THE THE GIFT OF l _Dr. Ja__is o, _ l ______ lte l'itriT~il^ PLANE AND SPHERICAL TRIGONOMETRY, SURVEYING AND TABLES BY G. A. WENTWORTH, A.M. AUTHOR OF A SERIES' OF TEXT-BOOKS IN MATHEMATICS REVISED EDITION BOSTON, U.S.A., AND LONDON GINN & COMPANY, PUBLISHERS 1895 Entered, according to Act of Congress, in the year 1882, by G. A. WENTWORTH in the Office of the Librarian of Congress, at Washington. Copyright, 1895, by G. A. WENTWORTH. I I PREFACE. IN preparing this work the aim has been to furnish just so much of Trigonometry as is actually taught in our best schools and colleges. Consequently, all investigations that are important only for the special student have been omitted, except the development of functions in series. The principles have been unfolded with the utmost brevity consistent with simplicity and clearness, and interesting problems have been selected with a view to awaken a real love for the study. Much time and labor have been spent in devising the simplest proofs for the propositions, and in exhibiting the best methods of arranging the logarithmic work. The object of the work on Surveying is to present this subject in a clear and intelligible way, according to the best methods in actual use; and also to present it in so small a compass that students in general may find the time to acquire a competent knowledge of this very interesting and important study. The author is under particular obligation for assistance to G. A. Hill, A.M., of Cambridge, Mass., to Prof. James L. Patterson, of Schenectady, N.Y., to Dr. F. N. Cole, of Ann Arbor, Mich., and to Prof. S. F. Norris, of Baltimore, Md. G. A. WENTWORTH. EXETER, N.H., July, 1895 C ON TENTS. PLANE TRIGONOMETRY. CHAPTER I. FUNCTIONS OF ACUTE ANGLES: Angular measure, page 1; trigonometric functions, 3; representation of functions by lines, 7; changes in the functions as the angle changes, 10; functions of complementary angles, 11; relations of the functions of an angle, 12; formulas for finding all the other functions of an angle, when one function of the angle is given, 15; functions of 45~, 30~, 60~, 17. CHAPTER II. THE RIGHT TRIANGLE: Given parts of a triangle, 19. Solutions without logarithms, 19; Case I., when an acute angle and the hypotenuse are given, 19; Case II., when an acute angle and the opposite leg are given, 20; Case III., when an acute angle and an adjacent leg are given, 20; Case IV., when the hypotenuse and a leg are given, 21; Case V., when the two legs are given, 21. General method of solving a right triangle, 22; solutions by logarithms, 24; area of the right triangle, 26; the isosceles triangle, 31; the regular polygon, 33. CHAPTER III. GONIOMETRY: Definition of goniometry, 36; angles of any magnitude, 36; general definitions of the functions of angles, 37; algebraic signs of the functions, 39; functions of a variable angle, 40; functions of angles greater than 360~, 42; formulas for acute angles extended to all angles, 43; reduction of the function of all angles to the functions of angles in the first quadrant, 46; functions of angles that differ by 90~, 48; functions of a negative angle, 49; functions of the sum of two angles, 51; functions of the difference of two angles, 53; functions of twice an angle, 55; functions of half an angle, 55; sums and differences of functions, 56. CHAPTER IV. THE OBLIQUE TRIANGLE: Law of sines, 60; law of cosines, 62; law of tangents, 64. Solutions: Case I., when one side and two angles are given, 64; Case II., vi TRIGONOMETRY. when two sides and the angle opposite to one of them are given, 66; Case III., when two sides and the included angle are given, 71; Case IV., when the three sides are given, 74; area of a triangle, 78-79. CHAPTER V. MISCELLANEOUS EXAMPLES: Plane Trigonometry, 82-99; goniometry, 99-105. EXAMINATION PAPERS, 106-116. CHAPTER VI. CONSTRUCTION OF TABLES: Logarithms, 117; exponential and logarithmic series, 120; trigonometric functions of small angles, 125; Simpson's method of constructing a trigonometric table, 127; De Moivre's theorem, 128; expansion of sin x, cos x, and tan x, in infinite series, 132. SPHERICAL TRIGONOMETRY. CHAPTER VII. THE RIGHT SPHERICAL TRIANGLE: Introduction, 135; formulas relating to right spherical triangles, 137; Napier's rules, 141. Solutions: Case I., when the two legs are given, 142; Case II., when the hypotenuse and a leg are given, 142; Case III., when a leg and the opposite angle are given, 143; Case IV., when a leg and an adjacent angle are given, 143; Case V., when the hypotenuse and an oblique angle are given, 144; Case VI., when the two oblique angles are given, 144. The isosceles spherical triangle, 149. CHAPTER VIII. THE OBLIQUE SPHERICAL TRIANGLE: Fundamental formulas, 150; formulas for half angles and sides, 152; Gauss's equations and Napier's analogies, 154. Solutions: Case I., when two sides and the included angle are given, 156; Case II., when two angles and the included side are given, 158; Case III., when two sides and an angle opposite to one of them are given, 160; Case IV., when two angles and a side opposite to one of them are given, 162; Case V., when the three sides are given, 163; Case VI., when the three angles are given, 164. Area of a spherical triangle, 166. CHAPTER IX. APPLICATIONS OF SPHERICAL TRIGONOMETRY: To reduce an angle measured in space to the horizon, 170; to find the distance between two places on the earth's surface, when the latitudes of the places and the difference in their longitudes are known, 171; the celestial sphere, 171; spherical co-ordinates, 174; the astronomical triangle, 176; astronomical problems, 177-185. CONTENTS. vii SURVEYING. CHAPTER I. DEFINITIONS. INSTRUMENTS AND THEIR USES: Definitions, 135; instruments for measuring lines, 136; chaining, 136; obstacles to chaining, 138; the surveyor's compass, 141; uses of the compass, 143; verniers, 145; the surveyor's transit, 149; uses of the transit, 150; the theodolite, 150; the railroad compass, 150; plotting, 153. CHAPTER II. LAND SURVEYING: Determination of areas, 155; rectangular surveying, 159; field notes, computation, and plotting, 160; supplying omissions, 164; irregular boundaries, 164; obstructions, 164; modification of the rectangular method, 167; variation of the needle, 168; methods of establishing a true meridian, 170; dividing land, 173; United States public lands, 176; Burt's solar compass, 177; laying out the public lands, 179; Plane-table surveying, 181; the three-point problem, 186. CHAPTER III. TRIANGULATION: Introductory remarks, 187; the measurement of base lines, 188; the measurement of angles, 189. CHAPTER IV. LEVELLING: Definitions, 190; the Y level, 191; the levelling-rod, 191; differ-.ence of level, 192; levelling for section, 195; substitutes for the Y level, 198; topographical levelling, 200. CHAPTER V. RAILROAD SURVEYING: General remarks, 202; cross-section work, 202; railroad curves, 203. PLANE TRIGONOMETRY. CHAPTER I. TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES. ~ 1. ANGULAR MEASURE. As lengths are measured in terms of various conventional units, as the foot, meter, etc., so different units for measuring angles are employed, or have been proposed. In the common or sexagesimal system the circumference of a circle is divided into 360 equal parts. The angle at the centre subtended by each of these parts is taken as the unit angle and is called a degree. The degree is subdivided into 60 minutes, and the minute into 60 seconds. A right angle is equal to 90 degrees. NOTE. The sexagesimal system was invented by the early Babylonian astronomers in conformity with their year of 360 days. In the circular system an arc of a circle is laid off equal in length to the radius. The angle at the centre subtended by this arc is taken as the unit angle and is called a radian. The number of radians in 360~ is equal to the number of times the length of the radius is contained in the circumference. It is proved in Geometry that this number is 2r ( r= 3.1416) for all circles; therefore the radian is the same angle in all circles. 2 TRIGONOMETRY. Since the circumference of a circle is 2 X times the radius, 27r radians = 360~, and r radians =180~; 180~ 180~ Therefore, 1 radian - = 11 = 57~ 17' 45" rr 3.1416 and 1 degree — 0 radian = 0.017453 radian. 180 By the last two equations the measure of an angle can be changed from radians to degrees or from degrees to radians. 1800 Thus, 2 radians = 2 X - =2 X (57~ 17' 45") 114~ 35' 30". NOTE. The circular system came into use early in the last century. It is found more convenient in the higher mathematics, where the radians are simply expressed as numbers. Thus the angle r means 7r radians, and the angle 3 means 3 radians. On the introduction of the metric system of weights and measures at the close of the last century, it was proposed to divide the right angle into 100 equal parts called grades, which were to be taken as units. The grade was subdivided into 100 minutes and the minute into 100 seconds. This French or centesimal system, however, never came into actual use. EXERCISE I. [Assume r = 3.1416.] 1. Reduce the following angles to circular measure, expressing the results as fractions of 7. 60~, 45~, 150~, 195~, 11~ 15', 123~ 45', 37~ 30'. 2 3 2. How many degrees are there in 3r radians? 3 r radians? 8 Xr radians? 1- 7r radians? 7 7r radians? 5 15 7 r radians?- radians? ~ radians? 8 16 15 3. What decimal part of a radian is 1~? 1'? 4. How many seconds in a radian? TRIGONOMETRIC FUNCTIONS. 3 '5. Express in radians one of the interior angles of a regular octagon; dodecagon. 6. On a circle of 50 ft. radius an arc of 10 ft. is laid off; how many degrees does the arc subtend at the centre? 7. The earth's equatorial radius is approximately 3963 miles. If two points on the equator are 1000 miles apart, what is their difference in longitude? 8. If the difference in longitude of two points on the equator is 1~, what is the distance between them in miles? 9. What is the radius of a circle, if an arc of 1 foot subtends an angle of 1~ at the centre? 10. In how many hours is a point on the equator carried by the earth's rotation through a distance equal to the earth's radius? 11. The minute hand of a clock is 31 ft. long; how far does its extremity move in 25 minutes? [Take r =272.] 12. A wheel makes 15 revolutions a second; how long does it take to turn through 4 radians? [Take Xr = 22-.] ~ 2. THE TRIGONOMETRIC FUNCTIONS. The sides and angles of a plane triangle are so related that any three given parts, provided at least one of them is a side, determine the shape and the size of the triangle. Geometry shows how, from three such parts, to construct the triangle and find the values of the unknown parts. Trigonometry shows how to compute the unknown parts of a triangle from the numerical values of the given parts. Geometry shows in a general way that the sides and angles of a triangle are mutually dependent. Trigonometry begins by showing the exact nature of this dependence in the right triangle, and for this purpose employs the ratios of its sides. 4 TRIGONOMETRY. Let MAN (Fig. 1) be F/ p8 an acute angle. If from any points B. D, F...... in one of its sides M / perpendiculars BC, DE, FG,..... are let fall to the other side, then the right triangles ABC, ADE, AFG,..... thus formed have the angle A common, and are therefore mutually equiangular and similar. Hence, the ratios of their corresponding sides, pair by A U k' U FIG. 1. pair, are equal. That is '3 AC AE AG AC AEB AG AB- AD AF'; BC -DEE FG' These ratios, therefore, remain unchanged so long as the angle A remains unchanged. Hence, for every value of an acute angle A there are certain numbers that express the values of the ratios of the sides in all right triangles that have this acute angle A. There are altogether six different ratios: I. The ratio of the opposite leg to the.hypotenuse is called the Sine of A, and is written sin A. II. The ratio of the adjacent leg to the hypotenuse is called the Cosine of A, and written cos A. III. The ratio of the opposite leg to the adjacent leg is called the Tangent of A, and written tan A. IV. The ratio of the adjacent leg to the opposite leg is called the Cotangent of A, and written cot A. V. The ratio of the hypotenuse to the adjacent leg is called the Secant of A, and written sec A. VI. The ratio of the hypotenuse to the opposite leg is called the Cosecant of A, and written csc A. These six ratios are called the Trigonometric Functions of the angle A. TRIGONOMETRIC FUNCTIONS. 5 To these six ratios are often added the two following functions, which also depend only on the angle A: VII. The versed sine of A is 1 - cos A and is written vers A. VIII. The coversed sine of A is 1 —sin A and is written covers A. In the right triangle ABC (Fig. 2) let a, b, c denote the lengths of the sides opposite to the acute angles A, B, and the right angle C, respectively, these lengths being all expressed in terms of a common unit. Then, a opposite leg sin A -- COS c hypotenuse a opposite leg tan A adjacent leg cot c hypotenuse sec A= =- ego see b adjacent leg' b c-b vers A -- -- --. cov 6 C >B C., U A b A./ I C FIG. 2. b adjacent leg c hypotenuse b adjacent leg a opposite leg' c hypotenuse a opposite leg' a c-a 'ersA= — =- c. c c EXERCISE II. 1. What are the functions of the triangle ABC (Fig. 2)? 2. If A4 + B=- 90~, prove sin A = cos B, cos A = sin B, tan A - cot B, cot A =-tanB, the other acute angle B of sec A = csc B, csc A = see B, vers A = covers B, covers A = vers B. 6 TRIGONOMETRY. 3. Find the values of the functions of A, if a, b, c respectively have the following values: (i.) 3, 4, 5. (ii.) 5, 12, 13. (iii.) 8, 15, 17. (v.) 3.9, 8, 8.9. (iv.) 9, 40, 41. (vi.) 1.19, 1.20, 1.69. 4. What condition must be fulfilled by the lengths of the three lines a, b, c (Fig. 2) in order to make them the sides of a right triangle? Is this condition fulfilled in Example 3? 5. Find the values of the functions of A, if a, b, c respectively have the following values: (i.) 2mn, m2 — n2, m2- n2. (iii.) pqr, qrs, rsp.. 2xy. y- Y2_ mn mv nr (ii.) - x + y, ~ (iv.) n, nr X-Y [-y pq sq '-Ps x - y' x-y p' sg 's 6. Prove that the values of a, b, c, in (i.) and (ii.), Example 5, satisfy the condition necessary to make them the sides of a right triangle. 7. What equations of condition must be satisfied by the values of a, b, c, in (iii.) and (iv.), Example 5, in order that the values may represent the sides of a right triangle? Compute the functions of A and B when, 8. a =24, b 143. 9. a=0.264, c=0.265. 10. b-9.5, c=19.3. Compute the functions of A 14. a=2b. 15. a=-c. 11. a=-Vp2~~, 6=1V2p. 12. a = p2+pq, c=p+. 13. 6=2 pq, c= p+q. when, 16. a+b-6=c. 17. a-b=4 18. Find a if sin A = and c = 20.5. 19. Find b if cos A = 0.44 and c = 3.5. 20. Find a if tan A = -J and b =-2r. TRIGONOMETRIC FUNCTIONS. 21. Find b if cot A 4 and a -17. 22. Find c if sec A = 2 and b = 20. 23. Find c if csc A = 6.45 and a = 35.6. Construct a right triangle: given, 24. c=6, tan`=A. 26. b-2, sinA =0.6. 25. a=3.5, cosA=. 27. b 4, csc A = 4. 28. In a right triangle, c =2.5 miles, sin A = 0.6, cos A= 0.8; compute the legs. 29. Construct (with a protractor) the A 20~, 40~, and 70~; determine their functions by measuring the necessary lines, and compare the values obtained in this way with the more correct values given in the following table: sin cos tan cot sec csc 20~ 0.342 0.940 0.364 2.747 1.064 2.924 40~ 0.643 0.766 0.839 1.192 1.305 1.556 70~ 0.940 0.342 2.747 0.364 2.924 1.064 30. Find, by means of the above table, the legs of a right triangle if A = 20~, c = 1; also if A = 20, c = 4. 31. In a right triangle, given a =3 and c- 5; find the hypotenuse of a similar triangle in which a = 240,000 miles. 32. By dividing the length of a vertical rod by the length of its horizontal shadow, the tangent of the angle of elevation of the sun at the time of observation was found to be 0.82. How high is a tower, if the length of its horizontal shadow at the same time is 174.3 yards? ~ 3. REPRESENTATION OF THE FUNCTIONS BY LINES. The functions of an angle, being ratios, are numbers; but we may represent them by lines if we first choose a unit of length, and then construct right triangles, such that the 8 TRIGONOMETRY. denominators of the ratios shall be equal to this unit. The most convenient way to do this is as follows: A - o M B/ FIG. 3. About a point 0 (Fig. 3) as a centre, with a radius equal to one unit of length, describe a circle and draw two diameters AA' and BB' perpendicular to each other. The circle with radius equal to 1 is called a unit circle, AA' the horizontal, and BB' the vertical diameter. Let A OP be an acute angle, and let its value (in degrees, etc.) be denoted by x. We may regard the Z x as generated by a radius OP that revolves about 0 from the position OA to the position shown in the figure; viewed in this way, OP is called the moving radius. Draw PM _ to OA, PN I OB. In the rt. A OPM~ the hypotenuse OP = 1; therefore, sin x = PM; cos x = OM. Since PM is equal to ON, and ON is the projection of OP on BB', and since OM is the projection of OP on AA', therefore, in a unit circle, sin x = projection of moving radius on vertical diameter; cos x= projection of moving radius on horizontal diameter. Through A and B draw tangents to the circle meeting OP, produced in T and S, respectively; then, in the rt. A OAT, the leg OA = 1, and in the rt. A OBS, the leg OB = 1; while the Z OSB = L x. Therefore, tan x =AT; secx = OT; cot x BS; cse x = OS; vers x - AM; covers x = BN. These eight line values (as they may be termed) of the functions are all expressed in terms of the radius of the circle as a unit; and it is clear that as the angle varies in value the TRIGONOMETRIC FUNCTIONS. 9 line values of the functions will always remain equal numerically to the ratio values. Hence, in studying the changes in the functions as the angle is supposed to vary, we may employ the simpler line values instead of the ratio values. EXERCISE III. 1. Represent by lines the functions of a larger angle than that shown in Fig. 3. If x is an acute angle, show that 2. sin x is less than tan x. 3. sec x is greater than tan x. 4. cscx is greater than cotx. Construct the angle x if, 5. tan x -3. 7. cos x -. 9. sin x = 2 cos x. 6. csc x 2. 8. sin x = cos x. 10. 4 sin x - tan x. /11. Show that the sine of an angle is equal to one-half the chord of twice the angle. 12. Find x if sin x is equal to one-half the side of a regular inscribed decagon. 13. Given x and y, x + y being less than 90~; construct the value of sin (x + y) - sin x. 14. Given x and y, x-+-y being less than 90~; construct the value of tan (x + y) - sin (x + y) + tanx - sinx. Given an angle x; construct an angle y such that, 15. siny= 2sinx. 17. tany =3tanx. 16. cosy == cosx. 18. secy= cscx. 19. Show by construction that 2 sin A > sin 2 A. 20. Given two angles A and B, A +-B being less than 90~; show that sin (A + B) < (sin A + sin B). 21. Given sin x in a unit circle; find the length of a line corresponding in position to sin x in a circle whose radius is r. 22. In a right triangle, given the hypotenuse c, and also sin A - m, cos A= n; find the legs. 10 TRIGONOMETRY. ~ 4. CHANGES IN THE FUNCTIONS AS THE ANGLE CHANGES. If we suppose the / AOP, or x (Fig. 4) to increase gradually by the revolution of the moving radius OP about 0, the point P will move along the arc Tl AB towards B, T will move along the tangent AT away from A, S will move along the tangent BS towards t B, and M will move along the radius,, / / OA towards O. B SI^ a d S Hence, the lines PM, AT, OT will //\ T \gradually increase in length, and the lines OM, BS, OS will gradually decrease. That is, As an acute angle increases, its 0 - MM-S~T sine, tangent, and secant also increase, FIG. 4. while its cosine, cotangent, and cosecant decrease. On the other hand, if we suppose x to decrease gradually, the reverse changes in its functions will occur. If we suppose x to decrease to 0~, OP will coincide with OA and be parallel to BS. Therefore, PM and AT will vanish, OM will become equal to OA, while BS and OS will each be infinitely long, and be represented in value by the symbol oo. And if we suppose x to increase to 90~, OP will coincide with OB and be parallel to AT. Therefore, PM and OS will each be equal to OB, OM and BS will vanish, while AT and OT will each be infinite in length. Hence, as the angle x increases from 0~ to 90~, sinx increases from 0 to 1, cos x decreases from 1 to 0, tan x increases from 0 to oo, cot x decreases from oo to 0, secx increases from 1 to oo, csc x decreases from oo to 1. TRIGONOMETRIC FUNCTIONS. 11 The values of the functions of 0~ and of 90~ are the limiting values of the functions of an acute angle. It is evident that (disregarding the limiting values), Sines and cosines are always less than 1; Secants and cosecants are always greater than 1; Tangents and cotangents have all values between 0 and oo. REMARK. We are now able to understand why the sine, cosine, etc., of an angle are called functions of the angle. By a function of any magnitude is meant another magnitude which remains the same so long as the first magnitude remains the same, but changes in value for every change in the value of the first magnitude. This, as we now see, is the relation in which the sine, cosine, etc., of an angle stand to the angle. ~ 5. FUNCTIONS OF COMPLEMENTARY ANGLES. The general form of two complementary angles is A and 90~-A. In the rt. A ABC (Fig. 5), A + B =900; hence B = 90 - A. B Therefore (~ 2), sin A = cos B = cos (90~ - A), a cos A - sin B = sin (90~ - A), tan A = cot B = cot (90~ - A), cot A =tan B = tan (90~ - A), A bC sec A = csc B = csc (90~ - A), FIG. 5. csc A = sec B = sec (90~ - A), Therefore, Each function of an acute angle is equal to the co-named function of the complementary angle. NOTE. Cosine, cotangent, and cosecant are sometimes called cofunctions; the words are simply abbreviated forms of complement's sine, complement's tangent, and complement's secant. Hence, also, Any function of an angle between 45~ and 90~ may be found by taking the co-named function of the complementary angle between 0~ and 45~. 2 TRIGONOMETRY. EXERCISE IV. 1. Express the following functions as functions of the complementary angle: sin 30~. tan 89. csc 18~ 10'. cot 82~ 19'. cos 45~. cot 15~. cos 37~ 24'. cse 54 46'. 2. Express the following functions as functions of an angle less than 45~: sin 60~. tan 57~. cse 690 2'. cot 89~ 59', cos 75~. cot 84~. cos 85~ 39'. csc 4501'. 3. Given tan 30~ =- V/3; find cot 60~. 4. Given tan A = cot A; find A. 5. Given cos A = sin 2 A; find A. 6. Given sin A = cos 2 A; find A. 7. Given cos A = sin (45~ - A); find A. 8. Given cot 1 A = tan A; find A. 9. Given tan (45~ + A) = cot A; find A. 10. Find A if sin A cos 4 A. 11. Find A if cot A tan 8 A. 12. Find A if cot A - tan nA. ~ 6. RELATIONS OF THIE FUNCTIONS OF AN ANGLE. Formula [1]. Since (Fig. 5) a + 2= 2, therefore, a2 12 a 2 b c c —2 or Therefore (~ 2), (sin A)2 + (cos A)2 =1; or, as usually written for convenience, sin2A + cos2A = 1. [1] That is: The sum of the squares of the sine and the cosine of an angle is equal to unity. TRIGONOMETRIC FUNCTIONS. 13 Formula [1] enables us to find the cosine of an angle when the sine is known, and vice versa. The values of sin A and of cos A deduced from [1] are: sin A = /1 - cos2A, cos A = 1 - sin2A. Formula [2]. Since a b a c a ' C - X c c b b therefore (~ 2), tan A siA [2].cos A[ That is: The tangent of an angle is equal to the sine divided by the cosine. Formula [2] enables us to find the tangent of an angle when the sine and the cosine are known. Formula [3]. Since a c b e a b - — 1, -X 1, and X-=1, c a C b b a therefore (~ 2), sin A X csc A 1 cos A X sec A -1 [3] tanAX cotA- 1J That is: The sine and the cosecant of an angle, the cosine and secant,, and the tangent and cotangent, pair by pair, are reciprocals. The equations in [3] enable us to find an unknown function contained in any pair of these reciprocals when the other function in this pair is known. 14 TRIGONOMETRY. EXERCISE V. 1. Prove Formulas [1] - [3], using for the functions the line values in the unit circle given in ~ 3. Prove that 2. 1 + tan2A sec2A. 3. 1 + cot2A csc2A. NOTE. -Equations 2 and 3 should be remembered. cos A 4. cot A -- sin A 5. sin A sec A = tanA. 6. sinA cot A - cos A. 7. cos A cscA = cotA. 8. tanA cos A = sin A. 9. sinA secA cotA =1. 10. cos A csc A tan A- 1. 11. (1 — sin2A) tan2A. = sin2A. 12. i -- cos2A cot A = cos A. 13. (1 + tan2A) sin2A = tan2A. 14. csc2A (1 - sin2A)= cot2A. 15. tan2A cos2A + cos2A = 1. 16. (sin2A - cos2A) =1 - 4 sin2A cos2A. 17. (1- tan2A)2 = sec4A- 4 tan2A. sinA cos A 18. + = sec A csc A. cos A sin A 19. sin4A - cos4A = sin2A - cos2A. 20. sec A -cos A = sin A tan A. 21. csc A- sin A = cos A cot A. cosA 1+ sinA 1 -- sinA cos A TRIGONOMETRIC FUNCTIONS. 15 ~ 7. APPLICATION OF FORMULAS [1] - [3]. Formulas [1], [2], and [3] enable us, when any one function of an angle is given, to find all the others. A given value of any one function, therefore, determines all the others. EXAMPLE L Given sin A —; find the other functions. By [1], cos A= Vi-t = V = V5. By[2], tan A= - - 5= X - 3 3 3 3 <5 3 3 By [3], cot A= -, sec A -, csc A- EXAMPLE 2. Given tan A = 3; find the other functions. sin A By [2], cos A And by [1], sin2A + cos2A =1. If we solve these equations (regarding sin A and cos A as two unknown quantities), we find that, sin A =3 V/, cos A = /I. Then by [3], cot A = ', sec A = V10, csc A= V/10. EXAMPLE 3. Given sec A = m; find the other functions. By [3], cos A - By [I], sin A = - -,= 2- =-Vm2-_1. By [2], [3], tan A= Vm2-1, cot A= V, ~ 1' csc A \m2-_ 16 TRIGONOMETRY. EXERCISE VI. Find the values of the other functions, when 1. sin A-==. 5. tan A=. 9. csc A= 2. 2. sinA =0.8. 6. cot A=. 10. sin A -m. 2m 3. cosA- =. 7. cot A=0.5. 11. sin A= -- 1 + 1 m2 2 mn 4. cosA = 0.28. 8. sec A= 2. 12. cos A - 2 13. Given tan 45~ = 1; find the other functions of 45~. 14. Given sin 30~-=; find the other functions of 30~. 15. Given csc 60~ =- V3; find the other functions of 60~. 16. Given tan 15~ =2 - V3; find the other functions of 15~. 17. Given cot 22~ 30'= /2+1; find the other functions of 22~ 30'. 18. Given sin 0~ =0; find the other functions of 0~. 19. Given sin 90~ = 1; find the other functions of 90~. 20. Given tan 90~ = oo; find the other functions of 90~. 21. Express the values of all the other functions in terms of sin A. 22. Express the values of all the other functions in terms of cos A. 23. Express the values of all the other functions in terms of tan A. 24. Express the values of all the other functions in terms of cot A. 25. Given 2 sin A =cos A; find sin A and cos A. 26. Given 4 sin A = tan A; find sin A and tan A. 27. If sin A: cos A = 9: 40, find sin A and cos A. 28. Transform the quantity tan2A- +cot2A- sin2A - cos2A into a form containing only cos A. 29. Prove that sin A + cos A = (1+ tan A) cos A. 30. Prove that tan A - cot A = sec A X csc A. TRIGONOMETRIC FUNCTIONS. 17 ~ 8. FUNCTIONS OF 45~. Let ABC (Fig. 6) be an isosceles right triangle, in which the length of the hypotenuse AB is equal to 1; then AC is equal to BC, and the angle A is equal to 45~. Since ACP-2 + BC2 1, therefore 2 AC2 = 1, and AC V/ -=1 /2. Therefore (~ 2), sin 45~ =cos 45~ = /2. tan 45 - cot 45~ 1. see 45= csc 45~ - 2. ~ 9. FUNCTIONS OF 30~ AND 60~. Let ABC be an equilateral triangle, in which the length of each side is equal to 1; and let CD bisect the angle C. Then CD is perpendicular to AB and bisects AB; hence, AD =, and CD = /1- I= V 3= V-3. In the right triangle ADC, the angle ACD = 30~, and the angle CAD= 60~. Whence (~ 2), sin 30~ = cos 60~ =. cos 30~ = sin 60~ =- /3. 1 tan 30~ - cot 60~ - 1 - 3. V3 cot 30~ - tan 60~ = /3. _ 2 sec 30~ - csc 60~ = = f /V3. csc 30~ = sec 60~ - 2. C i L i, cr A FG. FIG. 7. 6 I The results for sine and cosine of 30~, 45~, and 60~ may be easily remembered by arranging them in the following form: 18 TRIGONOMETRY. 30~ 45~ 60~ /1 = 0.5 /-i VA2 iV3 V2 = 0.70711 i3 V2 V1 2 3= 0.86603 EXERCISE VII. Solve the following equations: 1. 2 cos x sec x. 7. 3 tan2 x - sec2 x - 1. 2. 4sinx =cscx. 8. tan x - cot = 2. 3. tan x =2 sin x. 9. sin2 x - cos x. 4. sec x - /2 tan x. 10. tan2- sec x= 1. 5. sin2 x =3 cos2 x. 11. sin x+V/3cosx=2. 6. 2 sin2 x + cos2x = _. 12. tan2 x + csc2 = 3. 13. 2 cos x sec x= 3. 14. cos2 x - sin2 x =sin x. 15. 2 sin x+ cot x 1 + 2 cos x. 16. sin2 x + tan2 x 3 cos2 x. 17. tan x + 2 cot x =- csc x. NOTE. Wentworth & Hill's Five-place trigonometric and logarithmic tables have full explanations, and directions for using them. Before proceeding to Chapter II. the student should learn how to use these tables. Table VI. is to be used in solutions without logarithms. This fourplace table contains the natural functions of angles at intervals of 1'. The decimal point must be inserted before each value given, except where it appears in the values of the table. CHAPTER II. THE RIGHT TRIANGLE. ~ 10. THE GIVEN PARTS. IN order to solve a right triangle, two parts besides the right angle must be given, one of them at least being a side. The two given parts may be: I. An acute angle and the hypotenuse. II. An acute angle and the opposite leg. III. An acute angle and the adjacent leg. IV. The hypotenuse and a leg. V. The two legs. ~ 11. SOLUTION WITHOUT LOGARITHMS. The following examples illustrate the process of solution when logarithms are not employed. CASE I. Given A=43~ 17', c=26; find B, a, b. B 1. B = 90~ - A 46' 43'. 2. sinA;.-.a= sinA. a b 3. - - COSA;.. b - Cos AC. A. G. FIG. 8. 20 TRIGONOMETRY. sin A= 0.6856 c — 26 41136 13712 a = 17.8256 cos A= 0.7280 c —= 26 43680 14560 b = 18.9280 CASE II. Given A =13~ 58', a =15.2; find B, b, c. B c/ a A/ - I a 1. B= 90 -A -760 2'. b 2. -=cotA;.. b acotA. a a a. 3. -=sinA;.'.cc sin A b FIG. 9. v cot A 4.0207 a 15.2 80414 201035 40207 b -61.11464 a 15.2, sin A =0.2414 0.2414)15.200(62.9 14484 7160 4828 c - 62.9 2332 CASE III. Given A =27~ 12', b=31; find B, a, c. B a A/\ F. 10.b Fia. 10. 1. B=90-A =62 48'. 2. -=tan A;.. a=btanA. b b 3. -=cosA;.'.c=-s C cos A THE RIGHT TRIANGLE. 21 tan A 0.5139 b= 31 5139 15417 a = 15.9309 b = 31, cos A = 0.8894 0.8894)31.000(34.8 26 682 43180 3 5576 c - 34.8 7604 CASE IV. Given a =47, c = 63; find A, B, b. 1. a. a 1. sin A=2. B =90~ A. 3. b =c2- =a= V( - a)(c + a). B c / a A y b FIG. 11. a =47, c =63 63)47.0(0.7460 441 290 252 sinA =0.7460 380.'.A =48 15 378 B 41~ 45' 2 c+a= 110 c- a - 16 660 110 b2 = 1760 b =- 1760 =41.95 CASE V. Given a =121, b = 37; find A, B, c. 1. tan A -- b 2. B 90- A. 3. c=-V2+b2. B A FI. FIG. 12. 22 TRIGONOMETRY. a =121, b= 37. a2= 14641 37)121(3.2703 b2= 1369 111 2=- 16010 100... = V16010 74 =126.5 tan A 3.2703 260. A —730 259 B =17~ 1 ~ 12. GENERAL METHOD OF SOLVING THE RIGHT TRIANGLE. From these five cases it appears that the general method of finding an unknown part in a right triangle is as follows: Choose from the equation A + B = 90~, and the equations that define the functions of the angles, an equation in which the required part only is unknown; solve this equation, if necessary, to find the value of the unknown part; then compute the value. NOTE. In Case IV., if the given sides (here a and c) are nearly alike in value, then A is near 90~, and its value cannot be accurately found from the tables, because the sines of large angles differ little in value (as is evident from Fig. 4). In this case it is better to find B first, by means of a formula proved later. See formula [18], ~ 30; viz., tan B = c - a. EXAMPLE. Given a = 49, c = 50; find A, B, b. c- a1=, c+a= 99. c-a= 1 c- -a c01010 +a99 = 0.01010 c+ a b2= 99 tan ~ B 0.1005 b = V99.. B = 5 44' = 9.95 B = 11~ 28' A = 78~ 32' THE RIGHT TRIANGLE. 23 EXERCISE VIII. 1. In Case II. give another way of finding c, after b has been found. 2. In Case III. give another way of finding c, after a has been found. 3. In Case IV. give another way of finding b, after the angles have been found. 4. In Case V. give another way of finding c, after the angles have been found. 5. Given B and c; find A, a, b. 6. Given B and b; find A, a, c. 7. Given B and a; find A, b, c. 8. Given b and c; find A, B, a. Solve the following right triangles: GIVEN. REQUIRED. 9 a= 3, b = 4. A = 36~ 52', B = 53~ 8', c = 5. 10 a= 7, c = 13. A = 28~18', B = 610 42', b = 10.954. 11 a= 5.3, A = 12~17'. B = 7743', b = 24.342, c = 24.918. 12 a = 10.4, B = 43~18'. A = 46042', b = 9.800, c = 14.290. 13 c = 26, A = 37042'. B = 52018', a = 15.900, b = 20.572. 14 c =140, B = 240 12'. A =65048', a = 127.694, b =57.386. 15 b = 19, c 23. A = 34018', B = 55042', a = 12.961. 16 b = 98, c = 135.2, A = 43~33', B = 46027', a = 93.139. 17 b 42.4, A = 32~ 14'. B = 57~46', a = 26.733, c = 50.124. 18 b = 200, B = 46 11'. A 43~49', a = 191.900, c = 277.160. 19 a= 95, b = 37. A = 68043', B = 21~ 17', c= 101.951. 20 a = 6, c 103. A = 321', B = 86~39', b = 102.825. 21 a 3.12, B=508'. A =84~52', b=0.280, c=3.133 22 a = 17, c 18. A = 70048', B = 19012', b = 5.916. 23 c = 57, A = 3829'. B = 51031', a = 35.471, b = 44.620. 24 a+ c-18, b 12. A 22~37', B-= 6723', a= 5, c= 13. 25 a+b-9. c=8. A=82018, B'7 42', b-7.07928..~ b = 1.072. 24 TRIGONOMETRY. ~ 13. SOLUTION BY LOGARITHMS. CASE I. Given A= 34~28', c 18.75; find B, a, b. iB 1. B=900-A=550321. 2. -_=sinA;..a - csinA. 3. -- os A;.. bc cos A. o * FIG. 13. log a = log c + log sin A log c = 1.27300 log sin A 9.75276-10 log a = 1.02576 a -- 10.61 log b = log c + log cos A log c = 1.27300 logcos A= 9.91617-10 log b = 1.18917 b 15.459 CASE II. Given A = 62~ 10', a= 78; find B, b, c. B cC a A I b FIG. 14. 1. 2. B 90 - A - 27~ 50'. - =cotoA;.'. b= a cotA. a log b log a log cot logb b = log a + log cot A = 1.89209 A- 9.72262 —10 = 1.61471 - 41.182 3. - -sinA... a = csinA and c.'. a c sin A, and c -sinA log c =loga+c ologsinA log a = 1.89209 cologsinA= 0.05340 logc = 1.94549 c = 88.204 THE RIGHT TRIANGLE. 25 CASE III. Given A = 50~ 2', b =88; find B, a, c. T2 1. B-90~ -A -39~ 58'. 2. - tanA;.-.a-btanA. b 3. - cos A. c.'. b ccosA, and c — cos A log a - log b + log tanA lo log b 1.94448 lo logtanA= 10.07670-10 cc loga = 2.02118 lo a = 105.0 CASE IP a CZ A b I ~ FIG. 15. gc =-logb-+cologcosA gb - 1.94448 4logcosA= 0.19223 gc = 2.13671 c =-137.0 7. Given c 58.40, a =47.55; find A, B, b. a 1. sinA — 2. B =90~- A. b 3. — =cotA;.-.b=acotA. a log sin A - log a + colog c log a =1.67715 cologc =8.23359-10 log sin A = 9.91074-10 - 54~ 31' -= 35 29' B AFIG. 1 FIG. 16. log b = log a +- log cot A log a = 1.67715 log cot A= 9.85300 -10 log b = 1.53015 b = 33.896 26 TRIGONOMETRY. CASE V. Given a = 40, b 27; find A, B, c. 13 a 1. tanA=- /y^~ ~~~ b c/ a- 2. B =90~-A. 3. -- sinA. A^ a A b.. a =csin A;.c FIG. 17. sin A log tan A = log a + colog b logc =loga+ cologsinA log a = 1.60206 log a = 1.60206 cologb = 8.56864-10 cologsinA= 0.08152 log tan A 10.17070 -10 log c = 1.68358 A-=55 59'c =48.259 B-34 1' NOTE. In Cases IV. and V. the unknown side may also be found from the equations (for Case IV.) b = c2 —2 - (c= a)(c - a); (for Case V.) c = Va2 + b2. These equations express the values of b and c directly in terms of the two given sides; and if the values of the sides are simple numbers (e.g. 5, 12, 13), it is often easier to find b or 6 in this way. But this value of c is not adapted to logarithms, and this value of b is not so readily worked out by logarithms as the value of b given under Case IV. See also ~ 12, Note. ~ 14. AREA OF THE RIGHT TRIANGLE. It is shown in Geometry that the area of a triangle is equal to one-half the product of the base by the altitude. Therefore, if a and b denote the legs of a right triangle, and F the area, F-= ab. By means of this formula the area may always be found when a and b are given or have been computed. THE RIGHT TRIANGLE. For example: Find the area, having given: CASE I. (~ 13). A =34~ 28', c -18.75. First find (as in ~ 13) log a and log b. log F= log a + log b + colog 2 loga = 1.02578 log b = 1.18915 colog2 = 9.69897 - 10 log F= 1.91390 F =82.016 CASE IV. (~ 13). a = 47.54, c= 58.40. First find (as in ~ 13) loga and log b. logF = loga + log b + colog 2 log a =1.67715 logb =1.53025 colog 2 = 9.69897 -10 logF =2.90637 F =806.06 EXERCISE IX. Solve the following triangles, finding the angles to the nearest minute: GIVEN: REQTUIRED: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a=6, c= 12. A = 60, b = 4. A = 30~, a = 3. a=4, b=4. a =2, c = 2.82843. c = 627, A = 23 30'. c =2280, A = 28~ 5'. c= 72.15, A=39~34'. c=, A = 36~. c = 200, B = 21 47'. c = 93.4, B= 76 25'. a = 637, A= 4035'. a = 48.532, A = 36~ 44'. a =0.0008, A = 86~. b = 50.937, B= 43~ 48'. b =2, B= 3~38'. A = 30~, B = 60~, B = 30~, c = 8, B = 60~, c = 6, A = B = 45~, c = 5.6568. A = B = 45~, b = 2. B = 66030', a =250.02, B=61~ 55', a= 1073.3, B =50~ 26', a=45.958, B = 54~, a = 0.58779 A=680 13', a =185.73, A =13~ 35', a=21.936, B=850 25', b = 7946, B=53~ 16', b=65.031, B = 4~, b = 0.00005 A =46~ 12', a=53.116, A = 86~ 22', a =31.497, b =10.392. a =- 6.9282. b =5.1961. b=- 575.0. b= 2011.6. b = 55.620., = 0.80902. b = 74.22. b = 90.788. c - 7971.5. c = 81.144. 59, c= 0.000802. c= 73.59. c = 31.560. 28 TRIGONOMETRY. GIVEN: REQUIRED: 17 a =992, B = 7619'. A==13041', b =4074.5, c= 4193.6. 18 a 73, B = 68052'. A-2108', =b188.86, c=202.47. 19 a-2.189, B=45025'. A=44035', b —2.2211, c=3.1185. 20 b 4, A=370 56'. B = 520 4', a =3.1176, c= 5.0714. 21 c =8590, a =4476. A = 31~ 24', B = 58~ 36', b = 7332.8. 22 c=86.53, a =71.78. A-56~ 3', B=33057', b=48.324. 23 c = 9.35, a =8.49. A =65~ 14', B= 24~ 46', b = 3.917. 24 c = 2194, b = 1312.7. A = 530 15', B = 36045', a = 1758. 25 c= 30.69, b= 18.256. A =53~30', B =36~30', a =24.67. 26 a=38.313, b= 19.522. A 63~, B=27~, c=43. 27 a-=1.2291, b=14.950. A= 4042', B-=85018', c=15. 28 a=415.38, b=62.080. A=81030', B= 8~30', c=420. 29 a=13.690, b=16.926. A =38058', B =5102', c= 21.769. 30 c=-91.92, a=2.19. A= 1~22', B=88038', b=91.894. Compute the unknown parts and also the area, having given: 31. a=5, 32. a -0.61 3 -33.b = 12, 34. a- 7, 35. b= 12, b = 6. 3, c = 70. c = 3. - = 18~ 14'. A == 29~ 8'. 36. c==68, A =69~ 54'. 37. c = 27, B — 44 4'. 38. a=47, B - 48 49'. 39. b =9, B = 34 44'. 40. c = 8.462, B = 86~ 4'. 41. Find the value of F in terms of c and A. 42. Find the value of F in terms of a and A. 43. Find the value of F in terms of b and A. 44. Find the value of F in terms of a and c. 45. Given F- =58, a= 10; solve the triangle. 46. Given F= 18 b- = 5; solve the triangle. 47. Given F-12, A = 29~; solve the triangle. 48. Given F-= 100, c- 22; solve the triangle. 49. Find the angles of a right triangle if the hypotenuse is equal to three times one of the legs. THE RIGHT TRIANGLE. 29 50. Find the legs of a right triangle if the hypotenuse = 6, and one angle is twice the other. 51. In a right triangle given c, and A - nB; find a and b. 52. In a right triangle the difference between the hypotenuse and the greater leg is equal to the difference between the two legs; find the angles. The angle of elevation of an object (or angle of depression, if the object is below the level of the observer) is the angle which a line from the eye to the object makes with a horizontal line in the same vertical plane. 53. At a horizontal distance of 120 feet from the foot of a steeple, the angle of elevation of the top was found to be 60~ 30'; find the height of the steeple. 54. From the top of a rock that rises vertically 326 feet out of the water, the angle of depression of a boat was found to be 24~; find the distance of the boat from the foot of the rock. 55. How far is a monument, in a level plain, fron the eye, if the height of the monument is 200 feet and the angle of elevation of the top 3~ 30'? 56. In order to find the breadth of a river a distance AB is measured along the bank, the point A being directly opposite a tree C on the other side. The angle ABC is also measured. If AB is 96 feet, and ABC is 21~ 14' find the breadth of the river. If ABC were 45~, what would be the breadth of the river? 57. Find the angle of elevation of the sun when a tower a feet high casts a horizontal shadow b feet long. Find the angle when a =120, b = 70. 58. How high is a tree that casts a horizontal shadow b feet in length when the angle of elevation of the sun is A~? Find the height of the tree when b 80, A = 50~. 30 TRIGONOMETRY. 59. What is the angle of elevation of an inclined plane if it rises 1 foot in a horizontal distance of 40 feet? 60. A ship is sailing due north-east with a velocity of 10 miles an hour. Find the rate at which she is moving due north, and also due east. 61. In front of a window 20 feet high is a flower-bed 6 feet wide. How long must a ladder be to reach from the edge of the bed to the window? 62. A ladder 40 feet long may be so placed that it will reach a window 33 feet high on one side of the street, and by turning it over without moving its foot it will reach a window 21 feet high on the other side. Find the breadth of the street. 63. From the top of a hill the angles of depression of two successive milestones, on a straight level road leading to the hill, are observed to be 5~ and 15~. Find the height of the hill. 64. A fort stands on a horizontal plain. The angle of elevation at a certain point on the plain is 30~, and at a point 100 feet nearer the fort it is 45~. How high is the fort? 65. From a certain point on the ground the angles of elevation of the belfry of a church and of the top of the steeple were found to be 40~ and 51~ respectively. From a point 300 feet farther off, on a horizontal line, the angle of elevation of the top of the steeple is found to be 33~ 45'. Find the distance from the belfry to the top of the steeple. 66. The angle of elevation of the top C of an inaccessible fort observed from a point A, is 12~. At a point B, 219 feet from A and on a line AB perpendicular to AC, the angle ABC is 61~ 45'. Find the height of the fort. THE ISOSCELES TRIANGLE. 31 ~ 15. THE ISOSCELES TRIANGLE. An isosceles triangle is divided by the perpendicular from the vertex to the base into two equal right triangles. Therefore, an isosceles triangle is determined by any two parts that determine one of these right triangles. Let the parts of an isosceles triangle ABC (Fig. 18), among which the altitude CD is to be included, be denoted as follows: C a - one of the equal sides, c - the base, h - the altitude, a/ 7 a A = one of the equal angles, C= the angle at the vertex. For example: Given a and c; re- A c D 1 /Sc B quired A, C, h. FIG.18. 1. cos == - —. a 2 a 2. C+ 2A =180~;.. C 180 - 2A= 2 (900- A). 3. h may be found by any one of the equations: c2 h 2 h2+ -j- 2= c si nA; si -===tanA; whence h = V(a - 1 c) (a+ ); = asinA; -- ctanA. The area F of the triangle may be found, when c and h are given or have been computed, by means of the formula F2 C ch 32 TRIGONOMETRY. EXERCISE X. Solve the following isosceles triangles, finding the angles to the nearest second: 1. Given a and A; find C, c, h. 2. Given a and C; find A, c, h. 3. Given c and A; find C, a, h. 4. Given c and C; find A, a, h. 5. Given A and A; find C, a, c. 6. Given h and C; findA, a, c. 7. Given a and h; find A, C, c. 8. Given c and h; findA, C, a. 9.I Given a = 14.3, c =11; find A, C, h. 10. Given a= 0.295, -A68 010'; find c, h,F. 11. Given c= 2.352, C 690 49'; find a, h, F. 12. Given A- 7.4847, A= 76~ 14'; find a, c, F. 13. Given a= 6.71, h ==6.60; find A, C, c. 14. Given c=9, h 20; find A, C, a. 15. Given c - 147, F=- 2572.5; find A, C, a, h. 16. Given hA=16.8, F=43.68, findA, C, a, c. 17. Find the value of F in terms of a and c. 18. Find the value of F in terms of a and C. 19. Find the value of F in terms of a and A. 20. Find the value of F in terms of h and C. 21. A barn is 40 X 80 feet, the pitch of the roof is 45~; find the length of the rafters and the area of both sides of the roof. 22. In a unit circle what is the length of the chord corresponding to the angle 45~ at the centre? 23. If the radius of a circle is 30, and the length of a chord is 44, find the angle at the centre. 24. Find the radius of a circle if a chord whose length is 5 subtends at the centre an angle of 133~ THE REGULAR POLYGON. 33 25. What is the angle at the centre of a circle if the corresponding chord is equal to - of the radius? 26. Find the area of a circular sector if the radius of the circle = 12, and the angle of the sector - 30~. ~ 16. THE REGULAR POLYGON. Lines drawn from the centre of a regular polygon (Fig. 19) to the vertices are radii of the circumscribed circle; and lines drawn from the centre to the middle points of the sides are radii of the inscribed circle. These lines divide the polygon into equal right triangles. Therefore, a regular polygon is determined by a right triangle whose sides are the radius of the circumscribed circle, the radius of the inscribed circle, and half of one side of the polygon. If the polygon has n sides, the angle of this right triangle at the centre is equal to 1 _360~\ 180~ 2 -V~ or ~Ao If, also, a side of the polygon, or one of the above-mentioned radii, is given, this triangle may be solved, and the solution gives the unknown parts of the polygon. Let, n = number of sides, / c c- length of one side, r radius of circumscribed circle, h = radius of inscribed circle, p = the perimeter, F= the area. h Then, by Geometry, F= ~^lp. FIG. 19. 34 TRIGONOMETRY, EXERCISE XI. 1. Given n- 10, c-1; find r, h, F. 2. Given n-=12, p=-70; find r, h, F. 3. Given n==18, r=l; find h, p, F. 4. Given n 20, r-20; find h, c, F. 5. Given n =8, h= 1; find r, c, F. 6. Given n-11, F=-20; find r, h, c. 7. Given n =7, F=7; find r, h, p. 8. Find the side of a regular decagon inscribed in a unit circle. 9. Find the side of a regular decagon circumscribed about a unit circle. 10. If the side of an inscribed regular hexagon is equal to 1, find the side of an inscribed regular dodecagon. 11. Given n and c, and let b denote the side of the inscribed regular polygon having 2n sides; find b in terms of n and c. 12. Compute the difference between the areas of a regular octagon and a regular nonagon if the perimeter of each is 16. 13. Compute the difference between the perimeters of a regular pentagon and a regular hexagon if the area of each is 12. 14. From a square whose side is equal to 1 the corners are cut away so that a regular octagon is left. Find the area of this octagon. 15. Find the area of a regular pentagon if its diagonals are each equal to 12. 16. The area of an inscribed regular pentagon is 33> 8; find the area of a regular polygon of 11 sides inscribed in the same circle. THE REGULAR POLYGON. 35 17. The perimeter of an equilateral triangle is 20; find the area of the inscribed circle. 18. The area of a regular polygon of 16 sides, inscribed in a circle, is 100; find the area of a regular polygon of 15 sides, inscribed in the same circle. 19. A regular dodecagon is circumscribed about a circle, the circumference of which is equal to 1; find the perimeter of the dodecagon. 20. The area of a regular polygon of 25 sides is equal to 40; find the area of the ring comprised between the circumferences of the inscribed and the circumscribed circles. CHAPTER III. GONIOMETRY. ~ 17. DEFINITION OF GONIOMETRY. IN order to prepare the way for the solution of an oblique triangle, we now proceed to extend the definitions of the trigonometric functions to angles of all magnitudes, and to deduce certain useful relations of the functions of different angles. That branch of Trigonometry which treats of trigonometric functions in general, and of their relations, is called Goniometry. ~ 18. ANGLES OF ANY MAGNITUDE. Let the radius OP of a circle (Fig. 20) generate an angle by turning about the centre 0. This angle will be measured by the arc described by the point P; B L and it may have any magnitude, because the arc described by P / may have any magnitude. Let the horizontal line OA be A the initial position of OP, and 0O }^- flet OP revolve in the direction shown by the arrow, or opposite to the way clock hands revolve. Let, also, the four quadrants into B' which the circle is divided by the FiG. 20. horizontal and vertical diameters AA', BB', be numbered I., II., III., IV., in the direction of the motion. GONIOMETRY. 37 During one revolution OP will form with OA all angles from 0~ to 360~. Any particular angle is said to be an angle of the quadrant in which OP lies; so that, Angles between 0~ and 90~ are angles of Quadrant I. Angles between 90~ and 180~ are angles of Quadrant II. Angles between 180~ and 270~ are angles of Quadrant III. Angles between 270' and 360~ are angles of Quadrant IV. If OP make another revolution, it will describe all angles from 360~ to 720~, and so on. If OP, instead of making another revolution in the direction of the arrow, be supposed to revolve backwards about 0, this backward motion tends to undo, or cancel, the original forward motion. Hence, the angle thus generated must be regarded as a negative angle; and this negative angle may, obviously, have any magnitude. Thus we arrive at the conception of an angle of any magnitude, positive or negative. ~ 19. GENERAL DEFINITIONS OF THE FUNCTIONS. The definitions of the trigonometric functions may be extended to all angles, by making the functions of any angle equal to the line values in a unit circle drawn for the angle.in question, as explained in ~ 4. But the lines that represent the sine, cosine, tangent, and cotangent must be regarded as negative, if they are opposite in direction to the lines that represent the corresponding functions of an angle in the first quadrant; and the lines that represent the secant and cosecant must be regarded as negative, if they are opposite in direction to the moving radius. Figs. 21-24 show the functions drawn for an angle AOP in each quadrant, taken in order. In constructing them, it must be remembered that the tangents to the circle are always drawn through A and B, never through A' or B'. Let the angle AOP be denoted by x; then, in each figure, 38 TRIGONOMETRY. the absolute values of the functions (that is, their values without regard to the signs + and -) are as follows: sin x MP. tan x = A T sec x = 01 COSx =OM, cotx=BS, cscX= OS. A' A'l B' B FIG. 21. FIG. 22. B B S, \[ X^0 ----- I [ -— ^ Xl ' 13' B' T FIG. 23. FIG. 24. Keeping in mind the position of the points A and B, we may define in words the first four functions of the angle x thus: sinx = the vertical projection of the moving radius; cosx = the horizontal projection of the moving radius; the distance measured along a tangent to the circle tanx-= from the beginning of the first quadrant to the moving radius produced; GONIOMETRY. 39 {the distance measured along a tangent to the circle cotxa == from the end of the first quadrant to the moving radius produced. Seex and esex are the distances from the centre of the circle measured along the moving radius produced to the tangent and cotangent, respectively. ~ 20. ALGEBRAIC SIGNS OF THE FUNCTIONS. The lengths of the lines, defined above as the functions of any angle, are expressed numerically in terms of the radius of the circle as the unit. But, before these lengths can be treated as algebraic quantities, they must have the sign + or - prefixed, according to the condition stated in ~ 19. The reason for this condition lies in that fundamental relation between algebraic and geometric magnitudes,. in virtue of which contrary signs in Algebra correspond to opposite directions in Geometry. The sine MP and the tangent AT always extend from the horizontal diameter, but sometimes upwards and sometimes downwards; the cosine OM and the cotangent BS always extend from the vertical diameter, but sometimes towards the right and sometimes towards the left. The functions of an angle in the first quadrant are assumed to be positive. Therefore,, 1. Sines and tangents extending from the horizontal diameter upwards, are positive; downwards, negative; 2. Cosines and cotangents extending from the vertical diameter towards the right, are positive; towards the left, are negative. The signs of the secant and cosecant are always made to agree with those of the cosine and sine, respectively. This agreement is secured if secants and cosecants extending from the centre, in the direction of the moving radius, are considered positive; in the opposite direction, negative. 40 TRIGONOMETRY. Hence, the signs of the functions for each quadrant are: In Quadrant I. all the functions are positive. In Quadrant II. the sine and cosecant only are positive. In Quadrant III. the tangent and cotangent only are positive. In Quadrant IV. the cosine and secant only are positive. ~ 21. FUNCTIONS OF A VARIABLE ANGLE. Let the angle x increase continuously from 0~ to 360~; what changes will the values of its functions undergo? It is easy, by reference to Fig. 25, to trace these changes throughout all the quadrants. FIG. 25. 1. The Sine. In the first quadrant, the sine MP increases from 0 to 1; in the second it remains positive, and decreases from 1 to 0; in the third it is negative, and increases in absolute value from 0 to 1; in the fourth it is negative, and decreases in absolute value from 1 to 0. GONIOMETRY. 41 2. The Cosine. In the first quadrant, the cosine OM11 decreases from 1 to 0; in the second it becomes negative, and increases in absolute value from 0 to 1; in the third it is negative, and decreases in absolute value from 1 to 0; in the fourth it is positive, and increases from 0 to 1. 3. The Tangent. In the first quadrant, the tangent AT increases from 0 to oo; in the second quadrant, as soon as the angle exceeds 90~ by the smallest conceivable amount, the moving radius OP', prolonged in the direction opposite to that of OP', will cut AT at a point T' situated very far below A; hence, the tangents of angles near 90~ in the second quadrant have very large negative values. As the angle increases, the tangent AT' continues negative, but diminishes in absolute value. When x = 180~, then T' coincides with A, and tan 180~ =-0. In the third quadrant, the tangent is positive, and increases from 0 to;o in the fourth it is negative, and decreases in absolute value from co to 0. 4. The Cotangent. In the first quadrant, the cotangent BS decreases from o to 0; in the second quadrant it is negative, and increases in absolute value from 0 to co; in the third and fourth quadrants it has the same sign, and undergoes the same changes as in the first and second quadrants, respectively. 5. The Secant. In the first quadrant, the secant OT increases from 1 to co; in the second it is negative (being measured in the direction opposite to that of OP'), and decreases in absolute value from oo to 1; in the third it is negative, and increases in absolute value from 1 to m; in the fourth it is positive, and decreases from ow to 1. 6. The Cosecant. In the first quadrant, the cosecant OS decreases from so to 1; in the second it is positive, and increases from 1 to oo; in the third it is negative, and decreases in absolute value from oo to 1; in the fourth it is negative, and increases in absolute value from 1 to o. 42 TRIGONOMETRY. The limiting values of the functions are as follows: 0o 90~ 180~ 270~ 360~ Sine...... 0 1 ~0 - 1 0 Cosine...... 1 0 -1 0 1 Tangent..... ~0 ~ - - 0 ~ oo ~ O0 Cotangent... c -c 0 - ~ c - 0 ~ oo Secant.......1._ c o 1 Cosecant.. ~... I oo-1 ~ coo Sines and cosines extend from +1 to -1; tangents and cotangents from + co to - oo; secants and cosecants from + oo to +1, and from — 1 to - ce. In the table given above the double sign ~ is placed before 0 and oo. From the preceding investigation it appears that the functions always change sign in passing through 0 and oo; and the sign + or - prefixed to 0 or oo simply shows the direction from-which the value is reached. Take, for example, tan 90~: The nearer an acute angle is to 90~, the greater the positive value of its tangent; and the nearer an obtuse angle is to 90~, the greater the negative value of its tangent. When the angle is 90~, OP (Fig. 25) is parallel to A T, and cannot meet it, But tan 90~ may be regarded as extending either in the positive or in the negative direction; and according to the view taken, it will be + oo or - oo. ~ 22. FUNCTIONS OF ANGLES LARGER THAN 360~. The functions of 360 + x are the same in sign and in absolute value as those of x; for the moving radius has the same position in both cases. If n is a positive integer, The functions of (n X 360~ + x) are the same as those of x. For example: The functions of 2200~(6 X 360 + 40~) are equal to the functions of 40~. GONIOMETRY. 43 ~ 23. EXTENSION OF FORMULAS [1]-[3] TO ALL ANGLES. The Formulas established for acute angles in ~ 6 hold true for all angles. Thus, Formula [1], sin2x + cos2x = 1, is universally true; for, whether MP and OM (Fig. 25) are positive or negative, _IP and O are always positive, a in each quadrant IP2- O- 0M2_ OP2 1. sinl x Also, [2] tan x —, cos x asin x X csc x =1, and [3] - cos x X sec x =1, Ltan x X cot x =1, are universally true; for they are in harmony with the algebraic signs of the functions, given at the end of ~ 20; and we have in each quadrant from the similar triangles OMP, OAT OBS, (Fig. 25) the proportions AT: OA=-JMP: OM, MP: OP= OB:OS, OM: OP OA: OT, AT: OA=OB: BS, which, by substituting 1 for the radius, and the right names for the other lines, are easily reduced to the above formulas. Formulas [1]-[3] enable us, from a given value of one function, to find the absolute values of the other five functions, and also the sign of the reciprocal function. But in order to determine the proper signs to be placed before the other four functions, we must know the quadrant to which the angle in question belongs; or the sign of any one of these four functions; for, by (~ 20) it will be seen that the signs of any two functions that are not reciprocals determine the quadrant to which the angle belongs. 44 TRIGONOMETRY. EXAMPLE. Given sin x =-+ -, and tan x negative; find the values of the other functions. Since sin x is positive, x must be an angle in Quadrants I. or II.; but, since tan x is negative, Quadrant I. is inadmissible. By [1], cos x = V 1-. = +. Since the angle is in Quadrant II. the minus sign must be taken, and we have cos x = -. By [2] and [3], tan x —, cot X ----, sec x=-, csc x= —. EXERCISE XII. 1. Construct the functions of an angle in Quadrant II. What are their signs? 2. Construct the functions of an angle in Quadrant III. What are their signs? 3. Construct the functions of an angle in Quadrant IV. What are their signs? 4. What are the signs of the functions of the following angles: 340~, 239~, 145~, 400~, 700~, 1200~, 3800~? 5. How many angles less than 360~ have the value of the sine equal to + -, and in what quadrants do they lie? 6. How many values less than 720~ can the angle x have if cos x + 23, and in what quadrants do they lie? 7. If we take into account only angles less than 180~, how many values can x have if sin x =-? if cos x =? if cos x --? if tanx =-? if cotx — 7? 8. Within what limits must the angle x lie if cos x = -- if cot x 4? if secx 80? if scx=- 3? (if x < 360~). 9. In what quadrant does an angle lie if sin and cosine are both negative? if cosine and tangent are both negative? if the cotangent is positive and the sine negative? GONIOMETRY. 45 10. Between 0~ and 3600~ how many angles are there whose sines have the absolute value 3? Of these sines how many are positive and how many negative? 11. In finding cos x by means of the equation cos x ~ V 1- sin2x, when must we choose the positive sign and when the negative sign? 12. Given cos x= —V -; find the other functions when x is an angle in Quadrant II. 13. Given tan x V 3; find the other functions when x is an angle in Quadrant III. 14. Given sec x =+ 7, and tan x negative find the other functions of x. 15. Given cot x=- 3; find all the possible values of the other functions. 16.. What functions of an angle of a triangle may be negative? In what case are they negative? 17. What functions.of an angle of a triangle determine the angle, and what functions fail to do so? 18. Why may cot 360~ be considered equal either to +- o or to- oo? 19. Obtain by means of Formulas [1]-[3] the other func'tions of the angles given: (i.) tan 90~ =-a. (iii.) cot 270~ =0. (ii.) cos 180~ - 1. (iv.) ese 360 = - o. 20. Find the values of sin 450~, tan 540~, cos 630~, cot 720~, sin 810~, csc 900~. 21. For what angle in each quadrant are the absolute values of the sine and cosine equal? Compute the values of the following expressions: 22. c sin 0~ + b cos 90~ - c tan 180~. 23. a cos 90~ - b tan 180~ + c cot 90~. 24. a sin 90~ - b cos 360~ + (a - b) cos 180~. 25. (a2 - b2) cos 360~ - 4 ab sin 270~. 46 TRIGONOMETRY. ~ 24. REDUCTION OF FUNCTIONS TO THE FIRST QUADRANT. In a unit circle (Fig. 26) draw two diameters PR and QSequally inclined to the horizon^-'-B tal diameter AA', or so that the p angles AOP, A'OQ, A'OR, and AOS shall be equal. From the N1 A \^ / 1Apoints P, Q, R, S let fall perpen-.'\ -- - A diculars to AA'; the four right triangles thus formed, with a R C \ f( common vertex at O, are equal; because they have equal hypoteB' nuses (radii of the circle) and FIG 26. equal acute angles at O. Therefore, the perpendiculars PM, QN, RNr, SM1, are equal. Now these four lines are the sines of the angles AOP, AOQ, AOR, and AOS, respectively. Therefore, in absolute value, sin AOP -sin AOQ- =sin AOR -sin AOS. And from ~ 23 it follows that in absolute value the cosines of these angles are also equal; and likewise the tangents, the cotangents, the secants, and the cosecants.* Hence, for every acute angle (A OP) there is an angle in each of the higher quadrants whose functions, in absolute value, are equal to those of this acute angle. Let Z AOP - x, / POB = y; then x+ y = 90~, and the functions of x are equal to the co-named functions of y (~ 5); and Z AOQ (in Quadrant II.) =180~ -x- 90~ +y, Z A OR (in Quadrant III.) - 180~ + x = 270 -y, A OS (in Quadrant IV.) =360~ - x = 270~ + y. Hence, prefixing the proper sign (~ 20), we have: * In future, secants, cosecants, versed sines, and coversed sines will be disregarded. Secants and cosecants may be found by [3], versed sines and coversed sines by VII. and VIII., page 5, if wanted, but they are seldom used in computations. GONIOMETRY. 47 Angle in Quadrant II. sin (180~ - x) = sin x. sin (90 + y) cos y. cos (180~- x) =- cos x. cos (90~ + y) -- sin y. tan (180 - x) =- tan x. tan (90 + y) =- cot y. cot (180~ - x) - cot x. cot (90 + y) - tan y. 1 Angle in Quadrant III. sin (180~ + x)= - sin x. sin (270~ - y) - cos y. cos (180~ + x) - cos x. cos (270 - y) =- sin y. tan (180~ + x) = tan. tan (270 - y) = cot y. cot (180~ + x) = cot x. cot (270~ - y) = tan y. Angle in Quadrant IV. sin (360~ - x) = - sin x. sin (270~ + y) - cos y. cos (360~ -x) = cosx. cos (270~ + y)= sin y. tan (360~- x) =-tan x. tan (270~ + y) - cot y. cot (360 - x) - cot x. cot (270~ + y) -tan y. REMARK. The tangents and cotangents may be found directly from the figure, or by formula [2]. It is evident from these formulas, 1. The functions of all angles can be reduced to the functions of angles not greater than 45~. 2. If an acute angle be added to or subtracted from 180~ or 360~, the functions of the resulting angle are equal in absolute value to the like-named functions of the acute angle; but if an acute angle be added to or subtracted from 90~ or 270~, the functions of the resulting angle are equal in absolute value to the co-named functions of the acute angle. 3. A given value of a sine or cosecant determines two supplementary angles, one acute, the other obtuse; a given value of any other function determines only one angle: acute if the value is positive, obtuse if the value is negative. [See functions of (180 ~- x). 48 TRIGONOMETRY. ~ 25. ANGLES WHOSE DIFFERENCE IS 90~. The general form of two such angles is x and 90~ + x, and they must lie in adjoining quadrants. The relations between their functions were found in ~ 24, B('' J ~but only for the case when x is p acute. These relations, however, may be shown to hold true for all values of x. A ' MA In a unit circle (Fig. 27) draw two diameters PR and QS per/\; \ / pendicular to each other, and let \,S fall to AA' the perpendiculars B' PM, QH,,RK, and SN. The FIG. 27. right triangles OMP, OHQ, OKR, and ONS are equal, because they have equal hypotenuses and equal acute angles POM, OQH, ROK, and OSN. Therefore, OM= QH= OK= NS, and PM1- OH -KR= ON. Hence, taking into account the algebraic sign, sin AOQ= cosAOP; sinAOS =- cos A OR; cos AOQ= -sin A OP; cos AOS - sin A OR; sinAOR== cosAOQ; sin (360~ +AOP)- cos AOS; cos AOR =- sin AOQ; cos(360~ + AOP)=- sin AOS. In all these equations, if x denote the angle on the righthand side, the angle on the left-hand side will be 90~-+x. Therefore, if x be an angle in any one of the four quadrants, sin (90~ + x) = cos x, tan (90~ + x) - cot x, cos (90~ + x) - sin x, cot (90~ + x) = - tan x. In like manner, it can be shown that all the formulas of ~ 24 hold true, whatever be the values of the angles x and y. Hence, in every case the algebraic sign of the function of the resulting angle will be the same as when x and y are both acute. GONIOMETRY. 49 ~ 26. FUNCTIONS OF A NEGATIVE ANGLE. If the angle A OP (Fig. 26) is denoted by x, the equal angle A OS, generated by a backward rotation of the moving radius from the initial position OA, will be denoted by -x. It is obvious that the position OS of the moving radius for this angle is identical with its position for the angle 360~-x. Therefore, the functions of the angle -x are the same as those of the angle 360~ -x; or (~ 24), sin (- x) = - sin x, cos (- x) = cosx, tan (- x) = - tan x, cot (- x) = - cot x, EXERCISE XIII. 1. Express sin250~ in terms of the functions of an acute angle less than 45~. Ans. sin 250 = sin (270~ - 20~) =- cos 20~. Express the following functions in terms of the functions of- angles less than 45~: 2. sin 172~. 3. cos 100~. 4. tan 125~. 5. cot 91~. 6. sec 110~. 7. csc 157~. 8. sin 204~. 9. cos 359. 10. tan 300~. 11. cot 264. 12. sec 244~. 13. csc 271~. 14. sin 163~ 49'. 15. cos 195~ 33'. 16. tan269~ 15'. 17. cot 139~ 17'. 18. sec 299~ 45'. 19. csc 92~ 25'. Express all the functions of the following negative angles in terms of those of positive angles less than 45~: 20. - 750. 21. -127~. 22. - 200~. 23. - 345~. 24. -52~ 37'. 25. — 196 54'. 26. Find the functions of 120~. HINT. 120~ = 180 - 60~, or, 120~ = 90~ + 30~; then apply ~ 24. 50 TRIGONOMETRY. Find the functions of the following angles: 27. 135~. 29. 210~. 31. 240~. 33. -30~. 28. 150~. 30. 225~. 32. 300~. 34. -225~. 35. Given sin x --- —, and cos x negative; find the other functions of x, and the value of x. 36. Given cot x= —3, and x in Quadrant II.; find the other functions of x, and the value of x. 37. Find the functions of 3540~. 38. What angles less than 360~ have a sine equal to —? a tangent equal to -V3? 39. Which 'of the angles mentioned in Examples 27-34 have a cosine equal to — A2? a cotangent equal to — 3? 40. What values of x between 0~ and 720~ will satisfy the equation sin x =+ i? 41. Find the other angle between 0~ and 360~ for which the corresponding function (sign included) has the same value as sin 12, cos 26~, tan 45~, cot 72~; sin 191~, cos 120~, tan 244~,cot 357~. 42. Given tan 238~ = 1.6; find sin 122~. 43. Given cos 333~ = 0.89; find tan 117~. Simplify the following expressions: 44. a cos (90 -x) + b cos (900 + x). 45. m cos (90~ — x) sin (90~ - x). 46. (a - b) tan (90~ - x) + (a + b) cot (90~ + x). 47. a2+ b2 — 2 ab cos (180~ -x). 48. sin (90~ + x) sin (180~ + x) + cos (90~ + x) cos (180~- x). 49. cos (180~+x) cos (270~-y) - sin (180~+x) sin (270- y). 50. tan x + tan(- y)-tan (180-). 51. For what values of x is the expression sin x +cos x positive, and for what values negative? Represent the result by shading the sectors corresponding to the negative values. 52. Answer the question of last example for sin x- cos x. 53. Find the functions of (x — 90~) in functions of x. 54. Find the functions of (x - 180) in functions of x. GONIOMETRY. 51 ~ 27. FUNCTIONS OF THE SUM OF Two ANGLES. In a unit circle (Fig. 28) let the angle AOB = x, the BOC=y; then the angle AOC C x+y. / In order to express sin (x + y) and cos (x +y) in terms of the sines and cosines of x and y, draw CFL OA, CD I_ OB, DEL OA, G D D)G CF; then CD sin y, OD = cosy, and the angle DCG- = the angle GDO- x. Also, ~ sin (x + y) CF- D-E+ CG. FIG. 28. DE i == sin x; hence, DE = sin x X OD sin x cos y. CG C=Dcos x; hence, CG== cos x X CD= cos x sin y. Therefore, sin (x y)sin x cos y cos x sin Therefore, sin (x 4 y) = sin x cos y + cos x sin y. ingle [4] Again, cos (x + y) - OF = OE -D G. OFE OD- cos x; hence, OE = cos x X OD - cos x cos y. ODG DG sin x; hence, DG = sin x X XCD sin x sin y. CD Therefore, cos (x + y) = cos x cos y - sin x sin y.. [5] In this proof x and y, and also the sum x + y, are assumed to be acute angles. If the sum x +y of the acute angles x and B y is obtuse, as in Fig. 29, the proof remains, word for word, \ \ the same as above, the only dif- Al \ ference being that the sign of F O E OF will be negative, as DG is FIG 29. now greater than OF. The above formulas, therefore, hold true for all acute angles x and y. 52 TRIGONOMETRY. If these formulas hold true for any two acute angles x and y, they hold true when one of the angles is increased by 90~. Thus, if for x we write x'-90~ + x, then, by ~ 25, sin (x' y)sin (90~ + x + Y) os (x + ), cos (x' + y) — cos (90~ x + y) - sin (x + y). Hence, by [5], sin (x' + y) = cos x cos y - sin x sin y, by [4], cos (x' + y)- sin x cos y - cos x sill y. Now, by ~ 25, cos x= sin (90~ + x) sin x', sin x - -os (90~ -+ ) - cos X'. Substitute these values of cos x and sin x, then y) s(x' + y) = sin x' cos + cos x' sin y, cos (x' + y) cos ax cos y- sin x' sin y. It follows that Formulas [4] and [5] hold true if either angle is repeatedly increased by 90~; therefore they apply to all angles whatever. By ~ 23, sin (x + y) sin x cos y + cos x sin y tan (x + y} - cos (x + y) cos x cosy - sinl x sin y If we divide each term of the numerator and denominator of the last fraction by cos x cos y, and again apply ~ 23, we obtain tan x + tan y tan (x+ y) 1-tanxtany In like manner, by dividing each term of the numerator and denominator of the value of cot (x + y) by sin x sin y, we obtain cot x cot y — [7 cot(x+y)= coty- - * [7] coty+cotx GONIOMETRY. 53 ~ 28. FUNCTIONS OF THE DIFFERENCE OF TWO ANGLES. In a unit circle (Fig. 30) let the angle AOB =x, then the angle A4OC x — y. In order to express sin (x — y) and cos (x -y) in terms of the D sines and cosines of x and y, draw CF 1 OA, CD _ OB, DE J_ OA, DG _ FC prolonged; then CD)sin y, OD =cos y, and the angle DCG==the angle EDC x. And, sin (x -y) = C1- = D - CG. 0 E COB-y; \. G FA.el C 1i. &U. DE 0J- sin; hence, DE sinx X OD sin x cosy. ODy^ CG CD- cos x; hence, CG cos x X CD =cos x sin y. Therefore, sin (x- y) =sinx cosy- cosx sin y. Again, cos (x - y) = OF == OE + D G. OK D = cos x; hence, OE cos x X OD cos x cosy. D G CD iC; hence, DG=sinx X CD=- sin x sing. CDilX [8] Therefore, cos (x- y) = cos x cos y + sin x sin y. [9] In this proof, both x and y are assumed to be acute angles; but, whatever be the values of x and y, the same method of proof will always lead to Formulas [8] and [9], when due regard is paid to the algebraic signs. The general application of these formulas may be at once shown by deducing them from the general formulas established in ~ 27, as follows: It is obvious that (x -- y = x. If we apply Formulas [4] and [5] to (x - y) + y, then 54 TRIGONOMETRY. sin l (x - y) + y } or sin x - sin (x - y) cos y + cos (x - y) sin y, cos (x - y)+ y r or cos x =os (x - y) cos - sin (x- y) sin y. Multiply the first equation by cos y, the second by sin y, sin x cos y= sin (x - y) cos2y + cos (x - ) sin y cos y, cos x sin y - sin (x - y) sin2y + cos (x - y) sin y cos y; whence, by subtraction, sin x cos y - cos x sin y = sin (x - y) (sin2y + cos2y). But sin2y + cos'y =1; therefore, by transposing, sin (x - y)= sin x cos y -cos x sin y. Again, if we multiply the first equation by sin y, the second equation by cos y, and add the results, we obtain, by reducing, cos (x - y) = cos x cos y + sin x sin y. Therefore, Formulas [8] and [9], like [4] and [5], from which they have been derived, are universally true. From [8] and [9], by proceeding as in ~ 27, we obtain x-y- -- tanx-tany [10] tan (x- ) -1 + tan x tan y cot x cot y + 1[11] cot (x - ) coty-cot [11] Formulas [4]-[11] may be combined as follows: sin (x ~ y) = sin x cos y ~ cos x sin y, cos (x y) = cos x cos y sin x sin y, tan x ~ tan y tan (x -~ y} = — ---—, n ( 1 tan x tan y cot (x cot x cot y = 1 cot y +_ cot x GONIOMETRY. 55 ~ 29. FUNCTIONS OF TWICE AN ANGLE. If y=x, Formulas [4]-[7], become sin 2 x=2 sinxcosx. [12] cos 2 x = coS2X -in. [13] 2 tan x cr t2X - 1 tan 2x= 1-tanx [14] cot2 x co [15] 1-tan2x 2cotx By these formulas the functions of twice an angle are found when the functions of the angle are given. ~ 30. FUNCTIONS OF HALF AN ANGLE. Take the formulas cos2x + sin2x r 1 cos2X -- Sin2 -cos 2 X 2 sin2x 1 - cos 2 x 2 cos2X = 1 + cos 2 x [1] [13] Subtract, Acd, Whence / - cos 2x 1- cos 2 x sin x Co --- _ cos X = -- ^1^ 2 2.. These values, if z is put for 2x, and hence ~ — for x, become cos z 1 6] cos -- 2 2 Hence, by division (~ 23), tanIZ== + /1-cosz [18] 1 - cosz 7 tan -T z == -I- 1 +Cos Z [18 coUtlz _ — \/1CSZ. [19] - 1- cosz By these formulas the functions of half an angle may be computed when the cosine of the entire angle is given. The proper sign to be placed before the root in each case depends on the quadrant in which the angle g z lies. (~ 20.) Let the student show from Formula [18] that tan1 2 B= / + (See page 22, Note.) c jca 56 TRIGONOMETRY. ~ 31. SUMS AND DIFFERENCES OF FUNCTIONS. From [4], [5], [8], and [9], by addition and subtraction: sin (x + y) + sin (x - y) 2 sin x cos y, sin (x + y) - sin (x - y) 2 cos x sin y, cos (x + y) + cos (x- y) = 2 cos x cos y, cos (x + y) - os (x - y) — 2 sin x sin y; or, by making x + y - A, and x-y B, and therefore, x = (A + ), and y - (A - B), sin A +sin B= 2sin (A +B)cos -(A-B). [20] sinA-sinB- 2cos (A+B)sin (A-B). [21] cos A + cos B = 2 cos 4 (A + B) cos i (A- B). [22] cosA-cos B —2 B)s(A B)in (A-B). [23] From [20] and [21], by division, we obtain sin A +sin B tan 2 (A + B) cot 2 (A - B); sin A -- sin B or, since cot (A — B) ta (A B) sin A + sin B tan (A+ - B) - - [24] sin A-sin B tan (A- B) EXERCISE XIV. 1. Find the value of sin (x + y) and cos (x + y), when sin x -35 COSX —4 Sin y = siX COSy - 1 =-, cosx==, siy=%, cosy =}. 2. Find sin (90~ - y) and cos (90 - y) by making x - 90~ in Formulas [8] and [9]. Find, by Formulas [4]-[11], the first four functions of: 3. 90~ +y. 8. 360~ -y. 13. -y. 4. 180~ -y. 9. 360 + y. 14. 45~-y. 5. 180~+y. 10. x —90~. 15. 45~ + y. 6. 2700 -y. 11. x —180o. 16. 30 + y. 7. 270~ + y. 12. x- 270~. 17. 60 — y. GONIOMETRY, 57 18. Find sin 3x in terms of sin x. 19. Find cos 3x in terms of cos x. 20. Given tan x= 1; find cos x. 21. Given cot x= 3; find sin x. 22. Given sin x - 0.2; find sin ix and cos ~x. 23. Given cos x = 0.5; find cos 2x and tan 2x. 24. Given tan 45~ = 1; find the functions of 22~ 30t 25. Given sin 30~ =0.5; find the functions of 15~. sin 33~ +- sin 3~ 26. Prove that tan 18~ = sin 33 cos 330 + cos 30 Prove the following formulas: 2 tan x sin x 27. sin 2 x- tan 29. tan x1 + tan2x 1 + cos x 1 -- tain2x sin x 28. cos 2x== - 30. cotx ---- 1 + tan2x 1- cos x 31. sin -x! cos -- V/1 + sin x. 32 tan x - tan y ta tan 32. = _+ tan x tan y. cot x ~ cot y 33. ta^ _1- 1tan x 33. tan(45 — x) t I + tan x If A, B, C are the angles of a triangle, prove that: 34. sin A + sin B + sin C = 4 cos I A cos I B cos - C. 35. cos A + cos B + cos C 1 + 4 sin I A sin I B sinl C. 36. tanA+tanB+ tan C - tanA X tanB X tan C. 37. cot I A + cot - B + cot - C =cot - A X cot -B X cot C. Change to forms more convenient for logarithmic computation: 38. cot x + tan x. 43. 1 + tan x tan y. 39. cotx - tanx. 44. 1 - tan x tan y. 40. cotx+- tany. 45. cotx coty+-1. 41. cot x -tan y. 46. cot x cot y- 1. 1 - cos 2x tan x - taniy 42. cos cot 7.cot 1 + cos 2x cot X + coty 58 TRIGONOMETRY. ~ 32. ANTI-TRIGONOMETRIC FUNCTIONS. If y is any trigonometric function of an angle x, then x is said to be the corresponding anti-trigonometric function of y. Thus, if y = sin x, x is the anti-sine of y, or inverse sine of y. The anti-trigonometric functions of y are written sin-l y, tan- y, sec- y, vers-ly, cos-'y, cot-ly, sc-1 y, covers-ly. These are read, the angle whose sine is y, etc. For example, sin 30~ =; hence 30~- sin-1. Similarly 90~ = cos- 0 = sin- l; and 45~ = tan-'1 = sin-l -; etc. V2 The symbol -~ must not be confused with the exponent - 1. Thus sin-1 x is a very different expression from —, which would be written sin x (sin x)-. On the Continent of Europe mathematical writers employ the notation arc sin, arc cos, etc., for sin-1, cos-~, etc. But the latter symbols are most common in England and America. There is an important difference between the trigonometric and the anti-trigonometric functions. When an angle is given, its functions are all completely determined; but when one of the functions is given the angle may have any one of an indefinite number of values. Thus, if sin y = -, y may be 30~, or 150~, or either of these increased or diminished by any integral multiple of 360~ or 27r, but cannot take any other values. Accordingly sin-l '=- 30~ ~ 2 nor, or 150~ -- 2 nwr, where n is any positive integer. Similarly, tan-1 = 45~0 2n7r or 225~ + 2 nr; i.e., tan- 1 = 45~ - nwr. Since one of the angles whose sine is x and one of the angles whose cosine is x together make 90~, and since similar relations hold for the tangent and cotangent, for the secant and cosecant, and for the versed sine and coversed sine, we have sx 7ir'r 7C sin-1 x + cos-1 x= - sec x + csc =, 2 2 GONIOMETRY. 59 7T' tan-lx + cot- x =- 2, vers-I x + covers -x =, 2 2 where it must be understood that each equation is true only for a particular choice of the various possible values of the functions. For example, if x is positive, and if the angles are always taken in the first quadrant, the equations are correct. EXEPCISE XV. 1. Find all the values of the following functions: sin'- V/3, tan- /<3, vers-', cos( —(-<2), csc-'(<2), tan-1 0, sec-12, cos- ( -4/3). 2. Prove that sin-1(-x)= —sin-lx; cos-l(-x)=-r-cos-lx. 3. If sin-lx + sin-ly 7r, prove that x = y. 4. If y sin-l', find tan y. 5. Prove that cos (sin-Ix) = - x2. 6. Prove that cos (2 sin-l x) 1 - 2 x2. 7. Prove that tan (tan-lx + tan- y) = — y 8. If x = V/, find all the values of sin-lx + cos-lx. 9. Prove that tan-l ( = sin- x. 10. Find the value of sin (tan-l'~). 11. Find the value of cot (2 sin-l1 ). 12. Find the value of sin (tan-l- + tan-l ). 13. If sin-1x =- 2 cos-lx, find x. 2x 14. Prove that tan (2 tan-' x) 1 x 2x 15. Prove that sin (2 tan-lx) =- x 1 + X2 CHAPTEEP IV. THE OBLIQUE TRIANGLE. ~ 33. LAW OF SINES. LET A, B, C denote the angles of a triangle ABC (Figs. 31 and 32), and a, b, c, respectively, the lengths of the opposite sides. Draw CD I_ AB, and meeting AB (Fig. 31) or AB produced (Fig. 32) at D. Let CD= h. 0 b a' a A c D B A C BD FIG. 31. FIG. 32. h Ill both figures, -= sillA. In Fig. 31, - sin B. In Fig. 32, - = sin (180 - B) = sin B. Therefore, whether h lies within or without the triangle, we obtain, by division, b sinB [25] b- sin B THE OBLIQUE TRIANGLE. 61 By drawing perpendiculars from the vertices A and B to the opposite sides we may obtain, in the same way, b sin B a sin A c sin C c sin C Hence the Law of Sines, which may be thus stated: -'The sides of a triangle are proportional to the sines of the opposite angles. If we regard these three equations as proportions, and take them by alternation, it will be evident that they may be written in the symmetrical form, a b c sin A sin B- sin C NOTE. Each of these equal ratios has a simple geometrical meaning which will appear if the Law of Sines is proved as follows: Circumscribe a circle about the triangle ABC (Fig. 33), and draw the radii OA, OB, OC; these radii divide the triangle into three isosceles triangles. Let R denote the radius. Draw 0 / \ L BC. By Geometry, the angle / BOC=2 A; hence, the angle B OM A, then BJV - R sin B O A Ca A — sin A...BCor a — 2sinA. \ In like manner, b = 2 sin B and c = 2 R sin C. Whence we FIG. 33. obtain a b C sil A si si sin C That is: The ratio of any side of a t)iangle to the sine of the opposite angle is numerically equal to the diameter of the cicumrscribec circle. 62 TRIGONOMETRY. ~ 34. LAW OF COSINES. This law gives the value of one side of a triangle in terms of the other two sides and the angle included between them. In Figs. 31 and 32, a2 h2 + BD2. In Fig. 31, BD c - AD; in Fig. 32, BD AD - c; in both cases, BD2 = -2 _ 2 c X AD + c2. Therefore, in all cases, a-2 h2 + AD + l 2 - 2c XAD. Now, Aj2 + AD2 b2, and AD -b cos A. Therefore, a2 = b2 + c2- 2 b cos A. [26] In like manner, it may be proved that b2 = a2+ -2 - 2 ac cos B, c2 a- 2+b2- 2ab COS C. The three formulas have precisely the same form, and the law may be stated as follows: The square of any side of a triangle is equal to the sum of the squares of the other two sides, diminisshed by twice their product into the cosine of the included angle. ~ 35. LAW OF TANGENTS. By ~ 33, a: b sin: sinl B; whence, by the Theory of Proportion, a - b sin A - sin B a+- b sin A + sin B But by [24], page 56, sin A - sin B tan (A - B) sin A + sin B tan - (A + B) Therefore, a- b tan (A- B) a +b tan 2 (A + B) THE OBLIQUE TRIANGLE. 63 By merely changing the letters, a - c tan I (A - C) b- c tan (B - C) a+c tan(A+ C) b+c tanc (B+ C) Hence the Law of Tangents: The difference of two sides of a triangle is to their sum as the tangent of half the difference of the opposite angles is to the tangent of half their cm. NOTE. If in [27] b > a, then B > A. The formula is still true, but to avoid negative quantities, the formula in this case should be written b- _a tan (B -A) b+- a tan (B +A) ExERcIs: XVI. 1. What do the formulas of ~ 33 become when one of the angles is a right angle? 2. Prove by means of the Law of Sines that the bisector of an angle of a triangle divides the opposite side into parts proportional to the adjacent sides. 3. What does Formula [26] become when A 90~? when A 0~? when A = 180~? What does the triangle become in each of these cases? NOTE. The case where A = 90~ explains why the theorem of ~ 34 is sometimes termed the Generalized Theorem of Pythagoras. 4. Prove (Figs. 31 and 32) that whether the angle B is acute or obtuse, c = a cos B + b cos A. What are the two symmetrical formulas obtained by changing the letters? What does the formula become when B 90~? 5. From the three following equations (found in the last exercise) prove the theorem of ~ 34: c = a cos B - b cos A, b= a cos C - c cosA, a b cos C + e cos B. HINT. Multiply the first equation by c, the second by b, the third by a; then from the first subtract the sum of the second and third. 64 TRIGONOMETRY. 6. In Formula [27] what is the maximum value of - (A-B)? 7. Find the form to which Formula [27] reduces, and describe the nature of the triangle, when (i.) C=90~; (ii.) A -B - 90~, and B =C. ~ 36. THE SOLUTION OF AN OBLIQUE TRIANGLE. The formulas established in ~~ 33-35, together with the equation A + B + C - 180, are sufficient for solving every case of an oblique triangle. The three parts that determine an oblique triangle may be: I. One side and two angles; II. Two sides and the angle opposite to one of these sides; III. Two sides and the included angle; IV. The three sides. Let A, B, C denote the angles, a, b, c the sides respectively. ~ 37. CASE I. Given one side a, and two angles A and B; find the remaining parts C, by and c. 1. C= 800- (A + B). b sin B asinB a s 2. -= —;.b r - snB. asin A sin A A sin A c sin C a sin C a 3. ---..c — -- sC. a sin A sin A sin A EXAMPLE. a = 24.31, A = 45~ 18', B = 22~ 11'. The work may be arranged as follows: a= 24.31 log a - 1.38578 =1.38578 A= 45 18' colog sin A=0.14825 — 0.14825 ]= 22~ 11' log sin B=9.57700 log sin C- 9.96556 A +B=- 67~ 29' log b= 1.11103 log c= 1.49959 C - 112~ 31' b =12.913 c = 31.593 NOTE. When - 10 is omitted after a logarithm or cologarithm, it must be remembered that the log or colog is 10 too large. THE OBLIQUE TRIANGLE. 65 EXERCISE XVII. 1. Given a= 500, find C = 123~ 12', 2. Given a = 795, find C = 55~ 20', 3. Given a= 804, find C 35~ 4', 4. Given c - 820, find C =25~ 12', 5. Given c=1005, find C- 47~ 14', 6. Given b=13.57, find A 108~ 50', 7. Given a= 6412, find B 56 56', 8. Given b= 999, find B = 77~, A-10~ 12', b= 2051.48, A 79~ 59', b 567.688, A=99~ 55', = 577.313, A= 12~ 49', b = 2276.63, A ==78~ 19', a 1340.6, B= 13~ 57', a = 53.276, A =70~ 55', b =5685.9, A - 37~ 58', a -630.77, B=46~ 36'; c — 2362.61. B =44~ 41'; c - 663.986. B 45~ 1'; c = 468.933. B =141~ 59'; c = 1573.89. B =54~ 27'; b- =1113.8. C - 57~ 13'; c - 47.324. C= 52~ 9'; c -5357.5. C- 650 2'; c — 929.48. 9. In order to determine the distance of a hostile fort A from a place B, a line BC and the angles ABC and BCA were measured, and found to be 322.55 yards, 60~ 34', and 56~ 10', respectively. Find the distance AB. 10. In making a survey by triangulation, the angles B and C of a triangle ABC were found to be 50~ 30' and 122~ 9', respectively, and the length BC is known to be 9 miles. Find AB and AC. 11. Two observers 5 miles apart on a plain, and facing each other, find that the angles of elevation of a balloon in the same vertical plane with themselves are 55~ and 58~, respectively. Find the distance from the balloon to each observer, and also the height of the balloon above the plain. 12. In a parallelogram given a diagonal d and the angles x and y which this diagonal makes with the sides. Find the sides. Find the sides if d = 11.237, x =19~ 1'9, and y = 42~ 54'. 66 TRIGONOMETRY. 13. A lighthouse was observed from a ship to bear N. 34~ E.; after sailing clue south 3 miles, it bore N. 23~ E. Find the distance from the lighthouse to the ship in both positions. NOTE. The phrase to bear N. 34~ E. means that the line of sight to the lighthouse is in the north-east quarter of the horizon, and makes, with a line due north, an angle of 34~. 14. In a trapezoid given the parallel sides a and b, and the angles x and y at the ends of one of the parallel sides. Find the non-parallel sides. Compute the results when ac= 15, b=7, x =70~, y=40~. Solve the following examples without using logarithms: 15. Given b = 7.07107, A =30~, C = 105; find a and c. 16. Given c - 9.562, AA 45~, B- 60~; find a and b. 17. The base of a triangle is 600 feet, and the angles at the base are 30~ and 120~. Find the other sides and the altitude. 18. Two angles of a triangle are, the one 20~, the other 40~. Find the ratio of the opposite sides. 19. The angles of a triangle are as 5: 10: 21, and the side opposite the smallest angle is 3. Find the other sides. 20. Given one side of a triangle equal to 27, the adjacent angles equal each to 30~. Find the radius of the circumscribed circle. (See ~ 33, Note.) ~ 38. CASE II. Given two sides a and b, and the angle A opposite to the side a; find the remaninig parts B, C, c. This case, like the preceding case, is solved by means of the Law of Sines. sinl B b. b sin A Since -—,sn therefore sin B= - sin A a a C = 1s80 - (A + B). THE OBLIQUE TRIANGLE. c sin C a sin C And since - - therefore c = a sin A sinA When an angle is determined by its sine it admits of two values, which are supplements of each other (~ 24); hence, either value of B may be taken unless excluded by the conditions of the problem. If a > b, then by Geometry A > B, and B must be acute whatever be the value of A; for a triangle can have only one obtuse angle. Hence, there is one, and only one, triangle that will satisfy the given conditions. If a = b, then by Geometry A = B; both A and B must be acute, and the required triangle is isosceles. If a < b, then by Geometry A < B, and A must be acute in order that the triangle may be possible. If A is acute, it is evident from Fig. 34, where Z BA C= A, / \Ca AC =b, CB = CB' == a, / that the two triangles A CB and ACB' will satisfy the A - given conditions, provided FG. 34. J? r FIG. 34. a is greater than the perpendicular CP; that is, provided a is greater than b sinA (~ 11). The angles ABC and AB'C are supplementary (since ABC = / BB'C); they are in fact the supplementary angles obtained from the formula b sin A sin B a If, however, a = sin A CP (Fig. 34), then sin B 1, B = 90~, and the triangle required is a right triangle. If a < b sin A, that is, < CP, then sin B> 1, and the triangle is impossible. 68 8TRIGONOMETRY. These results, for convenience, may be thus stated: Two solutions; if A is acute and the value of a lies between b and b sin A. NTo solution; if A is acute and a < b sin A; or if A is obtuse and a < b. One solution; in all other cases. The number of solutions can often be determined by inspection. In case of doubt, find the value of b sin A. Or we may proceed to compute log sin B. If log sin B = 0, the triangle required is a right triangle. If log sin B> 0, the triangle is impossible. If log sin B < 0, there is one solution when a > b; there are two solutions when a < b. When there are two solutions, let B', C' c', denote the unknown parts of the second triangle; then, = 180 - B, C' = 180 - (A + B') = -A,,a sin C' sin A EXAMPLES. 1. Given a =16, b 20, A 106~; find the remaining parts. In this case a <b, and A >90~; therefore the triangle is impossible. 2. Given a 36, b 80, A= 30~; find the remaining parts. Here we have bsinA = 80 x - = 40; so that a < b sinA, and the triangle is impossible. 3. Given a=72630, b= 117480, A-=80~0'50"; find B, C, c. a = 72630 colog a = 5.13888 Here log sin B > 0. b = 117480 logb = 5.06996.'. no solution. A 80~ 0' 50" log sin A = 9.99337 log sin B = 0.20221 THE OBLIQUE TRIANGLE. 69 4. Given a =13.2, b= 15.7, A - 57~ 13'15"; find B, C, c. a = 13.2 colog a - 8.87943 c = bcosA b= 15.7 logb =1.19590 logb = 1.19590 A = 57013'15" log sin A = 9.92467 log cos A = 9.73352 Here logsin B = 0, log sin B = 0.00000 log = 0.92942.. a right triangle. B = 90~ c = 8.5.. C - 32046'45" 5. Giten a- =767, b 242, A =36~ 53' 2"; find B C,,c. a = 767 colog a = 7.11520 log a = 2.88480 b = 242 log b = 2.38382 log sin C = 9.86970 A = 36~ 53' 2" log sin A = 9.77830 cologsin A = 0.22170 Here a> b, log sill B = 9.27732 log c = 2.97620 and log sin B < 0. B = 10~ 54' 58" c = 946.675.. one solution..-. C = 132 12' 0" 6. Given a == 177.01 b= 216.45, A 35~ 36'20"; find the other parts. a = 177.01 colog a = 7.75200 loga = 2.24800 2.24800 b = 216.45 log b = 2.33536 colog sinA = 0.23493 0.23493 A = 35 36' 20" log sin A = 9.76507 log sin C= 9.99462 9.23034 Here a < b, log sin B = 9.85243 logc = 2.47755 1.71327 and log sin B < 0. B = 45~ 23' 28" c = 300.29 or 51.674.-. two solutions. or 134~ 36' 32".. C = 99 0' 12" or 9~ 47' 8" EXERCISE XVIII. 1. Determine the number of solutions in following cases: each of the (i.) (ii.) (iii.) (iv.) (v.) (vi.) (vii.) a 80, a 50, a =40, a -13.4, a =70, a - 134.16, a =200, b 100, b - 100, b- 100, b- 11.46, b -75, b- 84.54, b =100, A 30~. A -30~. A -30~. A - 77~ 20'. A — 60~. B = 52~ 91 11". A = 30~. 70 TRIGONOMVETRY. 2. Given ac- 840, b =485, A 21~ 31'; find B- 12~ 13' 34", C 146~ 15'26", c= 1272.18. 3. Given a -9.399, b= 9.197, find -- 57~ 23' 40" C =: 2 1' 20", A =120~ 35'; c = 0.38525. 4. Given a= 91.06, find B- =41~ 13', b = 77.04, A -51~ 9' 6"; C =87~ 37' 54", c = 116.82. 5. Given a 55.55, b = 66.66, B - 77~ 44'40"; find A- =54~ 31' 13", C 47 44' 7", c 50.481. 6. Given a 309, find B = 24~ 57'54", B'= 1550 2'6", b = 360, A -21~ 14' 25"; C - 133047'41", c- 615.67, C'=3~ 43' 29", c' 55.41. 7. Given a 8.716, b - 9.787, A =38~ 14' 12"; find B=44~ 1'28", C =97~ 44'20", c -13.954, B'= 135058'32", C'- 5~ 47' 16", c'= 1.4203. 8. Given a 4.4, find B- 90~, = 5.21, A = 57~ 37' 17"; C 32~ 22' 43", c 2.79. 9. Given a- 34, b -22, find A = 51~ 18' 27", C = 98~ 21'33", A' =128~ 41'33", C' 20~ 58'27", 10. Given 6=19, c=18, find B= 16~ 43'13", A 147~ 27'47", B' = 163~ 16'47", A'= 0~ 54' 13", B 30~ 20'; c - 43.098, c'= 15.593. C= 15~ 49'; a =35.519, a' = 1.0415. 11. Given a =75, b 29, B- 16~ 15' 36"; find the difference between the areas of the two corresponding triangles without finding their areas separately. 12. Given in a parallelogram the side a, a diagonal d, and the angle A made by the two diagonals; find the other diagonal. Special case: a = 35, d = 63, A = 21~ 36' 30". THE OBLIQUE TRIANGLE. ~ 39. CASE III. Given two sides a and b and the included angle C; find the remaining parts, A, B and c. SOLUTION I. The angles A and B may both be found by means of Formula [27], ~ 35, which may be written a —b tan ( -- B -- X tan (A + B). a ~ b Since I (A + B) (180 - C), the value of -(A + B) is known; so that this equation enables us to find the value of (A —B). We then have ~ (A + B) + (A - B) A, and (A + B)- (A-B)=B. After A and B are known, the side c may be found by the Law of Sines, which gives its value in two ways, as follows: a sin C b sin C c -. or c -- sin A sin B SOLUTION II. The third side c may be found directly from the equation (~ 34) c = /a2 b2 -2 ab cos C; and then, by the Law of Sines, the following equations for computing the values of the angles A and B are obtained: sin C sin C sin A a X -- sinB b Xc G SOLUTION III. If, in the triangle AB C (Fig. 35), BD is drawn perpendicular to the side AC, then BD BD B tan A = — - Now BD= a sin C (~ 10), \a and DC- a cos C. a sin C --..tanA= — A b D b-a cosC FIG. 35. 72 TRIGONOMETRY. By merely changing the letters, b sin C tan B -- a -b os C It is not necessary, however, to use both formulas. When one angle, as A, has been found, the other, B, may be found from the relation A + B + C 180~. When the angles are known, the third side is found by the Law of Sines, as in Solution I. NOTE. When all three unknown parts are required, Solution I. is the most convenient in practice. When only the third side c is desired, Solution II. may be used to advantage, provided the values of a2 and b2 can be readily obtained without the aid of logarithms. But Solutions II. and III. are not adapted to logarithmic work. EXAMPLES. 1. Given a- 748, b =375, C=63~ 35' 30"; find A, B, and c. a + b 1123 a-b= 373 log (a- b)=2.57171 logb = 2.57403 (A+B) = 116~ 24' 30" colog(a +b)-6.94962 log sin C = 9.95214 (A+B) = 58 12' 15" logtanl(A+B)=0.20766 cologsinfB- 0.30073 ~(A-B) = 28~ 10' 52" log tan (A —B)=9.72899 log c= 2.82690 A= 86~ 23' 7" (A - B) = 28~ 10' 52" c= 671.27 B= 30~ 1'23" NOTE. In the above Example we use the angle B in finding the side c, rather than the angle A, because A is near 90~, and therefore its sine should be avoided. 2. Given a-4, c- 6, B- 60~; find the third side b. Here Solution II. may be used to advantage. We have b Va2 + c2- 2 ac cos B = 16 + 36 - 24 = 28; log28=1.44716, logV28 = 0.72358, 28 = 5.291,5; that is, b = 5.2915. THE OBLIQUE TRIANGLE.. EXERCISE XIX. 1. Given a=77.99, b 83.39, C=72~15'; find A = 51~ 15', B= 56 30', c = 95.24. 2. Given b = 872.5, c -632.7, A = 80~; find B -60~ 45', C= 39~ 15', a- 984.83. 3. Given a =17, b =12, C= 59~ 17'; find A = 77~ 12'53", B 43~ 30'7", c = 14.987. 4. Given b = V5, c= /3, A =35~ 53'; find B =93~ 28' 36", C= 50~ 38'24", a 1.313. 5. Given a=0.917, b= 0.312, C=33~ 7' 9"; find A = 132 18' 27", B = 14~ 34' 24', c = 0.67748. 6. Given a = 13.715, c = 11.214, B = 15~ 22' 36"; find A =118~ 55'49", C=45~ 41' 35", =4.1554. 7. Given b 3000.9, c = 1587.2, A =86~ 4' 4"; find B == 65 13' 51", C=28~ 42' 5", a=3297.2. 8. Given a= 4527, b = 3465, C= 660 6' 27"; find A = 68~ 29' 15", B = 45~ 24' 18", c =4449. 9. Given a=55.14, b =33.09, C= 30~ 24'; find A =117~ 24' 33", B-32~ 11'27", c= 31.431. 10. Given a= 47.99, b = 33.14, C = 175 19' 10"; find A = 2 46' 8", = 1~ 54' 42", c = 81.066. 11. If two sides of a triangle are each equal to 6, and the included angle is 60~, find the third side. 12. If two sides of a triangle are each equal to 6, and the included angle is 120~, find the third side. 13. Apply Solution I. to the case in which a is equal to b; that is, the case in which the triangle is isosceles. 14. If two sides of a triangle are 10 and 11, and the included angle is 50~, find the third side. 15. If two sides of a triangle are 43.301 and 25, and the included angle is 30~, find the third side. 16. In order to find the distance between two objects A and B separated by aswamp, a station C was chosen, and the 74 TRIGONOMETRY. distances CA = 3825 yards, CB = 3475.6 yards, together with the angle ACB= 62~ 31', were measured. Find the distance from A to B. 17. Two inaccessible objects A and B are each viewed from two stations C and D on the same side of AB and 562 yards apart. The angle ACB is 62~ 12', BCD 41~ 8', ADB 60~ 49', and ADC 34~ 51'; required the distance AB. 18. Two trains start at the same time from the same station, and move along straight tracks that form an angle of 30~, one train at the rate of 30 miles an hour, the other at the rate of 40 miles an hour. How far apart are the trains at the end of half an hour? 19. In a parallelogram given the two diagonals 5 and 6, and the angle that they form 49~ 18'. Find the sides. 20. In a triangle one angle = 139~ 54', and the sides forming the angle have the ratio 5: 9. Find the other two angles. ~ 40. CASE IV. Given the three sides a, b, c; find the angles A, B, C. The angles may be found directly from the formulas established in ~ 34. Thus, from the formula a2 = b2 - 2 be cos A b2 + C2- a2. we have cos A 2- - 2 be From this equation formulas adapted to logarithmic work are deduced as follows: For the sake of brevity, let a - b + c - 2s; then b + c-a 2= (s-a), a-b+c =2 (s-b), and a+b-c =2 (s-c). Then the value of 1 - cos A is b + 2 - a2 2 be -b2 - c + a2 a- (b-C)2 2 be 2 be 2 b _ (a+ b - c) (a- b+ c) _2 (s- b) (s - c) 2be be THE OBLIQUE TRIANGLE. 75 and the value of 1 + cos A is b+c2 - a 2 ++C2 - a2 (b + C)2- 2 (b+c- -ct) (b + c-c ) _2s(s - a) 2 be be But from Formulas [16] and [17], ~ 30, it follows that 1 -cos A - 2sinl2 A, and 1 + cos A - 2 cos2 - A. 2.. 22A2 s i 2 (s-b) (s- c) and 2 Cos A 2 s ( -a) -- be 2 - be whence sin lA (s b) (s -c) 28] cos 2 A^- 8 ( - b) [29] and by [2] tan A ( [30] s (s - a) By merely changing the letters, sin ~ B_= (s — a) (s — c) sill 1 (s a- a) (s -- b) si B=\ s -, Bsin2C= jb ac ab cosB _ s (s -b), Co, s (S - c) (s. —a) (s-c), (s-a)(s-b) tan4B-4( (-/ tanllC - s(s-c) 2 ^ 5 (5-b) 2 (S-C) There is then a choice of three different formulas for finding the value of each angle. If half the angle is very near 0~, the formula for the cosine will not give a very accurate result, because the cosines of angles near 0~ differ little in value; and the same holds true of the formula for the sine when half the angle is very near 90~. Hence, in the first case the formula for the sine, in the second that for the cosine, should be used. But, in general, the formulas for the tangent are to be preferred. 76 TRIGONOMETRY. It is not necessary to compute by the formulas more than two angles; for the third may then be found from the equation A+B+ C- 180~. There is this advantage, however, in computing all three angles by the formulas, that we may then use the sum of the angles as a test of the accuracy of the results. In case it is desired to compute all the angles, the formulas for the tangent may be put in a more convenient form. The value of tan — A may be written /I(s -a)(s-1) (s-c) o 1 s-a) (s- ) (s) - c) )2 or / (ss-a s - a S Hence, if we put /(s-a) (s-b) (s ) r [31] ^/ ------- - -, U[31 S r we have tan A A= [32] s a P. r Likewise, tan - B- tan - C s - s-b s-c EXAMPLES. 1. Given a 3.41, b =2.60, c-1.58; find the angles. Using Formula [30], and the corresponding formula for tan IB, we In ay arrange the work as follows: a 3.41 colog s= 9.42079 colog s= 9.42079- 10 b 2.60 colog (s- a) = 0.41454 log (s- a)= 9.58546- 10 c =1.58 log (s -b) = 0.07737 eolog(s- b)= 9.92263- 10 2s = 7.59 log (s -c)= 0.34537 log(s-c)= 0.34537 s=3.795 2)0.25807 2)19.27425- 20 s- a = 0.385 log tan A = 0.12903 log tan ~ B= 9.63713-10 s —b= 1.195 IA = 53023'20" JB= 23026'37" s-c = 2.215 A = 106~ 46' 40" B= 46053/14/.A + B = 153~ 39' 54", and C = 26~ 20' 6". THE OBLIQUE TRIANGLE. 77 2. Solve Example 1 by finding all three angles by the use of Formulas [31] and [32]. Here the work may be compactly arranged as follows, if we find logtan A, etc., by subtracting log(s - a), etc., from logr instead of adding the cologarithin: a= 3.41 b = 2.60 c- 1.58 2s - 7.59 s= 3.795 s - a = 0.38o s-b =1.195 s - c = 2.215 2 s = 7.590 (proo log (s- a) = 9.58546 log (s- b) = 0.07737 log (s- c) = 0.34537 colog s = 9.42079 logr2 =9.42899 logr = 9.71450 log tan A - 10.12903 log tan B 9.63713 log tan C = 9.36912 ~-A= 53~ 23'20" B = 23 26' 37" ~C= 13~10' 3" A = 106~ 46' 40" B = 46~ 53' 14" C = 26~ 20' 6" A + B + C = 180~ 0' 0" Proof, NOTE. Even if no mistakes are made in the work, the sum of the three angles found as above may differ very slightly from 180~ in consequence of the fact that logarithmic computation is at hest only a method of close approximation. When a difference of this kind exists it should be divided among the angles according to the probable amount of error for each angle. EXERCISE XX. Solve the following triangles, taking the three sides as the given parts: a b c A B C 1 51 65 20 38~ 52' 48" 126~ 52' 12" 14~ 15' 2 78 101 29 32~ 10' 54" 136~ 23' 50" 11~ 25' 16" 3 111 145 40 27~ 20 32"/ 143~ 7'48" 9~ 31' 40" 4 21 26 31 42~ 6'13" 56~ 6'36" 81~47'11" 5 19 34 49 16~ 25' 36" 30~ 24' 133~ 10' 24" 6 '43 50 57 46~ 49' 35" 570 59'44" 75~ 10' 41" 7 37 58 79 26~0 0'29" 430 25/ 20// 110 34' 11" 8 73 82 91 49~ 34' 58" 58~ 46' 58'- 71~ 38' 4" 9 14.493 55.4363 66.9129 8~ 20' 33~ 40/ 138~ 10 V5 V/6 V7 51~ 53' 12" 590 31'48" 680 35' 78 TRIGONOMETRY. 11. Given a =6, b=8, c-10; find the angles. 12. Given a-=6, b-6, 6= c10; find the angles. 13. Given a=6, 6=6, c=6; find the angles. 14. Given a = 6, b 5, c 12; find the angles. 15. Given a = 2, b =- V6, c = o- 1; find the angles. 16. Given a- =, 6b- 6, c = <V3 + 1; find the angles. 17. The distances between three cities A, B, and C are as follows: AB = 165 miles, ACC =72 miles, and BC = 185 miles. B is due east from A. In what direction is C from A? What two answers are admissible? 18. Under what visual angle is an object 7 feet long seen by an observer whose eye is 5 feet from one end of the object and 8 feet from the other end? 19. When Formula [28] is used for finding the value of an angle, why does the ambiguity that occurs in Case II. not exist? 20. If the sides of a triangle are 3, 4, and 6, find the sine of the largest angle. 21. Of three towns A, B, and C, A is 200 miles from B and 184 miles from C, B is 150 miles due north from C; how far is A north of C? ~ 41. AREA OF A TRIANGLE. CASE I. 1When two sides and the included angle are given: In the triangle ABC (Fig. 31 or 32), the area F=2 c X CD. By ~ 11, CD = a sin B. Therefore, F =- ac sin B. [33] Also, F= ab sin C and -= be sin A. CASE II. When a side and the two adjacent angles are given: By ~ 33, sinA: sin C:: a: c. a sin C Therefore, c s --- sin A THE OBLIQUE TRIANGLE. 79 Putting this value of c in Formula [33], we have a2 sin B sin C ^ sh^B+C)- [34] 2 sin (B +- C) CASE III. When the three sides of a triangle are given: By ~ 29, sinB-2 sin B Xcos~B. By substituting for sin ~ B and cos - B their values in ternms of the sides given in ~ 40, sin B = — Vs (s - a) (s - b) (s - c). ac By putting this value of sin B in [33], we have F= Vs (s - a) (s -b) (s- c). [35] CASE IV. When the three sides and the radius of the circumscribed circle, or the radius of the inscribed circle, are given: If B denote the radius of the circumscribed circle, we have, from ~ 33, b sin B = — 211 By putting this value of sin B in [33], we have abe 4 R [36] If r denote the radius of the inscribed circle, divide the triangle into three triangles by lines from the centre of this circle to the vertices; then the altitude of each of the three triangles is equal to r. Therefore, F = r(a b+c) rs. [37] By putting in this formula the value of F given in [35], -(s-a) (s- b) (s- c) whence r, in [31] ~ 40, is equal to the radius of the inscribed circle. 80 ITRIGONOMETRY. EXERCISE XXI. Find the area: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Given a =4474.5, Given b 21.66, Given a = 510, Given a - 408, Given a =-40, Given a = 624, Given b6 149, Given a - 215.9, Given b = 8, Given a - 7, b 2164.5, C=- 116~ 30' 20". c 36.94, A = 66~ 4'19". c=173, B= 162~ 30'28". b 41, c=401. b-13, c=37. b 205, c=445. A= 70~ 42' 30", B =39~ 18' 28". c 307.7, A = 25~ 9'31". C = 5, A = 60~. c-3, A-=60~. 11. Given a =60, B = 40~ 35' 12", area = 12; find the radius of the inscribed circle. 12. Obtain a formula for the area of a parallelogram in terms of two adjacent sides and the included angle. 13. Obtain a formula for the area of an isosceles trapezoid in terms of the two parallel sides and an acute angle. 14. Two sides and included angle of a triangle are 2416, 1712, and 30~; and two sides and included angle of another triangle are 1948, 2848, and 150~; find the sum of their areas. 15. The base of an isosceles triangle is 20, and its area is 100 - V/3 find its angles. 16. Show that the area of a quadrilateral is equal to one half the product of its diagonals into the sine of their included angle. EXERCISE XXII. 1. From a ship sailing down the English Channel the Eddystone was observed to bear N. 33~ 45' W.; and after the ship had sailed 18 miles S. 67~ 30' V. it bore N. 11~ 15' E. Find its distance from each position of the ship. THE OBLIQUE TRIANGLE. 81 2. Two objects, A and B, were observed from a ship to be at the same instant in a line bearing N. 15~ E. The ship then sailed north-west 5 miles, when it was found that A bore due east and B bore north-east. Find the distance from A to B. 3. A castle and a monument stand on the same horizontal plane. The angles of depression of the top and the bottom of the monument viewed from the top of the castle are 40~ and 80~; the height of the castle is 140 feet. Find the height of the monument. 4. If the sun's altitude is 60~, what angle must a stick make with the horizon in order that its shadow in a horizontal plane may be the longest possible? 5. If the sun's altitude is 30~, find the length of the longest shadow cast on a horizontal plane by a stick 10 feet in length. 6. In a circle with the radius 3 find the area of the part comprised between parallel chords whose lengths are 4 and 5. (Two solutions.) 7. A and B, two inaccessible objects in the same horizontal plane, are observed from a balloon at C, and from a point D directly under the balloon and in the same horizontal plane with A and B. If CD= 2000 yards, Z ACD -10~15'10", Z BCD = 6~ 7' 20", / ADB = 49~ 34' 50", find AB. 8. A and B are two objects whose distance, on account of intervening obstacles, cannot be directly measured. At the summit C of a hill, whose height above the common horizontal plane of the objects is known to be 517.3 yards, ZACB is found to be 15~ 13' 15". The angles of elevation of C viewed from A and B are 21~ 9' 18" and 23~ 15' 34" respectively. Find the distance from A to B. CHAPTER V. MISCELLANEOUS EXAMPLES. PROBLEMS IN PLANE TRIGONOMETRY. 1. TEE angular distance of any object from a horizontal plane, as observed at any point of that plane, is the angle which a line drawn from the object to the point of observation makes with the plane. If the object observed be situated above the horizontal plane (that is, if it is farther from the earth's centre than the plane is), its angular distance from the plane is called its angle of elevation. If the object be below the plane, its angular distance from the plane is called its angle of depression. These angles are evidently vertical angles. If two objects are in the same horizontal plane with the point of observation, the angular distance of one object from the other is called its bearing from that object. If two objects are not in the same horizontal plane with either each other or the point of observation, we may suppose vertical lines to be passed through the two objects, and to meet the horizontal plane of the point of observation in two points. The angular distance of these two points is the bearing of either of the objects from the other. It may also be called the horizontal distance of one object from the other. NOTE. "Problems in Plane Trigonometry " are selected from those published by Mr. Charles W. Sever, Cambridge, Mass. The full set can be obtained from him in pamphlet form. MISCELLANEOUS EXAMPLES. 83 RIGHT TRIANGLES. 2. The angle of elevation of a tower is 48~ 19' 14", and the distance of its base from the point of observation is 95 ft. Find the height of the tower, and the distance of its top from the point of observation. 3. From a mountain 1000 ft. high, the angle of depression of a ship is 77~ 35' 11". Find the distance of the ship from the summit of the mountain. 4. A flag-staff 90 ft. high, on a horizontal plane, casts a shadow of 117 ft. Find the altitude of the sun. 5. When the moon is setting at any place, the angle at the moon subtended by the earth's radius passing through that place is 57' 3". If the earth's radius is 3956.2 miles, what is the moon's distance from the earth's centre? 6. The angle at the earth's centre subtended by the sun's radius is 16' 2", and the sun's distance is 92,400,000 miles. Find the sun's diameter in miles. 7. The latitude of Cambridge, Mass., is 42~ 22' 49". What is the length of the radius of that parallel of latitude? 8. At what latitude is the circumference of the parallel of latitude half of that of the equator? 9. In a circle with a radius of 6.7 is inscribed a regular polygon of thirteen sides. Find the length of one of its sides. 10. A regular heptagon, one side of which is 5.73, is inscribed in a circle. Find the radius of the circle. 11. A tower 93.97 ft. high is situated on the bank of a river. The angle of depression of an object on the opposite bank is 25~ 12' 54". Find the breadth of the river. 84 TRIGONOMETRY. 12. From a tower 58 ft. high the angles of depression of two objects situated in the same horizontal line with the base of the tower, and on the same side, are 30~ 13' 18" and 45~ 46' 14". Find the distance between these two objects. 13. Standing directly in front of one corner of a flat-roofed house, which is 150 ft. in length, I observe that the horizontal angle which the length subtends has for its cosine V/-, and that the vertical angle subtended by its height has for its sine 3 -. What is the height of the house? V34 14. A regular pyramid, with a square base, has a lateral edge 150 ft. in length, and the length of a side of its baLse is 200 ft. Find the inclination of the face of the pyramid to the base. 15. From one edge of a ditch 36 ft. wide, the angle of elevation of a wall on the opposite edge is 62~ 39' 10". Find the length of a ladder which will reach from the point of observation to the top of the wall. 16. The top of a flag-staff has been broken off, and touches the ground at a distance of 15 ft. from the foot of the staff. The length of the broken part being 39 ft., find the whole length of the staff. 17. From a balloon, which is directly above one town, is observed the angle of depression of another town, 10~ 14' 9". The towns being 8 miles apart, find the height of the balloon. 18. From the top of a mountain 3 miles high the angle of depression of the most distant object which is visible on the earth's surface is found to be 2~ 13' 50". Find the diameter of the earth. 19. A ladder 40 ft. long reaches a window 33 ft. high, on one side of a street. Being turned over upon its foot, it reaches another window. 21 ft. high, on the opposite side of the street. Find the width of the street. MISCELLANEOUS EXAMPLES. 85 20. The height of a house subtends a right angle at a window on the other side of the street; and the elevation of the top of the house, from the same point, is 60~. The street is 30 ft. wide. How high is the house? 21. A lighthouse 54 ft. high is situated on a rock. The elevation of the top of the lighthouse, as observed from a ship, is 4~ 52', and the elevation of the top of the rock is 4~ 2'. Find the height of the rock, and its distance from the ship. 22. A man in a balloon observes the angle of depression of an object on the ground, bearing south, to be 35~ 30'; the balloon drifts 2{ miles east at the same height, when the angle of depression of the same object is 23~ 14'. Find the height of the balloon. 23. A man standing south of a tower, on the same horizontal plane, observes its elevation to be 54~ 16'; he goes east 100 yds., and then finds its elevation is 50~ 8'. Find the height of the tower. 24. The elevation of a tower at a place A south of it is 30~; and at a place B, west of A, and at a distance of a from it, the elevation is 18~. Show that the height of the tower is a <5-1 (2 ~- 2; the tangent of 18~ being (1 _ 5 V(2 + 2 V/5) V(10 + 2 V5) 25. A pole is fixed on the top of a mound, and the angles of elevation of the top and the bottom of the pole are 60~ and 30~ respectively. Prove that the length of the pole is twice the height of the mound. 26. At a distance (a) from the foot of a tower, the angle of elevation (A) of the top of the tower is the complement of the angle of elevation of a flag-staff on top of it. Show that the length of the staff is 2 a cot 2 A. 27. A line of tree level is a line every point of which is equally distant from the centre of the earth. A line drawn 86 TRIGONOMETRY. tangent to a line of true level at any point is a line of apparent level. If at any point both these lines are drawn, and extended one mile, find the distance they are then apart. 28. In Problem 2, determine the effect upon the computed height of the tower, of an error in either the angle of elevation or the measured distance. OBLIQUE TRIANGLES. 29. To determine the height of an inaccessible object situated on a horizontal plane, by observing its angles of elevation at two points in the same line with its base, and measuring the distance of these two points. 30. The angle of elevation of an inaccessible tower, situated on a horizontal plane, is 63~ 26'; at a point 500 ft. farther from the base of the tower the elevation of its top is 32~ 14'. Find the height of the tower. 31. A tower is situated on the bank of a river. From the opposite bank the angle of elevation of the tower is 60~ 13', and from a point 40 ft. more distant the elevation is 50~ 19'. Find the breadth of the river. 32. A ship sailing north sees two lighthouses 8 miles apart, in a line due west; after an hour's sailing, one lighthouse bears S.V., and the other S.S.W. Find the ship's rate. 33. To determine the height of an accessible object situated on an inclined plane. 34. At a distance of 40 ft. from the foot of a tower on an inclined plane, the tower subtends an angle of 41~ 19'; at a point 60 ft. farther away, the angle subtended by the tower is 23~ 45'. Find the height of the tower. 35. A tower makes an angle of 113~ 12' with the inclined plane on which it stands; and at a distance of 89 ft. from its base, measured down the plane, the angle subtended by the tower is 23~ 27'. Find the height of the tower. MISCELLANEOUS EXAMPLES. 87 36. From the top of a house 42 ft. high, the angle of elevation of the top of a pole is 14~ 13'; at the bottom of the house it is 23~ 19'. Find the height of the pole. 37. The sides of a triangle are 17, 21, 28; prove that the length of a line bisecting the greatest side and drawn from the opposite angle is 13. 38. A privateer, 10 miles S.W. of a harbor, sees a ship sail from it in a direction S. 80~ E., at a rate of 9 miles an hour. In what direction, and at what rate, must the privateer sail in order to come up with the ship in 1- hours? 39. A person goes 70 yds. up a slope of 1 in 3- from the edge of a river, and observes the angle of depression of an object on the opposite shore to be 24~. Find the breadth of the river. 40. The length of a lake subtends, at a certain point, an angle of 46~ 24', and the distances from this point to the two extremities of the lake are 346 and 290 ft. Find the length of the lake. 41. Two ships are a mile apart. The angular distance of the first ship from a fort on shore, as observed from the second ship, is 35~ 14' 10"; the angular distance of the second ship from the fort, observed from the first ship, is 42~ 11' 53". Find the distance in feet from each ship to the fort. 42. Along the bank of a river is drawn a base line of 500 feet. The angular distance of one end of this line from an object on the opposite side of the river, as observed from the other end of the line, is 53~; that of the second extremity from the same object, observed at the first, is 79~ 12'. Find the perpendicular breadth of the river. 43. A vertical tower stands on a declivity inclined 15~ to the horizon. A man ascends the declivity 80 ft. from the base of the tower, and finds the angle then subtended by the tower to be 30~. Find the height of the tower. 88 TRIGONOMETRY. 44. The angle subtended by a tower on an inclined plane is, at a certain point, 42~ 17'; 325 ft. farther down, it is 21~ 47'. The inclination of the plane is 8~ 53'. Find the height of the tower. 45. A cape bears north by east, as seen frbm a ship. The ship sails northwest 30 miles, and then the cape bears east. I-ow far is it from the second point of observation? 46. Two observers, stationed on opposite sides of a cloud, observe its angles of elevation to be 44~ 56' and 36~ 4'. Their distance from each other is 700 ft. What is the linear height of the cloud? 47. From a point B at the foot of a mountain, the elevation of the top A is 60~. After ascending the mountain one mile, at an inclination of 30~ to the horizon, and reaching a point C, the angle A CB is found to be 135~. Find the height of the mountain in feet. 48. From a ship two rocks are seen in the same right line with the ship, bearing N. 15~ E. After the ship has sailed northwest 5 miles, the first rock bears east, and the second northeast. Find the distance between the rocks. 49. From a window on a level with the bottom of a steeple the elevation of the steeple is 40~, and from a second window 18 ft. higher the elevation is 37~ 30'. Find the height of the steeple. 50. To determine the distance between two inaccessible objects by observing angles at the extremities of a line of known length. 51. Wishing to determine the distance between a church A and a tower B, on the opposite side of a river, I measure a line CD along the river (C being nearly opposite A), and observe the angles ACB, 58~ 20'; ACD, 95~ 20'; ADB, 53~ 30'; BDC, 98~ 45'. CD is 600 ft. What is the distance required? MISCELLANEOUS EXAMPLES. 89 52. Wishing to find the height of a summit A, I measure a horizontal base line CD, 440 yds. At C, the elevation of A is 37~ 18', and the horizontal angle between D and the summit is 76~ 18'; at D, the horizontal angle between C and the summit is 67~ 14'. Find the height. 53. A balloon is observed from two stations 3000 ft. apart. At the first station the horizontal angle of the balloon and the other station is 75~ 25', and the elevation of the balloon is 18~. The horizontal angle of the first station and the balloon, measured at the second station, is 64~ 30'. Find the height of the balloon. 54. Two forces, one of 410 pounds, and the other of 320 pounds, make an angle of 51~ 37'. Find the intensity and the direction of their resultant. 55. An unknown force, combined with one of 128 pounds, produces a resultant of 200 pounds, and this resultant makes an angle of 18~ 24' with the known force. Find the intensity and direction of the unknown force. 56. At two stations, the height of a kite subtends the same angle A. The angle which the line joining one station and the kite subtends at the other station is B; and the distance between the two stations is a. Show that the height of the kite is - a sin A sec B. 57. Two towers on a horizontal plane are 120 ft. apart. A person standing successively at their bases observes that the angular elevation of one is double that of the other; but, when he is half-way between them, the elevations are complementary. Prove that the heights of the towers are 90 and 40 ft. 58. To find the distance of an inaccessible point C from either of two points A and B, having no instruments to measure angles. Prolong CA to a, and CB to b, and join AB, Ab, and Ba. Measure AB, 500; aA, 100; aB, 560; bB, 100; and Ab, 550. 90 TRIGONOMETRY. 59. Two inaccessible points A and B, are visible from D, but no other point can be found whence both are visible. Take some p6int C, whence A and D can be seen, and measure CD, 200 ft.; ADC, 89~; ACD, 50~ 30'. Then take some point E, whence D and B are visible, and measure DE, 200; BDE, 54~ 30'; BED, 88~ 30'. At D measure ADB, 72~ 30'. Compute the distance AB. 60. To compute the horizontal distance between two inaccessible points A and B, when no point can be found whence both can be seen. Take two points C and D, distant 200 yds., so that A can be seen from C, and B from D. From C measure CF, 200 yds. to F, whence A can be seen; and from D measure DE, 200 yds. to E, whence B can be seen. Measure AFC, 83~; ACD, 53 30'; ACF, 54 31'; BDE, 54~ 30'; BDC, 156~ 25'; DEB, 88~ 30'. 61. A column in the north temperate zone is east-southeast of an observer, and at noon the extremity of its shadow is northeast of him. The shadow is 80 ft. in length, and the elevation of the column, at the observer's station, is 45~. Find the height of the column. 62. From the top of a hill the angles of depression of two objects situated in the horizontal plane of the base of the hill are 45~ and 30~; and the horizontal angle between the two objects is 30~. Show that the height of the hill is equal to the distance between the objects. 63. Wishing to know the breadth of a river from A to B, I take AC, 100 yds. in the prolongation of BA, and then take CD, 200 yds. at right angles to AC. The angle BDA is 37~ 18' 30". Find AB. 64. The sum of the sides of a triangle is 100. The angle at A is double that of B, and the angle at B is double that at C. Determine the sides. MISCELLANEOUS EXAMPLES. 91 65. If sin2A + 5 cos2A = 3, find A. 66. If sin2A =- m cos A - n, find cos A. 67. Given sin A = n sin B, and tan A = n tan B, find sin A and cos B. 68. If tan2A +4 sin2A =6, find A. 69. If sin A = sin 2 A, find A. 70. If tan 2 A = 3 tan A, find A. 71. Prove that tan 50~ + cot 50~ = 2 sec 10~. 72. Given a regular polygon of n sides, and calling one of them a, find expressions for the radii of the inscribed and the circumscribed circles in terms of n and a. If P, H, D are the sides of a regular inscribed pentagon, hexagon, and decagon, prove p2 =I t2+ D2. AREAS. 73. Obtain the formula for the area of a triangle, given two sides b, c, and the included angle A. 74. Obtain the formula for the area of a triangle, given two angles A and B, and included side c. 75. Obtain the formula for the area of a triangle, given the three sides. 76. If a is the side of an equilateral triangle, show that. a2V3 its area is 4 77. Two consecutive sides of a rectangle are 52.25 ch. and 38.24 ch. Find its area. 78. Two sides of a parallelogram are 59.8 ch. and 37.05 ch., and the included angle is 72~ 10'. Find the area. 79. Two sides of a parallelogram are 15.36 ch. and 11.46 ch., and the included angle is 47~ 30'. Find its area. 92 TRIGONOMETRY. 80. Two sides of a triangle are 12.38 ch. and 6.78 ch., and the included angle is 46~ 24'. Find the area. 81. Two sides of a triangle are 18.37 ch. and 13.44 ch., and they form a right angle. Find the area. 82. Two angles of a triangle are 76~ 54' and 57~ 33' 12", and the included side is 9 ch. Find the area. 83. Two sides of a triangle are 19.74 ch. and 17.34 ch. The first bears N. 82~ 30' W.; the second S. 24~ 15' E. Find the area. 84. The three sides of a triangle are 49 ch., 50.25 ch., and 25.69 ch. Find the area. 85. The three sides of a triangle are 10.64 ch., 12.28 ch., and 9 ch. Find the area. 86. The sides of a triangular field, of which the area is 14 acres, are in the ratio of 3, 5, 7. Find the sides. 87. In the quadrilateral ABCD we have AB, 17.22 ch,; AD, 7.45 ch.; CD, 14.10 ch.; B C, 5.25 ch.; and the diagonal AC, 15.04 ch. Find the area. 88. The diagonals of a quadrilateral are a and b, and they intersect at an angle D. Show that the area of the quadrilateral is I ab sin D. 89. The diagonals of a quadrilateral are 34 and 56, intersecting at an angle of 67~. Find the area. 90. The diagonals of a quadrilateral are 75 and 49, intersecting at an angle of 42~. Find the area. 91. Show that the area of a regular polygon of n sides, of n62 180~ which one is a, is -cot 4 n 92. One side of a regular pentagon is 25. Find the 'area. 93. One side of a regular hexagon is 32. Find the area. MISCELLANEOUS EXAMPLES. 93 94. One side of a regular decagon is 46. Find the area. 95. Find the'area of a circle whose circumference is 74 ft. 96. Find the area of a circle whose radius is 125 ft. 97. In a circle with a diameter of 125 ft. find the area of a sector with an arc of 22~. 98. In a circle with a radius of 44 ft. find the area of a sector with an arc of 25~. 99. In a circle with a diameter of 50 ft. find the area of a segment with an arc of 280~. 100. Find the area of a segment (less than a semicircle), of which the chord is 20, and the distance of the chord from the middle point of the smaller arc is 2. 101. If r is the radius of a circle, the area of a regular 1800 circumscribed polygon of n sides is nr'2 tann 3600 The area of a regular inscribed polygon is r2 sin 102. If a is a side of a regular polygon of n sides, the area ira2 1800 of the inscribed circle is cot2 4 nd 27ra 1800 The area of the circumscribed circle is 4 csc2 4 l 103. The area of a regular polygon inscribed in a circle is to that of the circumscribed polygon of the same number of sides as 3 to 4. Find the number of sides. 104. The area of a regular polygon inscribed in a circle is a gepmetric mean between the areas of an inscribed and a circumscribed regular polygon of half the number of sides. 105. The area of a circumscribed regular polygon is an harmonic mean between the areas of an inscribed regular 94 TRIGONOMETRY. polygon of the same number of sides, and of a circumscribed regular polygon of half that number. 106. The perimeter of a circumscribed regular triangle is double that of the inscribed regular triangle. 107. The square described about a circle is four-thirds the inscribed dodecagon. 108. Two sides of a triangle are 3 and 12, and the included angle is 30~. Find the hypotenuse of an isosceles right triangle of equal area. PLANE SAILING. 109. Plane Sailing is that branch of Navigation in which the surface of the earth is considered a plane. The problems which arise are therefore solved by the methods of Plane Trigonometry. The following definitions will explain the technical terms which are employed: The difference of latitude of two places is the arc of a meridian comprehended between the parallels of latitude passing through those places. The departure between two meridians is the arc of a parallel of latitude comprehended between those meridians. It evidently diminishes as the distance from the equator at which it is measured increases. When a ship sails in such a manner as to cross successive meridians at the same angle, it is said to sail on a rhiumb-line. The constant angle which this line makes with the meridians is called the cozrse, and the distaince between two places is measured on a rhumb-line. If we neglect the curvature of the earth, and consider the distance, departure, and difference of latitude of two places to MISCELLANEOUS EXAMPLES. be straight lines, lying in one plane, they will form a right triangle, called the triangle of plane sailing. If ABD be a plane triangle, right-angled at ), and AD represent the difference of latitude of A and Bi, DAB will be the course from A to B, AB the distance, and DB the departure, measured from B, between the meridian of A and that of B. 110. Taking the earth's equatorial diameter to be 7925.6 miles, find the length in feet of the are of one minute of a great circle.* 111. A ship sails from latitude 43~ 45' S., on a course N. by E., 2345 miles. Find the latitude reached, and the departure made. 112. A ship sails from latitude 1~ 45' N., on a course S.E. by E., and reaches latitude 2~ 31' S. Find the distance, and the departure. 113. A ship sails from latitude 13~ 17' S., on a course N.E. by E. 4 E., until the departure is 207 miles. Find the distance, and the latitude reached. 114. A ship sails on a course between S. and E., 244 miles, leaving latitude 2~ 52' S., and reaching latitude 5~ 8' S. Find the course, and the departure. 115. A ship sails from latitude 32~ 18' N., on a course between N. and W., making a distance of 344 miles, and a departure of 103 miles. Find the course, and the latitude reached. 116. A ship sails on a course between S. and E., making a difference of latitude 136 miles, and a departure 203 miles. Find the distance, and the course. 117. A ship sails due north 15 statute miles an hour, for one day. What is the distance, in a straight line, from the * The length of the arc of one minute of a great circle of the earth is called a geographical mile, or a knot. In the following problems, this is the distance meant by the term " mile," unless otherwise stated. 96 TRIGONOMETRY. point left to the point reached? (Take earth's radius, 3962.8 statute miles.) PARALLEL AND MIDDLE LATITUDE SAILING. 118. The difference of longitude of two places is the angle at the pole made by the meridians of these two places; or, it is the arc of the equator comprehended between these two meridians. 119. In Parallel Sailing, a vessel is supposed to sail on a parallel of latitude; that is, either due east or due west. The distance sailed is, in this case, evidently the departure made; and the difference of longitude made depends on the solution of the following problem: 120. Given the departure between any two meridians at any latitude, find the angle which those meridians make, or the difference of longitude of any point on one meridian from any point on the other. (The earth is considered to be a perfect sphere, and the solution depends on simple geometric and trigonometric principles. Cf. Problem 7.) The solution gives the following formula: Diff. long. = depart. x sec. lat. 121. A ship in latitude 42~16' N., longitude 72~ 16' W., sails due east a distance of 149 miles. What is the position of the point reached? 122. A ship in latitude 44~ 49' S., longitude 119~ 42' E., sails due west until it reaches longitude 117~ 16' E. Find the distance made. 123. In Middle Latitude Sailing, the departure between two places, not on the same parallel of latitude, is considered to be, approximately, the departure between the meridians of those places, measured on that parallel of latitude which lies midway between the parallels of the two places. Except in MISCELLANEOUS EXAMPLES. 97 very high latitudes or excessive runs, such an assumption produces no great error. By the formula of Art. 120, then, we shall haveDiff. long. = depart. X sec. mid. lat. 124. A ship leaves latitude 31~ 14' N., longitude 42~ 19' W., and sails E.N.E. 325 miles. Find the position reached. 125. Find t-ke bearing and distance of Cape Cod from Havana. (Cape Cod, 42~ 2' N., 70~ 3' W.; Havana, 23~ 9' N., 82~ 22' W.) 126. Leaving latitude 49~ 57' N., longitude 15~16' W., a ship sails between S. and W. till the departure is 194 miles, and the latitude is 47~ 18' N. Find the course, distance, and longitude reached. 127. Leaving latitude 42~ 30' N., longitude 58~ 51' W., a ship sails S.E. by S. 300 miles. Find the position reached. 128. Leaving latitude 49~ 57' N., longitude 30~ W., a ship sails S. 39~ W., and reaches latitude 47~ 44' N. Find the distance, and longitude reached. 129. Leaving latitude 37~ N., longitude 32~ 16'W., a ship sails between N. and W. 300 miles, and reaches latitude 41~ N. Find the course, and longitude reached. 130. Leaving latitude 50~ 10' S., longitude 30~ E., a ship sails E.S.E., making 160 miles' departure. Find the distance, and position reached. 131. Leaving latitude 49~ 30' N., longitude 25~ W., a ship sails between S. and E. 215 miles, making a departure of 167 miles. Find the course, and position reached. 132. Leaving latitude 43~ S., longitude 21~ W., a ship sails 273 miles, and reaches latitude 40~ 17' S. What are the two courses and longitudes, either one of which will satisfy the data? 98 TRIGONOMETRY. 133. Leaving latitude 17~ N., longitude 119~ E., a ship sails 219 miles, making a departure of 162 miles. What four sets of answers do we get? 134. A ship in latitude 30~ sails due east 360 statute miles. What is the shortest distance from the point left to the point reached? Solve the same problem for latitude 45~, 60~, etc. TRAVERSE SAILING. 135. Traverse Sailing is the application of the principles of Plane and Middle Latitude Sailing to cases when the ship sails from one point to another on two or more different courses. Each course is worked up by itself, and these independent results are combined, as may be seen in the solution of the following example: 136. Leaving latitude 37~ 16' S., longitude 18~ 42' W., a ship sails N.E. 104 miles, then N.N.W. 60 miles, then W. by S. 216 miles. Find the position reached, and its bearing and distance from the point left. We have, for the first course, difference of latitude 73.5 N., departure 73.5 E. We have, for the second course, difference of latitude, 55.4 N., departure 23 V. We have, for the third course, difference of latitude 42.1 S., departure 211.8 W. On the whole, then, the ship has made 128.9 miles of north latitude, and 42.1 miles of south latitude. The place reached is therefore on a parallel of latitude 86.8 miles to the north of the parallel left; that is, in latitude 35~ 49'.2 S. The departure is, in the same way, found to be 161.3 miles W.; and the middle latitude is 36~ 32'.6. With these data, MISCELLANEOUS EXAMPLES. 99. and the formula of Art. 123, we find the difference of longitude to be 201 miles, or 3~ 21' W. Hence the longitude reached is 22~ 3' W. With the difference of latitude 86.8 miles, and the departure 161.3 miles, we find the course to be N. 61~ 43' W., and the distance 183.2 miles. The ship has reached the same point that it would have reached, if it had sailed directly on a course N. 61~ 43' W., for a distance of 183.2 miles. 137. A ship leaves Cape Cod (Ex. 125), and sails A.E. by S. 114 miles, N. by E. 94 miles, W.N.W. 42 miles. Solve as in Ex. 136. 138. A ship leaves Cape of Good Hope (latitude 34~ 22' S., longitude 18~ 30' E.), and sails N.W. 126 miles, N. by E. 84 miles, W.S.W. 217 miles. Solve as in Ex. 136. PROBLEMS IN GONIOMETRY. Prove that 1. sin x + cos x= V/2 cos (x- 71). 2. si1n x-cos =- V2cos (x + 7r). 3. sinx+V3cosx==2sin(x+A-7). 4. sin (x - ) sin (x - 7r) sinx. 5. cos (x + T r) + cos (x - 7) = /3 cos x. 6. tan x+-secx tan( x+ -,r). 7. tanx+secx — sec x - tan x 1 - tan x cot x - 1 1 + tan x cot x + 1 sin x 1 + cos x 9. = —2 csc x. + cos x sin x 10. tanx+cotx=2csc2x. 12. 1+tanxtan2x sec2x. 11. cotx-tanx-2cot2x. 13. sec2x= se 2 -- sec2X 100 TRIGONOMETRY. Prove that 14. 2 sec 2 x sec (x + 45~) sec (x - 45~). cos x + sin x 15. tan2x+ sec2x C - scos x -- sin x 2 tan x 2 sin3x 16. sin2x= t- 17. 2sinx +sin2x= -- 1 + tan2x 1- cos x sin2 2 x- sin2x 18. sin 3x -- sin x 1. t 3 3 tan x - tan3x tan 2x +- tan x sin 3x 19. tan 3x= 20. 1 - 3 tan2x tan 2x - tan x sill x 21. sin (x + y)+ cos (x -y) -2 sin (x +- ) sin (y + 7r). 22. sin (x +y) - cos ( - y)- 2 sin (x- P 7) sin (y- - 7). sin (x +y) 23. tan x - tan y-= cos xacos y 24. t ( + ) - sin 2x + sin 2y 24. tan(x - y)= cos 2x +- cos2y 25. sin x + cos y tan [-(x+y) + 45 sin -cosy tllan -(x-y)- 4,5~ 26. sin 2x -sin 4x-2 sin 3x cos x. 27. sin 4x = 4.sin x cos x-8 sin cos x -= 8 cos3x sin x - 4 cos x sin x. 28. cos 4x = 1 - 8 cos2x + 8 cos4 =1 - 8 sin2x + 8 sin4. 29. cos 2x + cos 4 x 2 cos 3x cos 2x. 30. sin 3 x -sin x 2 cos 2 x sin x. 31. sin3x sin 3 x + cos3 cosx 3 x - cos3 2x. 32. cos4x - sinx = cos 2 x. 33. cos4x + sin4x 1 - 2 sin2 2 x. 34. cos6x - sin6x =- cos 2x (1 - sin2 cos2x). 35. cos6% + sin6x = 1 - 3 sin2x cos2x. sin 3x - sin 5x 36. = cot x. cos 3x -cos 5x MISCELLANEOUS EXAMPLES. 101 Prove that sin 3 x + sin 5 x 2 37. - 2 cos 2 x. sin x +- sin 3x 38. csc x -2 cot 2 x cos x = 2 sin x. 39. (sin 2x - sin 2 y) tan ( + y) = 2 (sin2 - sin2y). 40. (1 + cot x + tanx) (sin - cos x)= s - CSC csc X sec2w sin2 3x 41. sin sin3x +sin5 x -- in sin x 3 cos x + cos 3x cx. 42.. + - cot3x. 3 sin x - sin 3 x 43. sin 3 = 4 sins x sin (60~ + x) sin (60~ - ). 44. sin 4x = 2 sin x cos 3x + sill 2x. 45. sin + sin(x- r) + sin ( - ) = 0. 46. cos x sill(y - )+ cos y sin ( - x) + cos sin (x - y) =0. 47. cos (x + y) sill y cos (x + Z) sin z =sin (x + y) cos y - sin (x + z) cos z. 48. cos (x + y + ) + cos (X + y — Z)+ cos ( - y + ) + cos (y + - x)= 4 cos x cos y cos z. 49. sin(x + y) cos (x- y)+ sin (y + z) cos (y- n) + sin (z + x) cos (z - x) = sin 2 x + sin 2y + sin 2. sin 75~ + sin 15 = ta 6. 50. - -- tan 60~. sinT5- sin 15 51. cos 20~ + cos 100~ + cos 140~ = 0. 52. cos 36~ + sin 36 = V2 cos 9~. 53. tan 110 151' - 2 tan 22~ 30' - 4 tan 45~ = cot 11~ 15'. If A, B, C are the angles of a plane triangle, prove that 54. sin 2A +- sin 2B + sin 2 C =4 sinA sinB sin C. 55. cos 2A + cos 2B + cos 2 C- - 1 -4 cosA cosB cos C. 102 TRIGONOMETRY. If A, B, C are the angles of a plane triangle, prove that 3A 3B 3C 56. sin 3A + sin 3 B + sin 3 C -4 cos - cos - cos 2 2 2 2 57. cos2A + cos2B + cos2 C = -2 cos cos B cos C. If A + B + C = 90~, prove that 58. tan A tanB - tanB tan C -tan C tanA = 1. 59. si sin iB - sin2 C = 1 - 2 sinA sinB sin C. 60. sin 2 A +sin 2B + sin 2 C -4 cosA cosB cos C. 61. sin (sin- x + sin-ly) = x V - y2 + y 1 - x2. 62. tan (tan-l x + tan- y)= -y I - xy 2x 63. 2tan-lx —tan-l- 2 1 -X2' 64. 2 sin-x = sin-1 (2 V1 — x2). 65. 2 cos-lx=cos- (2 x2- 1). 3x X3 66. 3 tan-lx - tan-1-. 1 -- 32 67. sin-l _ =tan-l' x y y-x - z y 68. si - tan'1 - 69. tan-l 1 -- + tan- 1 2 - 4 tan'1 1-2x-+4x2 1 + 2 x — 4X2 2x2 70. sin-lxe= secVi1- X2 71. 2 sec-lx- tan-12 (2- 1) 2-X2 72. tan-l- +tan-l' 450. 73. tan-1 + tan-1 ' = tan-1 4 MISCELLANEOUS EXAMPLES. 103 Prove that 74. sin-'1 3 sin- 1 2- -- sin-1 6 3 1 4 75. sin-' \ sin- + n- --- 45~ 7582 s41 76. sec-'l + sec-'l -= 90~. 77. tan- (2 + V3) - tan-' (2 - 3) = sec- 2. 78. tan-l + tan-' -~ + tan-' 1 + tan-1 45~. 79. Given cos x =, find sin -x and cos x. 80. Given tan x, find tan -~x. 81. Given sin x + cos x- <i, find cos 2x. 82. Given tan 2x = 244, find sin x. 83. Given cos 3x — 23 find tan 2 x. 84. Given 2 cscx - cotx = -3, find sin x. 85. Find sin 18~, cos 36~. Solve the following equations: 86. sin x; 2 sin (~7r -+ X). 90. sin x+-cos 87. sin 2x= 2 cos x. 91. 4cos2x+ - 88. cos 2 x =2 sin x. 92. sin x -+sin' 89. sin x + cos x = 1. 93. sin2x=3s 94. ta x - tan 2x =tan 3x. 95. cot x -tan xsin x+ cosx. 96. tan2x= sin 2x. 99. sinx+sin2. 97. tanx-+cotx-tan2x. 100. sec2x+1: 1 -- tan x 98. = t = os2x. 101. tan2x-+ta: 1 + tanx 102. tan (7r + x) + tan ( 7r —x) =4. 103. V +sinxl- VI — sin x = 2 cosx. 2x =4 sin2 x. 3cosx = 1. 2x = sin 3x. in2x - cos2X. =x -cos2x. =2 cos x. n 3x -0. 104 TRIGONOMETRY. Solve the following equations: 104. tan x tan 3 x — - 105. sin (45~ + x) + cos (45 - ) =1. 106. tanx +secx-=a. 107. cos2x=a(1-cosx). 108. cos 2 x (1 -tan x) =a(1 + tanx). 109. sin6 x +- os6x = sin2 2 x. 110. cos 3x + 8 cos3x=0. 111. sec (x + 120~) + sec (x - 120~) = 2 cos x. 112. csc x cot x + /3. 114. cosx-cos 2 x =1. 113. 4cos2x +6sinx =5. 115. sin4x-sin22x=sinx. 116. 2 sin2x+ sin22x=2. 117. cos5 x+cos3x - cosx =0. 118. sec x -cot csc x —tan x. 119. tan2x + cot2x I-3. 120. sin4x-cos 3x sin2x. 121. sinx- -cosx -secx. 122. 2 cosx cos 3x +1 =0. 123. cos 3 x -2 cos 2 cos x =0. 124. tan 2x tanx =1. 125. sin (x + 12~) + sin (x - 8~) = sin 20~. 126. tan (600 + x) tan (600- x) =-2. 127. sin (x + 120~) + sin (x + 60~) =. 128. sin (x + 30~) sin (x - 30~) = ~. 129. sin4x + cos4x = ~. 131. tan (x + 30~) = 2 cos x. 130. sin4x —cos4x= —. 132. secx-2 tanx +-. 133. sin (x - y) = cos x, cos (x - y) - sin x. 134. tan x + tan y =; cot x + cot y -b. 135. sin (x + 12~) cos (x - 12~) -cos 33~ sin 57~. 136. sin-1x + sin-1 x = 120~. 137. tan-' x + tan-l 2 x = tan-13 /3. 138. sin-l x + 2 cos-l x = -7r. MISCELLANEOUS EXAMPLES. 105 Solve the following equations: 139. sin-lx+3 cos-lx-210~. 140. tan-x + 2 cot-1x= 135~. 141. tan-1 (x +1 ) + tan-1 (x -1) =tan- 2 x. 142. tan-i + tan-l x-2 r. x+ x —1 2x 143. tan-' 1 -600. 1 -- X2 Find the value of: 144. asecx + bcscx, when tanx =- 145. sin3x, when sin 2x = / - m2. CSC2x - se2x - 146. es2x - - seC2 when tan x = VT. csc 2X + seclc2 147. sin x, when tan2x + 3 cot2x =4. 148. cos x, when 5 tanx + secx 5. 149. seex, when tanx = - V2a+1 Simplify the following expressions: 50 (cos x + cos y)2 + (sin x + siny)2 cos - (x-y) 15 sin ( + 2y) - 2 sin (x + y) + sin x cos (+ 2 y) - 2 cos (x +y) + cosx 152. sin ( - ) + 2 sin + sin ( + ) sin (y - ) + 2 sin y+ sin (y+ z) cos 6x - cos 4x 153. sin 6 x + sin 4x 154. tan-l (2x + 1) + tan-' (2x -1). 1 1 1 i55. + + + 1 + sin x +1 + os2x 1 + sec2x 1 +-csc2x 156. 2 sec2x - sec4x - 2 csc2x + csc4x. ENTRANCE EXAMINATION PAPERS.* PLANE TRIGONOMETRY AND LOGARITHMS. I. (Cornell, June, 1889.) (One question may be omitted.) 1. Prove that cos c-O - sin 0; see (7r + 0) - - csc 0; tan (- 0) = - tan 0; csc (r - 0) = cs 0. 2. Draw the curve of tangents, and show the changes in the value of this function as the arc increases from 0~ to 360~. 3. In terms of functions of positive angles less than 45~, express the values of sin —250~, csc 7r, tan- J — r. Also find all the values of 0 in terms of a when cos 0 \/sin2a. 4. (a) Given cos x = 0.5, find cos 2x and tan 2 x. (b) Prove that vers (180~ - A) + vers (360~ - A) = 2. 5. Prove the check formulae: a- b: c cos - (A-B): sin C; a- b: c= sinl (A - B): cos C. * NOTE. In these papers, as in many text-books, the Greek letters a (alpha), p (bayta), y (gamma), a (delta), 0 (thayta), and 0 (phee), are occasionally used to denote angles. ENTRANCE EXAMINATION PAPERS. 107 6. In a right triangle, r (the hypotenuse) is given, and one acute angle is n times the other; find the sides about the right angle in terms of r and n. 7. The tower of McGraw Hall is 125 ft. high, and from its summit the angles of depression of the bases of two trees on the campus, which stand on the same level as the Hall, are respectively 57~ 44' and 16~ 59', and the angle subtended by the line joining the trees is 99~ 30'. Find the distance between the trees. II. (Cornell, Jlne, 1890.) (Omit one question.) 1. Prove that cot (- 0) - cot 0; csc -0 = csc 0; sin (r + 0) sin 0; sec co-0 csc 0; cos ( 7r + 0) - sin 0. 2. Show that in any plane triangle sill A = (s ) ( -- - 3. Find the value of sin ( ~ 0') in terms of sin 0, cos 0, Sill 0, and cos 0'. 4. Given tan 45~- =1; find all the functions of 22~ 30'. 5. Determine the number of solutions of each of the triangles: a=13.4, b=11.46, A=77~20'; c=58, a=75, C=60~; -6 109, a-94, A4 92~ 10'; c= 309, b=360, C= 21~ 14'25". 6. In a parallelogram, given side a, diagonal d, and the angle A formed by the diagonals; find the other diagonal and the other side. 7. A and B are two objects whose distance, on account of intervening obstacles, cannot be directly measured. At the summit of a hill, whose height above the common horizontal 108 TRIGONOMETRY. plane of the objects is known to be 517.3 yds., angle ACB is found to be 15~ 13' 15". The angles of elevation of C viewed from A and B are 21~ 9' 18" and 23~ 15' 34" respectively. Find the distance from A to B. III. (Cornell, September, 1891.) 1. Trace the value of tan 0 and that of csc 0, as 0 increases from 0~ to 360~. 2. (ca) Find the remaining functions of 0 when cos 0- - 3. (b) Determine all the values of 0 that will satisfy the relation cot 0 2 cos 0. 3. Prove the identity sin2 A -- cos2 A tan A - cot A -- Si- c A 2 cot 2 A. sin A cos A 4. Derive an expression for the sine of half an angle in a triangle in terms of the sides of the triangle. 5. Construct a figure and explain fully (giving formulae) how you would find the height above its base, and the distance from the observer, of an inaccessible vertical object that is visible from two points whose distance apart is known, and which can be seen from one another. 6. Given two sides of a plane triangle equal respectively to 121.34 and 216.7, and the included angle 47~ 21' 11", to find the remaining parts of the triangle. 7. In a right triangle, if the difference of the base and the perpendicular is 12 yds., and the angle at the base is 38~ 1' 8", what is the length of the hypotenuse? ENTRANCE EXAMINATION PAPERS. 109 IV. (Cornell, June, 1892.) 1. By means of an equilateral triangle, one of whose angles is bisected, find the numerical values of the functions of 30~ and 60~. 2. If 0 be any angle, prove that sin 0 = tan0: /1 + tan2 0, cos 0 csc' 0 -1: csc 0. 3. Prove that sin 0+ sin 01 —c 3. Prove that cos 0+ - - -c cot ~ (0-0'), where 0 and 0 COS 0 - COS 6' - V are any angles. 4. Find sin 2 0, cos 2 0, and tan 2 0, in terms of functions of 0. 5. Assuming the law of sines for a plane triangle, prove that (a + b): c = cos 1 (A - B): sin C, (a - b): c = sin I (A - 3): cos 2 C. 6. At 120 feet distance, and on a level with the foot of a steeple, the angle of elevation of the top is 62~ 27'; find the height. 7. Solve the plane triangle given the three sides, a 48.76, b 62.92, c= 80.24. V. (Harvard, June, 1889.) 1. In how many years will a sum of money double itself at 4 per cent., interest being compounded semi-annually? 2. Given sin2x = -- find sin 2 x and tan 2 x. 2 3. Find all values of x, under 360~, which satisfy the equation V8 cos 2 x 1 - 2 sin x. 110 TRIGONOMETRY. 4. What is always the value of 2 sin2x sin2y + 2 cos2 cos2y- cos 2 x cos 2y? 5. Find the area of a parallelogram, if its diagonals are 2 and 3, and intersect each other at an angle of 35~. 6. Find the bearing and distance from Cape Horn (55~ 55 S., 67~ 40' W.) to Falkland Island (51~ 40' S., 59~ W.). VI, (Hcarvard, June, 1890.) 1. In a certain system of logarithms 1.25 is the logarithm of. What is the base? Be careful to remember what 1.25 means. 2. Find the tangent of 3x in terms of the tangent of x. 3. One angle of a triangle is 35~, and one of the sides including this angle is 24. What are the smallest values the other sides can have? 4. Find all values of x, under 360~, which satisfy the equation tan 2x (tan2 - 1) = 2 sec2 - 6. 5. Two ships leave Cape Cod (42~ N., 70~ W.), one sailing E., the other sailing N.E. How many miles must each sail to reach longitude 65~ W.? 6. If A + B + C- 1800, find the value of tan A + tan B + tan C - tan A tan B tan C. ENTRANCE EXAMINATION PAPERS. 111 VII. (Harvard, September, 1891.) 1. What is the base, when log 0.008 - 1.5? 2. If cos (a-b) = 3 cos (a + b), find the value of sec( + seca sec b 3. The area of an oblique-angled triangle is 50. One angle is 30~, and a side adjacent to that angle is 12. Solve the triangle. 4. Find all values of x, less than 360~, which satisfy the equation sin 2x - cos x = cos2x. 5. Find, by Middle Latitude Sailing, the course and the distance from Cape Cod (Lat. 42~ 2' N., Long. 70~ 4' W.) to Fayal (Lat. 380 32' N., Long. 28~ 39' W.). 6. In any triangle ABC, prove tan I A, tan - B +- tan A tan I C + tan ~ B tan i- = 1. VIII. (Harvard, September, 1892.) (Take the questions in any order. One of the starred questions may be omitted.) 1. What is the base of a system of logarithms in which log (2j)- 2.33k? *2. Given the area of a right triangle, and the smallest angle, find the legs of the triangle in terms of the data. sin a tan a *3. Find a and b, given -= 2, and = V3. sin b tan b 112 TRIGONOMETRY. 4. One angle of an oblique-angled triangle is 45~, and an adjacent side is V/2. What is the smallest value which the opposite side can have? Solve the triangle when the opposite side is 4. 5. A ship leaves Cape Cod (42~ 2' N., 70~ 4' W.) and sails 200 knots on a course S. 40~ E. Find the latitude and longitude reached. 6. If 2 tan 2 a = tan 2 b sin 2 b, find the relation between the tangents of a and b. IX. (Harvard, June, 1893.) (Take the problems in any order. One of the starred problems may be omitted.) 1. What is the base of the system of logarithms, when log 3 =0.3976? *2. Solve the right-angled triangle in which one angle is 30~, and the difference of the legs is 4. *3. Find x, given sec x =2 tan x +2. *4. One angle of a triangle is double another angle. The side opposite the first angle is three-halves of the side opposite the second angle. Find the angles. 5. Find, by Middle Latitude sailing, thbe course and distance from Funchal (32~ 38' N., 16~ 54' W.) to Gibraltar (36~ 7' N., 5~ 21' W.). *6. Reduce to its simplest form cos 2 x tan (45~ + x) - sin 2 x. ENTRANCE EXAMINATION PAPERS, 113 X. (Harvard, September, 1893.) (One of the starred problems may be omitted.) 1. If the base of our system of logarithms were 20 instead of 10, what would be the logarithm of one tenth? *2. The area of a right triangle is 6, and the sum of the three sides is 12. Solve the triangle. *3. Reduce to its simplest form cos2 B + sin2 B cos 2 A - sin2A cos 2 B. *4. Two angles of a triangle are 40~ 14' and 60~ 37'. The sum of the two opposite sides is 10. Find these sides. 5. A ship leaves Cape of Good Hope (34~ 22' S., 18~ 30' E.), and sails N. 40~ W. to Latitude 30~ S. Find, by Middle Latitude Sailing, the Longitude reached and the distance sailed. *6. The base angles of a triangle are 22~ 30' and 112~ 30'. Find the ratio between the base and the height of the triangle. XI, (Harvard, June, 1894.) (Arrange your work neatly.) 1. What is meant by the logarithm of a number n in the system whose base is 8? What will be the logarithm of 4 in this system? 2. Establish the formula: sin x -=~ (1 2 cos ) 2-csx Which sign should be used when x lies in the first quadrant? When x lies in the second quadrant? 114 TRIGONOMETRY. 3. In a triangle two angles are equal to 32~ 47' and 49~ 28' respectively and the length of the included side is 0.072. Solve the triangle. 4. A circular tent 30 feet in diameter subtends at a certain point an angle of 15~. Find the distance of this point from the centre of the tent. 5. A ship leaves Latitude 42~ 2'N., Longitude 70~ 3' X., and sails N. 40~ E. a distance of 420 miles. Find by Middle Latitude Sailing the position reached. XII. (Sheffield Scientific School, September, 1892.) 1. Express an angle of 60~ in radians. 2. Represent geometrically the different trigonometric functions of an angle. State the signs of each function for each quadrant. 3. Express tan 9 and sec cb in terms of sin <. 4. Derive the formula sin a+ sin / =2 sin - (a- I /) Cos ~ (a- f). 5. Show that, if a, b and c are the sides of a triangle and A is the angle opposite the side a, then a2 = b2 + c2- 2 be cos A. 6. Given cos 2 x = 2 sin x, to find the value of sin x. 7. Given two sides of a triangle a-450.2, b=425.4, and the included angle C = 62~ 8'; find the remaining parts. XIII. (Sheffield Scientific School, June, 1893.) 1. Express an angle of 15~ in radians. 2. Write the simplest equivalents for sin (7r+-), tan (27r- ), cos ( 7r — ), sec (r + 4). ENTRANCE EXAMINATION PAPERS. 115 3. Express tan ( in terms of sin l, cos qb and cot <, respectively; and cos ( in terms of tan (, see b and cosec'(, respectively. 4. Show (a) that sil (a + 3) + sin (a- 3) 2 sin a cos /; (b) that cos (a +/) + cos (a - ) =2 Cos a cos. b2 + C2 - a2 5. Assume the formula cosa -- --- and show that sin2 a (s-b) (s c) when s= ( + b + c)., when s==^(a-b-c). 6. Obtain a formula for tan 1 a in terms of cos a. 7. The base of a triangle c=-556.7, and the two adjacent angles a=65~ 20'.2, /=-700 00'.5; calculate the area of the triangle. 8. Given 0 < a < 90~, and log cos a= 1.85254, to determine a.;XIV. (Sheffield Scientific School, September, 1893.) 1. Reduce an angle of 3.5 radians to degrees. 2. Define the different trigonometrical functions of an angle and give their algebraic signs for an angle in each quadrant. 3. Write simple equivalents for the following functions: sin (- a); cos (- a); tan (-r + a); see (3 r- a). 4. Express cosec a in terms, respectively, of sin a, cos a, tan a, cot a, sec a. 5. Reduce (cos a cos - sin a sin f)2+ (sin a cos /+ -cos a sin p)2 to its simplest equivalent. 6. Show that tal (7 a I-tan a 6. Show that tan 7 - _a a,- -— * \4 1 + tan a 116 TRIGONOMETtY. 7. The sum of two sides, a and b, of a triangle is 546.7 ft., the sum of the opposite angles, a and f/, is 124~, and sin a: sin f3 =-1.003; find the angles and sides of the triangle. 8. Given 0 < a<90~, and log cot a 0.03293, to determine a. XV. (Sheffield Scientific School, June, 1894.) 1. Express (a) an angle of 2 radians in degrees; (b) an angle of 30~ in radians. 2. Give simple equivalents for the following functions: tan(-x), cosec( —x), sin(x+ — 7r), sin (x — r), tan ( 7r- x), sin (2 r - x). 3. Given tan x =, to express sin x, cos x, cot x, sec x, and cosec x in terms of a and b. 4. Show that tan a ~ tan b sin (a — b) cos a cos b 5. Derive the formulae 1OS + cos a sin - cos a 2cos 1a-=~ 2, sin 2- 2 6. Given 180~ < < 270~, and log cot - = 0.3232, find p. 7. The sides of a triangle are a =32.5 ft., b -33.1 ft., c = 32.4 ft.: Calculate the area of the triangle and the angle C opposite the side c, using the following formulae: S -= Vpv (p - a) (p - b) (p - c) =- ab sin C, in which S denotes the area of the triangle, and p=-(a+b+c). CHAPTER VI. CONSTRUCTION OF TABLES. ~ 42. LOGARITHMS. Properties of Logarithms. Any positive number being selected as a base, the logarithm of any other positive number is the exponent of the power to which the base must be raised to produce the given number. Thus, if an- = N, then n = logaN. This is read, n is equal to log N to the base a. Let a be the base, M and N any positive numbers, m and n their logarithms to the base a; so that am AM, an -VM m - logaA n = logaN. Then, in any system of logarithms: 1. The logarithm of 1 is 0. For, a~ =1... O-=logl. 2. The logarithmi of the base itself is 1. For, a1 = a..'. 1 = oga. 3. The logarithm of the reciprocal of a positive number is the negative of the logarithm of the number. 1 1 For, if an =, then -- a.*. log ( n-logaN. 118 TRIGONOMETRY. 4. The logarithm of the product of two or more positive natmbers is found by adding together the logarithms of the several factors. For, M XNT= am X an - am + n... log, (M X N) = m + n= loga- log+ l N. Similarly for the product of three or more factors. 5. The logarithm of the quotient of two positive numbers is found by subtracting the logarithm of the divisor from the logarithm of the dividend. M am For, N _" = am-n..0loga (N) = - - n = logaf- logVN. 6. The logarithm of a power of a positive number is found by multiplying the logarithm of the number by the exponent of the power. For, N1 = (a) p = an. * log (Np) = np =p loga N. 7. The logarithm of the real positive value of a root of a positive number is found by dividing the logarithm of the number by the index of the root. n For, v'N a =..*. log /-N- n l10a _. r r Change of System. Logarithms to any base a may be converted into logarithms to any other base b as follows: Let Nbe any number, and let in = loga N and m = log NV. Then, N= an and =- bm. = a=.M CONSTRUCTION OF TABLES. 119 Taking logaritlhms to any base whatever, n log a = r- log b, or, log a X log Nlog X log, N, from which log, N may be found when log a, log b, and loga N are given; and conversely, loga N may be found when log a, log b, and logb N are given. Two Important Systems. Although the number of different systems of logarithms is unlimited, there are but two systems which are in common use. These are: 1. The common system, also called the Briggs, denary, or decimal system, of which the base is 10. 2. The natural system of which the base is the fixed value which the sum of the series 1 1 1 +.. 1 12+ 1.2.3+ 1.2.3.4 approaches as the number of terms is indefinitely increased. This fixed value, correct to seven places of decimals, is 2.7182818, and is denoted by the letter e. The common system is used in actual calculation; the natural system is used in the higher mathematics. EXERCISE XXIII. 1. Given log102=-0.30103, log03=0.47712, log107=0.84510 find log106, -log,,o4, logo121, log104, log1012, loglo5, logicO, logio~, logio- 7 loglo 2 2. With the data of example 1, find log210, log25, log35, log17, log5.3' 120 TRIGONOMETRY. 3. Given log1 e - 0.43429 find log,2, log,3, log,5, loge7, log,8, log, 9, loge,2, loge,5 log, 27, log, 7. 4. Find x from the equations 5x=12, 16x =10, 27 =4. ~ 43. EXPONENTIAL AND LOGARITHMIC SERIES. Exponential Series. By the binomial theorem ( +i) n + 1 2 n2 + 1 * 2 TO X q + 'I nx (nx - 1) (fx - 2) + x;- ~ ""' =1 + X + 1 +.-"'' (1) This equation is true for all real values of x, since the binomial theorem may readily be extended to the case of incommensurable exponents (College Algebra, ~ 264); it is, however, only true for values of n numerically greater than 1, since - must be numerically less than 1 (College Algebra, 7f ~ 375). As (1) is true for all values of x, it is true when x =1. (1+ )I+1+2 + 3- +..... (2) BU.t [('1 I)nL]x (' 1+1) nx But +(l+ =( +-) * lb n) n CONSTRUCTION OF TABLES. 121 Hence, from (1) and (2), J _ 1__ _I 2\2 ( —) sc x — x — ) =i+x,+ 2 + +13.. This last equation is true for all values of n numerically greater than 1. Taking the limits of the two members as n increases without limit we obtain (1+1+~_+~+ + + +..., (3) and this is true for all values of x. It is easily seen that both series are convergent for all values of x. The sum of the infinite series in parenthesis is the natural base e. Hence by (3), x2 x3 e =1+x+j+ - +i+-. (4) To calculate the value of e we proceed as follows: 1.000000 2 1.000000 3 0.500000 4 0.166667 5 0.041667 6 0.008333 7 0.001388 8 0.000198 9 0.000025 0.000003 Adding, e= 2.71828. To ten places, e 2.7182818284, 122 TRIGONOMETRY. Limit of 1 +~) By the binomial theorem, (1 + - -- 1 + x n (n-1) x 2 n 1'2 n2 n(n-1)(n-2) Xs T 12b 3 n 1_1 (1-1) ( I- _2) x + + + _____ _____ +.... This equation is true for all values of n greater than x (College Algebra, ~ 375). Take the limit as n increases without limit, x remaining finite; then limit x2 x3 n innimte ( + n)-1 + x + 3 + +' n infinite - _ -1-.. - ex limit ( 1 n infinite n Logarithmic Series. Let y= log,(1 + ); then 1 +x- _.. (, limit + J n infinite n If n is merely a large number, but not infinite, (1+7~ - +x+, where e is a variable number which approaches the limit 0, when n increases without limit. Hence 1 +n =-/1 + x+ E y-n/i:+x+ -n. CONSTRUCTION OF TABLES. 123 If now n becomes oo, and consequently e becomes 0, we have limit n - - Y n infinite X- j Assuming that x is less than 1, we may expand the righthand member of this equation by the binomial theorem. The result is limit 2 n l+-x+- — 1 -...-n n infinite 9? ( n n 2 limit I (l 1 1 X3 n infinitej'+ - J2\^n ^ j3 I 2 x2 2 3 x4 2 3 124 X2 X3 X4.. log, (1 + x) = 2 + 3 4I+ This series is known as the logarithnmic series. It is convergent only if x lies between -1 and + 1, or is equal to +1. Even within these limits it converges rather slowly, and for these reasons it is not well adapted to the computation of logarithms. A more convenient series is obtained in the following section. Calculation of Logarithms. The equation log" (1 + -+)..... (1) log,(l+ y)= y- J+^ - ~+ ' ([) holds true for all values of y numerically less than 1; therefore, if it holds true for any particular value of y less than 1, it will hold true when we put - y for y; this gives?/2 y 4 ()4 log,(l-y)= -y-2 4(2) 124 TRIGONOMETRY. Subtracting (2) from (1), since log, (1 + y) -log, (1 - Y) = log, ( _y) we find loge(t~) =2(+ ++...... 1 l_-y __- I Put y -2 1; thenl + + and loge ( + ) = loge ( + 1) - log 2 2+1 + 3 (2 +1)3+ (2 +1) This series is convergent for all positive values of z. Logarithms to any base a can be calculated by the series: loga ( + 1) - logaz - 2 1 + 1 + I + 42. log (2-+1 3 (2z+ )3+ 5(2z + )5 ) ~42. Calculate log,2 to five places of decimals. Let z=1; then z +1=2, 2z+ 1 = 3, 2 2 2 2 and log 2-= + + + 7 +..... 3 3 x 33 5 x 35 7 x 37 The work may be arranged as follows: 3 2.000000 9 0.666667 - 1 = 0.666667 9 0.074074 3 =0.024691 9 0.008230 5 =0.001646 9 0.000914 7 =0.000131 9 0.000102 9 = 0.000011 0.000011 - 11 = 0.000001 loge2 = 0.693147 NOTE. In calculating logarithms the accuracy of the work may be tested every time we come to a composite number by adding together the logarithms of the several factors. In fact, the logarithms of composite numbers are best found in this way, and only the logarithms of prime numbers need be computed by the series. CONSTRUCTION OF TABLES. 125 EXERCISE XXIV. 1. Calculate to five places of decimals loge3, log5, logg7. 2. Calculate to ten places of decimals log10O. 3. Calculate to five places of decimals log,02, log,0e, log011. ~ 44. TRIGONOMETRIC FUNCTIONS OF SMALL ANGLES. Let AOP be any angle less than 90~ and x its circular measure. Describe a circle of unit radius about 0 as a centre and take Z AOP' =-Z AOP. Draw the tangents to the circle at P and P', meeting OA in T. \ o - Then from Geometry chord PP' < arc PP' < PT'+ P', or, dividing by 2 FIG. 36. MP< arc AP <PT, or sin x < x < tan x. Hence, dividing by sin x 1 < < sec x, sin X sin x or 1> >cosx. (1) x sin x Then lies between cos x and 1. If now the angle x is constantly diminished, cos x approaches the value 1. sin x Accordingly, the limit of, as x approaches 0, is 1; x sin x or, in other words, if x is a very small angle differs from by a small value, which approaches as x approaches 1 by a small value e, which approaches 0 as x approaches 0. 126 TRIGONOMETRYo To find the sine and cosine of 1'. If x is the circular measure of 1', 2 5 3. 14159-1 -x - 302 - 3.14159+ - 0.00029088+, 300 x 00 10800 the next figure in x being either 7 or 8. Now sin x > 0 but <x; hence sin 1' lies between 0 and 0.000290889. Again cos 1' = — V - sin2 1' > /i - (0.0003)2 > 0.9999999. Hence cos 1' = 0.9999999+. But, from (1), sin x > x cos x.. sin 1'> 0.000290887 X 0.9999999 > 0.000290887 (1 - 0.0000001) > 0.000290887 - 0.000000000290887 > 0.000290886. Hence sin ' lies between 0.000290886 and 0.000290889; that is, to eight places of decimals sin 1' 0.00029088 +, the next figure being 6, 7, or 8. EXERCISE XXV. Given r = 3.1415926, 1. Compute sin 1', cos 1', and tan 1 to as many decimal places as possible. 2. Compute sin2' by the same method, and also by the formula sin 2 x =2 sin x cos x. To how many places do the two results agree? 3. Compute sin 1~ to four places of decimals. 4. From the formula cos x = 1- 2 sin2 2, show that X~2~~~~ ~ 2 cos x 1 -- CONSTRUCTION OF TABLES. 127 5. Show by aid of a table of natural sines that sin x and x agree to four places of decimals for all angles less than 4~ 40'. G. If the values of log x and log sin x agree to five decimal places, find from a table the greatest value x can have. ~ 45. SIMPSON'S METHOD OF CONSTRUCTING A TRIGONOMETRIC TABLE. By ~ 31 (Plane Trigonometry) we have sin (A + B) + sin (A - B) = 2 sin A cos B. If we put A=-x+2y, B=y, this becomes sin (x + 3y) + sin (x + y) = 2 sin (x + 2 y) cos y, or sin (x + 3y) = 2 sin (x +2 y) cos y -sin (x + y). Similarly cos (x + 3 y) =2 cos (x + 2 y) cos y -cos (x + y). (1) If y = ', the last two equations become sin (x + 3') = 2 sin (x + 2') cos 1' - sin (x + 1') cos (x + 3') = 2 cos (x + 2') cos 1'- cos (x + 1'). Hence, taking x successively equal to - 1', 0', 1', 2', we obtain sin 2'= 2 sin 1' cos 1', sin 3' = 2 sin 2' cos 1'- sin 1', sin 4' =2 sin 3'cos 1'- sin 2', cos 2'= 2 cos2 1 1, cos 3' = 2 cos 2' cos 1' - cos 1', cos 4' =2 cos 31 cos 1' - cos 2', Since the sin 1' and cos 1' are known, these equations enable us to compute step by step the sine and cosine of any angle. The tangent may then be found in each case as the quotient of the sine divided by the cosine. 128 TRIGONOMETRY. This process need be carried only as far as 30~. For sin (30~ + x) + sin (30~ - x) =2 sin 30~ cos x = cos x, cos (30~ + x) - cos (30~ - x) = - 2 sin 30~ sin x = - sin x,..sin (30 + x) = cos x - sin (30~ - x), cos (30~ + x) - sin x + cos (30~ - x). Moreover the sines and cosines need be calculated only to 45~, since sin (45~ + x) = cos (45 - x), cos (45 + x) = sin (45~ - x). In using this method the multiplication by cos 1', which occurs at each step, can be simplified by noting that cos 1' = 0.9999999 - 1 - 0.0000001. Simpson's method is superseded in actual practice by much more rapid and convenient processes in which we employ the expansions of the trigonometric functions in infinite series. EXERCISE XXVI. 1. Compute the sine and cosine of 6' to seven decimal places. 2. In the formula (1) let y=l~. Assuming sin 1~=0.017454 +, cos 1~= 0.999848 +, compute the sines and cosines from degree to degree as far as 4~. ~ 46. DE MOIVRE'S THEOREM. Expressions of the form cos x - i sin x, when i =/ —1, play an important part in modern analysis. Given two such expressions cos x + i sin x, cos y + i sin y, their product is (cos x + i sin x) (cos y + i sin y) = cos x cos y - sin x sin y + i (cos x sin y + sin x cos y) = cos (x + y) + i sin (x +y). CONSTRUCTION OF TABLES. 129 Hence, the product of two expressions of the form cos x + i sin x, cos y + i sin y is an expression of the same form in which x or y is replaced by x + y. In other words, the angle which enters into such a product is the sum of the angles of the factors. If x and y are equal, we have at once from the preceding (cos x + i sill X)2 COS 2 x + i sin 2x; and again (cos x + i sin x)3 (COS X + i sin x)2 (cos x + i sin x) = (cos 2x i sin 2x) (cos x + i sin x) - cos 3x +- i sin 3x Similarly (cos x + i sin x)4 = cos 4x + i sin 4x, and in general if n is a positive integer (cos x — i sin x) = cos nx + i sin nx. (1) Hence To raise the expression cos x + i sin x to the n th power when n is a positive integer, we have only to multiply the angle x by n. Again, if n is a positive integer as before, cos - + i sin cos x + i sin x n n/ x x.*. (cos x + i sin x-cos sinn 1 Since, however, x may be increased by any integral multiple of 2 7 without changing cos x + i sin x, it follows that all the n expressions x..x x+2r. x+27r cos-+ isin-, cos - - +isinn n n n x +47r.. 4-7r cos ---- + i sin, x..... n n Sx +(n- 1)27r +.. x+ (n - 1)27r cos -- - i+ sin n n 130 TRIGONOMETRY. are nth roots of cos x + i sin x. There are no other roots, since x-+-n27r. x +- n27 cos ----- sin n n (x \.. xx =cos + 27r +-isin (+ 2r ) =cos -+isin-, and cos -- +i silln n n ( wcos - +27r +)isin (+2-7+2)7r n\ n x + 27, x. - +27r =- cos --- + i sin n n and so on. Hence, if n is a positive integer, 1 (cos x + i sin x)nI x - 2 krr x + 2 k~cos+ +i zsinx + ( 0,1,2,..... n-). (2) n it From (1) and (2) it follows at once that if m and n are positive integers (cos x + i sin x) n - (cos x + i sin x)?} cos-(x +2 kr)+i sin-(x+ 2 k7) (k=0, 1,2,..... n- ). (3) Finally, if - is a negative fraction, 7n (cos x +i sin x)- n -- (cos x + i sin x)W c 1cos x - i si n x But -- cos x - i sin x (cos x - i sin x) (cos x - i sin x) cosr isinx cos x -- i sin x = cos x - i sin x, = cos (- x) + i sin (- x). CONSTRUCTION OF TABLES. 131 Hence m C m (cos x + i sin x)- n= cos (- x) +i sin (- x) }7 = cos - (- x + 2 7) + i sin (- x + 2 kr), n n (k= 0, 1, 2,..... n -) =cos ( — (x + 2 ) )+ sin - ( -x + 2 ), n it (k 0, 1, 2,.....- n ). (4) Consequently if n is a positive or negative integer or fraction (cos x i sin X = cos [n (x - 2 kr) ] + i sin [n (x + 2 77r)], (7 =0,, 2,..... n -). (5) EXAMPLE: Find the three cube roots of - 1. We have - 1 = cos 180~ + i sin 180~ 180~ + 2kw 180~0+ 2lcw.'. (- 1) = cos + i si (k = O, 1, 2). 3 3 For the three cube roots of - 1 we find therefore cos 60~ + i sin 60~, cos 180~ + i sin 180~, cos 300~ + i sin 300~, 1+iv-3 1-iV3 or 2 — 1, 2 ' 2 By aid of De Moivre's Theorem we may express sin nO and cos nO, when n is an integer, in terms of sin 0 and cos 0. Thus cos nO + i sin nO = (cos 0 + i sin 0)" -cosO+ incos"- 0 sin 0 +i2 n ) cos-2 sin2 + n3 n (n-1) (n ) —2) cosn3 0 Sin3 0.. Or, since i2=-1, i3-i, i4=+, cos n 0++ n- ic sin 0 n cos 0 sin 0 _ n ( ) (n- (n) (n -2 (n — 1) cos-2 0 sin2 O-i- CoSn-3 sin3 +..... 2 s 132 TRIGONOMETRY. Equating now the real parts and the imaginary parts separately, we obtain cos n 0 cos" 0 (- 2 )cos"2 0 sin2 0 + n — n ) (n-2) (n-3) cosn-4 0 sin4 0-..... sin n9= n cosn-1 0 sin 0 - n (n- ) (n 2) cos"-3 0 sin3 0 + n(n-l) (n-2) (n-3) (n-4) Cos-s 0 inso-... EXERCISE XXVII. 1. Find the six 6th roots of -1; of + 1. 2. Find the three cube roots of i. 3. Find the four 4th roots of - i. 4. Express sin 4 0 and cos 4 0 in terms of sin 0 and cos 0. ~ 47. EXPANSION OF SIN X, COS X, AND TAN X IN INFINITE SERIES. Let one radian be denoted simply by 1, and let cos 1 - i sin 1 = k. Then cos x + i sin x = (cos 1 + i sin 1) kx, and putting -x for x cos (- x) + i sin (- x) = cos x - i sin x - k-T. That is cos x+ i sin x - kx and cos x - i sin x - k-x By taking the sum and difference of these two equations, and dividing the sum by 2 and the difference by 2i, we have cos X =. (kZ + k-x), sin x - 1 (kx - k-). But kx = (elo~ )- = exlo, ck-x= e-xlogak CONSTEUCTION OF TABLES. 133 and exlogk — 1 + x lo g k- + X log k + +3 (log ) +..... 2 3 -^ l- log + x2 (log k)2 x3 (log )3....cosx- = (k + k-x) =1 + x2 (log )+a (l )4 og..).. sin xlok+ x3 (log g ((log +)5-+.) sinx 13 It only remains to find the value of k, and this can be obtained by dividing the last equation through by x and letting x approach 0 indefinitely, when we have limit (sin x\ 1 x O )T- = log+ k. But limit (sin x\ But XlO 0 x)=1..logk =i, k = ei. Therefore we have 2 v4 X 6 -cos I = - (+ e-i) =1- +- + +..... CosX 2 (i+ exi) - 2 14 + 1 3^ X5 X7 sin x = -(exi e-xi)= x - + - +". From the last two series we obtain by division sin x X3 2x5 17 x7 tan x = -- == X4- + -- + _ —.+. cos X 3 15 315 By the aid of these series the trigonometric functions of any angle are readily calculated. In the computation it must be remembered that x is the circular measure of the given angle. 134 TRIGONOMETRY. EXERCISE XXVIII. Verify by the series just obtained that 1. sin2x + cos2x- =1. 2. sin (- x) -sinx, cos (- x) =cosx. 3. sin 2 x- 2 sin x cos x. 4. cos 2 x =1- 2 sin2x. 5. Find the series for se x as far as the term containing the 6th power of x. 6. Find the series for x cot x, noting that x cot x -= cos x. sin X 7. Calculate sin 10~ and cos 10~ to 6 places of decimals. 8. Calculate tan 15~ to 6 places of decimals. From the exponential values of sin x and cos x show that 9. cos 3x = 4 cosx - 3 cos x. 10. sin 3 x -= 3 sin x -4 sin3x. SPHERICAL TRIGONOMETRY. CHAPTER VII. THE RIGHT SPHERICAL TRIANGLE. ~ 48. INTRODUCTION. THE object of Spherical Trigonometry is to show how spherical triangles are solved. To solve a spherical triangle is to compute any three of its parts when the other three parts are given. The sides of a spherical triangle are arcs of great circles. They are measured in degrees, minutes, and seconds, and therefore by the plane angles formed by radii of the sphere drawn to the vertices of the triangle. Hence, their measures are independent of the length of the radius, which may be assumed to have any convenient numerical value; as, for example, unity. The angles of the triangle are measured by the angles made by the planes of the sides. Each angle is also measured by the number of degrees in the arc of a great circle, described from the vertex of the angle as a pole, and included between its sides. The sides may have any values from 0~ to 360~; but in this work only sides that are less than 180~ will be considered. The angles may have any values from 0~ to 180~. If any two parts of a spherical triangle are either both less than 90~ or both greater than 90~, they are said to be alike in kind; but if one part is less than 90~, and the other part greater than 90~, they are said to be unlike in kind. 136 SPHERICAL TRIGONOMETRY. Spherical triangles are said to be isosceles, equilateral, equiangular, right, and oblique, under the same conditions as plane triangles. A right spherical triangle, however, may have one, two, or three right angles. When a spherical triangle has one or more of its sides equal to a quadrant, it is called a quadrantal triangle. It is shown in Solid Geometry, that in every spherical triangle I. If two sides of a spherical triangle are unequal, the angles opposite them are unequal, and the greater angle is opposite the greater side; and conversely. II. The sum of the sides is less than 360~. III. The sum of the angles is greater than 180~ and less than 540~. IV. If, from the vertices as poles, arcs of great circles are described, another spherical triangle is formed so related to the first triangle that the sides of each triangle are supplements of the angles opposite them in the other triangle. Two such triangles are called polar triangles, or supplemental triangles. Let A, B, C (Fig. 37) denote the angles of one triangle; A' a, b, c the sides opposite these angles respectively; and let A', B', C' and a', b', c' denote the corresponding sides and angles of the polar triangle. Then the above theorem gives the six following equations: A + a'- 180~, B + b' 180~, c C+ c'- 180~, A'+ a 180~, B' + b = 180~, C' -+ c = 180 a' FIG. 37. THE RIGHT SPHERICAL TRIANGLE. 137 EXERCISE XXIX. 1. The angles of a triangle are 70~, 80~, and 100~; find the sides of the polar triangle. 2. The sides of a triangle are 40~, 90~, and 125~; find the angles of the polar triangle. 3. Prove that the polar of a quadrantal triangle is a right triangle. 4. Prove that, if a triangle has three right angles, the sides of the triangle are quadrants. 5. Prove that, if a triangle has two right angles, the sides opposite these angles are quadrants, and the third angle is measured by the number of degrees in the opposite side. 6. How can the sides of a spherical triangle, given in degrees, be found in units of length, when the length of the radius of the sphere is known? 7. Find the lengths of the sides of the triangle in Example 2, if the radius of the sphere is 4 feet. ~ 49. FORMULAS RELATING TO RIGHT SPHERICAL TRIANGLES. As is evident from ~ 48, Examples 4 and 5, the only kind of right spherical triangle requiring further investigation is that which contains only one right angle. Let ABC (Fig. 38) be a right spherical triangle having only one right angle; and let A, B, C B denote the angles of the triangle; a, b, c, respectively, the opposite sides. Let C be the right angle; and for the / present suppose that each of the other parts is less than 90~, and that the radius 0 7 of the sphere is 1. A Let planes be passed through the sides, FIG. 38. intersecting in the radii OA, OB, and OC. 138 SPHERICAL TRIGONOMETRY. Also, let a plane perpendicular to OA be passed through B, cutting OA at E and OC at D. Draw BE, BD, and DE. BE and DE are each. _ to OA (Geom. ~ 462); therefore B /BED=A. The plane BDE is 1 to the plane AOC (Geom. ~ 518); hence BD, / | \ \ which is the intersection of the planes \ BDE and BOC, is I to the plane AOC 1 1) \ (Geom. ~ 520), therefore I to OC and 0 D DE..-\ l Now cos c = OE= OD X cos b, FIG. 39. and OD - cos a. Therefore, cos c - cos a cos b. [38] sin a = D - BE X sin A, and BE- sin c. Therefore, sin a = sin c sin A [39] changing letters, sin b = sin c sin B J A DE OE tan b o BE OE tan c Hence, cos A = tan b cot c 1 [ changing letters, cos B = tan a cot c J DE OD sin b sin b Again, cos A -BE s- C = osa sin BE sin c sin c sin b By substituting for sin its value from [39], we obtain cos A = cos a sin B ] changing letters, cos B = cos b sin A J [ ] DE BD cot A sin a cot A Also, sin b - - - - OD cos a cos a Hence, sin b= tan a cot A [42] changing letters, sin a tan b cot B If in [38] we substitute for cos a and cos b their values from [41], we obtain L- -'/ _ -...... & A_ ^ 2 A A&-T. r"A0O - UUS U-CUUL UULOD. L4~J NOTE. In order to deduce the second formulas in [39]-[42] geometrically, the auxiliary plane must be passed through A I to OB. THE RIGHT SPHERICAL TRIANGLE. 139 These ten formulas are sufficient for the solution of any right spherical triangle. In deducing these formulas, it has been assumed that all the parts of the triangle, except the right angle, are less than 90~. But the formulas also hold true when this hypothesis is not fulfilled. Let one of the legs a be greater than 90~, and construct a figure for this case (Fig. 39) in the same manner as Fig. 38. B~ B ~ \ W A A b FIG. 40. FIG. 41. The auxiliary plane BDE will now cut both CO and AO produced beyond the centre 0; and we have cos c - - OE - OD cos DOE = (- cos a) (- cos b) - COS a COS b. Likewise, the other formulas, [39]-[43], hold true in this case. Again, suppose that both the legs a and b are greater than 90~. In this case the plane BZDE (Fig. 40) will cut CO produced beyond 0, and A 0 between A and 0; and we have cos c- OE= -OD cos DOE = (- cos a) (- cos b) = cos a cos b, a result agreeing with [38]. And the remaining formulas may be easily shown to hold true. Like results follow in all cases; in other words, Formulas [38]-[43] are universally true. 140 SPHERICAL TRIGONOMETRY. EXERCISE XXX. 1. Prove, by aid of Formula [38], that the hypotenuse of a right spherical triangle is less than or greater than 90~, according as the two legs are alike or unlike in kind. 2. Prove, by aid of Formula [41], that in a right spherical triangle each leg and the opposite angle are always alike in kind. 3. What inferences may be drawn from Formulas [38]-[43] respecting the values of the other parts: (i.) if c- 90~; (ii.) if a - 90~; (iii.) if c = 90~ and a = 90~; (iv.) if a = 90~ and b- 90? Deduce from [38] - [43] and [18] - [23] the following formulas: 4. tan2 1 b tan I (- a) tan 2 (c + a). HINT. Use Formula [18] and substitute in it the value of cosb in [:38]. 5. tan2(45 — A) tan 1 (c-a)cot I ( + a). 6. tan2 2 B - sin (c- a) csc (c + a). 7. tan2 -= —cos (A +B)se (A B). 8. tan2 a tan [ (A + B)- 45~] tan [ (A -B) +45]. 9. tan2 (45~- -- ) =tan I (A -a) cot I (A + a). 10. tan2 (45 - b) = sin (A - a) csc (A - a). 11. tan2 (45~ - I B) tan (A - a) tan (A + a). Z vl 2 21~2\IILj THE RIGHT SPHERICAL TRIANGLE. 141 ~ 50. NAPIER'S RULES. The ten formulas deduced in ~ 49 express the relations between five parts of a right triangle, the three sides and the two oblique angles. All these relations may be shown to follow from two very useful Rules, devised by Baron Napier, the inventor of logarithms. For this purpose the right angle (not entering the formulas) is left out of account, and instead of the hypotenuse and the two oblique angles, their respective complements are employed; so that the five parts considered by the Rules are: a, b, co. A, co. c, co. B. Any one of these parts may be called a middle part; and then the two parts immediately adjacent are called adjacent parts, and the other two are called opposite parts. Rule I. The sine of the middle part is equal to the product of the tangents of the adjacent parts. Rule II. The sine of the middle part is equal to the product of the cosines of the opposite parts. These Rules are easily remembered by the expressions, tan. ad. and cos. op. The correctness of these Rules may be shown by taking each of the five parts as middle part, and com- o paring the resulting equations with the equations contained in Formulas [38]-[43]. For example, let co. c be taken as middle part, then co. A and co. B are the adjacent parts, and a and b the opposite parts, as is very plainly cO.c seen in Fig. 41. Then, by Napier's Rules: sin (co. c) = tan (co. A) tan (co. B), or cos c=cotA cot B; sin (co. c)= cos a cos b, o.e or cos c = cos a cos b; FIG. 42. results which agree with Formulas [38] and [43] respectively. 142 SPHERICAL TRIGONOMETRY. EXERCISE XXXI. 1. Show that Napier's Rules lead to the equations contained in Formulas [39], [40], [41], and [42]. 2. What will Napier's Rules become, if we take as the five parts of the triangle, the hypotenuse, the two oblique angles, and the complements of the two legs? ~ 51. SOLUTION OF RIGHT SPHERICAL TRIANGLES. By means of Formulas [38]-[43] we can solve a right triangle in all possible cases. In every case two parts besides the right angle must be given. CASE I. Given the two legs a and b. The solution is contained in Formulas [38] and [42]; viz.: cos c - cos a cos b, tan A - tan a csc b, tan B -=tan b csc a. For example, let a =27~ 28' 36", b = 51~ 12' 8"; then the solution by logarithms is as follows: log cos a 9.94802 log cos b= 9.79697 log cos c=9.74499 c 56~ 13'40" log tan a =9.71604 log tan b - 10.09477 log csc b =0.10826 log csc a = 0.33593 log tan A =9.82430 leg tan B =10.43070 A = 33~ 42' 50" B = 69~ 38' 54" CASE II. Given the hypotenuse c and the leg a. From Formulas [38], [39], and [40] we obtain cos b = cos c sec a, sin A = sin a csc c, cos B = tan a cot c. TIIE RIGH-T SPHERICAL TRIANGLE. 143 Although two angles in general correspond to sin A, one acute the other obtuse, yet in this case the indetermination is removed by the fact that A and a must be alike in kind (see Exercise XXX., Example 2). CASE III. Given the leg a and the opposite angle A. By means of Formulas [39], [41], and [42], we find, that sin c = sin a csc A, sin b = tan a cot A, sin B = sec a cos A; or, from [38] and [40], cos b = cos c sec a, cos B - tan a cot c. When c has been computed, b and B are determined by these values of their cosines; but, since c must be found from its sine, c may have in general two values b/ which are supplements of each other. This case, c therefore, really admits of two solutions. B In fact, if the sides b and c are extended until they meet in A' (Fig. 42), the two right triangles ABC and A'BC have the side a in common, and A' the angle A = A'. Also A'C =-180~- b, A'B FIG- 43 = 180~ - c, and A'BC = 180~ - B. CASE IV. Given the leg a and the adjacent angle B. Formulas [40], [41], and [42] give tan c - tan a sec B, tan b = sin a tan B, cos A - cos a sin B. 144 SPHERICAL TRIGONOMETRY. CASE V. Given the hypotenuse c and the angle A. From Formulas [39], [40], and [43] it follows that sin a = sin e sin A, tan b = tan c cos A, cot B- cos c tan A. Here a is determined by sin a, since a and A must be alike in kind (see Exercise XXX., Example 2). CASE VI. Given the two angles A and B. By means of Formulas [41] and [43] we obtain cos c - cot A cot B, cos a= cos A cse B, cos b - cos B cse A. NOTE 1. In Case I. (a and b given) the formula for computing c fails to give accurate results when c is very near 0~ or 180~; in this case it may be found with greater accuracy by first computing B, and then computing c, as in Case IV. NOTE 2. In Case II. (c and a given), if b is very near 0~ or 180~, it may be computed more accurately by means of the derived formula tan2 b = tan (c + a) tan (c - a). (Ex. 4, ~ 49.) And if A is so near 90~ that it cannot be found accurately in the Tables, it may be computed from the derived formula tan2 (45o - A) = tan (c- a) cot A (c + a). (Ex. 5, ~ 49.) In like manner, when B cannot be accurately found from its cosine we may make use of the formula tan2 i B = sil (c - a) csc (c + a). (Ex. 6, ~ 49.) NOTE 3. In Case III. (a and A given), when the formulas for the required parts do not give accurate results, we may employ the derived formulas tan2 (45~ - c) = tan (A -a) cot (A + a), (Ex. 9, ~ 49.) tan2 (45~ -- b) = sin (A - a) csc (A + a), (Ex. 10, ~ 49.) tan2 (45~ -i B) = tan (A - a) tan I (A + a). (Ex. 11, ~ 49.) THE RIGHT SPHERICAL TRIANGLE. 145 NOTE 4. In Case IV. (a and B given), if A is near 0~ or 180~, it may be more accurately found by first computing b and then finding A. NOTE 5. In Case V. (c and A given), if a is near 90~, it may be found by first computing b, and then computing a by means of Formula [42]. NOTE 6. In Case VI. (A and B given), for unfavorable values of the sides greater accuracy may be obtained by means of the derived formulas tan2 c = - cos (A + B) sec (A - B), (Ex. 7, ~ 49.) tan2 - a = tan [a (A + B) - 45~] tan [45~ + (A - B)], (Ex. 8, ~ 49.) tan2 -2 b = tan [ (A + B) - 45~] tan [45~ -(A - B)]. NOTE 7. In Cases I., IV., and V., the solution is always possible. In the other Cases, in order that the solution may be possible, it is necessary and sufficient that in Case II. sin a < sin c; in Case III., that a and A be alike in kind, and sin A > sin a; in Case VI., that A + B + C be> 180~, and the difference of A and B be < 90~. NOTE 8. It is easy to trace analogies between the formulas for solving right spherical triangles and those for solving right plane triangles. The former, in fact, become identical with the latter if we suppose the radius of the sphere to be infinite in length; in which case the cosines of the sides become each equal to 1, and the ratios of the sines of the sides and of the tangents of the sides must be taken as equal to the ratios of the sides themselves. NOTE 9. In solving spherical triangles, the algebraic sign of the functions must receive careful attention. If the sign of each function is written just above it, the sign of the function in the first member will be + or - according to the rule that like signs give + and unlike signs give -. If the function is a cos, tan, or cot, the + sign shows that the angle is less than 90~; the - sign shows that the angle is greater than 90~, and the supplement of the angle obtained from the table must be taken. If the function is a sine, since the sine of an angle and its supplement are the same, the acute angle obtained from the table and its supplement must be considered as solutions, unless there are other conditions that remove the ambiguity. For the conditions that remove the ambiguity, In case of right spherical triangles see examples 1 and 2 in Exercise XXX., and in case of oblique spherical triangles see I. of ~ 48. 146 SPHERICAL TRIGONOMETRY. NOTE 10. The solutions of a spherical triangle may conveniently be tested by substituting them in the formula containing the three required parts. If the formula required for any case is not remembered, it is always easy to find it by means of Napier's Rules. In applying these Rules we must choose for the middle part that one of the three parts considered - the two given and the one required-which will make the other two either adjacent parts or opposite parts. For example: given a and B; solve the triangle. First, represent the parts as in Fig. 43, and to prevent mistakes mark each of the given parts with a C\@O cross. To find b, take a as the middle part; then b and co. B are adjacent parts; and by Rule I., w\hn sin a = tan b cot B; co. \ a whence, tan b = sin a tan B. To find c, take co. B as middle part; then a and co. c are adjacent parts; and by Rule I., cos B- tan a cot c; co. 4 whence, tan c = tan a sec B. FIG. 44. To find A, take co. A as middle part; then a and co. B are the opposite parts; and by Rule II., cos A = cos a sin B. In like manner, every case of a right spherical triangle may be solved. EXERCISE XXXII. Solve the following right triangles, taking for the given parts in each case those printed in columns I. and II.: THE RIGHT SPHERICAL TRIANGLE. 147 I. II. III. IV. V. a b c A B 1 36~ 27' 43~ 32' 31" 54~ 20' 46~ 59' 43.2" 57~ 59'19.3" 2 86~ 40' 32~ 40' 87~ 11' 39.8" 88~ 11' 57.8" 32~ 42' 38.7" 3 50~ 36~ 54' 49" 59~ 4' 25.7" 63~ 15' 13.1" 44~ 26'21.6" 4 120~ 10' 150~ 59' 44" 63~ 55' 43.2" 105~ 44' 21.25" 147~ 19' 47.14" c a b A B 5 55~ 9' 32" 22~ 15' 7" 51~ 53' 270 28' 25.7" 73~ 27' 11.16" 6 23~ 49' 51" 14~ 16' 35" 19~ 17' 37~ 36' 49.4" 54~ 49' 23.3" 7 440 33' 17" 32~ 9' 17" 32~ 41' 49~ 20' 16.4" 50~ 19' 16" 8 97~ 13' 4" 132~ 14' 12" 79~ 13' 38.2" 131~ 43' 50" 81~ 58' 53.3" a A c b B 9 770 21' 50" 830 56' 40" 78~ 53' 20" 28~ 14' 31.3" 280 49' 57.4 101~ 6' 40" 151~ 45' 28.7" 151~ 10' 2.6" 10 770 21' 50" 400 40' 40" impossible. a B c b A 11 920 47' 32" 50~ 2' 1" 91~ 47' 40" 50~ 92~ 8' 23" 12 20 0' 55" 12~ 40' 2~ 3' 55.7" 0~ 27' 10.2" 770 20' 28.4" 13200 20' 20"2 38~ 10' 10" 25~ 14' 38.2" 15~ 16' 50.4" 540 35' 16.7" 14 54~ 30' 35~ 30' 59~ 51' 20.8" 30~ 8' 39.2" 700 17' 35" c A a b B 15 690 25' 11" 540 54' 42" 50~ 56~ 50' 49.3" 63~ 25' 4" 16 112~ 48' 560 11' 56" 50~ 127~ 4' 30" 120~ 3' 50" 17 460 40' 12" 37~ 46' 9" 26~ 27' 24" 39~ 57' 41.5" 62~ 0' 4" 18 118~ 40' 1" 1280 O' 4" 136~ 15' 32.3" 48~ 23' 38.4" 58~ 27' 4.3" A B a b c 19 63~ 15' 12" 135~ 33' 39" 50~ 0' 4" 143~ 5' 12" 120~ 55' 34.3" 20 116~ 43' 12" 116~ 31' 25" 120~ 10' 119~ 59' 46" 75~ 26' 58" 21 46~ 59' 42" 57~ 59' 17" 36~ 27' 43~ 32' 30" 54~ 20' 22 90~ 88~ 24' 35" 90~ 88~ 24' 35" 90~ NOTE. The values in the last three columns of example 9 cannot be combined promiscuously with those given in columns I. and II. Since a < 90~, with the value of b > 90 must be taken angle B > 90~ and c>90~; while with the value of b < 90~ must be taken, for the same reason, angle B < 90~ and c < 90~. Exercise XXX., 1 and 2. 148 SPHERICAL TRIGONOMETRY. 23. Define a quadrantal triangle, and show how its solution may be reduced to that of the right triangle. 24. Solve the quadrantal triangle whose sides are: a= 174~ 12' 49.1", b - 94~ 8'20", c = 90~. 25. Solve the quadrantal triangle in which c = 90~, A = 110~ 47' 50", B= 135~ 35' 34.5. 26. Given in a spherical triangle A, C, and c each equal to 90~; solve the triangle. 27. Given A = 60~, C =90~, and c= 90~; solve the triangle. 28. Given in a right spherical triangle, A =42~ 24' 9", B = 9~ 4' 11"; solve the triangle. 29. In a right spherical triangle, given a = 19~ 11 ', B = 126~ 54'; solve the triangle. 30. In a right spherical triangle, given c=50~, b=44~18'39"; solve the triangle. 31. In a right spherical triangle, given A = 156~ 20' 30", a= 65~ 15' 45"; solve the triangle. 32. If the legs a and b of a right spherical triangle are equal, prove that cos a = cot A = /cos c. 33. In a right spherical triangle prove that cos2 A X sin2 c= sin (c- a) sin (c- a). 34. In a right spherical triangle prove that tan a cos c = sin b cot B. 35. In a right spherical triangle prove that sin2 A = cos2 B + sin2 a sin2 B. 36. In a right spherical triangle prove that sin (b - c) = 2 cos2 - A cos b sin c. 37. In a right spherical triangle prove that sin (c - b) = 2 sin2 - A cos b sin c. 38. If, in a right spherical triangle, p denotes the arc of the great circle passing through the vertex of the right angle and perpendicular to the hypotenuse, m and n the segments of the hypotenuse made by this arc adjacent to the legs a and b, prove that (i.) tan2 a = tan c tan m, (ii.) sin2p = tan m tan n. THE EIGHT SPHERICAL TRIANGLE. 149 ~ 52. SOLUTION OF THE ISOSCELES SPHERICAL TRIANGLE. If an arc of a great circle is passed through the vertex of an isosceles spherical triangle and the middle point of its base, the triangle will be divided into two symmetrical right spherical triangles. In this way the solution of an isosceles spherical triangle may be reduced to that of a right spherical triangle. In a similar manner the solution of a regular spherical polygons may be reduced to that of a right spherical triangle. Arcs of great circles, passed through the centre of the polygon and its vertices, divide it into a series of equal isosceles triangles; and each one of these may be divided into two equal right triangles. EXERCISE XXXIII. 1. In an isosceles spherical triangle, given the base b and the side a; find A the angle at the base, B the angle at the vertex, and h the altitude. 2. In an equilateral spherical triangle, given the side a; find the angle A. 3. Given the side a of a regular spherical polygon of n sides; find the angle A of the polygon, the distance R from the centre of the polygon to one of its vertices, and the distance r from the centre to the middle point of one of its sides. 4. Compute the dihedral angles made by the faces of the five regular polyhedrons. 5. A spherical square is a regular spherical quadrilateral. Find the angle A of the square, having given the side a. *A regular spherical polygon is the polygon formed by the intersections of the spherical surface by the faces of a regular pyramid whose vertex is at the centre of the sphere. CHAPTER VIII. THE OBLIQUE SPHERICAL TRIANGLE. ~ 53. FUNDAMENTAL FORMULAS. LET ABC (Fig. 44) be an oblique spherical triangle, a, b, c ~C ~ its three sides, A, B, C the angles opposite to them, respectively.;/ V Through C draw an arc CD b/ | \ of a great circle, perpendicular / a to the side AB, meeting AB at / p \ D. For brevity let CD=p, AD=nm, BD= n, ZACD=x, Z BCD -y. A n t 1. By ~ 49 [38], in the right -D triangles BDC and ADC, FIG. 45. sinp - sin sin B, and sin ps = sin b sin A. Therefore, sin a sin B = sin b sin A similarly, sin a sin C = sin c sin A [44] and sin b sin C - sin c sin B These equations may also be written in the form of proportions sin a: sin b: sin c = sin A: sin B: sin C. That is, the sines of the sides of a spherical triangle are proportional to the sines of the opposite angles. In Fig. 44 the arc of the great circle CD cuts the side AB within the triangle. In case it cuts AB produced without the triangle, sin (180~-A), sin (180~-B), or sin(180~ — C), would THE OBLIQUE SPHERICAL TRIANGLE. be employed in the above proof instead of sin A, sin B, or sil C. These sines, however, are equal to sin A, sin B, and sin C, respectively, so that the Formulas [44] hold true in all cases. 2. In the right triangle BDC, by ~ 49 [38], cos a cosp cos n - cosp cos (c - n), or (~ 28) cos a = cosp cos c cos m + cosp sin c sin m. Now, cosp cos n cos b; [38] whence cosp - cos b sec m, and cos p sin nm - cos b tan m - cos b tan b cos A [40] - sin b cos A. Substituting these values of cosp cos m and cosp sin m in the value of cos a, we obtain cos a = cos b cos c + sin b sin c cos A and similarly, cos b cos a cos c + sin a sin c cos B [45] cos c - cos a cos b + sin a sin b cos CJ 3. In the right triangle ADC, by [41], cos A osp sin x -cosp sin (C- y), or (~ 28) cosA= cosp sin Ccos y - cosppcos Csiny. Now, cosp sin y= cos B; [41] whence, cosp = cos B cse y, and cosp cos y - cos B cot y - cos B tan B cos a [43] -- sin B cos a. Substituting these values of cosp sin y and cosp cos y in the value of cos A, we obtain cos A - cos B cos C + sin B sin C cos a and similarly, cos B = - cos A cos C + sin A sin C cosb [46] cos C - - Acos AcosB + sin A sin B cos c 152 SPHERICAL TRIGONOMETRY. Formulas [45] and [46] are also universally true; for the same equations are obtained when the arc CD cuts the side AB without the triangle. EXERCISE XXXIV. 1. What do Formulas [44] become if A = 90~? if B = 90? if C =90~? if a 90~? if A = B90~? if a b -90~? 2. What does the first of [45] become if A=0~? if A=90~? if A- 180? 3. From Formulas [45] deduce Formulas [46], by means of the relations between polar triangles (~ 48). ~ 54. FORMULAS FOR THE HALF ANGLES AND SIDES. From the first equation of [45], cos a - cos b cos c cos A = -- sin b sin c whence, 1- i b sin b sin c + cos cos c - cos a 1 - cos A = sill b sin c cos ( - c) - cos a. sill b sin c sin b sin c - cos b cos c +- cos a 1 + cos A =. sill b sin c cosa - cos (b + c) sin b sin c Hence, by ~ 30 [16] and [17], and ~ 31 [23], sin2 A = sin 1 (a + - c) sin (a- b + c) csc b csc c, cos2- A = sin - (a +- b + c) sin ( b+ c - a) csc b csc c. Now let (a+ b +) - s; whence, (b + c - ) - a, (a - + c) s -b, 2- (a + - c) s -c. THE OBLIQUE SPHERICAL TRIANGLE. 153 Then, by substitution and extraction of the square root, sin - A= Vsin (s-b) sin (s -c) csc b csc c cos I A = Vsin s sin (s - a) csc b csc c [47] tan A = Vcsc s csc (s -a) sin (s -b) sin (s-c) In like manner, it may be shown that sin B = Vsin (s - a) sin (s - c) csc a csc c cos B =- Vsin s sin (s -b) csc a csc c tan - B - Vcsc s csc (s - b) sin (s - a) sin (s - c) sin 2 C =- Vsin (s - a) sin (s - b) csc a csc b cos - C = Vsin s sin (s - c) csc a csc b tan I C Vcsc s csc (s- c) sin (s - a) sin (s - b) Again, from the first equation of [46], cos B cos C + cos A cos a =. — cos a-sin B sin C whence, sin B sin C- cos B cos C cos A 1 -- cos a -- sin B sin C sin B sin C + cos B cos C + cos A 1 +- cos a — sin.B sin C If we place -(A + B + C) -S, and proceed in the same manner as before, we obtain the following results: sin a = V-cos S cos (S - A) csc B csc C cos a = Vcos (S B) cos (S - C) csc B csc C [48] tan~ a -= /- cos S cos (S - A) sec (S - B) sec (S- C) 154 SPHERICAL TRIGONOMETRY. And, in like manner, sin - b — cos S cos (S -B) csc A csc C cos ~ b - Vcos (S - A) cos (S - C) csc A csc C tan b = -co S cos (S - B) sec (S- A) sec (S - C) sin - = V- cos S cos (S - C) csc A csc B cos ~ c = Vcos (S - A) cos (S - B) csc A csc B tan c - V- cos S cos (S -C) sec (S -A)sec (S -B) ~ 55. GAUSS' EQUATIONS AND NAPIER'S ANALOGIES. By ~ 27 [5], cos 4 (A + B) cos A cos - sin A sin B; or, by substituting for cos - A, cos — B, sin -A, sin - B, their values given in ~ 54, and reducing, cos. + B) /sin s sin (s - a) sin s sin (s - b) sin b sin c sin a sin c /sin (s -b) sin (s - c) X sin (s- a) sin (s —c) sin 6 sin c sin a sin c sin s - sill (s - c) X sin (s- a) sin (s - b) sin c sin a sin b This value, by applying ~~ 29 [12], 31 [21], and observing that the expression under the radical is equal to sin - C, becomes 2 sin I-c cos (s - -lc) cos -(A+ B)=- 2 (s n sin C; 2 2 sin c cos - c and this, by cancelling common factors, clearing of fractions, and observing that s — c = 2- (a + b), reduces to the form cos - (A + B) cos c = cos - (a + b) sin i C. By proceeding in like manner with the values of sin I (A + B), cos (A- B), and sin 1 (A - B), three analogous equations are obtained. THE OBLIQUE SPHERICAL TRIANGLE. 155 The four equations, cos1 (A +B) cos -c=os (a+b) sin - C sin (A + B) cos ~ c- cos (a -b) cos C 4 cos (A-B) sin c=sin -(a+b)sinC 4C sin (A- B) sin c-= sin I (a - b) cos 2 CJ are called Gauss's Equations. By dividing the second of Gauss's Equations by the first, the fourth by the third, the third by the first, and the fourth by the second, we obtain ta ( + ) — cos ~ (a - b) tan I (A+ B) 2 cot 2 C 2cos I (a + b) sin - (a - b) - cosKjA-B) tan - r cos (A+B) tan c tan (A - sin - B (A - B) tan n (a - b) - i tan I c | n2 (a sin- (A+ B) tan There will be other forms in each case, according as other elements of the triangle are used. These equations are called Napier's Analogies. In the first equation the factors cos (a -b) and cot 2 C are always positive: therefore, tan - (A + B) and cos ~ (a + b) must always have like signs. Hence, if a +- b < 180~, and therefore cos ~ (a + b) > 0, then, also, tan I (A + B) > 0, and therefore A + B < 180~. Similarly, it follows that if a + b > 180~, then, also, A + B > 180~. If a +- b = 180~, and therefore cos (a + b) = 0, then tan i (A + B) - o; whence ~ (A + B) 90~, and A + B = 180~. Conversely, it may be shown from the third equation, that a -b is less than, greater than, or equal to 180~, according as A + B is less than, greater than, or equal to 180~. 156 SPHERICAL TRIGONOMETRY. ~ 56. CASE I. Given two sides, a and b, and the included angle C. The angles A and B may be found by the first two of Napier's Analogies; viz.: = / A -~ OS (a - b) tan (A +B) = A (+ ) cot - C. ~~2 v cos 2 (a I- 2) sin '-(a -b) tan 2- (A - B) =. l cot 2- C 2 s / sin I- (a + b) 2 After A and B have been found, the side c may be found by [44] or by [50]; but it is better to use for this purpose Gauss's Equations, because they involve functions of the same angles that occur in working Napier's Analogies. Any one of the equations may be used; for example, from the first we have cos 2 c _ -(- A +C ) sin - C. 2 os ~ (A — +B) 2 EXAMPLE. a = 73~ 58' 54", b = 38~45' 0", C = 46 33/ 41/, log cost (a - b) = 9.97914 log sec i (a + b) =0.25658 log cot C =-0.36626 log tan- (A + B) = 0.60198 log sec (A + B) = 0.61515 log cos; (a + b) =9.74342 log sin i C = 9.59686 log cos c =9.95543 1 C = 25031' therefore, - (a - b) = 17~ 36' 57' (a + b) = 560 21'57" iC =23016'50.5" log sin i (a - b) = 9.48092 log csc (a + b) = 0.07956 log cot ~ C = 0.36626 log tan - (A - B) = 9.92674 (A + B) = 75~ 57'40.7" (A - B) = 40 11'25.6" A = 116~ 9' 6.3" B = 35~46'15.1' c =- 51~ 2' If the side c only is desired, it may be found from [45], without previously computing A and B. But the Formulas [45] are not adapted to logarithmic work. Instead of clanging them to forms suitable for logarithms, we may use the following method, which leads to the same results, and has the advantage that, in applying it, nothing has to be remembered except Napier's Rules: THE OBLIQUE SPHERICAL TRIANGLE. 157 Make the triangle (Fig. 45), as in ~ 53, equal to the sum (or the difference) of two right triangles. For this purpose, through B (or A, but not C) draw an arc of a great circle perpendicular to AC, cutting AC atD. Let BD= =, CD=m, / AD=n; and mark with crosses the given parts. A By Rule I., ^ cos C = tan cot a, b whence tan n -- tan a cos C.. FIG. 46. By Rule II., cos a =cos m cosp, whence cosp = cos a sec m. cos c - cos n cosp, whence cos- = cos c sec n. Therefore, cos c sec n = cos a sec m; or, since n =b - m, cos c =cos a sec cos (b -m). It is evident that c may be computed, with the aid of logarithms, from the two equations tan m -= tan a cos C, cos c - cos a sec m cos (b - ). EXAMPLE. Given a= 97~ 30'20", b 55 12'10", C 39 58'; find c. log tan a = 0.88025 (n) log cos a = 9.11602 (n) log cos C= 9.88447 log cos (b- w) - 9.85286 log tan n-= 0.76472 (n) log see m = 0.77103 (n) m= 99~ 45' 14" log cos c=- 9.73991 b-mn —44033' 4" c=- 56~ 40'20" EXERCISE XXXV. 1. Write formulas for finding, by Napier's Rules, the side a, when b, c, and A are given, and for finding the side b when a, c, and B are given. 158 SPHERICAL TRIGONOMETRY. 2. Given a = 88 12' 20", b -1240 7'17", C =500 2' 1"; find A = 63~ 15' 11", B - 132~ 17' 59", c = 59~ 4' 18". 3. Given a = 120 55' 35", b- 880 12'20", C = 47 42' 1"; find A = 129~ 58' 3", B = 63~ 15' 9", c =-55~ 52' 40". 4. Given b = 63~ 15' 12", c -47 42' 1", A = 59 4' 25"; find B = 880 12' 24", C- =550 52' 42", a- 50~ 1' 40". 5. Given b = 69 25' 11", c - 1090 46' 19", A = 540~ 54'42"; find B- 56~ 11' 57", C= 123~ 21' 12", a = 67~ 13'. ~ 57. CASE II. Given the side c and the two adjacent angles A and B. The sides a and b may be found by the third and fourth of Napier's Analogies, cos I (A -B) tan (a + b (A ) tan c, tanI (a -b) 2 ( B tan c, and then the angle C may be found by [44], by Napier's second Analogy, or by one of Gauss's equations, as, for instance, the second, which gives sin (A + B) cos C = coscos ) 2. cos ~(a - b) EXAMPLE. A = 107~47' 7" B = 38~ 58' 27" c - 51041' 14" log cos (A - B) = 9.91648 log sec (A + B) = 0. 54359 log tan 1 c = 9.68517 log tan (a + b) =0.14524 log sin 1 (A + B) = 9.98146 log see I (a- b) = 0.01703 log cos 1 c = 9.95423 log cos i C = 9.95272 ~ C = 26~ 14' 52.5"..1 (A - B) = 340~ 24' 20" (A + B) = 73022'47// 1 c = 25~ 50' 37" log sin (A - B) = 9. 75208 log sc 1 (A + B) = 0.01854 log tan 1 c = 9.68517 log tanll (a- b) - 9.45579 (a + b) = 54024' 24.4" 1 (a - b) = 15~ 56' 25.6" a - 70~ 20' 50" b =38~27'59" C - 520 29'45" THE OBLIQUE SPHERICAL TRIANGLE. 159 If the angle C alone is wanted, the best way is to decompose the triangle into two right triangles, and then apply Napier's Rules, as in Case I., when the side c alone is desired. Let (Fig. 46) ABD = x, Z CBD = y, BD =p; then, Rule I., B cos c = cot x cot A, whence cot x = tan A cos c. Rule II., / a cos A = cosp sin x, whence cos 2 - cos A csc x. cos C= cos p sin,, whence cos p = cos C csc'y. C Hence cos C= cos A cse x sin y FIG. 47. = cos A csc x sin (B - x). It is clear that C may be computed from the equations cot x = tan A cos c, cos C = cos A csc x sin (B -x). EXAMPLE. Given A = 35~ 46' 15", B 115~ 9' 7", c =51 2'; find C. log tan A = 9.85760 log cos A =9.90992 log cos c =9.79856 log sin (B - x) = 9.88122 log cot x = 9.65616 log csc x = 0.04055 x = 65~ 37' 35" log cos C- 9.83099.'.B-x -49 31 32" C- 47020'30" EXERCISE XXXVI. 1. What are the formulas for computing A when B, C, and a are given; and for computing B when A, C, and b are given? 2. Given A = 26~ 58' 46", B = 39~ 45' 10", c = 154~ 46' 48"; find a = 37~ 14' 10", b -= 121 28' 10", C= 161~ 22' 11". 3. Given A 128~ 41' 49", B= 107~ 33' 20", c- 124~ 12' 31"; find a - 125~ 41'44", b = 82~ 47' 34", C= 127~ 22'. 160 SPHERICAL TRIGONOMETRY. 4. Given B — 153~ 17' 6", C = 78~ 43'36", a =86~ 15'15"; find b = 152~ 43' 51", c 88~ 12' 21", A = 78~ 15' 48". 5. Given A = 125~ 41' 44", C= 82~ 47' 35", b - 52~ 37' 57"; find a = 128~ 41' 46", c = 107~ 33' 20",.B =- 55~ 47' 40". ~ 58. CASE III. Given two sides a and b, and the angle A opposite to a. The angle B is found from [44], whence we have sin B = sin A sin b csc a. When B has been found, C and c may be found from the fourth and the second of Napier's Analogies, from which we obtain tan c = sin (A + ) tan s (a - b), sin ~- (A - B) cot = n (+ ) tan (A - B). -V sin' (a - b) v / 'The third and first of Napier's Analogies may also be used. NOTE 1. Since B is determined from its sine, the problem in general has two solutions; and, moreover, in case sin B> 1, the problem is impossible. By geometric construction it may be shown, as in the corresponding case in Plane Trigonometry, under what conditions the problem really has two solutions, one solution, and no solution. But in practical applications a general knowledge of the shape of the triangle is known beforehand; so that it is easy to see, without special investigation, which solution (if any) corresponds to the circumstances of the question. It can be shown that there are two solutions, when A and a are alike in kind and sin b > sin a> sin A sin b; no solution when A and a are unlike in kind (including the case in which either A or a is 90~) and sin b is greater than or equal to sin a, or when sin a < sin A sin b; and one solution in every other case. NOTE 2. The side c or the angle C may be computed, without first finding B, by means of the formulas tan m = cos A tan b, and cos (c - m) = cos a sec b cos m, cot x = tan A cos b, and cos (C - x) = cot a tan b cos x. THE OBLIQUE SPHERICAL TRIANGLE. 161 These formulas may be obtained by resolution of the triangle into right triangles, and applying Napier's Rules; m is equal to that part of the side c included between the vertex A and the foot of the perpendicular from C, and x is equal to the corresponding portion of the angle C. NOTE 3. After the two values of B have been obtained, the number of solutions may readily be determined by ~ 48- I. If log sin B is positive, there will be no solution. EXAMPLE. Given a = 57~ 36', b = 31~ 12', A = 104~ 25' 30". In this case A > 90~, and a+ b<180~; therefore, A + B < 180~; hence, B< 90~, and only one solution. a + b = 88~ 50' a-b =26~26' A + B = 140~ 51' 53" A-B —= 67~59' 7" log sin (A + B) = 9.97416 log cse (A - B) = 0.25252 logtan ~ (a - b) = 9.37080 log tan = c 9.59748 c 21 35' 38" c = 43~11' 16" log sin A = 9.98609 logsinb = 9.71435 logcsca = 0.07349 log sin B = 9.77393 B = 360 27' 20" 2(a+b) =44~25' 2 (a - b) = 13~ 13' (A + B) = 70026'25" I (A -B) = 33~ 59' 5" logsin I (a + b) = 9.84502 log csc ~ (a - b) = 0.64086 logtan (A - B) = 9.82873 log cot - C = 0.31461 C = 25~ 51' 15" C = 510 42' 30" EXERCISE XXXVII. 1. Given a= 73~ 49' 38", b 120~ 53' 35", A = 88~ 52' 42"; find B = 116~ 42' 30", c -- 120 57' 27", C - 116~ 47' 4". 2. Given a = 150~ 57' 5", b = 134~ 15' 54",A 1 144~ 22'42"; find B1 = 120~ 47' 45", C 55~ 42' 8", C1 97~ 42' 55.4"; B2 = 59~ 12 15", C2 = 23~ 57'17.4", C =29~ 8'39". 3. Given a - 79~ 0' 54.5", b = 820 17' 4", A == 82~ 9' 25.8"; find B = 90~, c = 45~ 12' 19", C = 45~ 44'. 4. Given a 30~ 52' 36.6", b =31 9' 16", A = 87 34' 12"; show that the triangle is impossible. 162 SPHERICAL TRIGONOMETRY. ~ 59. CASE IV. Given two angles A and B, and the side a opposite to one of them. The side b is found from [44], whence sin b = sin a sin B csc A. The values of c and C may then be found by means of Napier's Analogies, the fourth and second of which give sin (A a B n tan c = s (A ) tan I (a - b), 2 sin (A - B),,,. cot c sin (a + b)-C si _(- tan (A - B). sin (a - b) NOTE 1. In this case the conditions for one, two, or no solutions can be deduced directly by the theory of polar triangles from the corresponding conditions of Case III. There are two solutions, when A and a are alike in kind and sin B > sin A > sin a sin B; no solution when A and a are unlike in kind (including the case in which either A or a is 90~) and sin B is greater than or equal to sin A, or when sin A < sin a sin B; and one solution in every other case. NOTE 2. By proceeding as indicated in Case III., Note 2, formulas for computing c or C, independent of the side b, may be found; viz.: tan m = tan a cos B, and sin (c - m) = cot A tan B sin m, cot x = cos a tan B, and sin (C - x) = cos A sec B sin x. In these formulas m = BD, x = Z BCD, D being the foot of the perpendicular from the vertex C. NOTE 3. As in Case III., only those values of b can be retained which are greater or less than a, according as B is greater or less than A. If log sin b is positive, the triangle is impossible. EXERCISE XXXVIII. 1. Given A - 110~ 10', B -= 133~ 18', a = 147~ 5' 32"; find b = 155~ 5' 18"', c = 33~ 1' 36", C =-70~ 20' 40". THE OBLIQUE SPHERICAL TRIANGLE. 163 2. Given A=113~ 39' 21", B= 123~ 40' 18", a=65~ 39'46"; find b = 124 7' 20", c - 159~ 50' 14", C 159~ 43' 34". 3. Given A=100~2' 11.3", B 98~ 30'28", a 95~ 20'38.7"; find b =90~, c = 147~ 41' 43", C = 148~ 5r33. 4. Given A = 24~ 33' 9", B =38~ 0' 12", a -65~ 20' 13"; show that the triangle is impossible. ~ 60. CASE V. Given the three sides, a, b, and c. The angles are computed by means of Formulas [47], and the corresponding formulas for the angles B and C. The formulas for the tangent are in general to be preferred. If we multiply the equation tan ~A = Vcsc s csc (s - a) sin (s - b) sin (s - c) by the equation 1 — sin (s - ) and put sin (s - a) \/csc s sin (s - a) sin (s - b) sin (s - c) = tan r, and also make analogous changes in the equations for tan2 B and tan ~ C, we obtain tan - A tan r csc (s - a), tan B = tan csc (s - b), tan - C = tan rcsc (s - c), which are the most convenient formulas to employ when all three angles have to be computed. EXAMPLE 1. a- 50~ 54' 32" b = 37~ 47' 18" c = 740 51' 50" 2 s =163~ 33' 40" s = 81~ 46'50" s-a = 30 52' 18" s-b = 43059'32" s- c= 6~55' 0" log csc s== 0.00448 log csc (s - a) = 0.28978 log sin (s- b)= 9.84171 logsin (s - c) =- 9.08072 2)19.21669 log tan A = 9.60835 A =22 5' 20" A = 440 10' 40" 164 SPHERICAL TRIGONOMETRY. EXAMPLE 2. a= 124~ 12' 31" b = 5418' 16" c = 970 12' 25" 2s= 275~ 43' 12" s = 137~ 51' 36" log sin (s - a) = 9.37293 log sin (s - b) = 9.99725 log sin (s - c) = 9.81390 logcsc s = 0.17331 log tan2r = 9.35739 log tan r = 9.67870 s-a= 13~ 39' 5" s- b= 83~ 33' 20" s - c = 40~ 39' 11" log tan A = 0.30577 log tan I B = 9.68145 log tan - C = 9.86480 1A= 63~ 41' 3.8" B = 25~ 39' 5.6" C = 36~ 13' 20.1" A = 127~ 22' 7" B = 510 18' 11" C = 72~ 26' 40" EXERCISE XXXIX. 1. Given a = 120~ 55' 35", b = 59~ 4' 25", = 106~ 10' 22"; find A = 116~ 44' 50", B = 63~ 15' 18", C - 91~ 7'22". 2. Given a= 500 12' 4", b =116~ 44'48", c =129~ 11'42"; find A = 59~ 4' 28", B = 94~ 23' 12", C = 120~ 4' 52". 3. Given a = 131~ 35' 4", b = 108~ 30'14", C = 84~ 46' 34"; find A = 132~ 14' 21", B = 110 10' 40", C = 99 42' 24". 4. Given a- 20~ 16'38", b-=56~ 19'40", c =66~ 20' 44"; find A =-20~ 9' 54", B = 55~ 52' 31", C = 114~ 20' 17". ~ 61. CASE VI. Given the three angles, A, B, and C. The sides are computed by means of Formulas [48]. The formulas for the tangents are in general to be preferred. If we multiply the equation tan la = /- cos S cos (S - a) sec (S - B) se (S- C) by the equation sec (S-A) se ( -- A), and put sec (S —A) /- cos S sec (S- A) sec (S - B) sec (S - C) = tan R, and also make analogous changes in the equations for tan 2b and tan -c, we obtain THE OBLIQUE SPHERICAL TRIANGLE. 165 tan 4 a -=tan R cos (S- A), tan b =tan R cos(S- ), tan - c = tan R cos (/- C), which are the most convenient formulas to use in case all three angles have to be computed. In Example 1, after we find the values of S, S- A, S — B, S- C, we write the formula for tan- a with the algebraic sign written above each function as follows: tan =- - -cos cos (S - A) sec (S - B) sec (S - C). EXAMPLE 1. A = 220~ log cos S = 9.53405 (n) B = 130~ log cos (S- A) = 9.93753 C = 150~ log sec (S - B) = 0.30103 (n) 2 S = 500~ log sec (S - C) = 0.76033 (n) S =250~ 2)0.53294 S-A = 30~ log tan a = 0.26647 S —B = 120~ a = 61~ 34' 6" S-C = 100~ a =123~ 8'12" NOTE. Here the effect, as regards algebraic sign, of three negative factors, is cancelled by the negative sign belonging to the whole product. In Example 2, after we find the values of 8, S-A, S- B, S - C, we write the formula for tan R with the algebraic sign written above each function as follows: tanR cos Sse -A)se (- B) se(S- ). tann R s eos S see(S A)see (S -B) see (S C). EXAMPLE 2. A = 20~ 9'56" B = 55~ 52' 32" C = 1140 20 14" 2 S 1900 22'42" log cos S = 8.95638 (n) log sec (S - A) = 0.58768 log sec (S - B) = 0.11143 log sec (S- C) = 0.02472 log tan2R = 9.68021 log tan R = 9.84010 S = 95~ 11' 21" S-A= 75~ 1'25" S- B = 39~18'49" S-C = —19~ 8' 53" log tan 1 a = 9.25242 log tan -1 b = 9.72867 log tan c = 9.81538 la =10~ 8'18.9" b - 28~ 9'50.4" c=33010'21.3" a = 20~ 16' 38" b = 56~ 19' 41" c = 66~ 20' 43" 166 SPHERICAL TRIGONOMETRY. EXERCISE XL. 1. Given A =130~, B -110~, C- 80~; find a 139~ 21' 22", b = 126~ 57' 52", c 56~ 51'48". 2. Given A = 59~ 55' 10", B = 85~ 36' 50", C= 59~ 55' 10"; find a - 51~ 17' 31", b -=64~ 2' 47", c =51~ 17' 31". 3. Given A = 102~ 14' 12", B = 54 32' 24", C 89~ 5' 46"; find a = 104~ 25'9", b =53~ 49'25", c= 970 44'19". 4. Given A = 4~ 23' 35", = 8 28'20", C=172~ 17'56"; find a 31~ 9' 14", b = 84~ 18' 28", c=115~ 10'. ~ 62. AREA OF A SPHERICAL TRIANGLE. I. VWhen the three angles, A, B, C, are given. Let R = radius of sphere, E = the spherical excess = A + B + C - 180~, F= area of triangle. Three planes passed through the centre of a sphere, each perpendicular to the other two planes, divide the surface of the sphere into eight tri-rectangular triangles. It is convenient to divide each of these eight triangles into 90 equal parts, and to call these parts spherical degrees. The surface of every sphere, therefore, contains 720 spherical degrees. Since in spherical degrees, the A ABC= E, and the entire surface of the sphere is equal to 720 spherical degrees,.~. A ABC: surface of the sphere = E: 720; or, since the surface of a sphere = 47r2, A ABC: 4 7r2= E: 720 whence 7r R2 E 18051] THE OBLIQUE SPHERICAL TRIANGLE. 167 II. When the three sides, a, b, c, are given. A formula for computing the area is deduced as follows: From the first of [49], cos (A + B) _ cos (a+ b) cos (900 - C) cos -c whence, by the Theory of Proportions, cos I (A+B) - cos (90~ — C) _ cos (a+b)-cos (c cos (A+ -B) + cos (90~- C) cos (a+b) +cos Now, in ~ 31, the division of [23] by [22] gives cos A - cos B (A-B) cos A cos B = - tan - (A + B) tan (A - B), cos A +- cos B (b) in which for A and B we may substitute any other two angular magnitudes, as for example, - (A + B) and (90 — C), or - (a + b) and ~ c. If we use in place of A and B the values g (A + B) and (90~ - C), the first side of equation (b) becomes cos ( (A + B) - cos (90~ - C). cos - (A + B) + cos (90~ - E C) and the second side becomes -tan I ( A+ - B+900 — C) tan ( A+ B- 900+ C); or, -tan (A+B- C+1800) tan (A +B+ C- 180~). If we remember that E-=A + B+ C- 180~, and observe that tan* (A+ B- C+180~) =tan (360~-2 C+A+B+ C- 180~) = tan(360 —2 C+E) = tan [90~- (2 C —E)] = cot (2 C- E), it will be evident that equation (b) may be written cos (A +- B) - cos (900- C) cos 2(A + B +os( 9 O ) — cot(2 C- ) tanE. (c) cos I(A+B) +cos (900-C) 2 2 -J" 168 SPHERICAL TRIGONOMETRY. If we substitute, in equation (b), for A and B, the values - (a - b) and ~ c, and also substitute s for - (a - b + c) and s - c for X (a + b - c), equation (b) will become cos } (a + ) - cos c. () --— r —i —^ = -- tan I s tan (s - c). (d) cos (a + b) + cosc 2 2 c a/ Comparing (a), (c), and (d), we obtain cot (2 C — E) tan E=- tan l s tan (s - c). (e) By beginning with the second equation of [49], and treating it in the same way, we obtain as the result, tank (2 C- E) ttan L E ( - tall an I (s - b). (f) By taking the product of (e) and (f), we obtain the elegant formula, tan2' E ta- an tan(s- a) tan (s —b) tan (s —c), [52] which is known as l'Huilier's Formula. By means of it E may be computed from the three sides, and then the area of the triangle may be found by [51]. III. In all other cases, the area may be found by first solving the triangle so far as to obtain the angles or the sides, whichever may be more convenient, and then applying [51] or [52]. EXAMPLE 1. A = 102~ 14' 12" log R2 = log R2. B = 54 32'24" log E = 5.37501 C= 89~ 5'46"log 4.68557 -10 2452' 22" log 648000 8557 10 E = 65 52 22" log F = 0.06058 + log R2 = 237142" F = 1.1497 R2 1800 = 648000" If, therefore, we know the radius of the sphere, we can express the area of a spherical triangle in the ordinary units of area. * See Wentworth & Hill's Tables, page 20. THE OBLIQUE SPHERICAL TRIANGLE. 169 EXAMPLE 2. a= 133~ 26' 19" b 64~ 50"53" c = 144~ 13' 45" 2s 342 30' 57" s-=171015'28.5" s-a= 37~49' 9.5" s - b = 106~ 24' 35.5" s- c= 27~ 1'43.5" s =85 37'44" (s - a) = 18 54' 35" (s - b) = 53~ 12' 18/' (s - c) = 130 30' 52" log tan ~ s = 1.11669 log tan (s - a) = 9.53474 log tan (s - b) = 0.12612 log tan ~ (s- c) = 9.38083 log tan2 = E 0.15838 log tan E = 0.07919 E= 50011'43" E = 2000 46' 52" EXERCISE XLI. 1. Given A = 84~ 20' 19", B = 27~ 22' 40", C 75~ 33'; find E =26159", F= 0.12682 R2. 2. Given a = 69~ 15' 6", = 120~ 42' 47", c 159~ 18' 33"; find E = 216~ 40' 28". 3. Given a =33 1' 45", b =155~ 5' 18", C =110~ 10'; find E= 133~ 48' 53". 4. Find the area of a triangle on the earth's surface (regarded as spherical), if each side of the triangle is equal to 1~. (Radius of earth = 3958 miles.) CHAPTER IX. APPLICATIONS OF SPHERICAL TRIGONOMETRY. ~ 63. PROBLEM. To reduce an angle measured in space to the horizon. Let 0 (Fig. 48) be the position of the observer on the ground, A OB h, the angle measured in O space, (for example, the angle pB 3between the tops of two church spires), OA' and OB' the projecN\ Ai tions of the sides of the angle upon the horizontal plane HIR, T —. —__ AAOA'-=m and BOB =n, the B' I/. angles of inclination of OA and On C / OB respectively to the horizon.. 'A/ Required the angle A'OB'=- x FIG. 48. R made by the projections on the horizon. The planes of, the angles of inclination AOA' and BOB' produced intersect in the line OC, which is perpendicular to the horizontal plane (Geom. ~ 520). From 0 as a centre describe a sphere, and let its surface cut the edges of the trihedral angle O-ABC in the points M, N, and P. In the spherical triangle MNP the three sides JIN= h, Ml P - 90~ - m, NP = 90~ - n, are known, and the spherical angle P is equal to the required angle x. From ~ 48 we obtain cos - x = </cos s cos (s - h) see m sec n, where 2 (m + n + h) = s. APPLICATIONS. 171 ~ 64. PROBLEM. To find the distance between two places on the earth's surface (regarded as spherical), given the latitudes of the places and the difference of their longitudes. Let Mi and N (Fig. 49) be the places; then their distance AiV is an arc of the great circle passing through the places. Let P be the pole, AB the equator. The arcs MR and NS are the \ latitudes of the places, and the arc RS, or the angle MFPlN, is A B the difference of their longi- tudes. Let MR a, NS =b, \ RS =-; then in the spherical triangle XJNP two sides, MP =90~- a, N P= 90~-b, and the FIG. 49. included angle MIPV= 1, are given, and we have (from ~ 56) tanm = cot a cos l, cos IMI= sin a sec n sin (b +- ). From these equations first find im, then the arc MN, and then reduce ilfN to geographical miles, of which there are 60 in each degree. ~ 65. THE CELESTIAL SPHERE. The Celestial Sphere is an imaginary sphere of indefinite radius, upon the concave surface of which all the heavenly bodies appear to be situated. The Celestial Equator, or Equinoctial, is the great circle in which the plane of the earth's equator produced intersects the surface of the celestial sphere. The Poles of the equinoctial are the points where the earth's axis produced cuts the surface of the celestial sphere. 172 SPHERICAL TRIGONOMETRY. The Celestial Meridian of an observer is the great circle in which the plane of his terrestrial meridian produced meets the surface of the celestial sphere. Hour Circles, or Circles of Declination, are great circles passing through the poles, and perpendicular to the equinoctial. The Horizon of an observer is the great circle in which a plane tangent to the earth's surface, at the place where he is, meets the surface of the celestial sphere. The Zenith of an observer is that pole of his horizon which is exactly above his head. Vertical Circles are great circles passing through the zenith of an observer, and perpendicular to his horizon. The vertical circle passing through the east and west points of the horizon is called the Prime Vertical; that passing through the north and south points coincides with the celestial meridian. The Ecliptic is a great circle of the celestial sphere, apparently traversed by the sun in one year from west to east, in consequence of the motion of the earth around the sun. The Equinoxes are the points where the ecliptic cuts the equinoctial. They are distinguished as the Vernal equinox and the Autumncal equinox; the sun in his annual journey passes through the former on March 21, and through the latter on September 21. Circles of Latitude are great circles passing through the poles of the ecliptic, and perpendicular to the plane of the ecliptic. The angle which the ecliptic makes with the equinoctial is called the obliquity of the ecliptic; it is equal to 23~ 27', nearly, and is often denoted by the letter e. These definitions are illustrated in Figs. 50 and 51. In Fig. 50, A VB U is the equinoctial, P and P' its poles, NPZS the celestial mleridian of an observer, NESW his horizon, Z his zenith, Jf a star, PMP' the hour circle passing through the star, ZJIfDZ the vertical through the star. APPLICATIONS. 173 In Fig. 51, A VBU represents the equinoctial, EVFU the ecliptic, P and Q their respective poles, V the vernal equinox, U the autumnal equinox, M a star, P1MBR the hour circle through the star, QMIT the circle of latitude through the star, and Z TVR ==e. Z p NV XSL- Ai l --- — \ S Z' FIG. 50. FIG. 51. The earth's diurnal motion causes all the heavenly bodies to appear to rotate from east to west at the uniform rate of 15~ per hour. If in Fig. 50 we conceive the observer placed at the centre 0, and his zenith, horizon, and celestial meridian fixed in position, and all the heavenly bodies rotating around PP' as an axis from east to west at the rate of 15~ per hour, we form a correct idea of the apparent diurnal motions of these bodies. When the sun or a star in its diurnal motion crosses the meridian, it is said to make a transit across the meridian; when it passes across the part NVFS of the horizon, it is said to set; and when it passes across the part NTES, it is said to rise (the effect of refraction being here neglected). Each star, as 3,1 describes daily a small circle of the sphere parallel to the equinoctial, and called the Diurnal Circle of the star. The nearer the star is to the pole the smaller is the diurnal circle; and if there were stars at the poles P and P', they would have no diurnal motion. To an observer north of 174 SPHERICAL TRIGONOMETRY. the equator, the north pole P is elezvated above the horizon (as shown in Fig. 50); to an observer south of the equator, the south pole P' is the elevated pole. ~ 66. SPHERICAL CO-ORDINATES. Several systems of fixing the position of a star on the surface of the celestial sphere at any instant are in use. In each system a great circle and its pole are taken as standards of reference, and the position of the star is determined by means of two quantities called its s/pherical co-ordinates. I. If the ho/rizon and the zenith are chosen, the co-ordinates of the star are called its altitude and its azimuth. The Altitude of a star is its angular distance, measured on a vertical circle, above the horizon. The complement of the altitude is called the Zenith Distance. The Azimuth of a star is the angle at the zenith formed by the meridian of the observer and the vertical circle passing through the star, and is measured therefore by an arc of the horizon. It is usually reckoned from the north point of the horizon in north latitudes, and from the south point in south latitudes; and east or west according as the star is east or west of the meridian. II. If the equinoctial and its pole are chosen, then the position of the star may be fixed by means of its declination and its hour angle. The Declination of a star is its angular distance from the equinoctial, measured on an hour circle. The angular distance of the star, measured on the hour circle, from the elevated pole, is called its Polar Distance. 'The declination of a star, like the latitude of a place on the earth's surface, may be either north or south; but, in practical problems, while latitude is always to be considered positive, declination, if of a different name from the latitude, must be regarded as negative. APPLICATIONS. 175 If the declination is negative, the polar distance is equal numerically to 90~ + the declination. The Hour Angle of a star is the angle at the pole formed by the meridian of the observer and the hour circle passing through the star. On account of the diurnal rotation, it is constantly changing at the rate of 15~ per hour. Hour angles are reckoned from the celestial meridian, positive towards the west, and negative towards the east. III. The equinoctial and its pole being still retained, we may employ as the co-ordinates of the star its declination and its right ascension. The Right Ascension of a star is the arc of the equinoctial included between the vernal equinox and the point where the hour circle of the star cuts the equinoctial. Right ascension is reckoned from the vernal equinox eastward from 0~ to 360~. IV. The ecliptic and its pole may be taken as the standards of reference. The co-ordinates of the star are then called its latitude and its longitude. The Latitude of a star is its angular distance from the ecliptic measured on a circle of latitude. The Longitude of a star is the arc of the ecliptic included between the vernal equinox and the point where the circle of latitude through the star cuts the ecliptic. For the star 2f (Fig. 50), let I= latitude of the observer, h = DM = the altitude of the star, = Z-M = the zenith distance of the star, a = / PZM-= the azinuth of the star, t = Z ZPM= the hour angle of the star, d RM = the declination of the star, p PM P = the polar distance of the star, r = FR - the right ascension of the star, u = MT (Fig. 51) = the latitude of the star, v = VT (Fig. 51) = the longitude of the star. 176 SPHERICAL TRIGONOMETRY. In many problems, a simple way of representing the magnitudes involved, is to project the sphere on the plane of the horizon, as shown in Fig. 52. NES W is the horizon, Z the zenith, NZS the meridian, WZE the prime vertical, WAE I~/ \ ~ the equinoctial projected on the W — / -Z E plane of the horizon, P the 1 M/^ / elevated pole, M a star, DM DX """ Q its altitude, ZM its zenith distance, / PZM its azimuth, MR AS^^ ~ ~its declination, PM its polar FrG. 52. distance, / ZPM its hour angle. ~ 67. THE ASTRONOMICAL TRIANGLE. The triangle ZPM (Figs. 50 and 52) is often called the astronomical triangle, on account of its importance in problems in Nautical Astronomy. The side PZ is equal to the complement of the latitude of the observer. For (Fig. 50) the angle ZOB between the zenith of the observer and the celestial equator is obviously equal to his latitude, and the angle POZ is the complement of ZOB. The arc NP being the complement of PZ, it follows that the altitude of the elevated pole is equal to the latitude of the place of observation. The triangle ZPM then (however much it may vary in shape for different positions of the star M) always contains the following five magnitudes: PZ = co-latitude of observer = 90~ - I, ZM== zenith distance of star = z, PZM= azimuth of star = a, PM= polar distance of star =p, ZPM-= hour angle of star ='t. APPLICATIONS. 177 A very simple relation exists between the hour angle of the sun and the local (apparent) time of day. Since the hourly rate at which the sun appears to move from east to west is 15~, and it is apparent noon when the sun is on the meridian of a place, it is evident that if hour angle = 0~, 15, - 15~, etc., time of day is noon, 1 o'clock P.M., 11 o'clock A.M., etc. In general, if t denote the absolute value of the hour angle, t t time of day = - P.M., or 12 - -A.M., according as the sun is west or east of the meridian. ~ 68. PROBLEM. Given the latitude of the observer and the altitude and azimuth of a star, to find its declination and its hour angle. In the triangle ZPM (Fig. 52), given PZ = 90~ - I = co-latitude, ZM = 90~ - h = co-altitude, / PZM- a = azimuth, to find PM= 90~ - d = polar distance, Z ZP1M = t = hour angle. Draw XQ _ to NS, and put ZQ = m, then, if a < 90~, PQ - 90~ - (I + m), and if a>90~, PQ=90~- ( -m); and, by Napier's Rules, cos a =- tan m tan h, sin d -= cos PQ cos 21Q, sin h = cos m cos MQ; whence, tan m= - cot h cos a, sin d = sin h sin (I ~- m) see m, in which the - sign is to be used if a> 90~. The hour angle may then be found by means of [44], whence we have sin t = sin a cos h sec d. 178 SPHERICAL TRIGONOMETRY. ~ 69. PROBLEM. To find the hour angle of a heavenly body when its declination, its altitude, and the latiN tude of the place are known. In the triangle ZPM (Fig. 53), given PZ = 90 - l, -W Z// - E PM= 90-d =p, M IQ /] ZM- 90 - h; I^D^ ' // required A\ = <^/ / ZPM= t. S If, in the first formula of FIG. 53. [47] sin ~A = Vsin (s -b) sin (s - c) csc b csc c, we put A t, a 90 - h, b-, c=90 -, we have s- b=- =90~- i-(+p + ), s- - ((+p- ), and the formula becomes sin -t = + [cos - (1 +p + h) sin - (1 p - ) sec 1 cscp], in which the - sign is to be taken when the body is east of the meridian. If the body is the sun, how can the local time be found when the hour angle has been computed? (See ~ 67.) APPLICATIONS. 179 ~ 70. PROBLEM. To find the altitude and azimuth of a celestial body, when its declination, its hour angle, and the latitude of the place are known. In the triangle ZPM (Fig. 53), given PZ - 90~ -, PM= 90 - d-=, / ZPM= t; required ZM = 90~ - h, Z PZM-=a. Here there are given two sides and the included angle. Placing PQ =m, and proceeding as in ~ 68, we obtain tan m - cot d cos t, sin h = sin (l + m) sin d sec m, tan a sec (I +- m) tan t sin m, in the last of which formulas a must be marked E. or W., to agree with the hour angle. ~ 71. PROBLEM. To find the latitude of the place when the altitude of a celestial body, its declination, and its hour angle are known. In the triangle ZPM (Fig. 53), given ZM= 90~ -h, PM- 90- d, Z ZPM1- t; required PZ = 90~ - 1. Let PQ=-m, ZQ=n. 180 SPHERICAL TRIGONOMETRY. Then, by Napier's Rules, cos t = tan m tan d, sin h cos n cos 1 Q, N sin d cos m cos MXQ; / p \ whence, tan m cot d cos t, W / Z_ E cos n = cos m sin h csc d, 1 Xl/ /j aland it is evident from the figure D - Q p/ that A\^^j-^~ I = =90~ - (mn + n), S in which the sign + or the sign FIG. 54. - is to be taken according as the body and the elevated pole are on the same side of the prime vertical or on opposite sides. In fact, both values of I may be possible for the same altitude and hour angle; but, unless n is very small, the two values will differ largely from each other, so that the observer has no difficulty in deciding which of them should be taken. ~ 72. PROBLEM. Given the declination, the right ascension of a star, and the obliquity of the ecliptic, to find the \ —" --- latitude and the longitude of the star. Q~-^ \ Let M (Fig. 55) be the star, P be the pole of the equinoctial, and Q the pole of the ecliptic. \ i\ Then, in the triangle PXQ, FIG. 55. M =given PQ =e=23 27,.PX 90 - d, Z MPQ = 90~ + r (see Fig. 51); required Q1M = 90~ - u and Z PQM1= 90~ - v (see Fig. 51). APPLICATIONS. 181 In this case, also, two sides and the included angle are given. Draw MI _1_ to PQ, and meeting it produced at H, and let P = — n. By Napier's Rules, sin r = tan n tan d, sin u = cos (e + n) cos MEl, sin d - cos n cos Mifl, sin (e + n) = tan v tan 2114, sin n = tan r tan MH; whence, tan n = cot d sin r, sin i = sin d cos (e + n) sec n, tan v = tan r sin (e + n) csc n. To avoid obtaining u from its sine we may proceed as follows: From the last two equations we have, by division, sin u = tan v cot (e + n) sin d cot r tan n. By taking MJI as middle part, successively, in the triangles JMQIH and MPH, we obtain cos u cos v - cos d cos r; whence, cos nt = sec v cos d cos r. From these values of sin u and cos u we obtain, by division, tan Zi = sin v cot (e + n) tan d csc r tan n. From the relation sin r = tan n tan d, it follows that tan d csc r tan n = 1. Therefore tan u = sin v cot (e + n), a formula by which u can be easily found after v has been computed. 182 SPHERICAL TRIGONOMETRY. EXERCISE XLII. 1. Find the dihedral angle made by adjacent lateral faces of a regular ten-sided pyramid; given the angle V= 18~, made at the vertex by two adjacent lateral edges. 2. Through the foot of a rod which makes the angle A with a plane, a straight line is drawn in the plane. This line makes the angle B with the projection of the rod upon the plane. What angle does this line make with the rod? 3. Find the volume V of an oblique parallelopipedon; given the three unequal edges a, b, c, and the three angles I, m, n, which the edges make with one another. 4. The continent of Asia has nearly the shape of an equilateral triangle, the vertices being the East Cape, Cape Romania, and the Promontory of Baba. Assuming each side of this triangle to be 4800 geographical miles, and the earth's radius to be 3440 geographical miles, find the area of the triangle: (i.) regarded as a plane triangle; (ii.) regarded as a spherical triangle. 5. A ship sails from a harbor in latitude 1, and keeps on the arc of a great circle. Her course (or angle between the direction in which she sails and the meridian) at starting is a. Find where she will cross the equator, her course at the equator, and the distance she has sailed. 6. Two places have the same latitude I, and their distance apart, measured on an arc of a great circle, is d. How much greater is the arc of the parallel of latitude between the places than the arc. of the great circle? Compute the results for 1 =45~, d =90~. 7. The shortest distance d between two places and their latitudes I and 1' are known. Find the difference between their longitudes. APPLICATIONS. 183 8. Given the latitudes and longitudes of three places on the earth's surface, and also the radius of the earth; show how to find the area of the spherical triangle formed by arcs of great circles passing through the places. 9. The distance between Paris and Berlin (that is, the arc of a great circle between these places) is equal to 472 geographical miles. The latitude of Paris is 48~ 50' 13"; that of Berlin, 52~ 30' 16". When it is noon at Paris what time is it at Berlin? NOTE. Owing to the apparent motion of the sun, the local time over the earth's surface at any instant varies at the rate of one hour for 15~ of longitude; and the more easterly the place, the later the local time. 10. The altitude of the pole being 45~, I see a star on the horizon and observe its azimuth to be 45~; find its polar distance. 11. Given the latitude I of the observer, and the declination d of the sun; find the local time (apparent solar time) of sunrise and sunset, and also the azimuth of the sun at these times (refraction being neglected). When and where does the sun rise on the longest day of the year (at which time d = + 23~ 27') in Boston (1= 42~ 21'), and what is the length of the day from sunrise to sunset? Also, find when and where the sun rises in Boston on the shortest day of the year (when d = - 23~ 27'), and the length of this day. 12. When is the solution of the problem in Example 11 impossible, and for what places is the solution impossible? 13. Given the latitude of a place and the sun's declination; find his altitude and azimuth at 6 o'clock A.M. (neglecting refraction). Compute the results for the longest day of the year at Munich (1 = 48~ 9'). 14. How does the altitude of the sun at 6 A.M. on a given day change as we go from the equator to the pole? At what 184 SPHERICAL TRIGONOMETRY. time of the year is it a maximum at a given place? (Given sin h = sin I sin d.) 15. Given the latitude of a place north of the equator, and the declination of the sun; find the time of day when the sun bears due east and due west. Compute the results for the longest day at St. Petersburg (I = 59~ 56'). 16. Apply the general result in Example 15 (cos t = cot I tan d) to the case when the days and nights are equal in length (that is, when d 0~). Why can the sun in summer never be due east before 6 A.M., or due west after 6 P.M.? How does the time of bearing due east and due west change with the declination of the sun? Apply the general result to the cases where 1 < d and I = d. What does it become at the north pole? 17. Given the sun's declination and his altitude when he bears due east; find the latitude of the observer. 18. At a point 0 in a horizontal plane A1IN a staff OA is fixed, so that its angle of inclination A OB with the plane is equal to the latitude of the place, 51~ 30' N., and the direction OB is due north. What angle will OB make with the shadow of OA on the plane, at 1 P.M., when the sun is on the equinoctial? 19.- What is the direction of a wall in latitude 52~ 30' N. which casts no shadow at 6 A.M. on the longest day of the year? 20. At a certain place the sun is observed to rise exactly in the north-east point on the longest day of the year; find the latitude of the place. 21. Find the latitude of the place at which the sun sets at 10 o'clock on the longest day. 22. To what does the general formula for the hour angle, in ~ 69, reduce when (i.) h = 0~, (ii.) I 0~ and d= 0~, (iii.) I or d 90~? APPLICATIONS. 185 23. What does the general formula for the azimuth of a celestial body, in ~ 70, become when t = 90~ = 6 hours? 24. Show that the formulas of ~ 71, if t= 90~, lead to the equation sin = sin hcscd; and that if d =0~, they lead to the equation cos I — sin h sec t. 25. Given latitude of place 52~ 30' 16", declination of star 38~, its hour angle 28~ 17' 15"; find its altitude. 26. Given latitude of place 51~ 19'20", polar distance of star 67~ 59' 5", its hour angle 15~ 8' 12"; find its altitude and its azimuth. 27. Given the declination of a star 7~ 54', its altitude 22~ 45' 12", its azimuth 129~ 45' 37"; find its hour angle and the latitude of the observer. 28. Given the longitude v of the sun, and the obliquity of the ecliptic e = 23~ 27'; find the declination d, and the right ascension r. 29. Given the obliquity of the ecliptic e 23~ 27', the latitude of a star 51~, its longitude 315~; find its declination and its right ascension. 30. Given the latitude of place 44~ 50' 14", the azimuth of a star 138~ 58' 43", and its hour angle 20~; find its declination. 31. Given latitude of place 51~ 31' 48", altitude of sun west of the meridian 35~ 14' 27", its declination + 21~ 27'; find the local apparent time. 32. Given the latitude of a place I, the polar distance p of a star, and its altitude h; find its azimuth a. APPENDIX. FORMULAS. PLANE TRIGONOMETRY. 1. sin2A + cos2A = 1. sin A 2. tanA = — s cosA ~ 23. sinA X cscA - 1. 3. cosAXsecA =1. tan A X cotA -. J 4. sin (x + y) - sin x cos y + cos x sin y. 5. cos (x + y)= cos x cos y - sin x sin y. tan x + tan y ~ 27. 6. tan (x y) ---- 1 -tan x tan y cot x cot y - 1 7. cot (x + y) cot cot cot y + cot x 8. sin (x -y) = sin x cos y -cos x sin y. 9. cos (x - y) == cos x cos y + sin x sin y. 10. tan (x- y) tanx- tany 28. 1 + tan x tan y cot x cot y + 1 11. cot(x - y) — cot — cotx coty-cotx 12. sin 2x = 2 sin x cos x..cos 2x cos2 - sin2x~29. 13. cos2x= cos2x -siln2x. J FORMULASo 187 2 tan x 14. tan2 2tan 14. tan 2x1 - tan2x ~29. cot2x- 1 15. cot2x-= cOt-x 2 cot x /I - cos ^ 16. sin~z —+ 2 17. cos ~ z- _+_ 1+ cos z. 17. cos =d - -- ~_____ > ~ ~ 30. / - cos. 18. tankz +1l+cosz -1+ cosz. 19. cot=z-+ -- cosz 20. sin A + sin B 2 sill (A + B) cOS (A - B). 21. sin A - sin B -- 2 cos (A + B) sin (A - B). 22. cosA + cos B -2 cos (A + B) cos (A - B). 31. 23. cosA - cosB = -2 sin - (A + B) sin I (A- B). 24 SinA + sin B _tan (A + B) sinA - sin B tan ~ (A - B) a sin A 25. T 33. b sin 26. a2= b2 +c2 — 2 becos A. ~ 34. a- b tan (A - B) 27. - 35. a + b tan - (A + B) 28. sinA=\ (s- )js c) ~40. 2 be 188 FORMULAS. 29. cos iA= - — (30. tanl=Xj(5b)(S-C). s0. tnA(s-c~) ~~ 40. 31. /(s )- (s - b)(s-c) = S r 32. tan A — s -a 33. F - = acsinB. a2sin B sin C 34. F 2 sin (B+ C) 35. F- Vs (s- ) (s- b) (s- c). 41. 36. F-ab 4R 37. =F-2 r(a+b+c)=- s. SPHERICAL TRIGONOMETRY 38. cos c = cos a cos b. sin a - sill c sin A. 39. sin b -- sin c sin B. 40. cos A - tan b cot c. cos B- tan a cot c. ~ 49. cos A - cos a sin B.. cos B -= cos b sin A. 42 { sin b - tan a cot A. sin a = tan b cot B. 43. cos c - cot A cot B. sin a sin B - sin b sinA. 44. sin a sin C=- sin c sinA. ~ 53. sin b sin C - sin c sin B. FORMULAS. 189 cos a = cos b cos c + sin b sin c cos A. 45. cos b cos a cos c + sin a sin ccos B. cos c = cos a cos b +- sin a sin b cos C. cosA - cos B cos C - sin B sin C cos a. 46. cos B = - cos A cos C - sinA sin C cos b. cos C = - cos A cos B + sinA sin B cos c. [ sin -= /sin (s — b) sin (s - c) esc b cse c. 47. cosA = V-in ssins sin (s a) ese b csc c. tan A = / cs s ese (s - a) sin (s - b) sin (s - c). sin -a= V-cos S os (S-A) esc B csc C. 48. - cosl a= Vcos (S- B) os (S — C) csc csc C. tan -a= - cos Scos(S-A)sec(S-B)sec(S-C). ~ 53. ~ 54. c (A + os (A B) cos c-==cos I (a + b) sin C. ' sin (A + B) os c =cos (a- )cos I C. 49 cos (A-) sin- si ( ) sin c sin(a s C. sin 1 (A — B) siln c -= sin (a - b) cos 2 C. 2 2 2 II \C, Vj-V3 50. tan~ (A +-B) 2C - b cot - C 1 0 - I -r\ cos ~I (c - b) tan (A B)=. +b ecot C. 2 tan (c sin ) (a + ) 2 cos I (A+ ) tan C c. tan ~ -13) sin (A -B) — Z ^ sin (A q- B) tan 2c. ~ 55. 51. F= 7rR2E 1 1SO 1002. '~tan2' [~~" - ta 1 ~ 62. 52. tan 21E —tanIstan1(s —a)tan(s —b)tanI(s —c). T y 2 y 2 -~ 190 FORMULAS. PROF. BLAKSLEE'S construction by which the direction ratios for plane right triangles give directly from a figure the analogies for a right trihedral or for a right spherical triangle. B3 The construction consists of two parts. (a) Lay off from the vertex Va unit's distance on each edge. (b) Pass through the three extremities of these distances three planes perpendicular to one of the edges, as VA. Now these three parallel planes will cut out three similar right triangles. The first being constructed in either of the two usual ways, the construction of the others is evident. Since the plane angles A1, A,, A, all equal the dihedral A, and the nine right triangles in the three faces give the values in the figure, we have: (1) sin A = sin a: sin h; (2) cos A = tan b: tan h; (3) tan A = tan a: sin b; (4) cos h = cos a: cos b; (5) sin A = cos B: cos b; similarly, sin B = sin b: sin h. similarly, cos B = tan a: tan h. similarly, tan B = tan b: sin a. (by 3) cot A cot B. sin B - cos A: cos a. NOTE. If a sphere of unit radius be described about V as a centre, the three faces will cut out a right spherical triangle, having the sides a, b, and h, and angles A, B, and H. The above formulas are thus seen to be the analogies of: FORMULAS. 191 (1) sin A = a: h; sin B = b: h. (2) cos A = b: h; cos B = a: h. (3) tanA = a: b; tanB = b: a. (4) h2 = a2 + b2; 1 = sin2 + cos2; 1 = cot A cot B. (5) sin A = cos B; sin B = cos A. Napier's rules give only the following, which follow from the analogies as numbered: By { sina = sinA sinh = tan b cotB 3 (1) sin b =sin B sin = tanacotA (3) ( cosA- sin B cos a = tan b cot h ) cosB = sinA cos b = tana cot h (2) (4) cos = cos a cos b = cotA cotB (4) THE GAUSS EQUATIONS. IV 2 3 1J cos ~ (A + B) cos I = cos -- (a + b) sin 2 C. sin - (A + B) cos - c = cos 1 (a - b) cos - C. cos ~ (A - B) sin - c = sin - (a + b) sin - C. sin I (A - B) sin c = sin (a - b) cos C. fi(A~+B) f 1c f (a+_b) f C. RULE I. sin in (I.) gives - in (3), and conversely. cos in (I.) gives + in (3), and conversely. RULE II. Functions have same names in (2) and (3). Functions have co-names in (4) and (1). SURVEYING. CHAPTER I. DEFINITIONS. INSTRUMENTS AND THEIR USES. ~ 1. DEFINITIONS. Surveying is the art of determining and representing distances, areas, and the relative position of points upon the surface of the earth. In plane surveying, the portion surveyed is considered as a plane. In geodetic surveying, the curvature of the earth is regarded. A Plumb-Line is a cord with a weight attached and freely suspended. A Vertical Line is a line having the direction of the plumbline. A Vertical Plane is a plane embracing a vertical line. A Horizontal Plane is a plane perpendicular to a vertical line. A Horizontal Line is a line in a horizontal plane. A Horizontal Angle is an angle the sides of which are in a horizontal plane. A Vertical Angle is an angle the sides of which are in a vertical plane. If one side of a vertical angle is horizontal, and the other ascends, it is an angle of elevation; if one side is horizontal, and the other descends, it is an angle of depression. The Magnetic Meridian is the direction which a bar-magnet assumes when freely supported in a horizontal position. 194 SURVEYING. The Magnetic Bearing of a line is the angle it makes with the magnetic meridian. Surveying commonly comprises three distinct operations; viz.: 1. The Field Measurements, or the process of determining by direct measurement certain lines and angles. 2. The Computation of the required parts from the measured lines and angles. 3. The Plotting, or representing on paper the measured and computed parts in relative extent and position. THE MEASUREMENT OF LINES. ~ 2. INSTRUMENTS FOR MEASURING LINES. The Gunter's Chain is generally employed in measuring land. It is 4 rods, or 66 feet, in length, and is divided into 100 links. Hence, links may be written as hundredths of a chain. The Engineer's Chain is employed in surveying railroads, canals, etc. It is 100 feet long, and is divided into 100 links. A Tape Measure, divided into feet and inches, is employed in measuring town-lots, cross-section work in railroad surveying, etc. In the United States Coast and Geodetic Survey, the meter is the unit; and, when great accuracy is required, rods placed end to end, and brought to a horizontal position by means of a spirit-level, are employed in measuring lines. ~ 3. CHAINING. Eleven tally-pins of iron or steel are used in chaining; also, one or more iron-shod poles called flag-staffs or range poles. A forward chainnan, or leader, and a hind chainman, or follower, are required. A flag-staff having been placed at the farther end of the line, or at some point in the line visible CHAINING. 195 from the beginning, the follower takes one end of the chain, and a pin which he thrusts into the ground at the beginning of the line. The leader moves forward in the direction of the flag-staff, with the other end of the chain and the remaining ten pins, until the word " halt" from the follower warns him that he has advanced nearly the length of the chain. At this signal he stops, and the follower, meanwhile having placed his end of the chain at the beginning of the line, directs the leader by the words "right" or "left" until the chain is exactly in line with the flag-staff. This being accomplished, and the chain stretched tightly in a horizontal position, the follower says, " down." The leader then puts in a tally-pin exactly at the end of the chain, and answers, "down"; after which the follower withdraws the pin at the beginning of the line, and the chainmen move forward until the follower nears the pin set by the leader. The follower again says, "halt," and the operation just described is repeated. This process is continued until the end of the line is reached. If the tally-pins in the hands of the leader are exhausted before the end of the line is reached, when he has placed ihe last pin in the ground, he waits until the follower comes up to him. The follower gives the leader the ten pins in his hand, and records the fact that ten chains have been measured. The measuring then proceeds as before. If the distance from the last pin to the end of the line is less than a chain, the leader places his end of the chain at the end of the line, and the follower stretches tightly such a part of the chain as is necessary to reach to the last pin, and the number of links is counted. The number of whole chains is indicated by the number of pins in the hands of the follower, the last pin remaining in the ground. In measuring, the chain must be held in a horizontal position. If the ground slopes, one end of the chain must be raised until the horizontal position is attained. By means of 196 SURVEYING. a plumb-line, or a slender staff, or, less accurately, by dropping a pin (heavy end downwards), the point vertically under the raised end of the chain may be determined. If the slope is considerable, half a chain or less may be used. To construct a perpendicular with a chain: 1. When the point through which the perpendicular is to pass is in the line: Let AB (Fig. 1) represent the line, and P the point. Measure from P to the right or left, PC = 40 links, and place a stake at C. Let one end of,'~ l the chain be held at P, and the end of,, the eightieth link at C then, taking the,/^ ~, chain at the end of the thirtieth link from P, draw it so that the portions DC FA C IGi P B and DP are tightly stretched, and place a stake at D. DP will be the perpendicular required. (Why?) 2. When the point is without the line: Let AB (Fig. 1) be the line, and D the point. Take C any point in the line, and stretch the chain between D and C; then, let the middle of thefpart of the chain between C and D be held in place, and swing the end at D around until it meets the line in P. DP will be the perpendicular required. (Why?) ~ 4. OBSTACLES TO CHAINING. I. When a tree, building, or other obstacle is encountered in measuring or extending a line, it may be passed by an offset in the following manner: To prolong the line AB' past a building 0 (Fig. 2). At B erect BE perpendicular to AB; A - B B' C C' D at E erect EF per'j__l_, ____. I pendicular to BE; E E' F F' at F erect FC - BE FIG. t2e. Cperpendicular to EF; then, CD perpendicular to FC will be in the required line, and AB + EF + CD = AD. By constructing two other perpendiculars, B'E' and F'C', the accuracy of the work will be increased. OBSTACLES TO CHAINING. 197 2. To measure across a body of water: Let it be required to measure the line ABCD (Fig. 3) crossing a river between B and C. Measure BE = 400 links; at E erect the perpendicular EF - 600 links; at B erect the perpendicular BG = 300 links. Place a stake at C, the intersection of AD and FG beyond the river. - - I // FIG. 3. Then BC 400 links. For, by similar triangles, EF: BG:: CE: CB. But EF 2 BG; hence, CE = 2 CB, and CB = BE = 400 links. EG and FG should be measured, in order to test the accuracy of the work. EG = FG = 500 links. Instead of the above distances, any convenient distances may be taken. For, if EF 2 BG, then CB BE, and EG = FG = ~B + G2. 3. To measure a line the end of which is invisible from the beginning, and intermediate points unknown: F A -_ / 'IG B I I ^Er"^ --- C- - FIG. 4. Let AB (Fig. 4) represent the line. Set up a flag-staff at D, beyond B and visible from A. From B let fall BC perpendicular to AD. Measure A C and BC. Then AB = AC + BC2. To find intermediate points on AB: At any point E on A C erect EF perpendicular to A C, and determine EG by the proportion AC: CB:: AE: EG. G will be a point on AB. The line AD is called a Random Line. 198 SURVEYING. THE MEASUREMENT OF ANGLES. ~ 5. THE SURVEYOR'S COMPASS.* The Surveyor's Compass is shown on the following page. The compass circle is divided into half-degrees, and is figured from 0~ to 90~ each way from the north and south points. In the centre of the compass circle is the pivot which supports the magnetic needle. The needle may be lifted from the pivot by a spring and pressed against the glass covering of the compass circle, when the instrument is not in use. The main plate moves around the compass circle through a small arc, read by the vernier, for the purpose of allowing for the variation of the needle (~ 23). The sight standards at the extremities of the main plate have fine slits nearly their whole length, with circular openings at intervals; on the edges of the north standard are tangent scales for reading vertical angles; and on the outside of the south standard are two eye-pieces at the same distance from the main plate as the zeros of the tangent scales, respectively. The telescopic sight (a recent improvement by the Messrs. Gurley), consists of a small telescope attached to the south standard. The main plate is furnished with two spirit levels at right angles, and turns horizontally upon the upper end of the ball spindle, the lower end of which rests in a spherical socket in the top of the tripod or Jacob's staff which supports the instrument. From the centre of the plate at the top of the tripod a plummet is suspended by which the centre of the compass can be placed directly over a definite point on the ground. *The instruments described on this and the following pages are adjusted by the maker. If they should require readjustment, full directions will be found in the manual furnished with the instruments. The manual published by Messrs. W. & L. E. GURLEY, Troy, N. Y., has been freely used, by permission, in describing these instruments. THE SURVEYOR'S COMPASS. NOTE. The letters E and W on the face of the compass are reversed from their true positions. The reason for this is that if the sights are turned towards the west, the north end of the needle is turned towards the letter W, and if the north end of the needle is turned towards E, the sights are turned towards the east. If the north end of the needle points exactly towards E or W, the sights will range east or west. INSTRUMENTS AND THEIR USES. 201 ~ 6. USES OF THE COMPASS. To take the bearing* of a line. Place the instrument so that the plummet will be directly over one end of the line, and level by pressing with the hands on the main plate until the bubbles are brought to the middle of the spirit levels. Turn the south end of the instrument toward you, and sight at the flag-staff at the other end of the line. Read the bearing from the north end of the needle. First, write N. or S. according as the north end of the needle is nearer N. or S. of the compass circle; secondly, write the number of degrees between the north end of the needle and the nearest zero mark; and thirdly, write E. or W. according as the north end of the needle is nearer E. or W. of the compass circle. In Fig. 5 the -bearing would be N. 45~ W. In Fig. 6 the bearing would be S. 45~ W. In Fig. 7 the bearing would be S. 30~ E. In Fig. 8 the bearing would be N. 60~ E. If the needle coincides with the N.S. or E.W. line, the bearing would be N., S., E., or W., according as the north end of the needle is over N., S., E., or W. As the compass circle is divided into half-degrees, the bearing may be deter- FIG. 5. FIG. 6. mined pretty accurately to quarter-degrees. / When a fence or other obstruction interferes with\ placing the instrument over the line, it may be placed at one side, the flag-staff FIG. 7. FIG. 8. being placed at an equal distance from the line. * The magnetic bearing is meant unless otherwise specified. 202 SURVEYING. Local Disturbances. Before a bearing is recorded, care should be exercised that the chain, pins, and other instruments that would affect the direction of the needle, are removed from the vicinity of the compass. Even after the greatest care in this respect is exercised, the direction of the needle is often affected by iron ore, ferruginous rocks, etc. Reverse Bearings. When the bearing of a line has been taken, the instrument should be removed to the other end of the line and the reverse bearing taken. The number of degrees should be the same as before, but the letters should be reversed. To take the bearing of a line one end of which cannot be seen from the other. Run a random line (~ 4, 3); then (Fig. 4), BC tan CAB = A; whence the angle CAB may be found. This angle combined with the bearing of the random line will give the bearing required. Another method will be given in ~ 19. To measure a horizontal angle by means of the needle. Place the compass over the vertex of the angle, take the bearing of each side separately, and combine these bearings. To measure angles of elevation. Bring the south end of the compass towards you, place the eye at the lower eye-piece, and with the hand hold a card on the front side of the north sight, so that its top edge will be at right angles to the divided edge and coincide with the zero mark; then, sighting over the top of the card, note upon a flag-staff the height cut by the line of sight; move the staff up the elevation, and carry the card along the edge of the sight until the line of sight again cuts the same height on the staff; read off the degrees of the tangent scale passed over by the card. To measure angles of depression. Proceed in the same manner as above, using the eye-piece and tangent scale on the opposite sides of the sights, and reading from the top of the sight. INSTRUMENTS AND THEIR USES. 203 ~ 7. VERNIERS. First form. Let AB (Fig. 9) represent a portion of a rod for measuring heights (~ 32). The graduation to feet and hundredths of a foot begins at the lower end, which rests on the ground when the rod is in use. The line extending nearly across the rod at the bottom _ B of the portion shown marks the beginning of the fourth foot. The face of the rod is 3 divided into four columns: in the first is written the number of feet; in the second, the number of tenths; and in the third, the g number of hundredths. 7 0^ It is evident that, with the arrangement 5_ just described, heights could be measured 3 only to hundredths of a foot. To enable us 4 to find the height more precisely, a contri- 1 vance called a Vernier is used. This is shown 1 7 at the right of the rod. It consists of a piece 8- 8 of metal or wood, the graduated part of 7 1 which is yo6 of a foot in length; and this 5 is divided into ten equal parts. Hence, one t division of the vernier -1 of 2 == 'of a foot; and one division of the vernier 1 exceeds one division of the rod by -r X - lT = -- TOlY- of a foot. The vernier slides along the face or side FIG. 9. of the rod. To use the vernier, place the lower end of the rod upon the ground, and move the vernier until its index or zero mark is opposite the point whose distance from the ground is desired. In the figure, the height of the index of the vernier is evidently 4.16 feet, increased by the distance of the index above the next lower line (4.16) of the rod. We shall now determine this distance. 204: SURVEYING. Observe which line of the vernier is exactly opposite a line of the rod. In this case, the line of the vernier numbered 7 is opposite a line of the rod. Then, since each division of the vernier exceeds each division of the rod by T-o o of a foot, 6 of the vernier is T-JO of a foot above the next lower line of the rod. 5 of the vernier is 3-r of a foot above the next lower line of the rod. 4 of the vernier is -fo3 of a foot above the next lower line of the rod. 3 of the vernier is -4 of a foot above the next lower line of the rod. 2 of the vernier is - of a foot above the next lower line of the rod. 1 of the vernier is -o-o of a foot above the next lower line of the rod. 0 of the vernier is -o- of a foot above the next lower line of the rod. Hence, the required reading is 4.16 + 0.007 4.167 feet. B In general, the following rule is evident: \, 7 -6 -i J M=ove the vernier until its zero line is at the 6 - 10 5 _ 9 required height; read the height to the nearest 4 = s hundredth below the index, and write in the 2_ 6 thousandths' place the number of the division 1 -— '- 5 line of the vernier which stands opposite any -s line of the rod. 1 Second form. In this form (Fig. 10) the G _ graduated part of the vernier is 906 of a foot 05 = in length, and is divided into ten equal parts. 3 -= Hence, one division of the vernier — = of 2 1 9 - a9 — of a foot; and one division of the i _ --- vernier is less than one division of the rod S = by - TdOO l- 9 oo of a foot. 7 =The height of the index of the vernier in 5 Fig. 10 is 4.16 feet, increased by the distance 4 = of the index from the next lower line (4.16) 2 of the rod. We shall now determine this 1 = distance. 9o We observe that the line of the vernier,- A numbered 7 stands exactly opposite the line IG. 1G. of the rod numbered 3. Hence, INSTRUMENTS AND THEIR USES. 205 6 of the vernier is T-, of a foot above the next lower line of the rod. 5 of the vernier is 2o-o of a foot above the next lower line of the rod. 4 of the vernier is T-o of a foot above the next lower line of the rod. 3 of the vernier is T-4 of a foot above the next lower line of the rod. 2 of the vernier is T-o5 of a foot above the next lower line of the rod. 1 of the vernier is o-T6 of a foot above the next lower line of the rod. 0 of the vernier is r-Q6 of a foot above the next lower line of the rod. Hence, the required reading is 4.16 + 0.007 4.167 feet; and the rule is evidently the same as for the first form. V/ 30 ZE 1 0RO' 0 4 FIG. 11. Compass Verniers. Let LL' (Fig. 11) represent the limb of the compass graduated to half-degrees, and VY' the vernier divided into thirty equal spaces, equal to twenty-nine spaces of the limb. Then one space of the vernier is less than one space of the limb by 1', and the reading may be obtained to single minutes. In Fig. 11 the index or zero of the vernier stands between 32~ and 32~ 30', and the line of the vernier marked 9 coincides with a line of the limb. Hence, the reading is 32~ 9'. When the index moves from the zero line of the limb in a direction contrary to that in which the numbers of the limb run, the number of minutes obtained as above must be subtracted from 30' to obtain the minutes required. If, however, the vernier be made double, that is, if it have thirty spaces on each side of the zero line, it is always read 206 SURVEYING. directly. The usual form of the double vernier, shown in Fig. 12, has only fifteen spaces on each side of the zero line. When the vernier is turned to the right less than 15' past a division line of the limb, read the lower figures on the left of the zero line at any coincidence; if moved more than 15' past a division line of the limb, read the upper figures on the right of the zero line at any coincidence; and vice versa. V1 - 20 25 30 25 20 vi l I 5- zero 5 \ 10 1\5 ze ro FIG. 12. Uses of the Compass Vernier. The most' important use of the vernier of the vernier compass is in setting off the variation of the needle (~ 23). If the variation of the needle at any place is known, by means of the vernier screw the compass circle may be turned through an arc equal to the variation. If the observer stands at the south end of the instrument, the vernier is turned to the right or left according as the variation is west or east. The compass will now give the bearings of the lines with the true meridian. In order to retrace the lines of an old survey, turn the sights in the direction of a known line, and move the vernier until the needle indicates the old bearing. The arc moved over by the vernier will indicate the change of variation since the time of the old survey. If no line is definitely known, the change of variation from the time of the old survey will give the arc to be set off. INSTRUMENTS AND THEIR USES. 207 ~ 8. THE SURVEYOR'S TRANSIT. This instrument is shown on page 17. The compass circle is similar to that of the compass. The vernierplate which carries the telescope has two verniers and moves entirely around the graduated limb of the main plate. The axis of the telescope carries a vertical circle which measures vertical angles to single minutes by means of a vernier. Under the telescope, and attached to it, is a spirit level by which horizontal lines may be run, or the difference of level between two stations found. The cross wires are two fine fibres of spider's web, or fine platinum wires, which extend across the tube of the telescope at right angles to each other; their intersection determines the optical axis or line of collimation of the telescope. The transit is levelled by four levelling screws which pass through a plate firmly fastened to the ball spindle, and rest in depressions on the upper side of the tripod plate. A quick centring head enables the surveyor to change the position of the vertical axis horizontally without moving the tripod; and a quick levelling head enables him to bring the transit quickly to an approximately level position by the pressure of the hands, after which the levelling screws are used; also, to change the position of the transit without changing the position of the tripod legs, so as to bring the plummet exactly over any point. To level the transit by the levelling screws. Turn the instrument until the spirit levels on the vernier plate are parallel to the vertical planes passing through opposite pairs of levelling screws. Take hold of opposite screw heads with the thumb and fore-finger of each hand, and turn both thumbs in or out as may be necessary to raise the lower side of the parallel plate and lower the other until the desired correction is made. To use the telescope. Both the eye-piece and the object glass may be moved in and out by a rack-and-pinion movement. The eye-piece must be moved until the cross wires are 208 SURVEYING. perfectly distinct; then a slight movement of the eye of the observer, from side to side, will produce no apparent change in the position of the threads upon the object. The object glass must be moved until the object is distinctly visible; and this must be repeated, if the distance of the object is changed. ~ 9. USES OF THE TRANSIT. The transit may be used for all the purposes indicated in ~ 6, but with much greater precision than the compass. The principal use, however, of the transit is in measuring horizontal angles by means of the graduated limb and verniers. To measure a horizontal angle with the transit. Place the transit over the vertex of the angle; level, and set the limb at zero. Turn the telescope in the direction of one of the sides of the angle, clamp to the spindle; unclamp the main plate, and turn the telescope until it is in the direction of the other side of the angle, and read the angle by the verniers. The object of the two verniers on the vernier plate is to correct any mistake that might arise from the want of exact coincidence in the centres of the verniers and the limb. The correct reading may be obtained by adding to the reading of one vernier the supplement of the reading of the other, and dividing by 2. By turning off a right angle by this method, perpendiculars may be constructed with greater facility than by the chain. ~ 10. THE THEODOLITE. The telescope of the transit can perform a complete revolution on its axis; whence the name transit. The theodolite differs from the transit chiefly in that its telescope cannot be so revolved. It is not much used in this country. ~ 11. THE RAILROAD COMPASS. This instrument has all the features of the ordinary compass, and has also a vernier plate and graduated limb for measuring horizontal angles. THE SURVEYOR'S TRANSIT. INSTRUMENTS AND THEIR USES. 211 ~ 12. PLOTTING. The principal plotting instruments are a ruler, pencil, straight-line pen, hair-spring dividers, diagonal scale, a right triangle of wood, and a circular protractor. A T-square will also be found convenient. A' h B' C' 10 9 8 7 5 4 A B FIG. 13. The Diagonal Scale. A portion of this scale is shown in Fig. 13. AB is the unit. AB and A'B' are divided into ten equal parts, and B is joined with h, the first division point to the left of B'; the first division point to the left of B is joined with the second to the left of B', etc. The part of the horizontal line numbered 1 intercepted between BB' and Bh is evidently -yl of y-1 =- - of the unit; the part of the horizontal line numbered 2 intercepted between BB' and Bh is 2 of the unit, etc. The method of using this scale is as follows: Let it be required to lay off the distance 1.43. Place one foot of the dividers at the intersection of the horizontal line numbered 3 and the diagonal numbered 4, and place the other foot at the intersection of the vertical line numbered 1 (CC') and the horizontal line numbered 3; the distance between the feet of the dividers will be the distance required. For, measuring along the horizontal line numbered 3, from CC' to BB' is 1; from BB' to Bh is 0.03; and from Bh to the diagonal numbered 4 is 0.4; and 1 + 0.03 + 0.4 1.43. 212 SURVEYING. The Circular Protractor. This instrument (Fig. 14) usually consists of a semi-circular piece of brass or german silver, having its arc divided into degrees and its centre marked. To lay off an angle with the protractor, place the centre over the vertex of the angle, and make the diameter coincide with the given side of the angle. Mark off the number of degrees in the given angle, and draw a line through this point and the vertex. FIG. 14. Some protractors have an arm which carries a vernier, by which angles may be constructed to single minutes. To draw thbrough a given point a line parallel to a given line, make one of the sides of a triangle coincide with the given line, and, placing a ruler against one of the other sides, move the triangle along the ruler until the first side passes through the given point; then draw a line along this side. To draw through a given point a line perpendicular to a given line, make the hypotenuse of a right triangle coincide with the given line, and, placing a ruler against one of the other sides of the triangle, revolve the triangle about the vertex of the right angle as a centre until its other perpendicular side is against the ruler; then move the triangle along the ruler until the hypotenuse passes through the given point, and draw a line along the hypotenuse. CHAPTER II. LAND SURVEYING. ~ 13. DEFINITION. Land Surveying is the art of measuring, laying out, and dividing land, and preparing a plot. ~ 14. DETERMINATION OF AREAS. The unit of land measure is the acre - 10 square chains 4 roods = 160 square rods, perches, or poles. Areas are referred to the horizontal plane, no allowance being made for inequalities of surface. For convenience of reference, the following rules for areas are given: Let A, B, and C be the angles of a triangle, and a, b, and c the opposite sides, respectively; and let s = ~ (a + b + c). Area of triangle ABC = - base X altitude [A] = bc sinA [B] a2 sin B sin C 2 sin(B + C) = s (s - (s) (s - (s - c). [D] Area of rectangle = base X altitude. Area of trapezoid -= sum of parallel sides X altitude. PROBLEM 1. To determine the area of a triangular field. Measure the necessary parts with a Gunter's chain, or with a chain and transit, and compute by formula [A], [B], [c], or [D]. 214 SURVEYING. PROBLEM 2. To find the area of a field having any number of straight sides. (a; Divide the field into triangles by diagonals; find the area of each triangle and take the sum. (b) Run a diagonal, and perpendiculars from the opposite vertices to this diagonal. The field is thus divided into right triangles, rectangles, and trapezoids, the areas of which may be found and the sum taken. G' \ E' ' H' K G — D F D D ~ FIG. 15. FIG. 16. PROBLEM 3. To find the area of a field having an irregular boundary line. (a) Let AGBCD (Fig. 15) represent a field having a stream AEFG HKB as a boundary line. Run the line AB. From E, F, G, H, and K, prominent points on the bank of the stream, let fall perpendiculars EE', FF', etc., upon AB. Regarding AE, EF, etc., as straight lines, the portion of the field cut off by AB is divided into right triangles, rectangles, and trapezoids, the necessary elements of which can be measured and the areas computed. The sum of these areas added to the area of ABCD will give the area required. (b) When the irregular boundary line crosses the straight line joining its extremities, as in Fig. 16, the areas of AEFH and HGB may be found separately, as in the preceding case. Then the area required = ABCD + 11GB - AEF. PROBLEM 4. To determine the area of a field from two interior stations. Let ABCD (Fig. 17) represent a field, and P and P' two stations within it. Measure PP' with great exactness. Measure the angles between PP' and the lines from P and P' to the corners of the field. DETERMINATION OF AREAS. 215 In the triangle PP'D, PP' and the angles P'PD and PP'D are known; hence, PD may be found. In like manner, PC may be found. Then in the triangle D PDC, PD, PC, and the angle DPC are \,' C known; hence, the area of PDC may be \ \ - computed. \ - '- / In like manner, the areas of all the triangles about P and P' may be deter- \ mined. \ - Area ABCD = PAD + PDC + PCB \\,'/ " + PBA. Also.B Area ABCD = P'AD + P'DC + P'CB A + P'BA. FIG. 17. PROBLEM 5. To determine the area of a field from two exterior stations. D) Let ABCD (Fig. 18) represent the field, and P and P' the stations. Determine the areas of the triangles PAD, PDC, PCB, and PBA, as / \ in the preceding problem./ / Area ABCD = PAD + PDC + PBC - A / ' PBA. Also, ' y -- \ Area ABCD = P'AD + P'DC + P'BA - i ---'. -z: P'BC. -P FIG. 18. EXERCISE I. 1. Required the area of a triangular field whose sides are respectively 13, 14, and 15 chains. 2. Required the area of a triangular field whose sides are respectively 20, 30, and 40 chains. 3. Required the area of a triangular field whose base is 12.60 chains, and altitude 6.40. 4. Required the area of a triangular field which has two sides 4.50 and 3.70 chains, respectively, and the included angle 60~. 5. Required the area of a field in the form of a trapezium, one of whose diagonals is 9 chains, and the two perpendiculars upon this diagonal from the opposite vertices 4.50 and 3.25 chains. 216 SUR VEYING. 6. Required the area of the field ABCDTF (Fig. 19), if AE =9.25 chains, FE'- 6.40 chains, BE = 13.75 chains, DD' ~~=F 7 chains, )D = 10 chains, CC' 4 chains, and AA' = 4.75 chains. 7. Required the area of the field ------- ABCDEF (Fig. 20), if ~Ar/ \' AF' 4 chains, iF' = 6 chains, B - ____ ' B E' =6.50 chains, AE' 9 chains, AD = 14 chains, AC'= 10 chains, C AB' -6.50 chains, BB' = 7 chains FIG. 19. CC'= 6.75 chains. 8. Required the area of the field AGBCD (Fig. 15), if the diagonal AC - 5, BB' (the perpenF E dicular from B to AC)= 1, DD' (the / '. perpendicular from D to AC) 1.60,.// _B__' L- D EYE' 0.25, 'F = 0.25, GG' =0.60, F' E / = 0.52, KK'0.54, AE 0.2, \ j / ^'Fl' 0.50, F'G' 0.45, G'H' = 0.45, _I'Kf' - 0.60, and K'B -0.40. B 9. Required the area of the field FIG. 20. FIG. 20 AGBCD (Fig. 16), if AD = 3, AC =5, AB =6, angle DA C 450, angle BAC= 30~, A' = 0.75, AF! 2.25, AI- 2.53, A G- 3.15, BE'- 0.60, FF'-= 0.40, and GG'= 0.75. 10. Determine the area of the field ABCD from two interior stations, P and P', if PP' - 1.50 chains, PP'C= 89~ 35', PP'D- 3490 45', P'PD= 165~ 40', PP'B =185~ 30', P'PB 3~ 35', P'PC = 303~ 15'. PP'A = 309~ 15', P'PA 1130 45' 11. Determine the area of the field ABCD from two exterior stations, P and P', if PP' = 1.50 chains, P'PB= 41~ 10', P'PD- =104~ 45', PP'B = 1320 15', P'PA= 55~ 45', PP'D- 66 45', PP'A =103~ 0'. P'PC= 77~ 20', PP'C.= 95~ 40', RECTANGULAR SURVEYING. 217 RECTANGULAR SURVEYING. ~ 15. DEFINITIONS. An East and West Line is a line perpendicular to the magnetic meridian. The Latitude of a line is the distance between the east and west lines through its extremities. The Departure of a line is the distance between the meridians through its extremities. NOTE. When a line extends north of the initial point the latitude is called a northing; when it extends south, a southing; when it extends east the departure is called an easting; when it extends west, a westing. The Meridian Distance of a point is its distance from a meridian. The Double Meridian Distance of a course is double the distance of the middle point of the course from the meridian. Let AB (Fig. 21) represent a line, and NAS the magnetic meridian. Let BB' be perpendicular to NlS. N The bearing of the line AB is the angle BAB'. The latitude of the line AB is AB'. The departure of the line AB is BB'. B' B The meridian distance of the point B is BB'. In the right triangle ABB', AB' = AB X cos BAB', and BB' - AB X sin BAB'. Hence, latitude = distance X cos of bearing, and departure = distance X sin of bearing. The latitudes and departures corresponding A to any distance and bearing may be found from the above formulas by means of a table of natural sines and cosines, or from "The Traverse Table." FIG. 21. * See Table VII. of Wentworth & Hill's Five-Place Logarithmic and Trigonometric Tables, 218 SURVEYING. ~ 16. FIELD NOTES, COMPUTATION, AND PLOTTING. The field notes are kept in a book provided for the purpose. The page is ruled in three columns, in the first of which is written the number of the station; in the second, the bearing of the side; and in the third, the length of the side. EXAMPLE 1. To survey the field ABCD (Fig. 22). FIELD NOTES. 1 N. 20~ E. 8.66 B E 2 S. 700 E. 5.00 3 S. 10~ E. 10.00 C' --- I 4 N.70~W. 10.00 (a) To obtain the field notes. Place the compass at A, the first station, and take the bearing of AB (~ 6); suppose it to be N. 20~ E. Write the result in the A \\ second column of the field notes opposite the number of the station. Measure AB = 8.66 chains, and write the result in the DT —, D --- —-- D third column of the field notes. Place the compass at B, and, after testing the bearing of AB (~ 6), take the bearing of BC, measure BC, and write the results in the S field notes; and so continue until the bearing FIG. 22. and length of each side have been recorded. (b) To compute the area. I. II. III. IV. V. VI. VII.VIII. I. X. X. Side. Bearing. Dist. N. S. E. W. M, D. D. M, D. N,A, SA. AB' BB BB' BB' 2ABB' AB N. 20~ E. 8.66 8.14.... 2.96... 2.96. 2.96 24.0944 B'C' CC" CC' BB'+CC' 2C'CBB BC S. 70~ E. 5.00.... 1.71 4.70... 7.66 10.62.... 18.1602 C'D' DD" DD' CC'+DD' 2D'DCC' CD S. 10~ E. 10.00....85 1.74.. 0.40 17.06.... 168.0410 )'A1 DD' DD' 2 ADnD' DA N. 70~ W. 10.00 3.42........9.40 0 9.40 32.1480 [33.66 11.56 11.561 9.40 1 1 56.24241 186.2012 FIELD NOTES. 219 The survey may begin at any corner of the field; but in computing the area, the field notes should be arranged so that the 186.2012 most eastern or most western station will stand 56.2424 first. For the sake of uniformity, we shall always 2 129.9588 begin with the most western station, and keep the 10 64.98 sq. chains. 6.498 acres. field on the right in passing around it..498 acres. The field notes occupy the first three of the eleven columns in the above tablet. Columns IV., V., VI., and VII. contain the latitudes and departures corresponding to the sides, and taken froln the Traverse Table. The lines represented by these numbers are indicated immediately above each number. Column VIII. contains the meridian distances of the points B, C, D, and A, taken in order. Column IX. contains the double meridian distances of the courses. Their composition is indicated by the letters immediately above the numbers. Column X. contains the products of the double meridian distances by the northings in the same line. The first number, 24.0944 = 2.96 x 8.14 = BB' x AB' - 2 area of the triangle ABB'; 32.1480 = 9.40 x 3.42 = DD' x AD' = 2 area of the triangle ADD'. Column XI. contains the products of the double meridian distances by the southings in the same line. The first number, 18.1602 = 10.62 x 1.71 = (BB' + CC') x B'C' = 2 area of the trapezoid C'CBB'; 168.0410 = 17.06 x 9.85 (CC' + DD') x D'C' - 2 area of the trapezoid D'DCC'. The sum of the north areas in column X. = 56.2424 = 2 (ABB' + ADD'). The sum of the south areas in column XI. = 186.2012 = 2 (C'CBB' + D'DCC'). But (C'CBB' + D'DCC') - (ABB' + ADD') = ABCD. Hence, 2(C'CBB' + D'DCC') - 2(ABB' + ADD') = 2ABCD; that is, 186.2012 - 56.2424 = 129.9588 = 2ABCD. Hence, area ABCD = ~ of 129.9588 = 64.9794 sq. ch. = 6.498 acres. (c) To make the plot. The plot or map may be drawn to any desired scale. If a line one inch in length in the plot represents a line one chain in length, the plot is said to be drawn to a scale of one chain to an inch. In this case the plot (Fig. 22) is drawn to a scale of eight chains to an inch. Draw the line NAS to represent the magnetic meridian, and lay off the first northing AB' = 8.14 (~ 12). Draw the indefinite line B'E per 220 SURVEYING. pendicular to NS and lay off B'B, the first easting = 2.96. Join A and B; then the line AB will represent the first side of the field. Through B draw BC" perpendicular to BB', and make BC" = 1.71, the first southing. Through C" draw C"C perpendicular to BC", and equal to 4.70, the second easting. Join B and C. The line BC will represent the second side of the field. Proceed in like manner until the field is completely represented. The extremity of the last line D'A, measured from D', should fall at A. This will be a test of the accuracy of the plot. By drawing the diagonal AC, and letting fall upon it perpendiculars from B and D, the quadrilateral ABCD is divided into two triangles, the bases and altitudes of which may be measured and the area computed approximately. Other methods of plotting will suggest themselves, but the method just explained is one of the best. Balancing the Work. In the survey, we pass entirely around the field; hence, we move just as far north as south. Therefore, the sumn of the northings should equal the sum of the southings. In like manner, the sum of the eastings should equal the sum of the westings. In this way the accuracy of the field work may be tested. In Example 1, the sum of the northings is equal to the sum of the southings, being 11.56 in each case; and the sum of the eastings is equal to the sum of the westings, being 9.40 in each case. Hence, the work balances. In actual practice the work seldom balances. When it does not balance, corrections are generally applied to the latitudes and departures, by the following rules: The perimeter of the field: any one side:: total error in latitude: correction;:: total error in departure: correction. If special difficulty has been experienced in taking a particular bearing, or in measuring a particular line, the corrections should be applied to the corresponding latitudes and departures. FIELD NOTES. 221 The amount of error allowable varies in the practice of cifferent surveyors, and according to the nature of the ground. An error of 1 link in 8 chains would not be considered too great on smooth, level ground; while, on rough ground, an error of 1 link in 2 or 3 chains might be allowed. If the error | is considerable, the field meas- ~ _? m.. t urellents should be repeated. Ca S 0.0a co Co 0 ~ c" EXAMPLE 2. Let it be re- c o9 o ' quired to survey the field AB k- 8 ' CDEF (Fig. 23).:: -T Co C) Co 0 1 N cc FIELD NOTES. 1 N. 73 30' W. 5.00 2 S. 16030'W. 5.00 3 N. 28030 'W. 7.07 4 N. 200 00'E. 11.18 5 S. 43030' E. 5.00 6 S. 13~ 30'E. 10.00 Io f Co -qo F t CO ^ fc0 CO * *. c -_ co ~.. Q c cO -- dC Ci - -.' f o.. KOi 3 r 0* 243.0888 81.4955 21161.5933 10 80.7967 8.0797 acres., 4 ' ' ' 4 C +; — 1;v EXPLANATION. The first station,. in the field notes is D, but we re- ___ b c" arrange the numbers in the tablet so co P ' that A stands first. The northings a. and southings balance, but the east- ings exceed the westings by 1 link. _. We apply the correction to the west-:: ( Ic ing 4.79 (the distance DE being in. *: doubt), making it 4.80, and rewrite all the latitudes and departures in the next four columns, incorporating the correction. In practice, the corrected numbers are written in red ink. 222 SURVEYING. The remainder of the computation does not require explanation. It will be seen that this method of computing areas is perfectly general. N ~ 17. SUPPLYING OMISSIONS. If, for any reason, the bearing and length of any side do not appear B' — _ --- B in the field notes, the latitude and departure of this side may be found in the following manner: C i*/~ - 0c -f C Find the latitudes and departures of the other sides as usual. The difference between the northings and southings will give the northing or southing of the unknown A \ side, and the difference between E' --- —-- the eastings and westings will give I ------ — _ the easting or westing of the un-,D"' Dp" D known side. '\,! If the length and bearing of the _J F' F F"' unknown side are desired, they may be found by solving the right triFTG. 23. angle, whose sides are the latitude and departure found by the method just explained, and whose hypotenuse is the length required. ~ 18. IRREGULAR BOUNDARIES. If a field have irregular boundaries, its area may be found by offsets, as explained in ~ 14, Prob. 3. ~ 19. OBSTRUCTIONS. If the end of a line be not visible from its beginning, or if the line be inaccessible, its length and bearing may be found as follows: OBSTRUCTIONS. 223 1. By means of a random line (~ 4, 3). 2. When it is impossible to run a random line, which is frequently the case on account of the extent of the obstruction, the following method may be used: ~V I Let AB (Fig. 24) represent an inaccessible line whose extremities A and B only are known, and B invisible from A. Set flag-staffs at convenient points, C and D. Find the bearings and lengths of A C, CD, and DB, and then proceed to find the latitude and departure of AB, as in ~ 17. A B C D S FIG. 24. have the following notes (see EXAMPLE. Suppose that we Fig. 24): SIDE. BEARING. DIST. N. S. E. W. AC S. 45~ E. 3.00 2.12 2.12 CD E. 3.50 3.50 DB N. 30~ E. 4.83 4.18 2.42 4.18 2.12 8.04 0 I B A C FIG. 25. 4.18 The northing of AB is 2.06, and the easting, 8.04; which 2.12 numbers may be entered in the tablet in the columns N. and E., 2.06 opposite the side AB. If the bearing and length of AB are required, construct the right triangle ABC (Fig. 25), making AC 8.04 and BC = 2.06. BC 2.06 tan BAC = = = =- 0.256. AC 8.04 Hence, the angle BAC = 14~ 22'. Also, AB = VAC2 + BC2 = </8.042 + 2.062 = 8.29. Therefore, the bearing and length of AB are N. 75~ 38' E. 8.29. NOTE. Keep all the decimal figures until the result is obtained; then reject all decimal figures but two, increasing the last decimal figure retained by 1, if the third decimal figure is 5 or greater than 5. 224 SURVEYING. EXERCISE II. In examples 5 and 6 detours were made on account of inaccessible sides (~ 19, 2). The notes of the detours are written in braces. 1. Sta. 1 2 3 4 5 Bearings, S. 75~ E. S. 15~ E. S. 75~ W. N. 45~ E. N.45~W. Dist, 6.00 4.00 6.93 5.00 5.19| Sta, 1 2[ 3 4 5 6 5. Bearings, S. 2015'E. N.51~45'W. S. 85000'W. S. 55~10'W. N. 3~45'E. S. 66~45'E. N.1500'E. S. 82~45/E. Dist. 9.68 2.39 6.47 1.62 6.39 1.70 4.98 6.03 8. Sta. Bearings. Dist. 1 N. 5~30'W. 6.08 2 S. 82030'W. 6.51 3 S. 3000'E. 5.33 4 E. 6.72 2. Sta. Bearings, Dist, 1 N. 45~ E. 10.00 2 S. 75~E. 11.55 3 S. 15~W. 18.21 4 N.45~W. 19.11 3. Sta. Bearings. Dit 1 3.00_~ 6. Sta. Bearings. S. 81020'W. I N.76~30'W. 2 N. 5~0(YE. 3 S. 8730'E. S. 7~00'E. 4 S. 27~000'E. S. 10~30'E. N.76045'W. 1 2 3 4 N. 15~E. N. 75E. S. 15~W. N. 750 W. 3.00 6.00 6.00 5.20 Dist. 4.28 2.67 8.68 5.54 1.79 1.94 5.35 1.70 Sta, 1 2 3 4 5 9. Bearings, Dist. N.20000'E. 4.621 N.73000E. 4.16 S. 45~15'E. 6.18k S. 38030/W. 8.00 Wanting. Wanting. 4. Sta, Bearings. 1 N.89~45'E. 2 S. 7000'W. 3 S. 2800O'E. 4 S. 0~45'E. 5 N.84~45'W, 6 N. 2~30'VW Dist, 4.94 2.30 1.52 2.57 5.11 5.79 7. Sta, Bearings. Dist, 1 N. 6015/W. 6.31 2 S. 81~5(0W. 4.06 3 S. 5000E. 5.86 4 N.88~30'E. 4.12 Sta, 1 2 3 4 5 6 7 8 9 10. Bearings. S. 3~00OE. S. 86045/W. S. 37~00'W. N.81000'W. N.6100'W. N.32~00'E. S. 75050/E. Dist. 4.23 4.78 2.00 7.45 2.17 8.68 6.38 S. 14045'W. 0.98 S. 7915'E. 4.52 RECTANGULAR METHOD. 225 ~ 20. MODIFICATION OF THE RECTANGULAR METHOD. The area of a field may be found by a modification of the rectangular method, if its sides and interior angles are known. Let A, B, C, D, represent the interior angles of the field ABCD (Fig. / 26). Let the side AB determine the B, direction of reference. // The bearing of AB, with reference / \ to AB, is e0~. / The bearing of B C, with reference / \ to AB, is the angle b = 180~ - B. / The bearing of CD, with reference A to AB, is the angle c = C- b. FIG. 26. The bearing of DA, with reference to AB, is the angle d-A. The area may now be computed by the rectangular method, regarding AB as the magnetic meridian. In practice, the exterior angles, when acute, are usually measured. As the interior angles may be measured with considerable accuracy by the transit, the latitudes and departures should be obtained by using a table of natural sines and cosines. EXERCISE III. 1. Find the area of the field ABCD, in which the angle A==120~, B —60~, C=150~, and D =30~; and the side AB =4 chains, BC =4 chains, CD - 6.928 chains, and DA - 8 chains. Keep three decimal places, and use the Traverse Table. 2. Find the area of the farm ABCDE, in which the angle A = 106 19', B = 99~ 40', C - 120~ 20', D 86~ 8', and E-= 127~33'; and the side AB=79.86 rods, BC=121.13 rods, CD = 90 rods, D =E- 100.65 rods, and EA = 100 rods. Use the table of natural sines and cosines, keeping two decimal places as usual. 226 SURVEYING. ~ 21. GENERAL REMARKS ON DETERMINING AREAS. Operations depending upon the reading of the magnetic needle must lack accuracy. Hence, when great accuracy is required (which is seldom the case in land surveying), the rectangular method (~~ 16-19) cannot be employed. The best results are obtained by the methods explained in ~~ 14 and 20, the horizontal angles being measured with the transit, and great care exercised in measuring the lines. ~ 22. THE VARIATION OF THE NEEDLE. The Magnetic Declination, or variation of the needle, at any place, is the angle which the magnetic meridian makes with the true meridian, or north and south line. The variation is east or west, according as the north end of the needle lies east or west of the true meridian. Western variation is indicated by the sign +, and eastern by the sign -. Irregular Variations are sudden deflections of the needle, which occur without apparent cause. They are sometimes accompanied by auroral displays and thunder storms, and are most frequent in years of greatest sun-spot activity. Solar-Diurnal Variation. North of the equator, the north end of the needle moves to the west, from 8 A.M. to 1.30 P.M., about 6' in winter and 11' in summer, and then returns gradually to its normal position. Secular Variation is a change in the same direction for about a century and a half; then in the opposite direction for about the same time. The line of no variation, or the Agonic Line, is a line joining those places at which the magnetic meridian coincides with the true meridian. In the United States, this line at present (1895) passes through Michigan, Ohio, Eastern Kentucky, the extreme southwest of Virginia, and the Carolinas. It is moving gradually westward, so that the variation is increasing TO ESTABLISH A TRUE MERIDIAN. 227 at places east of this line, and decreasing at places west of this line. East of this line the variation is westerly, and west of this line the variation is easterly. The table on pages 234 and 235, which has been prepared by permission from data furnished by the United States Coast and Geodetic Survey, shows the magnetic variation at different places in the United States and Canada for several years; also, the annual change for 1895. ~ 23. To ESTABLISH A TRUE MERIDIAN. This may be done as follows: 1. By means of Burt's Solar Compass (~ 25). 2. By observations of Polaris. The North Star or Polaris revolves about the pole at present at the distance of about 14~; hence, it is on the meridian twice in 23 h. 56 in. 4 s. (a sidereal day), once above the pole, called the upper culmination, and 11 h. 58 m. 2 s. later below the pole, called the lower culmination. It attains its greatest eastern or western elongation, or greatest distance from the meridian, 5 h. 59 i. 1 s. after the culmination. The following table gives the mean local time of the upper culmination of Polaris for 1895 at Washington, The time is growing later at the rate of about one minute in three years. MONTH. FIRST DAY. ELEVENTH DAY. TWENTY-FIRST DAY. H. I. H. M. H. IM. January... 6 35 P.. 5 55 P.M 5 16 P.M. February.. 4 32 P.r. 3 53 P.M. 3 14 P.M. March.... 2 42 P.M. 2 03 P.M. 1 23 P.M. April.... 12 40 P.M. 12 00 ri. 11 17 A. r. May..... 10 38 A.M. 9 59 A.M. 9 20 A.M. June..... 8 37 A.M. 7 57 A.M. 7 18 A. M. July..... 6 39 A.M. 6 00 A.M. 5 21 A.. August.... 4 38 A.M. 4 00 A.A.. 3 19 A.M. September. 2 36 A.M. 1 57 A.. 1 18 A.. October... 12 39 A.M. 11 59 P.M. 11 20. M. November.. 10 37 P.. 9 57 P.. 9 18 P.M. December. 8 39 P.M. 7 59 r.r. 7 20 P.M. 228 SURVEYING. The time of the upper culmination of Polaris may be found by means of the star Mizar, which is the middle one of the three stars in the handle of the Dipper (in the constellation of the Great Bear). It crosses the meridian at almost exactly the same time as Polaris. Suspend from a height of about 20 feet a plumb-line, placing the bob in a pail of water to lessen its vibrations. About 15 feet south of the plumb-line, upon a horizontal board firmly supported at a convenient height, place a compass sight fastened to a board a few inches square. At night, when Mizar by estimation approaches the meridian, place the compass sight in line with Polaris and the plumb-line, and move | * * it so as to keep it * Polaris. * * * in this line until the {Pole. * plumb-line also falls on Mizar (Fig. 27). Note the time; then (1895) fifty-one seconds later Polaris will Pole. be on the meridian.: *. This interval is ' * Polaris. * " + I* gradually increasing t at the rate of 21 secFIG. 27. onds a year. If the lower culmination takes place at night, the time may be found in a similar manner. When Mizar cannot conveniently be used, as in the spring, 8 Cassiopeiae may be employed. This is the star at the bottom of the first stroke of the NW frequently imagined to connect roughly the five brightest stars in Cassiopeia. In 1895 it culminates 1.75 minutes before Polaris, with an annual increase of the interval of 20 seconds. Instead of the compass sight, any upright with a small opening or slit may be used. TO ESTABLISH A TRUE MERIDIAN. 229 (a) To locate the true meridian by the position of Polaris at its culmination. 1. By using the apparatus described in finding the time of culmination. At the time of culmination bring Polaris, the plumb-line, and the compass sight into line. The compass sight and the plumb-line will give two points in the true meridian. 2. By means of the transit. Bring the telescope to bear on Polaris at the time of culmination, holding a light near to make the wires visible, if the observation is made at night. The telescope will then lie in the plane of the meridian, which may be marked by bringing the telescope to a horizontal position. (b) To locate the meridian by the position of Polaris at greatest elongation. The Azimuth of a star is the angle which the meridian plane makes with a vertical plane passing through the star and the zenith of observer. A star is said to be at its greatest elongation, when its vertical circle ZN (Fig. 28) is tangent to its diurnal circle, that is, perpendicular to the hour circle PN. Let Z (Fig. 28) represent the zenith of the place, P the pole, and N Polaris at its greatest elongation; that is, when its vertical circle ZN is perpendicular to the hour circle PN. Let ZP, ZN, and PN be arcs of great circles; then N will be a right angle. sin PN = cos (90~ - ZP) cos (90~ - Z). [Spher. Trig. ~ 47.] But ZP = the complement of the latitude. Hence, 90~ - ZP = the latitude of the place. Hence, sin PN = cos latitude X sin Z. sin P9 Hence, sill Z FIG. 28. cos latitude 230 SURVEYING. Hence, Z (the azimuth of Polaris) can be found if the latitude of the place and the greatest elongation of Polaris (PN) are known. The following table gives the mean value of the latter element for each year from 1895 to 1906. GREATEST ELONGATION OF POLARIS. 1895 10 15.1' 1899 1~ 13.8' 1903 1012.6' 1896 10 14.8' 1900 1~13.5' 1904 1~12.3' 1897 10 14.5' 1901 10 13.2' 1905 10 12.0' 1898 10 14.1' 1902 1 12.9' 1906 l~11.7' The greatest elongation of Polaris, or the polar distance, is given in the Nautical Almanac. The table gives this element for Jan. 1. It may be found for other dates by interpolation. To obtain a line in the direction of Polaris at greatest elongation. 1. By using the apparatus for finding the time of culmination. A few minutes before the time of greatest elongation (5 h. 59 m. 1 s. after culmination), place the compass sight in line with the plumb-line and Polaris, and keep it in line with these, by moving the board in the opposite direction, until the star begins to recede. At this moment the sight and plumbline are in the required line. 2. By means of the transit. A few minutes before the time of greatest elongation, bring the telescope to bear on the star, and follow it, keeping the vertical wire over the star until it begins to recede. The telescope will then be in the required line. To establish the meridian. Having the transit sighted in the direction of the line just found, turn it through an angle equal to the azimuth in the proper direction. DIVIDING LAND. 231 ~ 24. DIVIDING LAND. The surveyor must, for the most part, depend on his general knowledge of Geometry and Trigonometry, and his own ingenuity, for the solutions of problems that arise in dividing land. PROBLEM 1. To divide a triangular field into two parts having a given ratio, by a line through a A given vertex. Let ABC (Fig. 29) be the triangle, and A the given vertex. BD Divide BC at D, so that - equals the given DC ratio, and join A and D. ABD and ADC will be the parts required; for ABD: ADC:: BD: DC. C FIG. 29. PROBLEM 2. To cut off from a triangular field a given area, by a line parallel to the base. Let ABC (Fig. 30) be the triangle, and let DE be the division line re- quired. / — VABC: VADE::AB:AD. B - IADE.AD = AB ~1~' 0.. AD=AB\ ABC FIG. 30. PROBLEM 3. To divide a field into B C two parts having a given ratio, by a line through a given point in the perimeter. Let ABCDE (Fig. 31) represent the field, D P the given point, and PQ the required divi- P Q sion line. The areas of the whole field and of the required parts having been determined, run the line PD from P to a corner D, dividing A the field, as near as possible, as required. E Determine the area PBCD.,,,T. 1 dl a.q~, ~ 232 SURVEYING. The triangle PDQ represents the part which must be added to PBCD to make the required division. Area PDQ = I x PD x DQ x sinPDQ. 2 area PD Q Hence, DQ =prsiPpQ Hence, DQ D ) sin PDQ 2 area PDQ NOTE. DQ = 2 area PDQ This perpendicular from perpendicular from P on DE P on DE may be run and measured directly. PROBLEM 4. To divide a field into a given number of parts, so that access to a pond of water is B given to each. Let ABCDE (Fig. 32) represent the field, A / I \S and P the pond. Let it be required to divide /the field into four parts. Find the area of / the field and of each part. Let AP be one division line. Run PE,,/ /, / and find the area APE. Take the difference I / \ / between APE and the area of one of the | // |A / Irequired parts; this will give the area of the triangle PQE, from which QE may be found, Q /!' /? as in Problem 3. Join P and Q; PAQ will be one of the required parts. In like manner, PQR and PAS are determined; whence, FIG. 32. FIG. 32. PSR must be the fourth part required. EXERCISE IV. 1. From the square ABCD, containing 6 A. 1 R. 24 p., part off 3 A. by a line EF parallel to AB. 2. From the rectangle ABCD, containing 8 A. 1 i. 24 P., part off 2 A. 1 R. 32 P. by a line iEF parallel to AD = ch. Then, from the remainder of the rectangle, part off 2 A. 3 R. 25 p., by a line GH parallel to EB. 3. Part off 6 A. 3 n. 12 p. from a rectangle ABCD, containing 15 A., by a line EF parallel to AB; AD being 10 ch. 4. From a square ABCD, whose side is 9 ch., part off a triangle which shall contain 2 A. 1 n, 36 P., by a line BE1 drawn from B to the side AD. EXAMPLES. 233 5. From ABCD, representing a rectangle, whose length is 12.65 ch., and breadth 7.58 ch., part off a trapezoid which shall contain 7 A. 3 R. 24 p., by a line BE from B to DC. 6. In the triangleAB C, AB 12 ch., A C-10 ch., B C8 ch.; part off 1 A. 2 R. 16 P., below the line DE parallel to AB. 7. In the triangle ABC, AB = 26 ch., AC- 20 ch., and BC-=16 ch.; part off 6 A. 1 B. 24 p., below the line DE parallel to AB. 8. It is required to divide the triangular field ABC among three persons whose claims are as the numbers 2, 3, and 5, so that they may all have the use of a watering-place at C; AB = 10 ch., AC = 6.85 ch., and CB = 6.10 ch. 9. Divide the five-sided field ABCHE among three persons, X, Y, and Z, in proportion to their claims, X paying $500, Y paying $750, and Z paying $1000, so that each may have the use of an interior pond at P, the quality of the land being equal throughout. Given AB = 8.64 ch., BC - 8.27 ch., CH= 8.06 ch., ITE= 6.82 ch., and EJA = 9.90 ch. The perpendicular PD upon AB = 5.60 ch., PD' upon BC = 6.08 ch., PD'" upon CH=-4.80 ch., PD"' upon HE — 5.44 ch., and PD"" upon EA= 5.40 ch. Assume PH as the divisional fence between X's and Z's shares; it is required to determine the position of the fences PI and PN between X's and Y's shares and Y's and Z's shares, respectively. 10. Divide the triangular field ABC, whose sides AB, AC, and BC are 15, 12, and 10 ch., respectively, into three equal parts, by fences EG and DF parallel to BC, without finding the area of the field. 11. Divide the triangular field ABC, whose sides AB, BC, and AC are 22, 17, and 15 ch., respectively, among three persons, A, B, and C, by fences parallel to the base AB, so that A may have 3 A. above the line AB, B, 4 A. above A's share, and C, the remainder, ANNUAL PLACE. LATITUDE. LONGITUDE. VARIATION CHANGE _____ __ __ 1800. I 1820. I 1840. I 1860. I 1880. I 1890. I 1895. I 1895. Halifax N. S..... Eastport, Me. Bangor, Me. Provincetown, Mass. Portland, Me.. Portsmouth, N.H. Boston, Mass... Cambridge, Mass. Quebec, Canada. Providence, R.I.. Hartford, Conn. New Haven, Conn. Burlington, Vt. Williamstown, Mass. Montreal, Canada Albany, N.Y. New York, N.Y. New Brunswick, N.. Cape Henlopen, Del. Philadelphia, Pa. Cape Henry, Va.. Ithaca, N.Y. Baltimore, Md. Williamsburg, Va. Harrisburg, Pa.. Washington, D.C. Newbern, N.C. Buffalo, N.Y. Toronto, Canada... I I Deg. Min. 44 39.6 44 54.4 44 48.2 42 03.1 43 38.8 43 04.3 42 21.5 42 22.9 46 48.4 41 50.2 41 45.9 41 18.5 44 28.5 42 42.8 45 30.5 42 39.2 40 42.7 40 29.9 38 46.7 39 56.9 36 55.6 42 26.8 39 17.8 37 16.2 40 15.9 38 53.3 35 06 42 52.8 43 39.4 I Deg. Min. 63 35.3 66 59.2 68 46.9 70 11.3 70 16.6 70 42.5 71 03.9 71 07.7 71 14.5 71 23.8 72 40.4 72 55.7 73 12.0 73 13.4 73 34.6 73 45.8 74 00.4 74 26.8 75 05.0 75 09.0 76 00.4 76 28.9 76 37.0 76 42.4 76 52.9 77 00.6 77 02 78 53.5 79 23.5 I Degrees. Degrees. 15.9 17.4 13.2 14.8 10.9 12.1 7.2 8.2 8.50 9.46 7.6 8.3 6.90 7.78 7.10 7.97 12.1 12.3 6.46 6.71 5.10 5.58 4.7 5.0 7.2 7.78 5.7 6.3 8.0 7.9 5.5 6.02 4.3 4.61 2.54 3.43 0.8 1.1 2.1 2.44 0.24 0.25 3.3 2.7 0.64 0.88 -0.17 -0.22 0.0 0.8 -0.1 0.3 -2.1 -1.66 0.22 0.41.... o... Degrees. 18.9 16.4 13.7 9.61 10.82 9.55 9.01 9.29 13.8 8.24 6.59 5.95 8.90 7.4 9.4 7.07 5.61 4.66 2.00 3.46 0.82 3.1 1.70 0.38 2.2 1.19 -0.70 1.35 1.32 Degrees. 19.9 17.79 15.3 11.00 12.29 11.03 10.33 10.63 16.0 9.78 7.93 7.35 10.27 8.8 12.0 8.44 6.91 5.98 3.36 4.73 1.80 4.1 2.90 1.47 3.71 2.42 0.54 2.84 2.17 TE~~- I_ _ _- E____ ____ Degrees. 20.6 18.71 16.54 12.12 13.58 12.40 11.47 11.59 17.4 10.79 9.29 8.84 11.58 10.3 13.8 9.87 7.90 7.12 4.86 6.20 2.94 5.71 4.17 2.75 5.05 3.66 1.74 4.51 3.62 Degrees. 20.7 18.92 16.99 12.51 14.08 12.94 11.9 11.9 17.5 11.56 9.89 9.52 12.11 10.9 14.4 10.52 8.49 7.55 5.6 6.97 3.5 6.58 4.74 3.3 5.52 4.18 2.25 5.30 4.12 Degrees. 20.7 19.0 17.2 12.65 14.3 13.1 12.1 12.0 17.5 11.9 10.2 9.8 12.3 11.2 14.7 10.82 8.8 7.72 5.9 7.4 3.7 7.0 5.00 3.6 5.7 4.40 2.45 5.66 4.5 linutes. -0.2 0.2 1.7 1.5 2.2 2.5 1.9 1.2 -0.9 3.6 3.0 3.4 2.4 3.0 3.4 3.4 3.8 1.8 3.7 4.4 2.8 5.2 2.8 3.2 1.8 3.0 2.3 4.2 4.4 qd t^ VARIATION. CHANGE. PLACE. LATITUDE. LONGITUDE. ___________________ -A(_ 1800. 1 1820. 1 1840. 1860. 1880. 1890. 1895 Deg. Min. Deg. Min. Degrees. Degrees. Degrees. Degrees. Degr3es. Degrees. Degrees. Minutes. Charleston, S.C....32 46.6 79 55.8 -4.55 -4.05 -3.03 -1.73 -0.45 0.09 0.3 2.5 Pittsburg, Pa..... 40 27.6 80 00.8........ 0.18 1.26 2.49' 3.06 3.3 3.0 Erie, Pa... 42 07.8 80 05.4 -0.46 -0.39 0.36 1.60 2.99 3.62 3.9 3.2 Savannah, Ga.... 32 04.9 81 05.5.... -4.7 -4.2 -3.27 -2.06 -1.45 -1.2 3.4 Cleveland, 0. 41 30.4 81 41.5 -1.8 -1.4 -0.66 0.39 1.52 2.05 2.29 2.8 Key West, Fla... 24 33.5 81 48.5..... -6.86 -6.03 -4.85 -3.57 -2.96 -2.7 3.2 g Detroit, Mich... 42 20.0 83 03.0 -3.1 -2.84 -2.04 -0.93 0.23 0.74 0.96 2.5 W Sault Ste. Marie, Mich. 46 29.9 84 20.1 -0.5 -1.1 -1.04 -0.34 0.84 1.52 1.9 4.1 Cincinnati, 0.. 39 08.4 84 25.3 -4.89 -4.99 -4.51 -3.57 -2.39 -1.80 -1.53 3.3 g Grand Haven, Mich. 43 05.2 86 12.6.... -5.0 -5.2 -4.45 -2.73 -1.5 -1.0.... Nashville, Tenn.. 36 08.9 86 48.2.... -6.7 -6.9 -6.3 -5.13 -4.40 -4.0 4.7 Michigan City, Ind.. 41 43.4 86 54.4........ -5.4 -4.6 -3.5 -2.9 -2.6 3.4 o Pensacola, Fla. 30 20.8 87 18.3 -6.84 -7.50 -7.43 -6.65 -5.34 -4.59 -4.20 4.6 Chicago, Ill. 41 50.0 87 36.8.... -6.1 -6.2 -5.7 -4.52 -3.81 -3.45 4.4 Milwaukee, Wis.. 43 02.5 87 54.2............-6.9 -5.4 -4.5 -4.1 5.5 Mobile, Ala.... 30 41.4 88 02.5 -5.81 -6.71 -7.07 -6.75 -5.84 -5.23 -4.9 4.0 New Orleans, La...29 57.2 90 03.9 -7.12 -7.96 -8.16 -7.66 -6.59 -5.91 -5.6 4.3 St. Louis, Mo.. 38 38.0 90 12.2........ -8.6 -7.7 -6.4 -5.6 -5.3 4.3 Duluth, Minn. 46 45.5 92 04.5............ -10.02 -10.06 -9.9 -9.7.... Galveston, Tex.... 29 17.4 94 47............ -8.84 -8.16 -7.56 -7.20 4.5 m Omaha, Neb. 41 15.7 95 56.5..... -12.6- -12.33-11.47 -10.23 -9.56 -9.21 4.1 Austin, Tex.....30 16.4 97 44.2........ -10.7 -9.74 -8.80 -8.34 -8.1.... San Antonio, Tex...29 26.8 98 27.9..... -9.8 -10.29 -10.16 -9.43 -8.89 -8.59 3.7 Denver, Col. 39 45.3 104 59.5 -15.14 -14.52 -14.06 -13.8 3.4 Salt Lake City, Utah.40 46.1 111 53.8............-16.45 -16.58 -16.3 -16.0 3.2 San Diego, Cal. 32 42.1 117 14.3 -10.69 -11.79 -12.67 -13.21 -13.32 -13.22 -13.12 1.3 Seattle, Wash... 47 35.9 122 20.0.... — 21.8 -22.28 -22.25 -22.2 1.3 San Francisco, Cal..37 47.5 122 27.3 -13.6 -14.54 -15.42 -16.10 -16.51 -16.58 -16.59 0.1 C. Mendocino, Cal.. 40 26.3 124 24.3 — 15.1 -16.0 -16.9 -17.4 -17.69 -17.69 -17.7 0.6: 236 SURVEYING. ~ 25. UNITED STATES PUBLIC LANDS. Buirt's Solar Comp2ass. This instrument, which is exhibited on the following page, may be used for most of the purposes of a compass or transit. Its most important use, however, is to run north and south lines in laying out the public lands. A full description of the solar compass, with its principles, adjustments, and uses, forms the subject of a considerable volume, which should be in the hands of the surveyor who uses this instrument. The limits of our space will allow only a brief reference to its principal features. The main plate and standards resemble these parts of the compass. a is the latitude arc. b is the declination arc. A is an arm, on each end of which is a solar lens having its focus on a silvered plate on the other end. c is the hour arc. n is the needle-box, which has an arc of about 36~. To run a north and south line with the solar compass. Set off the declination of the sun on the declination arc. Set off the latitude of the place (which may be determined by this instrument) on the latitude arc. Set the instrument over the station, level, and turn the sights in a north and south direction, approximately, by the needle. Turn the solar lens toward the sun, and bring the sun's image between the equatorial lines on the silvered plate. Allowance being made for refraction, the sights will then indicate a true north and south line. / -Ii I! BU1RT'S SOLAR COMPASS LAYING OUT THE PUBLIC LANDS. 239 Laying Out the Public Lands. The public lands north of the Ohio River and west of the Mississippi are generally laid out in townships approximately six miles square. A Principal Meridian, or true north and south line, is first run by means of Burt's Solar Compass, and then an east and west line, called a Base Line. Parallels to the base line are run at intervals of six miles, and north and south lines at the same intervals. Thus N the tract would be divided into townships exactly six miles square, if it were not _ A for the convergence of the meridians on account of the- - --- curvature of the earth. The north and south lines, or meridians, are - _ __ called Range Lines. The east and west lines, or parallels, are called Town- FIG. 33. ship Lines. Let NS (Fig. 33) represent a principal meridian, and WE a base line; and let the other lines represent meridians and parallels at intervals of six miles. The small squares, A, B, C, etc., will represent townships. A would be designated thus: T. 3 N., R. 2 W.; that is, township three north, range two west; which means that the township is in the third tier north of the base line, and in the second tier west of the principal meridian. B and C, respectively, would be designated thus: T. 4 S., R. 3 W.; and T. 2 N., R. 2 E. 240 SURVEYING. The townships are divided into sections approximately one mile square, and the sections are divided 6 5 4 3 2 1 into quarter-sections. The township,. ---.-. section, and quarter-section corners are 7 8 9 10 11 12 7- - - - - - permanently marked. 18 17 16 15 14 13 - 17 16 1- - The sections are numbered, beginning 19 20 21 22 23 24 at the northeast corner, as in Fig. 34, 30 29 28 27 26 25 which represents a township divided 31 32 33 34 35 36 into sections. The quarter-sections are FIG. 34. designated, according to their position, as N. E., N.W., S. E., and S.W. Every fifth parallel is called a Standard Parallel or Correction Line. Let NS (Fig. 35) represent a principal meridian; WE a base line; rp, etc., meridians; ~N ~and ms the fifth parallel. If m h L k. Op equals six miles, mr will be less than six miles on account of the convergence of the meridians. Surveyors are instructed to make Op such a distance that mr shall be six miles; then mh, hk, etc., are taken similarly. At the correction lines north of W _0 __ _ __ ~E ms the same operation is IV 0.v~ ~repeated. S The township and section FIG. 35. lines are surveyed in such an order as to throw the errors on the north and outer townships and sections. If, in running a line, a navigable stream or a lake more than one mile in length is encountered, it is meandered by PLANE-TABLE SURVEYING. 241 marking the intersection of the line with the bank and running lines from this point along the bank to prominent points which are marked, and the lengths and bearings of the connecting lines recorded. Six principal meridians have been established and connected. In addition to these there are several independent meridians in the Western States and Territories which will in time be connected with each other and with the eastern system. ~ 26. PLANE-TABLE SURVEYING.* After the principal lines of a survey have been determined and plotted, the details of the plot may be filled in by means of the plane-table; or, when a plot only of a tract of land is desired, this instrument affords the most expeditious means of obtaining it. An approved form of the plane-table, as used in the United States Coast and Geodetic Survey, is shown in the plate on page 51. The Table-top is a board of well-seasoned wood, panelled with the grain at right angles to prevent warping, and supported at a convenient height by a Tripod and Levelling Head. The Alidade is a ruler of brass or steel supporting a telescope or sight standards, whose line of sight is parallel to a plane perpendicular to the lower side of the ruler, and embracing its fiducial edge. The Declinatoire consists of a detached rectangular box containing a magnetic needle which moves over an arc of about 10~ on each side of the 0. * In preparing this section the writer has consulted, by permission, the treatise on the plane-table by Mr. E. Hergesheimer, contained in the report for 1880 of the U.S. Coast and Geodetic Survey. 242 SURVEYING. Two spirit levels at right angles are attached to the ruler or to the declinatoire. In some instruments these are replaced by a circular level, or by a detached spring level. The paper upon which the plot is to be made or completed is fastened evenly to the board by clamps, the surplus paper being loosely rolled under the sides of the board. To place the table in position. This operation, which is sometimes called orienting the table, consists in placing the table so that the lines of the plot shall be parallel to the corresponding lines on the ground. This may be accomplished approximately by turning the table until the needle of the declinatoire indicates the same bearing as at a previous station, the edge of the declinatoire coinciding with the same line on the paper at both stations. If, however, the line connecting the station at which the instrument is placed with another station is already plotted, the table may be placed in position accurately by placing it over the station so that the plotted line is by estimation over and in the direction of the line on the ground; then making the edge of the ruler coincide with the plotted line, and turning the board until the line of sight bisects the signal at the other end of the line on the ground. To plot any point. Let ab on the paper represent the line AB on the ground; it is required to plot c, representing C on the ground. 1. By intersection. Place the table in position at A (Fig. 36), plumbing a over A, and ^~~~~C ~making the fiducial edge of the ruler pass through a; turn the alidade about a until the line of sight -.\.^ "bisects the signal at C, and draw a K- /...... — In line along the fiducial edge of the >' I dsI ruler. Place the table in position,'"/ ^ Ad at B, plumbing b over B, and repeat.b ' a b ' a the operation just described. c will A B be the intersection of the two lines FIG. 36. thus drawn. a a THE PLANE-TABLE. PLANE-TABLE SURVEYING. 245 2. By resection. Place the table in position at A (Fig. 37), and draw a line in the direction of C, as in the former case; then remove the instrument to C, place it in position by the line drawn from a, make the edge of the ruler pass through b, and turn the alidade about / b until B is in the line of sight. A b -/ a line drawn along the edge of the ruler will intersect the line from / a in c. / 3. By radiation. ' a _tB Place the table in position at A A (Fig. 38), and draw a line from a FIG. 37. toward C, as in the former cases. Measure AC, and lay off ac to the ~ same scale as ab. To plot a field ABCD..... 1. By radiation. c Set up the table at any point P, b -a and mark p on the paper over P. B 'a Draw indefinite lines from p to- F ward A, B, C..... Measure PA, PB., and lay off pa, pb,....., to a suitable scale, and join a and b, b and c, c and d, etc. 2. By progression. Set up the table at A, and draw a line from a toward B. Measure AB, and plot ab to a suitable scale. Set up the table in position at B, and in like manner determine and plot be, etc. 3. By intersection. Plot one side as a base line. Plot the other corners by the method of intersection, and join. 4. By resection. Plot one side as a base line. Plot the other corners by the method of resection, and join. 246 SURVEYING. The Three Point Problem. Let A, B, C represent three field stations plotted as a, b, c, respectively (Fig. 39); it is required to plot d representing a fourth field station D, visible from A, B, and C. tb //a % ---" —' F 3 FIG. 39. Place the table over D, level and orient approximately by the declinatoire. Determine d by resection as follows: Make the edge of the ruler pass through a and lie in the direction aA, and draw a line along the edge of the ruler. In like manner, draw lines through b toward B and through c toward C. If the table were oriented perfectly these lines would meet at the required point d, but ordinarily they will form the triangle of error, ab, ac, be. In this case, through a, b, and ab; a, c, and ac; and b, c, and be, respectively, draw circles; these circles will intersect in the required point d. For at the required point the sides ab, ac, be must subtend the same angle as at the points ab, ac, be, respectively. Hence, the required point d lies at the intersection of the three circles mentioned. The plane-table may now be oriented accurately. NOTE. The three point problem may be solved by fastening on the board a piece of tracing paper and marking the point d representing D, after which lines are drawn from d toward A, B, and C. The tracing paper is then moved until the lines thus drawn pass through a, b, c, respectively, when by pricking through d the point is determined on the plot below. CHAPTER III. TRIANGULATION.* ~ 27. INTRODUCTORY REMARKS. Geographical positions upon the surface of the earth are commonly determined by systems of triangles which connect a carefully determined base line with the points to be located. Let F (Fig. 40) represent a point whose position with reference to the base line AB is required. Connect AB with F by the series of triangles ABC, ACD, ADE, and DEF, so that a signal at C is visible from A and B, a signal at D visible from A and C, a signal FIG. 40. at E visible from A and D, and a signal at F visible from D and E. In the triangle ABC, the side AB is known, and the angles at A and B may be measured; hence, AC may be computed. In the triangle ACD, AC is known, and the angles at A and C may be measured; hence, AD may be computed. In like manner DE and EF or DF may be determined. DF, or some suitable line connected with DF, may be measured, and this result compared with the computed value to test the accuracy of the field measurements. * In preparing this chapter the writer has consulted, by permission, recent reports of the United States Coast and Geodetic Survey. 248 SURVEYING. Three orders of triangulation are recognized, viz.: Primary, in which the sides are from 20 to 150 miles in length. Secondary, in which the sides are from 5 to 40 miles in length, and which connect the primary with the tertiary. Tertiary, in which the sides are seldom over 5 miles in length, and which bring the survey down to such dimensions as to admit of the minor details being filled in by the compass and plane-table. ~ 28. THE MEASUREMENT OF BASE LINES. Base lines should be measured with a degree of accuracy corresponding to their importance. Suitable ground must be selected and cleared of all obstructions. Each extremity of the line may be marked by cross lines on the head of a copper tack driven into a stub which is sunk to the surface of the ground. Poles are set up in line about half a mile apart, the alignment being controlled by a transit placed over one end of the line. The preliminary measurement may be made with an iron wire about one-eighth of an inch in diameter and 60m in length. In measuring, the wire is brought into line by means of a transit set up in line not more than one-fourth of a mile in the rear. The end of each 60m is marked with pencil lines on a wooden bench whose legs are thrust into the ground after its position has been approximately determined. If the last measurement exceeds or falls short of the extremity of the line, the difference may be measured with the 20m chain. The final measurement is made with the base apparatus, which consists of bars 6m long, which are supported upon trestles when in use. These bars are placed end to end, and brought to a horizontal position, if this can be quickly accomplished; if not, the angle of inclination is taken by a sector, or a vertical offset is measured with the aid of a transit, so that the exact horizontal distance can be computed. MEASUREMENT OF ANGLES. 249 A thermometer is attached to each bar, so that the temperature of the bar may be noted and a correction for temperature applied. The method of measuring lines varies according to the required degree of accuracy in any particular case, but the brief description given above will give the student a general idea of the methods employed. ~ 29. THE MEASUREMENT OF ANGLES. Angles are measured by the transit with much greater accuracy than by the compass, since the reading of the plates of the transit is taken to minutes, and by means of microscopes to seconds, while the reading of the needle of the compass is to quarter or half-quarter degrees. In order to eliminate errors of observation, and errors arising from imperfect graduation of the circles, a large number of readings is made and their mean taken. Two methods are in use; viz., repetition and series. The method of repetition consists, essentially, in measuring the angles about a point singly, then taking two adjacent angles as a single angle, then three, etc.; thus "closing the horizon," or measuring the whole angular magnitude about a point in several different ways. The method of series consists, essentially, in taking the readings of an angle with the circle or limb of the transit in one position, then turning the circle through an arc and taking the readings of the same circle again, etc.; thus reading the angle from successive portions of the graduated circle. On account of the curvature of the earth, the sum of the three angles of a triangle upon its surface exceeds 180~. This spherical excess, as it is called, becomes appreciable only when the sides of the triangle are about 5 miles in length. To determine the angles of the rectilinear triangle having the same vertices, one-third of the spherical excess is deducted from each spherical angle. CHAPTER IV. LEVELLING. ~ 30. DEFINITIONS, CURVATURE, AND REFRACTION. A Level Surface is a surface parallel with the surface of still water; and is, therefore, slightly curved, owing to the spheroidal shape of the earth. A Level Line is a line in a level surface. Levelling is the process of finding the difference of level of two places, or the distance of one place above or below a level line through another place. The Line of Apparent Level of a place is a tangent to the level line at that place. Hence, the line of apparent level is perpendicular to the plumb-line. The Correction for Curvature is the deviation of the line of apparent level from the level line for any distance. Let t (Fig. 41) represent the line of apparent level of the place P, a the level line, d the diamet P ter of the earth; then c represents the coda >^ correction for curvature. To compute the correction for curvature: |.f \d 4 +t2= c(c+d). (Geom. ~ 348.) t2 a2 Therefore, c -- - c-d d approximately, since c is very small FiG. 41. compared with d, and t =a without FIG. 41. appreciable error. Since' d is constant (= 7920 miles, nearly), the correction for curvature varies as the square of the distance. THE LEVELLING ROD. 251 EXAMPLE. What is the correction for curvature for 1 mile? By substituting in the formula deduced above, a2 12 c- 92 mi. -= 8 in. d 7920 Hence, the correction for curvature for any i distance may be found in inches, approximately, by multiplying 8 by the square of the distance expressed in miles. NOTE. The effect of curvature is to make an object 6 appear lower than it really is; and the effect of refraction of light, caused by the greater density of the atmosphere 4 near the surface of the earth, is to make an object appear 3 higher than it really is. When both effects are taken into 2 a2 I account c is more correctly expressed by c = 5 of - ' ~ 31. THE Y LEVEL. 8 This instrument is shown on page 61. The telescope is about 20 inches in length, and rests on supports called Y's, fromi their shape. The spirit level is underneath the telescope, and attached to it. The levelling-head and tripod are similar to the same parts of the transit. ~ 32. THE LEVELLING ROD. The two ends of the Philadelphia levelling rod are shown in Fig. 42. The rod is made of two 9 pieces of wood, sliding upon each other, and held together in any position by a clamp. The front surface of the rod is graduated to 5 hundredths of a foot up to 7 feet. If a greater height than 7 feet is desired, the back part of the 2 rod is moved up until the target is at the required 1 height. When the rod is extended to full length, the front surface of the rear half reads from 7 to FIG. 42. 13 feet, so that the rod becomes a self-reading rod 13 feet long. 252 SURVEYING. The target slides along the front of the rod, and is held in place by two springs which press upon the sides of the rod. It has a square opening at the centre, through which the division line of the rod opposite to the horizontal line of the target may be seen. It carries a vernier by which heights may be read to thousandths of a foot (~ 7). ~ 33. DIFFERENCE OF LEVEL. To find the difference of level between two places visible from an intermediate place, when the difference of level does not exceed 13 feet. Let A and B (Fig. 43) represent the two places. Set the Y level at a station equally distant, or nearly so, from A and A' 4... 'B' B FIG. 43. B, but not necessarily on the line AB. Place the legs of the tripod firmly in the ground, and level over each opposite pair of levelling screws, successively. Let the rodman hold the levelling rod vertically at A. Bring the telescope to bear upon the rod (~ 8), and by signal direct the rodman to move the target until its horizontal line is in the line of apparent level of the telescope. Let the rodman now record the height AA' of the target. In like manner find BB'. The difference between AA' and BB' will be the difference of level required. If the instrument be equally distant from A and B, or nearly so, the curvature and the refraction on the two sides of the instrument balance, and no correction for curvature or refraction will be necessary. THE Y LEVEL. DIFFERENCE OF LEVEL. 255 If the instrument be set up at one station, and the rod at the other, the difference between the heights of the optical axis of the telescope and the target, corrected for curvature and refraction, will be the difference of level required. To find the difference of level of two places, one of which cannot be seen from the other, and both invisible from the same place; or, when the two places differ considerably in level. Let A and D (Fig. 44) represent the two places. Place the level midway between A and some intermediate station B. A"l FIG. 44. Find AA' and BB', as in the preceding case, and record the former as a back-sight and the latter as a fore-sight. Select another intermediate station C, and in like manner find the back-sight BB" and the fore-sight CC'; and so continue until the place D is reached. The difference between the sum of the fore-sights and the sum of the back-sights will be the difference of level required. For, the sum of the fore-sights = BB'+ CC' + DD' = BB"+ B'B"f+ CCG"+ C'C"+ DD'. The sum of the back-sights =AA'+BB"+ CC". Hence, the difference = B'B" + C'C" + DD' -AA' = A'A" - AA'= AA". 256 SURVEYING. ~ 34. LEVELLING FOR SECTION. The intersection of a vertical plane with the surface of the earth is called a Section or Profile. The term profile, however, usually designates the Plot or representation of the section on paper. Levelling for Section is levelling to obtain the data necessary for making a profile or plot of any required section. A profile is made for the purpose of exhibiting in a single view the inequalities of the surface of the ground for great distances along the line of some proposed improvement, such as a railroad, canal, or ditch, and thus facilitating the establishment of the proper grades. The data necessary for making a profile of any required section are, the heights of its different points above some assumed horizontal plane, called the Datum Plane, together with their horizontal distances apart or from the beginning of the section. The position of the datum plane is fixed with reference to some permanent object near the beginning of the section, called a Bench Mark, and, in order to avoid negative heights, is assumed at such a distance below this mark that all th6 points of the section shall be above it. The heights of the different points of the section above the datum plane are determined by means of the level and levelling-rod; and the horizontal length of the section is measured with an engineer's chain or tape, and divided into equal parts, one hundred feet in length, called Stations, marked by stakes numbered 0, 1, 2, 3, etc. Where the ground is very irregular, it may be necessary, besides taking sights at the regular stakes, to take occasional sights at points between them. If, for instance, at a point sixty feet in advance of stake 8 there is a sudden rise or fall in the surface, the height of this point would be determined and recorded as at stake 8.60. LEVELLING FOR SECTION. 257 The readings of the rod are ordinarily taken to the nearest tenth of a foot, except on' bench marks and points called turning points, where they are taken to thousandths of a foot. - A Turning Point is a point on which the last sight is taken just. before changing the position of the level, and the first sight from the o new position of the instrument. A turning point may be coincident with one of the stakes, but must always --- be a hard point, so that the foot of ----- the rod may stand at the same level for both readings. to To explain the method of obtaining the field notes necessary for ---- - making a profile, let 0, 1, 2, 3,..... 11 (Fig. 45) represent a portion of a section to be levelled and plotted. Establish a bench mark at or near the beginning of the line, measure the horizontal length of the section, and set stakes one hundred feet apart, o\ numbering them 0, 1, 2, 3, etc. Place the level at some point, as between 2 co and 3, and take the reading of the rod on the bench =4.832. Let PP' rep- resent the datum plane, say 15 feet --- below the bench mark, then 15 + 4.832 = 19.832 1 - /. will be the height of the line of sight AB, called the Height of the Instrument, above the datum plane. Now take the reading at 0 -5.2 = OA, and subtract the same from 19.832, which 258 SURVEYING. leaves 14.6 = OP, the height of the point 0 above the datum plane. Next take sights at 1, 2, 3, 3.40, and 4 equal respectively to 3.7, 3.0, 5.1, 4.8, and 8.3, and subtract the same from 19.832; the remainders 16.1, 16.8, 14.7, 15.0, and 11.5 will be the respective heights of the points 1, 2, 3, 3.40, and 4. Then, as it will be necessary to change the position of the instrument, select a point in the neighborhood of 4 suitable as a turning point (t.p. in the figure), and take a careful reading on it =8.480; subtract this from 19.832, and the remainder, 11.352, will be the height of the turning point. Now carry the instrument forward to a new position, as between 5 and 6, shown in the figure, while the rodman remains at t.p.; take a second reading on t.p. = 4.102, and add it to 11.352, the height of t.p. above PP'; the sum 15.454 will be the height of the instrument CD in its new position. Now take sight upon 5, 6, and 7, equal respectively to 4.9, 2.8, and 0.904; subtract these sights from 15.454, and the results 10.6, 12.7, and 14.550 will be the heights of the points 5, 6, and 7 respectively. The point 7, being suitable, is made a turning point, and the instrument is moved forward to a point between 9 and 10. The sight at 7=6.870 added to the height of 7 gives 21.420 as the height of the instrument EF in its new position. The readings at 8, 9, 10, and 11, which are respectively 5.4, 3.6, 5.8, and 9.0, subtracted from 21.420, will give the heights of these points, namely, 16.0, 17.8, 15.6, and 12.4. Proceed in like manner until the entire section is levelled, establishing bench marks at intervals along the line to serve as reference points for future operations. As wind and bright sunshine affect the accuracy of levelling, for careful work a calm and cloudy day should be chosen; and great pains be taken to hold the rod vertical and to manipulate the level properly. A record of the work described above is kept as follows: LEVELLING FOR SECTION. 259 STATION. + S. H.I. -S. H.S. REMARKS. B 4.832 15. Bench on rock 20 ft. 0 19.832 5.2 14.6 south of 0. 1 3.7 16.1 2 3.0 16.8 3 5.1 14.7 3 to 3.40 turnpike road. 3.40 4.8 15.0 4 8.3 11.5 t.p. 4.102 8.480 11.352 5 15.454 4.9 10.6 6 2.8 12.7 7 6.870 0.904 14.550 8 21.420 5.4 16.0 9 3.6 17.8 10 5.8 15.6 11 9.0 12.4 B Bench on oak stump 12 27 ft. N.E. of 12, etc. etc. The first column contains the numbers or names of all the points on which sights are taken. The second column contains the sight taken on the first bench mark, and the sight on each turning point taken immediately after the instrument has been moved to a new position. These are called Plus Sights (+ S.) because they are added to the heights of the points on which they 'are taken to obtain the height of the instrument given in the third column (H..). The fourth column contains all the readings except those recorded in the second column. These are called Minus Sights (- S.) because they are subtracted from the numbers in the third column to obtain all the numbers in the fifth column except the first, which is the assumed depth of the datum plane below the bench. The fifth column (H.S., height of surface) contains the required heights of all the points of the section named in the first column together with the heights of all benches and turning points. 260 SURVEYING. To find the difference of level between any two points of the section, we have only to take the difference between the numbers in the fifth column opposite these points. The real field notes are contained in the first, second, fourth, and last columns; the other columns may be filled after the field operations are completed. The field book may contain other columns; one for height of grade (H. G.), another for depth of cut (C.) and another for height of embankment or fill (F.). To plot the section. Draw a line PP' (Fig. 45), to represent the datum plane, and beginning at some point as P, lay off the distances 100, 200, 300, 340, 400 feet, etc., to the right, using some convenient scale, say 200 feet to the inch. At these points of division erect perpendiculars equal in length to the height of the points 0, 1, 2, 3.40, 4, etc., given in the fifth column of the above field notes, using in this case a larger scale, say 20 feet to the inch. Through the extremities of these perpendiculars draw the irregular line 0,1, 2, 3..... 11, and the result, with some explanatory figures, will be the required plot or profile. The making of a profile is much simplified by the use of profile paper, which may be had by the yard or roll. If a horizontal plot is required, the bearings of the different portions of the section must be taken. A plot should be made, if it will assist in properly understanding the field work, or if it is desirable for future reference in connection with the field notes. ~ 35. SUBSTITUTES FOR THE Y LEVEL. For many purposes not requiring accuracy, the following simple instruments in connection with a graduated rod will be found sufficient. The Plumb Level (Fig. 46) consists of two pieces of wood joined at right angles. A straight line is drawn on the SUBSTITUTES FOR THE Y LEVEL. 261 upright perpendicular to the upper edge of the crosshead. The instrument is fastened to a support by a screw through the centre of the cross-head. The upper edge of the crosshead is brought to a level by making the line on the upright coincide with a plumb-line. FIG. 46. FIG. 47. FIG. 48. A modified form is shown in Fig. 47. A carpenter's square is supported by a post, the top of which is split or sawed so as to receive the longer arm. The shorter arm is made vertical by a plumb-line which brings the longer arm to a level. The Water Level is shown in Fig. 48. The upright tubes are of glass, cemented into a connecting tube of any suitable material. The whole is nearly filled with water, and supported at a convenient height. The surface of the water in the uprights determines the level. By sighting along the upper surface of the block in which the Spirit Level is mounted for the use of mechanics, a level line may be obtained. EXERCISE V. 1. Find the difference of level of two places from the following field notes: back-sights, 5.2, 6.8, and 4.0; fore-sights, 8.1, 9.5, and 7.9. 2. Write the proper numbers in the third and fifth columns of the following table of field notes, and make a profile of the section: 262 SURVEYING. STATION. + S. H.I. -S. H.S. REMARKS. B 6.944 20 Bench on post 22 ft. 0 7.4 north of 0. 1 5.6 2 3.9 3 4.6 t.p. 3.855 5.513 4 4.9 5 3.5 6 1.2 3. Stake 0 of the following notes stands at the lowest point of a pond to be drained into a creek; stake 10 stands at the edge of the bank, and 10.25 at the bottom of the creek. Make a profile, draw the grade line through 0 and 10.25, and fill out the columns H. G. and C., the former to show the height of grade line above the datum, and the latter, the depth of cut at the several stakes necessary to construct the drain. STATION. + S. H.I. -S. H.S. H.G. C. REMARKS. B 6.000 25 Bench on rock 0 10.2 20.8 0.0 30 feet west of 1 5.3 5.3 stake 1. 2 4.6 3 4.0 4 6.8 5 4.572 7.090 6 3.9 7 2.0 8 4.9 9 4.3 10 4.5 10.25 11.8 Horizontal scale, 2 ch. = 1 in. Vertical scale, 20 ft. = 1 in. TOPOGRAPHICAL LEVELLING. 263 ~ 36. TOPOGRAPHICAL LEVELLING. The principal object of topographical surveying is to show the contour of the ground. This operation, called topographical levelling, is performed by representing on paper the curved lines in which parallel horizontal planes at uniform distances apart would meet the surface. It is evident that all points in the intersection of a horizontal plane with the surface of the ground are at the same level. Hence, it is only necessary to find points at the same level, and join these to determine a line of intersection. The method commonly employed will be understood by a reference to Fig. 49. The ground ABCD is divided into equal squares, and a numbered stake driven at each intersection. By means of a level and levelling rod the heights of the other stations above m and D, the lowest stations, are determined. A plot of the ground with the intersecting lines is then drawn, and the height of each station written as in the figure. D r C Suppose that the horizontal 5 planes are 2 feet apart; if the first passes through m and D, / / the second will pass through p, which is 2 feet above m; and since n is 3 feet above m, the p n 5 second plane will cut the line mn in a point s determined by the proportion mn: ms:: 3 2. __ o 3s In like manner the points t, q, A. 4t E FIo. 49. and r are determined. The irregular line tsp..... qr represents the intersection of the second horizontal plane with the surface of the ground. In like manner the intersections of the planes, respectively, 4, 6, and 8 feet above m are traced. The more rapid the change in level the nearer these lines will approach each other. CHAPTER V. RAILROAD SURVEYING. ~ 37. GENERAL REMARKS. When the general route of a railroad has been determined, a middle surface line is run with the transit. A profile of this line is determined, as in ~ 34. The levelling stations are commonly 1 chain (100 feet) apart. Places of different level are connected by a gradient line, which intersects the perpendiculars to the datum line at the levelling stations in points determined by simple proportion. Hence, the distance of each levelling station, above or below the level or gradient line which represents the position of the road bed, is known. ~ 38. CRoss SECTION WORK. A C' f D' B II I I _ '_, _ _, A' C f' D B' FIG. 50. Excavations. If the road bed lies below the surface, an excavation is made. Let A CDB (Fig. 50) represent a cross section of an excavation, f a point in the middle surface line, f' the corresponding point in the road bed, and CD the width of the excavation at the bottom. The slopes at the sides are commonly made so that AA! = A'C, and BB' = DB'. ff' and CD being known, the points A, B, C', and D' are readily determined by a level and tape measure. RAILROAD CURVES. 265 If from the area of the trapezoid ABB'A' the areas of the triangles AA'C and BB'D be deducted, the remainder will be the area of the cross section. In like manner the cross section at the next station may be determined. These two cross sections will be the bases of a frustum of a quadrangular pyramid whose volume will be the amount of the excavation, approximately. Embankments. If the road bed lies above the surface, an embankment is made, the cross section of which is like that of the excavation, but inverted. A' C f' D B' I ---A |-a - I I A c' FIG. 51. Fig. 51 represents the cross section of an embankment which is lettered so as to show its relation to Fig. 50. ~ 39. RAILROAD CURVES. When it is necessary to change the direction of a railroad, it is done gradually by a A curve, usually the arc of a circle. Let AF and AO (Fig. 52) represent two lines to B be thus connected. Take / any convenient distance F / AB =AE- t. The intersection of the perpendicu-,,e lars BC and EC deter- FIG. 52. mines the centre C, and the radius of curvature BC =-r. The length of the radius of curvature will depend on 266 SURVEYING. the angle A and the tangent AB. For, in the right triangle ABC, BC r tanBAC =, ortan A =AB t Hence, r = t tan -A. The degree of a railroad curve is the angle subtended at the centre of the curve by a chord of 100 feet. If D is the degree of a curve and r its radius, 50 sin D =- and r =50 csc ~D. For example, a 6~ curve has a radius of 955.37 feet. To Lay out the Curve. First Method. Le m/ B y'B ' I \ I C FIG. 53. m P B I I/. I \ I C FIG. 54. tt Bm (Fig. 53) represent a portion of the tangent. It is required to find mP, the perpendicular to the tangent meeting the curve at P. m.P = Bn CB- Cn. CD= r, But and CnCp = V -p V —r2- t2. Hence, mP- = r- Vr- t2. Second Method. It is required to find nP (Fig. 54) in the direction of the centre. mP=mC- PC. But m C= VB-C2 + 2 = Vr2 + t2. Hence, mP = Vr + t2- r. RAILROAD CURVES. 267 Third Method. Place transits at B and E (Fig. 55). Direct the telescope of the former to E, and of the latter to A. A Turn each toward the curve the same number of degrees, and mark P, the point of B - -- -9 intersection of the lines of / sight. P will be a point in FIG. 5 the circle to which AB and AE are tangents at B and E, respectively. Fourth Method. If the degree D of the curve is given and the tangent BA at B (Fig. 56), place the transit at B and direct toward A. Turn off successively the angles ABP, PBP', P'BP",..... each B, equal to ~D, and take DP, PP', PP",..... each 100 ft., the length of the tape. Then P, P', P",..... lie on the FIG. 56. required curve. If the angle A and the tangent distance BA = t are given, D can be found from the formulas sin 50 tan, n = c50 sin ~ D -- r=ttan~A,.. sin -D --- cot [A. r t \% TRANSIT WITH SOLAR ATTACHMENT. The circles shown in the cut are intended to represent in miniature circles supposed to be drawn upon the concave surface of the heavens. ANSWERS. PLANE TRIGONOMETRY. EXERCISE I. Z 55it 137~ it 1. 600~= 450 -, 150~=- O- 1950= -, 110 15 -3 4 6 1. 21 123~ 45' =, 37~ 30' -- 16 24 27It 3i t it 15 Z 7it 2. = 1200, 4 = 135, = 112~ 30', 16 168~ 45', 5= 840. 3 4 8 11 15 3. 1~ = 0.0174533-radian. 1' = 0.00029089 radian. 4. 1 radian = 206265". 7. 14~ 27' 27". 10. 3 hr. 49 min. 11 sec. 5. 3- 5 8. 69.167 miles. 11. 9 ft. 2 in. 4 6 6. 110 27' 33". 9. 57 ft. 3.55 in. 12. T sec. EXERCISE II. b a b a c c 1. sin B= -, cos.B=- tanB= -- cotB= - secB — csc B=c c a b a b 3. (i.)sin =, os =, (.) sin = ( etc. (v.) sin = 39, etc. tan = -, cot = 4 (iii.) sill= 8, etc. (vi.) sin = -}9, etc. sec = a, csc =, (iv.) sin =, etc. 4. The required condition is that a2 + b2 = c2. It is. 5. (i.) sin= 2+ n etc. (iii.) sin=, etc. - m2 + n2 s (ii.) sin =, x etc. (iv.) sin = — etc. x2 + y2 qr 7. In (iii.) p2q2 + q2S2 = p2s2; in (iv.) m2C2s2 + m72p2V2 = n2q2r2. 8. c = 145; whence, sin A -= -25 = cos B; cos A = 4 = sin B; tan A = 4 = cot B; cot A = -124 — tan B; sec A =4 - cscB; etc. IT~~~~~~~~~~; csc B;4 1et3. 2 TRIGONOMETRY. 9. b = 0.023; whence, tan A = cot B = —; cot A = tan =, etc. 10. a = 16.8; whence, sin A = 1-6 =- cos B, etc. p2 + q~2 11. c = p + q; whence, sin A = cos B; etc. 12. b = Vq (p + q); whence, tan A = -r = cot B; etc. 13. a = p - q; whence, sinA= = cos B; etc. 14. sin A = 2 V5 = 0.89443; etc. 15. sin A = -; etc. 16. sin A = 8 (5 + /7) = 0.95572; etc. 17. cos A = (V31 — 1) = 0.57097; sin A = (V + 1) = 0.82097; etc. 18. a= 12.3. 20. a= 9. 22. c= 40. 19. b = 1.54. 21. b = 68. 23. c= 229.62. 24. Construct a it. A with legs equal to 3 and 2 respectively; then construct a similar A with hypotenuse equal to 6. In like manner, 25, 26, 27, may be solved. 28. a = 1.5 miles; b = 2 miles. 31. 400,000 miles. 30. a = 0.342, b = 0.940; a = 1.368, b = 3.760. 32. 142.926 yards. EXERCISE III. 5. Through A (Fig. 3) draw a tangent, and take A T = 3; the angle A OT is the required angle. 6. From 0 ( Fig. 3) as a centre, with a radius = 2, describe an arc cutting at T the tangent drawn through B; the angle AOT is the required angle. 7. In Fig. 3, take OM= i, and erect MP 1 OA and intersecting the circumference at P; the angle POM is the required angle. 8. Since sin x = cos x, OM = PM (Fig. 3), and x = 45~; hence, construct x = 45~. 9. Construct a rt. A with one leg = twice the other; the angle opposite the longer leg is the required angle. 10. Divide OA (Fig. 3) into four equal parts; at the first point of division from O erect a perpendicular to meet the circumference at some point P. Join OP; the angle A OP is the required angle. 21. r sin x. 22. Leg adjacent to A = nc, leg opposite to A = mc. ANSWERS. 3 EXERCISE IV. 1. cos 60~. cot 1~. sec 71~ 50'. tan 7~ 41'. sin 45~. tan 75~. sin 52~ 36'. sec 35~ 14'. 2. cos 30~. cot 33~. sec 20~ 58'. tan 0~ 1'. sin 15~. tan 6~. sin 4~ 21'. sec 44~ 59'. 3. ~\/3 4. tan A = cotA = cot (90~ - A); hence, A = 90~ - A and A = 45~. 5. 30~. 7. 90~. 9. 22~ 30'. 11. 10~. 6. 30~. 8. 60~. 10. 18~. 900 12. 9 n+l EXERCISE VI. 1. cos A = -- tan A= 1, cot A=5 sec A-= 13, csc A=. 2. cosA=0.6, tanA=1.3333, cotA=0.75, secA=1.6667, cscA=1.25. 3. sin A= 1, tan A= 11 cotA= 6, secA= 6, cscA= 6. 6 1 60, S A 61 1 4. sin A=0.96, tan A=3.4285, cot A=0.29167, sec A=3.5714. 5. sin A=0.8, cos A=0.6, cotA=0.75, secA=1.6667, cscA=1.25. 6. sin A= /2, cos A = 2, tanA=l, secA= /2, cscA= 2. 7. tanA=2, sin A=0.90, cos A=0.45, sec A=2.22, cscA=l.ll. 8. cosA=, sin A=i/3, tanA= /3, cot A= V3, cscA= ft/3 9. sinA=~V/2, cosA=~ /2, tanA=l, cotA=l, secA=/2. m 1 10. cos A= /l-m2, tan A= - / - n2,m2 cotA= Vl1- 2. 1 - m2 m 1 - m2 2m 1 — m2 1 + m_ 2 11. cosA= 2 tan A=.2 cotA=, secA= 1 t+ m2 1 —m2 2m 1 —m2,2 - n2 2 - n2 m2 + n2 12. sin A=- - tan A= ~,secA= m2 + n2 2mnn 2mn 13. cot = 1, sin = /2, cos -= /2, sec = /2, csc =. 14. cos = - V 3, tan = -V, cot = V, sec = 2 V, csc = 2. 15. sin = i V3, cos =, tan = 3, cot = 3, sec = 2. 16. sin = ~V2-/3, cos = -V2 + 3, cot = 2 + 3. 17. sin = V2- -/2, cos= A/2 + V2, tan= 2 — 1. 18. cos = 1, tan = 0, cot = oo, sec = 1, csc = oo. 19. cos = 0, tan = oo, cot = 0, sec = o, csc = 1. 20. sin = 1, cos = 0, cot= 0, sec = oo, csc = 1. 4 TRIGONOMETRY. 21. cos A = Vi - sin2A, 22. sin A = V -- cos2A, sec A - cos cos A sinA 1 tan A=,1 — csc A= -- /1 - sin2 A sin A /1 - COS2 cos2A tan --, cot A — o cosA V 1-cos2 A 1 csc A = V/ - cos2A. tan A 1 1 23. sin A = cos A = A, cotA = q- 1+ ttaifA VI l+tan2A tanA sec A = %/1 + tan2A-, csc A = - + tan A 24. tanA = — ct csc A = +cot2A, sin A = 1 + cot cotA V/1 cot2 A cos A = - sec A = — V/1 + cot2 A cotA 25. sinA= V5, cosA= V5. 27. sinA=-T, cosA =. 26. sin A = V/5, tan A = 15. 281 -cos A + 3 cos4 A cos2 A - cos4 A EXERCISE VII. 1. x=45~. 2. x= 30~. 3. x = 0, or 60~. 4. x =45~. 5. x= 60~. 6. x = 45. 7. x =45~. 8.x = 450. 9. = 60~. 10. xa=0~. 11. = 30~. 12. x = 45~. 13. x = 0~, or 60. 14. x= 30~. 15. x = 30~, or 45~. 16. x = 45. 17. x = 600. EXERCISE YIII. b b 1. -=cosA;'.c= bc cos A a a 2. - = sin A;. c = c sin A 3. -= cos A;... b =c cos A. b b 4. -=cosA;.. c= c cos A 5. A = 90~ - B, a = c cos B, b = csin B. 6. A =90~-B, a = b cot B, b sin B 7. A =90~-B, b = a tan B, a cos B 8. cos A = - C B = 900 -A, a = c2 - b2. ANSWERS. 5 EXERCISE IX. 31. c =7.8112, 32. b= 69.997, 33. a =1.1886, 34. b =21.249, 35. a=6.6882, 36. a = 63.859, 37. a = 19.40, 38. b = 53.719, 39. a =12.981, 40. a00.58046, A = 39~ 48', A = 30' 12", A = 430 20', c = 22.372, c= 13.738, b= 23.369, b= 18.778, c =71.377, c = 15.796, b - 8.442, B = 50~ 12', F= 15. B = 89~ 29' 48", F = 21.525. B = 46~ 40', F 0.74876. B = 71~ 46', F = 74.371. B = 60~ 52', F = 40.129. B = 20~ 6', F = 746.15. A =45~ 56', F= 182.15. A = 41 11', F= 1262.4. A = 550 16', F= 58.416. A= 3~ 56', F = 2.4501. 43. F=- (b2tanA). 44. F = i (aVc2 - a2). 41. F = ~ (c2 sin A cos A). 42. F = - (a2 cot A). 45. b= 11.6, c —15.315, A= 40~ 45' 48", B = 49~ 14' 12". 46. a= 7.2, c= 8.7658, B = 34~ 46' 40" A = 55~ 13' 20". 47. a= 3.6474, b = 6.58, c= 7.5233, B = 61~. 48. a= 10.283, b= 19.449, A 27~ 52', B 62~ 8'. 49. 190 28' 17" and 70~ 31' 43". 7 A 59 44' 35". 50. 3 and 5.1961.. 58. 95.34. 900 51. a = c cos -- 59. 1~ 25' 56". n + 1 900 60. 7.0712 miles in each direction. b - c sin n + 1 61. 20.88 feet. 52. 36~ 52' 12" and 53~ 7' 48". 62. 56.65 feet. 53. 212.1 feet. 63. 228.63 yards. 54. 732.22 feet. 64. 136.6 feet. 55. 3270 feet. 65. 140 feet. 56. 37.3 feet, 96 feet. 66. 84.74 feet. 1. C=2 (90~-A), 2.A = (1800- C), 3. C=2 (9C~-A), EXERCISE X. c = 2 a cos A, c = 2 a cos A, a = c - 2 cosA, h = a sin A. h = a sin A. h= asinA. 6 TRIGONOMETRY. 4. A= (180~- C), 5. C=2 (900-A), 6. A = 2 (180~- C), 7. sin A = - a, 8. tanA= h i c, 9. A = 67~ 22' 50", 10. c = 0.21943, 11. a = 2.0555, 12. a = 7.706, 13. A = 79~ 36' 30", 14. A = 77~ 19' 11", 15. A =25~ 28', 16. A = 81~12' 9", 17. F= c C4 a2 c2. a= c 2 cosA, a = h. sin A, a = h sin A, C = 2 (90~-A), C =2 (90 - A), C = 45 14' 20", h = 0.27384, h= 1.6852, c = 3.6676, C = 20~ 47', C = 25 21' 38", C = 129 4', C = 17~ 35' 42", h = a sin A. c = 2 a cosA. c= 2 acosA. c = 2 acosA. a = h - sinA. h= 13.2. F = 0.03004. F= 1.9819. F= 13.725. c = 2.4206. a= 20.5. a = 81.40, h = 35. a= 17, c= 5.2. 22. 0.76537. 23. 94~ 20'. 24. 2.7261. 25. 380 56'33". 26. 37.699. 18. F= a2sin Ccos C. 19. F= a2 sin A cos A. 20. F= h2tan -C. 21. 28.284 feet, 4525.44 sq. feet. EXERCISE XI. 1. r= 1.618, 2. r=11.269, 3. h = 0.9848, 4. h=19.754, 5. r= 1.0824, 6. r=2.592, 7. r= 1.5994, 8. 0.6181. 9. 0.64984. 10. 0.51764. 11. b= 0 2 cos f h = 1.5388, F= 7.694. h = 10.886, F= 381.04. p = 6.2514, F= 3.0782. c = 6.2572, F= 1236. c = 0.82842, F= 3.3137. h = 2.4882, c = 1.4615. h=1.441, p=9.716. 12. 0.2238. 1 13. 0.31. 1 14. 0.82842. 1 15. 94.63. 2 16. 415..7. 11.636. L8. 99.64. L9. 1.0235. 0. 0.635. ANSWERS. EXERCISE XII. 5. Two angles: one in Quadrant I., the other in Quadrant II. 6. Four values: two in Quadrant I., two in Quadrant IV. 7. x may have two values in the first case, and one value in each of the other cases. 8. If cos x =-, x is between 90~ and 270~; if cot x = 4, x is between 0~ and 90~ or 180~ and 270~; if sec x = 80, x is between 0~ and 90~ or between 270~ and 360~; if csc x = - 3, x is between 180~ and 360~. 9. In Quadrant III.; in Quadrant II.; in Quadrant III. 10. 40 angles; 20 positive and 20 negative. 11. +, when x is known to be in Quadrant I. or IV.; -, when x is known to be in Quadrant II. or III. 14. sin = — V3, tan x =-4 V3, cot x = — /3, csc x = — /3. 15. sin x = ~, cos x = q V10, tan x = —, sec x = T V10. csc = V10. 16. The cosine, the tangent, the cotangent, and the secant are negative when the angle is obtuse. 17. Sine and cosecant leave it doubtful whether the angle is an acute angle or an obtuse angle; the other functions, if + determine an acute angle, if - an obtuse angle. 20. sin 450~ = sin (360~+90~)=sin 90~=1; tan 540~ = tan 180~ = 0; cos 630~ = cos 270 = 0; cot 720~ = cot 0 = oo; sin 810~ = sin 90~ = 1; csc 900~ = csc 180~ = oo. 21. 45~, 135~, 225~, 315~. 22. 0. 23. 0. 24. 0. 25. a2- b+ 4 ab. EXERCISE XIII. 2. sin 172~ = sin 8~. 8. sin 204~ = - sin 24~. 3. cos 100~ = -sin 10~. 9. cos 359~= cos.1. 4. tan 125~ = - cot 35~. 10. tan 300~ = - cot 30~. 5. cot 91~ = -tan 1~. 11. cot 264~ = tan 6~ 6. sec 110~ = - csc 20~. 12. sec 244~ = - csc 26~. 7. csc 1570 = csc 23~. 13. csc 271~ = - sec 1~. 8 TRIGONOMETRY. 14. sin 163~ 49' = sin 16~ 11'.:17. cot 139~ 17' = - cot 40~ 43'. 15. cos 195~ 33' = - cos 15~ 33'. 18. sec 299~ 45' = csc 29~ 45'. 16. tan 269~ 15' = cot 0~ 45'. 19. csc 92~ 25' = sec 2~ 25'. 20. sin (- 75) = - sin 75 = - cos 15~, cos(- 75~)= cos 75~= sin 15~, etc. 21. sin (-127~) = sin 127~= -cos 37~, cos (- 127~) = cos 127~ = - sin 37~, etc. 22. sin (- 200~) = sin 160~ = sin 20~, cos (-200~) = cos 200 = -cos20~, etc. 23. sin (- 345~) = - sin 345~ = sin 15~, cos(- 345~)= cos345~=- cos 15~, etc. 24. sin(- 52~ 37') = - sin 52~ 37' - cos 37 23', cos(- 52~ 37') = cos 52~ 37' = sin 37~ 23', etc. 25. sin (- 196~ 54') = - sin 196~ 54' = sin 16~ 54', cos (- 196~ 54') = cos 196~ 54' = - cos 16~ 54', etc. 26. sin 120~= i -3, cos 120~ - 0, etc. 27. sin 135~ = +- i2, cos 135~ -- 2, etc. 28. sin 150~ = + i, cos 150~ = - -/3, etc. 29. sin 210~ = -, cos 210 = - 1- 3, etc. 30. sin 225~ = - -1i/2, cos 225~ = - -7 V2, etc. 31. sin 240~ = - ~ /3, cos 240 = -, etc. 32. sin 300~ = - - 3, cos 300~ = + ~, etc. 33. sin (-30~) =- -, cos (- 30~) = + i 3, etc. 34. sin (- 225~) = + 1 v2, cos (- 225~) =- ^J2, etc. 35. cos x= - 72 or -, etc., x= 2250. 36. tan x - 7, sin x = C, cos x-= - — 3, x = 150~. 37. sin 3540~ = sin 300~ = - sin 60~ = - -/3, cos 3540~ =, ', etc. 38. 210~ and 330~; 120~ and 300~. 39. 135~, 225~, and — 225~; 150~ and -30~. 40. 30~, 150~, 390~, and 510~. 41. sin 168~, cos 334~, tan 225~, cot 252~, sin 349~, cos 240~, tan 64, cot 177~. 42. 0.848. (Hint: tan 238~ = tan 58~, sin 122~= sin 58~.) 43. - 1.952. 45. m sin x cos x. 44. (a -b) sin x. 46. (a - b) cot x - (a + b) tan x. ANSWERS. 9 47. a2 + b2 + 2 ab cos x. 49. cos x sin y- sin x cos y. 48. 0. 50. tan x. 51. Positive between x = 0~ and x = 135~, and between x = 315~ and x = 360~; negative between x = 135~ and x = 315~. 52. Positive between x = 45~ and x = 225~; negative between x = 0~ and x = 45~, and between x = 225~ and x = 360~. 53. sin (x - 90) = - cos, cos (x - 90~) = sin x, etc. 54. sin (x - 180~) =- sin x, cos (x - 180~) =- cos x, etc. Exercises 53 and 54 should be solved by drawing suitable figures, and employing a mode of proof similar to that used in ~ 24. EXERCISE XIV. 1. Si (, 2 c, sin(x ), C. 2. cos, sy. 3. sin ( 90~ + y) = cosy, cos ( 90~ + y) = - sin y, etc. 4. sin (180 - y) = sin y, cos (180~ - y) = - cosy, etc. 5. sin (180~ + y) = - siny, cos (180~ + y) = - cosy, etc. 6. sin (270~ - y) =- cosy, cos (270~ - y) = - siny, etc. 7. sin (270~ + y) = - cosy, cos (270~ + y) sin y, etc. 8. sin (360~ - y) = -- sin y, cos (360~ - y) cos y, etc. 9. sin (360~ + y) -- siny, cos (360~ + y) = cosy, etc. 10. sin (x- 90~) = - cos x, cos (x- 90~) = sin x, etc. 11. sin (x - 180~) = - sin x, cos (x - 180) =- cos x, etc. 12. sin (x - 270) = cos x, cos (x - 270~) =- sin x, etc. 13. sin (- y) = - sin y, cos (- y) = cos y, etc. 14. sin(45~-y)= -/2(cosy-siny), cos(45~-y)= - V2(cosy+siny), etc. 15. sin(45~+y) ~ /2(cosy+siny), cos(45~+y)= V2(cosy-siny), etc. 16. sin(300+y)=-(cosy+ \/3siny), cos(30~+y)=1/V3(cosy-siny), etc. 17. sin(60~-y)= (V33cosy-siny), cos(60 —y)s, (cos(6y+- V3siny), etc. 18. 3 sin x- 4 sin3x. 19. 4cos3x-3cosx. 20. 0. 21. ~ V3. 1 - 0./,\ d 10.9994 22. sin-, x = - 04 0.10051; cos'x = X /I-^41 = 0.99494. 223. 2 23. cos 2 x - tan2x -— 3. 10 TRIGONOMETRY. 24. sin 22~- = /2- V2 = 0.3827, cos 22~o = V2 + V = 0.9239. tan221~ = V2-1 = 0.4142, cot220~ - /2 + 1 = 2.4142. 25. sin 150 = 2 - V3 = 0.25885, cos 150 = V2 + -V = 0.96592. tanl5~ =2 —3 = 0.26799, cot 15~ = 2 + /3 = 3.7321. 27-33. The truth of these equations is to be established by expressing the given functions in terms of the same function of the same angle. Thus, in Example 27, sin 2 x = 2 sin cos x, sin x 1 and 2 tan x = 2 - 1 + tan2x = sec2x = cos X cos2x By making these substitutions in the given equation its truth will be evident. 34. sin A + sin B + sin C = sin A + sin B + sin [180 - (A + B)] = sin A + sin B + sin (A + B) =2sin (A+ B) cos (A-B)+2sin (A+ B)cosa (A +B) = 2 sin ~ (A + B) [cos I (A - B) + cos (A + B)] = 4 sin (A + B) cos cosos B, (see ~~ 30 and 31). But cos ~ C = cos [90~- (A + B)] = sin (A + B). Therefore, sin A + sin B + sin C = 4 cos 2 A cos 1 B cos C. 35. 36. Proof similar to that for 34. sin AcosB cos A sin B sin C tan A +- tan B +r tan C + + tanA+ tanB+tan cos A cos B cos A cos B cos C sin C _ sin sin C sin C cos A cos B sin C cos A cos B cos C cos A cos B os C - (cos A cos B + cos C) sin C _ [cos A cos B - cos (A + B)] sin C cos A cos B cos C cos A cos B cos C sin A sin B sin GC = sin A sin B sin C = tan A tan B tan C. cos A cos B cos C 37. Proof similar to that for 36. 38 2 sin 2 x 39. 2 cot 2x. 40. os (x-y) sin x cos y 41 cos (x + y) sin x cos y 42. tan2 x. 43 cos (x - y) cos x cos y 44. cos (x y) cos x cosy 4 cos ( - y) sin x sin y 46. cos (x y) sin x sin y 47. tan x tany. ANSWERS. 11 EXERCISE XV. 1. sin-'i =60~+2n7t or 120~+2n7t. tan --- =30~ + 2 n 7 or 210~ 2 n t. vers-1 = 60~ + 2 n. cos- -- = 135~ + 2 n or 225~+ 2 n. csc-l2 = 45~ + 2 n or 135~ + 2 n. tan-oo = 90 + 2 n t or 270~+ 2 n. sec-12 = 60~+ 2 n 7. cos-1 ( — 13) = 150~ + 2 n t or 210~ + 2 n 7.. 1 5. 12. -~2}/2. 4. 2 1 10. 13 2V2 I3 8. 00, 900, 1800. 11. ~ 7 13. x=0, or ~ /3. EXERCISE XVI. 1. If, for instance, B = 90~, [25] becomes = sin A. 3. a2=b2+c2, a2=b2+c2-2bc, a = b2 + 2 + 2 bc. 6. 90~ in each case. a-b 7. (i.) a b= tan (A - 45~), and a right triangle. (ii.) a + b = (a - b) (2 + /3), an isosceles triangle with the angles 30~, 30~, 120~. EXERCISE XVII. 9. 300. 10. AB = 59.564 miles. AC = 54.285 miles. 11. 4.6064 miles, 4.4494 miles, 3.7733 miles. 12. 4.1501 and 8.67. 13. 6.1433 miles and 8.7918 miles. 14. 8 and 5.472T. 15. a=5, c = 9.6592. 16. a -7, b- - j73. 1" ues, 600 feet and 1039.2 feet; altitude, 519.6 feet. 18. 855: 1607. 19. 5.438 and 6.857. 20. 15.588. 12 TRIGONOMETRY. EXERCISE XVIII. 12. The other diagonal = 124.617. 11. 420. EXERCISE XIX. 11. 6. 12. 10.392. 14. 8.9212. 15. 25. 16. 3800 yards. 17. 729.68 yards. 18. 10.266. 19. 5.0032 and 2.3385. 20. 26~ 0' 10' and 14~ 5' 50". EXERCISE XX. 11. A = 36~ 52' 12", B 530 7' 48", C = 90~. 16. 45~, 60~, 75~. 12. A =B = 330 33'27", C = 1120 53' 6". 17. 4 23'W. of N., orW. of S. 13. A = B= C= 60~. 18. 60~. 14. Impossible. 20. 0.88877. 15. 45~, 120~, 15~. 21. 54.516 miles. 1. 4333600. 2. 365.68. 3. 13260. 4. 8160. 5. 240. 6. 26208. 7. 15540. 8. 29450 or 6983. EXERCISE XXI. 9. 17.3204 10. 10.3923 11. 0.19952. 12. ab sil A. 13. (a2 - b2) tan A. 14. 2421000. 15. 30~, 30~, 120~. EXERCISE XXII. 1. 21.166 miles; 24.966 miles. 2. 6.3399 miles. 3. 119.29 feet. 4. 30~. 5. 20 feet. 6, 2.6247 or 21.4587 7. 276.14 yards. 8. 383.35 yards. ANSWERS. 13 MISCELLANEOUS EXAMPLES. 2. 106.70 feet; 142.86 feet. 3. 1023.9 feet. 4. 370 34' 5". 5. 238,400 miles. 6. 861,880 miles. 7. 2922.4 miles. 8. 600. 9. 3.2068. 10. 6.6031. 11. 199.56 feet. 12. 43.107 feet. 13. 45 feet. 14. 26~ 34'. 15. 78.367 feet. 16. 75 feet. 17. 1.4446 miles. 18. 3956.2 miles. 19. 56.649 feet. 20. 69.282 feet. 21. 260.20 feet; 3690.3 feet. 22. 1.3438 miles. 23. 235.80 yards. 27. 8 inches. 30. 460.46 feet. 31. 88.936 feet. 32. 13.657 miles. 34. 56.564 feet. 35. 51.595 feet. 36. 101,892 feet. 38. N. 76~ 56' E.; 13.938 miles an hr. 39. 442.11 yards. 40. 255.78 feet. 41. 3121.2 feet; 3633.5 feet. 42. 529.49 feet. 43. 41.411 feet. 44. 234.51 feet. 45. 25.433 miles. 46. 294.69 feet. 47. 12,492.6 feet. 48. 6.3397 miles. 49. 210.44 feet. 51. 757.50 feet. 52. 520.01 yards. 53. 1366.4 feet. 54. 658.36 pounds; 22~ 23' 47" with first force. 55. 88.326 pounds; 45~ 37' 16" with known force. 58. 500.16; 536.27. 59. 345.48 feet. 60. 345.25 yards. 61. 61.23 feet. 63. 307.77. 64. 19.8; 35.7; 44.5. 65. ~ 45~, ~ 135~. 66. cosA = -m 4( 67. sinA=J]'2 - _2 nC - m2 cos B =- ~\-M m \ 1-n2 68. ~ 60~, ~ 120~. 69. 0~, 180~, ~ 60~. 70. 0~, 30~, 180~, 2100. 74. 75. a 180~ 72. r= csc — 2 n a 180~ R=cot2 n 73. I bc sin A. 1 c2 sin A sin B csc(A + B). Vs (s- a) (s -b) (s- c). 14 TRIGONOMETRY. 77. 199 A. 3 R. 8. r. 78. 210 A. 3 R. 26 P. 79. 12 A. 3 R. 36 P. 80. 3 A. O R. 6 r. 81. 12 A. 1 R. 15 P. 82. 4 A. 2. 26 P. 83. 14 A. 2. 9 P. 84. 61 A. 2 R. 85. 4 A. 2 R. 26 P. 86. 13.93, 23.21, 32.50 ch. 87. 9 A. O R. 1 P. 89. 876.34. 90. 1229.5. 92. 1075.3. 93. 2660.4. 94. 16,281. 95. 435.76 sq. ft. 96. 49,088 sq. ft. 97. 750.12 sq. ft. 98. 422.38 sq. ft. 99. 1834.95 sq. ft. 100. 26.87. 103. 6. 108. 6. 110. 6086.4 feet. 111. 5~ 25'S.; 457.5 miles. 112. 460.8 miles; 383.1 miles. 113. 229 miles; lat. 11~ 39' S. 114. S. 560 7' 30" E.; 202.6 miles. 115. N. 170 25'W.; 37~ 46' N. 116. S. 56~ 11'E.; 244.3. 117. 359.87 miles. 121. Long. 68~ 55' W. 122. 103.6 miles. 124. 330 18' N.; 36~ 24' W. 125. N. 280 47'E.; 1293 miles. 126. S. 50~ 40' W.; 250.8; 20~ 9' W. 127. 38~ 21'N.; 55~ 12' W. 128. 171 miles; 32~ 44' W. 129. N. 36~ 52' W.; 36~ 8' W. 130. 173 miles; 51~ 16' S.; 34~ 13' E. 131. S. 50~ 58' E.; 47~ 15' N.; 20~ 49' W. 132. N. 530 20' E., 16~ 7' W.; or N. 53~ 20' W., 25~ 53' W. 133. N. 470 42.5'E., 190 27' N., 121~ 51'E.; or N. 470 42.5' W., 190 27'N., 116~ 9' E.; or S. 47~ 42.5' E., 14~ 33' N., 121~ 48' E.; or S. 470 42.5' W., 14~ 33' N., 116~ 12' E. 134. Lat. 30~, 359.82 miles; lat. 45~, 359.73 miles; lat. 60~, 359.60 miles. 137. N. 72~ 33' E.; 45 miles; 42~ 15' N., 69~ 5' W. 138. N. 72~ 4' W., 287 miles; 32~ 54' S., 13~ 2' E. ANSWERS. PROBLEMS IN GONIOMETRY. [The solutions here given are for angles less than 360~. 79. ~ 2 80. ~V 5-2. 81. ~tVs. 82. ~i, ~4. 83. ~ /2. 84.. -1i V5+1 85.., 4 4 86. x =- t, r. 87. x = 90~, 270~. 88. x = sin-1 \. 2 89. x = 0~, 90~. 90. x = 30~, sin-1 ( — ). 91. x =180~, cos-1-. 92. x = 0, 120~, 180~, 240~. 93. x = 450, 225~, tan- ( — ). 94. x = 00, ~ 60~, ~ 120~, 180~. 95. x = - 45~, 135~, 2 sin-' (2 2 - 2). 96. x = 00, 450, 180~, 2250. 97. x= cos —(~4i ). 98. = 0~, 45~, 90~, 180~, 225~, 270~. 99. = 0~, 180~, ~ sin-1. 100. x = 0, ~ 90~, ~ 120~. 101. x = 0~, ~ 36~, ~ 72~, ~ 108~, ~ 144~, 180~. 102. X = -~ i t, ~-it-. 103. s = 0~, ~ 60~, ~ 1200, 180~. 104. x = tan-1 V2. 105. x= —15~, 105~. 106. x= -2 cot-1 a. 107. xs- ( —a~- a2+ 8 a+ 8). 108. x = 45, 1350,) 108. x= —45~, 135~, sin-1 (1 - a). 109. x = ~ 30~, ~ 60~, ~ 120~, ~ 150~. 110. x= ~ 60~, ~ 90~, ~ 120~. 111. x= ~ 60~, ~ 90~, t 120~. 112. x = 120~. 113. x = 30~, 150~, sin-1l. 114. x = ~ 60~, ~ 90~. 115. x = 00, ~ 20~, ~ 100~, ~140~, t 180~. 116. x = ~ 45~, ~ 90~, ~ 135~. 117. x = ~ 30~, ~ 60~, ~ 90~, ~ 120~, ~ 150~. 118. x = 00, 45~, ~ 90~, 225~. 119. x = ~ 30~, ~ 60~, ~ 120~, ~ 150~. 120. x = ~ 30~, ~ 90~, ~ 150~. 121. x = 0~, 45~, 180~, 225~. 122. x = ~ 45~, ~ 60~, ~ 120~, ~ 135~. 123. x= 0, ~ 45~, ~ 135~. 124. x= ~ 30~, ~ 90~, ~ 150~. 16 TRIGONOMETRY. 125. x = 8~, 168~. 126. x tan-'. 127. x = 30~. 128. x = t 60~, ~ 120~. 129. x= I 30~, ~ 60~, ~ 120~, ~ 150~. 130. x = ~ sin-' -. 1 132. x = tan-1' 5J, -tan-1'. 133. y = - 90~, x indeterminate; x = 45~, y = 0; x= 135~, y = 180~; x = 225~, y = 00; x = 315~, y = 180~. ab / V/a2 b2 - 4 ab 134. x = tan- 2b 2b ab ~ a2 b2 - 4 ab y= tan-' 2 135. x = 45~, 225~. 136. x= ~1, 3~V. 137. x= —' - V 13 V3. 138. xa=W/3. 139. x=i. 140. x=l. 141. x 0, 1, -1. 142. x = ~ /V. 143. x - ' 144. (a + b)'1. 145. (1~ -2n). (-b m2 )(1 F2 m). 146. 8. 147. ~V, 1 V3. 148. 4, -. a + 1 149. V2a+ 1 150. 4. 151. tan (x + y). sin x 152. sm. sin y 153. cot 5 x. 154. tan-i2 x 1 - 2x2 155. 2. 156. cot2x - tan2 x. ENTRANCE EXAMINATION PAPERS. I. 900 6. r sin -, n+ 1 900 r cos- n+ 1 7. 475.27 feet. 4. sin = V2- \2, cos = V2 + V2, 5. (i.) one, (ii.) none, 7. 383.35 yards. II. tan = /2- 1, sec = V4 - 2 /2, cot = /2 + 1, csc = V/4 + 2 V2. (iii.) none, (iv.) two. ANiSWERS. i1 III. 1 2. (a) sin= ~, tan 0 = cot = V3, 2 sec =-, csc 0 = ~ 2. (b) 30~, 90~, 150~, 270~. 6. 161.42, 33~ 34/-5", 990 4' 43"/ 7. 69.812 yds. IV. 6. 230.03 feet. 7. A = 37~ 24' 58", B = 51~ 37' 52", C = 900 57' 10". V. 1. 17- years. 2. sin 2x = ~ m, tan 2 x = ~ /1 - m2 3. x = 210~, 330~, sin-1 7 VI. 4. 1. 5. 1.7208. 6. N. 50018' E., 399 miles. 1. 2. 3. 16. 4. 45 3 tan x - tan3x 5. Fi 1 -3 tan2 x Opposite side, any value; third 6. 0. side, 13.766. 0, 225~, tan-1 (- 2). rst ship, 223 miles; second ship, 306 miles. VII. 1. 25. 2. 2. 4. ~ 90~, 180~, sin-14. 5. S. 83~ 41' W.; 1907 miles. 3. 8.6814, 5-, 43~ 43' 10", 106~ 16' 50". VIII. 1. 27. 2. a = 2 Ftan A, b = /2 FcotA. 3. a = ~45~, ~ 135~; b = ~ 30~, ~ 150~. 4. Smallest value of opposite side, 1; 1.75, 53~ 7' 48", 81~ 52' 12" or 2.50, 126~ 52' 12", 80 7' 48". 5. 39~ 29' N., 67~ 14' W. 6. tan a = tan2 b or - cot2 b. 18 TRIGONOMETRY. IX. 1. 15.849. 2. a= 2(3 + 3), b = 2(V/3+ 1), c =4(\/V3+ 1). -4 ~V7 3. tan-1'3 4. 41~ 24' 35", 820 49' 9", 550 46' 16". 5. N. 760 2'.E.; 866 miles. 6. 1. X. 1. 1.23138. 2. a =4, b=3, c= 5, A =53~7'48". 3. cos2 A + 4 sin2 A sin2 B. 4. 5.743, 4.257. 5. 14 10' E.; 342 miles. 6. 2. 1. logs 4 —=3 3. 0.039345, 0.055226. XI. 4. 115 feet. 5. 470 24' N., 630 43' W. XII. 71 1. 3 3 a/ 3-I 6. 2 7. 452.34, 61~ 37' 30", 56~ 14' 30". XIII. 8. 45~ 24' 20". XIV. 1. 12 7. 188,280. 1. 200~ 32' 6". 7. a = 273.76, a = 62~ 9' 41", 8. 47~ 10' 12". b = 272.94, - = 61~ 50' 19", 5. 1. c = 256.65, y = 56~. XV. 1. (a) 114~ 35' 30", (b). 6. 222~ 52' 12". 7. 461.94; 59~ 11' 8". ANSWERS. 19 EXERCISE XXIII. 1. loglo6 =0.77815. loglo4 =0.60206. loglo - =1.69897. loglo = 0.02119. 2. log210 = 3.3226. log7 = - 0.3562 3. loge2 =0.69315. log, 7 =1.94591. loge = -0.4054 log, - - 2.1484 4. x = 1.5439. ( loglo 14 loglo 12 loglo log2 5 log 5 log, 3 loge 8 3. log 4 3. x = 0.83048. = 1.14613. = 1.07918. = 1.39794. = 2.3224. = - 2.3838. = 1.09861. = 2.07944. = - 0.22314. loglo 21 = 1.32222. loglo 5 = 0.69897. logio - = 1.89086. log3 5 = 1.4650. log0 5 log. 9 log. 7 - = 1.60944. = 2.19722. = 0.25952. x = 0.42061. EXERCISE XXIV. 1. loge3 = 1.09861. 2. logelO = 2.3025850930. 3. loglo2 = 0.30103. loge5 =1.60944. logioe = 0.43429. log, 7 = 1.94591. logo 11 = 1.04139. EXERCISE XXV. 1. sin 1' = 0.00029088820. cos 1' = 0.99999995769. tan 1' = 0.000290888012. 2. sin 2' = 0.000581776. 3. sin 1~ = 0.0174. EXERCISE XXVI. 6. 00 40' 9" 1. sin 6' = 0.0017453; 2. sin 2~ = 0.034902; sin 3~ = 0.052340; sin 4~ = 0.069762; cos 6' = 0.999992. cos 2~ = 0.999392, cos 3~ = 0.998632. cos 4~ = 0.997568. EXERCISE XXVII. 1. The 6 sixth roots of - 1 are: 3+ i,. -\3+i -/3-i _. /-i 2 2' - 2 2' ' 2 The 6 sixth roots of + 1 are: 1+ - 3 - 1 + /- 3 - 1 - - 3 1 - - 3 I1, 2 ', - 1, 2 ~2 2 2 2 20 SPHERICAL TRIGONOMETRY. V+ i - + i 2, -~ —, -i. 2 2 3. cos 671~ + i sin 67~~, cos 1571~ + i sin 157~~, cos 2471~ + i sin 2470~, cos 337i~ + i sin 3371~. 4. sin 4 0 = 4 cos3 0 sin 0 - 4 cos 0 sin3 0. cos 4 0 = cos4 0 - 6 cos2 0 sin'2 0 + sin4 0. EXERCISE XXVIII. X2 5 X4 61 Xs 5. secx =l+ +- + -.... 2 24 720 x2 2 X4 11 x6 6. x otx 1 — --- -.. 3 45 1890 7. sin 10~ = 0.173648, cos 10~ = 0.984808. 8. tan 15~ =0.267944. SPHERICAL TRIGONOMETRY. EXERCISE XXIX. 1. 110~, 100~, 80~. 2. 140~, 90~, 550. 7O, 2 r, -2-r. EXERCISE XXX. 3. (i.) Either a or b must be equal to 90~. (lii.) A = 90~, B =b. (ii.) A = 90~ and B b. (iv.) c = 90~, A = 90~, B =90~. EXERCISE XXXI. 2. I, The cosine of the middle part = the product of the cotangents of the adjacent parts. IL The cosine of the middle part- the product of the sines of the opposite parts. ANSWERS. I 21l EXERCISE XXXII..24. A = 175~ 57' 10", B 135~ 42' 50", C =135~ 34' 7". 25. C =104~ 41' 39", a = 104~ 53' 2", b = 1330 39' 48"o 26. a = 900~; band B are indeterminate. 27. a A = 60~, b= 90~, B= 90. 28. The triangle is impossible. 29. b = 130~ 41' 42", c = 71~ 27' 43", A = 112~ 57' 2". 30. a = 26~ 3' 51", A = 35~, B = 65~ 46' 7" 31. Impossible. EXERCISE XXXIII. 1 cos A = cot a tan b, sin B = csc a sin, cos h = cos a sec 2. sin A = I sec a... sc 1800 1800 3. sio a cs sin = sec cos-, sin sin a csc n n 180~ sin r =tan I a cot -- n 4. Tetrahedron, 70~ 31' 46"; octahedron, 109~ 28' 14"; icosahedron, 138~ 11' 36"; cube, 90~; dodecahedron, 116~ 33' 44". 5. cot ~ A = /cos a. EXERCISE XXXV. 1. (i.) tan m = tan b cosA, (ii.) tan m = tan c cos B, cos a = cos b sec m cos (c - m); cos b = cos c sec m cos (a - m). EXERCISE XXXVI. 1. (i.) cot x = tan B csc a, (ii.) cot z = tan C csc b, cos A = cos B csc x sin (C- x); cosB =cos C csc x sin (A - x). EXERCISE XLI. 4. 2066.5 square miles. 22 SPHERICAL TRIGONOMETRY. EXERCISE XLII. 1. If x denotes the angle required, sin - x = cos 18~ sec 9~, x = 148~ 42'. 2. cos x = cos A cos B. 3. Let w = the inclination of the edge c to the plane of a and b. Then it is easily shown that V= abc sin I sin w. Now, conceive a sphere constructed having for centre the vertex of the trihedral angle whose edges are a, b, c. The spherical triangle, whose vertices are the points where a, b, c meet the surface of this sphere, has for its sides 1, Im, n; and w is equal to the perpendicular arc from the side I to the opposite vertex. Let L, M, N denote the angles of this triangle. Then, by means of [39] and [48], we find that sin w = sin m sin N = 2 sin m sin I N cos I N 2 = sin V/sin s sin (s - ) sin (s - m) sin (s - n), sin 1 where s = (l+ m n- n); hence, V = 2 abc Vsin s sin (s - 1) sin (s - m) sin (s - n). 4. (i.) 9,976,500 square miles; (ii.) 13,316,560 square miles. 5. Let mn = longitude of point where the ship crosses the equator, B = her course at the equator, d = distance sailed. Then tan m = sin I tan a, cos B = cos I sin a, cot d = cot I cos a. 6. Let k = arc of the parallel between the places, x = difference required; then sin" k = sin ~ d sec 1. x = 90~(V2- 1). 7. tan i (m-m') = sec s sec (s - d) sin (s - 1) sin (s - '); where 2 s = 1 + I- + d, and in and m' are the longitudes of the places. 9. 44 min. past 12 o'clock. 10. 60~. 11. cos t = - tan d tan; time of sunrise = 12 - 1 o'clock A.M.; time of sunset = - o'clock P.M.; cos a = sin d sec 1. For longest day 15 at Boston: time of sunrise, 4 hrs. 26 min. 50 sec. A.M.; time of sunset, 7 hrs. 33 min. 10 sec. P.M. Azimuth of sun at these times, 57~ 25' 15"; length of day, 15 hrs. 6 min. 20 sec.; for shortest day, times of sunrise and sunset are 7 hrs. 33 min. 10 sec. A.M. and 4 hrs. 26 min. 50 sec. P.M.; azimuth of sun, 1220 34' 45"; length of day, 8 hrs. 53 min. 40 sec. 12. The problem is impossible when cot d < tan 1; that is, for places in the frigid zone. ANSWERS. 23 13. For the northern hemisphere and positive declination, sin h = sin I sin d, cot a = cos I tan d. Example: h = 17~ 14' 35", a = 73~ 51' 34" E. 14. The farther the place from the equator, the greater the sun's altitude at 6 A.M. in summer. At the equator it is 0~. At the north pole it is equal to the sun's declination. At a given place, the sun's altitude at 6 A.M. is a maximum on the longest day of the year, and then sin h = sin I sin e (where e = 23~ 27'). 15. cos t = cot 1 tan d. Times of bearing due east and due west are t t 12 - o'clock A.M., and o'clock P.Mr., respectively. 15 15 Example: 6 hrs. 58 min. A.M. and 5 hrs. 2 min. P.M. 16. When the days and nights are equal, d = 0~, cos t = 0, t = 90~; that is, sun is everywhere due east at 6 A.M., and due west at 6 P.M. Since I and d must both be less than 90~, cos t cannot be negative, therefore t cannot be greater than 90~. As d increases, t decreases; that is, the times in question both approach noon. If 1< d, then cos t>1l; therefore this case is impossible. If I = d, then cos t = 1, and t = 0~; that is, the times both coincide with noon. The explanation of this result is, that for d = I the sun at noon is in the zenith, and south of the prime vertical at every other time. And if I > d, the diurnal circle of the sun and the prime vertical of the place meet in two points which separate further and further as I increases. At the pole the prime vertical is indeterminate; but near the pole, t = 90~, and the sun is always east at 6 A.M. 17. sin 1 = sin d csc h. 18. 11~ 50' 35". 19. The bearing of the wall, reckoned from the north point of the horizon, is given by the equation cot x = cos I tan d; whence, for the given case, x 75~ 12' 38". 20. 55~ 45' 6" N. 21. 63~ 23' 41" N. or S. 22. (i.) cos t = - tan I cotp; (ii.) t = z; (iii.) the result is indeterminate. 23. cot a = cos I tan d. 28. sin d = sin e sin u, tan r = cos e tan u. 25. h = 65~ 37' 20". 29. d = 32~ 24' 12", r = 301~ 48' 17". 26. h = 58~ 25' 15", a = 1520 28'. 30. d = 20~ 48' 12". 27. t = 4542', 1 = 67~ 58' 54". 31. 3 hrs. 59 min. 27-2 sec. P.M. 32. Cos a =cos (- a + h + p) Cos I (1 + h -p) sec I sec h. c Vc ~L I j LVUZ\(II os SURVEYING. 1. 8 A. 64 p. 2. 29 A. 7 P. 3. 4 A. 5- P. 4. 115- P. EXERCISE I. 5. 3 A. 78 P. 6. 13 A. 6- p. 7. 11 A. 157 P. 8. 7.51925. 9. 13.0735. 10. 2 A. 58r P. 11. 4 A. 35 P. 1. 2 A. 26 P. 5. 2. 20 A. 12 P. 6. 3. 2 A. 54 P. 7. 4. 2 A. 151 P. 1. 2 A. 121 P. y EXERCISE II. 8 A. 54 p. 5 A. 42 P. 2 A. 78 P. 8. 3 A. 122 P. 9. 6 A. 2P. 10. 9A. 40P. EXERCISE III. 2. 98 A. 92 P. EXERCISE IV. 1. AE=3.75 ch. 2. AE = 3.50 ch.; EG = 3.42 ch. 3. AE = 4.55 ch. 4. AE = 5.50 ch. 5. CE = 4.456 ch. 6. AD = 2.275 ch,; BE = 1.82 ch. 7. AD= 4.51 ch.; BE = 3.61 ch. 8. The distances on AB are 2, 3, and 5 ch. 9. EM (on EA) 2.5087 ch.; AN (on AB) 6.439 ch. 10. LetEG >DF, rAE =12.247 ch. AG = 9.798 ch. then AD= 8.659 ch. AF= 6.928 ch. 11. Let DG > EF, r CG = 14.862 ch. CD =13.113 ch. then CF = 11.404 ch. CE-= 10.062 ch. ANSWERS. 25 EXERCISE V. 1. 9.5 feet. 2. Third column: 26.944 opposite 0; 25.286 opposite 4. Fifth column: 20, 19.5, 21.3, 23, 22.3, 21.431, 20.4, 21.8, 24.1. &/fface, IIC a I i i Datum Level, 0 1 2 3 4 5 6 3. Column H.G. 20.8, 20.4, 20.0, 19.6, etc. Column C. 0.0, 5.3, 6.4, 7.4, 5.0, 5.1, etc. o:: CO e e 0 r 01 a " Oc c 0,o0 co 1X c -c t- d c oo a) tCO t- o0 Surface -Gradient- 04 ft. per Sta, Datum Level, 0x - 2 r 4 _ 8 _ 0102 0 1 2 3 4 5 8 9 10 10.25 FIVE -PLACE LOGARITHMIC AND TRIGONOMETRIC TABLES ARRANGED BY G. A. WENTWORTH, A.M. AND G. A. HILL, A.M. BOSTON, U.S.A., AND LONDON PUBLISHED BY GINN & COMPANY 1895 Entered according to Act of Congress, in the year 1882, by G. A. WENTWORTH AND G. A. HILL in the office of the Librarian of Congress at Washington Copyright, 1895, by G. A. WENTWORTH and G. A, HILL, INTRODUCTION. 1. If the natural numbers are regarded as powers of ten, the exponents of the powers are the Common or Briggs Logarithms of the numbers. If A and B denote natural numbers, a and b their logarithms, then 10a = A, 10b = B; or, written in logarithmic form, log A = a, log B - b. 2. The logarithm of a product is found by adding the logarithms of its factors. For, A x B = Oa X b = la + b. Therefore, log (A X B) = a + b = log A + log B. 3. The logarithm of a quotient is found by subtracting the logarithm of the divisor from that of the dividend. A 10o For, - -- = 10a-b. Therefore, log A = a - b = log A - log B. 4. The logarithm of a power of a number is found by multiplying the logarithm of the number by the exponent of the power. For, An = (lOa)n = 10an. Therefore, log A= an = n log A. 5. The logarithm of the root of a number is found by dividing the logarithm of the number by the index of the root. For, / A = /1oa = 10n. Therefore, log A = a = log A. n n 6. The logarithms of 1, 10, 100, etc., and of 0.1, 0.01, 0.001, etc., are integral numbers. The logarithms of all other numbers are fractions. iv LOGARITHMS. For, 109 = 1, hence log 1 = 0; 10-1 = 0.1, hence log.1 = - 1; 101= 10, hence log 10 = 1; 10-2 = 0.01, hence log0.01= -2; 102 = 100, hence log 100 = 2; 10-3 = 0.001, hence log 0.001 = - 3; 103 = 1000, hence log 1000 = 3; and so on. If the number is between 1 and 10, the logarithm is between 0 and 1. If the number is between 10 and 100, the logarithm is between 1 and 2. If the number is between 100 and 1000, the logarithm is-between 2 and 3. If the number is between 1 and 0.1, the logarithm is between 0 and -1. If the number is between 0.1 and -0.01, the logarithm is between -1 and -2. If the number is between 0.01 and 0.001, the logarithm is between -2 and -3. And so on. 7. If the number is less than 1, the logarithm is negative (~ 6), but is written in such a form that the fractionalpart is alwayspositive. For the number may be regarded as the product of two factors, one of which lies between 1 and 10, and the other is a negative power of 10; the logarithm will then take the form of a difference whose minuend is a positive proper fraction, and whose subtrahend is a positive integral number. Thus, 0.48 4.8 x 0.1. Therefore (~ 2), log 0.48 = log 4.8 + log 0.1 = 0.68124 - 1. (Page 1.) Again, 0.0007 = 7 X 0.0001. Therefore, log 0.0007 = log 7 + log 0.0001 = 0.84510 - 4. The logarithm 0.84510-4 is often written 4.84510. 8. Every logarithm, therefore, consists of two parts: a positive or negative integral number, which is called the Characteristic, and a positive proper fraction, which is called the Mantissa. Thus, in the logarithm 3.52179, the integral number 3 is the characteristic, and the fraction.52179 the mantissa. In the logarithm 0.78254- 2, the integral number - 2 is the characteristic, and the fraction 0.78254 is the mantissa. 9. If the logarithm is negative, it is customary to change the form of the difference so that the subtrahend shall be 10 or a multiple of 10. This is done by adding to both minuend and subtrahend a number which will increase the subtrahend to 10 or a multiple of 10. Thus, the logarithm 0.78254 - 2 is changed to 8.78254 - 10 by adding 8 to both minuend and subtrahend. The logarithm 0.92737-13 is changed to 7.92737 - 20 by adding 7 to both minuend and subtrahend. 10. The following rules are derived from ~ 6:If the number is greater than 1, make the characteristic of the logarithm one unit less than the number of figures -on the left of the decimal point. If the number is less than 1, make the characteristic of the logarithm negative, and one unit more than the number of zeros between the decimal point and the first significant figure of the given number. INTRODUCTION. V If the characteristic of a given logarithm is positive, make the number of figures in the integral part of the corresponding number one more than the number of units in the characteristic. If the characteristic is negative, make the number of zeros between the decimal point and the first significant figure of the corresponding number one less than the nunber of units in the characteristic. Thus, the characteristic of log 7849.27 = 3; the characteristic of log 0.037 = - 2 = 8.00000 - 10. If the characteristic is 4, the corresponding number has five figures in its integral part. If the characteristic is - 3, that is, 7.00000 - 10, the corresponding fraction has two zeros between the decimal point and the first significant figure. 11. The logarithms of numbers that can be derived one from another by multiplication or division by an integral power of 10 have the same mantissa. For, multiplying or dividing a number by an integral power of 10 will increase or diminish its logarithm by the exponent of that power of 10; and since this exponent is an integer, the mantissa of the logarithm will be unaffected. Thus, log 4.6021 = 0.66296. (Page 9.) log 460.21 = log (4.6021 x 102) = log 4.6021 + log 102 = 0.66296 + 2 = 2.66296. log 460210 - log (4.6021 x 105) = log 4.6021 + log 105 = 0.66296 + 5 =.66296. log 0.046021 = log (4.6021 - 102) = log 4.6021 - log 102 = 0.66296 - 2 = 8.66296 - 10. TABLE I. 12. In this table (pp. 1-19) the vertical columns headed N contain the numbers, and the other columns the logarithms. On page 1 both the characteristic and the mantissa are printed. On pages 2-19 the mantissa only is printed. The fractional part of a logarithm can be expressed only approximately, and in a five-place table all figures that follow the fifth are rejected. Whenever the sixth figure is 5, or more, the fifth figure is increased by 1. The figure 5 is written when the value of the figure in the place in which it stands, together with the succeeding figures, is more than 41, but less than 5. Thus, if the mantissa of a logarithm written to seven places is 5328732, it is written in this table (a five-place table) 53287. If it is 5328751, it is written 53288. If it is 5328461 or 5328499, it is written in this table 53285. Again, if the mantissa is 5324981, it is written 53250; and if it is 4999967, it is written 50000. vi LOGARITHMS. This distinction between 5 and 5, in case it is desired to curtail still further the mantissas of logarithms, removes all doubt whether a 5 in the last given place, or in the last but one followed by a zero, should be simply rejected, or whether the rejection should lead us to increase the preceding figure by one unit. Thus, the mantissa 13925 when reduced to four places should be 1392; but 13925 should be 1393. To FIND THE LOGARITHM OF A GIVEN NUMBER. 13. If the given number consists of one or two significant figures, the logarithm is given on page 1. If zeros follow the significant figures, or if the number is a proper decimal fraction, the characteristic must be determined by ~ 10. 14. If the given number has three significant figures, it will be found in the column headed N (pp. 2-19), and the mantissa of its logarithm in the next column to the right, and on the same line. Thus, Page 2. log 145 = 2.16137, log 14500 = 4.16137. Page 14. log 716 = 2.85491, log 0.716 = 9.85491 - 10. 15. If the given number has four significant figures, the first three will be found in the column headed N, and the fourth at the top of the page in the line containing the figures 1, 2, 3, etc. The mantissa will be found in the column headed by the fourth figure, and on the same line with the first three figures. Thus, Page 15. log 7682 = 3.88547, log 76.85 =1.88564. Page 18. log 93280 = 4.96979, log 0.9468 = 9.97626 - 10. 16. If the given number has five or more significant figures, a process called interpolation is required. Interpolation is based on the assumption that between two consecutive mantissas of the table the change in the mantissa is directly proportional to the change in the number. Required the logarithm of 34237. The required mantissa is (~ 11) the same as the mantissa for 3423.7; therefore it will be found by adding to the mantissa of 3423 seven-tenths of the difference between the mantissas for 3423 and 3424. The mantissa for 3423 is 53441. The difference between the mantissas for 3423 and 3424 is 12. Hence, the mantissa for 3423.7 is 53441 + (0.7 X 12) - 53449. Therefore, the required logarithm of 34237 is 4.53449. INTRODIUCTION. vii Required the logarithm of 0.0015764. The required mantissa is the same as the mantissa for 1576.4; therefore it will be found by adding to the mantissa for 1576 four-tenths of the difference between the mantissas for 1576 and 1577. The mantissa for 1576 is 19756. The difference between the mantissas for 1576 and 1577 is 27. Hence, the mantissa for 1576.4 is 19756 + (0.4 x 27) = 19767. Therefore, the required logarithm of 0.0015764 is 7.19767 - 10. Required the logarithm of 32.6708. The required mantissa is the same as the mantissa for 3267.08; therefore it will be found by adding to the mantissa for 3267 eight-hundredths of the difference between the mantissas for 3267 and 3268. The mantissa for 3267 is 51415. The difference between the mantissas for 3267 and 3268 is 13. Hence, the mantissa for 3267.08 is 51415 + (0.08 x 13) = 51416. Therefore, the required logarithm of 32.6708 is 1.51416. 17. When the fraction of a unit in the part to be added to the mantissa for four figures is less than 0.5 it is to be neglected; when it is 0.5 or more than 0.5 it is to be taken as one unit. Thus, in the first example, the part to be added to the mantissa for 3423 is 8.4, and the.4 is rejected. In the second example, the part to be added to the mantissa for 1576 is 10.8, and 11 is added. To FIND THE ANTILOGARITHM; THAT IS, THE NUMBER CORRESPONDING TO A GIVEN LOGARITHM. 18. If the given mantissa can be found in the table, the first three figures of the required number will be found in the same line with the mantissa in the column headed N, and the fourth figure at the top of the column containing the mantissa. The position of the decimal point is determined by the charac, teristic (~ 10). Find the number corresponding to the logarithm 0.92002. Page 16. The number for the mantissa 92002 is 8318. The characteristic is 0; therefore, the required number is 8.318. Find the number corresponding to the logarithm 6.09167. Page 2. The number for the mantissa 09167 is 1235. The characteristic is 6; therefore, the required number is 1235000. Find the number corresponding to the logarithm 7.50325- 10. Page 6. The number for the mantissa 50325 is 3186. The characteristic is - 3; therefore, the required number is 0.003186. Viii LOGARITHMS. 19. If the given mantissa cannot be found in the table, find in the table the two adjacent mantissas between which the given mantissa lies, and the four figures corresponding to the smaller of these two mantissas will be the first four significant figures of the required number. If more than four figures are desired, they may be found by interpolation, as in the following examples: Find the number corresponding to the logarithm 1.48762. Here the two adjacent mantissas of the table, between which the given mantissa 48762 lies, are found to be (page 6) 48756 and 48770. The corresponding numbers are 3073 and 3074. The smaller of these, 3073, contains the first four significant figures of the required number. The difference between the two adjacent mantissas is 14, and the difference between the corresponding numbers is 1. The difference between the smaller of the two adjacent mantissas, 48756, and the given mantissa, 48762, is 6. Therefore, the number to be annexed to 3073 is T6 of 1 - 0.428, and the fifth significant figure of the required number is 4. Hence, the required number is 30.734. Find the number corresponding to the logarithm 7.82326- 10. The two adjacent mantissas between which 82326 lies are (page 13) 82321 and 82328. The number corresponding to the mantissa 82321 is 6656. The difference between the two adjacent mantissas is 7, and the difference between the corresponding numbers is 1. The difference between the smaller mantissa, 82321, and the given mantissa, 82326, is 5. Therefore, the number to be annexed to 6656 is - of 1 = 0.7, and the fifth significant figure of the required number is 7. Hence, the required number is 0.0066567. In using a five-place table the numbers corresponding to mantissas may be carried to five significant figures, and in the first part of the table to six figures.* 20. The logarithm of the reciprocal of a number is called the Cologarithm of the number. If A denotes any number, then colog A =log =log 1 -log A (~ 3) =-log A. Hence, the cologarithm of a number is equal to the logarithm of the number with the minus sign prefixed, which sign affects the entire logarithm, both characteriistic and mantissa. *In most tables of logarithms proportional parts are given as an aid to interpolation; but, after a little practice, the operation can be performed nearly as rapidly without them. Their omission allows a page with larger-faced type and more open spacing, and consequently less trying to the eyes. INTRODUCTION. ix In order to avoid a negative mantissa in the cologarithm, it is customary to substitute for - log A its equivalent (10 -log A) - 10. Hence, the cologarithm of a number is found by subtracting the logarithm of the number from 10, and then annexing -10 to the remainder. The best way to perform the subtraction is to begin on the left and subtract each figure of log A from 9 until we reach the last significant figure, which must be subtracted from 10. If log A is greater in absolute value than 10 and less than 20, then in order to avoid a negative mantissa, it is necessary to write -log A in the form (20- log A)- 20. So that, in this case, colog A is found by subtracting log A from 20, and then annexing - 20 to the remainder. Find the cologarithm of 4007. 10 -10 Page 8. log 4007= 3.60282 colog 4007= 6.39718- 10 Find the cologarithm of 103992000000. 20 -20 Page 2. log 103992000000 = 11.01700 colog 103992000000 = 8.98300- 20 If the characteristic of log A is negative, then the subtrahend, -10 or - 20, will vanish in finding the value of colog A. Find the cologarithm of 0.004007. 10 -10 log 0.004007 = 7.60282 - 10 colog 0.004007 = 2.39718 With practice, the cologarithm of a number can be taken from the table as rapidly as the logarithm itself. By using cologarithms the inconvenience of subtracting the logarithm of a divisor is avoided. For dividing by a number is equivalent to multiplying by its reciprocal. Hence, instead of subtracting the logarithm of a divisor its cologarithm may be added. x LOGARITHMS. EXERCISES. Find the logarithms of: 1. 6170. 4. 85.76. 7. 0.8694. 10. 67,3208. 2. 0.617. 5. 296.8. 8. 0.5908. 11. 18.5283. 3. 2867. 6. 7004. 9. 73243. 12. 0.0042003. Find the cologarithms of: 13. 72433. 16. 869.278. 19. 0.002403. 14. 802.376. 17. 154000. 20. 0.00077-7. 15. 15.7643. 18. 70.0426. 21. 0.051828. Find the antilogarithms of: 22. 2.47246. 25. 1.26784. 28. 9.79029 - 10. 23. 7.89081. 26. 3.79029. 29. 7.62328-10. 24. 2.91221. 27. 5.18752. 30. 6.15465-10. COMPUTATION BY LOGARITHMS. 21. (1) Find the value of x, if x = 72214 X 0.08203. Page 14. log 72214 = 4.85862 Page 16. log 0.08203 = 8.91397 - 10 By ~ 2. log x = 3.77259 Page 11. x = 5923.63 (2) Find the value of x, if x= 5250- 23487. Page 10. log 5250 = 3.72016 Page 4. colog 23487 = 5.62917 - 10 Page 4. log x = 9.34933 - 10 = log 0.22363.-. x = 0.22353 7.56 X 4667 X 567 (3) Find the value of x, if x 899.1 X 0.00337 X 23435' Page 15. log 7.56 = 0.87852 Page 9. log 4667 = 3.66904 Page 11. log 567 = 2.75358 Page 17. colog 899.1 = 7.04619- 10 Page 6. colog 0.00337 = 2.47237 Page 4. colog 23435 = 5.63013-10 Page 5. log x = 2.44983 = log 281.73,. x = 281.73. INTRODUCTION. xl (4) Find the cube of 376. Page 7. log 376 = 2.57519 Multiply by 3 (~ 4), 3 Page 10. log 3763 = 7.72557 = log 53158600.. 3763 = 53158600. (5) Find the square of 0.003278. Page 6. log0.003278 = 7.51561-10 2 Page 2. log 0.0032782 = 15.03122 - 20 = log 0.000010745.. 0.0032782 = 0.000010745. (6) Find the square root of 8322. Page 16. log 8322 = 3.92023 Divide by 2 (~ 5), 2)3.92023 log V8322 = 1.96012 = log 91.226..8322 = 91.226. If the given number is a proper fraction, its logarithm will have as a subtrahend 10 or a multiple of 10. In this case, before dividing the logarithm by the index of the root, both the subtrahend and the number preceding the mantissa should be increased by such a number as will make the subtrahend, when divided by the index of the root, 10 or a multiple of 10. (7) Find the square root of 0.000043641. Page 8. log 0.000043641 = 5.63989-10 10 -10 Divide by 2 (~ 5), 2)15.63989 - 20 Page 13. log V0.000043641 = 7.81995 - 10 =log 0.0066062.'. V0.000043641 = 0.0066062. (8) Find the sixth root of 0.076553. Page 15. log 0.076553 = 8.88397- 10 50 -50 Divide by 6 (~ 5), 6 6)58.88397-60 Page 13. log 6/0.076553 = 9.81400- 10 = log 0.65163.. /0.076553 = 0.65163. EXERCISES. Find by logarithms the value of: 45607 5.6123 2.567 1. 31045 2 0.01987 0.05786 xii LOGARITHMS. 0.06547 74.938 x 0.05938 4.657 x 0.03467 3.908 x 0.07189 0.0075389 x 0.0079 0.00907 x 0 009784 312 x 7.18 x 31.82 519 x 8.27 x 5.132 0.007 x 57.83 x 28.13.9.317 x 00.28 x 476.5 5.55 x 0.0007632 x 0.87654 2.79 X 0.0009524 X 1.46785 10.003457 x 43.387 x 99.2 X 0.00025 1 0.005824 x 15.724 x 1.38 x 0.00089 3/23.815 x 29.36 x 0.007 x 0.62487 11. 0.00072 x 9.236 x 5.924 x 3.0007 /3.1416 x 0.031416 x 0.0031416 12 1.7285 x 0.017285 x 0.0017285 TABLE II. 22. This table (page 20) contains the value of the number wr, its most useful combinations, and their logarithms. Find the length of an arc of 47~ 32' 57". in a unit circle. 47~ 32' 57" = 171177" log 171177 = 5.23344 log -1 = 4.68557 - 10 log arc 470 32' 57" = 9.91901 - 10 = log 0.82994.'. length of arc = 0.82994. Find the angle if the length of its arc in a unit circle = 0.54936. log 0.54936 = 9.73986 - 10 colog 1 = log a" = 5.31443 Jog angle = 5.05429 = log 113316.. angle = 113316" = 31~ 28' 36". INTRODUCTION. xiii 23. The relations between arcs and angles given in Table II. are readily deduced from the circular measure of an angle. In Circular Measure an angle is defined by the equation are angle = r., radius in which the word arc denotes the length of the are corresponding to the angle, when both are and radius are expressed in terms of the same linear unit. Since the arc and radius for a given angle in different circles vary in the same ratio, the value of the angle given by this equation is independent of the value of the radius. The angle which is measured by a radius-arc is called a Radian, and is the angular unit in circular measure. C __ Since C =2 7R, it follows that - 2 v, and = 7r. Therefore, R 1 If the arc = circumference, the angle = 2 r. If the arc = semicircumference, the angle = r. If the arc = quadrant, the angle = 2 7r. If the arc = radius, the angle = 1. Therefore, 7r = 180~, I r -= 90~, vr = 60~, 7r =45~, - 7r = 300 7r =22~~, and so on. Since 180~ in common measure equals 7r units in circular measure, 7r 1~ in common measure =80 units in circular measure; 180 180~ 1 unit in circular measure -- in common measure. 7r By means of these two equations, the value of an angle expressed in one measure may be changed to its value in the other measure. Thus, the angle whose arc is equal to the radius is an angle of 180~ 1 unit in circular measure, and is equal to, or 57~ 17' 45", very nearly. TABLE III. 24. This table (pp. 21-49) contains the logarithms of the trigo. nometric functions of angles. In order to avoid negative characteristics, the characteristic of every logarithm is printed 10 too large. Therefore, -10 is to be annexed to each logarithm. On pages 28-49 the characteristic remains the same throughout each column, and is printed at the top and the bottom of the column. xiv LOGARITHMS. But on pp. 30, 49, the characteristic changes one unit in value at the places marked with bars. Above these bars the proper characteristic is printed at the top, and below them at the bottom, of the column. 25. On pages 28-49 the log sin, log tan, log cot, and log cos, of 1~ to 89~, are given to every minute. Conversely, this part of the table gives the value of the angle to the nearest minute when log sin, log tan, log cot, or log cos is known, provided log sin or log cos lies between 8.24186 and 9.99993, and log tan or log cot lies between 8.24192 and 11.75808. If the exact value of the given logarithm of a function is not found in the table, the value nearest to it is to be taken, unless interpolation is employed as explained in ~ 26. If the angle is less than 45~ the number of degrees is printed at the top of the page, and the number of minutes in the column to the left of the columns containing the logarithm. If the angle is greater than 45~, the number of degrees is printed at the bottom of the page, and the number of minutes in the column to the right of the columns containing the logarithms. If the angle is less than 45~, the names of its functions are printed at the top of the page; if greater than 45~, at the bottom of the page. Thus, Page 38. log sin 21~ 37'= 9.56631-10. Page 45. log cot 36~ 53' = 10.12473 -10 = 0.12473. Page 37. logcos69~ 14' = 9.54969-10. Page 49. log tan 45~ 59' = 10.01491 - 10 = 0.01491. Page 48. If log cos = 9.87468 - 10, angle = 41~ 28'. Page 34. If log cot = 9.39353 - 10, angle = 76~ 6'. If log sin = 9.47760 - 10, the nearest log sin in the table is 9.47774 - 10 (page 36), and the angle corresponding to this value is 17~ 29'. If log tan = 0.76520 = 10.76520 - 10, the nearest log tan in the table is 10.76490 - 10 (page 32), and the angle corresponding to this value is 80~ 15'. 26. If it is desired to obtain the logarithms of the functions of angles that contain seconds, or to obtain the value of the angle in degrees, minutes, and seconds, from the logarithms of its functions, interpolation must be employed. Here it must be remembered that, The difference between two consecutive angles in the table is 60". Log sin and log tan increase as the angle increases; log cos and log cot diminish as the angle increases. INTRODUCTION. XV Find log tan 70~ 46' 8". Page 37. log tan 70~ 46' = 0.45731. The difference between the mantissas of log tan 70~ 46' and log tan 700 47' is 41, and - of 41 5. As the function is increasing, the 5 must be added to the figure in the fifth place of the mantissa 45731; and Therefore log tan 70~ 46' 8" = 0.45736. Find log cos 47~ 35' 4". Page 48. log cos 470 35'= 9.82899- 10. The difference between this mantissa and the mantissas of the next log cos is 14, and -4 of 14 = 1. As the function is decreasing, the 1 must be subtracted from the figure in the fifth place of the mantissa 82899; and Therefore log cos 470 35' 4" = 9.82898 - 10. Find the angle for which log sin = 9.45359 -10. Page 35. The mantissa of the nearest smaller log sin in the table is 45334. The angle corresponding to this value is 160 30'. The difference between 45334 and the given mantissa, 55359, is 25. The difference between 45334 and the next following mantissa, 45377, is 43, and 2 5 of 60" - 35". As the function is increasing, the 35" must be added to 16~ 30'; and the required angle is 16~ 30' 35". Find the angle for which log cot =0.73478. Page 32. The mantissa of the nearest smaller log cot in the table is 73415. The angle corresponding to this value is 100 27'. The difference between 73415 and the given mantissa is 63. The difference between 73415 and the next following mantissa is 71, and 6-I of 60" = 53". As the function is decreasing, the 53" must be subtracted from 10~ 27'; and the required angle is 100 26' 7". EXERCISES. Find 1. log sin 300 8' 9". 9. log tan 250 27' 47". 2. log sin 540 54' 40". 10. log cos 56~ 11' 57". 3. log cos 430 32' 31". 11. log cot 620 0' 4". 4. log cos 69~ 25' 11". 12. log cos 75~ 26' 58". 5. log tan 320 9' 17". 13. log tan 330 27' 13". 6. log tan 50 2' 2". 14. log cot 81~ 55' 24". 7. log cot 440 33' 17". 15. log tan 89~ 46' 35". 8. log cot 55~ 9' 32". 16. log tan 1~ 25' 56". xvi LOGARITHMS. Find the angle A if 17. log sin A = 9.70075. 25. log cos A = 9.40008. 18. log sin A= 9.91289. 26. log cot A = 9.78815. 19. log cosA = 9.86026. 27. log cos A = 9.34301. 20. log cosA = 9.54595. 28. log tan A = 10.52288. 21. log tanA = 9.79840. 29. log cot A = 9 65349. 22. log tanA = 10.07671. 30. log sin A = 8.39316. 23. log cot A = 10.00675. 31. log sin A = 8.06678. 24. log cot A = 9.84266. 32. log tanA = 8.11148. 27. If log sec or log csc of an angle is desired, it may be found from the table by the formulas, sec A == A; hence, log sec-A = colog cos A. cos A csc A = -i A; hence, log csc A = colog sin A. Page 31. log sec 8~ 28' = colog cos 8~ 28' = 0.00476. Page 42. log csc 59~ 36' 44" = colog sin 59~ 36' 44" = 0.06418. 28. If a given angle is between 0~ and 1~, or between 89~ and 90~; or, conversely, if a given log sin or log cos does not lie between the limits 8.24186 and 9.99993 in the table; or, if a given log tan or log cot does not lie between the limits 8.24192 and 11.75808 in the table; then pages 21-24 of Table III. must be used. On page 21, log sin of angles between 0~ and 0~ 3', or log cos of the complementary angles between 89~ 57' and 90~, are given to every second; for the angles between 0~ and 0~ 3', log tan = log sin, and log cos = 0.00000; for the angles between 89~ 57' and 90~, log cot = log cos, and log sin = 0.00000. On pages 22-24, log sin, log tan, and log cos of angles between 0~ and 1~, or log cos, log cot, and log sin of the complementary angles between 89~ and 90~, are given to every 10". Whenever log tan or log cot is not given, they may be found by the formulas, log tan = colog cot. log cot = colog tan. Conversely, if a given log tan or log cot is not contained in the table, then the colog must be found; this will be the log cot or log tan, as the case may be, and will be contained in the table. On pages 25-27 the logarithms of the functions of angles between 1~ and 2~, or between 88~ and 90~, are given in the manner employed on pages 22-24. These pages should be used if the angle lies between these limits, and if not only degrees and minutes, but degrees, minutes, and multiples of 10" are given or required. INTRODUCTION. xvii When the angle is between 0~ and 2~, or 88~ and 90~, and a greater degree of accuracy is desired than that given by the table, interpolation may be employed; but for these angles interpolation does not always give true results, and it is better to use Table IV. Find log tan 0~ 2' 47", and log cos 89~ 37' 20". Page 21. log tan 0~ 2' 47" = log sin 0~ 2' 47" = 6.90829 - 10. Page 23. log cos 89~ 37' 20" = 7.81911- 10. Find log cot 0~ 2' 15". 10 -10 Page 21. log tan 0~ 2' 15" = 6.81591-10 Therefore, log cot 0~ 2' 15" = 3.18409 Find log tan 89~ 38' 30". 10 -10 Page 23. log cot 89~ 38' 30" = 7.79617-10 Therefore, log tan 89~ 38' 30" = 2.20383 Find the angle for which log tan = 6.92090 -10. Page 21. The nearest log tan is 6.92110 - 10. The corresponding angle for which is 0~ 2' 52". Find the angle for which log cos = 7.70240 -10. Page 22. The nearest log cos is 7.70261 - 10. The corresponding angle for which is 89~ 42' 40". Find the angle for which log cot = 2.37368. This log cot is not contained in the table. The colog cot = 7.62632 - 10 = log tan. The log tan in the table nearest to this is (page 22) 7.62510- 10, and the angle corresponding to this value of log tan is 0~ 14' 30". 29. If an angle x is between 90~ and 360~, it follows, from formulas established in Trigonometry, that, between 90~ and 180~, between 180~ and 270~, log sin x-=log sin (180~- x), log sin x =log sin (x —180~)., log cos x = log cos (180~ - x)., log cos x = log cos (x - 1800),,, log tan x = log tan (180~ - x)., log tan x = log tan (x -180~), log cot x = log cot (180~ - x),; log cot x = log cot (x -180~); between 270~ and 360~, log sin x = log sin (360~ - x), log cos x= log cos (360~- x), log tan x = log tan (360~ - x), log cot x = log cot (360 - x),. Xviii LOGARITHMS. The letter n is placed (according to custom) after the logarithms of those functions which are negative in value. The above formulas show, without further explanation, how to find by means of Table III. the logarithms of the functions of any angle between 90~ and 360~. Thus, log sin 137~ 45' 22" = log sin 42~ 14' 38" = 9.82756 - 10. log cos 137~ 45' 22" = log,, cos 42~ 14' 38" = 9.86940, - 10. log tan 137~ 45' 22" = log, tan 42~ 14' 38" = 9.95815, - 10. log cot 137~ 45' 22" =log,, cot 42~ 14' 38" = 0.04185,,. log sin 209~ 32' 50" = log, sin 29~ 32' 50" = 9.69297, - 10. log cos 330~ 27' 10" = log cos 29~ 32' 50" = 9.93949 - 10. Conversely, to a given logarithm of a trigonometric function there correspond between 0~ and 360~ four angles, one angle in each quadrant, and so related that if x denote the acute angle, the other three angles are 180~ -x, 180~ + x, and 360~ - x. If besides the given logarithm it is known whether the function is positive or negative, the ambiguity is confined to two quadrants, therefore to two angles. Thus, if the log tan = 9.47451 - 10, the angles are 16~ 36' 17" in Quadrant I. and 196~ 36' 17" in Quadrant III.; but if the log tan = 9.47451,, - 10, the angles are 163~ 23' 43" in Quadrant II. and 343~ 23' 43" in Quadrant IV. To remove all ambiguity, further conditions are required, or a knowledge of the special circumstances connected with the problem in question. TABLE IY. 30. This table (page 50) must be used when great accuracy is desired in working with angles between 0~ and 2~, or between 889 and 90~. The values of S and T are such that when the angle a is expressed in seconds, S = log sin a - log a", T = log tan a - log a". Hence follow the formulas given on page 50. The values of S and T are printed with the characteristic 10 too large, and in using them -10 must always be annexed. Find log sin 0~ 58' 17". Find log cos 88~ 26' 41.2". 0~ 58' 17" = 3497" 90 ~- 88~ 26' 41.2" = 1~ 33' 18.8" log 3497 = 3.54370 = 5598.8" S = 4.68555- 10 log 5598.8 = 3.74809 log sin 0~ 58' 17" = 8.22925 - 10 S = 4.68552- 10 log cos 88~ 26' 41.2" = 8.43361 - 10 INTRODUCTION. xix Find log tan 0~ 52' 47.5". 00 52' 47.5" = 3167.5" log 3167.5 = 3.50072 T = 4.68561 - 10 log tan 0~ 52' 47.5" = 8.18633- 10 Find log tan 89~ 54' 37.362". 900 - 89~ 54' 37.362 - = 0~ 5' 22.638" = 322.638" log 322.638 =2.50871 T = 4.68558 - 10 log cot 89~ 54' 37.362" = 7.19429- 10 log tan 89~ 54' 37.362" = 2.80571 Find the angle, if log sin = 6.72306 -10. 6.72306 - 10 S = 4.68557- 10 Subtract, 2.03749 = log 109.015 109.015" = 00 1' 49.015". Find the angle for which log cot 1.67604. colog cot = 8.32396 - 10 T = 4.68564- 10 Subtract, 3.63832 = log 4348.3 4348.3" = 1~ 12' 28.3". Find the angle for which log tan = 1.55407. colog tan 8.44593 - 10 T 4.68569 - 10 Subtract, 3.76024 = log 5757.6 5757.6" - 1~ 35' 57.6", and 900 - 1~ 35' 57.6" = 88~ 24' 2.4". Therefore, the angle required is 88~ 24' 2.4". TABLE V. 31. This table (p. 51), containing the circumferences and areas of circles, does not require explanation. TABLE VI. 32. Table VI. (pp. 52-69) contains the natural sines, cosines, tangents, and cotangents of angles from 0~ to 90~, at intervals of 1'. If greater accuracy is desired it may be obtained by interpolation. NOTE. In preparing the preceding explanations, we have made free use of the Logarithmic Tables by F. G. Gauss. For Table VI. we are indebted to D. Carhart. TABLE VII. 33. This table (pp. 70-75) gives the latitude and departure to three places of decimals for distances from 1 to 10, corresponding to bearings from 0~ to 90~ at intervals of 15', XX LOGARITHMS. If the bearing does not exceed 45~ it is found in the left-hand column, and the designations of the columns under "Distance" are taken from the top of the page; but if the bearing exceeds 45~, it is found in the right-hand column, and the designations of the columns under "Distance" are taken from the bottom of the page. The method of using the table will be made plain by the following examples: — (1) Let it be required to find the latitude and departure of the course N. 35~ 15' E. 6 chains. On p. 75, left-hand column, look for 35~ 15'; opposite this bearing, in the vertical column headed "Distance 6," are found 4.900 and 3.463 under the headings "Latitude" and "Departure" respectively. Hence, latitude or northing = 4.900 chains, and departure or easting = 3.463 chains. (2) Let it be required to find the latitude and departure of the course S. 87~ W. 2 chains. As the bearing exceeds 45~, we look in the right-hand column of p. 70, and opposite 87~ in the column marked " Distance 2 " we find (taking the designations of the columns from the bottom of the page) latitude = 0.105 chains, and departure = 1.997 chains. Hence, latitude or southing= 0.105 chains, and departure or westing = 1.997 chains. (3) Let it be required to find the latitude and departure of the course N. 15~ 45' W. 27.36 chains. In this case we find the required numbers for each figure of the distance separately, arranging the work as in the following table. In practice, only the last columns under "Latitude " and "Departure " are written. DISTANCE. LATITUDE. DEPARTURE. 20 = 2 X 10 1.925 X 10 = 19.25 0.543 X 10 = 5.43 7 6.737 1.90 0.3 = 3 -10 2.887 -10 = 0.289 0.814 -10 = 0.081 0.06 = 6. 100 5.775 + 100 = 0.058 1.628 - 100 = 0.016 27.36 26.334 7.427 Hence, latitude = 26.334 chains, and departure = 7.427 chains. TABLE I, THE COMMON OR BRIGGS LOGARITHMS OF TIHE NATURAL NUMBERS From 1 to 10000. 1-100 N log 1. 00 000 2 0. 30 103 3 0. 47 712 4 0. 60 206 5 0. 69 897 6 0. 77 815 7 0.84510 8 0. 90 309 9 0. 95 424 10 1. 00000 11 1.04 139 12 1.07 918 13 -1.11394 14 1. 14 613 15 1.17 609 16 1.20412 17 1.23045 18 1. 25 527 19 1.27 875 20 1. 30 103 N log N log 21 1.32 222 22 1.34 242 23 1. 36 173 24 1.38 021 25 1.39 794 26 1.41497 27 1. 43 136 28 1. 44 716 29 1.46 240 30 1. 47 712 31 1.49136 32 1.50 515 33 1. 51 851 34 1. 53 148 35 1. 54 407 36 1.55 630 37 1.56820 38 1.57 978 39 1. 59 106 40 1. 60 206 N log N log 41 1. 61 278 42 1. 62 325 43 1. 63 347 44 1. 64345 45 1.65321 46 1.66276 47 1. 67 210 48 1.68 124 49 1.69 020 50 1.69 897 51 1.70757 52 1. 71 600 53 1. 72 428 54 1. 73 239 55 1. 74036 56 1. 74 819 57 1. 75 587 58 1. 76343 59 1. 77 085 60 1. 77 815 N log N log 61 1. 78 533 62 1.79 239 63 1. 79 934 64 1.80 618 65 1. 81 291 66 1. 81 954 67 1. 82 607 68 1.83 251 69 1. 83 885 70 1.84 510 71 1. 85 126 72 1. 85 733 73 1. 86 332 74 1.86 923 75 1. 87 506 76 1. 88 081 77 1. 88 649 78 1. 89 209 79 1. 89 763 80 1. 90 309 N log N log 81 1.90 849 82 1.91381 83 1. 91 908 84 1.92 428 85 1. 92 942 86 1.93450 87 1. 93 952 88 1. 94 448 89 1. 94 939 90 1.95 424 91 1. 95 904 92 1.96379 93 1. 96 848 94 1.97313 95 1. 97 772 96 1. 98 227 97 1.98677 98 1.99 123 99 1. 99 564 100 2. 00 000 N log 1-100 2 100-150 I 0 N 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 N 0 00 000 00 432 00 860 01 284 01 703 02 119 02 531 02 938 03 342 03 743 04 139 04 532 04 922 05 308 05 690 06 070 06 446 06819 07 188 07 555 07 918 08 279 08 636 08 991 09 342 09 691 10037 10 380 10721 11059 11 394 11 727 12057 12385 12 710 13 033 13 354 13 672 13 988 14 301 14 613 14 922 15 229 15 534 15 836 16 137 16435 16 732 17 026 17 319 17 609 0 1 00 043 00475 00 903 01326 01 745 02 160 02 572 02 979 03 383 03 782 04 179 04 571 04961 05 346 05 729 06 108 06 483 06 856 07 225 07 591 07 954 08314 08 672 09 026 09377 09 726 10072 10415 10755 11 093 2 00 087 00518 00 945 01 368 01 787 02 202 02 612 03 019 03 423 03 822 04 218 04 610 04 999 05 385 05 767 06 145 06 521 06 893 07 262 07 628 07 990 08350 08 707 09 061 09 412 09 760 10 106 10449 10 789 11 126 3 4 00130 00173 00561 00604 00988 01 030 01410 01452 01 828 01 870 02 243 02 284 02653 02694 03 060 03 100 03 463 03 503 03 862 03 902 04258 04297 04650 04 689 05 038 05 077 05 423 05 461 05 805 05 843 06183 06221 06558 06595 06930 06967 07298 07335 07 664 07 700 08027 08063 08386 08422 08 743 08 778 09096 09132 09447 09482 5 6 00217 00260 00647 00689 01 072 01 115 01 494 01 536 01912 01953 7 00 303 00 732 01 157 01 578 01995 8 00 346 00 775 01 199 01 620 02 036 02 449 02 857 03 262 03 663 04 060 9 00 389 00 817 01 242 01 662 02 078 02 490 02 898 - 03 302 03 703 04 100 02 325 02 735 03 141 03 543 03 941 04 336 04 727 05 115 05 500 05 881 06 258 06 633 07 004 07 372 07 737 08 099 08 458 08 814 09 167 09 517 09 864 10 209 10551 10 890 11 227 02366 02407 02 776 02 816 03 "L 03 222 03 583 03 623 03981 04021 04376 04415 04 454 04493 04 766 04805 04844 04883 05 154 05 192 05231 05 269 05 538 05 576 05 614 05 652 05 918 05 956 05 994 06 032' 06296 06333 06371 06408 06670 06707 06744 06781 07 041 07078 07 115 07 151 07408 07445 07482 07518 07 773 07 809 07 846 07 882 08 135 08 493 08 849 09 202 09552 09 899 10 243 10 585 10 924 11 261 09 795 10 140. 10 483 10 823 11 160 09 830 10 175 10517 10857 11 193 08 171 08 529 08 884 09237 09 587 09 934 10 278 10 619 10 958 11 294 11 628 11 959 12 287 12 613 12937 13 258 13 577 13 893 14 208 14520 08 207 08 243 08565 08600 08920 08955 09 272 09 307 09621 09656 11428 11461 11494 11528 11 760 1.1 793 11 826 11 860 12 090 12 123 12 156 12189, 12418 12450 12483 12516 12 743 12775 12808 12840 13 066 13 098 13 130 13 162 13 386 13 418 13 450 13 481 13 704 13 735 13 767 13 799 14019 14051 14082 14114 14333 14364 14395 14426 14644 14675 14 706 14 737 14953 14 983 15 014 15 045 15 259 15 290 15 320 15 351 15 564 15 594 15 625 15 655 15 866 15 897 15 927 15 957 16167 16 197 16227 16256 16465 16495 16524 16554 16761 16791 16820 16850 17056 17085 17 114 17 143 17348 17377 17406 17435 17638 17667 17696 17 725 1 2 3 4 11 561 11 594 11 893 11 926 12 222 12 254 12 548 12 581 12 872 12905 13 194 13 513 13 830 14 145 14457 14 768 15 076 15 381 15 685 15 987 16 286 16 584 16 879 17 173 17 464 13 226 13 545 13 862 14 176 14 489 09 968 10 312 10 653 10 992 11 327 11 661 11 992 12 320 12 646 12 969 13 290 13 609 13 925 14 239 14 551 14 860 15 168 15 473 15 776 16 077 16376 16 673 16 967 17 260 17 551 17 840 8 10 003 10 346 10 687 11025 11 361 11 694 12 024 12 352 12 678 13 001 13 322 13 640 13 956 14 270 14 582 14 891 15 198 15 503 15 806 16 107 16 406 16 702 16 997 17289 17 580 17869 9 ar 14 799 14 829 15 106 15 137 15412 15442 15 715 15 746 16 017 16 047 16316 16346 16613 16643 16 909 16 938 17 202 17 231 17 493 17 522 17754- 17.782 17811 5 6 7 I _ — - - - - _ -_- - - a - - I;w-., — - I - I, -----;1- - - - zZ, P 100-100 150-200 01 N 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 N 0 1 2 17609 17638 17667 17898 17926 17955 18184 18213 18241 18469 18498 18526 18752 18780 18 808 19 033 19 312 19 590 19 866 20 140 20 412 20 683 20 952 21 219 21 484 21 748 22 011 22 272 22 531 22 789 23 045 23 300 23 553 23 805 24055 24 304 24551 24 797 25 042 25 285 25 527 25 768 26 007 26 245 26 482 26 717 26 951 27 184 27 416 27 646 27 875 28 103 28 330 28556 28 780 29 003 29 226 29 447 29 667 29 885 19061 19089 19340 19368 19618 19645 19893 19921 20167 20194 20439 20466 20710 20737 20 978 21005 21 245 21 272 21511 21 537 21 775 21 801 22037 22063 22298 22324 22557 22583 22814 22840 23 070 23 096 23 325 23 350 23 578 23 603 23 830 23 855 24 080 24 105 3 4 17 696 17 725 17984 18013 18 270 18298 18554 18583 18837 18865 19 117 19 145 19396 19424 19 673 19 700 19948 19976 20 222 20 249 20493 20520 20 763 20 790 21 032 21059 21 299 21325 21 564 21 590 21 827 21854 22089 22115 22350 22376 22 608 22 634 22866 22891 24 329 24576 24 822 25 066 25 310 25 551 25 792 26 031 26 269 26 505 26741 26 975 27 207 27 439 27 669 27 898 28 126 28 353 28 578 28 803 29 026 29 248 29 469 29 688 29 907 24353 24 601 24 846 25 091 25 334 25 575 25 816 26 055 26 293 26 529 26 764 26 998 27 231 27 462 27 92 27 921 28 149 28 375 28 601 28 825 29 048 29 270 29 491 29 710 29 929 23 121 23 376 23 629 23 880 24 130 24378 24 625 24871 25 115 25 358 25 600 25 840 26 079 26316 26553 26 788 27021 27 254 27 485 27 715 27 944 28 171 28 398 28 623 28 847 29 070 29 292 29513 29 732 29 951 23 147 23 401 23 654 23 905 24 155 24 403 24650 24 895 25 139 25 382 25 624 25 864 26 102 26 340 26 576 26811 27045 27 277 27 508 27 738 27 967 28 194 28 421 28 646 28 870 29 092 29314 29535 29 754 29 973 5 17 754 18 041 18 327 18 611 18 893 19 173 19451 19 728 20 003 20 276 20 548 20817 21 085 21 352 21 617 21 880 22 141 22 401 22 660 22 917 23 172 23 426 23 679 23 930 24 180 24 428 24 674 24 920 25 164 25 406 25 648 25 888 26 126 26 364 26 600 26 834 27 068 27300 27 531 27 761 27 989 28217 28 443 28 668 28 892 29 115 29 336 29 557 29 776 29 994 30211 5 - 6 17 782 18 070 18 355 18639 18 921 7 17 811 18 099 18 384 18 667 18 949 19 201 19 229 19479 19507 19 756 19 783 20030 20058 20303 20330 20575 20602 20844 20871 21 112 21 139 21378 21405 21 643 21 669 21906 21932 22 167 22 194 22 427 22453 22686 22712 22943 22968 23 198 23 223 23452 23477 23 704 23 729 23 955 23 980 24 204 24 229 24 452 24 477 24699 24 724 24944 24969 25 188 25 212 25 431 25 455 25 672 25 696 25 912 25 935 26150 26174 26387 26411 26623 26647 26858 26881 27 091 27 114 27323 27346 27554 27577 27 784 27 807 28012 28035 28 240 28 262 28466 28488 28 691 28 713 28914 28937 29 137 29159 29358 29380 29579 29601 29 798 29 820 30016 30038 8 9 17840 17 869 18 127 18 156 18412 18 441 18696 18724 18977 19005 19257 19285 19535 19562 19811 19838 20085 20112 20358 20385 20629 20656 20 898 20925 21 165 21192 21431 21458 21 696 21722 21958 21985 22 220 22 246 22479 22505 22 737 22 763 22994 23019 23 249 23 274 23 502 23.528 23 754 23 779 24005 24030 24254 24279 24502 24527 24 748 24 773 24 993 25 018 25 237 25 261 25 479 25 503 25 720 25 744 25 959 25 983 26 198 26 221 26435 26458 26670 26694 26905 26928 27 138 27 161 27370 27393 27600 27623 27830 27852 28 058 28 081 28 285 28 307 28511 28533 28735 28758 28959 28981 29 181 29 203 29 403 29 425 29 623 29 645 29 842 29 863 30060 30081 30 103 30 125 30 146 30 168 30 190 0 1 2 3 4 30 233 6 30 255 7 30 276 8 30 298 9 0 - It I i I II I s - Ir I 150-200 4 200-250 N 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239. 240 241 242 243 244 245 246 247 248 249 250 N 0 1 2 30103 30 320 30 535 30 750 30963 31 175 31387 31 597 31 806 32015 32 222 32 428 32634 32 838 33 041 33 244 33 445 33 646 33 846 34 044 34 242 34439 34 635 34 830 35 025 35 218 35411 35 603 35 793 35 984 36 173 36361 36 549 36 736 36922 37107 37 291 37475 37 658 37 840 38 021 38 202 38382 38 561 38 739 38917 39 094 39 270 39 445 39 620 39 794 0 30125 30 341 30 557 30 771 30 984 31 197 31 408 31 618 31 827 32 035 32 243 32 449 32654 32858 33 062 33 264 33 465 33 666 33 866 34 064 34 262 34459 34655 34 850 35 044 35 238 35 430 35 622 35 813 36 003 36 192 36 380 36 568 36 754 36 940 37 125 37310 37 493 37676 37 858 38 039 38 220 38 399 38578 38 757 38934 39 111 39 287 39 463 39637 39811 1 30 146 30 363 30 578 30 792 31006 31218 31 429 31 639 31 848 32056 32263 32469 32675 32879 33 082 33 281 33 486 33 686 33 885 34 084 34 282 34479 34 674 34 869 35 064 35 257 35 449 35 641 35 832 36 021 36211 36399 36 586 36 773 36 959 37 144 37 328 37511 37694 37 876 38057 38 238 38417 38 596 38 775 38 952 39 129 39 305 39 480 39655 3 4 30168 30190 30384 30406 30600 30621 30814 30835 31027 31048 31 239 31260 31450 31471 31660 31681 31869 31890 32077 32098 32284 32305 32490 32510 32695 32715 32899 32919 33 102 33 122 33 304 33 325 33 506 33 526 33 706 33 726 33 905 33 925 34104 34 124 34301 34321 34498 34518 34694 34713 34889 34908 35 083 35 102 35276 35295 35 468 35 488 35 660 35 679 35 851 35 870 36040 36059 36229 36248 36418 36436 36605 36624 36791 36810 36977 36996 37 162 37 181 37346 37365 37530 37548 37 712 37 731 37894 37912 38075 38093 38256 38274 38435 38453 38 614 38 632 38792 38810 38 970 38 987 39 146 39 164 39322 39340 39498 39515 39672 39690 5 30211 30 428 30 643 30856 31069 31 281 31 492 31 702 31911 32 118 32325 32531 32 736 32 940 33 143 6 30 233 30449 30 664 30878 31 091 31 302 31 513 31723 31 931 32 139 32 346 32552 32 756 32 960 33 163 33 345 33 365 33 546 33 566 33 746 33 766 33 945 33965 34 143 34 163 34 341 34 361 34537 34557 34733 34753 34928 34947 35 122 35 141 7 30 255 30471 30 685 30 899 31 112 31 323 31 534 31 744 31 952 32 160 32 366 32572 32777 32 980 33 183 33 385 33 586 33 786 33 985 34 183 34 380 34577 34 772 34 967 35 160 35 353 35 545 35 736 35 927 36 116 36 305 36493 36680 36 866 37 051 37236 37 420 37 603 37 785 37967 38 148 38 328 38 507 38 686 38 863 39041 39217 39 393 39 568 39 742 8 30 276 30 492 30 707 30 920 31 133 31 345 31 555 31 765 31 973 32 181 32387 32 593 32 797 33 001 33 203 33 405 33 606 33 806 34 005 34 203 34 400 34 596 34 792 34 986 35 180 35 372 35 564 35 755 35 946 36 135 36324 36511 36698 36 884 37 070 37 254 37438 37 621 37 803 37985 38 166 38 346 38 525 38 703 38 881 39 058 39235 39410 39 585 39759 9 30 298 30 514 30 728 30 942 31154 31 366 31 576 31 785 31 994 32 201 32408 32613 32818 33021 33 224 33 425 33 626 33 S26 34 025 34 223 34 420 34616 34 811 35 005 35 199 35 392 35 583 35 774 35 965 36 154 36342 36 530 36 717 36 903 37 088 37273 37457 37639 37 822 38003 38 184 38 364 38 543 38 721 38 899 39076 39 252 39 428 39 602 39 777 35 315 35 507 35 698 35 889 36078 36 267 36455 36 642 36 829 37014 37199 37383 37566 37749 37931 38 112 38 292 38471 38 650 38 828 39005 39 182 39 358 39 533 39 707 39 881 5 35 334 35 526 35 717 35 908 36 097 36 286 36474 36 661 36 847 37033 37 218 37401 37 585 37 767 37949 38 130 38 310 38 489 38 668 38 846 39 023 39 199 39375 39550 39 724 39829 39846 39863 2 3 4 39898 39915 39933 39950 6 7 8 9 200-250 250-300 5 i e wwmwmm a N 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 N I l l [.. 0 1 39794 39811 39967 39985 40140 40157 40312 40329 40483 40500 40654 40 671 40824 40841 40993 41010 41 162 41 179 41330 41347 41497 41 514 41664 41681 41 830 41 847 41996 42012 42-160 42 177 42325 42341 42488 42504 42651 42667 42813 42830 42975 42991,,~ II th1..I ~1. 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I, 1,I 1._. 260-300 6 300-350 N 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 N 0 1 47 712 47 727 47857 47871 48001 48015 48 144 48159 48 287 48 302 48430 48444 48572 48586 48 714 48 728 48855 48869 48996 49010 49 136 49 150 49276 49290 49415 49429 49554 49568 49693 49707 49831 49 84 49969 49982 50 106 50120 50243 50256 50379 50393 50515 50529 50651 50664 50 786 50 799 50920 50934 51055 51068 51 188 51202 51322 51335 51455 51468 51587 51 601 51 720 51 733 51851 51865 51983 51996 52 114 52 127 52244 52257 52375 52388 52504 52517 52634 52647 52 763 52 776 52892 52905 53020 53033 53 148 53 161 53 275 53 288 53403 53415 53 529 53 542 53 656 53 668 53 782 53 794 53 908 53 920 54033 54045 1554 1 4 170 54283 54295 54407 54419 0 1 2 47 741 47 885 48 029 48 173 48316 48458 48 601 48 742 48 883 49024 49 164 49 304 49 443 49 582 49 721 49859 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88 440 88 497 88 553 88 610 4 87529 87 587 87 645 87 703 87 760 87 818 87 875 87 933 87 990 88 047 88 104 88 161 88 218 88 275 88 332 88 389 88 446 88 502 88 559 88615 88655 88660 88666 88672 88 711 88 717 88 722 88 728 88 767 88 773 88 779 88 784 88824 88829 88835 88840 88880 88885 88891 88897 88936 88941 88947 88953 88992 88997 89003 89009 89048 89053 89059 89064 89104 89109 89115 89120 89159 89165 89170 89176 89 215 89 221 89 226 89 232 89271 89276 89282 89287 89326 89332 89337 89343 89 382 89387 89393 89398 89437 89 443 89448 89454 89492 89498 89 504 89509 89548 89553 89559 89564 89603 89609 89614 89620 89658 89664 89669 89675 89713 89719 89724 89730 89 768 89 774 89 779 89 785 89823 89829 89834 89840 89878 89883 89889 89894 89933 89938 89 944 89949 89988 89 993 89998 90004 90042 90048 90053 90059 90097 90102 90108 90113 90151 90157 90162 90168 90206 90211 90217 90222 90260 90266 90271 90276 90314 90320 90325 90331 1 2 3 4 5 87 535 87 593 87 651 87 708 87 766 87 823 87 881 87 938 87 996 88 053 88 110 88 167 88 224 88 281 88 338 88 395 88 451 88 508 88 564 88 621 88 677 88 734 88 790 88 846 88 902 88 958 89 014 89 070 89 126 89 182 89 237 89 293 89 348 89 404 89 459 89 515 89 570 89 625 89 680 89735 89 790 89 845 89 900 89955 90009 90 064 90119 90173 90 227 90 282 90 336 5 6 87 541 87 599 87 656 87 714 87 772 87 829 87 887 87 944 88 001 88 058 88 116 88 173 88 230 88287 88 343 88 400 88 457 88 513 88 570 88 627 88 683 88 739 88 795 88852 88 908 88 964 89 020 89 076 89 131 89187 89 243 89 298 89 354 89 409 89465 89520 89 575 89 631 89 686 89 741 89 796 89 851 89 905 89 960 90015 90 069 90 124 90 179 90 233 90 287 90 342 6 7 8 87547 87552 87604 87610 87 662 87 668 87 720 87 726 87 777 87 783 87835 87841 87 892 87 898 87950 87955 88007 88013 88064 88070 88 121 88 127 88 178 88 184 88235 88 241 88 292 88 298 88349 88355 88406 88412 88417 88463 88468 88474 88519 88525 88530 88576 88581 88587 88632 88638 88643 88689 88694 88700 88745 88 750 88 756 88801 88807 88812 88 857 88 863 88 868 88913 88919 88925 88969 88975 88981 89025 89031 89037 89081 89087 89092 89 137 89 143 89 148 89193 89198 89204 89248 89254 89260 89304 89310 89315 89360 89365 89371 89415 89421 89426 89 470 89476 89481 89 526 89 531 89 537 89581 89586 89592 89636 89642 89647 89691 89697 89702 89746 89752 89757 89801 89807 89812 89856 89862 89867 89911 89916 89922 89966 89 971 89977 90020 90026 90031 90075 90080 90086 90 129 90135 90140 90 184 90 189 90 195 90238 90244 90249 90 293 90298 90304 90347 90352 90358 7 8 9 9 87 558 87 616 87 674 87 731 87 789 87 846 87 904 87 961 88 018 88 076 88 133 88 190 88 247 88 304 88 360 A~. 750- 800 800-850 N 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 N_ YLI I I I --- -I- - I I I I I I 0 0 90 309 90 363 90 417 90 472 90 526 90 580 90 634 90 687 90 741 90 795 90 849 90 902 90 956 91 009 91 062 91 116 91 169 91 222 91 275 91 328 91 381 91 434 91 487 91 540 91 593 91 645 91 698 91751 91 803 91 855 1 90314 90 369 90 423 90 477 90 531' 90 585 90 639 90 693 90 747 90 800 90 854 90 907 90 961 91 014 91 068 91 121 91 174 91 228 91 281 91 334 91 387 91 440 91 492 91 545 91 598 91 651 91 703 91 756 91 808 91 861 2 90 320 90 374 90 428 90 482 90 536 90 590 90 644 90 698 90752 90 806 90 859 90 913 90 966 91 020 91 073 91 126 91180 91 233 91 286 91 339 91392 91 445 91 498 91 551 91 603 91 656 91 709 91 761 91 814 91 866 91 918 91 971 92 023 92075 92 127 92 179 92 231 92 283 92335 92387 92 438 92 490 92 542 92 593 92 645 92 696 92 747 92 799 92 850 92 901 3 4 90325 90331 90380 90385 90434 90439 90488 90493 90542 90547 90596 90601 90650 90655 90 703 90 709 90757 90 763 90811 90816 90865 90870 90918 90924 90972 90977 91025 91030 91 078 91084 91 132 91137 91 185 91 190 91 238 91 243 91 291 91 297 91344 91350 91397 91403 91 450 91455 91 503 91 508 91 556 91561 91 609 91614 91 661 91 666 91 714 91 719 91 766 91 772 91 819 91 824 91 871 91 876 91 924 91929 91 976 91981 92 028 92 033 92080 92085 92 132 92 137 92 184 92 189 92 236 92 241 92288 92293 92 340 92 345 92 392 92 397 92443 92449 92 495 92 500 92 547 92 552 92 598 92 603 92 650 92 655 5 90 336 90 390 90 445 90 499 90 553 90 607 90 660 90714 90 768 90 822 90 875 90 929 90 982 91 036 91 089 91 142 91 196 91 249 91 302 91,355 91408 91 461 91 514 91 566 91 619 91 672 91 724 91 777 91 829 91 882 91 934 91 986 92 038 92 09] 92 143 92 195 92 247 92 298 92350 92 402 92 454 92 505 92557 92 609 92 660 92711 92 763 92 814 92 865 92 916 6 90 342 90 396 90450 90 504 90 558 90 612 90 666 90 720 90 773 90 827 90 881 90 934 90 988 91 041 91 094 91 148 91 201 91 254 91 307 91 360 91 413 91 466 91 519 91 572 91 624 91 677 91 730 91 782 91 834 91 887 91 939 91 991 92 044 92 096 92 148 92 200 92 252 92304 92 355 92 407 92 459 92 511 92 562 92 614 92 665 92 716 92 768 92 819 92 870 92 921 7 90 347 90 401 90 455 90 509 90 563 90617 90671 90 725 90 779 90 832 90 886 90 940 90 993 91 046 91 100 91 153 91 206 91 259 91 312 91 365 91418 91 471 91 524 91 577 91 630 91 682 91 735 91 787 91 840 91 892 91 944 91 997 92 049 92 101 92 153 92 205 92 257 92 309 92 361 92412 92 464 92 516 92 567 92 619 92 670 92 722 92 773 92 824 92 875 92 927 8 90 352 90 407 90 461 90 515 90 569 90 623 90 677 90 730 90 784 90 838 90 891 90 945 90 998 91 052 91 105 91 158 91 212 91 265 91 318 91 371 91 424 91.477 91 529 91 582 91 635 91 687 91 740 91 793 91 845 91 897 91 950 92 002 92 054 92 106 92 158 92210 92 262 92 314 92 366 92418 92 469 92 521 92572 92 624 92 675 92 727 92 778 92 829 92 881 92 932 9 90 358 90 412 90 466 90 520 90 574 90 628 90 682 90 736 90 789 90 843 90 897 90 950 91 004 91 057 91 110 91164 91 217 91 270 91 323 91 376 91 429 91482 91 535 91 587 91 640 91 693 91 745 91 798 91 850 91 903 91 955 92 007 92 059 92 111 92 163 92 215 92 267 92 319 92 371 92 423 92 474 92 526 92 578 92 629 92 681 92 732 92 783 92 834 92 886 92 937 4 91 908 91913 91 960 91965 92012 92018 92065 92070 92 117 92 122 92 169 92174 92 221 92 226 92 273 92 278 92324 92330 92376 92381 92 428 92 480 92 531 92 583 92 634 92 686 92 737 92 788 92 840 92891 92 433 92 485 92 536 92 588 92 639 92 691 92 742 92 793 92 845 92 896 92 701 92 752 92 804 92 855 92 906 92 706 92 758 92 809 92 860 92911 92942 92947 92952 92957 92962 0 1 2 3 4 92967 92973 92978 92983 92988 5 6 7 8 9 t. 800-850 850-900 N 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 N 0 92 942 92 993 93 044 93 095 93 146 93 197 93 247 93 298 93 349 93 399 93 450 93 500 93 551 93 601 93 651 93 702 93 752 93 802 93 852 93 902 93 952 94 002 94 052 94 101 94 151 94 201 94 250 94 300 94 349 94 399 1 92 947 92 998 93 049 93 100 93 151 93 202 93 252 93 303 93354 93 404 93 455 93 505 93 556 93 606 93 656 93 707 93 757 93 807 93 857 93 907 93 957 94 007 94 057 94 106 94 156 94 206 94 255 94 305 94354 94404 2 3 92952 92957 93 003 93 008 93054 93059 93 105 93 110 93 156 93 161 93 207 93 258 93 308 93 359 93 409 93 460 93 510 93 561 93 611 93 661 93 712 93 762 93 812 93 862 93 912 93 962 94 012 94 062 94 111 94 161 94 211 94 260 94 310 94359 94 409 94458 94 507 94557 94 606 94 655 94 704 94 753 94 802 94 851 94 900 93 212 93 263 93 313 93 364 93 414 93 465 93 515 93'566 93 616 93 666 93 717 93 767 93 817 93 867 93 917 93 967 94 017 94 067 94 116 94 166 94 216 94 265 94315 94364 94414 94 463 94 512 94 562 94 611 94 660 94 709 94 758 94 807 94856 94 905 4 92 962 93 013 93 064 93 115 93 166 93 217 93 268 93 318 93 369 93 420 93 470 93 520 93 571 93 621 93 671 93 722 93 772 93 822 93 872 93 922 93 972 94 022 94 072 94 121 94 171 94221 94 270 94 320 94 369 94 419 94 468 94 517 94 567 94 616 94 665 94 714 94 763 94 812 94 861 94910 94 959 95 007 95 056 95 105 95 153 95 202 95 250 95 299 95 347 95 395 5 92 967 93 018 93 069 93 120 93 171 93 222 93273 93 323 93374 93 425 93 475 93 526 93 576 93 626 93 676 93 727 93 777 93 827 93 877 93 927 93 977 94 027 94077 94 126 94 176 94 226 94 275 94 325 94 374 94 424 6 92 973 93 024 93 075 93 125 93 176 93 227 93 278 93 328 93 379 93 430 93 480 93 531 93 581 93 631 93 682 93 732 93 782 93 832 93 882 93 932 93 982 94 032 94 082 94 131 94 181 94 231 94 280 94 330 94 379 94 429 7 8 92978 92983 93 029 93 034 93 080 93 085 93 131 93 136 93 181 93 186 93 232 93 237 93 283 93 288 93 334 93 339 93 384 93 389 93 435 93 440 93485 93490 93 536 93 541 93 586 93 591 93 636 93 641 93 687 93 692 93 737 93 742 93 787 93 792 93 837 93 842 93 887 93 892 93'937 93 942 93 987 93 992 94037 94042 94086 94091 94 136 94 141 94 186 94 191 94 236 94 240 94 285 94 290 94 335 94 340 94384 94389 94433 94438 94 483 94 488 94532 94537 94581 94586 94630 94635 94680 94685 94 729 94 734 94 778 94 783 94827 94832 94876 94880 94924 94929 94973 94978 95 022 95 027 95 071 95 075 95 119 95 124 95 168 95 173 95 216 95 221 95 265 95 270 95 313 95 318 95 361 95 366 95 410 95 415 9 92 988 93 039 93 090 93 141 93 192 93 242 93 293 93 344 93 394 93 445 93 495 93 546 93 596 93 646 93 697 93 747 93 797 93 847 93 897 93 947 93 997 94 047 94 096 94 146 94 196 94 245 94 295 94 345 94 394 94 443 94 493 94 542 94 591 94 640 94 689 94 738 94 787 94836 94 885 94 934 94 983 95 032 95 080 95 129 95 177 95 226 95 274 95 323 95 371 95 419 95 468 9 1 94448 94453 94498 94503 94547 94552 94596 94601 94645 94650 94 694 94 699 94 743 94 748 94 792 94 797 94 841 94846 94 890 94 895 94473 94478 94522 94527 94 571 94576 94621 94626 94 670 94675 94719 94724 94 768 94 773 94817 94822 94866 94871 94915 94919 94939 94944 9 949 94954 94988 94993 94998 95 002 95 036 95 041 95 046 95 051 95 085 95 090 95 095 95 100 95 134 95 139 95 143 95 148 95 182 95 187 95 231 95 236 95 279 95 284 95 328 95 332 95 376 95 381 95 192 95 240 95 289 95 337 95 386 95 197 95 245 95 294 95 342 95 390 94 963 95 012 95 061 95 109 95 158 95 207 95 255 95 303 95 352 95 400 95 448 5 94 968 95 017 95 066 95 114 95 163 95 211 95 260 95 308 95 357 95 405 95 424 95 429 95 434 95 439 95 444 0 1 2 3 4 95 453 95 458 6 7 95 463 8 850-900 18 900-950 N 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 N 0 95 424 95 472 95 521 95 569 95 617 95 665 95 713 95 761 95 809 95 856 95 904 95 952 95 999 96 047 96 095 96 142 96 190 96 237 96 284 96 332 96 379 96 426 96 473 96 520 96 567 96 614 96 661 96 708 96 755 96 802 96 848 96 895 96 942 96 988 97 035 97 081 97 128 97 174 97 220 97 267 97 313 97 359 97 405 97 451 97 497 97543 97589 97635 97 681 97 727 97 772 0 1 95 429 95 477 95 525 95 574 95 622 95 670 95 718 95 766 95 813 95 861 95 909 95 957 96 004 96 052 96 099 96 147 96 194 96 242 96 289 96 336 96 384 96 431 96478 96 525 96 572 96 619 96 666 96 713 96 759 96 806 96 853 96 900 96 946 96 993 97 039 97 086 97 132 97 179 97 225 97 271 97 317 97 364 97 410 97 456 97502 97 548 97 594 97 640 97 685 97 731 97 777 1 2 3 4 95 434 95 439 95 444 95 482 95 487 95 492 95 530 95 535 95 540 95 578 95 583 95 588 95 626 95 631 95 636 95 674 95 722 95 770 95 818 95 866 95 914 95 961 96 009 96 057 96 104 96152 96 199 96 246 96 294 96341 96 388 96 435 96 483 96 530 96577 96 624 96 670 96 717 96 764 96811 96858 96 904 96 951 96 997 97 044 97 090 97 137 97 183 97 230 97 276 97 322 97 368 97 414 97 460 97 506 97 552 97 598 97 644 97 690 97 736 97 782 2 95 679 95 727 95 775 95 823 95 871 95 918 95 966 96 014 96 061 96 109 96 156 96 204 96 251 96 298 96 346 96 393 96 440 96 487 96 534 96 581 96 628 96 675 96 722 96 769 96 816 96 862 96 909 96956 97 002 97 049 97 095 97 142 97 188 97 234 97 280 97 327 97373 97 419 97 465 97 511 97 557 97 603 97 649 97 695 97 740 97 786 3 95 684 95 732 95 780 95 828 95 875 95 923 95 971 96 019 96 066 96 114 96 161 96 209 96 256 96 303 96350 96 398 96 445 96 492 96 539 96586 96633 96 680 96 727 96 774 96 820 96 867 96 914 96 960 97 007 97 053 97 100 97 146 97 192 97 239 97 285 97331 97 377 97 424 97 470 97 516 97 562 97 607 97 653 97 699 97 745 97 791 4 5 95 448 95 497 95 545 95 593 95 641 95 689 95 737 95 785 95 832 95 880 95 928 95 976 96 023 96 071 96 118 96 166 96 213 96 261 96 308 96 355 96 402 96 450 96 497 96 544 96 591 96 638 96 685 96 731 96 778 96 825 96 872 96 918 96 965 97 011 97 058 97 104 97 151 97 197 97 243 97 290 97 336 97 382 97 428 97 474 97 520 97 566 97 612 97 658 97 704 97 749 97 795 5 6 95 453 95 501 95 550 95 598 95 646 95 694 95 742 95 789 95 837 95 885 95 933 95 980 96 028 96 076 96 123 96 171 96 218 96 265 96 313 96 360 96 407 96 454 96 501 96 548 96 595 96 642 96 689 96 736 96 783 96 830 96 876 96 923 96 970 97 016 97 063 97 109 97 155 97 202 97 248 97 294 97 340 97 387 97 433 97 479 97 525 97 571 97 617 97 663 97 708 97 754 97 800 6 7 8 9 95 458 95 463 95 468 95 506 95 511 95 516 95 554 95 559 95 564 95 602 95 607 95 612 95 650 95 655 95 660. 95 698 95 703 95 708 95 746 95 751 95 756 95 794 95 799 95 804 95 842 95 847 95 852 95 890 95 895 95 899 95 938 95 942 95 947 95 985 95 990 95 995 96033 96038 96042 96080 96 085 96 090 96 128 96 133 96 137 96 175 96 180 96 185 96223 96227 96232 96270 96275 96280 96317 96322 96327 96365 96369 96374 96412 96417 96421 96459 96464 96468 96506 96511 96515 96553 96558 96562 96 600 96 605 96 609 96647 96652 96656 96694 96699 96703 96 741 96 745 96 750 96 788 96 792 96 797 96 834 96 839 96 844 96 881 96 886 96890 96928 96932 96937 96974 96979 96984 97 021 97 025 97 030 97067 97072 97077 97 114 97 118 97 123 97 160 97 165 97 169 97 206 97 211 97 216 97253 97257 97 262 97 299 97 304 97 308 97345 97350 97354 97 391 97 396 97 400 97437 97442 97447 97483 97488 97493 97 529 97 534 97 539 97575 97580 97585 97621 97626 97630 97667 97.672 97676 97 713 97 717 97 722 97 759 97 763 97 768 97804 97809 97813 7 8 9 900-950 960-1000 19 rl I --ri" ` -i-;cp-- 'g —"'- —ii ---- — _~ ~qll -I N 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992, 993 994 995 996 997 998 999 1000 N 0 1 97 772 97 777 97818 97823 97 864 97 868 97909 97914 97 955 97 959 2 97 782 97 827 97 873 97 918 97 964 3 97 786 97 832 97 877 97 923 97 968 4 97 791 97 836 97 882 97 928 97 973 98 000 98 046 98091 98005 98009 98050 98055 98096 98100 98 137 98 141 98 146 98 182 98 186 98 191 98227 98 272 98 318 98 363 98 408 98 453 98 498 98 543 98588 98 632 98 677 98 722 98 767 98 811 98856 98 900 98945 98 989 99 034 99 078 99 123 99 167 99 211 99 255 99 300 99 344 99388 99 432 99 476 99 520 99 564 99 607 99 651 99695 99 739 99 782 99 826 99 870 99 913 99 957 00 000 0 98232 98236 98277 98281 98322 98327 98367 98372 98412 98417 98457 98462 98 502 98 507 98547 98552 98592 98597 98637 98641 98682 98686 98726 98731 98771 98776 98816 98820 98 860 98 865 98905 98909 98949 98954 98994 98 998 99038 99043 99083 99 087 99 127 99131 99 171 99 176 99216 99220 99 260 99 264 99304 99308 99348 99352 99392 99396 99436 99441 99480 99484 99524 99528 99568 99572 99612 99616 99656 99660 99 699 99 704 99 743 99 747 99 787 99 791 99830 99835 99874 99878 99 917 99 922 99961 99965 98014 98019 98059 98064 98 105 98 109 98 150 98155 98 195 98 200 98 241 98 245 98 286 98 290 98331 98336 98376 98381 98 421 98 426 98466 98471 98511 98516 98556 98561 98 601 98 605 98646 98650 98691 98695 98 735 98 740 98 780 98 784 98825 98829 98869 98874 5 97 795 97 841 97 886 97932 97 978 98 023 98 068 98 114 98 159 98 204 98 250 98 295 98 340 98 385 98 430 98 475 98520 98 565 98 610 98 655 98 700 98 744 98 789 98 834 98 878 98 923 98 967 99012 99 056 99 100 99 145 99 189 99 233 99 277 99 322 99 366 99 410 99 454 99 498 99 542 99585 99 629 99 673 99 717 99 760 99 804 99 848 99 891 99 935 99 978 98028 98032 98073 98 078 98 118 98 123 98 164 98 168 98209 98214 98 254 98 259 98 299 98 304 98345 98349 98390 98394 98435 98439 98 480 98 484 98 525 98 529 98570 98574 98614 98619 98659 98664 6 97 800 97 845 97 891 97937 97 982 7 97 804 97 850 97 896 97 941 97 987 98 914 98 958 99 003 99 047 99 092 99 136 99 180 99 224 99 269 99 313 99357 99 401 99445 99 489 99 533 99 577 99 621 99 664 99 708 99752 99 795 99 839 99 883 99 926 99970 98 918 98 963 99 007 99 052 99 096 99 140 99 185 99 229 99 273 99 317 99 361 99 405 99 449 99 493 99 537 99 581 99 625 99 669 99 712 99756 99 800 99 843 99 887 99 930 99 974 98 704 98 749 98 793 98 838 98 883 98 927 98 972 99 016 99 061 99 105 99 149 99 193 99 238 99 282 99 326 99 370 99 414 99 458 99 502 99 546 99 590 99 634 99 677 99 721 99 765 99 808 99852 99 896 99 939 99 983 98 709 98 753 98 798 98 843 98 887 98 932 98 976 99 021 99 065 99 109 99 154 99 198 99 242 99 286 99 330 99 374 99 419 99 463 99506 99 550 99 594 99 638 99 682 99 726 99 769 99 813 99856 99 900 99 944 99 987 8 9 97 809 97 813 97855 97859 97 900 97 905 97 946 97 950 97 991 97996 98037 98 041 98082 98087 98 127 98 132 98 173 98177 98218 98223 98 263 98 268 98308 98313 98354 98358 98 399 98 403 98 444 98 448 98 489 98 493 98 534 98 538 98 579 98583 98 623 98628 98 668 98 673 98 713 98 717 98758 98762 98 802 98 807 98847 98851 98 892 98 896 98936 98941 98981 98985 99025 99029 99069 99074 99114 99118 99 158 99162 99 202 99 207 99247 99251 99291 99295 99335 99339 99379 99383 99423 99427 99467 99471 99511 99515 99555 99559 99599 99603 99642 99647 99686 99691 99 730 99 734 99 774 99 778 99817 99822 99861 99865 99904 99909 99 948 99 952 99991 99996 000004 00009 00013 00017 1 2 3 4 00022 00026 00030 00035 00039 5 6 7 8 9 0 950-1000 20 TABLE II. -LOGARITHMS OF CONSTANTS. log Circumference of the Circle in degrees............ = 360 2. 55 630 250 Circumference of the Circle in minutes............ = 21 600 4. 33 445 375 Circumference of the Circle in seconds............. = 1 296 000 6. 11 260 500 If the radius r = 1, half the Circuinference of the Circle is r = 3. 14 159 265 358 979 323 846 264 338 328................. 0. 49 714 987 Also: log log 2r= 6.28318531 0.79 817 987 2 = 9. 86 960 440 0. 99 429 975 4 wr = 12. 56 637 061 1. 09 920 986 1 =0.10 132 118 9. 00 570 025 - 10?7= 1.57079633 0.19 611988 72 2 1. 77 2435 3 0.24857 494 = 1. 04 719 755 0. 02 002 862 3 = 0. 56 418 958 9. 75 142 506 - 10 4 = 4.18 879 020 0. 62 208 861 /7 l3 1 = 0. 97 720 502 9. 98 998 569 - 10 =r 0. 78 539 816 9.89 508988 - 10 32 W7= 0.52359878 9.71899 862 - 10 =1.12837917 0. 05245 506 1 0. 31830 989 9. 50 285 013-10 -/ = 1.46459189 0. 16571662 7r = 0.15 915 494 9 20 182 013 - 10 = 0. 68 278 406 9. 83 428 338 - 10 0.15915494 9.20182013- 10 2r. = 0. 95 492 966 9. 97 997 138 - 10 r = 214 502 940 0. 33 143 325 r.3 4 =. 27323954 0.0491012 - 0.62 035 049 9. 79 263 713 - 10 0. 23 873 241 9.37 791 139 - 10 T = 0. 80 599 598 9.90 633 287- 10 _r4 7 ____ ____ ____________6__ I Arc a, whose length is equal to the radius r, is: in degrees. a~......= 57. 29 577 951~. 7r in minutes...... a..... 10800 = 3 437. 74 677' 7r in seconds...... a"..... = 206 264. 806I Ir Arc 2 a, whose length is equal to twice the radius, 2 r, is: in degrees...... 2 a~ 360........= 114. 59 155 903~ 7r 21 600 in minutes. a'..... 21. = 6 875. 49 354' 7r in seconds...... 1 2906.. 412 529. 612".. 7r If the radius r = 1, the length of the arc is: for 1 degree..... 0......... 01 745 329.. a~ 180 1 _ _ _ _ _ _ for 1 minute........... = 0. 00 029 089... a' 10 800 for 1 second..... -..... = 0. 000 485... a" 648 000 1 7r for ~ degree.... 0.00 872 665... 2 a -.... 60 for minute..... 1 -= -...... 00 014 544... 2 a' 21 600 for second....... 0. 00 000 242.. 2 a" 1 296 000 Sin 1" in the unit circle....................= 0. 00 000 485.. log 1. 75 812 263 3. 53 627 388 5. 31 442 513 2. 05 915 263 3.83 730 388 5. 61 545 513 8.24 187 737 — 10 6.46 372 61Z - 10 4. 68 557 487 — 10 7.94 084 737 - 10 6.16 269 612 - 10 4.38 454 487 - 10 4.68 557 487 - 10 0 I-Gl r I 21 TABLE III, THE LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS: From 0~ to 0~ 3', or 89~ 57' to 90~, for every second; From 0~ to 2~, or 88~ to 90~, for every ten seconds; From 1~ to 89~, for every minute. NOTE. To all the logarithms -10 is to be appended. log sin 0 log tan log sin log Sin 0 log cos =10, O0 000 Pr Of 1' 2'- I I 0t 1 2' I I 0 - 6.46 373 6.76 476 60 30 6.16270 6.63982 6.86 167 30 1 4. 68 557 6.47 090 6.76 836 59 31 6.17694 6.64 462 6. 86455 29 2 4. 98 660 6. 47 797 6. 77 193 58 32 6.19072 6.64936 6.86742 28 3 5.16 270 6.48 492 6. 77 548 57 33 6.20409 6.65 406 6. 87027 27 4 5.28 763 6. 49 175 6. 77 900 56 34 6.21 705 6.65 870 6.87310 26 5 5.38454 6.49849 6.78248 55 35 6.22964 6.66330 6.87591 25 6 5.46 373 6. 50 512 6. 78 595 54 36 6.24188 6.66785 6.87870 24 7 5.53 067 6. 51 165 6.78 938 53 37 6.25378 6.67235 6. 88 147 23 8 5.58 866 6.51 808 6.79278 52 38 6.26536 6.67680 6. 88423 22 9 5.63 982 6. 52 442 6.79 616 51 39 6.27664 6. 68 121 6.88697 21 10 5.68557 6.53067 6.79952 50 40 6.28 763 6. 68557 6.88969 20 11 5.72 697 6. 53 683 6.80 285 49 41 6.29836 6.68990 6.89240 19 12 5.76 476 6. 54 291 6.80 615 48 42 6.30882 6.69418 6.89509 18 13 5.79 952 6. 54 890 6. 80 943 47 43 6.31 904 6. 69841 6. 89776 17 14 5.83170 6.55481 6.81268 46 44 6.32903 6. 70261 6.90042 16 15 5.86167 6.56064 6.81591 45 45 6.33879 6.70676 6.90306 15 16 5.88969 6.56639 6.81911 44 46 6.34833 6.71088 6.90568 14 17 5. 91 602 6. 57 207 6. 82 230 43 47 6. 35 767 6. 71 496 6.90 829 13 18 5. 94 085 6. 57 767 6. 82 545 42 48 6.36 682 6. 71 900 6.91 088 12 19 5.96 433 6. 58 320 6. 82 859 41 49 6.37577 6.72300 6.91346 11 20 5.98660 6.58866 6.83170 40 50 6.38454 6.72697 6.91602 10 21 6.00779 6. 59 406 6.83 479 39 51 6.39315 6. 73090 6.91 857 9 22 6.02800 6.59939 6.83 786 38 52 6.40158 6. 73479 6. 92 110 8 23 6. 04 730 6. 60 465 6. 84 091 37 53 6. 40 985 6. 73 865 6. 92 362 7 24 6. 06 579 6.60 985 6.84 394 36 54 6.41 797 6. 74248 6.92612 6 25 6.08351 6.61499 6.84694 35 55 6.42594 6.74627 6.92861 5 26 6.10055 6. 62 007 6. 84 993 34 56 6.43376 6.75003 6.93 109 4 27 6.11694 6.62509 6.85289 33 57 6.44145 6.75 376 6.93355 3 28 6. 13 273 6. 63 006 6. 85 584 32 58 6.44900 6.75746 6.93 599 2 29 6.14 797 6.63 496 6. 85 876 31 59 6. 45 643 6. 76112 6.93843 1 30 6.16270 6.63982 6.86167 30 60 6.46373 6.76476 6.94085 0 " 59 8 57' t 58' 57' " I I I,a a log cot = log cos log sin = 10. 00 000 89~ log cos 22 0O 0 0 f ff, 0 0 10 20 30 40 50 1 0 10 20 30 40 50 2 0 10 20 30 40 50 3 0 10 20 30 40 50 4 0 10 20 30 40 50 5 0 10 20 30 40 50 6 0 10 20 30 40 50 7 0 10 20 30 40 50 8 0 10 20 30 40 50 9 0 10 20 30 40 50 100 p ff log sin 5.68 557 5.98660 6. 16 270 6. 28 763 6. 38 454 6.46 373 6. 53 067 6. 58 866 6. 63 982 6. 68 557 6. 72 697 6. 76 476 6. 79 952 6. 83 170 6. 86 167 6. 88 969 6.91 602 6. 94 085 6. 96 433 6. 98 660 7. 00 779 7. 02 800 7.04 730 7. 06 579 7. 08351 7.10055 7.11 694 7. 13 273 7. 14 797 7. 16 270 7.17 694 7. 19 072 7. 20 409 7. 21 705 7. 22 964 7. 24 188 7. 25 378 7. 26 536 7. 27 664 7. 28 763 7. 29 836 7. 30 882 7. 31 904 7.32 903 7. 33 879 7. 34 833 7. 35 767 7.36 682 7.37 577 7.38 454 7.39 314 7. 40 158 7.40 985 7. 41 797 7.42 594 7. 43 376 7.44 145 7. 44 900 7. 45 643 7. 46 373 log cos log tan 5. 68 557 5. 98 660 6. 16 270 6. 28 763 6. 38 454 6. 46 373 6. 53 067 6. 58 866 6. 63 982 6. 68 557 6. 72 697 6.76 476 6.79 952 6. 83 170 6. 86 167 6.'88 969 6. 91 602 6. 94 085 6. 96 433 6. 98 661 7. 00 779 7. 02 800 7. 04 730 7.06 579 7.08 352 7.10055 7. 11 694 7. 13 273 7.14 797 7. 16 270 7. 17 694 7. 19 073 7.20 409 7. 21 705 7. 22 964 7. 24 188 7. 25 378 7.26 536 7. 27 664 7. 28 764 7. 29 836 7. 30 882 7. 31 904 7. 32 903 7. 33 879 7. 34 833 7. 35 767 7.36 682 7. 37 577 7.38455 7. 39 315 7. 40 158 7. 40 985 7. 41 797 7. 42 594 7. 43 376 7. 44 145 7.44 900 7.45 643 7.46373 log cot log cos 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10 00000 10.00000 10.00000 1000000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 log sin yf f 060 50 40 30 20 10 059 50 40 30 20 10 058 50 40 30 20 10 057 50 40 30 20 10 056 50 40 30 20 10 055 50 40 30 20 10 054 50 40 30 20 10 053 50 40 30 20 10 052 50 40 30 20 10 051 50 40 30 20 10 050,.6; t!! r i ftt 100 10 20 30 40 50 110 10 20 30 40 50 120 10 20 30 40 50 130 10 20 30 40 50 140 10 20 30 40 50 150 10 20 30 40 50 160 10 20 30 40 50 170 10 20 30 40 50 180 10 20 30 40 50 190 10 20 30 40 50 200 f ft log sin 7. 46 373 7. 47 090 7.47 797 7. 48 491 7. 49 175 7.49 849 7. 50 512 7. 51 165 7. 51 808 7. 52 442 7.53 067 7. 53 683 7. 54 291 7. 54 890 7. 55 481 7. 56 064 7. 56 639 7. 57 206 7. 57 767 7. 58 320 7. 58 866 7. 59 406 7.59 939 7. 60 465 7. 60 985 7. 61 499 7. 62 007 7. 62 509 7. 63 006 7. 63 496 7.63 982 7.64 461 7. 64 936 7. 65 406 7.65 870 7.66 330 7. 66 784 7. 67 235 7. 67 680 7. 68 121 7. 68 557 7. 68 989 7. 69 417 7. 69 841 7. 70 261 7. 70 676 7. 71 088 7. 71 496 7. 71900 7. 72 300 7. 72 697 7. 73 090 7. 73 479 7. 73 865 7. 74 248 7. 74 627 7. 75 003 7. 75 376 7. 75 745 7. 76 112 7. 76 475 log cos log tan 7. 46 373 7. 47 091 7.47 797 7. 48 492 7.49 176 7. 49 849 7.50512 7. 51165 7.51809 7. 52 443 7. 53 067 7. 53 683 7.54 291 7. 54 890 7. 55 481 7. 56 064 7. 56 639 7. 57 207 7. 57 767 7. 58 320 7. 58 867 7. 59 406 7. 59 939 7. 60 466 7. 60 986 7. 61 500 7. 62 008 7.62 510 7. 63 006 7. 63 497 7. 63 982 7. 64 462 7. 64 937 7. 65 406 7. 65 871 7. 66 330 7. 66 785 7. 67 235 7. 67 680 7. 68 121 7. 68 558 7. 68 990 7. 69 418 7. 69 842 7. 70 261 7.70677 7. 71 088 7. 71 496 7. 71 900 7. 72 301 7. 72 697 7. 73 090 7. 73 480 7. 73 866 7. 74 248 7. 74 628 7.75 004 7. 75 377 7. 75 746 7.76 113 7. 76 476 log cot log cos 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 9.99999 9.9999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9.99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9.99 999 9. 99 999 9. 99 999 9. 99 999 log sin ff f 050 50 40 30 20 10 049 50 40 30 20 10 048 50 40 30 20 10 047 50 40 30 20 10 046 50 40 30 20 10 045 50 40 30 20 10 044 50 40 30 20 10 043 50 40 30 20 10 042 50 40 30 20 10 041 50 40 30 20 10 040 if p L I I. 0 L I. 0 89~ 0O I *! f! 200 10 20 30 40 50 210 10 20 30 40 50 220 10 20 30 40 50 230 10 20 30 40 50 240 10 20 30 40 50 250 10 20 30 40 50 260 10 20 30 40 50 270 10 20 30 40 50 280 10 20 30 40 50 290 10 20 30 40 50 300,,t log sin 7. 76 475 7. 76 836 7. 77 193 7. 77 548 7. 77 899 7. 78 248 7.78 594 7. 78 938 7. 79 278 7. 79 616 7.79 952 7. 80 284 7.80615 7. 80 942 7. 81 268 7. 81 591 7.81 911 7. 82 229 7. 82 545 7. 82 859 7. 83 170 7. 83 479 7. 83 786 7. 84 091 7. 84 393 7. 84 694 7. 84 992 7. 85 289 7. 85 583 7. 85 876 7. 86 166 7.86455 7. 86 741 7. 87 026 7. 87 309 7. 87 590 7. 87 870 7. 88 147 7. 88 423 7. 88 697 7. 88 969 7. 89 240 7. 89 509 7. 89 776 7. 90 041 7. 90 305 7. 90 568 7. 90 829 7. 91 088 7. 91 346 7. 91 602 7.91 857 7.92 110 7. 92 362 7. 92 612 7.92 861 7. 93 108 7. 93 354 7. 93 599 7. 93 842 7.94 084 log cos log tan 7. 76 476 7. 76 837 7. 77 194 7. 77 549 7. 77 900 7. 78 249 7. 78 595 7. 78 938 7. 79 279 7. 79 617 7. 79 952 7. 80 285 7. 80 615 7. 80 943 7. 81 269 7. 81 591 7. 81 912 7. 82 230 7. 82 546 7. 82 860 7. 83 171 7. 83 480 7. 83 787 7. 84 092 7. 84 394 7. 84 695 7. 84 994 7. 85 290 7. 85 584 7. 85 877 7. 86 167 7. 86 456 7. 86 743 7. 87 027 7. 87 310 7. 87 591 7. 87 871 7. 88 148 7. 88 424 7. 88 698 7. 88 970 7.89 241 7. 89 510 7. 89 777 7. 90 043 7. 90 307 7. 90 569 7. 90 830 7. 91 089 7. 91 347 7. 91 603 7. 91 858 7.92 111 7. 92 363 7. 92 613 7. 92 862 7.93 110 7. 93 356 7. 93 601 7. 93 844 7. 94 086 log cot log cos 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9.99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9.99 999 9. 99 999 9.99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99,999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 999 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 log sin 040 50 40 30 20 10 039 50 40 30 20 10 038 50 40 30 20 10 037 50 40 30 20 10 036 50 40 30 20 10 0 35 50 40 30 20 10 034 50 40 30 20 10 033 50 40 30 20 10 032 50 40 30 20 10 031 50 40 30 20 10 030 t tf I 300 10 20 30 40 50 310 10 20 30 40 50 320 10 20 30 40 50 330 10 20 30 40 50 340 10 20 30 40 50 35 0 10 20 30 40 50 36 0 10 20 30 40 50 370 10 20 30 40 50 380 10 20 30 40 50 390 10 20 30 40 50 400 t ff log sin 7. 94 084 7. 94 325 7. 94 564 7. 94 802 7. 95 039 7. 95 274 7. 95 508 7. 95 741 7. 95 973 7. 96 203 7. 96 432 7. 96 660 7. 96 887 7. 97 113 7. 97 337 7.97 560 7. 97 782 7. 98 003 7. 98 223 7. 98 442 7. 98 660 7. 98 876 7. 99 092 7.99 306 7. 99 520 7. 99 732 7. 99 943 8. 00 154 8. 00 363 8. 00 571 8.00 779 8. 00 985 8. 01 190 8. 01 395 8. 01 598 8. 01 801 8. 02 002 8. 02 203 8. 02 402 8.02601 8. 02 799 8. 02 996 8. 03 192 8. 03 387 8. 03 581 8.03 775 8. 03 967 8. 04 159 8. 04 350 8.04 540 8. 04 729 8.04 918 8. 05 105 8. 05 292 8. 05 478 8. 05 663 8. 05 848 8. 06 031 8. 06 214 8. 06 396 8. 06 578 log cos 0 II log tan 7. 94 086 7. 94 326 7. 94 566 7. 94 804 7. 95 040 7. 95 276 7.95 510 7. 95 743 7. 95 974 7. 96 205 7. 96 434 7. 96 662 7. 96 889 7. 97 114 7.97 339 7. 97 562 7. 97 784 7. 98 005 7. 98 225 7. 98 444 7. 98 662 7.98 878 7.99 094 7. 99 308 7. 99 522 7. 99 734 7. 99 946 8. 00 156 8. 00 365 8.00 574 8.00 781 8. 00 987 8.01 193 8. 01 397 8.01 600 8. 01 803 8. 02 004 8. 02 205 8. 02 405 8. 02 604 8. 02 801 8.02 998 8. 03 194 8. 03 390 8. 03 584 8. 03 777 8. 03 970 8. 04 162 8. 04 353 8. 04 543 8.04 732 8. 04 921 8. 05 108 8. 05 295 8. 05 481 8. 05 666 8. 05 851 8. 06 034 8. 06 217 8. 06 399 8.06 581 log cot log cos 9. 99 998 9.99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9.99 998 9. 99 998 9. 99 998 9.99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9.99 998 9. 99 998 9. 99 998 9.99 998 9.99 998 9. 99 998 9. 99 998 9.99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 998 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9.99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9.99 997 9. 99 997 9. 99 997 9. 99 997 9.99 997 9. 99 997 log sin ff f 030 50 40 30 20 10 029 50 40 30 20 10 028 50 40 30 20 10 027 50 40 30 20 10 026 50 40 30 20 10 025 50 40 30 20 10 024 50 40 30 20 10 023 50 40 30 20 10 022 50 40 30 20 10 021 50 40 30 20 10 020 ff f I 0 i m J 89~ 24 O0 0 a t " log sin 400 8.06 578 10 8.06 758 20 8. 06 938 30 8.07 117 40 8. 07 295 50 8.07 473 410 8.07 650 10 8.07 826 20 8.08 002 30 8.08 176 40 8. 08350 50 8.08 524 420 8.08696 10 8. 08 868 20 8.09 040 30 8.09 210 40 8.09 380 50 8.09550 430 8.09718 10 8.09 886 20 8.10054 30 8. 10 220 40 8.10 386 50 8. 10552 440 8.10 717 10 8.10881 20 8.11044 30 8.11 207 40 8.11370 50 8.11 531 450 8.11693 10 8.11853 20 8.12 013 30 8.12 172 40 8.12 331 50 8.12 489 460 8.12647 10 8.12 804 20 8.12 961 30 8.13 117 40 8.13 272 50 8.13 427 470 8.13581 10 8.13 735 20 8.13 888 30 -8.14 041 40 8.14 193 50 8.14 344 480 8.14 495 10 8.14 646 20 8. 14 796 30 8.14 945 40 8. 15 094 50 8. 15 243 490 8.15 391 10 8. 15 538 20 8. 15 685 30 8. 15 832 40 8.15 978 50 8. 16 123 500 8.16268 tt 'log cos log tan 8. 06 581 8. 06 761 8. 06 941 8. 07 120 8. 07 299 8. 07 476 8.07 653 8. 07 829 8. 08 005 8.08 180 8. 08 354 8. 08 527 8. 08 700 8. 08 872 8. 09 043 8. 09 214 8.09 384 8. 09 553 8. 09 722 8. 09 890 8. 10 057 8.10 224 8. 10 390 8. 10 555 8. 10 720 8. 10 884 8. 11 018 8.11211,8. 11 373 8. 11 535 8.11 696 8.11 857 8. 12 017 8.12 176 8. 12 335 8.12 493 8. 12 651 8. 12 808 8. 12 965 8.13 121 8. 13 276 8. 13 431 8. 13 585 8. 13 739 8. 13 892 8. 14 045 8. 14 197 8. 14 348 8.14 500 8. 14 650 8.14 800 8. 14 950 8. 15 099 8. 15 247 8. 15 395 8. 15 543 8.15 690 8. 15 836 8. 15 982 8. 16 128 8. 16 273 log cot log cos 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9.99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 997 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9.99 996 9.99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9.99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 996 9. 99 995 9. 99 995 9. 99 995 log sin it P 020 50 40 30 20 10 019 50 40 30 20 10 018 50 40 30 20 10 017 50 40 30 20 10 016 50 40 30 20 10 015 -50 40 30 20 10 014 50 40 30 20 10 013 50 40 30 20 10 012 50 40 30 20 10 011 50 40 30 20 10 010 ft, 0 iI [i 500 10 20 30 40 50 510 10 20 30 40 50 520 10 20 30 40 50 530 10 20 30 40 50 540 10 20 30 40 50 550 10 20 30 40 50 560 10 20 30 40 50 570 10 20 30 40 50 580 10 20 30 40 50 590 10 20 30 40 50 600 P rP log sin 8. 16 268 8.16 413 8. 16 557 8. 16 700 8.16 843 8. 16 986 8. 17 128 8.17 270 8.17411.8. 17 552 8.17 692 8. 17 832 8. 17 971 8. 18 110 8. 18 249 8.18 387 8. 18 524 8. 18 662 8.18 798 8.18 935 8. 19 071 8.19 206 8. 19 341 8.19476 8.19 610 8. 19 744 8. 19 877 8. 20 010 8. 20 143 8. 20 275 8. 20 407 8. 20 538 8. 20 669 8. 20 800 8. 20 930 8.21 060 8. 21 189 8. 21 319 8. 21 447 8. 21 576 8. 21 703 8. 21 831 8. 21 958 8. 22 085 8.22 211 8. 22 337 8. 22 463 8. 22 588 8. 22 713 8. 22 838 8. 22 962 8. 23 086 8. 23 210 8. 23 333 8. 23 456 8. 23 578 8. 23 700 8. 23 822 8. 23 944 8. 24 065 8. 24 186 log cos log tan 8. 16 273 8.16417 8. 16 561 8. 16 705 8.16 848 8.16 991 8.17 133 8.17 275 8.17416 8.17 557 8. 17 697 8. 17 837 8. 17 976 8. 18 115 8. 18 254 8.18 392 8. 18 530 8. 18 667 8. 18 804 8. 18 940 8.19 076 8. 19 212 8. 19 347 8.19 481 8. 19 616 8. 19 749 8.19 883 8. 20 016 8. 20 149 8. 20 281 8. 20 413 8. 20 544 8. 20 675 8. 20 806 8. 20 936 8.21 066 8. 21 195 8. 21 324 8. 21 453 8. 21 581 8. 21 709 8. 21 837 8. 21 964 8. 22 091 8. 22 217 8. 22 343 8. 22 469 8. 22 595 8. 22 720 8. 22 844 8. 22 968 8. 23 092 8. 23 216 8. 23 339 8. 23 462 8. 23 585 8. 23 707 8. 23 829 8. 23 950 8.24071 8. 24 192 log cot log cos t t 9.99995 010 9. 99 995 50 9. 99995 40 9.99 995 30 9.99 995 20 9. 99 995 10 9.99995 0 9 9.99 995 50 9.99 995 40 9.99 995 30 9.99 995 20 9.99 995 10 9.99995 0 8 9. 99 995 50 9.99995 40 9. 99 995 30 9.99 995 20 9.99 995 10 9. 99 995 0 7 9.99 995 50 9.99 995 40 9. 99 995 30 9.99 995 20 9.99995 10 9.99995 0 6 9.99 995 50 9.99 995 40 9.99 995 30 9.99995 20 9. 99 994 10 9. 99 994 0 5 9.99 994 50 9.99 994 40 9.99 994 30 9.99 994 20 9.99 994 10 9.99 994 0 4 9.99 994 50 9. 99 994 40 9:99 994 30 9.99 994 20 9.99 994 10 9.99 994 0 3 9. 99 994 50 9.99 994 40 9. 99 994 30 9.99 994 20 9.99 994 10 9.99 994 0 2 9.99 994 50 9.99 994 40 9.99 994 30 9.99 994 20 9. 99 994 10 9.99 994 0 1 9. 99 994 50 9.99 994 40 9.99 993 30 9.99 993 20 9.99 993 10 9.99 993 0 0 log sin t t I zm a 89~ 16 25 I 06iaMw 1^ | ~~~~~ I _,WY ICB -~-iL~Il I —S~~PV-U~l l BIR m ' log sin log tan log cos? ' log sin log tan log cos?? 0 0 8.24 186 8.24 192 9.99 993 060 100 8.30 879 8.30 888 9.99 991 050 10 8.24306 8.24313 9.99993 50 10 8.30983 8.30992 9.99 991 50 20 8.24426 8.24433 9.99993 40 20 8.31086 8.31095 9.99991 40 30 8.24546 8.24553 9.99993 30 30 8.31188 8.31198 9.99991 30 40 8.24665 8.24672 9.99993 20 40 8.31291 8.31300 9.99991 20 50 8.24785 8.24791 9.99993 10 50 8.31393 8.31403 9.99-991 10 1 0 8.24903 8.24910 9.99993 059 110 8.31495 8.31505 9.99991 049 10 8.25022 8.25029 9.99993 50 10 8.31 597 8.31606 9.99991 50 20 8.25140 8.25147 9.99993 40 20 8.31699 8.31 708 9.99 991 40 30 8.25258 8.25265 9.99 993 30 30 8.31800 8.31809 9.99991 30 40 8.25 375 8. 25 382 9.99 993 20 40 8.31 901 8.31 911 9. 99 991 20 50 8.25493 8.25500 9.99993 10 50 8.32002 8.32012 9.99991 10 2 0 8.25609 8.25616 9.99993 058 120 8.32103 8.32112 9.99990 048 10 8.25726 8.25733 9.99993 50 10 8.32203 8.32213 9.99 990 50 20 8. 25 842 8. 25 849 9. 99 993 40 20 8 32 303 8.32 313 9. 99 990 40 30 8.25 958 8.25 965 9.99993 30 30 8.32403 8.32413 9.99990 30 40 8.26074 8.26081 9.99993 20 40 8.32503 8.32513 9.99990 20 50 8.26189 8.26196 9.99993 10 50 8.32602 8.32612 9.99990 10 3 0 8.26 8.26312 9.993 30 8.32702 8.32711 9.999930 047 10 8.26419 8.26426 9.99993 50 10 8.32801 8.32811 9.99 990 50 20 8.26533 8.26541 9.99993 40 20 8.32899 8.32909 9. 99 990 40 30 8. 26 648 8. 26 655 9.99 993 30 30 8.32998 8.33008 9.99990 30 40 8.26761 8.26769 9.99993 20 40 8.33096 8.33106 9.99990 20 50 8.26 875 8. 26 882 9. 99 993 10 50 8.33 195 8.33 205 9. 99 990 10 4 0 8.26 988 8.26 996 9. 99 992 056 140 8.33292 8.33302 9.99990 046 10 8. 27101 8.27 109 9. 99 992 50 10 8. 33 390 8.33 400 9.99 990 50 20 8.27 214 8. 27221 9. 99992 40 20 8.33 488 8.33 498 9. 99 990 40 30 8.27 326 8.27 334 9.99 992 30 30 8.33 585 8.33 595 9.99 990 30 40 8.27 438 8.27 446 9. 99 992 20 40 8.33 682 8.33 692 9.99 990 20 50 8.27550 8. 27 558 9.99 992 10 50 8.33 779 8.33 789 9.99 990 10 5 0 8. 27 661 8.27 669 9.99 992 055 150 8.33875 8.33886 9.99990 045 10 8. 27 773 8.27 780 9.99992 50 10 8.33 972 8.33 982 9.99 990 50 20 8.27 883 8. 27 891 9.99992 40 20 8.34 068 8.34 078 9.99 990 40 30 8.27 994 8.28 002 9. 99 992 30 30 8.34 164 8.34 174 9.99990 30 40 8.28 104 8.28 112 9. 99 992 20 40 8.34 260 8.34 270 9.99 989 20 50 8.28 215 8.28 223 9.99 992 10 50 8.34 355 8.34 366 9. 99 989 10 6 0 8.28324 8.28332 9.99992 054 160 8.34450 8.34461 9.99989 044 10 8.28 434 8.28 442 9. 99 992 50 10 8.34 546 8.34 556 9.99 989 50 20 8.28 543 8.28 551 9. 99 992 40 20 8.34 640 8.34 651 9.99 989 40 30 8.28 652 8. 28 660 9.99 992 30 30 8.34 735 8.34 746 9.99 989 30 40 8.28 761 8.28 769 9. 99 992 20 40 8.34 830 8. 34840 9.99 989 20 50 8.28 869 8.28 877 9. 99 992 10 50 8.34 924 8.34 935 9. 99 989 10 7 0 8.28977 8.28986 9.99992 053 170 8.35018 8.35029 9.99989 043 10 8. 29 085 8.29 094 9.99 992 50 10 8.35 112 8. 35123 9. 99 989 50 20 8.29 193 8.29 201 9. 99 992 40 20 8.35 206 8. 35 217 9.99 989 40 30 8. 29 300 8. 29 309 9.99 992 30 30 8.35 299 8.35 310 9.99 989 30 40 8.29 407 8. 29 416 9.99 992 20 40 8.35 392 8.35 403 9.99 989 20 50 8.29 514 8.29 523 9.99 992 10 50 8.35 485 8.35 497 9.99 989 10 8 0 8. 29 621 8.29629 9. 99 992 052 180 8.35578 8.35590 9.99989 042 10 8.29 727 8. 29 736 9.99991 50 10 8.35 671 8.35 682 9.99989 50 20 8.29833 8.29 842 9.99991 40 20 8.35 764 8. 35 775 9. 99 989 40 30 8. 29939 8. 29947 9. 99 991 30 30 8.35 856 8.35 867 9.99989 30 40 8.30044 8. 30053 9. 99991 20 40 8. 35 948 8. 35 959 9.99989 20 50 8.30 150 8.30 158 9.99 991 10 50 8.36 040 8.36 051 9.99 989 10 9 0 8.30 255 8. 30 263 9.99 991 0 51 190 8.36 131 8.36 143 9.99 989 041 10 8.30359 8.30368 9.99 991 50 10 8.36 223 8.36 235 9. 99 988 50 20 8.30464 8.30473 9. 99 991 40 20 8.36314 8.36326 9.99 988 40 30 8.30568 8.30 577 9.99 991 30 30 8.36405 8.36417 9.99 988 30 40 8.30 672 8.30681 9.99 991 20 40 8.36 496 8.36508 9.99 988 20 50 8.30776 8.30785 9.99 991 10 50 8. 36 587 8.36599 9.99 988 10 100 8.30879 8.30888 9.99991 050 200 8.36678 8.36689 9.99988 040 t f log cos log cot log sin f I' log os log cot' log sin ~f 88~ 26 I rI.... - II - 1 [- v I p! 200 10 20 30 40 50 210 10 20 30 40 50 220 10 20 30 40 50 230 10 20 30 40 50 240 10 20 30 40 50 250 10 20 30 40 50 260 10 20 30 40 50 270 10 20 30 40 50 280 10 20 30 40 50 290 10 20 30 40 50 300 f it log sin 8. 36 678 8.36 768 8. 36 858 8. 36 948 8. 37 038 8. 37 128 8.37 217 8. 37 306 8. 37 395 8. 37 484 8.37 573 8.37 662 8. 37 750 8. 37 838 8. 37 926 8.38 014 8. 38 101 8. 38 189 8.38 276 8. 38 363 8. 38 450 8. 38 537 8.38 624 8. 38 710 8. 38 796 8. 38 882 8;38 968 8.39 054 8. 39 139 8. 39 225 8.39 310 8. 39 395 8.39 480 8. 39 565 8.39 649 8.39 734 8. 39 818 8.39 902 8. 39 986 8. 40 070 8. 40 153 8. 40 237 8. 40 320 8. 40 403 8. 40 486 8. 40 569 8.40 651 8. 40 734 8. 40 816 8.40 898 8. 40 980 — 8. 41 062 8.41 144 8. 41 225 8. 41 307 8. 41 388 8. 41 469 8.41 550 8. 41 631 8.41711 8. 41 792 log cos log tan 8. 36 689 8.36780 8. 36 870 8. 36 960 8.37 050 8. 37 140 8. 37 229 8.37 318 8. 37 408 8. 37 497 8. 37 585 8. 37 674 8. 37 762 8. 37 850 8. 37 938 8.38 026 8. 38 114 8. 38 202 8. 38 289 8.38 376 8. 38 463 8. 38 550 8. 38 636 8. 38 723 8. 38 809 8.38 895 8. 38 981 8.39 067 8. 39 153 8. 39 238 8.39 323 8. 39 408 8. 39 493 8.39 578 8.39 663 8. 39 747 8. 39 832 8.39 916 8. 40 000 8. 40 083 8. 40 167 8.40 251 8. 40 334 8.40 417 8. 40 500 8. 40 583 8. 40 665 8. 40 748 8. 40 830 8. 40 913 8.40 995 8. 41 077 8. 41158 8. 41 240 8. 41 321 8. 41 403 8. 41 484 8. 41 565 8. 41 646 8. 41 726 8. 41 807 log cot log cos 9. 99 988 9. 99 988 9. 99 988 9. 99 988 9. 99 988 9. 99 988 9. 99 988 9. 99 988 9. 99 988 9. 99 988 9. 99 988 9.99 988 9. 99 988 9. 99 988 9. 99 988 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9. 99 987 9.99 987 9. 99 987 9. 99 987 9.99 987 9. 99 987 9. 99 987 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 986 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 log sin,t f 040 50 40 30 20 10 039 50 40 30 20 10 0 38 50 40 30 20 10 037 50 40 30 20 10 036 50 40 30 20 10 035 50 40 30 20 10 034 50 40 30 20 10 033 50 40 30 20 10 032 50 40 30 20 10 031 50 40 30 20 10 030 f fp I Il L q- I I elll I I -I AB f tt 300 10 20 30 40 50 310 10 20 30 40 50 320 10 20 30 40 50 330 10 20 30 40 50 340 10 20 30 40 50 350 10 20 30 40 50 360 10 20 30 40 50 370 10 20 30 40 50 380 10 20 30 40 50 390 10 20 30 40 50 400 f fp log sin 8.41 792 8. 41 872 8.41 952 8. 42 032 8. 42 112 8. 42 192 8. 42 272 8.42351 8. 42 430 8. 42 510 8. 42 589 8. 42 667 8. 42 746 8. 42 825 8. 42 903 8. 42 982 8.43 060 8. 43 138 8. 43 216 8. 43 293 8. 43 371 8. 43 448 8. 43 526 8. 43 603 8. 43 680 8. 43 757 8. 43 834 8. 43 910 8.43 987 8. 44 063 8.44 139 8. 44 216 8. 44 292 8. 44 367 8. 44 443 8.44 519 8.44 594 8. 44 669 8. 44 745 8. 44 820 8. 44 895 8. 44 969 8. 45 044 8.45 119 8. 45 193 8. 45 267 8. 45 341 8. 45 415 8. 45 489 8. 45 563 8.45 637 8. 45 710 8. 45 784 8.45 857 8. 45 930 8. 46 003 8. 46 076 8. 46 149 8. 46 222 8. 46 294 8. 46 366 log cos log tan 8. 41 807 8. 41 887 8. 41 967 8. 42 048 8. 42 127 8. 42 207 8. 42 287 8. 42 366 8. 42 446 8. 42 525 8. 42 604 8.42 683 8. 42 762 8. 42 840 8. 42 919 8.42 997 8.-43 075 8. 43 154 8. 43 232 8. 43 309 8. 43 387 8. 43 464 8. 43 542 8. 43 619 8. 43 696 8. 43 773 8. 43 850 8. 43 927 8. 44 003 8. 44 080 8. 44 156 8. 44 232 8. 44 308 8. 44 384 8. 44 460 8. 44 536 8.44 611 8. 44 686 8. 44 762 8. 44 837 8. 44 912 8. 44 987 8. 45 061 8.45 136 8. 45 210 8. 45 285 8. 45 359 8. 45 433 8. 45 507 8. 45 581 8. 45 655 8. 45 728 8. 45 802 8. 45 875 8. 45 948 8. 46 021 8. 46 094 8.46 167 8. 46 240 8.46312 8. 46 385 log cot log cos 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9. 99 985 9.99 985 9. 99 985 9. 99 984 9. 99 984 9. 99 984 9.99 984 9. 99 984 9.99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 984 9. 99 983 9. 99 983 9. 99 983 9.99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 983 9. 99 982 9. 99 982 9. 99 982 9. 99 982 9. 99 982 9. 99 982 9. 99 982 9.99 982 9. 99 982 9. 99 982 9. 99 982 9. 99 982 9. 99 982 9.99 982 9.99 982 log sin It! 030 50 40 30 20 10 029 50 40 30 20 10 028 50 40 30 20 10 027 50 40 30 20 10 026 50 40 30 20 10 025 50 40 30 20 10 024 50 40 30 20 10 023 50 40 30 20 10 022 50 40 30 20 10 021 50 40 30 20 10 020!!F P m j m - h h -L, _ I, 1 J 880 0 27 400 10 20 30 40 50 410 10 20 30 40 50 420 10 20 30 40 50 430 10 20 30 40 50 440 10 20 30 40 50 450 10 20 30 40 50 460 10 20 30 40 50 470 10 20 30 40 50 480 10 20 30 40 50 490 10 20 30 40 50 500 f pt log sin 8.46 366 8. 46 439 8.46 511 8. 46 583 8. 46 655 8. 46 727 8. 46 799 8. 46 870 8. 46 942 8. 47 013 8. 47 084 8.47 155 8. 47 226 8. 47 297 8. 47 368 8. 47 439 8. 47 509 8. 47 580 8. 47 650 8. 47 720 8. 47 790 8. 47 860 8. 47 930 8. 48 000 8. 48 069 8. 48 139 8. 48 208 8.48 278 8. 48 347 8. 48 416 8. 48 485 8.48 554 8. 48 622 8. 48 691 8. 48 760 8. 48 828 8. 48 896 8. 48 965 8. 49 033 8. 49 101 8.49 169 8. 49 236 8. 49 304 8.49 372 8. 49 4-39 8. 49 506 8. 49 574 8. 49 641 8. 49 708 8. 49 775 8. 49 842 8. 49 908 8.49 975 8.50 042 8. 50 108 8. 50 174 8. 50 241 8. 50 307 8. 50373 8. 50 439 8. 50 504 log cos log tan 8. 46 385 8.46 457 8. 46 529 8. 46 602 8. 46 674 8. 46 745 8.46 817 8. 46 889 8. 46 960 8. 47 032 8. 47 103 8. 47 174 8. 47 245 8.47 316 8.47 387 8. 47 458 8. 47 528 8. 47 599 8. 47 669 8. 47 740 8. 47 810 8. 47 880 8. 47 950 8. 48 020 8. 48 090 8. 48 159 8. 48 228 8. 48 298 8. 48 367 8. 48 436 8. 48 505 8. 48 574 8. 48 643 8.48 711 8. 48 780 8. 48 849 8. 48 917 8. 48 985 8. 49 053 8. 49 121 8. 49 189 8. 49 257 8. 49 325 8. 49 393 8. 49 460 8. 49 528 8. 49 595 8. 49 662 8. 49 729 8. 49 796 8.49 863 8. 49 930 8. 49 997 8. 50 063 8.50 130 8. 50 196 8. 50 263 8. 50 329 8. 50 395 8.50 461 8. 50 527 log cot log cos 9. 99 982 9. 99 982 9. 99 982 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9.99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 981 9. 99 980 9. 99 980 9. 99 980 9. 99 980 9.99 980 9. 99 980 9. 99 980 9. 99 980 9. 99 980 9. 99 980 9. 99 980 9. 99 980 9. 99 980 9. 99 980 9. 99 980 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 979 9. 99 978 9. 99 978 9. 99 978 9. 99 978 9. 99 978 9. 99 978 9. 99 978 9. 99 978 9. 99 978 9. 99 978 9. 99 978 log sin O tt 020 50 40 30 20 10 019 50 40 30 20 10 018 50 40 30 20 10 017 50 40 30 20 10 016 50 40 30 20 10 015 50 40 30 20 10 014 50 40 30 20 10 013 50 40 30 20 10 012 50 40 30 20 10 011 50 40 30 20 10 010 t! p f ft 500 10 20 30 40 50 510 10 20 30 40 50 520 10 20 30 40 50 530 10 20 30 40 50 540 10 20 30 40 50 55 0 10 20 30 40 50 560 10 20 30 40 50 570 10 20 30 40 50 580 10 20 30 40 50 590 10 20 30 40 50 600 P P log sin 8. 50 504 8. 50 570 8. 50 636 8. 50 701 8. 50 767 8.50 832 8. 50 897 8. 50 963 8. 51 028 8. 51 092 8. 51157 8. 51 222 8. 51 287 8. 51351 8.51 416 8. 51 480 8. 51 544 8. 51 609 8. 51 673 8. 51 737 8. 51 801 8. 51 864 8.51 928 8. 51 992 8. 52 055 8. 52 119 8. 52 182 8. 52 245 8. 52 308 8. 52 371 8. 52 434 8. 52 497 8. 52 560 8. 52 623 8. 52 685 8. 52 748 8. 52 810 8. 52 872 8. 52 935 8. 52 997 8. 53 059 8. 53 121 8. 53 183 8. 53 245 8. 53 306 8. 53 368 8. 53 429 8. 53 491 8. 53 552 8. 53 614 8. 53 675 8. 53 736 8. 53 797 8. 53 858 8. 53 919 8.53 979 8. 54 040 8.54 101 8. 54 161 8. 54 222 8. 54 282 log cos log tan 8. 50 527 8. 50 593 8. 50 658 8. 50 724 8. 50 789 8.50 855 8.50 920 8. 50 985 8.51 050 8.51 115 8. 51 180 8. 51 245 8. 51 310 8. 51 374 8. 51 439 8.51 503 8. 51 568 8. 51 632 8. 51 696 8. 51 760 8. 51 824 8. 51 888 8. 51 952 8. 52 015 8. 52 079 8. 52 143 8. 52 206 8. 52 269 8. 52 332 8. 52 396 8.52459 8. 52 522 8. 52 584 8. 52 647 8. 52 710 8. 52 772 8. 52 835 8. 52 897 8. 52 960 8. 53 022 8. 53 084 8. 53 146 8. 53 208 8. 53 270 8. 53 332 8. 53 393 8. 53 455 8.53 516 8. 53 578 8. 53 639 8. 53 700 8. 53 762 8. 53 823 8. 53 884 8.53 945 8.54 005 8. 54 066 8.54 127 8. 54 187 8. 54 248 8. 54 308 log cot log cos 9. 99 978 9. 99 978 9. 99 978 9. 99 978 9. 99 977 9. 99 977 9. 99 977 9. 99 977 9. 99 977 9. 99 977 9.99 977 9. 99 977 9. 99 977 9. 99 977 9. 99 977 9. 99 977 9. 99 977 9. 99 977 9. 99 977 9. 99 976 9. 99 976 9. 99 976 9. 99 976 9. 99976 9. 99 976 9. 99 976 9. 99 976 9. 99 976 9. 99 976 9. 99 976 9. 99 976 9. 99 976 9. 99 976 9. 99 975 9. 99 975 9. 99 975 9. 99 975 9. 99 975 9. 99 975 9. 99 975 9. 99 975 9.99 975 9. 99 975 9. 99 975 9. 99975 9. 99 975 9. 99 975 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 9. 99 974 log sin ff p 010 50 40 30 20 10 0 9 50 40 30 20 10 08 50 40 30 20 10 0 7 50 40 30 20 10 0 6 50 40 30 20 10 0 5 50 40 30 20 10 0 4 50 40 30 20 10 0 3 50 40 30 20 10 0 2 50 40 30 20 10 0 1 50 40 30 20 10 O O 0 0 ppt 88~ 28 1~ * il 0 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 31 -32 33 34 43 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 1 log sin 8 24 186 24 903 25 609 26304 26 988 27 661 28 324 28977 29 621 30 255 30879 31 495 32 103 32 702 33 292 33 875 34450 35 018 35 578 36 131 36678 37 217 37 750 38 276 38 796 39 310 39818 40 320 40 816 41 307 41 792 42 272 42 746 43 216 43 680 44 139 44 594 45 044 45 489 45 930 46 366 46 799 47 226 47 650 48 069 48 485 48 896 49 304 49 708 50 108 50 501 50 897 51 287 51 673 52055 52434 52810 53 183 53 552 53 919 54 282 log log cos log tan 8 24 192 24 910 25 616 26312 26 996 27 669 28 332 28 986 29 629 30 263 30 888 31 505 32 112 32 711 33 302 33 886 34 461 35 029 35 590 36 143 36 689 37 229 37 762 38 289 38 809 39 323 39 832 40 334 40 830 41 321 41 807 42 287 42 762 43 232 43 696 44 156 44611 45 061 45 507 45 948 46 385 46 817 47245 47 669 48 089 48505 48917 49325 49 729 50 130 50 527 50 920 51310 51 696 52079 52 459 52 835 53 208 53 578 53945 54308 8 log cot log cot 11 75 808 75 090 74 384 73 688 73 004 72 331 71 668 71 014 70371 69 737 69 112 68 495 67 888 67 289 66 698 66 114 65 539 64 971 64 410 63 857 63 311 62 771 62 238 61 711 61 191 60677 60168 59 666 59 170 58679 58 193 57 713 57238 56 768 56304 55 844 55 389 54 939 54493 54 052 53 615 53 183 52 755 52331 51911 51 495 51 083 50 675 50 271 49 870 49473 49 080 48 690 48 304 47 921 47 541 47 165 46 792 46 422 46 055 45 692 11 log tan log cos 9 99 993. 99 993 99 993 99 993 99 992 99 992 99 992 99 992 99 992 99 991 99 991 99 991 99 990 99 990 99 990 99 990 99 989 99 989 99 989 99 989 99 988 99 988 99 988 99 987 99 987 99987 99986 99986 99986 99985 99 985 99 985 99 984 99 984 99 984 99 983 99 983 99 983 99 982 99 982 99 982 99981 99 981 99 981 99 980 99 980 99 979 99 979 99 979 99 978 99 978 99 977 99 977 99977 99 976 99 976 99 975 99 975 99974 99 974 99 974 9 log sin m_.i a 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 -5 I I 0 0 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 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 f log sin 8 54 282 54642 54 999 55 354 55 705 56 054 56 400 56 743 57 084 57421 57 757 58 089 58 419 58 747 59 072 59 395 59 715 60 033 60 349 60 662 60 973 61 282 61 589 61 894 62 196 62 497 62 795 63 091 63 385 63 678 63 968 64256 64 543 64827 65 110 65 391 65 670 65 947 66 223 66 497 66 769 67 039 67 308 67 575 67 841 68 104 68 367 68 627 68 886 69 144 69 400 69 654 69 907 70 159 70 409 70 658 70 905 71151 71 395 71 638 71 880 8 log cos log tan 8 54 308 54 669 55 027 55 382 55 734 56 083 56 429 56 773 57 114 57 452 57 788 58 121 58 451 58 779 59 105 59 428 59 749 60 068 60 384 60 698 61 009 61 319 61 626 61931 62 234 62 535 62 834 63 131 63 426 63 718 64 009 64 298 64 585 64 870 65 154 65 435 65 715 65 993 66 269 66 543 66 816 67 087 67356 67 624 67 890 68 154 68417 68 678 68 938 69 196 69 453 69 708 69 962 70 214 70 465 70 714 70 962 71 208 71453 71 697 71 940 8 log cot 0 log cot 11 45 692 45331 44 973 44618 44 266 43 917 43 571 43 227 42 886 42 548 42212 41 879 41 549 41 221 40 895 40572 40 251 39932 39616 39 302 38 991 38 681 38374 38 069 37 766 37465 37 166 36869 36574 36 282 35991 35 702 35415 35 130 34 846 34 565 34 285 34 007 33 731 33 457 33 184 32913 32 644 32376 32 110 31 846 31 583 31 322 31 062 30 804 30 547 30 292 30 038 29 786 29 535 29 286 29 038 28 792 28 547 28 303 28 060 11 log tan log cos 9 99 974 99 973 99 973 99 972 99 972 99971 99 971 99 970 99 970 99 969 99 969 99 968 99 968 99 967 99 967 99967 99 966 99 966 99 965 99 964 99 964 99 963 99 963 99 962 99 962 99 961 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23 82872 96028 03972 86844 37 24 82 041 94528 05 472 87513 36 24 82885 96 053 03 947 86 832 36 25 82 055 94 554 05 446 87501 35 25 82 899 96 078 03 922 86 821 35 26 82 069 94 579 05 421 87490 34 26 82 913 96 104 03 896 86 809 34 27 82 084 94 604 05 396 87 479 33 27 82 927 96 129 03 871 86 798 33 28 82 098 94 630 05 370 87468 32 28 82 941 96 155 03 845 86 786 32 29 82 112 94 655 05 345 87 457 31 29 82 955 96 180 03 820 86 775 31 30 82 126 94 681 05 319 87446 30 30 82 968 96 205 03 795 86 763 30 31 82 141 94 706 05294 87 434 29 31 82 982 96 231 03 769 86 752 29 32 82 155 94 732 05 268 87423 28 32 82 996 96 256 03 744 86 740 28 33 82 169 94 757 05 243 87412 27 33 83010 96 281 03 719 86 728 27 34 82184 94783 05 217 87401 26 34 83023 96307 03693 86717 26 35 82198 94808 05192 87390 25 35 83.037 96332 03668 86705 25 36 82 212 94834 05 166 87378 24 36 83 051 96357 03 643 86 694 24 37 82 226 94 859 05 141 87 367 23 37 83 065 96 383 03 617 86 682 23 38 82240 94884 05 116 87356 22 38 83078 96408 03 592 86670 22 39 82 255 94 910 05 090 87345 21 39 83 092 96433 03 567 86659 21 40 82 269 94 935 05 065 87334 20 40 83 106 96 459 03 541 86 647 20 41 82 283 94 961 05 039 87322 19 41 83 120 96484 03 516 86635 19 42 82 297 94 986 05 014 87311 18 42 83 133 96510 03 490 86 624 18 43 82'311 95 012 04 988 87300 17 43 83 147 96 535 03 465 86 612 17 44 82326 95 037 04 963 87288 16 44 83 161 96 560 03 440 86 600 16 45 82340 95 062 04938 87 277 15 45 83 174 96586 03 414 86589 15 46 82354 95 088 04912 87 266 14 46 83 188 96 611 03 389 86577 14 47 82368 95 113 04887 87255 13 47 83202 96636 03364 86565 13 48 82382 95 139 04 861 87 243 12 48 83 215 96 662 03 338 86554 12 49 82396 95 164 04836 87 232 11 49 83 229 96687 03313 86542 11 50 82410 95190 04810 87221 10 50 83242 96712 03288 86530 10 51 82424 95 215 04 785 87 209 9 51 83 256 96 738 03 262 86518 9 52 82 439 95 240 04 760 87 198 8 52 83 270 96 763 03 237 86507 8 53 82453 95 266 04734 87 187 7 53 83 283 96 788 03 212 86495 7 54 82 467 95 291 04 709 87 175 6 54 83 297 96 814 03 186 86483 6 55 82481 95317 04683 87164 5 55 83310 96839.03161 86472 5 56 82 495 95342 04 658 87 1-53 4 56 83324 96 864 03 136 86460 4 57 82509 95368 04632 87141 3 57 83338 96890 03110 86448 3 58 82523 95393 04 607 87 130 2 58 83 351 96915 03 085 86 436 2 59 82537 95 418 04582 87 119 1 59 83 365 96 940 03 060 86425 1 60 82551 95 444 04 556 87 107 0 60 83 378 96 966 03 034 86413 0 9 9 10 9, 9 9 10 9 t log cos log cot log tan log sin t log cos log cot log tan log sin 48~ 47~ 43~ 44~ 49 II f 0 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 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 ti log sin 9 83 378 83 392 83 405 83 419 83 432 83 446 83 459 83 473 83 486 83 500 83 513 83 527 83 540 83 554 83 567 83 581 83 594 83 608 83 621 83 634 83 648 83 661 83 674 83 688 83 701 83 715 83 728 83 741 83 755 83 768 83 781 83 795 83 808 83 821 83 834 83 848 83 861 83 874 83 887 83 901 83 914 83 927 83 940 83 954 83 967 83 980 83 993 84 006 84 020 84033 84 046 84 059 84 072 84 085 84 098 84 112 84 125 84 138 84 151 84 164 84 177 9 log cos log tan 9 96 966 96 991 97 016 97 042 97 067 97 092 97 118 97 143 97 168 97 193 97 219 97 244 97 269 97 295 97 320 97 345 97 371 97 396 97 421 97 447 97 472 97 497 97 523 97 548 97 573 97 598 97 624 97 649 97 674 97 700 97 725 97 750 97 776 97 801 97 826 97 851 97 877 97 902 97 927 97 953 97 978 98 003 98 029 98054 98 079 98 104 98 130 98 155 98 180 98 206 98 231 98 256 98 281 98 307 98 332 98 357 98 383 98 408 98 433 98 458 98 484 9 log cot log cot 10 03 034 03 009 02 984 02958 02 933 02 908 02 882 02 857 02 832 02 807 02 781 02 756 02 731 02 705 02 680 02 655 02 629 02 604 02 579 02 553 02 528 02503 02 477 02 452 02 427 02 402 02 376 02351 02 326 02 300 02 275 02 250 02 224 02 199 02 174 02 149 02 123 02 098 02 073 02 047 02 022 01 997 01 971 01 946 01 921 01 896 01 870. 01 845 01 820 01 794 01 769 01 744 01 719 01 693 01 668 01 643 01 617 01 592 01 567 01542 01 516 10 log tan log cos 9 86 413 86 401 86 389 86377 86 366 86354 86 342 86 330 86318 86 306 86 295 86 283 86 271 86 259 86 247 86 235 86 223 86211 86 200 86 188 86 176 86 164 86 152 86 140 86 128 86116 86 104 86 092 86 080 86 068 86 056 86 044 86032 86 020 86 008 85 996 85 984 85 972 85 960 85 948 85 936 85 924 85 912 85 900 85 888 85 876 85 864 85 851 85 839 85 827 85 815 85 803 85 791 85 779 85 766 85 754 85 742 85 730 85 718 85 706 85 693 9 log sin f 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38.37 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 11 10 9 8 '7 6 5 4 3 2 1 0 f a f 0 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 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 ~i4 log sin 9 84 177 84 190 84 203 84 216 84 229 84 242 84 255 84 269 84 282 84 295 84 308 84 321 84 334 84 347 84 360 84373 84 385 84 398 84411 84 424 84 437 84 450 84463 84 476 84 489 84 502 84515 84 528 84 540 84553 84 566 84 579 84 592 84 605 84 618 84 630 84 643 84 656 84 669 84 682 84 694 84 707 84 720 84 733 84 745 84 758 84 771 84 784 84 796 84 809 84 822 84 835 84 847 84 860 84 873 84 885 84 898 84911 84 923 84 936 84 949 9 log cos log tan log cot 9 10 98 484 01 516 98 509 01 491 98 534 01466 98 560 01440 98 585 01 415 98 610 01390 98 635 01365 98661 01339 98 686 01314 98711 01289 98 737 01 263 98 762 01 238 98 787 01 213 98 812 01 188 98 838 01 162 98 863 01137 98 888 01 112 98 913 01087 98 939 01061 98964 01036 98 989 01011 99015 00985 99040 00960 99065 00935 99090 00910 99116 00884 99 141 00859 99166 00834 99191 00809 99217 00783 99242 00758 99 267 00 733 99 293 00 707 99318 00682 99343 00657 99368 00632 99394 00606 99419 00581 99444 00556 99469 00531 99495 00505 99520 00480 99545 00455 99 570 00430 99596 00404 99621 00379 99646 00354 99672 00328 99697 00303 99722 00 278 99747' 00253 99773 00227 99 798 00 202 99 823 00 177 99 848 00152 99874 00126 99899 00101 99924 00 076 99949 00051 99975 00025 00000 00000 10 10 log cot log tan log cos 9 85 693 85 681 85 669 85 657 85 645 85 632 85 620 85 608 85 596 85 583 85 571 85 559 85 547 85 534 85 522 85 510 85 497 85 485 85 473 85 460 85 448 85 436 85 423 85 411 85 399 85 386 85 374 85 361 85 349 85 337 85 324 85 312 85 299 85 287 85 274 85 262 85 250 85 237 85 225 85 212 85 200 85 187 85 175 85 162 85 150 85 137 85 125 85 112 85 100 85 087 85 074 85 062 85 049 85 037 85 024 85 012 84 999 84 986 84 974 84 961 84 949 9 log sin f 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 _ _ I m. ~ ~ ~ ~ ~ 46 0 45~ 50 TABLE IV, I FOR DETERMINING WITH GREATER ACCURACY THAN CAN BE DONE BY MEANS OF TABLE III.: 1. log sin, log tan, and log cot, when the angle is between 0~ and 2~; 2. log cos, log tan, and log cot, when the angle is between 88~ and 90~; 3. The value of the angle when the logarithm of the function does not lie between the limits 8. 54 684 and 11. 45 316. FORMULAS FOR THE USE OF THE NUMBERS S AND T. I. When the angle a is between 0~ and 2~: log sin a = log a'" + S. log a" = log sin a -S, log tan a = log all + T. = log tan a - T, log cot a = colog tan a. = colog cot a - T. II. When the angle a is log cos a = log (90~ - a)"t + S. log cot a = log (90 ~- a)" + T. log tan a = colog cot a. between 88~ and 90~: log (900 - a)" = log cos a - S, = log cot a - T, = colog tan a - T, and a = 90~ - (90~ -a). VALUES OF S AND T. O 2409 3417 3 823 4190 4 840 5414 5932 6408 6633 6851 7 267 atl S log sin a 4. 68 557 8. 06 740 4. 68 556 8. 21 920 4.68 555 8. 26 795 4. 68 555 8.30 776 4. 68 554 8.37 038 4. 68 553 8. 41 904 4. 68 552 8.45 872 4. 68 551 8. 49 223 4. 68 550 8. 50 721 4. 68 550 8. 52 125 4. 68 549 8. 54 684 S log sin a ana 0 200 1 726 2432 2 976 3434 3838 4 204 4 540 4 699 4 853 5 146 T log tan a 4. 68 557 6. 98 660 4. 68 558 7. 92 263 4. 68 559 8. 07 156 4. 68 560 8.15 924 4. 68 561 8. 22 142 4. 68 562 8. 26 973 4. 68 563 8. 30 930 4. 68 564 8. 34 270 4. 68 565 8. 35 766 4. 68 565 8. 37 167 4. 68 566 8.39 713 5 146 5 424 5 689 5 941 6184 6417 6642 6859 7070 7173 7 274 T log tan a 8. 39 713 4. 68 567 8. 41 999 4. 68 568 8. 44 072 4.68 569 8. 45 955 4. 68 570 8. 47 697 4. 68 571 8. 49 305 4. 68 572 8. 50 802 4. 68 573 8.52 200 4. 68 574 8. 53 516 4. 68 575 8. 54 145 4. 68 575 8. 54 753 Y ac/ T log tan a a T log tan a TABLE V,- CIRCUMFERENCES AND AREAS OF CIRCLES, 51 If NV= the radius of the circle, the circumference = 2 7rX. If N = the radius of the circle, the area = 7rN2. If N = the circumference of the circle, the radius = -N. 27r If N = the circumference of the circle, the area = 1 N2. 47r N 0 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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5O 2 7rN 7riV2 '1 V N2 2 ' 4 " 0.00 0.0 0.000 0.00 6.28 3.1 0.159 0.08 12.57 12.6 0.318 0.32 18.85 28.3 0. 477 0. 72 25.13 50.3 0. 637 1.27 31.42 78.5 0.796 1.99 37.70 113. 1 0.955 2.86 43.98 153.9 1. 114 3.90 50. 27 201.1 1. 273 5.09 56. 55 254. 5 1. 432 6. 45 62.83 314.2 1. 592 7.96 69.12 380. 1 1. 751 9.63 75.40 452.4 1. 910 11.46 81.68 530.9 2. 069 13.45 87.96 615.8 2. 228 15.60 94.25 706. 9 2. 387 17. 90' 100. 53 80. 2 2. 546 20.37 106.81 907. 9 2. 706 23. 00 113.10 1 017. 9 2. 865 25.78 119.38 1134.1 3. 024 28.73 125.66 1256.6 3. 183 31.83 131.95 1 385.4 3.342 35.09 138. 23 1 520. 5 3. 501 38. 52 144.51 1661.9 3. 661 42.10 150.80 1 809.6 3. 820 45.84 157.08 1963. 5 3. 979 49. 74 163.36 -2123.7 4. 138 53.79 169.65 2290.2 4. 297 58. 01 175.93 2463.0 4. 456 62.39 182.21 2642.1 4. 615 66. 92 188.50 2827.4 4. 775 71.62 194. 78 3019.1 4. 934 76. 47 201.06 3217.0 5. 093 81.49 207.35 3421.2 5. 252 86. 66 213.63 3631.7 5. 411 91.99 219. 91 384+8.5 5. 570 97.48 226.19 4 071.5 5. 730 103.13 232.48 4300.8 5. 889 108.94 238. 76 4536.5 6. 048 114. 91 245.04 4778.4 6. 207 121.04 251.33 5026.5 6.366 127.32 257. 61 5 281. 0 6. 525 133. 77 263. 89 5541.8 6 658 140.37 270. 18 5 808. 8 6. 844 147. 14 276.46 6082.1 7. 003 154. 06 282.74 6361.7 7. 162 161.14 289.03 6 647.6 7. 321 168.39 295.31 6 939.8 7. 480 175.79 301.59 7 238. 2 7. 639 183.35 307. 88 7 543. 0 7. 799 191. 07 314. 16 7854. 0 7. 958 198. 94 N 50 51 52 53 54 55 56 57 58 59 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 90 91 92 93 94 95 96 97 98 99 100 2wN irN2 IN!N2 27 4w 314. 16 7854 7.96 198.94 320.44 8171 8. 12 206. 98 326. 73 8 495 8.28 215.18 333.01 8825 8.44 223.53 339.29 9161 8.59 232.05 345.58 9503 8.75 240.72 351.86 9852 8.91 249.55 358. 14 10 207 9. 07 258. 55 364. 42 10 568 9. 23 267.70 370. 71 10 936 9.39 277. 01 376.99 11 310 9.555 286.48 383.27 11 690 9.71 296.11 389.56 12 076 9.87 305.90 395.84 12 469 10.03 315.84 402.12 12 868 10. 19 325. 95 408.41 13 273 10.35 336.21 414.69 13 685 10.50 346.64 420. 97 14 103 10. 66 357. 22 427.26 14 527 10.82 367.97 433.54 14 957 10.98 378.87 439.82 15 394 11.14 389.93 446.11 15 837 11.30 401.15 452.39 16 286 11.46 412.53 458.67 16 742 11.62 424.07 464.96 17 203 11.78 435.77 471.24 17 671 11.94 447.62 477.52 18 146 12. 10 459. 64 483.81 18 627 12. 25 471.81 490. 09 19113 12.41 484.15 496.37 19 607 12.57 496.64 502.65 20106 12.73 509.30 508.94 20 612 12.89 522.11 515. 22 21 124 13. 05 535. 08 521. 50 21 642 13. 21 548. 21 527.79 22 167 13.37 561.50 534.07 22 698 13.53 574.995 540. 35 23 235 13. 69 588. 55 546.64 23 779 13.85 602.32 -552.92 24 328 14.01 616.25 559. 20 24 885 14. 16 630. 33 565. 49 25 447 14.32 644. 58 571.77 26 016 14.48 658.98 578.05 26 590 14.64 673.54 584.34 27.172 14.80 688.27 590. 62 27 759 14.96 703. 15 596.90 28 353 15.12 718.19 603.19 28 953 15.28 733.39 609.47 29 559 15.44 748.74 615.75 30 172 15.60 764.26 622. 04 30 791 15. 76 779. 94 628. 32 31 416 15. 92 795. 77 2 wyN N2.LN 2 4N2 2 7 4w7 N 2 wN wN2 -N 1 N2 N 2 24 7r 52 TABLE VI.- NATURAL FUNCTIONS. 00 1oo 20 30 40 r sin in os sin os sin os sin ossin cos 0 0000 1.000 0175 9998 0349 9994 0523 9986 0698 9976 60 1 0003 1.000 0177 9998 0352 9994 0526 9986 0700 9975 59 2 0006 1.000 0180 9998 0355 9994 0529 9986 0703 9975 58 3 0009 1.000 0183 9998 0358 9994 0532 9986 0706 9975 57 4 0012 1.000 0186 9998 0361 9993 0535 9986 0709 9975 56 5 0015 1.000 0189 9998 0364 9993 0538 9986 0712 9975 55 6 0017 1.000 0192 9998 0366 9993 0541 9985 0715 9974 54 7 0020 1.000 0195 9998 0369 9993 0544 9985 0718 9974 53 8 0023 1.000 0198 9998 0372 9993 0547 9985 0721 9974 52 9 0026..000 0201 9998 0375 9993 0550 9985 0724 9974 51 10 0029 1.000 0204 9998 0378 9993 0552 9985 0727 9974 50 1.1 0032 1.000 0207 9998 0381 9993 0555 9985 0729 9973 49 12 0035 1.000 0209 9998 0384 9993 0558 9984 0732 9973 48 13 0038 1.000 0212 9998 0387 9993 0561 9984 0735 9973 47 14 0041 1.000 0215 9998 0390 9992 0564 9984 0738 9973 46 15 0044 1.000 0218 9998 0393 9992 0567 9984 0741 9973 45 16 0047 1.000 0221 9998 0396 9992 0570 9984 0744 9972 44 17 0049 1.000 0224 9997 0398 9992 0573 9984 0747 9972 43 18 0052 1.000 0227 9997 0401 9992 0576 9983 0750 9972 42 19 0055 1.000 0230 9997 0404 9992 0579 9983 0753 9972 41 20 0058 1.000 0233 9997 0407 9992 0581 9983 0756 9971 40 21 0061 1.000 0236 9997 0410 9992 0584 9983 0758 9971 39 22 0064 1.000 0239 9997 0413 9991 0587 9983 0761 9971 38 23 0067 1.000 0241 9997 0416 9991 0590 9983 0764 9971 37 24 0070 1.000 0244 9997 0419 9991 0593 9982 0767 9971 36 25 0073 1.000 0247 9997 0422 9991 0596 9982 0770 9970 35 26 0076 1.000 0250 9997 0425 9991 0599 9982 0773 9970 34 27 0079 1.000 0253 9997 0427 9991 0602 9982 0776 9970 33 28 0081 1.000 0256 9997 0430 9991 0605 9982 0779 9970 32 29 0084 1.000 0259 9997 0433 9991 0608 9982 0782 9969 31 30 0087 1.000 0262 9997 0436 9990 0610 9981 0785 9969 30 31 0090 1.000 0265 9996 0439 9990 0613 9981 0787 9969 29 32 0093 1.000 0268 9996 0442 9990 0616 9981 0790 9969 28 33 0096 1.000 0270 9996 0445 9990 0619 9981 0793 9968 27 34 0099 1.000 0273 9996 0448 9990 0622 9981 0796 9968 26 35 0102 9999 0276 9996 0451 9990 0625 9980 0799 9968 25 36 0105 9999 0279 9996 0454 9990 0628 9980 0802 9968 24 37 0108 9999 0282 9996 0457 9990 0631 9980 0805 9968 23 38 0111 9999 0285 9996 0459 9989 0634 9980 0808 9967 22 39 0113 9999 0288 9996 0462 9989 0637 9980 0811 9967 21 40 0116 9999 0291 9996 0465 9989 0640 9980 0814 9967 20 41 0119 9999 0294 9996 0468 9989 0642 9979 0816 9967 19 42 0122 9999 0297 9996 0471 9989 0645 9979 0819 9966 18 43 0125 9999 0300 9996 0474 9989 0648 9979 0822 9966 17 44 0128 9999 0302 9995 0477 9989 0651 9979 0825 9966 16 45 0131 9999 0305 9995 0480 9988 0654 9979 0828 9966 15 46 0134 9999 0308 9995 0483 9988 0657 9978 0831 9965 14 47 0137 9999 0311 9995 0486 9988 0660 9978 0834 9965 13 48 0140 9999 0314 9995 0488 9988 0663 9978 0837.9965 12 49 0143 9999 0317 9995 0491 9988 0666 9978 0840 9965 11 50 0145 9999 0320 9995 0494 9988 0669 9978 0843 9964 10 51 0148 9999 0323 9995 0497 9988 0671 9977 0845 9964 9 52 0151 9999 0326k 9995 0500 9987 0674 9977 0848 9964 8 53 0154 9999. 0329 9995 0503 9987 0677 9977 0851 9964 7 54 0157 9999 0332 9995 0506 9987 0680 9977 0854 9963 6 55 0160 9999 0334 9994 0509 9987 0683 9977 0857 9963 5 56 0163 9999 0337 9994 0512 9987 0686 9976 0860 9963 4 57 0166 9999 0340 9994 0515 9987 0689 9976 0863 9963 3 58 0169 9999 0343 9994 0518 9987 0692 9976 0866 9962 2 59 0172 9999 0346 9994 0520 9986 0695 9976 0869 9962 1 60 0175 9999 0349 9994 0523 9986 0698 9976 0872 9962 0 cos sin cos sin cos sin cos sin Cos sin 890 880 870 86~ 85~0 -4 0 IATTURAL SINES AND COSINES. G a O 0 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 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 p 50 sin cos 0872 9962 0874 9962 0877 9961 0880 9961 0883 9961 0886 9961 0889 9960 0892 9960 0895 9960 0898 9960 0901 9959 0903 9959 0906 9959 0909 9959 0912 9958 0915 9958 0918 9958 0921 9958 0924 9957 0927 9957 0929 9957 0932 9956 0935 9956 0938 9956 0941 9956 0944 9955 0947 9955 0950 9955 0953 9955 0956 9954 0958 9954 0961 9954 0964 9953 0967 9953 0970 9953 0973 9953 0976 9952 0979 9952 0982 9952 0985 9951 0987 9951 0990 9951 0993 9951 0996 9950 0999 9950 1002 9950 1005 9949 1008 9949 1011 9949 1013 9949 1016 9948 1019 9948 1022 9948 1025 9947 1028 9947 1031 9947 1034 9946 1037 9946 1039 9946 1042 9946 1045 9945 cos sin 84o 6~ sill cos 1045 9945 1048 9945 1051 9945 1054 9944 1057 9944 1060 9944 1063 9943 1066 9943 1068 9943 1071 9942 1074 9942 1077 9942 1080 9942 1083 9941 1086 9941 1089 9941 1092 9940 1094 9940 1097 9940 1100 9939 1103 9939 1106 9939 1109 9938 1112 9938 1115 9938 1118 9937 1120 9937 1123 9937 1126 9936 1129 9936 1132 9936 1135 9935 1138 9935 1141 9935 1144 9934 1146 9934 1149 9934 1152 9933 1155 9933 1158 9933 1161 9932 1164 9932 1167 9932 1170 9931 1172 9931 1175 9931 1178 9930 1181 9930 1184 9930 1187 9929 1190 9929 1193 9929 1196 9928 1198 9928 1201 9928 1204 9927 1207 9927 1210 9927 1213 9926 1216 9926 1219 9925 cos sin 83~ 70 sin cos 1219 9925 1222 9925 1224 9925 1227 9924 1230 9924 1233 9924 1236 9923 1239 9923 1241 9923 1245 9922 1248 9922 1250 9922 1253 9921 1256 9921 1259 9920 1262 9920 1265 9920 1268 9919 1271 9919 1274 9919 1276 9918 1279 9918 1282 9917 1285 9917 1288 9917 1291 9916 1294 9916 1297 9916 1299 9915 1302 9915 1.305 9914 1308 9914 1311 9914 1314 9913 1317 9913 1320 9913 1323 9912 1325 9912 1328 9911 1331 9911 1334 9911 1337 9910 1340 9910 1343 9909 1346 9909 1349 9909 1351 9908 1354 9908 1357 9907 1360 9907 1363 9907. 1366 9906 1369 9906 1372 9905 1374 9905 1377 9905 1380 9904 1383 9904 1386 9903 1389 9903 1392 9903 cos sin 82~ 8~ sin Cos 1392 9903 1395 9902 1397 9902 1400 9901 1403 9901 1406 9901 1409 9900 1412 9900 1415 9899 1418 9899 1421 9899 1423 9898 1426 9898 1429 9897 1432 9897 1435 9897 1438 9896 1441 9896 1444 9895 1446 9895 1449 9894 1452 9894 1455 9894 1458 9893 1461 9893 1464 9892 1467 9892 1469 9891 1472 9891 1475 9891 1478 9890 1481 9890 1484 9889 1487 9889 1490 9888 1492 9888 1495 9888 1498 9887 1501 9887 1504 9886 1507 9886 1510 9885 1513 9885 1515 9884 1518 9884 1521 9884 1524 9883 1527 9883 1530 9882 1533 9882 1536 9881 1538 9881 1541 9880 1544 9880 1547 9880 1550 9879 1553 9879 1556 9878 1559 9878 1561 9877 1564 9877 cos sin 81~ 90 sin cos 1564 9877 1567 9876 1570 9876 1573 9876 1576 9875 1579 9875 1582 9874 1584 9874 1587 9873 1590 9873 1593 9872 1596 9872 1599 9871 1602 9871 1605 9870 1607 9870 1610 9869 1613 9869 1616 9869 1619 9868 1622 9868 1625 9867 1628 9867 1630 9866 1633 9866 1636 9865 1639 9865 1642 9864 1645 9864 1648 9863 1650 9863 1653 9862 1656 9862 1659 9861 1662 9861 1665 9860 1668 9860 1671 9859 1673 9859 1676 9859 1679 9858 1682 9858 1685 9857 1688 9857. 1691 9856 1693 9856 1696 9855 1699 9855 1702 9854 1705 9854 1708 9853 1711 9853 1714 9852 1716 9852 1719 9851 1722 9851 1725 9850 1728 9850 1731 9849 1734 9849 1736 9848 cos sin 80~ 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 15 4 3 2 1 0! W I NATURAL SINES AND COSINES. I,' 0 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 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! 100 sin cos 1736 9848 1739 9848 1742 9847 1745 9847 1748 9846 1751 9846 1754 9845 1757 9845 1759 9844 1762 9843 1765 9843 1768 9842 1771 9842 1774 9841 1777 9841 1779 9840 1782 9840 1785 9839 1788 9839 1791. 9838 1794 9838 1797 9837 1799 9837 1802 9836 1805 9836 1808 9835 1811 9835 1814 9834 1817 9834 1819 9833 1822 9833 1825 9832 1828 9831 1831 9831 1834 9830 1837 9830 1840 9829 1842 9829 1845 9828 1848 9828 1851 9827 1854 9827 1857 9826 1860 9826 1862 9825 1865 9825 1868 9824 1871 9823 1874 9823 1877 9822 1880 9822 1882 9821 1885 9821 1888 9820 ]891 9820 1894 9819 1897 9818 1900 9818 1902 9817 1905 9817 1908 9816 cos sin 79~ 11~ sin cos 1908 9816 1911 9816 1914 9815 1917 9815 1920 9814 1922 9813 1925 9813 1928 9812 1931 9812 1934 9811 1937 9811 1939 9810 1942 9810 1945 9809 1948 9808 1951 9808 1954 9807 1957 9807 1959 9806 1962 9806 1965 9805 1968 9804 1971 9804 1974 9803 1977 9803 1979 9802 1982 9802 1985 9801 1988 9800 1991 9800 1994 9799 1997 9799 1999 9798 2002 9798 2005 9797 2008 9796 2011 9796 2014 9795 2016 9795 2019 9794 2022 9793 2025 9793 2028 9792 2031 9792 2034 9791 2036 9790 2039 9790 2042 9789 2045 9789 2048 9788 2051 9787 2054 9787 2056 9786 2059 9786 2062 9785 2065 9784 2068 9784 2071 9783 2073 9783 2076 9782 2079 9781 cos sin 78~ 12~ sin cos 2079 9781 2082 9781 2085 9780 2088 9780 2090 9779 2093 9778 2096 9778 2099 9777 2102 9777 2105 9776 2108 9775 2110 9775 2113 9774 2116 9774 2119 9773 2122 9772 2125 9772 2127 9771 2130 9770 2133 9770 2136 9769 2139 9769 2142 9768 2145 9767 2147 9767 2150 9766 2153 9765 2156 9765 2159 9764 2162 9764 2164 9763 2167 9762 2170 9762 2173 9761 2176 9760 2179 9760 2181 9759 2184 9759 2187 9758 2190 9757 2193 9757 2196 9756 2198 9755 2201 9755 2204 9754 2207 9753 2210 9753 2213 9752 2215 9751 2218 9751 2221 9750 2224 9750 2227 9749 2230 9748 2233 9748 2235 9747 2238 9746 2241 9746 2244 9745 2247 9744 2250 9744 cos sin 770 13~ sin cos 2250 9744 2252 9743 2255 9742 2258 9742 2261 9741 2264 9740 2267 9740 2269 9739 2272 9738 2275 9738 2278 9737 2281 9736 2284 9736 2286 9735 2289 9734 2292 9734 2295 9733 2298 9732 2300 9732 2303 9731 2306 9730 2309 9730 2312 9729 2315 9728 2317 9728 2320 9727 2323 9726 2326 9726 2329 9725 2332 9724 2334 9724 2337 9723 2340 9722 2343 9722 2346 9721 2349 9720 2351 9720 2354 9719 2357 9718 2360 9718 2363 9717 2366 9716 2368 9715 2371 971.5 2374 9714 2377 9713 2380 9713 2383 9712 2385 9711 2388 9711 2391 9710 2394 9709 2397 9709 2399 9708 2402 9707 2405 9706 2408 9706 2411 9705 2414 9704 2416 9704 2419 9703 cos sin 76~ 140 sin cos 2419 9703 2422 9702 2425 9702 2428 9701 2431 9700 2433 9699 2436 9699 2439 9698 2442 9697 2445 9697 2447 9696 2450 9695 2453 9694 2456 9694 2459 9693 2462 9692 2464 9692 2467 9691 2470 9690 2473 9689 2476 9689 2478 9688 2481 9687 2484 9687 2487 9686 2490 9685 2493 9684 2495' 9684 2498 9683 2501 9682 2504 9681 2507 9681 2509 9680 2512 9679 2515 9679 2518 9678 2521 9677 2524 9676 2526 9676 2529 9675 2532.9674 2535 9673 2538 9673 2540 9672 2543 9671 2546'9670 2549-9670 2552 9669 2554 9668 2557 9667 2560' 967 2563 9666 2566 9665 2569 9665 2571 9664 2574 9663 2577 9662 2580 9662 2583 9661 2585 9660 2588 9659 cos sin 750 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0! _I_ I ~..... I NATURJAL SINES AND COSINES. 55 01 i p 0 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 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 __ 150 sin cos 2588 9659 2591 9659 2594 9658 2597 9657 2599 9656 2602 9655 2605 9655 2608 9654 2611 9653 2613 9652 2616 9652 2619 9651 2622 9650 2625 9649 2628 9649 2630 9648 2633 9647 2636 9646 2639 9646 2642 9645 2644 9644 2647 9643 2650 9642 2653 9642 2656 9641 2658 9640 2661 9639 2664 9639 2667 9638 2670 9637 2672 9636 2675 9636 2678 9635 2681 9634 2684 9633 2686 9632 2689 9632 2692 9631 2695 9630 2698 9629 2700 9628 2703 9628 2706 9627 2709 9626 2712 9625 2714 9625 2717 9624 2720 9623 2723 9622 2726 9621 2728 9621 2731 9620 2734 9619 2737 9618 2740 9617 2742 9617 2745 9616 2748 9615 2751 9614 2754 9613 2756 9613 cos sin 740 16~ sin cos 2756 9613 2759 9612 2762 9611 2165 9610 2768 9609 2770 9609 2773 9608 2776 9607 2779 9606 2782 9605 2784 9605 2787 9604 2790 9603 2793 9602 2795 9601 2798 9600 2801 9600 2804 9599 2807 9598 2809 9597 2812 9596 2815 9596 2818 9595 2821 9594 2823 9593 2826 9592 2829 9591 2832 9591 2835 9590 2837 9589 2840 9588 2843 9587 2846 9587 2849 9586 2851 9585 2854 9584 2857 9583 2860 9582 2862 9582 2865 9581 2868 9580 2871 9579 2874 9578 2876 9577 2879 9577 2882 9576 2885 9575 2888 9574 2890 9573 2893 9572 2896 9572 2899 9571 2901 9570 2904 9569 2907 9568 2910 9567 2913 9566 2915 9566 2918 9565 2921 9564 2924 9563 cos sin 730 170 sin cos 2924 9563 2926 9562 2929 9561 2932 9560 2935 9560 2938 9559 2940 9558 2943 9557 2946 9556 2949 9555 2952 9555 2954 9554 2957 9553 2960 9552 2963 9551 2965 9550 2968 9549 2971 9548 2974 9548 2977 9547 2979 9546 2982 9545 2985 9544 2988 9543 2990 9542 2993 9542 2996 9541 2999 9540 3002 9539 3004 9538 3007 9537 3010 9536 3013 9535 3015 9535 3018 9534 3021 9533 3024 9532-. 3026 9531 3029 9530 3032 9529 3035 9528 3038 9527 3040 9527 3043 9526 3046 9525 3049 9524 3051 9523 3054 9522 3057 9521 3060 9520 3062 9520 3065 9519 3068 9518 3071 9517 3074 9516 3076 9515 3079 9514 3082 9513 3085 9512 3087 9511 3090 9511 cos sin 720 18~ sin cos 3090 9511 3093 9510 3096 9509 3098 9508 3101 9507 3104 9506 3107 9505 3110 9504 3112 9503 3115 9502 3118 9502 3121 9501 3123 9500 3126 9499 3129 9498 3132 9497 3134 9496 3137 9495 3140 9494 3143 9493 3145 9492 3148 9492 3151 9491 3154 9490 3156 9489 3159 9488 3162 9487 3165 9486 3168 9485 3170 9484 3173 9483 3176 9482 3179 9481 3181 9480 3184 9480 3187 9479 3190 9478 3192 9477 3195 9476 3198 9475 3201 9474 3203 9473 3206 9472 3209 9471 3212 9470 3214 9469 3217 9468 3220 9467 3223 9466 3225 9466 3228 9465 3231 9464 3234 9463 3236 9462 3239 9461 3242 9460 3245 9459 3247 9458 3250 9457 3253 9456 3256 9455 cos sin 71~ 19~ sin cos 3256 9455 3258 9454 3261 9453 3264 9452 3267 9451 3269 9450 3272 9449 3275 9449 3278 9448 3280 9447 3283 9446 3286 9445 3289 9444 3291 9443 3294 9442 3297 9441 3300 9440 3302 9439 3305 9438 3308 9437 3311 9436 3313 9435 3316 9434 3319 9433 3322 9432 3324 9431 3327 9430 3330 9429 3333 9428 3335 9427 3338 9426 3341 9425 3344 9424 3346 9423 3349 9423 3352 9422 3355 9421 3357 9420 3360 9419 3363 9418 3365 9417 3368 9416 3371 9415 3374 9414 3376 9413 3379 9412 3382 9411 3385 9410 3387 9409 3390 94-08 3393 9407 3396 9406 3398 9405 3401 9404 3404 9403 3407 9402 3409 9401 3412 9400 3415 9399 3417 9398 3420 9397 cos sin 700 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0! 0 I.. q 1A~ATrJIAL SINES AlND COSN.ES. 0 i 0 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 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 f 200 sin cos 3420 9397 3423 9396 3426 9395 3428 9394 3431 9393 3434 9392 3437 9391 3439 9390 3442 9389 3445 9388 3448 9387 3450 9386 3453 9385 3456 9384 3458 9383 3461 9382 3464 9381 3467 9380 3469 9379 3472 9378 3475 9377 3478 9376 3480 9375 3483 9374 3486 9373 3488 9372 3491 9371 3494 9370 3497 9369 3499 9368 3502 9367 3505 9366 3508 9365 3510 9364 3513 9363 3516 9362 3518 9361 3521 9360 3524 9359 3527 9358 3529 9356 3532 9355 3535 9354 3537 9353 3540 9352 3543 9351 3546 9350 3548 9349 3551 9348 3554 9347 3557 9346 3559 9345 3562 9344 3565 9343 3567 9342 3570 9341 3573 9340 3576 9339 3578 9338 3581 9337 3584 9336 cos sin 69~ 21~ sin cos 3584 9336 3586 9335 3589 9334 3592 9333 3595 9332 3597 9331 3600 9330 3603 9328 3605 9327.3608 9326 3611 9325 3614 9324 3616 9323 3619 9322 3622 9321 3624 9320 3627 9319 3630 9318 3633 9317 3635 9316 3638 9315 3641 9314 3643 9313 3646 9312 3649 9311 3651 9309 3654 9308 3657 9307 3660 9306 3662 9305 3665 9304 3668 9303 3670 9302 3673 9301 3676 9300 3679 9299 3681 9298 3684 9297 3687 9296 3689 9295 3692 9293 3695 9292 3697 9291 3700 9290 3703 9289 3706 9288 3708 9287 3711 9286 3714 9285 3716 9284 3719 9283 3722 9282 3724 9281 3727 9279 3730 9278 3733 9277 3735 9276 3738 9275 3741 9274 3743 9273 3746 9272 cos sin 68~ sin cos 3746 9272 3749 9271 3751 9270 3754 9269 3757 9267 3760 9266 3762 9265 3765 9264 3768 9263 3770 9262 3773 9261 3776 9260 3778 9259 3781 9258 3784 9257 3786 9255 3789 9254 3792 9253 3795 9252 3797 9251 3800 9250 3803 9249 3805 9248 3808 9247 3811 9245 3813 9244 3816 9243 3819 9242 3821 9241 3824 9240 3827 9239 3830 9238 3832 9237 3835 9235 3838 9234 3840 9233 3843 9232 3846 9231 3848 9230 3851 9229 3854 9228 3856 9227 3859 9225 3862 9224 3864 9223 3867 9222 3870 9221 3872 9220 3875 9219 3878 9218 3881 9216 3883 9215 3886 9214 3889 9213 3891 9212 3894 9211 3897 9210 3899 9208 3902 9207 3905 9206 3907 9205 cos sill 670 23~ sin cos 3907 9205 3910 9204 3913 9203 3915 9202 3918 9200 3921 9199 3923 9198 3926 9197 3929 9196 3931 9195 3934 9194 3937 9192 3939 9191 3942 9190 3945 9189 3947 9188 3950 9187 3953 9186 3955 9184 3958 9183 3961 9182 3963 9181 3966 9180 3969 9179 3971 9178 3974 9176 3977 9175 3979 9174 3982 9173 3985 9172 3987 9171 3990 9169 3993 9168 3995 9167 3998 9166 4001 9165 4003 9164 4006 9162 4009 9161 4011 9160 4014 9159 4017 9158 4019 9157 4022 9155 4025 9154 4027 9153 4030 9152 4033 9151 4035 9150 4038 9148 4041 9147 4043 9146 4046 9145 4049 9144 4051 9143 4054 9141 4057 9140 4059 9139 4-062 9138 4065 9137 4067 9135 cos sin 66~ 24~ sin cos 4067 9135 4070 9134 4073 9133 4075 9132 4078 9131 4081 9130 4083 91.28 4086 9127 4089 9126 4091 9125 4094 91.24 4097 9122 4099 9121 4102,9120 4105 9119 4107 9118 4110 911.6 4112 9115 4115 9114 4118 9113 -4120 9112 4123 9110 4126 9109 4128 9108 4131 9107 4134 9106 4136 9104 4139 9103 4142 9102 4144 9101 4147 9100 4150 9098 4152 9097 4155 9096 4158 9095 4160 9094 4163 9092 4165 9091 4168 9090 4171 9088 4173 9088 4176 9086 4179 9085 4181 9084 4184 9083 4187 9081 4189 9080 4192 9079 4195 9078 4197 9077 4200 9075 4202 9074 4205 9073 4208 9072 4210 9070 4213 9069 4216 9068 4218 9067 4221 9066 4224 9064 4226 9063 cos sin 65~ f 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 p I Il Iua - Nq'ATtTRAL SINES AND COSINE8. k --— ~ --- ~ I — I I " O 0 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 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 60 250 sin cos 4226 9063 4229 9062 4231 9061 4234 9059 4237 9058 4239 9057 4242 9056 4245 9054 4247 9053 4250 9052 4253 9051 4255 9050 4258 9048 4260 9047 4263 9046 4266 9045 4268 9043 4271 9042 4274 9041 4276 9040 4279 9038 4281 9037 4284 9036 4287 9035 4289 9033 4292 9032 4295 9031 4297 9030 4300 9028 4302 9027 4305 9026 4308 9025 4310 9023 4313 9022 4316 9021 4318 9020 4321 9018 4323 9017 4326 9016 4329 9015 4331 9013 4334 9012 4337 9011 4339 9010 4342 9008 4344 9007 4347 9006 4350 9004 4352 9003 4355 9002 4358 9001 4360 8999 4363 8998 4365 8997 4368 8996 4371 8994 4373 8993 4376 8992 4378 8990 4381 8989 4384 8988 cos sin 64~ 260 Sill ('OS sin cos 4384 8988 4386 8987 4389 8985 4392 8984 4394 8983 4397 8982 4399 8980 4402 8979 4405 8978 4407 8976 4410 8975 4412 8974 4415 8973 4418 8971 4420 8970 4423 8969 4425 8967 4428 8966 4431 8965 4433 8964 4436 8962 4439 8961 4441 8960 4444 8958 4446 8957 4449 8956 4452 8955 4454 8953 4457 8952 4459 8951 4462 8949 4465 8948 4467 8947 4470 8945 4472 8944 4475 8943 4478 8942 4480 8940 4483 8939 4485 8938 4488 8936 4491 8935 4493 8934 4496 8932 4498 8931 4501 8930 4504 8928 4506 8927 4509 8926 4511 8925 4514 8923 4517 8922 4519 8921 4522 8919 4524 8918 4527 8917 4530 8915 4532 8914 4535 8913 4537 8911 4540 8910 cos sin 630 270 sin cos 4540 8910 4542 8909 4545 8907 4548 8906 4550 8905 4553 8903 4555 8902 4558 8901 4561 8899 4563 8898 4566 8897 4568 8895 4571 8894 4574 8893 4576 8892 4579 8890 4581 8889 4584 8888 4586 8886 4589 8885 4592 8884 4594 8882 4597 8881 4599 8879 4602 8878 4605 8877 4607 8875 4610 8874 4612 8873 4615 8871 4617 8870 4620 8869 4623 8867 4625 8866 4628 8865 4630 8863 4633 8862 4636 8861 4638 8859 4641 8858 4643 8857 4646 8855 4648 8854 4651 8853 4654 8851 4656 8850 4659 8849 4661 8847 4664 8846 4666 8844 4669 8843 4672 8842 4674 8840 4677 8839 4679 8838 4682 8836 4684 8835 4687 8834 4690 8832 4692 8831 4695 8829 cos sin 620 280 sin cos 4695 8829 4697 8828 4700 8827 4702 8825 4705 8824 4708 8823 4710 8821 4713 8820 4715 8819 4718 8817 4720 8816 4723 8814 4726 8813 4728 8812 4731 8810 4733 8809 4736 8808 4738 8806 4741 8805 4743 8803 4746 8802 4749 8801 4751 8799 4754 8798 4756 8796 4759 8795 4761 8794 4764 8792 4766 8791 4769 8790 4772 8788 4774 8787 4777 8785 4779 8784 4782 8783 4784 8781 4787 8780 4789 8778 4792 8777 4795 8776 4797 8774 4800 8773 4802 8771 4805 8770 4807 8769 4810 8767 4812 8766 4815 8764 4818 8763 4820 8762 4823 8760 4825 8759 4828 8757 4830 8756 4833 8755 4835 8753 4838 8752 4840 8750 4843 8749 4846 874-8 4848 8746 cos sin 610 290 sin cos 4848 8746 4851 8745 4853 8743 4856 8742 4858 8741 4861 8739 4863 8738 4866 8736 4868 8735 4871 8733 4874 8732 4876 8731 4879 8729 4881 8728 4884 8726 4886 8725 4889 8724 4891 8722 4894 8721 4896 8719 4899 8718 4901 8716 4904 8715 4907 8714 4909 8712 4912 8711 4914 8709 4917 8708 4919 8706 4922 8705 4924 8704 4927 8702 4929 8701 4932 8699 4934 8698 4937 8696 4939 8695 4942 8694 4944 8692 4947 8691 4950 8689 4952 8688 4955 8686 4957 8685 4960 8683 4962 8682 4965 8681 4967 8679 4970 8678 4972 8676 4975 8675 4977 8673 4980 8672 4982 8670 4985 8669 4987 8668 4990 8666 4992 8665 4995 8663 4997 8662 5000 8660 cos sin 600 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 f I is NATURAL SINES AND COSINES. a fi 0 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 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 f 30~ sin cos 5000 8660 5003 8659 5005 8657 5008 8656 5010 8654 5013 8653 5015 8652 5018 8650 5020 8649 5023 8647 5025 8646 5028 8644 5030 8643 5033 8641 5035 8640 5038 8638 5040 8637 5043 8635 5045 8634 5048 8632 5050 8631 5053 8630 5055 8628 5058 8627 5060 8625 5063 8624 5065 8622 5068 8621 5070 8619 5073 8618 5075 8616 5078 8615 5080 8613 5083 8612 5085 8610 5088 8609 5090 8607 5093 8606 5095 8604 5098 8603 5100 8601 5103 8600 5105 8599 5108 8597 5110 8596 5113 8594 5115 8593 5118 8591 5120 8590 5123 8588 5125 8587 5128 8585 5130 8584 5133 8582 5135 8581 5138 8579 5140 8578 5143 8576 5145 8575 5148 8573 5150 8572 cos sin 590 31~ sin cos 5150 8572 5153 8570 5155 8569 5158 8567 5160 8566 5163 8564 5165 8563 5168 8561 5170 8560 5173 8558 5175 8557 5178 8555 5180 8554 5183 8552 5185 8551 5188 8549 5190 8548 5193 8546 5195 8545 5198 8543 5200 8542 5203 8540 5205 8539 5208 8537 5210 8536 5213 8534 5215 8532 5218 8531 5220 8529 5223 8528 5225 8526 5227 8525 5230 8523 5232 8522 5235 8520 5237 8519 5240 8517 5242 8516 5245 8514 5247 8513 5250 8511 5252 8510 5255 8508 5257 8507 5260 8505 5262 8504 5265 8502 5267 8500 5270 8499 5272 8497 5275 8496 5277 8494 5279 8493 5282 8491.5284 8490 5287 8488 5289 8487 5292 8485 5294 8484 5297 8482 5299 8480 cos sin 580 32 sin cos, 5299 8480 5302 8479 5304 8477 5307 8476 5309 8474 5312 8473 5314 8471 5316 8470 5319 8468 5321 8467 5324 8465 5326 8463 5329 8462 5331 8460 5334 8459 5336 8457 5339 8456 5341 8454 5344 8453 5346 8451 5348 8450 5351 8448 5353 8446 5356 8445 5358 8443 5361 8442 5363 8440 5366 8439 5368 8437 5371 8435 5373 8434 5375 8432 5378 8431 5380 8429 5383 8428 5385 8426 5388 8425 5390 8423 5393 8421 5395 8420 5398 8418 5400 8417 5402 8415 5405 8414 5407 8412 5410 8410 5412 8409 5415 8407 5417 8406 5420 8404 5422 8403 5424 8401 5427 8399 5429 8398 5432 8396 5434 8395 5437 8393 5439 8391 5442 8390 5444 8388 5446 8387 cos sin 570 33 sin cos 5446 8387 5449 8385 5451 8384 5454 8382 5456 8380 5459 8379 5461 8377 5463 8376 5466 8374 5468 8372 5471 8371 5473 8369 5476 8368 5478 8366 5480 8364 5483 8363 5485 8361 5488 8360 5490 8358 5493 8356 5495 8355 5498 8353 5500 8352 5502 8350 5505 8348 5507 8347 5510 8345 5512 8344 5515 8342 5517 8340 5519 8339 5522 8337 5524 8336 5527 8334 5529 8332 5531 8331 5534 8329 5536 8328 5539 8326 5541 8324 5544 8323 5546 8321 5548 8320 5551 8318 5553 8316 5556 8315 5558 8313 5561 8311 5563 8310 5565 8308 5568 8307 5570 8305 5573 8303 5575 8302 5577 8300 5580 8299 5582 8297 5585 8295 5587 8294 5590 8292 5592 8290 cos sin 560 340 sin cos 5592 8290 5594 8289 5597 8287 5599 8285 5602 8284 5604 8282 5606 8281 5609 8279 5611 8277 5614 8276 5616 8274 5618 8272 5621 8271 5623 8269 5626 8268 5628 8266 5630 8264 5633 8263 5635 8261 5638 8259 5640 8258 5642 8256 5645.8254 5647 8253 5650 8251 5652 8249 5654 8248 5657 8246 5659 8245 5662 8243 5664 8241 5666 8240 5669 8238 5671 8236 5674 8235 5676 8233 5678 8231 5681 8230 5683 8228 5686 8226 5688 8225 5690 8223 5693 8221 5695 8220 5698 8218 5700 8216 5702 8215 5705 8213 5707 8211 5710 8210 5712 8208 5714 8207 5717 8205 5719 8203 5721 8202 5724 8200 5726 8198 5729 8197 5731 8195 5733 8193 5736 8192 cos sin 550 60 59 58 57 56 55 54. 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 " Ia 61 NATURAL SINES AND COSINES. 59 I. Ir., 0 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 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! 350 sin cos 5736 8192 5738 8190 5741 8188 5743 8187 5745 8185 5748 8183 5750 8181 5752 8180 5755 8178 5757 8176 5760 8175 5762 8173 5764 8171 5767 8170 5769 8168 5771 8166 5774 8165 5776 8163 5779 8161 5781 8160 5783 8158 5786 8156 5788 8155 5790 8153 5793 8151 5795 8150 5798 8148 5800 8146 5802 8145 5805 8143 5807 8141 5809 8139 5812 8138 5814 8136 5816 8134 5819 8133 5821 8131 5824 8129 5826 8128 5828 8126 5831 8124 5833 8123 5835 8121 5838 8119 5840 8117 5842 8116 5845 8114 5847 8112 5850 8111 5852 8109 5854 8107 5857 8106 5859- 8104 5861 8102 5864 8100 5866 8099 5868 8097 5871 8095 5873 8094 5875 8092 5878 8090 cos sin 540 36~ sin cos 5878 8090 5880 8088 5883 8087 5885 8085 5887 8083 5890 8082 5892 8080 5894 8078 5897 8076 5899 8075 5901 8073 5904 8071 5906 8070 5908 8068 5911 8066 5913 8064 5915 8063 5918 8061 5920 8059 5922 8058 5925 8056 5927 8054 5930 8052 5932 8051 5934 8049 5937 8047 5939 8045 5941 8044 5944 8042 5946 8040 5948 8039 5951 8037 5953 8035 5955 8033 5958 8032 5960 8030 5962 8028 5965 8026 5967 8025 5969 8023 5972 8021 5974 8020 5976 8018 5979 8016 5981 8014 5983 8013 5986 8011 5988 8009 5990 8007 5993 8006 5995 8004 5997 8002 6000 8000 6002 7999 6004 7997 6007 7995 6009 7993 6011 7992 6014 7990 6016 7988 6018 7986 cos sin 530 370 sin cos 6018 7986 6020 7985 6023 7983 6025 7981 6027. 7979 6030 7978 6032 7976 6034 7974 6037 7972 6039 7971 6041 7969 6044 7967 6046 7965 6048 7964 6051 7962 6053 7960 6055 7958 6058 7956 6060 7955 6062 7953 6065 7951 6067 7950 6069 7948 6071 7946 6074 7944 6076 7942 6078 7941 6081 7939 6083 7937 6085 7935 6088 7934 6090 7932 6092 7930 6095 7928 6097 7926 6099 7925 6101 7923 6104 7921 6106 7919 6108 7918 6111 7916 6113 7914 6115 7912 61.18 7910 6120 7909 6122 7907 6124 7905 6127 7903 6129 7902 6131 7900 6134 7898 6136 7896 6138 7894 6141 7893 6143 7891 6145 7889 6147 7887 6150 7885 6152 7884 6154 7882 6157 7880 cos sin 520 380 sin cos 6157 7880 6159 7878 6161 7877 6163 7875 6166 7873 6168 7871 6170 7869 6173 7868 6175 7866 6177 7864 6180 7862 6182 7860 6184 7859 6186 7857 6189 7855 6191 7853 6193 7851 6196 7850 6198 7848 6200 7346 6202 7844 6205 7842 6207 7841 6209 7839 6211 7837 6214 7835 6216 7833 6218 7832 6221 7830 6223 7828 6225 7826 6227 7824 6230 7822 6232 7821 6234 7819 6237 7817 6239 7815 6241 7813 6243 7812 6246 7810 6248 7808 6250 7806 6252 7804 6255 7802 6257 7801 6259 7799 6262 7797 6264 7795 6266 7793 6268 7792 6271 7790 6273 7788 6275 7786 6277 7784 6280 7782 6282 7781 6284 7779 6286 7777 6289 7775 6291 7773 6293 7771 cos sin 51~ 390 sin cos 6293 7771 6295 7770 6298 7768 6300 7766 6302 7764 6305 7762 6307 7760 6309 7759 6311 7757 6314 7755 6316 7753 6318 7751 6320 7749 6323 7748 6325 7746 6327 7744 6329 7742 6332 7740 6334 7738 6336 7737 6338 7735 6341 7733 6343 7731 6345 7729 6347 7727 6350 7725 6352 7724 6354 7722 6356 7720 6359 7718 6361 7716 6363 7714 6365 7713 6368 7711 6370 7709 6372 7707 6374 7705 9376 7703 6379 7701 6381 7700 6383 7698 6385 7696 6388 7694 6390 7692 6392 7690 6394 7688 6397 7687 6399 7685 6401 7683 6403 7681 6406 7679 6408 7677 6410 7675 6412 7674 6414 7672 6417 7670 6419 7668 6421 7666 6423 7664 6426 7662 6428 7660 cos sin 500 I f 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 f 60 NATURAL SINES AND COSINES. _II _ __ -_= I _I I I I_, -_ aI I 0 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 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 f 40~ sin cos 6428 7660 6430 7659 6432 7657 6435 7655 6437 7653 6439 7651 6441 7649 6443 7647 6446 7645 6448 7644 6450 7642 6452 7640 6455 7638 6457 7636 6459 7634 6461 7632 6463 7630 6466 7629 6468 7627 6470 7625 6472 7623 6475 7621 6477 7619 6479 7617 6481 7615 6483 7613 6486 7612 6488 7610 6490 7608 6492 7606 6494 7604 6497 7602 6499 7600 6501 7598 6503 7596 6506 7595 6508 7593 6510 7591 6512 7589 6514 7587 6517 7585 6519 7583 6521 7581 6523 7579 6525 7578 6528 7576 6530 7574 6532 7572 6534 7570 6536 7568 6539 7566 6541 7564 6543 7562 6545 7560 6547 7559 6550 7557 6552 7555 6554 7553 6556 7551 6558 7549 6561 7547 cos sin 490 41O sin cos 6561 7547 6563 7545 6565 7543 6567 7541 6569 7539 6572 7538 6574 7536 6576 7534 6578 7532 6580 7530 6583 7528 6585 7526 6587 7524 6589 7522 6591 7520 6593 7518 6596 7516 6598 7515 6600 7513 6602 7511 6604 7509 6607 7507 6609 7505 6611 7503 6613 7501 6615 7499 6617 7497 6620 7495 6622 7493 6624 7491 6626 7490 6628 7488 6631 7486 6633 7484 6635 7482 6637 7480 6639 7478 6641 7476 6644 7474 6646 7472 6648 7470 6650 7468 6652 7466 6654 7464 6657 7463 6659 7461 6661 7459 6663 7457 6665 7455 6667 7453 6670 7451 6672 7449 6674 7447 6676 7445 6678 7443 6680 7441 6683 7439 6685 7437 6687 7435 6689 7433 6691 7431 cos sin 480 42~ sin cos 6691 7431 6693 7430 6696 7428 6698 7426 6700 7424 6702 7422 6704 7420 6706 7418 6709 7416 6711 7414 6713 7412 6715 7410 6717 7408 6719 7406 6722 7404 6724 7402 6726 7400 6728 7398 6730 7396 6732 7394 6734 7392 6737 7390 6739 7388 6741 7387 6743 7385 6745 7383 6747 7381 6749 7379 6752 7377 6754 7375 6756 7373 6758 7371 6760 7369 6762 7367 6764 7365 6767 7363 6769 7361 6771 7359 6773 7357 6775 7355 6777 7253 6779 7351 6782 7349 6784 7347 6786 7345 6788 7343 6790 7341 6792 7339 6794 7337 6797 7335 6799 7333 6801 7331 6803 7329 6805 7327 6807 7325 6809 7323 6811 7321 6814 7319 6816 7318 6818 7316 6820 7314 cos sin 470 430 sin cos 6820 7314 6822 7312 6824 7310 6826 7308 6828 7306 6831 7304 6833 7302 6835 7300 6837 7298 6839 7296 6841 7294 6843 7292 6845 7290 6848 7288 6850 7286 6852 7284 6854 7282 6856 7280 6858 7278 6860 7276 6862 7274 6865 7272 6867 7270 6869 7268 6871 7266 6873 7264 6875 7262 6877 7260 6879 7258 6881 7256 6884 7254 6886 7252 6888 7250 6890 7248 6892 7246 6894 7244 6896 7242 6898 7240 6900 7238 6903 7236 6905 7234 6907 7232 6909 7230 6911 7228 6913 7226 6915 7224 6917 7222 6919 7220 6921 7218 6924 7216 6926 7214 6928 7212 6930 7210 6932 7208 6934 7206 6936 7203 6938 7201 6940 7199 6942 7197 6944 7195 6947 7193 cos sin 460 440 sin cos 6947 7193 6949 7191 6951 7189 6953 71.87 6955 7185 6957 7183 6959 7181 6961 7179 6963 7177 6965 7175 6967 7173 6970 7171 6972 7169 6974 7167 6976 7165 6978 7163 6980 7161 6982 7159 6984 7157 6986 7155 6988 7153 6990 7151 6992 7149 6995 7147 6997 7145 6999 7143 7001 7141 7003 7139 7005 7137 7007 7135 7009 7133 7011 7130 7013 7128 7015 7126 7017 7124 7019 7122 7022 7120 7024 7118 7026 7116 7028 7114 7030 7112 7032 7110 7034 7108 7036 7106 7038 7104 7040 7102 7042 7100 7044 7098 7046 7096 7048 7094 7050 7092 7053 7090 7055 7088 7057 7085 7059 7083 7061 7081 7063 7079 7065 7077 7067 7075 7069 7073 7071 7071 cos sin 450 I f 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39, 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0!b mI NATURAL TANGENTS AND COTANGENTS. 61 I;s 0 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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 4:5 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 l 00 tan cot 0000 Infinite 0003 3437.75 0006 1718.87 0009 1145.92 0012 859.436 0015 687.549 0017 572.957 0020 491.106 0023 429.718 0026 381.971 0029 343.774 0032 312.521 0035 286.478 0038 264.441 0041 245.552 0044 229.182 0047 214.858 0049 202.219 0052 190.984 0055 180.932 0058 171.885 0061 163.700 0064 156.259 0067 149.465 0070 143.237 0073 137.507 0076 132.219 0079 127.321 0081 122.774 0084 118.540 0087 114.589 0090 110.892 0093 107.426 0096 104.171 0099 101.107 0102 98.2179 0105 95.4895 0108 92.9085 0111 90.4633 0113 88.1436 0116 85.9398 0119 83.8435 0122 81.8470 0125 79.9434 0128 78.1263 0131 76.3900 0134 74.7292 0137 73.1390 0140 71.6151 0143 70.1533 0146 68.7501 0148 67.4019 0151 66.1055 0154 64.8580 0157 63.6567 0160 62.4992 0163 61.3829 0166 60.3058 0169 59.2659 0172 58.2612 0175 57.2900 cot tan 890 1~ tan cot 0175 57.2900 0177 56.3506 0180 55.4415 0183 54.5613 0186 53.7086 0189 52.8821 0192 52.0807 0195 51.3032 0198 50.5485 0201 49.8157 0204 49.1039 0207 48.4121 0209 47.7395 0212 47.0853 0215 46.4489 0218 45.8294 0221 45.2261 0224 44.6386 0227 44.0661 0230 43.5081 0233 42.9641 0236 42.4335 0239 41.9158 0241 41.4106 0244 40.9174 0247 40.4358 0250 39.9655 0253 39.5059 0256 39.0568 0259 38.6177 0262 38.1885 0265 37.7686 0268 37.3579 0271 36.9560 0274 36.5627 0276 36.1776 0279 35.8006 0282 35.4313 0285 35.0695 0288 34.7151 0291 34.3678 0294 34.0273 0297 33.6935 0300 33.3662 0303 33.0452 0306 32.7303 0308 32.4213 0311 32.1181 0314 31.8205 0317 31.5284 0320 31.2416 0323 30.9599 0326 30.6833 0329 30.4116 0332 30.1446 0335 29.8823 0338 29.6245 0340 29.3711 0343 29.1220 0346 28 8771 0349 28.6363 cot tan 88~ 20 tan cot 0349 28.6363 0352 28.3994 0355 28.1664 0358 27.9372 0361 27.7117 0364 27.4899 0367 27.2715 0370 27.0566 0373 26.8450 0375 26.6367 0378 26.4316 0381 26.2296 0384 26.0307 0387 25.8348 0390 25.6418 0393 25.4517 0396 25.2644 0399 25.0798 0402 24.8978 0405 24.7185 0407 24.5418 0410 24.3675 0413 24.1957 0416 24.0263 0419 23.8593 0422 23.6945 0425 23.5321 0428 23.3718 0431 23.2137 0434 23.0577 0437 22.9038 0440 22.7519 0442 22.6020 0445 22.4541 0448 22.3081 0451 22.1640 0454 22.0217 0457 21.8813 0460 21.7426 0463 21.6056 0466 21.4704 0469 21.3369 0472 21.2049 0475 21.0747 0477 20.9460 0480 20.8188 0483 20.6932 0486 20.5691 0489 20.4465 0492 20.3253 0495 20.2056 0498 20.0872 0501 19.9702 0504 19.8546 0507 19.7403 0509 19.6273 0512 19.5156 0515 19.4051 0518 19.2959 0521 19.1879 0524 19.0811 cot tan 870 30 tan cot 0524 19.0811 0527 18.9755 0530 18.8711 0533 18.7678 0536 18.6656 0539 18.5645 0542 18.4645 0544 18.3655 0547 18.2677 0550 18.1708 0553 18.0750 0556 17.9802 0559 17.8863 0562 17.7934 0565 17.7015 0568 17.6106 0571 17.5205 0574 17.4314 0577 17.3432 0580 17.2558 0582 17.1693 0585 17.0837 0588 16.9990 0591- 16.9150 0594 16.8319 0597 16.7496 0600 16.6681 0603 16.5874 0606 16.5075 0609 16.4283 0612 163499 0615 16.2722 0617 16.1952 0620 16.1190 0623 16.0435 0626 15.9687 0629 15.8945 0632 15.8211 0635 15.7483 0638 15.6762 0641 15.6048 0644 15.5340 0647 15.4638 0650 15.3943 0653 15.3254 0655 15.2571 0658 15.1893 0661 15.1222 0664 15.0557 0667 14.9898 0670 14.9244 0673 14.8596 0676 14.7954 0679 14.7317 0682 14.6685 0685 14.6059 0688 14.5438 0690 14.4823 0693 14.4212 0696 14.3607 0699 14.3007 cot tan 860 40 tan cot 0699 14.3007 0702 14.2411 0705 14.1821 0708 14.1235 0711 14.0655 0714 14.0079 0717 13.9507 0720 13.8940 0723 13.8378 0726 13.7821 0729 13.7267 0731 13.6719 0734 13.6174 0737 13.5634 0740 13.5098 0743 13.4566 0746 13.4039 0749 13.3515 0752 13.2996 0755 13.2480 0758 13.1969 0761 13.1461 0764 13.0958 0767 13.0458 0769 12.9962 0772 12.9469 0775 12.8981 0778 12.8496 0781 12.8014 0784 1.2.7536 0787 12.7062 0790 12.6591 0793 12.6124 0796 12.5660 0799 12.5199 0802 12.4742 0805 12.4288 0808 12.3838 0810 12.3390 0813 12.2946 0816 12.2505 0819 12.2067 0822 12.1632 0825 12.1201 0828 12.0772 0831 12.0346 0834 11.9923 0837 11.9504 0840 11.9087 0843 11.8673 0846 11.8262 0849 11.7853 0851 11.7448 0854 11.7045 0857 11.6645 0860 11.6248 0863 11.5853 0866 11.5461 0869 11.5072 0872 11.4685 0875 11.4301 cot tan 85~ 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 t I 0 J 6 2 NATURAL TANGENTS AND COTANGENTS. 50 6~ 70 8 90 tan cot tan cot tan cot tan cot tan cot 0 0875 11.4301 1051 9.5144 1228 8.1443 1405 7.1154 1584 6.3138 60 1 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 1411 7.0855 1590 6.2901 58 3 0884 11.3163 1060 9.4352 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 5 0890 11.2417 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 11.1681 1072 9.3315 1249 8.0095 1426 7.0117 1605 6.2316 53 8 0898 11.1316 1075 9.3060 1251 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 11 0907 11.0237 1083 9.2302 1260 7.9344 1438 6.9538 1617 6.1856 49 12 0910 10.9882 1086 9.2052 1263 7.9158 1441 6.9395 1620 6.1742 48 13 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 6.9110 1626 6.1515 46 15 0919 10.8829 1095 9.1309 1272 7.8606 1450 6.8969 1629 6.1402 45 16 0922 10.8483 1098 9.1065 1275 7.8424 1453 6.8828 1632 6.1290 44 17 0925 10.8139 1101 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 1107 9.0338 1284 7.7883 1462 6.8408 1641 6.0955 41 20 0934 10.7119 1110 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 10.6118 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 25 094S 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 1486 6.7313 1664 6.0080 33 28 0957 10.4491 1134 8.8225 1311 7.6301 1489 6.7179 1667 5.9972 32 29 0960 10.4172 1136 8.7996 1314 7.6129 1492 6.7045 1670 5.9865 31 30 0963 10.3854 1139 8.7769 1317 7.5958 1495 6.6912 1673 5.9758 30 31 0966 10.3538 1142 8.7542 1319 7.5787 1497 6.6779 1676 5.9651 29 32 0969 10.3224 1145 8.7317 1322 7.5618 1500 6.6646 1679 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 5.9333 26 35 0978 10.2294 1154 86648 1331 7.5113 1509 6.6252 1688 5.9228 25 36 0981 10.1988 1157 8.6427 1334 7.4947 1512 6.6122 1691 5.9124 24 37 0983 10.1683 1160 8.6208 1337 7.4781 1515 6.5992 1694 5.9019 23 38 0986 10.1381 1163 8.5989 1340 7.4615 1518 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 1169 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 1175 8.5126 1352 7.3962 1530 6.5350 1709 5.8502 18 43 1001 9.9893 1178 8.4913 1355 7.3800 1533 6.5223 1712 5.8400 17 44 1004 9.9601 1181 8.4701 1358 7.3639 1536 6.5097 1715 5.8298 16 45 1007 9.9310 1184 8.4490 1361 7.3479 1539 6.4971 1718 5.8197 15 46 1010 9.9021 1187 8.4280 1364 7.3319 1542 6.4846 1721 5.8095 14 47 1013 9.8734 1189 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 1019 9.8164 1195 8.3656 1373 7.2844 1551 6.4472 1730 5.7794 11 50 1022 9.7882 1198 8.3450 1376 7.2687 1554 6.4348 1733 5.7694 10 51 1025 9.7601 1201 8.3245 1379 7.2531 1557 6.4225 1736 5.7594 9 52 1028 9.7322 1204 8.3041 1382 7.2375 1560 6.4103 1739 5.7495 8 53 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 55 1036 9.6499 1213 8.2434 1391 7.1912 1569 6.3737 1748 5.7199 5 56 1039 9.6220 1216 8.2234' 1394 7.1759 1572 6.3617 1751 5.7101 4 57 1042 9.5949 1219 8.2035 1397 7.1607 1575 6.3496 1754 5.7004 3 58 1045 9.5679 1222 8.1837 1399 7.1455 1578 6.3376 1757 5.6906 2 59 1048 9.5411 1225 8.1640 1402 7.1304 1581 6.3257 1760 5.6809 1 60 1051 9.5144 1228 8.1443 1405 7.1154 1584 6.3138 1763 5.6713 0 cot tan cot tan cot tan cot tan cot tan 840 830 820 810 800 NATURAL TANGENTS AND COTANGENTS. 63 __ 10~ 110 120 13~ 14~ tan cot tan cot tan cot tan cot tan cot 0 1763 5.6713 1944 5.1446 2126 4.7046 2309 4.3315 2493 4.0108 60 1 1766 5.6617 1947 5.1366 2129 4.6979 2312 4.3257 2496 4.0058 59 2 1769 5.6521 1950 5.1286 2132 4.6912 2315 4.3200 2499 4.0009 58 3 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 56 5 1778 5.6234 1959 5.1049 2141 4.6712 2324 4.3029 2509 3.9861 55 6 1781 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 2333 4.2859 2518 3.9714 52 9 1790 5.5857 1971 5.0736 2153 4.6448 2336 4.2803 2521 3.9665 51 10 1793 5.5764 1974 5.0658 2156 4.6382 2339 4.2747 2524 3.9617 50 11 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 13 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 15 1808 5.5301 1989 5.0273 2171 4.6057 2355 4.2468 2540 3.9375 45 16 1811 5.5209 1992 5.0197 2174 4.5993 2358 4.2413 2543 3.9327 44 17 1814 5.5118 1995 5.0121 2177 4.5928 2361 4.2358 2546 3.9279 43 18 1817 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 37 24 1835 5.4486 2016 4.9594 2199 4.5483 2382 4.1976 2568 3.8947 36 25 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 2211 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 1853 5.3955 2035 4.9152 2217 4.5107 2401 4.1653 2586 3.8667 30 31 1856 5.3868 2038 4.9078 2220 4.5045 2404 4.1600 2589 3.8621 29 32 1859 5.3781 2941 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 2595 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 36 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 2611 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 3.8118 18 43 1893 5.2839 2074 4.8218 2257 4.4313 2441 4.0970 2627 3.8073 17 44 1896 5.2755 2077 4.8147 2260 4.4253 2444 4.0918 2630 3.8028 16 45 1899 5.2672 2080 4.8077 2263 4.4194 2447 4.0867 2633 3.7983 15 46 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 11 50 1914 5.2257 2095 4.7729 2278 4.3897 2462 4.0611 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 5.2011 2104 4.7522 2287 4.3721 2472 4.0459 2658 3.7627 7 54 1926 5.1929 2107 4.7453 2290 4.3662 2475 4.0408 2661 3.7583 6 55 1929 5.1848 2110 4.7385 2293 4.3604 2478 4.0358 2664 3.7539 5 56 1932 5.1767 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 58 1938 5.1606 2119 4.7181 2303 4.3430 2487 4.0207 2673 3.7408 2 59 1941 5.1526 2123 4.7114 2306 4.3372 2490 4.0158 2676 3.7364 1 60 1944 5.1446 2126 4.7046 2309 4.3315 2493 4.0108 2679 3.7321 0 _ cot tan cot tan cot tan cot tan cot tan t 7 790 78~ 77~ 76~ 75~ 64 4NATURAL TANGENTS AND COTANGENTS. 3 0 15o 16~ 170 18~ 19~ tan cot tan cot tan cot tan cot tan cot 0 2679 3.7321 2867 3.4874 3057 3.2709 3249 3.0777 3443 2.9042 60 1 2683 3.7277 2871 3.4836 3060 3.2675 3252 3.0746 3447 2.9015 59 2 2686 3.7234 2874 3.4798 3064 3.2641 3256 3.0716 3450 2.8987 58 3 2689 3.7191 2877 3.4760 3067 3.2607 3259 3.0686 3453 2.8960 57 4 2692 3.7148 2880 3.4722 3070 3.2573 3262 3.0655 3456 2.8933 56 5 2695 3.7105 2883 3.4684 3073 3.2539 3265 3.0625 3460 2.8905 55 6 2698 3.7062 2886 3.4646 3076 3.2506 3269 3.0595 3463 2.8878 54 7 2701 3.7019 2890 3.4608 3080 3.2472 3272 3.0565 3466 2.8851 53 8 2704 3.6976 2893 3.4570 3083 3.2438 3275 3.0535 3469 2.8824 52 9 2708 3.6933 2896 3.4533 3086 3.2405 3278 3.0505 3473 2.8797 51 10 2711 3.6891 2899 3.4495 3089 3.2371 3281 3.0475 3476 2.8770 50 11 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 13 2720 3.6764 2908 3.4383 3099 3.2272 3291 3.0385 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 45 16 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 3115 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 3508 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 3/5 26 2761 3.6222 2949 3.3904 3140 3.1845 3333 3.0003 3528 2.8344 34 27 2764 3.6181 2953 3.3868 3143 3.1813 3336 2.9974 3531 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 3544 2.8213 29 32 2780 3.5978 2968 3.3687 3159 3.1652 3352 2.9829 3548 2.8187 28 33 2783 3.5937 2972 3.3652 3163 3.1620 3356 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 36 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 2801 3.5696 2991 3.3438 3182 3.1429 3375 2.9629 3571 2.8006 21 40 2805 3.5656 2994 3.3402 3185 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 45 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 12 49 2833 3.5300 3022 3.3087 3214 3.1115 3408 2.9347 3604 2.7751 11 50 2836 3.5261 3026 3.3052 3217 3.1084 3411 2.9319 3607 2.7725 10 51 2839 3.5222 3029 3.3017 3220 3.1053 3414 2.9291 3610 2.7700 9 52 2842 3.5183 3032 3.2983 3223 3.1022 3417 2.9263 3613 2.7675 8 53 2845 3.5144 3035 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 6 55 2852 3.5067 3041 3.2880 3233 3.0930 3427 2.9180 3623 2.7600 5 56 2855 3.5028 3045 3.2845- 3236 3.0899 3430 2.9152 3627 2.7575 4 57 2858 3.4989 3048 3.2811 3240 3.0868 3434 2.9125 3630 2.7550 3 58 2861 3.4951 3051 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 1 60 2867 3.4874 3057 3.2709 3249 3.0777 3443 2.9042 3640 2.7475 0 cot tail cot tan cot tan cot tan cot tan p f 740 73~ 720 71~ 70~ t 1 0 NATURAL TANGENTS AND COTANGENTS. 65 _ 20~ 210 220 230 240 r tan cot tan cot tan cot tan cot tan cot 0 3640 2.7475 3839 2.6051 4040 2.4751 4245 2.3559 4452 2.2460 60 1 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 2.2408 57 4 3653 2.7376 385? 2.5961 4054 2.4668 4258 2.3483 4466 2.2390 56 5 3656 2.7351 3855 2.5938 4057 2.4648 4262 2.3464 4470 2.2373 55 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 3872 2.5826 4074 2.4545 4279 2.3369 4487 2.2286 50 11 3676 2.7204 3875 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 13 3683 2.7155 3882 2.5759 4084 2.4484 4289 2.3313 4498 2.2234 47 14 3686 2.7130 3885 2.5737 4088 2.4464 4293 2.3294 4501 2.2216. 46 15 3689 2.7106 3889 2.5715 4091 2.4443 4296 2.3276 4505 2.2199 45 16 3693 2.7082 3892 2.5693 4095 2.4423 4300 2.32577 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 3716 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 26 3726 2.6841 3926 2.5473 4129 2.4222 4334 2.3072 4543 2.2011 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 4135 2.4182 4341 2.3035 4550 2.1977 32 29 3736 2.6770 3936 2.5408 4139 2.4162 4345 2.3017 4554 2.1960 31 30 3739 2.6746 3939 2.5386 4142 2.4142 4348 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 2 5 36 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 38 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 2.6511 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 3782 2.6441 3983 2.5108 4187 2.3886 4393 2.2763 4603 2.1725 17 44 3785 2.6418 3986 2.5086 4190 2.3867 4397 2.2745 4607 2.1708 16 45 3789 2.6395 3990 2.5065 4193 2.3847 4400 2.2727 4610 2.1692 15 46 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 4411 2.2673 4621 2.1642 12 49 3802 2.6302 4003 2.4981 4207 2.3770 4414 2.2655 4624 2.1625 11 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 9 52 3812 2.6233 4013 2.4918 4217 2.3712 4424 2.2602 4635 2.1576 8 53 3815 2.6210 4017 2.4897 4221 2.3693 4428 2.2584 4638 2.1560 7 54 3819 2.6187 4020 2.4876 4224 2.3673 4431 2.2566 4642 2.1543 6 55 3822 2.6165 4023 2.4855 4228 2.3654 4435 2.2549 4645 2.1527 5 56 3825 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 58 3832 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 1 60 3839 2.6051 4040 2.4751 4245 2.3559 4452 2.2460 4663 2.1445 0 cot tan cot tan cot tan cot tan cot tan t 6690 68~ 670 660 650 t i i,, i i i... - 66 'NATURAL TANGENTS AND COTANGENTS. p _ 9250 26~ 270 28~ 290 tan cot tan cot tan cot tan cot tan cot O 4663 2.1445 4877 2.0503 5095 1.9626 5317 1.8807 5543 1.8040 60 1 4667 2.1429 4881 2.0488 5099 1.9612 5321 1.8794 5547 1.8028 59 2 4670 2.1413 4885 2.0473 5103 1.9598 5325 1.8781 5551 1.8016 58 3 4674 2.1396 4888 2.0458 5106 1.9584 5328 1.8768 5555 1.8003 57 4 4677 2.1380 4892 2.0443 5110 1.9570 5332 1.8755 5558 1.7991 56 5 4681 2.1364 4895 2.0428 5114 1.9556 5336 1.8741 5562 1.7979 55 6 4684 2.1348 4899 2.0413 5117 1.9542 5340 1.8728 5566 1.7966 54 7 4688 2.1332 4903 2.0398 5121 1.9528 '5343 1.8715 5570 1.7954 53 8 4691 2.1315 4906 2.0383 5125 1.9514 5347 1.8702 5574 1.7942 52 9 4695 2.1299 4910 2.0368 5128 1.9500 5351 1.8689 5577 1.7930 51 10 4699 2.1283 4913 2.0353 5132 1.9486 5354 1.8676 5581 1.7917 50 11 4702 2.1267 4917 2.0338 5136 1.9472 5358 1.8663 5585 1.7905 49 12 4706 2.1251 4921 2.0323 5139 1.9458 5362 1.8650 5589 1.7893 48 13 4709 2.1235 4924 2.0308 5143 1.9444 5366 1.8637 5593 1.7881 47 14 4713 2.1219 4928 2.0293 5147 1.9430 5369 1.8624 5596 1.7868 46 15 4716 2.1203 4931 2.0278 5150 1.9416 5373 1.8611 5600 1.7856 45 16 4720 2.1187 4935 2.0263 5154 1.9402 5377 1.8598 5604 1.7844 44 17 4723 2.1171 4939 2.0248 5158 1.9388 5381 1.8585 5608 1.7832 43 18 4727 2.1155 4942 2.0233 5161 1.9375 5384 1.8572 5612 1.7820 42 19 4731 2.1139 4946 2.0219 5165 1.9361 5388 1.8559 5616 1.7808 41 20 4734 2.1123 4950 2.0204 5169 1.9347 5392 1.8546 5619 1.7796 40 21 4738 2.1]107 4953 2.0189 5172 1.9333 5396 1.8533 5623 1.7783 39 22 4741 2.1092 4957 2.0174 5176 1.9319 5399 1.8520 5627 1.7771 38 23 4745 2.1076 4960 2.0160 5180 1.9306 5403 1.8507 5631 1.7759 37 24 4748 2.1060 4964 2.0145 5184 1.9292 5407 1.8495 5635 1.7747 36 25 4752 2.1044 4968 2.0130 5187 1.9278 5411 1.8482 5639 1.7735 35 26 4755 2.1028 4971 2.0115 5191 1.9265 5415 1.8469 5642 1.7723 34 27 4759 2.1013 4975 2.0101 5195 1.9251 5418 1.8456 5646 1.7711 33 28 4763 2.0997 4979 2.0086 5198 1.9237 5422 1.8443 5650 1.7699 32 29 4766 2.0981 4982 2.0072 5202 1.9223 5426 1.8430 5654 1.7687 31 30 4770 2.0965 4986 2.0057 5206 1.9210 5430 1.8418 5658 1.7675 30 31 4773 2.0950 4989 2.0042 5209 1.9196 5433 1.8405 5662 1.7663 29 32 4777 2.0934 4993 2.0028 5213 1.9183 5437 1.8392 5665 1.7651 28 33 4780 2.0918 4997 2.0013 5217 1.9169 5441 1.8379 5669 1.7639 27 34 4784 2.0903 5000 1.9999 5220 1.9155 5445 1.8367 5673 1.7627 26 35 4788 2.0887 5004 1.9984 5224 1.9142 5448 1.8354 5677 1.7615 2,5 36 4791 2.0872 5008 1.9970 5228 1.9128 5452 1.8341 5681 1.7603 24 37 4795 2.0856 5011 1.9955 5232 1.9115 5456 1.8329 5685.1.7591 23 38 4798 2.0840 5015 1.9941 5235 1.9101 5460 1.8316 5688 1.7579 22 39 4802 2.0825 5019 1.9926 5239 1.9088 5464 1.8303 5692 1.7567 21 40 4806 2.0809 5022 1.9912 5243 1.9074 5467 1.8291 5696 1.7556 20 41. 4809 2.0794 5026 1.9897 5246 1.9061 5471 1.8278 5700 1.7544 19 42 4813 2.0778 5029 1.9883 5250 1.9047 5475 1.8265 5704 1.7532 18 43 4816 2.0763 5033 1.9868 5254 1.9034 5479 1.8253 5708 1.7520 17 44 4820 2.0748 5037 1.9854 5258 1.9020 5482 1.8240 5712 1.7508' 16 45 4823 2.0732 5040 1.9840 5261 1.9007 5486 1.8228 5715 1.7496 15 46 4827 2.0717 5044 1.9825 5265 1.8993 5490 1.8215 5719 1.7485 14 47 4831 2.0701 5048 1.9811 5269 1.8980 5494 1.8202 5723 1.7473 13 48 4834 2.0686 5051 1.9797 5272 1.8967 5498 1.8190 5727 1.7461 12 49 4838 2.0671 5055 1.9782 5276 1.8953 5501 1.8177 5731 1.7449 11 50 4841 2.0655 5059 1.9768 5280 1.8940 5505 1.8165 5735 1.7437 10 51 4845 2.0640 5062 1.9754 5284 1.8927 5509 1.8152 5739 1.7426 9 52 4849 2.0625 5066 1.9740 5287 1.8913 5513 1.8140 5743 1.7414 8 53 4852 2.0609 5070 1.9725 5291 1.8900 5517 1.8127 5746 1.7402 7 54 4856 2.0594 5073 1.9711 5295 1.8887 5520 1.8115 5750 1.7391 6 55 4859 2.0579 5077 1.9697 5298 1.8873 5524 1.8103 5754 1.7379 5 56 4863 2.0564 5081 1.9683 5302 1.8860 5528 1.8090 5758 1.7367 4 57 4867 2.0549 5084 1.9669 5306 1.8847 5532 1.8078 5762 1.7355 3 58 4870 2.0533 5088 1.9654 5310 1.8834 5535 1.8065 5766 1.7344 2 59 4874 2.0518 5092 1.9640 5313 1.8820 5539 1.8053 5770 1.7332 1 60 4877 2.0503 5095 1.9626 5317 1.8807 5543 1.8040 5774 1.7321 0 cot tan cot tan cot tan cot tan cot tan _,7~ 640 630 62~ 610 600 NATURAL TANGENTS AND COTANGENTS. 67 0 0 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 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 30~ 31~ tan cot tan cot 5774 1.7321 6009 1.6643 5777 1.7309 6013 1.6632 5781 1.7297 6017 1.6621 5785 1.7286 6020 1.6610 5789 1.7274 6024 1.6599 5793 1.7262 6028 1.6588 5797 1.7251 6032 1.6577 5801 1.7239 6036 1.6566 5805 1.7228 6040 1.6555 5808 1.7216 6044 1.6545 5812 1.7205 6048 1.6534 5816 1.7193 6052 1.6523 5820 1.7182 6056 1.6512 5824 1.7170 6060 1.6501 5828 1.7159 6064 1.6490 5832 1.7147 6068 1.6479 5836 1.7136 6072 1.6469 5840 1.7124 6076 1.6458 5844 1.7113 6080 1.6447 5847 1.7102 6084 1.6436 5851 1.7090 6088 1.6426 5855 1.7079 6092 1.6415 5859 1.7067 6096 1.6404 5863 1.7056 6100 1.6393 5867 1.7045 6104 1.6383 5871 1.7033 6108 1.6372 5875 1.7022 6112 1.6361 5879 1.7011 6116 1.6351 5883 1.6999 6120 ].6340 5887 1.6988 6124 1.6329 5890 1.6977 6128 1.6319 5894 1.6965 6132 1.6308 5898 1.6954 6136 1.6297 5902 1.6943 6140 1.6287 5906 1.6932 6144 1.6276 591]0 1.6920 6148 1.6265 5914 1.6909 6152 1 6255 5918 1.6898 6156 1.6244 5922 1.6887 6160 1.6234 5926 1.6875 6164 1.6223 5930 1.6864 6168 1.6212 5934 1.6853 6172 1.6202 5938 1.6842 6176 1.6191 5942 1.6831 6180 1.6181 5945 1.6820 6184 1.6170 5949 1.6808 6188 1.6160 5953 1.6797 6192 1.6149 5957 1.6786 6196 1.6139 5961 1.6775 6200 1.6128 5965 1.6764 6204 1.6118 5969 1.6753 6208 1.6107 5973 1.6742 6212 1.6097 5977 1.6731 6216 1.6087 5981 1.6720 6220 1.6076 5985 1.6709 6224 1.6066 5989 1.6698 6228 1.6055 5993 1.6687 6233 1.6045 5997 1.6676 6237 1.6034 6001 1.6665 6241 1.6024 6005 1.6654 6245 1.6014 6009 1.6643 6249 1.6003 cot tan cot tan 59~ 58~ 32~ 330 tan cot tan cot tan cot 6249 1.6003 6494 1.5399 6745 1.4826 6253 1.5993 6498 1.5389 6749 1.4816 6257 1.5983 6502 1.5379 6754 1.4807 6261 1.5972 6506 1.5369 6758 1.4798 6265 1.5962 6511 1.5359 6762 1.4788 6269 1.5952 6515 1.5350 6766 1.4779 6273 1.5941 6519 1.5340 6771 1.4770 6277 1.5931 6523 1.5330 6775 1.4761 6281 1.5921 6527 1.5320 6779 1.4751 6285 1.5911 6531 1.5311 6783 1.4742 6289 1.5900 6536 1.5301 6787 1.4733 6293 1.5890 6540 1.5291 6792 1.4724 6297 1.5880 6544 1.5282 6796 1.4715 6301 1.5869 6548 1.5272 6800 1.4705 6305 1.5859 6552 1.5262 6805 1.4696 6310 1.5849 6556 1.5253 6809 1.4687 6314 1.5839 6560 1.5243 6813 1.4678 6318 1.5829 6565 1.5233 6817 1.4669 6322 1.5818 6569 1.5224 6822 1.4659 6326 1.5808 6573 1.5214 6826 1.4650 6330 1.5798 6577 1.5204 6830 1.4641 6334 1.5788 6581 1.5195 6834 1.4632 6338 1.5778 6585 1.5185 6839 1.4623 6342 1.5768 6590 ].5175 6843 1.4614 6346 1.5757 6594 1.5166 6847 1.4605 6350 1.5747 6598 1.5156 6851 1.4596 6354 1.5737 6602 1.5147 6856 1.4586 6358 1.5727 6606 1.5137 6860 1.4577 6363 1.5717 6610 1.5127 6864 1.4568 6367 1.5707 6615 1.5118 6869 1.4559 6371 1.5697 6619 1.5108 6873 1.4550 6375 1.5687 6623 1.5099 6877 1.4541 6379 1.5677 6627 1.5089 6881 1.4532 6383 1.5667 6631 1.5080 6886 1.4523 6387 1.5657 6636 1.5070 6890 1.4514 6391 1.5647 6640 1.5061 6894 1.4505 6395 1.5637 6644 1.5051 6899 1.4496 6399 1.5627 6648 1.5042 6903 1.4487 6403 1.5617 6652 1.5032 6907 1.4478 6408 1.5607 6657 1.5023 6911 1.4469 6412 1.5597 6661 1.5013 6916 1.4460 6416 1.5587 6665 1.5004 6920 1.4451 6420 1.5577 6669 1.4994 6924 1.4442 6424 1.5567 6673 1.4985 6929 1.4433 6428 1.5557 6678 1.4975 6933 1.4424 6432 1.5547 6682 1.4966 6937 1.4415 6436 1.5537 6686 1.4957 6942 1.4406 6440 1.5527 6690 1.4947 6946 1.4397 6445 1.5517 6694 1.4938 6950 1.4388 6449 1.5507 6699 1.4928 6954 1.4379 6453 1.5497 6703 1.4919 6959 1.4370 6457 1.5487 6707 1.4910 6963 1.4361 6461 1.5477 6711 1.4900 6967 1.4352 6465 1.5468 6716 1.4891 6972 1.4344 6469 1.5458 6720 1.4882 6976 1.4335 6473 1.5448 6724 1.4872 6980 1.4326 6478 1.5438 6728 1.4863 6985 1.4317 6482 1.5428 6732 1.4854 6989 1.4308 6486 1.5418 6737 1.4844 6993 1.4299 6490 1.5408 6741 1.4835 6998 1.4290 6494 1.5399 6745 1.4826 7002 1.4281 cot tan cot tan cot tan 570 56~ 55~ 340 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 I c NATJURAL TANGENTS AND COTANGENTS. m r 0 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 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 _ 350 tan cot 7002 1.4281 7006 1.4273 7011 1.4264 7015 1.4255 7019 1.4246 7024 1.4237 7028 1.4229 7032 1.4220 7037 1.4211 7041 1.4202 7046 1.4193 7050 1.4185 7054 1.4176 7059 1.4167 7063 1.4158 7067 1.4150 7072 1.4141 7076 1.4132 7080 1.4124 7085 1.4115 7089 1.4106 7094 1.4097 7098 1.4089 7102 1.4080 7107 1.4071 7111 1.4063 7115 1.4054 7120 1.4045 7124 1.4037 7129 1.4028 7133 1.4019 7137 1.4011 7142 1.4002 7146 1.3994 7151 1.3985, 7155 1.3976 7159 1.3968 7164 1.3959 7168 1.3951 7173 1.3942 7177 1.3934 7181 1.3925 7186 1.3916 7190 1.3908 7195 1.3899 7199 1.3891 7203 1.3882 7208 1.3874 7212 1.3865 7217 1.3857 7221 1.3848 7226 1.3840 7230 1.3831 7234 1.3823 7239 1.3814 7243 1.3806 7248 1.3798 7252 1.3789 7257 1.3781 7261 1.3772 7265 1.3764 cot tan 540 360 370 tan cot tan cot 7265 1.3764 7536 1.3270 7270 1.3755 7540 1.3262 7274 1.3747 7545 1.3254 7279 1.3739 7549 1.3246 7283 1.3730 7554 1.3238 7288 1.3722 7558 1.3230 7292 1.3713 7563 1.3222 7297 1.3705 7568 1.3214 7301 1.3697 7572 1.3206 7306 1.3688 7577 1.3198 7310 1.3680 7581 1.3190 7314 1.3672 7586 1.3182 7319 1.3663 7590 1.3175 7323 1.3655 7595 1.3167 7328 1.3647 7600 1.3159 7332 1.3638 7604 1.3151 7337 1.3630 7609 1.3143 7341 1.3622 7613 1.3135 7346 1.3613 7618 1.3127 7350 1.3605 7623 1.3119 7355 1.3597 7627 1.3111 7359 1.3588 7632 1.3103 7364 1.3580 7636 1.3095 7368 1.3572 7641 1.3087 7373 1.3564 7646 1.3079 7377 1.3555 7650 1.3072 7382 1.3547 7655 1.3064 7386 1.3539 7659 1.3056 7391 1.3531 7664 1.3048 7395 1.3522 7669 1.3040 7400 1.3514 7673 1.3032 7404 1.3506 7678 1.3024 7409 1.3498 7683 1.3017 7413 1.3490 7687 1.3009 7418 1.3481 7692 1.3001 7422 1.3473 7696 1.2993 7427 1.3465 7701 1.2985 7431 1.3457 7706 1.2977 7436 1.3449 7710 1.2970 7440 1.3440 7715 1.2962 7445 1.3432 7720 1.2954 7449 1.3424 7724 1.2946 7454 1.3416 7729 1.2938 7458 1.3408 7734 1.2931 7463 1.3400 7738 1.2923 7467 '1.3392 7743 1.2915 7472 1.3384 7747 1.2907 7476 1.3375 7752 1.2900 7481 1.3367 7757 1.2892 7485 1.3359 7761 1.2884 7490 1.3351 7766 1.2876 7495 1.3343 7771 1.2869 7499 1.3335 7775 1.2861 7504 1.3327 7780 1.2853 7508 1.3319 7785 1.2846 7513 1.3311 7789 1.2838 7517 1.3303 7794 1.2830 7522 1.3295 7799 1.2822 7526 1.3287 7803 1.2815 7531 1.3278 7808 1.2807 7536 1.3270 7813 1.2799 cot tan cot tan 530 520 380 tan cot 7813 1.2799 7818 1.2792 7822 1.2784 7827 1.2776 7832 1.2769 7836 1.2761 7841 1.2753 7846 1.2746 7850 1.2738 7855 1.2731 7860 1.2723 7865 1.2715 7869 1.2708 7874 1.2700 7879 1.2693 7883 1.2685 7888 1.2677 7893 1.2670 7898 1.2662 7902 1.2655 7907 1.2647 7912 1.2640 7916 1.2632 7921 1.2624 7926 1.2617 7931 1.2609 7935 1.2602 7940 1.2594 7945 1.2587 7950 1.2579 7954 1.2572 7959 1.2564 7964 1.2557 7969 1.2549 7973 1.2542 7978 1.2534 7983 1.2527 7988 1.2519 7992 1.2512 7997 1.2504 '8002 1.2497 8007 1.2489 8012 1.2482 8016 1.2475 8021. 1.2467 8026 1.2460 8031 1.2452 8035 1.2445 804-0 1.2437 8045 1.2430 8050 1.2423 8055 1.2415 8059 1.2408 8064 1.2401 8069 1.2393 8074 1.2386 8079 1.2378 8083 1.2371 8088 1.2364 8093 1.2356 8098 1.2349 cot tan 510 390 tan cot 8098 1.2349 8103 1.2342 8107 1.2334 8112 1.2327 8117 1.2320 8122 1.2312 8127 1.2305 8132 1.2298 8136 1.2290 814-1 1.2283 8146 1.2276 8151 1.2268 8156 1.2261 8161 1.2254 8165 1.2247 8170 1.2239 8175 1.2232 8180 1.2225 8185 1.2218 8190 1.2210 8195 1.2203 8199 1.2196 8204 1.2189 8209 1.2181 8214 1.2174 8219 1.2167 8224 1.2160 8229 1.2153 8234 1.2145 8238 1.2138 8243 1.2131 8248 1.2124 8253 1.2117 8258 1.2109 8263 1.2102 8268 1.2095 8273 1.2088 8278 1.2081 8283 1.2074 8287 1.2066 8292 1.2059 8297 1.2052 8302 1.2045 8307 1.2038 8312 1.2031 8317 1.2024 8322 1.2017 8327 1.2009 8332 1.2002 8337 1.1995 8342 1.1988 8346 1.1981 8351 1.1974 8356 1.1967 8361 1.1960 8366 1.1953 8371 1.1946 8376 1.1939 8381 1.1932 8386 1.1925 8391 1.1918 cot tan 500 f 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 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 11 10 9 8 7 6 5 4 3 2 1 0 f I I I _ _.. _, i NATURAL TANGENTS AND COTANGENTS. 0 400 410 420 430 44 r tan cot tan cot tan cot tan cot tan cot 0 8391 1.1918 8693 1.1504 9004 1.1106 9325 1.0724 9657 1.0355 60 1 8396 1.1910 8698 1.1497 9009 1.1100 9331 1.0717 9663 1.0349 59 2 8401 1.1903 8703 1.1490 9015 1.1093 9336 1.0711 9668 1.0343 58 3 8406 1.1896 8708 1.1483 9020 1.1087 9341 1.0705 9674 1.0337 57 4 8411 1.1889 8713 1.1477 9025 1.1060 9347 1.0699 9679 1.0331 56 5 8416 1.1882 8718 1.1470 9030 1.1074 9352 1.0692 9685 1.0325 5,5 6 8421 1.1875 8724 1.1463 9036 1.1067 9358 1.0686 9691 1.0319 54 7 8426 1.1868 8729 1.1456 9041 1.1061 9363 1.0680 9696 1.0313 53 8 8431 1.1861 8734 1.1450 9046 1.1054 9369 1.0674 9702 1.0307 52 9 8436 1.1854 8739 1.1443 9052 1.1048 9374 1.0668 9708 1.0301 51 10 8441 1.1847 8744 1.1436 9057 1.1041 9380 1.0661 9713 1.0295 50 11 8446 1.1840 8749 1.1430 9062 1.1035 9385 1.0655 9719 1.0289 49 12 8451 1.1833 8754 1.1423 9067 1.1028 9391 1.0649 9725 1.0283 48 13 8456 1.1826 8759 1.1416 9073 1.1022 9396 1.0643 9730 1.0277 47 14 8461 1.1819 8765 1.1410 9078 1.1016 9402 1.0637 9736 1.0271 46 15 8466 1.1.812 8770 1.1403 9083 1.1009 9407 1.0630 9742 1.0265 45 16 8471 1.1806 8775 1.1396 9089 1.1003 9413 1.0624 9747 1.0259 44 17 8476 1.1799 8780 1.1389 9094 1.0996 9418 1.0618 9753 1.0253 43 18 8481 1.1792 8785 1.1383 9099 1.0990 9424 1.0612 9759 1.0247 42 19 8486 1.1785 8790 1.1376 9105 1.0983 9429 1.0606 9764 1.0241 41 20 8491 1.1778 8796 1.1369 9110 1.0977 9435 1.0599 9770 1.0235 40 21 8496 1.1771 8801 1.1363 9115 1.0971 9440 1.0593 9776 1.0230 39 22 8501 1.1764 8806 1.1356 9121 1.0964 9446 1.0587 9781 1.0224 38 23 8506 1.1757 8811 1.1349 9126 1.0958 9451 1.0581 9787 1.0218 37 24 8511 1.1750 8816 1.1343 9131 1.0951 9457 1.0575 9793 1.0212 36 25 8516 1.1743 8821 1.1336 9137 1.0945 9462 1.0569 9798 1.0206 3,5 26 8521 1.1736 8827 1.1329 9142 1.0939 9468 1.0562 9804 1.0200 34 27 8526 1.1729 8832 1.1323 9147 1.0932 9473 1.0556 9810 1.0194 33 28 8531 1.1722 8837 ].1316 9153 1.0926 9479 1.0550 9816 1.0188 32 29 8536 1.1715 8842 1.1310 9158 1.0919 9484 1.0544 9821 1.0182 31 30 8541 1.1708 8847 1.1303 9163 1.0913 9490 1.0538 9827 1.0176 30 31 8546 1.1702 8852 1.1296 9169 1.0907 9495 1.0532 9833 1.0170 29 32 8551 1.1695 8858 1.1290 9174 1.0900 9501 1.0526 9838 1.0164 28 33 8556 1.1688 8863 1.1283 9179 1.0894 9506 1.0519 9844 1.0158 27 34 8561 1.1681 8868 1.1276 9185 1.0888 9512 1.0513 9850 1.0152 26 35 8566 1.1674 8873 1.1270 9190 1.0881 9517 1.0507 9856 1.0147 25 36 8571 1.1667 8878 1.1263 9195 1.0875 9523 1.0501 9861 1.0141 24 37 8576 1.1660 8884 1.1257 9201 1.0869 9528 1.0495 9867 1.0135 23 38 8581 1.1653 8889 1.1250 9206 1.0862 9534 1.0489 9873 1.0129 22 39 8586 1.1647 8894 1.1243 9212 1.0856 9540 1.0483 9879 1.0123 21 40 8591 1.1640 8899 1.1237 9217 1.0850 9545 1.0477 9884 1.0117 20 41 8596 1.1633 8904 1.1230 9222 1.0843 9551 1.0470 9890 1.0111 19 42 8601 1.1626 8910 1.1224 9228 1.0837 9556 1.0464 9896 1.0105 18 43 8606 1.1619 8915 1.1217 9233 1.0831 9562 1.0458 9902 1.0099 17 44 8611 1.1612 8920 1.1211 9239 1.0824 9567 1.0452 9907 1.0094 16 45 8617 1.1606 8925 1.1204 9244 1.0818 9573 1.0446 9913 1.0088 1 5 46 8622 1.1599 8931 1.1197 9249 1.0812 9578 1.0440 9919 1.0082 14 47 8627 1.1592 8936 1.1191 9255 1.0805 9584 1.0434 9925 1.0076 13 48 8632 1.1585 8941 1.1184 9260 1.0799 9590 1.0428 9930 1.0070 12 49 8637 1.1578 8946 1.1178 9266 1.0793 9595 1.0422 9936 1.0064 11 50 8642 1.1571 8952 1.1171 9271 1.0786 9601 1.0416 9942 1.0058 10 51 8647 1.1565 8957 1.1165 9276 1.0780 9606 1.0410 9948 1.0052 9 52 8652 1.1558 8962 1.1158 9282 1.0774 9612 1.0404 9954 1.0047 8 53 8657 1.1551 8967 1.1152 9287 1.0768 9618 1.0398 9959 1.0041 7 54 8662 1.1544 8972 1.1145 9293 1.0761 9623 1.0392 9965 1.0035 6 55 8667 1.1538 8978 1.1139 9298 1.0755 9629 1.0385 9971 1.0029 5 56 8672 1.1531 8983 1.1132 9303 1.0749 9634 1.0379 9977 1.0023 4 57 8678 1.1524 8988 1.1126 9309 1.0742 9640 1.0373 9983 1.0017 3 58 8683 1.1517 8994 1.1119 9314 1.0736 9646 1.0367 9988 1.0012 2 59 8688 1.1510 8999 1.1113 9320 1.0730 9651 1.0361 9994 1.0006 1 60 8693 1.1504 9004 1.1106 9325 1.0724 9657 1.0355 1.000 1.0000 0 cot tan cot tan cot tan cot tan cot tan t - 490 480 470 460 450 f..I.... ii- - * TABLE VII. -TRAVERSE TABLE. 0 -I. I Bearing. o f 015 30 45 1 0 15 30 45 2 0 15 30 45 3 0 15 30 45 4 0 15 30 45 0 15 30 45 6 0 15 30 45 7 0 15 30 45 8 0 15 30 45 9 0 15 30 45 10 0 15 30 45 11 0 15 30 45 12 0 15 30 45 13 0 15 30 45 14 0 15 30 45 15 0 0 f Bearing, Distance 1. Lat. Dep, 1.000 0.004 1.000 0.009 1.000 0.013 1.000 0.017 1.000 0.022 1.000 0.026 1.000 0.031 0.999 0.035 0.999 0.039 0.999 0.044 0.999 0.048 0.999 0.052 0.998 0.057 0.998 0.061 0.998 0.065 0.998 0.070 0.997 0.074 0.997 0.078 0.997 0.083 0.996 0.087 0.996 0.092 0.995 0.096 0.995 0.100 0.995 0.105 0.994 0.109 0.994 0.113 0.993 0.118 0.993 0.122 0.992 0.126 0.991 0.131 0.991 0.135 0.990 0.139 0.990 0.143 0.989 0.148 0.988 0.152 0.988 0.156 0.987 0.161 0.986 0.165 0.986 0.169 0.985 0.174 0.984 0.178 0.983 0.182 0.982 0.187 0.98?, 0.191 0.981 0.195 0.980 0.199 0.979 0.204 0.978 0.208 0.977 0.212 0.976 0.216 0.975 0.221 0.974 0.225 0.973 0.229 0.972 0.233 0 971 0.238 0.970 0.242 0.969 0.246 0.968 0.250 0.967 0.255 0.966 0.259 Dep. Lat. Distance 1. Distance 2. Distance 3. Distance 4. Lat. Dep. Lat, Dep. Lat. Dep. 2.000 0.009 3.000 0.013 4.000 0.017 2.000 0.017 3.000 0.026 4.000 0.035 2.000 0.026 3.000 0.039 4.000 0.052 2.000 0.035 3.000 0.052 3.999 0.070 2.000 0.044 2.999 0.065 3.999 0.087 1.999 0.052 2.999 0.079 3.999 0.105 1.999 0.061 2.999 0.092 3.998 0.122 1.999 0.070 2.998 0.105 3.998 0.140 1.998 0.079 2.998 0.118 3.997 0.157 1.998 0.087 2.997 0.131 3.996 0.174 1.998 0.096 2.997 0.144 3.995 0.192 1.997 0.105 2.996 0.157 3.995 0.209 1.997 0.113 2.995 0.170 3.994 0.227 1,996 0.122 2.994 0.183 3.993 0.244 1.996 0.131 2.994 0.196 3.991 0.262 1.995 0.140 2.993 0.209 3.990 0.279 1.995 0.148 2.992 0.222 3.989 0.296 1.994 0.157 2.991 0.235 3.988 0.314 1.993 0.166 2.990 0.248 3.986 0.331 1.992 0.174 2.989 0.261 3.985 0.349 1.992 0.183 2.987 0.275 3.983 0.366 1.991 0.192 2.986 0.288 3.982 0.383 1.990 0.200 2.985 0.301 3.980 0.401 1.989 0.209 2.984 0.314 3.978 0.418 1.988 0.218 2.982 0.327 3.976 0.435 1.987 0.226 2.981 0.340 3.974 0.453 1.986 0.235 2.979 0.353 3.972 0.470 1.985 0.244 2.978 0.366 3.970 0.487 1.984 0.252 2.976 0.379 3.968 0.505 1.983 0.261 2.974 0.392 3.966 0.522 1.982 0.270 2.973 0.405 3.963 0.539 1.981 0.278 2.971 0.418 3.961 0.557 1.979 0.287 2.969 0.430 3.959 0.574 1.978 0.296 2.967 0.443 3.956 0.591 1.977 0.304 2.965 0.456 3.953 0.608 1.975 0.313 2.963 0.469 3.951 0.626 1.974 0.321 2.961 0.482 3.948 0.643 1.973 0.330 2.959 0.495 3.945 0.660 1.971 0.339 2.957 0.508 3.942 0.677 1.970 0.347 2.954 0.521 3.939 0.695 1.968 0.356 2.952 0.534 3.936 0.712 1.967 0.364 2.950 0.547 3.933 0.729 1.965 0.373 2.947 0.560 3.930 0.746 1.963 0.382 2.945 0.572 3.927 0.763 1.962 0.390 2.942 0.585 3.923 0.780 1.960 0.399 2.940 0.598 3.920 0.797 1.958 0.407 2.937 0.611 3.916 0.815 -1.956 0.416 2.934 0.624 3.913 0.832 1.954 0.424 2.932 0.637 3.909 0.849 1.953 0.433 2.929 0.649 3.905 0.866 1.951 0.441 2.926 0.662 3.901 0.883 1.949 0.450 2.923 0.675 3.897 0.900 1.947 0.458 2.920 0.688 3.894 0.917 1.945 0.467 2.917 0.700 3.889 0.934 1.943 0.475 2.914 0.713 3.885 0.951 1.941 0.484 2.911 0.726 3.881 0.968 1.938 0.492 2.908 0.738 3.877 0.985 1.936 0.501 2.904 0.751 3.873 1.002 1.934 0.509 2.901 0.764 3.868 1.018 1.932 0.518 2.898 0.776 3.864 1.035 Dep. Lat. Dep. Lat. Dep, Lat. Distance 2. Distance 3. Distance 4. Distance 5. Bearing. f, Lat. Dep. 5.000 0.022 5.000 0.044 5.000 0.065 4.999 0.087 4.999 0.109 4.998 0.131 4.998 0.153 4.997 0.174 4.996 0.196 4.995 0.218 4.994 0.240 4.993 0.262 4.992 0.283 4.991 0.305 4.989 0.327 4.988 0.349 4.986 0.371 4.985 0.392 4.983 0.414 4.981 0.436 4.979 0.458 4.977 0.479 4.975 0.501 4.973 0.523 4.970 0.544 4.968 0.566 4.965 0.588 4.963 0.609 4.960 0.631 4.957 0.653 4.954 0.674 4.951 0.696 4.948 0.717 4.945 0.739 4.942 0.761 4.938 0.782 4.935 0.804 4.931 0.825 4.928 0.847 4.924 0.868 4.920 0.890 4.916 0.911 4.912 0.933 4.908 0.954 4.904 0.975 4.900 0.997 4.895 1.018 4.891 1.040 4886 1.061 4.881 1.082 4.877 1.103 4.872 1.125 4.867 1.146 4.862 1.167 4.857 1.188 4.851 1.210 4.846 1.231 4.841 1.252 4.835 1.273 4.830 1.294 Dep. Lat. Distance 5. 0! 89 45 30 15 89 0 45 30 15 88 0 45 30 15 87 0 45 30 15 86 0 45 30 15 85 0 45 30 15 84 0 45 30 15 83 0 45 30 15 82 0 45 30 15 81 0 45 30 15 80 0 45 30 15 79 0 45 30 15 78 0 45 30 15 77 0 45 30 15 76 0 45 30 15 75 0 O f Bearing. L I I 75~-90~ O0-o_15 71 Bearing, Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. o ~ Lat. Dep, Lat. Dep. Lat. Dep. Lat. Dep. Lat, Dep. 0 15 6.000 0.026 7.000 0.031 8.000 0.035 9.000 0.039 10.000 0.044 89 45 30 6.000 0.052 7.000 0.061 8.000 0.070 9.000 0.079 10.000 0.087 30 45 5.999 0.079 6.999 0.092 7.999 0.105 8.999 0.118 9.999 0.131 15 1 0 5.999 0.105 6.999 0.122 7.999 0.140 8.999 0.157 9.999 0.175 89 0 15 5.999 0.131 6.998 0.153 7.998 0.175 8.998 0.196 9.998 0.218 45 30 5.998 0.157 6.998 0.183 7.997 0.209 8.997 0.236 9.997 0.262 30 45 5.997 0.183 6.997 0.214 7.996 0.244 8.996 0.275 9.995 0.305 15 2 0 5.996 0.209 6.996 0.244 7.995 0.279 8.995 0.314 9.994 0.349 88 0 15 5.995 0.236 6.995 0.275 7.994 0.314 8.993 0.353 9.992 0.393 45 30 5.994 0.262 6.993 0.305 7.992 0.349 8.991 0.393 9.991 0.436 30 45 5.993 0.288 6.992 0.336 7.991 0.384 8.990 0.432 9.989 0.480 15 3 0 5.992 0.314 6.990 0.366 7.989 0.419 8.988 0.471 9.986 0.523 87 0 15 5.990 0.340 6.989 0.397 7.987 0.454 8.986 0.510 9.984 0.567 45 30 5.989 0.366 6.987 0.427 7.985 0.488 8.983 0.549 9.981 0.611 30 45 5.987 0.392 6.985 0.458 7.983 0.523 8.981 0.589 9.979 0.654 15 4 0 5.985 0.419 6.983 0.488 7.981 0.558 8.978 0.628 9.976 0.698 86 0 15 5.984 0.445 6.981 0.519 7.978 0.593 8.975 0.667 9.973 0.741 45 30 5.982 0.471 6.978 0.549 7.975 0.628 8.972 0.706 9.969 0.785 30 45 5.979 0.497 6.976 0.580 7.973 0.662 8.969 0.745 9.966 0.828 15 5 0 5.977 0.523 6.973 0.610 7.970 0.697 8.966 0.784 9.962 0.872 85 0 15 5.975 0.549 6.971 0.641 7.966 0.732 8.962 0.824 9.958 0.915 45 30 5.972 0.575 6.968 0.671 7.963 0.767 8.959 0.863 9.954 0.959 30 45 5.970 0.601 6.965 0.701 7.960 0.802 8.955 0.902 9.950 1.002 15 6 0 5.967 0.627 6.962 0.732 7.956 0.836 8.951 0.941 9.945 1.045 84 0 15 5.964 0.653 6.958 0.762 7.952 0.871 8.947 0.980 9.941 1.089 45 30 5.961 0.679 6.955 0.792 7.949 0.906 8.942 1.019 9.936 1.132 30 45 5.958 0.705 6.951 0.823 7.945 0.940 8.938 1.058 9.931 1.175 15 7 0 5.955 0.731 6.948 0.853 7.940 0.975 8.933 1.097 9.926 1.219 83 0 15 5.952 0.757 6.944 0.883 7.936 1.010 8.928 1.136 9.920 1.262. 45 30 5.949 0.783 6.940 0.914 7.932 1.044 8.923 1.175 9.914 1.305 30 45 5.945 0.809 6.936 0.944 7.927 1.079 8.918 1.214 9.909 1.349 15 8 0 5.942 0.835 6.932 0.974 7.922 1.113 8.912 1.253 9.903 1.392 82 0 15 5.938 0.861 6.928 1.004 7.917 1.148 8.907 1.291 9.897 1.435 45 30 5.934 0.887 6.923 1.035 7.912 1.182 8.901 1.330 9.890 1.478 30 45 5.930 0.913 6.919 1.065 7.907 1.217 8.895 1.369 9.884 1.521 15 9 0 5.926 0.939 6.914 1.095 7.902 1.251 8.889 1.408 9.877 1.564 81 0 15 5.922 0.964 6.909 1.125 7.896 1.286 8.883 1.447 9.870 1.607 45 30 5.918 0.990 6.904 1.155 7.890 1.320 8.877 1.485 9.863 1.651 30 45 5.913 1.016 6.899 1.185 7.884 1.355 8.870 1.524 9.856 1.694 15 10 0 5.909 1.042 6.894 1.216 7.878 1.389 8.863 1.563 9.848 1.737 80 0 15 5.904 1.068 6.888 1.246 7.872 1.424 8.856 1.601 9.840 1.779 45 30 5.900 1.093 6.883 1.276 7.866 1.458 8.849 1.640 9.833 1.822 30 45 5.895 1.119 6.877 1.306 7.860 1.492 8.842 1.679 9.825 1.865 15 11 0 5.890 1.145 6.871 1.336 7.853 1.526 8.835 1.717 9.816 1.908 79 0 15 5.885 1.171 6.866 1.366 7.846 1.561 8.827 1.756 9.808 1.951 45 30 5.880 1.196 6.859 1.396 7.839 1.595 8.819 1.794 9.799 1.994 30 45 5.874 1.222 6.853 1.425 7.832 1.629 8.811 1.833 9.791 2.036 15 12 0 5.869 1.247 6.847 1.455 7.825 1.663 8.803 1.871 9.782 2.079 78 0 15 5.863 1.273 6.841 1.485 7.818 1.697 8.795 1.910 9.772 2.122 45 30 5.858 1.299 6.834 1.515 7.810 1.732 8.787 1.948 9.763 2.164 30 45 5.852 1.324 6.827 1.545 7.803 1.766 8.778 1.986 9.753 2.207 15 13 0 5.846 1.350 6.821 1.575 7.795 1.800 8.769 2.025 9.744 2.250 77 0 15 5.840 1.375 6.814 1.604 7.787 1.834 8.760 2.063 9.734 2.292 45 30 5.834 1.401 6.807 1.634 7.779 1.868 8.751 2.101 9.724 2.335 30 45 5.828 1.426 6.799 1.664 7.771 1.902 8.742 2.139 9.713 2.377 15 14 0 5.822 1.452 6.792 1.693 7.762 1.935 8.733 2.177 9.703 2.419 76 0 15 5.815 1.477 6.785 1.723 7.754 1.969 8.723 2.215 9.692 2.462 45 30 5.809 1.502 6.777 1.753 7.745 2.003 8.713 2.253 9.682 2.504 30 45 5.802 1.528 6.769 1.782 7.736 2.037 8.703 2.291 9.671 2.546 15 15 0 5.796 1.553 6.761 1.812 7.727 2.071 8.693 2.329 9.659 2.588 75 0 o Dep. Lat Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 0 Bearing. Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. 75~- 90Q 150-30~ 0 It Bearing. o f 15 15 30 45 16 0 15 30 45 17 0 15 30 45 18 0 15 30 45 19 0 15 30 45 20 0 15 30 45 21 0 15 30 45 22 0 15 30 45 23 0 15 30 45 24 0 15 30 45 25 0 15 30 45 26 0 15 30 45 27 0 15 30 45 28 0 15 30 45 29 0 15 30 45 30 0 o f Bearing, Distance 1. Lat. Dep, 0.965 0.263 0.964 0.267 0.962 0.271 0.961 0.276 0.960 0.280 0.959 0.284 0.958 0.288 0.956 0.292 0.955 0.297 0.954 0.301 0.952 0.305 0.951 0.309 0.950 0.313 0.948 0.317 0.947 0.321 0.946 0.326 0.944 0.330 0.943 0.334 0.941 0.338 0.940 0.342 0.938 0.346 0.937 0.350 0.935 0.354 0.934 0.358 0.932 0.362 0.930 0.367 0.929 0.371 0.927 0.375 0.926 0.379 0.924 0.383 0.922 0.387 0.921 0.391 0.919 0.395 0.917 0.399 0.915 0.403 0.914 0.407 0.912 0.411 0.910 0.415 0.908 0.419 0.906 0.423 0.9040 0.427 0.903' 0.431 0.901 0.434 0.899 0.438 0.897 0.442 0.895 0.446 0.893 0.450 0.891 0.454 0.889 0.458 0.887 0.462 0.885 0.466 0.883 0.469 0.881 0.473 0879 0.477 0.877 0.481 0.875 0.485 0.872 0.489 0.870 0.492 0.868 0.496 0.866 0.500 Dep. Lat, Distance 1. Distance 2. Lat. Dep. 1.930 0.526 1.927 0.534 1.925 0.543 1.923 0.551 1.920 0.560 1.918 0.568 1.915 0.576 1.913 0.585 1.910 0.593 1.907 0.601 1.905 0.610 1.902 0.618 1.899 0.626 1.897 0.635 1.894 0.643 1.891 0.651 1.888 0.659 1.885 0.668 1.882 0.676 1.879 0.684 1.876 0.692 1.873 0.700 1.870 0.709 1.867 0.717 1.864 0.725 1.861 0.733 1.858 0.741 1.854 0.749 1.851 0.757 1.848 0.765 1.844 0.773 1.841 0.781 1.838 0.789 1.834 0.797 1.831 0.805 1.827 0.813 1.824 0.821 1.820 0.829 1.816 0.837 1.813 0.845 1.809 0.853 1.805 0.861 1.801 0.869 1.798 0.877 1.794 0.885 1.790 0892 1.786 0.900 1.782 0.908 1.778 0.916 1.774 0.923 1.770 0.931 1.766 0.939 1.762 0.947 1.758 0.954 1.753 0.962 1.749 0.970 1.745 0.977 1.741 0.985 1.736 0.992 1.732 1.000 Dep. Lat. Distance 2. Distance 3. Distance 4. Lat. Dep. Lat. Dep. 2.894 0.789 3.859 1.052 2.891 0.802 3.855 1.069 2.887 0.814 3.850 1.086 2.884 0.827 3.845 1.103 2.880 0.839 3.840 1.119 2.876 0.852 3.835 1.136 2.873 0.865 3.830 1.153 2.869 0.877 3.825 1.169 2.865 0.890 3.820 1.186 2.861 0.902 3.815 1.203 2.857 0.915 3.810 1.220 2.853 0.927 3.804 1.236 2.849 0.939 3.799 1.253 2.845 0.952 3.793 1.269 2.841 0.964 3.788 1.286 2.837 0.977 3.782 1.302 2.832 0.989 3.776 1.319 2.828 1.001 3.771 1.335 2.824 1.014 3.765 1.352 2.819 1.026 3.759 1.368 2.815 1.038 3.753 1.384 2.810 1.051 3.747 1.401 2.805 1.063 3.741 1.417 2.801 1.075 3.734 1.433 2.796 1.087 3.728 1.450 2.791 1.100 3.722 1.466 2.786 1.112 3.715 1.482 2.782 1.124 3.709 1.498 2.777 1.136 3.702 1.515 2.772 1.148 3.696 1.531 2.767 1.160 3.689 1.547 2.762 1.172 3.682 1.563 2.756 1.184 3.675 1.579 2.751 1.196 3.668 1.595 2.746 1.208 3.661 1.611 2.741 1.220 3.654 1.627 2.735 1.232 3.647 1.643 2.730 1.244 3.640 1.659 2.724 1.256 3.633 1.675 2.719 1.268 3.625 1.690 2.713 1.280 3.618 1.706 2.708 1.292 3.610 1.722 2.702 1.303 3.603 1.738 2.696 1.315 3.595 1.753 2.691 1.327 3.587 1.769 2.685 1.339 3.580 1.785 2.679 1.350 3.572 1.800 2.673 1.362 3.564 1.816 2.667 1.374 3.556 1.831 2.661 1.385 3.548 1.847 2.655 1.397 3.540 1.862 2.649 1.408 3.532 1.878 2.643 1.420 3.524 1.893 2.636 1.431- 3.515 1.909 2.630 1.443 3.507 1.924 2.624 1.454 3.498 1.939 2.617 1.466 3.490 1.954 2.611 1.477. 3.481 1.970 2.605 1.489 3.473 1.985 2.598 1.500 3.464 2.000 Dep. Lat, Dep. Lat. Distance 3. Distance 4. Distance 5. Bearing. Lat. Dep, o 4.824 1.315 74 45 4.818 1.336 30 4.812 1.357 15 4.806 1.378 74 0 4.800 1.399 45 4.794 1.420 30 4.788 1.441 15 4.782 1.462 73 0 4.775 1.483 45 4.769 1.504 30 4.762 1.524 15 4.755 1.545 72 0 4.748 1.566 45 4.742 1.587 30 4.735 1.607 15 4.728 1.628 71 0 4.720 1.648 45 4.713 1.669 30 4.706 1.690 15 4.698 1.710 70 0 4.691 1.731 45 4.683 1.751 30 4.676 1.771 15 4.668 1.792 69 0 4.660 1.812 45 4.652 1.833 30 4.644 1.853 15 4.636 1.873 68 0 4.628 1.893 45 4.619 1.913 30 4.611 1.934 15 4.603 1.954 67 0 4.594 1.974 45 4.585 1.994 30 4.577 2.014 15 4.568 2.034 66 0 4.559 2.054 45 4.550 2.073 30 4.541 2.093 15 4.532 2.113 65 0 4.522 2.133 45 4.513 2.153 30 4.503 2.172 15 4.494 2.192 64 0 4.484 2.211 45 4.475 2.231 30 4.465 2.250 15 4.455 2.270 63 0 4.445 2.289 45 4.435 2.309 30 4.425 2.328 15 4.415 2.347 62 0 4.404 2.367 45 4.394 2.386 30 4.384 2.405 15 4.373 2.424 61 0 4.362 2.443 45 4.352 2.462 30 4.341 2.481 15 4.330 2.500 60 0 Dep. Lat. Distance 5. Bearing. I.r -- --- - --- --- -~ --- —- -- -- -- 60~-75~ 15o -30~ 73 Bearing. Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. o ' Lat. Dep Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 15 15 5.789 1.578 6.754 1.841 7.718 2.104 8.683 2.367 9.648 2.630 74 45 30 5.782 1.603 6.745 1.871 7.709 2.138 8.673 2.405 9.636 2.672 30 45 5.775 1.629 6.737 1.900 7.700 2.172 8.662 2.443 9.625 2.714 15 16 0 5.768 1.654 6.729 1.929 7.690 2.205 8.651 2.481 9.613 2.756 74 0 15 5.760 1.679 6.720 1.959 7.680 2.239 8.640 2.518 9.601 2.798 45 30 5.753 1.704 6.712 1.988 7.671 2.272 8.629 2.556 9.588 2.840 30 45 5.745 1.729 6.703 2.017 7.661 2.306 8.618 2.594 9.576 2.882 15 17 0 5.738 1.754 6.694 2.047 7.650 2.339 8.607 2.631 9.563 2.924 73 0 15 5.730 1.779 6.685 2.076 7.640 2.372 8.595 2.669 9.550 2.965 45 30 5.722 1.804 6.676 2.105 7.630 2.406 8.583 2.706 9.537 3.007 30 45 5.714 1.829 6.667 2.134 7.619 2.439 8.572 2.744 9.524 3.049 15 18 0 5.706 1.854 6.657 2.163 7.608 2.472 8.560 2.781 9.511 3.090 72 0 15 5.698 1.879 6.648 2.192 7.598 2.505 8.547 2.818 9.497 3.132 45 30 5.690 1.904 6.638 2.221 7.587 2.538 8.535 2.856 9.483 3.173 30 45 5.682 1.929 6.629 2.250 7.575 2.572 8.522 2.893 9.469 3.214 15 19 0 5.673 1.953 6.619 2.279 7.564 2.605 8.510 2.930 9.455 3.256 71 0 15 5.665 1.978 6.609 2.308 7.553 2.638 8.497 2.967 9.441 3.297 45 30 5.656 2.003 6.598 2.337 7.541 2.670 8.484 3.004 9.426 3.338 30 45 5.647 2.028 6.588 2.365 7.529 2.703 8.471 3.041 9.412 3.379 15 20 0 5.638 2.052 6.578 2.394 7.518 2.736 8.457 3.078 9.397 3.420 70 0 15 5.629 2.077 6.567 2.423 7.506 2.769 8.444 3.115 9.382 3.461 45 30 5.620 2.101 6.557 2.451 7.493 2.802 8.430 3.152 9.367 3.502 30 45 5.611 2.126 6.546 2.480 7.481 2.834 8.416 3.189 9.351 3.543 15 21 0 5.601 2.150 6.535 2.509 7.469 2.867 8.402 3.225 9.336 3.584 69 0 15 5.592 2.175 6.524 2.537 7.456 2.900 8.388 3.262 9.320 3.624 45 30 5.582 2.199 6.513 2.566 7.443 2.932 8.374 3.299 9.304 3.665 30 45 5.573 2.223 6.502 2.594 7.430 2.964 8.359 3.335 9.288 3.706 15 22 0 5.563 2.248 6.490 2.622 7.417 2.997 8.345 3.371 9.272 3.746 68 0 15 5.553 2.272 6.479 2.651 7.404 3.029 8.330 3.408 9.255 3.787 45 30 5.543 2.296 6.467 2.679 7.391 3.061 8.315 3.444 9.239 3.827 30 45 5.533 2.320 6.455 2.707 7.378 3.094 8.300 3.480 9.222 3.867 15 23 0 5.523 2.344 6.444 2.735 7.364 3.126 8.285 3.517 9.205 3.907 67 0 15 5.513 2.368 6.432 2.763 7.350 3.158 8.269 3.553 9.188 3.947 45 30 5.502 2.392 6.419 2.791 7.336 3.190 8.254 3.589 9.171 3.988 30 45 5.492 2.416 6.407 2.819 7.322 3.222 8.238 3.625 9.153 4.028 15 24 0 5.481 2.440 6.395 2.847 7.308 3.254 8.222 3.661 9.136 4.067 66 0 15 5.471 2.464 6.382 2.875 7.294 3.286 8.206 3.696 9.118 4.107 45 30 5.460 2.488 6.370 2.903 7.280 3.318 8.190 3.732 9.100 4.147 30 45 5.449 2.512 6.357 2.931 7.265 3.349 8.173 3.768 9.081 4.187 15 25 0 5.438 2.536 6.344 2.958 7.250 3.381 8.157 3.804 9.063 4.226 65 0 15 5.427 2.559 6.331 2.986 7.236 3.413 8.140 3.839 9.045 4.266 45 30 5.416 2.583 6.318 3.014 7.221 3.444 8.123 3.875 9.026 4.305 30 45 5.404 2.607 6.305 3.041 7.206 3.476 8.106 3.910 9.007 4.345 15 26 0 5.393 2.630 6.292 3.069 7.190 3.507 8.089 3.945 8.988 4.384 64 0 15 5.381 2.654 6.278 3.096 7.175 3.538 8.072 3.981 8.969 4.423 45 30 5.370 2.677 6.265 3.123 7.160 3.570 8.054 4.016 8.949 4.462 30 45 5.358 2.701 6.251 3.151 7.144 3.601 8.037 4.051 8.930 4.501 15 27 0 5.346 2.724 6.237 3.178 7.128 3.632 8.019 4.086 8910 4.540 63 0 15 5.334 2.747 6.223 3.205 7.112 3.663 8.001 4.121 8.890 4.579 45 30 5.322 2.770 6.209 3.232 7.096 3.694 7.983 4.156 8.870 4.618 30 45 5.310 2.794 6.195 3.259 7.080 3.725 7.965 4.190 8.850 4.656 15 28 0 5.298 2.817 6.181 3.286 7.064 3.756 7.947 4.225 8.829 4.695 62 0 15 5.285 2.840 6.166 3.313 7.047 3.787 7.928 4.260 S.809 4.733 45 30 5.273 2.863 6.152 3.340 7.031 3.817 7.909 4.294 8.788 4.772 30 45 5.260 2.886 6.137 3.367 7.014 3.848 7.891 4.329 8.767 4.810 15 29 0 5.248 2.909 6.122 3.394 6.997 3.878 7.872 4.363 8.746 4.848 61 0 15 5.235 2.932 6.107 3.420 6.980 3.909 7.852 4.398 8.725 4.886 45 30 5.222 2.955 6.093 3.447 6.963 3.939 7.833 4.432 8.704 4.924 30 45 5.209 2.977 6.077 3.474 6.946 3.970 7.814 4.466 8.682 4.962 15 30 0 5.196 3.000 6.062 3.500 6.928 4.000 7.794 4.500 8.660 5.000 60 0 o ' Dep. Lat. p LatDep, La ep Lt. Dep. Lat. Dep. Lat. o Bearing. Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. I 60 -75~ 74 300 4o5 0 Bearing. Distance 1. Distance 2. Distance 3. Distance 4. Distance. 5 Bearing. 1 o I Lat, Dep. Lat, Dep. 30 15 30 45 31 0 15 30 45 32 0 15 30 45 33 0 15 30 45 34 0 15 30 45 35 0 15 30 45 36 0 15 30 45 37 0 15 30 45 38 0 15 30 45 39 0 15 30 45 40 0 15 30 45 41 0 15 30 45 42 0 15 30 45 43 0 15 30 45 44 0 15 30 45 45 0 o f 0.864 0.504 0.862 0.508 0.859 0.511 0.857 0.515 0.855 0.519 0.853 0.522 0.850 0.526 0.848 0.530 0.846 0.534 0.843 0.537 0.841 0.541 0.839 0.545 0.836 0.548 0.834 0.552 0.831 0.556 0.829 0.559 0.827 0.563 0.824 0.566 0.822 0.570 0.819 0.574 0.817 0.577 0.814 0.581 0.812 0.584 0.809 0.588 0.806 0.591 0.804 0.595 0.801 0.598 0.799 0.602 0.796 0.605 0.793 0.609 0.791 0.612 0.788 0.616 0.785 0.619 0.783 0.623 0.780 0.626 0.777 0.629 0.774 0.633 0.772 0.636 0.769 0.639 0.766 0.643 0.763 0.646 0.760 0.649 0.758 0.653 0.755 0.656 0.752 0.659 0.749 0.663 0.746 0.666 0.743 0.669 0.740 0.672 0.737 0.676 0.734 0.679 0.731 0.682 0.728 0.685 0.725 0.688 0.722 0.692 0.719 0.695 0.716 0.698 0.713 0.701 0.710 0.704 0.707 0.707 Dep. Lat. 1.728 1.008 1.723 1.015 1.719 1.023 1.714 1.030 1.710 1.038 1.705 1.045 1.701 1.052 1.696 1.060 1.691 1.067 1.687 1.075 1.682 1.082 1.677 1.089 1.673 1.097 1.668 1.104 1.663 1.111 1.658 1.118 1.653 1.126 1.648 1.133 1.643 1.140 1.638 1.147 1.633 1.154 1.628 1.161 1.623 1.168 1.618 1.176 1.613 1.183 1.608 1.190 1.603 1.197 1.597 1.204 1.592 1.211 1.587 1.218 1.581 1.224 1.576 1.231 1.571 1.238 1.565 1.245 1.560 1.252 1.554 1.259 1.549 1.265 1.543 1.272 1.538 1.279 1.532 1.286 1.526 1.292 1.521 1.299 1.515 1.306 1.509 1.312 1.504 1.319 1.498 1.325 1.492 1.332 1.486 1.338 1.480 1.345 1.475 1.351 1.469 1.358 1.463 1.364 1.457 1.370 1.451 1.377 1.445 1.383 1.439 1.389 1.433 1.396 1.427 1.402 1.420 1.408 1.414 1.414 Dep. Lat. Lat. Dep. 2.592 1.511 2.585 1.523 2.578 1.534 2.572 1.545 2.565 1.556 2.558 1.567 2.551 1.579 2.544 1.590 2.537 1.601 2.530 1.612 2.523 1.623 2.516 1.634 2.509 1.645 2.502 1.656 2.494 1.667 2.487 1.678 2.480 1.688 2.472 1.699 2.465 1.710 2.457 1.721 2.450 1.731' 2.442 1.742 2.435 1.753 2.427 1.763 2.419 1.774 2.412 1.784 2.404 1.795 2.396 1.805 2.388 1.816 2.380 1.826 2.372 1.837 2.364 1.847 2.356 1.857 2.348 1.868 2.340 1.878 2.331 1.888 2.323 1.898 2.315 1.908 2.307 1.918 2.298 1.928 2.290 1.938 2.281 1.948 2.273 1.958 2.264 1.968 2.256 1.978 2.247 1.988 2.238 1.998 2.229 2.007 2.221 2.017 2.212 2.027 2.203 2.036 2.194 2.046 2.185 2.056 2.176 2.065 2.167 2.075 2.158 2.084 2.149 2.093 2.140 2.103 2.131 2.112 2.121 2.121 Dep. Lat. Lat. Dep. 3.455 2.015 3.447 2.030 3.438 2.045 3.429 2.060 3.420 2.075 3.411 2.090 3.401 2.105 3.392 2.120 3.383 2.134 3.374 2.149 3.364 2.164 3.355 2.179 3.345 2.193 3.336 2.208 3.326 2.222 3.316 2.237 3.306 2.251 3.297 2.266 3.287 2.280 3.277 2.294 3.267 2.309 3.257 2.323 3.246 2.337 3.236 2.351 3.226 2.365 3.215 2.379 3.205 2.393 3.195 2.407 3.184 2.421 3.173 2.435 3.163 2.449 3.152 2.463 3.141 2.476 3.130 2.490 3.120 2.504 3.109 2.517 3.098 2.531 3.086 2.544 3.075 2.558 3.064 2.571 3.053 2.584 3.042 2.598 3.030 2.611 3.019 2.624 3.007 2.637 2.996 2.650 2.984 2.664 2.973 2.677 2.961 2.689 2.949 2.702 2.937 2.715 2.925 2.728 2.913 2.741 2.901 2.753 2.889 2.766 2.877 2.779 2.865 2.791 2.853 2.804 2.841 2.816 2.828 2.828 Dep. Lat. Lat. Dep. 4.319 2.519 4.308 2.538 4.297 2,556 4.286 2.575 4.275 2.594 4.263 2.612 4.252 2.631 4.240 2.650 4.229 2.668 4.217 2.686 4.205 2.705 4.193 2.723 4.181 2.741 4.169 2.760 4.157 2.778 4.145 2.796 4.133 2.814 4.121 2.832 4.108 2.850 4.096 2.868 4.083 2.886 4.071 2.904 4.058 2.921 4.045 2.939 4.032 2.957 4.019 2.974 4.006 2.992 3.993 3.009 3.980 3.026 3.967 3.044 3.953 3.061 3.940 3.078 3.927 3.095 3.913 3.113 3.899 3.130 3.886 3.147 3.872 3.164 3.858 3.180 3.844 3.197 3.830 3.214 3.816 3.231 3.802 3.247 3.788 3.264 3.774 3.280 3.759 3.297 3.745 3.313 3.730 3.329 3.716 3.346 3.701 3.362 3.686 3.378 3.672 3.394 3,657 3.410 3.642 3.426 3.627 3.442 3.612 3.458 3.597 3.473 3.582 3.489 3.566 3.505 3.551 3.520 3.536 3.536 Dep, Lat. o! 59 45 30 15 59 0 45 30 15 58 0 45 30 15 57 0 45 30 15 56 0 45 30 15 55 0 45 30 15 54 0 45 30 15 53 0 45 30 15 52 0 45 30 15 51 0 45 30 15 50 0 45 30 15 49 0 45 30 15 48 0 45 30 15 47 0 45 30 15 46 0 45 30 15 45 0 O t Bearing. Distance 1. Distance 2. Distance 3. Distance 4. Distance 5. Bearing. Beaing I_ I I a 45~-60~ 300 — 45 75 I L7 L I L Bearing. o! 30 15 30 45 31 0 15 30 45 32 0 15 30 45 33 0 15 30 45 34 0 15 30 45 35 0 15 30 45 36 0 15 30 45 37 0 15 30 45 38 0 15 30 45 39 0 15 30 45 40 0 15 30 45 41 0 15 30 45 42 0 15 30 45 43 0 15 30 45 44 0 15 30 45 45 0 o f Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing, I I I i I I i I I I I I I I I I I I I I I I I I I Lat. Dep, Lat. Dep. Lat. Dep. 5.183 3.023 6.047 3.526 6.911 4.030 5.170 3.045 6.031 3.553 6.893 4.060 5.156 3.068 6.016 3.579 6.875 4.090 5.143 3.090 6.000 3.605 6.857 4.120 5.129 3.113 5.984 3.631 6.839 4.150 5.116 3.135 5.968 3.657 6.821 4.180 5.102 3.157 5.952 3.683 6.803 4.210 5.088 3.180 5.936 3.709 6.784 4.239 5.074 3.202 5.920 3.735 6.766 4.269 5.060 3.224 5.904 3.761 6.747 4.298 5.046 3.246 5.887 3.787 6.728 4.328 5.032 3.268 5.871 3.812 6.709 4.357 5.018 3.290 5.854 3.838 6.690 4.386 5.003 3.312 5.837 3.864 6.671 4.416 4.989 3.333 5.820 3.889 6.652 4.445 4.974 3.355 5.803 3.914 6.632 4.474 4.960 3.377 5.786 3.940 6.613 4.502 4.945 3.398 5.769 3.965 6.593 4.531 4.930 3.420 5.752 3.990 6.573 4.560 4.915 3.441 5.734 4.015 6.553 4.589 4.900 3.463 5.716 4.040 6.533 4.617 4.885 3.484 5.699 4.065 6.513 4.646 4.869 3.505 5.681 4.090 6.493 4.674 4.854 3.527 5.663 4.115 6.472 4.702 4.839 3.548 5.645 4.139 6.452 4.730 4.823 3.569 5.627 4.164 6.431 4.759 4.808 3.590 5.609 4.188 6.410 4.787 4.792 3.611 5.590 4.213 6.389 4.815 4.776 3.632 5.572 4.237 6.368 4.842 4.760 3.653 5.554 4.261 6.347 4.870 4.744 3.673 5.535 4.286 6.326 4.898 4.728 3.694 5.516 4.310 6.304 4.925 4.712 3.715 5.497 4.334 6.283 4.953 4.696 3.735 5.478 4.358 6.261 4.980 4.679 3.756 5.459 4.381 6.239 5.007 4.663 3.776 5.440 4.405 6.217 5.035 4.646 3.796 5.421 4.429 6.195 5.062 4.630 3.816 5.401 4.453 6.173 5.089 4.613 3.837 5.382 4.476 6.151 5.116 4.596 3.857 5.362 4.500 6.128 5.142 4.579 3.877 5.343 4.523 6.106 5.169 4.562 3.897 5.323 4.546 6.083 5.196 4.545 3.917 5.303 4.569 6.061 5.222 4.528 3.936 5.283 4.592 6.038 5.248 4.511 3.956 5.263 4.615 6.015 5.275 4.494 3.976 5.243 4.638 5.992 5.301 4.476 3.995 5.222 4.661 5.968 5.327 4.459 4.015 5.202 4.684 5.945 5.353 4.441 4.034 5.182 4.707 5.922 5.379 4.424 4.054 5.161 4.729 5.898 5.405 4.406 4.073 5.140 4.752 5.875 5.430 4.388 4.092 5.119 4.774 5.851 5.456 4.370 4.111 5.099 4.796 5.827 5.481 4.352 4.130 5.078 4.818 5.803 5.507 4.334 4.149 5.057 4.841 5.779 5.532 4.316 4.168 5.035 4.863 5.755 5.557 4.298 4.187 5.014 4.885 5.730 5.582 4.280 4.206 4.993 4.906 5.706 5.607 4.261 4.224 4.971 4.928 5.681 5.632 4.243 4.243 4.950 4.950 5.657 5.657 Lat. Dep. Lat. Dep. 7.775 4.534 8.638 5.038 7.755 4.568 8.616 5.075 7.735 4.602 8.594 5.113 7.715 4.635 8.572 5.150 7.694 4.669 8.549 5.188 7.674 4.702 8.526 5.225 7.653 4.736 8.504 5.262 7.632 4.769 8.481 5.299 7.612 4.802 8.457 5.336 7.591 4.836 8.434 5.373 7.569 4.869 8.410 5.410 7.548 4.902 8.387 5.446 7.527 4.935 8.363 5.483 7.505 4.967 8.339 5.519 7.483 5.000 8.315 5.556 7.461 5.033 8.290 5.592 7.439 5.065 8.266 5.628 7.417 5.098 8.241 5.664 7.395 5.130 8.217 5.700 7.372 5.162 8.192 5.736 7.350 5.194 8.166 5.772 7.327 5.226 8.141 5.807 7.304 5.258 8.116 5.843 7.281 5.290 8.090 5.878 7.258 5.322 8.064 5.913 7.235 5.353 8.039 5.948 7.211 5.385 8.013 5.983 7.188 5.416 7.986 6.018 7.164 5.448 7.960 6.053 7.140 5.479 7.934 6.088 7.116 5.510 7.907 6.122 7.092 5.541 7.880 6.157 7.068 5.572 7.853 6.191 7.043 5.603 7.826 6.225 7.019 5.633 7.799 6.259 6.994 5.664 7.772 6.293 6.970 5.694 7.744 6.327 6.945 5.725 7.716 6.361 6.920 5.755 7.688 6.394 6.894 5.785 7.660 6.428 6.869 '5.815 7.632 6.461 6.844 5.845 7.604 6.495 6.818 5.875 7.576 6.528 6.792 5.905 7.547 6.561 6.767 5.934 7.518 6.594 6.741 5.964 7.490 6.626 6.715 5.993 7.461 6.659 6.688 6.022 7.431 6.691 6.662 6.051 7.402 6.724 6.635 6.080 7.373 6.756 6.609 6.109 7.343 6.788 6.582 6.138 7.314 6.820 6.555 6.167 7.284 6.852 6.528 6.195 7.254 6.884 6.501 6.224 7.224 6.915 6.474 6.252 7.193 6.947 6.447 6.280 7.163 6.978 6.419 6.308 7.133 7.009 6.392 6.336 7.102 7.040 6.364 6.364 7.071 7.071 0! 59 45 30 15 59 0 45 30 15 58 0 45 30 15 57 0 45 30 15 50 0 45 30 15 55 0 45 30 15 54 0 45 30 15 53 0 45 30 15 52 0 45 30 15 51 0 45 30 15 50 0 45 30 15 49 0 45 30 15 48 0 45 30 15 47 0 45 30 15 406 0 45 30 15 45 0 o f Bearing. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. [] Bearing, IDistance 6. Distance 7. Distance 8. Distance 9. Distance 10. 45~-60~