45.776937 2.82 903770 157.873167 439.126833 15 46.777106 282.903676 157.873430 39.126570 14 47.777275 82.903581.873694.126306 13 48.777444.903487.5.873957 39.126043 12 49.777613 2.903392 58.874220 9 125780 11 2.81 1.68 4.39 50 9.777781 2 81 9.903298 1 58 9.874484.39 0.125516 10 51.777950 281.903203 158.874747 39 125253 9 52.778119 2.903108.875010 39 124990 8 53.778287 2.81.903014 1.58.875273 124727 7 54.778455 2.902919 158.875537 438 124463 6 55.778624 280.90224 18.875800 433.124200 5 56.778792 2.902729.5.876063 438 123937 4 2.70719 1.59 4.38.124993 57.778960 2.80 902634 1.58.876326..123674 3 58.77912 22.80.902539.59.876589.123411 2 59.779295 279 902444.876852 38 123148 1 60.779463._.902349.5.877114.122886 0 M. Cosine.. 11". Sine. D. 11. Cotang. D. 11". Tang. M. 5t 260 ~9 1%6o ~~*~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~878243 134 COSINES, TANGENTS, AND COTANGENTS. 211 373 142 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.779463 279 9.902349 1. 9.877114 4. 0.122886 60 1.779631 79.9(12253.877377 4 3.122623 59 2.779798.902158 1.59 877640..122360 58 2.79 1.59 4,3 3.7 279966.9203.5.877903 4.38.122097 57 4.780133 79.901967 1.9.878165..121835 56 5.780300 78.901872 59.87842.3.121572 55 6.780467.901776 19.87691.121309 54 7.780634 278.90161.878953.121047 53 2.78 6 4.385 8.780801.901585.879216 4.120784 52 9.780968 78 901490 160.879478.120522 51 10 9.781134 278 9.901394 1 60 9.879741 4 0.120259 50 11.781301 277.901298 60.880003..119997 49 12.781468 27.901202 1.880265 4..119735 48 2.77 1.60 4.37.119472 47 13.781634 277.901106 60.880528 47.119472 47 14.781800 27.901010 16.880790 4..119210 46 15.781966 77.900914 60.881052 7.118948 45 16.782132 277.900818 1 881314 4..1186S6 44 17.782298 276.900722 10.881577.118423 43 18.782464.7.900626.881839 4.37.118161 42 19.782630 276.900529 61.882101 437.11799 41 20 9.782796 76 9.900433 61 9.882363 0.117637 40 21.782961 276.900337 1 6.882625.117375 39 22.783127 26.900240 161.882887.117113 38 23.783292.900144.883148 43.116852 37 24.783458.900047 61.83410 36.116590 36 25.783623 75.899951 61.883672 436 116328 35 26.783788.899854 16.883934 4..116066 34 27.783953 25.899757 1.61.884196 36.115 33 28.784118.75 899660.6.884457.1553 32 2275 8845 4.36.115543 32 29.784282.899564 884719 36.11521 31 30 9.784447 9.899467 62 9.884980 4 3 0.115020 30 96 2.74 1.62 4,36 4 29 31.784612.74.899370 162.885242.114758 29 31 ]',,~ ].114496 28 32.784776.899273 62.885504 4 36.114496 28 33.784941 2.4.899176.885765 436.114235 27 34.785105 2.74.899078 62.886026.113974 26 35.785269 2.73.898981 62.886288 36.113712 2 36.785433 273.898884 1 62.886549.113451 24 37.785597 2.73.898787 162.886811 4..11319 23 38.785761 23.898689 162.887072 4.3.112928 22 39.785925 23.898592 1..887333 435.112667 21 2:73 1:62 435 3 2 40 9.786089 2 73 9.898494 1 63 9.887594 4.3 0.112406 20 41.786252 27.898397 13.887855. 5.112145 19 42.786416 27.898299 16.888116.3.1114 18 43.786579 272.898202 63.888378 4.3.111622 17 44.786742 72.898104 13.888639 5.111361 16 45.786906 72.898006 163.888900 4.1111(0( 1 46.787069 27.897908.63.889161 3.11083 14 47.787232 272.897810 13.889421.110579 13 48.787395 2.7.897712.6.889682 4 3.110318 12 49.787557 71.897614 163.889943 4.3.110057 1I 2.7861.63 4:35 50 9.787720 2 7 9.897516 1 64 9.890204 435 0.109796 10 51.787883.71.897418 64.890465 43.109535 9 52.788045 2.71.897320 1.6.890725.109275 S 53.788208 71.897222 164.890986.109014 7 54.788370 2.7.897123 1.64 891247 434.108753 6 55.788532 70.897025 14.891507 44.8493 56.788694 2.7.896926 6.891768..108232 4 57.788856 270.896828.64.892028 4.107972 3 58.789018 70.896729 164.892289.107711 2 59.789180 270.896631 1.64 892549.107451 1 60 9342.895.789389.892810.107190 0 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1.". Tang. M. 55.78_ tsrG~~.7 [ 8725.6'43 212 TABLE XIII. LOGARITHMIC SINES, 3801 1410 M. Sine. D. 1". Cosine. D. 1". Tang. D. l. Cotang. M. 0 9.789342' 269 9.896532 165 9.892810 44 0.107190 60 1 789504 269.896433 165.893070 4.34.106930 59 2.789665 69.896335 1.65.893331 4.34.106669 58 3.789827 269.896236.5.893591.106409 57 2.69 1.65 4.34.789988 2.69 896137 65.893851 4.34.106149 56 5.790149 269.896038 1.65.894111 4.34 105889 55 6.790310 895939 65 89432 4.105628 54 7.790471 268.895840 1.65 894632 4.105368 53.790632..89541 5. 8 790632.895741.65.89492 4.3.105108 52 9.790793 2.895641.895152.104848 51 2.68.8954.33 10 9.790954 9.895542 9.895412 4.33 0.104588 50 11.791115 26.895443 66.895672.104328 49 12.791275 26.895343 66.895932.104068 48 13.791436 26.895244 166.896192.103808 47 14.791596 26.895145 66.896452 4. 103548 46 15.791757 26.895045 16.896712.103288 45 16.791917.894945.896971 4.33.103029 44 17.792077 2.894846 66.897231 4.33 102769 43 18.792237 2.6.894746 1.6.897491.102509 42 19.792397 2.89446 1.66.897751 4.33.102249 41 20 9.792557 266 9.894546 1.67 9.898010 4.3 0.101990 40 21.792716 26.894446 6.898270 43.101730 39 22 792876 266.894346 1.67 898530.101470 38 23.793035.894246 67.898789.101211 37 24.793195 266.894146 167.899049 433 100951 36 26.793514 893946 1.899568.100432 34 4965 77*91.6 7. 89 4.33 ~ 01 1!37 27.793673 2.65.893846 1.67.89927 4.3.100173 33 28.793832 26.893745 17.900087 4.099913 32 29.793991 2.893645 1.9346 32. 099654 31 30 9.794150 65 9.893544 168 9.900605 32 0.099395 30 31.794308 264.893444 1.6.900864.099136 29 32.794467 26.893343 16.901124 43.098876 28 33.794626 264.893243 1.901383 43.098617 27 34 2794784 24.893142.901642.098358 26 35.794942 264.893041.68.901901 432.098099 25 36.795101 2.892940 168.902160.097840 24 37.795259.892839 68.902420 32.097580 23 2.6 5 1.68 4.32 38.795417 2.892739 1.902679 4.097321 22 39. 795575 ^2.892638.902938.097062 21 2. 893 44 1.68 4.32 40 9.795733 263 9.892536 69 9.903197 3 0.096803 20 41.795891 23.892435 69.903456.096544 19 42.796049 2.892334 169 903714 31.096286 18 43.796206.892233 903973.096027 17 44.796364 2.892132 169.922 431.095768 16 45.796521 2.892030.69 904491 4.31.095509 15 46.796679 2.6.891929 1.6.904750 4.31.095250 14 47.796836.891827 69 905008 31.094992 13 48.796993 262.891726 169.905267 431.094733 12 49.797150 2.61.891624 1.6.905526 4:31.094474 11 50 9.797307 261 9.891523 170 9.905785 4.31 0.094215 10 51.797464 2.6.891421 1.70.906043.093957 9 52.797621 261.891319 70.906302 4.31.093698 8 53.797777.891217 70.906560 4.31.093440 7 9.60 19 4.31 54.797934 61.891115 70.906819.093181 6 55.79S091 261.891013 170 907077.092923 5 56.798247 261.890911 170 907336.092664 4 57.798403.80809 70.907594 431.092406 3 58.798560.890707 907853.092147 2 2.60.8900 170 4.31 59.798716.890605 170.908111 4.31.091889 1 60.798872.890503.908369.091631 0 M. Cosine. D. I". Sine. D. 1". Cotang. D. 1". Tang. M. 1280 51 COSINES, TANGENTS, AND COTANGENTS. 21c 390 140C M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.793872 9.890503 9.908369 30 0.091631 60 1.799028 60 890100 1.71 908628.091372 59 2.799184 2.60.890298.71 903886 3.091114 58 3.799339.890195.909144 30.090856 57 4.79995 259.890093.71.909402 430.090598 56 5.799651.889990 1 1.909660.090340 55 2.59 1.71 4.30 6.799806 2..889888 909918. 090082 54 7.799962 2.59 889785 71.910177.089823 53 8.800117 2.59.889682 1.910435.089565 52 9.800272 2.59.889579 71.910693 30.089307 51 2.59 1.71 4.30 10 9.800427. 9.889477.72 9.910951 4 0.089049 50 1.800582 2.58 889374 72.911209.088791 49 12.800737 2.58 889271 1.72.911467 430.088533 48 13.800892 2.58.889168 1.7.911725.088275 47 14.801047 2.58 889064 1.72 91192 4..038018 46 15.801201 2.58 888961 172.912240.3.087760 45 16.801356 2.58.88858 1.72.912498 4.30.087502 44 17.801511 2.5.88755 1..912756.087244 43 2.57 1.72 913014 4.30 18.801665 2.57.888651 1.7.913014 4.086986 42 19.81819 2.57.888548 1.72 913271.086729 41 20 9.801973 27 9.888444 9.913529 429 0.086471 40 21.802128 25.888341.913787 429.086213 39 22.802282 25.888237 73.914044 429.085956 38 23.802436 256.888134 1.73.914302 49.085698 37 24.802.589 26.888030.3 914560 29.085440 36 25.802743 256.887926 73.914817 29.085183 35 26.802897 2.5.887822 1.95075 429.084925 34 27.803050 56.887718 73.915332 4.084668 33 28.803204 2.56.87614 173.915590 429.084410 32 29.803357 2.55.887510 1:74.915847 4:29.084153 31 30 9.803511 255 9.887406 74 9.916104 4.2 0.083896 30 31.803664 255.887302 1.916362 29.083638 29 32.803817 255.887198 1.74.916619 29.083381 28 33.803970 887093.74.916877 29.083123 27 34.804123 255 886989 174.917134 4.082866 26 35*.804276 255.886885 1.99171 429.02609 25 36,804428..886780 1..917648 4:29.082352 24 37.804581 54.886676 1.74.917906.082094 23 38.804734 54.886571 74.918163 42.081837 22 39.804836.886466.918420.081580 21 2.54 1.75 4.29 40 9.805039 54 9.886362 75 9.918677 4 28 0.081323 20 41.805191 254.886257 175 918934 28.081066 19 42.805343 2.54.886152 1.75.919191 428.080809 18 43.805495 25.886047 1.7.919448 4.080552 17 44.805647 25.885942 1.75 919705 28.080295 16 45.805799.5.885837 1.5.919962 428.080038 15 46.805951 23.885732 75 920219 428.079781 14 47.806103 253.885627 175 920476 28.079524 13 48.806254 253.885522 1.75.920733 428.079267 12 49.806406 2:52.885416 176.920990 4:28.079010 11 50 9.806557 252 9.885311 76 9.921247 428 0.078753 10 51.806709.885205 176.921503 428.078497 9 52.806860.52. 885100 1.76.921760 428.078240 8 53.807011 2.52 84994 176 922017 2.077983 7 54.807163 252.884889 1.76.922274 4.28.077726 6 55.807314 2.5.884783 1.7.922530 428.077470 5 56.807465 2.5.884677 76.922787 4.077213 4 57.807615 251 884572 1.76.923044 4.2.076956 3 58.807766 2.5 884466 1.7.923300 28.076700 2 59.807917.51 884360.77 923557 428.076443 1 60.808067.884254.923814.076186 0 M. Cosine. D. 1". Sine. D. 1". otang. D. 1". Tang. M. 1293 A4 214 TABLE XIII. LOGARITHMIC SINES, 400 1390 M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.808067 9.884254 7 9.923814 0.076186 60 1.808218 261.884148.924070.075930 59 2.808368 251.884042 177.924327 42.075673 58 2.51 1.77 4.27 3.808519 250.883936 77.924583 27.075417 57 4.808669 2.883829.924840 27.075160 56 5.808819 250.883723'.925096 427.074904 55 6.808969 250.883617 177.925352 27.074648 54 7.809119 50.883510 77.925609 47.074391 53 8.809269.883404 8.925865.074135 52 9.809419 250.883297 78.926122 427.073878 51 2.50 1.78 4.27 10 9.809569 9.883191 9.926378 0.073622 50 11 809718 2.49 883084 1.8 926634.27 073366 49 12.809868 2.49 882977 1.78 926890 4.27 073110 48 13.810017 2.49.882871 1.7.927147 4.2 072853 47 14.810167 2.49 882764 1.78 927403 4.27 072597 46 15.810316 2.49 882657.78.927659 4.27 072341 45 16.810465 2.4.882550 78 927915 4.27 072085 44 17.810614 2..882443 1.7.928171.2 071829 43 2.48 1.79 4.27 18.810763 2.48.882336 79.928427 4.27 071573 42 19.810912 48.882229 1.92S64 27 071316 41 2.48 1.79 4.27 20 9.811061 2 9.88-2121 9 9.928940 7 0.071060 40 21.811210 248.882014 1.79.929196 427.070804 39 22.811358 2.48.881907 179.929452 4.070548 38 23.811507 247.881799 179.929708 4.27.070292 37 24.811655 247.881692 179.929964 4.27.070036 36 25.811804 2.881584 179.930220 2.069780 35 26.811952 2.47.881477 1..930475 4.2.069525 34 27.812100 47 881369.930731 4.26 069269 33 28.812248.881261 80.930987 4.26.069013 32 29.88196 881153.931243 4.26.068757 31 30 9.812544 6 9.881046. 9.931499 6 0.068501 30 31 312692 246.880938 1..931755 4.26.068245 29.067990 28 32.812840 2.46.880830 1.80.932010 4.26 067990 28 33.812988 246.880722 1.80.932266 4.26.067734 27 34.813135 246.880613 180.93222 4.26 067478 26 35.813283 2.46.880505 1..932778 4.26.067222 25 36.813430 2.46 880397 1.80.933033 4.26 066967 24 37.813578 2.46.880289 1.81.933289 4.26.066711 23 38.813725 245.880180 1.81.933545 26.066455 22 39.813872 25.880072 11.933800 4.066200 21 2.45 1.81 4.26 40 9.814019 5 9.879963 1 9.9.34056 6 0.065944 20 41.814166 245.879855 181.934311 426.065689 19 42.814313 45.879746 8.934567 26.065433 18 43.814460 245.879637 1.1.934822 4.6.065178 17 44.814607 2.4.879529 181.935078 426.064922 16 45.814753 244.879420 1.81.935333 4.26.064667 15 46.814900 244.879311 1.8.935589 4.26.064411 14 47.815046 244.879202 182.935844 426.064156 13 48.815193 44.879093.936100 26.063900 12 49.815339 2.44 878984 2.936355 4.26.063645 11 2.44 1.82 4.26 50 9.815485 44 9.878875 2 9.936611 6 0.063389 10 51 815632 2.4.878766 182.936866 26.063134 9 52.815778 243.878656.82.937121 4.26.062879 8 53.815924 2.43.878547 1.82.937377.062623 7 54.816069 2.43.878438 1.82.937632 4.25 062368 6 55.816215 243.878328 1.8.937887 4.2.062113 5 56.816361 243.878219 83.938142 425 061858 4 57.816507 43.878109 1..938398 4.5.061602 3 58.816652 2.877999 183.938653.061347 2 59.816798 242.877890 183.938908 425 061092 1 2.42 1.83 4.25 60.816943 2 877780.939163.060837 0 M. Cosine. D.. Sine. D. 1". Cotang. D. ". Tang. M. 102 A490 COSINES, TANGENTS, AND COTANGENTS. 215 410 1380 3. Sine. D. ". Cosine. D. 1". Tang. D. I. Cotang. M. 0 9.816943 242 9.877780 9.939163 425 0.060837 60 1.817088 242.877670 83.939418 25.060582 59 2.817233 42.877560 83.939673 425.060327 58 3.817379 2 42.877450 183.939928 425.060072 57 4.817524 242.877340 184.940183 425.059817 56 5.817668 21.877230 1..940439 425.059561 55 6.817813 241.877120 184.940694 4.5.059306 54 7.817958 2.41.877010 84 940949 425.059051 53 8.818103 2'41.876899 184.941204 425.058796 52 9.818247 2 41.876789! 84.941459 425.058541 51 10 9.818392.876678 9.941713 0.058287 50 I1.818536 2.1.876568 1.84.941968 4.5.058032 49 12.818681 24.876457 184.942223 425.057777 48 13.818825 40.876347 1.84.942478 4.25.057522 47 14.818969 2.0.876236 1.8.942733 425.057267 46 15.819113.4.876125 1.8.942988 4.25.057012 45 16.819257 2.40.876014.5.943243 4..056757 44 17.819401 2.40 875904 1.85 943498 4.25.056502 43 18.819545 2.40 875793.85.943752 4.25.056248 42 19.819689 2.9.875682 1.85.944007 25.055993 41 20 9.819832 2 39 9.875571 1 85 9.944262 4 25 0.055738 40 21.819976 239.875459 1-8.944517 4.25.055483 39 22.820120 29.875348 185.944771.05229 38 23.820263.875237.945026.054974 37 24.820406 2.39.875126.86.945281 4.24.054719 36 25.820550 239.875014 86.945535 4.24.054465 35 26.820693.38.874903 186.945790 4.24.054210 34 27.820836 238.874791 186.946045 4 4.053955 33 28.820979 2.874680 1.946299 24.053701 32 29.821122 38.87456 1' 4655.44 053446 31 30 9.821265 238 9.874456 16 9.946808 424 0.053192 30 31.821407 38.874344 86 947063 24.052937 29 32.821550 238.874232 1.8.947318 424.052682 28 33.821693 237.874121 187.947572 424.052428 27 34.821835 37.874009 87.947827 24.052173 26 35.821977 237.873896 17.948081 24.051919 25 36.822120 7.873784. 948335.051665 24 37.822262 2.37.873672 187.948590 4.24.051410 23 38.822404 2.37.873560 187.948844 4..051156 22 39.822546 23.873448 1 949099.050901 21 2.37 1.87 4.24 40 9.822688 2 37 9.873335 87 9.949353 24 0.050647 20 41.822830 236.873223 1.88.949608.050392 19 42.822972.873110.88 949862 24.050138 18 43.823114 236.872998..950116 4.4.049884 17 44.823255 2.36.872885 1.88.950371.24.049629 16 45.823397 36.872772.88.950625 24.049375 15 46.823539 236.872659.88.950879 424.049121 14 47.823680 2.36.872547 1.88.951133 424.048867 13 48.823821 23.872434..951388 424.048612 12 49.823963 2.872321 88.951642.048358 11 4.836 2.35 1.88 424 50 9.824104 23 9.872208 9 9.951896 4 0.048104 10.9 1.8.952150.047850 9 51.824245 2.35 872095 I.89 952150 4.24.047850 9 52.824386 2.35.871981 1.89.952405 4.24.047595 8 53.824527 2.5.871868.89.952659 424.047341 7 54.824668 35.871755 1.8.952913 24.047087 6 55.824808 2.3.871641 1.89.953167..046833 5 56.82949 234.871528.89.953421 4.24.046579 4 57.825090 234.871414 1.89 953675 423.046325 3 58.825230 34.871301 1..953929 423.046071 2 59.82537 71187 1..954183 42.045817 1 60.825511 2.871073..954437.045563 0 M. Cosine. D.. Sine. D. 1". Cotang. D.1". Tang. M. 1310 4ia 216 TABLE XIII. LOGARITHMIC SINES, 412o 1370 M. Sine. D. 1". Cosine. D. ". Tang. D. 1". Cotang. M. 0 9.825511 234 9.871073 190 9.954437 423 0.045563 60 1.825651 234.870960 10.954691.045309 59 2.825791 2.870846 90.954946 43.045054 58 3.825931 233.870732.955200 3.044800 5 4.826071 2.870618 10.9515454 23.044546 56 5.826211 233.870504 190.9557085 423.044292 55 6.826351 2.33.870504 1 90 4.23.4429255 2.33.870390 1.90 6.826351 233.870390 9.955961 423.044039 54 7.826491 2-33.870276.956215.043785 53 8.82 1.90 4.23 8.826631.870161 1.9.56469 23.043531 52 9.826770 2.33.870047 1.91.956723 423.043277 51 10 9.826910 232 9.869933 191 9.956977 4.23 0.04302.3 50 11.827049 32.869818 1.957231.042769 49 12.827189 2.3.869704..957485 23.042515 48 13.827328 232.869589 1.957739 423.042261 47 14.827467 232.869474 91.957993.042007 46 15.827606 232.869360 191.958247 423.041753 45 16.827745 232.869245 9.958500.041500 44 17.827884 23.869130 192.95754 423.041246 43 18.828023.31.869015 92.99008 423.040992 42 19.828162 23.868900.959262.040738 41 2.31 1.92 4.23 20 9.828301.31 9.868785 192 9.959516 423 0.040484 40 21.828439 23.863670 92.959769 423.040231 39 22.828578 21.868555 192.960023 423.039977 38 23.828716 231.86440 192.960277 423.039723 37 24.828855 2.868324 192.960530 23.039470 36 25.828993 2.863209 192.960784 423.039216 35 26.829131 30.86093.961038 423.038962 34 27.829269 1.93 4.23 27.829269 230.867978 93.961292 423.038708 33 28.829407 230.867862 193.961545 423.038455 32 29.829545 2:30.867747 1:23.961799 4:23.038201 31 30 9.829683 2 3 9.867631 9.962052 0.037948 30 31.829821.867515.962306 23.037694 29 2.30 1.93 4.23 32.829959 2.29.867399 1.9.962560 423.037440 28 33.830097 229.867283 193.962813.037187 27 34.830234 229.867167 9.963067 423.036933 26 35 830372 29.867051 194.963320 423.036680 25 36.830509 2.866935.963574.036426 24 2.29 8 1.9 4 4.23 37.830646 229.866819 1.94.963328 423.036172 23 38.830784 22.866703 14.964081 23.035919 22 39.830921 229.866586 4.964335 4.23 035665 21 2.29 2 1.94 4.23 40 9.831058 2.28 9.866470 94 9.964588.22 0.035412 20 41.831195 28.866353 1.964842.035158 19 42.831332 228.866237 1.96. 095.034905 18 43.831469 28.866120 194.965349 22 034651 17 44.831606 228.866004 195.965602 422.034393 16 46.831879 2.2.865770.966109 4 033891 14 47..832015 1.95 4.22 47.832015 227.865653.95.966362 422.033638 13 48.832152 27.865536 195.96666 22.033384 12 49.832288 2:27.865419 1:95.966369 4:22.033131 11 50 9.832425 227 9.865302 1 9 9.967123 422 0.032877 10 51.832561 227.865185 1.967376.032624 9 52.832697.865063.967629 422.032371 8 53.832333 227.864950 1.96.967883 422.032117 7 51.832969 227.864833 1.968136 22.031864 6 55.833105 226.864716 19.968389 4.031611 5 56.833241 2.2.864593.9.968643 422.03137 4 57.833377 2.2.864481 96.968896 22.031104 3 58.833512 2.2.864363.969149 22.030851 2 59.833618 22.864245.969403 422.030597 1 60.833783..864127..969656 030344 0 M. Cosine. D. 1". Sine. D. 1". Cotang. D. 1". Tang. M. 1329 470 COSINES, TANGENTS, AND COTANGENTS. 21*7 1.30 136c M. Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.833783 226 9.864127 196 9.969656 4.22 0.030344 60 1.833919 226.864010 197.969909 22.030091 59 2.834054 225.863892 1.970162 22.029838 58 3.834189 225.863774.970416 22.029584 57 4.834325 2 863656.970669.029331 56 2.25 ~ 1.97 422 5.834460 225.863538 97.970922 22.029078 55 6.834595 2.863419 97.971175 22.028825 54 7.834730 225.863301 1.971429 22 028571 53 8.834865.863183 97.971682 22.028318 52 9.834999 2.25.863064 97.971935 22.028065 51 10 9.835134 9.862946 9.972188 0.027812 50 11.835269 2.24 862827 1.98 972441 4.22.027559 49 12.835403 2.24 862709 1.98 972695.027305 48 13.835538 2.24 862590 1.98.972948 4.22.027052 47 2.24 1.97 4.22 14.835672 2.24 862471 1.98.973201 4.026799 46 15.835807 2.862353 1.973454 4.22.026546 45 16.835691 2.24 98.026293 44 16.835941 2.24 862234 98.973707 422.026293 44 17.836075 2.2.862115.973960 4.026040 43 18.836209 23.861996 98.974213 4.22.025787 42 19.836343.861877 1.974466 4.22.025534 41 2.23 1.99 4.22 20 9.836477 22 9.861758 9.974720 0.025280 40 21.836611 23.861638 99.974973 22.025027 39 22.836745 223.861519 199.975226 22.024774 38 23.836878.861400.975479.024521 37 24.837012 2.23 861280 1.99.975732 4.22.024268 36 2-00 1.99 4.22 25.837146 23.861161.975985.024015 36 2.22 861041 1.99. 422 26.837279 2.861041.99.976238 4.023762 34 27.837412 2.860922.0.976491 4.023509 33 28.837546 2.2 860802 2.00.976744 22.023256 32 29.837679 2.22.860682.976997 22.023003 31 30 9.837812 2.22 9.860562 2.00 9.977250 422 0.022750 30 31.837945 2.22.860442 200.977503 22.022497 29 32.838078 2.860322 2.977756 22.022244 28 2.21 8620 4.22 33.838211 2.860202.978009 22.021991 27 34.838344 2.21.860082 200.978262 422.021738 26 35.838477 2.21 859962.978515 22.021485 25 36.838610 2.21 859842 1.978768 22.021232 24 37.838742 2.21 859721 201.979021 422.020979 23 38.83875.859601.979274 22.020726 22 2.21 2.2 3 4.22 39.839007 2.859480.979527,020473 21 2.21 2.01 4.22 40 9.839140 2.21 9.859360 201 9.979780 422 0.020220 20 41.839272 2.20 859239 201.980033 422.019967 19 42.839404 2.20.859119 01.980286 22.019714 18 43.839536 2.20 858998 2.01.980538 422.019462 17 44.839668.858877.980791 422.019209 16 45.839800 2.20 858756 2.02.981044 421.018956 15 46.839932 2.2 858635.981297 21.018703 14 47.840064 2.20 858514 2.02.981550.018450 13 421.82.040 4.21 48.840196 2..858393 2.02.931803 421.018197 12 49.840328 2.19 858272 2.02.982056.017944 11 2.19 7 2.02 4.21 50 9.840459 2.19 9.858151 202 9.982309 4.21 0.0176)1 10 51.840591 2.19.858029 202.982562, 4.21.017438 9 52.840722.857908.02 982814 21.017186 8 53.840854.857786 03.983067 21.016933 7 54.840985 2.19 857665.983320 21.016680 6 55.841116 2.19.857543 203.93573 421.016427 5 56.841247 2.1 857422.983826 421.016174 4 57.841378 2.18.857300 2.03.984079 421.015921 3 58.841509 2.18.857178 2.03.984332 21.015668 2 60.841771.856934.984837.015163 0 M. Cosine. D.I". Sine. D. 1". Cotang. D. 1". Tang. M. 1330 t 218 TABLE XIII. LOGARITHMIC SINES, &C. 440 1356 M. Sine. D. 1".' Cosine. D. 1". Tang. D. 1". Cotang. M. 0 9.841771 2.18 9.856934 203 9.9,84837 4.21 0.015163 60 1.841902 18.856812 204.985090 21.014910 59 2.842033 218.856690 204.985343 421.014657 58 3.842163 218.856568 204.985596 421.014404 57 4.842294 217.856446 2.04.985848 4.21.014152 56 5.842424 2.17.856323.986101 4.21.013899 55 6.842555.856201 04.986354 21.013646 54 7 842685 17.856078 204.986607 421.013393 53 8.842815 2.17.855956 2.04.986860 421.013140 52 9.842946 2.855833 04.987112 21.012888 51 10 9.843076 2.17 9.855711 2.05 9.987365 4.21 0.012635 50 11.843206 217.855588 05.987618 21.012332 49 12.843336 2.855465 05.987871 21.012129 48 13.843466 216.855342 205.988123 21.011877 47 14.843595 216.855219 205.988376 421.011624 46 15.843725 2.16.855096 205.988629 2.011371 45 16.843855.854973.988882.011118 44 2.16 2.05 4.24 17.843984 216.854850 205.989134 4.010866 43 18.844114 216.854727 206.989387. 421.010613 42 19.844243 21.854603 06.989640 21.010360 41 20 9.844372 215 9.854480 206 9.989893 421 0.010107 40 21.844502.854356 06.990145 21.009855 39 22.844631 215.854233 206.990393 421.009602 38 23.844760 215.854109 06.99051 21.009349 37 24.844889 215.853986 206.990903 421.009097 36 25.845018 15.853862 06.991156 41.008844 35 26.845147 2.853738 0.991409 421.008591 34 27.845276 15.853614 2.07.991662 21.008338.33 28.845405 2.853490 07.991914 21.008086 32 29.845533 2.1.853366 07.992167 21.007833 31 30 9.845662 214 9.853242 07 9.992420 4.21 0.007580 30 31.845790 214.853118 0.992672 421.007328 29 2.14 2.06 4.21 28 32.845919 214.852994.992925 21.007075 28 33.846047 214.852869 07.993178 21.006822 34.846175 214.852745 07.993431 21.006569 26 35.846304 14.852620.08.99363 21.006317 25 36.846432.852496 08 993936 21.006064 24 37.846560 13.852371 08.994189 21.005811 23 38.846688 2.852247.994441.005559 22 39.846816 2.852122 08 994694 21.005306 21 40 9.846944 213 9.851997 208 9.994947 421 0.005053 20 41.847071 213.851872 2.995199 21.004801 19 2.13 2.08 4.21 17 42.847199 2.851747 2.995452 42.004548 18 43.847327 213.851622 209.995705 421.004295 17 44 47454.851497.995957 21.004043 16 45.847582 212.851372 09.996210 21.003790 15 46.847709 212.851246 209.996463 4:21.003537 14 47.847836 2.851121 2.996715 4.21.003285 13 48.847964 12.850996.996968 4.21.003032 12 49.848091 2.12.850870.997221 4.21.002779 11 50 9.848218 9.850745 9.997473 0.002527 10 51.848345 212.850619.997726 42.002274 9 52.848472.850493.997979.002021 8 53.848599.850368 2.10..98231 4.21.001769 7 882 2.11. 4 2.210.9877 4.21 55.848852 211.850116 10.998737.001263 5 56.848979 2.11.849990 2.10.998989 4.21.001011 4 57.849106 211.849864 210.999242 4.21 000758 3 58.849232 11.849738.999495 4.21 000505 2 59.849359 11.849611 211.0999747 4.21 000253 1 60.849485.849485 0. 00 000000 0 M. Cosine. D.1". Sine. D. i"'. Cotang. D.1".1 Tang. I M. 1340 4~0 TABLE XIV. NATURAL SINES AND COSINES 220 TABLE XIV. NATURAL SINES AND COSINES. 0o lo 20 30 40 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. 0.00000 One..001745.99985.03490.99939.05234.99863.06976 99756 60 1.00029 One..01774.99984,03519.99938.05263.99861.07005.99754 59 2.00058 One..01803.99984.03548.99937.05292.99860.07034.99752 58 3.00087 One..01832.99983.03577.99936 05321.99858.07063.99750 57 4.00116 One..01862.99983.03606.99935.05350.99857.07092.99748 56 5.00145 One..01891.99982.03635.99934.05379.99855.07121.99746 55 6.00175 One..01920.99982.C(664.99933.05408.99854.07150.99744 54 7.00201 One..01949.99981.03693.99932.05437.99852.07179.99742 53 8.00233 One..01978.99980.03723.99931.05466.99851.07208.99740 52 9.00262 One..02007.99980.03752.99930.05495.99849.07237.99738 51 10.00291 One..02036.99979.03781.99929.05524.99847.07266.99736 50 11.00320.99999.02065.99979.03810.99927.05553.99846.07295.99734 49 12.00349.99999.02094.99978.03839.99926.05582.99844.07324.99731 48 13.00378.99999.02123.99977.03868.99925.05611.99842.07353.99729 47 14.00407.99999.02152.99977.03897.99924.05640.99841.07382.99727 46 15.00436.99999.02181.99976.03926.99923.05669.99839.07411.99725 45 16.0046.5.99999.02211.99976.03955.99922.05698.99838.07440.99723 44 17.00495.99999.02240.99975.03984.99921.05727.99836.07469.99721 43 18.00524.99999.02269.99974.04013.99919.05756.99834.07498.99719 42 19.00553.99993.02298.99974.04042.99918.05785.99833.07527.99716 41 20.00582.99993.02327.99973.04071.99917.05814.99831.07556.99714 40 21.00611.99998.02356.99972.04100.99916.05844.99829.07585.99712 39 22.00640.99998.02335.99972.04129.99915.05873.99827.07614.99710 38 23.00669.99993.02414.99971.04159.99913.05902.99826.07643.99708 37 21.00698 99993.08443.99970.04188.99912.05931.99824.07672.99705 36 25.00727.99997.02472.99969.04217.99911.05960.99822.07701.99703 35 26.00756.99997.02501.99969.04246.99910.05989.99821.07730.99701 34 27.00785-).99997.02530.99968.04275.99909.06018.99819.07759.99699 33 23.00814.99997.02560.99967.04304.99907.06047.99817.07788.99696 32 29.00844.99996.02589.99966.04333.99906.06076.99815.07817.99694 31 30.00373.99996.02618.99966.04362.99905.06105.99813.07846.99692 30 31 00902.99996.02647.99965.04391.99904.06134.99312.07875.9968 29 32.00931.99996.02676.99964.04420.99902.06163.99310.07904.99687 28 33.00960.99995.02705.99963.04449.99901.06192.99808.07933.99685 27 34.00939.99995.02734.99963.04478.99900.06221.99806.07962.99683 26 35.01018.99995.02763.99962.04507.99898.06250.99804.07991.99680 25 36.01047.99995.02792.99961.04536.99397.06279.99803.08020.99678 24 37.01076.99994.02821.99960.04565.99396.06308.99801.08049.99676 23 33.01105.99994.02350.99959.04594.99894.06337.99799.08078.99673 22 39.01134.99994.02379.99959.04623.99393.06:366.99797.08107.99671 21 40.01164.99993.02908.99958.04653.99392.06395.99795.08136.99668 20 41.01193.99993.02938.99957.04682.99890.06424.99793.08165.99666 19 42.01222.99993.02967.99956.04711.99889.06453.99792.08194.99664 18 43.0125.1.99992.02996.99955.04740.99388.06482.99790.03223.99661 17 44.01230.99992.03025.99954.04769.99886.06511.99788.08252.99659 16 45.01309.99991.03054.99953.04798.99835.06540.99786.08281.99657 15 46.01338.99991.03083.99952.04827.99883.06569.99784.08310.996.54 14 47.01367.99991.03112.99952.04856.99382.06598.99782.03339.99652 13 43.01.396.99990.03141.99951.04885.99831.06627.99780.08363.99649 12 49.01425.99990.03170.99950.04914.99879.06656.99778.08397.99647 11 50.014.4.99939.03199.99949.04943.99978.06685.99776.08426.99644 10 51.01483.99989.03223.99948.04972.99876.06714.99774.08455.99642 9 52.01513.99989.03257.99947.05001.99875.06743.99772.08484.99639 8 53.01542.99988 03236.99946.05030.99873.06773.99770.08513.99637 7 54.01571.99938.03316.99945.05059.99872.06302.99763.08542.99635 6 55.01600.99937 03345.99944.05088.99870.06331.99766.08571.99632 5 56.01629.99937.03374.99943.05117.99869.06360.99764 08600.99630 4 57.016;58.99986.03403.99942.05146.99867.06.389.99762.08629.99627 3 58.01687.99986.03432.99941.05175.99866.06918.99760.08658.99625 2 59.01716.99935.03461.99940.05205.99864.06947.99758.08687.99622 1 60.01745.99985.03490.99939.05234.99863.06976.99756.08716.99619 0 M. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. M. 893 88D 870 86go s85 _...... _....~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE XIV. AATURAL SINES AND COSINES. 221 o50 60 7o 80 90 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. 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 2.08774.99614.10511.99446.12245.99248.13975.99019.15701.9S760 58 3.08803.99612.10540.99443.12274.99244.14004.99015.15730.98755 57 4.0883 1.99609.10569.99440.12302.99240.14033.99011.15758.98751 56 5.08860.99607.10597.99437. 12331.99237.14061.99006.15787.98746 55 6.08889.99604.10626.99434.12360.99233.14090.99002.15816.98741 54 7.08918.99602.10655.99431.12389.99230.14119.98998.15845.98737 53 8.08947.99599.10684.99428.12418.99226.14148.98994.15873.98732 52 9.08976.99596.10713.93424.12447.99222.14177.98990.15902.98728 51 10.09005.99594.10742.99421.12476.99219.14205.98986.15931.98723 50 11.09034.99591.10771.99418.12504.99215.14234.98932.15959.98718 49 12.09063.99583.10800.99415.12533.99211.14263.98978.15988.98714 48 13.09092.99586.10829.99412.12562.99208.14292.98973.16017.98709 47 14.09121.99583.10858.99409.12591.99204.14320.98969.16046.98704 46 15.09150.99580.10387.99406.12620.99200.14349.98965.16074.98700 45 16.09179.99578.10916.9910*2.12649.99197.14378.98961.16103.98695 44 17.09208.99575.10945.99399.12678.99193.14407.98957.16132.98690 43 18.09237.99572.10973.99396.12706.99189.14436.98953.16160.98686 42 19.09266.99570.11002.99393.12735.99186.14464.98948.16189.98681 41 20.09295.99567.11031.99390.12764.99182.14493.98944.16218.98676 40 21.09324.99564.11060.993S6.12793.99178.14522.98940.16246.98671 39 22.09353.99562.11089.99383.12822.99175.14551.98936.16275.98667.38 23.09332.99559.11118.99380.12851.99171.14580.98931.16304.98662 37 24.09411.99556.11147.99377.12880.99167.14608.98927.16333.98657 36 25.09440.99553.11176.99374.12903.99163.14637.98923.16361.98652 35 26.09469.99551.11205.99370.12937.99160.14666.98919.16390.98648 34 27.09498.9Q.54.11234.99367.12966.99156.14695.98914.16419.98643 33 23.09527.99545.11263.99364.12995.99152.14723.98910.16447.98638 32 29.09556.99542.11291.99360.13024.99148.14752.98906.16476.98633 31 30.09585.99540.11320.99357.13053.99144.14781.98902.16505.98629 30 31.09614.99537.11349.99354.13081.99141.14810.98897.16533.98624 29 32.09642.99534.11378.99351.13110.99137.14838.98893.16562.98619 28 33.09671.99531.11407.99347.13139.99133.14867.98889.16591.98614 27 34.09700.99528.11436.99344.13168.99129.14896.98884.16620.98609 26 35.09729.99526.11465.99341.13197.99125.14925.98880.16648.98604 25 36.09758.99523.11494.99337.13226.99122.14954.98876.16677.98600 24 37.09787.99520.11523.99334.13254.99118.14982.98871.16706.98595 23 38.09816.99517.11552.99331.13283.99114.15011.98867.16734.98590 22 39.09345.99514.11580.99327.13.312.99110.15040.98863.16763.98585 21 40.09374.99511.11609.99324.13341.99106.15069.98858.16792.98580 20 41.09903.99508.11638.99320.13370.99102.15097.98854.16820.98575 19 42.09932.99506.11667.99317.13399.99098.15126.98849.16849.98570 18 43.09961.99503.11696.99314.13427.99094.15155.98845.168378.98565 17 44.09990.99509.11725.99310.13456.99091.15184.98841.16906.98561 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 47.10077.99491.11812.99300.13543.99079.15270.98827.16992.98546 13 48.10106.9948. 11840.99297. 13572.99075. 15299.98323. 17021.9541 1 2 49.10135.99435.11869.99293.13600.99071.15327.98818.17050.98536 11 50.10164.99482.11898.99290.13629.99067.15356.98814.17078.98531 10 51.10192.99479.11927.99286.13658.99063.15385.98809.17107.98526 9 52.10221.99476.11956.99283.13637.99059.15414.98805.17136.98521 8 53.102:50.99473.11935.99279.13716.99055.15442.98800.17164.98516 7 54.10279.99470.12014.99276.13744.99051.15471.98796.17193.98511 6 55.10338.99467.12043.99272.13773.99047 15500.98791.17222.93506 5 56.10337.99464.12071.99269.13802.99043.15529.93787.17250.98501 4 57.10366.99461.12100.99265 13331.99039.15557.98782.17279.98496 3 58.10395.99458.12129.99262.13860.99035.15586.98778.17308.98491 2 59.10424.99455.12158.99258.13839.99031.15615.98773.17.336.98436 1 60.10453.99452.12187.99255.13917.99027.15643.98769.17365.9 1 M Cosin. Sine. Cosin. SiSo Sine. Cosin. Sine. Cosin s. Sine. M. 840 830 820 810 800 222 TABLE XIV. NATURAL SINES AND COSINES. 100 110 120 o 130 140 I. Sine. Cosn. Sine Cosi.Sine. CosinSine Cosin Sine.. S.Cosin. M. 0.17365.98481.19081.98163.20791.97815.22495.97437.24192.97030 60 1.17393.98476.19109.98157.20820.97809.22523.97430.24220.97023 59 2.17422.98471.19138.98152.20848.97803.22552.97424.24249.97015 58 3.17451.98466.19167.98146.20877.97797.22580.97417.24277.97008 57 4.17479.98461.19195.98140.20905.97791.22608.97411.24305.97001 56 5.17508.98455.19224.98135.20933.97784.22637.97404.24333.96994 55 6.17537.98450.19252.98129.20962.97778.22665.97398.24362.96987 54 7.17565.98445.19281.98124.20990.97772.22693.97391.24390.96980 53 8.17594.98440.19309.98118.21019.97766.22722.97384.24418.96973 52 9.17623.98435.19338.98112.21047.97760.22750.97378.24446.96966 51 10.17651.98430.19366.98107.21076.97754.22778.97371.24474.96959 50 11.17630.93425.19395.98101.21104.97748.22807.97365.24503.96952 49 12.17708.98420.19423.98096.21132.97742.22835.97358.24531.96945 48 13.17737.98414.19452.98090.21161.97735.22863.97351.24559.96937 47 14.17766.93409.19481.98034.21189.97729.22892 97345.24587.96930 46 15.17794.98404.19509.98079.21218.97723.22920.97338.24615.96923 45 16.17823.98.399.19533.98073.21246.97717.22948.97331.24644.96916 44 17.17852.98394.19566.98067.21275.97711.22977.97325.24672.96909 43 18.17330.98389.19595.98061.21303.97705.23005.97318.24700.96902 42 19.17909.98383.19623.98056.21331.97698.23033.97311.24728.96894 41 20.17937.98378.19652.93050.21360.97692.23062.97304.24756.96887 40 21.17966.93373.19680.98044.21388.97686.23090.97298.24784.96880 39 22.17995.98368.19709.98039.21417.97680.23118.97291.24813.96873.38 23.18023.98362.19737.98033.21445.97673.23146.97284.24841.96866 37 24.18052.98357.19766.98027.21474.97667.23176.97278.24869.96858 36 25.18081.98352.19794.98021.21502.97661.23203.97271.24897.96851 35 26.18109.93347.19823.98016.21530.97655.23231.97264.24925.96844 34 27.18133.98341.19851.98010.21559.97648.23260.97257.24954.96837 33 28.18166.98336.19830.98004.21587.97642.23288.97251.24982.96829 32 29.18195.98331.19908.97998.21616.97636.2.3316.97244.25010.96822 31 30.18224.93325.19937.97992.21644.97630.23345.97237.25038.96815 30 31.18252.98320.19965.97987.21672.97623.23373.97230.25066.96807 29 32.18281.98315.19994.97981.21701.97617.23401.97223.25094.96800 28 33.18309.98310.20022.97975.21729.97611.23429.97217.25122.96793 27 34.18338.98304.20051.97969.21758.97604.23458.97210.25151.96786 26 35.18367.98299.20079.97963.21786.97598.23486.97203.25179.96778 25 36.18395.98294.20108.97958.21814.97592.23514.97196.25207.96771 24 37.18424.93288.20136.97952.21843.97585.23542.97189.25235.96764 23 33.18452.98283.20165.97946.21871.97579.23571.97182.25263.96756 22 39.18481.98277.20193.97940.21899.97573.23599.97176.25291.96749 21 40.18509.98272.20222.97934.21928.97566.23627.97169.25320.96742 20 41.18538.98267.20250.97928.21956.97560.23656.97162.25348.96734 19 42.18567.98261.20279.97922.21985.97553.23684.97155.25376.96727 18 43.18595.93256.20307.97916.22013.97547.23712.97148.25404.96719 17 44.18624.98250.20336.97910.22041.97541.23740.97141.25432.96712 16 45.18652.93245.20364.97905.22070.97534.23769.97134.25460.96705 15 46.18681.98240.20393.97899.22098.97528.23797.97127.25488.96697 14 47.18710.93234.20421.97893.22126.97521.23325.97120.25516.96690 13 48.18733.98229.20450.97887.22155.97515.23553.97113.25545.96682 12 49.18767.98223.20478.97881.22183.97508 23882.97106.25573.966751 1 50.18795.93218.20507.97875.22212.97502.23910.97100.25601.96667 10 51.18824.98212.20535.97869.22240.97496.2393S.97093.25629.96660 9 52.18852.93207.20563.97863.22268.97489.23966.97086.25657.96653 8 53.18881.98201.20592.97857.22297.97483.23995.97079.25685.96645 7 54.18910.98196.20620.97851.22325.97476.24023.97072.25713.96638 6 55.18938.98190.20649.97845.22353.97470.24051.97065.25741.96630 5 56,18967.98185.20677.97839.22382.97463.24079.97058.25769.96623 4 57.18995.93179.20706.97833.22410.97457.24108.97051.25798.96615 3 58.19024.98174.20734.978270 78.22438.97450.24136.97044.25826.96608 2 59.19052.98168.20763.97821.22467.97444.24164.97037.25854.96600 01 60.19081.98163.20791.97815.22495.97437.24192.97030.25882.96593 0. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosir. Sine. Cosin. Sine. M 79- 780 770 76Q 7-50 TABLE XIV. NATURAL SINES AND COSINES. 223 150 160 10 1803 190 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. 0.25882.96593.27564.96126.29237.95630.30902.95106.32557.94552 60 11.25910.96585.27592.96118.29265.95622.30929.95097.32584.94542 59 2.25938.96578.27620.96110.29293.95613.30957.95088.32612.94533 58 3.25966.96570.27648.96102.29321.95605.30985.95079.32639.94523 57 4.25994.96562.27676.96094.29348.95596.31012.95070.32667.94514 56 5.26022.96555.27704.96036.29376.95588.31040.95061.32694.94504 55 6.26050.96547.27731.96078.29404.95579.31063.95052.32722.94495 54 7.26079.96540.27759.96070.29432.95571.31095.95043.32749.94485 53 8.26107.96532.27787.96062.29460.95562.31123.95033.32777.94476 52 9.26135.96524.27815.96054.29487.95554.31151.95024.32804.94466 51 10.26163.96517.27843.96046.29515.95545.31178.95015.32832.94457 50 11.26191.96.509.27871.96037.29543.95536.31206.95006.32859.94447 49 12.26219.96502.27899.96029.29571.9552S.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 15.26303.96479.27983.96005.29654.95502.31316.94970.32969.94409 45 16.26331.96471.28011.95997.29682.95493.31344.94961.32997.94399 44 17.26359.96463.28039.95989.29710.95485.31372.94952.33024.94390 43 18.26337.96456.28067.95981.29737.95476.31399.94943.33051.94380 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 21.26471.96433.28150.95956.29821.95450.31482.94915.33134.94351 39 22.26500.96425.28178.95948.29849.95441.31510.94906.33161.94342.33 2.3.26528.96417.28206.95940.29876.95433.31537.94897.33189.94332 37 24.26556.96410.28234.95931.29904 95424.31565.94888.33216.94322 36 25.26534.96402.28262.95923.29932.95415.31593.94878.33244.94313 35 26.26612.96394.28290.95915'29960.95407.31620.94869.33271.94303 34 27.26640.96336.28318.95907.29987.95398.31648.94860.33298.94293 33 28.26668.96379.28346.95898.30015.95389.31675.94851.33326.94284 32 29.26696.96371.28374.95890.30043.95380.31703.94842.33353.94274 31 30.26724.96363.28402.95382.30071.95372.31730.94832.33381.94264 30 31.26752.96355.28429.95874.30098.95363.31758.94823.33408.94254 29 32.26780.96347.28457.95865.30126.95354.31786.94814.33436.94245 28 33.26303.96340.28485.95857.30154.95345.31813.94805.33463.94235 27 34.26836.96332.28513.95849.30182.95337.31841.94795.33490.94225 26 35.26864.96324.28541.95841.30209.95328.31868.94786.33518.94215 25 36.26892.96316.28569.95832.30237.95319.31896.94777.33545.94206 24 37.26920.96308.28597.95324.30265.95310.31923.94768.33573.94196 23 3i.26948.96301.28625.95816.30292.95301.31951.94758.33600.94186 22 39.26976.96293.28652.95807.30320.95293.31979.94749.33627.94176 21 40.27004.96285.28680.95799.30348.95284.32006.94740.33655.94167 20 41.27032.96277.28708.95791.30376.95275.32034.94730.33682.94157 19 42.27060.96269.28736.95782.30403.95266.32061.94721.33710.94147 18 43.27083.96261.28764.95774.30431.95257.32089.94712.33737.94137 17 44.27116.96253.28792.95766.30459.95248.32116.94702.33764.94127 16 45.27144.96246.28820.95757.30486.95240.32144.94693.33792.94118 15 46.27172.96238.28847.95749.30514.95231.32171.94634.33819.94103 14 47.27200.96230.23875.95740.30542.95222.32199.94674.33846.94098 13 48.27228.96222.28903.95732.30570.95213.32227.94665.33874.94088 12 49.27256.96214.28931.95724.30597.95204 32254.94656.33901.94078 11 50.27234.96206.28959.95715.30625.95195.32282.94646.33929.94068 10 51.27312.96198.28987.95707.30653.95186.32309.94637.33956.94058 9 52.27340.96190.29015.95698.30680.95177.32337.94627.33933.94049 8 53.27363.96182.29042.95690.30708.95168.32364.94618.34011.94039 7 54.27396.96174.29070.95681.30736.95159.32392.94609.34038.94029 6 55.27421.96166.29098.95673.30763.95150.32419.94599.34065.94019 5 56.27452.96158.29126.95664.30791.95142.32447.94590.34093.94009 4 57.27480.96150.29154.95656.30819.95133.32474.94580.34120.93999 3 58.27508.96142.29182.95647.30846.95124.32502.94571.34147.93989 2 59.27536.96134.29209.95639.30374.95115.32529.94561.34175.93979 1 60.27564.96126.29237.95630.30902.95106.32557.94552.34202.93969 0 M. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. M. 740 73D 720 i 710 700 224 TABLE XIV. NATURAL SINES AND COSINES. 20D 210 22~ 230 240 __ M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. 0.34202.93969.35337.93358.37461.9-2718.39073.92050.40674.91355 60 1.34229.93959.35864.93348.37488.92707.39100.92039.40700.91343 59 2.34257.93949.35891.93337.37515.92697.39127.92028.40727.91331 58 3.34284.93939.35918.93327.37542.926S6.39153.92016.40753.91319 57 4 34311.93929.35915.93316.37569.92675.39180.92005.40780.91307 56 5.34339.93919.35973.93306.37595.92664.39207.91994.40806.91295 55 6.34366.93909.36000.93295.37622.92653.39234.91982.40S33.91283 54 7.34393.93899.36027.93285.37649.92642.3926U.91971.40860.91272 53 8.34421.93889.36054.93274.37676.92631.39237.91959.40886.91260 52 9.34448.93879.36031.932f4.37703.92620.39314.91948.40913.91248 51 10.34475.93369.36108.93253.37730.92609.39341.91936.40939.91236 50 I 1.34503.93859.36135.93243.37757.92593.39367.91925.40966.91224 49 12.34530.93349.36162.93232.37784.92587.39394.91914.40992.91212 48 13.34557.93339.36190.93222.37811.972576.39421.91902.41019.91200 47 14.34584.93829.36217.93211.3783.92565.39448.91891.41045.911 8S 46 15.31612.93819.36244.93201.37865.925.354.39474.91879.41072.91176 45 16.34639.93809.36'271.93190.37892.92543.39501.91868.41098.91164 44 17.34666.9.3799.36298.93180.37919.92532.39523.91856.41125.91152 43 1.34694.93789.36325.93169.37946.92321.39555.91845.41151.91140 42 19.34721.93779.36352.93159.37973.92510.39.581.91833.41178.91128 41 20.31748.93769.36379.93148.37999.92499.39608.91822.41204.91116 40 21.34775.93759.361t)6.93137.33026.92488.39635.91810.41231.91104 39 22.34803.93748.36434.93127.38053.92477.39661.91799.41257.91092.38 23.34330.93738.36461.93116.3308(.92466.39688.91787.41234.91080 37 21.34357.93723.36438.93106.38107.92455.39715.91775.41310.91068 36 25.34834.93718.36515.93095.33134.92444.39741.91764.41337.91056 35 26.34912.93708.36512.93034.38161.92432.39768.91752.41363.91044 34 27.31939.93693.36569.93074.33188.92421.39795.91741.41390.91032 33 23.34966.93633.36596.93063.38215.92410.39322.91729.41416.91020 32 29.34993.93677.36623.93052.38241.92399.39848.91718.41443.91008 31 30.35021.93667.36650.93042.33263.92338.39875.91706.41469.90996 30 31.3504-.93657.36677.93031.33295.92377.39902.91694.41496.90984 29 32.3.075.93647.35704.93020.33.322.92366.39928.91683.41522.90972 28 33.35102.93637.3q731.93010.33349.92.355.39955.91671.41549.90960 27 34.35130.93626.36758.92999.33376.92343.39982.91660.41575.90948 26 35.35157.93616.367335.92983.38403.92332.40003.91648.41602.90936 25 36.35184.93606.36812.92978.384:30.92321.40035.91636.41628.90924 24 37.3211.93596.36339.92967.38456.92310.40062.91625.41655.90911 23 33.35239.93585.36367.92956.33433.92299.40088.91613.41681.90899 22 39.3.5266.93575.36894.92945.33510.92837.40115.91601.41707.90887 21 40.3.5253.93565.36921.92935.38537.92276.40141.91590.41734.90875 20 41.353320.93555.36948.92924.38.564.92265.40169.91578.41760.90863 19 42.35347.93544.36975.92913.33591.92254.40195.91566.41787.90851 18 43.35375.93.534.37002.92902.38617.92243.40'221.91555.41813.90839 17 44.3.5402.93524.37029.92392.38644.92231.40248.91543.41840.90826 16 45.35429.93514.37056.92881.38671.92220.40275.91.531.41866.90814 15 46.354.56.93503.37083.92'70.33698.92299.40301.91519.41892.90802 14 47.35434.93493.37110.92859.38725.92198.40323.91508.41919.90790 13 48.35511.93433.37137.92849.38752.92186.40355.91496.41945.90778 12 49.33533.93472.37164.92838.33778.92175 40381.91484.41972.90766 11 50.335565.93462.37191.92827.3.8805.92164.40408.91472.41998.90753 10 51.35.592.934.37218.92816 8832.2152.40434.91461.42024.90741 9 52.3.619.9:3441.37245.92805.38859.92141.40461.91449.42051.90729 8 53.3 647.93431.37272.92794.33886.92130.40488.91437.42077.90717 7 54.35674.93421).37209.92784.38912.92119.40514.91425.42104.90704 6 55.3571.37 93411.37326.92773.38939.92107.40541.91414.42130.90692 5 56.35723.93400.37353.92762.38966.92096.40567.91402.42156.90680 4 57.35755.9339.37380.92751.38993.92035.40594.91390.42183.9!668 3 53.35782.93379.37407.92740.39020.9-2073.40621.91378.42209.90655 2 59.35810.93368.37434.92729.39046.92062.40647.91366.42235.90643 1 60.35337.93358.37461.92718.36073.92050.40674.91355.42262.99631 O M. Cosin. Sine. ine. Co sin. Sine. Cosin. Sine. Sine. Csin.S M. I 690 68 60 1 60 650 TABLE X1V. NATURAL SINES AND COSINES. 225 2590 26 27c 2s8: 290 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M..42262.90631.43837 89879.45399.89101.46947.88295.48481.87462 60 1.42288.90618.43863.89367.45425.89087.46973.88281.48506.87448 59 2.42315.90306.43389.89354.45451.89074.46999.88267.48532.87434 58 3.42341.90594.43916.89841.45477.89061.47024.88254.48557.87420 57 4.42367.9082.43942.89328.45503.89048.47050.88240.48583.87406 56 5.42394.90569.43968.89816.45529.89035.47076.88226.48608.87391 55 6.42420.90557.43994.89803.45554.89021.47101.88213.48634.87377 54 7 42446.90545.44020.89790.45580.89008.47127.88199.48659.87363 53 8.42473.90532.44046.89777.45606.88995.47153.88185.48634.87349 52 9.42499.90520.44072.89764.45632.88981.47178.88172.48710.87335 51 10.42525.90507.44098.89752.45658.88968.47204.88158.48735.87321 50 11.42552.90495.44124.89739.45684.88955.47229.88144.48761.87306 49 12.42578.90483.44151.89726.45710.88942.47255.88130.48786.87292 48 13.42604.90470.44177.89713.45736.88928.47281.88117.48811.87278 47 14.42631.90458.44203.89700.45762.88915.47306.88103.48837.87264 46 15.42657.90446.44229.89687.45787.88902.47332.88089.48862.87250 45 16.42683.90433.44255.89674.45813.88888.47358.88075.4883.87235 44 17.42709.90421.44281.89662.45839.88875.47383.88062.48913.87221 43 18.42736.90408.44307.89649.45865.88862.47409.88048.48938.87207 42 19.42762.90396.44333.89636.45891.88848.47434.88034.48964.87193 41 20.42788.90383.44359.89623 4.5917.88835.47460.88020.48989.87178 40 21.42815.90371.44385.89610.45942.88822.47486.88006.49014.87164 39 22.42841.90358.44411.89597.45968.88808.47511.87993.49040.87150.38 23.42867 90346.44437.89584.45994.88795.47537.87979.49065.87136 37 24.42894.90334.44464.89571.46020.88782.47562.87965.49090.87121 36 25.429-20.90321.44490.89558.46046.88768.47588.87951.49116.87107 35 26.42946.90309.44516.89545.46072.88755.47614.87937.49141.87093 34 27.42972.90296;44542.89532.46097.88741.47639.87923.49166.87079 33 28.42999.90284.44568.89519.46123.88728.47665.87909.49192.87064 32 29.43025.90271.44594.89506.46149.88715.47690.87896.49217.87050 31 30.43051.90259.44620.89493.46175.88701.47716.87882.49242.87036 30 31.43077.90246.44646.89480.46201.88688.47741.87868.49268.87021 29 32.43104.90233.44672.89467'.46226.88674.47767.87854.49293.87007 28 33.43130.90221.44698.89454.46252.88661.47793.87840.49318.86993 27 34 43156.90208.44724.89441.46278.88647.47818.87826.49344.86978 26 35.43182.90196.44750.89428.46304.88634.47844.87812.49369.86964 25 36.43209.90183.44776.89415.4630.88620.47869.87798.49394.86949 24 37.43235.90171.44802.89402.46355.88607.47895.87784.49419.86935 23 38.43261.90158.44828.89389.46381.88593.47920.87770.49445.86921 22 39.43287.90146.44854.89376.46407.88580.47946.87756.49470.86906 21 40.43313.90133.44880.89363.46433.88566.47971.87743.49495.86892 20 4 1.43340.90120.44906.89350.46458.88553.47997.87729.49521.86878 19 42.43.366.90108.44932.89337.46484.88539.48022.87715.49546.86863 18 43.43392.90095.44958.89324.46510.88526.48048.87701.49571.86849 17 44.43418.90082.44984.89311.46536.88512.48073.87687.49596.86834 16 45.43445.90070.45010.89298.46561.88499.48099.87673.49622.86820 15 46.43471.90057.45036.89285.46587.88485.48124.87659.49647.86805 14 47.43497.90045.45062.89272.46613.88472.48150.87645.49672.86791 13 48.43523.90032.45088.89259.46639.88458.48175.87631.49697.86777 12 49.43549.90019.45114.89245.46664.88445.48201.87617.49723.86762 11 50.43575.90007.45140.89232.4690.88431 48226.87603.49748.86748 10 51.43602.89994.45166.89219.46716.88417.48252.87589.49773.867.33 9 52.43628.89981.45192.89206.46742.88404.48277.87575.49798.86719 8 53.43654.89968.45218.89193.46767.88390.48303.87561.49824.86704 7 54.43680.89956.45243.89180.46793.88377.48328.87546.49849.86690 6 55.43706.89943.45269.89167.46819.88363.48354.87532.49874.86675 5 56.43733.89930.45295.89153.46844.88349.48379.87518.49899.86661 4 57.43759.89918.45321.89140.46870.88336.48405.87504.49924.86646 3 68.43785.89905.45347.89127.46896.88322.48430.87490.49950.86632 2 59.43811.89892.45373.89114.46921.88308.48466.87476.49975 86617 1 60.43837.89879.45399.89101.46947.88295.48481.87462.50000.86603 O M. C S Co in; Cos ne. CCo dn. Sine. Cosin. Sine. Cosin. Sne. M. 640 1 630 o620 610 60 0 11 226 TABLE XIV. NATURAL SINES AND COSINES. 300 310 3o2 330 340 a. Sine. Cosin. Sine. Cosin Sine. i i osin. Sine. Cosin. Sine. Cosin. M..50000.86603.51504.85717.52992.84805.54464.83867.55919.82904 60 I.50025.86588. 1529.85702.63017.84789.54488.83851 55943.82887 59 2.50J50.86573.61554.8687.63041.84774.54513.83835.55968.82871 58 3.50076-.86559.51579.85672.63066.84759.54537.83819.55992.82855 57 50101.86544.6.51604.8667.53091.84743.54561.83804.56016.82839 56,.50126.86530.51628.85642.53115.84728.54586.83788.56040.82822 55 6.50151.86515.61663.85627.53140.84712.54610.83772.56064.82806 54 7.50176.86501.51678.85612.53164.84697.54635.83756.56088.82790 53 s.50201.86486.51703.85597.53189.84681.54659.83740.56112.82773 52 9.50227.86471.61728.85582.63214.84666.54653.83724.56136.82757 51 10.50252.86457.51753.85567.53238.84650.54708.83708.56160.82741 50 1 50277.86442.51778.85551.53263.84635.54732.83692.56184.82724 49 12.50302.86427.51803.85536.53288.84619.54756.83676.56208.82708 48 1b.50327.86413.51828.85521.53312.81604.54781.83660.56232.82692 47 14.50352.86398.51852.85596.53337.8-588.54805.83645.56256.82675 46 15.50377.86384.51877.85411.53361.84573.54829.83629.56280.82659 45 16.50403.86369.51902.85476.53386.84557.54854.83613.56305.82643 44 17.5042|.86354.51927.85461.53411.84542,54878.83597.56329.82626 43 1.50453.86340.51952.85446. 534:35.S41526.54902.83581.56353.82610 42 19.50478.86325.51977.854.:1.5.3460.84511.54927.83565.56377.82593 41 20.50503.85310.52002.854: 6.53484.84495.54951.83549.56401.82577 40 2.50528.862.w5.52026.85401.53509.84480.54975.83533.56425.82561 39 22.50553.86281.52051.85385.53534.84164.54999.83517.56449.82544 38 23.50578.86266.52076.85370.53558.84448.55024.83501.56473.82528 37 24.50603.86251.52101.85355.53583.84433.55048.83485.56497.82511 36 25.50628.86237.521 6.8534) 1.53607.84417.55072.83469.56521.82495 35 26.50654.86282.52151.85325.53632.84a102.55097.83453.56545.82478 34 27.50679.86207.52175.85310.53656.8438:6.55121.83437.56569.82462 33 2S.50701.86192.52200.85294.5368i.84370.55145.83421.56593.82446 32 29.50729.86178.52225.85279.53705.84355.55169.83405.56617.82429 31 30.50754.86163.52o25.85264.53730.84339.55194.83-389.56641.82413 30 31.50779.86148.52275.85249.53754.84324.55218.83373.56665.82396 29 32.50804.86133.52299.85234.53779.84308.55212.83356.56689.82380 28 33.50329.86119.52324.85218.53804.84292.55266.83340.56713.82363 27 31.508.54.86101.52349.85203.53828.84277.55291.83:324.56736.82.347 26 35.50379.86089.52374.85188.53353.84261.55315.83308.56760.82330 25 36.50904.86074.52399.85173.53377.84245.55339.83292.56784.82314 24 37.50929.86059.52423.85157.53902.84230.55363.83276.56808.82297 23 ~3S.50954.86046.52448.85142.53926.84214.55388.83260.56832.82281 22 39.50979.86030.52473.85127.53951.84198.55412.83244.56856.82264 21 40.51004.86015.52498.85112.53975.84182.55436.83228.56880.82248 20 41.51029.86000.52522.85096.54000.84167.55460.83212.56904.82231 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 44.51104.85956.52597.85051.54073.84120.55533.83163.56976.82181 16 45.51129.85941.52621.85035.54097.84104.55557.83147.57000.82165 15 46.51154.85926.52646.85020.54122.84088.55581.83131.57024.82148 14 47.51179.85911.52671.85005.54146.84072.55605.83115.57047.82132 13 48.51204.85896.52696.84989.54171.84057.55630.83098.57071.82115 12 49.51229.85881.52720.84974.54195.84041.55654.83082.57095.820981 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 52.51304.85836.52794.84928.54269.83994.55726.830.34.57167.82048 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 55.51379.85792.52569.84882.54342.83946.55799.82985.5723S.81999 5 56.51404.85777.52393.84866.54366.83930.55823.82969.57262.81982 4 57.51429.85762.52918.84851.54391.83915.55847.82953.57286.81965 3 58.51454.85747.52943.84836.54415.83899.55871.82936.57310.81949 2 59.51479.85732.52967.84820.54440.83883.55895.82920.57334.81932 1 60.51504.85717.52992.84805 54464.83s67.55919.82904.57358.81915 0 M. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. M. 590 583 570 560 5 0 TABLE X1V. NATURAL SINES AND COSINES. 227 350 360 37o 380 390 M. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. M. 0.57358.81915.58779.80902.60182.79864.61566.78801.62932.77715 60 1.57381.81899.58802.80885.60205.79846.61589.78783.62955.77696 59 2.57405.81882.58826.80867.60228.79829.61612.78765.62977.77678 58 3.57429.81865.58849.80850.60251.79811.61635.78747.63000.77660 57 4.57453.81848.58873.80833.60274.79793.61958.78729.63022.77641 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 8.57548.81782.58967.80765.60367.79723.61749.78658.63113.77568 52 9.57572.81765.58990.80748.60390.79706.61772.78640.63135.77550 51 10.57596.81748.59014.80730.60414.79688.61795.78622.63158.77531 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 15.57715.81664.59131.80644.60529.79600.61909.78532.63271.77439 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 20.57833.81580.59248.80558.60645.79512.62024.78442.63383.77347 40 21.57857.81563.59272.80541.60668.79494.62046.78424.63406.77329 39 22.57881.81546.59295.80524.60691.79477.62069.78405.63428.77310 38 23.57904.81530.59318.80507.60714.79459.62092.78387.63451.77292 37 24.57928.81513.59342.80489.60738.79441.62115.78369.63473.77273 36 -25.57952.81496.59365.80472.60761.79424.62138.78351.63496.77255 35 26.57976.81479.59389.80455.60784.79406.62160.78333.63518.77236 34 27.57999.81462.59412.80438.60807.79388.62183.78315.63540.77218 33 28.58023.81445.59436.80420.60830.79371.62206.78297.63563.77199 32 29.58047.81428.59459.80403.60853.79353.62229.78279.63585.77181 31 30.58070.81412.59482.80386.60876.79335.62251.78261.63608.77162 30 31.58094.81395.59506.80368.60899.79318.62274.78243.63630.77144 29 32.58118.81378.59529.80351.60922.79300.62297.78225.63653.77125 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 36.58212.81310.59622.80282.61015.79229.62388.78152.63742.77051 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 43.58378.81191.59786.80160.61176.79105.62.547.78025.63899.76921 17 44.58401.81174.59809.80143.61199.79087.62570.78007.63922.76903 16 45.58425.81157.59832.80125.61222.79069.62592.77988.63944.76884 15 46.58449.81140.59856.80108.61245.79051.62615.77970.63966.76866 14 47.58472.81123.59879.80091.61268.79033.62638.77952.63989.76847 13 48.58496.81106.59902.80073.61291.79016.62660.77934.64011.76828 12 49.58519.81089.59926.80056.61314.78998 62683.77916.64033.76810 11 50.58543.81072.59949.80038.61337.78980.62706.77897.64056.76791 10 51.58567.81055.59972.80021.61360.78962.62728.77879.64078.76772 9 52.58590.81038.59995.80003.61383.78944.62751.77861.64100.76754 8 53.58614.81021.60019.79986.61406.78926.62774.77843.64123.76735 7 54.58637.81004.60042.79968.61429.78908.62796.77824.64145.76717 6 55.58661.80987.60065.79951.61451.78891.62819.77806.64167.76698 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. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. Cosin. Sine. M. - 4. 5 530 52o 510 500 228 TABIE XIV. NATURALI SINES AND COSINES. 400 410 4go 430 440 M. Sine. Cosin. Sins. Cosin.inei Co Sine. Cosin. Sine. Cosin. M. 0.64279.76604.65606.75471.66913.74314.68200.73135.69466.71934 60 1.64301.76586.65628.7452.66935.74295.68221.73116.69487.71914 59 2.64323.76567.65650.75433.66956.74276.68242.73096.69508.71894 58 3.64346.76548.65672.75414.66978.74256.68264.73076.69529.71873 57 4 64368.76530.65694.75395.66999.74237.68285.73056.69549.71853 56 5.64390.76511.65716.75375.67021.74217.68306.73036.69570.71833 55 6.64412.76492.65738.75356.67043.74198.68327.73016.69591.71813 54 7.64435.76473.65759.75337.67064.74178.68349.72996.69612.71792 53 8.64457.76455.65781.75318.67086.74159.68370.72976.69633.71772 52 9.64479.76436.65803.75299.67107.74139.68391.72957.69654.71752 51 10.64501.76417.65825.75280.67129.74120.68412.72937.69675.71732 50 11.64524.76398.65847.75261.67151.74100.68434.72917.69696.71711 49 12.64546.76380.65869.75241.67172.74080.68455.72897.69717.71691 48 13.61568.76361.65891.75222.67194.74061.68476.72877.69737.71671 47 14.64590.76342.65913.75203.67215.74041.68497.72857.69758.71650 46 15.64612.76323.65935.75184.67237.74022.68518.72837.69779.71630 45 16.64635.76304.65956.75165.67258.74002.68539.72817.69800.71610 44 17.64657.76286.65978.75146.67280.73983.68561.72797.69821.71590 43 18.64679.76267.66(00.75126.67301.73963.68582.72777.69842.71569 42 19.64701.76248.66022.75107.67323.73944.68603.72757.69862.71549 41 20.64723.76229.66044.75088.67344.73924.68624.72737.69883.71529 40 21.64746.76210.66066.75069.67366.73904.68645.72717.69904.71508 39 22.64768.76192.66088.75050.67387.73885.68666.72697.69925.71488 38 23.64790.76173.66109.75030.67409.73865.68688.72677.69946.71468 37 24.64812.76154.66131.75011.67430.73846.68709.72657.69966.71447 36 25.64834.76135.66153.74992.67452.73826.68730.72637.69987.71427 35 26.64856.76116.66175.74973.67473.73806.68751.72617.70008.71407 34 27.64878.76097.66197.74953.67495.73787.68772.72597.70029.71386 33 28.64901.76078.66218.74934.67516 73767.68793.72577.70049.71366 32 29.64923.76059.66240.74915.67538.73747.68814.72557.70070.71345 31 30.64945.76041.66262.74896.67559.73728.68835.72537.70091.71325 30 31.64967.76022.66'284.74876.67580.73708.68857.72517.70112.71305 29 32.64989.76003.66306.74857.67602.73688.68878.72497.70132.71284 28 33.65011.75984.66327.74838.67623.73669.68899.72477.70153.71264 27 34.65033.75965.66319.74818.67645.73649.68920.72457.70174.71243 26 35.65055.75946.66371.74799.67666.73629.68941.72437.70195.71223 25 36.65077.75927.66393.74780.67688.73610.68962.72417.70215.71203 24 37.65100.75908.66414 74760.67709.73590.68983.72397.70236.71182 23 38.65122.75389.66436.74741.67730.73570.69004.72377.70257.71162 22 39.65144.75870.66458.74722.67752.73551.69025.72357.70277.71141 21 40.65166.75851.66480.74703.67773.73531.69046.72337.70298.71121 20 41.65188.75832.66501.74683.67795.73511.69067.72317.70319.71100 19 42.65210.75813.66523.74664.67816.73491.69088.72297.70339.71080 18 43.65232.75794.66545.74644.67837.73472.69109.72277.70360.71059 17 44.65254.75775.66566.74625.67859.73452.69130.72257.70381.71039 16 45.65276.75756.66588.74606.67880.73432.69151.72236.70401.71019 15 46.65293.75738.66610.74586.67901.73413.69172.72216.70422.70998 14 47.65320.75719.66632.74567.67923.73393.69193.72196.70443.70978 13 48.65342.75700.66653.74548.67944.73373.69214.72176.70463.70957 12 49.65364.75680.66675.74528.67965.73353.69235.72156.70484.70937 11 50.65386.7.5661.66697.74509.67987.73333.69256.72136.70505.70916 10 51.65408.75642.66718.74489.68008.73.314.69277.72116.70525.70896 9 52.65430.75623.66740.74470.68029.73294.69298.72095.70546.70875 8 53.65452.75601.66762.74451.68051.73274.69319.72075.70567.70855 7 54.65474.75585.66783.74431.68072.73254.69340.72055.70587.70834 6 55.65496.75566.66805.74412.68093.73234.69361.72035.70608.70813 5 56.65518.75547.66827.74392.68115.73215.69382.72015.70628.70793 4 57.65540.75528.66848.74373.68136.73195.69403.71995.70649.70772 3 58.65562.75509.66870.74353.68157.73175.69424.71974.70670.70752 2 59.65584.75490.66891.74334.68179.73155.69445.71954.70690.70731 1 60.65606.75471.66913.74314.68200.73135.69466.71934.70711.70711 0 M.Cosn Sin. Cosi Sine. I-'. Sin Cosin. Sine. Cosin. Sine... 493 48o 470 460 1 450 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 230 TABLE XV. NATURAL TANGENTS AND COTANGENTS. I o o.20 30 J _M. Tng. Cotang.Ta ng. Cotang. Tng. Cotang. Tang. 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 572.957.01920 52.0807.03667 27.2715.95416.18.4645 54 7.00204 491.106.01949 51.3032.03696 27.0566 05445 18.3655 53 8.00233 429.718.01978 50.5485.03725 26.8450 05474 18.2677 52 9.00262 381.971.02007 49.8157.03754 26.6367.05503 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 16.00465 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..00669 149.465.024165 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 23.3718.060-29 16.5874 33 28.00815 122.774.02560 39.0568.04308 23.2137.06058 16.5075 32 29.00844 118.540.02589 38.6177.04337 23.0577.06087 16.4283 31 301.00873 114.589.02619 38.1885.04366 22.9038.06116 16.3499 30 31.00902 110.892.02648 37.7686.04395 22.7519.06145 16.2722 Ji 32.00931 107.426.02677 37.3579.04424 22.6020.06175 16.1952 28 33.00960 104.171.02706 36.9560.04454 22.4541.06204 16.1190 27 34.00989 101.107.02735 36.5627.04483 22.3081.06233 16.0435 26 35.01018 98.2179.02764 36.1776.04512 22.1640.06262 15.9687 25 36.01047 95.4895.02793 35.8006.04541 22.0217.06291 15.8945 24 37.01076 92.9085.02822 35.4313.04570 21.8813.06321 15.8211 23 38.01105 90.4633.02851 35.0695.04599 21.7426.06350 15.7483 22 39.01135 88.1436.02881 34.7151.04628 21.6()56.06379 15.6762 21 40.01164 85.9398.02910 34.3678.04658 21.4704.06408 15.6048 20 41.01193 83.8435.02939 34.0273.04687 21.3369.06437 15.5340 19 42.01222 81.8470.02968 33.6935.04716 21.2049.06467 15.4638 18 43.01251 79.9434.02997 33.3662.04745 21.0747.06496 15.3943 17 44.01280 78.1263.03026 33.0452 04774 20.9460.06525 15.3254 16 45.01309 76.3900.03055 32.7303.04803 W 188.06554 152571 15 46.01338 74.7292.03084 32.4213.04833 20.6932.06584 15.1893 14 47.01367 73.1390.03114 32.1181.04862 20.5691.06613 15.1222 13 48 01396 71.6151.03143 31.8205.04891 20.4465.06642 15. 557 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 61.3829.03376 29.6245.05124 19.5156.06876 14.5438 4 57.01658 60.3058.03405 29.3711.05153 19.4051.06905 14.4823 3 58.01687 59.2659.03434 29.1220.05182 19.2959.06934 14.4212 2 59.01716 58.2612.03463 28.8771.05212 19.1879.06963 14.3607 1 60.01746 57.2900.03492 28.6363.05241 19.0811.06993 14.3007 0 M. Cotang. Tang. Cotang ngTang. Cotang.Tang. Cotang. Tang. M. 1 890 o880 870 860 _R o... Ig. TABLE XV. NATURAL TANGENTS AND COTANGENTS. 231 4:0 50 60 70 If. Tang Cotang Tang. Cotang. Tang. Cotang. Tang. Cotang. M. O.06993 14.3007.08749 11.4301.10510 9.51436.12278 8.14435 60 1.07022 14.2411.08778 11.3919.10540 9.48781.12308 8.12481 59 2.07051 14.1821.08807 11.3540.10569 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 13.8940.08954 11.1681.10716 9.33155.12485 8.00948 53 8.07227 13.8378.08983 11.1316.10746 9.30599.12515 7.99058 52 9.07256 13.7821.09013 11.0954.10775 9.28058.12544 7.97176 51 10.07285 13.7267.09042 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 22.07636 13.0958.09394 10.6450.11158 8.96227.12929 7.73480 38 23.07665 13.0458.09423 10.6118.11187 8.93867.12958 7.71715 37 24.07695 12.9962.09453 10.5789.11217 8.91520.12988 7.69957 36 25.07724 12.9469.09482 10.5462.11246 8.89185.13017 7.68208 35 26.07753 12.8981.09511 10.5136.11276 8.86862.13047 7.66466 34 27.07782 12.8496.09541 10.4813.11305 8.84551.13076 7.64732 33 28.07812 12.8014 -09570 10.4491.11335 8.82252.13106 7.63005 32 29.07841 12.7536.09600 10.4172.11364 8.79964.13136 7.61287 31 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 8.66482.13313.7.51132 25 36.08046 12.4288.09805 10.1988.11570 8.64275.13343 7.49465 24 37.08075 12.3538.09834 10.1683.11600 8.62078.13372 7.47806 23 38.03104 12.3390.09864 10.1381.11629 8.59893.13402 7.46154 22 39.08134 12.2946.09893 10.1080.11659 8.57718.13432 7.44509 21 40.08163 12.2505.09923 10.0780.11688 8.55555.13461 7.42871 20 41.08192 12.2067.09952 10.0483.11718 8.53402.13491 7.41240 19 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 8.24345.139()6 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 7.160(71 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 60.08749 11.4301.10510 9.51436.12278 8.14435.14054 7.11537 O M. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. IM. i 850 84 830 82o 232 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 80 90 100 110 M. Tang. Cotoag. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.14054 7.11537.15838 6.31375.17633 5.67128.19438 6.14455 60 1.14084 7.10038 15868 6.30189.17663 5.66165.19468 5.13658 59 2.14113 7.08546.15898 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 55 6.14232 7.02637.16017 6.24321.17813 5.61397.19619 5.09704 54 7.14262 6.91174.16047 6.23160.17843 5.60452.19649 5.08921 53 8.14291 6.99718.16077 6.22003.17873 5.59511.19680 5.08139 52 9.14321 6.98268.16107 6.20851.17903 5.58573.19710 5.07360 51 10.14351 6.96823.16137 6.19703.17933 5.57638.19740 5.06584 50 11.14381 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 6.89688.16286 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 6.84082.16405 6.09552.18203 5.49356.20012 4.99695 41 20.14648 6.82694.16435 6.08444.18233 5.48451.20042 4.98940 40 21.14678 6.81312.16465 6.07340.18263 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 6.75838.16585 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 5.95448.18694 5.37805.20406 4.90056 28 33.15034 6.65144.16824 5.94390.18624 5.36936.20436 4.89330 27 34.15064 6.63831.16854 5.93335.18654 5.36070.20466 4.88605 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 6.59921.16944 5.90191.18745 5.33487.20557 4.86444 23 38.15183 6.58627.16974 5.89151.18775 5.3-2631.20588 4.85727 22 39.15213 6.57339.17004 5.88114.18805 5.31778.20618 4.85013 21 40.15243 6.56055.17033 5.87080.18835 5.30928.20648 4.84300 20 41.15272 6.54777.17063 5.86051.18865 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 45.15391 6.49710.17183 5.81966.18986 5.26715.2000 4.80769 15 46.15421 6.48456.17213 5.80953.19016 5.25880.20830 4.80068 14 47.15451 -6.47206.17243 5.79944.19046 5.25048.20861 4.79370 13 48.15481 6.45961.17273 5.78938.19076 5.24218.20891 4.78673 12 49.15511 6.44720.17303 5.77936.19106 5.23391.20921 4.77978 11 50.15540 6.43484.17333 5.76937.19136 5.22566.20952 4.77286 10 51.15570 6.42253.17363 5.75941.19166 5.21744.20982 4.76595 9 52.15600 6.41026.17393 5.74949.19197 5.20925.21013 4.75906 8 53.15630 6.39804.17423 5.73960.19227 5.20107.21043 4.75219 7 54.15660 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 5.17671.21134 4.73170 4 57.15749 6.34961.17543 5.70037.19347 5.16863.21164 4.72490 3 58.15779 6.33761.17573 5.69064.19378 5.16058.21195 4.71813 2 59.15809 6.32566.17603 5.68094.19408 5.15256.21225 4.71137 1 60.15838 6.31375.1763.3 5.67128.19438 5.14455.21256 4.70463 0 M. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. 810o 800 T790 78o TABLE XV. NATURAL TANGENTS AND COTANGENTS. 233 120 130 140 150 M. Tang. Cotang. Tang. Cdotng. Tang. Cotang. Tasg. Cotang. M. 0 21256 4.70463.23087 4.33148.24933 4.01078.26795 3.73205 60 1.21286 4.69791.23117 4.32573.24964 4.00582.26826 3.72771 59 2.21316 4.69121.23148 4.32001.24095 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 463171.23424 4.26911.25273 3.95680.27138 3.68485 49 12.21621 4.62518.23455 4.26352.25304 3.95196.27169 3.68061 48 13.21651 4.61868.23485 4.25795.25335 3.94713.27201 3.67638 47 14.21682 4.61219.23516 4.25239.25366 3.94232.27232 3.67217 46 15.21712 4.60572.23547 4.24685.25397 3.93751.27263 3.66796 45 16.21743 4.59927.23578 4.24132.25428 3.93271.27294 3.66376 44 17.21773 4.59283.23608 4.23580.25459 3.92793.27326 3.65957 43 18.21804 4.58641.23639 4.23030.25490 3.92316.27357 3.65538 42 19.21834 4.58001.23670 4.22481.25521 3.91839.27388 3.65121 41 20.21864 4.57363.23700 4.21933.25552 3.91364.27419 3.64705 40 21.21895 4.56726.23731 4.21387.25583 3.90890.27451 3.64289 39 22.21925 4.56091.23762 4.20842.25614 3.90417.27482 3.63874 38 23.21956 4.55458.23793 4.20298.25645 3.89945.27513 3.63461 37 24.21986 4.54826.23823 4.19756.25676 3.89474.27545 3.63048 36 25.22017 4.54196.23854 4.19215.25707 3.89004.27576 3.62636 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 4.14934.25955 3.85284.27826 3.59370 27 34.22292 4.48600.24131 4.14405.25986 3.84824.27858 3.58966 26 35.22322 4.47986.24162 4.13877.26017 3.84364.27889 3.58562 25 36.22353 4.47374.24193 4.13350.26048 3.83906.27921 3.58160 24 37.22383 4.46764.24223 4.12825.26079 3.83449.27952 3.57758 23 38.22414 4.46155.24254 4.12301.26110 3.82992.27983 3.57357 22 39.22444 4.45548.24285 4.11778.26141 3.82537.28015 3.56957 21 40.22475 4.44942.24316 4.11256.26172 3.82083.28046 3.56557 20 41.22505 4.44338.24347 4.10736.26203 3.81630.28077 3.56159 19 42.22536 4.43735.24377 4.10216.26235 3.81177.28109 3.55761 18 43.22567 4.43134.24408 4.09699.26266 3.80726.28140 3.55364 17 44.22597 4.42534.24439 4.09182.26297 3.80276.28172 3.54968 16 45.22628 4.41936.24470 4.08666.26328 3.79827.28203 3.54573 15 46.22658 4.41340.24501 4.08152.26359 3.79378.28234 3.54179 14 47.22689 4.40745.24532 4.07639.26390 3.78931.28266 3.53785 13 48.22719 4.40152.24562 4.07127.26421 3.78485.28297 3.53393 12 49.22750 4.39560.24593 4.06616.26452 3.78040.28329 3.53001 11 50.22781 4.38969.24624 4.06107.26483 3.77595.28360 3.52609 10 51.22811 4.38381.24655 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 4.36040.24778 4.03578.26639 3.75388.28517 3.50666 5 56.22964 4.35459.24809 4.03076.26670 3.74950.28549 3.50279 4 57.22995 4.34879.24840 4.02574.26701 3.74512.28580 3.49894 3 58.23026 4.34300.24871 4.02074.26733 3.74075.28612 3.49509 2 59.23056 4.3;3723.24902 4.01576.26764 3.78640.28643 3.49125 1 601.23087 4.33148.24933 4.01078.26795 3.73205.28675 3.48741 0 M..Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. I {?7- 760 75o 740 234 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 160~ 17~ 180 190 M. Tang. Cotang. Tang. Cooang. 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 6.28864 3.46458.30764 3.25055.32685 3.05950.34628 2.88783 54 7.28895 3.46080.30796 3.24719.32717 3.05649.34661 2.88511 53 8.23927 3.45703.30828 3.24383.32749 3.05349.34693 2.88240 02 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 1.29021 3.44576.30923 3.23381.32846 3.04450.34791 2.87430 49 12.29053 3.44202.30955 3.23048.32878 3.04152.34824 2.87161 48 13.29084 3.43829.30987 3.22715.32911 3.03854.34856 2.86892 47 14.29116 3.43456.31019 3.22384.32943 3.03556.34889 2.86624 46 15.29147 3.43084.31051 3.2'2053.32975 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 3.40869.31242 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 3.00028.35281 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 28 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 3.35443.31722 3.15240.33654 2.97144.35608 2.80833 24 37.29843 3.35087.31754 3.14922.33686 2.96858.35641 2.80574 23 38.29875 3.34732.31786 3.14605.33718 2.96573.35674 2.80316 22 39.29906 3.34377.31818 3.14288.33751 2.96288.35707 2.80059 21 I40.29938 3.34023.31850 3.13972.33783 2.96004.35740 2.79802 20 41.29970 3.33670.31882 3.13656.33816 2.95721.35772 2.79545 19 42.30001 3.33317.31914 3.13341.33848 2.95437.35805 2.79289 18 43.30033 3.32965.31946 3.1.3027.33881 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 49.30224 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 52.30315 3.29829.32235 3.10223.34173 2.92632.36134 2.76750 8 53.30351 3.29483..32267 3.09914.34205 2.92.354.36167 2.76498 7 54.30382 3.29139.32299 3.09606.34238 2.92076.36199 2.76247 6 55.30414 3.28795.32331 3.09293.34270 2.91799.36232 2.75996 5 56.30446 3.28452.32363 3.08991.34303 2.91523.36265 2.75746 4 57.30478 3.28109.32396 3.08685.34335 2.91246.36298 2.75496 3 58.30509 3.27767.32428 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 M. Clotang. Tang. ot ang. an. Tang. Cotang. Tang. CotangM. 730 720 710 703 TABLE XV. NATURAL TANGENTS AND COTA.iGENTS. 235 Q200 1o 2_____ 2 230 j M. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.36397 2.74748.38386 2.60509.40403 2.47509.42447 2.35585 60 1.36430 2.74499.38420 2.60283.40436 2.47302.42482 2.35395 59 2.36463 2.74251.38453 2.60057.40470 2.47095.42516 2.35205 58 3.36496 2.74004.38487 2.59831.40504 2.46888.42551 2.35015 57 4.36529 2.73756.38520 2.59606.40538 2.46682.42585 2.34825 56 5.36562 2.73509.38553 2.59381.40572 2.46476.42619 2.34636 55 6.36595 2.73263.38587 2.59156.40606 2.46270.42654 2.34447 54 7.36628 2.73017.38620 2.58932.40640 2.46065.42688 2.34258 53 8.36661 2.72771.38654 2.58708.40674 2.45860.42722 2.34069 52 9.36694 2.72526.38687 2.58484.40707 2.45655.42757 2.33881 51 10:36727 2.72281.38721 2.58261.40741 2.45451.42791 2.33693 50 11.36760 2.72036.38754 2.58038.40775 2.45246.42826 2.33505 49 12.36793 2.71792.38787 2.57815.40809 2.45043.42860 2.33317 48 13.36826 2.71548.38821 2.57593.40843 2.44839.42894 2.33130 47 14.36859 2.71305.38854 2.57371.40877 2.44636.42929 2.32943 46 15.36892 2.71062.38888 2.57150.40911 2.44433.42963 2.32756 45 16.36925 2.70819.38921 2.56928.40945 2.44230.42998 2.32570 44 17.36958 2.70577.38955 2.56707.40979 2.44027.43032 2.32383 43 18.36991 2.70335.38988 2.56487.41013 2.43825.43067 2.32197 42 19.37024 2.70094.39022 2.56266.41047 2.43623.43101 2.32012 41 20.37057 2.69853.39055 2.56046.41081 2.43422.43136 2.31826 40 21.37090 2.69612.39089 2.5.5827.41115 2.43220.43170 2.31641 39 22.37123 2.69371.39122 2.55608.41149 2.43019.43205 2.31456 38 23.37157 2.69131.39156 2.55389.41183 2.42819.43239 2.31271 37 24.37190 2.68892.39190 2.55170.41217 2.42618.43274 2.31086 36 25.37223 2.68653.39223 2.54952.41251 2.42418.43308 2.30902 35 26.37256 2.68414.39257 2.54734.41285 2.42218.43343 2.30718 34 27.37289 2.68175.39290 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 2.54082.41387 2.41620.43447 2.30167 31 30.37388 2.67462.39391 2.53865.41421 2.41421.43481 2.29984 30 31.37422 2.67225.39425 2.53648.41455 2.41223.43516 2.29801 29 32.37455 2.66989.39458 2.53432.41490 2.41025.43550 2.29619 28 33.37488 2.66752.39492 2.53217.41524 2.40827.43585 2.29437 27 34.37521 2.66516.39526 2.53001.41558 2.40629.43620 2.29254 26 35.37554 2.66281.39559 2.52786.41592 2.40432.43654 2.29073 25 36.37588 2.66046.39593 2.52571.41626 2.40235.43689 2.28891 24 37.37621 2.65811.39626 2.52.357.41660 2.40038.43724 2.28710 23 38.37654 2.65576.39660 2.52142.41694 2.39841.43758 2.28528 22 39.37687 2.65342.39694 2.51929.41728 2.39645.43793 2.28348 21 40.37720 2.65109.39727 2.51715.41763 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 2.51289.41831 2.39058.43897 2.27806 18 43.37820 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 2.63483.39963 2.50229.42002 2.38084.44071 2.26909 13 48.37986 2.6.3252.39997 250018.42036 2.37891.44105 2.26730 12 49.38020 2.63021.40031 2.49807.42070 2.37697.44140 2.26552 11 50.38053 2.62791.40065 2.49597.42105 2.37504.44175 2.26374 10 51.38086 2.62561.40098 2.49386.42139 2.37311.44210 2.26196 9 52.38120 2.62332.40132 2.49177.42173 2.37118.44244 2.26018 8 53.38153 2.62103.40166 2.4S967.42207 2.36925.44279 2.25840 7 54.38186 2.61874.40200 2.48758.42242 2.36733.44314 2.25663 6 55.33220 2.61646.40234 2.48549.42276 2.36541.44349 2.25486 5 56.33253 2.61418.40267 2.48340.42310 2.36349.44384 2.25309 4 57.38286 2.61190.40301 2.48132.42345 2.36158.44418 2.25132 3 58.38320 2.60963.40.335 2.47924.42379' 2.35967.44453 2.24956 2 59.33353 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. I Tang Cotang. Tang. Cotang. Tang..;93 | 68 67D 660 I'-._ 236 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 240 50 260aeo ~I. Tang. Cotang. Tang. Cotang. Tang. Cotng. Tg. Tang. Cotang. IM. 0.44523 2.24604.46631 2.14451.48773 2.05030.50953 1.96261 60 1.44558 2.24428.46666 2.14288.48809 2.04879.50989 1.96120 59 2.44593 2.24252.46702 2.14125.48845 2.04728.51026 1.95979 58 3.44627 2.24077.46737 2.13963.48881 2.;04577.51063 1.95838 57 4.44662 2.23902.46772 2.13801.48917 2.04426.51099 1.95698 56 5.44697 2.23727.46808 2.13639.48953 2.04276.51136 1.95557 55 6.44732 2.23553.46343 2.13477 48989 2.04125.51173 1.95417 54 7.44767 2.23378.46879 2.13316.49026 2.03975.51209 1.95277 53 8.44802 2.23204.46914 2.13154.49062 2.03825.51246 1.95137 52 9.44837 2.23030.46950 2.12993.49098 2.03675.51283 1.94997 51 10.44872 2.22857.46985 2.12832.49134 2.03526.51319 1.94858 50 11.44907 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 1.93608 41 20.45222 2.21132.47341 2.11233.49495 2.02039.51688 1.93470 40 21.45257 2.20961.47377 2.11075.49532 2.01891.51724 1.93332 39 22.45292 2.20790.47412 2.10916.49568 2.01743.51761 1.93195 38 23.45327 2.20619.47448 2.10758.49604 2.01596.51798 1.93057 37 24.45362 2.20449.47483 2.10600.49640 2.01449.51835 1.929d0 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 2.00423.520591 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 2.00131.5S168 1.91690 27 34.45713 2.18755.47840 2.09028.50004 1.99986.52205 1.91554 26 35.45748 2.18587.47876 2.08872.50040 1.99841.52242 1.91418 25 36.45784 2.18419.47912 2.08716.50076 1.99695.52279 1.91282 24 37.45819 2.18251.47948 2.08560.50113 1.99550.52316 1.91147 23 38.45854 2.18084.47984 2.08405.50149 1.99406.52353 1.91012 22 39.45889 2.17916.48019 2.08250.50185 1.99261.52390 1.90876 21 40.45924 2.17749.48055 2.08094.50222 1.99116.52427 1.90741 20 41.45960 2.17582.48091 2.07939.50258 1.98972.52464 1.90607 19 42.45995 2.17416.48127 2.07785.50295 1.98828.52501 1.90472 18 43.46030 2.17249.48163 2.07630.50331 1.98684.52538 1.90337 17 44.46065 2.17083.48198 2.07476.50368 1.98S40.52575 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 2.16420.48.342 2.06860.50514 1.97966.52724 1.89667 12 49.46242 2.16255.48378 2.06706.50550 1.97823.52761 1.89533 11 50.46277 2.16090.48414 2.06553.50587 1.97681.52798 1.89400 10 51.46312 2.15925.48450 2.06400.50623 1.97538.52836 1.89266 9 52.46348 2.15760.48486 2.06247.50660 1 97395 52873 1.89133 8 53.46.383 2.15596.48521 2.06094.50696 1.97253.52910 1.89000 7 54.46418 2.15432.48557 2.05942.50733 1.97111.52947 1.88867 6 55.46454 2.15268.48593 2.05790.50769 1.96969.52985 1.88734 5 56.46489 2.15104.48629 2.05637.50806 1.96827.53022 1.88602 4 57.46525 2.14940.48665.2.05485.50843 1.96685.53059 1.88469 3 58.46560 2.14777.48701 2.05333.50879 1.96544.53096 1.88337 2 59.46595 2.14614.48737 2.05182.50916 1.96402.53134 1.88205 1 60.46631 2.14451.48773 2.05030.50953 1.96261.53171 1.88(73 0 M. Cotang. Tang. Cotng. Tang. Cotang. Tang. Cotang. Tang. M. 650 S 640 630 620 tABLE XV. NATURAL TANGENTS AND COTANGENTS. 237 280 2k90 _ 300 310 M1 Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.53171 1.88073.55431 1.80405.57735 1.73205.60086 1.66428 60 1.53208 1.87941.55469 1.80281.57774 1.73089.60126 1.66318 59 2.53246 1.87809.55507 1.80158.57813 1.72973.60165 1.66209 58 3.53283 1.87677.55545 1.80034.57851 1.72857.60205 1.66099 57 4.53320 1.87546.55583 1.79911.57890 1.72741.6245 1.65990 56 5.53358 1.87415.55621 1.79788.57929 1.72625.6084 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 1.86499.55888 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 1.78563.58318 1.71473.60681 1.64795 45 16.63769 1.85979.56041 1.78441.58357 1.71358.60721 1.64687 44 17.53807 1.85850.56079 1.78319.58396 1.71244.60761 1.64579 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 1.70901.60881 1.64256 40 21.53957 1.85333.56232 1.77834.58552 1.70787.60921 1.64148 39 22.53995 1.85204.56270 1.77713.58591 1.70673.60960 1.64041 38 23.54032 1.85075.56309 1.77592.58631 1.70560.61000 1.63934 37 24.54070 1.84946.56347 1.77471.58670 1.70446.61040 1.63826 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 29.542.58 1.84305.56539 1.76869.58865 1.69879.61240 1.63292 31 30.54296 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 34.54446 1.83667.56731 1.76271.59061 1.69316.61440 1.62760 26 35.54484 1.83540.56769 1.76151.59101 1.69203.61480 1.62654 25 36.54522 1.83413.56808 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 1.83033.56923 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 1.82150.57193 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 1.74492.59651 1.67641.62043 1.61179 11 50.55051 1.81649.57348 1.74375.59691 1.67530.62083 1.61074 10 51.55089 1.81524.57386 1.74257.59730 1.67419.62124 1.60970 9 52.55127 1.81399.57425 1.74140.59770 1.67309 62164 1.60865 8 53.55165 1.81274.57464 1.74022.59809 1.67198.62204 1.60761 7 54.55203 1.81150.57503 1.73905.59849 1.67088.62245 1.60657 6 55.55241 1.81025.57541 1.73788.59888 1.66978.62285 1.60553 5 56.55279 1.80901.57580 1.73671.59928 1.66867.62325 1.60449 4 57.55317 1.80777.57619 1.73555.59967 1.66757.62366 1.60345 3 58.55355 1.80653.57657 1.73438.60007 1.66647.62406 1.60241 2 59.55393 1.80529.57696 1.73321.60046 1.66538.62446 1.60137 1 60.55431 1.80405.57735 1.73205.60086 1.66428.62487 1.60033 0 M. Cotang. T oang. Tang. Cotang. Tang. Cotang. Tang. M. 610fflci 60s 5903 80 238 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 32o 330 340 350 M Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. 0.62487 1.60033.64941 1.53986.67451 1.48256.70021 1.42815 60 1.62527 1.59930.64982 1.53888.67493 1.48163.70064 1.42726 59 2.62568 1.59826.65024 1.53791.67536 1.48070.70107 1.42638 58 3.62608 1.59723.65065 1.53693.67578 1.47977.70151 1.42550 57 4.62649 1.F9620.65106 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 1.59105.65314 1.53107.67832 1.47422.70412 1.42022 51 10.62892 1.59002.65355 1.53010.67875 1.47330.70455 1.41934 50 11.62933 1.58900.65397 1.52913.67917 1.47238 70499 1.41847 49 12.62973 1.58797.65438 1.52816.67960 1.47146.70542 1.41759 48 13.63014 1.58695.65480 1.52719.68002 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 1.46686.70760 1.41322 43 18.63217 1.58184.65638 1.52235.68215 1.46595.70804 1.41235 42 19.63258 1.58083.6.5729 1.52139.68258 1.46503.70848 1.41148 41 20.63299 1.57981.65771 1.52043.68301 1.46411.70891 1.41061 40 21.63340 1.57879.65813 1.51946.68343 1.46320.70935 1.40974 39 22.63380 1.57778.65854 1.51850.69386 1.46229.70979 1.40887 38 23.63421 1.57676.65896 1.51754.68429 1.46137.71023 1.40800 37 24.63462 1.57575.65938 1.51658.68471 1.46046.71066 1.40714 36 25.63503 1.57474.65980 1.51562.68514 1.45955.71110 1.40627 35 26.63544 1.57372.66021 1.51466.68557 1.45864.71154 1.40540 34 27.63584 1.57271.66063 1.51370.68600 1.45773.71198 1.40454 33 28.63625 1.57170.66105 1.51275.68642 1.45682.71242 1.40367 32 29.63666 1.57069.66147 1.51179.68685 1.45592.71285 1.40281 31 30.63707 1.56969.66189 1.51084.68728 1.45501.71329 1.40195 30 31.63748 1.56868.66230 1.50988.68771 1.45410.71373 1.40109 29 32.63789 1.56767.66272..50893.68814 1.45320.71417 1.40022 28 33.63830 1.56667.66314 1.50797.68857 1.45229.71461 1.39936 27 34.63871 1.56566.66356 1.50702.68900 1.45139.71505 1.39850 26 35.63912 1.56466.66398I.50607.68942 1.45049.71549 1.39764 25 36.63953 1.56366.66440 1.50512.68985 1.44958.71593 1.39679 24 37.63994 1.56265.66482 1.50417.69028 1.44868.71637 1.39593 23 38.64035 1.56165.66524 1.50322.69071 1.44778.71681 1.39507 22 39.64076 1.56065.66566 1.50228.69114 1.44688 71725 1.39421 21 40.64117 1.55966.66608 1.50133.69157 1.44598 71769 1L39336 20 41.64158 1.55866.66650 1.50038.69200 1.44508 71813 1.39250 19 42.64199 1.55766.66692 1.49944.69243 1.44418.71857 1.39165 18 43.64240 1.55666.66734 1.49849.69286 1.44329.71901 1.39079 17 44.64281 1.55567.66776 1.49755.69329 1.44239.71946 1.38994 16 45.64322 1.55467.66E18 1.49661.69372 1.44149.71990 1.38909 15 46.64363 1.55368.66860 1.49566.69416 1.44000.72034 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 1.55071.66986 1.49284.69545 1.43792.72167 1.38568 11 50.64528 1.54972.67028 1.49190.69588 1.43703.72211 1.38484 10 51.64569 1.54873.67071 1.49097.69631 1.43614.72255 1.38399 9 52.64610 1.54774.67113 1.49003.69675 1.43525.72299 1.3S314 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 1.48256.7C021 1.42815.72654 1.37638 0 M. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. 570 5660 550 540 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 239 360 3T70 380 390 M. Tang Coang. 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 4.72332 1.37302.75538 1.32384.78316 1.27658.81171 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 1.36967.75721 1.32064.78504 1.27.382.81364 1.22904 52 9.73055 1.36883.75767 131984.78551 1.27306.81413 1.22831 51 10.73100 1.36800.75812 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 1.36383.76042.31507.78834 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 1.256319.82044 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 1.35554.76502 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 1.25792.82385 1.21382 31 30.73996 1.35142.76733 1.30323.79544 1.25717.82434 1.21310 30 31.74041 1.35060.76779 1.30244.79591 1.25642.82483 1.21238 29 32.74086 1.34978.76825 1.30166.79639 1.25567.82531 1.21166 28 33.74131 1.34896.76871 1.30087.79686 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 1.29853.79829 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 43.74533 1.34079.77335 1.29307.80163 1.24746.83071 1.20379 17 44.74623 1.33998.77382 1.29229.80211 1,24672.83120 1.20308 16 45.74674 1.33916.77428 1.29152.80258 1.24597.83169 1.20237 15 46.74719 1.33835.77475 1.29074.80306 1.24523.83218 1.20166 14 47.74764 1.33754.77521 1.28997.80354 1.24449.83268 1.20095 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 1.33511.77661 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 1.23379.80738 1.23858.83662 1.19528 5 56.75173 1.33026.77941 1.28302.80786 1.23784.83712 1.19457 4 57.75219 1.32946.77988 1.28225.80834 1.23710.83761 1.19387 3 58.75264 1.32865.78035 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. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. I 530 520 510 500 240 TABLE XV. NATURAL TANGENTS AND COTANGENTS. 400 410 42o 430 M. Tang. Cotang. Tang. Cotang. Tang. Cotang Tang. Cotang. M. 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 1.06862 54 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 11.84457 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 1.14028.90834 1.10091.94071 1.06303 45 16.84706 1.18055.87749' 1.13961.90887 1.10027.94125 1.06241 44 17.84756 1.17986.87801 1.13894.90940 1.09963.94180 1.06179 43 18.84806 1.17916.87852 1.13828.90993 1.09899.94235 1.06117 42 19.84856 1.17846.87904 1.13761.91046 1.09834.94290 1.06056 41 20.84906 1.17777.87955 1.13694.91099 1.09770.94345 1.05994 40 21.84956 1.17708.88007 1.13627.91153 1.09706.94400 1.05932 39 22.85006 1.17638.88059 1.13561.91206 1.09642.94455 1.05870 38 23.85057 1.17569.88110 1.13494.91259 1.09578.94510 1.05809 37 24.85107 1.17500.88162 1.13428.91313 1.09514.94565 1.05747 36 25.85157 1.17430.88214 1.13361.91366 1.09450.94620 1.05685 35 26.85207 1.17361.88265 1.13295.91419 1.09386.94676 1.05624 34 27.85257 1.17292.88317 1.13228.91473 1.09322.94731 1.05562 33 28.85308 1.17223.88369 1.13162.91526 1.09258.94786 1.05501 32 29.85358 1.17154.88421 1.13096.91580 1.09195.94841 1.05439 31 30.85408 1.17085.88473 1.13029.91633 1.09131.94896 1.05378 30 31.8.5458 1.17016.88524 1.12963.91687 1.09067.94952 1.05317 29 32.85509 1.16947.88576 1.12897.91740 1.09003.95007 1.05255 28 33.85559 1.16878.88623 1.12831.91794 1.08940.95062 1.05194 27 34.85609 1.16809.88680 1.12765.91847 1.08876.95118 1.05133 26 35.85660 1.16741.88732 1.12699.91901 1.03813.95173 1.05072 25 36.85710 1.16672.88784 1.12633.91955 1.08749.95229 1.05010 24 37.85761 1.16603.88836 1.12567.92008 1.03686.95284 1.04949 23 38.85811 1.16535.88888 1.12501.92062 1.08622.95340 1.04888 22 39.85862 1.16466.88940 1.12435.92116 1.08559.95395 1.04827 21 40.85912 1.16398.88992 1.12369.92170 1.08496.95451 1.04766 20 41.85963 1.16329.89045 1.12303.92224 1.08432.95506 1.04705 19 42.86014 1.16261.89097 1.12238.92277 -1.08369.95562 1.04644 18 43.86064 1.16192.89149 1.12172.92.331 1.08306.95618 1.04583 17 44.86115 1.16124.89201 1.12106.92385 1.08243.95673 1.04522 16 45.86166 1.16056.89253 1.12041.92439 1.08179.95729 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 11 50.86419 1.15715.89515 1.11713.92709 1.07864.96008 1.04158 10 51.86470 1.15647.89567 1.11648.92763 1.07801.96064 1.04097 9 52.86521 1.15579.89620 1.11582.92817 1.07738.96120 1.04036 8 53.86572 1.15511.89672 1.11517.92872 1.07676.96176 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 56.86725 1.15308.89830 1.11321.93034 1.07487.96344 1.03794 4 57.86776 1.15240.89883 1.11256.93088 1.07425.96400 1.03734 3 58.86327 1.15172.89935 1.11-191.93143 1.07362.96457 1.03674 2 59.86878 1.15104.89988 1.11126.93197 1.07299.96513 1.03613 1 60.86929 1.15037.90010 1.11061.93252 1.07237.96569 1.03553 0 M. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. I 490 480 470 460 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 67 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 1.01998 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 61 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 1.02892 49 31.98327 1.01702 29 51.99478 1.00525 9 12.97246 1.02832 48 32.98384 1.01642 28 52.99536 1.00467 8 13.97302 1.02772 47 33.98441 1.015S3 27 53.99594 1.00408 7 14.97359 1.02713 46 34.98499 1.01524 26 54.99652 1.003.50 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 56.99768 1.00233 4 17.97529 1.02533 43 37.98671 1.01347 23 57.99826 1.00175 3 18.975S6 1.02474 42 33.98728 1.01288 22 58.99884 1.00116 2 19.97643 1.02414 41 39.98786 1.01229 21 59.99942 1.00058 1 20.97700 1.02355 40 40.98843 1.01170 20 60 1.00000 1.00000 0 M. Cotang. Tang. 9M. M. Cotang. Tang. M 9. M. Cotang. Tang. Y. 1 450 450 4 50 242 TABLE XVI. RISE PER MILE OF VARIOUS GRADES. TABLE XVI. RISE PER MILE OF VARIOUS GRADES. Grade Rise per Grade Riseper Grade Rise per Grade Rise per per Mile per Mile. per Mile Mile. Ptation. Station. Station. taion..01.523.41 21.648.81 42.768 1.21 63.888.02 1. 0.6.42 22.176.82 43.296 1.22 64.416.03 1.584.43 22.704.83 43.821 1.23 64.944.04 2.112.44 23.232.84 44.352 1.24 65.472.05 2.640.45 2:3.760.85 44.880 1.25 66.000.06 3.168.46 24.288.86 45.408 1.26 66.528.07 3.696.47 24.816.87 45.936 1.27 67.056.08 4.224.48 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 50.688 1.36 71.808.17 8.976.57 30.096.97 51.216 1.37 72.336.18 9.504.58 30.624.98 51.744 1.38 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 1.01 53.328 1.41 74.448.22 11.616.62 32.736 l.d 53.856 1.42 74.976.23 12.144.63 33.264 1.03 54.384 1.43 75.504.24 12.672.64 33.792 1.04 54.912 1.44 76.032.25 13.200.65 34.320 1.05 55.440 1.45 76.560.26 13.728.66 34.848 1.06 55.968 1.46 77.088.27 14.256.67 35.376 1.07 56.496 1.47 77.616.28 14.784.68 35.904 1.08 57.024 1.48 78.144.29 15.312.69 36.432 1.09 57.552 1.49 78.672.30 15.840.70 36.960 1.10 58.080 1.50 79.200.31 16.368.71 37.488 1.11 58.608 1.51 79.728.32 16.896.72 38.016 1.12 59.136 1.52 80.256.33 17.424.73 38.544 1.13 59.664 1.53 80.784.34 17.952.74 39.072 1.14 60.192 1.54 81.312.35 18.480.75 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 41.184.18 62.304 1.58 83.424.39 20.692.79 41.712 1.19 62.832 1.59 83.952.40 21.120.80 42.240 1.20 63.360 1.60 84.480 TABLE XVI. RISE PER MILE OF VARIOUS GRADES. 243 araae Grade l rde per Gra Gd Rise per Grd Rise per G eRise per Grad e ise per Rise per per Mile. per Mile. per ile. per Mile. Station. tation. tation. 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 88.1.76 1.87 98.736 2.70 142.560 4.70 248.160 1.68 88.704 1.88 99.264 2.80 147.840 4.80 253.440 1.69 89.232 1.89 99.792 2.90 153.120 4.90 258.720 1.70 89.760 1.90 100.320 3.00 158.400 5.00 264.000 1.71 90.288 1.91 100.848 3.10 163.680 5.10 269.280 1.72 90.816 1.92 101.376 3.20 168.960 5.20 274.560 1.73 91.344 1.93 101.904 3.30 174.240 5.30 279.840 1.74 91.872 1.94 10-2.432 3.40 179.520 5.40 285.120 1.75 92.400 1.95 102.960.;0 184.800 5.50 290.40U 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 HIE END