IN MEMORIAM FLORIAN CAJORI PLANE TRIGONOMETRY BY ARTHUR GRAHAM HALL, Ph.D. (Leipzig) Pbofessor of Mathematics University of Michigan FRED GOODRICH FRINK, M.S. (Chicago) Professor of Railway Engineering University of Oregon NEW YORK HENRY HOLT AND COMPANY Copyright, 1W9, BY HENRY HOLT AND COMPANY. CAJORI NarfajDoli \$xt9S J. 8. Gushing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE This book has had its origin in the desire of the authors to meet the mutual demands of mathematicians and engineers for a treatment that shall more completely supply the needs of the technological student. It is believed that this has been done by enriching the subject with applications to physics and engineering, in such a way as to increase its value at the same time to the general student. The present volume is, moreover, based upon a preliminary edition actually used for several terms in the class- room. In view of the peculiar situation of trigonometry in the cur- riculum, the course has been kept of the usual length. The topics have been arranged, however, in the order of increasing difficulty, by postponing the more abstract but no less essential study of the functions of the general angle, until after the arith- metical solution of triangles. The abundance of exercises and problems will give the teacher large opportunity for selection. The discussion of the slide rule is inserted because of the increasing employment of this useful instrument. The authors gratefully acknowledge their indebtedness to Professor E. J. Townsend and Professor H. L. Rietz, of the University of Illinois, and Professor A. Ziwet and Professor J. L. Markley, of the University of Michigan, for valuable criticisms and suggestions. ARTHUR G. HALL. FRED G. FRINK. Ann Arbor, January, 1909. iii CONTENTS PAET I PLANE TBIQONOMETBY CHAPTER I GEOMETRIC NOTIONS ARTICLE PAOS 1. General statement 1 2. Directed line segments 1 3. Positive and negative angles 2 Exercise I . . . . ^ 3 4. Rectangular coordinates 3 Exercise II 6 CHAPTER II THE ACUTE ANGLE 6. The purpose of trigonometry 6 6. Definitions of the trigonometric functions . 6 7. Relations between the ratios 7 8. Signs and limitations in value 7 Exercise III 8 9. Fundamental relations 9 Exercise IV 10 10. Functions of complementary angles 11 11. Functions of 45°, 30°, 60° 12 12. Functions of 0° and 90° •. . . .12 Exercise V 13 13. Variation of the trigonometric functions as the angle varies ... 14 14. Inverse trigonometric functions 15 Exercise VI 16 16. Orthogonal projection .• . . 16 Exercise VII 18 CHAPTER III RIGHT TRIANGLES 16. Laws for solution 19 17. Area of right triangles . 20 VI CONTENTS ARTICLE PAGK 18. Method of solution 20 19. Trigonometric tables 22 20. Errors and checks 23 Exercise VIII 25 21. 22. 23. 24. 25. 27. 28. 29. 30. 31. CHAPTER IV LOGARITHMS Definition of a logarithm . . . 29 Laws of combination , . . .30 Common logarithms 31 Characteristic ............ 32 Mantissa 33 Exercise IX . 34 Interpolation 1 ... 35 Numbers from logarithms 35 Cologarithms . .36 Logarithms of trigonometric functions 36 Exercise X . 37 The slide rule 39 Exercise XI .41 Right triangles solved by logarithms .41 Exercise XII 44 CHAPTER V THE OBTUSE ANGLE 32. Definitions of the trigonometric functions of obtuse angle 33. Signs and limitations in value 34. Fundamental relations 35. Variation 36. Functions of 180° 37. Functions of supplementary angles 38. Functions of (90° -fa) Exercise XIII 47 47 48 48 48 48 49 50 CHAPTER VI OBLIQUE TRIANGLES 39. Formulas for solution 52 40. Law of projections 53 41. Law of sines ............ 63 42. Law of cosines 53 43. Law of tangents 64 44. Angles in terms of sides 65 CONTENTS vii ARTICLE PAGE 45. Area of oblique triangles ....,,.... 55 46. Numerical solution . . ' 56 47. Case I. Two angles and one side 57 48. Case II. Two sides and an opposite angle 58 49. Case III. Two sides and the included angle 61 50. Case IV. Three sides 61 51. Composition and resolution of forces. Equilibrium . . . . . 63 Exercise XIV 66 CHAPTER VII THE GENERAL ANGLE 52. General definition of an angle 70 53. Axes, quadrants, etc. ' . . . . . . . . . .71 54. Definitions of the trigonometric functions ....... 71 55. Signs and limitations in value 72 Exercise XV 73 56. Variation of the trigonometric functions ....... 74 57. Graphs of the trigonometric functions 75 58. Functions of 270° and 360° 78 Exercise XVI 79 69. Fundamental relations 79 60. Line representations of the trigonometric functions . . . . . 80 Exercise XVII 82 61. Periodicity of the trigonometric functions 83 62. Functions of (k • ^± a) , . . .83 Exercise XVIII 87 CHAPTER VIII FUNCTIONS OF TWO ANGLES 63. Formulas for sin (a + /3) and cos (a + /S) . . = . . . .89 64. Extension of the addition formulas 90 Exercise XIX 91 65. Subtraction formulas 91 66. Formulas for tan (a ±/3), cot (a ± (S) . . . . . . .92 Exercise XX .... 92 67. Functions of twice an angle 93 68. Functions of half an angle . . . , . . . . .93 Exercise XXI .... . . . . . . . .94 69. Conversion formulas for products 95 Exercise XXII 96 70. Conversion formulas for sums and differences 97 71. Multiple angles 97 Exercise XXIII 97 viii CONTENTS CHAPTER IX ANALYTIC TRIGONOMETRY AETIOLE PAGE 6 72. Limits of - — t, and - — -, as 6 approaches zero 99 sin d tan 0' Examples 101 73. De Moivre's theorem 101 Examples 103 74. Graphical representation of complex numbers 103 Examples 108 75. Exponential values of the trigonometric functions 109 Examples 110 76. Hyperbolic functions 110 77. Exponential and trigonometric series Ill Examples 115 78. Computation of trigonometric tables ........ 116 79. Proportional parts 116 80. General inverse functions . . . . , . , . . .117 81. Logarithmic values of inverse functions 118 Examples 119 Review Exercises ' . . 120 Formulas 130 Answers 137 Index , . 147 TRIGONOMETRIC AND LOGARITHMIC TABLES I. Common logarithms of numbers 3 II. Logarithms of the trigonometric functions . . . . . .25 III. Natural trigonometric functions ........ 71 IV. Squares and square roots ..... o .... 91 TRIGONOMETRY GREEK ALPHABET Letters Names Letters Names Letters Names A o Alpha ' I I Iota P P Rho B^ Beta K/c Kappa S(r s Sigma Ty Gamma AX Lambda Tt Tau A5 Delta Mm Mu Tu Upsilon Ee Epsilon N p Nu $0 Phi Zf Zeta S^ Xi Xx Chi H^ Eta Oo Omicron ^,/. Psi ee Theta Htt Pi w Omega TRIGONOMETRY PART I PLANE TRIGONOMETRY CHAPTER I GEOMETRIC NOTIONS 1. General statement. It is assumed that the student is well versed in those theorems of elementary geometry concerning angles, arcs, and triangles. It is particularly desirable that he be familiar with the measurement of angles and with the proper- ties of similar triangles. While the review thus suggested is left to the student, certain more advanced geometric ideas are treated in the remaining arti- cles of this chapter. Throughout the course the student should make continual, careful, and intelligent use of such drawing instruments as are included in the equipment at technical schools. In case such sets are not available, as in more general classes, there should be pro- vided at least a straightedge, with graduated scale, a protractor, and a pair of compasses or dividers. 2. Directed line segments. A point which moves from one position to a second, without changing its direction of motion, traces a directed line segment. Directed line segments are always read with regard to their direction, from the initial extremity to the terminal extremity. Two line segments are equal if they have the same length and direction, whether their lines are coincident or parallel. Either of two line segments having the same length but opposite direc- tions is said to be the negative of the other. If one direction is taken as positive, the opposite direction is negative. 1 GEOMETRIC NOTIONS iXiius, An FigCli,. E F K H HK^ BA = - AB, Fio.i. CD='2BA = -2AB. If ^ is the initial point and B the terminal point, the line segment is read AB, and in this notation the positive or negative direction of the segment is expressed without the aid of a prefixed + or — . In case the segment is represented by a single symbol, as the letter «, the direction must be indicated in some further manner, as by a prefixed + or — , or by an arrowhead in the figure. Two line segments lying in the same line are added by placing the initial point of the second upon the terminal point of the first, each retaining its proper direction. The sum is the segment extending from the initial point of the first to the termi- nal point of the second. Subtraction is performed by reversing the direction of the subtrahend and adding. Line segments having the same or opposite directions may all be transferred to a common line. Their addition and subtraction thus cor- respond exactly to the algebraic addition and subtraction of posi- tive and negative numbers. If A, B^ denote three points arranged in any order along a straight line, then AB^BQ=AQ, and AB^BC^CA = ^. 3. Positive and negative angles. If a line rotates (in a plane) about one of its points, an angle is generated, of which the origi- nal position of the line is the initial side and the final position the terminal side. A distinction may be made according as the rota- tion is clockwise or counter-clockwise about the vertex. The counter-clockwise direction is chosen as positive. Angles are always read with regard to their direction of rotation ; thus if OA is the initial side and OB the terminal side, tlie angle is read AOB. This notation includes the direction or sign of the angle, and the -f- or — should not be prefixed. In case the angle is represented by a single symbol, as by the Greek letter a, the direction must be indicated in some further manner, as by a pre- fixed -f or — , or by a curved arrow in the figure. POSITIVE AND NEGATIVE ANGLES 3 Just as with line segments, reversing the direction multiplies the angle by - 1 ; thus BOA = -AOB. Two angles are added by placing them in the same plane with a common vertex, the initial side of the second coincident with the terminal side of the first, each retaining its own direction. The sum is the angle from the initial side of the first to the terminal side of the second. Subtraction is performed by reversing the subtraliend and adding. In Fig. 2, AOB + BOO=AOC, AOO-BOC=AOB. ^:;^^ EXERCISE I Solve the following problems graphically : 1. On a train running 40 miles an hour, a man walks 4 miles an hour. Find the speed of the man with reference to the ground, (a) if he walks toward the front ; (6) if he walks toward the rear of the train. 2. The man's speed with reference to the ground is 10 miles an hour. What is the speed of the train (a) if he is walking 5 miles an hour toward the front; (b) if lie is running 8 miles an hour toward the rear? 3. On June 1 the price of corn was 50 cents, and during the succeeding ten days it fluctuated as follows : rose 2 cts., rose 3, fell 1, fell 2, fell 5, fell 3, rose 2, rose 2, rose 3, rose 1. Find the price on June 11. 4. During a football game the progress of the ball from the middle of the field was north 40 yards, south 25, south 5, south 10, south 30, north 50, north 10, north 20. Find the resulting position of the ball. Combine graphically, using a protractor : 5. 45° + 30° ; 90° + 45° ; 40° + 35° + 50°. 6. 60° - 45° ; 90° - 50° ; 180° - 120°. 7. 30° + 80° + 55° ; 40° + 60° - 30° ; 60° - 20° + 70° - 90°. 8. 40° - 70° + 15°; 65° -f 15° - 90°; 75° - 180°. 4. Rectangular coordinates. If two mutually perpendicular straight lines are chosen, and a positive direction on each, the position of any point in their plane is determined by giving its perpendicular distances from these fixed lines. The two lines are called the axes of coordinates, and are usually taken so that one GEOMETRIC NOTIONS r^^ O^^-x. -^x ^A is horizontal and the other vertical. The point of intersection of the axes is called the origin. The two determining data for any point are called its coordinates. The horizontal distance from the, axis Oy to the point is the abscissa of the point, and the vertical .y distance from the axis OX to the point is the ordinate of the point. The point ^^-jP^ C^^ whose abscissa is x and J^ ordinate y is denoted by the notation (a:, ^). Be- cause it is convenient to lay off the abscissa of a point upon the axis OX and the ordinate upon the axis OY^ these axes are referred to ^^^- ^- as the axes of abscissas and ordinates respectively. When x denotes the abscissa and y the. ordinate of the point, the axes may be referred to as the X-axis and the iF-axis respectively. The distance from the origin to the point is called the radius vector of the point. It is known whenever the abscissa and the ordinate are given, since the three form, respectively, the hypote- nuse, base, and altitude of a right triangle. The abscissa of a point should always be read from the F-axis to the point. The direction from left to right is chosen as posi- tive. Therefore all points at the right of the y"-axis have positive abscissas, and all points at the left, negative abscissas. The ordi- nate of a point is always read from the JT-axis to the point. The upward direction is chosen as positive. Hence all points above the X-axis have positive ordinates, and all points below, negative ordinates. The radius vector is always read from the origin to the point, and is always considered positive. It will be noticed that the abscissa and the ordinate are equal to the projections of the radius vector on the X-axis and P"-axis, respectively; these projections will henceforth be used inter- changeably for the coordinates themselves.* * The foot of the perpendicular dropped from a point upon a given line is said to be the orthogonal or orthographic projection of the point on the line. The projec- tion of a line segment on a given line is the segment from the projection of the ini- tial point of the given segment to that of the terminal point. This kind of projec- tion will be used exclusively throughout this book, unless otherwise expressly stated. RECTANGULAR COORDINATES The two axes divide the whole plane into four portions, known as the first, second, third, and fourth quadrants, beginning with the upper right-hand quadrant and numbering counter-clockwise about the origin. If two points, P and Q, lie in a line through the origin, their coordi- nates, with the radii vectores, form two similar triangles. If the abscissa, ordinate, and radius vector of P are X, y, V, respectively, those of Q are Jcx, ky^ kv. Fig. 4. EXERCISE II 1. Plot the points (2, 3), (-3, 5), (-2, -4), (1, -3), (3, 0), (0, 4), (-5,0), (0, -2), (0,0). 2. Plot the points (3, 2), (6, 4), (12, 8). 3. Plot the points (0, 5), (4, 3), (-3, 4), (0, -5). 4. Find both graphically and by computation the radius vector of each point in examples 1, 2, 3. In what quadrant does each point lie ? 5. Describe the location of all points whose abscissas equal 3 ; whose ordi- nates equal 5 ; whose radii vectores equal 6. 6. Write the coordinates of the vertices of a square of side a, if the axes of coordinates are taken along two sides ; along its diagonals. 7. The hour hand of a clock is 5 inches long. Find the coordinates of its tip referred to the horizontal and vertical diameters of its face at three o'clock ; at six ; at eight ; at half-past ten. 8. Compare the location of the points (2, 3), (3, 2), ( — 2, 3), ( — 2, —3), (3, —2). Describe the directions of their radii vectores. 9. Find the distance from (2, 5) to (6, 9) ; from (-3, 2) to (4, 5). 10. Find the direction of the line joining (6, 1) to (8, 3) ; (4, 1) to (1, 4) ; (6, 3) to (1,3); (-2,4) to (1,1). 11. A man starts from the southwest corner of a government township and goes in turn to the northwest corner of section 36 ; northwest corner section 22 ; southeast corner section 3 ; northeast corner section 5 ; thence to the point of beginning. Show by sketch the shortest route along section lines, and com- pute the cross-country distances. 12. A city is laid out in checker-board fashion. The streets are eight to the mile and look to the cardinal points of the compass. It is proposed to in- troduce two diagonal (45"") streets extending through the busiest corner to the outskirts. What distances will be saved thereby in driving from this corner to each of the points specified below? Nine blocks east and six blocks north; 5 W. and 7 S. : 10 W. and 10 N. : 1 E. and 14 S. CHAPTER II THE ACUTE ANGLE 5. Purpose of trigonometry. One of the principal objects of trigonometry is the computation of triangles. We have learned from elementary geometry that a triangle is determined when we know any three of its parts (sides and angles), at least one of them being a side. These data enable us to construct the triangle ; but elementary geometry does not teach us how to calculate the re- maining parts. The reason is that sides and angles are expressed in different units. It is, therefore, desirable to measure angles not only in degrees, but also by means of lines, or rather by means of ratios of lines. This can be done as shown in the following articles. 6. Definitions of the trigonometric functions.. Suppose the acute angle A OB ( = «) to be placed on a system of axes of coor- dinates with its vertex at the origin and its initial side OA extending along the X-axis toward the right. Its terminal side OB will fall in the first quadrant. (See Fig. 5.) Any point P in its terminal side possesses one and only one set of related distances (two coordinates and radius vector). 1 ^ > X Its abscissa x(^=OM)^ its ordinate ^ (=zMP), and its radius vector V (= OF) are all positive and con- tY O M Fig. 5. nected by the relation o . o o •^ TT -\- y^ — v^. The six ratios between these three distances are of frequent occurrence and prime importance. They serve, indeed, the pur- pose mentioned in Art. 5, and are given the following names and accompanying abbreviations : ^ = sine of a = sin a, = cosine of a = cos a, = tangent of a = tan a, = cosecant of a = esc a, = secant of a = sec a, = cotangent of a = cot a. FUNCTIONAL RELATION 7. Relations between the ratios and the angle. The three re- lated distances of any other point P^ in the terminal side of the angle a are connected with the determining distances of P by the relation , , , X y V It follows that the values of the six ratios defined in Art. 6 do not depend at all upon the particular choice of the point in the terminal side of the angle, but are determined solely and definitely by the size of the angle. A number that is determined in value by the value of some other number is said to be a function of that other number. The six ratios are therefore called trigonometric functions of the angle. This relation between the ratios and the angle is, moreover, a mutual one, such that a knowledge of one of the ratios leads to a knowledge of the angle.* Thus if we have given tan a = J, we can construct the angle a as follows : On the system of axes OX and OF plot the point P whose coordinates are (3, 2), using any convenient scale. Draw the line OA from the origin through the point P ; then is XOA the required angle a, in consequence of the definitions of Art. 6. It appears still further that from the knowledge of any one of the six trigonometric functions the remaining five can be found. Thus in the foregoing example we know by the * Pythagorean proposition that on the scale employed the hypotenuse or radius vector is V9 H- 4 = Vl3. We have then at once sin a cos a Vl3 3 2 tan„ = -. Vl3 seca = -— , cot '^= -, VT3 CSC « = —-—. V13 The properties and relations of these functions and their more immediate applications in pure and applied mathematics constitute the subject-matter of trigonometry. The science takes its name from its origin in the attempts of the ancient peoples to measure triangles. 8. Signs and limitations in value. When any acute angle is placed on the axes of coordinates in the manner prescribed in *Tbis statement, as well as some others in the present chapter, will require some modification when we extend the consideration to angles greater than 90°. 8 THE ACUTE ANGLE Art. 6, its terminal side will always fall in the first quadrant. The abscissa and ordinate, as well as the radius vector, of every point in the terminal side will all be positive. It follows that their ratios, comprising the six trigonometric functions named in Art. 6, are all positive. In no case can the abscissa or the ordinate of a point be greater than the radius vector. Indeed, save in the exceptional cases con- sidered in Art. 12, they are less than the radius vector. Conse- quently, sin a and cos a cannot be greater than unity and sec a and CSC a cannot be less than unity. EXERCISE III By careful construction and measurement find the approximate values of the following functions : 1. cos 60°. 2. tan 30°. 3. esc 45°. 4. cot 35°. 5. sin 20°. 6. sec 50°. Construct each of the following angles and find by measure- ment the values of all its functions, given 7. sin a = f . 8. cos a = y\. 9. tan a = ^\. 10. cot a = y«3. 11. sec a = ^^. 12. esc a = f i. 13. cos a = ^f . 14. sin a - |^. 15. For what angle is the tangent equal to the cotangent ? For what angle is the sine equal to the cosine ? 16. Show that the direct functions (sin a, tan a, sec a) are greater or less than the corresponding complementary functions (cos a, cot a, esc a) respec- tively, according as the angle a is greater or less than 45°. 17. Can sin a and tana be equal? When do they approach equality? 18. Show that tan a • cot a does not depend on a. Show that the same is true of sin a • esc a. 19. Show that cos a • see a does not depend on a. Show the same for csc2 « - eot^ a. 20. Does sin2 a + eos^ a depend on a ? Does see^ a — tan^ a ? 21. Before reading Art. 11, find the values of the sine, cosine, and tangent of 30°, 45°, and 60°. 9. Fundamental relations. The Pythagorean theorem a;2 -J- ?/2 _ ^2^ FUNDAMENTAL RELATIONS 9 and the definitions of Art. 6 yield certain fundamental relations between the six trigonometric functions of a single angle. Thus, we have directly from the definitions sma . ^ ^ sec a = -, (2^ cos a cota = - — -• (3) tana Again, by division, tena = ^"^, (4) cos a and cota=^^^. (5) sin a Dividing by v^ both members of the equation 2/2 -f ^2 _ ^2^ we have {t\ j^{^ ^x whence sin^ a + cos^ a = 1 . (6) Dividing, in like manner, by a^ and by y^ respectively, we obtain tan^ a + 1 = sec^ a, (T) cot^ a + 1 = csc^ a. (8) . These eight equations constitute the fundamental relations of trigonometry. Of these the fifth, seventh, and eighth may be derived algebraically from the other five. By means of these equations it is possible to express all six functions in terms of any one of them. If the value of any one is given, the values of the others can be found. Simple numerical examples of the latter kind are more quickly solved by the geometrical method of Art. 7. Example i. To find the remaining functions of the acute angle whose tangent is -f^. (1) G-eometric Method. The right triangle OMP (Art. 6, ^^^' 5) has sides of relative length ;«/ = 5, a; = 12, whence on the same scale v = 13. Thus the sine is -j^, etc. 10 THE ACUTE ANGLE (2) Analytic Method. Given tana = ^5_^ Then by the for- mulas of Art. 9, 1 12 ,. ^- 13. 13 cot a = = — ; sec a = Vl + tiin^ « - zj^ , tan « 5 12 CSC a = VI + cot^ CL— r 1 12 cos a = = —7 ; sm a sec a 13 esc a 18 Example 2. To express all the functions of a in terms of sec «. By the use of the appropriate formulas of Art. 9, we obtain 1 . /^ 9 - Vsec^ « — ] ; cosa = ; sina = VI — cos^a = sec a sec a 1 sec a csca = -r— - = — =^=i' tan«= Vsec2«-l; sin « Vsec^a— 1 cot 06 = tana Vsec^a-l Example 3. Verify the following relation by reducing the first member to the second : tanSyg 1 o ^ — 1 = sec /3. sec /3 — 1 By means of the formulas of Art. 9, we have tan'-^ ^ -< sec2 yS — 1^ o,ii o W^ -1 = S — -- 1 = sec/3+1 -1 = sec ^. sec p — 1 sec p — 1 EXERCISE IV By means of the formulas of Art. 9, find the values of the remaining functions of each of the following angles, given 1. sina = -iV5. 2. cot/3 = ff- 3. cosy = ||. 4. tany^^^. Express all the functions of a in terms of 5. tan a. 6. cos a. 7 cot a. 8. sin a. Find the values of the following expressions: ft tan a + cot a i 9 9. , when cos a = — tana — cot a 41 , r, sec a — cos a i • 12 10. , when sm a = — tan a — sin a 13 FUNCTIONS OF COMPLEMENTARY ANGLES 11 11. ^ ^^"^^ + cotB, when tanB=—- 1 + cos/3 ^ ^ 21 12. cos ^ • tan /? + sin y8 . cot y8, when sec /? = ||. Express the following in terms of a single function 13. CSC a cot a + tan a in terms of cos a- , - sin B , 1 + COS B • . ^ ^ 14. ^=-— H =!^^ — —^ m terms of esc B. 1 + cos /? sni y8 ^ 15. sec y — tan y in terms of sin y. 1 1 16. + 1 — sin y 1 + sin y Verify the following identities : 17. cos4y8-sin4yS = 2cos2^-l. 18. cos a ' tan a = sin a. cot2a in terms of tan y. 19. 20. cos'^ a. 1 + cot2 a ^ + L_ tan^yg+l cot2y8 + l I. 10. Functions of complementary angles. If, in Fig. 6, Z XOjS is constructed equal to Z.AOY^ XOB (=fi} and XOA (= a) are complementary. The triangles OM'F' and Oi)[fP are similar, the pairs of corresponding sides being v' and v^ x' and «/, ?/' and x. We have then sin (90°- a) = sin ^ = cos «, cos (90° — a) — cos ^ = , = ^ = sin a, v' V tan (90° - «) = tan ^ = -^^ = - = cot «, cot (90°- a) = cot /?= -. = ^= tan a, y ^ 12 THE ACUTE ANGLE sec (90° - «) = sec /? = -^ CSC a, CSC (90' a) = CSC 8= —. = -= sec a. The prefix " co " is thus seen to be the abbreviation of the word "complement's." The general theorem may be stated as follows : Ant/ trigonometric function of an acute angle is equal to the corresponding co function of its complementary angle. Thus, sin 72° = cos 18°, cot 54° = tan 36°, etc. 11. Functions of 45°, 30°, 60°. The exact values of the func- tions of certain angles are readily found. (1) If, in Fig. 7, Z XOA = 45°, the triangle OMF is isosceles, so that x — y = We have at once >X V2 sin 45° = cos 45° = \-\/%. tan 45° = cot 45° = 1, sec 45° = CSC 45° = V2. >X Fig. 7. (2) If, 'in Fig. 8, AXOA = ^0\ and "{Y /. XOQ is constructed equal to —30°, the triangle QOP is equilateral, so that y = ^v, x = \^lv. We have, at once, sin 30° = |, cos 30° = I V3, tan 30° = 1 V3, cot 30° = V3,. sec 30°=|V3, CSC 30° = 2. (3) In like manner to (2), or by Art. 10, we obtain sin 60°= J V3, COS 60° =1, tan 60° = V3, cot 60° = l VB, sec 60° = 2, CSC 60° = | V3. 12. Functions of 0° and 90°. (1) As the Z XOA (see Fig. 9) decreases so as to approach the limit zero, the abscissa x will approach equality to the radius FUNCTIOA^S OF 0° AND QO'^ 13 vector V. If, at the same time, the radius vector remains finite, the ordinate ?/ will approach the limit zero. It should be noticed that the cosecant varies in such a manner that its denominator approaches the limit zero while its numera- tor remains constantly equal to the finite number v, so that the value of the cosecant increases without limit as the angle approaches zero. It is then said to become infinite and is represented by the symbol oo. The cotangent also becomes infinite as the angle approaches zero, since its numerator, although changing, never exceeds v. We have, then, sin 0° = 0, tan 0° = 0, sec 0° = 1, (2) In like manner, we obtain sin 90° = 1, cos 90° =0, tan 90°= 00, cot 90° = 0, sec 90° = 00, CSC 90° = 1. Example. Solve the equation sec^ 7 — V3 tan 7 = 1. Expressing wholly in terms of tan 7, tan^ ry -f 1 — V3 tan 7—1 = 0, tan^ 7 — V3 tan 7 = 0, tan 7=0 and V3. ■^1 1^ p (2) '1 1 y • I J \i X Fig. 10. we have or whence Then, by Arts. 11 and 12, 7 = 0° and 60°. EXERCISE V 1. Express in terms of an angle less than 45° the functions of 75". 2. Express in terms of an angle less than 45° the functions of 65°. 14 THE ACUTE ANGLE Verify the following for « = 30° ; also for a = 45° ; 3. sin 2 a = 2 sin a cos a. 4. cos 2 a = cos^ a — sin^ a. 2 tan a 5. tan2a = 6. cot 2 a = 1 — tan^ a cot2 a-\ 2 cot a Notice that sin 2 a does not equal 2 sin a, cos 3 a does not equal 3 cos a, etc. Verify for a = 30° : 7. sin 3 a = 3 sin a — 4 sin* a. 8. cos 3 a = 4 cos^ a — 3 cos a. Evaluate the following expressions : 9. sin 60° cos 30° - cos 60° sin 30°. 10. cos 60° cos 30° + sin 60° sin 30°. 11. csc2 45° + sin 60° tan 30°. 12. sin 60° tan 45° - sec 30° sin2 45° cot 60°. Solve each of the following equations for some one function of a and find the angle in degrees. Verify the results by substitu- tion in the given equation. 13. tan a - 3 cot a = 0. 14. sec a + 2 cos a = 3. 15. 4 sin2 a + 3 cot^ a = 4. 16. since + 3cosa = 2V2. 17. A ladder 22 feet long leans against a wall, its foot being 11 feet away from the wall. Find (a) the angle formed by the ladder with the ground; (6) the height of the top of the ladder above the ground. 18. The diagonal of a rectangle makes an angle of 30° with the long side. If the length of this side is 14 inches, what is the length of the short side of the rectangle ? of the diagonal ? 19. The side of a conical pile of sand makes an angle of 45° with the floor. If the height from the floor is 12 feet, what is the area of the base ? 20. A guy rope (assumed to be straight) has a length of 60 feet and extends from the top of a mast 30 feet high to the ground. Find the angle between the rope and the mast. 13. Variation of the trigonometric functions as the angle varies. Suppose the angle 6 to vary continuously from 0° to 90°. The VARIATION. INVERSE FUNCTIONS 15 terminal side revolves about the origin from the position OX to the position OY. li v remains constant, y will increase from to V, while X will decrease from v to 0. Consequently, the nu- y merator of the fraction -(= sin ^) increases from to v. while the V denominator remains constant. Hence, while 6 increases from 0° to 90°, sin 6 increases from to 1. The numerator of the fraction - ( = cos 6) decreases from v to V 0, while the denominator remains constantly equal to v. Hence, while increases from 0° to 90°, cos decreases from 1 to 0. The numerator of the fraction -(= tan 0^ increases from to X ^ V, while the denominator decreases from V to 0. Hence, while 6 in- creases from 0° to 90°, tan ^, be- ginning with zero, increases with- out limit as 6 approaches 90°. We express this by saying that tan 6 varies from to qo. The student should trace care- fully the variation of the other trigonometric functions and compare the results with the values found in Arts. 11 and 12. Article 7 should be read again at this point. 14. Inverse trigonometric functions. The same functional re- lation is expressed by the two statements, " m is the sine of the acute angle a" and "a is the acute angle whose sine is m." The corresponding symbolic notations are m = sin a, a = arcsin m,* with the understanding that a is an acute angle and that m is a positive number not greater than unity. A similar symbolic relation holds for the other trigonometric functions. It is fre- quently read " arc-sine m," or "anti-sine m," since two mutually inverse functions are said each to be the anti-function of the other. * This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, a = sin-i m, is still found in English and American texts. The notation a = inv sin m is perhaps better still on account of its general applicability. (See Art. 80.) 16 THE ACUTE ANGLE The inverse notation is convenient for the statement of prob- lems. The purposes of interpretation and manipulation are better served by transforming to the corresponding direct notation. Example. Find the value of sin (90° — arccot ^^2)- In the direct notation the example reads : Given cot a = -^^^ find sin (90° - a). Then, by Arts. 10 and 9, sin (90° — a) = cos a = ^\. EXERCISE VI 1. Trace the variation in value of sec 6. 2. Trace the variation in value of esc 0. Find the values of the following : 3. tan (cos-iyV)- 7. sec (90°- arcsec 2). 4. sin (arccot |). 8. esc (90°— arccsc V2). 5. cos (90°-arctan ^\). 9. sin (2 tan-i 1). 6. cot (90°- sin-i if). 10. cos (2 sin-i I). Solve the following equations : 11. 2sin2/? + 3cos/8-3 = 0. 12. sec y8 - 2 tan )8 = 0. 13. tan^(2sin^-l)(secj8-V2) = 0. 14. sin y8 (2 cos j8 - V'3) (tan yS - 1) = 0. Verify the identities : 15. sin^ a + cos* a + sin^ a cos^ a = sin* a — sin^ a + 1. 16. (esc a — cot a) (esc a + cot a) = 1. 17. (tan « + cot a) (sin a -cos a) = 1. 18. 1 - tan* a = 2 sec^ a - sec%. 19. sin« a + cos^ a = 1 - 3 sin^ a cos^ a. 20. cos* a - sin* a = l -2 sin^ a. 15. Orthogonal projection. In accordance with the definitions of Art. 4 (see note, page 4) it follows that the projection of a line segment on any line is equal to the length of the line segment multiplied by the cosine of the angle formed by the line segment with the line of projection. Thus, in Fig. 12, the projection of AB on RS is M]V= AB cos a. ORTHOGONAL PROJECTION IT In like manner, the projection of AB on a line perpendicular to MS (i.e. making + 90° with MS) has the value AB sin a. These projections are called the components of the line segment AB along and at right angles to the direction MS. In physics, line segments are often used to represent quanti- ties that have direction as well as magnitude ; for example, forces, velocities, accelerations. The components of the line segment used to represent a force represent components of the force ; like- wise for a line segment representing a velocity, acceleration, or moment. Suppose, for example, that the line segment AB, Fig. 13, represents a force applied to the block m resting on a liorizon- tal plane. This segment has the component UB parallel to the plane and the compo- nent FB perpendicular to the plane. Segment FB represents a force component F^ parallel to the plane, which tends to move the block along the plane ; seg- ment FB represents a force component F.^ per- pendicular to the plane and tending to produce pressure between the block and plane. Denoting by F the force represented by AB, we have F^ = F cos a, Fy = F sin a. Example. At a given instant a point is moving in a direc- tion at an angle of 30° with a given horizontal line with a velocity of 20 feet per second. Find the component of the velocity along the line. Taking the given line as the JT-axis, we have for the compo- nent v^ v^ = v COS 30°= 20 X I V3 = 17.321 feet per second. The component along a line perpendicular to the given hori- zontal line in the plane of motion is Vy = v sin 30°= 20 x | = 10 feet per second. 18 THE ACUTE ANGLE EXERCISE VII The student should draw appropriate figures for each of the following exercises. 1. Find the projections of a line segment 8.5 inches in length on the X- and Z-axes, (a) when the segment makes an angle of 45° with the Z-axis ; (b) when it makes an angle of 60° with the F-axis. 2. A crank 16 inches long rotates in a vertical plane. When the crank makes an angle of 30° with the horizontal diameter of the circle described by the moving end, what is the distance of the moving end, (a) from the hori- zontal diameter? (h) from the vertical diameter ? 3. If in Fig. 13 the force jP denoted by AB is 40 pounds, find the compo- nents F^ and Fy, (a) when a = 30° ; (b) when a = 45°. Discuss the cases a =r 0° and a = 90°. 4. A steamer is moving at a speed of 18 miles per hour in a direction north of east, making an angle of 30° with an east and west line. At what rate is the steamer sailing eastward ? at what rate northward ? 5. A guy wire exerts a pull of 3000 pounds on its anchorage and makes an angle of 30° with the ground. Find the component of this force, (a) along the ground; (b) vertical. 6. The eastward and northward components of the velocity of a moving body are found to be 1^^= 12 miles per hour and Vj,= 12 V3 miles per hour, respec- tively. Find (a) the magnitude and (b) the direction of the body's velocity. CHAPTER ITT RIGHT TRIANGLES 16. Laws for solution. If, in a right triangle, two independent parts are known, in addition to the right angle, the three remain- ing parts can be found. Thus two given parts, at least one of which is a side, determine a right triangle. The formulas needed in all cases to effect this solution are five in number. Two are the state- ments of well-known geometric theo- rems, while the other three are the immediate consequences of the defini- tions of the trigonometric ratios con- tained in Art. 6. In Fig. 14 let ACB be a right triangle, right-angled at 0. We shall denote the interior angles at the ver- tices by a, yS, 7, and the lengths of the sides opposite them by a, 5, c, respectively. Note that = 254 X 1V3 = 254 X .86605 = 219.976T. The check formula gives 6 = 127x1.7321 = 219.9767. Example 2. Given « = 39.00, 6 = 80.00, to find c, a, /9, and the area. As before, we may pass immediately to the second stage. Now c is given directly in terms of a and b by the formula c?= Va^ + h^ 22 RIGHT TRIANGLES If we are to avoid the use of this formula, we must first find a and /8, and then get c by means of one of these angles. We use the forms : I a = a "V /9 = 90°- C = a sm a A = \ah. c = h sin l3' and for the check We obtain, then tan « = 39 -^ 80 == .4875, a = 25° 59^ as found from Table III, /3= 90° -25° 59' =64° 01', ^ = 39 _j. .4381 = 89.01, ^ = 1 X 39 X 80 = 1560, and for the check c=SO-^ .8989 = 89.00, showing a difference of .01. On account of the simplicity of the numbers, we may, by using the formula c^ = a^-]- P, find, exactly, c = 89. Explain the accumulation of errors and, hence, the reason for rule of procedure (jB). Examples, l. Given c = 42, ^ = arcsin .28 ; find a and h, 2. Given 5 = 27, « = tan~i .75 ; find a and c. 3. Given a = 300, a = cos"^ .45 ; find c and b. 4. Given c= 200, a— arccot 1.12; find a and h. 19. Trigonometric tables. In the first example worked in the preceding article, the functions of 30° had been determined in Art. 11. In the second example, however, the value of tan a was not one of those previously ascertained, and the value of a was not recognizable from its tangent. For convenience of reference the numerical values of the sines, cosines, tangents, and cotangents of all angles differing by intervals of one minute from 0° to 90° have been collected in Table III, on pages 71-89. The arrangement is simple and plain. The degree numbers from 0° to 44° occur at the top of the page, with the minutes running down the left margin. TABLES. ERRORS AND CHECKS 23 The numerical values of the functions, computed to four decimal places, are placed in columns under the names of the functions. Since sin (90° — a)— cos a, and tan (90° — ce) = cot a, the space required may be reduced one half. The degree numbers of angles from 45° to 90° are printed at the bottom of the pages in reversed order, the minutes run up the right margin, and the names of the functions are in reversed order at the bottom. For the present the student need not concern himself with smaller divisions of the angle than the minute. Further refine- ment is attained by a method to be described in Art. 26. Table IV, on pages 91-93, contains the squares of numbers less than 1000 and, by interpolation, of numbers up to 9999. The first page gives directly the squares of numbers from 1 to 100. On the second and third pages the tens and units digits of the number to be squared are in the left margin, while the hundreds digits are at the tops of the several columns. The last two figures of the square are in the column at the right under U., opposite the tens and units digits ; the first three, or four, figures of the square are in the same line in the column under the hundreds digit. In the right margin are the last two figures of the tabular difference used in interpolation, to which must be prefixed the remainder obtained by subtracting the first three, or four, figures of the square from those in the same column immediately beneath, or that remainder diminished by 1 when the asterisk (*) is present. The use of the table is best shown by illustration. Examples, i. 3282 = 107,584. 2. 475.3 = 4752 4- .3 X 951 = 225,625 + 285 = 225,910.* 3. 28.372 =28.32 + . 07 x567 = 800.89 + 3.97 = 80 1.56. Square roots are extracted by reversing the process ; thus. 4. V27556 = 166. 5. V658,037 = V657,72l + 316 ^ 1623 = 811 + .2 = 811.2. 20. Errors and checks. The results obtained are not always, nor even usually, exactly correct. The deviations from the true values are of two sorts, mistakes and errors, and a sharp distinc- tion must be made between them. * This result is, of course, only approximately correct. The true result may be obtained as follows : 475.32 = 4752 + .3 X (475.3 + 475) = 225,625 + 285.09 = 225,910.09. 24 RIGHT TRIANGLES The data for problems arising in actual practice are derived from observations made with instruments for measuring lengths, angles, etc. Mistakes may arise from a false reading of the observing instru- ment, a misapprehension of the problem, the employment of the wrong formula, faulty addition, etc. ^ They are never allowable or excusable. On the other hand, instruments are so constructed as to yield results only to a certain degree of precision, which should be ascertained for each instrument. Moreover, observation is per- formed by the human apparatus, eyes, ears, etc., and a certain per- sonal equation, an anticipation or lagging in sight or hearing, is always present, varying with personal fitness and experience. Methods of eliminating instrumental errors, so as to obtain the maximum precision possible with the instruments used, are given in standard works on engineering instruments. Again, the arith- metical calculation involves the trigonometric ratios, which are, in general, non-terminating decimal fractions, while their values in the mathematical tables are computed only to a certain number of decimal places. Errors, therefore, will always be present ; but every precaution should be taken to keep the errors due to com- putation well within the limits of error of the observed data and desired results fixed by the nature of the problem. In both observation and solution, certain additional processes are employed, to avoid, or to reveal, mistakes. These processes are known as checks and vary with the nature of the problem. While no general rules for checks can be laid down, a frequent practice in the solution of triangles is to make use of a formula connecting the required parts, just found, noting if the results are within the range of allowable error. The size of this allowable error should be known for each table. As a check to arithmetical computation, graphical construction is well understood and strongly advised. As a means of avoiding the grosser mistakes, a free-hand sketch will frequently suffice by guiding the student to a reasonable interpretation of data, and indicating possible results. A drawing constructed to scale will further aid by yielding values more or less approximate, approaching those obtained by computation. Carried a step farther as regards accuracy, by the use of pre- cise instruments, the graphical construction often attains to the PROBLEMS INVOLVING RIGHT TRIANGLES 25 dignity of an independent solution, with results falling within the limits prescribed by the physical conditions of the problem. There is no better evidence of careful work than the record- ing of a reasonable error obtained by the comparison of two methods. In practical work the allowable per cent of error becomes an important consideration. EXERCISE VIII Find the missing parts of the following triangles, using the natural trigonometric functions, Table IIL a /3 a h c A 1. 2.5° 10' 34 2. 52° 20' 73 3. 61° 15' 243 4. 78° 35' 521 5. 21° 25' 235 6. 72° 45' 720 7. 80° 30' 1200 8. 17° 30' 1500 9. 240 360 10. 381 715 11. 521 630 12. 840 1400 13. 648 864 14. 595 600 15. 215 385 16. 2111 1234 17. 95 7980 18. 264 30360 19. 74° 20' 1225 20. 24° 50' 843 21. In the same vertical plane the distances shown in Fig. 15 were meas- ured in feet along the surface of the ground. The distances of the « ^ ^ ^ ^ f^R- different points below the instru- ment, as measured by a rod, are given also in feet. The vertical scale is exaggerated for clearness. What is the horizontal distance from B to Gl (Check by a table of squares and square roots.) 26 RIGHT TRIANGLES 22. A line surveyed across a ridge is 1500 feet in horizontal length. Stakes are set 100 feet apart horizontally by level chaining. By leveling, the elevations of the surface at the different stakes is obtained as follows : 730.2, 735.9, 739.7, 743.4, 750.1, 751.8, 760.7, 764.1, 764.3, 765.8, 765.0, 763.2, 758.3, 750.2, 743.1, 740.2. What length of wire will be required for fencing along this line? (Check by a table of squares and square roots.) 23. If a gravel roof slopes one half inch to the horizontal foot, what angle does it make with the horizon? 24. If the face of a wall has a batter or inclination of one inch in one ver- tical foot, what is its angle with the vertical ? 25. What is the angle of ascent of a railway built on a 2 per cent grade (i.e. 2 vertical feet to 100 horizontal feet) ? C 26. The pitch of a roof is the ratio — . (See Fig. 16.) What is the in- clination to the horizon of a roof with •^ pitch, I pitch, I pitch? 27. What is the pitch of a roof slop- ing to the horizon at 15°, 30°, 45° ? 28. What is the inclination to the horizon of the corner or hip rafter of a pyramidal roof whose pitches are ^? 29. What is the inclination from the vertical of the corner edge of a wall, both of its faces having a batter of ^^^ ? 30. At what angle does a railway slope if it has a grade of 0.25%, 0.5%, 2.5%? 31. At what angle must a cog railway ascend in order to rise 2640 feet in one horizontal mile ? 32. A battleship known to be 341 feet long subtends an angle of 3° 20' when presenting its broadside to a fort on shore. For what distance should guns be sighted when trained upon it ? (Note that the isosceles triangle hav- ing the length of the ship for its base is separable into two right triangles.) 33. In planning the stairway for a house it is desired that the riser, or vertical distance between steps, shall be 7 inches, and the treads, or horizontal distances between faces, 11 inches. What will be the angle of inclination of the hand rail? 34. Taking the data of the preceding problem, what will be the length of the hand rail if straight, provided the height between floors is 11 feet 8 inches? 35. A cylindrical water tower whose external diameter is 25 feet subtends a horizontal angle of 5° 30' as viewed from a distance. How far is its center from the instrument? (Note that we have a triangle that is right-angled when the line of sight is tangent. The base is the radius of the tower and the opposite angle is half of the one observed.) PROBLEMS INVOLVING RIGHT TRIANGLES 27 36. What horizontal angle would be subtended, at a distance of 2 miles^ by a vertical cylindrical gas receiver 60 feet in diameter ? (See note to problem 35.) 37. The end of a pendulum 34 inches long swings through an arc of 3| inches. Find the angle through which the pendulum swings. 38. When vertically over a village, a balloon's angle of inclination, as viewed from 9 miles distant, was 15° 20'. Assuming the surface of the country to be fairly level, what was the height of the balloon ? 39. A flagstaff 110 feet high is covered by a vertical angle of 12° 30' at a point approximately on a level with its center. How far is the observer from the staff? 40. The data of a preliminary survey are as follows: AB = 240.9 feet. BC = 310.7 feet. CZ> = 611.5 feet. DE = 237.2 feet. J5:i^= 528.0 feet. Considering A, Fig. 17, as the origin of coordinates and AB a,s the axis of abscissas, it is required to compute coordinates for all points given, thus providing for the accurate mapping of the survey. 41. Find the missing parts and area of the following isos- celes triangles (see Fig. 18 for lettering) Angle at 5 = 62° 11' left. Angle at C = 55° 50' left. Angle at D = 43° 42' right. Angle a.tE = 51° 23' right. Fig. 17. 35°, a = 42; « = 72°, & = 12o; 350, & = 180; /3 = 54°, a = 360 ; 51° 26', & = 480; a = 640, b = 840. 42. Find the lengths of the chords of the follow- ing arcs in terms of the radius: 30°, 36°, 40°, 45°, 60°, 75°, 90°, 120°. Compute, given R = 100. 43. Express in terms of the sine and radius the relation between the chord of an arc and the chord of half the arc. 44. Express in trigonometric form the most important relations between the radius R of the circumscribed circle, the radius r of the inscribed circle, the side s, and the number of sides n of a regular polygon. 28 RIGHT TRIANGLES 45. Compute and tabulate the perimeter and the circumferences of the circum- and in-circles of a regular polygon of n sides for n = 4, 8, 16, 32, given 72 = 10. 46. Compute and tabulate the area of a regular polygon of n sides and of its circum- and in-circles for n = 4, 8, 16, 32, given R = 10. 47. Repeat example 45 for n == 6, 12, 24, 48. 48. Repeat example 46 for n = 6, 12, 24, 48. 49. A body is acted upon by three forces of magnitudes 20, 40, 60, parallel to the sides of an equilateral triangle. Resolve these forces along two perpen- dicular axes, then combine, and thus find the magnitude and direction of the resultant. 50. A body situated at one vertex of a regular hexagon is acted upon by five forces represented in magnitude and direction by the vectors drawn to the five other vertices. Resolve along and perpendicular to the diameter through the point and find the magnitude and direction of the resultant. 51. A point describes a circle with uniform speed. Determine the position of its projection upon a diameter in terms of its angular displacement from that diameter. 52. A point describes a circle of radius 30 feet at a rate of 4 revolutions per minute. Find the position of its projection upon a diameter at the end of 5 seconds after passing the extremity of that diameter. 53. Determine the components of the vertical acceleration g along and perpendicular to a plane inclined at an angle a to the horizon. 54. If ^r = 32, find the acceleration along and perpendicular to a plane whose inclination to the horizontal is 30°, 15°, 10°, 5°. 55. A man weighing 150 pounds stands midway on a 30-foot ladder whose foot is 10 feet horizontally from the vertical wall against which it leans. Find the normal (perpendicular) pressure on the ladder and the force tending to cause him to slide along the ladder. 56. Find the components along the X- and F-axes of a force of 65 pounds making an angle of 28° 13' with the Z-axis. 57. A steamer is sailing in such a way that its speed due east is 12 miles per hour and its speed due south is 14 miles per hour. Find the direction of the steamer's course and the speed in that course. 58. In an oblique triangle, angle B = 45°, angle C = 32°, and side b = 16. Find side c. (Suggestion. Draw the perpendicular from the vertex A upon the opposite side.) Attempt to deduce a general relation between the func- tions of the acute angles of au oblique triangle and the opposite sides. CHAPTER IV LOGARITHMS 21. Definition of a logarithm. If we have given we can find the product of 5Q and 79 without performing the operation of multiplication, provided we know in advance the powers of 10. For, we have from the general laws governing exponents, 56 X 79 = 10i-^4«i^ X W'^''^' __ -j^Ql.74819+1.89763 =:103-«4'^82^4424. It will be seen that the process of multiplication has been replaced by the simpler one of addition. Many other processes in computation can be simplified in a similar manner ; for example, if we wish to find the cube root of a number, say 89.1, we have 89.1 = 101-94988^ and consequently • V89.1 = (10i-94988y = 100-64996 ^ 4.466+. In this case the extraction of a root has been accomplished by the simple process of division. In order to extend this method we must know all of the powers of some convenient number. The exponents involved are called logarithms, and the number raised to a power is referred to as the base of the logarithmic system. We may define a logarithm more exactly as follows : If a is any number and x and n are so related that «^ = n^ then X is^,«^lled the logarithm of n to the base a ; that is, a logarithm is the index of the power to which the base must be raised to obtain the given number. This relation is denoted symbolically by writing X = log„ n, and is read ^'•x is equal to the logarithm of n to the base a." 29 30 LOGARITHMS Thus 3 is the logarithm of 8 to the base 2, since 2^ = 8 ; and in the illustrations given above, 1.74819 is the logarithm of 66 to the base 10, etc. The two statements a^ = n, x = logo n are inverse to each other, just as are the relations sin x and arcsin x^ etc., of Art. 14. Exercise. Find by inspection log3 27, log5.625, log8 32, log,. 04. . . . , The logarithm of a number to itself as base is unity, since 71^ = 71. The logarithm of 1 to any base other than zero is zero, since a^ = 1. In conformity with the definition just laid down, it follows that, if two numbers are equal, their logarithms to the same base are equal. It is also true conversely, that if the logarithms of two numbers to the same base are equal, the numbers are equal.* If the base is real and positive, real logarithms produce only positive numbers. If the base is real and negative, even loga- rithms produce positive numbers ; odd logarithms, negative num- bers. For this reason only real positive bases are chosen in prac- tice, and only positive numbers are combined by the aid of their logarithms. The sign of the result is ascertained entirely apart from the numerical computation. 22. Laws of combination. Logarithms are important in trigo- nometry and elsewhere as labor-saving devices in calculations with numbers containing many digits. Only so much of the theory of logarithms as is necessary for this purpose will be developed in the present chapter. The laws of combination of numbers by the aid of their loga- rithms follow at once from the definition of the preceding article. I. The logarithm of the product of two factors is equal to the sum of their logarithms, all to the same hase. For, if a; = log„ n and y = log^ m we may write • n = a^ and m = a^. * In the theory of analytic functions a broader definition of the logarithm is laid down, and the statement just made requires modification. LAWS OF COMBINATION 31 Multiplying, we have, by the exponential law, nm = a^+^, whence, loga nm ^x-\-y= logc* ^ + lo&a ^- (1 ) This law may evidently be extended to any finite number of factors. II. The logarithm of the quotient is equal to the logaritJim of the dividend minus the logarithm of the divisor^ all to the same base. For, if a; = log^ri and y = log^ m, we may write as before, n = a*, m = a^. n Dividing, we have — = a^"^, m whence, log„ ^— j =x — y = log« n— log„ m. (2) Manifestly log« f — j = — log^ m. III. The logarithm of the power of a number is equal to the loga- rithm of the number multiplied by the index of the power. For, if a; = log^ n, then n = a^. Hence, n^ = {a^y = a^"^ or, log« (nP) =px = p loga n- (3) IV. The logarithm of the root of a number is equal to the loga- rithm of the number divided by the index of the root. For, if :r = log^ n, then n = a^. Extracting the ^'th root of both members, we get v n = aQ, whence, log^^^ = - = i log„ n. (4) q <1 23. Common logarithms. Any number may be used as a base of a system of logarithms. For certain purposes the so-called natural system of logarithms, which has for its base the number e = 2.71828183 •••, has advantages. For the purposes of ordinary numerical computation, however, it is most convenient to employ for the base of the system of logarithms, 10, the base of the universally adopted system of numeration. 32 LOGARITHMS The common logarithms of all exact integral powers of 10 are positive integers ; for instance logio (1000000) = logio(W) = 6 log,, 10 = 6. The logarithms of reciprocals of integral powers of 10 are negative integers ; thus logio (.00001) =logi„ (10-0 = -51og,„10 = -5. The losrarithms of numbers situated between two consecutive integral powers of 10, say between 10^ and 10*+^, lie between k and k + 1, where k is any integer, positive or negative. Thus 103 < 2417 < 104, whence, , 3 < log^^ 2417 < 4, or, logjQ 2417 = 3 + a number lying between and 1. The logarithms of numbers greater than the base consist of an integer plus a proper fraction. The fractional part is written decimally, calculated to a number of decimal places, depending on the degree of accuracy desired in the use of the table. The integral part of the logarithm is called the characteristic; the decimal fraction, its mantissa. Hereafter, in this book, except in Chapter IX, we shall have to do only with common logarithms and, unless otherwise expressly stated, log n will denote logjQ n. 24. Characteristic. If one number is equal to another number multiplied by a factor which is a power of 10, the logarithms of the two numbers differ by an integer. For log (10* xn')= log (10^ + log n = k + log n. Example. log 34000 = 3 + log 34 = 4 + log 3.4, etc. Every number containing one digit at the left of the decimal point lies between 10^ and 10^. The characteristic of its logarithm CHARACTERISTIC. MANTISSA 33 is therefore 0. The cipher should always be written to indicate that the characteristic has not been overlooked. Every number containing k digits at the left of the decimal point is 10*"^ times a number with one digit at the left. The characteristic is therefore k — \. We have then the following rule for the characteristic : The characteristic of the logarithm of any number greater than unity is one less tlian the number of digits at the left of the decimal point. Should the number be less than unity, move the decimal point ten places to the right (thus multiplying by 10^^) and apply the same rule as before, then write — 10 after the logarithm for correction. Thus log 7.12 = 0.85248, log 71200 = log (10* X 7.12) = 4.85248, log .00712 = log (10-10 X 71200000) = log (10-10 X 107 X 7.12) = 7.85248-10. The positive part of the last characteristic is seen to be the difference found by subtracting from 9 the number of ciphers immediately following the decimal point in the number. The characteristic of the logarithm of any number less than unity is found by subtracting from 9 the number of ciphers between the decimal point and the first significant digit, then affixing —10. 25. Mantissa. We have seen that moving the decimal point in the number merely changes the characteristic of the logarithm, leaving its mantissa unaltered. The mantissa depends solely upon the sequence of significant digits. In the tables given, the logarithms are computed to five deci- mal places (see pp. 1-21), and the mantissas alone for all numbers from 100 to 9999 are given, arranged in the following manner : Running down the left margin, under iV, are to be found the first three digits of the number. In the next, (open) column occur the first two figures of the mantissa. In the next ten columns are the remaining three figures of the mantissa arranged under the fourth digit of the number at the top of the columns. 34 • LOGARITHMS Thus to find the mantissa of log 3814, we select the row having 381 in the left margin. The first two figures of the mantissa, 58, are found in the first column. The three remaining figures, 138, are found in the column headed 4, the fourth digit of the number, giving the mantissa .58138. To avoid repetition, the first two figures, 58, are not printed in every line, but are to be used from 3802 to 3890, inclusive. The prefixed asterisk, *006, denotes that the mantissa of 3891 is .59006, not .58006. EXERCISE IX 1. Find by inspection logg 16, log,, 27, log^ ^^. 2. Find by inspection logg 81, logg 32, logo; 9. 3. What numbers correspond to the following logarithms to base 4 : 0, 1, 2,2.5,3, -2,-3? 4. What numbers correspond to the following logarithms to base 8 : 0, 1,H, -I, -2? 5. Find by logarithms: («)^; (P) '^^^f^iM. 6. Find («) VtW; (b) \/W7 ; (c) \/9l. ^ Find^ 18 X V240 X 753 72 X Vim X 200 3/ , Find (^xV720xl5Y^ V2x V480x 248/ 9. Find { "^^ ] * , where k = 1.41. \ 65 / 10. Solve for x: ^^ = 24. 11. Solve for x : 6* = 25. The amount A attained by a principal P at interest at the rate r com- pounded annually for n years is A =P(1 +r)~. 12. Find the amount of $ 3680 at 4 per cent in 6 years. 13. Find the principal which, in 7 years at 5 per cent, amounts to ^ 5820. 14. At what rate will ^ 5000 amount to $7500 in 8 years ? 15. In how many years will $ 86,500 amount to $ 129,600 at 3^ per cent ? 16. If a city increases its population I each year, in how many years will it double its size ? INTERPOLATION 35 26. Interpolation. It will be shown in Art. 79 that the differ- ence in the logarithms of two numbers is approximately propor- tional to the difference in the numbers provided these differ- ences are small. Thus, approximately, log 51473 - log 51470 ^ 51473 - 51470 ^ 3 log 51480 - log 51470 51480-51470 10* We have, then, log 51473 = log 51470 + ^\ (log 51480 - log 51470). Introducing the values from Table I, log 51473 = 4.71155 + ^^^ (4. 71164 - 4.71155) = 4.71155 4-. 3 X .00009 = 4. 71155 +.00003 = 4.71158. The difference .00009, or omitting the denominator, the 9 is called the tabular difference corresponding to the logarithm of 5147. Note that the added difference is computed to the nearest fifth decimal place. This process is called interpolation by the principle of pro- portional parts. To facilitate interpolation, tables of proportional parts are inserted in the logarithmic tables in the column headed P.P. At the top of each of the P.P. tables is the tabular differ- ence and under this is the number to be added corresponding to the digit at the left. For example log 38.25 = 1.58263 log 38.26 = 1.58274. The difference is .00011 and in the P.P. column is a table headed 11. Suppose now that log 38.257 is required. Opposite 7 under 11 is found 7.7 ; hence 8 is to be added in the fifth deci- mal place, giving log 38.257 = 1.58271. 27. Numbers from given logarithms. The inverse process of finding the number corresponding to a given logarithm is best explained by an illustration. Given the logarithm 3.84235. Only the mantissa need be considered at first, as the characteristic merely determines the position of the decimal point in the number. 36 LOGARITHMS Looking for 84 in the first column after the margin, we find it corresponding to numbers from 692 to 707. The nearest tabular number (mantissa) smaller than 235 is 230, corresponding to the number 6955. The difference is 5, while the tabular difference, found by subtracting 230 from 236, is 6. We have now the pro- portion for the next digit, n _5 ^ 10~6' so that the next digit is found by dividing 50 by 6. It is inad- visable to carry the interpolation beyond one additional digit. Since 50 -^ 6 = 8 • + • • •, we have found the desired number to be 6955.8. The decimal point is placed after the fourth digit accord- ing to the rule for the characteristic, the given characteristic being 3. Should the logarithm be followed by — 10, the decimal point must finally be moved ten places to the left. 28. Cologarithms. The logarithms of divisors have to be sub- tracted. Subtraction, however, can be avoided and the logarith- mic computation of a succession of multiplications and divisions effected by a single addition process. There is no advantage in using cologarithms when but two factors are involved. When, however, more than two are involved, instead of dividing by the denominator or divisor factors, we may multiply by their recipro- cals, obviously a legitimate substitution. Now log— = — log m = (10 — log m) — 10. m This logarithm, (10 — log m) — 10, is called the cologarithm of m, written cologm. It may be written down immediately from the table by beginning at the left and subtracting each figure from 9, until the last figure, which must be subtracted from 10. Thus log 28.24 = 1.45086 and colog 28.24 = 8.54914 -10. 29. Logarithms of trigonometric functions. Logarithms of the trigonometric functions are arranged in Table II in the same manner as are the natural functions, or true numerical values of the functions. Logarithmic secants and cosecants need not be printed, since they are the cologarithms of the cosines and sines. The sines and cosines of angles and the tangents of angles less LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 37 than 45° are numerically less than unity. In conformity with Art. 24, therefore, their logarithms are written in the augmented ^^^^^' log sin 6d° 21^ = 9. 95850 - 10. The — 10 is not printed in the table but it is always understood. The positive portion of the characteristic is printed in the table. Usage differs with respect to printing the logarithmic tangents of angles greater than 45°. Engineering and physical instruments are usually graduated to minutes or larger divisions of the angle, so that it is not feasible to carry the interpolation farther than to tenths of minutes. The tables of functions and of proportional parts printed in connection with this book are arranged with this in view. Astronomic observations justify carrying the interpolation to seconds, and astronomers use for this purpose tables computed to seven or more decimal places. For example, log sin 29° 37' = 9.69890 - 10, log sin 29° 38' = 9. 69412 - 10. The difference is .00022, and in the P.P. column is a table headed 22. Suppose now that log sin 29° 37.4' is required. Opposite 4 under 22 is found 8.8 ; hence 9 is to be added in the fifth decimal place, giving log sin 39° 37.4' = 9.69399 - 10. EXERCISE X 1. Find from the table the logarithms of 72484, 619.25, 695 x 10^ .00064375, 3 x lO^i. 2. Find from the table the logarithms of 91386, 14.295, 321 x 10^, .000078541, 2 x lO^*. 3. Find the numbers whose logarithms are 3.71295, 12.61242, 8.21312 - 10. 4. Find the numbers whose logarithms are 4.21382, 11.75153, 6.13579 - 10. 5. Find Young's modulus of elasticity from the formula Y= — ^, if m = 4932.5, g = 980, I = 110.5, tt = 3.1416, r = .25, s = .3. '^'' ^ 6. Find the radius of the sun if its mass is 2.03 x 10^^ grams, and its average density is 1.41, knowing that mass = volume x density. 7. The radius r of each of two equal, tangent, iron spheres which attract each other with a force of 1 gram's weight, is given by the formula 4r2 i22' 38 LOGARITHMS in which the density of iron p = 7.5, the mass of the earth ikf = 6.14 x 10*^ grams, and the radius of the earth R — 6.37 x 10^ cm., while ir = 3.1416. Solve for r and compute by logarithms. 8. Solve example 7 for spheres of lead with density p = 11.3. 9. The population of a county increases each year by 12.5 per cent of the number at the beginning of the year. If its population Jan, 1, 1776, was 2.5 X 106, what will it be Dec. 31, 1926? 10. If the number of births and deaths per annum are 3.5 per cent and 1.2 per cent respectively of the population at the beginning of each year, and* the population on Jan. 1, 1830, was 5 x 10^ find the population Jan. 1, 1905. 11. Find from the tables log sin 25° 32.3', log cot 71° 18.6', colog cos 16° 29.2'. 12. Find from the tables log cos 19° 25.7', log tan 31° 16.2', colog sin 65° 12.8'. 13. Find the angles corresponding to log cos a = 9.31723, log cot y8 = 9.16251, log tan y = 0.61253. 14. Find the angles corresponding to log sin a = 9.63152, log tati (3 = 9.71728, log cot y = 0.15382. 15. Francis deduces the following formula for the discharge over a weir, q = 3.01 bH^-^, in which q is the discharge in cubic feet per second, b the breadth of the crest, and H the head of water. Find by logarithms the discharge when 6 = 3.5 and// =1.2. 16. A common formula for finding the diameter of a water pipe is m d = 0.479 h in which /is a friction factor, I the length of the pipe, q the discharge, and h the head. Find d when /= 0.02, I = 500, ^ = 5, ^ = 10. 17. The discharge from a triangular weir is given as ^ = c I'V V2 g H^, in which c is a constant, g the acceleration of gravity, and H the head. Find q when g = 32.2, H = 1.2, c = 0.592. 18. The formula for velocity head is h = 0.01555 V^. Find ?i when V = b. 19. The elevation of the outer rail on what is known as a one-degree railwav curve to resist centrifugal force is sometimes given by the formula e = 0.00066 V^, e being in inches and V the speed of the train in miles per hour. When F = 45, comjmte e. 20. Another expression for the relation of the preceding problem is qY-2 c = '^ • Here e is in feet, g is the gauge of the track, V is the speed in feet per second, and R is the radius of the curve. Given g = 4.71, V = 66, R = 5730, compute e. THE SLIDE RULE 39 21. The difference between the base and hypotenuse of a right triangle is given by c — a = , and when a and c are nearly equal, approximately by yi c -\- a c — a = — . 2c Find the per cent of error introduced by the second method when the angle between a and c is 15°. 22. If a = length of a short circular arc and c = its chord, then approxi- mately a — c = . Given a = ^ and R — 100, compute the value of this difference. 23. The relation between the pressure and volume of air expanding under certain conditions is pj??j ^-^^ — pv^*\ where p^ and v^ are initial values. If p^ = 40, v^ = 5.5, find V when j9 = 24 ; also when p = 16. 24. The relation between the initial and final temperatures and pressures is given by the equation With ^j = 60 and the other data as in Ex. 23, find the final temperatures for p = 24: and j9 = 16, respectively. 30. The slide rule. The principles of logarithmic computation are conveniently illustrated by means of the slide rule, now widely used in performing mechanically such operations as admit of the use of logarithms. A brief description of this instrument will be found profitable at th^s stage, and its use by the student as a ready check upon the numerical solution of problems is strongly recommended. As will be seen by an inspection of the simplified diagram of Fig. 19, the rule is essentially a device for adding and A B Rule C 1 2 3 4 5 6 7 a Slide h 1 1 — I — I — I — 3 4 5 6789 10 Fig. 19. subtracting logarithms, thereby giving a wide range of computa- tions. In the figure the point 6X on the "slide" is set opposite the point B on the "rule." If both scales, which are alike, are so divided that AB is equal, or proportional, to log 2 and ab to log 3, then on the rule opposite b on the slide gives the distance ^(7 equal, or proportional, to log 6. That is, log 2 + log 3 = log (2x3) = log 6. Similarly by subtraction, AC — ab = AB ; that is log 6 — log 3 = log 2. 40 LOGARITHMS The point a of the slide is called the index^ hence we have the following rules for simple operations. 1. To multiply two numbers, set the index opposite one num- ber on the rule and opposite the other number on the slide read the product on the rule. 2. To divide one number by another, set the divisor on the slide opposite the dividend on the rule and read the quotient on the rule opposite the index. In the instrument as actually constructed, * Fig. 20, there are four scales denoted respectively by A, B, C, and D, of which scales B and C are on the Fig. 20. slide. For convenience in compound operations the rule is provided with a runner r by means of which a setting of the slide may be preserved while the slide is moved to a new position. The following example will illustrate the manipulation of slide and runner. Example 1. Find 6^^^115x27. 14.6 X 342 Set 14.6 on C scale opposite 63 on D scale ; move runner to 115 on C scale ; move 342 on C scale to runner, and opposite 27 on C scale read result on D scale. In this, as in all slide-rule computations, the decimal point must be located by inspection. On the lower side of the slide are three scales, the outer of which are marked S and Irrespectively. The following examples illustrate the use of these scales. Example 2. Find 36 sin 22^ Set 22 on the S scale opposite the mark on the slot in the right-end of the rule ; then opposite the end of the A scale can be read on the B scale the natu- ral sine of 22°. Now opposite 36 on the A scale read the result on the B scale. Example 3. Find 26.5 tan 13° 15'. Reverse the slide and set 13° 15' on the T" scale opposite the mark on the slot ; then opposite the end of the B scale can be read on the D scale the natu- ral tangent of 13° 15'. Set the runner at this point and replace the slide with the index at the runner. Opposite 26.5 on the C scale read the required prod- uct on the D scale. Example 4. Find 56i''. Set the index of C scale opposite 56 on D scale and opposite the mark on the under side of the right-hand end of the rule read 748 on the middle scale * A more detailed description of the slide rule is not within the scope of this book. A manual describing fully the use of tlie instrument can be had of any firm selling slide rules. LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 41 of the lower side of the slide. This reading is the mantissa of the logarithm of 56. The characteristic 1 must be supplied as usual. Now in the usual way- find 1.3 X 1.748; that is, put index to 1.748 on D scale and opposite 1.3 on C scale read the product 2.272. This is the logarithm of SG^-^. Set the mantissa 272 on the logarithm scale opposite the mark on the rule and read 118.7 on the D scale opposite the index. EXERCISE XI 1 17- /I ^ \ 64 X 37 ,,. 193 . . 0.05 x 137 x 62 1. liind (a) (b) — ; (c) • ^ 163 ^ ^ 67 X 2.1 ^ ^ 14 X 28 X 6.5 2. Find (a) 127 sin 24°, (6) 0.32 sin 72°, (c) 16.5 cos 35°. 3. Find (a) 37 tan 8° 20', {h) 1.35 tan 40° 10'. 4. Find («) 11^2^32:, (/.) 35.5 ?H^^ ^ ^ 64 ^ ^ sin 47° 5. Find (a) 28^ {h) y/^^, (c) 7.311-27. 31. Right triangles solved by logarithms. — It is now possible, with the aid of the logarithmic tables, to solve right triangles the numerical values of whose parts contain more digits than those given in Chapter III, without entailing laborious multiplications and divisions. Example 1. Given a = 51.72, jS = 73° 46^ Solving the proper formulas for the unknown parts, we have a ^ = B' cos p h=a tan /3, A = \a^ tan ^, h — c cos a, check. Sum of angles =90° 00^ ^= 73° 46^ «=16°14^ log «= 1.71366 log cos /3= 9.44646 -10 log (? = 2.26720 .•.c= 185.01 42 LOGARITHMS log a = 1.71366 log tan 13 = 0.53587 log 6 = 2.24953 .-. 5 = 177.64 21og«= 3.42732 log tan /3= 0.53587 colog 2 = 9.69897-10 log ^ = 13.66216 -10 .-. J. = 4593.67 Check logc= 2.26720 log cos a = 9.98233-10 log 5 = 12.24953 -10 .-.5 = 177.64 Note that log a^=2 log a. In solving, first write all the forms needed for the complete solution ; secondly, look up and write in all the needed logarithms of the data from the tables ; thirdly, per- form the additions and subtractions ; lastly, from the logarithmic results find the numbers. Then log cos /3, log tan /3, and log cos a ( = log sin )S) can all be found from once turning to the angle 73° 46'. A form of computation sometimes used is given below. It has the advantage of being more compact than the usual form, and furthermore the logarithms of the data stand close to the data, thus permitting easy verification of results or correction of mistakes. ^^^^^ a=51.72 log 1.71366 logl. 71366 /3 = 73° 46' log cos 9.44646-10 log tan 0.53587 c = 185.01 log 2.26720 2 ^ = 4593.7 log 2.26720 5 = 177.64 « = 16°14' 5 = 177.64 log 2.24953 log cos 9.98233- 10 log 2.24953 log 3.42732 log tan 0.53587 colog 0.69897 -10 log 3.66216 LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 43 Example 2. Given 5 = 7124.5, c = 9365.4. We have, cos a = 1 c ^ = : 90° - a. a = ■ e sin a, A = : ^ he sin a, a = •■ h tan a, check. log 5 = : 3.85275 logc = : 3.97153 log cos a = : 9.88122- 10 a = : 40° 28.4^ = : 49° 31.6' loge^ : 3.97153 log sin a = : 9.81231- -10 loga = = 13.78384- -10 a = : 6079.2 Check log 5 = : 3.85275 log tan a = : 9.93109- -10 log a = 13.78384 -10 a =6079.2 The following is the compact arrangement of the computation : Check b = 7124.5 log 3.85275 log 3.85275 c = 9365.4 log 3.97153 log 3.97153 a = 40° 28.4^ log cos 9.88122-10 log sin 9.81231- 10 log tan 9.93109- 10 13 = 49° 31.6^ a = 6079.2 log 3.78384 a = 6079.2 log 3.78384 It appears that the Pythagorean proposition, a^ + 5^ = c^, is not used because it is not adapted to the use of logarithms. It might be used in this case, however, in the form 44 LOGARITHMS EXERCISE XII Find the missing parts of the following triangles, using loga- rithms. (The work may be checked with a slide rule.) a fi a b c A 1. 63° 2584 2. 7531 8642 3. 75° 15.2' 965.24 4. 47.193 3972.6 5. 7.3298 6.1032 6. 18° 25.5' 32.96 7. 132.97 985.27 8. 53.215 13.712 9. 65983 72916 10. 29° 50.2' 10.207 11. 25° 17.4' 382.97 12. .00020 .00037 13. 63° 12.7' 7.1436 14. .07154 .09127 15. 35° 16.4' .62961 16. 35° 16.8' 41658 17. .00615 .00415 18. 80° 12.5' 5.2108 19. .00729 .01625 20. 25° 18.2' 1729.3 The examples 1-20 of Exercise VIII may also be solved by logarithms and the results compared with those there obtained. 21. Find the radius of the circle inscribed in a regular pentagon whose side is 12 feet. 22. Find the side of a regular pentagon inscribed in a circle whose radius is 15 feet 7 inches. 23. Find the area of a regular octagon whose circumscribed circle has a diameter of 10 feet. 24. A tower 120 feet high throws a shadow 69.2 feet long upon the plane of its base. What is the angle of inclination of the sun? 25. The top of a certain lighthouse is known to be 73 feet above the water. From a boat the angle between the top and its reflection is measured as 6° 45'. How far is the boat from the light ? 26. Two trains leave a station at the same time, one going north at the rate of 30 miles per hour, and the other east at the rate of 40 miles per hour. PROBLEMS INVOLVING RIGHT TRIANGLES 45 How far apart will they be in 20 minutes, and what is the direction of the line joining them ? 27. Show that if a is the side of a regular polygon of n sides, the area of 1 1 S()° the polygon is given by ^4 = - d^n cot 28. Show that if r is the radius of a circle, then the area of a regular cir- cumscribed polygon of n sides is A = rhi tan n 29. Find a value for the area of an inscribed polygon corresponding to that given above. 30. Taking the moon's diameter as 31' 20" and its distance from the earth as 239,000 miles, what is its diameter in miles ? 31. At what distance may a mountain 4 miles high be seen across a plain, the earth being taken as a sphere of 4000 miles radius? 32. If the sun's diameter is taken at 866,000 miles and its distance from the earth as 93,000,000 miles, what angle should it subtend at the center of the earth. 33. An approximate formula for the distance from the midpoint of a cir- cular arc to the midpoint of its chord is m = - — — — ■, 4 100 in which I is the length of the chord in feet and a the deflection or circumferential angle subtended by a base or chord of 100 feet. Find m for Z = 30, a = 2°. 34. If / is the angle of intersection between two tangents to a circle of radius i2, the distance T from a point of tangency to the point of intersection is given hj T = Rcod. Find T ior R = 3000 feet, and / = 22° 52'. 35. The length of a chord is given by 2 72 sin i 7, in which / is the central angle. Find the chord length for R = 2000, / = 12° 13'. 36. A river which obstructs chaining on a survey is passed by tri- angulation. The line ^J5,'Fig. 22, is measured 200 feet perpendicular to AC, and the angle ADC found to be 35° 27'. AVhat is the distance AC 2 37. With an instru- ment at A, Fig. 23, a level line of sight passes 6 ft. above the top of a wall as measured on a rod. The angles of depression * to the top and bottom of the vertical face are respectively, 2° 31' and 42° 16'. What is the height of the wall? * The angles of elevation and depression of an object measure respectively its angular distance above or below the horizon of the observer. Fig Fig. 22. Fig. 23. 46 LOGARITHMS AC DAB Fig. 24. 14° 41' find AD and DB. 38. In order to obtain both the horizontal and vertical distances to an inaccessible point, the solution of two triangles may be necessary. Fig. 24 represents two views of the problem. Wishing the distances AD and BD, first lay out the base line A C of any convenient length per- pendicular to AB. Measure the angle A CD and compute AD. Next from AD and the angle DAB^ the angles of elevation, compute DB. Having ^C = 300 ft., ACB = Ql°d4:', and CHAPTER V THE OBTUSE ANGLE 32. Definitions of the trigonometric functions of obtuse angles. If an obtuse angle (^.e. an angle greater than 90° and less than 180°) is placed on the axes of coordinates in the same manner as was the acute angle in Art. 6, the terminal line will extend into the second quadrant. The trigonometric functions of such angles are defined exactly as in Art. 6. Thus in Fig. 25, sin a I cos a tan a , etc. 4F 33. Signs and limitations in value. The abscissas of all points in OA (Fig. 25) are negative, while their ordinates and radii vectores are positive. It is evident, therefore, that some of the defining ratios are negative. In accordance with the law of signs in algebraic division, we find that the sines and cosecants of all obtuse angles are positive, while their cosines, secants, tangents, and cotangents are negative. The student should verify each of these statements in de- tail and become unhesitatingly familiar with these fundamental facts. P\irthermore, the sine and cosine cannot be numerically greater than unity and the secant and cosecant cannot be numerically less than unity. 47 ^X Fig. 25. 48 THE OBTUSE ANGLE Query. What are the limitations in value of the tangent and co- tangent ? 34. Fundamental relations. If the effects of the law of signs are traced, it will be seen that all the relations of Art. 9 hold also for functions of an obtuse angle without any modifications. 35. Variation. As the angle 6 varies from 90° to 180°, while V remains constant, x is always negative and varies from to — v, and y is positive and varies from v to 0. Consequently, as 6 increases from 90° to 180°, sin 6 decreases from 1 to 0, cos 6 decreases (algebraically) from to — 1, tan 6 increases from — oo to 0, cot 6 decreases from to — oo, sec 6 increases from — oo to — 1, esc 6 increases from 1 to oo. The terms positive infinity and negative infinity require careful consideration. If 6 varies continuously from 89° to 90°, tan varies in such a way as to exceed in magnitude any previously assigned definite value, however large. As it is positive for all values of 9 in the first quadrant, it is consequently said to become positively infinite (+oo). If 6 varies continuously from 91° to 90°, tan 6 varies so as to exceed numerically any previously assigned definite value. As it is, however, always negative for values of 6 in the second quadrant, it is said to become negatively infinite (-co). The plus or minus sign written before the symbol 00 merely indicates whether the trigonometric function increases numerically without limit through a positive or a negative set of values. 36. Functions of i8o°. As 6 approaches 180°, v remaining constant, x approaches — v and y approaches 0. We have, \Y then, sin 180° = 0, cos 180° = - 1, tan 180° = 0, cot 180° = 00, sec 180° = - 1, FIG. 26. cscl80°=oc. 37. Functions of supplementary angles. Two angles are called supplementary if their sum is 180°. Thus, in Fig. 27, a and fi are FUNCTIONS OF (180^^ - a) AND (90°+ a) 49 supplementary, and yS=180 — «, a being acute. The triangles OMP and ONQ are similar, but ON is negative. The pairs of corresponding sides are v and v\ x and r^:^ y and yK Hence we have sin (180° - a) = sin /3 cos (180° - a) = cos yS = ^ v' V v' sm a. = — cos a, tan (180° - a) = tan ^ Similarly : cot (180° - a) = - cot a, sec (180° — a) = — sec a, esc (180° — a) = CSC a. As a consequence of the relation sin (180° — «) = sin a, two values exist for arcsin m, the one acute, the other obtuse, and supplemental to each other. Y y _ y — tan a. Y P^^ "^-^^ V ^ y' ^' J f3 3^ y I ST X' X M Fig. 27. In case m = 1, the two values are identical. Fig. 28. QdERY. Is this also true of arccos m, arctan tw, etc. ? 38. Functions of (90° + a). In y Fig. 28, ^ = 90° + a, a being acute. -^f-^^ The triangles OMP and ONQ are similar, but the pairs of homologous sides are v^and v\ x and y, y and a;', while 2:' is negative. We thus obtain sin (90° 4- a) = sin /8 = ^ = - = cos a, cos (90° + a) = cos /8 _^_ ^_ sma, tan (90° + a) = tan /3 - = — cot a. 50 THE OBTUSE ANGLE In like manner, cot (90° + a) = - tan a, sec (90° + a) = — CSC a, esc (90° + a)= sec a. EXERCISE XIII 1. Find the values of the functions of 135°. (See Art. 11.) 2. Find the values of the functions of 150°. (See Art. 11.) 3. Find the value of sin [cos-^(— |f)], tan (csc-if ^), cos [arctan (— :^)], the angles being of the second quadrant. 4. Find the value of cos (arccos — ^j), sin [tan-i ( — j^^)], cot (arcsin ff), the angles being of the second quadrant. 5. Express in terms of an angle less than 45°, cos 160°, tan 130°, sec 150°. 6. Express in terms of an angle less than 45°, sin 170°, esc 95^ cot 140°. 7. Verify for a = 60°, the ftquatiiuiig - ^c/& ^^^c/^S sin 2 a = 2 sin a cos a, cos 2 a = 2 cos^ a — 1. 8. Verify for a = 45°, the equations sin 3 a = 3 sin ct — 4 sin^ a, cos 3 a = 4 cos^ a — 3 cos a. • 9. Verify for a = 120°, the equations ^^^ - ^ -. ,- + cos a cos -'—'»' tanl«:=V^-"Q^^^^-^»^^. 2 ^ 1 + cos a sin a 10. Verify for a = 120°, the equations i«=4 cot- ^--^1- + cos«_l + cosa cos a sm a 11. Verify for a = 120°, (3 = 30°, the equations sin (« + ^) = sin a cos /3 + cos a sin ^, cos (a- 13) = cos a cos )8 + sin a sin /S. 12. Verify for a = 120°, /3 = 60°, the equations sin (a — )8) = sin a cos /3 — cos a sin y8, cos (a -\- f3) = cos a cos /3 — sin a sin ^. FUNCTIONS OF OBTUSE ANGLES 51 13. Fill in the proper values in the following table for handy reference :" a sin a cos a tan a cot a sec a CSC a 0° 30=^ i 45° lV2 60° W3 90° 1 120° iV3 135° iV2 150° i 180° CHAPTER VI OBLIQUE TRIANGLES 39. Formulas for solution. In the oblique triangle ABC, Fig. 29, let the angles be denoted by a, ^, 7, and the lengths of the opposite sides by a, 5, c?, as in the figure. The relation a 4- p + 7 = 180° al- ways exists, and consequently when two of the angles are known, the third is determined. Five of the six parts of the triangle still remain to be found ; namely, the three sides and two angles. It has been shown in elementary geometry that if any three independent parts are given, the triangle is determined and the remaining parts can be found. Then two formulas, in addi- tion to the one just stated, are sufficient for the complete solution. It is, nevertheless, convenient to express the relations between the sides and angles in a variety of forms. Those given in the following pages are selected on the score of utility. They fall into sets of three each. From any one of each set the other two may be written by cyclic advance of the letters involved ; i.e. by changing a into 6, h into c, c into a, and at the same time a into ^, /3 into 7, 7 into a. The legitimacy of this process and the truth of the resulting formulas appear from the consideration that no distinction is made as to any one side or any one angle. Any side and its opposite angle can be exchanged for any other pair. The cyclic advance affords a convenient systematic method of writing all possible forms. From any one of these sets, as for instance that of Art. 40, or that of Art. 42, all the other sets may be derived by purely analytical processes. An independent geometric proof is given of each, however. The derivation by the analytic method suggested will afford a valuable review exercise after Chapter VIII has been studied. 62 LAWS FOR OBLIQUE TRIANGLES 53 40. Law of projections. If, in Fig. 30, the perpendicular CD is drawn from to AB^ the portions AB and BB are respectively the projections on the side AB of the other two sides AC and CB. Con- sequently, by Art. 15, we have Fig. 30. AB ==ACco^a-\-CB cos /3, or c = & cos a + a cos p. By drawing the perpendicular from A and B in turn, we get a zz c cos p + 6 cos "y, 6 = a cos "Y + c cos a. By cyclic advance of the letters the first formula is transformed into the second, the second into the third, and the third into the first. 41. Law of sines. Connect the circumcenter K in Fig. 31 with the vertices, A^ B, C, and the midpoints, X, M, iV, of the sides. Then is Z BK0 = 2 «, Z CKA = 2 /3, Z.AKB=2y. (Why?) In the right triangle KLO^ /. LKO = a, and LC=^a. Denoting the cir- cumradius by R^ Art. 16 gives R sin a. Fig. 31. J- a The other right triangles give likewise ^b = R sin yS, ^ c = R sin y. Equating the values of 2 R, we obtain the law of sines ; namely, a _ b _ c sin a sin p sin 'y The cyclic symmetry is apparent. The student should draw the figure and give the proof in case one angle of the triangle is obtuse. 42. Law of cosines. In Fig. 32 the perpendicular jt? drawn from (7 divides the opposite side c into two portions m and w, and the 54 OBLIQUE TRIANGLES whole triangle into two right triangles ADC and BDO, In the latter triangles, we have, by Art. 16, a^ = n^ -\-p^ = ((? — 771)2 -{-p^ Fig. 32. or a^ — ^2 ^ ^2 _ 2 hc COS a. Proper changes in the figure yield 62 = c'^ + a^ -2ac cos p, c2 = a2 _f. ^2 -2 ah cos 7. These again may be written by cyclic advance of the letters. Useful forms for writinsr these laws are : cos a = 52+^2- a^ 2 be cos /3 = c2-fa2- h'^ 2 ao i->r»o i\/ — a'^+P- -6'2 lab 43. Law of tangents. lu Fig. 33 draw AE the bisector of the angle at J., and BF and CD perpendicular to it from the other vertices. Then ABAF=^DAC=^a, while Z DCE= Z. EBF= 90° - Z BEF = 90° - (Z ABE + Z ^^ JS') = K« + ^ + 7)-(^ + -|«) = K7-/3). Again, i)^= ^^+ DE=AF- AD. From the right triangles in the figure we get AF~AD Fig. 33. (e- b) cos J a ^^^2^7 Z:^; ^^ ^^^ FB + CD FB^-CD ((? + *) sin i« or taiil(v-p) c + 6 cot ^ a. LAWS FOR OBLIQUE TRIANGLES 55 The forms tani (a-v) = ^-^cotlp, a tani(p-a)=— --cot|Y, & + « may be obtained from suitably altered figures or by cyclic advance. The formula may be written symmetrically tanK7-y3) ^g-^ tan i (7 + yg) c-^h Ji b>c, the first formula will stand tan i (/? - ry) = -Zl cot I a. Similar changes may occur in the other two. 44. Angles in terms of the sides. Construct the inscribed circle, Fig. 3-4, and denote its radius by r. Denoting the perim- eter a-{-b-{- c by 2 s, we have AE=AF=s-a, BD = BF=s-h, CD=CE = s- c. Consequently, by Art. 16, V V tan J a = , tan \ p = -, tan \ y = The value of r in terms of the three sides is derived in the corollary of Art. 45, thus completing this theorem. 45. Area of oblique triangles. (1) By elementary geometry, we have (see Fig. 35) Introducing the value of p found by Art. 16, we get the formula ^ = 1 6c sin a. 56 OBLIQUE TRIANGLES with the cognate forms A= ^ca sin p, ^ = | ab sin -y. (2) Squaring both members of the formula just derived, we obtain, with the aid of readily justifiable transformations and sub- stitutions. = i5V(l-cos2 a) , = — (l + coscc) • |(1- - cos a) her. ^h^ + e^- -> !(■ b'^^c^- 2 5c .■) 2bc-hb^ + c^- aP' 2bc- 52_,.2 4.^2 4 4 b-\-c + a b + c— a a- -b-{- c a-\- b — c 2 2 = s(s — «)(s — 5)(s — c). Whence we have the desired formula A = Vs(s — a)(s — h) (s — c). (3) If r is the radius of the inscribed circle, we have, by elementary geometry, A = rs. Corollary. Equating the values of A found in (2) and (3), and solving for r, we get ^ s the result needed to complete the theorem of Art. 44. 46. Numerical solution. The formulas of Arts. 40 and 42 are not adapted to the employment of logarithms. They are useful, however, in case the numerical values of the sides contain few digits. The solution of oblique triangles falls into four well-defined cases, according as the three given parts consist of I. Two angles and one side. II. Two sides and an angle opposite one of them. III. Two sides and the included angle. IV. Three sides. NUMERICAL SOLUTION. CASE I 57 Each of these three cases with a model solution is discussed in detail in the following articles. 47. Case I. Given two angles and one side. Let the given parts be a, j3, a. The solution is effected by means of the formulas of Arts. 39 and 41. Solving for the unknown parts, we have 7 = 180°-(« + /3), , a sin y8 Example. sin a ' a sin 7 c=—. -, sin a t, , b sin 7 formula c = —. — ~ . sm l3 Given a = 47° 13.2' ^=65° 24.5' a = 43.176 sum of angles = 180° « 4-/5 = 112° 37.7' .-. 7= 67° 22.3' loga= 1.63524 log sin /3= 9.95871- -10 cologsina= 0.13433 log 5 = 11.72828 - -10 .-. 5 = 53.491 loga= 1.63524 log sin 7= 9.96522- -10 cologsina= 0.13433 log c= 11.73479- n^ .-. c = 54.299 Check log 5= 1.72828 log sin 7= 9.96522- 10 cologsiny8= 0.04129 log (? = 11.73479- -10 /. (?= 54.299 58 ^ OBLIQUE TRIANGLES The compact form of computation is as follows ; log 1.63524 a = 43.176 ^ ^ 65° 24.5' a = 47° 13.2' b = 53.491 y = 67° 22.3 c = 54.299 log 1.63524 log sin 9.95871 - 10 colog sin 0.13433 log b 1.72828 coiog sin 0.13433 log sin 9.9652g - 10 Check colog sin 0.04129 log 1.72828 log sin 9.96522 log c 1.73479 Examples Find the remaining three parts, given 1. ;8 = 65°15.5', y = 81° 24.6', 2. /? = 38°37.4', 7 = 75° 32.8', 3. a= 48° 29.2', y= 115° 33.8', ^^— 4. a = 68° 41.5', y = 110° 16.5', 10 log c 1.73479 b = 724.32. c = 129.63. a = 14.829. c = 9.4326. 48. Case II. Given two sides and an angle opposite one. Let the given parts be a, 5, a. The solution is effected by the formulas of Arts. 39 and 41. Solving, we have . ^ b sin a sin/3= — , a sin 7 c = with the check formula (? = sm a 5 sin 7 sin y8 An ambiguity arises in this case, however, since to any value of the sine correspond two supplementary angles, one acute, the other obtuse. Thus we also have /3' = 180°-/3, i = 180°- (a + /30. c' a sin 7' sin a ' e' h sin 7' siuiS' CASE II 59 The nature of this ambiguity will appear from the construction of the triangles with the given parts. If the given angle a is acute, there will be no solution, one solution, or two solutions, according as the free end of a (see Fig. 36), swinging about .>— . A\ T^L Fig. 36. (7, meets the line AL in no points, one point, or two points ; i.e. as a is shorter than (72), the perpendicular from upon AL, longer than AO^ or intermediate between CD and AC For a = CD there is a single right triangle ; and for a = AC, a single isosceles triangle. When a is right or obtuse, there is no solution or one solution, according as a is shorter or longer than AC. These results may be tabulated for reference. «<90' 'a 90^ a :^ 6, no solution, > 5, one solution. If we proceed with the numerical work, without previously testing the number of solutions possible, the case of a single solution will appear from the fact that a -\- jS^ > 180°. (Whence a + (180° - /9) > 180°, or a - yS > 0, or ^ < a.) When there is no solution, we shall get log sin fi>0 ; i.e. its augmented character- istic will be 10 or greater. A preliminary free-hand sketch will ordinarily serve to determine the number of possible solutions. Example i. Given a = 3541, b = 4017, a = 61° 27'. 60 OBLIQUE TRIANGLES By careful arrangement of the work, we can determine the number of solutions by inspection. Check log &=4017 az=61°27' h sin a a = 3541 ^= 85° ir a-f;8=146°38' 7 = 33° 22' c=2217.16 c = 2217.16 3.60390 log sin 9.94369-10 log 3.54759 log 3.54913 log sin 9.99846 colog sin 0.05631 log 3.54913 log sin 9.74036-10 log 3.34580 log 3.60390 colog sin 0.00154 log sin 9.74036-10 log 3.34580 From the logarithms of 6, a, and h sin a it is seen that h sin a h. The solution is effected by the formulas of Arts. 43, 39, and 41. Solving, we have tan 1 (« — /3) = I(« + ^) = 90°-i7, a sin 7 cot J 7, with the check formula Example. Given y = 78° 15' a = .745 6 = .231 a-/; =.514 log 9.71096- -10 a + 6=. 976 colog 0.01055 ^=39° 7.5' 2 log cot 0.08969 ^^^=32° 55.3' 2 ^^±^=50° 52.5' 2 log tan 9.81120 - -10 a = 83° 47.8' )8 = 17°57.2' c = . 73368 c = . 73367 sm a h sin 7 smy3 a =.745, 5 = .231, 7 = 78°15^ log sin 9.99080- 10 log 9.87216-10 colog sin 0.00255 log 19.86551-20 Examples Find the unknown parts, given 1. 6 = 284.12, c = 361.26, a = 125° 32'. 2. c = 395.71, a = 482.33, ^ = 137° 21'. 3. a = .06351, c = .10329, /8 = 83° 29.4.' 4. c = .00397, h = .00513, a = 68° 21.8^ log sin 9.99080-10 log 9.36361 - 10 colog sin 0.51109 19.86550-20 50. Case IV. Given the three sides. The given parts are a, 5, G. 62 OBLIQUE TRIANGLES The solution is effected by the formulas of Art. 44, with the formula for r from Art. 45. We have at once s = 1 (a -f 6 + 0' ^_J(«-«)(«-*)(« -0 tan -« = , etc. 2 s — a « + /3 + 7 = 180°, serves as a check formula. Example i. Given a =.05341, 5 =.06217, tf=. 03482. Then 2 s = .15040 s= .07520 colog 1.12378 s-a=. 02179 log 8.33826-10 s- 5 = .01303 log 8.11494-10 «_ ^ = .04038 log 8.60617-10 r2 log 16.18315-20 r log 8.09157-10 ^=29° 32.3' log tan 9.75331-10 1 = 43° 27.6' log tan 9.97663-10 ^=17° 0.1' log tan 9.48540-10 a= 59° 4.6' • y8= 86° 55.2' 7= 34° 0.2' sum of angles =180° 0' When the three sides are given and only one angle is required, say /S, the two appropriate formulas may be combined into one, as tan -^ _^l(8 — a)(8 — c) sCs-h) CASE IV. COMPOSITION OF FORCES 63 Example 2. Given a= 35, h= 64, c= 73. Then 2 s = 172 s= 86 colog 8.06550-10 s-a= 51 log 1.70757 8-h= 22 colog 8.65758-10 8-e= 13 log 1.11394 2)19.54459-20 i;8=30°37.4' log tan 9.7723U-10 ;e=61°14.8' Examples Find the angles of the following triangles : 1. a = 6123, ^> = 7148, c = 6815. 2. a = 12,545, 5=8612, ^=10,353. 3. a = .05431, 5 = .03714, 6'=. 06513. -- — 4. a = .006152, b = .008174, c = .007534. 5. ^. = 72,584, 5 = 125,217, c?= 36,925. 6. a = 13,579, 6 = 35,791, (? = 24,680; find /3. 7. a = 80,812, h = 37,194, e = 43,618. 8. « = 36,925, 5 = 25,814, c= 14,703; find 7. Find the areas in examples 1 and 2. 51. Composition and resolution of forces. Equilibrium. In mechanics the solution of oblique triangles is frequently required in problems relating to the composition and resolution of forces, velocities, and other directed quantities. In this article will be stated, without proof, some of the laws governing the combination of such quantities, showing the appli- cation of trigonometry to certain of the problems involved. Suppose the line segments AB and JL(7, P'ig. 37, to represent in magnitude and direction two forces acting at a point J., and in- cluding between their lines of action the angle <^. 64 OBLIQUE TRIANGLES Complete the parallelogram ABBQ. The diagonal AB^ drawn from the point A^ is the line segment representing the resultant of the two given forces, i.e. the sin- gle force that will produce the same effect as the two given forces. The process of finding the resultant of two or more given forces is called the composition of forces. Conversely, the two line segments AB and AC may be taken as the components of AB. Thus the two Fig. 37. ^ forces AB and AC, acting together at A, produce the same effect as the single force AB. The pro- cess of finding two or more forces equivalent to a given force is called the resolution of the force into its components. Since the segment BB is equal and parallel to AC, it follows that the resultant and the two components form a closed triangle ABB, and the relation between the forces may be obtained by solving this triangle. Note that the angle ABB is the supple- ment of the angle <^, so that by Art 37, cos ABB = — cos (/). Example 1. Find the resultant of two forces of 320 dynes and 400 dynes, respectively, acting on a common point, at an angle of 54° 28^ In the triangle ABB, Fig. 37, we have given two sides and the included angle. If only the magnitude of the resultant is desired, it may be obtained by the law of cosines. Art. 42. Thus we obtain „ « AB = ^\A^ -f ^(7+2 AB AC ■ cos c^j. If the angle formed by the resultant with its components is also required, the logarithmic computation may be effected as in Case III, Art. 49. Example 2. Resolve a force of 40 pounds into components making angles of 32° and 74° 20 with its line of action. Referring to Fig. 37, we have ^D = 40, Z ^^2> = 32°, and Z i>^ (7= Z ^i)^ = 74° 20' . Denoting the sides opposite the angles A, B, B, respectively, by a, h, d, we have from the law of sines, ,sin^ J ysini) a — b -, d = o— — — • sin B sin ^ EQUILIBRIUM OF FORCES 65 Hence the components may be computed. Three forces are in equilibrium when the resultant of any two forces is equal and opposite to the third. Thus in Fig. 37, if the direction of the force AD is reversed, it and the forces AB and AQ will be in equilibrium. The necessary conditions that three forces shall be in equilibrium are : 1. Their lines of action shall lie in the same plane. 2. Their lines of action shall meet in a point. 3. The line segments representing the three forces when laid off in order shall form a triangle. In Fig. 38 the forces a, 5, and c applied at a common point are in equilibrium. The angles between the lines of action are de- noted by J., B^ C^ as indicated. When the forces are laid off to form the triangle, the angles of the triangle are seen to be the supplements of the corresponding angles A, B^ O. That is, a = 180° — J., whence sin a = sin A, /3 = 180° - B, whence sin /S = sin B. etc. etc. From the law of sines. Therefore, a h c sin a sin ^ sin 7 a h c sin A sin B sin O 66 OBLIQUE TRIANGLES EXERCISE XIV Find the unknown parts of the following triangles : a 18 y a b c 1. 62° 35' 82916 59278 2. 75290 92841 69289 3. 25° 36.2' 68° 13.5' 3.9168 4. 55° 55.4' .25317 .36291 5. 69° 17.5' 329.12 689.12 6. 100° 10' 62198 29322 7. .0000713 .0000987 .0001255 8. 61° 15.2' 49° 16.3' 58.291 9. 120° 50.2' 2.8315 4.1217 10. 38° 17.2' 21.992 50.715 11. 150° 24.2' .038251 .047319 12. 58° 06.5' 57.15 67.31 13. 75° 19.3' 70° 29.2' 658.42 14. 100.05 200.07 150.08 15. 126° 26.4' .0021868 .0032292 16. 10° 32.8' 25.317 37.293 17. 50010 70020 90030 18. 48° 25.3' 56° 34.5' 7219.2 19. 120° 15' 62158 75292 20. 90° 00' 725.63 617.25 Solve the following triangles, given 21. a = 2500, c = 2125, A = 208,690. 22.. ft = 103.5, c = 90, A = 4586.7. 23. a = 73° 10', b = 753, A = 74,803. 24. ^ = 57° 25', c = 57.65, A = 3055.7. 25. Find the areas in examples 1, 9, 17. 26. Find the areas in examples 2, 4, 14. 27. Determine the magnitude and direction of the resultant of two forces of magnitudes a and h, if their lines of action include an angle . 28. Carry out the computation of example 27 in the following cases : a = 20, & = 36, <^ = 45° ; a = 300, b = 540, <^ = 64°; a = 75, 6 =3 60, <^ = 145° ; a = 250, b = 320, <^ = 120°. 29. Find the directions of three forces in equilibrium if a = 7, 6 = 10, c = 15; also if a = 24, 6 == 36, c = 42. EXERCISES 67 30. Referring to Figure i a = 695, 6 = 483, = 155°: a ' solve completely and interpret physically when 720, b = 840, B = 100°. = 135, 31. Solve and interpret when a = 1200, 5 = 135°, C = 150° h = 142, c = 95. 32. Resolve a force of magnitude 84 into two equal components making an angle of 60° with each other. 33. Resolve a force of magnitude 240 into two components of 120 and 180 each and find the directions of the components. 34. Determine the formula for one side of a quadrilateral in terms of the other three sides and their included angles. Compute for a = 10,b = 12, c = 15, ^6 = 135°, 6c = 60°. Query. How many given parts serve to determine the remaining parts of a quadrilateral? 35. Given the four sides and one angle of a quadrilateral, determine the other angles and the diagonals. Compute for a = QO, b = 72, c = 90, d = 100, Q = 120°. 36. Given three angles and two sides of a quadrilateral, determine the remaining sides. Compute for a = 630, b = 500, ab = 100°, be = 80°, cd= 60°. 37. Find the angles and the lengths of the sides of a regular pentagram, or five-pointed star, inscribed in a circle of radius 8. 38. Compute the volume for each foot in depth of a horizontal cylindrical tank of length 30 feet and radius 6 feet. 39. Having measured the following data, ^A = 80° 30', B = 72° 15', and c = 232.5 feet, compute the inaccessi- ble distance b (Fig. 39). 40. Compute the dis- tance a across a lake. Fig. 40, having measured A, B, and c, which are respectively 51° 20', 72° 40' and , 3420.5 feet. Fig. 41. 41. A being invisible from C, find the distance b through a forest, having measured a = 1037 feet, c = 1208 feet, B = 69° 25'. 42. In Fig. 42, BC, the distance of the foot of a wall below the instrument is 12.3 feet, 6 and a, the angles of elevation and depression, are 15° 20' and 21° 15', respectively. Find the height of the wall and its distance from the instrument. Fig. 42. 68 OBLIQUE TRIANGLES Fig. 43. Fig. 44. 43. A pole BC, Fig. 43, is 12 feet long and leans two feet from a vertical toward the instrument at ^ . If the angles of elevation of the top and bottom are respectively 37° 15' and 11° 50', what are the horizontal and vertical distances from the instrument to the foot of the pole? 44. It is desired to find the horizontal distance and eleva- tion of the inaccessible point B, Fig. 44, with reference to an instrument at A. Having laid out a base line A C, 250 feet long, the angles at A and C are found to be 87° 10' and 73° 51', respectively, and from A the angular elevation of B is 11° 32'. ,(7 45. Given 5 = 110° 05', ^ J5: = .4 7) = 200 f eet, DE = 125 feet, and ^5 = 632 feet; find the distance AC to be laid off, and the inaccessible distance BC (Fig. 45). 46. From measure- ments we have (Fig. 46) AB = Qm feet, BAC = 10° 40', BAD = Q2° 30', ABD = 65° 32', ABC = 89° 25'. Find the inacces- sible distances AD F^«- ^^^ and DC, and the angle between DC and AB. 47. From the instrument at A (Fig. 47) the angles of elevation to the top and base of the vertical wall are 15° 12' and 1° 23', respec- tively. A base line AB \& measured 75 feet toward the wall down a plane inclined 8° 16', and from B the angle of elevation to the top of the wall is 37° 46'. Compute the height of the wall and its horizontal distance from A. Fig. 47. 48. It is required to prolong the line AB (Fig. 48) beyond an obstacle. At B is made an angle 52° 20' to the right and at C an angle of 110° 00' to the left, BC being 210 feet. Compute the proper distance CD and angle to the right at Z), also the inaccessible distance BD. Note that by mak- ing B = D = 60° and C = 120°, then BC=CD = BD and all computations are avoided. 49. Having but one point C (Fig. 49) from which both inaccessible points A and B are visible, we are required to find the inaccessible distances AC EXERCISES 69 and AD and the angle between AB and DC. ADC = 87° 42', DC A = 60° 32', DCE = 170° 05', BCE = 41° 20', CEB = 111° 35', DC = 365.2 feet, C^^ = 410.7 feet. 50. It is required to ascertain the length and j) position of an in- A_, ,B Fig. 50. accessible line AB (Fig. 50), its ex- tremities not being visible from a common point beyond the obstacles. By chaining we have CD = 210.7 feet, DE = 390.4 feet, EF = 173.5 feet. Then the follow- ing angles are measured : A CD = 83° 41', CDE = 19° 12' left (180°-19° 12'), CDA = 79° 49', FEB = 53° 20', DEF = 42° 03' left, EFB = 115° 27'. In order to locate points suitably upon a map, find lengths AB, AD, and BE. /77777my 51. A tower 115 feet high casts a shadow 157 feet long upon a walk which slopes downward Fig- 51. from its base at the rate of 1 in 10. What is the elevation of the sun above the horizon? CHAPTER YII THE GENERAL ANGLE Only those parts of trigonometry that are necessary for the solution of triangles have been developed thus far. In this and the following chapters are considered some of the more important topics of another phase of trigonometry that is no less essential for the further study of pure and applied mathematics. 52. General definition of an angle. If a straight line rotates about one of its points, remaining always in the same plane, it generates an angle. The angle is measured by the amount of ro- tation by which the line is brought from its original position into its terminal position. For the small rotation leading to acute and obtuse angles this definition agrees with the customary elementary definition, the knowledge of which has been presupposed in the foregoing chapters. As in Art. 3, counterclockwise rotation generates positive angles ; clockwise rotation, negative. In the sexagesimal system of angle measurement the standard unit is the angle produced by one complete rotation of the generating line. This angle is divided into 360 equal parts called degrees^ the degree into 60 minutes, and the minute into 60 seconds. In the circular system the standard unit is the radian^ the angle produced by such a rotation that each point in the generat- ing line describes an arc equal in length to its radius. Angu- lar magnitudes are stated in radians and decimal fractions thereof. Instruments are graduated and tables printed in accordance with the sexagesimal system, which is used in practical numerical calculations. Astronomers, however, employ decimal fractions of seconds, while engineers make use of tenths of minutes and deci- mal divisions of degrees. In theoretical discussions the radian system is commonly employed. Hereafter, in this book, the two systems will be used interchangeably. 70 DEFINITIONS OF TRIGONOMETRIC FUNCTIONS 71 .^^-^ Since the circumference of a circle is equal to 2 tt times its radius, where 7r= 3.14159---, we may write the following relations between the two systems : 2 IT radians = 360° 1 radian = 57.29578°. . = 57° 17' 44.8'' and, in general, the number of degrees in any angle is equal to 180 the number of radians multiplied by • , while the number of TT radians is equal to the number of degrees multiplied by -^. Thus the straight angle is tt radians ; the right angle, — radians. A If the radius of the circle is represented by r, the arc by a, and the angle, in radians, by a, we have the important relation a — vQ^. 53. Axes, quadrants, etc. Let the two axes of coordinates be assumed as in Art. 4 ; and, as in Art. 6, let the angle be placed upon the axis, its vertex at the origin, and its initial line extending along the X-axis toward the right. The sign and magnitude of the angle will determine the position of the terminal line, causing it to coincide with one of the axes or to fall in one of the quadrants. An angle is said to be of the first, second, third, or fourth quadrant according as its terminal line falls in that quadrant. While the acute angle is of the first quadrant, the converse is by no means necessarily true. The ter- minal line of every angle, however large, must coincide with the terminal line of some positive angle less than 360° (see Fig. 52). For the purpose of trigonometry as developed in the present chapter, for every angle, positive or negative, and of any mag- nitude, may be substituted a positive angle less than 360°. 54. Definitions of the trigonometric functions. The trigono- metric functions of angles of any size are defined identically as in 72 THE GENERAL ANGLE Art. 6. Thus for all positions of the terminal line, Fig. 53, y . X - = sm a, - = cos a, V V y X — = tan a, — = cot a, X y V V - = sec a, - = CSC a. X y W O X M p\a {a) Q>) (c) Fig. 53. (d) 55. Signs and limitations in value. The abscissas are positive for all points in the first and fourth quadrants, negative for those in the second and third. Ordinates are positive for all points in the first and second quadrants, negative for those in the third and fourth. The radius vector is, by agreement, considered positive for all points. In conformity with the sign law of algebra, the functions of angles of the different quadrants will have signs as displayed in the following table : Quad. Sine Cosine Tangent Cotangent Secant Cosecant I + + + + + + II + - — — - + III - — + + — — IV - + — — + - It will be noticed that for angles of the first quadrant all six functions are positive. In each of the other quadrants one pair of mutually reciprocal functions are positive, the other two pairs are negative. These positive pairs run as follows : second quadrant, sine and cosecant : third quadrant, tangent and cotangent : fourth quadrant, cosine and secant. SIGNS AND LIMITATIONS IN VALUE 73 The student should establish these statements regarding the signs of the functions and memorize them. Since the lengths of the abscissa and ordinate can never exceed that of the radius vector, it follows that the sine and cosine can never be numerically greater than unity, and the secant and cosecant can never be numerically less than unity. The tangent and cotangent can have numerical values either greater or less than unity. EXERCISE XV 1. Express in degrees, minutes, and seconds the angles — , — , '-~^, o,r« o« 3^ ■ 4 3 6 2. Express in radians the angles 30°, 15°, 45°, 120°, 240°, 300°, 450°. /-^r' In a circle of radius 60 cm., what is the length of the arc which sub- tends at the center the angle 30°, 60°, ^, ^ ? ^ ' ' 3 ' 4 4. In a circle of radius 10 inches, what is the circular measure of the angle subtended by an arc whose length is 10, 5, 20, 5 ir inches? 5. A friction gear consists of two tangent wheels, whose radii are 8 and 12 inches, respectively. The smaller wheel makes 4 revolutions per second. Find the number of revolutions per second made by the larger, the angular velocity of each, and the linear velocity of a point on the circumference of each. If the larger wheel is attached to the rear axle of an automobile whose rear wheel has a diameter of 30 inches, find the speed of progress of the machine. 6. The diameters of the front and rear sprocket wheels of a bicycle are 10 inches and 4 inches, respectively, and the diameter of the rear wheel is 28 inches. Find the rate of pedaling when the bicycle is traveling 12 miles per hour, the corresponding angular velocities of the two sprocket wheels, and the linear velocity of the chain. 7. Determine the quadrant to which each of the following angles belongs : 210°, 465°, 745°, - 830°, ^, i^, -^. 3 ' 4 ' 3 8. Determine the signs of the functions of the following angles: 240°, 330°, 400°, ^, -'Le, 6^. 3 4' 9. Show that the quadrant to which an angle belongs is determined if the signs of any two non-reciprocal functions are given. 10. To what quadrant does an angle belong if its sine and tangent are negative ; its secant and cotangent positive ; sine and secant negative ; tangent and cosine positive ? 74 THE GENERAL ANGLE 11. Determine the quadrants of the following angles: sin"i|; arccos — j\; arctanf; cot-^ — j\. 12. Determine the quadrants of the following angles : sin-i I = cot-i — f ; arccos — j% = arccsc i|. 13. For what values of a is sin a — cos a positive ? 14. For what values of cc is tan a — cot a negative ? 15-20. Find the missing values in the following table : z sin cos tan cot sec CSC QlFAD. a 7 8 it -II --A — 15 II III III IV ¥ IV ^ -¥ III 56. Variation of the trigonometric functions. A change in the angle will produce a corresponding change in the values of the coordinates and in their ratios. If, for convenience, the chosen point in the terminal line of the angle is maintained at a constant distance from the vertex, the radius vector will retain the constant value + V. As the angle increases continuously from 0° to 360°, the abscissa and ordinate vary continuously between the limits — v and 4- v. As 6 increases from 0° to 90°, x is positive and decreases from V to ; as 6 increases from 90° to 180°, x is negative and decreases (algebraically) from to — v ; as ^ increases from 180° to 270°, X is negative and increases from —v to ; and as 6 increases from 270° to 360°, x is positive and increases from to v. As 6 increases from 0° to 90°, i/ is positive and increases from to y ; as ^ increases from 90° to 180°, ^ is positive and decreases from V to ; as ^ increases from 180° to 270°, y is negative and decreases from to —V ; as increases from 270° to 360°, i/ is negative and increases from — v to 0. Upon introducing these varying values into the ratio definitions, we are enabled to trace the variation of the trigonometric functions. We see, for example, that as increases from 0° to 360°, tan 6 continually increases algebraically, changing sign from negative to positive through the value as ^ passes through 0°, GRAPHS OF THE TRIGONOMETRIC FUNCTIONS 75 180°, and 360°, and from positive to negative by becoming infinite as 6 passes through 90° and 270°. There is an infinite discon- tinuity in tan 6>, for 6 = 90° and d = 270°. Query. Which of the trigonometric functions other than the tangent become infinite and therefore discontinuous ? The student should trace the variation of each function in detail, stating the narrative verbally. 57, Graphs of the trigonometric functions. The whole behavior of each function can be conveniently represented by means of the graphical method already introduced in Art. 4. Assume a pair Fig. 54. Graph of sin 6. of axes of coordinates, as in Art. 4, and along the JT-axis to the right lay off equal spaces corresponding to the number of degrees in the angle 6. At each point in the JT-axis erect a perpendicular whose length is proportional to the value of the sine of that angle. Each point thus determined has the property that its abscissa represents the angle 6 and its ordinate the corresponding value of sin 6. Now having located a sufficient number of points, draw through them a smooth curve. It will be seen that the value, sign, and variation of the sign at each instant is fully exhibited by the ordinate, position, and inclination of the curve or graph. The same may be done for each of the functions. The graphs of the different functions are here presented. The student should trace carefully the intimate and exact cor- respondence of the graphical and the verbal narratives. 76 THE GENERAL ANGLE ^T Fig. 55. Graph of cos d. -h. O Fig. 56. Graph of tan d. GRAPHS 77 ^Y X Fig. 57. Graph of cot d. /^Y O Fig. 58. Graph of sec 6. 78 THE GENERAL ANGLE AY _7C_ -7t 2 2 371 2 ^ 2 X Fig. 59. Graph of esc d. 58. Functions of 270° and 360°. By the method of limits em- ployed in Art. 12, we get the following sets of values : sin 270° = - 1, cos 270° = 0, tan 270° = oo, cot 270° = 0, ■ sec 270° = 00. CSC 270° = - 1. sin 360° = 0, cos 360° = 1, tan 360° = 0, cot 360°= oo, sec 360° = 1, CSC 360° = oo. Here oo is used as before to denote the value of a fraction whose numerator remains finite while its denominator approaches zero. The sign + or — is prefixed to the symbol oo according as tlie variable becomes oo through a positive or a negative sequence of values. In the light of this discussion the values of the functions oih y. —(k any integer) may be tabulated, the upper of the pair of double signs arising when the angle approaches the critical value from below. e sin 9 COS tan 9 cot 6 sec B esc B TO + 1 TO Too + 1 Too f -fl ±0 ±cc ±0 ±co + 1 TT ±0 -1 TO Too - 1 ±GO It -1 TO ±cc ±0 Too -1 27r ±0 + 1 TO T 00 + 1 Too FUNDAMENTAL RELATIONS 79 EXERCISE XVI n 1. Trace the variation, as d varies, (a) of sin 2 6\ (b) of cot -• n 2. Trace the variation, as ^ varies, (a) of tan 2^; (h) of cos - • id 3. Draw the graph of cos 2 B. 4. Draw the graph of sin 3 B. 5. In what points will a horizontal line \ unit above the JT-axis intersect the graph of sin Q ? Explain the significance of the result. 6. In what points will a horizontal line 1 unit above the X-axis intersect the graph of tan B1 Explain. 7. If the graphs of tan B and cot B are drawn on the same axes to the same scale, where will they intersect? What is the significance? 8. If the graphs of sin B and cos B are drawn on the same axes to the same scale, where will they intersect? What is the significance ? 9. Construct the graph of logj^x, taking values of the number x as abscis- sas and the corresponding logarithms as ordinates. 59. Fundamental relations. Just as in Art. 9 we find, by in- spection, (1) (2) (3) by division, tan a = ^*L^^ (^4) (5) sm a by virtue of the Pythagorean proposition, sin^ a + cos^ a = 1, (6) tan^ a + 1 = sec^ a, (T) cot^ a -f- 1 = csc^ a. (8) The student should prove that all these formulas conform, for angles in all quadrants, to the algebraic law of signs. CSC a 1 sina^ sec a 1 cos a cot a 1 . tan a' tana sin a cos a nnt CL _ cos a . 80 THE GENERAL ANGLE 60. Line representations of the trigonometric functions. As the names tangent and secant indicate, the trigonometric functions were originally defined as certain lines measured in terms of a standard unit line. The adoption of the abstract ratios, as in this book, is of comparatively recent date. It is both interesting and advantageous to know the line representations and sliow that they lead to the same science of trigonometry as do the ratio deti- nitions. The line representations most frequently used involve the use of a unit circle, i.e. a circle of radius unity. It is evident that we may replace each of the defining ratios of Art. 54 by an equal ratio so chosen that its denominator is positive unity. The value of the ratio will be equal to that of the numerator. In other words, if a positive unit radius is taken as the denominator, the length and sign of the numerator will represent the function in magnitude and sign. We have, then, simply to select six lines whose ratios to the radius agree with the definitions of Art. 54. The ratio of the subtended arc to the radius is, by Art. 52, the circular measure of the angle. Suppose, then, a circle of unit radius drawn with its center at the origin of coordinates. The angle is placed upon the axes just as in Art. 6, and from the point P of intersection of the terminal line with the circle, perpendiculars MP and NP are drawn to the two axes. From the two points Jl and ^ where the positive axes cut the circle, tangents ^2^ and BS are drawn meeting the terminal line (pro- duced if necessary) in the points T and S. Since P, T, jS, Figs. 60-63, lie in the terminal line, we have, at once, in accordance with Art. 54 (or Art. 6) : MP NP AT , BS tan a = -—- , cot « = -— , OA OB OT OS sec«=^, csc« = -^. But by construction, OP=OA=OB = 1. LINE REPRESENTATIONS 81 These denominators may then be suppressed and the functions represented graphically as indicated below : t \^ B ^ \ \ A , 1 M 1 \ 1 ^•\r Fig. 60 Fig. 61. Fig. 62. Fia. 63. sin a — MP, cos a = NP, tan a = AT, cot a = BS, sec a= OT, CSC a = OS. Moreover, the angle, in radians, is represented as follows arc AP OP arc AP. According to the modern view, the line is not the function, but by its length and direction represents the function in magnitude and sign. Note that the line representing the tangent is always drawn from the point A and that representing the cotangent from B. All the lines are read from the axes to the terminal line. Hori- zontal lines are positive toward the right, negative toward the left. Vertical lines are positive upward, negative downward. 82 THE GENERAL ANGLE By means of the Pythagorean proposition, and the theorems concerning similar triangles, the fundamental relations given in the preceding article, as also the limitations of value stated in Art. 55, are readily established. So, also, the subsequent theorems of trigonometry may be interpreted by means of the line represen- tation of the trigonometric functions. This graphic interpretation frequently presents special advantages. This is the case, for ex- ample, in the investigation of the variation of the functions con- sidered in Art. 5Q. So, too, the construction of the graphs of the functions as treated in Art. 57 is facilitated, since the lengths of the defining lines may be transferred by the use of dividers. EXERCISE XVil Find the values of the following expressions : 1. cos^ a — sin^ a, when a = arctan ( — |), in the 2d quadrant. 2. • H , when a = sec-i(— 3), in the 3d quadrant. 1 — tan a 1 — cot ct 3. '- -, when a = arcsin(— 14), in the 4th- quadrant. tana - sec a + 1 \ o^/ ^ 4. 1 , when a = cos-^M, in the 4th quadrant. CSC a — cot a esc a + cot a Solve the following equations, finding all the angles less than 2 TT that satisfy each equation : 5. cos/? =^. 6. tan y8 = - Vs. 7. sin 2 a = - ^V^. 8. cot 3 a =: 1. 9. 4 sin2 a — 4 cos a — 1 = 0. 10. 3tan2y8-l = 0. 11. 2 sin ^ cos ^ - sin ^ = 0. 12. 2 sin a + \/3 tan a =0. In exercises 13-24, verify the given identities by transforming the first member into the second. 13. (sin a + cos a) (cot a + tan a) = sec « + esc a. 14. (sec a — cos a) (esc a — sin a) = sin a cos a. 15. ^^"^ + ^^^^ ^tan«cot)8. cot a + tan ft ' PERIODICITY 83 16. (r cos Oy -{- (r sin cos <^)2 + (r sin ^ sin y = r\ ^_ tan a — tan y8 ^ cot a cot ^ + 1 _ ^ 1 + tan a tan ^ cot y8 — cot a 18. CSC a (sec a — 1) — cot a (1 — cos a) = tan a — sin a. 19. (sin a cos 13 — cos a sin ^)^ + (cos a cos ^ + sin a sin ^)2 = 1. 20. sec ct CSC a (1 — 2 cos'^ a) + cot a = tan a. 21. (sin a cos /? + cos « sin f^)'^ + (cos ot cos /? — sin a sin I3y = l. oo o 2 (l-tan2a)2 . » 22. sec- a csc^ ct — ^^ ^ = 4. tan-^ a 23. (cos a + V— 1 sin a) (cos a — V — 1 sin a) = 1. 24. (cos a + V— 1 sin a)^ + (cos a — V— 1 sin «)2 z= 4 cos^ a — 2. 25. By means of Fig. 60 show that, when is acute and measured in radians, sec > tan 0>0> sin 0. 26. By means of Fig. 60 show that, when 6 is acute and measured in radians, esc ^ > cot ^ > ( '^ — ^ j > cos ^. 61. Periodicity of the trigonometric functions. It was pointed out, in Art. 53, that if two angles differing by an integral multiple of 360° are placed on the axes, their terminal lines coincide. As an immediate consequence, it follows that corresponding functions of the two angles are identical. Thus we may write sin (2 kir -^ a) = sin a, and, in general, F(2kiT -\- a) = jP(a), where F denotes the same function in both members of the equa- tion, and k is an integer. 62. Functions of k^±a . Precisely as in Art. 10, 37, and 38, we may express the functions of the angles ± a, 90° ± a, 180° ± a, 270° ± a, 360° ± a, and other similarly compounded angles in terms of the functions of cc, no matter what the quadrant of the angle a. Because of the periodicity brought out in the preced- ing article, it is not necessary to carry the investigation beyond the five multiples of the right angle mentioned ; indeed, the fifth reduces to the first. On account of the double signs and the pos- sibility^ of a belonging to any one of the four quadrants, there exist thirty-two distinct cases. The demonstration is tlie same 84 THE GENERAL ANGLE for all cases, involving the same proportionality of sides of similar triangles and the same question of agreement or opposition of signs. The working out of the proof in three characteristic instances should be sufficient to enable the student to do the same for any and all cases. The theorem is, however, somewhat elusive, and the student can completely master it and render it an infallible instrument only by actual careful construction and proof of most of the cases. Upon first study it may be well to limit considera- tion to the cases in which a is of the first quadrant. Let it be required first to express the functions of (180° -f- a) in terms of functions of a, when a is an angle of the first quad- rant. If, in Fig. 64, Z XOA = a, thenZXOB=/3=lS0°-\-a. The two triangles OMP and OJSfQ are similar, the pairs of correspond- ing sides being v and v', x and x\ and y and y. Notice also that x^ and ?/' are negative, all the other sides being positive. Giv- ing due attention to signs, we Fig 64. may write : sin (180° + a) = sin /3 = y sin a, cos (180° + a) = cos /5 = = — cos OJ, V tan (180°+ a) =tan /5 = ^ = ^ = tan a, cot (180° + a) = cot /3 = - = - = cot «, y y sec (180°+ a) = sec /? sec a, esc (180° + «) = CSC /3 = ^ y - = — CSC a. y Again, let it be required to express the functions of (270® — a) in terms of functions of «, when a is of the first quadrant. In Fig. Qb, Z XOA = «. Z XOB = /3 = 270° - a. The two triangles OMP and ONQ are similar, the pairs of corresponding sides now FUNCTIONS OF [..|±«] being v and v\ x and y\ and y and x\ The sides x^ and negative, all the others positive. We may then write; sin(270°-«) = sinyS = cos(270°-a)=cos)8 = tan(270°-«)=tan^ = cot(270°-«)=coty8 = sec(270°-«)=secy8 = csc(270°-a) = csc,/S = As a third and especially important instance, let us find the functions of — a, when a is of the second quadrant. In Fig. m, XOA = «, XOB= /3= - a. The two triangles OMP and ONQ are similar, the pairs of correspond- ing sides being v and v', x and x\ y and y\ while a;, a;', and ^' are ^ negative. We then have, as before : p^^ ^g sin ( — a ) = sin ,8 = ^ = — ^= — sin a, 85 are y v' - X V ~ - COS a. x' v' - V - sin a. y' x' x y = cot a. x' y' = y X = tan a. v[ _ V _ - CSC a. x' y v' y' = — V _ X • sec a. X X cos ( — a) = COS ^ = — = - = cos a, v' V tan ( — a) = tan /9 = '-^ cot ( — a) = cot yS = — = tana. — cot a, sec ( — a ) = sec ytj = — = - = sec a, CSC ( — a) = CSC /3 = -r = = — esc a. v' X 86 THE GENERAL ANGLE It will be noticed that whenever the number of right angles involved is even the pairs of corresponding sides are v and v\x and x' ^ y and y^ ; while whenever the number of right angles is odd the pairs of corresponding sides are v and v\ x and y\ y and x' . Thus we have the theorem : Any function of an even number of right angles plus or minus a is numerically equal to the same func- tion of a; any function of an odd number of right angles plus or minus a is numerically equal to the cor- responding co-function of a; the agreement or opposition of signs is to be determined from the quadrants of a and of the compound angle. It may easily be verified that in all cases this agreement or opposition of signs is the same as when a is of first quadrant. The general theorem may also be stated as follows : If the sum or difference of two angles is an even number of right angles^ the functions of the one are numerically equal to the same functions of the other. If the sum or difference of two angles is an odd number of right angles^ the functions of the one are numerically equal to the corresponding co-functio7is of the other. The agreement or opposition of signs is to be determined from the quadrants of the two angles. The significance of the theorem is made clear by application to an example: Required to find the value of cos (810° + a). Here 810° = 9 x 90°, an odd number of right angles. When a is considered as of the first quadrant (and its functions consequently positive), the compound angle (810° + a) is of the second quadrant and hence its cosine is negative. The required relation is, there- fore, cos (810° + a) = — sin «, which holds for all values of a. Again, to find the value of tan 1230°. We have 1230° = 14 X 90° - 30°, and is of second quadrant. Then tan 1230° = _ tan 30° = - — . V3 The student may, if he prefers, construct the figure and proceed as in the demonstration just given. FUNCTIONS OF Fa; • | ± a"] 87 As a consequence of these relations, it follows that to every inverse function correspond two angles, lying between and 2 tt. Thus arc sin a = a and ir — a^ arc cos h = a and 2 tt — a, arc tan c= a and tt + a, arc cot d= a and tt + «, arc sec e = a and 2 tt — «, arc esc/ = a and tt — a. These statements should be verified by the student. EXERCISE XVIII Express in terms of a positive angle less than 45° : 1. sin 700^ 4. cot - 35°. 2. cos 260°. 5. CSC 930". 3. tan436^ 6. sec 1400°. Find the value of cos a -}- sin a and of tan a — cot a when a has the value 7. ?. 10. '^'^. 6 (j 8. -2j-. 11 iijr 3 ■ 3 * 9. 1^. 12. -^. 4 4 Find all the values between 0° and 360° of 13. arctaii V3. 16. arcsec 2. 14. csc-i(- V2). 17. arccot(-l). 15. arccos (- .5). 18. sin-i (- ^ V3). Find the value of 19. sin 480° sin 690° + cos ( - 420°) cos 600°. 20. tan 840° cot 420° + tan (- 300°) cot (- 120°). 21. tan llr tan ii5 + cot ( - 11^) cot ( - t'^) . 22. sin 1|^ cos ( - ^) - sin If cos ( - ^ 88 THE GENERAL ANGLE 23. If sin 200° 30' = - .35, find cos 830° 30'. 24. If tan 558° 26' = i, find cot 468° 26'. 25. If cot 520° = - a, find sin 160°. 26. If cos 590° = - m, find tan 850°. 27. Express cos (a — 90°) as a function of a. 28. Express sin (a — 180°) as a function of a. 29. Express tan (a — 360°) as a function of a. 30. Express cot (a — 270)° as a function of a. CHAPTER VIII FUNCTIONS OF TWO ANGLES 63. Formulas for sin (a + p) and cos (a4- p). Suppose a and y8 to be acute angles. In Fig. 67 (ct + /3) is acute ; in Fig. 68 (a + yS) is obtuse. The following demonstration applies to both figures. Let ZXOA = a, ZAOB = 0', then ZX0B = a + /3. From P, a point in 0J5, draw Pil[f perpendicular to OX, PQ perpendic- M N Fig. 67. >X Fig. 68. ular to OA^ and from Q draw §iV perpendicular to OX, and QR perpendicular to MP. The angle RPQ == a and PP = QP cos a, PQ =z QP sin a, by Art. J6. By the same article. MP=OP s'm (a + /3). Also MP = MR + RP = NQ-\-RP = 0§ sin a + ^jP cos a = OP sin a cos yS + OP cos a sin y8. Equating the two values of MP and dividing through by the common factor OP, we have the theorem sin (a + p) = sin a cos P + cos a sin p. 80 0) 90 FUNCTIONS OF TWO ANGLES In like manner Oi)^f= OP cos («+)S), and also OM=^ON-MN= ON- RQ = OQ cos a — QP sin a , = OP cos a cos /S — OP sin a sin p. Hence the companion theorem cos (a + p) = cos a cos p — sin a sin p. (2) These are called the addition formulas and are fundamental in trigonometry. 64. Extension of addition formulas. The two formulas of the last article were proved only for angles both of the first quadrant. It remains to be shown that they hold when a and /S denote any angles. First, let a be an angle of the second quadrant. Then (^ (= a — 90°) is an angle of the first quadrant. Now a = 90° + <^, so tliat, by Art. 38, sin a = cos <^, cos a = — sin + /S) = cos (^ COS /3 — sin sin /S, = cos a cos yS — sin a sin y8. The formulas are therefore true when one angle is of the first and the other of the second quadrant. By adding 90° succes- sively to each of the angles, the formulas are established for two positive angles of all quadrants. If one of the angles is negative, it can be augmented by such an integral multiple of 360° as to produce a positive angle possessing the same functions. The addition formulas are, therefore, true for angles of any size. ADDITION AND SUBTRACTION FORMULAS 91 EXERCISE XIX Evaluate the addition formulas for 1. a = 60°, /? = 30°. 3. a = 240°, 13 = 150°. 2. a = 45°, /? = 90°. 4. a = 300°, ^ = 150°. . 5. « = arctanf, ^ = arccos (— -^j), a first quadrant, ft second. 6. a = sin-i (— j^^), /3 = cot-^ 5^, a fourth quadrant, /? third. Find the value of 7. cos (^ + a) cos (1 + /«) - sill (| + «) sin (| + )»). 8. sin (7 + a) cos(^+/8) + cos (| + a) sin (| + ysV 9. sin (1 + n) a cos (1 - n) « + cos (1 + n) « sin (1 — n) a. 10. cos (1 + n) a cos (1 — n) a — sin (1 + n) a sin (1 — n) a. 11. sin (^ + <^) cos {6 - ) + cos (^ + <^) sin (^ - <^). 12. cos (^ — <^) cos ^ — sin (^ — <^) sin <^. 13. Evaluate the addition formulas for a = 60°, ^ = 45°, and thus find sin 105°, cos 105°, sin 15°, cos 15°. 14. Evaluate the addition formulas for a = 45°, ^ = 30°, and thus find sin 75°, cos 75°, sin 15°, cos 15°. 65. Subtraction formulas. In the addition formulas replace /3 by — fi. We have sin (a — yg) = sin a cos (—/?) + cos a sin (— yS). But by Art. 62, sin (— y8) = — sin y5, cos (— jS') = cos y5. Making this substitution, we have sin (a — P) = sin a cos p — cos a sin p. (1) In like manner cos (a — yS) = cos a cos (— yS) — sin a sin (— yS), or, by the same substitution, cos (a — p) = cos a cos p + sin a sin p. (2) 92 FUNCTIONS OF TWO ANGLES 66. Formulas for tan (a ± p), cot (a ± p). From Arts. 59 and 63 we have . . , a\ sin (a 4- yS) tan(a + /3)= ) ^ ^( cos (« + /3) _ sin a cos /8 + cos a sin /3 cos a cos yS — sin a sin yS sin a cos /3 cos « sin ^ cos a cos /8 cos a cos /S cos a cos yS _ sin a sin /S cos ct cos ^ cos ct cos ff or, finally, -I ^ . ON tan a + tan B ^^ ^ tan (a + P) = ;; ^ • (1) ^ ^^ 1-tanatanP ^ ^ In like manner we may derive X / o\ tan a — tan B ^^;. tan (a — P) = -^- (2) ^^ 1 + tanatanP ^ ^ cot(« + ^)=22^fi±|l, Sin (ct + yS) _ COS a cos /6 — sin a sin /3 sin a cos fi + cos a sin yS' cos ct cos /3 sin cc sin y8 __ sin a sin y8 sin a sin ^ sin a cos ^5 cos a sin yS' sin a sin jS sin a sin /3 Again, or Likewise ^ ^^ cot p + cot a ^ ^ J. r ox cot a cot p + 1 ^ . ^ cot (a — P) = ~ ^ — - • (4) ^ cot p - cot a ^ ^ EXERCISE XX 1. Demonstrate geometrically the formula for sin (« — y8), when a > (3^ both acute. 2. Demonstrate geometrically the formula for cos (a — (3), when a > /?, both acute. FUNCTIONS OF 2 a AND ^ 93 Evaluate the formulas of Art. ^^ for 3. a = 60°, y8 = 120°. 5. a = arcsin ^V, ^ = arctan - ^f . 4. a = 240°, y8 = 150°. 6. a = cot-i |^, ^ = cos-i ||. Evaluate the formulas of Art. ^^ for 7. a = 330^ ^ = 150°. 8. a = 210°, )8 = 300°. 9. a = cos-i y5_, ^ :^ tan-i ( - f ). 10. a - arccot \\, /? = arctan f Q. 11. Find the functions of 15° by putting a = 45°, (3 = 30°. 12. Find the functions of 15° by putting a = 60°, /S = 45°. Show that 13. sin (a + (3) sin (a- /3) = sin-^ a — sin^ (3 = cos^ /? — cos^ a. 14. cos (a + /3) cos (a - yS) = cos^ a — sin^ ^ =: cos^ (3 — sin^ a. Expand by successive applications of the formulas : 15. sin(« + /?+y). 17. tan(« + ^ + y). 16. cos(a + j8 + y). 18. cot(a+^ + y). Show that 19. sin(^+ « J - sin f "^ - a j = sina. ( - — a y = \/3 cos a. 6 ■ / ■ \6 / 67. Functions of twice an angle. In the addition formulas of Arts. 63 and 6Q, place jS = a. We then obtain sin 2 a = 2 sin a cos a, (1) cos 2 a = cos^ a — sin^ a, (2) = l-2sin2a, (2 a) = 2cos2a-l. (2?0 . n 2 tan a ^ox tan2a = - —- (3) 1 — tan^ a . o cot^ a — 1 ^ , X cot 2 a = ~ (4) 2 cot a 68. Functions of half an angle. From Art. 67 we may write cos 2 /3 = 1 - 2 sin2 /8, 94 FUNCTIONS OF TWO ANGLES and solving for sin/3, sin ^ = Vi(l-cos2yS). Now placing 2 /S = a, so that y^ = ^, we obtain sin I a = Vi (1 - cos a). (1) Similarly cos 2 yS = 2 cos2 y8 - 1 ; so that cos y8 = Vi(l + COs2y8), and, with the same substitution, cos I a = Vi(l + cosa) . (2) Dividing the first formula by the second, we get . 1 ^ /I — cos a >-Q\ taii-a = ^— , (d) 2 ^ 1 + cos a and inverting, .1 ^ /I + COS a ^^N cot a = X|- (4) a ^ 1 — cos a Rationalizing the numerators of the last two formulas, we get other useful forms, ^-^^ 1 1 — cos a ,r>, tan -a = -. , (5) 2 since ^^i. 1 1 4- cos a y^«>, cot - a = — : (6) 2 sin a Query. — Why are formulas (1) to (4) ambiguous in sign, while (5) and ((>) are apparently not ? EXERCISE XXI Find the values of 1. The functions of 60° from those of 30°. 2. The functions of 120° from those of 60°. 3. The functions of 75° from those of 150°. « 4. The functions of 15° from those of 30°. Find the values of the functions of 5. 2 arctan :f%. 8. I arctan \\l, 6. 2 cos-i (— j\). 9. arcsin ^^ + 2arccot f. 7. J sin-i ( — M)' ^0- arctan ^^ — 2 arccos f . CONVERSION FORMULAS 95 Transform the first member into the second : 11. l+sin^-cos2^ ^^^^^^ cos ^ + sin 2 ^ 12. l+cos^ + cos2^^^^^^ sin 6 + sin 2 6 13. (VT+sina+ Vl — sin a)2z=4cos2ia. 14. ( Vl + sin a — Vl —sin a) 2 = 4 sin^ i a. 15. tan f - + a J - tan f - - a J = 2 tan 2 a. 16. cot [- + a^ - cot [ - - a] = - 2 tan 2 a. Find the values of a which satisfy the following equations 17. (2 + V3)(l-sin2a) -2cos22a = 0. 18. sin 2 a + 2 cos 2 a = 1. 19. 4 sec2 2 a + tan 2a = l. 20. CSC 2 a + cot 2 a = 2. Show that 21. tan-i ^^^ = cos-i ^ 3 Va;2 - 4 a: + 13 2 a: + — — ^i^:^^::^^::::^ — arccsc — --- y/x'^ + 2 x - 3 2 9 a: + 1 22. arctan — " — arccsc - 23. Find sin (^- 2 tan-i '\-- \-x \ l+xj' 24. Find sin fsin-i m 4- tan"!^^-^^ — ^V \ ml 25. Find sin f arccos (1 — a) — 2 arctan -^ — ^^ — | • 26. Find cos (arccos (1 — 2 a) — 2 arcsin Va). 69. Conversion formulas for products. Adding the two first formulas of Arts. 63 and 65, we have sin (« + ,5) + sin (« — yS) = 2 sin a cos /3, or, reversing and dividing by 2, sin a cos p = 1 [sin (a + p) + sin (a - p)] . (1) If we subtract, instead of adding, we get sin (a + yS) — sin (a — /S) = 2 cos a sin /3, or cos a sin p = i [sin (a + P) - sin (a - p)]. (2) 96 FUNCTIONS OF TWO ANGLES Treating the two second formulas in like manner, we obtain cos a COS p =1 [cos (a + p) + cos (a — p)], (3) and smasmp = -l [cos(a + p) -cos(a- P)]. (4) By means of these formulas, products of sines and cosines are expressed as sums or differences. By successive applications higher powers and products are reducible to expressions linear in sines and cosines. The same transformations may often be effected by application of the formulas of Art. 67, written in the form sin a cos a = | sin 2 a, (5) sin^ a = J (1 — cos 2 a), (6) cos2a= I (l + cos2a). (7) EXERCISE XXII Reduce the following products to linear expressions : 1. sin 5 a cos S (^. 6. sin a cos^ u- 2. cos 6 a sin 4 cc. 7. cos^ a. 3. sin 7 a sin 3 a. 8. sin^ a. 4. cos 2 a cos 5 a. 9. cos^ a sin^ a. 5. sin^ a cos a. 10. sin^ a cos^ a. Show that 11. cos a sin (/? — y) + cos /S sin (y — a) + cos y sin (« — ^) = 0. 12. sin(^ - y) sin (« - 8) + sin(y - a) sin (/3 - 8) + sin (a-/3) sin (y- 8) = 0. 13. sin — cos^+ sin^cosi^ = 0. 5 5 5 5 14. 2 cos^cos^ + sin^ + cos^ = 0. 8 4 8 8 Solve for a, making use of Art. 10. 15. COS (50'^ + a) sin (50° -a)- cos (40° - a) sin (40° + a) = 0. 16. sin (70° + a) sin (70° - a) + sin (20° + a) sin (20° - a) = 0. Solve for a, making use of Art. 69. 17. cos 3 a + cos 9 a = 0. 18. sin 5 a — sin 10 a = 0. CONVERSION FORMULAS 97 19. cos (a + 0) cos (a- 6) + cos (3 a + 6) cos (3 a - 6) = cos 2 0. 20. sin (a + 6) cos (a - ^) + sin (3 a + ^) cos (3 ct - ^) = sin 2 0. 70. Conversion formulas for sums and differences. In the process of deriving the formulas of the last article, before revers- ing and dividing by 2, substitute a + /3 = ^, a— (3 = 6^ so that We then obtain the following formulas : sincj) + sin 6 = 2 sin ^ ^^^ T '> (^) sine))- siii6 =2cos^ sin ^ , (2) coscf) + cos6 =2 cos^-— — cos^— — , (3) cos<)) — cosG = - 2sin^-- — ^i^^^^ — (^) These formulas serve to effect transformations converse to those mentioned in Art. 69. 71. Multiple angles. In the formula for sin (a + /3) put y8 = 2 a. Then sin 3 a = sin a cos 2 a + cos a sin 2 a = sin a — 2 sin^ ce + 2 sin a cos^ a = 3 sin a — 4 sin"^ a. Again, cos 3 a = cos a cos 2 a — sin a sin 2 a = 2 cos^ a — cos a — 2 sin^ « cos a = 4 cos^a — 3 cos a. In like manner the other functions of 3 ct and, by repeating the process, the functions of any integral multiple of a may be ex- pressed in terms of functions of a. EXERCISE XXIII Show that 1. ?i5A«±^iEij? = tan5a. cos 6 a + cos 4 a 2 co^3^t-_cos^^^^^^^ sin 3 a + sin 5 a 98 FUNCTIONS OF TWO ANGLES « sin 7 « — sin 5 a , 3. = tan a. cos 7 a + cos 5 a . cos 4 a — cos 2 « j. o 4. - — : — -— = - tan 3 a. sin 4 « — sm 2 cj 5 sina-sin^^^^^^gM:^^^^a-^^ sin ct + sin |8 2 2 6 cos« + co^^_^^^«+^^^^«-^ cos a — cos )8 2 2 ^ cos 3 ^ + 2 cos 5 ^ + cos 7 _ , ^ ^ sin 3 ^ + 2 sin 5 ^ + sin 7 ^ ~ 8 sin ^ - 2 sin 4 ^ + sin 7 6 _. .q ' cos ^ - 2 cos 4 ^ + cos 7 ^ ~ Solve the following equations : 9. cos 6 + cos 5 ^ = cos 3 ^. 10. sin ^ + sin 5^ = sin 3^. 11. sin 2 ^ + 2 sin 4 ^ + sin 6 ^ = 0. 12. cos3^ + 2cos4^+cos5^ = 0. Derive the formulas for : 13. cot 3 a. (In terms of cot a.) 14. tan 3 a. (In terms of tan a.) 15. sin 4 a. 16. cos 4 a. Solve the equations : 17. sin 3 a = V2 sin 2 a. 18. V3cos3a + 2sin2a = 0. 19. cos 3 a = cos a cos 2 a. 20. sin 3 a = sin a cos 2 a. CHAPTER IX ANALYTIC TRIGONOMETRY The foregoing chapters constitute an introduction to the elementary principles of trigonometry. The student ought now to be prepared for a more advanced study of the theory of the trigonometric functions, which may be entitled analytic trigonometry. It is beyond the scope of this book to consider more than a few of the most important topics which might be discussed under this head. For a more extended treatment the student is referred to the treatises by Henrici and Treut- lein, Hobson, Lock, Loney, Todhunter, and others, and, of course, to articles in the various mathematical journals. 72. Limits of 6/sin 6 and 6/tan 9 as 6 approaches zero. Let 6 be an acute angle measured in radians. Con- struct, as in Fig. 69, the angle XOP = ^, repeated symmetrically as XOQ. Draw through P the arc PAQ with center 0, the chord PMQ, and the broken or double tangent PTQ, Then AP OP = 6, MP OP = sin 6^ TP OP = tan 6, By elementary geometry, PMQ < PAQ < PTQ. Whence, dividing by 2 and by OP, sin S < e < tan 6. Dividing equation (1) through by sin 6, wo have e (1) 1 < sin e < sec 6. Now in Art. 12 it was proved that as 6 approaches the limit 0, cos B and its reciprocal sec 6 approach the limit 1. Thus, the value of ^/sin 6 is always intermediate between 1 and a number that approaches the limit 1, as approaches 0. The ratio ^/sin 6 99 100 ANALYTIC TRIGONOMETRY must, therefore, approach the limit 1 at the same time. This is expressed symbolically by writing Again, dividing equation (1) through by tan 0, we get cos e < — ^ < 1. tan u Now as approaches 0, cos 6 approaches 1, and hence, as before, O/tand approaches the limit 1 at the same time. Symbolically, 0=0 Vtan 0/ Note. — Since sin 6 and tan d both approach along with 6, it might seem that they therefore approach equality, and then the theorems would follow. The fallacy of assuming that the limiting form - has the value 1 will appear on considering the following instances. The circumference and area of a circle approach zero simul- taneously with the radius. We have, however, the general relations Circumference 2 Trr ^ ^ ^ 6.28318 ••., Radius r Area _ irr^ Radius r = Trr = 3.14159 ...r. Now when r approaches the limit 0, the limit of the first ratio is the constant 2 tt, and the limit of the*second ratio is 0. The limiting form - will be discussed at length in calculus. (See Townsend and Goodenough's ''First Course in Calculus," Art. 13.) Example. If is increased by an angle S, let it be required to determine the limit of the ratio of the consequent increase in sin 6 to the increment 3 of 6, as that increment 8 approaches zero. By Art. 70, we have 2cosf(9 + |^sin I s in ((9 + 3) -sin (9 _ V 27 2 = cos (6 + sin - S\ 2 2/ 8 2 De moivrp:'S theorem lOi Now when B approaches 0, cos ( ^ + ^ ) approaches cos 6 and . 8 sm- — -— approaches 1. o 2 Hence lim sin ((9 + 3) - sin 3 ^ ^^^ ^ 6 = g It will be noticed that the numerator and denominator approach simultaneously, but that the limit of the value of their ratio is a number somewhere between — 1 and + 1, and depending upon the value of 0. Examples In like manner find the limits as 3=0, of , cos (6 + 8) — cos 8 n, sec (0 -\- 8) — sec ^c< -r« • j. r • \ 2. ^^ — ^^—^ • (Suggestion. Express in terms of cosme.) « CSC (0 -\-S) - CSC $ 3. U 4 tan (0 + 8) — tan (Suggestion. Express in terms of sine and 8 cosine.) 5 cot (^ + 8) -cot^ 73. De Moivre's theorem. If we adopt the customary nota- tion z = V— 1, so that P = —1, we have, on performing the mul- tiplication, (cos a+ i sin a) (cos /3 -f ^ sin /3) = cos a cos /3 — sin a sin ^ + { (sin a cos /3 4- cos a sin /3) = cos (« -1- /3) + « sin ((X + /3), (1) a relation which holds for all values of a and /3, whether positive or negative. Putting y8 = a, we get (cos a-\-i sin a)^ = cos 2a + i sin 2 a. Again, putting /3 = 2 a in equation (1) and making use of the relation just established, we get (cos a+ i sin a)^ = (cos a-{-i sin cc) (cos 2a + i sin 2 a) = cos 3 a -f i sin 3 a. 102 ANALYTIC TRIGONOMETRY Repetition of this process proves the relation (cos a H- ^ sin a)" = cos na + i sin na (2) for all positive, integral values of n. It, is evident^ ypon multiplying, that ''"'' rcr.«. '(^QQs ^_|_ ^- gii^ ^^(^cosa — z since) = 1, whence (cos a 4- i sin a)~^ = cos a — i sin a. Suppose 71 to be a negative integer. Let n = — m^ where m is a positive integer. Now (cos P — i sin /3)""' = (cos yS + ^ sin yS)"" = cos myS + ^ sin m^. Substituting m = — n and yS = — a, we get (cos a + z sin «)" = cos Tia + i sin Tia, true also for negative integral values of n. Suppose 71 to be a fraction, either positive or negative. Let 71 = -, where r and s are integers. Now r 1 (cos yS + z sin y8) * = (cos ryS + ^ sin rp~) ' . Raising both members to the sth power, r (cos Sy8 + i sin s/3) * = cos r/3 + i sin ry8. Introducing - = ti, and putting sfS = a, so that r^= - - s^=na, s ' s . we get (cos a + ^ sin a)" = cos 7i« + i sin 7i«. This relation, therefore, holds for all rational values of n. By an argument involving the method of limits it can be proved also for all irrational values of n. This is De Moivre's theorem, an instrument of great importance in some branches of mathematics. Example. An illustration of its use is afforded by applying it to tlie derivation of the formulas for the sines and cosines of multiple angles. Thus cos 3 a -h ^ sin 3 a = (cos a -f- 1 sin a)^ = cos^ a 4- 3 ^ cos^ « sin « — 3 cos a sin^ a— i sin^ a. COMPLEX NUMBERS 103 On equating the real terms on each side, and also the imagi- nary terms, separately, we have at once cos 3 a = cos^ a — 3 cos a sin^ a = 4 cos^ ct — 3 cos a. sin 3 a = 3 cos^ « sin a — sin^ a = 3 sin a — 4 sin^ a. The functions of 4 « and of higher multiples of a are as readily found. The simplicity and beauty of the method appears on comparison with that of Art. 71. Examples 1. Show that cos«+^'s;"« ^ cos («-/?) + / sin (« - S). cos (3 +isin^ 2. Show that ( cos "^ ^ 1- i sin ^ — ) = cos cc + i sin a. \ n n J 3. Show that f cos ^ ^^ "^ " + f sin - ^'^ + ^ V = cos a + « sin a, where /[: is \ n n J any integer. 2 y^TT 4- C£ 4. Show that the angle — — ~ — has n different values as k takes the suc- n cessive values, 0, 1, 2, • • • n — 1 (n being a positive integer). Show also that for all integral values of k outside these limits, the terminal sides of the angles coincide with those of the n angles already found. 5. Since cos -\- i sin = 1, find the ?i different nth roots of 1, of which all but one are imaginary. Making use of the tables of natural sines and cosines compute for n = 2, 3, 4, 6. 6. Since cos tt + « sin tt = — 1, find the n different nth roots of — 1, of which all but one are imaginary when n is odd, and all imaginary when n is even. Compute for n = 2, 3, 4, 6. 74. Graphical representation of complex numbers. An interest- ing application of De Moivre's theorem is found in the graphical representation of complex numbers, devised by Wessel, a Danisli mathematician, and published by Argand in 1608. The treatment of this topic belongs rather to the courses in algebra and function theory. (See Rietz and Crathorne's "Algebra.") Only so much of the rudiments of the method will be developed here as possess a trigonometric interest. A pure imaginary is an indicated square root of a negative number. A complex number is an indicated sum of a real number 104 ANALYTIC TRIGONOMETRY and a pure imaginary. All pure imaginaries can be expressed in the form «/^, and all complex numbers in the form x + yi. Here ^ = V— 1, so that z^ = — 1 ; while x and y are real numbers, either rational or irrational. Argand's method makes use of a pair of mutually perpendicu- lar axes. The Argand diagram must not, however, be confused with the Cartesian scheme of coordinates. All real numbers, rational or irrational, are represented by dis- tances from the origin to points in the horizontal axis, called now the axes of reals, positive to the right, negative to the left. To every real number corresponds a point in this axis, and conversely, to every point in this axis corresponds a real number. Thus there is said to be a one-to-one correspondence between the totality of real numbers and the totality of points in the line. All pure imaginaries are represented by distances from the origin to points in the vertical axis, now called the axis of imagi- naries, points above and below the origin giving, respectively, positive and negative coefficients for the imaginary unit factor i = V— 1. Here again there exists a one-to-one correspondence between the totality of pure imaginaries and the totality of points in the vertical axis. Notice that the origin alone, of all points in the plane, is on both axes. The number zero belongs to both systems. With this single exception, no pure imaginary can equal a real number, since the directions of the two axes are essentially different. In order to represent the complex number x + yi recourse must be had to the method of adding coplanar but non-collinear directed line segments employed in the graphical composition and resolution of forces in physics. Since directed line segments may undergo translation, the segment yi may be placed with its initial point upon the terminus of the segment x. The complex number is therefore represented by the right line segment (radius vector) v from the origin to the resulting terminus of the segment yi. For y—^ we have real numbers, for a: = we have pure imaginaries. As the lengths of the horizontal segment x and the vertical segment ^^ measure respectively the magnitudes of the reals and the pure imaginaries, so the length of the radius vector v may be said to measure the absolute magnitude of the complex number v — x-\- yi. This is called the absolute or numerical value of v, and is denoted by the letter r. Evidently all points on the unit circle about the origin possess the absolute value 1. COMPLEX NUMBERS 105 The directed line segment, or radius vector, v makes in general an oblique angle with the axis of reals, and its direction is deter- mined by the angle it forms with the positive axis of reals. This angle is denoted by ^, and is called the amplitude of the complex number. All points lying on the same radius have a common amplitude, while radii vectores extending from the origin in opposite directions have amplitudes differing by tt. All positive real numbers have the amplitude ; negative reals, ir ; pure TT 3 TT imagmaries, — or -— -. The right triangle formed by a;, ^, and v yields the relations 6 = arctan-1 X \Y x = r cos ^, 3/ = ^ sin 6, We may write interchangeably, v^ ov x + yi^ or r (cos ^ + ^ sin ^). The expression cos 6 -\-i sin d consequently denotes a unit segment (complex unit) with the amplitude ^, while r is a purely arith- metical factor. Conjugate complex numbers, x -f yi and x — y% evidently have the same absolute value and amplitudes which are negatives of each other. Addition is effected graphically by placing the initial point of the second segment upon the terminus of the first and con- necting the initial point of the first to the terminus of the second. Thus in Fig. 70, = ^1 + %i + ^2 + % = (^1 + ^2) +^'(^1 + ^2)- The values of r and d in terms of /-j, r^^ 6^ and 0^ are readily deter- mined, but exhibit little of present interest. Suffice it to point out that r < rj + 9-2, e^e^ + e^. Subtraction reduces at once to addition on reversing the sub- trahend segment. Fig. 70. 106 ANALYTIC TRIGONOMETRY On attacking the problem of multiplication, we must define the product of a directed rectilinear segment by the imaginary unit { as a segment of equal length turned through a positive right angle. Thus v = x-{-iy =r (cos 6 -\-i sin ^) multiplied by i gives = — y -\-ix=r cos ( ^ + ^ ) + 2 sin ( ^ + ^ j • 2 7 V2 The absolute value is unchanged, while the amplitude is in- ir creased by — • This is consistent with the original scheme of rep- A resentation, since reals multiplied by i give pure imaginaries, and these multiplied by i give — 1 times the original, i.e. the original radius vector reversed. Multiplying a directed segment by a positive real number simply stretches it, multiplying its length and leaving its direc- tion unchanged. Multiplying V =x-\-iy = r (cos -\-i sin ^) by k^ we get v' = hv = kx -\- iky = kr (cos -\-i sin ^). The absolute value is multiplied by the factor k, while the ampli- tude is unchanged. Multiplication of one complex number by another is effected by combining the two processes just described, applying the asso- ciative and distributive laws. Thus v = v^'V^= (^1 H- iyi) ' (a^2 + ^^2) = ^1 (^2 + ^>2) + *>i (^2 + %) = (.V2 - ViVi) + ^^1^2 + ^2^1) • Using the other notation and applying De Moivre's theorem, v = v^ • V2 = r^ (cos ^j + i sin ^j) • r^ (cos 0^ + i sin 0^^ = r^r^ ' [cos ((9i + 0^-) + i sin ((9^ -hO^)-]. Figure 71 illustrates the multiplication of 5 — 2 ^ by 2 4- 3 ^. The product is shown to be 16 + 11 i. We have then the law that the absolute value of the product of two complex numbers equals the product of their absolute values, while the amplitude of the product equals the sum of their amplitudes. COMPLEX NUMBERS 107 The inverse process of division is readily performed, with the result or V = -1 = ^2 X, + ly^ _ x^x^^ + y^y^ . x^y^ - x^y^ ^2 + ^Vl ^2 + ^2 r^(cos 0^ + ^ sin Q^ r^ (cos ^2 + ^ sin ^2) ' + ^ + ^2^ i; = ^ [cos (6'i - 6>2) + ^ sin ((9i - 6>2)] • ^2 The absolute value of the quotient is equal to the quotient of the absolute values, while the amplitude of the quotient is equal to the difference of the amplitudes. We have further, vz=v{' = r-^ (cos nO^ + i sin nO^. The absolute value of the power is equal to the power of the abso- lute value, while the amplitude of the power is equal to the ampli- tude of the number multiplied by the index of the power. Here " power " is used to de- note the result of affect- ing the number by the exponent n^ whatever the value of n. This includes both involu- tion and evolution. In particular let n be the reciprocal of a positive integer m. Fig. 71. V- m/ Then cos—i + m mj But Vj is just as well and exactly represented by r [cos (2 ^TT + ^) + i sin (2 kir + ^)], where h is any integer. Thus the mth root just found is only one of an infinite number, all given by the form m/-r 2kir + 0. Vr. cos =^— i I + i sin ^kir m ±6,1 ^ J 108 ANALYTIC TRIGONOMETRY in which k assumes all integral values. This form gives m dif- ferent values for the root, corresponding to A; = 0, 1, 2, ••• m — 1. All the others are repetitions of these m roots, since the terminal sides of all the other amplitude angles will coincide with the ter- minal sides of the m amplitudes specified. Hence every complex number has m different mth roots, whose common absolute value is the arithmetical mth root of the absolute value of the number, while their amplitudes have the m different values, e. ^TT + e^ 4 7r + 6>^ 2fm-l)7r + (9^ m m m m all less than 2 tt. In the special case of any positive real number a^j, whose am- plitude is therefore zero, we obtain m different mth roots with the common absolute value Vr^, which is called the principal value of Va^j, and the m different amplitudes, /^ 2 TT 4 7r Gtt 2(m — l)7r u, , , , ••• • m m m m Only one of these is real, the first, and it is called the principal mth root of the positive real number. The student should construct figures to illustrate the foregoing theorems. Still another analytic notation for complex numbers will be brought out in Art. 75. Examples 1. Represent by Argand's diagrams^the numbers 2, —3, 3i, — 4i, 3 + 5t, 4 - 3 1, - 2 + i, - 5 - 3 i, 4 + VZ^ Vs - VIT?. 2. Write the numbers the termini of whose radii vectores have the Carte- sian coordinates (3,4), (-3,2)^ (7,-3), (-5,-2), (6,0), (0,5), (-2,0), (0, -6), (0,0), (V^, V5). 3. Find the absolute values and the amplitudes (expressed in degrees and minutes) of the numbers in examples 1 and 2. 4. Describe the situation of the number points which have : (1) the common absolute value 3 ; (2) the common amplitude 30° ; (3) the amplitudes 45° and 225°. 5. Perform graphically: (3 + 40 + (7 - 2 ; (-3 + 2/) + (6 - 3 {) ; (7 - 3 - (4 + 2 i) ; (3 - 2 - (- 6 - 3 i) ; (5 + 2 i) + (3 - 4 i) - (6 - 3 {). 6. Perform graphically, taking the first factor in each case as the multi- plier : 3.(5 + 2 0; I -(3 + 50; 2 t- (6-30; -4.(2+50; -6 i . (3+ 2 0; (4+ 2 • (3 + 4 ; (3 + 4 -(4 + 20. EULER'S EXPONENTIAL VALUES 109 21 + ; 6 - 17 1 7. Construct the quotient of 3 + 2f 4-3t 8. Construct: (3 + 202; l-l^i^V-, (l-iV'dy;i\ 9. Find by construction: V7-24i; \/- 119 + 120 i; (-5 + 12i)^' ^/ITl; ^; ^16. 10, Write the general solution of the binomial equation : x" — a" = 0. 11. Find all the roots of the equations a:^ — 1 = ; x'^ -{- 1 — 0; x^ — 1 = ; a;3 - 8 = 0. 75. Exponential values of the trigonometric functions. The first form of De Moivre's theorem, Art. 73, Eq. (1), may be written symbolically, which is read, function of a times (the same) function of /9 equals (the same) function of (a + /3) ; or, the product of the (same) functions of two numbers equals the (same) function of the sum of the two nuifibers. Now this is identically the characteristic relation or law governing the exponential function, that is, a function of the form a^ ; thus. For reasons discussed in Art. 77, it is found that instead of the more general function a% we must place cos a-\-i sin a = e'% (1) where e = 2.71828183 ••• is the base of the Naperian, or natural, system of logarithms given in Art. 23. Note that the law of exponents, derived for positive integral exponents, and assumed to hold also for negative, fractional, and irrational exponents, is still further assumed for exponents which are pure imaginaries and complex numbers. As in the former cases, the significance must be determined in conformity to the action of the assumed law. Indeed, the law defines the function. Since cos a — i sin a = ;— : , we have also cos a + ^ sin a cos a—i sin a = er^°-. (2) Adding and dividing by 2, we obtain cosa = ; Ko) 110 ANALYTIC TRIGONOMETRY again, subtracting and dividing by 2 ^^ sm a = — ^j— . (4) These values were first given by Euler in 1743. Starting from these two exponential values as fundamental definitions, and de- fining further , sin rt , 1 1 1 tana = , cota = , seca = , csca = cos a tan a cos a sm a it is possible to develop all the laws and formulas of trigonometry as contained in Arts. 59 and 63-71, quite apart from any geo- metric meaning attached to the functions or their argument a. The analogous derivation of those trignometric theorems de- pendent on the periodicity of the trigonometric functions involves the periodicity of the logarithm, and is therefore postponed until the later mathematical study of the student. A third notation for complex numbers now becomes manifest.; for v = x + iy = r (cos 6 -\- i sin ^) = re^^. The consequent theorems regarding the absolute values and am- plitudes of products, quotients, powers, and roots follow readily, and should be worked out by the student. Examples 1. Find the exponential values of tan a, cot a, sec a, esc a. 2. Derive from the exponential values the laws sin^ a + cos^ a = 1, etc., of Art. 59. 3. Derive from the exponential values the formulas of Arts. 63-71. 4. Derive from the exponential notation the laws for the absolute values and amplitudes of products, quotients, powers, and roots of complex numbers. 76. Hyperbolic functions. Closely allied to Euler's forms of the last article are the two interesting and important forms. 2 2 They are called, by analogy, the hyperbolic cosine and hyperbolic sine. Thus, employing the customary notation, cosh a = , sinha= — HYPERBOLIC FUNCTIONS 111 The remaining hyperbolic functions are defined from these, just as in Art. 75 : tanh a = ^IBIL^, coth a = —1—, sech a = -—, csch a = ___. cosh a tanli a cosh a sinh a A very simple relation exists between the hyperbolic and the circular (^i.e. ordinary trigonometric) functions. Evidently cosh a = cos ^a, sinh a = — i sin m, tanh a = — ^ tan ia ; and conversely, cos a — cosh za, sin a = — i sinh m, tan a = — { tanh za. To each formula of Chapter YIII corresponds a formula for the hyperbolic functions, which may be deduced either directly from the exponential definitions, or by substituting the values just given in the formulas for the circular functions. The student should derive these formulas by both methods. The analogue to De Moivre's theorem is (cosh a -h sinh a)" = cosh na -\- sinh na. Cosh a and sinli a possess an imaginary period 2 7^^, since gM _ ^u+2kni^ jf. being any integer. (See treatises on the theory of functions.) 77. Exponential and trigonometric series. In the present article values in the form of infinite series will be derived for certain exponential, logarithmic, and trigonometric functions. In the proof, however, the use of the binomial formula and the manipulation of the series introduce a lack of rigor requiring ex- tended consideration in the subsequent courses in algebra, the calculus, and the theory of functions. (1) Exponential series. Expanding by the binomial formula,* n) ~ ll'^^ 2! ^ 3! ' n^ *The symbol \k, or k !, is used to denote the product 1 • 2 • 3 ••■ A:, where k is any positive integer, and is read " factorial A;". 112 ANALYTIC TRIGONOMETRY .2 (2) Now as n becomes infinite, the binomial factors [ 1 ), f 1 — — ), etc., all approach the common limit 1, and we shall have, in the limit, ,- = limrA+^Y1=l + ^ + ^ + |^+.... (1) n=^[\ nJ J 1! 2! 3! This series is convergent for all finite values of x. (See Rietz and Crathorne's "Algebra.") For a; = 1 we get 1,1,1.1,1, ^=^ + l! + 2!-^3! + T!-^- The terms diminish rapidly in value and, when expressed deci- mally, the value of e is found to be 2.71828183 •••. The series for e^ is valid also for negative and imaginary values of X ; thus, substituting successively —x, ix, and —ix for a;, we have * =^-T! + 2!-8T+-' /y>^ /"Y*^ /^O /y% /y»t> /y*0 /y»7 I ^ =^-2! + 4!-6! + - + ln-^ + 6!-f!+-} _,> -, x^ , x^ x^ , .r X a^ , x^ x'^ , ~] ^ ='-2! + 4!-0l+--iT!-^ + 5!-7!+-J- (2) Logarithmic series. From the expansion just obtained for e"^ can be derived a series for log, (1 + ^). Since* ^t^ = e^l«ge«, we have ^^ = 1 + ^ (log, ^) + |y (loge ^)^ + ^y (log, uy + -- -. Placing u=l + t/f (l+y)-=l + ^log,(l+y) + ^[log,(l + y)P + ^Doge(l+y)?+-- * Let w = u*. Taking logarithms to base e, we have loge w = x log« u. Now taking exponentials to base e, lo = w* = e* log*". TRIGONOMETRIC SERIES 113 Expanding the first member by the binomial formula, (1+^)^ = 1 + — y+ ^ ^^ ^ f + -^ ^y^ -f+ •••• Picking out and equating the coefficients of x in the two expres- sions, the required expansion is obtained, log«(l+2/) = f-f + |-J+-. (8) This series is convergent for — 1 < ?/ ^ + 1. If w = loga u^ we have a"' = u; wlience, taking logarithms to base g, w log^ a = logg u. Therefore w = log« u = • log. u, log, a Substituting from (3) we see that (3) Trigonometric series. From De Moivre's theorem cos mO -\-i sin mO = (cos 6 -\-i sin ^)"*. Expanding b}^ the binomial formula and- separating the real terms from the imaginary, cos mO + i sin md = cos"^ 6 - ^^^ ~ ^^ cos'^-s ^ sin2 . w(m— l)(m — 2)(m— 3) m-A a - 4. a + — ^^ — ^ cos"^ ^ a sin^ a •" 4! -f ^ f — - cos"* 1 6 sm 6 — — ^ ^ ^ cos"*-3 gm^ j^ ... \ Equating separately the real and the imaginary parts, cos mO = cos"* - ^^^"-^^ cos"*-^ sin2 (9 4- •••, sinmu==—j cos"* ^^sm^ ^^ /y^ ^cos"* ^osin^6-{- •••. Place now m$ = a, so that ^ = — ; co.„ = cos"Q-^(^cos-Qsi,.Q+..., sm 1 ! \wy Vm/ m(m-l)(m-r2) 3! -<^)^''<3^-- 114 ANALYTIC TRIGONOMETRY Now let approach the limit zero and m become infinite, while still obeying the condition that mO = a, where a remains finite. By Art. 12, cos—, cos^— , etc., approach the limit 1 as m becomes mm . . infinite, and in the calculus the same is shown for cos"*f — j, cos^^-if-Y etc. Again, m sm — = •, \mj u ■ m(m-l)sin2('"'\=a(«-6>)./'^Y, m{rn - 1) (m - 2) sin^/"-^ = «(« - ^)(« - 2 l9) • f^}^\\ etc. Since li^^^ ^ = 1, the limits approached by these expressions, as ^ = 0, are a, a\ a^, etc. Making these substitutions, we obtain, in the limit, ^ «2 «4 ^6 . cos« = l-- + jj-^j+..., (6) '^"" = r!-3!+6T-^+-- <^^> These series are convergent for all values of a. Tan a may also be expanded into the series *'^"" = i + F + i5 + W+-- ('> It will be noticed that the series for cos a contains only even powers of a, while those for sin a and tan a contain only the odd powers of a. (See the third example worked out in Art. 62.) The assumption of Art. 75 may now be justified. For, on sub- stituting for e'^, e"'^ sin a:, and cos a; their expansions in series, we obtain cos x-\-i sin x = e'^, cos ic — ^ sin a; = e~"^, and cos x — ^rx 4- e-'' 2 ^ix - e-'^ sma: 2i COMPUTATION OF TABLES • 115 (4) Hyperbolic series. From the analogous relations the expansions for the hyperbolic functions are readily obtained. ri^i 1 e'^ + e~'^ 1 , ^2 ^4 ^6 Ihus cosha = -^^=l + - + - + -+ .-, (8) Examples 1. By substituting — a for a, find the series for sin(— a), cos(— a), tan (— a), and by comparison, verify the corresponding relations of Art. 62. 2. By substituting ia for a, verify the relations of Art. 76, cosh a — cos la, etc. 3. Using two terms of the expansions for sin a and cos«, and retaining only powers of a below the fifth, obtain an approximate verification of the fol- lowing formulas : sin^a + cos^a = 1, sin (a + /?) = •••, cos (a + y8)= •••, sin 2 a= ••-, cos 2 a = •••• 4. Do as required in example 3 for the hyperbolic functions. 5. Repeat examples 3 and 4, using three terms of the series and retaining powers of a below the seventh, thus arriving at a closer approximation. 78. Computation of trigonometric tables. The numerical values of the sine, cosine, and otlier trigonometric functions of angles from 0° to 90°, as tabulated in Table III, may be calculated by means of various trigonometric formulas, or better, by the use of the series derived in Art. 77. Euler gave the following series, carrying the computation to 28 decimal places and a corresponding number of terms : Place a = m ' -^ in the series for sin a and cos a ; whence sin ^w. 1^ = 1.570 796m -0.615 964^3 + 0.079 693 m^ - 0.004 682 m'^ -f- 0.000 160 m^- 0.000 004m" + , cos C^ • I) = 1-000 000 - 1.233 700 m^ + 0.253 699 m^ - 0.020 863 m^ -{- 0.000 919 m^ - 0.000 025 m^^ + 116 * ANALYTIC TRIGONOMETRY We need to calculate the sines and cosines of angles up to 45° only, so that m is a fraction and always less than |-. The terms of the series converge rapidly and a few terms suffice to give values correct to a small number of decimal places. More extended discussion of this topic may be found in Hob- son's "Trigonometry," Chap. IX; Todhunter's "Plane Trigonome- try," Chap. X ; and other advanced treatises on trigonometry. The series for log (1 + ^) converges too slowly for convenient calculation, but a modified form is easily obtained. ^ Manifestly /2 ~ 8 log (1 - ^) = - ^ - |- ^ and log l^f = 1^'g (! + «/)- log (1 - ^) — 9 I + -8+5 + } Place y = , Avhence ^ = ; then 2i; + l 1 — y ^ log !i±i = 2^-1-+ 1 + 1 : + ...), or log (v + 1) = lo^ V -h 2r — - — H h — + •••T This series converges rapidly and by it log 2 can be computed from log 1 = 0, log 3 from log 2, etc. Logarithms of composite numbers can be checked by adding the logarithms of their factors. 79. Proportional parts. In using the logarithmic and trigo- nometric tables it Avas assumed, as stated in Art. 2^, that for small differences in the number, the differences in the logarithm are proportional to the differences in the number, and that like- wise, for small differences in the angle, the differences in the sine (or other trigonometric function, or logarithmic function) are pro- portional to the differences in the angle. We have log (a; + 8) - log a; = log^±^ = log f 1 + -^ X ' \ xj ^S_j5^ J3 ^ X 2X^ SX^ 4:X^ __ 8 (Approximately for small X values of 3.) PROPORTIONAL PARTS. INVERSE FUNCTIONS 117 Therefore, we have approximately log ix-\-8^)-lo^x ^ S^ /^ = ^1 , (13 log (x + 82) - log X hjx S2' for small differences. Again, sin (3 -}- |9) - sin (9 = 2 cos [o + -") sin | = cos Q • 3. (Approximately for small values of 3.) Hence, approximately sin (0 4- 3j) — sin \ cos ^ \ sin (^ + ^2) — sin Q h^ cos d h^ (2) For the other functions, the proof follows exactly similar lines, and can easily be supplied by the student. Full discussion along this line may be found in Loney's " Plane Trigonometry," Chap. XXX ; Lock's " Higher Trigonometry," Chap. VIII ; Hobson's " Trigonometry," Chap. IX ; etc. 80. General inverse functions. In Art. 14 only acute angles were under consideration, so that the relations m = sin a, a = arcsin m, expressed a one-to-one correspondence. In other words, under the condition that 0°<«<90°, 05m 3 T 19. Find the value of ^^^ " + ^"^ ^ ii a = arcsin - and of 2d quadrant. 1 - tan « l-cot« 5 20. Find the value of tan /? + sec y8 - 1 if ^ ^ arccsc f - — ) and of 3d quadrant. tan ^ - sec /? + 1 ^ V 6 J 21. Find the value of tan^g - sin^ct .^ ^ ^ arccot f - I] and of 4th quad- rant. '^^'^ ^ 2; 22. Find the value of 1 + cosy -2 secy -^ arctan-^ and of 1st quad- rant. 3 + cosy + 2 secy 40 23. Express ^^^^ " + ^^^^ ^ in terms of tan a. sec2 a + csc2 a 24. Express 4-sin^-3csc^ j^ ^^^^^ ^^ ^^^^^ sin /8 - 1 - 6 CSC /^ ^ 25. Express Q - cos «) (1 -f sec «) -^ ^^^^^ ^^ ^^^ ^ (1 -sin a)(l -fcsca) 26. In the following identity transform the first member into the second, (1 + tan y) (cosy- cot y) ^ _ ^^^ (1 + cot y) (sin y — tan y) 122 REVIEW EXERCISES 27. Show that (l-tana)(l + cot(x) ^ _ ^^ (1 + tan a)(l - cotw) 28. Show that sin^ /g + cos^ /S siii« ^ - cos^ ^ ^ _ ^ ^ .^^, ^ ^^^, ^^ Solve the following equations and find the angle in degrees : 29. 4sin2y- tan^ynzO. 30. 2 tan2 a - sec a = 4:. 31. 4 CSC ;8 + cot2 y8 = 5. 32. tan2y + 3cot2y = 4. 33. sin2 a + sin^ /3 = I, cos"^ a + cos^ (3 = 0. 34. 2 cos2 a + sin2 y8 = 2, sin a + cos^ y8 = 0. 35. For what range of values of a between and 2 tt is sin a + cos a posi- tive ; negative ? 36. For what range of values of (3 between and 2 rr is tan jS — cot/3 posi- tive; negative? 37. Show that tan y + cot y must always be numerically greater than unity. 38. Trace the variation of sin^ ^ as ^ varies from to 2 tt.. 39. Trace the variation of cos^ ^ as ^ varies from to 2 tt. 40. Trace the variation of 1 — sin ^ as ^ varies from to 2 tt. 41. Trace the variation of 1 — cos ^ as ^ varies from to 2 tt. 42. Find by inspection logg .625, log8i27, loggg .008. 43. What numbers correspond to the following logarithms to base 9 : — 3, -2, -1.5, -1,0, .5, 1,2,3? 44. In the formula W=^^^p^v^ [( ~)"^ ~ "^T ^^^^^ ^ives the work of an air compressor, find W when n = 1.3, p^ = 14.7, p2 = 72, vi = 6. 45. Work the following with the slide rule : ,. .72x137x14 . .J. fl20y-^ . ., 42 sin 27° . ^^^ 372x778 =' ^'M42J =^ ^'^-13^- = ^ 46. Solve for a: : 52^ = 6; 8»=-i = 7. Query. Does the result depend on the base of the system of logarithms used? 47. Solve for x: 32« - 4 • 3'' + 3 = 0. 48. Find the amount of $2000 in 5 years at 4% compound interest. 49. At what rate, compound interest, will $45,000 amount in 8 years to $60,000? 50. In how many years will a city become three times its original size if it increases | each year ? REVIEW EXERCISES 123 51. Derive the formulas of Art. 42 from those of Art. 40. [Suggestion. Multiply respectively by a, b, and - c and add.] 52. Derive the formulas of Art. 40 from those of Art. 42. [Suggestion. Solve for a cos /8 and b cos a and add.] 53. From the law of sines, Art. 41, show that ^^^-^ = sm^-smy ^ b + c sin 13 + sin y 54. By applying the formulas of Art. 70 to the result obtained in example 53, derive the law of tangents of Art. 48. 55. From the formulas of Arts. 42 and 68 derive the results sin ^^ = J5ZS5ZI) ; COS « = JiSZ3 ; tan ^ = JKEMlZs) . 2 ^ be ' 2 > 6c 2 > s(s-a) 56. Draw the graph of sin ^ + cos ^ and thus trace its variation. What values of cause the given expression to assume maximum values ; minimum? 57. Draw the graph of tan + cot and thus trace its variation. What values of make the given expression a maximum ; a minimum ? 58. Draw the graph of arcsin u and trace its variation. [Suggestion. Lay off the values of u as abcissas, of arcsin u as ordinates.] 59. Draw the graph of arccos u and trace its variation. 60. Draw the graph of arctan u and trace its variation. 61. Draw the graph of arccot u and trace its variation. What discon- tinuities are exhibited by the functions of examples 58-61 ? 62. Find from the table the values of cos 625° 12' ; of sin 238° 25'; of tan 324° 6'; of cot 921° 32'. 63. Find without reference to the table the value of cos 285° cos 345° + sin 195° sin 465°. 64. Find without reference to the table the value of tan 205° cot 335° + tan 295° cot 115°. 65. Find all the values between and 2 tt of arcsin ( — ) ; arccos ( ] : arctan - . V 2/' V V2/ 3 66. Find all the values of a between and 2 tt if sin 3 a Z3 A ; if tan 2 a= - \/3 ; if cos - = — ; if cot - = - 1. V2 '2 V2 3 67. Find the value of sin (2 a + /3) if a = arcsin | and of 2d quadrant, ft = arctan | and of 1st quadrant. 68. Find the value of tan (3 a + 2 /?) if a = arcsin | and of 2d quadrant, P = arccos y\ and of 4th quadrant. 69. Derive the formula for sin (« + /8 + y) in terms of sine and cosine. 70. Derive the formula for cos (^ + (S -\- y) in terms of sine and cosine. 124 REVIEW EXERCISES 71. Derive the formula for tan (« + ^8 + y) in terras of tangent. Discuss the results of examples 69-71 in case a + ft + y = ir. 72. In the results of examples 69-71 put ft = y — a, and thus obtain the formulas for sin 3 a, cos 3 a, and tan 3 a. 73. Find the value of cos (3 a — 2 /3) if « = arccos ( — jV) and of 3d quad- rant and ^ = arctan | and of 1st quadrant. 74. Find the value of cot {^ a — ft) \i a — arcsin f and of 1st quadrant and ft = arccos (- ^f) and of 2d quadrant. 75. Find the value of sec (a + /8) if a = arctan ^^ and ft = arcsin ^V? both of 1st quadrant. 2 tan a 76. Show that sin 2 = 77. Show that cos 2 a -- 1 + tan'-^ a 1 - tan2 a 1 + tan2 a 78. Show that tan ( 45° — ^ ) =: esc a + cot a. 79. Show that cot (45° — - J =csc a — cot a. Transform into products or quotients the following expres- sions (80-84) : 80. cot a + tan a. 81. cot a — tan a. 82. 1 + tan a tan ft. 83. cot a — tan ft. g^ cot a + cot ft tan a + tan ft If a + /3 + 7 = TT, show that (85-87) 85. sin a + sin /? + sin y = 4 cos - cos ^ cos ^. [Suggestion. Apply A Ji Ji Art. 70 to the first two terms and Art. 67 to the third term.] 86. cos a + cos ft + cos y = 1 + 4 sin ^ sin ^ sin ^ • Jd Z 2i 87. tan a + tan ft + tan y = tan a tan y8 tan y. (See example 71.) 88. How does it appear from example 87 that either all three angles of a triangle are acute or else two are acute and one obtuse? (Consider the signs.) 89. How does it appear from example 87 that if one angle of a triangle is obtuse, it is numerically nearer 90° than either of the acute angles? In the following equations find the angle : 90. tan 2 a tan « = 1. 91. sin (60° - ft) - sin (60° + yg) = f . SECONDARY TRIGONOMETRIC FUNCTIONS 125 92. cos 6 y — cos 2 y = 0. 93. r sin ^ = 8, r cos ^ = 15. 94. r sin 6 cos = S, r sin 6 sin <^ = 4, r cos 6 = 12. 95. Show that sin ^ sin ^ sin ^ sin i^ = A. 5 5 5 5 16 96. Show that cos — cos |^ cos ^ cos i^ = -i; • 15 15 lo 15 lb 97. Show that 2 sin ^ = ± VI + sin a ± VI - sin a. 98. Show that 2 cos ^ = ± Vl + sin a =f VI - sin a. 99. The formula for the horizontal range of a projectile fired at an eleva- tion a with a muzzle speed u, is — sin 2 a. Show that the maximum range 9 is attained for an elevation of 45*^. 100. A triangle is formed by two given sides of constant length b and c, including a variable angle a. For what value of a is the third side a maxi- mum ; the area a maximum? SECONDARY TRIGONOMETRIC FUNCTIONS In addition to the trigonometric functions defined in Art. 6, 32, and 54, there are certain other expressions which are also functions of the angle. While of less importance than the six primary functions, an investigation of their properties will be valuable, not only for the results obtained, but as a review of the fundamental principles of trigonometry. We may define, then, versed sine a = vers a = 1 — cos a, coversed sine a = covers a = 1 — sin a, exsecant a = exsec a = sec a — 1, excosecant a = excsc a = esc a— 1, 101. By reference to Fig. 53, show that vers a = 1 , V covers « = 1 — - V exsec a = - — 1 , excsc a = ' — 1 . V 102. By reference to Figs. 60-63, show that, in line representations, vers a = MA, covers a = NB, exsec a = PT, excsc a = PS. 103. Determine the signs and limitations in value of each of the four secondary functions in the different quadrants. 126 REVIEW EXERCISES 104. Show that if the quadrant of the angle and the value of any one of its ten functions are given, the values of the other nine can be found. 105. Find the values of the four secondary functions, given : a = arcsin ( — ^%"j-) and of '3d quadrant ; jS = arccos |§ and of 4th quadrant ; y = arctan (— -^^j) and of 2d quadrant; 8 = arccot ff and of 1st quadrant. 106. Find all ten functions, given: a = arcvers |° and of 4th quadrant; ^ = arccovers t| and of 2d quadrant; y = arcexsec ^ and of 1st quadrant ; 8 — arcexcsc 2^^ and of 3d quadrant. 107. Trace the variation of each of the secondary functions as the angle varies from to 2 tt. 108. Draw the graph of each of the secondary functions. What discon- tinuities, if any, are present. 109. Find the secondarv functions oi k •-, for ^ = 1, 2, ••• 8. 4 110. Find the secondary functions of ^' .--, for ^ = 1, 2, ••• 12. 111. Verify the relations of Art. 61 for the secondary functions. 112. Determine the relations analogous to those of Art. C2 affecting the secondary functions. [Suggestion. Use Art. 60.] 113. By means of the cosine series. Art. 77, Eq. (.5), show that lim Y^^— — 0. PXSPO 114. From the preceding example, show that lim — '-^ — = 0. 115. Show that vers a + covers a exsec a + excsc a 2 vers a covers a vers cc — covers a exsec a — excsc a vers a — covers a 116. Show that exsec^ /? + 2 exsec fi = tan^ p. 117. Solve and find y in degrees : 2 vers y (2 — vers y) = 1. 118. Solve and find 8 in degrees : tan^ 8 + exsec 8 = 4. 119. Show that, if a is the angle at the center of a circle of radius r, the ordinate at the middle of the chord is given by the formula m = r vers -. Find m for r = 1433, a = 11° 32'. ^ 120. If T is the intersection angle of two tangents to a circle of radius r, the shortest distance of their point of intersection from the arc is given by the formula d = r exsec -. Find d for r = 5730, t = 5° 32'. 121. Reduce the first member to the second in the identity (exsec a + vers a) (excsc a + covers a) = sin a cos a. 122. Show that vers 2 a = 2 sin^ a. 123. Show that exsec 2 «= _2_sm^-a__^ 1-2 sin2 a 124. Show that excsc 2 a cos 2 a = tan a. REVIEW EXERCISES 127 125. At points in a straight line ordinates are erected siich that for each point (x, y), x = vers (arcsin y). Show that the graph thus determined is a circle tangent to the F-axis at the origin. 126. At each point in a circular arc the radius is extended an amount equal to the exsecant (in line values) of the arc measured from a fixed point in it. What is the graph thus determined? 127. Two equal circles have their centers in the same horizontal line. Show that the horizontal distance between two points in the neighboring arcs is equal to twice the versed sine (in line values) of the arc, measured from the point of tangency. 128. Two tangents to a circle intersect at an angle t. Show that the dis- tance of the point of intersection from the midpoint of the chord of contact equals exsec t + vers t (in line values). 129. Show how the versed sine is of practical use in staking out a circular railroad track passing through three given points. 130. Show how the exsecant is of practical use in staking out a circular railroad spur of given radius branching tangentially from a straight track. Compute the missing parts of the following triangles, distinguishing right from oblique : "A '3 a b c 131. 37° 42.8' 90° 6244.8 132. 72° 25.6' 90° 64.863 133. 90° 375.84 296.57 - 134. 54° 36.9' 24.465 42.850 ^135. 136° 36.8' 36902 37490 136. 68° 51.5' 90° 7532.8 137. 90° 396.45 531.53 138. 90° .005428 .006395 139. 148° 24' 7.4536 5.3648 140. .038456 .028638 .051524 141. 125° 34.6' 35° 25.3' 2584.6 142. 24° 36.8' 2.4657 3.6542 143. 80° 04.5' 90° 30.007 144. 94° 46.8' 34.086 52.475 145. .93274 .40586 .63208 146. 76° 46.3' 85° 38.7' 8.4637 147. 90° 29846 53857 148. 29° 57.4' 43° 52.6' 64.475 149. 17° 46.8' .39475 .29478 — 150. 36875 28467 48542 128 REVIEW EXERCISES 151. In a given triangle a = 280, c = 420, y = 38° ; find the radius of the circumscribed circle. 152. In a given circle a = 63, & = 81, y = 54°; find the lengths of the bisec- tors of the interior angles. 153. The sides of a given triangle are 220, 350, 440 ; find the lengths of the three medians. 154. In a given triangle b = 340, a = 48°, y = 63°; find the lengths of the radii of the inscribed and of the three escribed circles. 155. A boat drifts in a stream whose current runs 4 miles an hour due east under a breeze of 10 miles an hour from the southwest. Determine the motion during 35 minutes, if the resistance reduces the effect of the wind 30 %. 156. Three forces of 1800, 2200, and 2700 dynes are in equilibrium ; find the angles they make with one another. 157. A helical spring is fastened to the door 16 inches from the axis of the hinges, and to the jamb 4 inches from the same line in the same horizontal plane. Find the length of the spring when the door is closed, open at 30°, 45°, 70°, 90°, 120°. Neglect the thickness of the door. 158. A cable 30 feet long is suspended from the tops of two vertical poles 20 feet apart and 15 and 18 feet high, and bears a load of 200 pounds hanging from it by a trolley. Find the position of the trolley when at rest, and the lengths, inclinations to the horizon, and (common) tensions of the segments of the cable. Neglect the weight of the cable. 159. Let the data be as in the preceding example, save that the load hangs from a ring knotted at the center of the cable ; find the inclinations to the hori- zon and the (unequal) tensions of the segments. Solve when the ring is knotted at a point 12 feet from the lower end of the cable. 160. The eye is 40 inches in front of a mirror and an object appears to be 35 inches back of it, while the line of sight makes an angle of 48° with the mirror. Find the distance and direction of the object from the eye. (Note. The angles of incidence and reflection are equal.) 161. The line from the eye to the object recedes from the mirror at an angle of 32°, while the object is 36 inches from the eye and 12 inches from the mirror. Find the angles of incidence and reflection, and the point of reflection. 162. Two railway tracks intersect at an angle of 75°, and are connected by a circular " Y " of 800 feet radius lying in the obtuse angle and tangent to the two tracks. Find the distances of the points of tangency from the crossing and the length of the " Y ". 163. Two railway tracks, intersecting at an angle of 62°, are joined by a circular " Y " in the acute angle and tangent to the two tracks at points 900 feet from the crossing. Find the radius of the " Y " and its length. REVIEW EXERCISES 129 Fig. 72. 164. In setting a door frame 6 feet wide and 8 feet high, the vertical side is found to be 2 inches (horizontally) out of plumb. Find the angles of the paral- lelogram and the lengths of the diagonals. Is the diagonal of the true rectan- gle the average (arithmetic mean) of these two ? 165. The staking out of a certain building requires the setting of stakes at the vertices of a rectangle 32 x 48 feet. A test of the trial setting shows the figure to be a parallelogram whose sides are as given above, but whose diagonals dilfer by 9 inches. Find the angle through which the longer sides must be swung to correct distortion, and the chord of the arc through which the back corners must be moved. 166. A triangular roof truss is 80 feet long and divided into 8 equal seg- ments, by vertical members, as shown in Fig. 72. The height being 25 feet, what are the lengths and inclinations to the horizon of the various members? 167. The triangular roof truss shown in Fig. 73 is 60 feet long and 20 feet high. The bottom chord and rafters are divided into equal segments. Find the lengths and inclinations of the members. 168. In order to determine the exact location of the point G (Fig. 74), a base line AB \s laid oif due north and south, measuring precisely 130 rods. Convenient intermediate stations are chosen, and angles meas- ured as follows: ^^0=^36° 35', BA C = 61° 10', BAD = 43° 54', CAD = 17° 16', DCE = 66° 36', CDE = 47° 41', EDF=55°4:H',DEF = 73° 12', FEG = 5S°32\ B EFG = 10° 28'. Com- pute the distances com- posing the sides of the triangles in the figure. 169. By projecting the distances AC^ CE, EG (Fig. 74), perpen- dicular and parallel to A B, compute the east- erly and southerly dis- tances of G from A; find thence the direct Fig. 74. distance and direction of G from A. Fig. 73. FORMULAS GENERAL TRIGONOMETRY 1 CSC a — sec « = cot a — tan a = cot a = sill a sin^a + cos^ a = 1. tan^ a + I = sec^ a. cot^H- 1 = csc2 a. 2 TT^ = 360°. F(2k7r + a) = F(a), k an integer. sm a 1 cos a 1 tan a sin « cos a cos a Ffk ± a] = ±F (a), k an even integer. Fik'-± a]= ± co-F (a), ^ an odd integer, sin (a ± /3) = sin a cos yQ ± cos a sin yS. . cos (a ± y6) = cos a cos /3 =F sin a sin /3. tan(«±^)= tan«±tanff ^ 1 =F tan at'dn 8 cot(«±ff)=«"t«C"t/3Tl. cot y8 ± cot a 130 GENERAL TRIGONOMETRY 131 sin 2 a = 2 sin a cos a. cos 2 a = cos^ a — sin^ a = 2 cos^ a — 1 = 1 — 2 sin^ «, 2 tan a tan 2 a = cot 2 « = 1 — tan^ a cot^ ce — 1 2 cot a sin \a= V| (1 — cos a). cos I a = V| {\ 4- cos a). .1 /l — cos a 1 — cos a tan - a = \— = — : 2 ^ 1 H- cos a sm a ,1 /I + cos « 1 + cos a cot - a=V— ^- =— ^^ 2 ^ 1 — cos a sm a sin a cos yS = | [sin (ct + y8) + sin (a — /8)]. cos « sin /3 = J [sin (a -f- ^) — sin (a ^ /S)] . cos a cos yS = I [cos (« + /S) + cos (a — 6)] . sin a sin /3 = — -| [cos (ot + yS) — cos (a — ^)]. sin a cos « = |^ sin 2 a. cos^ a = 1 (1 + cos 2 a). sin^ a = 1 (1 — cos 2 cj). sin a -\- sin p = 2 sin — — ^ cos — --^ • sin a — sm /3 = 2 cos — ^— ^ sm — --^» cos a + COS p = 2 cos — —-^ cos — — ^ • /o o- a-\- S . a— ^ cos a — cos p = — 2 sin — ^ sm — —^ RIGHT TRIANGLES 6^2 _|. ^2 ^ ^, a + ^ = 90°. a . ^ - = sin a = cos p. 132 FORMULAS = cos a = sin y8. a . - = tan a COtyS. A= ^ ab = ^bcsin a= I c^ sin a cos a = ^ c^ gj^ 2 a. OBLIQUE TRIANGLES « + ye + 7=180°. (? = 5 cos a-\- a cos yS, etc. a _ h _ ^ (>S^- «) cos (S-fi) cos (a^- 7) ^ cos aS' tan 1 (/3 + 7) = ^^^i^l^cot 1 a, etc. cos -^- (0 + c) tan 1 (5 - ^) = ^"' 2 (f — ll tan 1 «, etc. 2^ sin 1(^4- 7) tan 1 (5 + c?) = ^"^t ^^~^^ tan J a, etc. - V cos 1 (^ - 7) ANALYTIC TRIGONOMETRY lim 0=0 lim 0=0 ' e 1 _sin ^J • e - tan ^ = 1. = 1. (cos a-\-{ sin a) (cos ^+ i sin /3) = cos (a + /3) + i sin (a + /3). (cos « + i sin a)*^ = cos na 4- ^ sin na. cos a + ^ sin a = e**". cos « — I sin a = e~**. COS a = • . 134 ANALYTIC TRIGONOMETRY. CONSTANTS sm a = — — 2^ cosh a + sinh a = g*. cosh a — sinh a = e~"-. cosh « = sinh a = 2> iog(i+.)=f-|Vf-|V-. C0S«=l-- + jj-^+-. tan « = - + — + — ^ + 1 8 15 315 cosh.= l + - + -+^4 CONSTANTS 7r = 3.14159265 •••. 7r^ = 180°. l^ = 5T.295779e5° ••• =57° 17'44.8'^... g = 2.7182818285 ••.. Mode 10 = — ^ = .4342944819 .... CONSTANTS 135 1 inch = 2.54001 ••• centimeters. 1 foot = .3048 .-. meters. 1 mile= 1.60935 ••• kilometers. 1 centimeter = .3937 ••• inches. 1 meter = 3.28083 feet = 1.09361 yards. 1 kilometer = .62137 miles. g= 32.086528 + .171293 sin^t^ feet per second per second. = 9.779886 + .05221 sin^ (^ meters per second per second at sea level for latitude ^. ANSWERS TO EXERCISES (Answers are omitted in case their knowledge would detract from the value of the exercise.) Exercise II 6. (0, 0), (a, 0), (a, a), (0, a); (^ V2, o), ("' | Vs), (-|V2,o), (0,-|V2). 7. (5, 0), (0, - 5), (-4.33, -2.5), (-3.54, 3.54). 9. 5.6569; 7.6158. 11. Cross country distances, in miles : 5.099; 2.828; 2.236; 2.236; 6.325. 12. Distances saved, in yards : 773.2 ; 644.4 ; 1288.7 ; 128.9. Exercise IV 9. 1681 11. If. 13. cos a. 15. Jl-siny ^l + sin y 1519 12. If. 14. 2csc^. 10. f. Exercise V 16. 2(l+tan2y). 9. h 13. 60°. 17. (a) 60°; (b) 19.05 ft. 10. W3. 14. 0° and 60°. 18. 8.08 ; 16.17. 11. i- 15. 60° and 90°. 19. 452.39. 12. K3V3- -2). 16. 45°. 20. 60°. Exercise VI 11. 0° and .60°. 12. 30°. 13. 0°, 30°, and 45°. 14. 0°, 30°, and 45°. Exercise VII 1. (a) 6.7,6.7; (b) 8.23,4.75. 4. 15.6; 9. 2. (a) 8; (6) 13.86. 5. (a) 2598.16; (b) 1500. 3. (a) 20, 34.64 ; (6) 28.28, 28.28. 6. 24 miles per hour, 30° east of north. ' Article 18 1. a = 40.32, & = 11.76. 3. 6 = 151.5, c = 381.6. 2. a = 20.25, c = 33.75. 4. a = 133.2, b = 149.2. 137 138 ANSWERS TO EXERCISES Exercise VIII (These results were ol)tained with four-place tables.) 1. ^ = 64° 50', a = 14.46, h = 30.77. 2. p = 37° 40', a = 57.79, b = 44.61. 3. a = 28° 45', a = 116.88, b = 213.04. 4. a = 11° 25', a = 103.11, 6 = 510.68. 5. (3 = 68° 35', b = 599.13, c = 643.66. • 6. /3 = 17° 15', b = 223.56, c = 753.93. 7. a = 9° 30', b = 7170.96, c = 7272.73. 8. a = 72° 30', a = 4757.40, c = 4988.36. 9. a = 41° 49', /3 = 48° 11', 6 = 268.33. 10. a = 32° 12', (3= 57° 48', 6 = 605.03. 11. a = 34° 13', 13 = 55° 47', a = 354.25. 12. a = 53° 8', ^ = 36° 52', a = 1120. 13. a = 36° 52', ^ = 53° 8', c = 1080. 14. « = 44°46', )8 = 45°14', c = 845.07. ' 15. a = 29° 11', ^ = 60° 49', c = 440.94. 16. a = 59° 41', 13 = 30° 19', c = 2445.55. 17. a = 29° 29', (3 = 60° 31', b = 168.00, c = 193.00. 18. a = 41° 04', f3 = 48° 56', a = 230.00, c = 350.13. 19. (3 = 15° 40', a = 93.47, b = 26.21, c = 97.08-. 20. a = 65° 10', a = 60.35, 6 = 27.93, c = 66.50. 21. 200.1 ft. 23. 2° 23'. 25. 1° 9'. 22. 1501.73 ft. 24. 4° 46'. 26. 33° 41', 26° 34', 45°. 27. .134 pitch, .2887 pitch, | pitch. 28. 19° 28' inclination. 31. 26° 31'. 34. 21.73 ft. = 21 ft. 8| in. 29. 8° 3' inclination. 32. 5859.71 ft. 35. 260.4 ft. 30. 0° 9', 0° 17', 1° 26'. 33. 32° 28'. 36. 0° 20'. 37. 5° 54' ( = .1029 radians). Note that .1029 = sin 5° 52.5', an approxima- tion. See Arts. 72 and 77 (3). 38. 2.468 miles. 39. 502.2 ft. 40. ^=(0,0), 5 = (240.9,0), C= (385.9, 274.8), D = (98.7, 814.6), E = (162.8, 1043.0), F= (649.1, 1248.8). 41. 13 = 110°, b = 68.81, A = 828.81; ^ = 36°, a = 202.27, A = 12,641.88; a = 75° 06', p = 29° 48', A = 30,441.6 ; a = 63°, b = 326.88, A = 52,425.01 ; a = 64° 17', a = 553.12, A = 119,594.88; a = 41° 1', ^ = 97° 58', A = 202,809.6. 42. 51.76, 61.80, 68.40, 76.54, 1, 121.76, 141.42, 173.20. 43. 45. 5 = 2scos?. 4 44. s = 2Rsin'''' = 2rt.u'''' n n n 2'7rR cos 180°. n e = 62.83, P4= 56.57, 0^= 44.43; P8 = 61.23, C8 = 58.05; P,e = 62.43, c,«= 61.62; P32 = 62.72, C32 = 62.53. Ap ^nW^&ui 180° 180° 1 po . 360° cos = - nR^ sm n n 2 n Ai = tt/^^ cos n 2 \ 360°\ , ^P4 = 200.00, .4,4 = 157.08; Ap^ = 282.84, ^,-8 = 268.15; ^P16 = 306.16, A^Q = 302.21 ; ^P32 = 312.16, ^,32 = 311.14. ^6- 60.00, c,. = 54.41 ; ^2 = 62.11, c,. = 60.69 ; ^^24 = 62.64, c.„ = 62.29 ; ^48 = 62.78, c,« = 62.69. 1, ^^6 = 259.80, J,g= 235.62; Aj,^^ = 300.00, .4 ,-,2 = 293.11; ^,24 = 310.56, ^/ = 308.80; A,^ = 313.20, ^,-^8 = 312.81. ANSWERS TO EXERCISES 139 46. Ac=7rR% ^,. = 314.16, 47. C = 62.83, 48. ^. = 314.16 49. Z = -30, F=- 17.321, 22 = 34.641, 30° south of west. 50. A' = 6 r, F = 0, R = 6 r, due east. 51. Distance from center = r cos 0. 52. x= — 15. 53. Component along plane = g sin a, component perpendicular to plane = g cos a. 54. 16, 27.71; 8.28, 30.91; 5.56, 31.51; 2.79, 31.88. 55. 50 pounds pressure, 141.42 pounds along ladder. 56. Z = 57.28, y= 30.73. 57. i2 = 18.44, ^ = 49° 24'. 58. c= 11.99. Exercise IX 1. 4,1.5, -2.5. 4. 1,8, 16, i,^V 2. 4, if. 5. (a) .4724; (b) .01614. 3. 1, 4, 16, 32, 64, tV, ^?- 6. (a) 28.16; (b) .01913; (c) 2.465. 7. 17.978. 9. 524.9. 11. a: = 1.79. 13. $4136.09. 15. 11.6 years. 8. .76252. 10. a; = 2.29. 12. $4656.20. 14. 5.2%. 16. 3.8 years. Exercise X 1. 4.86024, 2.79187, 9.84198, 5.80872 - 10, 21.47712. 2. 4.96088, 1.15518, 11.50651, 5.89510 - 10, 24.30103. 3. 516.35, 4.0966 x 10^2, .016335. 4. 16361, 5.64325 x 10", .00013671. 5. 9,067,800,000. 7. 88.594 cm. 9. 13,231 x lO^o. 6. 7.0048 X 1010 cm. 8. 71.68 cm. 10. 2,754,100. 11. 9.63459 - 10, 9.52928 - 10, 0.01824. 13. 78° 01.1', 81° 43.7', 76° 17.1'. 12. 9.97454 - 10, 9.78340 - 10, 0.04197. 14. 25° 20.7', 27° 32.6', 35° 3.6'. 15. 13.861. 16. .91186. 17. 3.9968. 18. .38875. 19. 1.3365 inches. 22. .074765. 20. .1111 foot (=1.3332 inches). 23. 6:711; 8.381. 21. 1.7% less than the true value. 24. - 11.85; - 61.38. 140 ANSWERS TO EXERCISES Exercise XII 1. ^ = 27^ a = 2302.3, h = 1173.1. 2. cc = 60" 37.6', /3 = 29'^ 22.4', h = 4238.9. 3. ^ = 14° 44.8', b = 254.07, c = 998.12. 4. a = 15° 39.6', ^ = 74° 20.4', b = 168.36, c = 174.85. 5. a = 50° 13.1', (3 = 39° 46.9', c = 9.5378. 6. ^ z:. 71° 34.5', a = 10.417, ^> = 31.271. 7. a = 83° 38.4', y8 = 6° 21.6', 6 = 14.82, c = 133.79. 8. a = 75° 33', ^ = 14° 27', c = 54.953. 9. a = 64° 48.5', /S = 25° 11.5', 6 = 31,037. 10. ;8 rr 60° 9.8', a = 5.854, c = 11.766. 11. /? = 64° 42.6', a = 19.023, b = 40.264, c = 44.531. 12. a = 28° 23.6', (3 = 61° 36.4', c = .00042. 13. a = 26° 47.3', a = 3.2196, & = 6.3769. 14. a = 38° 23.3', ft = 51° 36.7', a = .056677. 15. a = 54° 43.6', b = .44535, c = .77120. 16. a = 54° 43.2', a =: 242.79, b = 343.16, c = 420.37. 17. a = 55° 59.3', 13 = 34° 0.7', c = .0074192. 18. a := 9° 47.5', a = .89928, c = 5.2878. 19. a = 63° 20.7', /3 = 26° 39.3', a = .014523. 20. a = 64° 41.8', a = 1563.4, 6 = 739.12. 21. 22. 8.2583 feet. 18 feet 3.8 inches. 23. 141.42 square feet. 24. 60° 1.8'. 25. 1237.8 feet. 26. 16| miles, 36° 52.2' north of west. 29. nR^ sin'-^cos'^- n n 30. 2177.4. 31. 32. 33. 34. 35. 36. 37. 38. 178.8 miles. 0° 32'. .078523 feet. 14834 feet. 425.64 feet. 142.4 feet. 118.1 feet. 554.06; 145.17. Article 47 1. a = 33° 19.9', a = 438.23, c = 788.58. 2. a = 65° 49.8', a = 122.13, b = 885.60. 3. (3 = 15° 57.0', b = 5.442, c = 17.865. — 4. ^ = 1° 02.0', a = 9.368, b = .18134. Article 48 3. a = 57° .59.9', y = 23° 36.6', c = 29.526. 4. ^ = 13° 55.6', y = 35° 30.7', b = 135.96. ^ (a = 104° 31.3', 13 = 40° 2.9', a = 5889.9 ; ^ - 4° 37.1', ^' = 139° 57.1', a' = 489.8. 94° 17.9', y = 47° 13.3', a = 207,810; 6. ^ (a [ a' = 47° 4.6', y' = 132° 46.7', a' = 152,600. ANSWERS TO EXERCISES 141 Article 49 1. ^ = 23° 42.8', y =3o° 45.2', a = 450.35. 2. a = 23° 31.8', y = 19° 7.2', h = 818.54. 3. a = 33° 17.5', y = 63° 13.1', b = .11496. 4. ^ = 66° 27.0', y = 45° 11.2', a = .005202. Article 50 66= '49.4', y = z6r *13.4'; A = 1.9181 X 101 42° '51.8', y = = 54= •51.8'; A = 4.4175 X 10^. 34° 45.4', 7 = = 88^ '45.8'. 72= ' 33.2', ^y-- = QV ^33.4'. r? 7. Impossible. Why? 8. y = 8< ' 58.3' '. 1. a = 51° 57.2', y8 2. a = 82° 16.4', /3 3. a = 56° 28.8', ^ 4. « = 45° 53.4', 13 5. Impossible. Why? 6. IS= 136° 39.8'. Exercise XIV 1. /8 = 74° 0.3', y = 43° 24.7', a = 76,568. 2. a = 52° 56.6', y8 = 79^^ 47.8', y = 47° 15.6'. 3. y = 86° 10.3', b = 8.4172, c = 9.0436. 4. a = 43° 29.3', y = 80° 35.3', 6 = .30470. 5. Impossible. Why? 6. a = 52° 11.2', y = 27° 38.8', a = 49,921. 7. a = 34° 32.1', ^ = 51° 41.8', y = 93° 46.1'. 8. y = 69° 28.5', a = 67,439, c = 72.037. 9. a = 23° 34.1', jS = 35° 35.7', c = 6.0804. 10. a = 15° 35.2', y = 126° 7.6', c = 66.113. 11. (3 = 13° 11.7', y = 16° 24.1', a = .082764. I ^ = 32° 8.5', y = 89° 45', b = 34.993 ; ■ i /?' = 31° 38.5', y' = 90° 15', b' = 36.210. 13. a = 34° 11. .5', a = 382.48, c = 641.52. 14. a = 28° 57.0', /? =: 104° 28.6', y = 46° 34.4'. 15. a = 21° 13.9', y = 32° 19.7', b = .0048578. 16. a = 162° 18.9', y = 7° 08.3', a = 61.896. 17. a = 33° 33.1', /? = 50° 42.0', y = 95° 44.9'. 18. a = 75° 0.2', a = 8355.2, b = 6470.6. 19. a = 45° 29.5', /S = 14° 15.5', 6 = 2146.7. 20. a = 49° 36.8', ^ = 40° 23.2', c = 952.67. 21. a = 151° 56.6', /8 = 4° 30.4', y = 23° 38.0', b = 416.45. 22. a = 80° 0.0', ^ = 54° 45.2', y = 45° 14.8', a = 124.81. 23. /? =: 90° 50', y = 16° 0', a = 720.81, c = 207.58. 24. a = 95° 26.6', y = 27° 8.4', a = 125.81, b = 106.49. 25. 2.1815 X 109; 5.0105; 1,742,040,000. 26. 2.567 X 109; .038051; 7270.3. 27. Case III. 142 ANSWERS TO EXERCISES 28. 29. 30. a = 15° 45', yS a = 2r 52.6 , fi a = 91° 54.7', 13 a = 47° 59.5', (3 r A = 156° 55.6', B = 145° 57.2', C = 57° 7.2' ; t ^ = 145° 13.6', B = 121° 11.4', C = 93° 35.0'. 29° 15', c = 52.1 ; 42° 7.4', c = 723.6; 53° 5.3', c = 43.042 ; 72° 0.5', c = 291.38. (A 31. ^ 63° 25.9', B = 141° 34.1', c = 328.4 ; 122° 25.4', C = 137° 34.6', c = 575.41. 75°, & = 878.48, cz= 621.17; L A = 113° 58.6', B = 106° 2.2', C = 139° 59.2'. 32. a = h = 48.5. 33. ^ = 151° 2.7', B = 133° 25.9', C = 75° 31.4'. 34. d^ = 0.2 + ^2 + c2 _ 2 a& cos ah -2 be cos kr + 2 ac cos (a6 + k-) ; d = 12.98 35. be = 84° 03.5', cS = 75° 53.0', (/a = 82° 03.5', ab - cd = 109.28, 6c - da = 114.47. 36. c = 1001.1, (/ = 568.6. 37. « rz 36°, s = 15.217. „„ 6 - 38. V = 6 30 (6 - h) \/l2 A - A- : F^ = 135.0, V^ = 371. F3 = 663.3, F4 = 990.0, F5 = 1338.0, F^ = 1696.4, F^ = 2054.8, Fg = 2402.8, F9 = 2729.5, Fio = 3021.1, F„ = 3257.8, V^^ = 3392.8. 39. 6 = 483.4. 40. a = 3221.5. 41. b = 1286. 42. Distance = 31.63, total height = 20.97. 43. Distance = 24.24, height = 5.08. 44. AD = 738.2, DB = 150.6. 45. AC = 1075.1, BC = 679.5. 46. ^D = 1460, DC = 678, angle BXC at left = 17° 27.5'. 47. Distance = 135.74, height = 36.602. 48. D = 57° 40', CD = 196.73, BD = 233.55. 49. AD = 603.94, ^C = 693.12, BC = 838.82, 5^ = 595.76, AB = 867.48, angle CXB at left = 4° 27.7'. 50. AC = 730.17, ^Z> = 737.37, BE = 805.40, BF = 715.52, ^5 = 841.67. 51. 39° 54'. Exercise XV 1. 45°, 60°, 150°, 112° 30', 171° 53' 14.4", 42° 58' 18.6". 2. 77-^ TT^ TT^ 27r^ ilT^ 5^^ StT^ 6'12'4'3'.3'3'2* 3. 31.416 cm., 62.832 cm., 125.664 cm., 47.124 cm. 4. 1^, l\ 2^ ^^ 2 2 5. Smaller sprocket : 4 revolutions per second, angular velocity = 8 tt radi- ans per second, linear velocity of circumference = 201.06 inches per second; 167r larger sprocket: f revolutions per second, angular velocity 3 radians per second, linear velocity of circumference = 201 .06 inches per second. Speed of machine = 20.944 feet per second = 14.28 miles per hour. ANSWERS TO EXERCISES 143 6. Linear velocity of chain and of circumferences of both sprockets = 75.43 inches per second ; angular velocity of larger sprocket = 15.09 radians per sec- ond, of smaller sprocket = 37.76 radians per second ; smaller sprocket makes 5.1 revolutions per second. 13. I<«- ■ - (!■)■ (V , -1), etc. 6. (-;,l).etc. ' fe f )• { 5 TT V2\ , Exercise XVII 1. f. 7. 120°, 150°, 300°, 330°. 2. l4-2\^ 3 8. 15°, 75°, 135°, 9. 60°, 300°. 195°, 255°, 315°. 3. 5 10. 30°, 150°, 210< ^ 330°. 4. -If- 11. 0°, 60°, 180°, 300°, 360°. 5. 60°, 300°. 12. 0°, 150°, 180°, 210°, 360°. 6. 120°, 300°. Exercise XVIII 1. - sin 20°. 9. 0,0. 22. 0. 2. 3. - sin 10°. cot 14°. 10. 1 + V3 2V3 2 ' 3 • 23. - .35. 24. -1. 4. 5. 6. 7. - cot 35°. - CSC 30°. sec 40°. 1 + V3 2V3 2 ' 3 • 11. 12. 19. V3 - 1 2V3 2 ' 3 • 0,0. 1 + V3 4 25. + VI - a\ 26. ^l--^ m 27. sin«. 28. — sin a. 8. 1 + V3 2V3 2 ' 3 • 20. 21. 0. 0. 29. tan a. 30. - tan a. Exercise XIX 5. 6. 7. a + ^ = sin-i If = cos-i - |1, cc + ft z= arcsin — ||f = arccos -sin(a + y8). 9. II quadrant. — Iff, III quadrant, sin 2 a. 11. sin 2 6. 8. cos (a + y8). 10 . cos 2 a. 12. cos^. 13. 105°- arcsin ^" + 4 V6 1 \/2 - V6 = arccos ; 4. 15= ' = 105° 90" = arcsin^ 7^ = 4 \/6+V2 arccos J 144 ANSWERS TO EXERCISES 14. 75° = arcsm^ + ^^ = arccos^-^: 15° = 90° - 75° = arcsin^-^ = arccos^^ + ^ = arccos- 4 4 Exercise XX 11. tan 15° = 2 - V3, cot 15° == 2 + V3. 15. sin (a + /8 + y) = sin a cos y8 cos y + cos a sin y8 cos y + cos a cos jS sin y — sin a sin y8 sin y. 16. cos (a+ /3 + y) = cos ct cos ^ cos y — sin a sin y8 cos y — sin a cos ^ sin y — cos a sin /? sin y. T „ , . , a , N tan a + tan B + tan v — tan « tan (3 tan v 17. tan (a-\-a-\-y)=: ^^ ? 1- ti _L_. 1 — tan ytJ tan y — tan y tan a — tan a tan /3 18. cot (a + B+y)= ^ot^^oty + cotycotg+cotctcoty ^ '"^ cot ct cot /? cot y — cot (/ — cot /5 — cot y Note the symmetry in the last four formulas. Exercise XXI 5. 2 a =: sin-i yV/o "= cos-i ifif. 6. 2 a = arcsin ± ilg = arccos — ^i|. Explain the signs. 7. sin ^ a = ± f and ± f , cos J =i ± f and ± f. a s4«=.^^^ ana ±i^:eoa„..l^- and .^. 9. sin (a + 2 ;8) = f|f and - ||f, cos (a + 2 ;g; .. -^%% and - |04. 10. sin (a- 2 ft) =± |f | and T ff |, cos (a - 2 ^) = T^W and T |tf. 17. a = 30°, 45°, 60°, 210°, 225°, 240°. 18. a = 90^ 270°, and I arccos f . 19. a = 67° 30', 157° 30', 247° 30', 337° 30', and ^ arctan f . 20. a = 90°, 270°, and i cos-i f. 23. 2 x. 24. 1. 25. 0. 26. 1. Exercise XXII 1. 1 [sin 8 a + sin 2 a]. 8. ^ [3 - 4 cos 2 a + cos 4 a]. 2. ^ [sin 10 a - sin 2 a]. 9. -^ [3 sin 2 a - sin 6 a]. 3. ^ [cos 4 a - cos 10 a]. 10. ^^ [1 - cos 4 a]. 4. 1 [cos 7 « + cos 3 a]. ^5^ ;t . J; [/t = 0, 1, 2, 3, 4]. 5. i [cos a - cos 3 a] . 4 > > » > j 6. K2sin2a + sin4a]. • royfc+n!!:. p/t - 1 9 31 7. i[3 + 4cos2a+cos4a]. ^^' ^"^ + ^^4' L^-«'1'-^J- 17. (2^+1)^; [/j=0, l,2,...ll]and(3A:+l)|; [A: = 0, 1, 2, 3]. 18. (2A: + 1)^; [^ = 0, 1,2, ...29]and(2^-+l)^; [^ = 0, 1, 2, ...9]. 19. A:.|; [A: = 0, 1, 2, ...7]. 20. ^.|; [^^ = 0, 1, 2, ... 7]. ANSWERS TO EXERCISES 145 Exercise XXIII 9. (2A: + 1)^; [/^ - 0, 1, 2, ...]. 13. ^0^3 a - 3 cot oc . "^ ^6^ ' ' ' J 3cot2a-l kir ^n kir -. ^ 3 tan a — tan^ a 10. — . 11. ^. 14. 3 12. TT and (2^ + 1) 3 4 1-3 tan'-^ a 15. 4 sin a cos^ a — 4 sin^ a cos a. 16. 8 cos* « - 8 cos2 a - 3. 17. 0°, 15°, 105°, 180°, 255°, 345^. (See Exercise XIX, examples 13 and 14.) 18. a = 60°, 90°, 120°, 270°, and arcsin ^. 19. — • 20. ^. 2V3 2 2 Article 72 1. - sin 0. 2. sec tan 0. 3. - esc cot ^. 4. sec^ 6. 5. - csc^ (9. Article 73 5. cos^-+ I sin——; ±1; 1, ^ ; ±1, ±i; ±1, ^ _ (2;t + l)7r, . . (2^+l)7r , . . 1± V^^ 6. cos ^^ — 1^ I sin -^^ ^— ; ±i: — 1, : n n 2 ' ± 1 ±« . ± V3±i — Ti — ; ±h — 7^ Article 74 6. Products: 15 + 6i; -5 + 3t; 6 + 12 {; _8-20^; 12-18/; 4+22i; 4 + 22 i. 7. Quotients: 5-3*; 3 - 2 i. 8. Results: 4 + 12i; 1; - 8; 1. 9. Roots: 4 - 3 ^ 3 + 2 t; - 46 + 9 i ; ±21 10. ^/a [cos ?^ + i sin ?^] ; [>l- = 0, 1, 2, ... (n - 1)]. 11. ±1; ±i; 1, ^\^ '^ 2, -l±V-3. INDEX [Eeferences are to articles, except where otherwise indicated.] Addition formulas, 63, 64. Ambiguous case of oblique triangles, 48. Angle, general definition of, 52. Angles, positive and negative, 3. Answers to exercises, page 135. Area, laws for oblique triangles, 45. right triangles, 17. Checks, 20. Common logarithms of numbers. Table I, page 3. Complementary angles, functions of, 10. Complex numbers, graphical methods of representation and combination, 74. Composition of forces, 51. Conversion formulas for products, 69. Conversion formulas for sums and differ- ences, 70. Coordinates, 4. Definitions of the trigonometric functions, 32, 54. De Moivre's theorem, 73. Directed rectilinear segments, 2. Drawing instruments, 1. Equilibrium of forces, 51. Errors, 20. Exponential values of the trigonometric functions, 75. Formulas for tan (a:t0), cot (a i |3), 66. Formulas, list of, page 130. Functions of 0°, 90°, 12. of 180°, 36. of 270°, 360°, 58. of 30°, 45°, 60°, 11. of (90° + a), 38. of (^~±a],62. of half an angle, 68. of twice an angle, 67. Fundamental relations between the func- tions of a single angle, 9, 34, 59. General inverse functions, 80. Graphs of the trigonometric functions, 57. Greek alphabet, page x. Half an angle, functions of, 68. Hyperbolic functions, 76. Infinity, definition of, 12, 35, 58. Inverse functions, logarithmic values of, 81. Inverse trigonometric functions, 14. Law for angles in terms of sides, 44. Law of cosines, 42. Law of projections, 40. Laws of sines, 41. Law of tangents, 43. Laws of area, oblique triangles, 45. right triangles, 17. Laws for solution of oblique triangles, 39-44. right triangles, 16. Limitations in value of the trigonometric functions, 8, 33, 55. Limits of 6»/sin e and 0/ta.n e, 72. Line representations of the trigonometric functions, 60. List of formulas, page 130. Logarithms, characteristic, 24. cologarithms, 28. common system, 23. definition of, 21. interpolation, 26. laws of combination, 22. mantissa, 25. numbers from logarithms, 27. of numbers, Table I, page 3. of trigonometric functions. Table 1\ page 25. Method of solution of triangles, 18. Multiple angles, 71. 147 148 INDEX Natural trigonometric functions, Table III, page 71. Oblique triangles, area, 45. laws for solution, 3f)-44. Oblique triangles, solution of, 46-50. Orthogonal projection, 4 (note), 15. Periodicity of the trigonometric functions, 61. Proportional parts, theory of, 79. Purpose of trigonometry, 5. Relation between the ratios and the angle, 7. Resolution of forces, 51. Right triangles, area of, 17. laws for solution, 16. solution by logarithms, 31. solution by natural functions, 18. Series, exponential, logarithmic, trigono- metric, hyperbolic, 77. Signs of the trigonometric functions, 8, 33, 55. Slide rule, 30. Solution of oblique triangles, 46-50. Solution of right triangles, by logarithmic functions, 31. by natural functions, 18. Squares of numbers. Table IV, page 91. Subtraction formulas, 65. Supplementary angles, functions of, 37. Table I. Common Logarithms of Num- bers, page 3. Table II. Logarithms of the Trigonometric Functions, page 25. Table III. Natural Trigonometric Func- tions, pagre 71. Table IV. Squares of Numbers, page 91. Trigonometric functions, definitions of, for acute angles, 6. for obtuse angles, 32. for the general angle, 54. logarithms of, Table II, page 25. natural, Table III, page 71. Trigonometric tables, pages 1-93. computation of, 78. description of, 19. Trigonometry, purpose of, 5. Twice an angle, functions of, 67. Variations of the trigonometric functions, 13, 35, 56. TABLES -/ (- TABLE I COMMON LOGARITHMS OF NUMBERS N. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Lo^. Infinity. o.oo ooo 0.30 103 0.47 712 0.60 206 0.69 897 0.77 815 0.84 510 0.90 309 0.95424 1.04 139 1.07 918 I. II 394 1. 14 613 1. 17 609 1.20 412 1.23045 1.25 527 1.27875 1.30 103 I .32 222 I 34 242 1 36 173 1 . 38 02 1 1.39 794 1. 41 497 1.43 136 1.44 716 1 . 46 240 N. 1.47 712 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Log. 1.47 712 .49 136 .50515 .51851 •53 148 .54 4-'7 • 55630 . 56 820 . 57 978 .59 106 60 206 .61 278 .62 325 .63 347 .64345 .65 321 .66 276 .67 210 .68 124 .69 020 ,69 897 .70757 .71 600 .72 428 .73239 .74036 .74819 •75 587 76343 .77085 1.77 815 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 Log. 1.77 815 78 533 .79239 .79934 .80618 .81 291 .81 954 . 82 607 .83251 .83885 .84 510 .85 126 .85733 .86332 .86923 .87 506 .88081 .88649 . 89 209 .89763 90309 90849 91 381 91 908 92 428 92942 93450 93952 94448 94 939 90 1.95424 3 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 Log. 1.95424 1.95 904 1.96 379 1 . 96 848 1.97 313 1.97 772 1.98 227 1.98677 1 .99 123 1.99 564 2 . 00 000 2.00 432 2 . 00 860 2.01 284 2.01 703 2.02 119 2.02 531 2.02 938 2.03 342 2.03743 2.04139 2.04 532 2.04 922 2.05 308 2.05 690 2.06070 2.06446 2.06 819 2.07 188 2.07 555 N. 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 120 2.07918 160 2.17609 Log. 2.07 918 2.08 279 2.08 636 2.08 991 2.09 342 2.09 691 2.10 037 2.10 380 2.10 721 2. 11 059 2. II 394 2. 1 1 727 2.12 057 2.12 385 2.12 710 2.13033 2.13354 2.13672 2.13988 2.14301 2. 14 613 2.14 922 2.15 229 2.15 534 2.15836 2.16 137 2.16435 2.16 732 2.17 026 2.17 319 TABLE I N. 100 01 02 03 04 05 06 07 08 09 110 11 12 13 14 15 16 17 18 19 120 21 22 23 24 25 26 27 28 29 130 31 32 33 34 35 36 37 38 39 140 41 42 43 44 45 46 47 48 49 150 N. O oo ooo 432 860 01 284 703 02 119 531 938 03 342 743 043 04 139 532 922 05 308 690 06 070 446 819 07 188 555 918 08 279 636 991 09 342 691 10 037 380 721 11 059 394 727 12 057 385 710 13 033 354 672 988 I4_30i^ 613 922 15 229 534 836 16 137 435 732 17 026 319 609 475 903 326 745 160 572 979 383 782 179 571 961 346 729 108 483 856 225 591 087 518 945 368 202 612 *oi9 423 822 218 954 314 672 *026 377 726 072 415 755 093 428 760 090 418 743 066 386 704 *oi9 333 644 953 259 564 866 167 465 761 056 348 638 610 999 385 767 145 521 893 262 628 990 350 707 *o6i 412 760 106 449 789 126 461 793 123 450 775 098 418 735 ♦051 364 130 561 988 410 828 243 653 *o6o 463 862 258 650 ♦038 423 805 183 558 930 298 664 *027 386 743 *o96 447 795 140 483 823 160 494 826 156 483 808 130 450 767 *o82 395 675 706 983 *oH 290 594 897 197 495 791 085 377 667 320 625 927 227 524 820 114 406 "696 73 604 030 452 870 284 694 *ioo 503 902 297 689 *077 461 843 221 595 967 335 700 063 422 *I32 482 830 175 517 857 193 528 860 189 516 840 162 481 799 *ii4 426 737 '045 351 655 957 256 554 850 143 435 725 217 647 ♦072 494 912, 325 735 '141 543 941 689 *ii5 536 953 366 776 *i8i 583 981 336 727 500 881 258 633 '004 372 737 ^099 458 814 *i67 517 864 209 551 890 227 561 893 222 548 872 194 513 830 *i45 457 768 ♦076 381 685 987 286 584 879 173 464 754 260 732 *i57 578 995 407 816 *222 623 021 376 766 *I54 538 918 296 670 *04i 408 773 35 493 849 552 899 243 585 924 261 594 926 254 581 905 226 545 862 *I76 489 799 *io6 412 715 *oi7 316 613 909 202 493 782 303 415 805 *I92 576 956 333 707 *o78 445 809 *i7i 529 884 '237 587 934 278 619 958 294 628 959 287 613 937 258 577 893 ^208 520 829 *I37 442 746 *o47 346 643 938 231 522 "sTT S 9 Prop. Pts. 346 775 *I99 620 ♦036 449 857 *262 663 *o6o 454 817 ♦242 662 =078 490 898 *302 703 ICX) 493 844 ♦231 614 994 371 744 = 115 482 846 *207 565 920 '272 621 968 312 653 992 327 661 992 320 646 969 290 609 925 *239 551 860 *i68 473 *o77 376 673 967 260 551 840 389 883 ♦269 652 ♦032 408 781 *i5i 518 882 =1=243 600 955 *307 656 *oo3 346 687 *02 5 361 694 *024 352 678 *OOI 322 640 956 *270 582 "89? ♦198 503 806 *io7 406 702 997 289 580 9 44 43 4* 4-4 4 3 4 8 8 8 6 8 13 2 12 9 12 17 6 17 2 16 22 21 5 21. 26 4 25 8 25- 30 8 30 I 29 35 2 34 4 33 39 6 38 7 37- 41 40 39 12 o 16 o 20 o 61240 7 28.0 8 32.0 9I36.0 37 3-7 7-4 II. I 14-8 18.5 22.2 25 9 29.6 33-3 3 9 7 II 7 15 6 19 5 23 4 27 3 31-2 35 » 36 3-6 7- 35 34 33 3-5 3 4 3 7 6.8 6. 10 5 10 2 9 14 136 13 17 5 17.0 16. 21 20.4 19 24 5 238 23 28 27 2 26. 31 5 30.5 29- 3a 31 30 16.0 19 2 22.4 25.6 28-8 3-0 6.C 9.0 12 o 15 5 »50 18 6 18 o 21 7 24 8 27 9 21 .0 24 o 27.0 Prop. Pts. LOGARITHMS OF NUMBERS N. O 1 2 3 4 5 6 y S 1 9 Prop. Pts. 150 51 17 609 898 638 926 667 955 696 984 725 *oi3 754 782 811 840 869 *04i *o7o *o99 *I27 *i56 29 28 52 iS 184 2n 241 270 298 327 355 3^4 412 441 53 469 498 526 554 583 611 639 667 696 724 1 2 2.9 5 8 2.8 5.6 54 752 780 808 «37 865 «93 921 949 977 *oo5 3 8.7 8.4 55 19033 061 089 117 145 173 201 229 257 285 4 n.6 11.2 56 312 340 368 396 424 451 479 507 535 562 5 14.5 14.0 57 590 618 645 673 700 728 756 783 811 838 6 17.4 16.8 58 893 921 948 976 *oo3 ♦030 *o58 *o85 *II2 '/ 20.3 19.6 59 160 61 20 140 167 194 222 249 276 303 330 602 358 629 385 656 8 9 23.2 26.1 27 22.4 25.2 26 412 683 439 466 493 520 548 575 710 737 763 790 817 844 871 898 925 62 952 978 *oo5 *032 *o59 *o85 *II2 *I39 *i65 *,92 1 2.7 5 4 2.6 5.2 63 21 219 245 272 299 325 352 378 405 431 458 64 484 511 537 564 590 617 643 669 696 722 3 8.1 7.8 65 748 11^ 801 827 854 880 906 932 958 985 4 10.8 10.4 66 22 on 037 063 089 115 141 167 194 220 246 5 13.5 13.0 67 272 298 324 350 376 401 427 453 479 505 6 7 8 9 16.2 18.9 21.6 24 3 15.6 18.2 20.8 9.^ A. 68 531 557 5«3 608 634 660 686 712 737 .7^3 69 170 71 789 814 840 866 891 917 943 ,968 994 *oi9 23 045 070 096 121 147 172 ' 198 223 249 274 300 32s 350 376 401 426 452 477 502 528 25 72 553 603 629 654 679 704 729 754 779 1 2 5 73 805 830 «55 880 905 930 955 989 *oo5 *o3o 2 5.0 74 24 055 080 los 130 155 180 204 229 254 279 3 7.5 75 304 329 35^ V^ 403 428 452 477 502 527 4 10.0 76 551 576 601 625 650 674 699 724 748 773 5 6 7 12.5 15.0 17.5 77 797 822 846 871 895 920 944 969 993 *oi8 78 25 042 066 091 115 139 164 188 212 237 261 8 20.0 79 180 81 285 527 768 310 334 358 3«2 406 ^648^ 888~ 431 672 912 455 479 503 9 22.5 551 575 600 624 864 696 720 744 983 . . 1 792 816 840 935 959 224: 23 82 26 007 031 055 079 102 126 150 174 198 221 1 2.4 2.3 83 245 269 293 316 340 3^4 3^7 411 435 458 2 4.8 4.6 84 482 505 529 553 576 600 623 647 670 694 3 7.2 6.9 9.2 85 717 741 764 788 811 8s4 858 881 90s 928 4 9.6 86 951 975 998 *02I *o45 *o68 *o9i Hl^ *i38 *i6i 5 6 12.0 14 4 11.5 13.8 87 27 184 207 231 254 277 300 323 346 370 393 7 16.8 16.1 88 416 439 462 485 S08 531 554 577 600 623 8 19.2 18.4 89 190 91 646 669 692 715 738 761 989 217 784 807 830 852 9 21.6 20.7 J71 28 103 898 126" 921 149 944 967 *OI2 *o35 *o58 *o8i .. .. 1 171 194 240 262 285 307 22 Zi 21 92 330 353 375 398 421 443 466 488 511 533 1 2.2 2.1 93 556 578 601 623 646 668 691 713 735 758 2 4.4 4.2 94 780 803 825 847 870 892 914 937 959 981 3 4 b.b 8 8 6.3 ft 4 95 29 003 026 048 070 092 115 137 159 181 203 ;=; n fi 10.5 12.6 96 226 248 270 292 314 336 35« 380 403 425 6 13.2 97 447 469 491 513 535 557 579 601 623 64s 7 15.4 14.7 98 667 688 710 732 754 776 798 820 842 863 8 17.6 16.8 99 200 885 30 103 907 929 951 973 994 211 *oi6 *o38 *o6o *o8i 9 19.8 18.9 125 146 168 190 233 255 276 298 1 2 3 4 5 6 7 « «| Prop. Pts. TABLE I N. 1 2 3 4 5 6 7 8 9 Prop. Pt8. 200 01 02 03 30 103 320 535 750 12? 146 168 190 211 233 255 276 298 341 557 771 363 578 792 384 600 814 406 621 835 428 643 856 449 664 878 471 685 899 492 707 920 5'1 728 042 1 2 3 4 5 22 2.2 4.4 6.6 8.8 11.0 21 2.1 4.2 6.3 8,4 10.5 04 05 06 963 31 175 387 984 408 *oo6 218 429 *027 239 450 *048 260 471 *o69 281 492 ♦091 302 513 *II2 323 534 ♦133 345 555 *I54 366 576 07 08 09 210 11 12 13 597 32 015 618 827 035 639 848 056 "263- 660 869 077 284 681 890 098 305 702 118 723 931 139 744 952 160 765 973 181 785 994 201 6 7 8 9 13.2 15.4 17.6 19.8 12.6 14.7 16.8 18.9 222 243 325 346 366 387 408 '613 818 *02I 428 634 838 449 654 858 469 675 879 490 695 899 510 715 919 531 736 940 756 960 572 777 980 593 797 *OOI 1 20 2.0 4.0 14 15 16 33 041 244 445 062 264 465 082 284 486 102 304 506 122 526 143 546 163 365 566 183 385 586 203 224 425 626 3 4 5 6.0 8.0 10.0 17 18 19 220 21 22 23 646 846 34 044 666 866 064 686 885 084 706 90s 104 726 925 124 746 945 143 766 965 163 786 985 183 806 *oo5 203 826 *02 5 223 6 7 8 9 1 2 12.0 14.0 16.0 18.0 19 1.9 3.8 242 439 635 830 262 282 301 321 341 361 380 400 420 616 811 *oo5 459 850 479 674 869 498 518 908 537 733 928 557 753 947 577 772 967 596 986 24 25 26 35 025 218 411 044 238 430 064 257 449 083 276 468 102 488 122 315 507 141 334 526 160 353 545 180 372 564 199 392 583- 3 4 5 6 7 8 9 5.7 7.6 9.5 11.4 13.3 15.2 17.1 27 28 29 230 31 32 33 603 984 622 813 *oo3 641 832 *02I 660 851 *o4o 679 870 *o59 698 889 *o78 717 908 *o97 736 927 *ii6 755 946 *i35 774 965 *I54 36 173 192 211 229 248 267 286 305 324 342 .. 1 361 549 736 380 568 754 399 586 773 418 605 791 436 624 810 642 829 474 661 847 493 680 866 511 698 884 530 717 903 1 2 1» 1.8 3.6 34 35 36 922 37 107 291 940 125 310 959 144 328 977 162 346 996 181 365 *oi4 If. *033 218 401 *o5i 236 420 ♦070 254 438 *o88 273 457 3 4 5 6 5.4 7.2 9.0 10 8 37 38 H9 240 41 42 43 475 658 840 493 676 858 1" 694 876 530 712 894 548 731 912 566 749 931 767 949 603 967 621 803 985 639 822 *oo3 7 8 9 1 2 12.6 14.4 16.2 17 1.7 3.4 5.1 6.8 8.5 10.2 38 021 039 057 075 093 112 292 471 650 130 148 166 184 202 561 220 399 578 238 417 596 256 614 274 632 489 668 328 507 686 346 525 703 364 543 721 44 45 46 739 39 094 757 934 III 775 952 129 792 970 146 810 987 164 828 *oo5 182 846 *023 199 863 ^^04 1 217 881 ♦058 235 899 ♦076 252 3 4 5 6 47 48 49 250 270 445 620 287 463 637 305 480 655 498 672 340 515 690 358 533 707 375 550 724 III 742 410 585 759 428 602 777 7 8 9 11.9 13.6 15.3 794 811 829 846 863 881 898 915 933 950 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 7 N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 250 51 52 52 39 794 811 829 846 863 881 898 915 933 950 967 40 140 312 985 157 329 ♦002 175 346 ♦019 192 364 *037 209 381 *o54 226 398 *o7i 243 415 *o88 261 432 *io6 278 449 *I23 295 466 1 2 3 4 5 18 1.8 3.6 5.4 7.2 9 54 55 56 483 654 824 500 671 841 518 688 858 535 705 875 552 722 892 569 739 909 586 756 926 603 773 943 620 790 960 637 807 976 57 58 59 260 61 62 63 993 41 162 330 497 *OIO 179 347 *027 196 363 *o44 212 380 l47 *o6i 229 397 564 *o78 246 414 581 *o95 263 430 *iii 280 447 *I28 296 464 *i45 313 481 647 6 7 8 9 1 2 10.8 12.6 14.4 16.2 17 1.7 3.4 514 531 597 614 631 664 830 996 681 847 *OI2 697 863 *029 714 880 *o45 731 896 ♦062 747 913 *o78 764 929 *o95 780 946 *iii 797 963 *I27 814 979 *i44 64 65 66 42 160 325 488 177 341 504 193 357 521 210 374 537 226 390 553 406 570 259 423 586 275 439 602 292 455 619 308 472 635 3 4 5.1 6.8 S.5 67 68 69 270 71 72 73 651 813 975 667 830 991 684 846 ♦008 700 862 *024 716 878 *040 732 894 ♦056 217 749 911 *072 765 927 ♦088 781 943 *io4 797 959 *I20 6 7 8 9 1 2 10.2 11.9 13.6 15.3 10 1.6 3.2 43 136 152 169 185 201 233 249 265 281 297 457 616 313 473 632 329 489 648 345 505 664 361 521 680 377 537 696 393 553 712 409 569 727 425 584 743 441 600 759 74 75 76 775 933 44 091 791 949 107 807 965 122 823 981 138 838 996 154 854 *OI2 170 870 *028 185 886 *o44 201 902 *059 217 917 *o75 232 3 4 5 6 7 8 9 4.8 6.4 8.0 9.6 11.2 12.8 14.4 77 78 79 280 81 82 83 248 404 560 264 420 576 279 43^^ 592 295 451 607 467 623 326 483 638 342 498 654 358 514 669 373 529 685 389 545 700 716 731 747 762 778 793 809 824 840 855_ *OIO 163 317 .. 1 871 45 025 179 886 040 194 902 056 209 917 071 225 932 086 240 948 102 255 963 117 271 979 133 286 994 148 301 1 2 15 1.5 3.0 84 85 86 332 484 637 347 500 652 362 515 667 378 530 682 393 545 697 408 561 712 423 576 728 439 591 743 454 606 758 469 621 773 3 4 5 6 4.5 6.0 7.5 9 87 88 89 290 91 92 93 788 939 46 090 803 954 105 818 969 120 834 984 135 849 *ooo 150 864 *oi5 165 879 894 *o45 195 909 *o6o 210 924 *o75 225 7 8 9 10.5 12.0 13.5 240 "389" 687 255 404 553 702 270 285 300 315 330 345 359 374 .. 1 419 568 716 434 583 731 449 598 746 464 613 761 479 627 776 494 642 790 509 657 805 672 820 1 2 3 4 5 6 J.* 1.4 2.8 94 95 96 835 982 47 129 850 997 144 864 *OI2 159 879 ♦026 173 894 *04i 188 909 *o56 202 923 *o7o 217 938 +085 232 953 *IOO 246 967 *ii4 261 4.2 5.6 70 8.4 97 98 99 300 276 422 567 290 436 582 305 451 596 319 465 611 334 480 625 770 349 494 640 784 363 509 654 799 378 524 669 I13 392 538 683 828 407 553 698 842 7 8 9 9.8 11.2 12.6 712 727 741 756 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 8 TABLE I N. 1 2 9 4 5 6 7 8 9 Prop. Pts. 300 01 02 03 47 712 727 741 756 770 784 799 813 828 842 c^57 48 001 144 871 015 159 885 029 173 yoo 044 187 914 058 202 929 073 216 943 C87 230 958 lOI 244 972 116 259 986 130 273 15 04 Go OG 287 430 572 302 444 586 316 458 601 330 473 615 344 487 629 359 643 373 657 387 401 544 686 416 558 700 1 2 3 1.5 3.0 4.5 07 08 09 310 11 12 13 996 728 869 *0I0 742 883 *024 756 897 *o38 770 911 ♦052 785 926 *o66 799 940 *o8o 813 954 *094 827 968 *io8 841 982 *I22 4 5 6 7 8 9 6.0 7.5 9.0 10.5 12.0 13.6 49 136 150 164 178 192 206 220 234 248 262 402 541 679 276 415 554 290 429 568 304 443 582 318 457 596 332 471 610 346 485 624 3^ 638 374 513 651 388 527 665 14 15 16 693 831 969 707 721 859 996 734 872 *OIO 748 886 *024 762 900 *o37 776 914 ♦051 790 927 *o65 803 941 *o79 817 955 ♦092 1 14 1.4 17 18 19 320 21 22 23 50 106 243 379 515 120 256 393 529 133 270 406 284 420 161 297 433 174 311 447 188 325 461 202 338 474 215 352 229 365 501 2 3 4 5 6 7 8 9 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 542 556 569 583 596 610 623 637 651 786 920 664 799 934 678 813 947 691 826 961 705 840 974 718 853 987 732 866 *OOI 745 880 *oi4 759 893 *028 772 907 *04i 24 25 26 51 055 188 322 068 202 335 081 215 348 095 228 362 108 242 375 121 255 388 135 268 402 148 282 415 162. 255 428 175 308 441- 27 28 29 830 31 32 33 455 587 720 468 601 733 481 614 746 495 627 759 508 640 772 654 786 534 667 799 548 680 812 693 825 574 706 838 1 2 3 4 5 6 7 13 1.3 2.6 3.9 5.2 6.5 7.8 9.1 851 865 878 891 904 917 930 943 957 970 983 52 114 244 996 127 257 *oo9 140 270 *022 III *o35 166 297 *048 179 310 *o6i 192 323 *o75 205 336 *o88 218 349 *IOI 231 362 34 35 36 375 504 634 388 517 647 401 530 660 414 543 673 427 lit 440 569 699 453 582 711 466 595 724 479 608 737 492 621 750 8 9 10.4 11.7 37 38 39 340 41 42 43 763 892 53 020 148 776 905 033 789 917 046 802 930 058 815 943 071 827 956 084 840 969 097 853 982 no 866 994 122 879 *oo7 135 1 2 3 4 5 12 1.2 2.4 3.6 4.8 6 161 173 186 199 212 224 237 250 263 275 403 529 288 415 542 301 428 555 314 441 567 326 453 580 339 466 593 352 479 605 364 491 618 377 504 631 390 643 44 45 46 656 782 908 668 794 920 681 807 933 694 820 945 706 832 958 719 845 970 732 7s 744 870 995 757 882 *cx)8 769 895 *020 6 7 8 7.2 8.4 9.6 47 48 49 350 54 033 158 283 045 170 295 058 183 307 070 195 320 083 208 332 095 220 345 108 233 357 120 245 370 i 145 270 394 9 10.8 407 419 432 444 456 469 481 494 506 518 N. 1 2 3 4 5 6 7 s 9 Prop. Pts. LOGARITHMS OF NUMBERS 9 N. 1 2 3 4 5 6 7 S 9 Prop. Pts. 350 61 52 53 54 407 419 432 444 456 469 481 494 506 518 531 654 777 543 667 790 555 679 802 568 691 814 580 704 827 716 839 605 728 851 617 741 864 630 753 876 642 765 888 13 54 55 56 900 55 023 145 913 035 157 925 047 169 937 060 182 949 072 194 962 084 206 974 096 218 986 108 230 998 121 242 *oii 133 255 1 2 3 1.3 2.6 3.9 57 58 59 360 61 62 63 267 388 509 279 400 522 291 413 534 303 425 546 315 437 558 328 449 570 691 340 461 582 352 473 594 364 376 497 618 4 5 6 7 8 9 5.2 6.5 7.8 9.1 10.4 11.7 630 871 991 642 654 666 678 703 715 727 739 763 883 *oo3 775 895 *oi5 787 907 *027 799 919 *o38 811 931 *o5o 823 943 *o62 835 955 *o74 847 967 ♦086 859 979 ♦098 64 65 66 56 no 229 348 122 241 360 134 253 372 146 265 384 158 277 .396 170 289 407 182 301 419 194 312 431 205 324 443 217 336 455 1 12 1.2 67 68 69 370 71 72 73 467 585 703 478 597 714 490 608 726 502 620 738 5H 632 750 526 644 761 656 773 549 667 785 561 679 797 573 691 808 2 3 4 5 6 7 8 9 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 820 832 844 855 867 879 891 902 914 +031 148 264 926 *043 159 276 937 57 054 171 949 066 183 961 078 194 972 089 206 984 lOI 217 996 113 229 *oo8 124 241 *oi9 136 252 74 75 76 287 403 519 299 415 530 310 426 542 322 438 553 334 449 565 461 576 357 473 588 368 484 600 380 496 611 392 507 623 77 78 79 380 81 82 83 634 749 864 978 646 761 875 657 772 887 669 784 898 680 795 910 692 807 921 703 818 933 830 944 726 841 955 852 967 *o8i 195 309 422 1 2 3 4 5 6 7 11 1.1 2.2 3.3 4.4 5.5 6.6 7.7 990 *OOI *oi3 *024 *o35 *047 ♦058 ♦070 58 092 206 320 104 218 331 115 229 343 127 240 354 138 252 365 149 26^ 377 161 274 388 172 286 399 184 297 410 84 85 86 433 546 659 444 557 670 456 569 681 467 580 692 478 591 704 490 602 715 501 614 726 512 625 737 524 636 749 647 760 8 9 8.8 9.9 87 88 89 390 91 92 93 771 883 995 59 106 782 894 *oo6 T18 794 906 *oi7 805 917 *028 816 928 ♦040 827 939 ♦051 838 950 *o62 850 961 ♦073 861 973 ♦084 872 984 ♦095 1 2 3 4 5 10 1.0 2.0 3.0 4.0 5 129 140 151 162 173 184 195 207 318 428 539 218 329 439 229 340 450 240 461 362 472 262 373 483 273 384 494 284 395 506 7J> 517 306 417 528 94 95 96 550 660 770 671 780 572 682 791 693 802 594 704 813 605 824 616 726 835 627 737 846 638 748 857 649 6 7 8 6.0 7.0 8.0 97 98 99 400 N. 60 097 "206 890 ?o1 901 *OIO 119 912 *02I 130 923 ♦032 141 934 *o43 152 945 "054 163 956 *o65 173 966 +076 184 977 *o86 195 9 9.0 217 228 239 249 260 271 282 293 304 1 2 3 4 5 7 S 9 Prop. Pts. 10 TABLE I N. O 1 2 3 4 5 6 7 S 9 Prop. Pta. 400 01 02 03 6o 2o6 217 228 239 249 260 271 282 293 304 3H 423 531 325 433 541 336 444 552 347 455 563 358 466 574 369 477 584 379 487 595 390 498 606 401 509 617 412 520 627 04 05 06 638 746 853 649 756 863 660 767 874 670 778 885 681 788 895 692 799 906 703 810 917 713 82 F 927 724 83/ 938 P5 842 949 11 07 08 09 410 11 12 13 959 61 066 172 970 077 183 981 087 194 991 098 204 *002 109 215 *oi3 119 225 ♦023 130 236 *034 140 247 *o45 151 257 *o55 162 268 1 2 3 4 5 6 7 8 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 278 289 300 310 321 331 342 352 363 374 384 490 595 395 500 606 405 511 616 416 521 627 426 532 637 437 542 648 448 553 658 458 563 669 469 574 679 479 584 690 14 15 16 700 805 909 711 815 920 721 826 930 731 836 941 742 847 951 752 857 962 763 868 972 773 878 982 784 888 993 794 899 *cx)3 9 9.9 17 18 19 420 21 22 23 62 014 118 221 024 128 232 034 138 242 045 149 252 055 159 263 066 170 273 076 180 284 086 190 294 097 201 304 107 211 315 325 335 346 356 366 377 387 397 408 418 428 634 439 542 644 449 655 459 665 469 572 675 480 583 685 490 593 696 500 603 706 613 716 521 624 726 1 2 1.0 2.0 24 25 26 737 839 941 747 849 951 757 859 961 767 870 972 778 880 982 788 890 992 798 900 *002 808 910 *OI2 818 921 *022 829 931 *o33 3 4 5 6 3.0 4.0 5.0 6 27 28 29 430 31 32 33 63 043 144 246 053 155 256 063 165 266 073 175 276 083 185 286 094 195 296 104 205 306 114 215 317 124 225 327 134 236 337 7 8 9 7.0 8.0 9.0 347 357 367 377 387 397 407 417 428 438 639 739 448 548 649 458 659 468 568 669 478 579 679 488 589 689 498 599 699 508 609 709 619 719 629 729 34 35 36 749 849 949 759 859 959 769 869 969 779 879 979 789 889 988 799 899 998 809 909 *oo8 819 919 *oi8 829 929 *028 839 939 *038 9 37 38 39 440 41 42 43 64 048 147 246 058 157 256 068 167 266 078 177 276 088 187 286 098 197 296 108 207 306 118 217 316 128 227 326 137 237 335 1 2 3 4 5 6 7 8 0.9 1.8 2.7 3.6 4.5 5.4 6 3 7.2 345 355 365 375 385 395 404 414 424 434 444 542 640 454 650 464 562 660 473 572 670 483 582 680 493 591 689 503 601 699 611 709 621 719 532 631 729 44 45 46 738 836 933 748 846 943 758 856 953 768 865 963 777 875 972 787 885 982 797 895 992 807 904 *002 816 914 826 924 ♦021 9 8.1 47 48 49 450 65 031 128 225 040 137 234 050 147 244 060 157 254 070 167 263 079 176 273 369 089 186 283 099 196 292 108 205 302 118 215 312 408 321 331 341 350 360 379 389 398 N. i 2 3 4 5 6 7 § Prop. Pts. LOGARITHMS OF NUMBERS 11 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 450 51 52 53 65 321 331 341 350 360 369 379 389 398 408 418 514 610 427 523 619 437 533 629 447 543 639 456 552 648 466 562 658 475 571 667 485 581 677 495 504 600 696 54 55 56 706 801 896 811 906 725 820 916 734 830 925 744 839 935 753 849 944 763 858 954 772 868 963 782 877 973 792 887 982 10 57 58 59 4G0 61 62 63 992 66 087 181 276 *OOI 096 191 *OII 106 200 *020 115 210 *o3o 124 219 *039 134 229 *o49 143 238 ♦058 153 247 *o68 162 257 *o77 172 266 1 2 3 4 5 6 7 8 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 285 295 304 3H 408 502 596 323 332 342 351 361 455 549 642 370 464 558 380 474 567 389 483 577 398 492 586 417 427 521 614 436 530 624 445 633 64 65 66 652 745 839 661 755 848 671 764 857 680 773 867 689 783 876 699 792 885 708 801 894 717 811 904 727 820 913 736 829 922 9 9.0 67 68 69 470 71 72 73 932 67 025 117 941 034 127 950 043 136 960 052 145 969 062 154 978 071 164 987 080 173 997 089 182 *oo6 099 191 201 210 219 228 237 247 256 265 274 284 293 1 2 3 4 6 6 9 0.9 1.8 2.7 3.6 4.5 5.4 302 486- 311 403 495 321 413 504 330 422 514 339 431 523 348 440 532 357 449 541 367 459 550 376 468 560 385 477 569 74 75 76 578 669 761 587 679 770 596 688 779 605 697 78S 614 706 797 624 715 806 633 724 815 642 733 825 651 742 834 660 752 843 77 78 79 480 81 82 83 852 68 034 861 952 043 870 961 052 879 970 061 888 979 070 897 988 079 906 997 088 916 *oo6 097 925 *oi5 106 934 *024 115 7 8 9 6.3 7.2 8.1 124 215 305 395 133 142 151 160 169 178 187 196 205 224 314 404 233 323 413 242 332 422 251 341 431 260 350 440 269 359 449 278 368 458 287 377 467 296 386 476 84 85 86 485 574 664 673 502 592 681 511 601 690 520 610 699 529 538 628 717 547 637 726 Pi 646 735 655 744 8 87 88 89 490 91 92 93 753 842 931 762 851 940 771 860 949 780 869 958 878 966 797 886 975 806 895 984 815 904 993 824 913 *002 833 922 *9ii 1 2 3 4 5 6 7 8 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 69 020 028 037 046 055 064 073 082 090 099 108 197 285 117 205 294 126 214 302 135 223 311 144 232 320 152 241 329 161 249 338 170 258 346 179 267 355 188 276 364 94 95 96 373 461 548 469 557 390 478 566 399 487 574 408 417 504 592 425 513 601 434 522 609 443 452 627 9 7.2 97 98 99 500 636 723 810 644 732 819 653 740 827 662 749 836 671 758 845 932 679 767 854 688 775 862 697 784 871 705 III 714 801 888 897 906 914 923 940 949 958 966 975 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 12 TABLE I N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 600 01 02 03 69 897 984 70 070 157 906 914 923 932 940 949 958 966 975 992 079 165 *OOI 088 174 *OIO 096 183 *oi8 105 191 *027 114 200 ♦036 122 209 *044 131 217 *o53 140 226 *o62 148 234 04 05 06 243 329 415 252 338 424 260 346 432 269 355 441 278 364 449 286 458 295 467 303 389 475 312 321 406 492 9 07 08 09 510 11 12 13 501 586 672 757 842 927 71 012 509 595 680 518 603 689 612 697 P5 621 706 791 876 961 046 544 629 714 800 552 638 723 808 561 646 731 569 655 740 578 663 749 1 2 3 4 5 3 7 8 0.9 1.8 2.7 3 6 4.5 0.4 6.3 7.2 766 851 935 020 774 859 944 029 783 868 952 037 817 825 834 885 969 054 893 978 063 902 986 071 910 995 079 919 *oo3 088 U 15 16 265 105 189 273 "3 282 122 206 290 130 214 299 139 223 307 147 231 315 i9'5 240 324 248 332 172 257 341 9 8.1 17 18 19 520 21 22 23 349 433 517 357 441 525 366 450 533 374 458 542 383 466 550 391 475 559 399 483 567 408 492 575 416 500 584 592 8 10.8 21.6 600 684 767 850 609 617 625 634 642 650 659 667 675 692 775 858 700 784 867 709 792 875 717 800 883 725 809 892 734 817 900 742 825 908 750 834 917 759 842 925 24 25 26 933 72 016 099 941 024 107 950 032 115 958 041 123 966 049 132 975 057 140 983 066 148 991 074 156 999 082 165 *oo8 090 173 4 5 6 3.2 4.0 4.8 27 28 29 530 31 32 33 181 263 346 428 189 272 354 198 280 362 206 288 370 214 296 378 222 304 387 230 313 395 239 321 403 247 329 411 255 337 419 7 8 9 5.6 6.4 7.2 436 444 452 460 469 477 485 493 501 509 591 673 518 599 681 607 689 534 616 697 1^^ 624 705 550 632 713 558 640 722 567 648 730 575 656 738 583 665 746 34 35 36 ^54 835 916 762 843 925 770 852 933 779 860 941 787 868 949 795 876 957 803 884 965 811 892 973 819 900 981 827 908 989 7 37 38 39 540 41 42 43 997 73 078 159 239 *oo6 086 107 *oi4 094 175 *022 102 183 ♦030 III 191 *o38 119 199 *o46 127 207 *o54 135 215 *o62 143 223 ♦070 151 231 1 2 3 4 5 6 7 8 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5 6 247 255 263 272 280 288 296 304 312 320 400 480 328 408 488 336 416 496 344 424 504 352 432 512 360 440 520 368 448 528 376 456 536 384 464 544 392 472 552 44 45 46 560 719 568 648 727 576 656 735 584 664 743 592 672 751 600 679 759 608 687 767 616 695 775 624 703 783 632 711 791 9 6.3 47 48 49 650 957 807 886 965 815 894 973 823 902 981 830 910 989 838 918 997 846 926 *cx>5 854 933 *oi3 862 941 *020 870 949 *028 74 036 044 052 060 068 076 084 092 099 107 N. 1 2 3 4 5 6 7 S 9 Prop. Pts. LOGARITHMS OF NUMBERS 13 N. O 1 2 3 4 5 6 7 S 9 Prop. Pt8. 51 52 53 74 036 044 052 060 068 076 084 092 099 107 115 194 273 123 202 280 131 210 288 139 218 296 147 225 304 155 233 312 162 241 320 170 249 327 178 257 335 186 265 343 54 55 56 351 429 507 359 437 515 367 445 523 374 453 531 382 4.61 539 390 468 547 398 476 554 406 484 562 414 492 570 421 500 578 57 58 59 560 61 62 63 IS 741 593 671 749 601 679 757 609 687 764 617 695 772 624 702 780 632 710 788 640 718 796 648 726 803 656 733 811 889 819 827 834 842 850 858 865 873 881 896 974 75 051 904 981 059 912 989 066 920 997 074 927 *oo5 082 935 *OI2 089 943 *020 097 950 *028 105 958 *o35 113 966 *043 120 1 2 0.8 1.6 64 65 66 128 205 282 136 213 289 143 220 297 151 228 305 159 236 312 166 243 320 174 328 182 259 335 189 266 343 197 274 351 3 4 5 6 2.4 3.2 4.0 4.8 67 68 69 670 71 72 73 358 435 5" 587 366 442 519 595 374 450 526 458 534 389 465 542 618 397 473 549 626 404 481 557 633 412 488 565 420 496 572 427 504 580 7 8 9 5.6 6.4 7.2 603 610 641 648 656 664 740 815 671 747 823 679 831 686 762 838 694 770 846 702 778 853 709 785 861 717 868 724 800 876 732 808 884 74 75 76 891 967 76 042 899 974 050 906 982 057 914 989 065 921 997 072 929 *oo5 080 937 *OI2 087 944 *020 095 952 *027 103 959 *o35 no 77 78 79 580 81 82 83 118 193 268 125 200 275 III 283 140 215 290 148 223 298 155 230 305 1% 313 170 245 320 178 253 328 185 260 335 343 350 358 365 373 380 388 395 403 410 418 492 567 425 500 574 433 507 582 440 515 589 448 522 597 455 530 604 462 537 612 470 545 619 477 III 486 559 634 1 2 7 0.7 1.4 84 85 86 87 88 89 590 91 92 93 641 716 790 864 938 77 012 649 723 797 871 945 019 656 730 805 879 026 664 If. 886 960 034 671 819 893 967 041 678 753 827 901 975 048 686 760 834 908 982 056 842 916 989 063 701 775 849 923 997 070 708 782 856 930 *oo4 078 3 4 5 6 7 8 9 2.1 2.8 3.5 4.2 4.9 5.6 6.3 085 093 100 107 115 122 129 137 144 151 159 232 305 166 240 313 173 247 320 181 254 327 188 262 335 269 342 203 276 349 210 283 357 217 291 364 225 298 371 94 95 96 379 452 525 386 459 532 393 466 539 401 474 546 408 481 554 415 488 561 422 430 503 576 437 583 444 517 590 97 98 99 670 743 605 677 750 612 685 757 619 692 764 627 699 772 634 706 779 641 714 786 648 721 793 866 656 728 801 663 735 GOO 815 822 830 837 844 851 859 873 880 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 14 TABLE I N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 600 01 02 03 n 815 887 960 78 032 822 830 837 844 851 859 866 873 880 895 967 039 902 974 046 909 981 053 916 988 061 924 996 068 931 *oo3 075 938 *OIO 082 945 *oi7 089 952 *025 097 04 05 06 104 176 247 III 183 254 118 190 262 125 197 269 132 204 276 140 211 283 147 219 290 226 297 161 233 305 168 240 312 8 07 08 09 GIO 11 12 13 319 390 462 326 398 469 333 405 476 340 412 483 347 419 490 355 426 497 362 433 504 369 440 512 376 447 519 383 455 526 1 2 3 4 6 6 7 8 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 533 540 547 554 561 569 576 583 590 597 604 675 746 611 682 753 618 689 760 625 696 767 633 704 774 640 711 781 647 718 789 654 725 796 661 732 803 668 739 810 14 15 16 817 888 958 824 895 965 831 902 972 838 909 979 845 916 986 852 923 993 859 930 *ooo 866 937 *oo7 873 944 *oi4 880 951 *02I 9 7.2 17 18 19 620 21 22 23 79 029 099 169 036 106 176 043 113 183 050 120 190 057 127 197 064 134 204 274 071 141 211 078 148 218 085 155 225 092 162 232 239 246 253 260 267 281 288 295 302 309 379 449 316 386 456 323 393 463 330 400 470 337 407 477 344 414 484 351 421 491 358 428 498 365 435 505 372 442 511 1 2 7 0.7 1.4 24 25 26 518 588 657 525 595 664 532 602 671 539 609 678 546 616 685 553 623 692 560 630 699 637 706 574 644 713 650 720 3 4 5 2.1 2.8 3.5 4 9 27 28 29 630 31 32 33 727 796 865 734 803 872 810 879 748 817 886 754 824 893 761 831 900 768 837 906 775 844 913 782 851 920 789 858 927 7 8 9 4.9 5.6 6.3 934 941 948 955 962 030 168 969 975 982 989 996 80 003 072 140 010 079 147 017 085 154 024 092 161 037 106 175 044 051 120 188 058 127 195 06s 134 202 34 35 36 209 277 346 216 284 353 223 291 359 298 366 236 305 373 243 312 380 250 387 257 325 393 264 332 400 271 339 407 6 37 38 39 640 41 42 43 414 482 550 421 489 557 428 496 564 434 502 570 441 509 577 448 584 652 455 523 659 462 530 598 468 536 604 475 543 611 1 2 3 4 5 6 7 8 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4 8 618 625 632 638 645 665 672 679 686 754 821 693 760 828 699 767 835 706 774 841 713 781 848 720 787 855 726 794 862 733 801 868 740 808 875 747 814 882 44 45 46 889 956 81 023 895 963 030 902 969 037 909 976 043 916 983 050 922 990 057 929 996 064 936 *oo3 070 943 *oio 077 949 ♦017 084 9 5.4 47 48 49 650 090 158 224 097 164 231 104 171 238 III 178 245 117 184 251 124 258 131 198 265 137 204 271 144 211 278 218 285 291 298 305 3" 318 325 331 338 345 351 N. 1 2 3 4 5 6 7 8 9 Prop Pts. LOGAEITHMS OF NUMBERS 15 N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 650 51 62 53 8 1 291 298 305 311 318 325 331 338 345 351 358 425 491 365 498 438 505 378 445 511 385 518 391 458 525 398 465 531 405 471 538 411 478 544 418 485 551 54 55 56 |5» 624 690 564 631 697 637 704 578 644 710 584 651 717 591 657 723 598 664 730 604 671 737 611 677 743 617 684 750 57 58 59 660 61 62 63 823 889 763 829 895 770 836 902 776 842 908 783 849 915 790 856 921 796 862 928 803 869 935 809 875 941 816 882 948 954 82 020 086 151 961 968 974 981 987 994 *CXXD *oo7 *oi4 027 092 158 033 099 164 040 105 171 046 112 178 053 184 060 125 191 066 132 197 073 138 204 079 145 210 1 2 0.7 1.4 64 65 66 217 282 347 223 289 354 230 295 360 236 302 367 Itl 373 249 315 380 256 387 263 328 393 269 334 400 276 406 3 4 5 6 2.1 2.8 3.5 4.2 67 68 69 670 71 72 73 413 478 543 419 484 549 426 491 556 432 497 562 439 504 569 445 510 575 452 582 458 Pi 465 530 595 471 536 601 7 8 9 4.9 5.6 6 3 607 672 802 614 620 627 633 640 646 653 659 666 679 III 685 750 814 692 756 821 698 763 827 705 769 834 711 776 840 718 782 847 724 789 853 730 795 860 74 75 76 866 930 995 872 937 *OOI 879 943 *cx)8 885 950 *oi4 892 956 *020 898 963 *027 905 969 *033 911 975 *04o 918 982 *o46 924 988 ♦052 77 78 79 680 81 82 83 83 059 123 187 065 129 193 072 136 200 078 142 206 085 149 213 091 155 219 097 161 225 104 168 232 no 174 238 117 181 245 251 257 264 270 276 283 289 296 302 308 315 378 442 321 385 448 327 391 455 334 398 461 340 404 467 347 410 474 353 417 480 359 423 487 366 429 493 372 436 499 1 2 0.6 1.2 84 85 86 506 569 632 512 575 639 518 645 525 588 651 531 594 658 537 601 664 544 607 670 |5° 613 677 556 620 683 563 626 689 3 4 5 6 1.8 2.4 3.0 3.6 87 88 89 690 91 92 93 696 759 822 702 765 828 708 771 835 778 841 721 784 ^47 727 790 853 734 797 860 740 803 866 746 809 872 879 7 8 9 4.2 4.8 5.4 885 891 897 904 910 916 923 929 935 942 *oo4 067 130 948 84 on 073 954 017 080 960 023 086 967 029 092 973 036 098 979 042 105 985 048 III 992 055 117 061 123 94 95 96 198 261 142 205 267 148 211 273 155 28c 161 223 167 230 292 298 180 242 305 186 248 311 192 255 317 97 98 99 700 N. 323 448 330 392 454 336 398 460 342 404 466 348 410 473 354 417 479 361 485 367 429 491 373 435 497 379 4.^2 504 510 516 522 528 535 541 547 553 559 566 1 2 3 4 5 6 7 8 9 Prop. Pts. 16 TABLE 1 N. O 1 « 3 4 5 6 7 S 9 Prop. Pts. 700 01 02 03 84 510 572 634 696 516 522 528 535 541 547 553 559 566 628 689 751 640 702 584 646 708 590 652 714 597 658 720 603 665 726 609 671 733 677 739 621 683 745 04 05 06 ^57 819 880 763 825 887 770 831 893 776 837 899 782 844 905 788 850 911 794 856 917 800 862 924 807 868 930 813 874 936 7 07 08 09 710 11 12 13 942 85003 065 948 009 071 954 016 077 960 022 083 967 028 089 973 034 095 979 040 lOI 985 046 107 991 052 114 997 058 120 1 2 3 4 5 6 7 8 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 126 132 138 144 150 156 217 278 339 163 169 175 181 248 309 193 254 315 199 260 321 Tel 327 211 272 333 224 285 345 230 291 352 236 297 358 242 303 364 14 15 16 370 431 491 376 437 497 382 443 503 388 449 509 394 455 516 400 461 522 406 467 528 412 473 534 418 479 540 425 485 546 9 6.3 17 18 19 720 21 22 23 552 612 673 558 618 679 564 625 685 570 631 691 576 697 582 643 703 588 649 709 594 655 715 600 661 721 606 667 727 733 739 745 751 757 763 769 775 781 788 794 854 914 800 860 920 806 866 926 812 872 932 818 878 938 824 884 944 830 890 950 836 896 956 842 902 962 848 908 968 1 2 6 0.6 1.2 24 25 26 974 86 034 094 980 040 100 986 046 106 992 052 112 998 058 118 *oo4 064 124 *OIO 070 130 *oi6 076 136 *022 082 141 *028 088 147 3 4 5 1.8 2.4 3.0 27 28 29 730 31 32 33 153 213 273 332 159 219 279 338 165 225 285 171 231 291 177 237 297 183 243 303 189 249 308 195 255 3H 201 261 320 207 267 326 6 7 8 9 3.0 4.2 4.8 5.4 344 350 356 362 368 374 380 386 445 504 564 392 451 510 390 457 516 404 463 522 410 469 528 415 475 534 421 481 540 427 487 546 433 493 552 439 499 558 34 35 36 570 629 688 576 635 694 581 641 700 587 646 705 593 652 711 658 717 605 664 723 611 670 729 617 676 735 623 682 741 5 37 38 39 740 41 42 43 747 806 864 923 753 812 870 759 817 876 764 823 882 770 829 888 776 894 782 841 900 788 847 906 794 853 911 800 859 917 1 2 3 t 6 6 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 929 935 941 947 953 958 964 970 976 982 87 04.0 099 988 046 105 994 052 III 058 116 *oo5 064 122 *OII 070 128 *oi7 075 134 *023 081 140 *029 087 146 *o35 093 151 44 45 46 '57 216 274 163 221 280 169 227 286 175 233 291 181 239 297 186 245 303 192 251 309 198 256 315 204 262 320 210 268 326 9 4.5 47 48 49 750 332 390 448 396 454 344 402 460 408 466 355 413 471 361 419 477 367_ 483 373 489 379 437 495 384 442 500 506 512 518 523 529 535 541 547 552 558 N. 1 2 3 4 5 6 7 S 9 Prop. Pts. LOGAEITHMS OF NUMBERS 17 N. 1 2 3 4 5 535 593 651 708 6 7 8 9 Prop. Pts. 750 51 52 53 87 506 512 518 523 529 541 547 552 558 564 622 679 570 628 685 633 691 581 639 697 587 645 703 599 656 714 604 662 720 610 668 726 616 674 731 54 55 56 737 795 852 800 858 806 864 812 869 760 818 875 766 823 881 772 829 887 777 f35 892 783 841 898 789 846 904 57 58 59 760 61 62 63 910 967 88 024 915 973 030 978 036 927 984 041 933 990 047 938 996 053 944 *OOI 058 950 ♦007 064 955 *oi3 070 961 *oi8 076 081 087 093 098 104 no 116 121 127 133 190 247 304 138 195 252 144 201 258 150 207 264 156 213 270 161 218 275 167 224 281 173 230 287 178 235 292 184 241 298 1 2 0.6 1.2 64 65 66 3?? 366 423 315 372 429 321 377 434 326 383 440 389 446 338 395 451 343 400 457 349 406 463 355 412 468 360 417 474 3 4 5 6 1.8 2.4 3.0 3 6 67 68 69 770 71 72 73 480 536 593 485 542 598 491 604 497 553 610 502 |59 615 508 564 621 513 570 627 519 632 5^5 581 638 F 643 7 8 9 4.2 4.8 5.4 649 655 660 666 672 (>77 683 689 694 700 705 762 818 711 767 824 717 773 829 722 779 835 728 784 840 734 790 846 739 795 852 801 857 750 807 863 756 812 868 74 75 76 874 880 936 992 885 941 997 891 947 *oo3 897 953 *oc9 902 958 *oi4 908 964 *020 913 969 *025 919 975 ♦031 925 98. *037 77 78 79 780 81 82 83 89 042 098 154 048 104 159 053 109 165 059 115 170 064 120 176 070 126 182 076 131 081 137 193 087 143 198 092 148 204 260 209 265 ?76 215 271 326 382 221 226 232 237 243 248 254 276 332 387 282 337 393 287 343 398 293 404 298 354 409 304 360 415 310 365 421 315 426 1 2 5 0.5 1.0 84 85 86 542 437 492 548 443 498 553 448 504 559 454 509 564 459 515 570 465 520 575 470 526 581 476 We 481 537 592 3 4 5 p 1.5 2.0 2.5 3 87 88 89 790 91 92 93 597 708 603 658 713 609 664 719 614 669 724 620 675 730 625 680 735 631 686 741 636 691 746 642 697 752 647 702 757 7 8 9 3.5 4.0 4.5 763 768 774 779 785 790 796 801 807 812 818 873 927 823 878 933 829 883 938 834 889 944 840 894 949 845 900 955 851 905 960 856 911 966 862 916 971 867 922 977 94 95 96 982 90 037 091 988 042 097 993 048 102 998 "III *oo4 059 i'3 *oo9 064 119 *oi5 069 124 *020 075 129 *026 080 135 *o3i 086 140 97 98 99 800 146 200 255 '53 206 260 157 211 266 162 217 271 168 222 276 173 227 282 179 233 287 184 238 293 189 244 298 195 249 304 309 314 320 325 331 336 342 347 352 358 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 18 TABLE I N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 800 01 02 03 90 309 363 417 472 314 320 325 331 336 342 347 352 358 369 423 477 374 428 482 380 434 488 385 439 493 390 445 499 396 450 504 401 455 509 407 461 515 412 466 520 04 05 06 526 580 634 531 585 639 536 590 644 596 650 601 655 553 607 660 |5^ 612 666 f3 617 671 569 623 677 574 628 682 07 08 09 810 11 12 13 687 741 795 693 747 800 698 752 806 703 811 709 763 816 714 768 822 875 720 773 827 881 725 779 832 886 730 784 838 -89T 736 789 843 897 849 854 859 865 870 902 956 91 009 907 961 014 ^6^ 020 918 972 025 924 977 030 929 982 036 041 940 993 046 945 998 052 950 *oo4 057 1 2 0.6 1.2 14 15 16 062 116 169 068 121 174 180 078 185 084 137 190 089 142 196 094 148 201 100 153 206 105 158 212 no 164 217 3 4 5 6 1.8 2.4 3.0 3 6 17 18 19 820 21 22 23 222 275 328 228 281 334 233 286 339 238 291 344 243 297 350 249 302 355 254 307 360 259 312 365 265 318 371 270 323 376 7 8 9 4.2 4.8 5.4 381 387 392 397 403 408 413 418 424 429 434 487 540 440 492 545 445 498 551 450 503 556 455 508 561 461 514 566 466 519 572 471 524 577 477 582 482 535 587 24 25 26 593 645 698 598 651 703 603 656 709 609 661 714 614 666 719 619 672 724 624 677 730 630 682 735 635 687 740 640 693 745 27 28 29 830 31 32 33 751 803 855 756 808 861 761 814 866 766 819 871 772 824 876 777 829 882 782 834 887 7^7 840 892 793 845 897 798 850 903 908 913 918 924 929 934 939 944 950 955 960 92 012 065 965 018 070 971 023 075 976 028 080 981 033 085 986 038 091 991 044 096 997 049 lOI *002 054 106 *oo7 059 III 1 2 6 0.5 1.0 34 35 36 117 169 221 122 174 226 127 179 231 132 184 236 137 189 241 143 195 247 148 200 252 153 205 257 158 210 262 163 215 267 3 4 5 1.5 2.0 2.5 3 37 38 39 840 41 42 43 273 324 376 278 330 381 283 335 387 288 340 392 293 345 397 298 350 402 304 355 407 309 361 412 3H 366 418 319 371 423 7 8 9 3.5 4.0 4.5 428 433 438 443 449 454 459 464 469 474 480 531 583 485 536 588 490 542 593 495 547 598 500 603 505 557 609 562 614 516 567 619 521 572 624 526 578 629 44: 45 46 634 686 737 639 691 742 7A7 650 701 752 655 706 758 660 711 763 665 716 768 670 722 773 675 727 77S 681 732 783 47 48 19 850 N. 788 840 891 793 845 896 799 850 901 804 855 906 809 860 911 814 865, 916 819 870 921 824 875 927 829 881 932 834 886 937 942 947 952 957 962 967 973 978 983 988 1 2 3 4 5 6 7 8 9 Prop. VU. LOG ARIl HMS OF NUMBERS 19 N. 1 2 3 4 5 6 7 § 9 Prop. ris. 850 51 52 53 92 942 947 952 957 962 967 973 978 983 988 993 93 044 095 998 049 100 *oo3 054 105 *oo8 059 no *oi3 064 115 *oi8 069 120 *024 075 125 *029 080 131 *oi! 136 *039 090 141 54 55 5G 146 197 247 151 202 252 156 207 258 161 212 263 166 217 268 171 222 273 176 227 278 181 232 283 186 192 242 293 6 57 58 59 860 61 62 63 298 349 399 303 354 404 308 359 409 313 364 414 369 420 323 374 425 328 379 430 334 384 435 339 389 44.0 344 394 445 1 2 3 4 5 6 7 8 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 450 500 III 455 460 465 470 475 526 576 626 480 485 490 495 505 556 606 510 611 515 566 616 520 571 621 531 636 541 591 641 546 596 646 64 65 66 651 702 752 656 707 757 661 712 762 666 717 767 671 722 772 676 727 777 682 732 782 687 787 692 742 792 697 747 797 9 5.4 67 68 69 870 71 72 73 802 852 902 807 857 907 812 862 912 817 867 917 822 872 922 827 877 927 832 882 932 837 887 937 842 892 942 847 897 947 952 957 962 967 972 977 982 987 992 997 1 2 3 4 5 6 5 0.5 1.0 1.5 2.0 2.5 3.0 94 002 052 lOI 007 106 012 062 III 017 067 116 022 072 121 027 077 126 131 086 136 042 091 141 047 096 146 74 75 76 151 201 250 156 206 255 161 211 260 166 216 265 171 221 270 176 226 275 181 186 285 191 240 290 196 245 295 77 78 79 880 81 82 83 300 349 399 448 305 354 404 310 359 409 364 414 320 369 419 325 374 424 330 379 429 335 384 433 340 438 345 394 443 7 8 9 3.5 4.0 4.5 453 458 463 468 473 478 483 488 493 498 547 596 503 552 601 507 557 606 512 611 567 616 522 621 527 576 626 532 630 537 586 635 r42 591 640 84 85 86 645 694 743 650 699 748 655 704 753 660 709 758 665 714 763 670 719 768 675 724 773 680 729 778 685 734 783 689 738 787 4 87 88 89 890 91 92 93 792 841 890 797 846 895 802 851 900 807 856 905 812 861 910 817 866 915 822 871 919 827 876 924 880 929 836 885 934 1 2 3 4 5 6 7 8 4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 939 988 95 036 085 944 993 041 090 949 954 959 *oo7 056 105 963 *OI2 061 109 968 973 978 983 998 046 095 *002 051 100 ♦017 066 114 *022 071 119 *027 075 124 ♦032 080 129 94 95 96 231 139 187 236 143 192 240 148 197 245 153 202 250 158 207 255 163 211 260 168 216 265 173 221 270 177 226 274 9 3.6 97 98 99 900 279 ^;6 284 332 381 289 337 386 294 342 390 299 347 395 303 352 400 308 357 405 361 410 318 366 415 323 371 419 424 429 434 439 444 448 453 458 463 468 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 20 TABLE I ■, N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 900 01 02 03 95 424 429 434 439 444 448 453 458 463 468 472 569 477 525 574 482 530 578 487 535 583 492 540 588 497 545 593 501 598 506 554 602 511 559 607 564 612 04 05 06 617 665 713 622 670 718 626 674 722 631 679 727 636 684 732 641 6S9 737 646 694 742 650 698 746 655 703 751 660 708 756 07 08 09 910 11 12 13 761 809 856 904 766 813 861 818 866 823 871 780 828 875 785 880 789 885 794 842 890 799 847 895 804 852 899 909 914 918 923 928 933 938 942 947 952 999 96 047 957 *oo4 052 961 *oo9 057 966 *oi4 061 971 ♦019 066 976 *023 071 980 *028 076 985 *o33 080 ,995 *042 090 1 2 0.5 1.0 14 15 IG 095 142 190 099 147 194 104 152 199 109 156 204 114 161 209 118 166 213 123 218 128 175 223 133 180 227 137 185 232 3 4 5 6 1.5 2.0 2.5 3.0 17 18 19 920 21 22 23 237 284 332 379 426 473 520 1% 336 246 294 341 251 298 346 256 303 350 261 308 355 402 450 497 544 265 360 270 317 365 275 322 369 280 327 374 7 8 9 3.5 4.0 4.5 384 388 393 398 407 412 417 421 478 525 435 483 530 440 487 534 445 492 539 454 548 459 506 553 464 558 468 562 24 25 26 567 614 661 572 619 666 577 624 670 581 628 675 586 633 680 591 638 685 P5 642 689 600 647 694 605 652 699 609 656 703 27 28 29 930 31 32 33 708 755 802 848 713 759 806 717 764 811 722 ^^? 816 727 774 820 825 736 783 830 741 788 834 745 792 839 750 844 853 858 862 ^7 872 876 881 886 890 895 942 988 900 946 993 904 951 997 909 956 *002 914 *oo7 918 965 *OII 923 970 *oi6 928 974 *02I 932 979 *025 ♦030 1 2 4 0.4 0.8 34 35 36 97 035 081 128 132 044 090 137 049 095 142 053 100 146 058 104 151 063 109 155 067 114 160 072 118 165 077 123 169 3 4 5 6 1.2 1.6 2.0 2.4 37 38 39 940 41 42 43 174 220 267 .79 225 271 183 230 276 188 192 197 243 290 202 248 294 206 253 299 211 257 304 216 262 308 7 8 9 2.8 3.2 3.6 313 359 405 451 317 322 327 331 336 340 345 350 354 400 447 493 364 410 456 368 414 460 373 419 465 377 424 470 382 428 474 387 433 479 391 437 483 396 442 488 44 45 46 497 589 502 548 594 506 552 598 511 603 562 607 •520 566 612 525 571 617 529 575 621 534 580 626 630 47 48 49 950 635 681 727 640 685 731 644 690 736 649 695 740 653 699 745 658 704 749 663 708 754 667 713 759 672 717 763 676 722 768 772 m 782 786 791 795 800 804 809 813 N. 1 2 «|4 5 6 7 8 9 Prop. PtR. LOGARITHMS OF NUMBERS 21 N. o 1 2 3 4 5 6 7 8 9 Prop. Pts. 050 51 52 53 97 772 777 782 786 791 795 800 804 809 813 8i8 864 909 823 868 914 827 873 918 832 ^77 923 836 882 928 841 886 932 845 891 937 850 896 941 855 900 946 859 905 950 54 55 56 0^55 98 000 046 959 005 050 964 009 055 968 014 059 973 019 064 978 023 068 982 028 073 987 032 078 991 21 996 041 087 57 58 59 960 61 62 63 091 096 141 186 100 146 191 105 150 195 109 155 200 114 159 204 118 164 209 123 168 214 127 173 218 132 177 223 227 232 236 241 245 250 254 259 263 268 272 363 277 322 367 281 327 372 286 331 376 290 336 381 295 340 385 299 345 390 304 349 394 308 354 399 313 358 403 1 2 5 0.5 1.0 64 65 66 408 453 498 412 457 502 417 462 507 421 466 511 426 471 516 430 475 520 480 525 439 484 529 444 489 534 448 .493 538 3 4 5 6 1.5 2.0 2.5 3.0 67 68 69 970 71 72 73 543 588 632 547 592 ^57 552 597 641 601 646 561 605 650 610 655 570 614 659 619 664 579 668 583 628 673 7 8 9 3.5 4.0 4.5 677 682 686 691 695 700 744 789 834 704 709 713 717 722 767 811 726 771 816 731 820 825 740 784 829 749 III 798 843 802 847 762 807 851 74 75 76 856 900 945 860 905 949 865 909 954 869 914 958 874 918 963 878 923 967 883 927 972 ^2>7 932 976 892 936 981 896 941 985 77 78 79 080 81 82 83 989 99 034 078 994 038 083 998 043 087 *oo3 047 092 *oo7 052 096 *OI2 056 100 145 189 233 277 ^=016 061 105 *02I 065 109 *025 069 114 *029 074 118 123 127 131 136 140 149 154 158 162 167 , 211 255 171 216 260 176 220 264 180 224 269 185 229 273 193 282 198 242 286 202 247 291 207 251 295 1 2 4 0.4 0.8 84 85 86 300 344 388 304 348 392 308 396 313 357 401 317 361 405 322 366 410 326 370 414 330 374 419 335 379 423 339 383 427 3 4 5 1.2 1.6 2.0 2 4 87 88 89 090 91 92 93 432 476 520 436 480 524 441 484 528 445 489 533 449 493 537 454 498 542 458 502 546 590 463 506 550 467 511 555 471 515 559 603 7 8 9 2.8 3.2 3.6 564 568 572 577 581 585 629 673 717 594 599 607 695 612 656 699 616 660 704 621 664 708 625 669 712 614 677 721 it 726 642 686 730 647 691 734 94 95 96 739 782 826 743 787 830 747 791 835 752 795 839 756 800 843 760 804 848 765 808 852 769 813 856 774 817 861 822 865 97 98 99 1000 870 9'3 957 874 917 961 878 922 965 883 926 970 887 930 974 891 935 978 896 939 983 900 944 987 904 948 991 909 952 996 00 coo 004 009 013 017 022 026 030 035 039 N. 1 2 3 4 5 6 7 § 9 Prop. Pts. 22 TABLE I N 1 2 3 4 5 6 7 S 9 Prop. Pts. 1000 1001 1002 1003 ooo ooo ~434 868 ooi 301 043 477 911 344 087 130 174 217 260 304 347 391 521 388 564 998 431 608 ♦041 474 651 ♦084 517 694 ♦128 561 738 *i7i 604 781 ♦214 647 824 ♦258 690 44 1004 1005 1006 734 002 166 598 777 209 641 820 252 684 863 296 727 907 339 771 950 382 814 993 857 ♦036 468 900 *o8o 512 943 *I23 555 986 1 2 3 4.4 8.8 13.2 1007 1008 1009 1010 1011 1012 1013 003 029 461 891 073 504 934 116 547 977 159 590 *020 202 633 *o63 245 676 *io6 536 288 719 *i49 331 762 *I92 374 805 *235 417 848 *278 4 5 6 7 8 9 17.6 22.0 26.4 30.8 35.2 39.6 004 321 364 407 450 493 579 622 665 708 751 005 180 609 794 223 652 837 266 695 880 309 738 923 352 966 395 824 *oo9 438 867 *052 481 909 *o95 524 952 *i38 567 995 1014 1015 1016 006 038 466 894 081 509 936 124 552 979 166 594 *022 209 637 *o65 252 680 *io7 295 723 *i5o 338 *i93 808 ♦236 f3 851 *278 1 43 4.3 1017 1018 1019 1020 1021 1022 1023 007 321 748 008 174 600 oop 026 451 S76 364 790 217 406 833 259 449 876 302 492 918 345 961 387 577 *oo4 430 620 *o46 472 662 *o89 515 705 *I32 558 2 3 4 5 6 7 8 9 8.6 12.9 17.2 21.5 25.8 30.1 34.4 38.7 643 685 728 770 813 856 898 941 983 068 493 918 III 536 961 578 *oo3 196 621 *o45 238 663 *o88 281 706 *i30 323 748 *i73 366 791 *2I5 408 833 *258 1024 1025 1026 010 300 724 on 147 342 766 190 232 427 851 274 470 893 317 936 359 978 401 597 *020 444 639 *o63 486 681 *io5 528 1027 1028 1029 1030 1031 1032 1033 570 993 012 415 *035 458 655 ''078 5cx> 697 *I20 542 740 *l62 584 782 *204 626 824 *247 669 866 *289 711 909 *33i 753 951 *373 795 1 2 I 6 7 42 4.2 8.4 12.6 16.8 21.0 25.2 29.4 837 879 922 964 *oo6 ♦048 ♦090 *I32 *I74 *2I7 013 259 680 014 ICX) 301 722 142 343 764 184 226 427 848 268 469 890 310 511 932 352 553 974 395 596 *oi6 437 638 ♦058 479 1034 1035 1036 521 940 015 360 563 982 402 605 *024 444 647 *o66 485 689 *io8 527 730 *i5o 569 772 *I92 611 814 *234 653 856 ♦276 695 898 *3i8 737 8 9 33.6 37.8 1037 1038 1039 1040 1041 1042 1043 779 016 197 616 017 033 821 239 657 863 281 699 904 323 741 946 365 783 988 407 824 +030 448 866 ♦072 490 908 *ii4 532 950 ♦156 574 992 1 2 3 4 5 41 4.1 8.2 12.3 16.4 20 5 075 117 159 200 242 284 326 ! 367 409 451 86S 018 284 492 909 326 534 368 576 993 409 618 "034 451 659 ♦076 492 701 *ii8 534 743 *I59 576 784 *20I 617 826 *243 659 1044 1045 1046 700 019 116 532 742 158 573 784 199 615 825 241 656 867 282 698 908 324 739 950 366 781 992 407 822 *o33 449 864 *o75 490 905 6 7 8 24.6 28.7 32.8 1047 10 i8 1019 1050 947 020 361 775 988 403 817 *030 444 858 *o7i 486 900 *ii3 527 941 *i54 568 982 "3^6 *I95 610 *024 *237 651 *o65 *278 693 *io7 ♦320 734 *i48 561 9 36.9 021 189 231 272 313 355 437 479 520 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 23 N. 1 2 3 4 5 6 7 S 9 Prop. i'U, 1050 1051 1052 1053 02I 189 231 272 313 355 396 437 479 520 561 974 387 799 603 022 016 428 644 057 470 685 098 511 727 140 552 768 181 593 809 222 635 851 263 676 892 305 717 933 346 758 42 1054 1055 1056 841 882 294 705 923 335 746 964 376 787 ♦005 417 828 *047 458 870 *o88 499 911 *I29 541 952 *i7o 582 993 *2II 623 *o34 1 2 3 4.2 8.4 12.6 1057 1058 1059 1000 1061 1062 1063 024 075 486 896 025 306 116 527 937 157 568 978 198 609 ♦019 239 650 *o6o 280 691 *IOI 321 732 *I42 363 773 *i83 404 814 *224 445 855 ♦265 674 4 5 6 7 8 9 16.8 21,0 25.2 29.4 33.6 37.8 347 388 429 470 511 552 593 634 026 125 533 756 165 574 797 206 615 838 247 656 879 288 697 920 329 737 961 370 778 *002 819 *o43 452 860 ♦084 492 901 1064 1065 1066 942 027 350 757 982 390 798 *023 431 839 *o64 472 879 *io5 513 920 *i46 961 ♦186 594 *002 *227 635 *042 *268 676 *o83 *309 716 *I24 1 41 4.1 1067 1068 1069 1070 1071 1072 1073 028 164 _978 02Q 384 789 030 195 600 205 612 *oi8 246 653 *o59 287 693 *ICX) 327 734 ♦140 368 775 *i8i 409 815 *22I 449 856 *262 490 896 ♦303 531 937 *343 749 2 3 4 5 6 7 8 9 8.2 12.3 16.4 20.5 24.6 28.7 32.8 36.9 424 830 640 465 871" 276 681 506 546 587 627 668 708 316 721 952 357 762 992 397 802 *o33 438 843 *o73 478 883 *ii4 519 923 *i54 559 964 1074 1075 1076 031 004 408 812 045 449 853 085 489 893 126 530 933 166 570 974 206 610 *oi4 247 651 *o54 287 691 *o95 328 732 *i35 368 772 *i75 1077 1078 1079 1080 1081 1082 1083 032 216 619 033 021 256 659 062 296 699 102 337 740 142 377 780 182 4^7 820 223 458 860 263 498 901 J03 705 538 941 343 745 578 981 384 785 1 2 3 4 5 6 7 40 4.0 8.0 12.0 10.0 20.0 24.0 28.0 424 464 504 544 585 625 665 826 034 227 628 866 267 669 906 308 709 946 348 749 986 388 789 *027 428 829 *o67 468 869 *io7 50S 909 *I47 548 949 *i87 588 989 1084 1085 1086 035 029 430 830 069 470 870 109 510 910 149 550 950 190 590 990 230 630 ♦030 270 670 ♦070 310 710 *IIO 350 750 *i5o 390 790 ♦190 8 9 32.0 36.0 1087 1088 1089 1000 1091 1092 1093 036 230 629 037 028 269 669 068 309 709 108 349 749 148 389 789 187 429 828 227 469 868 267 509 908 307 549 948 347 III 387 785 *i83 580 978 1 2 3 4 5 39 3.9 7.8 11.7 15.6 19 5 426 466 506 546 944 342 739 586 984 382 779 626 665 705 745 038 223 620 865 262 660 904 302 700 *024 421 819 *o64 461 859 *io3 501 898 *I43 541 938 1094 1095 1096 039 017 057 454 850 097 493 890 136 533 929 176 969 216 612 ♦009 ?55 652 ♦048 295 692 *o88 335 731 *I27 374 771 *i67 6 7 8 23.4 27.3 31.2 1097 1098 1099 1100 040 207 602 998 246 642 *037 286 681 *o77 325 721 *ii6 761 ♦156 405 800 *i95 590 444 840 *235 630 484 879 *274 669 523 919 *3i4 708 563 958 "353 748 9 35.1 041 393 432 472 511 551 N. 1 2 3 4 5 6 7 § Prop. Pts. TABLE II LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE 25 26 TABLE II 0^ 2 3 _4_ 5 6 7 8 _9_ 10 II 12 13 ;i 17 i8 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 II 49 60 51 52 53 il ii II Jl 00 L. Sin. 6.46373 6.76476 6.94085 7.06 579 16 270 24 188 30882 36682 41 797 46373 50512 54291 57767 60985 63982 66 784 69417 71 900 74248 76475 78594 80615 82545 84393 86 166 87870 89509 91 088 92 612 94 084 95508 96 §87 98223 99520 8.00779 8 . 02 002 8.03 192 8.04350 8.05478 8.06578 8.07650 8.08696 8.09 718 8.10 717 II 693 12647 13 581 14495 15 391 8.16268 8.17 128 8.17971 8.18798 8.19 610 8 . 20 407 8.21 189 8.21 958 8.22 713 8.23456 8.24 186 L. Cos, 30103 17609 12494 9691 7918 6694 5800 5"5 4576 4139 3779 3476 3218 2997 2802 2633 2483 2348 2227 2119 2021 1930 1848 1773 1704 1639 1579 1524 1472 1424 1379 1336 X297 1259 1223 1190 1158 1128 IIOO 1072 1046 I022 999 976 954 934 914 896 877 860 843 827 812 797 782 769 755 743 .730 L. Tang, c. d. L, Cotg 46373 76476 94085 06579 16 270 24 188 30882 36682 41 797 46373 50512 54291 57767 60986 63982 66785 69418 71 900 74248 76476 78595 80615 82 546 84394 86167 87871 89 510 91 089 92613 94 086 95510 96889 98 225 99522 00 781 02 004 03 194 04353 05481 06581 07653 08 700 09 722 10 720 11 696 12 651 13585 14500 15395 16 273 17 133 17976 18804 19 616 20413 21 195 21 964 22 720 23462 8.24 192 L. Cotg. 30103 17609 12494 9691 7918 6694 5800 5"5 4576 4139 3779 3476 3219 2996 2803 2633 2482 2348 2228 21 19 2020 193 1 1848 1773 1704 1639 1579 1524 1473 1424 1379 1336 1297 "59 1223 1x90 "59 Z128 ixoo 1072 X047 X022 998 976 955 934 915 895 878 860 843 828 812 797 782 769 756 742 730 c. d. 3 53627 3 23 524 3 05915 2.93421 2.83 730 2.75 812 2.69 118 2.63318 2 58 203 2.53627 2 . 49 488 2 45 709 2 42233 2.39014 2.36018 2 33215 2 30 582 2 28 100 2 25 752 2 23 524 2 21 405 2 19385 7 454 [5 606 13833 12 129 10 490 08 911 07387 05914 04 490 03 III 01 775 00478 99219 97996 96806 95647 94519 93419 92347 91 300 90278 89280 88304 87349 86415 85 500 84605 83727 82867 82 024 81 196 80384 79587 78805 78036 77 280 76538 75808 L. Tang, 89^ L. Cos. 000 000 000 000 000 000 000 000 000 oco 000 000 000 000 000 00 000 00 000 99 999 99 999 99 999 99 999 99 999 99 999 99 999 99 999 99 999 99 999 99 999 99 999 99998 99998 99998 99998 09998 99998 99998 99998 99997 99 997 99 997 99997 99 997 99 997 99997 99996 99996 99996 99996 99996 99996 99 995 99 995 99 995 99 995 99 995 99 994 99 994 99 994 99994 99 994 99 993 L. Sin. 35 34 33 32 3» 30 29 28 27 26 Prop. Pts. 3476 3218 .X 348 322 .2 695 644 •3 1043 965 •4 1390 1287 •5 1738 X609 2803 2633 1 .1 280 263 .2 560 527 •3 841 790 •4 1121 1053 •5 1401 X316 2227 203I .1 223 202 .2 445 404 •3 668 606 •4 891 808 •5 IXX3 xoxo 1704 1579 .1 170 158 .2 341 316 •3 5" 474 ■4 682 632 •5 852 789 1379 1297 .X 138 130 .2 276 259 •3 414 389 •4 552 519 •5 690 649 XX58 1x00 X16 110 232 220 347 330 463 440 579 550 999 954 .1 100 95 .2 200 191 •3 300 286 •4 400 382 •5 500 477 877 843 .1 88 84 .2 »75 169 •3 263 253 •4 351 337 •5 438 422 782 755 .1 78 75 .a 156 i5» •3 235 226 • 4 313 302 •5 391 377 2997 300 599 899 1 199 1498 2483 24& 497 745 993 1242 1848 185 370 554 739 924 147a H7 294 442 589 736 1223 122 245 367 489 612 X046 105 209 314 418 523 914 91 X83 274 366 457 8ia 81 162 244 325 406 730 73 146 219 29a 365 Prop. Pte. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 27 L. Sill. \ 2 3 _4_ I 7 8 _9_ 10 12 13 il. ;i 17 t8 20 21 22 23 24 26 27 28 29 31 32 33 34 36 37 38 ii 40 41 42 43 44 45" 46 47 48 60 51 52 53 il. ^i II 59 60 24 186 24903 25 609 26304 26988 27 661 28 324 28977 29 621 30255 30879 31 495 32 103 32 702 33 292 33875 34450 35018 35578 36 131 36678 37217 37750 38276 38796 39310 39818 40320 40 816 41 307 41 792 42272 42 746 43216 43 680 44 139 44 594 45 044 45489 45930 46 366 46 799 47 226 47650 48 069 48485 48896 49304 49 708 50 108 50504 50897 51 287 51 673 52055 52434 52 810 53183 53552 53919 54282 L. Cos. 717 706 695 684 673 663 653 644 634 624 616 608 599 590 583 575 568 560 553 547 539 533 526 520 514 508 502 496 491 485 480 474 470 464 459 455 450 445 441 436 433 427 424 419 416 411 408 404 400 396 393 390 386 382 379 376 373 369 367 363 Tang. 24 192 24 910 25 616 26 312 26 996 27 669 28 332 28986 29 629 30263 30888 31 505 32 112 32 711 33302 33886 34461 35029 35590 36143 36689 37229 37762 38289 38809 39323 39832 40334 40830 41 321 41 S07 42 287 42 762 43232 43696 44 156 44 611 45 061 45507 45948 46385 46817 47245 47669 48089 48505 48917 49325 49 729 50 130 50527 50 920 51 310 51 696 52079 52459 52835 53208 53578 53 945 54308 L. Cotg. c.d. 718 706 696 684 673 663 654 643 634 625 617 607 599 591 584 575 568 561 553 546 540 533 527 520 514 509 502 496 491 486 480 475 470 464 460 455 450 446 441 437 432 428 424 420 416 412 408 404 401 397 393 390 386 383 380 376 373 370 367 363 C.d. L. Cotg. 75808 75090 74384 73688 73004 72331 71 668 71 014 70371 69 737 69 112 68495 67888 67 289 66698 66 114 65539 64971 64 410 63857 63 3" 62771 62238 61 711 61 191 60 677 60168 59 666 59 170 58679 58193 57713 57238 56768 56304 55844 55389 54 939 54 493 54052 53615 53183 52755 52331 51 9" 51495 51083 50675 50 271 49870 49 473 49 080 48 690 48304 47921 47541 47165 46 792 46 422 46055 .45692 L. Tang. 88° L. Cos. 99 993 99 993 99 993 99 993 99992 99992 99992 99992 99992 99991 99991 99991 99990 99990 99990 99990 99989 99989 99989 99989 99988 99988 99988 99987 99987 99987 99 986 99 986 99 986 99985 99985 99985 99984 99984 99984 99983 99983 99983 99 982 99982 99982 99981 99981 99981 99 980 99 980 99 979 99 979 99979 99978 99978 99977 99 977 99977 99976 99976 99 975 99 975 99 974 99 974 9 99 974 L. Sin, GO 59 58 57 _5i 55 54 53 52 _51 50 49 48 47 46 45 44 43 42 _iL 40 39 38 _3^ 35 34 33 32 30 29 28 27 26 25 24 23 22 21 20" 19 18 17 16 15 14 13 12 II To" 9 8 7 6 Prop. Pte. 717 695 .1 71.7 69.5 .2 M3-4 139.0 •3 215. 1 208.5 •4 286.8 278.0 • 5 358.5 347-5 653 634 .1 65.3 63-4 .2 130.6 126.8 •3 195-9 190.2 •4 261.2 253.6 •5 326.5 3170 599 583 59-9 58.3 119. 8 n6.6 179.7 174.9 239-6 233-2 299-5 291-5 553 539 .1 55-3 53-9 .2 no. 6 107.8 ■3 165.9 161.7 •4 221.2 215.6 •5 276.5 269.5 514 503 51.4 50.2 102.8 100.4 154-2 150.6 205.6 200.8 257.0 251.0 480 470 .1 48 47 .2 96 94 •3 144 141 •4 192 188 •5 240 235 450 440 .X 45 44 .2 90 88 •3 135 132 •4 180 176 •5 225 220 430 410 .1 42 4» .2 84 82 •3 126 123 •4 168 164 •5 210 205 390 380 39 38 78 76 117 114 156 152 >95 190 673 67.3 134.6 201 9 269.2 336.5 616 61.6 123.2 184.8 246.4 308.0 56.8 113.6 170.4 227.2 284.0 536 52.6 105.2 157.8 210.4 263.0 490 49 98 U7 196 245 460 46 92 138 184 230 430 43 86 129 172 215 400 40 80 120 160 300 37 74 III 148 Prop. Pts. 28 TABLE II 2' 3 ±^ l I 9_ 10 II 12 13 :i ii. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 11 35 36 37 3« 39. 40 41 42 43 44 46 47 48 49 50 51 52 53 54 II 57 58 59 GO L. Sin. 8 54 282 8 . 54 642 8.54999 8-55 354 8 . 56 054 8 . 56 400 8.56743 8.57084 8.57421 8.57757 8.58089 8.58419 8.58747 8.59072 8 59 395 8.59715 8 . 60 033 8.60349 8.60662 8.60973 8.61 282 8.61 589 8.61 894 8.62 196 8 . 62 49 7 8.62 795 8.63 091 8.6338c 8.6367 8.63968 8.64256 8.64543 8.64827 8.65 no 8.65391 8.65670 8.65947 8 . 66 223 8.66 497 8.66 769 8 ■ 67 039 8.67.308 8.67575 8.67841 8.68 104 8.68367 8.68627 8.68 886 8.69 144 8 . 69 400 8 . 69 654 8.69 907 8.70 159 8 . 70 409 8.70658 8 70 905 8.71 151 8.71 395 8.71638 8.71 880 L. Cos. 360 357 355 351 349 346 343 341 337 336 332 330 328 325 323 320 318 316 313 3" 309 307 305 303 301 298 296 294 293 290 288 287 284 283 281 279 277 276 274 272 270 269 267 266 263 263 260 259 258 256 254 253 252 250 249 247 246 244 243 242 L. Tang. c.d. 8.54308 8 . 54 669 8.55027 8.55382 8 55 734 8 56~oS3" 8.56 429 8.56773 8.57 "4 8.57452 8.57788 8 58 121 8 58451 8 58779 8.59 105 8.59428 8 59 749 8.60068 8.60384 8.60698 8.61 009 8.61 319 8.61 626 8.61 931 8.62 234 8.62535 8.62834 8.63 131 8.63426 8.63718 8 . 64 009 8.64298 8.64585 8.64870 8.65 154 8.65435 8.65 715 8.65993 8.66269 8.66543 8.66816 8.67087 8.67356 8.67624 8 67890 8.68 154 8.68417 8.68678 8.68938 8.69 196 8.69453 8.69 708 8 . 69 962 8.70 214 8 . 70 465 8.70714 8 . 70 962 8.71 208 8.71453 8.71 697 8.71 940 L. Cotff. 361 358 355 352 349 346 344 341 338 336 333 330 328 326 323 321 319 316 3H 3" 310 307 305 303 301 299 297 295 292 291 289 287 285 284 281 280 278 276 274 273 271 269 268 266 264 263 261 260 258 257 255 254 252 251 249 248 246 245 244 243 L. Cotg. 1.45692 1-45 331 1.44 973 I .44 618 1 . 44 266 1-43917 I -43 571 1.43227 1.42886 1 . 42 548 I .42 212 1. 41 879 I -41 549 1 .41 221 1.40895 1.40572 1 .40 251 1-39932 1.39 616 1-39302 •38991 .38681 -38374 .38069 .37766 •27 ^§ .37 166 • 36869 -36574 . 36 282 1-35 991 1.35 702 1-35415 1-35 130 1.34846 1-34565 1-34285 1.34007 1-33 731 I 33 457 I 33 184 I 32913 1.32644 1.32376 I 32 no C.d, 1. 31 846 I -31 583 1. 31 322 1 .31 062 I 30 804 I 30547 1 . 30 292 1 . 30 038 I . 29 786 1-29 535 1 . 29 286 1 . 29 038 I . 28 792 1.28547 1.28303 1.28060 L. Tang. 87^ L. Cos. 99 974 99 973 99 973 99972 99972 99971 99971 99970 99970 99969 99969 99 968 99 968 99967 99967 99967 99 966 99 966 99965 99964 99964 99963 99963 99 962 99 962 99961 99 961 99 960 99 960 99 959 99 959 99958 99958 99 957 99956 99956 99 955 9 99 955 9 99 954 9-99 954 99 953 99952 99952 99951 99951 99950 99 949 99 949 99948 99948 99 947 99946 99946 99 945 99 944 99 944 99 943 99942 99942 99941 99940 L. Sin. GO 59 58 57 55 54 53 52 il 50 49 48 47 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II To 9 8 7 6 Prop. Pte. 360 350 36 35 72 70 108 105 144 140 180 175 216 210 • 7 252 245 .8 288 280 •9 324 315 330 320 1 33 32 3 66 64 3 99 96 4 132 128 5 165 160 6 108 192 7 231 224 8 264 256 9 297 288 300 290 385 30 29 28. 60 58 57 90 87 85- 120 116 114. 150 145 142. .6 180 174 171. •7 210 203 199. .8 240 232 228. •9 270 261 256 380 375 270 I 28.0 27-5 27- .2 56.0 55-0 54- •3 84.0 82.5 81. •4 112. IIO.O 108. • 5 140.0 137-5 135- .6 168.0 165.0 162. -7 196.0 192.5 189. .8 224.0 220.0 216. •9 252.0 247 -5 243- 365 ■ 26.5 •53-0 •79-5 106.0 132-S 159.0 1.85 5 212 o 260 .26.0 .52.0 .78.0 104.0 130.0 156.0 182.0 208.0 234.0 250 245 .25.0 •24.5 .50 -49 •75 •73 5 100 198 125 122 5 .6 150 »47 '75 171 5 200 196 •9 225 320 5 Prof). Pts. LOGARITHMS OP THE TRIGONOMETRIC FUNCTIONS 29 3° 2 3 j4 I 7 8 9 10 II 12 13 J4_ ;i \i Ji_ 20 21 22 23 24 26 27 28 29 30 31 32 33 36 37 38 40 41 42 43 44 45 46 47 48 19. 50 51 52 53 54. 55 56 II 59 (JO L. Sin. 8 74 226 8.74454 8.74680 8 . 74 906 ■ 75 130 71 880 72 120 72359 72597 72834 73069 73303 73 535 73767 73 997 75 353 75 575 75 795 76015 76234 76451 76 667 76883 77097 77310 77522 77 733 77 943 78152 78360 78568 78774 78979 79183 79386 79588 79789 79990 80189 80388 80585 80782 80978 81 173 81367 8.B1 560 8.81 752 8.81 944 8.82 134 8.82324 82513 82 701 82888 83075 83261 83446 83630 83813 83996 84177 8.84358 L. Cos. 240 239 238 237 23s 234 232 233 230 229 228 226 226 224 223 222 220 220 219 217 216 216 214 213 212 211 210 209 208 208 206 205 204 203 20-i. 201 201 199 199 197 197 196 194 193 192 192 190 190 189 188 187 187 186 185 184 183 183 181 L. Tang. c. d. 71 940 72 181 72 420 72659 72 896 73366 73 600 73832 74063 74292 74521 74748 74 974 75 199 75423 75645 75867 76087 76306 76525 76742 76958 77173 77387 77 600 77 811 78 022 78232 78441 78649 78855 79 061 8 79 266 8.79470 79673 79875 80076 80 277 80476 80 674 80872 81 068 81 264 81 459 8.81 653 8 81 846 8 82038 8 82 230 8 82 420 8 82610 8 82 799 8 82987 883175 8.83361 8 83547 8 83732 8 83 916 8 84 100 8 84282 8,84464 L. Cotg. 241 239 239 237 236 234 234 232 231 229 229 227 226 225 224 222 222 220 219 219 217 216 215 214 213 211 209 208 206 206 205 204 203 202 201 20I 199 198 I9& 196 196 194 193 192 192 190 190 189 186 186 185 184 184 182 L. Cotg. 1 . 28 060 1.27 819 1.27 580 I 27341 I .27 104 1.26868 1.26634 1 . 26 400 1.26 168 1-25937 I . 25 708 I 25 479 I .25 252 I .25 026 1 . 24 801 I 24577 1-24355 I 24 133 1-23913 1.23694 I 23 475 1.23258 1.23042 1.22 827 I .22 613 1 . 22 400 I .22 189 1. 21 978 1. 21 768 I-21 559 I 21 351 I 21 145 1.20939 1.20734 1.20530 c. d. 1 . 20 327 I .20 125 I 19 924 I 19 723 I 19 524 1. 19 326 1. 19 128 1 . 18 932 1. 18 736 1. 18 541 18347 18 154 17 962 17770 17580 1. 17 390 1.17 201 1.17013 1. 16 825 I 16 639 I 16453 1 . 16 268 1 . 16 084 1 . 1 5 900 1.15 718 15 536 L. Tang. 80° L. Cos. 9 99940 9 99940 9 99 939 9 99938 9 99938 9 99 937 9 99936 9.99936 9 99 935 9 99 934 9 99 934 9 99 933 9.99932 9 99932 9 99931 9 99930 9 99 929 9.99929 9.99928 9.99927 9.99926 9.99926 9-99925 9-99924 9.99923 9.99923 9.99922 9 99921 9.99920 9.99920 9.99919 9.99918 9.99917 9 99917 9.99916 9 99915 9.99914 9 99913 9 99913 9.99912 9 99911 9 99910 9.99909 9.99909 9.99908 9,99907 9.99906 9 99905 9 99904 9 99 904 9.99903 9.99902 9.99901 9.99900 9.99899 9.99898 9.99898 9.99897 9 99896 9 99895 9.99894 L. Sin. Prop. Pts. 238 334 sag .1 23.8 23 -4 22 .2 47.6 46. « 45 •3 71.4 70.2 68 •4 95-2 93-6 91 •5 119.0 117. 114 .6 142.8 140.4 137 •7 166.6 163.8 160 .8 190.4 187.2 183 •9 214.2 210.6 206 225 220 .1 22.5 22.0 2 45.0 44.0 •3 67.5 66.0 •4 90.0 88.0 •5 112.5 IIO.O .6 135 -o 132.0 ■7 157-5 I54-0 .8 180.0 176.0 9 202.5 198.0 212 208 21.2 20.8 42.4 41.6 63.6 62.4 84.8 83.2 106.0 104.0 127.2 124.8 -7 148.4 145.6 .8 169.6 166.4 •9 190.8 187.2 201 197 .1 20.1 19.7 .2 40.2 39-4 •3 60.3 59-1 •4 80.4 78.8 •5 100.5 98.5 .6 120.6 118.2 •7 140.7 137-9 .8 160.8 157-6 •9 180.9 J77-3 i8g 18.9 37-8 56.7 75 6 94-5 "3 4 132.3 151.2 170.1 185 18.5 37-0 55-5 74 -o 92.5 III.O 129.5 148.0 166. 5 43a 1 0.4 0.3 0.2 c 2 0.8 0.6 0.4 o 3 1.2 0.9 0.6 o 4 1.6 1.2 o.'^ o 5 2.0 1.5 1.0 o 6 2.4 1.8 1.2 o 7 2.8 2.1 1.4 o 8 3.2 2.4 1.6 o 9 3.6 2.7 1.8 o 216 21.6 43-2 64.8 86.4 108.0 129.6 151.2 172.8 194.4 204 20.4 40.8 61.2 81.6 102.0 122.4 142.8 163.2 183 6 193 19-3 38.6 57-9 77.2 96.5 1158 135 I 154-4 173-7 181 18. 1 36.2 54-3 72.4 905 108.6 126.7 144.8 162.9 Prop. Pts. 30 TABLE II _9 10 II 12 13 ii_ \i 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 il 36 37 3« 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L. Sin. 8.84358 8 84539 8.84 718 8 84897 8 85 075 8 85 252 8.85429 8 85 605 8 85 780 8 85955 8 86 128 8 86 301 8 86 474 8 86 645 8 86816 8 86 987 8 87 156 8 87325 8 87494 8 87661 8 87 829 8 87 995 8 88 161 8 88 326 8 88 490 8 88 654 8 88817 8.88980 8 89 142 8 89 304 8 89 464 8 89 625 8 89 784 8.89943 8 90 102 8 90 260 8 90 417 8 90574 8 90 730 8 90 885 8 91 040 8 91 195 8 91 349 8 91 502 8 91 655 8 91 807 8 91.959 8 92 no 8 92 261 8 92 411 8 92 561 8 92 710 8 92 859 8 93 007 8 93 ^54 8 93 301 8 93448 8 93 594 8.93 740 8 93 885 8.94030 L. Cos. 179 179 178 177 177 176 >75 175 173 173 173 171 171 171 169 169 169 167 168 166 166 165 164 164 163 163 162 162 160 161 159 159 159 158 157 157 156 155 '55 155 154 153 153 152 152 151 151 150 150 149 149 148 147 147 147 146 146 H5 145 L. Tang. 8.84464 8.84646 8.84826 8.85006 8.85 185 c.d. 8 85 363 8.85 540 8.85717 8.85893 8.86 o6q 8 86 243 8.86417 8.86591 8.86 763 8.86935 8.87 106 8.87277 8.87447 8.87616 8 87 785 8.87953 8.88 120 8.88287 8.88453 8.88618 8.88783 8.88948 8.89 III 8.89274 8.89437 8 89598 8.89 760 8 . 89 920 8 . 90 080 8 . 90 240 8.90399 8 90557 8 90715 8.90872 8 91 029 8.91 185 8.91 340 8.91 495 8 91 650 8.91 803 91 957 o 92 no 8.92 262 8.92 414 8.92 565 8.92 716 8.92866 8.93 016 8.93 165 8 93 S13 8 93 462 8 9^ 609 8 93 756 8 93903 8 94 049 8 94 '95 182 180 180 179 178 177 177 176 176 174 ^74 174 172 172 171 171 170 169 169 168 167 167 1 66 165 165 165 163 163 163 i6i 162 160 160 160 159 158 158 157 157 156 155 155 155 153 154 153 152 152 151 151 150 ISO 149 I J*8 149 I X47 147 147 I 146 j 146 L. Cotg. I 15 536 I 15 354 I 15 174 1 . 14 994 I 14 815 1. 14 637 I 14 460 I 14 283 I 14 107 I 13 931 13757 13583 13409 13237 13065 1 . 12 894 I 12 723 I 12 553 I 12 384 1 . 12 215 12047 n 880 II 713 II 547 11382 I II 217 III 052 1 . 10 889 1 . 10 726 I 10 563 1 . 10 402 1 . 10 240 1 . 10 080 1 . 09 920 I . 09 760 I 09 601 1.09443 I 09 285 I .09 128 1.08 971 I 08815 1 .08 660 1.08505 I 08350 I .08 197 I 08 043 1.07 890 1.07738 1.07586 I 07 435 1 .07 284 I 07 134 I 06 984 1.06 835 1.06687 To6~538" 1.06 391 I 06 244 1 .06 097 I 059 51 I 05 805 L. Cos. L. Cotg. ic. d.l L. Tang. 85^ 99894 99893 99892 99891 99891 99 890 99889 99888 99887 99886 99885 99884 99883 99882 99881 99880 99879 99879 99878 99877 99876 99875 99874 99873 99872 99871 99 870 99 869 99868 99867 99866 99865 99 864 99863 99 862 . 99 861 9 99 860 . 99859 9 99 858 9 99 857 9 99 856 9 99855 99854 99853 99852 99851 99850 99848 99847 99 846 99845 99844 99843 99842 99841 99 840 99839 99838 99837 99836 99834 L. Sin. GO 59 58 57 55 54 53 52 51 50 49 48 47 45 43 42 41 40 39 38 36 35 34 33 32 31 30 29 28 27 26 Prop. Pts. 181 179 I 18.1 17.9 .2 36.2 35-8 •3 54-3 53-7 •4 72.4 71.6 •5 90-5 89.5 .6 108.6 107.4 •7 126.7 125.3 .8 144.8 143-2 •9 162.9 161. 1 175 X73 .1 17-5 17-3 .2 35.0 34-6 •3 52-5 51-9 •4 70.0 69.2 •5 87.5 86.5 .6 105.0 103.8 ■7 122.5 121. 1 .8 140.0 138.4 •9 157-5 155.7 168 166 I .16.8 16.6 3 33-6 33-2 •3 50-4 49-8 •4 67.2 66.4 •5 84.0 83.0 .6 100.8 99-6 •7 117. 6 116.2 .8 134-4 132.8 •9 151-2 149.4 162 159 16.2 159 32-4 31-8 48.6 47-7 64.8 63.6 81.0 79-5 6 97-2 95-4 •7 "3-4 III. 3 .8 129.6 127.2 •9 145.8 I43-I 155 153 I 15 5 153 2 31-0 30.6 3 46.5 45-9 4 62.0 61.2 5 77-5 76.5 .6 93-0 91.8 •7 108.5 107.1 .8 124.0 122.4 •9 139-5 '37-7 149 147 14.9 M-7 29.8 29.4 44-7 59-6 44.1 58.8 .6 74-5 894 73-5 S8.a •7 .8 104 3 119.2 102.9 1176 •9 134 -I 132 3 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 31 2 3 _4 5' 6 7 8 9 10 II 12 13 14 15 i6 17 i8 20 21 22 23 26 27 28 29_ 30 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 J9 60 L. Sin. 94030 94 174 94317 94461 94603 94 746 94887 95029 95 170 95310 95450 95589 95 728 95867 96 005 96 143 96 280 96417 96553 96 689 96825 96 960 97095 97229 97363 97496 97629 97762 97894 98 026 98157 98288 98419 98549 98679 98808 98937 99 066 99 194 99322 99450 99 577 99 704 99830 99956 00 082 00 207 00332 00 456 00 581 00 704 00828 00951 01 074 01 196 01 318 01 440 01 561 01 682 01 803 01 923 L. Cos. L. Tang. 94195 94340 94485 94630 94 773 94917 95 060 95 202 95 344 954S6 95627 95 767 95 908 96 047 96 187 96325 96 464 96 602 96 739 96877 97013 97 150 97285 97421 97556 97691 97825 97 959 98 092 98 225 98358 98 490 98 622 98753 98884 99015 99 145 99275 99405 99 534 99 662 99 791 99919 00 046 00 174 00 301 00 427 00553 00 679 00 805 00 930 01 055 01 179 01 303 01 427 01 550 01 673 01 796 01 918 02 040 02 162 Cotg. c.d. c.d. L. Cotg. 1.05 805 1 . 05 660 I 05 515 1.05 370 05 227 05 083 04 940 04 798 04 656 04514 04373 04233 04 092 03953 03813 03675 03536 03398 03 261 03 123 02 987 02 850 02 715 02579 02444 02 309 02 175 02 041 01 908 01 775 01 642 01 510 01378 01 247 01 116 1 . 00 985 00 855 I 00 725 I 00595 I 00 466 I 00338 1 . 00 209 I 00 081 o 99 954 o 99 826 0.99699 o 99 573 0.99447 0.99321 0.99 195 o . 99 070 o 98945 0.98821 0.98 697 0.98573 o . 95 450 0.98327 o . 98 204 o . 98 082 0.97 960 0.97 L. Tang. 84° I .Cos. 60 It 11 55 54 53 52 51 9 9 9 9 9 99834 99833 99832 99831 99830 9 9 9 9 9 99829 99828 99827 99825 99824 9 9 9 9 9 99823 99822 99821 99 820 99819 50 49 48 47 46 45 44 43 42 41 40 1 9 9 9 9 9 99817 99 816 99815 99814 99813 9 9 9 9 9 99812 99 810 99809 99808 99807 9 9 9 9 9 99806 99804 99803 99802 99801 35 34 33 32 31 30 29 28 11 9 9 9 9 9 99800 99798 99 797 99796 99 795 9 9 9 9 9 99 793 99792 99791 99790 99788 25 24 23 22 21 9 9 9 9 9 997S7 99786 99785 99783 99782 20 19 18 \l IS 14 13 12 II 9 9 9 9 9 99781 99780 99778 99777 99776 9 9 9 9 9 99 775 99 773 99772 99771 99769 10 7 6 9 9 9 9 9 99768 99767 99765 99 764 99763 5 4 3 2 I 9 99761 I .Sin. f Prop. Pts. 145 X43 141 14.5 14.3 14. ap.o 28.6 28. 43-5 42.9 42 58.0 57-2 56 72-5 71-3 70 6 87.0 85.8 84 7 101.5 100. 1 98. 8 116.0 114.4 112. 9 130.5 128.7 126. 139 ,^3-9 27 41 55 69 83 97 138 13 ■ 27 41 55 69 82 96 no .•24 135 133 .1 13.5 13.3 3 27.0 26.6 3 40.5 39-9 4 540 53-2 5 675 66.5 6 81.0 79.8 7 94-5 93-1 8 108.0 106.4 9 121.5 119.7 139 128 .1 12.9 12.8 .2 25.8 25.6 •3 38.7 38.4 •4 51.6 51.2 •5 645 64.0 .6 77-4 76.8 •7 90-3 89.6 .8 103.2 t02.4 •9 116. 1 115 2 135 133 .t "•5 12.3 .2 25 24.6 3 37 5 36.9 4 50 49.2 •5 62 5 61.5 6 75 73-8 7 87 5 86.1 8 100 98.4 9 112 5 110.7 X3I xao .1 12. 1 12.0 .2 24.2 «4.o •3 36.3 36.0 .4 48.4 48.0 •5 60.5 60.0 .6 72.6 72.0 7 847 84.0 .8 96.8 96.0 •9 108.9 108.0 136 13 27 40 54 131 13-1 26.2 39-3 52.4 65.5 78.6 91.7 104.8 117.9 136 12.6 25.2 37-8 50.4 63.0 75-6 88.2 100.8 "34 122 12.2 24.4 36.6 48.8 61.0 73-2 85.4 97.6 109.8 z 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Prop. Pts. 32 TABLE II 6^ __9_ 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 il 26 27 28 29 31 32 33 34 36 37 38 39 40 41 42 43 44 46 47 48 49 50 51 52 53 il. 55 56 57 58 i9_ 60 L. Sin. 01 923 02 043 02 163 02 283 02 402 02 520 02 639 02 757 02 874 02 992 03 109 03 226 03342 03458 03574 03 690 03 805 03 920 04034 04 149 04 262 04376 04 490 04 603 04 715 04828 04 940 05 052 05 164 05275 05386 05497 05 607 05 717 05 827 05937 06 046 06155 06 264 06372 06 481 06 589 06 696 06 804 06 911 07018 07 124 07231 07337 07442 07548 07653 07758 07863 07968 08 072 08 176 08280 08383 08486 08589 L. Cos. d. L. Tang. 02 162 02 283 02 404 02 525 02 645 02 766 02885 03005 03 124 03 242 03 361 03479 03597 03 714 03832 03948 04 065 04 181 04297 04413 04 528 04 643 04758 04873 04987 05 lOI 05214 05 328 05441 05553 05 666 05 778 05 890 06 002 06 113 06 224 06335 06445 06 556 06666 06775 06885 06 994 07 103 07 211 07 320 07428 07536 07643 07751 07858 07964 08 071 08 177 08 283 08389 08495 08 600 08 705 08810 08 914 d« L. Cotg. c. d. c. d, L. Cotg. 0.97838 0.97717 0.97 596 0.97475 0.97355 0.97234 0.97 115 0.96995 0.96 876 0.96 758 0.96639 0.96 521 o . 96 403 o . 96 286 0.96 168 0.96 052 0.95935 o 95 819 0.95 703 o 95587 0.95472 o 95 357 0.95 242 0.95 127 o 95013 0.94899 0.94 786 0.94 672 o 94 559 0.94447 0.94334 o 94 222 0.94 no 0.93998 0.93887 o 93 776 0.93665 o 93 555 o- 93 444 o 93 334 0.93 225 o 93 "5 o . 93 006 0.92 897 0.92 789 0.92 680 0.92 572 0.92 464 0.92357 0,92 249 0.92 142 o . 92 036 0.91 929 0.91 823 0.91 717 0.91 oil o 91 505 0.91 400 0.91 295 0.91 190 0.91 086 L. Tang, 83° L. Cos. 99 761 99760 99 759 99 757 99756 99 755 99 753 99 752 99 751 99 749 99 748 99 747 99 745 99 744 99 742 99741 99 740 99 738 99 737 99736 99 734 99 733 99731 99 730 99 728 99727 99 726 99 724 99 723 99721 99 720 99718 99717 99 716 99714 99 713 99 711 99710 99 708 99707 99705 99 704 99702 99 701 99699 99 698 99 696 99695 99693 99 692 9.99690 9.99689 9.99687 9 . 99 686 9 99684 99683 99 681 99 680 99678 99677 9 99675 L. Sin. Prop. Pts. lai xao I 12.1 12.0 .3 24.2 24.0 3 36.3 36.0 4 48.4 48.0 5 60.5 60.0 6 72.6 72.0 7 84.7 84.0 8 96.8 96.0 9 108.9 108.0 118 117 .1 II. 8 II. 7 .2 236 23 4 .3 35-4 35 I •4 47.2 46 8 •5 59.0 58 5 .6 70.8 70 2 •7 82.6 81 9 .8 94-4 93 6 ■9 106.2 los 3 "5 114 .1 "•5 II. 4 .2 33.0 22.8 •3 34-5 34-2 '4 46.0 45-6 .5 57-5 570 .6 69.0 68.4 • 7 80.5 79.8 .8 92.0 91.2 .9 103.5 102.6 iia III 1 .1 II. 2 II. I .2 22 4 22.2 •3 33 6 33-3 .4 44 8 44-4 •5 56 55-5 .6 67 2 66.6 •7 78 4 77-7 .8 89 6 88.8 •9 100 8 99.9 109 108 lOJ .1 10.9 10.8 10 .3 31 8 21.6 21 •3 32 7 32.4 32 •4 43 6 43-2 42 •5 54 5 54 -o 53 .6 65 4 64.8 64 •7 76 3 75.6 74 .8 87 2 86.4 85 •9 98 I 97.2 96 X06 IC.5 I 10.6 10.5 .3 21.2 21. •3 31-8 31 5 •4 42.4 42.0 •5 .6 53 63.6 52 5 63.0 .7 .8 74.2 84.8 73-5 84.0 •9 95-4 94-5 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 33 L. Sin. 9_ 10 II 12 13 \l 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 46 47 48 49 50 51 52 53 II II 59 GO 08589 08 692 08795 08897 08 999 09 lOI 09 202 09304 09405 09 506 09 606 09 707 09 807 09907 0006 o 106 o 205 0304 0402 o 501 0599 0697 0795 0893 o 990 I 087 1 184 1 281 1377 I 474 I 570 I 666 I 761 1857 1952 2047 2 142 2 236 2331 2425 2519 2 612 2 706 2 799 2 892 2985 3078 3 171 3263 3 355 3 447 3 539 3630 3 722 3813 3904 3 994 4085 4175 4 266 4356 L. Cos. 103 103 102 102 1 03 101 1 03 lOI lOI 100 lOI 100 100 •99 100 99 99 98 99 98 98 98 98 97 97 97 97 96 97 96 96 95 96 95 95 95 94 95 94 94 93 94 93 93 93 93 93 92 92 92 92 91 92 91 91 90 91 90 91 90 L. Tang. c. d. 08 914 09 019 09 123 09 227 09330 09434 09537 09 640 09 742 09845 09947 o 049 o 150 o 252 0353 0454 o 656 0756 0856 0956 I 056 1 155 1 254 I 353 452 649 747 845 1943 2 040 2138 2235 2332 2428 2525 2 621 2 717 2813 2 909 3004 3099 3 194 3289 3384 3478 3 573 3667 3 761 3854 3948 4041 4 134 4227 4320 4412 4504 4 597 4688 4780 L. Cotgr. c. d. L. Cotg. 0.91 086 0.90 981 0.90 877 0.90 773 0.90 670 o . 90 566 0.90463 o . 90 360 0.90 258 0.90155 0.90053 0.89 951 o . 89 850 0.89 748 o . 89 647 0.89 546 0.89445 0.89344 o . 89 244 0.89 144 o . 89 044 o . 88 944 0.88845 0.88 746 0.88647 0.88548 o . 88 449 0.88351 0.88253 0.88 155 0.88057 0.87960 c. 87 862 0.87 765 0.87668 0.87572 0.87475 0.87379 0.87283 0.87 187 0.87091 0.86 996 0.86 901 0.86806 0.86 711 0.86616 0.86 522 0.86 427 0.86333 o . 86 239 0.86 146 0.86 052 0.85959 0.85 866 0.85 773 0.85 680 0.85588 0.85 496 0.85403 0.85 312 0.85 220 L. Tang. 82° L* Cos. 9.99667 9 . 99 666 9.99664 9.99663 9.99 661 99675 99674 99672 99670 99 669 99659 99658 99656 99655 99653 99651 99650 99 648 99647 99645 99643 99642 99 640 99638 99637 99635 99633 99632 99630 99 629 99627 99625 99624 99 622 99 620 99 618 99617 99615 99613 99 612 99 610 99 608 99607 99605 99603 9.99 601 9.99 600 9-99 598 99596 99 595 99 593 99591 99589 99588 99586 99584 99582 99581 99 579 99 577 9-99 575 L. Sin. 60 59 58 57 _5i 55 54 53 52 _5i 50 49 48 47 _46 45 44 43 42 41 40 39 38 _36 35 34 33 32 _3i 30 29 28 27 26 Prop. Pt8. 105 104 I IO-5 10.4 .2 21.0 20.8 •3 31 5 31.2 •4 42.0 41.6 .5 52. s 52.0 .6 63.0 62.4 •7 73-5 72.8 .8 84.0 832 •9 94 5 93 6 102 lOI I 10.2 10. I 2 20.4 20.2 3 •4 i 30.6 40.8 61 .2 303 40.4 I l\:t £i ■9 9.. 8 90.9 9S ) 98 I 9 9 9 ? 19 8 19. •3 29 7 29. •4 39 6 39- 49 5 49- .6 59 4 58. 7 69 3 68. 8 79 2 78. 9 89 I 88. 97 9 7 19 29 38 48 58 67 77 87 94 9-4 18.8 28.2 37 6 47.0 56.4 65.8 75-2 84.6 9e ' 1 9.6 19.2 28.8 38.4 48.0 57-6 67.2 76.8 86.4 93 9 3 18 6 27 9 37 2 46 55 1 65 I 74 4 83 7 103 10.3 20.6 309 41 .2 6i'8 72.1 82.4 92.7 100 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 4 2 95 9 5 19 o 285 380 47-5 91 90 .1 9 ^ 9.0 .2 18 2 18.0 •3 27 3 27.0 •4 ?6 4 36.0 .5 45 5 45 .6 54 b .54 • 7 63 7 63.0 .8 r. 8 72.0 9 9 81.0 92 9.2 18.4 27.6 36.8 46.0 64.4 73 6 82.8 a 0.2 0.4 0.6 0.8 i.o 1.2 1-4 1.6 1.8 Prop. Pts. 34 TABLE II 8° L. Sin, 10 II 12 13 14 15 i6 17 i8 i9_ 20 21 22 23 24 26 27 28 30 31 32 33 34 36 37 38 39 40 41 42 43 44 46 47 48 j49 50 51 52 53 54 59 60 E 4356 4445 4535 4624 4 714 4803 4891 4 980 5069 5 J57 5245 5 333 5421 5508 5596 5683 5 770 5857 5 944 6030 6 116 6 203 6289 6374 6 460 6545 6631 6 716 6801 6 886 6970 7055 7 139 7223 7307 7391 7 474 7558 7641 7724 7807 7890 7 973 8055 8137 6 220 8 302 8383 8465 8547 8628 8 709 8790 8871 8952 9033 9 "3 9 193 9273 9 353 9 19433 L. Cos, d. 89 L. Tan^. 4780 4872 4963 5054 5 145 5236 5327 5417 5 "^08 5598 5 688 5 777 5867 6 046 6135 6 224 6312 6 401 6489 6577 6665 6753 6841 6928 7 016 7103 7 190 7277 7363 7450 7536 7 622 7708 7 794 7880 7965 8051 8 136 8221 8306 8391 8475 8560 8644 8728 8812 8896 8979 9063 9 146 9 229 9312 9 395 9478 9561 9643 9725 9807 9889 9.19971 L. Cotg. c. d c.d. L. Cotg. 0.85 220 o 85 128 o 85037 o 84 946 0.84855 o 84 764 o . 84 673 084583 o . 84 492 o . 84 402 84312 84223 84133 84 044 83954 83865 83776 83688 83599 83 5" 0.83423 o 83335 0.83247 0.83 159 o . 83 072 0.82984 0.82897 0.82810 o 82 723 o 82 637 0.82 550 o . 82 464 0.82378 0.82 292 o 82 206 0.82 120 o 82 035 o 81 949 o 81 864 o 81 779 o 81 694 0.81 609 o 81 525 o 81 440 0.81 356 0.81 272 0.81 188 0.81 104 o 81 021 o 80 937 o 80 854 o 80 771 o 80688 o . 80 605 0.80 522 o . 80 439 0.80357 0.80275 0.80 193 0.80 III . 80 029 L. T,aiig. 8r L. Cos. 9 99 575 9 99 574 9 99 572 9 99 570 9 99 568 99566 99565 99563 99561 99 559 99 557 99556 99 554 99552 99550 99548 99 546 99 545 99 543 99541 99 539 99 537 99 535 99 533 99532 99530 99528 99526 99524 99522 99520 99518 99517 99515 99513 99 5" 99509 99507 99505 99503 99501 99 499 99 497 99 495 99 494 99492 99490 99488 99 486 99484 99482 99 480 99478 99476 99 474 99472 99470 99 468 99 466 99464 99462 L. Sin, GO 59 58 57 _5i 55 54 53 52 _51 50 49 48 47 _46^ 45 44 43 42 41 40 39 38 37 36 Prop. Pts. 9i 91 1 I 9.2 9.1I 2 18 4 2 3 27 6 27 3 4 36 8 36 4 <; 46 45 5 6 55 2 S4 6 7 64 4 63 7 8 73 6 72 8 9 82 8 81 9 90 9.0 18.0 27.0 36.0 45.0 54 o 63.0 72.0 81.0 89 87 17 26 34 43 52 60 69 78 85 17 25 34 42 51 59 68 76 83 8 16 24 33 41 49 58 66 74 8.8 17.6 26.4 35 2 44 o 52.8 61.6 70.4 79 2 86 i.6 17.2 25.8 34 4 43 o 51 6 60.2 68.8 77 4 84 8.4 16.8 25.2 33-6 42.0 50 4 58.8 67 2 75.6 16 24 32 41 49 57 65 6 738 81 80 I 8.1 8.0 2 16 2 16.0 3 24 3 24.0 4 32 4 32.0 s 40 5 40.0 6 48 6 48.0 7 S6 7 56.0 8 64 8 64.0 9 72 9 72.0 3 0.2 0.4 0.6 0.8 1 .0 1.2 1-4 1.6 1.8 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 35 9° 1 L. Sin. d. L. Tan^. c.d. L. Cot^. L. Cos. Prop. Pts. 9 19433 80 9 19 971 82 80 029 9 99462 (JO I 9 19 513 9.20053 79 947 9.99460 59 83 81 80 2 9 19592 80 9 20 134 82 79 866 9 99 458 58 .1 8.2 8.1 8.0 3 9 19672 79 79 9 20 216 79 784 9 99 456 57 2 16.4 16.2 16.0 4 5 9 19 751 9 19 830 9 20 297 81 81 0.79703 9 99 454 56 .3 55 -4 24.6 24.3 24.0 32.8 32.4 32.0 9 20 378 . 79 622 9 99 452 6 9 19909 9 20 459 79541 9 99 450 54 -5 41 .0 40.5 40.0 7 9 19988 79 9 20 540 . 79 460 9 99 448 53 ^ 49.2 48.6 48.0 8 9 20 067 79 78 78 9 20 621 80 81 0.79379 9 99446 52 •/ 57.4 56.7 56.0 9 10 9 20 145 9 20 701 9 20 782 79 299 9 99 444 51 .^ 50 5 65 6 64.8 64.0 73.8 72.9 72.0 9 20 223 79 218 9 99 442 II 9 20 302 78 9 20 862 80 79 138 9 99440 49 79 78 12 9 20 380 9 20 942 . 79 058 9 99438 48 .1 7.9 7-8 13 9 20 458 78 9 21 022 80 0.78978 9 99436 47 .2 15.8 15.6 14 9 20535 77 78 9.21 102 80 0.78898 9 99 434 46 45 .3 23.7 4 31 6 23 4 31.2 9 20 613 9 21 182 0.78818 9 99432 i6 9.20691 78 9 21 261 79 0.78739 9 99 429 44 5 39-5 2^2 17 9 20 768 77 9 21 341 80 0.78 659 9 99427 43 6 47.4 46.8 i8 9 ■ 20 845 77 9 21 420 79 0.78 580 9 99 425 42 l I^^ 54 6 19 20 9 20 922 77 77 9 21499 79 79 0.78 501 9 99423 41 40 .8 63.2 •9l 71 I 62.4 70.2 9.20999 9 21578 0.78422 9 99 421 21 9.21 076 77 9 21 657 79 78343 9.99419 39 77 76 22 9 21 153 77 76 9 21 736 79 0.78264 9.99417 38 .1 7-7 7.6 23 9.21 229 9 21 814 78 0.78 186 9 99415 37 .2 15 4 ^5-^ 24 9.21 306 77 76 9 21893 79 78 0.78 107 9 99413 36 35 ■3 23.1 .4 30.8 22.8 30.4 2S 9.21 382 9 21 971 . 78 029 9 99 411 26 9.21 458 76 9 22049 78 0.77951 9.99409 34 l ^l^ 38. 45-6 27 9 21 534 76 9 22 127 78 0.77873 9.99407 33 .6 46.2 28 9.21 610 76 9 22 205 78 0.77795 9.99404 32 I in 9 693 ^:8 68.4 29 30 9.21 685 75 76 9 22 283 78 78 0.77717 9 99402 31 80 9.21 761 9 22 361 0.77639 9.99400 31 9.21 836 75 9 22438 77 0.77562 9 99398 29 75 74 32 9.21 912 76 9 22 516 78 0.77484 9.99396 28 .1 75 7-4 33 9.21 987 75 9 22593 77 0.77407 9 99 394 27 .2 15.0 14 8 34 35 9 22 062 75 75 9 22 670 77 77 0.77330 9 99392 26 25 •3 22.5 •4 30 22.2 29.6 9.22 137 9 22747 0.77253 9 99 390 3^ 9.22 211 74 9 22 824 77 0.77176 9 99 388 24 •5 37 5 .6 45.0 37 37 9 22 286 75 9 22 901 77 0.77099 9 99385 23 Tii 3« 9.22 361 75 9 22977 7b 0.77023 9 99383 22 .8 60.0 ■ 9 67 5 39 40- 9 22435 74 74 9 23054 77 76 0.76946 9 99381 21 20 Iti 9 22509 9 23 130 0.76 870 9 99 379 41 9.22583 74 9 23 206 76 0.76 794 9 99 377 19 73 72 42 9 22657 74 9 23283 77 0.76 717 9 99 375 18 • I 7-3 7.2 43 9.22 731 74 9 23359 7b 0.76 641 9 99 372 17 .2 14.6 144 44 9.22805 74 73 9 23435 7b 75 0.76565 9 99370 16 15 .3 21.9 4 29.2 5 36.5 6 43 8 7 51 I .8 58 4 •9 65.7 21.6 28.8 36.0 43-2 50 4 57 6 64.8 45 9 22878 9 23510 0.76490 9.99368 46 9.22952 74 9 23 586 7b 0.76414 9.99366 14 ' 47 9 23025 73 9 23661 75 0.76339 9 99364 13 ,4« 9 23 098 73 9 23737 7b . 76 263 9 99362 12 49 60 9 23 171 73 73 9 23812 15 75 76 188 9 99 359 II 10 9 . 23 244 9 23887 0.76 113 999 357 51 9 23317 73 9 23962 75 0.76038 9 99 355 ?> T 71 3 3 52 9 23390 73 9 24037 75 0.75963 0.75888 9 99 353 8 I 7.1 c .6 0.4 .9 0.6 .2 0.8 5 '0 53 9.23462 72 9 24 112 75 9 99351 7 1 14.2 c 55 9 23535 73 72 9 24 186 74 75 0.75 814 9-99 348 6 3 5 1 4 6 21.3 c 28.4 I 35-5 I 42.6 I 49 7 2 56.8 2 63.9 2 9.23607 9 24 261 0.75739 9 -99 346 5t> 9 23679 72 9 24335 74 0.75665 9 99 344 .8 1 .2 H 9 23 752 73 9 24410 75 0.75590 9 99342 I -7 8 .1 1.4 .4 1.6 7 I 8 5^ 9.23823 71 9 24484 74 75516 9 99340 59 60 9 23895 72 72 9 24 55« 74 74 0.75442 9 99 337 ' 9.23967 9.24632 0.75368 9 99 335 ^ L. Cos. d. L. Cotg. c.d. L. Tan^. L. Sin. t Prop. Pts. 1 80° 1 36 TABLE II 10^ _9_ 10 II 12 13 14 15 16 17 18 19 20 21 22 23 ^_ '25 26 27 28 29 30 31 32 33 36 37 38 39 40 41 42 43 44 46 47 48 49_ 50 51 52 53 51 55 56 57 58 19. (io ]j. Siii< 23967 24039 24 no 24 181 24253 24324 24395 24 466 24536 24 607 24677 24 748 24818 24888 24958 25 028 25 098 25 168 25237 25 307 d* L. Tan^. c. d 25376 25445 25514 25583 25652 25 721 25 790 25858 25927 25995 26 063 26 131 26 r9'9 26 267 26335 26403 26 470 26538 26 605 26 672 26739 26806 26873 26 940 27 007 27073 27 140 27 206 27273 27 339 27405 27471 27537 27 602 27668 27 734 27 799 27864 27930 27995 28 060 L. Cos. 72 71 71 72 71 71 71 70 71 70 71 70 70 70 70 70 70 69 70 69 69 69 69 69 69 69 68 69 68 68 68 68 68 68 68 67 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 65 . 66 66 6S 65 66 65 65 24 632 24 706 ,24779 24853 24 926 .25 000 25073 .25 146 25 219 25 292 25365 25437 25510 25582 25655 •25 727 25 799 .25 871 •25943 .26 015 26086 26 158 26 229 26 301 26372 26443 26514 26585 26 655 26 726 26797 26867 26937 27 008 27078 27 148 27 218 27288 27357 27427 9 9 9 9_ 9 9 9 9 9. 9 9 9 9 9_ 9 9 9 9 9_ 9 9 9 9 _9 9 9 9 9 _9 9 9 9 9 J9 9 9 9 9 _9 L. Cot^. 2,7496 27 566 27635 27 704 27773 .27 842 .27911 27 980 28 049 28 117 ,28 186 .28254 28323 28391 ,28459 28 527 28 595 ,28662 28 730 ,28 798 ,28865 74 73 74 73 74 73 73 73 73 73 72 73 72 73 72 72 72 72 72 71 72 71 72 71 71 71 71 70 71 71 70 70 71 70 70 70 70 69 70 69 70 69 69 69 69 69 69 69 68 69 68 69 68 68 68 68 67 68 68 67 c. d. L. Cotg. 0.75368 0.75294 0.75 221 0.75 147 0.75074 o . 75 000 0.74927 0.74854 0.74781 o . 74 708 0.74635 0.74563 0.74490 0.74418 0.74345 0.74273 0.74 201 0.74 129 0.74057 0.73985 0.73914 0.73842 0.73 771 0.73699 0.73 628 0.73557 0.73486 0.73415 0.73345 0.73274 0.73203 0.73 133 0.73063 o . 72 992 0.72 922 L. Cos. 0.72 852 0.72 782 0.72 712 0.72643 0.72573 o . 72 504 0.72434 0.72 365 o . 72 296 0.72 227 0.72 155 o . 72 089 o . 72 020 0.71 951 0.71 883 0.71 814 0.71 746 0.71 677 0.71 609 0.71 541 0.71 473 0.71 405 0.71 338 0.71 270 0.71 202 0.71 135 L. Tang. 79^ 9 99335 9 99333 9 99331 9.99328 9 99326 9 99 9 99 9 99 9 99 9 99 324 322 319 317 315 9 99 9.99 9 99 9 99 9 99 3^3 310 308 306 304 9.99301 9.99299 9.99297 9.99294 9,99292 9.99290 9 . 99 288 9.99285 9.99283 9.99 281 9.99278 9.99276 9.99274 9.99271 9.99269 9.99267 9.99264 9 . 99 262 9 . 99 260 9-99 257 9 99255 9.99252 9.99250 9.99248 9 99245 99243 99241 99238 99236 99233 9.99231 9.99229 9.99 226 9 99224 9.99 221 9.99219 9.99217 9,99214 9.99 212 9 99209 9.99207 9.99204 9 99 202 9 . 99 200 9 99 197 9 99 195 L. Sin. 60 59 58 56 45 44 43 42 _1L 40 39 38 _3i 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20^ ^9 18 17 16 Prop. Pis. 74 73 I ^i 7. 2 14.8 14 •3 22.2 21. 4 29.6 29 .5 37.0 36. .6 44.4 43 ■7 51.8 51 .8 59.2 58 •9 66.6 165 7a I 7.2 2 14.4 3 21,6 4 28.8 36. c .6 43 2 .7 .50.4 .8 57-6 ■9 64.8 70 7.0 14,0 21 .0 28.0 35 o 42.0 49 o 56,0 63.0 68 6.8 13-6 20.4 27.2 34 o 40.8 47.6 54-4 61.2 3 03 0.6 0.9 I .2 66 I 6.6 2 13.2 3 4 19.8 26.4 .7 .8 330 39 6 46.2 52.8 •9 59 4 4 7 71 71 14 21 28 35 42 49 7 56.8 63 9 69 69 13 '^ 20, 27 34 41 48 55 62 67 6. 13 20. 26. 33 40.2 46.9 53 6 60.3 65 6.5 13.0 26.0 32 5 39 o 45 5 52.0 58.5 a 0.2 0.4 06 08 1 .0 12 14 16 1.8 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 37 11 7 8 _9_ 10 12 13 \l 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 46 47 48 49 60 51 52 53 ii 55 56 U 60 L. Sin. 9 28 060 9 28 125 9.28 190 9.28254 9,28319 9 • 28 384 9 2S 448 9.21 512 9.28577 9 28 641 9.28 705 9.28 769 9 28 833 9 . 28 896 9 28 960 9 29 024 9.29 087 9.29 150 9.29 214 9.29277 9 29340 9.29403 9 . 29 466 9.29529 9.29591 9.29654 9.29 716 9 29 779 9.29841 9.29903 9 . 29 966 9 . 30 028 9.30090 930 151 9.30213 9 30275 9 30336 9 30398 9 30459 9 30521 9.30 582 9 30643 9,30 704 9 30 765 9 . 30 826 9.30887 9 30947 9 31 008 9.31 068 9 31 ^29 9.31 189 9 31 250 9-31 310 9 31 370 9 31 430 9.31 490 9 31 549 9 31 609 9,31 669 9 31 728 9.31 788 L. Cos. 65 65 64 65 65 64 64 65 64 64 64 64 63 64 64 63 63 64 63 63 63 63 63 62 63 62 63 62 62 63 62 62 61 62 62 61 62 61 62 61 61 61 61 61 61 60 61 60 61 60 6i 60 60 60 60 59 60 60 59 60 L. Tang, c. d 28865 28933 29 000 29 067 29 134 29 201 29 268 29335 29 402 29 468 29535 29 601 29668 29734 29 800 29866 29 932 29998 30 064 30 130 30195 30 261 30326 30391 30457 30 522 30587 30652 30717 30 782 30 846 30 91 1 30975 31 040 31 104 31 168 31 233 31 297 31 361 31425 • 31 489 31 616 31 679 31 743 31 806 31 870 31933 31996 32059 32 122 32185 32248 323" 32373 32436 32498 32561 32623 32685 32747 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 66 66 66 65 66 65 65 66 65 65 65 65 65 64 65 64 65 64 64 65 64 64 64 64 63 64 63 64 63 64 63 63 63 63 63 63 63 62 63 62 61 62 62 62 L. Cotg. 0.71 135 0.71 067 0,71 000 o 70933 o . 70 866 0.70799 o . 70 732 o . 70 665 0.70 598 0.70532 o . 70 465 0.70399. 0.70332 o . 70 266 o . 70 200 0,70 134 o . 70 068 o . 70 002 0.69936 o 69 870 o . 69 805 0.69739 0.69 674 0.69 609 0.69543 0.69 478 0.69413 0.69348 o . 69 283 o 69 218 0.69 154 o . 69 089 o . 69 025 0.68 960 0.68896 L. Cos. L. Cotg. Ic. d. 0.68832 0.68 767 0.68 703 o 68 639 0.68575 0.68 511 0.68448 0.68384 o 68321 0.68 257 0.68 194 0.68 130 0.68067 o . 68 004 o 67 941 0.67878 0.67815 o 67 752 o 67 689 0.67 627 0.67 564 0.67 502 o 67439 0.67377 o 67315 67253 L. Tang. 78^ 9 99 195 9.99 192 9.99 190 9.99 187 9 99 185 9.99 182 9.99 180 9.99 177 9 99 175 9.99 172 9.99 170 9.99 167 9.99 165 9 99 162 9 99 160 9 99 9 99 9 99 9 99 9 99 157 155 152 150 147 9 99 9 99 9 99 9 99 9 99 145 142 140 137 135 9 99 9 99 9 99 9 99 9 99 132 130 127 124 122 9 99 9 99 9 99 9 99 9 99 9 99 9 99 9-99 9 99 9 99 106 104 lOI 099 096 9 99 9 99 9 99 9 99 9 99 093 091 088 086 083 9 99 9 99 9 99 9 99 9 99 080 078 075 072 076 9 99 9 99 9 99 9 99 9 99 067 064 062 059 056 9 99054 9,99051 9.99048 9.99046 9 99043 9 99 040 L. Sin. 60 59 58 55 54 53 52 _5L 50 49 48 47 46 25 24 23 22 21 20 19 18 17 16 Prop, rts. 68 6.8 13.6 20 4 27 2 34 o 40.8 47 6 54-4 61 2 66 6.6 13 2 10 8 26 33 39 46 52 59-4 64 6.4 12.8 19.2 25.6 32.0 38.4 44-8 51 2 576 63 62 12 24 31 37 43 49 55 60 6.0 12.0 18.0 24.0 30,0 36.0 42 o 48.0 54 oi 67 07 13 4 20 I 26 8 33 5 40.2 46 9 60.3 65 65 13 19 26 32 39 45 52 58 63 12.6 189 25 2 3^ 5 37 8 44 I 50 4 567 61 6 12 18 24 36 42 48 54 59 5 9 II 17 23 3 .1 03 .2 0.6 •3 0.9 4 1.2 I 'A 7 2. 1 8 2.4 9 2.7 a 0.2 0.4 0.6 0.8 1.0 1.2 14 16 1.8 Prop. Pts. 38 TABLE II 12° 1 / L. Sin. d. L.Taug. c.d.| L. Cotg. L. Cos. 60 Prop. Pts. 9.31 788 9 32 747 63 0.67253 9.99040 9 31 847 9.32 810 0.67 190 9.99038 59 63 6a 2 9.31 907 9-32872 0.67 128 9 99035 58 I 63 6.2 3 9.31 966 59 9 32933 62 0.67067 9.99032 57 .2 12 6 12 4 _± 9.32025 59 9-32995 62 62 0.67005 9.99030 56 55 .3 18 9 18.6 .4 25 2 24.8 9.32084 9-33 057 0.66943 9 99027 6 9 32 143 59 9 33 "9 fi-W 0.66881 9 99 024 54 .5 31 5 31 7 9.32 202 59 9 33 180 62 61 0.66820 9.99022 S3 -6 37 8 37 2 8 9.32 261 59 58 59 9 33242 0.66 758 9.99019 52 -7 44-1 43.4 9 10 9 32319 9 33303 62 0.66697 9.99016 51 .8 50.4 49 6 •9 567 55-8 9 32378 9 33 365 0.66635 9.99013 50 II 9 32437 59 58 58 9 33426 61 0.66 574 9.99 on 49 61 60 12 9 32495 9 33487 61 0.66 513 9 99 008 48 .1 6.1 6.0 13 9 32553 9 33548 61 0.66452 9 99 005 47 .2 12.2 12 14 15 9 32 612 59 58 58 58 58 58 58 58 9.33609 61 61 61 61 60 61 0.66 391 9 99002 46 .3 18.3 18.0 .4 24.4 24.0 9 32 670 9-33670 0.66330 9 99000 45 16 9 32 728 9 33 731 0.66269 9 98 997 44 I ^2i ^2 ° 17 932786 9-33 792 0.66208 9 98 994 43 .6 36.6 36.0 18 9.32844 9 33853 0.66 147 9 98991 42 .7 42.7 42.0 .8 48.8 48.0 •9 54-9 540 19 20 9.32902 9 33913 0.66087 998989 41 40 9.32960 9 33 974 . 66 026 9.98986 21 9.33018 9 34034 At 0.65 966 9-98983 39 59 22 9 33075 57 S8 9 34095 0.65905 9.98980 38 •' ^1 2S 9 33 133 9-34155 0.65 845 9.98978 37 .2 II. 8 24 25 9 33 190 57 58 9 34215 61 60 60 60 60 60 0.65 785 9 98975 36 35 .3 17 7 •4 23.6 •5 29-5 •6 35 4 9 33248 9.34276 0.65 724 9 98972 26 9 33305 57 9 34336 0.65 664 9 98969 34 27 9 33362 57 58 9 34396 0.65 604 9.98967 33 28 9-33 420 9 34456 0.65 544 9 98964 32 • 7 41 3 .8 47 2 9 53 I 58 57 1 5.8 5.7 2 II. 6 II. 4 29 30 9-33 477 57 57 9 34516 0.65 484 9.98 961 3^ 30 9-33 534 9 34 576 0.65 424 9.98958 V 9 33591 57 9 34635 59 60 60 0.65 365 9 98955 29 32 9-33 647 56 9 34695 0.65 305 9 98953 28 33 9 33 704 57 9 34 755 0.65245 9.98950 27 34 9-33761 57 57 56 9 34814 59 60 0.65 186 9.98947 26 ,3 17.4 171 .4 23.2 22.8 .5 29.0 28,5 .6 34.8 34 2 7 40 6 39 9 .8 46.4 45.6 .9 52 2 51.3 56 55 •I 56 55 3S 9 33818 9 34874 0.65 126 9 98944 25 3^ 9 33874 9 34 933 59 0.65 067 9.98941 24 37 9 33931 57 56 56 57 9 34992 59 0.65 008 9.98938 23 3« 39 40 9 33987 9 -34 043 9 35051 9 35 "I 59 60 59 0.64949 0.64889 9-98936 9 98933 22 21 20 9 34 100 9 35 170 0.64830 9.98930 41 9 34156 56 9 35229 59 0.64 771 9.98927 19 42 9.34212 56 9 35288 59 0.64 712 9 98924 18 43 9 . 34 268 56 56 56 56 9 35 347 59 58 59 0.64653 9.98 921 17 3 16. 8i 16 5 .4 22 4 22.0 ,5 28 27.5 .6 33.6 33.0 -7 39-2 38.5 8 44.8 44.0 44 45 9 34324 9 35 405 0.64595 9 98919 lb 15 9-34380 9 35 464 64 536 9 98 916 46 9-34436 9 35 523 59 58 0.64477 9 98913 14 47 9-34 491 55 9 35 581 0.64419 9 98 910 »3 48 9-34 547 56 9-35 640 59 58 59 58 58 58 58 S8 58 58 58 58 57 . 64 360 9 98907 12 49 50 9.34602 55 56 935698 . 64 302 9.98904 II 10 .9 50 4 49 5 3 3 9-34658 9 35 757 0.64243 9.98901 51 9-34713 55 9 35815 0.64 185 9.98898 9 I 0.3 0.2 .2 0.6 0.4 3 0.9 0.6 .4 1.2 08 •5 15 10 .6 1.8 1.2 52 9-34769 56 9 35873 0.64 127 9.98896 8 S3 9.34824 55 9 35931 . 64 069 9.98893 7 54 9 34879 55 55 9 35989 0.64 on 9.98890 6 5 9 34 934 9.36047 0.63953 9 98887 56 9.34989 55 9 36 105 63 895 9 98 884 4 57 9-35044 55 9.36 163 0.63837 9.98881 3 7 2.1 14 S« 9-35099 55 9.36221 0.63 779 9.98878 2 .8 24 16 59 or 9 35 154 55 55 9.36279 0.63 721 9.98875 I .9 2.7 1.8 9 35209 9 36336 . 63 664 9.98872 L. Cos. d. |l. Cotg. c.d. L. Taug. L. Sin. Prop. Pts. 77° 1 LOGARITHMS OF THE TKIGONOMETRIC FUNCTIONS 39 13° 1 L. Sin. 54 55 55 L. Tang. c.d. L. Cotg. L. Cos. 60 Prop. Pts. 1 9 35209 9 36 336 58 58 57 0.63 664 9.98872 1 I 9 35263 9 36 394 0.63 606 998869 S9 58 57 2 9 35318 9 36452 63 548 9 98867 58 • I 5.^ ? 5 7 3 II 4 \ 17 I I 22 8 3 9 35 373 9 36 509 63 491 9.98864 57 .2 II. ( 4 9 35427 54 9.36566 9.36624 58 63434 9 98861 56 S5 •3 17- •4 23. i 9 35481 63 376 9 98858 6 9 35 536 9.36681 0.63319 9 98855 S4 •5 29.: D 28 5 7 9 35 590 9 36738 63 262 9 98 852 SS .6 34. { ^ 34 2 8 9 35644 9 36 795 63 205 9 98 849 S2 .7 40 6 39.9 1 9 10 9 "9 35 698 54 9.36852 57 63 148 9 . 98 846 51 50 .8 46 . •9 52.^ \ 45-6 2 51 3 35752 9 36909 0.63091 9 98843 II 9 35 806 9 36966 0.63034 9 98 840 49 56 55 12 9.35860 9 37023 57 0.62977 9 98837 48 .1 5 < ^ 5 5 13 9 35914 9 37 080 0.62 920 9.98834 47 .2 II.: 2 II. 14 IS 9 35968 54 9 37 137 57 56 62863 9 98 831 46 45 .3 16.8 16.5 .4 22.4 22.0 9 36022 9 37 193 0.62 807 9.98828 i6 9 36075 53 9 37250 57 56 0.62 750 9 98825 44 •5 28.0 27 5 17 9 36 129 9 37306 . 62 694 9.98822 43 •6 33( ^ 330 i8 9 36 182 53 9 37363 56 57 56 56 56 56 56 0.62 637 9.98819 42 ■9 50-^ \ 38.5 19 20 9 36236 53 9 37419 0.62 581 9.98816 41 40 5 44.0 ^ 49-5 9 36289 9 37 476 0.62 524 9.98813 21 9 36342 9 37532 0.62468 9.98810 39 54 22 9 36395 937588 0.62 412 9.98807 38 .1 5 4 23 9 36449 54 9 37644 0.62 356 9.98804 37 .2 10.8 1 24 2S 9 36502 53 9.37700 0.62 300 9.98801 36 35 •3 I •4 2 62 16 9 36555 9 37756 62 244 9.98798 26 9 36 608 53 9 37812 56 0.62 188 9 98 795 34 •5 27.0 27 9 ^6 660 52 9 37 868 56 0.62 132 9 98 792 33 t ^li 28 9 36713 53 9 37924 56 56 55 0.62 076 9.98789 32 ■I V 29 9 36766 53 53 9 37980 . 62 020 9 98 786 31 30 .8 4 •9 4 3 2 8.6 9 36819 9 38035 0.61 965 9.98783 31 9 36871 52 9.38091 56 0.61 909 9 . 98 780 29 53 5a 32 9 36924 53 938 147 50 0.61 853 9.98 777 28 •I 5' ; 52 33 9 36976 52 9 38 202 55 0.61 798 9.98 774 27 .2 10. ( ) 10.4 34 9 37028 52 53 9 38257 55 56 0.61 743 9.98771 26 •3 15 S ) 156 20.8 26.0 35 9 37081 9 38313 61 687 9.98768 25 .4 21.2 • 5 26. c .6 31-^ 3^ 9 37 133 37185 52 9 38 368 55 61 632 9.98765 24 37 9 52 9 38 423 55 0.61 577 9.98 762 23 ; 31.2 36 4 \ 416 ' 46.8 3« 9 37 237 52 9 38 479 56 0.61 521 9 98 759 22 •7 371 .8 42.4 •9 47 y 39 40 9 37289 52 52 9 38534 55 55 0.61 466 9.98 756 21 9 37341 9 38 589 0.61 411 9 98 753 20 41 9 37 393 52 9 38644 55 61 356 9.98750 19 51 4 42 9 37 445 52 9 38 699 55 0.61 301 9.98 746 18 • I 51 0I 43 9 37 497 52 9 38 754 55 0.61 246 9 98 743 17 .2 10.2 44 4.S 9 37 549 52 51 9 38 808 54 55 0.61 192 9.98740 16 •3 15 3 .4 20.4 .6 30. e 9 45 5 3 1.2 1.6 2 37600 9 38 863 9.38918 0.61 137 9 98 737 15 46 9 37 652 52 55 0.61 082 9 98734 14 It 47 9 37703 51 9.38972 54 0.61 028 9.98 731 13 48 9 37 755 52 9 39 027 55 0.60973 9.98 728 12 11 49 60 9 37806 51 52 9.39082 55 54 0.60 918 9.98725 9.98 722 II 10 9 37858 9 39 136 . 60 864 51 9 37909 51 9 39 190 54 0.60 810 9 98 719 9 0.2 52 9 37960 51 9 39245 55 0.60 755 9 98 715 8 .2 0.6 53 9 38 on ^' 9.39299 54 0.60 701 9 98 712 7 0.6 08 I.O 54 55 9 38062 51 51 9 39 353 54 54 0.60 647 9 98 709 6 5 3 09 .4 12 9 38 113 9-39 407 0.60593 9 98 706 5^ 9 38 164 SI 9.39461 54 0.60539 9 98 703 4 1.2 ■■^^ 9 38215 51 9 39515 54 0.60485 9.98 700 3 .7 2.1 8 2.4 \i 5« 9 38 266 51 9 39569 54 60 43 1 9.98697 2 59 60_ 9 383^7 51 51 939623 54 54 0.60377 9.98694 I 9 2.7 1.8 9.38368 9.39677 0.60323 9.98690 L. Cos. d. L. Cotgr. C.d. L. Tangr. L. Sin. f Prop. Pts. ' 1 76^ 1 40 TABLE II 14 9_ 10 II 12 13 \l 18 i9_ 20 21 22 23 24^ 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 46 47 48 49_ 50 51 52 53 il II 60 L. Sin. d. 38368 38418 38469 38519 38570 38 620 38 670 38721 38771 38821 38871 38921 38971 39021 39071 39 121 39 170 39 220 39270 39319 39369 39418 39467 39566 39615 39664 39713 39 762 39 811 39860 39909 39958 40 006 40055 40 103 40 152 40 200 40249 40297 40346 40394 40 442 40 490 40538 40 586 40634 40 682 40730 40778 40 825 40873 40 921 4c 968 41 016 41 063 41 III 41 158 41 205 41 252 41 300 L. Cos. d. L. Tang. 39677 39 731 39785 39838 39892 39 945 39 999 40 052 40 106 40159 40 212 40 266 40319 40372 40425 40478 40531 40584 40 636 40 689 40742 40795 40847 40 900 40952 41 005 41057 41 109 41 161 41 214 41 266 41 318 41 370 41 422 41 474 41 526 41 578 41 629 41 681 41 733 41 784 41 836 41 887 41939 41 990 42 041 42093 42 144 42 195 42 246 42297 42348 42399 42450 42501 42552 42 603 42653 42 704 42755 9 . 42 805 L. Cotg. c.d. c.d, L. Cotg. o . 60 323 o . 60 269 0.60 215 0.60 162 0.60 108 0.60 055 0.60 001 0.59948 0.59894 0.59841 0.59 788 0.59734 0.59 681 0.59 628 059 575 0.59 522 0.59469 0.59 416 0.59364 0.59311 0.59258 o . 59 205 0.59153 0.59 100 0.59048 0.58995 0.58943 0.58891 0.58839 0.58 786 0.58 734 0.58682 o . 58 630 0.58578 0.58 526 0.58474 o . 58 422 0.58371 0.58319 0.58 267 0.58 216 0.58 164 0.58 113 0.58061 0.58 010 0.57959 0.57907 0.57856 0.57805 o 57 754 0.57 703 0.57652 0.57 601 0.57550 o. 57 499 0.57448 0.57397 0.57347 0.57 296 0.57245 0.57195 L. Tang. 75° L. Cos. 98 690 98687 98684 98681 98678 98675 98671 98668 98665 98662 98659 98656 98652 98 649 98646 98643 98 640 98636 98633 98630 98 627 98 623 98 620 98617 98614 98 610 98 607 98 604 98 601 98597 98594 98591 98588 98584 98581 98578 98574 98571 98568 98565 98561 98558 98555 98551 98548 98545 98541 98538 98535 98531 98528 98525 98521 98518 98515 98 511 98508 98505 98 501 98498 9.98494 L. Sin. d. 60 59 58 55 54 53 52 IL 50 49 48 47 _46_ 45 44 43 42 41 40 39 38 35 34 33 32 31 25 24 23 22 21 20 19 18 17 16 Prop. Pts. 54 53 I 5i 5- 2 10.8 10. 3 16.2 15- 4 21.6 21 . 27.0 26. 6 32.4 31 . 7 37.8 37- .8 48 6 42. •9 47 52 5- 10. 15- 20. 26. 41.6 46.8 50 50 10.0 15 o 20.0 25.0 30.0 350 40.0 45 o 4 0.4 o 8 1.2 1.6 2.0 2.4 2.8 36 48 47 I 4.8 4 2 9.6 9 3 14.4 14 4 19 2 18. S 24.0 23 6 28.8 28. 7 336 32 8 384 37 9 43 2 42 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 41 15^ L. Sin. I 2 3 I I 10 II 12 13 14 ;i 17 i8 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 J.9 60 42 001 42047 42093 42 140 42 186 42 232 42 278 42324 42370 42 416 42 461 42507 42553 42599 42644 42 690 42735 42 781 42 826 42872 42917 42 962 43008 43053 43098 43 143 4318S 43233 43278 43323 43367 43412 43 457 43502 43 546 43591 43 635 43 680 43 724 43 769 43813 43857 43901 43946 43990 44034 47 47 47 47 47 47 46 47 47 46 47 46 47 46 47 46 46 47 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 46 45 45 46 45 45 45 45 45 45 45 44 45 45 45 44 45 44 45 44 45 44 44 44 45 44 44 L. Tang. C. d. 42 805 42856 42 906 42957 43007 43 057 43 108 43 158 43 208 43258 43308 43358 43408 43458 43508 43558 43607 43657 43 707 43756 43 806 43855 43905 43 954 44 004 44053 44 102 44 151 44 201 44250 44299 44348 44 397 44446 44 495 44 544 44592 44641 44 690 44738 44787 44836 44884 44 933 44981 45029 45078 45 126 45 174 45 222 45 271 45319 45367 45415 45463 45 5" 45 559 45 606 45654 45 702 9 45 750 5» 50 51 50 50 51 50 50 50 50 50 50 50 50 50 49 50 50 49 50 49 50 49 50 49 49 49 50 49 49 49 49 49 49 49 48 49 49 48 49 49 48 49 48 48 49 48 48 48 49 48 48 48 48 48 48 47 48 48 48 L. Cotg. 0.57 195 0.57 144 0.57094 0.57043 0.56993 o 56943 o . 56 892 o . 56 842 0.56 792 0.56 742 o 56 692 o . 56 642 0.56 592 0.56542 0.56492 0.56 442 o 56393 0.56343 0.56 293 0.56 244 0.56 194 o 56 145 0.56 095 0.56 046 o 55996 55 947 55898 55849 55 799 55 750 o 55 701 o 55652 0.55603 o 55 554 o 55505 o 55456 0.55 408 o 55 359 o 55310 0.55 262 o 55213 0.55 164 o 55 116 0.55067 0.55019 0.54971 0.54922 0.54874 o . 54 826 o 54778 0.54 729 0.54 681 0.54633 o 54585 o 54 537 0.54489 0.54441 0.54394 0.54346 o . 54 298 0.54250 L. Cos. I d. I L. Cotg. c. d. L. Tang. 740 L. Cos. 9 98 4Q4 9.98491 9 98488 9 98484 9 98 481 98477 98474 98471 98467 98 464 98 460 98457 98453 98450 98447 98443 98 440 98436 98433 98429 98 426 98 422 98419 98415 98 412 98 409 98405 98 402 98398 98395 98391 98388 98384 98381 98377 98373 98370 98366 98363 98359 98356 98352 98349 98345 98342 98338 98334 98331 98327 98324 98317 98313 98309 98306 98 302 98299 98295 98 291 98288 9 . 98 284 L. Sin, d. 60 59 58 55 54 53 52 51 50 49 48 47 _46_ 45 44 43 42 41 40 39 38 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 Prop. Pts. 5t .1 51 .2 10.2 3 15 3 4 20.4 •5 25 5 6 30 6 7 35-7 8 40 8 9 45 9 49 48 .1 4 9 4 .2 9.8 9 •3 14.7 14 •4 19,6 19 . 5 24 5 24 .6 29.4 28 7 34-3 33 .8 39 2 38 9 44 I 43 45 46 I 4 7 4- 2 9 4 9 3 4 Is 8 \l 5 11 5 23- .6 2 27 7 32 9 32 .8 37 6 36. 9 42 3 41 43 1 I 4 51 2 9 3 13 5 4 18 5 22 5 6 27 ■ 7 31 5 .8 36 9 40 5 Prop. Pts. 42 TABLE II 16 9_ 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 il 55 56 II -59. 60 L. Sin. 9.44472 9.44516 9 44 559 9.44 602 9.44646 45 120 45 163 45 206 45 249 45 292 44034 44078 44 122 44 166 44 210 44253 44297 44341 44385 44428 44 689 44 733 44776 44819 44 862 44905 44948 44992 45035 45077 45 334 45 377 45419 45 462 45504 45 547 45589 45632 45674 45 716 45 758 45 801 45843 45885 45927 45969 46 on 46053 46095 46 136 46 178 46 220 46 262 46303 46345 46 386 46 428 46 469 46 511 46552 9 46594 L. Cos. d. L. Tang:. 45 750 45 797 45845 45892 45 940 45987 46 035 46 082 46 130 46 177 46 224 46271 46319 46 366 46413 46 460 46507 46554 46 601 46648 46 694 46741 46 788 46835 46881 46 928 46975 47 021 47 068 47 114 47 160 47207 47253 47299 47346 47392 47438 47484 47530 47576 47 622 47668 47 714 47760 47 806 47852 47897 47 943 47989 48035 48080 48 126 48 171 48217 48262 48307 48353 48398 48443 48489 48534 L. Cotg. c.d. 47 48 47 48 47 48 47 48 47 47 47 48 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 46 46 47 46 46 47 46 46 46 46 46 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 45 46 43 45 46 45 c. d, L. Cotg. 0.54250 0.54203 o 54 155 0.54 108 o . 54 060 o 54013 0.53965 0.53918 0.53870 0.53823 0.53 776 0.53 729 0.53681 0.53634 o- 53 587 o 53540 0.53493 0.53446 0.53399 0.53352 0.53306 0.53259 0.53 212 0.53 165 0.53 "9 0.53 072 0.53025 0.52979 0.52 932 0.52886 0.52 840 0.52 793 0.52 747 0.52 701 0.52654 o . 52 608 0.52 562 0.52 516 0.52 470 0,52424 0.52378 0.52332 0.52 286 0.52 240 0.52 194 0.52 148 0.52 103 0.52057 0.52 on o 51 965 0.51 920 0.51 874 0.51 829 0.51 783 0.51 738 o 51 693 0.51 647 0.51 602 o 51 557 05^ 5" 0.51 466 L. Tang. 73^ L. Cos. 98284 98281 98277 98273 98 270 98266 98 262 98259 9 98255 9 9825 1 9 98 248 98 244 98 240 98237 98233 98 229 98 226 98 222 98218 98215 98 211 98 207 98 204 98 200 98 196 98 192 98 189 98 185 98 i8i 98 177 98 174 98 170 98 166 98 162 98 159 98 155 98 151 98 147 08 144 98 140 98 136 98 132 98 129 98 125 98 121 98 117 98 113 98 no 98 106 98 102 98098 98 094 98 090 98087 9 8083 98079 98075 98071 98 067 98063 9 . 98 060 L. Sin." (>0 It 1 55 54 53 52 11 50 49 48 47 i5_ 45 44 43 42 41 40 39 38 37 Ji 35 34 33 32 31 30 29. 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II To 9 8 7 6 Prop. Pte. 48 47 I 4.8 4 2 9.6 9 3 14.4 H 4 19 2 18. S 24.0 23 6 28.8 28. 7 ,336 32 8 38 4 37 9 43 2 42. At 45 I 4.6 4 2 9 2 9 3 13 8 IT, 4 18 4 18. 5 23 22. .6 27 6 27 • 7 32 2 31 .8 36 8 36. 9 41 4 40 44 43 I 4 4 4 2 8.8 8. 3 13 2 12. 4 17.6 17 .q 22.0 21. .6 26.4 25 • 7 30.8 30 .8 35-2 34- 9 39-6 38. 43 ! I 2 3 4 4.2 8.4 12.6 16.8 21 .0 6 25.2 7 8 9 29 4 336 37 ^ 4 I 2 0.4 0.8 3 4 S 1.2 1.6 2.0 .6 .1 2.4 2.8 32 9 3-6 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 43 17' L. Sin. I 2 3 _4 i 7 8 _9_ 10 I 12 13 :i 17 i8 ii. 20 21 22 23 24 25 26 27 28 29 31 32 33 36 37 38 39 40 41 42 43 44 46 47 48 50 51 52 53 il 60 46594 46635 46 676 46 717 46758 46 8cx) 46 841 46882 46923 46 964 47005 47045 47 086 47 127 47 168 47209 47249 47290 47330 47371 47 411 47452 47492 47 533 47 573 47613 47654 47694 47 734 47 774 47814 47854 47894 47 934 47 974 48 014 48054 48 094 48 133 48173 48213 48252 48 292 48332 48371 48 411 48450 48 490 48529 48568 48607 48647 48686 48 ,725 48 764 48803 48842 48881 48 920 48959 48998 41 41 41 41 42 41 41 41 41 41 40 41 41 41 41 40 41 40 41 40 41 40 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 39 40 40 39 40 40 39 40 39 40 39 39 39 40 39 39 39 39 39 39 39 39 39 L. Cos. I d. L. Tang. c. d. 48534 48579 48 624 48669 48714 48759 48804 48849 48894 48939 48984 49029 49073 49 118 49 163 49207 49252 49296 49341 49385 49430 49 474 49 519 49563 49607 49652 49 696 49 740 49 784 49828 49872 49916 49960 50 004 50 048 50 092 50136 50 180 50223 50 267 503" 50355 50398 50442 50485 9 50 529 9 50572 50616 50659 50703 50746 50789 50833 50876 509^9 50 962 51 005 51 048 51 092 51 135 51 178 45 45 45 45 45 45 45 45 45 45 45 44 45 45 44 45 44 45 44 45 44 45 44 44 45 44 44 44 44 44 44 44 44 44 44 44 44 43 44 44 44 43 44 43 44 43 44 43 44 43 43 44 43 43 43 43 43 44 43 43 L. Cotg. 0.51 466 0.51 421 0.51 376 0-51331 0.51 286 0.51 241 0.51 196 o 51 151 o 51 106 0.51 061 o 51 010 o 50971 0.50 927 0.50882 0.50837 0.50793 0.50748 0.50 704 0.50 659 0.50615 0.50 570 o 50 526 0.50481 o 50 437 0.50393 o 50 348 o . 50 304 o . 50 260 0.50 216 0.50 172 o. 50 128 o . 50 084 o . 50 040 0.49 996 0.49952 0.49 908 o . 49 864 o . 49 820 0.49 777 0.49 733 o 49 689 0.49 645 o 49 602 0.49 558 0.49515 0.49471 0.49 428 0.49384 0.49 341 0.49 297 0.49254 0.49 211 0.49 167 0.49 124 0.49 081 0.49038 o 48 995 o 48 952 0.48 908 o 48 865 0.48822 L. Cotg. c. d. L. Tang. 72° L. Cos. 98060 98 056 98 052 98048 98 044 98 040 98 036 98 032 98 029 98025 98021 98017 98013 98 009 98 005 98 001 97997 97 993 97989 97986 97982 97978 97 974 97970 97966 97962 97958 97 954 97950 97946 97942 97938 97 934 97930 97 926 97 922 97918 97914 97910 97906 97902 97898 97894 97890 9.97886 9.97882 , 97 878 9.97874 9.97870 97866 97861 97857 97853 97849 97S45 97841 97837 97833 97829 97825 9782] L. Sin. d. 60 58 57 55 54 53 52 _5L 50 49 48 47 45 44 43 42 41 40 39 38 35 34 33 32 30 29 28 27 26 Prop. Pts. 45 I 4.5 2 9.0 3 13.5 4 18.0 22.5 .6 27.0 7 31-5 8 36.0 9 40.5 43 1 .1 4 3l .2 8 6 •3 12 9 •4 17 2 I 21 25 I :l 30 I 34 4 9 38 7 41 I 4-1 2 8.2 3 12.3 4 16.4 S 20. s 6 24.6 7 28.7 .8 32.8 9 36.9 39 I 39 2 7-8 3 II. 7 4 15.6 5 19 5 6 23 4 7 27 -3 8 31.2 9 35 I 4 .1 0.4 .2 0.8 ■3 1.2 .4 1.6 •5 2.0 .6 2.4 7 2.8 8 3-2 9 36 4 4 8.8 13.2 17.6 22.0 26.4 30.8 35 2 39-6 42 4.2 8.4 12.6 16 8 21 .0 25.2 29.4 33-6 37-8 40 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 5 05 1 .0 15 2.0 2.5 30 3-5 4.0 4 5 3 0.6 0.9 1.2 2.1 2.4 2.7 Prop. Pts. 44 TABLE II 18^ I 2 3 4 I 7 8 _9_ 10 II 12 13 14 15 16 17 18 i9_ 20 21 22 23 24 25 26 27 28 29 30 31 32 35 31 36 37 38 J9_ 40 41 42 43 44 46 47 48 49_ 50 5^ 52 53 ■>! ]^ i9. (JO L. Sin. 9 48998 9 49037 49076 49 115 49 153 49 192 49231 49269 49308 49 347 9 49385 9.49424 9 49462 9 49 500 9 49 539 9 49 577 9.49615 9 49654 49692 49 730 49768 49 806 49844 49882 49920 49958 49996 50034 50072 50 no 50 148 9.50185 9.50 223 9.50 261 9 50 298 T 50336" 9 50 374 9 50 411 9 50449 50486 9 50523 9 50 561 50598 50635 50673 50 710 50747 50784 50 821 50858 50 896 50933 50970 51 007 51 043 9 9 9 9 _9_ 9.51 080 9 SI "7 9 51 154 9.51 191 51 227 9.51 264 L. CoSu 39 39 39 38 39 39 38 39 39 38 39 38 38 39 38 38 39 38 38 38 38 38 38 38 38 38 38 38 38 38 37 38 3*? 37 38 38 37 38 37 37 38 37 37 38 37 37 37 37 37 38 37 37 37 36 37 37 37 37 36 37 L. Tang. 178 221 264 306 349 392 435 478 520 563 606 648 691 776 819 861 903 946 52031 52073 52 115 52157 52 200 52 242 52284 52 326 52368 52410 52452 52494 52536 52578 52 620 52 661 52703 52745 52 787 52 829 52 870 52 912 52953 52995 53037 53078 53 120 53 161 53 202 53244 53285 53327 53368 53409 53450 53492 53 533 53 574 53615 53656 9 53697 L. Cotg. c. d c.d. L. Cotg. o . 48 822 0.48 779 0.48 736 0.48 694 0.48 651 0.48608 0.48 565 0.48 522 o . 48 480 0.48437 0.48394 0.48352 o 48 309 0.48 266 o 48 224 0.48 181 0.48 139 o 48 097 0.48 054 0.48 012 0.47969 0.47927 0.47885 0.47843 0.47 800 0.47 758 0.47716 0.47674 0.47 632 0.47590 0.47548 0.47 506 0.47464 0.47 422 0.47380 0.47 339 0.47297 0.47255 0.47213 0.47 171 0.47 130 0.47088 0.47047 0.47 005 0.46963 0.46 922 0.46880 o . 46 839 0.46 798 0.46 756 0.46715 0.46 673 o . 46 632 0.46 591 0.46 550 o . 46 508 0.46 467 o . 46 426 0.46 385 0.46344 0.46303 L. Tang. 71° L. Cos. d. 9.97821 60 9 97817 59 9 97812 58 9 97808 57 9 97804 56 55 9 97800 9 97796 S4 9 97792 53 9 97788 52 9 97 784 51 50 9 97 779 9-97 775 49 9 97771 48 9 97767 47 9 97763 46 45 9 97 759 9 97 754 44 9 97 750 43 9 97746 42 9 97742 ^ 41 40 9 97738 9 97 734 39 9 97729 38 9 97725 37 9 97721 36 3S 9 97717 9 97713 34 9 97708 33 9 97704 32 9 97700 31 30 9 97696 9 97691 29 9 97687 28 9 97683 27 9 97679 26 25 9 97674 9 97670 24 9 97666 23 9 97662 22 9 97657 21 20 9 97653 9 97649 19 9 97645 18 9 97640 17 9 97636 16 15 9 97632 9 97628 14 9 97623 13 9 97619 12 9 97615 II 10 9 97 610 9 97606 9 9 97602 8 9 97 597 7 9 97 593 b S 9 97589 9 97584 4 9 97580 3 9 97576 2 9 97571 I 9 97567 L. Sin. d. f Prop. Pte. 43 .1 4 3 2 8.6 3 12.9 4 17.2 1 l\i 7 30 I 8 34-4 9 387 4a 4.2 37-8 41 41 8.2 12.3 16.4 20.5 24.6 28.7 32.8 369 39 1 I 3 9 2 7.8 3 II 7 4 15 6 5 19 5 b 23 4 7 27 3 8 31 2 9 35 I 3^ ' 1 I 3-7| .2 7 4 3 II I 4 14 8 •S 18 5 6 22 2 • 7 25 9 8 29 6 9 33 3 38 3-8 76 II 4 15 2 19 o 22.8 26.6 304 34-2 36 3-6 7.2 10.8 14.4 18.0 21.6 25.2 28.8 32 4 5 I 05 2 1.0 3 15 4 2.0 S 25 6 30 7 3-5 8 4.0 9 4-5 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 45 19^ 9_ 10 12 13 \i 17 i8 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 46 47 48 49_ 50 51 52 53 il 59_ 60 L. Sin. 264 301 338 374 411 447 484 520 557 593 629 666 702 738 774 811 847 883 919 955 51 991 52 027 52063 52099 52 135 52 171 52 207 52 242 52 278 52314 52350 52385 52421 52456 52492 52527 52563 52598 52634 52 669 52 705 52 740 52 775 52 811 52846 52881 52 916 52951 52 986 53.0^ 53056 53092 53 126 53 161 53 196 53231 53 266 53301 53336 53370 9 53405 L. Cos. 37 37 36 37 36 37 36 37 36 36 37 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 35 36 35 36 35 36 35 36 35 35 36 35 35 35 35 35 35 35 36 34 35 35 35 35 35 35 34 35 L. Tang. 53697 53738 53 779 53820 53861 c.d. 53902 53 943 53984 54025 54065 54 106 54147 54187 54228 54269 54 329 54350 54390 54431 54471 54512 54552 54 593 54633 54673 54714 54 754 54 794 54835 54875 54915 54 955 54 995 55035 55075 55 115 55 155 55 195 55235 55275 55315 55 355 55 395 55 434 55 474 55514 55 554 55 593 55633 55673 55712 55752 55 791 55831 55870 55910 55 949 55989 56028 56 067 56 107 41 41 41 41 41 41 41 41 40 41 41 40 41 41 40 41 40 41 40 41 40 41 40 40 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 40 40 40 39 40 40 40 39 40 40 39 40 39 40 39 40 39 40 39 39 40 L. Cotg. 0.46 303 o . 46 262 0.46 221 0,46 180 0.46 139 o . 46 098 0.46057 0.46 016 0.45 975 0.45 935 0.45 894 0.45 853 0.45 813 0.45 772 0.45 731 0.45 691 0.45 650 0.45 610 0.45 569 0.45 529 0.45 488 0.45 448 0.45 407 0.45 367 0.45327 0.45 286 0.45 246 0.45 206 0.45 165 0.45 125 0.45 085 0.45045 0.45 005 0.44965 0.44925 0.44885 0.44845 o . 44 805 0.44765 0.44 725 o . 44 685 0.44645 o . 44 605 0.44 566 0.44 526 o . 44 486 0.44446 0.44407 0.44367 0.44327 0.44 288 o 44 248 o . 44 209 0.44 169 0.44 130 o . 44 090 o . 44 05 1 0.44 on o 43972 o 43 933 0.43 893 L. Cotg. Ic. d. L. Tang. 70° L. Cos. 97567 97563 97558 97 554 97550 97 545 97541 97536 97532 97528 97523 97519 97515 97510 97506 97501 97 497 97492 97488 97484 97 479 97 475 97470 97466 97461 97 457 97 453 97448 97 444 97 439 97 435 97430 97426 97421 97417 97412 97408 97403 97 399 97 394 97390 97385 97381 97376 97372 97367 97363 97358 97 353 97349 97 344 97340 97 335 97331 97326 97322 97317 97312 97308 97303 9.97299 L. Sin. d. 60 59 58 57 55 54 53 52 _51 50 49 48 47 45 44 43 42 41 40 39 38 ,36 35 34 33 32 _3i 30 29 28 27 26 Prop. Pis. 4» .1 4.1 .2 8.2 •3 12.3 4 16.4 •5 20.5 6 24 6 •7 28.7 .8 32.8 9 369 39 .1 3- .2 7- •3 II. • 4 15 •5 19 .6 23 • 7 27. .8 31 9 35- 37 I 3-7 2 7-4 •3 II .1 ■4 14.8 18.5 .6 22.2 •7 25 -9 .8 29.6 9 33-3 3« 1 I 3 51 .2 7 •3 10 5 •4 14 . 5 17 5 .6 21 •7 24 5 .8 28 o| 9 31 51 5 .1 05 .2 1.0 •3 15 •4 2.0 •5 2.5 .6 30 • 7 3 5 .8 4.0 •9 4 5 40 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32 o 36.0 36 36 7.2 10.8 14.4 18.0 21 .6 25.2 28,8 32 4 34 3 4 6 8 10.2 13 6 17.0 20 4 23 8 27 2 30.6 4 0.4 0.8 1 .2 1.6 20 2.4 2.8 36 Prop. Pts. 46 TABLE II 20^ I 2 3 I 7 8 10 II 12 13 '4 15 i6 17 i8 i2. 20 21 22 23 24 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 46 47 48 49 60 51 52 53 il 60 L. Sin. 9 53405 9 53440 9 53 475 9 53509 9 53 544 9 53 578 9 53613 9 53647 9 53 682 9 53 716 9 53 751 9 53 785 9 53819 9 53854 9 53 888 53922 53 957 53991 54025 54059 54093 54 127 54 161 54 195 54229 54263 54297 54331 54365 54 399 54 433 54466 54500 54 534 54567 54601 54635 54668 54702 54 735 54769 54 802 54836 54869 54903 54936 54969 55003 55036 55069 55 102 55 136 55 169 55 202 55235 55268 55301 55 334 55367 55400 55 433 L. Cos. 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 34 34 34 34 34 34 34 34 34 34 34 34 33 34 34 33 34 34 33 34 33 34 33 34 33 34 33 33 34 33 33 33 34 33 33 33 33 33 33 33 33 33 L. Tang. 56 107 56 146 56185 56 224 56 264 56303 56342 56381 56 420 56459 56498 56576 56615 56654 56693 56732 56771 56810 56849 56887 56 926 56965 57004 57042 57081 57 120 57158 57 197 57235 57274 57312 57351 57389 57428 57466 57504 57 543 57581 57619 57658 57696 57 734 57 772 57810 57849 57887 57925 57963 58 001 58039 58077 58 115 58153 58 191 58 229 58267 58304 58342 58380 58418 d. L. Cotg. c. d c.d. L. Cotg. 0.43893 0.43 854 0.43815 0.43 776 0.43 736 0.43 697 0.43 658 0.43 619 0.43 580 0.43 541 0.43 502 0.43463 0.43424 0.43 385 0.43346 0.43 307 0.43 268 o 43 229 o 43 190 o 43 151 0.43 "3 0.43074 o 43035 o 42 996 o 42 958 0.42 919 o 42 880 o 42 842 o . 42 803 o 42 765 0.42 726 o 42688 o . 42 649 o 42 611 0.42 572 0.42 534 0.42 496 0.42457 0.42 419 0.42381 0.42 342 0.42 304 o . 42 266 o 42 228 o 42 190 0.42 151 0.42 113 o 42075 o 42037 0,41 999 o 41 961 o 41 923 0.41 885 0.41 847 0.41 809 o 41 771 o 41 733 o 41 696 o 41 658 o 41 620 0.41 582 L. Tang. 69° L. Cos. 9 97276 9.97271 9.97 266 9.97 262 9-97 257 97 206 97 201 97 196 97 192 97 187 97299 97294 97289 97285 97 280 97252 97248 97243 97238 97234 97229 97224 97 220 97215 97 210 97 182 97 178 97 173 97 168 97 163 97 159 97 154 97 149 97 145 97 140 97 135 97 130 97 126 97 121 97 116 97 III 97 107 97 102 97097 97092 9.97087 9.97083 9.97078 9 97073 9 97 068 97063 97059 97054 97049 97044 97039 97035 97030 97025 97 020 9.97015 L. Sin. 60 59 58 57 55 54 53 52 51 50 49 48 47 45 44 43 42 41 40 39 38 37 35 34 33 32 _31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II To I 7 6 Prop. Pte. 40 .1 .2 Vo 3 12.0 4 16.0 c 20.0 ^ 24.0 28.0 9 32.0 36.0 39 3 9 78 II 7 156 19 5 38 I 38 2 7.6 3 II 4 4 15 2 5 19 6 22.8 7 26 6 8 30 4 9 34 2 37 3-7 7-4 II . I 14.8 18.5 22.2 25.9 29.6 33-3 35 3 5 7.0 10.5 14.0 17 5 21 .0 24 5 28.0 31 5 34 33 I 3-4 3. 2 6.8 6. 3 10.2 9 4 13 6 13. 5 17 16. 6 20 4 19. 7 23.8 23- 8 27 2 26. 9 30.6 29. 5 I 05 2 I.O 3 15 •4 2.0 .5 2 S .6 30 •7 3 5 .8 4.0 9 4 5 4 0.4 0.8 1.2 1.6 2 2.4 28 ^l 36 Prop, Pt8. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 47 21 9_ 10 12 13 ;i 17 i8 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L. Sin. 9 55 433 9 55466 9 55 499 9 55532 9 55 564 9 55 597 9 55 630 9 55663 9 55 695 9 55 728 55 761 55 793 55 858 55 891 55923 55 956 55988 56021 56053 56085 56 118 56150 56 182 56215 56247 56279 563" 56343 56375 56 408 56440 56472 56504 56536 56568 56599 56631 56663 56695 56727 56759 56 790 56822 56854 56886 56917 56949 56980 57012 57044 57075 57 107 57138 57 169 9.57201 9 57232 9 57264 9 57 295 9 57 326 9-57 358 L. Cos. L. Tang«t€. i> L. Cotg 58418 58455 58493 58531 58569 58606 58644 58681 58719 58757 58794 58832 58869 58907 58944 58981 59019 59056 59094 59 131 59 168 59205 59243 59 280 59317 59 354 59391 59429 59466 59503 59540 59 577 59614 59651 59688 59725 59762 59 799 59835 59872 59909 59946 59983 60 019 60 056 60093 60 130 60 166 60 203 60 240 60 276 60 313 60349 60386 60422 60459 60495 60532 60568 60 605 60 641 d. L. Cotg. 37 38 38 38 37 38 37 38 38 37 38 37 38 37 37 38 37 38 37 37 37 38 37 37 37 37 38 37 37 37 37 37 37 37 37 37 37 36 37 37 37 37 36 37 37 37 36 37 37 36 37 36 37 36 37 36 37 36 37 36 41 582 41 545 41 507 41 469 41 431 41 394 41 356 41 319 41 281 41 243 41 206 41 168 41 131 41093 41 056 41 019 40 981 40944 40 906 . 40 869 40832 40795 40757 40 720 40683 40 646 40 609 40571 40534 40497 40 460 40423 40 386 40349 40312 40275 40238 40 201 40 165 40 128 40 091 40054 40 017 39981 39 944 L. Cos. 39907 39870 39834 39 797 39 760 39 724 39687 39651 39614 39578 39 541 39505 39468 39432 39 395 9.97015 9.97 010 9.97005 9 97 001 9 96 996 9 96 991 9 96 986 9 . q6 942 9.96937 96932 96927 96 922 39 359 c. d. L. Tang. 68° 96981 96 976 96971 96 966 96 962 96957 96952 96947 96917 96 912 96907 96 903 96898 96893 96888 96883 96878 96873 96868 96863 96858 96853 96848 d. Q6843 96838 96833 96828 96823 96818 96813 96808 96803 96798 96 793 96788 96783 96778 96 772 96 767 96 762 96757 96 752 96 747 96 742 96737 96 732 96 727 96 722 9.96717 L. Sin. 60 59 58 55 54 53 52 _5i_ 50 49 48 47 45 44 43 42 41 40 39 38 _36 35 34 33 32 31 Prop. Pts. 25 24 23 22 21 20^ 19 18 17 16 10 I 7 6 38 I 3 8 2 76 3 11 4 4 15 2 5 19 6 22 8 7 26 6 8 30 4 9 342 37 3 7 1 \ II . I 14 8 185 22 2 25 9 29 6 33 3 36 33 I 36 3 2 72 6. 3 10 8 9 4 S 18 \l 6 21.6 19 •7 25.2 23 .8 28.8 26 9 32.4 29 33 64 96 12 8 16 o 19 2 22.4 25 6 28.8 31 .2 6.2 3 •4 i 9 3 12.4 • 7 .8 9 21.7 24.8 27.9 5 I 05 2 I.O 3 15 4 20 5 2.5 6 30 7 3-5 8 4.0 9 4 5 6 0.6 1.2 I 2.4 3-6 4.2 4.8 5-4 4 0.4 O. I 2 1.6 2.0 24 2.8 36 Prop. Pts. 48 TABLE II 22' I 2 3 _4 I 7 8 _9_ 10 II 12 13 ii_ ;i 17 i8 19 20 21 22 23 24 26 27 28 29 30 31 32 33 36 38 39 40 41 42 43 44 46 47 48 49 50 51 52 53 54. II II 59 60 L. Sin. 57358 57389 57420 57451 57482 57514 57 545 57576 57607 57638 57669 57700 57731 57762 57 793 57824 57855 57885 57916 57 947 57978 58008 58039 58 070 58 loi 58 131 58 162 58 192 58223 58253 58284 58314 58345 58375 58 406 58436 58467 58497 58527 58557 58588 58618 58648 58678 58709 58 739 58769 58799 58829 58859 58889 58919 58949 58979 59009 59039 59069 59098 59128 59 158 59188 L. Cos. 31 31 31 31 32 31 31 3» 31 31 31 31 31 31 31 31 30 31 31 31 30 31 31 31 30 31 30 31 30 31 30 31 30 31 30 35^ 30 30 30 31 30 30 30 31 30 30 30 30 30 30 30 30 30 30 30 30 29 L. Tangr. 9.60 641 9.60 677 9.60 714 9 60 750 9 . 60 786 9 . 60 823 9 60 859 9 60 895 9.60 931 9 60 967 9.6 9.6 ^i 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 96 9.6 9.6 004 040 076 112 148 184 220 256 292 328 364 400 436 472 508 544 579 615 651 687 722 758 794 830 865 901 936 972 9 , 62 008 9.62043 9 . 62 079 9.62 114 9.62 150 9.62 185 9.62 221 9.62 256. 9 . 62 292 9.62327 9 . 62 362 9.62 398 9 62 433 9 . 62 468 9.62 504 9 62539 9 62 574 c. d. L. Cotg. 9.62 609 9.62 645 9 . 62 680 9.62 715 9.62 750 9.62 785 L. Cotg. 36 37 36 36 37 36 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 36 36 35 36 35 36 36 35 36 35 36 35 36 35 36 35 35 36 35 35 36 35 35 35 36 35 35 35 35 c. d. 0.39359 0.39323 0.39 286 0.39 250 0.39214 0.39 177 0.39 141 0.39 105 o . 39 069 0.39033 o 38 996 o . 38 960 0.38924 0.38888 0.38852 0.38816 0.38 780 0.38 744 0.38 708 0.38672 0.38636 o . 38 600 o . 38 564 0.38528 0.38492 0.38456 0.38 421 0.38385 0.38349 0.38313 0.38278 0.38 242 o . 38 206 0.38 170 0.38135 0.38099 o . 38 064 0.38028 0.37992 0.37957 0.37921 0.37886 0.37850 0.37815 o 37 779 0.37 744 0.37708 0.37673 0.37 638 0.37 602 o 37567 0.37532 0.37496 0.37461 o 37 426 0.37391 o 37 355 o 37320 0.37285 o 37250 o 37215 li. Tang. 67° L. Cos, 96717 96 711 96 706 96 701 96 696 96 691 96686 96681 96 676 96 670 96 665 96 660 96655 96 650 96645 96 640 96634 96 629 96 624 96 619 96 614 96608 96603 96598 96593 96588 96 582 96577 96572 96567 96 562 96556 96551 96546 96541 96535 96530 96525 96 520 96514 96509 96504 96 498 96493 96488 96483 96477 96472 96467 96 461 96456 96451 96445 96 440 9^35. 96 429 96424 96419 96413 96 408 9 96403 L. Sin. 60 59 58 1 55 54 53 52 S]_ 50 49 48 47 45 44 43 42 41 40 39 38 _3i 35 34 33 32 31 Prop. Pts. 35 1 .1 3-71 .2 7 4 •3 II I 4 14 8 .5 18 5 6 22 2 7 25 9 .8 29 6 9 33 3 36 36 7.2 10.8 14.4 18.0 21.6 25.2 28.8 32 4 35 3 5 7.0 10.5 14.0 17 5 21 o 24 5 28.0 31 5 32 I 32 2 6.4 3 9.6 4 12.8 5 16.0 .6 19.2 7 22.4 .8 25.6 28.8 9 30 29 I 2 3.0 6.0 2 3 4 9.0 12.0 II . I 15 18.0 14 17 I 9 21.0 20 24.0 27 23 26 6 0.6 1.2 1.8 24 36 4.2 4.8 5 4 31 3^ 6.2 9 3 12.4 \ll 21.7 24.8 27.9 5 05 I I 2.0 2 5 30 3 5 4.0 4 5 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 49 23° 1 L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. 9 59 188 9.62785 0.37215 9.96403 6 ■fiO" I 9 59 218 9 62820 35 35 36 35 0.37 180 9 96397 59 2 9 59247 9 62855 0.37 145 9 96392 5 58 36 3 7 10 35 6 3-5 2 7.C 8 10.5 4. 14.0 3 9 59277 30 30 29 9 62890 0.37 no 9 96 387 57 _4 9 59307 9 62 926 0.37074 9 96381 5 6 56 55 2 3 9 59336 9 62 961 0.37039 9 96376 6 9 59366 9 62 996 37004 9 96370 54 \/\ 7 9 59396 9 63031 0.36969 9 96365 53 5 t8 17.1; 8 9 59425 9 63 066 35 34 35 0.36934 9 96360 5 6 5 52 6 21 6 21 9 10 9 59 455 29 9 63 lOI . 36 S99 9 96354 51 50 7 8 li. 2 24.5 8 28.0 9 59484 9 63135 0.36865 9 96349 II 9 59514 9 63 170 . 36 830 9 96343 96338 49 9 32. 4 31 5 12 9 59 543 9 63205 0.36 795 9 48 1 i.S 9 59 573 9 63 240 35 35 0.36 760 9 96333 6 5 6 47 1 14 15 9 59602 30 9 63275 0.36725 9 96327 46 45 .1 34 9 59632 9 63310 0.36690 9 96322 16 9 59 661 9 63345 ■lA 0.36655 9 96316 44 .2 6.8 17 9 59690 29 9 63379 0.36 621 9 96 311 5 6 43 •3 [O 2 18 9 59 720 30 9 63414 0.36586 9 96305 42 4 [36 19 20 9 59 749 29 9 63449 35 0.36551 9 96300 5 6 41 40 i: 17.0 20.4 238 30.6 9 59778 9 63484 0.36516 9 96294 21 9 59808 9 63519 0.36481 9 96289 5 39 22 9 59^7 29 9 63 553 0.36447 9 96284 5 6 38 23 9 59866 29 9 35 0.36412 9 96278 37 ■y 1 . 24 2S 9 59895 29 29 9 63623 34 0.36377 9 96273 5 6 36 35 1 9 59924 9 63657 0.36343 9 96267 30 29 2b 9 59 954 9 63692 0.36 308 9 96 262 5 34 27 9 59983 29 9 63 726 34 0.36274 9 96256 33 i. 5.8 8.7 II. 6 14. ■; 28 9 60012 29 9 63 761 0.36239 9 96251 5 32 29 30 9 60 041 29 29 9 63796 34 . 36 204 9 96245 5 6 31 30 i 4 9 12. 15 . 9 . 60 070 9 63830 0.36 170 9 96 240 31 9 60099 9 63 865 0.36 135 9 96234 29 18. 17 4 32 9 60128 9 63899 0.36 lOI 9 96 229 5 6 28 I 21 . 20 3 23 2 33 9 60157 29 9 63934 0.36066 9 96 223 96 218 27 ?./[ 34 35 9 60186 29 29 9 63968 35 0.36032 9 5 6 26 25 9 27. 26 I 9 60215 9 64 003 0.35997 9 96 212 3^^ 9 60244 29 9 64037 0.35963 9 96 207 5 6 24 1 37 9 60273 29 9 64 072 35 35 928 9 96 201 23 38 3« 9 60302 29 9 64 106 34 0.35894 9 96 196 5 6 5 22 T 2 8 39 40 9 60331 29 28 9 64 140 35 0.35860 9 96 190 21 20 .2 3 56 8.4 9 60359 9 64 175 35825 9 96 185 41 9 60388 29 9 64 209 34 0.35 791 9 96179 19 .4 II .2 42 9 60417 29 9 64243 64278 0.35 757 9 96 174 5 18 s 14 43 9 60 446 29 28 29 9 0.35 722 9 96168 17 6 16 8 44 9 60474 9 64312 34 0.35 688 9 96 162 5 16 15 • 7 .8 196 22 4 45 9 60503 9 64 346 o- 35 654 9 96 157 46 9 60 532 29 9 64 381 0.35619 9 96 151 14 •9 - 25 2 47 9 60 561 29 28 9.64415 o- 35 585 9 96 146 5 6 13 1 48 9 60589 9 64449 34 0.35551 9 96 140 12 1 49 50 9 60618 29 28 9 64483 34 34 0.35517 9 96135 5 6 II 10 6 5 6 0.5 9 60 646 9 64517 35483 9 96 129 51 9 60675 29 9 64552 35 0.35448 9 96 123 9 2 I 2 I.O 52 9 60704 29 9 64586 34 0.35414 9 96 118 5 8 3 I 8 1.5 53 9 60732 9 64 620 34 0.35380 9 96 112 7 4 2 ^ 2.0 54 55 9 60 761 29 28 9 64654 34 34 0.35 346 9 96 107 5 6 6 5 3 < 3 5 3.0 9 60789 9 64688 0.35312 9 96 lOI 56 9 60818 29 9 64 722 34 0.35 278 9 96095 4 7 4 I 3 5 i 40 57 9 60846 9 64756 34 0.35 244 9 96 090 5 3 8 4^ 5« 9 60875 29 9 64790 34 0.35 210 9 96084 2 9 b-' ^ 4 5 59 60_ 9 60903 28 9 64824 34 34 0.35 176 9 96079 5 6 I 9 60931 9 64858 0.35 142 9 96073 L. Cos. d. L. Cotgr. c!T L. Tang. L. Sin. d. t Prop. Pt8. 66^ 50 TABLE II 24' 2 3 1 7 8 9 10 II 12 13 \l 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 J4 35 3^ 37 38 39 40 41 42 43 44 46 47 48 49 60 51 52 53 ii II 57 58 Ji. 60 L. Sin. 6093: 60 960 60988 016 045. 073 lOI 129 158 186 214 242 270 298 326 354 382 411 438 466 494 522 578 606 634 662 689 717 745 773 800 828 856 883 911 939 966 994 62 021 62 049 62 076 62 104 62 131 62 159 d. 62 186 62 214 62 241 62 268 62 296 62323 62 350 62377 62 405 62432 62459 62486 62513 62541 62568 62595 L. Cos. 29 27 28 28 28 28 28 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 27 28 37 28 27 28 L. Tang. c.d. 64858 . 64 892 . 64 926 .64 960 64 994 65 028 65 062 65 096 197 231 265 . 299 65333 65 366 9 . 65 400 9 65 434 9.65467 9 65 501 65535 65568 65 602 65 636 65 669 65 703 65 736 65 770 65 803 65837 130 164 65870 65 904 65937 65 971 66 004 66038 66071 66 104 66 138 66 171 66 204 66238 66 271 66 304 66337 9.66371 9 . 66 404 9.66437 9 . 66 470 9 66 503 9 66537 9 66 570 9 . 66 603 9 . 66 636 9 66 669 9.66 702 9 66735 9 66 768 9.66 801 9.66834 66867 L. Cotg. L. Cotg. 0.35 142 0.35 108 o 35074 0.35 040 0.35 006 o 34 972 0.34938 0.34904 0.34870 o 34 836 o 34 803 0.34 769 0.34735 0.34 701 0.34667 0.34634 o . 34 600 0.34566 0.34533 0.34499 0.34465 0.34432 0.34398 0.34364 0.34331 0.34297 o . 34 264 0.34230 0.34 197 0.34 163 0.34 130 0.34096 o . 34 063 o . 34 029 0.33996 0.33 962 0.33929 0:33896 0.33 862 0.33829 0.33 796 0.33 762 0.33 729 0.33696 o 33663 0.33 629 0.33 596 0.33563 033 530 0.33497 0.33463 0.33430 0.33397 0.33364 0.33 331 c.d. 0.33 298 o 33 265 o 33 232 0.33 199 0.33 166 0.33 133 L. Tang. 65° L. Cos, 9.96073 9 . 96 067 9 . 96 062 9.96 056 9 , 96 050 9 96 045 9 96 039 9 96 034 9 96 028 9 96 022 9.96 017 9.96 on 9 . 96 005 9.96 000 9 95 994 9.95988 9.95982 9 95 977 9 95971 9 95965 9 95 960 9 95 954 9 95948 9 95942 9 95 937 9 95931 9 95925 9.95920 9 95 914 9 95908 9 95 902 9 95897 9 95 891 9.95885 9 95879 9-95873 9 95 868 9.95 862 9 95856 9 95850 9-95 844 9 95839 9 95833 9.95827 9.95821 9 95815 9.95 810 9.95 804 9 95 798 9 95 792 9 95 786 9 95 780 9 95 775 9 95 769 9 95 763 9 95 757 9 95 751 9 95 745 9 95 739 9 95 733 9 95 728 L. Sin, d. 60 59 58 1 55 54 53 52 iL 50 49 48 47 45 44 43 42 41 40 It 1 35 34 33 32 _3i 30 29 28 27 26 Prop. Pte. 34 33 I ^A J. 2 6.8 6. 3 10 2 9 4 13 6 13 5 17.0 16 6 20,4 19 7 23.8 23 8 27.2 26 9 30.6 29 29 , I 2.9 .2 5.8 •3 8.7 .4 II. 6 •5 145 .6 17 4 •7 20.3 .8 23.2 •9 26.1 a8 .1 2.8 .2 5 6 .3 8.4 •4 II. 2 14.0 6 16.8 i 19.6 22.4 •9 25.2 37 .1 2.7 .2 5 4 •3 8. 1 •4 10.8 •S 13 5 .6 16.2 i 18.9 21 6 9 24 3 6 .1 0.6 .2 12 •3 1.8 4 2.4 . ^ 3 c 6 3-6 7 4.2 8 4-8 9 5 4 5 05 1 .0 15 2.0 2.5 30 3 5 4.0 4-5 Prop. Pts, LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 51 25" L. Siu. _9_ 10 II 12 13 !i i8 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 36 37 3« 39 40 41 42 43 44 46 47 48 j49 50 51 52 53 H 55 56 II GO 62595 62 622 62 649 62 676 62 703 62 730 62 757 62 784 62 811 62838 62865 62 892 62918 62945 62 972 62 999 63 026 63 052 63079 63 106 63 133 63213 63239 63 266 63 292 633^9 63345 63372 63398 63425 63451 63478 63 504 63531 63557 63 610 63636 63 662 63 689 63 715 63 741 63 767 63 794 63820 63 846 63872 63898 63924 63950 63976 64 002 64 028 64054 64 080 64 106 64 132 64 158 64 184 L. Cos. d. L. Tang. 66867 66 900 66933 66966 66 999 67032 67065 67098 67 131 67 163 67 196 67 229 67 262 67295 67327 67360 67393 67426 67458 67491 67524 67556 67589 67 622 67654 67687 67719 67752 67785 67817 67850 67882 67915 67947 67 980 68012 68 044 68077 68 109 68 142 68 174 68206 68239 68271 68303 68336 68368 68 400 68432 68465 68497 68529 68561 68593 68626 68658 68690 68 722 68754 68 786 68818 Cotgr. L. Cotg. 0-33 133 0.33 100 0.33067 0.33034 0.33001 o 32 968 0.32935 0.32 902 o . 32 869 0.32837 o . 32 804 0.32 771 0.32 738 0.32 705 0.32673 o . 32 640 0.32 607 0.32 574 0.32 542 0.32 509 0.32476 0.32444 0.32 4n 0.32378 0.32346 0.32313 0.32 281 0.32 248 0.32 215 0.32 183 0.32 150 0.32 118 o . 32 085 0.32 053 o . 32 020 0.31 988 0.31 956 o 31 923 0.31 891 0.31 858 0.31 826 o 31 794 0.31 761 0.31 729 0.31 697 0.31 664 0.31 632 0.31 600 0.31 568 o. 31535 0.31 503 0.31 471 o 31 439 0.31 407 0.31 374 342 310 278 246 214 0.31 182 c. d. L. Tang. 64^ L. Cos. 9.95698 9.95692 9.95 686 9.95 680 9 95674 95 728 95 722 95 716 95 710 95 704 95668 95663 95657 95651 95645 95639 95633 95627 95 621 95615 95609 95603 95 597 95591 95585 95 579 95 573 95567 95561 95 555 95 549 95 543 95 537 95531 95525 95519 95513 95507 95500 95 494 95488 95482 95476 95470 95464 95458 95452 95446 95440 95 434 95427 95421 95415 95409 95403 95 397 95391 95384 95378 95372 9.95366 d. L. Sin. I d. 60 59 58 57 55 54 53 52 _51 50 49 48 47 _46_ 45 44 43 42 41 40 39 38 _3^ 35 34 33 32 31 25 24 23 22 21 w 19 17 16 15 14 13 12 II To i 7 6 Prop. Pts. 33 I 3 3 2 6.6 3 9.9 4 13.2 16.5 6 19.8 7 8 26.4 9 29.7 33 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 27 2.7 i;t 10.8 16.2 .8.9 21.6 24-3 a6 2.6 7.8 10.4 lie 18 2 20 8 23 4 7 0.7 1-4 2.1 2.8 3 5 4 2 4-9 l^ 6.3 .1 6 0.6 .2 12 •3 I 8 •4 2 4 it i t:i •9 5-4 5 0.5 i.o 1-5 2 o 2.5 30 3-5 4.0 4-5 Prop. Pts. 52 TABLE II 26' 2 3 ± I 7 8 10 II 12 13 14_ IS i6 17 i8 II. 20 21 22 23 26 27 28 30 31 32 33 Ji 39 40 41 42 43 44 46 47 48 49 50 51 52 53 54. _59 GO L. Sill. 64 184 64 210 64236 64 262 64288 64313 64339 64365 64391 64417 9 64442 9 64 468 9 64 494 9 64519 9 64545 964 571 9.64596 9 . 64 622 9.64647 9.64673 9 . 64 698 9.64724 9.64 749 9 64775 9 . 64 800 9 . 64 826 9.64 851 9,64877 9 . 64 902 9 64927 9 64953 9.64978 9.65003 9 . 65 029 9 65054 9.65 079 9 65 104 9.65 130 9 65 155 9.65 180 9 . 65 205 9.65 230 9 65255 9.65 281 9.65 306 9 65331 9 65356 9 65381 9 65 406 9 65431 d. 9 65456 9.65 481 9.65 506 9 65531 9 65556 9 65 580 9 65 605 9 65 630 9 65655 9 65 680 9 65705 L. Cos. 26 26 26 26 25 26 26 26 26 25 26 26 25 26 26 25 26 25 26 25 26 25 26 25 26 25 26 25 25 26 25 25 26 25 25 25 26 25 25 25 25 25 26 25 25 25 25 25 25 25 25 25 25 25 24 25 25 25 25 25 L. Tang. c.d. d. 9.68818 9.68850 9.68882 9.68 914 9 . 68 946 9.68978 9.69 010 9 . 69 042 9.69074 9.69 106 9.69 138 9.69 170 9 . 69 202 9 69234 9 . 69 266 9 . 69 298 9.69329 9.69361 9 69 393 9.69425 9 69 457 9 . 69 488 9.69 520 9 69 552 9.69584 9 69 615 9.69647 9.69679 9.69 710 9.69742 9.69 774 9 , 69 805 9.69837 9.69868 9 . 69 900 9.69932 9.69963 9 69995 9 70 026 9 . 70 058 9 . 70 089 9 70 121 9.70 152 9.70 184 9 70 2 1 5 9.70247 9.70 278 9.70309 9 70341 9.70372 9.70404 9 70435 9 70 466 9 70498 9 70529 9 . 70 560 9.70592 9 70 623 9.70654 9 . 70 685 9.70717 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 31 32 32 32 32 31 32 32 32 31 32 32 31 32 32 31 32 31 32 32 31 32 3» 32 31 32 31 32 31 32 31 31 32 31 32 31 31 32 31 31 32 31 31 31 32 L. Cotg. 0.31 0.31 [82 - % 0.31 118 o 31 086 o 3» 054 o 31 022 o . 30 990 0.30 958 o . 30 926 0.30894 o . 30 862 o . 30 830 0.30 798 0.30 766 o 30 734 0.30 702 0.30 671 o . 30 639 0.30 607 o 30575 0.30543 0.30 512 o . 30 480 0.30448 o 30 416 0.30 385 o. 30 353 0.30 321 o . 30 290 0.30258 o . 30 226 0.30 195 0.30 163 0.30 132 0.30 100 L. Cos. o . 30 068 o 300^7 o . 30 005 0.29974 o . 29 942 0.29 911 0.29 879 o . 29 848 0.29 816 0.29 785 0.29753 0.29 722 0.29 691 0.29 659 0.29 628 o . 29 596 0.29 565 0.29534 0.29 502 0.29471 o . 29 440 o . 29 408 0.29377 0.29 346 0.29315 0.29 283 L. Cotg. led. I L. Tang. 68° 9 95 366 9 95360 9 95 354 9 95348 9 95341 9 95335 9 95 329 9 95323 9 95317 9 95 310 9 95304 9.95298 9.95292 9 95 286 9 95 279 9 95273 9.95 267 9 95 261 9 95254 9 95 248 9 95242 9 95236 9 95229 9 95223 9,95217 9 95 211 9.95 204 9.95 198 9 95 192 9 95 185 9 95 179 9 95 173 9 95 167 9 95 160 9 95 154 9.95 148 9 95 141 9 95 135 9.95 129 9 95 122 9 95 "6 9 95 "o 9 95 103 9,95097 9.95090 9.95084 9.95078 9.95071 9.95065 9 95059 9.95052 9.95046 9 95039 9 95033 9.95027 9.95 020 9.95014 9.95007 9 95001 9 94 995 9,94988 L. Sin, d. 60 59 58 57 55 54 53 52 51 50 49 48 47 45 44 43 42 41 40 39 38 36 25 24 23 22 21 20 19 18 17 16 Prop. Pts. 3a 3 2 6 4 9 6 12 8 16 19 2 22 4 % 6 8 26 .1 2 .2 5 •3 7 ■4 10 .5 13 .6 ■7 W .8 20 •9 23 3» 3.1 6,2 9 3 12 4 21 7 24 8 27.9 as 25 50 7-5 10.0 12 5 15 o 17 5 20.0 22.5 a4 24 8 2 6 o 4 8 2 6 7 I 07 2 14 •3 2 I •4 28 i 3 5 42 I S 6 9 6.3 6 2 8 4 o 6 2 8 54 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 53 w)7o 9_ 10 I 12 13 H \l \l 20 21 22 23 24 26 27 28 29 31 32 33 31 36 37 38 39L 40 41 42 43 44 46 47 48 49 60 51 52 53 ii 55 56 57 58 _59 60 L. Sin, 65 705 65 729 65 754 65 779 65 804 65828 65853 65878 65 902 65927 65952 65976 66 001 66 025 66 050 66075 66 099 66 124 66 148 66 173 9.66 197 9.66 221 9 66 246 9 66 270 9 . 66 295 66319 66343 66368 66392 66 416 66 441 66465 66489 66513 66 537, 66 562 66586 66 610 66 634 66658 66682 66 706 66 731 66755 66 779 66803 66827 66851 66875 66899 66 922 66 946 66 970 66 994 67018 67 042 67 066 67 090 67 "3 67137 67 i6i L. Cos. 24 25 2S 24 25 24 25 25 24 25 24 25 24 24 25 24 25 24 24 25 24 24 25 24 24 24 24 25 24 24 24 24 24 24 25 24 24 24 24 24 24 24 23 24 24 24 24 24 24 24 23 24 24 L. Tang. c. d, 70717 70748 70779 70 810 70 841 70873 70904 70935 70 966 70997 71 028 71 059 71 090 71 121 71 153 71 184 71 215 71 246 71 277 71 308 71 339 71 370 71 401 71 431 71 462 71 493 71 524 71 555 71586 71 617 71 648 71 679 71 709 71 740 71 771 71 802 71833 71863 71 894 71 925 72 017 72 048 72 078 72 109 72 140 72 170 72 201 72231 72 262 72293 72323 72354 72384 72415 72445 72476 72 506 72537 72567 Cotg. 3» 31 31 31 32 31 31 31 31 31 31 31 31 32 31 31 31 31 31 31 31 31 30 31 31 31 31 31 31 31 31 30 31 31 31 31 30 31 31 30 31 31 31 30 31 31 30 31 30 31 31 30 31 30 31 3? 31 30 31 30 L. Cotg. 0.29 283 0.29 252 0.29 221 0.29 790 0.29 159 0.29 127 o . 29 096 o . 29 065 0.29034 o . 29 003 0.28 972 0.28 941 0.28 910 0.28879 0.28847 0.28816 0.28 785 0.28754 0.28 723 o . 28 692 0.28661 o . 28 630 o . 28 599 0.28 569 0.28538 0.28 507 o 28 476 0.28 445 0.28 414 0.28383 0.28 352 0.28 321 0.28 291 0.28 260 0.28 229 0.28 198 0.28 167 0.28 137 0.28 106 0.28075 o . 28 045 0.28 014 0.27983 0.27 952 0.27 922 o 27 891 0.27 860 0.27 830 0.27 799 0.27 769 c.d, 0.27 738 o 27 707 o 27 677 o 27 646 0.27 616 0.27585 0.27555 0.27 524 0.27494 0.27463 0.27433 L. Tang. 62^ L. Cos. 94891 94885 94878 94871 94865 94988 94982 94 975 94969 94962 94956 94 949 94 943 94936 94930 94923 94917 94 91 1 94904 94898 94858 94852 94845 94839 94832 94 826 94819 94813 94 806 94 799 94 793 94786 94780 94 773 94767 94 760 94 753 94 747 94740 94 734 94727 94720 94714 94707 94700 94694 94687 94 680 94674 94667 94 660 94654 94647 94 640 94634 94627 94 620 94614 94607 94 600 9-94 593 L. Sin. d. 60 59 58 57 55 54 53 52 11 60 It 47 J^ 45 44 43 42 41 40 39 38 _3i 35 34 33 32 31 Prop. Pts. 2M I 2 3 4 32 6.4 9.6 12.8 16 6 19 2 8^ 22.4 25.6 9 28.8 SI II 9-3 12.4 155 18.6 21 .7 24.8 27.9 30 30 6 o 90 12.0 15 o 18.0 21.0 24.0 27.0 as I 2.5 2 50 3 7 5 4 10. •5 12.5 .6 15 .7 17-5 .8 20.0 9 22.5 H 2.4 4.8 7.2 9.6 12.0 14.4 16.8 19 2 21.6 6.9 9.2 ii? 18.4 20 7 7 I 0.7 2 14 3 2. I 4 2.8 5 3 5 6 4.2 I 56 9 6.3 6 0.6 I .2 1.8 24 36 4.2 48 5 4 Prop. Pfs. 54 TABLE II 28^ 9 67 i6i I 2 3 i 7 8 9 10 II 12 13 II ;i 17 18 19 20 21 22 23 ii 26 27 28 80 31 L. Sin. 67185 67208 67 232 67256 67280 67303 67327 67350 67374 9.67398 9 67421 9 67445 9 67 468 9 67 492 9 67515 9 67539 0.67 562 9.67586 9 67 609 9 67 633 9 67 656 9 . 67 680 9.67 703 9 67 726 9 67 750 9 67 773 9.67 796 9 67 820 9 67 843 9 67866 ^_ 9 67 890 32 967913 33 967936 34 9 67959 36 I 39 40 41 42 43 44 46 47 48 49 60 51 52 53 ii 59 60 9 67 982 9 68006 9 68 029 9 68 052 9 68 075 9.68098 9 68 121 9 68 144 9 68 167 9 68 190 9 68 213 9 68 237 9 . 68 260 9 68 283 9 68 305 9 68 328 9 68351 9 68 374 9 68 397 9 68 420 "9 68443 9.68466 9.68489 9.68 512 9-68 534 9 68 557 L. Cos. 24 23 24 24 24 23 24 23 24 24 23 24 23 24 23 24 23 24 23 24 23 24 23 23 24 23 23 24 23 23 24 23 23 23 23 24 23 23 23 23 23 23 23 23 23 24 23 23 23 23 23 23 23 23 23 23 23 23 23 23 L. Tang. 72567 72598 72 628 72659 72689 9 72 720 9 72 750 9 72 780 9 72 811 72 841 c.d. 72 872 72 902 72932 72 963 72993 73023 73 114 73 144 73 175 73205 73235 73265 73295 73326 73356 73386 73416 73446 73476 73507 73537 9 73567 9 12, 597 9 73 627 9 73 657 9.73687 73 717 73 747 73 777 73807 73837 73867 73897 73927 73 957 73987 74017 74047 9.74077 9 74 107 9 74 137 9 74 166 9 74 196 74 226 74256 74286 74316 74 345 9 74 375 31 30 3» 30 31 30 30 31 30 31 30 30 31 30 30 31 30 30 30 31 30 30 30 30 31 30 30 30 30 30 31 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 30 30 29 30 L. Cotg. 0.27433 0.27 402 0.27 372 o 27341 0.27 311 0.27 280 0.27 250 0.27 220 0.27 189 0.27 159 0.27 128 0.27 098 0.27 068 0.27 037 0.27 007 0.26 977 o . 26 946 0.26 916 0.26886 0.26856 0.26 825 0.26 795 0.26 765 0.26 735 0.26 705 0.26 674 o . 26 644 0.26 614 0.26 584 0.26554 0.26 524 o . 26 493 o . 26 463 0.26433 o . 26 403 L. Cos. 0.26373 0.26343 0.26 313 o . 26 283 0.26 253 o . 26 223 0.26 193 0.26 163 o 26 133 0.26 103 o 26 073 o 26 043 0.26 013 o 25983 o 25953 25923 25893 25863 25 834 25 804 0.25 774 o 25 744 0.25 714 o 25 684 0.25655 0.25 625 L. Cotg. c. d. L. Tang 61^ 9 94 593 9 94 587 9 94 580 9 94 573 9 94567 9 94 560 9 94553 9-94 546 9 94540 9 94 533 9 94526 9 94519 9 94513 9 94506 9 94 499 9 94492 9 94485 9 94 479 9.94472 9 94465 9 94458 9-94 451 9 94 445 9-94 438 9 94431 9 94424 9 94417 9.94410 9.94404 9 94 397 d. 9 94390 9 94383 9 94376 9 94369 9.94362 9 94 355 9 94 349 9 94342 9-94 335 9.94328 9 94321 9 94314 9 94307 9.94300 9 94293 9 . 94 286 9.94279 9 94273 9 94 266 9 94259 9 94252 9 94245 9 94238 9 94 231 9 94224 9 94217 9 94 210 9 94 203 9 94 196 9 94 189 9.94 182 L. Sin. 60 1 55 54 53 52 SL 60 49 48 47 45 44 43 42 _4i_ 40' 39 38 36 35 34 33 32 30 29 28 27 26 Prop. Pts. 31 1 I .2 C 3 4 1 9 3 12.4 •7 .8 9 21.7 24.8 27.9 30 30 6 O 9 o 12 C 18 O 21 .0 24.0 27.0 29 29 58 8.7 II 6 14 5 17 4 20.3 23.2 26.1 34 I 2.4 2 4.8 3 7.2 4 9.6 • S 12 .6 14 4 .7 16.8 .8 19.2 9 21.6 23 6.9 92 II. 5 «3 8 16 I 18 4 20.7 7 I 0.7 2 14 3 2.1 4 2 8 5 3 5 6 42 I S 6 9 6.3 sa 2.2 4 4 6.6 8 8 no 13 2 176 19.8 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 55 29° 1 L. Sin. d. L. Tang. c.d. 1. Cotg. L. Cos. d. Prop. Pts. 9 68557 23 9 74 375 . 25 625 9.94 182 ■fiO" I 9 68580 9 74405 0.25595 9 94175 94168 S9 2 9 68603 9 74 435 0.25565 9 S8 30 3 9 68625 23 23 9 74465 0.25535 9 94 161 S7 _4 5 9 68648 9 74 494 30 25 506 9 94 154 56 SS 2 7 io 9 12.0 9 68671 9 74524 0.25476 9 94147 6 9 68694 9 74 554 . 25 446 9 94140 S4 4 7 9 68716 9 74 583 29 0.25417 9 94133 S3 5 15 18.0 8 9 68739 23 22 9 74613 30 0.25387 9 94126 S2 6 9 10 9 68 762 9 74 643 30 0.25357 9 94 119 51 50 I 21.0 24.0 9 68 784 9 74673 0.25327 9 94 112 II 9 68 807 9 74702 29 0.25 298 9 94 105 ! 1 49 9 27.0 12 9 68 829 9 74732 0.25 268 9 94098 48 1 13 9 68 852 23 22 9 74762 30 0.25 238 9 94090 47 14 IS 9 68 875 9 74791 30 . 25 209 9 94083 46 4S 1 29 2.9 9 68 897 9 74821 0.25 179 9 94076 I i6 9 68 920 9 74851 30 0.25 149 9 94069 44 2 17 9 68942 9 74880 29 0.25 120 9 94062 43 3 8-7 II. 6 i8 9 68965 9 74910 30 0.25 090 9 94055 42 4 19 2r 9 68987 23 9 74 939 29 30 0.25 061 9 94048 41 40 •s^ 145 17 4 20.3 '41 9 69 010 9 74969 0.25 031 9 94041 21 9 69032 9 74998 0.25 002 9 94034 7 39 22 9 69055 9 75028 0.24972 9 94027 38 23 9 69077 9 75058 30 0.24942 9 94020 37 24 25 9 69 100 22 9 75087 29 30 0.24913 9 94012 36 3S 1 9 69 122 9 75 117 0.24883 9 94005 23 26 9 69 144 9 75 146 29 0.24 854 9 93998 34 T 1:1 69 9.2 27 9 69 167 9 75 176 . 24 824 9 93991 33 28 9 69 189 9 75 205 29 0.24795 9 93984 32 •3 • 4 29 30 9 69 212 22 9 75235 30 29 0.24765 9 93 977 31 30 9 69234 9 75264 0.24736 9 93970 31 9 69256 9 75294 30 0.24 706 9 93963 29 6 32 9 69279 23 9 75323 29 0.24677 9 93 955 28 y 16. 1 33 9 69301 9 75 353 30 0.24647 9 93948 27 g 18.4 34 3S 9 69323 22 9 75382 29 29 0.24 618 9 93 941 26 25 .9 20.7 9 69345 69 368 9 75 411 0.24589 9 93 934 1 36 9 9 75441 30 0.24559 9 93927 24 1 37 9 69390 9 75470 29 0.24530 9 93920 23 aa 3« 9 69412 9 75 500 30 . 24 500 9 93912 22 . I 2.2 39 40 9 9 69434 22 9 75529 29 29 0.24471 9 93905 21 20 .2 .3 44 6.6 69456 9 •75558 0.24442 9 93898 41 9 69479 23 9 75588 30 0.24412 9 93?9i 19 • 4 8.8 42 9 69501 9 75617 29 0.24383 9 93884 18 II. 43 9 •69 523 9 75647 30 0.24353 9 93876 17 6 13.2 44 45 9 69545 22 9 • 75 676 29 29 0.24324 9 93869 16 15 :l 17.6 9 69567 9 •75 705 0.24295 9 93 862 46 9 .69589 9 • 75 735 30 0.24 265 9 93 855 14 9 19.8 47 9 69 611 9 ■ 75 764 29 0.24236 9 93847 13 1 48 9 69633 9 •75 793 29 . 24 207 9 93840 12 1 49 50 _9 9 69655 22 9 .75822 29 30 0.24 178 9 93833 II 10 I 8 0.8 7 0.7 .69677 9 .75852 0.24 148 9 .93826 SI 9 .69699 9 .75881 29 0.24 119 9 93819 9 2 lb 14 S2 9 .69 721 9 75910 29 . 24 090 9 93 811 8 3 2.4 2.1 S3 9 69 743 9 75 939 29 0.24061 9 93804 7 4 32 2.8 54 SS 9 69 765 22 9 75 969 30 29 . 24 03 1 9 93 797 b 5 I It 73 3 b 4.2 49 9 69787 9 75998 . 24 002 9 93789 S6 9 69809 9 .76027 29 0.23973 9 93 782 4 57 9 69831 9 76 056 29 0.23944 9 W 3 SB 9 69853 9 . 76 086 30 0.23914 9 8 7 2 y 1 59 9 .69875 22 9 .76115 29 29 0.23885 9 93760 I 9 .69897 9 76144 0.23856 9 93 753 L. Cos. ~ L. Cotg. c.d. L. Tangi L. Sin. d. / Prop. Pts. 60^ 1 56 TABLE II 30^ I 2 3 i 7 8 _9_ 10 II 12 13 M 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 36 37 3« 39 40 41 42 43 44 46 47 48 49 60 51 52 53 il i2. 00 L. Sin, 69897 69919 69941 69963 69 984 70 006 70028 70 050 70 072 70093 70 115 70137 70159 70 180 70 202 70 224 70245 70 267 76288 70310 70332 70353 70375 70396 70418 70439 70461 70 482 70504 70525 70547 70568 70590 70 611 70633 70654 70675 70697 70 718 70 739 70 761 70 782 70803 70 824 70 846 70867 70888 70 909 70931 70952 70973 70994 71 015 71 036 71058 71 079 71 100 71 121 71 142 71 163 71 184 L. Cos. d. L. Tally. 76 144 76 173 76 202 76231 76 261 76 290 76319 76348 76377 76 406 76435 76464 76493 76 522 76551 76580 76 609 76639 76668 76697 76725 76 754 76 1^2> 76812 76841 76870 76899 76928 76957 76986 77015 77044 77073 77 101 77 130 77 159 77188 77217 77246 77274 77303 77332 77361 7739c 77418 77 447 77476 77505 77 533 77562 77591 77619 77648 77677 77706 77 734 77 763 77791 77 820 77^^49 77877 d. L. Cotg. c. d. L. Cotg. o 23 856 o 23 827 0.23 798 0.23 769 0.23 739 0.23 710 0.23 681 0.23 652 0.23 623 0.23 594 0.23565 0.23 536 0.23 507 0.23478 0.23449 o . 23 420 0.23391 0.23 361 0.23332 0.23303 0.23 275 0.23 246 0.23 217 0.23 188 o 23 159 0.23 130 0.23 lOI 0.23 072 0.23043 0.23014 . 22 985 O 22 956 O 22 927 O 22 899 0.22 870 c. d. O 22 841 O 22 812 O 22 783 0.22 754 0.22 726 0.22 697 . 22 668 . 22 639 0.22 610 O 22 582 0.22553 0.22 524 . 22 495 O 22 467 . 22 438 0.22 409 O 22 381 O 22 352 . 22 323 O 22 294 0.22 266 0.22 237 . 22 209 O 22 180 0.22 151 0.22 123 L. Tang. 59^ L. Cos. d. 60 9 93 753 9 93 746 59 9 93738 58 9 93 731 S7 9 93 724 7 56 9 93 717 55 9 93 709 54 9 93 702 S^ 9 93695 52 9 93687 51 50 9 93 680 9 93 673 49 9 93665 48 9 93 658 47 9 93650 46 4S 9 93643 9 93636 44 9 93 628 4^ 9 93621 42 9 93614 41 40 9 93606 9 93 599 39 9 93591 S8 9 93584 37 9 93 577 36 35 9 93569 9 93562 34 9 93 554 33 9 93 547 32 9 93 539 31 30 9 93532 9 93525 29 9 93517 28 9 93510 27 9 93502 26 25 9 93 495 9 93487 24 9 93480 23 9 93472 22 9 93465 21 20 9 93 457 9 93450 19 9 93442 18 9 93 435 17 9 93427 16 15 9 93420 9 93412 14 9 93405 13 9 93 397 12 9 93390 II 10 9 93382 9 93 375 9 9 93367 8 9 93360 7 9 93352 6 5 9 93 344 9 9Z2,i1 4 9 93329 3 9 93322 d. 2 9 93314 9 93307 L. Sin. Prop. Pts. 30 I 3 2 6.0 3 9.0 4 12.0 15.0 6 18 7 21.0 8 24.0 9 27.0 2.9 8 7 6 5 4 5 8 II 14 17 20.3 23.2 26.1 28 28 8 .1 0.8 .2 r 6 3 2.4 4 32 1; 4.0 6 48 i l\ 9 7-2! 23 2 2 4 4 6 6 8.8 II. o 13.2 17 6 19.8 ai 2.1 42 8.4 12.6 14 7 16 8 18.9 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 57 31° 1 L. Sin. d. L. Tang. c. d. L. Cotg. L. Cos. d. Prop. Pts. 9 71 184 977877 ,„ 0.22 123 9 93307 g 60 I 9 71 205 9 77 906 29 28 . 22 094 9.93299 3 59 2 9 71 226 9 77 935 . 22 065 9 93291 58 33 ^ 9 71 247 9 77 963 0.22037 9 93 284 Q 57 T 2 9 9 71 268 21 9 77 992 28 29 28 22 008 9 93276 7 8 56 55 2 3 9 71 289 9 78 020 0,21 980 9.93269 6 9 71 310 21 9 78049 0.21 951 9.93261 8 54 4 II 6 7 9 71 331 9 78 077 21 923 9-93 253 53 14 5 8 9 10 9 71 352 9 71 373 21 20 9 78 106 9 78135 29 28 0.21 894 0.21 865 9.93246 9-93 238 8 8 52 51 60 7 8 17 4 20 3 23.2 9 71 393 9 78 163 21 837 9 93230 II 9 71 414 9.78 192 28 21 808 9.93223 8 49 9 26.1 12 9 71 435 9 78 220 31 780 9 93215 8 .48 1 n 9 71 456 9 78 249 21 751 9.93207 47 15 9 71 477 21 9.78277 29 28 21 723 9 93200 8 8 46 45 28 2 8 ^.6 8.4 9 71 498 9.78306 0.21 694 9 93 192 I 16 9 71 519 9 78 334 0.21 666 9 93 184 44 2 •7 9 71 539 9 78363 29 28 28 29 28 21 637 9 93 177 8 8 7 8 8 43 3 18 9 71 560 ^ 9.78391 0.21 609 9 93 169 42 4 II .2 19 20 9 71 581 21 9.78419 0.21 581 9 93 161 41 40 I 14.0 16.8 19 6 22 4 25.2 9 71 602 9.78448 0.21 552 9 93 154 21 9 71 622 9.78476 0.21 524 9 93 146 39 22 9 71 643 9 78505 29 28 0.21 495 9 93 138 38 23 9 71664 9 78533 0.21 467 9 93 131 7 8 8 37 •y 24 2S 9 71685 20 9.78562 29 28 28 0.21 438 9 93 123 36 35 1 9 71 705 9 78 590 0.21 410 9 93 "5 26 9 71 726 9.78618 0.21 382 9.93 108 7 34 2 I 27 9 71 747 9 78 647 29 28 0.21 353 9.93 100 8 8 7 8 8 8 33 4.2 8.4 10 5 28 9 71 767 9.78675 0.21 325 9.93092 32 3 • 4 .5 29 30 9 71788 21 9 78 704 29 28 28 0.21 296 9.93084 31 30 9 71 809 9.78732 0.21 268 9.93077 31 9 71 829 9,78 760 0.21 240 9.93069 29 .6 12.6 32 9 71 850 9,78789 29 28 28 29 28 28 0.21 211 9.93061 28 .7 14 7 ,1S 9 71 870 9.78817 0.21 183 9-93 053 27 .8 16.8 34 3S 9 71 891 20 9.78845 0.21 155 9.93046 8 26 25 9 18.9 9 71 911 9.78874 0.21 126 9-93 038 1 ^^ 9 71 932 9.78902 0.21 098 9.93030 24 1 37 9 71 952 9.78930 0.21 070 9.93022 8 23 • 20 3« 9 71 973 9 78959 29 28 28 28 0.21 041 9.93014 22 J 2.0 39 40 9 71 994 20 9 78987 0.21 013 9.93007 8 Q 21 20 .2 ■ 3 40 6.0 9 72014 9.79015 . 20 985 9.92999 41 9 72034 9 79043 0.20957 9.92991 Q 19 • 4 8.0 42 9 72055 9 • 79 072 29 0.20 928 9.92983 18 .5 10.0 43 9 72075 9.79 100 . 20 900 9.92976 8 8 8 Q 17 .6 12.0 _4£_ 4S 9 72096 20 9.79 128 28 0.20 872 9.92 968 16 15 •7 .8 14.0 16.0 18.0 9 72 116 9 79156 9 79 185 . 20 844 9.92960 46 9 72 137 29 28 0.20 815 9.92952 14 •9 47 9 72 157 9.79213 0.20 787 9.92944 D 13 1 48 9 72 177 9 79 241 0.20759 9.92936 12 1 49 50 9 72 198 20 9 79 269 28 28 20 731 9.92929 7 8 8 8 8 8 8 10 .1 8 0.8 7 07 9 .72218 9.79297 0.20 703 9.92-921 51 9 .72238 9 79 326 29 . 20 674 9 92913 9 2 I.b 14 •ia 9 72 259 9 79 354 . 20 646 9.92905 8 3 2.4 2. I S3 9 .72 279 9.79382 28 0.20618 9.92897 7 4 32 2 8 54 SS 9 72 299 21 9.79410 28 28 . 20 590 9 92 889 6 5 6 4.0 3 b 4 2 4 9 9 72 320 9 -79 438 0.20 562 9.92881 ,S6 9 72 340 9.79466 28 0.20534 9 92874 7 4 I S7 9.72360 9 79 495 29 . 20 505 9.92866 3 Sb 9 72 381 9 79523 28 0.20477 9.92858 2 y 1 / ■^ *' Ji 59 00 9 72 401 20 9 79551 28 28 . 20 449 9 92 850 8 I 9.72421 9 79 579 0.20421 9 . 92 842 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. Prop. Pts. 58^ 1 58 TABLE II 32° 1 — L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. 60 Prop. Pte. 9.72421 9-79 579 28 0.20 421 9.92842 8 8 8 8 7 8 9.72441 9.79607 0.20393 9.92834 59 2 9 72 461 9 79635 28 0.20 365 9.92 826 58 29 38 2.9 2.8 58 5-6 8.7 8.4 3 4 s 9 72482 9 72 502 20 20 9.79663 9.79691 28 28 28 20337 20 309 9.92 818 9 92 810 u 55 2 3 9.72522 9.79719 0.20 281 9 92 803 6 9.72542 9-79 747 0.20 253 9 92 795 8 54 4 II .6 II .2 7 9 72 562 9 79 776 28 0.20 224 9 92787 53 14. c I4.0 8 9 72582 9.79804 28 0.20 196 9 92 779 52 6 17 4 16 8 9 10 9 . 72 602 20 9.79832 28 0.20 168 9.92771 8 8 8 8 8 8 8 8 8 51 50 •7 8 20.3 19.6 23 2 22.4 9 72 622 9 . 79 860 0.20 140 9 92763 II 9.72643 9.79888 28 0.20 112 9 92 755 49 9 26.] I252 12 9 72 663 9.79916 28 0.20084 9 92 747 48 1 M 9 72683 9-79 944 28 0.20 056 9 92 739 47 1 14 9 72 703 20 9.79972 28 28 . 20 028 9.92 731 46 45 I 37 2.7 9 72 723 9.80000 0.20000 9.92 723 i6 9 72 743 9.80028 28 0.19972 9.92 715 44 .2 5 4 17 9.72 763 9 . 80 056 19944 9.92 707 43 3 8.1 i8 9 72 783 9.80084 28 28 28 0.19 916 9.92699 42 4 10. '9 20 9.72803 20 9.80 112 0.19888 9.92691 8 8 8 8 8 8 8 8 8 8 8 8 8 41 40 I lit 9.72823 9.80 140 0.19 860 9.92683 21 9.72843 9.80 168 0.19 832 9 92675 39 22 9 72863 20 9.80195 0. 19 805 9,92667 38 9 24.3 2^ 9 72883 9.80223 28 19777 9 92659 37 25 9.72902 20 9.80251 28 ,0 0.19749 9 92651 36 35 9.72922 9.80279 0.19 721 9.92643 21 1 90 1 26 9.72942 9.80307 _Q 19 693 9 92635 34 J I 20 27 9.72962 9-80335 28 28 28 0,19 665 9.92627 33 4 6. 8. 10. 2 40 3 6.0 4 8.0 5 10.0 28 29 30 9.72982 9.73002 20 20 9.80363 9.80391 0.19637 0.19 609 9 92 619 9.92 611 32 31 30 3 4 9.73022 9.80419 0.19 581 9 92 603 ^i 9.73041 9.80447 o- 19 553 9 92 595 29- 5 12. 6 12.0 32 9.73061 9.80474 27 28 0.19526 9.92587 28 .7 14 7 14 33 9.73081 9.80 502 0.19498 9 92579 8 8 8 27 .8 16. 8 16.0 34 35 9 73 loi 20 9.80530 28 28 0.19470 9.92571 26 25 9 18. 9 18.0 9 73 121 9.80558 0.19442 9 92563 36 9 73 140 9.80586 28 28 27 28 28 „Q 0.19 414 9 92555 24 1 37 9.73 160 9.80 614 0.19386 9.92546 9 8 8 8 8 8 8 8 8 23 19 1 9 3« 39 40 41 9.73180 9.73200 20 20 9 . 80 642 9.80669 o. 19 358 0.19 331 9 92538 9 92530 22 21 20 19 .1 .2 3 .4 3 5- 7- 9 0.9 8 1.8 7 2.7 6 3.6 9.73219 9 -73 239 9.80697 9-80725 0.19303 0.19275 9.92522 9.92514 42 9 73259 9-80753 9.80 781 28 0.19247 9.92506 18 .5 9 5 4 5 43 9.73278 0.19 219 9.92498 17 .6 II 4 5 4 44 4S 9.73298 20 9.80808 28 28 28 0.19 192 9.92490 16 15 ■I 13 15 3 6.3 2 7.2 9 73318 9.80836 0.19 164 9 92 482 46 9 73 337 ^9 9.80864 0.19 136 9 92473 9 8 8 8 8 8 8 14 9 17 I 8.1 47 9 73 357 9.80892 0.19 108 9.92465 13 1 48 9-73 377 9.80919 27 28 28 28 0.19 081 9 92457 12 1 49 50 9 73396 ^9 20 9.80947 0.19053 9.92449 II 10 I 8 0. 7 8 0.7 9 73416 9.80975 0.19025 9.92441 51 9 73 435 19 9.81 003 0.18997 9 92433 9 .2 I. 6 1.4 52 9 73 455 9.81 030 27 28 28 27 28 28 0. 18 970 9.92425 8 3 2. 4 21 S3 9 73 474 ^9 9.81 058 9.81 086 0.18942 9.92416 9 8 8 8 8 7 -4 3 2 28 54 9 73 494 19 0.18914 9 . 92 408 6 5 i 4 4 3 5 8 42 6 49 \ is 9 73513 9.81 113 0.18887 9.92400 56 9 73 533 9 81 141 0.18859 9.92392 4 ■'J .8 6: ^S^ 9 73552 *9 9.81 169 0.18 831 9 92384 3 9-73 572 9.81 196 27 28 28 0.18804 9 92376 2 y y 59 60_ 9 73591 19 20 9 81 224 0.18 776 9.92367 9 8 I _0^ 9.73 611 9.81 252 0.18 748 9 92 359 L. Cos. d. L. Cotg. Ic. d. L. Tang. L. Sin. d. Prop. Pte. 57° 1 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 59 33° 1 L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. "60" Prop. Pt8. 9-73611 19 9 81 252 27 0.18 748 9 92359 g I 9 73630 20 9 81 279 28 0.18 721 9 92351 8 59 2 9-73 650 IQ 9 81 307 28 0.18 693 9 92343 58 38 87 2.7 5 4 3 9.73669 ao 981335 27 28 28 0.18665 9 92335 57 T 2.8 5.6 8.4 4 9.73689 19 19 9 81 362 0.18638 9 92326 8 8 8 56 55 .2 .3 9 73 708 9.81390 18 610 9 92318 6 9 73727 20 9 81 418 0.18 582 9,92310 54 4 II. 2 10.8 7 9 73 747 19 9.81445 28 0.18555 9,92302 9 8 53 14.0 13 5 8 9 73 766 9 81473 0.18527 9.92 293 52 .6 16.8 16.2 9 9 73785 20 9 81 500 28 0.18 500 9 92285 8 51 •7 19.6 18 9 10 9 73805 10 9.81 528 28 0.18472 9.92277 8 50 .8 22.4 21.6 II 9 73 824 9 81 556 27 28 0.18444 9.92269 9 8 49 9 25.2 243 12 9 73843 20 9 81 583 0.18 417 9.92 260 48 1 n 9 73863 19 19 9 81 611 0.18389 9.92252 47 15 9 73 882 9.81 638 28 0.18362 9 92244 9 8 46 45 1 30 2.0 9 73901 9.81 666 0.18334 9 92235 I .16 9.73921 19 9 81 693 28 0.18 307 9 92227 8 44 8.0 17 9 73940 9 81 721 0.18 279 9.92219 8 43 •3 18 9 73 959 19 9.81 748 27 28 27 ,0 0.18 252 9 92 211 42 •4 19 20 9 73978 ^9 19 20 9.81 776 0. 18 224 9 92 202 9 8 g 41 40 '\ 10.0 12 14 16 9 73 997 9 81 803 0.18 197 9.92 194 21 9 74017 19 19 19 19 9 81 831 0.18 169 9.92 186 39 22 9.74056 9 81 858 28 0.18 142 9.92 177 8 38 n 18 23 9 74055 9.81 886 0.18 114 9.92 169 37 24 25 9 74 074 9 81 913 28 0.18087 9.92 161 9 8 _36_ 35 1 9 74093 9 81 941 0. 18 059 9.92152 19 26 9 74 "3 19 19 19 19 9 81 968 28 0. 18 032 9.92 144 8 34 ¥ 3:1 n 27 28 29 30 9 74 132 9 74 151 9.74170 9 81 996 9 82 023 9 82 051 27 28 27 28 0.18004 0.17977 0.17949 9.92 136 9.92 127 9.92 119 9 8 8 33 32 31 30 .2 •3 •4 • S 9 74 189 9.82078 0.17922 9.92 III 31 9.74208 19 9.82 106 0.17894 9.92 102 29 .6 I r 1 32 9.74227 19 9 82133 28 0.17867 9.92094 8 28 .7 i^.\ 1 33 9.74246 ^9 9.82 161 0.17839 9.92086 27 .8 ) 2 34 35 9.74265 19 »9 9 82 188 27 27 28 0.17 812 9.92077 9 8 26 25 9 I I 9,74284 9 82 215 0.17785 9.92069 1 36 9 74303 19 9,82243 0.17757 9.92060 8 24 1 37 9.74322 9 82 270 28 0.17730 9.92052 8 23 18 3« 9 74341 9.82 298 0.17 702 9.92044 22 1 I 8 39 40 9 74 360 19 9 82325 27 27 28 0,17675 9 92035 8 21 20 2 .3 36 5 4 9 74 379 9.82352 0.17648 9.92027 41 9 74398 9 82 380 0.17 620 9.92 018 19 .4 7.2 42 9 74417 ^9 9 82 407 27 28 0.17593 9.92 010 18 9.0 43 9 74436 ^9 9.82435 0.17565 0.17538 9 92 002 17 6 10.8 44 4S 9 74 455 »9 »9 9 . 82 462 27 27 28 9 91 993 9 8 lb 15 i 12.6 14.4 9 74 474 9.82489 0.175" 9.91 985 46 9 74 493 19 9.82517 0,17483 9,91976 9 8 14 •9 16.2 47 9 74512 »9 9.82 544 27 0.17456 9.91 968 13 1 48 9 74 53^ 19 18 9 82571 27 28 27 0.17429 9 91 959 8 12 1 49 50 9 74 549 19 9.82599 0.17 401 9 91 951 9 II 10 . I 9 0.9 8 0.8 9 • 74 568 9.82 626 0.17374 9 91 942 51 9 74 587 19 9.82653 27 0.17347 9 9i 934 9 .2 1.8 1.6 52 9.74606 19 9.82681 0.17319 9.91 925 8 8 •3 2.7 2.4 S3 9 74 625 19 9.82 708 27 0.17 292 9.91 917 7 .4 36 32 54 55 9.74644 19 t8 9 82735 27 27 0.17265 9,91 908 9 8 6 5 i I 9 4.5 5 4 6.3 7.2 8.1 n 72 9.74662 9.82 762 0.17 238 9.91 900 56 9 74681 19 9.82 790 0. 17 210 9.91 891 8 9 8 4 57 58 9 . 74 700 9 74 719 19 19 18 19 9 82817 9.82844 27 27 0.17 183 17 156 9,91883 9,91 874 3 2 59 60 9 74 737 9.82871 27 28 0.17 129 9,91 866 9 I 9 74756 9.82899 0.17 lOI 9 91 857 L. Cos. T" L. Cotg. ™d. L. Tang* L. Sin. d. / Prop. Pts. 56° 1 60 TABLE II 34° 1 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. 60 Prop. Pts. "F 9-74 756 9.82 899 0.17 lOI 9 91 857 8 I 9 74 775 9 82 926 0.17074 9.91 849 5Q 2 3 9 9 74 794 74812 18 9 9 82953 82 980 27 28 0.17047 0.17 020 9.91 840 9.91 832 9 8 58 57 28 2.8 5.6 8.4 27 4 9 74831 19 18 9 83008 27 0.16 992 9.91 823 9 8 56 55 .2 •3 .4 2.7 9 74850 9 83035 0. 16 965 9.91 815 6 9 74868 9 83062 0.16938 9.91 806 9 8 54 II 2 10 8 7 9 74887 9 83089 28 0.16911 9.91 798 ^^ 14.0 8 9 74906 18 9 ^Z 117 27 27 0.16883 9.91 789 9 8 9 52 t6 8 9 10 9 74924 19 9 83 144 0.16856 9.91 781 51 oO •7 .8 19.6 22.4 189 21.6 9 74 943 9 83 171 0.16 829 9.91 772 II 9 74961 9- 83198 0.16 802 9.91 763 9 8 4Q .9 25.2 24 3 12 9 74980 9 83225 0.16775 9 91 755 48 n 9 74 999 18 9 l^ ^p 28 0. 16 748 9 91 746 9 8 9 47 14 15 9 75017 19 t8 9 2,z 280 27 0.16 720 9 91 738 46 45 ! 26 26 9 75036 9 83307 0.16 693 9 91 729 .1 16 9 75054 9 83334 0. 16 666 9.91 720 9 8 44 .2 M 17 9 75073 18 9 83 361 0.16 639 9.91 712 43 •3 18 9 75091 9 83 388 0. 16 612 9.91 703 9 8 9 42 .4 10.4 19 ^0" 9 75 no 18 9 83415 27 28 0.16 585 9.91 695 41 40 i 13.0 ;§2 9 75128 9 83442 0. 16 558 9.91 686 21 9 75 147 18 9 83470 27 27 27 27 0.16 530 9.91677 9 8 30 20 8 22 9 75 165 9 83497 0.16 503 9.91 669 38 23 4 23 9 75184 ^9 9 83524 0.16 476 9.91 660 9 37 •y 24 2S 9 75202 19 9 83 551 16449 9,91 651 9 8 36 3S 1 9 75221 9 83 578 0. 16 422 9.91 643 X9 26 9 75239 9 83605 0.16395 0.16368 9.91 634 9 34 ^ I - 1 27 9 75258 19 18 9 83632 9.91 625 9 8 33 \ 28 9 75276 18 19 t8 9 83659 83686 0.16 341 9.91 617 32 3 •4 >.7 r6 ) 5 29 30 9 75294 9 27 0.16 314 9.91 608 9 9 8 31 30 9 75313 9 83 713 0.16287 9-91 599 31 9 75331 9 83740 28 0. 16 260 9 91 591 29 (y I .4 32 9 75350 18 18 19 9 83 768 0.16232 9 91 582 9 28 7 I' r'i 33 9 75368 9 83 795 0,16 205 9 91 573 9 27 8 1 52 7.1 34 35 9 75386 9 83822 27 0.16 178 9 .91 565 9 26 25 9 9 75405 9 83 849 0.16 151 9 91 556 1 Z^ 9 75423 18 9 83876 0.16 124 9 91 547 9 24 1 37 9 75441 ,Q 9 83 903 0.16 097 9 91 538 9 8 23 18 3« 9 75 459 9 83930 0.16 070 9 91 530 22 I 8 39 40 9 75478 ^9 18 9 83957 27 0. 16 043 9.91 521 9 9 8 21 20 .2 .3 36 5 4 9 75496 9 83984 16 016 9.91 512 41 9 75514 9 84 on 0.15989 9.91 504 19 .4 72 42 9 75 533 9 84038 0.15 962 9 91 495 9 18 .«; 9.0 43 9 75551 18 18 18 9 84065 o- 15 935 9.91 486 9 17 .6 10.8 44 4.S 9 75569 9 84092 27 0.15908 9.91477 9 8 16 i 12 6 14 4 9 75587 9 84 119 0.15 881 9.91 469 4b 9 75605 9 84 146 27 0.15854 9.91 460 9 14 9 16.2 47 9 75624 19 18 9 84173 27 0.15827 9 91 451 9 13 1 48 9 75 ^f 9 .84200 27 0.15 800 9.91 442 9 12 1 49 50 51 9 75660 18 18 9 84227 27 27 26, 0.15 HZ 9 91 433 9 8 9 II 10 9 .2 9 8 08 I 6 9 9 75678 75696 9 9 '84280 0.15 746 0.15 720 9.91425 9 91 416 52 9 75 7M 9 .84307 27 9.91 407 9 8 •3 z 7 24 53 9 • 75 733 19 18 18 18 18 18 18 18 9 84334 27 0.15666 9.91 398 9 7 4 3 6 3 2 54 55 9 •75 751 9 .84361 27 27 0.15639 9.91 389 9 8 6 S i 4 5 5 4 63 4 48 5 6 64 9 •75769 9 84388 0. 15 612 9.91 381 50 9 .75787 9 84415 27 0.15585 9.91 372 9 4 i i^ 9 .75805 9 .84442 27 0.15 558 9 91 363 9 3 l\ 9 .75823 9 .84469 27 15 531 9 91 354 9 2 9 7^ 59 G0_ 9 .75841 9 .84496 27 27 0.15504 9 91 345 9 9 9 75859 9 ■84523 0.15477 9 91 336 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. Prop. Pts. 55° 1 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 61 85^ 2 3 I 7 8 _9_ 10 II 12 13 iil ;i 17 18 i9. 20 21 22 23 24 25 26 27 28 _29 30 31 32 33 ^\ 36 37 38 _39 40 41 42 43 44 J^ 47 48 49 50 51 52 53 54 55 56 57 58 ii. 60 L. Sin. 9 75 859 9 75877 9 75895 9 75913 9 75931 9 75 949 9 75 967 9 -75 985 9 . 76 003 9.76 021 9.76039 9.76057 9.76075 9.76093 9.76 III 9.76 129 9.76 146 9.76 164 9.76 182 9 76 200 9 76 218 9.76236 9 76253 9 76271 9 76 289 9 76 307 9 76 324 9.76342 9.76360 9.76378 9 76395 9 76413 9 76431 9.76448 9 . 76 466 9.76484 9.76501 9.76519 9 76537 9 76554 9.76572 9.76590 9 . 76 607 9 . 76 625 9 . 76 642 9 . 76 660 9.76677 9.76695 9.76 712 9 76 730 9.76747 9 76765 9 76 782 9 . 76 800 9.76 817 9 76835 9.76852 9 76 870 9.76887 9 76904 9 . 76 922 L. Cos. d. L. Tang. c. d. 9-84 5£3 9.84550 9.84576 9,84 603 9.84630 9.84657 9 84 684 9.84 711 9 84738 9 84 764 9.84791 9.84818 o . 84 845 9.84872 9.84899 9.84925 9.84952 9.84979 9 . 85 006 9 85033 9 85059 9.85086 9 85 113 9.85 140 9.85 166 9 85 193 9 85 220 9.85247 9 85273 9 85 300 9 85327 9 85354 9 85 380 9 85407 9 85434 9 . 85 460 9.85487 9 85514 9 85540 9 85567 9 85 594 9.85 620 9 85647 9.85674 9.85 700 9,85 727 9 85 754 9.85 780 9 85 807 9 85834 9.85 860 9.85887 9 85 913 9 85 940 9 85 967 9 85 993 9 86 020 9 86 046 9 86 073 9 86 100 9.86 126 L. Cotg. c. d 27 26 27 27 27 27 27 27 26 27 27 27 27 27 26 27 27 27 27 26 27 27 27 26 27 27 27 26 27 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 26 27 27 26 27 26 27 27 36 L. Cotg. 0.15477 0.15450 C.15 424 c. 15 397 0.15370 15343 15 316 15289 15 262 15236 0.15 209 0.15 182 o 15 155 0.15 128 o. 15 lOI o. 15 075 0.15 048 o. 15 021 0.14994 o. 14 967 0.14 941 0.14 914 0.14887 o . 14 860 0.14834 o. 14 807 0.14 780 0.14753 0.14 727 o. 14 700 0.14673 0.14 646 o. 14 620 o 14593 o. 14 566 o. o. o o o. 14540 14 513 14 486 14 460 14433 o. 14 406 0.14 380 o 14353 0.14326 0.14300 L. Cos. 0.14273 o. 14 246 o. 14 220 0.14 193 o. 14 166 0.14 140 0.14 113 o. 14 087 o. 14 060 0.14033 o. 14 007 o. 13 980 0.13954 0.13927 0.13 900 [3874 L. Tang. 54° 91 336 91 328 91 319 91 310 91 301 91 292 91 283 91 274 91 266 91 257 91 248 91 239 91 230 91 221 91 212 91 203 91 194 91 185 91 176 91 167 91 158 91 149 91 141 91 132 91 123 9 91 114 9 91 105 9 91 096 9.91 087 9.91 078 d. 91 069 91 060 91 051 91 042 91033 91 023 91 014 91 005 90 996 90987 90978 90 969 90 960 90951 90942 90933 90924 90915 90 906 90 896 90887 90878 90 869 90 860 90851 90 842 90 832 90 823 90 814 90805 9.90796 L. Sin. 60 59 58 55 54 53 52 51 50 49 48 47 _46_ 45 44 43 42 41 40 39 38 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20" 19 18 17 16 15 14 13 12 II To 9 8 7 6 Prop. Pts. 37 I 2.7 2 5 4 3 8. 1 4 10.8 . 5 13 5 .6 16.2 •7 18.9 .8 21.6 9 243 26 2.6 5-2 7.8 10.4 13.0 15 6 18 2 20,8 23 4 18 1.8 36 5 4 7.2 9.0 10 8 12 6 14.4 16.2 17 17 3 4 \\ 85 10.2 11.9 136 15-3 10 i.o 20 30 40 60 7.0 8.0 90 9 .1 0.9 .2 1.8 •3 27 •4 36 i 4 5 5 4 I 63 7 2 9 8.1 8 08 1.6 24 3 2 7 2 Prop. Pts. 62 TABLE II 36^ 9_ 10 ;i i8 i2_ 20 21 22 23 24 26 27 28 29 80 31 32 33 34 II II 40 41 42 43 44 46 47 48 49 60 51 52 53 ii 55 50 II GO L. Sill. 76922 76939 76957 76974 76991 77009 77026 77043 77061 77078 77095 77 112 77130 77147 77164 77 181 77199 77216 77233 77250 77268 772S5 77302 77319 77336 77 353 77370 77 3^7 77405 77422 77 439 77456 77 473 77490 77507 77 5-4 77541 77558 77 575 77592 77G09 77626 77643 77 CGo 77677 77694 77 711 77728 77 744 77761 77 77^ 77 795 77812 77829 77846 77862 77879 77896 77913 77930 77946 I L. Cos. I <1 17 18 J7 J7 18 »7 »7 18 17 17 17 18 17 17 17 x8 17 17 17 18 17 17 17 17 17 17 17 18 17 17 17 17 17 17 «7 17 17 17 17 17 17 17 17 17 16 17 Tangr. c.d. 86 ,86 126 o. '53 ,86 179 86206 86232 ,86259 ,86285 86312 86338 .86365 .86392 1. 86418 1.86445 1.86 471 1.86498 1.86524 '•^5551 1.86577 1.86603 86 630 9.86656 9.86683 9.86 709 9.86736 9.86 762 9.86 789 9.86815 9.86842 9.86868 9.86894 9.86921 9.86947 9.86974 9.87000 9.87027 9-87053 9.87079 9.87 106 9.87 132 9-87 158 9.87 185 9.87 211 9.87238 9.87264 9.87290 9.87317 9-87343 9.87369 9.87396 9.87422 9.87448 9-87475 9.87501 9.87527 9-87554 9.87580 9.87606 9-87633 9.87659 9.87685 9.8771 L. Cotg, 27 26 27 26 27 26 27 26 »7 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 26 27 26 26 27 26 27 26 26 27 26 26 27 26 26 27 26 26 27 26 26 27 26 26 L« Cotg. 0.13874 0.13847 0.13 821 o. 13 794 0.13768 o. 13 741 0.13 715 0.13688 0.13 662 0-13635 0.13 608 0.13582 0.13555 0.13529 0.13502 0.13476 0.13449 0.13423 0.13397 0.13370 13344 ^3317 13 291 13264 13 238 0.13211 0.13 185 0.13 158 0.13 132 0.13 106 0.13 079 0.13053 0.13026 0.13000 0.12973 0.12 947 0.12 921 0.12 894 0.12 868 0.12 842 0.12 815 0.12 789 0.12 762 0.12 736 0.12 710 c.d. 0.12683 0.12 657 0.12 631 0.12 604 0.12578 0.12 552 0.12 525 0.12 499 0.12473 0.12 446 0.12 420 0.12394 0.12367 0.12 341 0.12 315 [2 289 L. Tang. 53° L, Cos. 90657 90648 90639 ^ ,90 630 9.90 620 90 796 90787 90777 90 768 90759 90750 90741 90731 90 722 90713 90704 90 694 90685 90676 90667 90 611 90 602 90592 90583 90574 90565 90555 90546 90537 90527 90518 90509 90499 90490 90480 90471 90462 90452 90443 90 434 90424 90415 90405 90396 90386 90377 90368 90358 90349 90339 90330 90320 90 311 90301 90292 90 282 90273 90 263 90254 90 244 9 90 235 L. Sin. GO It I 55 54 53 52 JL 60 49 48 46 45 44 43 42 41 40 39 38 36 35 34 33 32 31 Prop. Pis. 27 36 I 2.7 26 •2 -3 il 11 •4 10.8 10.4 . 1; 13-5 16.2 13.0 .6 IS. 6 • 7 18. q .8 21.6 20.8 .9 243 23-4 18 1.8 36 5 4 7.2 9° 10.8 12.6 14.4 16.2 17 17 3-4 u 8.5 10.2 II. 9 136 15-3 x6 1.6 3 2 4.8 6.4 80 96 112 12 8 14.4 10 .1 I.O .2 2.0 •3 •4 i 30 4.0 i 7.0 8.0 -9 9.0 9 09 I 2 3 4 I Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 63 37° 1 L. Sin. d. L. Tang. c.d.iL. Cotg. 1 L. Cos. d. 60 Froi». V\s, 9 77946 9 87 711 0. 12 289 9 90 235 I 9 77963 9 87 738 26 0.12 262 9 90225 S9 2 9 77980 9 87 764 26 0.12 236 9 90 216 S8 ^ 9 77 997 9 87790 0.12 210 9 90 206 S7 27 ,J_ 9 78013 '7 9 87817 26 0. 12 183 9 90197 10 56 ss .2 •3 .4 9 78030 9 87 843 0.12 157 9 90 187 6 9 78047 i5 9 87869 36 12 131 9 90178 S4 10 8 7 9 78 063 9 87895 12 105 9 90 168 S3 ,5 'I 5 16 2 8 9 78080 9 87922 r>(\ 0.12078 9 90 159 9 S2 ,6 9 9 78097 [6 9 87948 26 26 0.12 052 9 90 149 10 51 50 :l 18.9 21.6 9 78 113 9 87974 0. 12 026 9 90139 1 1 9 78 130 9 88000 r%m 0.12000 9 90130 49 •9 24.3 12 9 78 147 [6 9 88027 on 973 9 90 120 48 1 n 9 78163 9 S°53 o.ii 947 9 90 III 47 1 •4 9 78 180 7 9 88079 26 26 o.ii 921 9 90 lOI 10 46 4S .1 36 26 IS 9 78 197 9 11 '°5 O.II 895 9 90091 i6 9 78213 9 88 131 O.II 869 9 90 082 9 44 .2 ^l '7 9 78230 6 9 88158 27 26 26 26 26 on 842 9 90072 43 •3 78 i8 9 78 246 9 Z?> 184 O.II 816 9 90063 9 42 •4 10.4 19 20 9 78 263 , 7 6 9 88210 on 790 9 90053 10 41 40 i 15 6 18 2 20 8 23 4 9 78280 9 88236 on 764 9 90 043 21 9 . 78 296 9 88262 on 738 9 90034 9 3Q ■ 7 .8 22 9 78313 9 88289 26 26 26 26 0. II 711 9 90024 38 2^ 9 78329 9 88315 on 685 9 90014 37 •y 24 2S 9 78346 16 9 88341 II 659 9 90003 10 36 3S 1 9 78362 9 88367 on 633 9 8999s 17 1-7 26 9 78379 16 9 88393 on 607 9 8998s 34 . 27 9 78395 9 88420 27 26 O.II 580 9 89976 9 33 28 9 78412 ^1 [6 '7 [6 9 88446 on 554 9 89966 32 34 29 BO 9 78428 9 88472 36 26 26 0. II 528 9 89956 9 31 80 •3 •4 10.2 9 78445 9 88498 on 502 9 89947 ^i 9 78461 9 SS524 oil 476 9 89937 29 ^2 9 78478 9 88 550 oil 450 9 89927 28 i II. 9 136 IS.:? 1^ 9 78494 ,£. 9 88577 27 26 26 0,11423 9 89918 9 27 34 9 78510 17 t6 9 88603 II 397 9 89908 10 26 2S .q 9 78527 9 88629 on 371 9 89898 1 ^6 9 78543 9 88 6^5 88681 26 26 O.II 345 9 89888 24 1 V 9 78560 [6 [6 '7 [6 9 II 319 9 89879 •9 23 16 ^« 9 78576 9 88707 II 293 9 89869 22 ^ I 6 39 40 9 78592 9 88733 26 II 267 9 89859 10 21 20 .2 .3 U 9 78609 9 88759 88786 II 241 9 89849 41 9 78625 9 27 0. II 214 9 89840 9 19 .4 6.4 42 9 78642 16 t6 t7 [6 t6 [6 9 88812 on 188 9 89830 18 .1; 8.0 43 9 78658 9 88838 II 162 9 89820 17 .6 9.6 44 4S* 9 78674 9 88864 26 26 II 136 9 89810 9 16 IS :l n.2 12.8 9 78691 9 88890 0. II no 9 89801 46 9 78707 9 88916 on 084 9 89791 14 .9 14.4 47 9 78723 9 88942 26 O.II 058 9 89781 13 1 48 9 78739 9 88968 O.II 032 9 89771 12 1 49 50 _9 9 78756 78772 16 [6 9 88994 26 O.II 006 9 89761 9 II 10 .1 zo I.O 9 0.9 9 89 020 . 10 980 9 89752 SI 9 78788 9 89046 26 0. 10 954 9 89 742 9 .2 2.0 1.8 S2 9 78805 I/' [6 16 6 6 9 89073 27 0. 10 927 9 89732 8 3 30 2.7 S3 9 78821 9 89099 10 901 9 89722 7 4 4.0 36 54. •>s 9 9 78837 9 89125 26 26 0.10875 9 89712 10 6 5 i 5 6 4.5 V. 78 85.3 9 89 151 0.10 849 9 89 702 S6 9 78 869 9 89 1 77 26 0.10823 9 89693 9 4- i 8^o S7 9 78886 16 6 [6 9 89 203 26 o.io 797 9 89 683 3 S« 9.78902 9 89 229 0. 10 771 9 89673 2 ■9 9.0 59 60 9 78918 9 89255 26 0.10 745 9 89663 10 I 9 78934 9 89281 0. 10 719 9 89653 L. Cos. 4 i. L. Cotg. c."(i. L. TaiifiT. L. Sin. d. / Prop. Fts. 52^ 1 64 TABLE II 38' I 2 3 _4 I 7 8 _9_ 10 II 12 13 ;i 17 i8 i9_ 20 21 22 23 24 26 27 28 29 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49. 50 51 52 53 il ^^ GO L. Sin, 78934 78 QW 78967 78983 78999 79015 79031 79047 79063 79079 79095 79 III 79 128 79 144 79 160 79176 79 192 79 208 79224 79240 79256 79272 79288 79304 79319 79 335 79351 79367 79383 79 399 79415 79431 79 447 79463 79478 79 494 79510 79526 79542 79558 79 573 79589 79605 79 621 79636 79652 79668 79684 79699 79715 79731 79 746 79 762 79778 79 793 79809 79825 79840 79856 79872 79887 L. Cos. L. Taiig. c. d. 89281 89307 89333 89359 89385 89 411 89437 89463 89489 89515 89541 89567 89593 89 619 89645 89671 89697 89723 89749 89775 89801 89827 89853 89879 89905 89931 89957 89983 90 009 90035 90061 90 086 90 112 90 138 90 164 90 190 90 216 90 242 90 268 90294 90 320 90346 90371 90397 90423 90449 90475 90 501 90527 90553 9.90578 9 . 90 604 90630 9.90 9 90 656 682 90 708 90734 90759 90785 90 811 9.90837 36 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 26 26 26 25 26 26 26 26 26 26 25 26 26 L. Cotg. Ic. (1 L. Cotg. o.io 719 o.ic 693 o ic 667 o 10 64£ o. 10 615 O.IO 589 o. 10 563 0.10537 O.IO 511 o. 10 485 0.10459 0.10433 o. 10 407 O.IO 381 0.10355 0.10329 o . 10 303 O.IO 277 O.IO 251 o. 10 225 o. 10 199 O.IO 173 O.IO 147 O. 10 121 0.10095 O.IO 069 o. 10 043 O.IOOI7 0.09 991 o • 09 965 0.09939 0.09 914 0.09888 o . 09 862 0.09 836 0.09 810 0.09 784 0.09 758 0.09 732 0.09 706 o . 09 680 0.09 654 o . 09 629 o . 09 603 0.09577 0.09551 0.09 525 0.09499 0.09473 0.09447 o . 09 422 0.09 396 0.09 370 0.09344 0.09 318 0.09 292 0.09 266 o . 09 241 o 09 2 1 5 o 09 189 0.09 163 L. Tang. 5r L. Cos. 89653 89643 89633 89 624 89 614 89 604 89594 89584 89574 89564 89554 89544 89534 89524 89514 89504 89495 89485 89475 89465 89455 89445 89435 89425 89415 89405 89395 89385 89375 89364 89354 89344 89334 89324 89314 89304 89294 89284 89274 89 264 89254 89244 89233 89 223 89213 89203 89193 8918:5 89173 89 Ib2 89152 89 142 89 132 89 122 89 112 89 lOI 89 091 89081 89071 89 060 9 . 89 050 L. Sin. 60 59 58 57 55 54 53 52 51 50 49 48 47 _46_ 45 44 43 42 41 40 39 38 36 35 34 33 32 Jl 30 29 28 27 26 Prop. Vis. 26 I 26 2 S-2 3 7.8 4 10.4 5 13 6 15.6 7 18 2 8 20.8 9 23 4 17 2 1 ■ 3 3 4 6 8. .6 10. .7 II .8 13 9 15 25 25 50 7 5 10 o 12 5 15 o 17 5 20.0 22.5 16 I 1.6 2 32 3 4.8 4 S4 .1^ 8 ol .6 96 • 7 II. 2 .8 12.8 9 14-4 XI I I. 2 2 3 3 •4 4 I 7 I 7 .8 •9 8 9 zo 9 10 20 I 30 2 4.0 3 SO 4 6.0 S 7.0 6 8.0 I 90 15 30 6.0 7 5 9.0 10. 5 12.0 ^3 5 Prop. Pis. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 65 39° 1 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. 60 Prop. Pts. 9 79887 16 9.90837 26 0.09 163 9 89 050 ■ 1 I 9 79903 TC 9 90«b3 26 0.09 137 9 89 040 S9 1 2 3 _4_ 9 9 9 79918 79 934 79950 16 16 15 16 9 9 9 90889 90914 90940 2S 26 26 26 0.09 III 0,09086 . 09 060 9 9 9 89 030 89 020 89009 10 II 10 58 .1 .2 3 4 26 2.6 li 10.4 ill 208 9 79965 9 90966 0.09034 9 88999 6 9 79981 15 16 9 90992 oA 0.09008 9 88989 54 7 9 79996 9 91 018 0.08982 9 88978 S^ 8 9 80012 15 16 16 9 91043 26 26 0.08957 9 88968 52 9 10 9 80027 9 91 069 0.08 931 9 88958 10 51 50 I 9 80043 9 91095 0.08905 9 88948 II 9 80058 9 91 121 26 0.08879 9 88937 49 9 23 4 12 9 80074 9 91 147 0.08853 9 88927 48 n 9 80089 16 9 91 172 25 26 26 26 26 0.08828 9 88917 47 1 14 15 9 80 105 IS 16 9 91 198 0.08802 9 88906 10 46 45 25 25 9 80 120 9 91 224 0.08 776 9 88896 16 9 80 136 9 91 250 0.08 750 9 88 886 44 .2 50 17 9 80 151 9 91 276 0.08 724 9 88875 43 •3 7-5 18 9 80166 16 9 91 301 25 26 26 26 0.08 699 9 88865 42 •4 10.0 19 20 9 80 182 15 16 9 91 327 0.08 673 9 88855 II 41 40 i 12.5 15 9 80197 9 91353 0.08 647 9 88844 21 9 80213 9 91 379 0.08 621 9 88834 39 i 17 5 22 9 80228 16 9 91404 25 26 26 26 0.08 596 9 88824 10 38 20.0 2S 9 80244 9 91430 0.08 570 9 88813 37 •9 22.5 24 2"^ ^9 "9 80 259 15 16 9 91456 0.08 544 9 88803 10 36 35 1 80274 9 91482 0.08518 9 88793 16 26 9 80290 9 91 507 26 26 26 25 26 26 26 0.08493 9 88782 34 1.6 27 9 80305 9 91 533 0.08467 9 88772 33 28 9 80320 16 15 9 91 559 0.08 441 9 88761 32 8 29 30 9 8033^ 9 91585 0.08 415 9 88751 10 10 31 80 ■3 •4 9 80351 9 91 610 0.08 390 9 88741 31 9 80366 16 9 91 636 0.08 364 9 88730 11 9.6 112 32 9 80382 9 91 662 0.08338 9 88 720 10 28 i 33 9 80397 9 91688 08 312 9 88 709 II 27 12 8 34 3S 9 80412 16 9 91 713 25 26 26 26 0.08287 9 88699 10 II 26 25 .g 9 . . lA A 9 80428 9 91 739 0.08 261 9 88 688 36 9 80443 9 91 765 0.08 235 9 88678 10 24 1 37 9 80458 9 91 791 08 209 9 88 668 10 23 X5 IS 30 38 9 80473 IS 16 15 9 91 816 25 0.08 184 9 88 657 II 22 39 40 9 80489 9 91842 26 0.08 158 9 88 647 10 11 21 20 .2 3 .4 9 80504 9 91 868 08 132 9 88636 41 9 80519 15 9 91 893 25 26 26 26 2S 26 0.08 107 9 88626 10 IQ 42 9 80534 15 9 91 919 0.08081 9 88615 II 18 7 5 43 9 80550 9 91 945 0.08 055 9 88605 17 5 90 44 4S 9 80565 IS 9 91 971 . 08 029 9 88594 10 16 15 i 10.5 12 9 80580 9 91 996 . 08 004 9 88584 46 9 80595 *s 9 92 022 0.07978 9 88 573 II 14 9 13 5 47 9 80610 IS 9 92 048 0.07952 9 88 S63 10 n 1 48 9 80625 IS 16 ^5 9 92073 25 26 26 0.07927 9 88 552 II 12 1 49 60 9 80641 9 92099 0.07 901 9 88542 II II 10 .1 II I . I 10 1.0 9 80656 9 92 125 0.07875 9 88531 SI 9 80671 ^s 9 92 150 2S 0.07850 9 88521 10 p .2 2.2 2.0 52 9 80686 IS 9 92176 26 0.07 824 9 88510 II 8 3 3 3 30 S3 9 80 701 9 92 202 0.07 798 9 88499 I' 7 4 4 4 4.0 54 SS 9 80 716 15 15 9 92227 2S 26 0.07 773 9 88489 10 II 6 S I U 50 6.C 9 80731 9 92253 0.07 747 9 88478 S6 9 80 746 15 16 9 92279 0.07 721 9 88468 10 4 :i ii 7.0 8.C S7 9 80 762 9 92304 2S 0.07 696 Q 88457 IX 3 S8 9 80 777 15 9 92330 0.07 670 9 88447 10 2 •9 9 9 9.0 59 9 80792 15 15 9 9 92356 92381 2S 0.07 644 9 88 436 II II I 9 80 807 0.07 619 9.88425 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. f Prop. Pis. 50^ 1 66 TABLE II 40° 1 t L. Sin. d. L. Tang. c. d. L. Cotg. L. Cos. d. w Prop. Pts. 9.80807 9.92381 0.07 619 9.88425 I 9 80 822 9 92407 0.07593 9 88415 S9 2 9 80837 9 92433 0.07567 9 88404 S8 26 3 9 80852 9 92458 25 0.07 542 9 88 394 S7 T 2 6 4 9 80867 9 92484 26 0.07 516 9 88383 II 56 55 2 3 '7s 9 80882 9 92510 0.07490 9 88 372 6 9 80 897 9 92535 25 0.07465 9 88362 54 4 10 4 7 9 80 912 9 92561 26 0.07439 9- 88351 53 13 8 9 80927 9 92587 0.07413 9. 88340 52 (5 15 6 9 10 9 80942 9 92 612 25 26 0.07388 9 88330 II 51 50 7 8 182 20 8 9 80957 9 92 638 0.07362 9 88319 II 9 80972 9 92 663 25 26 26 0.07337 9 88308 49 9 23 4 12 9 80987 9 92689 0.07 311 9 88298 48 1 n 9 81 002 9 92 715 0.07 285 9- 88287 47 1 14 9 81 017 9 92 740 26 26 0.07 260 9 88276 10 46 45 25 25 9 81 032 9 92766 0.07234 9 88266 1 i6 9 81 047 9 92 792 0.07 208 9 88255 44 2 50 17 9 81 061 9 92817 25 26 0.07 183 9. 88244 43 3 7 5 i8 9 81 076 9 92843 0.07 157 9 88234 42 4 10 19 20" 9 81 091 9 92868 25 26 26 0.07132 9 88223 II 41 40 I 12.5 15 17 5 20 9 81 106 9 92894 0.07 106 9 88212 21 9 81 121 9 92 920 0.07 080 9 88201 39 22 9 81 136 9 92945 25 26 0.07055 9 88 191 ° 38 n 22.5 2^ 9 81 151 9 92971 0.07029 9 88180 37 24 2S 9 81 166 9 92996 25 26 26 0.07004 9 88 169 II 36 35 1 9 81 180 9 93022 0.06978 9 88158 IS 26 9 81 195 9 93048 0.06952 9 88 148 34 ^ 15 3 4 5 6 7 5 27 9 81 210 9 93073 25 26 0.06927 9 88137 33 2 28 9 81 225 9 93099 06 901 9 88 126 32 •3 •4 • S 29 30 _9 9 81 240 81 254 9 93 124 25 26 06876 9 88 115 10 31 30 9 93 150 0.06 850 9 %% 105 SI 9 81 269 9 93 175 25 0.06825 9 88094 29. .6 90 32 9 81 284 9 93201 26 0.06 799 9 88083 28 .7 10 5 33 9 81 299 9 93227 0.06 773 06 748 9 88072 27 .8 12 34 3S 9 81 314 9 93252 25 26 9 88061 10 26 25 9 13 5 9 81 328 9 93278 0.06 722 9 88051 1 3^ 9 81343 9 93303 25 06 697 9 88040 24 1 37 9 81358 9 93329 06 671 9 88029 23 14 3« 9 81372 9 93 354 25 06 646 9 88018 22 I 14 2 8 4 2 39 40 9 ,81 387 9 93380 26 06 620 9 88007 11 21 20 .2 .3 9 81 402 9 93406 0.06594 9 87996 A\ 9,81 417 9 93431 25 0.06 569 9 87985 19 • 4 56 42 9 81 431 9 93 457 26 . 06 543 9 87975 18 .5 7 43 9 81 446 9 93482 25 06 518 9 87964 17 .6 84 44 4S 9 81 461 9 93508 26 25 06 492 9 87953 II 16 ^5 •7 .8 98 II 2 9^1 475 9 93 533 0.06467 9 87942 46 9 81 490 9 93 559 26 0.06 441 9 87931 14 9 12 6 47 9 81 505 9 93584 25 06 416 9 87 920 13 1 48 9 81 519 9 93 610 0.06390 9 87 909 12 1 49 50 9 «i 534 9 93636 26 25 06 364 9 87 898 II II 10 I XI II 10 ID 9 81 549 9 93661 06339 9 87 887 .S» 9 8' 563 9 «• 578 9 93687 26 06313 9 87877 9 .2 22 2 S2 9 93712 25 0.06288 9 87866 8 3 3 3 3 S3 9 81 592 9 93 738 26 0.06 262 9 87855 7 4 4 4 4 54 9 81 607 9 93 763 25 26 0.06 237 9 87844 6 5 .^5 IS 5S 9 81 622 9 93 789 0.06 211 9 ^l^ZZ 5 6 6 6 5^ 9 81 636 9 93814 25 0.06 186 9 87822 4 .8 1^ ^0 ^s^ 9 81 651 9 93 840 26 0.06 160 9 87 811 3 9 81 665 9 93865 25 0.06 135 9.87800 2 y 9 9 90 59 60 9 81 680 9 93891 26 25 0.06 109 9 87789 II 9 81 694 9 93916 06 084 9 87 778 L. Cos. L. Cotg. c. d. L. Tang. L. Sin. f Prop. Pt8. 49^ 1 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 67 41 9_ 10 II 12 \i i8 20 21 22 23 24 25 26 27 28 f9 30 31 32 33 Jl 36 37 38 39 40 41 42 43 44 46 47 48 49 50 51 52 53 _54 55 5^ 00 L. Sin, 694 709 723 738 752 767 781 796 810 825 839 854 868 882 897 911 926 940 955 969 983 998 82 012 82 026 82 041 82055 82 069 82084 82098 82 112 82 126 82 141 82155 82 169 82 184 82 198 82 212 82 226 82 240 82255 82 269 82283 82 297 82 311 82_326_ 82340 82354 82368 82382 82396 82 410 82 424 82439 82453 82467 82481 82495 82 509 82523 82537 82551 L. Cos. d. Tang. 93916 93942 93 967 93 993 94 018 94044 94 069 94095 94 120 94 146 94 171 94 197 94 222 94248 94273 94299 94324 94350 94 375 94401 94 426 94452 94 477 94503 94528 94 554 94 579 94 604 94630 94655 94 681 94 706 94 732 94 757 94783 94808 94834 94859 94884 94910 94 935 94961 94 986 95 012 95037 95 062 95088 95 113 95 139 95 164 95 190 95215 95 240 95 266 95291 95317 95 342 95368 95 393 95 418 c. d. 95 444 Cotff. c. d. L. Cotg. o . 06 084 0.06 058 0.06033 0.06 007 0.05 982 0.05 956 0.05931 0.05 905 0.05 880 0.05 854 0.05 829 0.05 803 o 05 778 o 05 752 0.05 727 o 05 701 o 05 676 0.05 650 0.05 625 0.05 599 0.05 574 0.05 548 0.05 523 0.05 497 0.05 472 0.05 446 0.05 421 o 05 396 0.05 370 005 345 0.05 319 0.05 294 0.05 268 0.05 243 6.05 217 0.05 192 0.05 166 0.05 141 o 05 116 o . 05 090 0.05 065 o 05 039 o 05 014 o 04 988 o 04963 o 04 938 o 04 912 o 04 887 0.04 861 o . 04 836 0.04 5IO 0.04 785 o 04 760 o 04 734 0.04 709 o . 04 683 o . 04 658 o . 04 632 0.04 607 o . 04 582 0.04 556 L. Tang. 48° L. Cos. 87778 87767 87756 87745 87734 87723 87712 87 701 87 69c 87679 87668 87657 87646 87635 87624 87613 87601 87590 87579 87 568 87557 87546 87535 87524 87513 87501 87490 87479 87468 87457 87446 87434 87423 87 412 87 401 87390 87378 87367 87356 9 87 345 9 87 334 9 87322 9 873" 9 87300 9.87288 87277 87266 87255 87243 87232 87221 87 209 87 198 9 87 187 9 87 175 9 87 164 9 87 153 9 87 141 9 87 130 9 87 119 9.87 107 L. Sin, d. «0 59 58 57 56 55 54 53 52 _5L 50 49 48 47 46 45 44 43 42 41 40 39 38 36 35 34 33 32 31 25 24 23 22 21 20 ^9 18 17 16 15 14 13 12 II 9 8 7 6 Prop. Pts. 6 2 8 4 o 18.2 20.8 23 4 25 25 50 7 5 10.0 12.5 15 o 17 5 20.0 22.5 15 IS 30 6.0 7-5 9.0 10.5 12.0 13 5 X4 14 8 2 6 o 4 8 2 6 13 .1 12 .2 2.4 3 36 • 4 48 6.0 .6 7.2 • 7 8.4 .8 9.6 •9 0.8 Prop. Pts. 68 TABLE II 42^ 2 3 A 5 6 7 8 9^ 10 II 12 13 14 11 17 i8 i9_ 20 21 22 23 24 26 27 28 29 30 31 32 33 34_ 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 I II 60 L. Sin. 82551 82565 82 579 82 593 82 607 82621 82 635 82 649 82663 82677 82 69_i 82 705 82 719 82 733 82 747 82 76_i 82775 82788 82802 82816 82830 82844 82858 82872 8288=: 82899 82913 82 927 82 941 82955 82968 82982 82 996 83 010 83023 83037 83051 83065 83078 83092 83 106 83 120 83133 83 147 83 161 83 174 83 188 83 202 83215 83 229 83242 83256 83 270 83283 83297 83310 83324 83338 83351 83365 983378 Cos. d. L. Tang. 95 444 95 469 95 495 95 520 95 545 95 571 95596 95 622 95647 95 672 95698 95 723 95 748 95 774 95 799 95 825 95850 95875 95901 95 926 95952 95 977 96 002 96 028 96053 96 078 96 104 96 129 96155 96 180 96 205 96231 96 256 96281 96307 96332 96357 96383 96 408 96433 96459 96 484 96 510 96535 96 560 96586 96 611 96636 96 662 96687 96 712 96738 96 763 96788 96 814 96839 96864 96 890 96915 96 940 c.d, 96 966 d. I L. Cotg. 25 26 25 25 26 25 26 25 25 26 25 25 26 25 26 25 25 26 25 26 25 25 26 25 25 26 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 c.d. L. Cotg. 0.04 556 o 04531 o . 04 505 o 04 480 o 04 455 o 04 429 o 04 404 0,04378 0.04353 0.04 328 o . 04 302 0.04 277 o 04 252 o . 04 226 0.04 201 0.04 175 0.04 150 0.04 125 o . 04 099 o . 04 074 o . 04 048 o . 04 023 0.03 998 0.03 972 0.03947 0.03 922 0.03 896 0.03871 0.03 845 0.03 820 0.03 795 0.03 769 0.03 744 0.03 719 0.03 693 0.03 668 0.03643 o 03617 0.03 592 0.03 567 0.03541 0.03 516 o . 03 490 o . 03 465 0.03440 0.03 414 0.03389 0.03364 0.03338 0.03313 0.03 288 0.03 262 0.03237 0.03 212 0.03 186 0.03 161 o 03 136 o 03 no o 03 085 o , 03 060 0.03034 L. Tang. 470 I . Cos. d • 60 9,87 107 9 87096 , 59 9 87085 58 9 87073 57 _9_ 9 87062 2 56 55 87 050 9 87039 54 9 87028 5^ 9 87016 52 9 87005 2 51 50 9 86993 9 86982 ^ 49 9 86970 48 9 86959 47 9 86947 I 46 45 9 86936 9 86924 44 9 86913 43 9 86902 42 9 86890 I 41 40 9 86879 9 86867 39 9 86 855 38 q 86844 37 9 86832 ^ 36_ 35 9 86821 9 86809 M 9 86798 ^ S3 9 86786 ^2 9 86775 2 31 30 9 86763 9 86752 ' 29- 9 86 740 28 9 86728 27 9 86717 2 26 25 9 86705 9 86694 ' 24 9 86682 23 9 86670 22 9 86659 2 21 20 9 86647 9 86 63s 19 9 86624 18 9 86612 17 9 86600 I 16 15 9 86589 9 86577 14 9 86 565 13 9 86554 12 9 86542 2 II 10 9 86530 9 86518 9 9 86507 8 9 86495 7 9 86483 6 5 9 86472 9 86460 4 9 86448 3 9 86436 2 9 86 425 9 86413 I .. Sin. i . / Prop. Pts, 36 2 6 25 25 o 5 5 7 10.0 12.5 14 14 2.8 13 .1 1 . .2 2 •3 3 • 4 5 .5 6. 6 7 7 9 .8 10. 9 II . 12 .1 I 2 .2 3 •4 3I 4.8 6.0 9 II I 2.2 3 3 4 4 II 7 7 8 8 9 9 Prop. Pts. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 69 43^ L. Sin. 7 8 9_ 10 II 12 13 lA 15 i6 17 i8 i9_ 20 21 22 23 24 26 27 28 29 31 32 33 34 36 37 38 J9 40 41 42 43 _44_ 46 47 48 49 50 51 52 53 id 55 56 57 58 i9_ 60 983378 9 83392 9 83405 9.83419 9 83432 9.83446 9 83 459 9 83473 9 83 486 9 83 500 9 83513 9 83 527 9 83540 83554 83567 83581 83594 83608 83621 83,634^ 83648 83661 83674 83688 83701 83715 83728 83741 83755 83768 83781 83795 83808 83821 83834 83848 83861 83874 83887 83901 83914 83927 83940 83954 83.967 83 83993 84 006 84 020 84033 9 . 84 046 9 84 059 9 84 072 9 84 085 9 84 098 84 112 84 125 84138 84 151 84 164 9 84177 L. Cos. L. Tang. 9 96 966 9 96 991 9 97 016 9 97042 9.97067 97092 97 118 97 143 97168 97 193 97219 97244 97269 97295 97320 97 345 97371 97396 97421 97 447 97472 97 497 97523 97548 97 573 97598 97624 97649 97674 97 700 97725 97750 97776 97 801 97 826 97851 97877 97902 97927 97 953 97978 98 003 98 029 98054 98079 98 104 98 130 98 155 98 180 98 206 9 98 231 9 98 256 9 98 281 9 98307 9 98 332 9 98357 998383 9 . 98 408 9 98433 9.98458 9.98484 L. Cotg. c. (I c. d. L. Cotg. 0.03034 o 03 009 o . 02 984 0.02 958 0.02933 0.02 908 o 02 882 0.02 857 0.02 832 o 02 807 02 781 02 756 02 731 02 705 02 680 02 655 02 629 02 604 0.02 579 0.02 553 0.02 528 0.02 503 0.02 477 o . 02 452 0.02 427 o . 02 402 0.02 376 o 02 351 o . 02 326 o 02 300 o 02 275 o . 02 250 0.02 224 o 02 199 0.02 174 o 02 149 o 02 123 o . 02 098 o . 02 073 0.02 047 o . 02 022 o.oi 997 o.oi 971 o.oi 946 o.oi 921 O.OI 896 O.OI 870 0.01 845 O.OI 820 O.OI 794 O.OI 769 O.OI 744 O.OI 719 O.OI 693 O.OI 668 O.OI 643 o 01 617 o 01 592 O.OI 567 O.OI 542 O.OI 516 L. Tang. 46° L . Cos. d. 9 86413 60" 9 86401 59 9 9 86389 86377 2 58 57 9 86366 2 2 56 55 S4 9 9 86354 86342 9 86330 ■^^ 9 86318 '^2 9 86306 51 50 9 86295 9 86283 49 9 86271 48 9 86259 47 9 86247 2 46 45 9 86235 9 8b 223 44 9 86 211 43 9 86200 42 9 86188 2 41 40 9 86176 9 86 164 i:^ 39 9 86152 38 9 86 140 37 9 86128 [2 36 35 9 86 116 9 86 104 34 9 86092 33 9 86080 32 9 86068 [2 31 30 9 86056 9 86044 29 9 86032 28 9 86020 27 9 86008 12 26 25 9 85996 9 85984 24 9 85972 23 9 85 960 22 9 85948 C2 21 20 9 85936 9 85924 19 9 85 912 18 9 85900 17 9 85888 [2 16 IS 9 85876 9 85 864 14 9 85851 3 13 9 85839 12 9 85827 2 II 10 9 85815 9 85 803 9 9 85 791 8 9 85 779 7 ■9 85 766 3 2 6 5 9 85754 9 85 742 2 4 9 85 730 3 9 85718 2 . 2 9 85706 2 I 9 85693 13 L. Sin. d. / Prop. Pts. 26 I 2. 2 5 3 7- 4 10 13 6 7 18 8 20 9 23 25 I 25 2 50 3 7-5 4 10.0 5 12 s 6 15 7 175 8 20.0 9 22.5 14 ,1 1-4 .2 2.8 •3 4.2 •4 5-6 •5 7.0 .6 .8.4 •7 98 .8 II. 2 •9 12.6 13 I I . 2 2 3 3 4 5 . 6. 6 7- 7 9 8 10 9 II. 12 II .1 1.2 I. .2 2.4 2. 3 36 3 •4 4.8 4- 5 6.0 5- 6 7.2 6. 7 8.4 7- 8 9.6 8. 9 10.8 9- Prop. Pts. 70 TABLE II 44° 1 / L. Sin. d. L. Tang. C.d. L. Cotg. L. Cos. d I. 60 Prop. Pts. 9 84 177 9.98484 0.01 516 985693 , I 9 84 190 13 9 98509 25 0.01 491 9 85 681 SP 2 9 84203 84216 *3 9 98534 25 0.01 466 985669 58 9fi S 9 9 98 560 0.01 440 9 85 657 S7 , 2 6 78 4 s 9 84229 »3 *3 998585 25 0.01 415 9.85645 ; 3 56 55 2 3 .4 9 84242 9 98 610 001 390 985632 6 9 84 2.S5 »3 9 98635 01 365 9 85 620 54 10 4 7 9 84269 9 98 661 01339 9.85608 ' 5S I IT, 8 9 84 282 13 9 98686 01 314 985596 ; 52 15 6 9 10 9 84295 13 13 9 98 711 26 0.01 289 985583 ; J 2 51 50 7 8 18 2 20 8 9 84 308 9 98737 01 263 985571 II 9 84 321 »3 9 98 762 01 238 985559 , 49 9 23 4 12 9 84 334 9 98 787 01 213 01 188 985547 , 48 1 IS 9 84 347 9 98812 26 25 985534 ■ 47 1 14 IS 9 84 360 13 13 _9_ 9 98838 9886^ 98888 0.01 162 9.85522 ; 2 46 45 .1 25 2 5 9 84 373 0.01 137 985510 i6 9 84385 9.84398 9 0.01 112 985497 3 44 .2 5.0 17 13 9 98913 25 26 0.01 087 9.85485 43 •3 7 5 i8 9 84 411 13 9 98939 01 061 985473 , 42 •4 10 19 20 9 84424 13 13 9 98964 25 25 26 01 036 9 85 460 3 3 41 40 ■ i .7 .8 •9 12 5 15 17 5 20.0 22 5 9 84 437 9 98989 o.oi on 985448 , 21 9 84 450 13 9 99015 0.00 985 985436 , 39 22 9.84463 9.84476 9 99040 . 00 960 985423 , 3 38 23 9 99065 0.00935 9-85411 37 24 2S 9 84 489 13 9 99090 26 0.00910 9 85 399 , 3 36 35 1 9 84 502 9 99 116 0.00 884 9.85386 14 26 984515 *3 9 99 141 25 0.00859 985374 , 34 .1 27 9 84528 >3 9 99 166 25 0.00834 9.85361 3 33 42 56 7.0 28 9 84540 9 99 191 25 26 25 0.00809 985349 32 •3 •4 29 30 9 84553 13 13 9 99217 0.00 783 9 85 337 ■3 31 30 9 84566 9 99242 0.00 758 9 85324 31 9 84579 *3 9 99267 25 26 0.00 733 9 85312 29 8 4 32 9 84592 9 99293 99318 0.00 707 9.85299 28 .7 9.8 33 9 84605 13 9 25 0.00 682 9.85287 27 8 II .2 34 9 84618 13 12 9 99 343 25 25 26 0.00657 9.85274 13 t2 26 25 .9 12.6 3S 9 84630 9 99368 0.00632 9 . 85 262 36 9 84643 84656 13 9 99 394 0.00606 9.85250 24 1 37 9 13 9 99419 25 0.00 581 9 85237 IJ 23 13 38 9 84669 13 9 99 444 25 0.00 556 9. 85 225 22 I I 1 39 40 9 84682 13 12 9 99469 25 26 0.00531 9.85 212 13 t2 21 20 .2 .3 3 9 9 84694 9 99 495 . 00 505 9.85 200 41 9 84 707 »3 9 99520 25 . 00 480 9 85 187 13 19 .4 5.2 42 9 84 720 9 99 545 25 0.00455 9 85175 18 M 43 9 84733 13 9 99570 25 26 25 0.00430 9 85 162 13 17 .6 44 4S 9 84745 13 9 99596 . 00 404 9 85 150 '3 16 15 • 7 .8 91 10.4 9 84758 9 99621 0.00379 9 85 137 46 9 84771 »3 9 99646 25 0.00354 9.85 125 14 9 II. 7 :i 9 84784 13 9 99672 00 328 9.85 112 13 13 1 9 84796 9 99697 25 00 303 9.85 100 12 1 49 60 9 84 809 13 13 9 99722 25 25 0.00 278 0.00 253 9.85087 13 3 II 10 la 12 9 84822 9 99 747 9.85 074 SI 9 84835 13 9 99 773 26 0.00 227 9 . 85 062 9 .2 2.4 S2 9 84847 9 99 798 25 . 00 202 9,85049 3 8 •3 3^ S3 9 84860 13 9 99823 25 0.00177 9 85037 7 •4 4.8 54 SS 9 84873 13 12 9 99848 25 26 0.00 152 9 85 024 3 2 6 5 i 6.0 96 10.8 9 84885 84898 9 99874 0.00 126 9.85 012 S6 9 13 9 99899 25 0.00 lOI 984999 ; 3 4 ■7 .8 ■>7 9 84 911 13 9 99924 25 0.00076 984986 ' 3 3 S8 9 84923 9 99 949 25 0.00 051 9.84974 2 •9 59 00 9 84936 13 13 9 99 975 26 25 0.00 025 9.84961 ] 3 2 _0_ 9 84949 00 000 0.00000 9.84949 L. Cos. d. L. Cotg. c.d. L. Tanjr. L. Sin. d [. Prop. Pts. 45° TABLE III NATURAL TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE 71 72 TABLE m ©° 1-- 30 30 ■ — 4» 1 t N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. 60 59 58 55 54 53 52 51 50 % 47 46 45 44 43 42 41 40 35 34 33 32 31 30 27 26 25 24 23 22 21 20 19 18 '\l 15 14 13 12 I 2 3 4 1 9 lO II 12 ~^ 14 \l 19 20 21 22 23 26 27 28 29 30 .00000 .00029 .ocx)58 .00087 .00116 .00145 .00175 I. 00000 I .00000 I .00000 I. 00000 I .00000 I. 00000 I. 00000 ■01745 .01774 .01803 .01832 .01862 .0189J .01920 •99985 .99984 .99984 .99983 ■99983 .99982 .99982 •03490 •03519 ■03548 ■03577 .03606 .03664 •99939 .99938 ■99937 ■99936 ■99935 •99934 •99933 •05234 .05263 .05292 ■05321 •05350 •05379 .05408 .99863 99861 .99860 .99858 ■99857 ■99855 .99854 .06976 ■07005 ■07034 .07063 .07092 .07121 .07150 ■99756 ■99754 ■99752 ■99750 ■99748 ■99746 ■99744 .00204 .00233 .00262 .00291 .00320 .00349 I. 00000 I. 00000 I. 00000 I. 00000 .99999 .99999 .01949 .01978 .02007 .02036 .02065 .02094 .99981 .99980 .99980 .99979 99979 ■99978 .03693 .03723 ■03752 ■03781 .03S10 •03839 .99932 99931 .99930 .99929 .99927 .99926 ■05437 .05466 ■05495 ■05524 ■05553 •05582 ■99852 .99851 .99849 •99847 .99846 .99844 . .07208 ■07237 .07266 .07295 •07324 .99742 .99740 ■99738 .99736 ■99734 •99731 .00378 .00407 .00436 .00465 .00495 .00524 .99999 .99999 .99999 .99999 •99999 .99999 .02123 .02152 .02181 .02211 .02240 .02269 .99977 •99977 .99976 .99976 •99975 ■99974 .03868 .03897 .03926 •03955 .03984 .04013 .99925 .99924 .99923 .99922 .99921 .99919 .05611 .05640 .05669 .05698 .05727 ■05756 .99842 .99841 .99839 .99838 .99836 .99834 •07353 .07382 .07411 .07440 .07469 .07498 .99729 .99727 •99725 •99723 .99721 .99719 •00553 .00582 .00611 .00640 .00669 .00698 •99998 .99998 .99998 •99998 .99998 .99998 .02298 .02327 .02356 .02385 .02414 .02443 .99974 ■99973 .99972 .99972 .99971 .99970 .04042 .04071 .04100 .04129 .04159 .04188 .99918 .99917 .99916 ■99915 ■99913 .99912 ■05785 .05814 .05844 ■05873 .05902 ■05931 99833 .99831 .99829 .99827 .99826 .99824 ■07527 ■07556 ■07585 .07614 .07643 .07672 .99716 .99714 .99712 .99710 .99708 ■99705 .00727 .00756 .00785 .00814 .00844 .00873 .99997 .99997 .99997 •99997 .99996 .99996 .02472 .02501 .02530 .02560 .02589 .02618 .99969 .99967 .99966 .99966 .99965 .99964 .99963 .99963 .99962 .99961 .04217 .04246 .04275 .04304 •04333 .04362 .99911 .99910 .99909 .99907 .99906 ■99905 .05960 .05989 .06018 .06047 .06076 .06105 .06163 .06192 .06221 .06250 .06279 .99822 .99821 .99819 .99817 .99815 .99813 .07701 •07730 •07759 .07788 .07817 .07846 .99703 .99701 .99699 .99696 .99694 .99692 31 32 33 34 % 39 40 41 42 43 44 % 49 50 51 52 53 54 55 56 .00902 .00931 .00960 .00989 .01018 .01047 .99996 .99996 •99995 •99995 •99995 •99995 .02647 .02676 .02705 •02734 .02763 .02792 .04391 .04420 .04449 .04478 •04507 •04536 .99904 .99902 .99901 .99900 .99898 .99897 .99812 .99810 .99808 .99806 .99804 .99803 ■07875 .07904 ■07933 .07962 .07991 .08020 .99689 ■99687 .99685 .99683 .99680 .99678 .01076 .01105 •01134 .01164 .01193 .01222 .99994 .99994 ■99994 •99993 •99993 ■99993 .02821 .02850 .02879 .02908 .02938 .02967 .99960 ■99959 •99959 .99958 ■99957 ■99956 ■04565 ■04594 .04623 •04653 .04682 .04711 .99896 .99894 .99893 .99892 .99890 .99889 .06308 ^06366 ■06395 .06424 •06453 .99801 .99799 .99797 •99795 •99793 .99792 .08049 .08078 .08107 .08136 .08165 08194 .99676 ■99673 .99671 .99668 .99666 .99664 .99661 •99659 ■99657 .99654 .99652 .99649 .01251 .01280 .01309 •01338 .01367 .01396 .99992 .99992 .99991 .99991 .99991 .99990 .02996 .03025 03054 .03083 .03112 .03141 •99955 ■99954 ■99953 .99952 .99952 ■99951 .04740 .04769 .04798 .04827 .04856 .04885 .99888 .99886 ■99885 ■99883 .99882 .99881 .06482 .06511 .06540 .06569 .06598 .06627 .06656 .06685 .06714 .06743 ■06773 .06802 .06831 .06860 .06889 .06918 .06947 .06976 .99790 .99788 •99786 .99784 .99782 .99780 ■99778 .99776 ■99774 ■99772 ■99770 .99768 .08223 .08252 .08281 .08310 ■08339 .08368 .01425 .01454 .01483 •01513 .01542 .01571 .99989 .99989 .99989 .03170 .03199 .03228 .03286 .03316 •99950 •99949 .99948 •99947 .99946 ■99945 .04914 •04943 .04972 .05001 .05030 •05059 .99879 .99878 .99876 .99875 ■99873 .99872 .08397 .08426 •08455 .08484 •08513 .08542 .99647 •99644 .99642 •99639 •99637 •99635 II 10 I 5 4 3 2 I .01600 .01629 .01658 .C1687 .01716 •01745 .99987 .99987 .99986 .99986 99985 ■99985 •Q3345 •03374 ■03403 •03432 .03461 .03490 .99944 •99943 .99942 .99941 .99940 99939 .05088 .05117 .05146 ■05175 .05205 ■05234 .99870 .99869 .99867 .99866 .99864 ■99863 .99766 .99764 .99762 .99760 •99758 99756 .08571 .08600 .08629 .08658 .08687 .08716 ■99632 ■99630 .99627 .99625 .99622 .99619 N. cos. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine 89° SS** 87° 86° 85° TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE 73 ^^■^ 5« 1 6° 1 7- 1 8» 1 »• 1 60 11 55 54 53 52 51 50 It % 45 44 43 42 o I 2 3 4 i I 9 lO i2 13 14 \l \l 19 20 21 22 23 24 1 29 30 31 32 33 34 11 11 39 40 41 42 NT. sine ] ST. COS. N. sine ] ST. COS. NT. sine] ST. COS. ^r. sine|] ST. COS. N. sine N. COS. .08716 .08745 .08774 .08803 .08831 .08860 .08889 .99619 .99617 .99614 .99612 .99609 .99607 .99604 . 10482 .10511 ■ 10540 ■ 10569 •10597 . 10626 •99452 •99449 •99446 •99443 •99440 99437 •99434 .12187 .12216 .12245 .12274 .12302 •12331 .12360 •99255 .99251 .99248 .99244 .99240 .99237 ■99233 •I39I7 .13946 •13975 .14004 • 14033 .14061 . 14090 .99027 .99023 .99019 .99015 .99011 .99006 .99002 •15643 .15672 .15701 •15730 .15758 .15787 .15816 .98769 .98764 .98760 •98755 .98751 .98746 .98741 .08918 .08947 .08976 .09005 .09034 .09063 .99602 •99599 .99596 •99594 .99591 .99588 .10713 .10742 .10771 . 10800 •99431 .99428 .99424 .99421 .99418 ■99415 .12389 .12418 .12447 .12476 .12504 •12533 .99230 .99226 .99222 .99219 .99215 .99211 .99208 .99204 .99200 .99197 •99193 .99189 .14119 .14148 .14177 • 14205 .14234 .14263 .98998 .98994 .98990 .98986 .98982 .98978 .15845 •15873 .15902 •15931 •98737 •98732 .98728 .98723 .98718 .98714 .09092 .09121 .09150 .09179 .09208 .09237 .99586 •99583 .99580 •99578 •99575 •99572 . 10829 '10887 .10916 .10945 .10973 .99412 •99409 .99406 .99402 •99399 •99396 .12562 .12591 .12620 .12649 .12678 .12706 . 14292 .14320 ■14349 •14378 .14407 .14436 •98973 .98969 .98965 .98961 •98957 •98953 .16017 .16046 .16074 .16103 .16132 .16160 .98709 .98704 .98700 .98695 .09266 .09295 .09324 •09353 .09382 .09411 •99570 •99567 •99564 .99562 •99559 •99556 .11002 .11031 .11060 . 1 1089 .IIII8 .11147 •99393 •99390 .99386 •99383 .99380 •99377 ■12735 .12764 .12793 .12822 .12851 .12880 .99186 .99182 .99178 •99175 .99171 .99167 .14464 •14493 .14522 .14551 .14580 . 14608 .98948 .98944 .98940 .98936 .98931 .98927 .16189 .16218 .16246 .16275 .16304 •16333 .98681 .98676 .98671 .98667 .98662 •98657 41 40 P .09440 .09469 .09498 .09527 •09556 ■09585 ■99553 •99551 .99548 •99545 .99542 .99540 . II 1 76 .11205 .11234 .11263 .11291 .11320 •99374 •99370 ■99367 .99364 •99360 •99357 .12908 .12937 .12966 .12995 .13024 •13053 .99163 .99160 .99156 .99152 .99148 .99144 .14666 .14695 •14723 •14752 .14781 ■98923 .98919 .98914 .98910 .98906 .98902 .16361 .16390 .16419 .16447 .16476 •16505 .98652 .98648 •98643 .98638 •98633 .98629 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 \l 15 14 13 12 .09614 .09642 .09671 .09700 .09729 .09758 ■99537 •99534 •99531 .99528 .99526 •99523 •I 1349 .11378 .11407 .11436 .11465 .11494 •99354 •99351 •99347 •99344 •99341 •99337 .13081 .13110 •13197 .13226 .99141 •99137 •99133 .99.129 .99125 .99122 .14810 • 14838 .14867 .14896 .14925 • 14954 .98897 .98884 .98880 .98876 •16533 .16562 .16591 .16620 .16648 .16677 .98624 .98619 .98614 .98609 .98604 .98600 .09787 .09816 .09845 .09874 .09903 .09932 .99520 •99517 •99514 •9951 1 .99508 .99506 •11523 •II552 .11580 .11609 .11638 .11667 •99334 •99331 .99327 •99324 .99320 ■99317 •13254 •13283 •13312 •13341 •13370 •13399 .99118 .99114 .99110 .99106 .99102 .99098 .14982 .15011 .15040 .15069 •15097 .15126 .98871 .98867 .98863 .98858 .98854 .98849 .16706 •16734 .16763 .16792 .16820 .16849 •98595 .98590 .98585 .98580 •98575 .98570 43 44 11 .09961 .09990 .10019 .10048 .10077 .10106 •99503 •99500 .99497 •99494 .99491 .99488 .11696 .11725 ■11754 .11783 .11812 .11840 ■99314 .99310 •99307 •99303 .99300 .99297 •13427 •13456 •13485 •13514 •13543 •13572 •99094 .99091 .99087 •99083 •99079 •99075 •I5I55 .15184 .15212 .15241 .15270 •15299 .98845 .98841 .98836 .98832 .98827 .98823 .16878 .16906 •16935 .16964 .16992 .17021 .98565 .98561 .98556 .98551 .98546 .98541 49 50 51 52 53 54 55 56 11 i. .10135 .10164 .10192 .10221 .10250 .10279 •99485 .99482 .99479 .99476 •99473 ■99470 .11869 .11898 .11927 .11956 .11985 .12014 •99293 .99290 99286 .99283 .99279 .99276 .13600 .13629 .13716 •13744 .99071 .99067 .99063 .99059 •99055 .99051 •15327 •15356 •15385 •I54I4 .15442 •I547I .98818 .98814 .98809 .98805 .98800 •98796 .17050 .17078 .17107 .17136 .17164 •17193 •98536 •98531 .98526 .98521 .98516 .98511 10 I 5 4 3 2 .10308 •10395 . 10424 •10453 ■99467 .99464 .99461 .99458 ■99455 ■99452 .12043 .12071 .12100 .12129 .12158 .12187 .99272 .99269 •99265 .99262 .99258 99255 .13889 •13917 .99047 .99043 .99039 •99035 .99031 .99027 •15500 •15529 •'5557 .15586 •15615 •15643 .98791 •98787 .98782 .98778 •98773 .98769 . 1 7222 .17250 .17279 .17308 •17336 •17365 .98506 .98501 .98496 .98481 N. COS. N. sine N. COS. M. sine N. COS. N.,sine N. COS. N. sine N. COS. N. sine / 84'* 1 §3° 1 82° 1 81° 1 80° 74 TABLE III t o I 2 3 4 I I 9 lO II 12 14 \l 17 i8 10° j If 19' 1 1»» .4^ N. sine N. COS. N. sine N. COS. N. sine N. COS. !^. sine N. COS. N. sine N. COS. 60 59 58 57 56 55 54 53 52 51 50 It 47 46 45 44 43 42 17365 17393 .17422 17451 ■17479 .17508 •17537 .98481 .98476 .98471 .98466 .98461 98455 •98450 .19081 .19109 .19138 .19167 •19195 . 19224 .19252 .98163 .98157 .98152 .98146 .98140 98.35 .98129 .98124 .98118 .98112 .98107 .98101 .98096 20791 . 20820 .20848 .20877 .20905 .20933 .20962 •97815 .97809 .97803 .97797 .97791 •97784 •97778 •22495 •22523 •22552 .22580 .22608 .22637 .22665 •97437 •97430 .97424 •97417 .97411 •97404 •97398 .24192 .24220 .24249 •24277 •24305 •24333 .24362 .97030 •97023 •97015 .97008 .97001 ■96994 .96987 ■17565 •17594 .17623 .17651 .17680 .17708 .98445 .98440 •98435 .98430 .98425 .98420 .19281 .19309 •19338 .19366 •19395 .19423 .20990 .21019 .21047 .21076 .21104 .21132 .97772 .97766 .97760 •97754 .97748 .97742 .22693 .22722 .22750 •22778 .22807 .22835 •97391 •97384 •97378 •97371 •97365 •97358 •24390 .24418 .24446 .24474 •24503 •24531 .96980 •96973 .96966 •96959 .96952 .96945 .17766 •17794 .17823 .17852 .17880 .98414 .98409 .98404 •98399 tut .19452 .19481 .19509 .19566 •19595 .98090 .98084 .98079 .98073 .98067 .98061 .21161 .21189 .21218 .21246 .21275 .21303 ■97735 •97729 •97723 .97717 .97711 •97705 .22863 .22892 .22920 .22948 .22977 •23005 •97351 •97345 •97338 •97331 •97325 •97318 •24559 .24587 •24615 .24644 .24672 .24700 •96937 .96930 .96923 .96916 .96909 .96902 19 20 21 22 23 _^ - 25 26 27 28 29 30 . 31 32 33 34 It .17909 •17937 .17966 •17995 .18023 .18052 •98383 ■98373 :98368 .98362 •98357 .19623 .19680 .19709 .19766 .98056 .98050 .98044 •98039 •98033 .98027 •21331 .21360 .21388 .21417 .21445 .21474 .97698 .97692 .97686 .97680 .97667 •23033 .23062 .23090 .23118 .23146 •23175 973" •97304 .97298 .97291 .97284 •97278 .24728 .24756 .24784 .24813 .24841 .24869 ! 9688 7 .96880 •96873 .96866 .96858 41 40 It 11 .18081 .18109 .18138 .18166 .18195 .18224 •98352 •98347 .98341 •98336 •98331 •98325 .19794 .19823 .19908 •19937 .98021 .98016 .98010 .98004 .97998 .97992 .21502 •21530 •21559 .21587 .21616 .21644 .97661 .97648 .97642 .97636 •97630 .23203 .23260 .23288 .23316 •23345 .97271 .97264 •97257 •97251 .97244 •97237 .24897 .24925 .24954 .24982 .25010 .25038 .96851 .96844 •96837 .96829 .96822 .96815 35 34 33 32 31 30_ 29 28 27 26 25 24 .18252 .18281 .18367 •18395 .98320 •98315 .98310 •98304 .98299 .98294 .19965 .19994 .20022 .20051 .20079 .20108 .97987 .97981 •97975 .97969 •97963 •97958 .21672 .21701 .21729 .21758 .21786 .21814 •97623 .97617 .97611 .97604 •97598 •97592 23373 .23401 •23429 •23458 •23486 •23514 .97230 •97223 .97217 .97210 •97203 .97196 .25066 .25094 .25122 •25151 •25179 •25207 .96807 .96800 till •96778 .96771 11 39 40 41 42 .18424 ^18481 •IP .18567 .98288 •98283 .98277 .98272 .98267 .98261 .20136 .20165 •20193 .20222 .20250 .20279 •97952 •97946 .97940 •97934 .97928 .97922 •21843 .21871 .21899 .21928 .21956 .21985 •97585 •97579 •97573 •97566 .97560 •97553 •23542 •23571 •23599 .23627 .97189 .97182 .97176 .97169 .97162 •97155 •25235 .25263 .25291 •25320 •25348 •25376 .96764 .967.56 •96749 .96742 •96734 .96727 23 22 21 20 19 18 17 16 15 H 13 12 II 10 I 5 4 3 2 I 43 44 45 1 49 50 51 52 53 54 11 .18652 .18681 18710 .18738 .98256 .98250 •98245 .98240 .98234 .98229 .20307 .20336 20364 •20393 .20421 .20450 .97916 .97910 .97899 •97893 •97887 .22013 .22041 .22070 .22098 .22126 •22155 •97547 •97541 •97534 •97528 •97521 •97515 .23712 .23740 .23769 :2^^^5^ •23853 .97148 .97141 ■97134 .97127 .97120 •97"3 •25404 •25432 .25460 .25488 •25516 •25545 .96719 .96712 •96705 .96697 .96690 .96682 .18767 ' 18795 . 18824 .18852 .18881 .18910 .98223 .98218 .98212 .98207 .98201 .98196 .20478 .20507 •20535 .20563 .20592 .20620 .97881 •97875 .97869 •97863 •97857 ■97851 .22183 .22212 .22240 .22268 .22297 •22325 .97508 .97502 .97496 .97489 •97483 .97476 .23882 .23910 .23966 •23995 .24023 .97106 .97100 .97093 .97086 •97079 .97072 •25573 .25601 .25629 ■.25685 •25713 .96675 .96667 .96660 .96645 .96638 .18938 .18967 .18995 .19024 .19052 .19081 .98190 .98185 .98179 tit .98163 .20649 .20677 .20706 .20734 .20763 .20791 •97845 •97839 •97833 •97827 .97821 97815 •22353 .22382 .22410 .22438 .22467 .22495 1 .97470 97463 •97457 .97450 •97444 97437 .24051 .24079 .24108 .24136 .24164 .24192 .97065 •97058 •97051 ■97044 •97037 .97030 •25741 .25769 '25826 •25854 .25882 .96630 .96623 & 96600 ■96593 N. COS. N. sine M. COS. N. sine N. cos.|N. sine N. COS. N. sine N. CCS. N. sine / 1 79- 1 78^ yyo 76° 75° TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE 75 o I 2 3 4 16° j 16" 1 17° 1§" 19° 60 It 11 55 54 53 52 51 50 tt % 45 44 43 42 N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. .25S82 25910 .25966 •25994 .26022 .26050 96593 96585 96578 .96570 .96562 96555 •96547 .27564 .27592 .27620 .27648 .27676 27704 27731 .96126 .96118 .96110 .96102 .96078 29237 .29265 .29293 .29321 .29348 .29376 .29404 •95630 .95622 •95613 •95605 •95596 •95588 •95579 .30902 .30929 •30957 •30985 .31012 .31040 .31068 .95106 •95079 .95070 .95061 .95052 •32557 •32584 .32612 .32694 .32722 •94552 •94542 •94533 •94523 •94514 •94504 •94495 I 9 lO II 12 .26079 .26107 .26135 .26163 .26191 .26219 96540 •96532 .96524 •96517 .96509 .96502 ■27759 .27787 •27815 27843 .27871 .27899 .96070 .96062 .96054 .96046 .96037 .96029 .29432 .29460 •29487 •29515 •29543 ■29571 •95571 •95562 •95554 ■95545 ■95536 •95528 •31095 •31123 •31151 .31178 .31206 •31233 •95043 •95033 .95024 •95015 .95006 .94997 •32749 .32777 .32804 •32832 .94485 .94476 .94466 •94457 •94447 •94438 13 14 \i 18 19 20 21 22 23 24 .26247 .26275 .26303 •26331 •26359 .26387 •96494 .96486 .96479 .96471 .96463 •96456 .27927 •27955 .27983 .28011 .28039 .28067 .96021 .96013 .96005 •95997 .95989 .95981 .29599 .29626 .29710 ■29737 •95519 •95511 •95502 •95493 •95485 •95476 .31261 .31289 •31316 •31344 •31372 •31399 .94988 •94979 .94970 .94961 .94952 •94943 .32914 .32942 •32969 •32997 •33024 •33051 .94428 .94418 .94409 •94399 .94390 .94380 •26415 •26443 .26471 .26500 .26528 .26556 .96448 .Q6440 •96433 .96425 .96417 .96410 .28095 .28123 .28150 .28178 .28206 •28234 ■95972 .95964 •95956 .95948 •95940 ■95931 .29765 •29793 .29821 .29849 .29876 .29904 •95467 •95459 •95450 •95441 ■95433 •95424 •31427 •31454 .31482 •31510 •31537 •31565 •94933 •94924 •94915 .94906 '.lists •33079 .33106 •33134 •33161 •33189 .33216 •94370 .94361 •94351 .94342 •94332 .94322 41 40 39 1 25 26 li 29 30 3J 32 33 34 35 36 11 39 40 41 43 44 45 46 47 48 .26584 .26612 ^26696 .26724 .26752 .26780 .26808 .26836 .26864 .26892 .96402 •96394 .96386 96379 •96371 96363 •96355 •96347 .96340 .96332 •96324 .96316 .28262 .28290 .28318 .28346 •28374 .28402 ■95923 •95915 •95907 .95898 .95890 .95882 ■29932 .29960 •29987 ■30015 ■30043 .30071 .30098 .30126 ■30154 .30182 .30209 .30237 •95415 •95389 •95380 •95372 •31593 .31620 .31648 •31675 •31703 •31730 .94860 .94851 .94842 .94832 •33244 •33271 •33298 •33326 •33353 •33381 •94313 •94303 •94293 .94284 •94274 .94264 •94254 •94245 •94235 .94225 .94215 .94206 35 34 33 32 31 30 27 26 25 24 23 22 21 20 \l 15 14 13 12 II 10 \ 5 4 3 2 I .28429 •28457 .28485 .28513 .28569 •95874 •95865 •95857 •95849 .95841 ■95832 •95363 ■95354 ■95345 95337 ■95328 95319 •31758 .31786 .31813 .31841 .31868 •31896 •94823 .94814 •94805 •94795 .94786 ■94777 •33408 •33436 •33463 •33490 •33518 •33545 .26920 .26948 .26976 .27004 .27032 .27060 .96308 •96301 .96293 .96285 .96277 .96269 .28597 .28625 .28652 .28680 .28708 •28736 •95816 •95807 •95799 •95791 .95782 .30265 .30292 ■30320 •30348 .30376 ■30403 95310 •95301 •95293 .95284 •95275 .95266 •31923 •31951 •31979 .32006 •32034 .32061 .94768 •94758 .94749 .94740 •94730 .94721 •33573 .33600 .33627 •33710 .94196 .94186 .94176 .94167 •94157 .94147 .27088 .27116 .27144 .27172 .27200 .27228 .96261 •96253 .96246 .96238 .96230 .96222 .28764 .28792 28820 .28847 .28875 .28903 .95766 •95757 •95749 •95740 ■95732 ■30431 ■30459 .30486 ■30514 •30542 •30570 ■95257 .95248 .95240 ■95231 ■95222 95213 .32089 .32116 .32144 .32171 .32199 .32227 .94712 .94702 •94693 .94684 .94674 •94665 •33737 •33764 •33792 •33819 •33846 •33874 •94137 .94127 .94118 .94108 49 50 51 11 54- 11 11 .27256 .27284 .27312 .27340 .27368 .27396 .96214 .96206 .96198 .96190 .96182 .96174 •28931 .29015 .29042 .29070 •95724 •95715 •95707 95698 .95690 .95681 •30597 .30625 .30708 •30736 ■95204 •^5195 .95186 .95168 •95159 •32254 .32282 .32309 •32337 •32364 .32392 •94656 .94646 •94637 .94627 .94618 .94609 •33901 •33929 •33956 •33983 .54011 ■34038 .94078 .94068 •94058 •94049 •94039 .94029 .27424 .27452 .27480 .27508 •27536 .27564 .96166 .96158 .96150 .96142 .96126 .29098 .29126 .29154 29182 .29209 .29237 •95664 •95656 •95647 95639 •95630 .30763 .30791 .30819 .30846 .30874 .30902 ■95150 ■95142 ■95133 ■95124 95"5 .95106 .32419 •32447 •32474 .32502 •32529 •32557 •94599 •94590 .94580 •94571 .94561 •94552 -34065 •34093 .34120 •34147 •34175 .34202 •94019 .94009 •93999 •93989 •93979 .93969 N. COS. N. sine N. COS. N sine N. COS. N. sine N. COS. N. sine N. COS. N. sine / 74" 1 73° 1 72" 71" 1 70" 76 TABLE III f o I 2 3 4 "I 9 lO II 12 20° 1 «» 22° 1 23° 1 24° 60 59 58 11 55 il_ 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 11 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 \t \l 15 14 13 12 10 I 5 4 3 2 I N. sine ^T. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. .34202 .34229 .34284 •343" •34339 •34366 .93969 •93959 93949 ■93939 .93929 •93919 •93909 •35891 ■35918 •35945 •35973 .36000 •93358 •93348 •93337 •93327 •93316 •93306 •93295 •37461 .37488 •37515 •37542 •37569 •37595 .37622 .92718 .92707 .92697 .92686 .92664 •92653 •39073 .39100 •39127 •39153 .39180 .39207 ■39234 .92050 .92039 92028 .92016 .92005 .91994 .91982 .40674 .40700 .40727 •40753 .40780 .40806 •40833 .40860 .40886 .40913 .40992 •91355 91343 91331 •91319 •91307 .9 [295 .91283 •34393 .34421 .34448 •34475 •34503 •34530 •93899 .93889 •93879 .93869 •93859 •93849 .36027 .36108 ■93285 •93274 •93264 •93253 •93243 .93232 •37649 .37676 ■37703 •37730 •37757 •37784 .92642 .92631 .92620 .92609 .92598 •92587 .39260 .39287 •39314 •39341 •39367 ■39394 .91971 •91959 .91948 .91936 .91925 .91914 .91272 .91260 .91248 .91236 .91224 .91212 13 14 •34557 •345^4 .34612 •34639 .34666 .34694 •93839 •93829 .93819 •93809 •93799 ■93789 .36190 .36217 •36244 .36271 .36298 •36325 .93222 .93211 •93201 .93180 .93169 .37811 •37838 •37865 .37892 •37919 •37946 .92576 .92565 •92554 •92543 •92532 .92521 .39421 .39448 •39474 •39501 •39528 •39555 .91902 .91891 .91879 .91868 .91856 .91845 .41019 .41045 .41072 .41098 .41125 .41151 .91200 .91188 .91176 .91164 .91152 .91140 19 20 21 22 23 24 •34721 •34748 •34775 •34803 •34830 •34857 •93779 •93769 •93759 •93748 •93738 .93728 •36352 •36379 .36406 .36461 .36488 •93159 .93148 •93137 •93127 .93116 .93106 •37973 .38107 .92510 •92499 .92488 .92477 .92466 •92455 •39581 .39608 .39688 •39715 •91833 .91822 .91810 .91799 .91787 •91775 .41178 .41204 .41231 •41257 .41284 .41310 .91128 .91116 .91104 .91092 .91080 .91068 29 30 31 32 33 34 P 39 40 41 42 43 44 45 46 ti 49 50 51 52 53 54 55 56 ,34884 .34912 •34939 .34966 •34993 •35021 •93718 .93708 .93698 .93688 .93677 .93667 •36515 •36542 .36569 .36596 •^^?^ .36650 •93095 .93084 •93074 .93063 •93052 .93042 .38188 .38215 .38241 .38268 •92444 .92432 .92421 .92410 •39741 •39768 •39795 .39822 •39848 •39875 .91764 •91752 .91741 .91729 .91718 .91706 •41337 •41363 .41390 .41416 •41443 .41469 .91056 .91044 .91032 .91020 .91008 .90996 •3^048 •35075 •35102 •35130 •35157 •35184 •93657 •93647 •93637 .93626 .93616 .93606 •36677 •36704 •36731 •36758 •93031 .93020 .93010 .92978 •38295 .38322 •38349 •38376 •38403 •38430 ■$t .38510 •38537 .38564 •38591 ■92377 .92366 •92355 •92343 .92332 .92321 .92310 .92299 .92287 .92276 .92265 •92254 •39902 •39928 •39955 .39982 .40008 •40035 .91694 .91683 .91671 .91660 .91648 .91636 .41496 .41522 •41549 •41575 .41602 .41628 .90984 .90972 .90960 .90948 .90936 .90924 •35211 •35239 .35266 •35293 •35320 •35347 •93596 •93585 •93575 •93565 •93555 •93544 •36894 .36921 .36948 •36975 .92967 .92956 •92945 •92935 .92924 .92913 .40062 .40088 .40115 .40141 .40168 .40195 .91625 .91613 .91601 .91590 .91566 tell .41707 •41734 .41760 .41787 .90911 .90899 .90887 •90875 .90863 .90851 •35375 •35402 •35429 •35456 •35484 •355" •93534 •93524 •93514 •93503 •93493 •93483 .37002 .37029 •37056 •37083 .37110 •37137 .92902 .92892 .92881 .92870 •92859 .92849 .38617 .38644 .38671 .38698 •38725 •38752 .92243 .92231 .92220 .92209 .92198 .92186 .40221 .40248 .40275 .40301 .40328 •40355 •91555 •91543 91531 91519 .91508 .91496 .41813 .41840 .41866 .41892 .41919 •41945 •90839 .90826 .90814 .90802 .90790 ■90778 •35538 •35565 •35592 •35619 •35647 •35674 •93472 •93462 •93452 •93441 •93431 .93420 •37164 •37191 .37218 •37245 .37272 •37299 .92838 .92827 .92816 •92805 .92794 .92784 .38912 •92175 .92164 .92152 .92141 .92130 .92119 .40381 .40408 .40434 .40461 .40488 .40514 .91484 .91472 .91461 .91449 •91437 .91425 .41972 .41998 .42024 •42051 •42077 .42104 .90766 •90753 .90741 9^729 .90717 .90704 .90692 .90680 .90668 90655 90643 .90631 •35701 •35728 •35755 •35782 •35810 •35837 .93410 •93400 •93389 •93379 •93368 •93358 •37326 •37353 •37380 •37407 •37434 •37461 .92773 .92762 .92751 .92740 .92729 .92718 •38939 .38966 •38993 .39020 .39046 •39073 .92107 .92096 .92085 .92073 .92062 .92050 .40541 .40567 .40594 .40621 .40647 .40674 .91414 .91402 .91390 .91366 •91355 •42130 .42156 .42183 .42209 •42235 .42262 N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine / 69° 1 68° 67° 1 66° 1 65° TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE 77 9 35° 1 26° 1 sr 1 28° 1 29° 1 60 59 58 57 56 55 54 53 52 51 50 :i 47 46 45 44 43 42 41 40 35 34 33 32 31 30 29 28 V, 25 24 23 22 21 20 N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. o I 2 3 4 c l_ 9 lO 12 13 14 :i 17 18 .42262 .42288 •42315 •42341 •42367 .42394 .42420 .90631 .90618 .90606 .90594 .90582 .90569 •90557 •43863 ■43889 .43916 ■43942 .43968 •43994 .89879 .89867 .89854 .89841 .89828 .89816 .89803 45399 •45425 •45451 ■45477 •45503 •45529 ■45554 .89101 .89087 .89074 .89061 .89048 ■89035 .89021 .46947 •46973 •46999 .47024 .47050 .47076 .47101 .88295 .88281 .88267 .88254 .88240 .88226 .88213 .48481 .48506 ■48532 ■48557 ■48583 .48608 .48634 ■48659 .48684 .48710 ■48735 .48761 .48786 .87462 .87448 ■87434 .87420 .87406 ■87391 ■87377 •87363 •87349 •87335 .87321 •87306 .87292 .42446 •42473 •42499 •42525 •42552 .42578 •90545 •90532 .90520 .90507 .90495 .90483 .44020 .44046 .44072 .44098 .44124 •44151 •44177 ■44203 .44229 ■44255 .44281 •44307 .89790 •89777 .89764 •89752 ■89739 .89726 .45580 .45606 •45632 ■45658 ■45684 •45710 .89008 .88995 .88981 .88968 .88955 .88942 .47127 •47153 .47178 .47204 .47229 •47255 .88199 .88185 .88172 .88158 .88144 .88130 .42604 •42631 .42709 •42736 .90470 .90458 .90446 •90433 .90421 .90408 ■89713 .89700 .89687 .89674 .89662 .89649 •45736 •45762 •45787 •45813 •45839 •45865 .88928 •88915 : 88888 .88875 .88862 .47281 .47306 •47332 •47358 ■47383 ■47409 .88ii7 .88103 .88089 .88075 .88062 .88048 .48811 •48837 .48862 .48888 •48913 •48938 .87278 .87264 .87250 •87235 .87221 .87207 19 20 21 22 23 24 .42762 .42788 •42815 .42841 .42867 •42894 •90396 •90383 •90371 •90358 .90346 •90334 •44333 •44359 •44385 .44411 •44437 •44464 .89636 .89623 .89610 •89597 .89584 .89571 .45891 •45917 •45942 .45968 •45994 .46020 .88848 •88835 .88822 .88808 :i5i ■47434 .47460 ■47486 ■475" ■47537 ■47562 •47588 .47614 •47639 .47665 .47690 .47716 .88034 .88020 .88006 ■87993 .87979 .87965 .48964 .48989 .49014 .49040 •49065 .49090 •87193 .87178 .87164 .87150 .87136 .87121 25 26 27 28 29 3c 31 32 33 34 % 39 40 41 42 43 44 47 48 49 50 51 52 53 54 .42920 .42946 .42972 .42999 •43025 •43051 .90321 .90309 .90296 .90284 .90271 •90259 ■44490 .44516 ■44542 •44568 •44594 .44620 .89558 ■89545 ■89532 .89519 .89506 .89493 .46046 .46072 .46097 .46123 .46149 •46175 .88768 •88755 .88741 .88728 .88715 .88701 •87951 •87937 li ".87882 .49116 .49141 .49166 .49192 .49217 .49242 .87107 .87093 .87079 .87064 .87050 •87036 •43077 .43104 •43130 •43156 .43182 .43209 .90245 .90233 .90221 .90208 .90196 .90183 .44646 •44672 .44698 .44724 •44750 .44776 .89480 .89467 ■89454 .89441 .89428 .89415 .46201 .46226 .46252 .46278 •46304 •46330 .88688 .88674 .88661 .88647 .88634 .88620 •47741 •47767 .47844 •47869 •47895 .47920 .47946 •47971 •47997 .48022 .87868 •87854 .87840 .87826 .87812 •87798 •87784 .87770 .87756 ■87743 .87729 .87715 .49268 .49293 .49318 •49344 •49369 •49394 .87021 .87007 •86993 .86978 .86964 .86949 •43235 .43261 .43287 •43313 ■43340 •43366 .90171 .90158 .90146 •90133 .90120 .90108 .44802 .44828 .44854 .44880 .44906 .44932 ■44958 .44984 45010 •45036 ■45062 .45088 .89402 .89389 ■89376 •89363 •89350 •89337 .89324 .89311 .89298 .89285 .89272 .89259 •46355 .46381 .46407 •46433 .46458 .46484 .88607 .88566 .88539 .49419 •49445 •49470 •49495 .49521 ■49546 •86935 .86921 .86878 .86863 •43392 .43418 •43445 •43471 •43497 •43523 .90095 .90082 .90070 .90057 .90045 .90032 .46510 •46536 •46561 .46587 .46613 ■46639 .88526 .88512 .88499 .88485 .88472 .88458 .48048 •48073 .48099 .48124 .48150 •48175 .87701 .87687 ■87673 ■87659 •^645 .87631 ■49571 ■49596 .49622 .49647 .49672 .49697 ■49723 ■49748 •49773 .49798 .49824 •49849 .86849 .86834 .8682c .86805 .86791 .86777 15 14 13 12 11 10 7 6 •43549 •43575 .43602 •43628 .90019 .90007 :& .89968 .89956 •89943 •89930 .89918 ■89905 .89892 •89879 ■45114 .45140 .45166 .45192 .45218 ■45243 ■89245 ■89232 .89219 .89206 .89193 .89180 .46664 .46690 .46716 .46742 .46767 ■46793 .88445 .88431 .88417 .88404 .88390 •88377 .48201 .48226 .48252 .48277 ■48303 .48328 .87617 .87603 .87589 •87575 .87561 .87546 .86762 .86748 ■86733 .86719 .86704 .86690 •43706 43733 43759 .43785 .43811 •43837 •45269 •45295 •45321 •45347 •45373 •4539S .89167 ■89153 .89140 .89127 .89114 .89101 .46819 .46844 .4687c .46896 .46921 ■46947 .88363 .88349 ■88336 .88322 .88308 .88295 •48354 •48379 .48405 .4843c ■48456 .48481 •87532 .87518 .87504 .87490 •87476 .87462 .49874 .49899 .49924 .49950 •49975 .5000c .86675 .86661 .86646 .86632 .86617 .86603 5 4 3 2 I N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine / fij^ «;i° 62° 61° 60° ^^__ 78 TABLE III SO'' 1 31° 1 92^ 1 33° 34° o I 2 3 4 I I 9 lO II 12 13 14 i6 N. sine ] ST. COS. N. sine ] N. COS. N. sine N. COS. N. sine NT. COS. N. sine N. COS. 60 ^^ 55 54 53 52 51 50 4I 47 46 45 44 43 42 50000 •5^5025 .50050 .50076 50101 50126 50151 .86603 .86588 8^573 86559 86544 86530 .86515 •51504 •51529 •51554 •5'579 •5'653 •85717 .85702 .85687 .85672 85657 •85642 .85627 .85612 •85597 .85582 .85567 .85551 •85536 52992 •530*7 •53091 53115 •53140 .84805 •84789 .84774 .84759 •84743 .84728 .84712 •54513 •54537 .54561 .54586 .54610 .83867 •83851 •83835 .83819 ■83772 •55919 :^ •55992 .56016 .56040 .56064 .82871 .82855 .82839 .82822 .82806 .50176 .50201 .50227 .50252 .50277 .50302 .86501 .86486 .86471 .86457 .86442 .86427 .51678 •51703 .51728 5'753 .51778 .51803 •53164 53189 .84697 .84681 .84666 .84650 •84635 .84619 ■54635 .54708 54732 ■54756 83756 .83740 83724 .83708 .83692 ■83676 .56088 .56112 itt .56184 .56208 .82790 82773 •82757 .82741 mil 50327 50352 •50377 •50403 .50428 •50453 86413 .86398 •86384 .86369 •86354 .86340 .51828 .51852 •51877 .51902 ,51927 •51952 .85521 .85506 .85491 .85476 .85461 .85446 •53312 •53337 •53361 •53386 •534" •53435 .84604 .84588 •84573 •84557 .84542 .84526 .54781 .54805 •54829 •54854 ■54878 ■54902 .83660 •83645 .83629 •83613 •83597 .83581 tit •56305 •56329 •56353 .82692 .82675 .82659 .82643 .82626 .82610 19 20 21 22 23 24 ^1 27 28 29 30 .50478 •50503 .50528 50553 50578 .50603 .86325 .86310 .86295 .86281 .86266 .86251 51977 .52002 .52026 52051 .52076 52101 •85431 .85416 •85401 ■85385 •85370 •85355 •53460 •53484 •53509 •53534 •53558 •53583. •53607 •53705 •53730 .84511 •84495 .84480 .84464 .84448 •84433 ■54927 •54951 •54975 ■54999 ■55024 ■55048 •83565 •83549 •83533 •83517 •83501 .83485 •56377 .56401 .56425 •56449 •56473 •56497 •82593 .82577 .82561 •82544 .82528 .82511 41 40 '^ 1? 35 34 33 32 31 30 29 28 V, 25 24 23 22 21 20 18 17 16 15 14 13 12 II 10 i I 5 4 3 2 I .50628 .50654 50679 •50704 .50729 •50754 86237 .86222 .86207 .86192 .86178 .86163 .52126 52151 •52175 .52200 .52225 •52250 •85340 •85325 .85310 .85294 .85279 .85264 .84417 .84402 .84386 •84370 •84355 •84339 ■55072 ■55097 .55121 •55145 •55169 •55194 •83469 •83453 •83437 .83421 •83405 •83389 •56521 .56641 •82495 .82478 .82462 .82446 .82429 .82413 3 32 33 34 11 •50779 .50804 . 50829 .50854 .50879 •50904 .86148 •86133 .86119 .86104 .86089 .86074 •52275 •52299 •52324 52349 •52374 •52399 •85249 .85234 .85218 .85203 .85188 •85173 •53754 •53828 •53853 •53877 .84292 •84277 .84261 .84245 •55218 .55266 •55291 55315 •55339 •83373 •83356 •83340 •83324 .83308 .83292 .56665 .56689 •56713 .56760 .56784 .82363 .82347 .82330 •82314 11 39 40 41 42 .50929 ■50954 •50979 .51004 .51029 •51054 .86059 .86045 .86030 86015 .86000 .85985 •52448 •52473 •52498 •52522 •52547 .85157 .85142 .85127 .85112 .85096 .85081 •53902 •53926 •53951 •53975 .54000 •54024 .84230 .84214 .84198 .84182 .84167 .84151 •55412 •55436 •55460 •55484 •83276 .83260 •83244 .83228 .83212 •83195 .56808 •56832 ■IS .82297 .82281 .82264 .82248 .82231 .82214 43 44 tl 47 48 49 50 51 52 53 54 55 56 •51079 .51104 •51129 •51154 •51179 .51204 .85970 .85956 ■85941 .85926 .85911 .85896 •52572 •52597 52621 .52646 .52671 .52696 .85066 .85051 •85035 .85020 •85005 .84989 •54049 •54073 •54097 •54122 .54146 •54171 •84135 .84120 .84104 .84088 .84072 •84057 •55509 •55533 •55557 .55581 •55605 •55630 •83179 •83163 •83147 •83J31 ; 83098 .57000 .57024 •57047 •57071 .82198 .82181 .82165 .82148 .82132 •82115 .51229 •51254 •51279 •51304 •51329 •51354 •51379 .51404 .51429 •5H54 •5H79 •51504 .85881 .85866 .85851 •85836 .85821 .85806 .85792 •85777 .85762 •85747 •85732 .85717 .52720 •52745 .52770 •52794 .52819 .52844 •84974 .84959 •84943 .84928 .84913 .84897 •54195 .54220 •54244 .54269 •54293 •54317 .84041 .84025 .84009 •83994 •83978 .83962 •55654 •55678 •55702 •55726 •55750 •55775 .83082 .83066 •83050 •83034 •83017 .83001 •57095 •57119 •57143 •57167 •57I9I •57215 .82098 .82082 .82065 .82048 .82032 .82015 .52869 •52893 .52918 •52943 .52967 .52992 .84882 .84866 .84851 .84836 .84820 .84805 •54366 ■54391 •54415 •54440 ■54464 •83946 •83930 i%l .83883 .83867 •55799 .55823 •55847 .55871 •55895 •55919 .82985 .82969 •82953 .82936 .82920 .82904 •57238 .57262 .57286 •57310 •57334 •57358 .81999 .81982 .81965 .81949 .81932 .81915 N COS. N. sine N. COS. Kfrgirie N. COS. N. sine N. COS. M. sine N. COS. N. sine p 59" 1 68** 1 .«» 1 56° 55' TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE 79 ss*' 1 36° 37° 1 38° 1 39° 9 N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. cos. O I 2 3 4 i 7 8 9 lO 12 13 14 15 16 57358 •57381 ■57405 57429 •57453 •57477 ■57501 •81882 .81865 •81848 .81832 •81815 •58779 •58802 .58826 .58849 .■|896 .58920 .80867 .80850 .80833 .80816 .80799 60182 . 60205 .60228 .60251 •60274 .60298 .60321 •79864 •79846 .79829 •79811 •79793 •79776 •79758 .61566 .61612 61635 .61658 .61681 .61704 .78801 •78783 .78765 •78747 •78729 •78711 •78694 .62932 •62955 .62977 •63000 •63022 •63045 •63068 •77715 . 77606 60 59 .77678 .77660 .77641 •77623 •77605 58 55 54 53 52 51 50 49 48 57524 •57548 •57572 •57596 •57619 •57643 •81798 •81782 •81765 •81748 •81731 •81714 ■58943 .58967 ■58990 .59014 ■59037 .59061 .80782 .80765 .80748 •80730 •80713 .80696 .60344 .60367 .60390 .60414 .60437 .60460 •79741 •79723 •79706 •79688 •79671 •79653 .61726 •61749 .61772 •61795 .61818 .61841 •78676 •78658 • 78640 .78622 .78604 •78586 •63090 •63113 •63135 •63158 .63180 .63203 •77586 •77568 •77550 •77531 •77513 • 77494 •57667 •57691 •57715 •57762 .57786 •81698 .81681 •81664 •81647 •81631 •81614 .59084 .59108 ■59131 •59154 •59178 •59201 .80679 •80662 .80644 80627 .80610 •80593 •60529 •60576 •60599 •79635 .79618 •79600 •79583 •79565 •79547 .61864 .61887 •61909 •61932 •61955 .61978 • 78568 •78550 ■78532 •78514 •78496 •78478 .63225 •63248 .63271 IP •63338 •77476 •77458 •77439 •77421 .77402 •77384 45 44 43 42 41 40 % 35 34 33 32 31 30 19 20 21 22 23 24 ^i 27 28 29 30 31 32 33 34 35 36 39 40 41 42 43 44 t 47 48 •57810 •57833 •57904 •57928 •81597 •81580 •81563 •81546 .81530 •81513 •59225 .59248 •59272 •59295 •59318 59342 .80576 .80558 .8054J .80524 •80507 •80489 .60622 .60645 •60668 •60691 •60714 •60738. •79530 •79512 •79494 •79477 •79459 •79441 .62001 .62024 .62046 •62069 •62092 •62115 •78460 •78442 .78424 .78405 •78387 .78369 .63361 •63383 .63406 .63428 •63451 •63473 .77366 •77347 .77329 .77.310 .77292 •77273 •57952 •57976 •57999 •58023 •58047 •58070 •81496 81479 .81462 • 81445 •81428 .81412 •593^5 59389 .59412 •59436 .80472 •80455 •80438 .80420 .80403 •80386 •60761 •60784 .60807 .60830 •60853 •60876 •79424 .79406 •79388 •79371 •79353 •79335 •79318 .79300 . 79282 .79264 .79247 .79229 .79211 •79193 .79176 •79158 .79140 .79122 •62138 •62160 .62183 •62206 .62229 .62251 •78351 •78333 •78315 .78297 .78279 .78261 •63496 •63518 •63540 •63563 •77255 •77236 .77218 •77199 .77181 .77162 .58141 .58165 .58189 .58212 •81395 .81378 .81361 .81344 .81327 .81310 •59506 •59529 •59552 •59576 •59599 •59622 •80368 •80351 .80264 .80247 •80230 .80212 •80195 .80178 •60899 .60922 •60945 .60968 .60991 .61015 •61038 •61061 .61084 .61107 .61130 •61153 .62274 •62297 •62320 .62342 ^62388 .78243 .78225 .78206 .78188 .78170 .78152 .63630 •63653 ^63698 .63720 •63742 •77144 •77125 .77107 .77088 .77070 •77051 29 25 24 23 22 21 20 19 .58236 .58260 •58283 •58307 •58330 •58354 .81293 .81276 .81259 .81242 •81225 •81208 •59646 •59669 •59693 •59716 •59739 •59763 .62411 •62433 •62456 •62479 •62502 •62524 •78134 .78116 .78098 •78079 .78061 •78043 •63765 •63787 .63810 •63832 •63854 •63877 •77033 .77014 .76996 .76977 •76959 .76940 •58378 .58401 •58425 .58449 .58472 .58496 •81191 .81174 .81157 .81140 .81123 .81106 •59786 •59809 59832 •59856 •59879 •59902 •80160 •80143 •80125 •80108 •80091 ■80073 .61176 .61199 .61222 •61245 61268 .61291 •79105 .79087 .79069 •79051 •79033 •79016 •62547 •62570 •62592 •62615 •62638 •62660 .78025 .78007 •77988 •77970 •77952 •77934 •63899 .63922 .63944 .63966 .63989 .64011 .76921 .76903 .76884 . 76866 .76847 •76828 15 14 13 12 11 10 \ 5 4 3 2 I 49 50 51 52 53 54 •58519 •58637 .81089 .81072 •81055 .81038 .81021 .81004 •59926 •59949 •59972 •59995 •60019 .60042 .80056 .80038 .80021 •80003 .79986 •79968 •79951 •79934 .79916 •79899 •79881 •79864 .61314 .61383 .61406 .61429 •61451 .61474 •61497 •61520 •61543 .61566 •78998 •78980 .78962 .78944 .78926 .78908 •62683 .62706 .62728 •62751 •62774 •62796 .77916 •77897 •77879 .77861 •77843 .77824 •64033 •64056 •64078 .64100 •64123 •64145 •76810 •76791 •76772 •76754 •76735 .76717 .76698 .76679 •76661 •76642 •76623 .76604 •58661 .58684 .58708 58731 •58755 58779 .80987 .80970 •80953 •80936 •80919 •80902 •60065 •60089 .60112 •60135 •60158 •60182 .78891 •78873 •78855 •78837 .78819 .78801 •62819 .62842 •62864 •62887 •62909 •62932 .77806 .77788 .77769 •77751 ■nnz •77715 •64167 .64190 .64212 .64234 .64256 .64279 N. COS. N. sine N. COS. ] V. sine N. COS. N^. sine N. COS. ] V. sme N^. COS. ] V. sine 54" 1 53° 1 52° 1 51° 1 50° 1 80 TABLE III o I 2 3 4 1 9 lO II 12 33 \l 18 40- 1 41** 1 42° 1 «» 1 44° 60 It % 55 54 53 52 51 50 It N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. .64279 .64301 64323 64346 .64368 .64390 .64412 .76604 .76586 .76567 .76548 •76530 .76511 .76492 .65606 .65628 •65650 .65672 •65694 .65716 •65738 •75471 •75452 •75433 •75414 •75395 •75375 •75356 66913 •66935 .66956 .66978 .66999 .67021 •67043 .67064 .67086 .67107 .67129 .67151 .67172 •74314 •74295 •74276 •74256 •74237 •74217 .74198 .74178 •74159 •74139 .74120 .74100 .74080 .68200 .68221 .68242 .68264 .68285 .68306 •68327 •73135 .73116 .73096 ■73076 •73056 •73036 .73016 .69466 •69487 .69508 •69529 •69549 .69570 .69591 •71934 .71914 .71894 ■71873 •71853 •71833 .71813 •64435 •64457 •64479 .64501 .64524 .64546 •76473 ■76455 .76436 .76417 .76398 .76380 •65759 •65803 .65825 .65847 .65869 ■7S337 •75318 •75299 .75280 •75261 •75241 •68349 .68370 •68391 .68412 •68434 •68455 .68476" •68497 .68518 •68539 .68561 .68582 .72996 .72976 •72957 .72937 .72917 .72897 .69612 .69633 .69654 •69675 .69696 .69717 .71792 .71772 •71752 .71732 .71711 .71691 .64568 .64590 .64612 ■64635 .64657 .64679 •76361 •76342 •76323 .76304 .76286 .76267 .65891 •65913 •65956 .65978 .66000 .75222 •75203 •75184 •75165 •75146 .75126 .67194 .67215 •67237 .67258 .67280 .67301 .74061 .74041 .74022 .74002 •73983 •73963 .72877 .72857 •72837 .72817 •72797 ■72777 •69737 •69758 .69779 .69800 .69821 .69842 .71671 .71650 .71630 .71610 •71590 •71569 % 45 44 43 42 19 20 21 22 23 24 11 11 29 3c 31 32 33 34 35 36 11 39 40 41 42 43 44 :i 47 48 49 50 51 52 53 54 55 56 11 .64701 .64723 .64746 .64768 .64790 .64812 .76248 .76229 .76210 .76192 .76173 •76154 •76135 .76116 .76097 .76078 •76059 .76041 .76022 .76003 •75984 •75965 •75946 •75927 .75908 .75889 •75870 .75851 •75832 •75813 .66022 .66044 .66066 .66088 .66109 .66131 .75069 •75050 •75030 .75011 •67323 67344 •67366 .67409 •67430 •73944 •73924 •73904 ■73885 •73865 .73846 .68603 .68624 .68645 .68666 .68688 .68709 ■72757 ■72737 •72717 .72697 .72677 .72657 .69862 •69883 .69904 .69946 .69966 •71549 •71529 .71508 .71488 .71468 •71447 41 40 35 34 33 32 31 30 29 28 % 25 24 23 22 21 20 19 \l 15 14 13 12 II 10 i 7 6 .64834 64856 64878 .64901 64923 64945 64967 .64989 .65011 •65033 ■65055 ■65077 .65100 .65122 .65144 .65166 .65188 .65210 66175 .66197 .66218 .66240 .66262 •74992 •74973 •74953 •74934 •74896 .67452 ■67473 ■67538 •67559 .73826 •73806 •73787 •73767 •73747 •73728 •68730 .68751 .68772 68793 .68814 ■68835, .72637 .72617 •72597 •72577 •72557 •72537 •69987 .70008 .70029 .70049 .70070 .70091 .71427 .71407 .71386 .71366 ■71345 ■71325 .66284 .66306 •66327 •66349 •66393 .66414 .66436 .66458 .66480 .66501 •66523 .74876 •74857 ■.Ifsfs .67580 .67602 .67623 .67688 .73708 •73688 73669 •73649 .73629 .73610 .68857 .68878 .68899 .68920 .68941 .68962 •72517 .72497 •72477 •72457 •72437 .72417 .70112 .70132 •70153 .70174 •70195 •70215 •71305 .71284 .71264 •71243 .71223 .71203 .74760 •74741 .74722 •74703 ■74683 .74664 .67709 .67730 .67752 •67773 •67795 .67816 •73590 •73570 •73551 •73531 •735" •73491 .68983 .69004 .69025 .69046 .69067 .69088 •72397 ■72377 ■72357 ■72337 .72317 .72297 .70236 •70257 .70277 .70298 .70319 •70339 .71182 .71162 .71141 .71121 .71100 .71080 .65232 •65254 •65276 .65298 •65320 •65342 •75794 •75775 •75756 •75738 •75719 ■75700 "66566 66588 .66610 .66632 •66653 .74644 .74606 •74586 •74567 •74548 •67837 .67901 •67923 .67944 •73472 •73452 •73432 •73413 •73393 ■73373 .69109 .69130 .69151 .69172 .69193 .69214 .72277 •72257 .72236 .72216 .72196 .72176 .70360 .70381 .70401 .70422 •70443 •70463 •71059 .71039 .71019 .70998 •70978 •70957 •65364 .65386 .65408 •65430 •65452 •65474 .75680 .75661 •75642 •75623 .75604 •75585 .66675 .66697 .66718 .66740 .66762 •66783 .74528 •74509 •74489 •74470 •74451 •74431 .67965 .68029 .68051 .68072 •73353 •73333 •73314 •73294 •73274 •73254 •69235 .69256 .69277 .69298 .69319 .69340 .72156 •72136 .72116 .72095 •72075 •72055 .70484 •70505 •70525 .70546 •70567 •70587 .70937 .70916 .70896 .70875 •70855 .70834 •65496 .65518 •65540 65562 .65584 .65606 •75566 •75547 •75528 •75509 •75490 •75471 .66805 .66827 .66848 .66870 .66891 •66913 .74412 •74392 •74373 •74353 •74334 •74314 .68093 .68115 .68136 .68157 .68179 .68200 •73234 •73215 •73195 •73175 •73155 •73135 •69361 .69382 .69403 .69424 •72035 •72015 •71995 •71974 •71954 71934 .70608 .70628 .70649 .70670 .70690 .70711 .70813 •70793 .70772 •70752 •70731 .70711 5 4 3 2 I t N. COS. ] NT. sine f^. COS. ] N". sine NT. COS. ] N". sine ^J. CO.S. ^ ST. sine N'. cos. ] V. sine 49° 1 4§*' 1 47° 1 46° 1 45- 1 NATURAL TANGENTS AND COTANGENTS 81 / 0^^ 1° 2^" 3° 40 / Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg I 2 3 4 0000 Infinite 0003 3437-75 0006 1718.87 0009 1145.92 0012 859.436 0175 57.2900 0177 56.3506 0180 55.4415 0183 54.5613 0186 53.7086 0349 28.6363 0352 28.3994 0355 28.1664 0358 27.9372 0361 27.7117 0524 19.0811 0527 18.9755 0530 18.8711 0533 18.7678 0536 18.6656 0699 14.3007 0702 14.2411 070^ 14.1821 0708 14.1235 0711 14.0655 60 5 6 7 8 9 0015 687.549 0017 572957 0020 491.106 0023 429.718 0026 381.971 0189 52.8821 0192 52.0807 0195 51.3032 0198 50.5485 0201 49.8157 0364 27.4899 0367 27.2715 0370 27.0566 0373 26.8450 0375 26.6367 0539 18.5645 0542 18.4645 0544 18.3655 0547 18.2677 0550 18.1708 0714 14.0079 0717 13-9507 0720 13.8940 0723 138378 0726 13.7821 55 54 53 52 51 10 II 12 13 14 0029 343-774 0032 312.521 0035 286.478 0038 264.441 0041 245.552 0204 49.1039 0207 48.4121 0209 47.7395 0212 47.0853 0215 46.4489 0378 26.4316 0381 26.2296 0384 26.0307 0387 25.8348 0390 25.6418 0553 18.0750 0556 17.9802 OS59 17.8863 0562 17.7934 ^565 17-7015 0729 13.7267 0731 13.6719 0734 13-6174 0737 13-5634 0740 13.5098 50 49 48 47 46 17 i8 19 0044 229.152 0047 214.858 0049 202,219 0052 190.984 0055 180.932 0218 45.8294 0221 45.2261 0224 44.6386 0227 44.0661 0230 43.5081 0393 25.4517 0396 25.2644 0399 25.0798 0402 24.8978 0405 24.7185 0568 17.6106 0571 17.5205 0574 1 7.43 H 0577 17.3432 0580 17.2558 0743 13.4566 0746 13.4039 0749 13.3515 0752 13.2996 0755 13.2480 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 0058 171.885 0061 163.700 0064 156.259 0067 149.465 0070 143.237 0233 42.9641 0236 42.4335 0239 41-9158 0241 41.4106 0244 40.9174 0407 24.5418 0410 23.3675 0413 24.1957 0416 24.0263 0419 23.8593 0582 17.1693 0585 17.0837 0588 16.9990 0591 16.9150 0594 16.8319 0758 13.1969 0761 13.1461 0764 13.0958 0767 13.0458 0769 12.9962 40 39 38 31 36 0073 137-507 0076 132.219 0079 127.321 0081 122.774 0084 118.540 0247 40.4358 0250 39.9655 0253 39.5059 0256 39.0568 0259 38.6177 0422 23.6945 0425 23.5321 0428 23.3718 0431 23.2137 0434 23.0577 0597 16.7496 0600 16.6681 0603 16.5874 0606 16.5075 0609 16.4283 0772 12.9469 0775 12.8981 0778 12.8496 0781 12.8014 0784 12.7536 35 34 32, 32 31 30 31 32 34 0087 114-589 0090 110.892 0093 107.426 0096 104.171 0099 101.107 0262 38.1885 0265 37.7686 0268 37-3579 0271 36.9560 0274 36.5627 0437 22.9038 0440 22.7519 0442 22.6020 0445 22.4541 0448 22.3081 0612 16.3499 0615 16.2722 0617 16.1952 0620 16.1190 0623 16.0435 0787 12.7062 0790 12.6591 0793 12.6124 0796 12.5660 0799 12.5199 30 29 28 27 26 35 36 39 0102 98.2179 0105 95.4895 0108 92.9085 oiii 904633 01 13 88.1436 0276 36.1776 0279 35.8006 0282 35-4313 0285 35-0695 0288 34.7151 0451 22.1640 0454 22.0217 0457 21.8813 0460 21.7426 0463 21.6056 0626 15.9687 0629 15.8945 0632 15.8211 0635 15.7483 0638 15.6762 0802 12.4742 0805 12.4288 0808 12.3838 0810 12.3390 0813 12.2946 25 24 23 22 21 40 41 42 43 44 01 16 85.9398 01 19 83.8435 0122 81.8470 0125 79.9434 0128 78.1263 0291 34.3678 0294 34.0273 0297 336935 0300 33.3662 0303 33-0452 0466 21.4704 0469 21.3369 0472 21.2049 0475 21.0747 0477 20.9460 0641 15.6048 0644 15-5340 0647 15.4638 0650 15.3943 0653 15.3254 0816 12.2505 0819 12.2067 0822 12.1632 0825 12.1201 0828 12.0772 20 19 18 17 i6 45 46 47 48 49 0131 76.3900 0134 74.7292 0137 73.1390 0140 71.6151 0143 70.1533 0306 32.7303 0308 32.4213 0311 32.1181 0314 31.8205 0317 31.5284 0480 20.8188 0483 20.6932 0486 20.5691 0489 20.4465 0492 20.3253 0655 15.2571 0658 15.1893 0661 15.1222 0664 15.0557 0667 14.9898 0831 12.0346 0834 11.9923 0837 11.9504 0840 11.9087 0843 11.8673 15 14 13 12 11 50 51 52 53 54 59 0146 68.7501 0148 67.4019 0151 66.1055 0154 64.8580 0157 63.6567 0320 31.2416 0323 30.9599 0326 30.6833 0329 30.4116 0332 30.1446 0495 20.2056 0498 20.0872 0501 19.9702 0504 19.8546 0507 19.7403 0670 14.9244 0673 14-8596 0676 14.7954 0679 14-7317 0682 14.6685 0846 11.8262 0849 11.7853 0851 11.7448 0854 11.7045 0857 11.6645 10 I 7 6 5 4 3 2 I 0160 62.4992 0163 61,3829 0166 60.3058 0169 59.2659 0172 58.2612 0335 29.8823 0338 29.6245 0340 29.3711 0343 29.1220 0346 28.8771 0509 19.6273 OCil2 19.5156 0515 19.4051 0518 19.2959 0521 19.1879 0685 14.6059 0688 14.5438 0690 14.4823 0693 14.4212 0696 14.3607 0860 11.6248 0863 11.5853 0866 11.5461 0869 11.5072 0872 11.4685 60 0175 57.2900 0349 28.6363 0524 19.0811 0699 14.3007 0875 11.4301 (.otg Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang / 8^0 8H0 H7^ i H(i^ 85^ / 82 TABLE III / 50 0° 7" 8'-^ 90 / Tang Cotg Tang totg Tang Cotg Tang Cotg Tang Cotg 60 0875 1 1. 4301 1051 9.5144 1228 8.1443 1405 7.1154 1584 6.3138 I 0878 11.3919 1054 9-4878 1231 8.1248 1408 7.1004 1587 6.3019 59 2 0881 11.3540 1057 9.4614 1234 8.1054 141 1 7.0855 1590 6.2901 58 ^ 0884 1 1. 3 1 63 1060 94352 1237 8.0860 1414 7.0706 1593 6.2783 57 4 0887 11.2789 1063 9.4090 1240 8.0667 1417 7-0558 1596 6.2666 56 s 0890 1 1.24 1 7 1066 9-3831 1243 8.0476 1420 7.0410 1599 6.2549 55 6 0892 11.2048 1069 9-3572 1246 8.0285 1423 7.0264 1602 6.2432 54 7 0895 1 1. 1 681 1072 9-3315 1249 8.0095 1426 7.0117 1605 6.2316 53 8 0898 11.1316 1075 9.3060 1 25 1 7.9906 1429 6.9972 1608 6.2200 52 9 0901 11.0954 1078 9.2806 1254 7.9718 1432 6.9827 1611 6.2085 51 10 0904 11.0594 1080 9-2553 1257 7.9530 1435 6.9682 1614 6.1970 50 II 0907 11.0237 1083 9.2302 1260 7.9344 1438 6.9538 1617 6.1856 49 12 9910 10.9882 1086 9.2052 1263 7.9158 1441 6.9395 1620 6.1742 48 1,3 0913 10.9529 1089 9.1803 1266 7.8973 1444 6.9252 1623 6.1628 47 14 0916 10.9178 1092 9-1555 1269 7.8789 1447 6.91 10 1626 6.1515 46 IS 0919 10.8829 1095 9.1309 1272 7.8606 1450 6.8969 1629 6.1402 45 i6 0922 10.8483 1098 9.1065 1275 7.8424 1453 6.8828 1632 6.1290 44 17 0925 10.8139 IIOI 9.0821 1278 7.8243 1456 6.8687 1635 6.1178 43 18 0928 10.7797 1104 9-0579 1281 7.8062 1459 6.8548 1638 6.1066 42 19 0931 10.7457 1 107 9-0338 1284 7.7883 1462 6.8408 1641 6-0955 41 20 0934 10.71 19 mo 9.0098 1287 7.7704 1465 6.8269 1644 6.0844 40 21 0936 10.6783 1113 8.9860 1290 7.7525 1468 6.8131 1647 6.0734 39 22 0939 10.6450 1116 8.9623 1293 7.7348 1471 6.7994 1650 6.0624 38 23 0942 10.61 18 1119 8.9387 1296 7.7171 1474 6.7856 1653 6.0514 37 24 0945 10.5789 1122 8.9152 1299 7.6996 1477 6.7720 1655 6.0405 36 2S 0948 10.5462 1125 8.8919 1302 7.6821 1480 6.7584 1658 6.0296 35 26 0951 10.5136 1128 8.8686 1305 7.6647 1483 6.7448 1661 6.0188 34 27 0954 10.4813 1131 8.8455 1308 7.6473 i486 6.7313 1664 6.0080 33 28 0957 10.4491 1 134 8.8225 1 31 1 7.6301 1489 6.7179 1667 5-9972 32 29 0960 10.4172 1 136 8.7996 1314 7.6129 1492 6.7045 1670 5.9865 31 30 0963 10.3854 1 139 8.7769 1317 7-5958 1495 6.0912 Tell 5-9758 30 31 0966 10.3538 1 142 8.7542 1319 7.5787 1497 6.6779 5-9651 29 32 0969 10.3224 1 145 8.7317 1322 7.5618 1500 6.6646 1 679 5.9545 28 33 0972 10,2913 1148 8.7093 1325 7.5449 1503 6.6514 1682 5.9439 27 34 0975 10.2602 1151 8.6870 1328 7.5281 1506 6.6383 1685 1688 5-9333 5.9228 26 25 3S 0978 10.2294 1154 8.6648 1331 7.5113 1509 6.6252 36 0981 10.1988 1157 8.6427 1334 7.4947 1512 6.6122 1691 5.9124 24 37 0983 10.1683 1 160 8.6208 1337 7.4781 1515 6.5992 1694 5-9019 23 38 0986 10.1381 1163 8.5989 1340 7.4615 15 18 6.5863 1697 5.8915 22 39 0989 10.1080 1166 8-5772 1343 7.4451 1521 6.5734 1700 5.8811 21 40 0992 10.0780 1 169 8.5555 1346 7.4287 1524 6.5606 1703 5.8708 20 41 0995 10.0483 1172 8.5340 1349 7.4124 1527 6.5478 1706 5.8605 19 42 0998 10.0187 1 175 8.5126 1352 7.3962 1530 6.5350 1709 5.8502 18 43 looi 9.9893 1178 8.4913 1355 7.3800 1533 6.5223 1712 5.8400 17 44 1004 9.9601 8.4701 1358 7.3639 1536 6.5097 1715 6.8298 16 15 45 1007 9.9310 1 184 8.4490 1361 7.3479 1539 6.4971 1718 58197 46 loio 9.9021 1187 8.4280 1364 7.3319 1542 6.4846 1721 5-8095 14 47 IOI3 9-8734 1 189 8.4071 1367 7.3160 1545 6.4721 1724 5-7994 13 48 1016 9.8448 1192 8.3863 1370 7.3002 1548 6.4596 1727 5-7894 12 49 IOI9 9.8164 "95 8.3656 1373 7.2844 1551 6.4472 1730 5-7794 II 50 1022 9.7882 1 198 8.3450 1376 7.2687 1554 6.4348 1733 5-7694 10 SI 1025 9.7601 1 201 8.3245 1379 7.2531 1557 6.4225 1736 5-7594 9 S2 1028 9-7322 1204 8.3041 1382 7.2375 1560 6.4103 1739 5-7495 8 S3 1030 9.7044 1207 8.2838 1385 7.2220 1563 6.3980 1742 5.7396 7 54 1033 9-6768 1210 8.2636 1388 7.2066 1566 6.3859 1745 5.7297 6 5S 1036 9-6499 1213 8.2434 1391 7.1912 1569 6.3737 1748 5.7199 5 5^ 1039 9.6220 1216 8.2234 1394 7.1759 1572 6.3617 1751 5.7101 4 S7 1042 9-5949 1219 8.2035 1397 7.1607 1575 6.3496 1754 5.7004 3 5« 1045 9.5679 1222 8.1837 1399 7.1455 1578 6.3376 1757 5.6906 2 59 1048 9-541 1 1225 8.1640 1402 7.1304 1581 6.3257 1760 5.6809 I 00 1051 9-5144 1228 8.1443 1405 7.1 154 1584 6.3138 Cotg Tang 1763 Cotg 5-6713 Tang Cotg Tang Cotg Tang Cotg Tang L. 84° 83° 82^ 81" 80" / NATURAL TANGENTS AND COTANGENTS 83 / 10^ 1 11^ 12° 13° 14° / Taug Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg 60 ^I^Z 5-6713 1944 5.1446 2126 4.7046 2309 4-3315 2493 4.0108 I 1766 5.6617 1947 5.1366 2129 4-6979 2312 4-3257 2496 4.0058 5? 2 1769 5-6521 1950 5.1286 2132 4.6912 2315 4.3200 2499 4.0009 58 ^ 1772 5-6425 1953 5.1207 2135 4.6845 2318 4-3143 2503 3-9959 57 4 1775 5-6330 1956 5.1128 2138 4.6779 2321 4.3086 2506 3.9910 5^ S 1778 5-6234 1959 5-I049 2141 4.6712 2324 4-3029 2509 3-9861 55 6 I78I 5.6140 1962 5-0970 2144 4.6646 2327 4-2972 2512 3-9812 54 7 1784 5.6045 1965 5.0892 2147 4.6580 2330 4.2916 2515 3-9763 53 8 1787 5-5951 1968 5.0814 2150 4.6514 233.3 4-2859 2518 3-9714 52 9 10 1790 5-5857 1971 5-0736 2153 4.6448 2336 4.2803 2521 3.9665 51 1793 5-5764 1974 5.0658 2156 4.6382 2339 4.2747 2524 3-9617 50 II 1796 5-5671 1977 5-0581 2159 4-6317 2342 4.2691 2527 3-9568 49 12 1799 5-5578 1980 5-0504 2162 4.6252 2345 4-2635 2530 3.9520 48 i,S 1802 5-5485 1983 5.0427 2165 4.6187 2349 4.2580 2533 3-9471 47 14 1805 5-5393 1986 5-0350 2168 4.6122 2352 4.2524 2537 3-9423 46 IS 1808 5-5301 1989 5-0273 2171 4-6057 2355 4.2468 2540 3-9375 45 i6 I8II 5-5209 1992 50197 2174 4.5993 2358 4.2413 2543 3-9327 44 17 I8I4 5.5118 1995 5.0121 2177 4-5928 2361 4.2358 2546 3-9279 43 18 I8I7 5.5026 1998 5-0045 2180 4-5864 2364 4.2303 2549 3-9232 42 19 1820 5-4936 2001 4.9969 2183 4.5800 2367 4.2248 2552 3.9184 41 20 1823 5-4845 2004 4.9894 2186 4-5736 2370 4.2193 2555 3-9136 40 21 1826 5-4755 2007 4.9819 2189 4.5673 2373 4.2139 2558 3.9089 39 22 1829 5-4665 2010 4.9744 2193 4.5609 2376 4.2084 2561 3.9042 38 23 1832 5-4575 2013 4.9669 2196 4-5546 2379 4.2030 2564 3-8995 31 24 i«35 5-4486 2016 4-9594 2199 4.5483 2382 4.1976 2568 3-8947 36 2S 1838 5-4397 2019 4.9520 2202 4.5420 2385 4.1922 2571 3.8900 35 26 1841 5-4308 2022 4-9446 2205 4.5357 2388 4.1868 2574 3-8854 34 27 1844 5-4219 2025 4-9372 2208 4.5294 2392 4.1814 2577 3-8807 33 28 1847 5-4131 2028 4.9298 221 1 4.5232 2395 4.1760 2580 3-8760 32 29 1850 5-4043 2031 4.9225 2214 4.5169 2398 4.1706 2583 3-8714 31 30 i8S3 5-3955, 2035 4.9152 2217 4-5107 2401 4.1653 2586 38667 30 31 1856 5.3868 2038 4.9078 2220 4.5045 2404 4.1600 2589 3.8621 29 32 i«59 5-3781 2041 4.9006 2223 4.4983 2407 4-1547 2592 3-8575 28 33 1862 5-3694 2044 4-8933 2226 4.4922 2410 4.1493 2.595 3-8528 27 34 1865 5-3607 2047 4.8860 2229 4.4860 2413 4.1441 2599 3.8482 26 35 1868 5-3521 2050 4.8788 2232 4.4799 2416 4.1388 2602 3-8436 25 3^ 1871 5-3435 2053 4.8716 2235 4.4737 2419 4.1335 2605 3-8391 24 37 1874 5-3349 2056 4.8644 2238 4.4676 2422 4.1282 2608 3-8345 23 38 1877 5-3263 2059 4.8573 2241 4-4615 2425 4.1230 261 1 3-8299 22 39 1880 5-3178 2062 4-8501 2244 4-4555 2428 4.1178 2614 3-8254 21 40 1883 5-3093 2065 4-8430 2247 4.4494 2432 4.1126 2617 3.8208 20 41 1887 5-3008 2068 4-8359 2251 4.4434 2435 4.1074 2620 3-8163 19 42 1890 5-2924 2071 4.8288 2254 4-4374 2438 4.1022 2623 3.81 18 18 43 1893 5-2839 2074 4.8218 2257 4.4313 2441 4.0970 2627 3-8073 17 44 45 1896 1899 5-2755 2077 4.8147 2260 4-4253 2444 4.0918 2630 3.8028 16 5.2672 2080 4-8077 2263 4.4194 2447 4.0867 2633 3-7983 15 4b 1902 5.2588 2083 4.8007 2266 4-4134 2450 4-0815 2636 3-7938 14 47 1905 5-2505 2086 4-7937 2269 4.4075 2453 4.0764 2639 3-7893 13 48 1908 5-2422 2089 4.7867 2272 4.4015 2456 4.0713 2642 3-7848 12 49 1911 5-2339 2092 4.7798 2275 4.3956 2459 4.0662 2645 3.7804 II 50 1914 5-2257 2095 4-7729 2278 4.3897 2462 4.061 1 2648 3.7760 10 51 1917 5-2174 2098 4-7659 2281 4.3838 2465 4.0560 2651 3-7715 9 52 1920 5.2092 2101 4-7591 2284 4.3779 2469 4.0509 2655 3-7671 8 53 1923 5.2C1 1 2104 4.7522 2287 4-3721 2472 4.0459 2658 3.7627 7 54 55 1926 5.1929 2107 4-7453 2290 4.3662 2475 4.0408 2661 3-7583 b 1929 5.1848 2110 4.7385 2293 4.3604 2478 4.0358 2664 3-7539 '5 5<^ 1932 ^•^lll 2113 4-7317 2296 4-3546 2481 4.0308 2667 3-7495 4 57 1935 5.1686 2116 4.7249 2299 4-3488 2484 4.0257 2670 3-7451 3 5« i93« 5.1606 2119 4.7181 2303 4.3430 2487 4.0207 2673 3.7408 2 59 1 941 5-1526 2123 4.7114 2306 4.3372 2490 4.0158 2676 3.7364 I 00 1944 5.1446 2126 4.7046 2309 4.3315 2493 4.0108 2679 3-7321 Cotg Tan^ Cotg Tang Cotg Tang Cotg Tang Cotg Tang / 7 0° 78° 77 ' 7 6^ 7 5° f 84 TABLE III / 15° 1(>° 17° 18^ 193 / ian^ Cot^ Tang Cotg Tang Cotg Tang Cotg Tang Cotg 2679 3.7321 2867 3-4874 3057 3-2709 3249 3.0777 3443 2.9042 60 I 2683 3-7277 2871 34836 3060 3-2675 3252 3.0746 3447 2.9015 S9 2 2686 3-7234 2874 3-4798 3064 3.2641 3256 3.0716 3450 2.8987 S8 3 2689 3-7191 2877 3.4760 3067 3.2607 3259 3.0686 3453 2.8960 S7 4 2692 3-7148 2880 3.4722 3070 3.2573 3262 3.0655 3456 2-8933 '^6 5 2695 3-7105 2883 3-4684 .3073 32539 3265 3.0625 3460 2.8905 SS 6 2698 3.7062 2886 3-4646 3076 3.2506 3269 3.0595 3463 2.8878 S4 7 2701 3-7019 2890 3.4608 3080 3-2472 3272 3.0565 3466 2.8851 S3 8 2704 3.6976 2893 34570 .3083 3.2438 3275 3.0535 3469 2.8824 S2 9 2708 3-6933 2896 3-4533 3086 3-2405 3278 3-0505 3473 2.8797 51 50 10 2711 3.6891 2899 3-4495 3089 3-2371 3281 3.0475 .3476 2.8770 II 2714 3.6848 2902 3-4458 3092 3-2338 3285 3.0445 3479 2.8743 49 12 2717 3.6806 2905 3.4420 3096 3.2305 3288 3.0415 .3482 2.8716 48 1.3 2720 3.6764 2908 3-4383 3099 3.2272 3291 30385 .3486 2.8689 47 14 2723 3.6722 2912 3-4346 3102 3-2238 3294 3-0356 3489 2.8662 46 15 2726 3.6680 2915 3-4308 3105 3.2205 3298 3.0326 3492 2.8636 4S lb 2729 3.6638 2918 3-4271 3108 3.2172 3301 3.0296 3495 2.8609 44 17 2733 3.6596 2921 3-4234 3111 3.2139 3304 3.0267 3499 2.8582 43 18 2736 3-6554 2924 3-4197 3"5 3.2106 3307 3.0237 3502 2.8556 42 19 2739 3.6512 2927 3.4160 3118 3.2073 3310 3.0208 3505 2.8529 41 20 2742 3.6470 2931 3.4124 3121 3.2041 3314 3.0178 3So8 2.8502 40 21 2745 3.6429 2934 3.4087 3124 3.2008 3317 3.0149 3512 2.8476 39 22 2748 3.6387 2937 3.4050 3127 3.1975 3320 3.0120 3515 2.8449 38 23 2751 3-6346 2940 3-4014 3131 3.1943 3323 3.0090 3518 2.8423 37 24 2754 3-6305 2943 3-3977 3134 3.1910 3327 3.0061 3522 2.8397 36 25 2758 3.6264 2946 3-3941 3137 3.1878 3330 3.0032 .3525 2.8370 3S 26 2761 3.6222 2949 3-3904 3140 3.1845 3333 3.0003 3528 2.8344 34 27 2764 3.6181 29S3 3-3868 3143 3-1813 3336 2.9974 3S3I 2.8318 33 28 2767 3.6140 2956 3.3832 3147 3.1780 3339 2-9945 3535 2.8291 32 29 2770 3.6100 2959 3-3796 3150 3-1748 3343 2.9916 3538 2.8265 31 30 2773 3-6059 2962 3.3759 3153 3.1716 3346 2.9887 3541 2.8239 30 31 2776 3.6018 2965 3.3723 3156 3.1684 3349 2.9858 3S44 2.8213 29 32 2780 3-5978 2968 3.3687 3I.S9 3-1652 3352 2.9829 3548 2.8187 28 ^^ 2783 3-5937 2972 3.3652 3163 3.1620 3.356 2.9800 3551 2.8161 27 34 2786 3.5897 2975 3-3616 3166 3-1588 3359 2.9772 3554 2.8135 26 35 2789 3-5856 2978 3-3580 3169 3-1556 3362 2.9743 3558 2.8109 25 3b 2792 3.5816 2981 3.3544 3172 3.1524 3365 2.9714 3561 2.8083 24 37 2795 3-5776 2984 3.3509 3175 3.1492 3369 2.9686 3564 2.8057 23 38 2798 3-5736 2987 3-3473 3179 3.1460 .3372 2.9657 3567 2.8032 22 39 40 2801 3.5696 2991 3.3438 3182 3.1429 3375 2.9629 3571 2.8006 21 2805 3.5656 2994 3.3402 318s 3.1397 3378 2.9600 3574 2.7980 20 41 2808 3-5616 2997 3.3367 3188 3.1366 3382 2.9572 3577 2.7955 19 42 2811 3-5576 3000 3-3332 3191 3.1334 3385 2.9544 3581 2.7929 18 43 2814 3-5536 3003 3.3297 3195 3.1303 3388 2.9515 3584 2.7903 17 44 2817 3-5497 3006 3.3261 3198 3.1271 3391 2.9487 3587 2.7878 16 4S 2820 3-5457 3010 3.3226 3201 3.1240 3395 2.9459 3590 2.7852 15 46 2823 3.5418 3013 3.3191 3204 3.1209 3398 2.9431 3594 2.7827 14 47 2827 3-5379 3016 3.3156 3207 3.1178 3401 2.9403 3597 2.7801 13 48 2830 3-5339 3019 3.3122 3211 3-1146 3404 2.9375 3600 2.7776 1 2 49 2833 3.5300 3022 3.3087 3214 3-i"5 3408 2.9347 3604 2.7751 II 50 2836 3.5261 3026 3-3052 3217 3.1084 3411 2.9319 3607 2.7725 10 SI 2839 3.5222 3029 3-3017 3220 3.1053 3414 2.9291 3610 2.7700 9 S2 2842 3.5183 3032 3-2983 3223 3.1022 3417 2.9263 3613 2-7675 8 S3 2845 3-5144 .S03S 3.2948 3227 3.0991 3421 2-9235 3617 2.7650 7 54 2849 3-5105 3038 3-2914 3230 3.0961 3424 2.9208 3620 2.7625 b ss 2852 3-5067 3041 3.2880 3233 3-0930 3427 2.9180 3623 2.7600 5 S6 2855 3-5028 3045 3-2845 3236 3.0899 3430 2.9152 3627 2-7575 4 S7 28s8 3.4989 3048 3.281 1 3240 3.0868 3434 2.9125 3630 2.7550 3 S8 2861 3-4951 .SOS I 3-2777 3243 3-0838 3437 2.9097 3633 2.7525 2 59 2864 3.4912 3054 3-2743 3246 3.0807 3440 2.9070 3636 2.7500 I GO 2867 3.4874 3057 3.2709 3249 3.0777 3443 2.9042 3640 2.7475 Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang / 74° 1 73^ 1 7 2° 71° 7 0° 1 / 1 NATURAL TANGENTS AND COTANGENTS 85 / 20° 21° 2 52° 23° 24° / Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg 3640 2.7475 3839 2.6051 4040 2.4751 4245 2.3559 4452 2.2460 60 I 3643 2.7450 3842 2.6028 4044 2.4730 4248 2.3539 4456 2.2443 59 2 3646 2.7425 3845 2.6006 4047 2.4709 4252 2.3520 4459 2.2425 58 3 3650 2.7400 3849 2.5983 4050 2.4689 4255 2.3501 4463 2.240S 57 4 S 3653 2.7376 3852 2.5961 4054 2.4668 4258 2.3483 4466 2.2390 56 55 3656 2.7351 3855 2.5938 4057 2.4648 4262 2.3464 4470 2.2373 6 3659 2.7326 3859 2.5916 4061 2.4627 4265 2.3445 4473 2.2355 54 7 3663 2.7302 3862 2.5893 4064 2.4606 4269 2.3426 4477 2.2338 53 8 3666 2.7277 .3865 2.5871 4067 2.4586 4272 2.3407 4480 2.2320 52 9 3669 2.7253 3869 2.5848 4071 2.4566 4276 2:3388 4484 2.2303 51 10 3673 2.7228 .S872 2.5826 4074 2.4545 4279 2.3369 4487 2.2286 50 II 367b 2.7204 .S875 2.5804 4078 2.4525 4283 2.3351 4491 2.2268 49 12 3679 2.7179 3879 2.5782 4081 2.4504 4286 2.3332 4494 2.2251 48 1.3 3(>^3 2.7155 3882 2-5759 4084 2.4484 4289 2.3313 4498 2.2234 47 14 3b86 2.7130 3885 2.5737 4088 2.4464 4293 2.3294 4501 2.2216 46 IS 3689 2.7106 3889 2-5715 4091 2.4443 4296 2.3276 4505 2.2199 45 I6 3693 2.7082 3892 2.5693 4095 2.4423 4300 2.3257 4508 2.2182 44 17 3696 2.7058 3895 2.5671 4098 2.4403 4303 2.3238 4512 2.2165 43 18 3699 2.7034 3899 2.5649 4101 2.4383 4307 2.3220 4515 2.2148 42 19 3702 2.7009 3902 2.5627 4105 2.4362 4310 2.3201 4519 2,2130 41 20 3706 2.6985 3906 2.5605 4108 2.4342 4314 2.3183 4522 2.2113 40 21 3709 2.6961 3909 2.5533 4111 2.4322 4317 2.3164 4526 2.2096 39 22 3712 2.6937 3912 2.5561 4115 2.4302 4320 2.3146 4529 2.2079 38 23 37^^ 2.6913 3916 2.5539 4118 2.4282 4324 2.3127 4533 2.2062 37 24 3719 2.6889 3919 2.5517 4122 2.4262 4327 2.3109 4536 2.2045 36 25 3722 2.6865 3922 2.5495 4125 2.4242 4331 2.3090 4540 2.2028 35 2b 3726 2.6841 3926 2.5473 4129 2.4222 4334 2.3072 4543 2.201 1 34 27 3729 2.6818 3929 2.5452 4132 2.4202 4338 2.3053 4547 2.1994 33 28 3732 2.6794 .3932 2.5430 413s 2.4182 4341 2.3035 4S50 2.1977 32 29 373^ 2.6770 3936 2.5408 4139 2.4162 4345 2.3017 4554 2 i960 31 30 3739 2.6746 3939 2.5386 4142 2.4142 4.348 2.2998 4557 2.1943 30 31 3742 2.6723 3942 2.5365 4146 2.4122 4352 2.2980 4561 2.1926 29 32 3745 2.6699 3946 2.5343 4149 2.4102 4355 2.2962 4564 2.1909 28 33 3749 2.6675 3949 2.5322 4152 2.4083 4359 2.2944 4568 2.1892 27 34 3752 2.6652 3953 2.5300 4156 2.4063 4362 2.2925 4571 2.1876 26 35 3755 2.6628 3956 2.5279 4159 2.4043 4365 2.2907 4575 2.1859 25 3^ 3759 2.6605 3959 2.5257 4163 2.4023 4369 2.2889 4578 2.1842 24 37 3762 2.6581 3963 2.5236 4166 2.4004 4372 2.2871 4582 2.1825 23 3« 3765 2.6558 3966 2.5214 4169 2.3984 4376 2.2853 4585 2.1808 22 39 3769 2.6534 3969 2.5193 4173 2.3964 4379 2.2835 4589 2.1792 21 40 3772 2.65 II 3973 2.5172 4176 2.3945 4383 2.2817 4592 2.1775 20 41 3775 2.6488 3976 2.5150 4180 2.3925 4386 2.2799 4596 2.1758 19 42 3779 2.6464 3979 2.5129 4183 2.3906 4390 2.2781 4599 2.1742 18 43 37^2 2.6441 3983 2.5108 4187 2.3886 4393 2.2763 4603 2.1725 17 44 37^5 2.6418 3986 2.5086 4190 2.3867 4397 2.2745 4607 2.1708 16 15 45 3789 2.6395 3990 2.5065 4193 2.3847 4400 2.2727 4610 2.1692 4b 3792 2.6371 3993 2.5044 4197 2.3828 4404 2.2709 4614 2.1675 14 47 3795 2.6348 3996 2.5023 4200 2.3808 4407 2.2691 4617 2.1659 13 48 3799 2.6325 4000 2.5002 4204 2.3789 441 1 2.2673 4621 2,1642 12 49 3802 2.6302 4003 2.4981 4207 2.3770 4414 2.2655 4624 2.1625 II 50 3805 2.6279 4006 2.4960 4210 2.3750 4417 2.2637 4628 2.1609 10 51 3809 2,6256 4010 2.4939 4214 2.3731 4421 2.2620 4631 2.1592 ? 52 3812 2.6233 4013 2.4918 4217 2.3712 4424 2.2602 4635 2.1576 8 53 3«i5 2.6210 4017 2.4897 4221 2.3693 4428 2.2584 4638 2.1560 7 54 3«i9 2.6187 4020 2.4876 4224 2.3673 4431 2.2566 4642 2.1543 b 55 3822 2.6165 4023 2.4855 4228 2.3654 4435 2.2549 4645 2.1527 5 5t> 3«25 2.6142 4027 2.4834 4231 2.3635 4438 2.2531 4649 2.1510 4 57 3829 2.6119 4030 2.4813 4234 2.3616 4442 2.2513 4652 2.1494 3 5« 3«32 2.6096 4033 2.4792 4238 2.3597 4445 2.2496 4656 2.1478 2 59 3835 2.6074 4037 2.4772 4241 2.3578 4449 2.2478 4660 2.1461 I 60 3839 2.6051 4040 2.4751 4245 2.3559 4452 2.2460 4663 2.1445 Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang / 69° 68° 67° 66° 1 d 5° / 86 TABLE TTT / 25° 26° 270 28° 29° / 2 3 4 Tang totg Tang Cotg Tang Cotg Tang Cotg Tang Cotg 4663 2.1445 4667 2.1429 4670 2. 14 1 3 4674 2.1396 4677 2.1380 487"7 2.0503 4881 2.0488 4885 2.0473 4888 2.0458 4892 2.0443 5095 1.9626 5099 1. 96 1 2 5103 1.9598 5106 1.9584 5110 1.9570 5317 1.8807 5321 1.8794 5325 1.8781 5328 1.8768 5332 1.8755 5543 1.8040 5547 1.8028 5551 1. 8016 5555 1-8003 5558 1.7991 60 59 58 57 56 7 8 9 4681 2.1364 4684 2. 1 348 4688 2.1332 4691 2.1315 4695 2.1299 4895 2.0428 4899 2.0413 4903 2.0398 4906 2.0383 4910 2.0368 51 14 1.9556 51 17 1.9542 5121 1.9528 5125 1.9514 5128 1.9500 5336 1.8741 5340 1.8728 5343 1-8715 5347 1.8702 5351 1.8689 5562 1.7979 5566 1.7966 5570 1.7954 5574 1.7942 5577 1-7930 55 54 53 52 51 10 II 12 13 14 4699 2.1283 4702 2.1267 4706 2. 1 25 1 4709 2.1235 4713 2.1219 4913 2.0353 4917 2.0338 4921 2.0323 4924 2.0308 4928 2.0293 5132 1.9486 5136 1.9472 5139 1.9458 5143 1.9444 5147 1.9430 5354 1.8676 5358 1.8663 5362 1.8650 5366 1.8637 5369 1.8624 5581 1.7917 5585 1.7905 5589 1-7893 5593 1.7881 5596 1.7868 50 49 48 11 17 19 4716 2.1203 4720 2.1 187 4723 2.1 171 4727 2.1155 4731 2.1 139 4931 2.0278 4935 2.0263 4939 2.0248 4942 2.0233 4946 2.0219 5150 1.9416 5154 1.9402 5158 1.9388 5161 1.9375 5165 1.9361 5373 1861 1 5377 1-8598 5381 1.8585 5384 1.8572 5388 1.8559 5600 1.7856 5604 1.7844 5608 1.7832 5612 1.7820 5616 1.7808 45 44 43 42 41 20 21 22 23 24 4734 2. 1 1 23 4738 2.1 107 4741 2.1092 4745 2.1076 4748 2.1060 4950 2.0204 4953 2.0189 4957 2.0174 4960 2.0160 4964 2.0145 5169 1.9347 5172 1.9333 5176 1.9319 5180 1.9306 5184 1.9292 5392 1.8546 5396 1.8533 5399 1.8520 5403 1.8507 5407 1.8495 5619 1.7796 5623 1.7783 5627 1.7771 5631 1.7759 5635 1.7747 40 38 37 36 27 28 29 4752 2.1044 4755 2.1028 4759 2.1013 4763 2.0997 4766 2.0981 4968 2.0130 4971 2.0115 4975 2.0101 4979 2.0086 4982 2.0072 5187 1.9278 5191 1.9265 5^95 1-9251 5198 1.9237 5202 1.9223 541 1 1.8482 5415 1.8469 5418 1.8456 5422 1.8443 5426 1.8430 5639 1.7735 5642 1.7723 5646 1.7711 5650 1.7699 5654 1.7687 35 34 33 32 31 30 31 32 33 34 4770 2.0965 4/ 73 2.0950 4777 2.0934 4780 2.0918 4784 2.0903 4986 2.0057 4989 2.0042 4993 2.0028 4997 2.0013 5000 1.9999 5206 1.9210 5209 1.9196 5213 1.9183 5217 1.9169 5220 1.9155 5430 1. 84 1 8 5433 1.8405 5437 1.8392 5441 1.8379 5445 1.8367 5658 1.7675 5662 1.7663 5665 1.7651 5669 1.7639 5673 1.7627 30 29 28 27 26 39 4788 2.0887 4791 2.0872 4795 2.0856 4798 2.0840 4802 2.0825 5004 1.9984 5008 1.9970 50" 1-9955 5015 1. 9941 5019 1.9926 5224 1.9142 5228 1.9128 5232 1.9115 5235 1.9101 5239 1.9088 5448 1.8354 5452 1.8341 5456 1.8329 5460 1. 83 1 6 5464 1.8303 5677 1.7615 5681 1.7603 5685 1.7591 5688 1.7579 5692 1.7567 25 24 23 22 21 40 41 42 43 44 4806 2.0809 4809 2.0794 4813 2.0778 4816 2.0763 4820 2.0748 5022 1. 9912 5026 1.9897 5029 1.9883 5033 1.9868 5037 1.9854 5243 1.9074 5426 1. 9061 5250 1.9047 5254 1.9034 5258 1.9020 5467 1. 8291 5471 1.8278 5475 1-8265 5479 1.8253 5482 1.8240 5696 1.7556 5700 1.7544 5704 1.7532 5708 1.7520 5712 1.7508 20 19 18 17 16 45 46 47 48 49 4823 2.0732 4827 2.0717 4831 2.0701 4834 2.0686 4838 2.0671 5040 1 .9840 5044 J. 9825 5048 1. 98 II 5051 1.9797 5055 1.9782 5261 1.9007 5265 1.8993 5269 1.8980 5272 1.8967 5276 1.8953 5486 1.8228 5490 1.8215 5494 1.8202 5498 1. 8 1 90 5501 1.8177 5715 1.7496 5719 1.7485 5723 1.7473 5727 1. 7461 5731 1.7449 15 14 13 12 II 50 51 52 53 54 4841 2.0655 4845 2.0640 4849 2.0625 4852 2.0609 4856 2.0594 5059 1.9768 5062 1.9754 5066 1.9740 5070 1.9725 5073 1.9711 5280 1.8940 5284 1.8927 5287 1.8913 5291 1.8900 5295 1.8887 5505 1-8165 5509 1.8152 5513 1.8140 5517 1.8127 5520 1.8115 5735 1-7437 5739 1.7426 5743 1-7414 5746 1.7402 5750 1.7391 10 9 8 7 6 59 4859 2.0579 4863 2.0564 4867 2.0549 4870 2.0533 4874 2.0518 5077 1.9697 5081 1.9683 5084 1.9669 5088 1.9654 5092 1.9640 5298 1.8873 5302 1.8860 5306 1.8847 5310 1.8834 5313 1.8820 5524 1.8103 5528 1.8090 5532 1.8078 5535 1.8065 5539 1.8053 5754 1-7379 5758 1.7367 5762 1.7355 5766 1.7344 5770 1.7332 5 4 3 2 I 60 4877 2.0503 5095 1.9626 5317 1.8807 5543 1.8040 5774 1-7321 Cotff Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang / 64° 63° 620 61° 60° / 1 NATURAL TANGENTS AND COTANGENTS 87 / 30° 31° 32° 38° 34° / 1 Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg I 2 3 4 5 6 7 8 9 5774 1.7321 5777 1-7309 5781 1.7297 5785 1.7286 5789 1.7274 6009 1.6643 6013 1.6632 6017 1.6621 6020 1. 66 10 6024 1.6599 6249 1.6003 6253 1.5993 6257 1.5983 6261 1.5972 6265 1.5962 6494 1.5399 6498 1.5389 6502 1.5379 6506 1.5369 6511 1-5359 6745 1.4826 6749 1.4816 6754 1.4807 6758 1.4798 6762 1.4788 60 59 58 57 56 5793 1-7262 5797 1-7251 5801 1.7239 5805 1.7228 5808 1.7216 6028 1.6588 6032 1.6577 6036 1.6566 6040 1.6555 6044 1.6545 6269 1.5952 6273 1.5941 6277 1.5931 6281 1.5921 6285 1. 59 1 1 6515 1-5350 6519 1.5340 6523 1-5330 6527 1-5320 6531 1-53" 6766 1.4779 6771 1.4770 6775 1.4761 6779 1.4751 6783 1.4742 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 10 II 12 13 5812 1.7205 5816 1.7193 5820 1.7182 5824 1.7170 5828 1.7159 6048 1.6534 6052 1.6523 6056 1.65 1 2 6060 1. 6501 6064 1.6490 6289 1.5900 6293 1.5890 6297 1.5880 6301 1.5869 6305 1.5859 6536 1.5301 6540 1.5291 6544 1.5282 6548 1.5272 6552 1.5262 6787 1.4733 6792 1.4724 6796 1.4715 6800 1.4705 6805 1.4696 15 i6 17 i8 19 5832 1. 7147 5836 1,7136 5840 1.7124 5844 1.7113 5847 1. 7 102 6068 1.6479 6072 1.6469 6076 1.6458 6080 1.6447 6084 1.6436 6310 1.5849 6314 1.5839 6318 1.5829 6322 • 1.5818 6326 1.5808 6556 1.5253 6560 1.5243 6565 1.5233 6569 1.5224 6573 1.5214 6809 1.4687 6813 1.4678 6817 1.4669 6822 1.4659 6826 1.4650 20 21 22 23 24 5851 1.7090 5S55 1-7079 5859 1.7067 5863 1.7056 5S67 1.7045 6088 1.6426 6092 1. 641 5 6096 1.6404 6100 1.6393 6104 1.6383 6330 1.5798 6334 1-5788 6338 1.5778 6342 1.5768 6346 1.5757 6577 1.5204 6581 1.5195 6585 1.5185 6590 1.5 1 75 6594 1.5 166 6830 1. 464 1 6834 1.4632 6839 1.4623 6843 1. 46 14 6847 1.4605 25 26 27 28 29 5871 1.7033 5875 1.7022 5879 1.7011 5883 1.6999 5887 1.6988 6108 1.6372 6112 1.6361 61 16 1.6351 6120 1.6340 6124 1.6329 6350 1.5747 6354 1.5737 6358 1.5727 6363 1-5717 6367 1-5707 6598 1.5156 6602 1.5 147 6606 1.5 137 6610 1. 5127 6615 1.5118 6851 1.4596 6856 1.4586 6860 1.4577 6864 1.4568 6869 1.4559 35 34 33 32 31 30 31 32 33 34 5890 1.6977 5894 1.6965 5898 1.6954 5902 1.6943 5906 1.6932 6128 1.6319 6132 1.6308 6136 1.6297 6140 1.6287 6144 1.6276 6371 1.5697 6375 1-5687 6379 1-5677 6383 1.5667 6387 1.5657 6619 1.5108, 6623 1.5099 6627 1.5089 6631 1.5080 6636 1.5070 6873 1.4550 6877 1.4541 6881 1.4532 6886 1.4523 6890 1.45 14 30 29 28 27 26 35 36 37 38 39 5910 1.6920 5914 1.6909 5918 1.6898 S922 1.6887 5926 1.6875 6148 1.6265 6152 1.6255 6156 1.62,1/1 6160 1.6234 6164 1.6223 6391 1.5647 6395 1-5637 6399 1-5627 6403 1. 56 1 7 6408 1.5607 6640 1. 506 1 6644 1.505 1 6648 1.5042 6652 1.5032 6657 1.5023 6894 1.4505 6899 1.4496 6903 1.4487 6907 1.4478 691 1 1.4469 25 24 23 22 21 40 41 42 43 44 5930 1.6864 5934 1.6853 5938 1.6842 5942 1.6831 5945 1.6820 6168 1.6212 6172 1.6202 6176 1.6191 6i8o 1.6181 6184 1. 6170 6412 1.5597 6416 1.5587 6420 1.5577 6424 1.5567 6428 1.5557 6661 1.5013 6665 1.5004 6669 1.4994 6673 1.4985 6678 1.4975 6916 1.4460 6920 1.445 1 6924 1.4442 6929 1.4433 6933 1-4424 20 19 18 17 16 45 46 47 48 49 50 51 52 53 54 5949 1.6808 5953 1-6797 5957 1-6786 5961 1.6775 5965 1.6764 6188 1.6160 6192 1.6149 6196 1.6139 6200 1. 6 1 28 6204 1. 61 18 6432 1.5547 6436 1.5537 6440 1.5527 6445 1-55 1 7 6449 1-5507 6682 1.4966 6686 1.4957 6690 1.4947 6694 1.4938 6699 1.4928 6937 1.4415 6942 1.4406 6946 1.4397 6950 1.4388 6954 1-4379 15 14 13 12 5969 1.6753 5973 1-6742 5977 1-6731 5981 1.6720 5985 1.6709 6208 1.6107 6212 1.6097 6216 1.6087 6220 1.6076 6224 1.6066 6453 1-5497 6457 1.5487 6461 1.5477 6465 1.5468 6469 1-5458 6703 1.4919 6707 1. 49 10 6711 1.4900 6716 1. 489 1 6720 1.4882 6959 1-4370 6963 1.4361 6967 1-4352 6972 1.4344 6976 1-4335 10 9 8 7 6 55 56 11 59 5989 1.6698 5993 1.6687 5997 1-6676 6001 1.6665 6005 1.6654 6228 1.6055 6233 1.6045 6237 1-6034 6241 1.6024 6245 1. 6014 6473 1-5448 6478 1.5438 6482 1.5428 6486 1.5418 6490 1.5408 6724 1.4872 6728 1.4863 6732 1.4854 6737 1.4844 6741 1.4835 6980 1.4326 6985 1.43 1 7 6989 1.4308 6993 1-4299 6998 1.4290 5 4 3 2 GO 6009 1.6643 6249 1.6003 6494 1-5399 6745 1.4826 7002 1. 4281 Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang / 51)° 5S° 57° 56° 55° / 88 TABLE III / 35° 36° 37° 38° 31)° / Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg I 2 3 4 7002 1. 428 1 7006 1.4273 70 II 1.4264 7015 1.4255 7019 1.4246 7265 1.3764 7270 1.3755 7274 1.3747 7279 1-3739 7283 1.3730 7536 1.3270 7540 1.3262 7545 1.3254 7549 1.3246 7554 1.3238 7813 1.2799 7818 1.2792 7822 1.2784 7827 1.2776 7832 1.2769 S098 1.2349 8103 1.2342 8107 1.2334 81 12 1.2327 8117 1.2320 60 59 58 I 7 8 9 7024 1.4237 7028 1.4229 7032 1.4220 7037 1.4211 7041 1.4202 7288 1.3722 7292 1.3713 7297 1.3705 7301 1.3697 7306 1.3688 7558 1.3230 7563 1.3222 7568 1.3214 7572 1.3206 7577 1.3198 7836 1.2761 7841 1.2753 7846 1.2746 7850 1.2738 7855 1-2731 8122 1. 2312 8127 1.2305 8132 1.2298 8136 1.2290 8141 1.2283 55 54 53 52 51 10 II 12 13 14 7046 1. 41 93 7050 1. 4 1 85 7054 1.4176 7059 1.4167 7063 1. 41 58 7310 1.3680 7314 1.3672 7319 1.3663 7323 1.3655 7328 1.3647 7581 1.3190 7586 1.3182 7590 1.3175 7595 1.3167 7600 1. 3 1 59 7860 1.2723 7865 1.2715 7869 1.2708 7874 1.2700 7879 1.2693 8146 1.2276 8151 1.2268 8156 1. 2261 8i6i 1.2254 8165 1.2247 60 49 48 47 46 19 7067 1.4150 7072 1.4141 7076 1.4132 7080 1. 4 1 24 7085 1.4115 7332 1.3638 7337 1.3630 7341 1.3622 7346 1.3613 7350 1.3605 7604 1.3151 7609 1.3143 7613 1.3135 7618 1.3127 7623 1.3119 7883 1.2685 7888 1.2677 7893 1.2670 7898 1.2662 7902 1.2655 8170 1.2239 8175 1.2232 8180 1.2225 8185 1.2218 8190 1. 2210 45 44 43 42 41 20 21 22 23 24 7089 1. 4 1 06 7094 1.4097 7098 1.4089 7102 1.4080 7107 1.4071 7355 1-3597 7359 1.3588 7364 1.3580 7368 1.3572 7373 1-3564 7627 1.3111 7632 1. 3103 7636 1.3095 7641 1.3087 7646 1.3079 7907 1.2647 7912 1.2640 7916 1.2632 7921 1.2624 7926 1.2617 8195 1.2203 8199 1. 2196 8204 1. 21 89 8209 1.2181 8214 1.2174 40 39 38 27 36 25 26 27 28 29 71 II 1.4063 71 15 1.4054 7120 1.4045 7124 1.4037 7129 1.4028 7377 1.3555 7382 1.3547 7386 1.3539 7391 1.3531 7395 1.3522 7650 1.3072 7655 1.3064 7659 1.3056 7664 1.3048 7669 1.3040 7931 1.2609 7935 1.2602 7940 1.2594 7945 1.2587 7950 1.2579 8219 1. 2167 8224 1.2 1 60 8229 1.2153 8234 1.2145 8238 1.2138 35 34 32 31 30 31 32 zz 34 7133 1.4019 7137 1.4011 7142 1.4002 7146 1.3994 7151 1.3985 7400 1.35 14 7404 1.3506 7409 1.3498 7413 1.3490 7418 1. 3481 7673 1.3032 7678 1.3024 7683 1.3017 7687 1.3009 7692 1.3001 7954 1.2572 7959 1.2564 7964 1.2557 7969 1.2549 7973 1.2542 8243 1.2131 8248 1. 21 24 8253 1.2117 8258 1.2109 8263 1.2102 30 29 28 27 26 37 38 39 7155 1-3976 7159 1.3968 7164 1.3959 7168 1.3951 7173 1.3942 7422 1.3473 7427 1.3465 7431 1.3457 7436 1.3449 7440 1.3440 7696 1.2993 7701 1.2985 7706 1.2977 7710 1.2970 7715 1.2962 7978 1.2534 7983 1.2527 7988 1.2519 7992 1.25 1 2 7997 1.2504 8268 1.2095 8273 1.2088 8278 1.2081 8283 1.2074 8287 1.2066 25 24 23 22 21 40 41 42 43 44 7177 1.3934 7181 1.3925 7186 1.3916 7190 1.3908 7195 1-3899 7445 1.3432 7449 1.3424 7454 1.3416 7458 1.3408 7463 1.3400 7720 1.2954 7724 1.2946 7729 1.2938 7734 1.2931 7738 1.2923 8002 1.2497 8007 1.2489 8012 1.2482 8016 1.2475 8021 1.2467 8292 1.2059 8297 1.2052 8302 1.2045 8307 1.2038 8312 1.203 1 20 »9 18 17 16 45 46 47 48 49 7199 1.3891 7203 1.3882 7208 1.3874 7212 1.3865 7217 1.3857 7467 1.3392 7472 1.3384 7476 1.3375 7481 1.3367 7485 1.3359 7743 1.2915 7747 1.2907 7752 1.2900 7757 1.2892 7761 1.2884 8026 1.2460 8031 1.2452 8035 1.2445 8040 1.2437 8045 1.2430 8317 1.2024 8322 1. 2017 8327 1.2009 8332 1.2002 8337 1.1995 15 14 13 12 II 50 51 52 53 54 7221 1.3848 7226 1.3840 7230 1.3831 7234 1.3823 7239 1.3814 7490 1.335 1 7495 1-3343 7499 1.3335 7504 1.3327 7508 1.3319 7766 1.2876 7771 1.2869 7775 1.2861 7780 1.2853 7785 1.2846 8050 1.2423 8055 1.2415 8059 1.2408 8064 1. 240 1 8069 1.2393 8342 1. 1 988 8346 1.1981 8351 1. 1974 8356 1. 1967 8361 1. 1960 10 9 8 7 6 55 56 59 7243 1.3806 7248 1.3798 7252 1.3789 7257 1.3781 7261 1.3772 7513 i-2>2,^i 7517 1.3303 7522 1.3295 7526 1.3287 7531 1.3278 7789 1.2838 7794 1.2830 7799 1.2822 7803 1.28 1 5 7808 1.2807 8074 1.2386 8079 1.2378 8083 1. 237 1 8088 1.2364 8093 1.2356 8366 1. 1953 8371 1. 1946 8376 1. 1939 8381 1. 1932 8386 1. 1925 5 4 3 2 60 7265 1.3764 7536 1.3270 7813 1.2799 8098 1.2349 8391 1.1918 Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang _ / 54° 53° 52° 51° 50° zl NATURAL TANGENTS AND COTANGENTS / 40° 41° 42o 43° 44° / Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg 60 59 58 11 I 2 3 4 8391 1.1918 8396 1.1910 8401 1. 1903 8406 1. 1 896 8411 1. 1889 8693 1.1504 8698 1.1497 8703 1.1490 8708 1.1483 8713 1.1477 9004 I.I 106 9009 I.I 100 9015 1.1093 9020 1.1087 9025 1.1080 9325 1.0724 9331 1.0717 9336 1.0711 9341 1.0705 9347 1.0699 9657 1-0355 9663 1.0349 9668 1.0343 9674 1-0337 9679 1.0331 1, 9 8416 1. 1882 8421 I. 1875 8426 1. 1 868 8431 1.1861 8436 1. 1 854 8718 1.1470 8724 1.1463 8729 1. 1456 8734 1.1450 8739 1. 1443 9030 1.1074 9036 1.1067 9041 1.1061 9046 1. 1 054 9052 1. 1048 9352 1.0692 9358 1.0686 9363 1.0680 9369 1.0674 9374 1.0668 9685 1.0325 9691 1.0319 9696 1. 03 1 3 9702 1.0307 9708 1. 030 1 55 54 53 52 51 50 49 48 47 46 10 II 12 13 14 8441 1. 1 847 8446 1. 1 840 8451 1.1833 8456 1. 1826 8461 1.1819 8744 1.1436 8749 1.1430 8754 1.1423 8759 1.1416 8765 1.1410 9057 1.1041 9062 1.1035 9067 1.1028 9073 1.1022 9078 1. 1016 9380 1.0661 9385 1.0655 9391 1.0649 9396 1.0643 9402 1.0637 9713 1.0295 9719 1.0289 9725 1.0283 9730 1.0277 9736 1.0271 15 16 17 18 19 8466 1.1812 8471 1. 1806 8476 I.I 799 8481 1. 1792 8486 1.17^5 8770 1.1403 8775 1-1396 8780 1.1389 8785 1.1383 8790 1.1376 9083 1.1009 9089 1. 1 003 9094 1.0996 9099 1.0990 9105 1.0983 9407 1.0630 9413 1.0624 9418 1.0618 9424 1. 06 1 2 9429 1.0606 9742 1.0265 9747 1-0259 9753 1-0253 9759 1.0247 9764 1.0241 45 44 43 42 41 20 21 22 23 24 8491 I. 1778 8496 1.1771 8501 1.1764 8506 1. 1757 8511 1. 1750 8796 1. 1 369 8801 1. 1363 8806 1.1356 8811 1.1349 8816 1.1343 9110 1.0977 9115 1.0971 91 21 1.0964 9126 1.0958 9131 1.0951 9435 1-0599 9440 1.0593 9446 1.0587 9451 1.0581 9457 1-0575 9770 1.0235 9776 1.0230 9781 1.0224 9787 1.0218 9793 1. 02 1 2 40 39 3^ 11 25 26 27 28 29 8516 1. 1743 8521 1. 1736 8526 1. 1729 8531 1. 1722 8536 1.1715 8821 1. 1336 8827 1.1329 8832 1.1323 8837 1.1316 8842 1.1310 9137 1.0945 9142 1.0939 9147 1.0932 9153 1.0926 9158 1.0919 9462 1.0569 9468 1.0562 9473 1-0556 9479 1-0550 9484 1.0544 9798 1.0206 9804 1.0200 9810 1.0194 9816 1.0188 9821 1.0182 35 34 33 32 31 30 31 32 33 34 8541 I. 1708 8546 I.I 702 8551 1-1695 8556 1. 1688 8561 1.1681 8847 1.1303 8852 1.1296 8858 1.1290 8863 1.1283 8868 1.1276 9163 1.0913 9169 1.0907 9174 1.0900 9179 1.0894 9185 1.0888 9490 1.0538 9495 1-0532 9501 1.0526 9506 1.0519 9512 1.0513 9827 1.0176 9833 1.0170 9838 1. 01 64 9844 1. 01 58 9850 1.0152 30 29 28 27 26 37 38 39 8566 1.1674 8571 1.1667 8576 1.1660 8581 I. 1653 8586 1. 1 647 8873 1.1270 8878 1.1263 8884 1.1257 8889 1. 1250 8894 1.1243 9190 1.0881 9195 1.0875 9201 1.0869 9206 1.0862 9212 1.0856 9517 1.0507 9523 1.0501 9528 1.0495 9534 1.0489 9540 1.0483 9856 1.0147 9861 1.0141 9867 1.0135 9873 1.0129 9879 1.0123 25 24 23 22 21 40 41 42 43 44 i 49 8591 I. 1640 8596 I. 1633 8601 1. 1 626 8606 1. 1619 8611 1. 1612 8899 1-1237 8904 I.I 230 8910 1.1224 8915 1.1217 8920 1.1211 9217 1.0850 9222 1.0843 9228 1.0837 9233 1. 083 1 9239 1.0824 9545 1-0477 9551 1.0470 9556 1.0464 9562 1.0458 9567 1.0452 9884 1.0117 9890 l.OIIl 9896 1.0105 9902 1.0099 9907 1.0094 20 19 18 17 16 8617 1. 1606 8622 1. 1599 8627 1. 1592 8632 1.1585 8637 1.1578 8925 1.1204 8931 1.1197 8936 1.1191 8941 I.I 184 8946 1.1178 9244 1. 08 1 8 9249 1. 08 1 2 9255 1.0805 9260 1.0799 9266 1.0793 9573 1.0446 9578 1.0440 9584 1.0434 9590 1.0428 9595 1.0422 9913 1.0088 9919 1.0082 9925 1.0076 9930 1.0070 9936 1.0064 15 14 13 12 II 50 51 52 53 54 8642 1.1571 8647 1. 1 565 8652 1.1558 8657 1.1551 8662 1.1544 8952 1.1171 8957 1.1165 8962 1.1158 8967 1.1152 8972 1.1145 9271 1.0786 9276 1.0780 9282 1.0774 9287 1.0768 9293 1.0761 9601 1.0416 9606 1. 04 10 9612 1.0404 9618 1.0398 9623 1.0392 9942 1.0058 9948 1.0052 9954 1.0047 9959 I -004 1 9965 1.0035 10 i 7 6 57 58 59 8667 1. 1538 8672 1.1531 8678 1.1524 8683 1.1517 8688 1. 1510 8978 I.I 139 8983 1.1132 8988 1.1126 8994 1.1119 8999 I-III3 9298 1.0755 9303 1-0749 9309 1.0742 9314 1.0736 9320 1.0730 9629 1.0385 9634 1-0379 9640 1.0373 9646 1.0367 9651 1.0361 9971 1.0029 9977 1.0023 9983 1.0017 9988 1. 00 1 2 9994 1.OC06 5 4 3 2 I (>0 8693 1.1504 9004 I.I 106 9325 1.0724 9657 1-0355 1000 1. 0000 Cotg Tang Cotg Tang Cotg Tang Cotg Tang Cotg Tang / 490 48° 47° 46° 45° / TABLE IV * SQUARES OF NUMBERS No. Square. No. Square. No. Square. No. Square. No. Square. I O 20 21 400 40 41 1600 m 61 3600 80 81 64CX) I 441 1681 3721 6561 2 4 22 484 42 1764 62 3844 82 6724 3 9 23 529 43 1849 63 3969 83 6889 4 i6 24 576 44 1936 64 4096 84 7056 5 25 25 625 45 2025 65 4225 85 7225 6 36 26 676 46 2116 66 4356 86 7396 7 49 27 729 47 2209 67 4489 87 7569 8 64 28 784 48 2304 68 4624 88 7744 9 10 1 1 81 29 30 31 841 49 50 51 2401 69 70 71 4761 89 90 91 7921 8100 100 900 2500 4900 121 961 2601 5041 8281 12 144 32 1024 52 2704 72 5184 92 8464 '3 169 33 1089 53 2809 73 5329 93 8649 14 196 34 1 1 56 54 2916 74 5476 94 8836 '5 22s 35 1225 55 3025 75 5625 95 9025 i6 256 36 1296 56 3136 76 5776 96 9216 17 289 37 1369 57' 3249 77 5929 97 9409 i8 324 38 1444 58' 3364 78 6084 98 9604 «9 •20 361 400 39 40 1521 59 GO 3481 79 80 6241 99 100 9801 1600 3600 6400 lOOOO 91 92 TABLE IV 00 !♦♦ 244 $♦♦ 4^# &♦♦ 6#^ T4^ §♦♦ 9^^ u 00 Diff' I 100 400 900 1600 2500 3600 4900 6400 8100 OI 02 03 102 104 106 404 408 412 906 912 918 1608 1616 1624 2510 2520 2530 3612 4914 4928 4942 6416 6432 6448 8118 8136 8154 01 04 09 3 5 7 04 108 no 112 416 420 424 924 930 936 1632 1640 1648 2540 2550 2560 3648 3660 3672 4956 6464 6480 6496 8172 8190 8208 16 25 36 9 II 13 11 09 114 116 118 428 432 436 04.2 948 954 1656 1664 1672 2570 2580 2590 3696 3708 4998 5012 5026 6512 6528 6544 8226 8244 8262 1'. IS 17 19* 10 121 441 961 1681 2601 3721 5041 6561 8281 00 21 II 12 13 123 125 127 445 449 453 967 973 979 1689 1697 1705 2611 2621 2631 3733 3745 3757 5055 5069 5083 6577 8299 8317 8335 21 44 69 23 25 27 14 129 132 134 til 466 985 992 998 1713 1722 1730 2641 2652 2662 3769 3782 3794 5097 5112 5126 6625 6642 6658 8353 8372 8390 96 5^ 29* 31 33 19 136 139 141 470 475 479 1004 ion 1017 1738 1747 1755 2672 2683 2693 3806 3819 3^3^ 5140 5155 51^9 6691 6707 8408 8427 8445 89 6i 35* 37 39* 20 144 484 1024 1764 2704 3844 5184 6724 8464 00 4» 21 22 23 146 148 151 488 492 407 1030 1036 1043 1772 17S0 1789 2714 2724 2735 3881 5198 5212 5227 6756 ^773 8482 8500 8519 41 84 29 43 45* 47 24 153 lit 501 506 510 1049 1056 1062 1797 1806 1814 2745 2766 3893 3906 3918 5241 5256 5270 6789 6806 6822 8537 8556 8574 76 25 76 49* 5» 53* 11 29 161 166 515 519 524 1069 1075 1082 1823 1831 1840 2777 2787 2798 3931 3943 3956 5285 5299 53H 6839 6855 6872 8630 29 84 41 55 51* 59* 30 169 529 1089 1849 2809 3969 5329 6889 8649 00 61 31 32 33 171 III 542 1095 1102 1 108 1857 1866 1874 2819 2830 2840 3981 3994 4006 5343 5358 5372 6905 6922 6938 8667 8686 8704 61 89 63* 65 67* 34 35 36 184 547 l^ "15 1122 1128 1883 1892 1900 2862 2872 4019 4032 4044 5387 5402 5416 6955 C972 6988 8723 8742 8760 56 11 69* 7» 73* 39 187 190 193 571 "35 1142 1 149 1909 1918 1927 2883 2894 2905 4057 4070 4083 5431 5446 5461 7005 7022 7039 8779 8798 8817 69 44 21 75* 77* 79* 40 196 576 1 156 1936 2916 4096 5476 7056 8836 00 81 41 42 43 198 201 204 580 5«5 590 1 162 1 1 76 1944 1953 1962 2926 2937 2948 4108 4121 4134 5490 5505 5520 7072 7089 7106 8854 8S73 8892 81 64 49 83* 85* 87^ 44 45 46 207 210 213 595 605 1 183 1 190 1197 1971 1980 1989 2959 2970 2981 4147 4160 4173 5535 5550 5565 7123 7140 7157 891 1 8930 8949 36 2 89* 91" 93* 49 216 219 222 610 620 1204 I2n 1218 1998 2007 2016 2992 3003 3014 4186 4199 4212 5580 7174 7191 7208 8968 8987 9006 09 04 01 95* 97* 99* 50 225 625 1225 2025 3025 4225 S62S 7225 9025 00 SQUAKES OF NUMBERS 93 60 !♦♦ 24^ 3## 4## 5## €♦♦ 74^ «♦♦ 94^ U 00 Diff. z 225 625 1225 2025 3025 4225 5625 7225 9025 51 52 53 228 231 234 630 640 1232 1239 1246 2034 2043 2052 3036 3047 3058 4238 4251 5640 5655 5670 7242 7259 7276 9044 9063 9082 01 04 09 3 5 7 54 55 5^^ 237 240 243 645 650 655 1253 12G0 1267 2061 2070 2079 3069 3080 3091 4277 4290 4303 5685 5700 5715 7293 7310 7327 9101 9120 9139 16 25 36 9 11 13' 11 59 246 249 252 660 665 670 1274 12S1 1288 20S8 2097 2106 3102 3113 3124 4316 4329 4342 5730 5760 7344 7361 7378 9158 9177 9196 49 Si 15 17 19* 00 256 676 1296 2116 3136 4356 5776 7396 9216 00 21 61 62 63 259 262 265 68i 686 691* 1303 1310 1317 2125 2134 2143 3147 3158 3169 4369 4382 4395 ^^6 5821 7413 7430 7447 9235 9254 9273 21 69 23 25 27 ^4 268 272 275 696 702 707 1324 1332 1339 2152 2162 2171 3180 3192 3203 4408 4422 4435 5836 ^5^7 7464 7482 7499 9292 9312 9331 96 25 56 29* 31 33 69 278 282 285 712 718 723 1346 1354 1361 2180 2190 2199 3214 3226 3237 4448 4462 4475 5882 5898 5913 7516 7534 7551 9350 9370 9389 89 24 61 35* 37 39* 70 289 729 1369 2209 3249 4489 5929 7569 9409 00 41 71 72 73 292 295 299 734 739 745 1376 1383 1391 2218 2227 2237 3260 3271 3283 4502 4515 4529 5944 5959 5975 7586 7603 7621 9428 9447 9467 29 43 45" 47 74 302 306 309 750 It 1398 1406 1413 2246 2265 3294 3306 3317 4542 4556 4569 5990 6021 7638 7656 7673 9486 9506 9525 76 49* 51 S3* 77 78 79 113 316 320 767 772 778 1421 1428 1436 2275 2284 2294 3329 3340 3352 4583 4596 4610 6037 6052 6ob8 7691 7708 7726 9545 9564 9584 41 55 57* 59* 80 324 784 1444 2304 3364 4624 6084 7744 9604 00 61 81 82 83 327 331 334 789 1451 2313 2323 2332 3375 4637 4651 4664 6099 6115 6130 7761 ,7779 7796 9623 9662 61 It 63* 67* 338 342 345 806 812 817 1482 1489 2342 2361 3410 3422 3433 4678 4692 4705 6146 6162 6177 7814 7832 7849 9682 9702 9721 56 25 96 69* 71 73* 89 349 353 357 823 829 835 1497 1505 1513 2371 2381 2391 3445 3457 3469 4719 4733 4747 6193 6209 6225 7867 7885 7903 9741 9761 9781 69 44 21 75* 77* 79* 90 361 841 1521 2401 3481 4761 6241 7921 9801 00 81 91 92 93 372 846 852 858 1528 1536 1544 2410 2420 2430 3492 4774 4788 6256 6272 6288 7938 7956 7974 9820 9840 9860 81 64 49 83* 85* 87* 94 384 864 870 876 1560 1568 2440 2450 2460 3528 3540 3552 4816 4830 4844 6304 6320 6336 7992 8010 8028 9880 9900 9920 36 89* 9x* 93* 99 388 392 396 882 888 894 1592 2470 2480 2490 3564 4858 6pi 6384 8046 8064 8082 9940 9960 9980 09 04 01 99* 100 400 900 1600 2500 3600 4900 6400 8100 lOOOO 00 MATHEMATICAL SERIES While this series has been planned to meet the needs of the student who is preparing for engineer- ing work, it is hoped that it will serve equally well the purposes of those schools where mathematics is taken as an element in a liberal education. In order that the applications introduced may be of such char- acter as to interest the general student and to train the prospective engineer in the kind of work which he is most likely to meet, it has been the policy of the editors to select as joint authors of each text, a mathematician and a trained engineer or physicist. The problems as well as the applications intro- duced in the text are of such a character as to draw upon the student's general information which will be of use to him later in the application of mathe- matics. Without sacrificing the value of mathe- matical study as a discipline, it is the purpose of the series so to correlate the mathematics with the phy- sical applications as to stimulate the interest and train the student to use his mathematics as a means of investigation and stating the laws of physical phenomena. The following texts have appeared : I. Calculus. By E. J. TowNSEND, Professor of Mathematics in the University of Illinois, and G. A. Goodenough, Professor of Mechanical Engineering, University of Illinois. $2.50. II. College Algebra. By H. L. RiETZ, Assistant Professor of Mathematics in the University of Illinois, and Dr. A. R. Crathorne, Asso- ciate in Mathematics in the University of Illinois. $1.40. III. Trigonometry. By A. G. Hall, Professor of Mathematics in the Uni- versity of Michigan, and F. G. Frink, Professor of Railway Engineering in the University of Oregon. $1.25. HENRY HOLT AND COMPANY NEW YORK CHICAGO ENGINEERING BOOKS Hoskins's Hydraulics. By L. M. HosKiNS, Professor in Leland Stanford Uni- versity. 8vo. 271 pp. $2.50. A comprehensive text-book, intended for the fundamental course in the subject usually offered in schools of engineering, but somewhat more com- pact in treatment than the ordinary treatise now available. Russell's Text-book on Hydraulics. By George E. Russell, Assistant Professor of Civil En- gineering, Massachusetts Institute of Technology. viii + 183 pp. 8vo. $2.50. This book is designed primarily for classroom use rather than for reference for practicing engi- neers. It avoids discussion of specialized topics which are taught separately with special books and devotes itself to the consideration of the more com- mon and important subjects. At the end of each chapter are given problems to illustrate the appli- cation of the principles just preceding. Benjamin's Machine Design. By Charles H. Benjamin, Professor in Purdue Uni- versity. i2mo. 202 pp. $2.00. Machinery : — We know of no v^^ork on Machine Design which can be more heartily recommended to the average student than this. . . . The work has the characteristics of Professor Benjamin's writing in general ; that is, clearness and simplicity. It is brought up to date, containing, for example, a summary of the paper on the collapsing strength of lap-welded steel tubes presented by Professor Stewart before the spring meeting of the A. S. M. E. in 1906. Leffler's The Elastic Arch. With special reference to the Reinforced Concrete Arch. By Burton R. Leffler, Engineer of Bridges on the Lake Shore and Michigan Southern Railway. viii-}-57 pp. i2mo. $1.00. HENRY HOLT AND COMPANY NEW YORK CHICAGO 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 290ct59lii n^C'D LP PCTisigsg 8JuI'6UM RECn 1 D JUN 841961 FEB 7 1969 3 8 RECEIVED — M^*^ 25 '69 -8 AM LOAN DEPT. MAR 6 1975 4 REC. CIR. MAR 6 '75 KTTt 1^ Wl jtECCIR- J>Pli?9 31 LD 21A-50m-4,'59 (A1724sl0)476B General Librsuy University of California Berkeley /'^ 5'S %3 - t^ '^ZZ^^ 918r8e< - • ^^^5"^^ //2r THE UNIVERSITY OF CALIFORNIA LIBRARY W // ^ J ^,r <^-^/^