/"< IN MEMORIAM FLORIAN CAJORl r i-i-t.^ (—-^y^^X^L^ -^ Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementsofplanesOOhunjrich THE ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY •y^)^' THE ELEMENTS OF PLANE AND SPHERICAL TEIGONOMETRY BY JOHN GALE HUN n AND CHARLES RANALD MacINNES J . > » » Neto ¥otfe THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1911 Copyright, 1911 By the MACMILLAN COMPANY Set up, electrotyped and printed Aug. 1911 CAJORI Press of The new Era Printing company LANCASTER. PA. PREFACE. The subject matter of the chapters of this book devoted to spherical trigonometry has been used in pamphlet form for the last four years in Princeton University, and that of the chapters on plane trigonometry for the last three years. The aim of the authors has been to present in as brief and clear a manner as possible the essentials of a short course in trigonometry. It has been found that the plane trigonometry may be covered in about thirty recitations, and the spherical trigonometry in somewhat less than this time. It has been thought advisable to devote some time to drawing the graphs of simple equations in polar coordinates. The reason for this is two-fold. Firstly, because such problems aid in giving the student a clearer idea of the way in which the trigo- nometric functions vary as the angle is changed ; and secondly, because of a very common lack of sufficient knowledge of polar coordinates on the part of students beginning the study of calculus. The fact that the trigonometric functions are ratios of line segments is emphasized, and their representation by means of lengths of lines is used as little as is conveniently possible. Certain of the proofs of theorems are shorter than in many text books, and, it is hoped, thereby made more clear; notably the proofs of the formulae for sin (A =1= B) and cos (A =*= B). The logarithmic tables were taken from Crockett's five place tables, and the proof sheets carefully compared with Albrecht's tables. It is hoped that few errors will be found. The explana- tion of the tables will be found at the end of the book. It is designed that the matter there contained be given as lessons, and for this reason the student is given examples involving the difficulties usually encountered by the beginner. The authors wish to thank the members of the mathematical department of Princeton University who have kindly suggested several changes from the form used in the pamphlet editions. Princeton, N. J., March 1, 1911. 918239 CONTENTS. Chapter I 1 Chapter II 23 Chapter III. Trigonometric Analysis 30 Chapter IV 53 Chapter V. Introductory Review 66 Chapter VI. Relations between the Sides and Angles of a Spherical Triangle 73 Chapter VII. The Right Spherical Triangle. ... 80 Chapter VIII. The Solution of Oblique Triangles. . 86 Chapter IX. Other Formulae Relating to Spherical Triangles 98 Table I. Logarithms of Numbers from 1 to 10,000 . 103 Table II. Logarithms of the Trigonometric Functions. 123 Table III. Natural Trigonometric Functions. . . .169 Explanation of Tables 193 vu ELEMENTS OF PLANE TRIGONOMETRY. CHAPTER I. 1. Positive and Negative Quantities. It is of fundamental importance for the student of trigonometry to realize at the outset the meaning of negative as well as of positive quantities. That a negative result may have an actual meaning is shown by the following examples : Let us suppose that a man has x dollars and that he owes y dollars. Then the difference x — y denotes the number of dollars he will have after paying his debts. If now this difference happens to be negative, it will indicate the number of dollars he will still owe after he has paid out all of his cash. As a second example consider the following: A man A is observed to pass a certain inn at exactly noon, walking at the rate of a miles per hour. At two o'clock another man B passes the same point at the rate of h miles per hour. How far will they be beyond the inn when B overtakes A? Let the required distance be denoted by x. A will take x/a hours to Yv^alk the distance, and B will take xjh hours. Also since A has a two-hour start on J5, the latter must walk the x miles in two hours less than A requires. We therefore have the equation - = - - 2 h a and hence 2ah X = b — a Now suppose that A walks two miles and B four miles per hour. Then B will overtake A 2.2.4/(4— 2) or eight miles beyond the inn. If on the other hand A walks three miles and B two miles per hour, they will be together 2.3.2/(2 — 3) or — 12 miles beyond the inn. The interpretation of this 1 1 A B C c* s> B o 2 ELEMENTS OF PLANE TRIGONOMETRY. negative result is evidently that they were together twelve miles before they came to the inn. That this is true is easily verified as follows: Suppose they are together at this place. It will take A 12/3 or four hours to reach the inn, and B 12/2 or six hours. Hence, if A gets there at noon, B will arrive at two o'clock as given in the problem. 2. Positive and Negative Lines. The second problem leads us naturally to the idea of positive and negative directions along a fine; that is, it brought out the fact that if we assume one direction of a line to be positive, the opposite direction algebraically is jTjQ I negative. Let A and B be two points of a line. By the symbol AB we shall mean not only the distance between A and B, but also the direction from A to B. The easiest way in which to think of the symbol AB is to consider it as expressing a motion along the line from A to B. Let us now have a third point C on the line. By the expression AB -\- BC we mean a motion from A to 5 and thence to C. The total motion is therefore merely from A to C. This is expressed by the equation AB + BC = AC. It is to be carefully observed that this equation is true no matter in what order the points A, B, and C may occur on the line. In par- ticular we have AB + BA = A A = 0, or AB = — BA. 3. Positive and Negative Angles. Let OX be a fixed line and let the positive direction be from to X. Also let a second line OP, whose positive direction is from to P, rotate about in the plane of the paper. The two lines OX and OP are said to form the angle XOP. Just as the symbol AB denotes not only the distance between A and B, but also the direction from A to B, so the symbol XOP denotes not only the size of the angle formed by the lines OX and OP, but also the direction of rotation from OX into OP. We may consider XOP as denoting a turning of the line OX through a given angle in a given direction. Let us now have another line OQ through 0. The symbol XOP + POQ then denotes a rotation of OX into the position of OP and thence into that of OQ. ELEMENTS OF PLANE TRIGONOMETRY. >X The total rotation is merely from OX into OQ, and therefore we have the equation XOP + POQ = XOQ. In particular XOP + POX = XOX = 0, or XOP = - POX. In the angle XOP, OX is known as the initial line and OP as the terminal line. We may choose either direction of rotation as positive. It is however customary to take as positive the counter-clockwise rota- tion, that is, a rotation in the direction opposite to that of the hands of a clock. Thus in Fig. 2 XOP, XOQ, and POQ are positive angles, while POX, QOX, and QOP are negative angles. 4. Measurement of Angles. Two systems of measuring angles are in common use. They are known as the sexagesimal system and the radian or circular measure system. In the sexagesimal system a right angle is divided into ninety equal parts called degrees. Each degree is in turn divided into sixty equal parts called minutes, and finally each minute is divided into sixty equal parts called seconds. To avoid writing degrees, minutes and seconds, the symbols °, ', " are employed. Thus 69° 34' 26'' is read sixty-nine degrees, thirty- four minutes and twenty-six sec- onds. In the radian system the unit angle is that angle which at the centre of a circle subtends an arc equal in length to the radius. This unit angle is known as a radian. In Fig. 3 let AOP be an angle at the center of a circle of radius If this angle is equal to a Fig. 3. radian, we have AOP r. circumference 360° = arc AP = r : 27rr, and hence AOP = one radian = 360°/27r = 57.2° -f. ELEMENTS OF PLANE TRIGONOMETRY. If we draw a circle of any radius r with the centre at the vertex of a given angle, this angle will contain as many radians as the arc subtended by it contains r. Hence the radian measure of an angle is the ratio of the length of the arc to the radius. Thus the arc subtended by 360° is 27rr, and therefore the radian measure of 360° is 27rr/r or 27r. Similarly the radian measure of 90° is 7r/2, that of 60° is 7r/3, etc. 5. Exercise I. L Express each of the following angles in radians: 30°, 150°, 750°, 45°, 260°, - 75°. 2. Express each of the following angles in degrees, minutes, and seconds f tt, ^-tt, yx, f tt, — y^x. 3. Find the angle subtended at the centre of a circle of radius 3 inches by an arc of 5 inches. 4. If the radius of the earth is 3,960 miles, find the angle at the centre between radii drawn to two points of the surface 200 miles apart. 6. Coordinates. Let X'OX and Y'OY be two perpendicular lines intersecting at 0. These lines divide the plane into four parts called quadrants. The portion of the plane above X'OX and to the right of Y'OY is called the first quadrant. The second quadrant includes the portion of the plane above X'OX and to the left of Y'OY, the third quadrant that below X'OX and to the left of Y'OY, and finally the fourth quadrant that be- low X'OX and to the right of Y'OY. By long usage it has become cus- tomary to consider the directions OX and OY as positive. In what fol- lows we shall therefore consider horizontal lines as positive when drawn to the right, and vertical lines as positive when drawn upwards. Y N P 7-- y ^ "V X X 'M ^ A V. Y' Fig. 4. ELEMENTS OF PLANE TRIGONOMETRY. Let now P be any point of the plane, and let MP and NP be the perpendicular distances of P from OX and OF respectively. If the point P be given, the distances NP and MP (or the equi- valent pair OM and MP) can be measured. Moreover OM and MP cannot have the same pair of values for two different points. This is obviously true if the two points lie in the same quadrant. If on the other hand they lie in different quadrants, one or both of the quantities OM and MP must have different signs in the two quadrants. For convenience we shall denote the horizontal distance OM by X, and the vertical distance MP by y. The two numbers X and y are called the coordinates of the point P. If P be in the first or fourth quadrant OM is measured to the right. Hence x is positive if P be in the first or fourth quadrant. If P be in the second or third quadrant OM is measured to the left, and hence x is negative if P be in either of these quadrants. Similarly y is positive in the first and second quadrants, for in these quadrants MP is measured upwards. In the third and fourth quadrants MP is measured downwards, and therefore in these quadrants y is negative. Example. On a figure draw the points for which x = 2, y = S; x=-2, y = S; x=-2, y=-S; x = 2, y=-3. To construct the first of these points we must make 0M = x = 2. Then M is two units to the right of on the line OX. Also MP = y = S. We must then draw MP perpendicular to - OX at M and mark off three units above M on this line. To construct the second point we must make OM = x= —2. Then M is two units to the left of on the line OX'. The required point is then three units above M. The other two points may be similarly constructed. They will be found to lie in the X — 2 2/=3 x-^2 x^"i 2/ = 3 !r=2 y — 3 Y Fig. 5. 6 ELEMENTS OF PLANE TRIGONOMETRY. third and fourth quadrants respectively. The four points evidently form the vertices of a rectangle whose centre is at 0. Example. Construct the points for which (a) x = l, y = 2', (b) x=-3, y = 5; (c) x=-3, y=-S; (d) x = 6, y=-5, 7. Trigonometric Functions. In general one quantity is said to be a function of another quantity if the value of the former depends only upon that of the latter. As a simple example suppose that a man is walking at the rate of four miles per hour, and that he walks for t hours. If x denote the dis- tance he walks, we evidently have the relation x = 4^. The value of X depends only upon that of t. Hence we say that a; is a func- tion of t. Let us now as in page 4 have two perpendicular lines X'OX and YVY. With OX as initial line draw the angle XOP. This angle is said to be in the first, second, third, or fourth quadrant according as the terminal line OP falls in one or other Fig. 6. of these quadrants. In any one of these cases take a point P on the terminal hne of the angle, and let fall the perpendicular MP from P upon OX. Also let Pi be any other point on OP, and let the corresponding perpendicular be MiPi. Any one of the triangles OMP, OMiPi, etc., is called a reference triangle for the angle XOP. The two right triangles OMP and OMiPi are similar, and hence the ratio of any two sides of the triangle OMP is equal ELEMENTS OF PLANE TRIGONOMETRY. 7 to the ratio of the corresponding sides of the triangle OMiPi. Hence the values of these ratios depend not upon the position of the point P upon OP, but only upon the size of the angle XOP. They are then functions of this angle. Let the angle XOP be denoted by 6. Also let r denote the length of OP, and X and y the values of OM and MP respectively. From the three quantities x, y, r we can form ratios all of which we have just seen are functions of the angle 6. We define these func- tions as follows: ,^ MP ' n y sme 01 6 = ^^^ or sm 9 = -, OP r ' r. OM . X cosme ot = y^^ or cos = - , OP r MP V tangent of ^ = ^^^7 or tan 6 = -, cotangent oi 6 = j^ or cot ^ = -, ^ fa OP . r secant of 6 = :p:r^rf or sec d = -, OM X cosecant oi 6 = irrp or esc 6 = ~, , . en 1 OM . .. X versed sme oi B = I — t^^^ or vers 6 = i . OP r 8. The Signs of the Trigonometric Functions. Since the positive direction of the terminal line of 6 is from to P, the value of r is in all cases positive. The signs of the trigonometric functions will then depend only upon the signs of x and y. The Sine. The sine will be positive whenever y is positive; that is, sin ^ is a positive number if 6 be an angle in the first or second quadrant. In the third and fourth quadrants y is nega- tive and therefore sin ^ is a negative number if 6 be an angle in the third or fourth quadrant. The Cosine. The cosine of 6 will be positive or negative ac- cording as X is positive or negative. Hence cos 6 is positive in the first and fourth quadrants, and negative in the second and third. The Tangent. The tangent of will be positive when and only 8 ELEMENTS OF PLANE TRIGONOMETRY. when X and y are of the same sign, for then only can their ratio be positive. Hence tan 6 is positive in the first and third quad- rants, and negative in the second and fourth. Let the student similarly consider the cases of the cotangent secant, and cosecant. We may arrange the results of this sec- tion in the form of a table as follows: sine, cosecant, cosine, secant, tangent, cotangent. I II Ill + + + — — + — + IV + 9. Given One Function of an Angle to Find all the Other Functions of This Angle. Consider the following problem. Given that d is an angle in the second quadrant and that cos 6 = — f , find all the other functions of d. We have cos = - = r In constructing the reference triangle for 6 we may take for OP any convenient length. Hence we may put r = 5, and so the conditions of our problem give a; = — 3. We shall now try to construct an angle 9 in the second quadrant such that X = — S and r = 5. On XVX layoff OM = x=-S. Then M is the point three units to the left of 0. At M erect a perpendicular to XVX. With as centre draw the arc of a circle of radius 5 cutting the per- pendicular at P. The angle 6 = XOP satisfies the conditions of the problem. The length of MP may be calculated from the relation MP^ = OP'' - OMK This gives MP^ = 25 - 9 = 16, and therefore MP = ± 4. But MP is drawn in the positive direction and therefore MP = 4. r P< / \tV ^ ^ \ y— -3 \ o Fig. 7. ELEMENTS OF PLANE TRIGONOMETRY. Hence we have sin0 r 5' 2/4' X 4 3' cot = - = — . , sec ^ = - = — -. 2/4' X S In an exactly similar way any problem of this type may be solved. 10. Exercise II. 1. Find all the functions of 6 when given that (a) 6 is an angle in the third quadrant and tan ^ = f . (6) d is an angle in the second quadrant and sec ^ = — yf . (c) is an angle in the fourth quadrant and sin ^ = — |. {d) cos 6 = — ^ and sin 6 is negative. (e) sin ^ = 1^ and sec 6 is negative (/) vers 0=1 and tan 6 is positive. (g) tan = — 3 and cos 6 is positive. (h) CSC 6 = -/- and sec 6 is negative. (i) sin = — Y and vers is greater than 1. (j) vers 0=1 and esc is negative. 2. If is an angle in the fourth quadrant and tan = — |, find the value of (sin + 4 cos d) (vers (9 - 2 tan ey. 3. If is an angle in the second quadrant and sin = f , find the value of i^tan d + vers 6 (cos 6 — 2 sec 6). 4. State which of the following problems are possible, and in those cases which have solutions find all the functions oie: (a) cosO = — f , tan = f , -l/5 (6) sec = 1, tan d = — - — , (c) sin = — I, CSC = 2, (c?) CSC = i, (e) CSC = 3, cos = ^-. o 5. Determine the signs of the following quantities: sin 150°, tan 310°, CSC 50°, sec 123°, sec -^, cot — . 11. Variation in Size of the Trigonometric Functions. With 10 ELEMENTS OF PLANE TRIGONOMETRY. the vertex of the angle XOP as centre draw a circle of unit radius. Let this circle cut OX, OP, and OY in A, P and B respectively. At A and B draw tangents to the circle to meet OP produced in T and S respectively. Fig. 8. We have from the similar triangles OMP, OAT, and SBO MP^MP OP ~ 1 MP AT sin B = tarn 6 sec a = = MP, AT ^ OM OM ^,, cos (9 = ^ = -^ = OM, OM BS ■DO OM~OA~ 1 =^^' ^^^^=^^ = 5i"T "^^' OP^^OT^OT _ _ OP ^OS^OS OM OA 1 ~^^' ^^^^~ MP OB 1 = 0S. vers d = 1 -cos 6 = 1- OM = OA-OM = MA, The numerical value of the ratio sin 6 is equal to the number of units of length of the line MP, where the radius of the circle is taken as this unit; and similarly for the other functions. Observe that the signs of the functions of any angle may be read off from this figure. Such a function is positive or negative according as the line corresponding to it is drawn in the positive or negative direction. Thus if B be an angle in the second quadrant, AT is drawn downwards. Hence the tangent of an ELEMENTS OF PLANE TRIGONOMETRY. 11 angle in the second quadrant is negative. These results will be found to agree with those of page 8. Let us now consider the changes in the values of the trigo- nometric functions as the angle B increases from 0° to 360°. The Sine. As B approaches 0° the length of the line MP ap- proaches zero. We have therefore sin 0° = 0. As ^ increases from 0° to 90° MP continually increases from to OB or 1. Therefore sin 90° = 1, and the sine of any angle in the first quadrant is a "positive number less than unity. Now as B increases from 90° to 180° MP decreases from 1 to remaining positive. Hence sin 180° = 0, and the sine of an angle in the second quadrant is a positive number less than unity. Similarly as B increases from 180° to 270° MP becomes negative and decreases from to — 1. Then sin 270° = — 1, and the sine of an angle in the third quadrant is a negative number between and — 1. In a similar manner, as B increases from 270° to 360° MP remains negative and increases in value from — 1 to 0. Hence sin 360° = 0, and the sine of an angle in the fourth quadrant is a negative number between — 1 and 0. The Tangent. When B = 0° we have AT = 0, and hence tan 0° = 0. AsB increases from 0° to 90° AT increases without limit. For as B approaches 90° the line OT approaches the position of a parallel to AT. Hence when B = 90°, AT = tan 90° = 00 . The tangent of an angle in the first quadrant is therefore a positive number, and may have any value between and 00 . As B enters the second quadrant A T becomes nega- tive, and as the angle approaches 180° the length oi AT ap- proaches 0. Hence the tangent of an angle in the second quadrant is a negative number, and may have any value between — 00 and 0. In the third quadrant the tangent again becomes positive, and increases in value from at 180° to oo at 270°. Finally as B increases from 270° to 360° tan B becomes negative and increases in value from — oo at 270° to at 360°. 12. Exercise III. 1. What angle between 0° and 360° has the greatest sine? The least sine? The greatest cosine? 2. As B increases from 0° to 360° explain how the following functions vary: (a) The cosine, (b) the cotangent, (c) the secant, (d) the cosecant, (e) the versed sine. 12 ELEMENTS OF PLANE TRIGONOMETRY. 3. For what values of 6 is sin 6 = cos 6? sin ^ = — cos 6? 4. Given an angle in the first quadrant, show on a figure how to draw an angle in the second quadrant such that the sines of the two angles shall be equal. Could the same problem be given with cosines instead of sines? 5. Given an angle 6 in the first quadrant, in what quadrant must x 34 ELEMENTS OF PLANE TRIGONOMETRY. M'P' OM sin (90° -e) = ~^pr = op == ^^^ ^> mc^o M OM' MP . ^ cos (90° -6) = ^pT = ^p = sin 6, tan (90 -^) = om^ = Sp cot0. Find cot (90° - 6), sec (90° - d), and esc (90° - d). 180°-^. Let ^ be the angle XOP) then if P'OM'= 6, XOP' will be 180° - 6. Draw the tri- angles of reference. The right triangles 0PM and OP'M' are equal in all respects. That is, -^ M'P' = MP and OM' = - OM. Then • none M ^'^' ^^ sm (180° -e) = ^pT = -^ cos (180° - e) = tan (180° - 0) = OM^ OP' '' WP'^ OM' OM OP MP OM sin Oj = — cos 0, — tan B. Find cot (180° - B), sec (180° - B) and esc (180° - B). — B. Again, in Fig. 28, the triangles of reference are equal in all respects and we have MP' = - MP. Then sin (- B) = cos (- B) = tan (- B) = MP' OP' = MP OP = — sin Bf OM OP' = OM OP = cos B, MP' OM = MP OM = - tan B. Fig. 28. ELEMENTS OF PLANE TRIGONOMETRY. 35 The solution of any question of this type depends on our abiUty to draw the angles and their triangles of reference and then to properly pair the sides of these triangles, attention be- ing paid to the signs. Since logarithmic tables give only acute angles it is necessary to use one of these relations in order to find the logarithm of a function of an angle greater than 90°. The most useful ones are those for 180° - 6. In the examples worked out 6 was taken as an acute angle; the same results would be obtained no matter what the size of B, the method of proof being the same in every case. As an example, let us take 90° -\- B^ 6 being an obtuse angle such as XOP. Draw POP' = 90°; then XOP' = 90° + B. Since POP' = 90° and MOP' + OP'M' = 90°, by taking away the common part M'OP' we have that POM = OP'M', There- fore the triangles OPM and OP'M' are equal in all respects; in particular OM' = - MP and M'P' = OM. Then Fig. 29. sin (90° + B) = M'P^ OP' OM OP cos B, cos (90° + (9) = 0M[ OP' MP . ^ , -^ = - sm B, etc. The results of all these are included in the following rule: Any Junction of 90° ± B {or 270° ± B) is equal to its co-function of B, and any function of — B, 180° ± or 360° ± B is equal to the same function of B, the proper sign being used in each case. To fix the sign, suppose B to he acute and use the sign of the function in question for the quadrant in which the compound angle falls. Thus to get tan (270° + (9), we say that with B acute 270° + B would be in the fourth quadrant, in which the tangent is negative. Hence tan (270° + 0) = - cot B. 36 ELEMENTS OF PLANE TRIGONOMETRY. 32. Exercise VIII. 1. By a figure, prove that sin (90° + ^) = cos 6 and that cos (90° + e) = - sin d. Find tan (90° + 6). 2. Find sin (180° + 6), cos (180° + d), and tan (180° +(9) in terms of d. 3. Find tan 160° 10' in terms of 19° 50'. 4. Find log sin 132° 20', log tan 200° 15', log cos 312° 18' and log sin 150°. 5. Prove that sin 160° = cos 70° and that cos 110° = sin 200°. 6. Given sin 140° = a/b find sin 40°, cos 130° and cos 310°. 7. Prove that the tangent and cotangent are periodic func- tions of period 180° [p. 12]. 33. Inverse Functions. If sin ^ = a and it is desired to draw attention to 6, one would use the phrase "6 is an angle whose sine is a." This is written d = arcsin a.* Similarly 6 = arctan h means that 6 is an angle whose tangent is h. Such functions are called inverse trigonometric functions; they are also called circular functions. For a given value of d only one value of a is possible; but on the other hand for a given value of a, there are infinitely many values of 6. Thus if ^ = arcsin ^, 6 is 30° or 150° or any angle obtained from these by adding multiples of 360°. Similarly arctan t/3 is 60° or any angle obtained from 60° by adding multiples of 180°. Of all these values, the smallest positive one is called the principal value and is the one to be used in the following examples: 1. If arcsin i = 0, find cot 6. If arcsin i = 6, then sin = J. To construct 6, draw a * The symbol sin~* a is also commonly used for this purpose. ELEMENTS OF PLANE TRIGONOMETRY. 37 right angle and on one arm lay off MP = 1 ; with P as centre and radius 3 draw an arc cutting the other arm at 0. Then MOP = e and OM = i/9 - 1 = 21/2^ Hence cot (9 = 2i/2. 2. If arccot (— i) = 6, find sin 6. Since arccot (— i) =6, then cot d = — \ and must be an obtuse angle. To construct 6, lay off OM = — 1 and at M erect MP = 2. Then XOP = d and OP = i/5. Hence sin (9 = 2/1/5. 34. Exercise IX. 1. Construct arccos (— |), arctan |, arcsin (— i). 2. Prove arcsin f = arctan 2/i/21. 3. Prove arccos (- f) = arctan (- i/33/4). 4. Find tan (arcsin |); cos [arccos (— i)]. 5. If arccos {2x^ - 2x) = 27r/3, find x. 6. Find arcsin 0, arctan 1, arccos ^, arctan go , arcsin (— 4l/^), arccos 1. 35. Orthogonal Projection. AB being any line segment and PQ any line, draw A A' and BB' perpendicular to PQ, Then A'B' is the orthogonal projection of AS on PQ. Through A draw AE parallel to PQ and produce BA to meet PQ in L. Then J^AS = B'LB = d, and AE = A'B'. Since AE-MB = cos 6 we have AE = AS cos ^ and therefore A'B' = AB cos 6- Hence the projection of any segment AB on a line is AB times the cosine of the angle between the lines. 38 ELEMENTS OF PLANE TRIGONOMETRY. Project the sides of the triangle ABC (in Fig. 33) on a line PQ. Since A'B'+B'C'= A'C and A'C'-^ C'B'= A'B' we see 4 A' Fig. 32. B' that the sum of the projections on any line of two sides of a triangle is equal to the projection on that line of the third side. 36. Functions of the Sum or Difference of Two Angles. To prove sin {A + B) = siaA cos B + cos A sin By cos {A-{-B) = cos i4 cos 5 — sin i4 sin B, The following proofs are valid no matter what may be the size of A and B. For the student to see this, however, is harder than to limit the present proofs to the case where A and B are acute and later to show that they may be extended to cover all cases. First Method. The following proofs apply whether A-\-B is acute or obtuse. Draw XOX' = A and X'OP = S. Then XOP = A+B. From P draw PQR perpendicular to 0X\ ELEMENTS OF PLANE TRIGONOMETRY. 39 By Art. 35, the projection on any line of OP is equal to the sum of the projections on that line of OQ and QP. Projecting on OX gives r But Hence OP cos (A+B) = 0Q cos A + QP cos XRP, cos XRP = cos (90°+ A) = - sin A. OP cos (A +5) =00 cos A-QP sin A. Dividing by OP, cos {A-^B)=OQlOP cos A-QPJOP sin A, and therefore cos (A+P) = cos B cos A — sin B sin A. Q. E. D. In the second figure, projecting OP on OF gives OP cos [(A +5) -90°], which is equal to OP cos [90° — (A +P)]. Hence in either figure, projecting OP, OQ and QP on OF gives OP cos [90°-(A+B)] = OQ cos (90°- A)+OP cos {XRQ-90°) Hence OP sin {A+B)=OQ sin A+QP cos A, and dividing by OP sin (A+P) = cos B sin A+sin P cos A. Q. E. D. Second Method. Draw XOX'' = A and X''OP = P; then XOP = A -\- B. To get triangles of reference for A, P and A + P, take any point P on the terminal line of A + P and 40 ELEMENTS OF PLANE TRIGONOMETRY. draw PM perpendicular to OX. P being also on the terminal line of B, draw PQ perpendicular to OX"; and Q being on the Fig. 35. terminal line of A draw QH perpendicular to OX. Through Q draw KQX' parallel to OX. Then X'QP is 90°+A. . ,. , p. MP MK-\-KP HQ , KP sm(A+B)=^ = -^^-=^ + ^. But HQ = OQ sin A, KP = QP sin (90°+A)=QP cos A. .'. sin (A+B) =sin A ~p + cos A QP OP = sin A cos B+cos A sin B. Q. E. D. Ai rA^m ^^ OH+HM OH , QK Also cos (A+5)= ^-^- = -^^ -OP^OP' But OH = OQ cos A, QK = QP cos (90°+A) = -QP sin A. . . cos {A-f-B) = cos A Yyp — sm A ^yp^ = cos A cos B — sin A sin B. Q. E. D. 37. To prove sin (i4 — 5) = sin A cos B — cos A sin 5 cos {A — B) = cos A cos ^ + sin A sin 5. Suppose J5+5' = 90°, so that B' also is acute. ELEMENTS OF PLANE TRIGONOMETRY. 41 Then sin (A-5)=sin(A+5'-90°) = -sin(90-A-BO = -cos {A+B') = —cos A cos B'+sin A sin B' = -cos A cos (90°-B)+sin A sin (90° -5) = —cos A sin B + sin A cos B. Q. E. D. Similarly cos {A - B) = cos (A + B' - 90°) = cos (90 - A - B') = sin (A + BO = sin A cos 5' + cos A sin 5' = sin A cos (90° - 5) + cos A sin (90° - B) = sin A sin 5 + cos A cos B. Q. E. D. 38. That these formulae are true when A and B are not acute may be shown in any special case as follows : Suppose A to be an angle in the third quadrant and B to be one in the second. Then A can be expressed in the form 180° + A' where A' will be an acute angle. Similarly B = 90° + B\ B' being acute. Then sin (A + 5) = sin (270° + A' + B') = - cos {A' + B') § 31. = — cos A' cos B' + sin A' sin B' = - cos (A - 180°) cos {B - 90°) + sin (A - 180°) sin (5 - 90°) = cos A sin J5 + sin A cos B. In like manner cos {A - B) = cos (90° ■\- A' - B') = - sin (A' - BO = — sin A' cos B' + cos A' sin B' = - sin (A - 180°) cos {B - 90°) + cos (A - 180°) sin {B - 90°) = sin A sin B + cos A cos B. 39. From the preceding formulae we get sin (A + B) sin A cos B + cos A sin B tan (A + B) = cos (A + B) cos A cos B — sin A sin B' 42 ELEMENTS OF PLANE TRIGONOMETRY. Dividing numerator and denominator by cos A cos B, we obtain sin A cos B cos A sin B cos A cos -B cos A cos B cos A cos B sin A sin 5 .-. tan (A + 5) = cos A COS 5 COS A cos 5 tan A 4- tan ^ I — tani4tan5 Similarly it can be proved that ... o\ tan i4 — tan B tan (i4 - 5) = I + tan A tan B' 40. Exercise X. Apply the formulae of §§ 36-39 to the following examples, simplifying when possible : 1. sin (180° - d), 2. cos (90° + 6). 3. tan (180° + 6). 4. sin (2A + B). 5. cos (90° - e). 6. tan (0° - 6). 7. By expanding cos (6 — 6) find cos 0°. 8. Similarly find sin 0° and tan 0°. 9. From tan (45° + 45°) find tan 90°. 10. From sin (45° + 30°) find sin 75°. 11. Find sin 15° and cos 15°. 12. If sin A = f and cos B = A find sin (A + B), cos (A — B) and tan (A + J5), A and B being acute. 13. If cos A = — f and sin B = \ find cos (A + B), sin (A — B), and tan {A — B), A and B being obtuse. 14. If A = arcsin \ and B = arccos (— f) find sin (A — B). 15. Prove the formula for sin (A — B) when A and B are in the third quadrant. 16. Prove the formula for cos (A + B) when A is in the second quadrant and B in the fourth. 41. Functions of an Angle in Terms of Half the Angle. sin 26 = sin {B -\- 6) = sin 6 cos 6 -\- cos d sin d .'. sin 26 = 2 sin 6 cos B. cos 2B = cos {B + B) = cos2 ^ - sin2 6. , - 2 tan ^ tan 2B = 7 — — -. I - tan2 B ELEMENTS I OF PLANE TRIGONOMETRY. Similarly sin d = ^^^(2 + 2) = = 2 . e sm - cos e '2 cos^ = a 2 tan- Z 1 - tan^l e 2 tan^ = 43 42. Functions of an Angle in Terms of Double the Angle. Since cos- 6 — sin^ 6 = cos 26 and cos^ 6 + sin^ 6=1 we get by adding and subtracting 2 cos2 = 1+ cos 26 and 2 sin^ ^ = i — cos 26, From cos^ - — sin^ - = cos 6 and cos^ ^ + sin^ ^ = 1 we can get in the same way a fi 2 cos^ - = 1 + COS and 2 sin^ - = 1 — cos ^. z z Hence cos|=-JL±^^; 2 \ 2 ' sm- , 1 , <9 2 . 1 - cos ^ and tan- = . 1 u -4\ 2 ^„ e \i + cosfl cos 2 43. It is well to be able to state any of these formulae in words. For example, the first one in Art. 41 is: the sine of any angle is twice the sine of half the angle times the cosine of half the angle. This enables us to write sm = 2 sm ^ cos ^, z z ' rA ^ n^ ^ • A+ B A+B . sm {A + B) = 2 sm — ^r — cos — ^r — , etc. 44 ELEMENTS OF PLANE TRIGONOMETRY. 44. Factoring Formulae. To prove sin^ + sm9 = 2 sin cos -, sin - sin = 2 cos ^sin ^, 2 2 ' cos + COS = 2 cos^ ^cos ~y , e ^- 4> . e - 4> COS B — cos = — 2 sin — sin^ ~, 22 From sin {A -\- B) = sin A cos B + cos A sin B and sin (A — 5) = sin A cos 5 — cos A sin B we get sin {A -\- B) -\- sin (A — B) = 2 sin A cos 5 and sin {A -{- B) — sin (A — 5) = 2 cos A sin 5. Similarly cos (A + B) + cos (A — J5) = 2 cos A cos B, cos (A + B) — cos (A — B) = — 2 sin A sin B. Put A + B = ^, A - B = Then 2A = ^ + and A (a) (^) (c) ^ + 2B = 61 - and B = ^-2^. Using these values for A and B in (a), (6), (c) and {d) we get the desired formulae. 45. Exercise XI. 1. Given arctan (— f) = ^ find sin 612. Since arctan (— f) 6, tan e = — \ and B must be in the second quadrant. ELEMENTS OF PLANE TRIGONOMETRY. 45 Cut off OM = - 4: and draw MP = 3. Then XOP = d and OP = 5. From Art. 42, 2 siri^ (9/2 = 1 - cos ^ = 1 - (- |) = f. Hence sin ^/2 = ± 1/^%; but since lies between 90° and 180°, 6/2 lies between 45° and 90° and the positive sign must be taken. So sin 6/2 = Vju = AV 10. 2. Given 6 = arcsin ( - |) find sin 26, cos 6/2, sin 6/2. 3. If A = arctan f find sin 2 A and tan 2 A. 4. By aid of Art. 42 find the functions of 22° 30'. 5. Given cos 6 = — j, 6 being in the third quadrant, find cos 6/2, tan 6/2, tan 26. 6. If arctan ^ = A and arcsin (— |) = B, find sin (2A + B) and cos {2 A — B). 7. If arccot (— J2) = ^j find tan A/2, sin A/2 and tan 2A. 8. If sin A = If and cos B = — f , A and B being in the second quadrant, find tan (A — B), cos A/2, tan B/2. Factor: 9. sin 5 A + sin A. 10. cos 3 A + cos A. 11. sin 80° - sin 70°. 12. cos 40° - cos 10°. 13. cos (A + J5) + cos (A - B). 14. sin (3A + B) - sin (A + SB). 15. sin 55° + cos 65°. 16. cos 15° - sin 35° 17. cos {26 + <^) + cos (2(9 - 0). 18. cos A — cos 3A. ,^ . SA+B . A +3J5 19. sm — - — - — sm — . 20. sin (A + 5 - C) - sin (A - B + C). 21. cos (A + 5/4) + cos (A/2 + 35/4). 22. If a + j8 = 45° prove that (1 + tana)(l + tan|8) = 2; also that 2 sin a cos a = (cos jS + sin /8)(cos jS — sin /3). 23. If tan 6 = '^ and cot = V prove that tan (20 + 0) = |. 24. If a: = arcsin l/i/5 and cot 1/ = 3 prove that a: + 2/ = 45°. 46. Identical Equations. 1. Prove cos A cos 15 46 ELEMENTS OF PLANE TRIGONOMETRY. Changing to sines and cosines sin A sin B _ sin {A + B ) cos A cos B ~~ cos A cos B' Clearing of fractions sin A cos B + cos A sin B = sin (A + B). But this is true by Art. 36; and since the steps may be reversed the original relation is also true. 2. Prove tan 26 = tsLU 6 + tan sec 2d Changing to sines and cosines sin 2d _ sin ^ sin 6 1 cos 20 ~ cos 6 cos cos 20' Clearing of fractions sin 20 cos ^ = sin cos 20 + sin 0, Transposing sin 20 cos —sin cos 20 = sin 6 But the left side is sin (20 — 0). Hence the identity has been proved. 3. Prove sin 50-2 sin 3(9 + sin ^ ^ ^^ ^ ^ iTT— i a — tan 66. cos 50 — 2 cos S0 + cos The left side is equal to (sin 50 + sin 0) - 2 sin 3(9 ^ 2 sin 30 cos 2(9-2 sin 30 (cos 50 + cos 0) - 2 cos 30 ~ 2 cos 30 cos 20-2 cos 30 _ 2 sin 30 [cos 20 - 1] - 2 cos 30 [cos 20 - 1] sin 30 ^ o^ ^ -n -n. = ^ = tan 30. Q. E. D. cos 30 ^ 4. If A, B, and C are the angles of a triangle prove ABC sin A + sin B + sin C = 4 cos ^ cos ^ cos ^. A + B, 2 ^ C 2 ^90°, sin 2 ~ cos - and cos A +B 2 ^sm^. (sin A + sin B) + sin C = 2 sin A+B A- 2 ^^^ 2 \-2 sm 2 cos 2 ELEMENTS OF PLANE TRIGONOMETRY. 47 ^ C A-B.^ A+B C = 2 cos - cos — h 2 cos — - — cos ^ ^ C\ A - B ^ A + Bl s 2 cos 2 [cos — h cos — 2 — I = 2 COS 2 [2 cos - cos -J (by § 44) ABC r^ ^ r^ = 4 cos "2 cos 2 cos -^. Q. E. D. 47. Exercise XII. Prove the following identities: 1. tan A — tan B = sin {A — B)/cos A cos B. 2. sin (A + B) cos (A - B) + cos (A + B) sin (A - B) = sin 2 A. 3. sin (A + J5) sin (A - B) = sin^ A - sin^ B. 4. cos (n + 1) A cos A + sin (n + 1) A sin A = cos nA. 5. cos (A + 5) cos {A - B) = cos^ A - sin^ B. tan(A + B)+tan(A-B) _ ^* 1 - tan (A + 5) tan (A - 5) " ^^"^ '^^• 7. sin ?i^ cos ^ = sin (n + 1)^ — cos nO sin d. 8. tan 2a; + sec 2x = (cos a; + sin a;)/(cos x — sin x). „ sin 3A + sin 2A + sin A ^ ^ . 9. 5-7—1 ^sr:r~i A — tan 2A. cos 3A + cos 2A + cos A 10. (cot e + l)/(cot - 1) = (1 + sin 26/)/cos 20. 11. sin 30/sin 6 — cos 30/cos = 2. 12. 2 sin 3A cos A = sin 4A + sin 2A. 13. cos A — cos 3A = 2 sin 2A sin A. 14. cos 30 = 2 cos 40 cos — cos 50. ^ p sin 8A — sin 6A + sin 4A — sin 2A . _ . 1^- E71 TTi — , n r. A — ~ cot oA. cos 8A — cos 6A + cos 4A — cos 2A 16. sin 20 = 2 tan 0/(1 + tan^ 0). 17. cos = (1 - tan2 0/2)/(l + tan^ 0/2). 18. (tan A + tan B)/(cot A + cot B) = tan A tan B. 19. (tan A - tan 5)/(cot A + tan B) = tan A tan {A - B). 20. 1 + tan A tan 5 = cos (A — J5)/cos A cos B. 21. tan (A + B) tan (^ - B) ^ T-L^TCb - 22. tan (7r/4 + 0) = (l + tan 0)/(l - tan 0). 23. cot (7r/4 + 0) = (cot - l)/(cot + 1). 48 ELEMENTS OF PLANE TRIGONOMETRY. 24. tan2 (7r/4 - A) = (1 - sin 2A)/(1 + sin 2A). 25. 1 + tan 2e tan 6 = sec 20. 26. cos2 A + cos2 (A + 60°) + cos^ (A - 60°) = 3/2. 27 tang cot ^ * tan - tan 30 "•" cot0 - cot30 ~ * ^^* tan 30 + tan d ~ cot 30 + cot 6 ~ ^^* ^"^^ 29. sin 30 cos + cos 50 sin 30 = cos 30 sin + sin 50 cos 30. 30. 2 sin (7r/4 + A) sin (x/4 - A) = I - vers 2A. 31. (cos 20 + cos 60)/(vers 60 - vers 20) = cot 20 esc 40 - 1. 32. 1 - vers 20 = (cot^ - l)/(cot2 0+1). „„ 2 + sin - vers ^ 33. r— ^-, — = cot -. sm + vers 2 vers 70 — vers _ cos + cos 70 _ ^ vers 50 — vers 30 cos 30 + cos 50 ~" „^ vers 20 vers ^ ^ ^ 35- • ozi • ^ - tan tan -. sm 20 sm 2 A + J5 A — B 36. tan — ^ — — tan — ^— = 2 sin B/(cos A + cos 5). ^^ sin (g - i8) sin (j(3 - 7) | ■ sin (7 - «) _ ^ sin a sin |8 sin j3 sin 7 sin 7 sin o: 38. tan q;/(1 — cot 2a tan «) = sin 2a. sin (g + /3) + sin (a — jg) ^ tan^ sin (a + /3) — sin (a — /3) ~ tan jS * In examples 40^9 A, B, and C are the angles of a triangle. 40. sin (A + 5) - sin C. 41. cos (A + 5) = - cos C. ,^ . A + B C 42. sm — z — = cos - . B+ C .A 43. cos — - — - = sm ^ . 44. sin 2 A + sin 25 + sin 2C = 4 sin A sin B sin C. ABC 45. sin A + sin 5 — sin C = 4 sin -- sin - cos - . ABC 46. cos A + cos 5 + cos C = 4 sin ^ sin ^ sin - + 1. 47. tan A + tan B + tan C = tan A tan B tan C. 48. tan — tan - + tan ^ tan - + tan ^ tan — = 1. ELEMENTS OF PLANE TRIGONOMETRY. 49 49. If sin A = 2 cos B sin C prove that the triangle is isosceles. 48. Conditional Equations. Throughout the work equalities have been met with which are true for any value of the angles involved. These have varied from simple relations like tan d = sin 0/cos 6 and sin^ 6 + cos^ 6 = 1 to the more complicated ones of the previous article. They are identical equations or more briefly identities. Equalities which are true only for special values of the angles are conditional equations, often called merely equations. To solve these is the present problem. Find all the values of 6 between 0° and 360° which satisfy: 1. sin2 + cos ^ = 1. Changing into one function we get 1 — cos^ 6 + cos ^ = 1, — cos^ 6 + cos ^ = 0, - cos ^(cos - 1) = 0. .*. cos 6 = 0, and 6 = 90° or 270°, or cos ^ = 1, and 6 = 0°. 2. 2 cos = sin 6 + 1. Substituting for cos 6 its value in terms of sin 6 2i/l - sin2 (9 = sin + 1, 4(1 - sin2 6) = sin2 6 + 2 sin 6 -\- 1, 5 sin2 ^ + 2 sin ^ - 3 = 0, (5sin^ - 3)(sin(9 + 1) = 0, .-. sin ^ = - 1 and 6 = 270°, or sin <9 = f and 6 = 36° 52' 12'' or 143° 7' 48". Since both sides of this equation were squared extraneous roots were introduced. It is necessary to test the angles found to see which suit the original equation. It is seen immediately that 143° 7' 48" does not satisfy the equation, the left side being negative and the right positive in that case. 3. sin 26 cos 6 = sin 6 Hence 2 sin 6 cos^ = sin 6, 315°. .*. sin = and 6 = 0° or or 180°, 2 cos2 (9 = 1, cos = ± l/v/2 and 6 = 4. cos 36 + cos = sin 3 ^ + sin 6. 4 45°, 135° 50 ELEMENTS OF PLANE TRIGONOMETRY. By Art. 44 2 cos 2d cos ^ = 2 sin 26 cos 0. .'. COS0 = and d = 90° or 270°, or cos 26 = sin 26, that is, tan 26 = 1. Hence 26 = 45°, 225°, 405°, 585°. 6 = 22° 30', 112° 30', 202° 30', 292° 30'. The above examples illustrate some of the methods of solving trigonometric equations. In all of them b}^ factoring the equation values of some function of an angle are obtained. In the first two, everything is expressed in terms of a single function. This is the most common method. Having in any way found the value of one function it remains to find the values of the angle. For this it is very useful to remember that any function of 6 has the same numerical value as the same function of 180° ^ 6 and 360° — 6, two of the four angles giving positive values, the remaining two negative ones. Thus if cos ^ = — ^, we look for the acute angle whose cosine is ^. This is 60°. Since cos 6 is negative, 6 must be in the second or third quadrants; it is therefore 180° ± 60°, that is, 120° or 240°. The student must guard against cancelling out a factor without keeping account of the corresponding roots. Thus in example 3 a common error is to merely cancel out sin 6 and reduce the equation to 2 cos^ ^ = 1. Also if it be found necessary to square both sides of an equation A = B, thus obtaining A^ = B^, the solutions will include those oi A = — B, in addition to the ones desired. In this case the angles should be substituted in the original equation and those dis- carded which do not satisfy it. In example (2) an instance of this was shown. 49. Exercise XIII. Find all values of between 0° and 360° which satisfy: 1. cos + sec ^ = |. 2. sin ^ = 1 — vers 6. 3. sin 6 = tan 6. 4. 2 cos ^ = 3 tan 6. 5. tan2 ^ + 3 cot^ 6 = 4. 6.s3 COS0 = 2 sin2^. 7. COS 26 + sin2 = f . ELEMENTS OF PLANE TRIGONOMETRY. 51 . 8. sin 5^ + sin Sd = cos 0. 9. sin 2^ = 3 tan ^ - 3 tan vers 2d. 10. sin 6(9 = sin 40 - sin 20. 11. cos 3^ + cos 20 + cos = 0. 12. cot 2^ + tan ^ + 1 = 0. 13. sin 20 + sin l the triangle is impossible since sin A cannot be greater than 1. ELEMENTS OF PLANE TRIGONOMETRY. 59 From the law of sines 6 sir C = -T— Sin , _ 5 sin C ~ sin B log h = 12.57771 - 10 - log sin' B = 9.88599 - 10 2.69172 2.69172 + log sin C = 9.97950 - -10 + log sin C = 9.08081 - 10 log c = 2.67122 logc' = 1.77253 .-. c = 469.05 .-. c' = 59.229 Hence the solutions are A = 57° IV 30'' A' = 122° 48' 30"] C = 72° 32' 5" or C = 6° 55' 5" c = 469.05 c' = 59.229. . Fig. 42. In the above example then, there are two triangles which satisfy the conditions. The solution of every problem under this case proceeds as this one did till we try to get C In certain examples A' -\- B will be found to be greater than 180°. It is evidently impossible to find C in that case and ther0> is only one solution. This will occur when the angle given is opposite the larger of the sides given. To recall the geometric construction in this case will be a help. Construct Z B and on one arm lay off BC = a. Then with C as centre and h as radius draw a circle to cut BD. If b < a two points A and A' will be found (provided h is long enough to reach BD. See footnote, p. 58). If 6>a only one point A will be found. 59. Case III. Given Two Sides and the Included Angle. If a, h, and C be known we can find a — h, a -\- h and ^{A -\- B) and then by aid of a formula of Art. 52, l(A — B) may be calculated. This with |(A + B) will give A and B. c may then be found by the law of sines. 60 ELEMENTS OF PLANE TRIGONOMETRY. Example. Given a = 42.38, 6 = 35 and C = 43° 14' 40" solve the triangle. Formulae: tan 2 = a-b^ A+B = a + 6*^ 2 ' asinC sm A a = 42.38 log a = 11.62716 - 10 b = 35 — log sin A = 9.99565 - 10 a-b = 7.38 1.63151 a + b = 77.38 + log sin C = 9.83576 - 10 A + B = 1S0-C = 136° 45' 20" logc = 1.46727 UA +B) = 68° 22' 40'' .-. c = 29.327 log (a — 6) = 0.86806 + log tan ^{A + B) = 0.40189 11.26995 - 10 - log (a + 6) = 1.88863 log tan i(A - 5) == 9.38132 - 10 .-. i(A - B) = 13° 31' 45" i(A +B) = 68° 22' 40" Adding, A = 81° 54' 25" Subtracting, B = 54° 50' 55" Hence the solution is A = 81° 54' 25", B = 54° 50' 55" and c = 29.327. 60. Case IV. Given Three Sides. If a, b, and c are given A/2, J5/2 and C/2 may be found from the formulae of Art. 54. ^ Example. Given a = 0.312, b = 0.423 and c = 0.342, find A, B and C. Formulae : r ^ B r . C r tan^ - 2 s — c a = 0.312 log {s - a) = 9.35507 - 10 b = 0.423 + log is -b) = 9.06258 - 10 c = 0.342 + log (s - c) = 9.29336 - 10 2s = 1.077 27.71101 - 30 s = 0.5385 - log s = 9.73119 - 10 s - a = 0.2265 2117.97982 - 20 s - b = 0.1155 I log r = 8.98991 - 10 s - c = 0.1965 s = 0.5385 ELEMENTS OF PLANE TRIGONOMETRY. 61 log r = 18.98991 - 20 - log (s- a) = 9.35507 - 10 log tan A/2 = 9.63484 - 10 .-. A/2 = 23° 20' A = 46° 40', log r = 18.98991 - 20 log {s -h) = 9.06258 - 10 log tan B/2 = 9.92733 - 10 .'. 5/2 = 40° 13' 45" B = 80° 27' 30", log r = 18.98991 - 20 log (s- c) = 9.29336 - 10 log tan (7/2 = 9.69655 - 10 .-. (7/2 = 26° 26' 15" C = 52° 52' 30". 61. Example 1. Each of two ships A and B, 415 yards apart, measures the horizontal angle subtended by a cliff and the other ship; the angles are 48° 17' and 110° 10' respectively. If the angle of elevation of the cliff from A is 15° 24' what is the height of the cliff? C is the top of the cliff and CC a vertical line, C being at the water level. Then it is given that AB = 415, zC'AB = 48° 17', ZC'BA = 110° 10', and ZCAC' = 15° 24'. It is required to find CC\ In the right triangle ACC we can find CC if AC be known. But AC may be found from the triangle ABC^ since three parts of this triangle are given. Use the law of sines sin B/AC = sin C'/A 5 re- membering that C = 180 - (A + B) = 21° 33'. Solving, AB sin B 415 sin 110° 10' Fig. 43. AC = sinC sin 21° 33' 62 ELEMENTS OF PLANE TRIGONOMETRY. log 415 = 12.61805 - log sin 21° 33' = 9-56504 3.0530T + log sin 110° 10' = 9.97252 .-. log AC = 3.02553 In the right triangle ACC 10 10 10 CC , = tan 15° 24'. AC .-. CC = AC tan 15° 24'. log AC = 3.02553 + log tan 15° 24' = 9.44004-10 log CC = 2.46557 .'. CC = 292.13. Example 2. Fig. 44. From a tower 80 ft. high, objects A and B in a plane are found to have angles of depression 12° 52' 30" and 10° 41' 25" respectively; the horizontal angle at C subtended by A and B is 43° 14' 40". Find the dis- tance between A and B. C being the top of the tower and C the foot, we have given CC = 80, ZCAC = 12° 52' 30". / CBC = 10° 41' 25" and zACB = 43° 14' 40". In the right triangles ACC and BCC AC = 80 cot 12° 52' 30", BC = 80 cot 10° 41' 25". log 80 = 1.90309 + log cot 12° 52' 30" = 0.64098 log AC = 2.54407 .-. AC = 350. log 80 = 1.90309 + log cot 10° 41' 25" = 0.72405 log BC = 2.62714 ,'.BC = 423.78. 1. A = 40° 20' 25'', 2. A = 52° 18' 25", 3. A = 57° 18', 4. a = 87.24, 5. a = 73 53, 6. a = 763.2, 7. a = 41 38, ELEMENTS OF PLANE TRIGONOMETRY. 63 Now in the triangle ABC we know three parts. Hence AB may be found as in Art. 59. 62. Exercise XIV. Solve the triangles in which the following parts are given: B = 75° 15' 35", a = 315.7. C = 62° 18' 50", h = 42.72. a = 3.715, h = 4.285 h = 73.58, C = 48° 17'. c = 81.27, B = 72° 19'. h = 653.5, c = 827.2. b = 51.71, c = 47.82. 8. A and 5 are two points 258 ft. apart and C a point such that at A the angle subtended by BC is 41° 18', at B the angle subtended by AC is 72° 21' 5"; find the distance between A and C. 9. A and B are points 400 ft. apart taken on the edge of a river and C is a stone on the opposite edge. If the angles CAB and CBA are 72° 20' 10" and 65° 10' 40" respectively find the width of the river. 10. The angle of elevation of an object from the foot of a hill is 32° 15'; after going 317 yds. up the hill away from the object the observer finds himself on a level with it. If the slope of the hill is 15°, find the distance from the foot of the hill to the object. 11. A cliff known to be 450 ft. high is observed to be due north of a boat and at an elevation of 30°. After going a dis- tance northeast the boat found the elevation to be 35°. How far did it go? 12. From a point on one side of a muskeg a man measures 450 yds. east; then northwest a distance of 652 yds. How far is he from the starting place? 13. The sides AB, BC and CA of a triangle are 250, 300 and 350 ft. respectively. BC is produced 275 ft. to the point D. What is the angle at A subtended by CDl 14. At each of two places 400 ft. apart the elevation of a kite is found to be 27° 15'. The horizontal angle at one place subtended by the kite and the other is 50° 20'. Find the height of the kite. 64 ELEMENTS OF PLANE TRIGONOMETRY. 15. At noon the shadow of a cloud falls 450 ft. to the north of an observer; if the elevation of the cloud (which is south of the observer) be 75° 12' and of the sun be 62° 8' what is the vertical height of the cloud? 16. In 15 what would be the height were the cloud north of the observer? 17. A lighthouse is 75 ft. high and has an elevation 25° 19' from one point; from another point 120 ft. from the first the elevation is 28° 23' 40". What is the horizontal angle at the lighthouse subtended by the two points? 18. The elevation of a balloon due north of a station is 30°; at a place one mile southeast of the first the horizontal angle subtended by the balloon and the first place is 20° 15'. Find the height of the balloon. 19. The elevation of a rock is 47°; after walking 1,000 ft. toward it up a hill inclined at 32° to the level the elevation is 77°. Find the height of the rock above the first point. 20. From the foot of a tree in a level field a line 200 ft. is measured and the elevation of the top of the tree is found to be 18° 20'. From this point a line 325 ft. long is measured and the elevation of the top is then found to be 22° 30' 25". Find the angle between the two lines which were measured. 21. A triangular piece of ground is found to be 37 ft. 6 in. by 42 ft. 3 in. by 58 ft. 3 in. Find the angle opposite the shortest side. 22. From a tower 75 ft. high the angles of depression of two objects are 24° 29' and 31° 58'; the horizontal angle subtended by the objects is 51° 42'. Find the distance between the objects (supposed to be level with the foot of the tower). 23. From a mountain top 3,200 ft. above sea level ships are observed, one east, the other southeast; the angles of de- pression are 12° 43' and 15° 37'. Find the distance between the ships, and the direction from one ship to the other. 24. Two towns A and B are four miles apart ; from a balloon above A the depression of B is 15° 6' and when the balloon is above B the depression of A is 19° 30'. Find how much the balloon has risen. 25. The horizontal distance between two points A and B ELEMENTS OF PLANE TRIGONOMETRY. 65 is a mile and a half. The horizontal angle at A subtended by B and the top of a mountain is 65° 18' 25" and that at B sub- tended by A and the top is 74° 29' 40''. The angle of elevation at A of the top of the mountain is 18° 28' 55". If the height above sea level of A is 2,500 ft. find the height of the mountain. 26. A flagstaff 78 ft. high stands on the face of a hill whose inclination to the horizon is 34°. At a point down the hill from the flagstaff, the angle of elevation of its top is 57° 28'; find the distance from the observer to the foot of the flagstaff. 27. At a point up the hill from the flagstaff in (26), the de- pression of its top is 18° 53'. Find the distance of the observer from the top of the flagstaff. 28. Two stations A and 5 on a level plane are 785.4 ft. apart. At A the elevation of an aeroplane C is 38° 19' 25", and the horizontal angle at A subtended by B and C is 41° 30' 20". At B the horizontal angle subtended by A and C is 63° 18' 20". Find the height of the aeroplane. 29. The observer at B in (28) makes the elevation of C 46° 49' at the same time that the other observations were taken. Find the height of C using this observation. CHAPTER V. SPHERICAL TRIGONOMETRY. Introductory Review. 63. Those definitions and theorems of sohd geometry which are essential to the study of spherical trigonometry will be stated or briefly discussed in this chapter. 64. Diedral Angles. Let ABC and ABD be two planes intersecting in the line AB. The figure formed by these planes at their intersection is called a diedral angle. The line AB is called the edge of the diedral angle. Let P be any point on the edge, and let PE and PF be lines drawn perpendicular to AS in the planes ABC and ABD respectively. The angle EPF Fig. 45. Fig. 46. is called the plane angle of the diedral angle, and is the measure of the diedral angle. 65. Triedral Angles. Let ABC, ACD, and ADB be three planes meeting in the common point A. The figure formed by these planes is called a triedral angle. The point A is called the vertex; the intersections of the planes AB, AC, AD are called the edges; the portions of the planes between the edges are called the faces; the angles formed at the vertex by the edges BAC, CAD, DAB are called the face angles; and the diedral angles formed at the edges are called the diedral angles of the triedral angle. 66. The Sphere. A spherical surface is a surface all points 66 ELEMENTS OF SPHERICAL TRIGONOMETRY. 67 of which are equidistant from a point called the centre. A sphere is a solid bounded by a spherical surface. Every plane section of a sphere is a circle. A great circle on a sphere is a circle whose plane passes through the centre of the sphere. All other circles on a sphere are called small circles. Let A and B be two points on the surface of a sphere, and let AB he the shorter arc of the great circle passing through these points. Then AB is the shortest path that can be drawn on the surface of the sphere between A and B. The length of this arc is defined as the distance from A to B measured on the surface of the sphere. Let be the centre of the sphere, and draw OA and OB. Then, since on the same or equal circles equal arcs subtend equal angles at the centre, the angle AOB may be tsken as the measure of the arc AB. The angle AOB is called the angular distance be- tween A and B. Produce BO to meet the sphere again in Bi. Let the radius of the sphere be -^ ^» ^ Fig. 47. r, and let the angle AOB (meas- ured in degrees) be 6°. Then, from plane geometry, we have arc AB : arc BBi = : 180 But arc BBi = a semicircle = irr. Therefore ^'•^ ^^ = iS- Thus, given the angular distance 6° and the radius of the sphere^ the actual distance on the surface of the sphere may be com- puted. Two points are said to be at a quadrant's distance when their angular distance is 90° The poles of any circle on a sphere are the extremities of the diameter of the sphere drawn perpendicular to the plane of the circle. Each pole of a great circle is at a quadrant's distance from every point of the circle 67. Spherical Angles. The angle between two circles on the 68 ELEMENTS OF SPHERICAL TRIGONOMETRY. Fig. 48. surface of a sphere is defined as the angle between the tangents to the circles at their point of intersection. The angle formed by the arcs of two great circles is called a spherical angle. Let AB and AC be arcs of two great circles on a sphere whose centre is at 0, and let AS and AT be the tangents to these circles at A. Then by definition the spherical angle is measured by the angle TAS. But since a tangent to a circle is perpendicular to the radius drawn to the point of contact, AT and AS are perpen- dicular to OA, and hence TAS is the measure of the diedral angle T-OA-S or B-OA-C, A spheri- cal angle is therejore measured hy the plane angle of the diedral angle formed hy the planes of the circles. Draw OB and OC perpendicular to OA at in the planes AOB and AOC respectively. Then the angles TAS and BOC are equal. But BOC is measured by the great circle arc BC. Hence, since A is the pole of the great circle arc BC, a spherical angle is measured hy the arc of the great circle of which the vertex is a pole, and which is intercepted hetween the sides of the angle. 68. Spherical Triangles. The figure bounded by the arcs of three great circles is called a spherical triangle. Let ABC be a spherical triangle on a sphere whose centre is at 0. Draw the radii OA, OB and OC. Then the figure formed by the planes OAB, OAC, and OBC is a triedral angle. The angles of the spherical triangle are measured by the plane angles of the diedral angles of this triedral angle. Also the sides of the spherical triangle are measured by the face angles of the triedral angles. For, by definition the angular ELEMENTS OF SPHERICAL TRIGONOMETRY. 69 distance AB is the angle A 05, subtended by AB at the centre. Hence with any spherical triangle there is associated a triedral angle at the centre of the sphere, such that the angles of the triangle are measured by the diedral angles of the triedral angle, and the sides of the triangle by the face angles of the triedral angle. This complete correspondence between the spherical triangle and the triedral angle at the centre of the sphere is of fundamental importance in what follows. 69. It is usual to assume that each side of a spherical triangle is less than 180°. This assumption causes no loss of generality. For let AB{D)C he SL spherical triangle in which the side BC is greater than 180°. Complete the great circle BDCE. Then in the triangle AB{E)C each of the sides Is less than 180°. We may then consider this triangle instead of the original one, and at the end of our investigations we may return to the triangle AB{D)C by taking the sup- plements of the angles B and C of AB{E)C, for the corresponding angles oiAB(D)C, and by takins for BDC the difference between 360° and BEC. It is easily shown (by producing BA to meet BDC in D) that the angle A is greater than 180° if the side opposite to it, BDC, is greater than 180°, and conversely. The sum of any two sides of a spherical triangle is greater than the third side. This follows from the corresponding theorem concerning the face angles of a convex triedral angle. The sum of the sides of a spherical triangle is less than four right angles, since the sum of the face angles of a convex triedral angle is less than four right angles. 70. Polar Triangles. Let arcs of great circles be drawn with the vertices of a spherical triangle ABC as poles, and let C be that intersection of the circles drawn with A and B as poles which lies on the same side of the arc A 5 as does C, and similarly for the other two intersections B' and A'. The triangle A'B'C is said to be the polar triangle of ABC, 70 ELEMENTS OF SPHERICAL TRIGONOMETRY. If the first of two spherical triangles is the polar of the second, then the second is also the polar of the first. In two polar triangles each angle of one is the supplement of the side opposite to it in the other. The latter statement may be proved as follows. Let ABC and A'B'C be two polar triangles. To prove that A = 180° - a\ etc. Produce the arcs AB and AC to meet B'C in M and N re- spectively. Since B' is the pole of the arc ACN, B'N is a qua- drant. Similarly MC is a qua- drant. Hence B'N + MC = 180°. That is, B'N + NC + MN = 180°. But B'N + N'C = B'C = a', and MN is the measure of the angle A, (page ""• 68). Therefore a' + A = 180°, or A = 180° - a', etc. 71. The sum of the angles of a spherical triangle is greater than two and less than six right angles. Let ABC be a spherical triangle. To prove that 180° 180°. Similarly, A -\- B + C = 540° - [something greater than 0°], that is, A + B + C < 540°. Therefore 180° N 2 sin U2s — c) cos - cos \{A — B ) _ ^^ ^ 2 A B sin s cos 2 cos - Upon dividing (2) by (3) and noting that 2s — c= (a + 6 + c) — c = a-{-h, we have cos K^ + B) ^ ''°^K« + &)sin | cosKA-B) ,inKa + 6)cos| = cot |(a + 6) tan-. (3) Therefore cosjOM-BJ _ _J^ V(a) cos-i(i4-5) tan-i(a + 6)* From (IV) we also have 2 _ sin (s — 6) B sin(s — a)' tan 2 • By replacing each tangent by the ratio of the sine to the cosine, this equation becomes . A B sm- COS-pr , f ,x 2 2 _ sm (s — o) ... A .""B"sin(s-a)* ^ ^ cos ^ sm - If now we apply to (4) the same series of operations that we have above applied to (1), we obtain sini(A-B) tani(a-6)* ^"^^ The details of this reduction are left for the student. Since V(a) is true for any spherical triangle, it is true for the polar triangle A'B'C of a given triangle ABC. That is, cosKA^ + gQ ^ ^^""2 .. cos \{A' - B') tan \{a' + 6')' ELEMENTS OF SPHERICAL TRIGONOMETRY. 79 But, i(A' + 50 = i(180° - a + 180° -h) = 180° - i(a + 6), ■|(A' - B') = i(180° - a - 180° + 6) = - ^(a - 5), i(a' + 6') = i(180° - A + 180° -h) = 180° - -i(A + B), and 1^ = 4(180°- C) = 90°- |. Substituting these values in (5) we have cos4(a-6) tani(A + 5)' ^^ Similarly, by aid of the polar triangle, V(6) gives rise tc • 1 / . L\ cot - sini(a + 6) _ 2 ^ . sini(a-6) taniiA-BY ^^""^ The four equations found in this article are known as Napier's analogies. Each of these equations, upon permuting the letters, gives rise to two new equations. In all, then, there are twelve such equations of which a, h, c and d are the types. 80. On Species. Two angles are said to be of the same species if they are both acute or both obtuse, and of different species if one is acute and the other obtuse. The following law of species is of great importance and should be carefully memorized. One half the sum of any two sides of a spherical triangle and one half the sum of the two opposite angles are of the same species. For, from V(a), cosUA + B) ^ ^^"^2 cos ^{A — B) tan ^(a + b)' Since each side and angle of the spherical triangle is less than 180° [Art. 69, page 69], i(A + B) and 4(a + b) are each less than 180°, and ^{A — B) and c/2 are each less than 90°. Hence, in the above equation, the denominator on the left and the numerator on the right are each positive. Then cos \{A + B) and tan -|(a + 6) are either both positive or both negative, and therefore \{A -\- B) and \{a -\- b) are of the same species . CHAPTER VII. THE RIGHT SPHERICAL TRIANGLE. 81. A spherical triangle one of whose angles is a right angle is called a right spherical triangle. We shall prove [Art. 85] that if, in addition to the right angle, two other parts be given the triangle can be solved. For this purpose we deduce from formulae of the preceding chapter ten relations each involving a different set of three parts. These formulae can be conveniently recalled by means of Napier^s rule of circular parts. By the circular parts of a right spherical triangle we mean the two sides about the right angle and the complements of the hypotenuse and the other two angles. Place these five parts as in the accompanying figure, being careful to arrange them in the order in which they occur in the triangle. It will be noticed that on this figure any three parts can be classed as either a middle and two adjacent parts, or a middle and two op- posite parts. 82. Napier's Rule. The sine of the middle part is equal to the product of the tangents of the adjacent parts, and to the product of the cosines of the opposite parts. Since there are ten combinations of five things taken three at a time, this rule will give us ten formulae. For example, let ABC be a right spherical triangle right-angled at C. Then taking b, co-A, co-B the middle part is co-B and the other two are the opposite parts. Therefore, Fig. 56. sin (90° - B) = cos (90° - A) cos 6, or cos B sin A cos b. 80 ELEMENTS OF SPHERICAL TRIGONOMETRY. 81 Also taking a, co-c, co-B, the middle part is co-B and the others are the adjacent parts. Therefore, sin (90° - B) = tan a tan (90° - c), or cos B = tan a cot c The following table contains all the formulae so obtained : cos c = cos a cos 6, (1) cos c = cot A cot B, (2) cos A = sin B cos a, (3) cos B = sin A cos h, (4) sin a = sin c sin A, (5) sin 6 = sin c sin B, (6) sin a = tan 6 cot Bj (7) sin 6 = tan a cot A, (8) cos A = tan h cot c, (9) cos 5 = tan a cot c. (10) 83. These relations may be deduced as follows: Upon setting C = 90°, the last equation of (I), page 74, becomes cos c = cos a cos b. (1) The last equation of (II), page 75, gives = — cos A cos 5 + sin A sin B cos c, and therefore, cos c = cot A cot ^. (2) The first two equations of (II), page 75, give cos A = sin 5 cos a, (3) and cos B = sin A cos h (4) The sine proportion (III), Art. 77, becomes sin A sin B 1 and hence and sin a sin b sin c sin a = sin c sin A, sin b = sin c sin B, sin ( sin 6 sin A (5) (6) sin B Or, upon substituting for sin A its value found from (4), sin a = tan b cot B, (7) and similarly sin b = tan a cot A. (8) From 8, tan a = sin 6 tan A 6 82 ELEMENTS OF SPHERICAL TRIGONOMETRY. or sing _ sin h sin A cos a cos A Substituting in this equation the values of sin a and cos a found from 5 and 1, we have sin c sin A cos h _ sin h sin A cos c " cos A or cos A = tan h cot c, (9) and similarly cos B = tan a cot c. (10) 84. Laws of Species for the right Spherical Triangle. We shall prove the following two laws of species for the right spheri- cal triangle. 1. A side of a right spherical triangle and the angle opposite it are of the same species. 2. If the hypotenuse of a right spherical triangle he acute j the two other sides are of the same species, and if the hypotenuse be obtuse, the two other sides are of different species. Proof of 1. sin b = tan a cot A. (8) Since h is less than 180°, sin b is positive. Then tan a and cot A are of the same sign, and hence a and A are of the same species. Proof of 2. cos c = COS a cos b. (1) If c is less than 90°, cos c is positive. Then cos a and cos b are of the same sign, and hence a and b are of the same species. On the other hand, if c is greater than 90°, cos c is negative, then cos a and cos b are of opposite signs, and hence a and b are of different species. 85. The Solution of the right Spherical Triangle. When two parts of a right spherical triangle are given the remaining parts may be computed by means of Napier's rule of circular parts and the laws of species. The method in detail will be illustrated by the following examples. When a relation involves a trigonometric function of an obtuse ELEMENTS OF SPHERICAL TRIGONOMETRY. 83 angle we replace the angle by its supplement. This substitution in any one of the formulae (I)-(IO) will at most give the wrong sign to the desired function, thus leaving it uncertain as to whether the desired angle is acute or obtuse. The laws of species are intended to remove this ambiguity Ex. 1. a = 47° 30' 40", c = 120° 20' 30''. Applying Napier's rule to acA, acB, acb and solving each equa- tion for the unknown part, we have . sin a r> X J. T cos c smA= ~. — ) cos ^ = tan a cot c, cos o = sm c cos a Since c is obtuse a and h are of different species. Then h and B are obtuse, and A is acute. log sin a = 9.86771 log tan a = 0.03812 log cos c = 9.70342 log sin c = 9.93603 log cot c = 9.76740 log cos a = 9.82959 log sin A = 9.93168 Tog cos 5= 9.80552 log cos h = 9.87383 A = 58° 41' 55" 5 = 180° - 50° 16' 50" 6 = 180° - 41° 35' 35" = 129° 43' 10" = 138° 24' 25" Check:* Napier's rules applied to A, B,b give cos B = cos h sin A. log cos h = 9.87383 log sin A = 9.93168 log cos B = 9.80551 Ex. 2. a = 103° 12', A = 97° 24'. Napier's rules give . , , ,4 . T) cos A . sin a sm = tan a cot A, sm B = , sm c = ~ — 7- cos a sm A log tan a = 0.62977 log cos A = 9.10990 log sin a = 9.98837 log cot A = 9.11353 log cos a = 9.35860 log sin A = 9.99637 log sin b = 9.74330 logTin B = 9.75130 log sin c = 9.99200 Since a is obtuse it follows that, if h be also obtuse, c must * To test the accuracy of the arithmetical work it is necessary to use as a check some formula which has not already been used. In any right spherical triangle the most convenient relation for this purpose is that one which involves the three required parts. 84 ELEMENTS OF SPHERICAL TRIGONOMETRY: be acute, and, if b be acute, c must be obtuse. The laws of species give no means of telling whether b is acute or obtuse. In fact there are two triangles with the given parts a and A, which together form a lune of angle A, as in the accompanying figure. The two solutions are then (1) (2) b = 33° 37' 25'', B = 34° 20' 5", c = 100° 58', 6' = 180° - b = 146° 22' 35", B' = 180° - B = 145° 39' 55", Fig. 57. c' = 180° - c = 79° 2'. Check: sin 6 = sin c sin B. log sin c = 9.99200 log sin B = 9.75130 log sin b = 9.74330 86. Exercise XV. Solve the following spherical triangles in which C = 90*^ 1. 3. a = 53° 48' 10", 2. a = 120° 40' 5", 6 = 73° 12' 20". c = 78° 22' 25". a = 23° 59' 55", 4. A = 27° 54' 5", c = 133° 24' 50". b = 100° 58' 10". b = 27° 50' 30", 6. B = 78° 53' 55", A = 114° 14' 25". b = 40° 12' 10". A = 53° 54' 55", 8. a = 127° 54' 10", c = 127° 12' 10". A = 105° 50' 25". A = 110° 13' 20", 10. b = 35° 55' 30", B = 73° 13' 35". c = 143° 14' 10". a = 29° 13' 35", ^ 12. a = 154° 42' 10", b = 54° 14' 25". c = 148° 14' 20". 7. A = 11. 87. Quadrantal Triangles. A spherical triangle is said to be quadrantal if one of its sides is equal to a quadrant, that is, equal to 90°. In the spherical triangle ABC let c = 90°. Then in the polar triangle A'B'C we have C = 90°. When any two parts of the quadrantal triangle in addition to c are given, this triangle may be solved by the methods explained in the foregoing ELEMENTS OF SPHERICAL TRIGONOMETRY. 85 articles; we may find the unknown parts of the (right) polar triangle, and, by subtracting these from 180°, find the required parts of the quadrantal triangle. Quadrantal triangles may also be solved directly as follows. Napier's rule, as stated for the right triangle, also holds for quadrantal triangles provided we take as the circular parts A, B, 90° - a, - (90° - C), 90° - 6. The laws of species for quadrantal triangles are: 1. A side of a quadrantal triangle and the angle opposite it are of the same species. 2. If the angle opposite the quadrant he acute, the two other angles are of different species, and if this angle he ohtuse the two other angles are of the same species. The proofs of these laws of species and of the application of Napier's rule to quadrantal triangles are left as exercises to the student. 88. Exercise XVI. Solve the following spherical triangles in which c = 90°: 1. a = 50° 49' 25", 2. A = 103° 25' 55", B = 73° 12' 35". C = 79° 15' 5". 3. 6 = 18° 28' 10", 4. a = 100° 13' 20", C = 55° 58' 30" A = 123° 16' 45". CHAPTER VIII. THE SOLUTION OF OBLIQUE TRIANGLES. 89. In the study of solid geometry we have learned that two triangles are equal or symmetrical if 1) Three sides of one are equal respectively to three sides of the other. 2) Three angles of one are equal respectively to three angles of the other. 3) Two sides and the included angle of one are equal respec- tively to two sides and the included angle of the other. 4) A side and the adjacent angles of one are equal respectively to a side and the adjacent angles of the other. That is, if the three parts mentioned in any one of these cases be given, the remaining parts are fully determined. It is then natural to suppose that, given such a set of three parts, the remaining three parts may be computed. In addition to these four cases we shall find that, when given two sides and an angle opposite one of them, or two angles and a side opposite one of them, there will be, at most, two distinct triangles having these parts, and that the remaining parts of each triangle may be computed. 90. We shall now show how to find the unknown parts of a spherical triangle of which we are given 1) The three sides, a, h, c. 2) The three angles, A, B, C. 3) Two sides and the included angle, e. g., a, b, C. 4) A side and the two adjacent angles, e. g., c, A, B. 5) Two sides and the angle opposite one of them, e. g., a, &, A. 6) Two angles and the side opposite one of them, e. g., A, 5, a. 91. Case (1). Given the Three Sides, a, h, c. The half angles may be determined by the formulae (IV), page 77. It is important to observe that, since the angles are less than 180°, each of the half angles is less than 90°. It therefore is not necessary to use the laws of species. After 86 ELEMENTS OF SPHERICAL TRIGONOMETRY. 87 finding the three angles the accuracy of the results should be tested by substituting the angles in some relation that has not already been used. For this the sine proportion (III), page 75, is the most convenient. Ex. 1. a = 58°, h = 80°, c = 96°. Formulae : ^ A K ^ B K ^ C K tan — = -. — ; r, tan -7^ = - — 7 — r^ , tan 7^ = sin (s— a)' 2 sin (s — 6)' 2 sin (s—cy K = f sin (g — o) sin (s — h) sin {s — c) \ sin s a = 58° b = 80° log sin (s - a) = 9.93307 A 96° log sin {s-h)= 9.77946 ^^g tan ~ = 9.72542 c = 2s = 234° log sin {s - c) = 9.55433 B s = 117° 9.26686 log ism— = 9.87903 s - a = 59° log sin g = 9.94988 ^ s -h = 37° log X2 = 9.31698 log tan -^ = 0.10416 s - c = 21° log K = 9.65849 s = 117° ~ = 27° 59' 10", A = 55° 58' 20'', ^ = 37° 7' 20", 5 = 74° 14' 40", ~ = 51° 48' 20", C = 103° 36' 40". ^, , , sin A sin 5 sin C Check : a = — = -; — r = -. . sm a sm sm c log sin A = 9.91843 log sin B = 9.98337 log sin C = 9.98763 log sin a = 9.92842 log sin h = 9.99335 log sin c = 9.99761 log d = 9.99001 ■ log d = 9.99001 log d = 9.99002 Ex. 2. a = 53' 12' 35", h = 75° 14' 25", c = 69° 27' 20". 92. Case (2). Given the Three Angles, A, B, C. Since the angles of a spherical triangle are the supplements of the corresponding sides of the polar triangle, we may find the sides of the polar triangle. Then, by the method of case (1), we may compute the angles of the polar triangle. The supplements of these angles will then be the required sides of the original triangle. 88 ELEMENTS OF SPHERICAL TRIGONOMETRY. Ex. 1. A = 78° 40' log sin (s - a)' = 9.98338 B = 63° 50' log sin (s - 6/ = 9.93495 C = 46° 20' log sin (s - c)' = 9.82481 . , 9.74314 In the polar triangle . , o oo«k.i // = 101° 20' log sm s' = 8.88654 ?, _l1.o-,n' log K'^ = 0.85660 ,. = 133040^ log Z'= 0.42830 2s-^^350^' logtan4-' = 0.44492, g/ _ X75° 35' -^ ^ = 70° 15' 10"-, A' = 140° 30' 20"-, ~ = 72° 11' 50"+, B' = 144° 23' 40"+, £1 ~ = 76° 0' 30"-, C = 152° 1'-. Hence, in the original triangle, a = 39° 29' 40", h = 35° 36' 20", c = 27° 59'. ^i_ , sin A sin B sin C Check: = ~. — r = -^ . sm a sm sm c log sin A = 9.99145 log sin B = 9.95304 log sin C = 9.85936 log sin a = 9.80346 log sin b = 9.76507 log sin c = 0.67137 0.18799 0.18797 0.18798 Ex. 2. A = 105° 14' 20", B = 80° 0' 10", C = 68° 23' 35". 93. Case (3) . Given Two Sides and the Included Angle, e. g., a, h, C, Solving the relations (c), (d), and (6), Art. 79, for tan i(A -\-B), tan i{A — B), and tan c/2 respectively, we have (J cos i{a — h) cot^ tan |(A + B) = tt — r-^^r , ^ cos -JCa + h) ' C sin ^(a — 6) cot ^ tan UA — B) = r—^. — ^-r^ — » ELEMENTS OF SPHERICAL TRIGONOMETRY. 89 , . c sin i(A + B) tan -^(a — h) and tan ^ = -. — ^Ta d\ • 2 sin^(A — B) By the first two of these relations we may compute half the sum, and half the difference of the unknown angles, and so can find these angles. This being done, the third side, c, may be found by means of the last relation above. The accuracy of the results may then be tested by the sine proportion. Observe that ^{A — B) and c/2 are always acute, and that therefore the law of species need not be used in finding the values of these angles. On the other hand, \(A -\- B) may be obtuse. In fact, this angle is acute or obtuse according as ^{a + h) is acute or obtuse, page 79. Ex. 1: c = 40° 20' |(a + c) = 70° 25' a = 100° 30' i(a - c) = 30° 5' B = 46° 40' B/2 = 23° 20' Formulae: cos ^{a — c) cot -X tan i(A + C) = , , , , , ^^ ^ cos I (a+ c) ' sin I (a — c) cot — taniU-C)= ,in^(^ + ,) , h _ sin |(A + C) tanj- (a — c) ^^2 ~ sin i{A - C) log cos i(a - c) = 9.93717 log sin i{a - c) = 9.70006 log cot J5/2 = 0.36516 0.36516 0.30233 0.06522 log cos iia + c) = 9.52527 log sin i{a + c) = 9.97412 log tan i(A + C) = 0.77706 log tan i{A - C) = 0.09110 i(A + C) = 80° 30' 50"+. i(A - C) = 50° 58'-. Hence, A = 131° 28' 50", C = 29° 32' 50". log sin i(A + C) = 9.99402 log tan |(a - c) = 9.76290 9.75692 log sin i(A - C) = 9.89030 log tan 6/2 = 9.86662 90 ELEMENTS OF SPHERICAL TRIGONOMETRY. 6/2 = 36° 20' 15''- and hence, b = 72° 40' 30"-. ^, , sin A sin B sin C Check: - — = ^—7 = -. . sm a sm sin c log sin A = 9.87459 log sin B = 9.86176 log sin C = 9.69297 log sin a = 9.99267 log sip 6 = 9^984 log sin c = 9.81106 9.88192 9.88192 9.88191 Ex. 2. h = 109° 12' 10", c = 131° 18' 25", A = 46° 14' 55". 94. Case (4). Given a Side and the Two Adjacent Angles, e. g., c, A, B. The solution in this case is exactly similar to that of III. Ex. 1: Formulse: C=110°40' j,^ p., a B = 100°36' , ,, ^,, c osKC-B)tan- a = 76° 38' ^^^ *^^ + ^) = "cosKC + 5) " ' |((7 + 5) = 105° 38' sin \{C - B) tan ^ i(C - B) = 5° 2' tan 4(0 - 6) = - ,i,x(c + ^) > 1 = 38° 19' cot 4 = BJrL Kc+6)tani(C-B) ^ 2 2 sin \{c — 0) log cos i(C - B) = 9.99832 log sin \(C- B) = 8.94317 log tan a/2 = 9.89775 9.89775 9.89607 8.84092 log cos 4(C + 5) = 9.43053 log sin i(C + 5) = 9.98363 log tan -i(c + &) = 0.46554 log tan Kc - &) = 8.85729 i(c + 6) = 180° - 71° 6' 5"+ \{c - h) =4°7'5"- = 108° 53' 55"- c = 113° 1'-, 6 = 104° 46' 50". log sin |(c + 6) = 9.97593 log tan \{C - B) = 8.94485 8.92078 log sin ijc -h) = 8.85617 log cot A/2 = 0.06461 4 = 40° 45' 15" - , A = 81° 30' 30" - . ^, , sin A sin B sin C Check: —. = — — 7 = ~. — • sin a sm o sm c ELEMENTS OF SPHERICAL TRIGONOMETRY. 91 log sin A = 9.99521 log sin B = 9.99252 log sin C = 9.97111 log sin a = 9.98807 log sin b = 9.98539 log sin c = 9.96397 0.00714 0.00713 0.00714 Ex. 2. A = 59° 19' 15'', B = 76° 14' 15", c = 130° 14' 50". 95. Case (5). Given two Sides and the Angle Opposite One of Them, e. g., a, 6, A. From III, page 75, . ^ sin & sin A smij = -. . sm a We may then compute the value of log sin B. But, since sin B = sin (180° — 5), we must apply the law of species to tell whether B is acute or obtuse. Denote the acute angle, found in the tables from the value of log sin B, by B, and the corre- sponding obtuse angle ( = 180° — B) by 5'. If B is to be a possible solution, the law of species, page 79, states that i(a + 6) and ^{A + B) must be of the same species. Similarly, if B' is a possible solution, |(a + 6) and -1{A + 5') must be of the same species. Hence, this law will show whether there be one solution, two solutions, or no solution. Having found B, we may compute the remaining parts, c and C, by Napier's analogies (6) and {d), page 78. We may then check by the sine proportion. Ex. 1. a = 30° 20', h = 46° 30', A = 36° 40'. sin h sin A Formulae: sin B = sm a C ^ sin ^{h + a) tan jjB - A) ^^^2 smiib-a) c _ sin i{B -\- A) tan |(b — a) ^^2 " sin i{B - A) • log sin b = 9.86056 log sin A = 9.77609 9.63665 log sin a = 9.70332 log sin B = 9.93333 B = 59° 3' 30", B' = 180° - 59° 3' 30" = 120° 56' 30". 92 ELEMENTS OF SPHERICAL TRIGONOMETRY. i(6 + a) = 38° 25' i(5 - A) = 11° 11' 45'' i(6 -a) = 8° 5', i(B' + A) = 78° 48' 15" i{B + A)= 47° 51'45", i(5' - A) = 42° 8' 15" There are two solutions since both |(5 + A) and |(B' + A) are of the same species as |-(6 + a). First Solution. Second Solution, log sin i{h + a) = 9.79335 9.79335 log tan i{B - A) = 9.29651 9.95653 9.08986 log sin i{h - a) = 9.14803 log cot C/2 = 9.94183 ^ = 48° 49' C = 97° 39' log sin i(B + A) = 9.87013 log tan 4(6 - a) = 9.15236 30" 14° 2' 35"- 9.02249 9.28817 log tan c/2 = 9.73432 log sin i(B - A) ~ = 28° 28' 30' 9.74988 9A4803 0.60185 C 2 C'=28°5' 10"- 9.99166 9.1523 6 9.14402 9.82666 9.31736 ^ = 11° 43' 55" - c' = 23° 27' 50" -. c = 56° 57' Hence the solutions are (1) B = 59° 3' 30" (2) B' = 120° 56' 30" C = 97° 39' C = 28° 5' 10" - c = 56° 57' + c' = 23° 27' 50" - p, , ^ sing _ sinjC _ sin C sin b sin c sin c' ' log sin B = 9.93333 log sin C = 9.99612 log sin C = 9.67284 log sin b = 9.86056 log sin c = 9.92335 log sin c' = 9.60007 0.07277 0.07277 0.07277 Ex. 2. 6 = 32° 18' 10", c = 50° 14' 15", C == 48° 12' 10". 96. Case (6). Given Two Angles and the Side Opposite One of Them, e. g., A, B, a. The solution of this case is exactly similar to that of Case V. * It is not necessary to use the ratio sin A Ism a since this was used in the solution. ELEMENTS OF SPHERICAL TRIGONOMETRY. 93 Ex. 1. B = 69°, C = 132°, b = 65°. Formulae: sin C sin h sm c = : — 5 — , sm B A sin i{c-\-h) ism UC - B) ^^^ 2 ~ sin Kc - ?>) a sin -KC' + 5) tan -i(c - 6) ^^2" sin UC-B) log sin C = 9.87107 log sin h = 9.95728 9.82835 log sin B = 9.97015 T'V.i-i'r* log sin c = 9.85820 inen, c = = 46° 10' 25'' c' = 180° - 46° 10' 25" = 133° 49' 35" -i(c' + h) = 99° 24' 50" - 4(c' -h) = 34° 24' 50" - and ^(C + B) = 100° 30' i(C - B) = 31° 30' i(c + 6) = 55° 35' 15" - Of the two values, c and c', the latter alone is a solution since it only obeys the law of species. Then, as there can be but one solution, we may omit accents and take c = 133° 49' 35". Then, i(c + 6) = 99° 24' 50" - i(c -h) = 34° 24' 50" - log sin i(C + B) = 9.99267 |(C + B) = 100° 30' i(C - B) = 31° 30' log sin i(c + 6) = 9.99411 log tan i(C - B) = 9.78732 9.78143 log sin -i(c - 6) = 9.75218 log cot A/2 = 0.02925 I = 43° 4' 20", A = 86° 8' 40", log tan |(c - 6) = 9.83573 9.82840 log sin i(C - B) = 9.71809 log tan a/2 = 0.11031 = 52° 12' -, a Check: a = 104° 24'- sin A sin C sm a log sin A = 9.99902 log sin a = 9.98614 0.01288 sm c log sin C log sin c 9.87107 9.85820 0.01287 94 ELEMENTS OF SPHERICAL TRIGONOMETRY. ' Ex. 2. A = 70° 14' 15'^ B = 59° 12' 25'', b = 51° 18' 35' 97. Exercise XVI. Solve the following spherical triangles. 3. A = 5. a = 7. a = 9. A = 11. c = 13. A = 15. A = 17. a = 48° 35' 2. A = 105° 24' 10" b = 77° 23' b = 110° 5' 15" c = 80° 56' c = 80° 37' 35" A = 120° 56' 25" 4. a = 29° 15' 25" B = 80° 32' 10" c = 50° 29' 55" C = 100° 27' 50" B = 47° 48' 50" a = 103° 22' 50" 6. a = 58° 32' 50" B = 76° 13' 25" b = 42° 23' 10" C = 37° 58' 55" A = 60° 36' 35" a = 102° 14' 10" 8. A = 70° 25' 10" b = 74° 14' 35" B = 50° 46' 25" c = 118° 29' 25" C = 80° 39' 55" A = 105° 35' 35" 10. A = 123° 34' 50" B = 120° 23' 10" B = 78° 22' 35" c = 74° 24' 25" b = 98° 29' 25" c = 73° 14' 10" 12. A = 58° 58' 55" b = 60° 12' 10" B = 36° 49' 45" B = 40° 28' 35" C = 96° 37' 35" A = 167° 43' 35" 14. a = 76° 56' 55" B = 103° 25' 25" c = 48° 48' 50" c = 70° 45' 50" B = 39° 32' 35" A = 80° 54' 55" 16. a = 47° 54' 55" B = 90° b = 36° 25' 50" c = 47° 16' 10" c = 80° 19' 10" b = 54° 51' 35" 18. B = 54° 51' 25" a = 79° 22' 25" A = 79° 22' 25" C = 28° 29' 55" c = 28° 29' 55" 98. Geographical Problems. The position of a point on the earth's surface is determined if the latitude and longitude of the point be given. By the aid of spherical trigonometry we may find the distance between two points whose latitudes and longitudes are known, and also the bearing of either point from the other. Since the shortest distance between two points is along the arc of a great circle, a navigator sails as nearly as possible upon the great circle arc from the point of departure to the desired destination. The ELEMENTS OF SPHERICAL TRIGONOMETRY. 95 course, when sailing upon a great circle arc, is continually changing except in the special cases of sailing along the equator or a meridian. If, on the other hand, a ship kept upon the same course, for example N. 25° E., it would sail in a spiral slowly approaching one of the poles. Such a path upon the earth's surface is known as a loxodrome or rhumb line. 99. In the following table the latitudes and longitudes of several cities are given: Baltimore 39° 17' N., 76° 37' W. Boston 42°2rN., 71° 4' W. Cape Town 33° 56' S., 18° 26' E. Halifax 44° 40' N., 63° 35' W. Honolulu 21° 18' N., 157° 55' W. Liverpool 53° 24' N., 3° 4' W. New York 40° 43' N., 74° W. San Francisco 37° 48' N., 122° 24' W. 100. Exercise XVII. In the following problems the earth is assumed to be spherical, and the radius is taken as 3960 miles. 1. Find the distance from Halifax to Liverpool, and the bearing of each place from the other. Also find the course of a ship sailing from Halifax to Liverpool when it crosses the meridian 55° W., the latitude of this point, and the distance the ship has sailed. Let G, L and H denote respectively Greenwich, Liverpool, and Halifax. Then GNL = 3° 4', LN = 90° - 53° 24' = 36° 36', GNH = 63° 35', HN = 90° - 44° 40' = 45° 20'. .-. LNH = 60° 31'. Then we are given two sides and the included angle of the triangle HNL. We can then solve for the angles NHL and 96 ELEMENTS OF SPHERICAL TRIGONOMETRY. HLN and for the side HL by the method of Art. 93. This gives NHL = 54° 54' 20'', HLN = 77° 25' 25", HL = 39° 22' 5" = 39.368°. Then the distance from Halifax to Liverpool is , 39.368 X TT X 3960 __, .. ,. . _, d = TgQ = 2721 miles. [Art. 66] The bearing of Halifax from Liverpool is N. 77° 25' 25", W., and that of Liverpool from Halifax is N. 54° 54' 20" E. To solve the last part of the problem consider the triangle HNA, where AN is the meridian of 55° W. In this triangle HN = 45° 20', NHA = 54° 54' 20", HNA = 63° 35' - 55° = 8° 35'. We have then a side and the two adjacent angles, and can therefore solve for the remaining parts by the method of Art. 94. Hence. AN = 41° 38' 20", HAN = 118° 51' 25", HA = 6° 57' 40" = 6.961°. Therefore the latitude of the point is 48° 21' 40", the course is N. 180° - 118° 51' 25" E., or N. 61° 8' 35" E., and the distance sailed is given by , 6.961 X TT X 3960 .^, . ., a = :r^ = 481 + miles. 2. A ship sails from Halifax to Liverpool along the arc of a great circle. Find her latitude and longitude after she has sailed one thousand miles. Find also her course at this point. 3. Find the distance from New York to San Francisco. 4. A ship sails from San Francisco to Honolulu. Find the course she is steering when she has covered half the distance. 5. A ship sails from Boston to Cape Town. Find the total distance sailed, and her course when 1500 miles from Cape Town. ELEMENTS OF SPHERICAL TRIGONOMETRY. 97 6. After sailing 1000 miles from Halifax a ship crosses the parallel of latitude of 40°. What will be the position of the ship (that is, the latitude and longitude) after she has sailed 800 miles further on the same great circle. 7. Find the area in square miles of the triangle whose vertices are Baltimore, Cape Town, and Liverpool. CHAPTER IX. OTHER FORMULiE RELATING TO SPHERICAL TRIANGLES. 101. From IV, page 77, we have tan 7: = -r-7 ^r. (1) 2 sin(s — a) Since this relation is true for any spherical triangle it is true for the polar triangle of a given triangle. We may then replace A' by 180° - a, a' by 180° - A, etc. Let2S = A-\- B + C, so that s' = 1(540° - [A -\- B + C]) = 270° - S. Then j^{- A -\- B + C) = S - A, etc. Hence s' - a' = 90° - {S - A), Similarly s' - h' = 90° - {S - B), and s' - c' = 90° - {S - C). Therefore ^, ^ I sin (s' — a') sin (s' — bQ sin js' — c') \ sin s' / cos (S - A) cos (>S - g) cos {S - C) ^i - coss Now let , , — cos S =4 (cos {S - A) cos {S - B) cos {S - C) so that «■ = I- The equation (1) then becomes a 1 cot - = 2 /b cos (^ - A) 98 ELEMENTS OF SPHERICAL TRIGONOMETRY. 99 or Similarly tan^ = k cos {S — A). tan ^ = A; cos (*S — 5), VI and tan » = A; cos {S C). By means of these equations a triangle in which the three angles are given may be solved directly. 102. Radius of the Circumscribed Circle. Let Ai, Bi, and Ci be the middle points of the sides of the spherical triangle ABC. At Ai draw the arc of a great circle perpendicular to the side BC, and similarly draw arcs through Bi and Ci perpendicular to the corresponding sides of ABC. These three arcs will meet in a point 0. [This may be proved in a manner entirely analogous to that employed for the corresponding theo- rem in plane geometry.] Draw the great circle arcs A, B and OC. The right triangles OAiB and OAiC are symmetrical since BAi = AiC and OAi is common to the two triangles. Hence the angles AiBO and AiCO are equal. Similarly BiAO ■■ Now 2S = A + B + C, = {OABi + OACi) + (OBAi + OBCi) + {OCAi + OCB,), = (OABi + OCBO + (OACi + OBCi) + {OBAi + OCAi), = 20ABi + 20ACi + 20BAi, = 2A+ 20BAi. Therefore OBAi = S - A. Then in the right triangle OBAi we have OBAi = S — A, BAi = a/2 and OB = r, where r is the radius of the circum- scribed circle. Applying Napier's rule to this triangle, we have Fig. 59. BiCO,SindCiAO=CiBO. 100 ELEMENTS OF SPHERICAL TRIGONOMETRY. or, cos OBAi = cot OB tan AiB, cos {S — A) = cot r tan -. Substituting for tan a/2 its value from VI, Art. 101, we have cos {S — A) = cot r-k cos {S — A). Hence it follows that tanr = fc = ^ — cos/S cos {S - A) cos {S - B) cos {S - C) VII By this formula we may find the radius r, that is the polar distance, of the small circle circumscribed to the spherical triangle. 103. Radius of the In- scribed Circle. Draw arcs of great circles bisecting the angles of the spherical tri- angle. These arcs meet in a point 0. Through draw the great circle arc OAi per- pendicular to the side BC, and similarly OBi and OCi perpendicular to CA and AB respectively. The right tri- angles OACi and OABi are equal since the angles OACi and OABx are equal and the hypotenuse OA is common. Then ABi = ACi. Similarly BCi = BAi and CAi = CBi. Now, 2s = a + 5 + c, = {BA, + A^C) 4- (CBi + B,A) + {ACi + C^B), = {BA, + BCi) + {A^C + CB,) + {B,A + C,A), = 2BAi + 2AiC + 2BiA, = 2a + 2BiA. Then 5iA = s - a. We then have in the right triangle OBiA, BiA = s — a, OABi = A/2 and 05i = R, where 72 is the radius of the inscribed ELEMENTS OF SPHERICAL TRIGONOMETRY. 101 circle. Applying Napier's rule to this triangle we have A sin {8 — a) = tan R cot—, or A tan R = sin (s — a) tan — , (1) = sin (s — a) - 2' K sin (s — a)' Therefore, tan 72 = Z = ^Mn(s-a)sin(s-b)sin(8-c)^ yjjl \ sin s 104. Escribed Circles. Let us produce the sides AB and AC of the spherical triangle ABC to meet again in A\, The circle inscribed in the triangle AiBC is said to be an escribed circle of ABC. Denote the radius of this circle by R^. Ob- viously there are two other escribed circles of ABC. The radii of these circles will be denoted by Rj^ and R^. Applying equation (1) of the last article to the triangle AiBC J we have tan 72, = tan ^ sin [i(a + 180° -b + 180° - c) - a], 2 K sin (s — a) K sin s sin [180° - \{a + h-\- c)], Hence, sin (s — a)' K sin s tan R^ = sin (s — ay , „ K sin s -.„ tan Rj^ = - — -. TT, IX. sm (s — h) , ^ i^ sin s tan R„ = sin (s — c) TABLE I, LOGARITHMS OF NUMBERS FROM 1 TO 10,000. 103 104 N. L. 100 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 00 000 . 432 ^860 J 284 )703 J 119 531 1 938 <342 1 743 139 532 922 308 690 070 446 819 188 555 918 279 636 991 342 691 037 12 1^27 1^ ^ 057 12 1^^^ ^^ ^ 033 354 13 1^^2 U [988 ^* ^ 301 00 01 01 02 02 03 03 04 04 05 05 06 06 07 07 08 08 09 09 10 17 613 922 229 534 836 1 -^137 435 732 026 319 609 N. L. 043 475 903 326 745 160 572 979 383 782 179 571 961 346 729 108 483 856 225 591 954 314 672 '026 377 726 072 415 755 093 428 760 090 418 743 066 386 704 ^019 333 644 953 259 564 866 167 465 761 056 348 638 218 610 999 385 767 145 521 893 262 628 990 350 707 *061 412 760 106 449 789 126 461 793 123 450 775 098 418 735 *051 364 675 983 290 594 897 197 495 791 085 377 667 130 561 988 410 828 243 653 *060 463 862 258 650 ^038 423 805 183 558 930 298 664 =027 386 743 *096 447 795 140 483 823 160 494 826 156 483 808 130 450 767 *082 395 706 '014 320 625 927 227 524 820 114 406 696 173 604 *030 452 870 284 694 *100 503 902 297 689 W7 461 843 221 595 967 335 700 *063 422 778 *132 482 830 175 517 857 193 528 860 189 516 840 162 481 799 *114 426 737 ^045 351 655 957 256 554 850 143 435 725 217 647 ^072 494 912 325 735 141 543 941 336 727 =115 500 881 258 633 =004 372 737 099 458 814 167 517 864 209 551 890 227 561 893 222 548 872 194 513 830 =145 457 768 =076 381 685 987 286 584 879 173 464 754 260 689 ni5 536 953 366 776 181 583 981 376 766 154 538 918 296 670 *041 408 773 135 493 849 *202 552 899 243 585 924 261 594 926 254 581 905 226 545 862 176 489 799 =106 412 715 =017 316 613 909 202 493 782 303 732 *157 578 995 407 816 *222 623 *021 415 805 192 576 956 333 707 =078 445 809 171 628 959 287 613 937 258 577 893 =208 520 829 137 442 746 =047 346 643 938 231 522 811 346 775 *199 620 *036 449 857 *262 663 * 060 454 844 ^231 614 994 371 744 115 482 846 ^207 565 920 *272 621 968 312 653 992 327 661 992 320 646 969 290 609 925 =239 551 860 168 473 776 *077 376 673 967 260 551 840 389 817 *242 662 *078 490 898 *302 703 *100 493 S83 ^269 652 ^032 408 781 151 518 882 243 600 955 *307 656 *003 346 687 *025 361 694 '024 352 678 =001 322 640 956 =270 582 891 198 503 806 *107 406 702 997 289 580 869 P.P. 44 4-4 8.8 13.2 17.6 22.0 26.4 30.8 35-2 39-6 43 42 17.2 16.8 21.5 2r.o 25.8 25 2 30.1 29.4 34-4133-6 38-737-S 12.3 16.4 20.5 24.6 28.7 32.8 9 '36.9 40 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 39 3-9 7.8 11.7 15-6 19-5 23-4 27-3 36.0I35.1 14.4 18.0 21.6 25-9 25-2 29.6 28.8 34-2 33-3 32-4 33 3-3 6.6 9.9 13.2 16.5 19.8 23.1 26.4 29.7 35 34 3-5 7.0 3-4 6.8 io.,s 10.2 14.0 136 17-5 17.0 21.0 20.4 24-5 28.0 23.« 27.2 31-5 30.6 1 31 30 3-0 6.0 9.0 12.0 15.0 18.0 21.0 24.8 24.0 27.9I27.0 P.P. 105 N. L. 1 2 3 4 5 6 7 8 9 P.P. 150 151 152 153 17 609 638 667 696 725 754 782 *070 355 639 811 *099 384 667 840 *127 412 696 869 *156 441 724 1^ 1898 18/184 469 926 213 498 955 241 526 984 270 554 *013 298 583 *041 327 611 I 2 29 2.9 5.8 28 2.8 5.6 154 155 156 i^}752 ^^ ^ 033 312 780 061 340 808 089 368 837 117 396 865 145 424 893 173 451 921 201 479 949 229 507 977 257 535 *005 285 562 3 4 5 6 8.7 11.6 14-5 17.4 8.4 II. 2 14.0 16.8 157 158 159 160 19 1 ^^^ ^^866 ^" J 140 412 618 893 167 439 645 921 194 673 948 222 700 976 249 520 728 *003 276 548 756 *030 303 575 783 *058 330 602 811 *085 358 629 838 *112 385 656 7 8 9 20.3 23.2 26.1 22.4 25.2 466 493 1 161 162 163 2^^219 710 978 245 737 *005 272 763 *032 299 •790 *059 325 817 *085 352 844 *112 378 871 *139 405 898 *165 431 925 *192 458 I 2 27 2.7 5-4 26 2.6 5-2 164 165 166 484 748 22 Oil 511 775 037 537 801 063 564 827 089 590 854 115 617 880 141 643 906 167 669 932 194 696 958 220 722 985 246 3 4 5 6 8.1 10.8 13.S 16.2 7.8 10.4 13.0 15.6 167 168 169 170 171 172 173 272 22.531 045 300 23.553 24} 805 %55 304 it]"' 285 527 298 557 814 070 324 583 840 096 350 608 866 121 376 629 880 376 634 891 147 401 654 905 401 660 917 427 686 943 453 712 968 479 737 994 505 763 *019 i 9 iS.y 21.6 24.3 20.8 23./ 172 426 679 930 198 452 704 955 223 477 729 980 249 502 754 *005 274 528 779 *030 325 578 830 350 603 855 25 1 2.S 2 5.0 174 175 176 080 329 576 105 353 601 130 378 625 155 403 650 180 428 674 204 452 699 229 477 724 254 502 748 279 527 773 3 7.5 4 1 0.0 5 12.5 6 is.o 177 178 179 822 066 310 551 846 091 334 575 871 115 358 600 895 139 382 624 920 164 406 944 188 431 969 212 455 993 237 479 *018 261 503 717.S 8 20.0 9 22.5 180 648 672 696 720 744 181 182 183 768 26 007 245 792 031 269 816 055 293 840 079 316 864 102 340 888 126 364 912 150 387 935 174 411 959 198 435 983 221 458 I 2 24 2.4 4.8 7.2 9.6 I2.C 14.4 23 2.3 4 6 184 185 186 482 'Hill 416 330 556 505 741 975 529 764 998 553 788 *021 576 811 *045 600 834 *068 623 858 *091 647 881 *114 670 905 *138 694 928 n6i 3 4 5 6 6.9 9.2 ii.S 13.8 187 188 189 190 207 439 669 898 231 462 692 921 254 485 715 944 277 508 738 967 300 531 761 989 323 554 784 *012 346 577 807 *035 370 600 830 *058 393 623 852 *081 I 9 i6.g 19.2 2l.t 16.1 18.4 20.7 191 192 193 126 353 578 149 375 601 171 398 623 194 421 646 217 443 668 240 466 691 262 488 713 285 511 J35 307 533 758 I 22 2.2 21 2.1 194 195 196 780 29 003 226 803 026 248 825 048 270 847 070 292 870 092 314 892 115 336 914 137 358 937 159 380 959 181 403 981 203 425 3 4 5 6 8.8 II.C 8.4 lo.s 12.6 197 198 199 447 2n885 3,1 885 469 688 907 491 710 929 513 732 951 535 754 973 557 776 994 579 798 *016 601 820 *038 623 842 *060 645 863 *081 7 8 9 15.4 n.t 19.8 14.7 16.8 18.9 200 125 146 168 3 190 211 233 255 276 1 298 N. L. 1 2 4 5 6 7 8 9 P.P. 106 ^ N. L. 1 2 3 4 5 211 428 643 856 6 7 8 9 P.P. 200 30 103 125 146 168 190 233 255 276 298 514 728 942 201 202 203 320 535 341 557 771 363 578 792 384 600 814 406 621 835 449 664 878 471 685 899 492 707 920 I 2 22 21 2.2 2.1 4.4 4.2 204 205 206 ^n963 ^^^ 175 387 984 197 408 *006 218 429 *027 239 450 *048 260 471 *069 281 492 *091 302 513 *112 323 534 *133 345 555 *154 366 576 3 4 5 6 6.6 6.3 8.8 8.4 ii.o 10.5 13.2 12.6 207 208 209 210 211 212 213 597 806 32 015 222 618 827 035 243 639 848 056 263 660 869 077 284 681 890 098 305 510 715 919 702 911 118 723 931 139 744 952 160 765 973 181 785 994 201 7 8 9 15.4 14-7 17.6 16.8 19.8 18.9 325 531 736 940 346 366 387 408 II 428 041 244 445 449 654 858 469 675 879 490 695 899 552 756 960 572 777 980 593 797 *001 613 818 *021 I 2 20 2.0 4.0 214 215 216 062 264 465 082 284 486 102 304 506 122 325 526 143 345 546 163 365 566 183 385 586 203 405 606 224 425 626 3 4 6.0 8.0 lO.O 12.0 217 218 219 33 . 646 ^^ ^ 044 242 666 866 064 262 686 885 084 282 706 905 104 301 726 925 124 321 518 713 908 746 945 143 341 537 733 928 766 965 163 786 985 183 806 *005 203 826 *025 223 7 8 9 14.0 16.0 18.0 220 221 222 223 361 557 753 947 380 577 772 967 400 596 792 986 420 616 811 *005 II 439 025 218 411 459 655 850 479 674 869 498 694 889 I 2 19 1.9 3.8 224 225 226 044 238 430 064 257 449 083 276 468 102 295 488 122 315 507 141 334 526 160 353 545 180 372 564 199 392 583 3 4 S 6 5.7 7.6 9-5 11.4 227 228 229 603 35 ) '^^^ ii. ^984 622 813 *003 192 641 832 *021 660 851 *040 679 870 *059 248 436 624 810 698 889 *078 267 455 642 829 717 908 *097 286 474 661 847 736 927 *116 305 493 680 866 755 946 *135 324 511 698 884 774 965 *154 342 530 717 903 7 8 9 13.3 15.2 17.1 230 211 229 II 231 232 233 361 549 2^ 1 ^^^ 380 568 754 399 586 773 418 605 791 I 18 1.8 3 6 234 235 236 ^^922 37 i JQ7 291 940 125 310 959 144 328 977 162 346 996 181 365 *014 199 383 *033 218 401 *051 236 420 *070 254 438 *088 273 457 3 4 1 5.4 7.2 9.0 10.8 237 238 239 . 475 37 1 6^^ 38 } 840 493 676 858 511 694 876 057 530 712 894 075 256 435 614 548 731 912 093 274 453 632 566 749 931 112 292 471 650 585 767 949 130 603 785 967 148 621 803 985 166 346 525 703 639 822 *003 184 364 543 721 7 8 9 12.6 240 241 242 243 ^^^21 202 382 561 039 II 220 399 578 238 417 596 310 489 668 328 507 686 I 17 1.7 244 245 246 ^n9i7 757 934 111 775 952 129 792 970 146 810 987 164 828 *005 182 846 *023 199 863 *041 217 881 *058 235 899 *076 252 3 4 5 6 3-4 tl 8.5 10.2 247 248 249 270 445 620 287 463 637 305 480 655 322 498 672 340 515 690 863 358 533 707 881 375 550 724 898 393 568 742 915 410 585 759 933 428 602 777 950 7 8 9 11.9 13.6 15.3 250 794 811 829 846 N. L. 1 2 3 4 5 6 7 8 9 P.P. 107 ' N. L. 1 2 3 4 5 6 7 8 9 P.P. 250 251 252 1 253 39 794 391967 40/140 312 811 829 846 863 881 898 *071 243 415 915 *088 261 432 933 *106 278 449 950 *123 295 466 985 157 329 *002 175 346 *019 192 364 *037 209 381 *054 226 398 I 2 18 1.8 3.6 5.4 7.2 9.0 10.8 254 1 255 256 483 654 in.^24 500 671 841 518 688 858 535 705 875 552 722 892 569 739 909 586 756 926 603 773 943 620 790 960 637 807 976 3 4 1 257 258 259 in 993 '^^^ 162 330 *010 179 347 *027 196 363 *044 212 380 *061 229 397 *078 246 414 *095 263 430 *111 280 447 *128 296 464 *145 313 481 7 8 9 12.6 Hi 260 497 514 531 547 564 581 597 614 631 647 II 261 262 263 664 Ai 1 830 11} 9^1 160 325 488 681 847 *012 697 863 *029 714 880 *045 731 896 *062 747 913 *078 764 929 *095 780 946 *111 797 963 *127 814 979 *144 I 17 1.7 3.4 5.1 6.8 8.5 10.2 264 265 266 177 341 504 193 357 521 210 374 537 226 390 553 243 406 570 259 423 586 275 439 602 292 455 619 308 472 635 3 4 5 6 267 268 269 270 271 272 273 651 .^.813 297 457 616 667 830 991 152 684 846 *008 169 700 862 *024 185 716 878 *040 201 732 894 *056 217 749 911 *072 233 765 927 *088 249 781 943 no4 265 797 959 *120 281 7 I I 11.9 13.6 15.3 16 1.6 313 473 632 329 489 648 345 505 664 361 521 680 377 537 696 393 553 712 409 569 727 425 584 743 441 600 759 274 275 276 14)933 ^^J 091 791 949 107 807 965 122 823 981 138 838 996 154 854 *012 170 870 *028 185 886 *044 201 902 *059 217 917 *075 232 3 4 5 6 P 8.0 9.6 277 278 279 280 281 282 283 248 404 560 44T ^^^ 264 420 576 731 279 436 592 747 295 451 607 762 311 467 623 778 326 483 638 342 498 654 358 514 669 824 373 529 685 840 389 545 700 855 *010 163 317 7 8 9 11.2 12.8 14.4 793 809 II 1^871 ^5^025 179 886 040 194 902 056 209 917 071 225 932 086 240 948 102 255 963 117 271 979 133 286 994 148 301 I 15 1. 5 284 285 286 332 484 637 347 500 652 362 515 667 378 530 682 393 545 697 408 561 712 423 576 728 439 591 743 454 606 758 469 621 773 3 4 5 6 7.S 9.0 287 288 289 290 291 292 293 ^^939 803 954 105 818 969 120 270 419 568 716 834 984 135 285 434 583 731 849 *000 150 864 *015 165 879 *030 180 330 894 *045 195 345 909 *060 210 359 •509 657 805 924 *075 225 374 523 672 820 7 8 9 lo.s 12.0 13.5 240 255 300 449 598 746 315 464 613 761 II 389 538 687 404 553 702 479 627 776 494 642 790 I 14 1.4 2.8 4.2 5.6 7.0 8.4 294 295 296 ^^^ 129 850 997 144 864 *012 159 879 *026 173 894 *041 188 909 *056 202 923 *070 217 938 *085 232 953 *100 246 967 *114 261 3 4 5 6 297 298 299 276 422 567 290 436 582 305 451 596 319 465 611 334 480 625 349 494 640 363 509 654 378 524 669 392 538 683 407 553 698 7 8 9 9.8 II. 2 12.6 300 712 727 741 756 770 784 799 813 828 842 N. L. 1 2 3 4 5 6 7 8 9 P.P. 108 N. L. 1 2 3 4 5 6 7 8 9 P.P. 300 47 712 727 741 756 770 784 799 813 828 842 301 302 303 857 48 001 144 871 015 159 885 029 173 900 044 187 914 058 202 929 073 216 943 087 230 958 101 244 972 116 259 986 130 273 15 304 305 306 287 430 572 302 444 586 316 458 601 330 473 615 344 487 629 359 501 643 373 515 657 387 530 671 401 544 686 416 558 700 2 3 4 I 5 3.0 ti 307 308 309 310 714 ,g.855 276 415 554 728 869 *010 150 742 883 *024 756 897 *038 770 911 *052 785 926 *066 799 940 *080 813 954 *094 827 968 *108 248 841 982 *122 262 6 7 8 9 9.0 10.5 I2.0 13-5 164 178 192 206 220 234 311 312 313 290 429 568 304 443 582 318 457 596 332 471 610 346 485 624 360 499 638 374 513 651 388 527 665 402 541 679 314 315 316 693 243 379 515 707 845 982 721 859 996 734 872 *010 748 886 *024 762 900 *037 776 914 *051 790 927 *065 803 941 *079 817 955 *092 I 14 1.4 317 318 319 120 256 393 529 133 270 406 542 147 284 420 161 297 433 174 311 447 188 325 461 202 338 474 610 215 352 488 623 229 365 501 637 2 3 4 l 7 2.8 4.2 5.6 7.0 8.4 9.8 320 556 569 583 596 321 322 323 651 188 322 664 799 934 678 813 947 691 826 961 705 840 974 718 853 987 732 866 *001 745 880 *014 759 893 *028 772 907 *041 8 9 II. 2 12.6 324 325 326 068 202 335 081 215 348 095 228 362 108 242 375 121 255 388 135 268 402 148 282 415 162 295 428 175 308 441 327 328 329 455 587 720 468 601 733 481 614 746 495 627 759 508 640 772 521 654 786 917 *048 179 310 534 667 799 930 *061 192 323 548 680 812 943 561 693 825 574 706 838 I 2 3 4 S 6 7 13 li 3.9 5.2 6.5 7.8 9.1 330 51 /^' 52 1 983 244 865 996 127 257 878 *009 140 270 891 *022 153 284 904 *035 166 297 957 970 331 332 333 *075 205 336 *088 218 349 *101 231 362 334 335 336 375 504 634 388 517 647 401 530 660 414 543 673 427 556 686 440 569 699 453 582 711 466 595 724 479 608 737 492 621 750 8 9 10.4 II.7 337 338 339 340 776 905 033 789 917 046 173 301 428 555 802 930 058 815 943 071 827 956 084 212 339 466 593 840 969 097 224 352 479 605 853 982 110 866 994 122 879 *007 135 263 390 517 643 148 161 186 199 237 364 491 618 250 377 504 631 341 342 343 275 403 529 288 415 542 314 441 567 326 453 580 I 2 3 12 1.2 344 345 346 656 53 1 ^82 ^ 033 158 283 668 794 920 681 807 933 694 820 945 706 832 958 719 845 970 732 857 983 744 870 995 757 882 *008 769 895 *020 4 5 6 7 8 9 7.2 10.8 347 348 349 045 170 295 058 183 307 432 070 195 320 444 083 208 332 456 095 220 345 108 233 357 120 245 370 133 258 382 506 145 270 394 518 350 407 419 469 481 494 N. L. 1 2 3 4 5 6 7 8 9 P.P. 109 N. L. 1 2 3 4 5 6 7 8 9 P.P. i 350 54 407 1 419 432 444 456 469 481 494 506 518 351 352 353 531 654 ,,777 543 667 790 555 679 802 568 691 814 580 704 827 593 716 839 605 728 851 617 741 864 630 753 876 642 765 888 13 354 355 356 -t \ 900 ^^ • 023 145 913 035 157 925 047 169 937 060 182 949 072 194 962 084 206 974 096 218 986 108 230 998 121 242 *011 133 255 I 2 3 4 1.3 2.6 3-9 5.2 357 358 359 360 361 362 363 267 388 509 630 279 400 522 642 291 413 534 654 775 895 *015 303 425 546 315 437 558 328 449 570 340 461 582 352 473 594 715 835 955 *074 364 485 606 727 847 967 *086 376 497 618 739 859 979 *098 7 8 6-5 7.8 9.1 10.4 666 678 691 811 931 *050 703 823 943 *062 751 ?A ^991 229 348 763 883 *003 787 907 *027 799 919 *038 364 365 366 122 241 360 134 253 372 146 265 384 158 277 396 170 289 407 182 301 419 194 312 431 205 324 443 217 336 455 12 • 367 368 369 370 371 372 373 467 585 703 57} 937 ^^ ^ 054 171 478 597 714 832 490 608 726 844 502 620 738 855 972 089 206 514 632 750 867 526 644 761 879 538 656 773 549 667 785 561 679 797 914 573 691 808 926 *043 159 276 2 3 4 7 8 9 2.4 4.8 6.0 7.2 8.4 9.6 10.8 891 *008 124 241 902 949 066 183 961 078 194 984 101 217 996 113 229 *019 136 252 *031 148 264 374 375 376 287 403 519 299 415 530 310 426 542 322 438 553 334 449 565 345 461 576 357 473 588 368 484 600 380 496 611 392 507 623 377 378 379 380 381 382 383 634 749 57/^^ 646 761 875 657 772 887 *001 115 229 343 669 784 898 *013 127 240 354 680 795 910 692 807 921 703 818 933 *047 161 274 388 715 830 944 726 841 955 738 852 967 *081 195 309 422 I 2 3 4 S 6 7 11 I.I 2.2 3-3 4.4 U 7.7 -}978 990 *024 138 252 365 *035 149 263 377 *058 172 286 399 *070 184 297 410 ^ 092 206 320 104 218 331 384 385 386 433 546 659 444 557 670 456 569 681 467 580 692 478 591 704 490 602 715 501 614 726 512 625 737 524 636 749 535 647 760 8 9 8.8 9.9 387 388 389 390 391 392 393 771 782 894 *006 794 906 *017 805 917 *028 816 928 *040 827 939 *051 838 950 *062 850 961 *073 861 973 *084 872 984 *095 1 in ^ 106 118 129 140 151 162 173 184 295 406 517 195 207 218 329 439 229 340 450 240 351 461 251 362 472 262 373 483 273 384 494 284 395 506 306 417 528 318 428 539 I 2 3 I.O 2.0 3.0 394 395 396 550 660 770 561 671 780 572 682 791 583 693 802 594 704 813 605 715 824 616 726 835 627 737 846 638 748 857 649 759 868 1 7 8 7.0 8.0 397 398 399 890 999 108 901 *010 119 912 *021 130 923 *032 141 934 *043 152 945 *054 163 956 *065 173 966 *076 184 977 *086 195 9 9.0 [400 206 1 217 228 239 249 260 271 282 293 304 N. L. 1 2 3 4 5 6 7 8 9 P.P. 110 N. L. 1 2 3 4 5 6 7 8 9 P.P. 400 60 206 217 228 239 249 260 271 282 293 304 401 402 403 314 423 531 325 433 541 336 444 552 347 455 563 358 466 574 369 477 584 379 487 595 390 498 606 401 509 617 412 520 627 404 405 406 638 746 r. 853 m}959 61^066 172 649 756 863 660 767 874 670 778 885 681 788 895 692 799 906 703 810 917 713 821 927 724 831 938 735 842 949 I 11 I.I 407 408 409 970 077 183 981 087 194 991 098 204 *002 109 215 *013 119 225 *023 130 236 *034 140 247 *045 151 257 *055 162 268 2 3 4 5 6 7 8 9 2.2 3-3 4-4 5-5 6.6 7-7 410 411 412 413 278 384 490 595 289 300 310 321 331 342 352 458 563 669 363 469 574 679 374 479 584 690 395 500 606 405 511 616 416 521 627 426 532 637 437 542 648 448 553 658 8.8 9.9 414 415 416 700 Ai ^ 805 909 014 118 221 711 815 920 721 826 930 731 836 941 742 847 951 752 857 962 763 868 972 773 878 982 784 888 993 794 899 *003 417 418 419 420 421 422 423 024 128 232 034 138 242 045 149 252 055 159 263 066 170 273 377 076 180 284 086 190 294 397 097 201 304 408 107 211 315 418 325 335 346 356 366 387 428 531 634 439 542 644 449 552 655 459 562 665 469 572 675 480 583 685 490 593 696 500 603 706 511 613 716 521 624 726 I 10 I.O 424 425 426 737 Ao ^ 839 62 1 941 63 J 043 144 246 347 448 548 649 747 849 951 757 859 961 767 870 972 778 880 982 788 890 992 798 900 *002 808 910 *012 818 921 *022 829 931 *033 3 4 3-0 4.0 5.0 6.0 427 428 429 430 431 432 433 053 155 256 357 458 558 659 063 165 266 073 175 276 083 185 286 387 488 589 689 094 195 296 104 205 306 407 508 609 709 114 215 317 417 518 619 719 124 225 327 134 236 337 7 8 9 7.0 8.0 9.0 367 377 397 498 599 699 428 528 629 729 438 538 639 739 468 568 669 478 579 679 434 435 436 749 A2 ^ 849 }949 '048 147 246 345 444 542 640 759 859 959 769 869 969 779 879 979 789 889 988 799 899 998 809 909 *008 819 919 *018 829 929 *028 839 939 *038 437 438 439 058 157 256 068 167 266 078 177 276 088 187 286 385 098 197 296 108 207 306 118 217 316 128 227 326 137 237 335 I 2 3 4 5 6 9 0.9 440 355 365 375 395 404 414 424 434 1.8 441 442 443 454 552 650 464 562 660 473 572 670 483 582 680 493 591 689 503 601 699 513 611 709 523 621 719 532 631 729 3.6 4-5 5.4 6.3 444 445 446 738 64)836 ''^031 128 225 321 748 846 943 758 856 953 768 865 963 777 875 972 787 885 982 797 895 992 807 904 *002 816 914 *011 826 924 *021 8 9 7.2 8.1 447 448 449 450 040 137 234 050 147 244 060 157 254 350 3 070 167 263 079 176 273 089 186 283 099 196 292 108 205 302 398 118 215 312 408 331 341 360 369 379 389 N. L. 1 2 4 5 6 7 8 9 P.P. Ill N. L. 1 2 3 4 5 6 7 8 9 P.P. 450 65 321 331 341 437 533 629 350 360 369 466 562 658 379 475 571 667 389 398 408 504 600 696 451 452 453 418 514 610 427 523 619 447 543 639 456 552 648 485 581 677 495 591 686 454 455 456 706 801 715 811 906 725 820 916 734 830 925 744 839 935 753 849 944 763 858 954 772 868 963 782 877 973 792 887 982 I 10 I.O 457 458 459 181 276 *001 096 191 ♦Oil 106 200 *020 115 210 *030 124 219 *039 134 229 323 *049 143 238 332 *058 153 247 342 *068 162 257 *077 172 266 3 4 1 7 8 9 3.0 4.0 il 7.0 8.0 9.0 460 285 395 304 314 351 361 461 462 463 370 464 558 380 474 567 389 483 577 398 492 586 408 502 596 417 511 605 427 521 614 436 530 624 445 539 633 455 549 642 464 465 466 652 745 ^7 1^32 ^^^025 117 210 302 394 486 661 755 848 671 764 857 680 773 867 689 783 876 699 792 885 708 801 894 717 811 904 727 820 913 736 829 922 • 467 468 469 941 034 127 950 043 136 960 052 145 969 062 154 978 071 164 987 080 173 997 089 182 *006 099 191 *015 108 201 470 219 228 237 247 256 265 274 367 459 550 284 293 471 472 473 311 403 495 321 413 504 330 422 514 339 431 523 348 440 532 357 449 541 376 468 560 385 477 569 I 9 0.9 1.8 474 475 476 578 669 761 587 679 770 596 688 779 605 697 788 614 706 797 624 715 806 633 724 815 642 733 825 651 742 834 660 752 843 3 4 1 4-S 5.4 477 478 479 A7 ■» 852 681^43 ^^ ^ 034 861 952 043 870 961 052 879 970 061 888 979 070 897 988 079 906 997 088 916 *006 097 925 *015 106 196 287 377 467 934 *024 115 205 296 386 476 7 8 9 6.3 7.2 8.1 480 124 215 305 395 133 142 151 160 169 260 350 440 178 187 481 482 483 224 314 404 233 323 413 242 332 422 251 341 431 269 359 449 278 368 458 484 485 486 485 574 664 494 533 673 502 592 681 511 601 690 520 610 699 529 619 708 538 628 717 547 637 726 556 646 735 565 655 744 487 488 489 490 753 ^Q 1 842 «}931 020 108 197 285 762 851 940 771 860 949 780 869 958 789 878 966 797 886 975 806 895 984 815 904 993 824 913 *002 833 922 *011 I 2 3 4 5 6 8 0.8 1.6 2.4 3.2 4.0 4.8 028 037 046 055 064 073 082 090 099 188 276 364 491 492 493 117 205 294 126 214 302 135 223 311 . 144 232 320 152 241 329 161 249 338 170 258 346 179 267 355 494 495 496 373 461 548 381 469 557 390 478 566 399 487 574 408 496 583 417 504 592 425 513 601 434 522 609 443 531 618 452 539 627 7 8 9 5.6 6.4 7.2 497 498 499 636 723 810 644 732 819 653 740 827 662 749 836 671 758 845 679 767 854 688 775 862 697 784 871 705 793 880 714 801 888 500 897 906 914 923 932 940 949 958 966 975 N. L. 1 2 3 4 5 6 7 8 9 P.P. 112 N. L. 1 2 3 4 5 6 7 8 9 P.P. 500 69 897 69)984 70/070 157 906 914 923 932 940 949 958 966 975 *062 148 234 501 502 503 992 079 165 *001 088 174 *010 096 183 *018 105 191 *027 114 200 *036 122 209 *044 131 217 *053 140 226 504 505 506 243 329 415 252 338 424 260 346 432 269 355 441 278 364 449 286 372 458 295 381 467 303 389 475 312 398 484 321 406 492 I 2 9 0.9 1.8 507 508 509 510 501 586 672 509 595 680 518 603 689 526 612 697 535 621 706 544 629 714 552 638 723 561 646 731 817 902 986 071 569 655 740 825 910 995 079 578 663 749 834 919 *003 088 3 4 5 6 7 8 9 4-5 5-4 6.3 7.2 8.1 757 766 774 783 791 800 808 511 512 513 70-. 842 ^^^012 851 935 020 859 944 029 868 952 037 876 961 046 885 969 054 893 978 063 514 515 516 096 181 265 105 189 273 113 198 282 122 206 290 130 214 299 139 223 307 147 231 315 155 240 324 164 248 332 172 257 341 517 518 519 349 433 517 357 441 525 366 450 533 374 458 542 383 466 550 391 475 559 399 483 567 408 492 575 416 500 584 425 508 592 520 600 609 617 625 634 642 650 659 667 675 521 522 523 684 767 7n'^' 72} 933 ^2^016 099 692 775 858 700 784 867 709 792 875 717 800 725 809 892 734 817 900 742 825 908 750 834 917 759 842 925 I 2 8 0.8 1.6 524 525 526 941 024 107 950 032 115 958 041 123 966 049 132 975 057 140 983 066 148 991 074 156 999 082 165 *008 090 173 3 4 5 6 2.4 3.2 527 528 529 181 263 346 189 272 354 198 280 362 206 288 370 214 296 378 222 304 387 230 313 395 239 321 403 247 329 411 255 337 419 7 8 9 15 7-2 530 428 436 444 452 460 469 477 485 493 501 531 532 533 509 591 673 518 599 681 526 607 689 534 616 697 542 624 705 550 632 713 558 640 722 567 648 730 575 656 738 583 665 746 534 535 536 754 835 ^^^078 159 762 843 925 770 852 933 779 860 941 787 868 949 795 876 957 803 884 965 811 892 973 819 900 981 827 908 9.89 537 538 539 *006 086 167 *014 094 175 *022 102 183 *030 111 191 *038 119 199 *046 127 207 *054 135 215 *062 143 223 *070 151 231 2 3 4 S 6 7 0.7 1.4 2.1 2.8 3-5 4.2 540 239 247 255 263 272 280 288 296 304 312 541 542 543 320 400 480 328 408 488 336 416 496 344 424 504 352 432 512 360 440 520 368 448 528 376 456 536 384 464 544 392 472 552 544 545 546 560 640 719 568 648 727 576 656 735 584 664 743 592 672 751 600 679 759 608 687 767 616 695 775 624 703 783 632 711 791 7 8 9 4.9 547 548 549 550 799 73.878 74)957 036 807 886 965 044 815 894 973 052 823 902 981 060 830 910 989 068 838 918 997 846 926 *005 854 933 *013 092 862 941 *020 099 870 949 *028 107 076 084 N. L. 1 2 3 4 5 6 7 8 9 P.P. 113 N. L. 1 2 1 3 4 5 6 7 8 9 P.P. 550 74 036 044 052 i 060 068 076 084 092 099 107 551 552 553 115 194 273 123 202 280 131 210 288 139 218 296 147 225 304 155 233 312 162 241 320 170 249 327 178 257 335 186 265 343 554 555 556 351 429 507 359 437 515 367 445 523 374 453 531 382 461 539 390 468 547 398 476 554 406 484 562 414 492 570 421 500 578 557 558 559 586 663 741 593 671 749 601 679 757 834 609 687 764 617 695 772 624 702 780 858 935 *012 089 632 710 788 865 943 *020 097 640 718 796 873 648 726 803 881 958 *035 113 656 733 811 889 966 *043 120 560 819 827 842 920 997 074 850 927 *005 082 I 2 3 4 8 0.8 1.6 2.4 3.2 561 562 563 -,.896 75} ^^4 '^^051 904 981 059 912 989 066 950 *028 105 564 565 566 128 205 282 136 213 289 143 220 297 151 228 305 159 236 312 166 243 320 174 251 328 182 259 335 189 266 343 197 274 351 6 7 8 9 '4 5.6 6.4 7.2 567 568 569 358 435 511 587 366 442 519 374 450 526 381 458 534 389 465 542 397 473 549 404 481 557 412 488 565 420 496 572 648 427 504 580 656 570 595 603 610 618 626 633 641 571 572 573 664 740 815 671 747 823 679 755 831 686 762 838 694 770 846 702 778 853 709 785 861 717 793 868 724 800 876 732 808 884 574 575 576 75 1 ^^1 7^1 967 ^^^042 899 974 050 906 982 057 914 989 065 921 997 072 929 *005 080 937 *012 087 944 *020 095 952 *027 103 959 *035 110 577 578 579 118 193 268 125 200 275 133 208 283 140 215 290 148 223 298 155 230 305 163 238 313 170 245 320 178 253 328 185 260 335 580 343 350 358 365 373 380 388 395 403 410 581 582 583 418 492 567 425 500 574 433 507 582 440 515 589 448 522 597 455 530 604 462 537 612 470 545 619 477 552 626 485 559 634 584 585 586 641 716 790 649 723 797 656 730 805 664 738 812 671 745 819 678 753 827 686 760 834 693 768 842 701 775 849 708 782 856 I 2 3 7 0.7 1.4 2.1 587 588 589 7^ ^ 864 77} 938 ^^^012 871 945 019 879 953 026 886 960 034 893 967 041 901 975 048 908 982 056 916 989 063 923 997 070 930 *004 078 4 1 7 8 9 2.8 3.5 4.2 4.9 590 085 093 100 107 115 122 129 137 144 151 591 592 593 159 232 305 166 240 313 173 247 320 181 254 327 188 262 335 195 269 342 203 276 349 210 283 357 217 291 364 225 298 371 594 595 596 379 452 525 386 459 532 393 466 539 401 474 546 408 481 554 415 488 561 422 495 568 430 503 576 437 510 583 444 517 590 597 598 599 597 670 743 605 677 750 612 685 757 619 692 764 627 699 772 634 706 779 641 714 786 648 721 793 656 728 801 663 735 808 600 815 822 830 837 844 851 859 866 873 880 N. L. 1 2 3 4 5 6 7 8 9 P.P. 114 In. L. 1 2 3 4 5 6 7 8 9 P.P. 600 77 815 822 830 837 844 916 988 061 851 924 996 068 859 931 *003 075 866 938 *010 082 873 945 *017 089 880 952 *025 097 601 602 603 77 . 887 ^U960 '»J032 895 967 039 902 974 046 909 981 053 604 605 606 104 176 247 111 183 254 118 190 262 125 197 269 132 204 276 140 211 283 147 219 290 154 226 297 161 233 305 168 240 312 J 8 8 607 608 609 319 390 462 533 326 398 469 540 333 405 476 340 412 483 347 419 490 355 426 497 362 433 504 369 440 512 376 447 519 383 455 526 2 3 4 S 6 7 8 9 1.6 2.4 3-2 4.0 4.8 7.2 1 1 610 611 612 613 547 554 561 569 576 583 590 597 604 675 746 611 682 753 618 689 760 625 696 767 633 704 774 640 711 781 647 718 789 654 725 796 661 732 803 668 739 810 614 615 616 817 78 -1 88^ ^}958 ^ 029 099 169 824 895 965 831 902 972 S38 909 979 845 916 986 852 923 993 859 930 *000 866 937 *007 873 944 *014 880 951 *021 617 618 619 036 106 176 043 113 183 050 120 190 057 127 197 267 337 407 477 064 134 204 071 141 211 078 148 218 085 155 225 092 162 232 302 372 442 511 620 239 309 379 449 246 253 260 274 344 414 484 281 351 421 491 288 295 621 622 623 316 386 456 323 393 463 330 400 470 358 428 498 365 435 505 I 2 7 0.7 l.A 624 625 626 518 588 657 525 595 664 532 602 671 539 609 678 546 616 685 553 623 692 560 630 699 567 637 706 574 644 713 581 650 720 3 4 1 2.1 2.8 3-5 4.2 627 628 629 727 796 865 734 803 872 741 810 879 748 817 886 754 824 893 761 831 900 768 837 906 775 844 913 982 782 851 920 989 789 858 927 996 065 134 202 7 8 9 4.9 630 934 941 948 955 962 969 975 • 631 632 633 80 003 072 140 010 079 147 017 085 154 024 092 161 030 099 168 037 106 175 044 113 182 051 120 188 058 127 195 634 635 636 209 277 346 216 284 353 223 291 359 229 298 366 236 305 373 243 312 380 250 318 387 257 325 393 264 332 400 271 339 407 637 638 639 640 641 642 643 414 482 550 421 489 557 428 496 564 434 502 570 441 509 577 448 516 584 455 523 591 462 530 598 468 536 604 475 543 611 I 2 3 4 5 6 7 6 0.6 1.2 1.8 2.4 3.0 3.6 4.2 618 625 632 638 645 652 659 665 672 679 686 754 821 693 760 828 699 767 835 706 774 841 713 781 848 720 787 855 726 794 862 733 801 868 740 808 875 747 814 882 644 645 646 so 1 889 ^1 J 023 895 963 030 902 969 037 909 976 043 916 983 050 922 990 057 929 996 064 936 *003 070 943 *010 077 949 *017 084 8 9 4.8 5-4 647 648 649 090 158 224 097 164 231 104 171 238 111 178 245 311 117 184 251 124 191 258 131 198 265 137 204 271 144 211 278 151 218 285 351 650 291 298 305 318 325 331 338 345 N. L. 1 2 3 4 5 6 7 8 9 P.P. 115 N. L. 1 2 3 4 5 6 7 8 9 P.P. 650 81 291 298 305 311 318 325 331 338 345 351 651 652 653 358 425 491 365 431 498 371 438 505 378 445 511 385 451 518 391 458 525 398 465 531 405 471 538 411 478 544 418 485 551 654 655 656 558 624 690 564 631 697 571 637 704 578 644 710 584 651 717 591 657 723 598 664 730 604 671 737 611 677 743 617 684 750 657 658 659 660 757 823 763 829 895 770 836 902 776 842 908 783 849 915 790 856 921 796 862 928 803 869 935 809 875 941 816 882 948 I 2 3 4 S 6 1.4 2.1 2.8 3-5 4.2 8 }95^ 961 968 974 981 987 994 *000 *007 *014 661 662 663 ^^^020 086 151 027 092 158 033 099 164 040 105 171 046 112 178 053 119 184 060 125 191 066 132 197 073 138 204 079 145 210 664 665 666 217 282 347 223 289 354 230 295 360 236 302 367 243 308 373 249 315 380 256 321 387 263 328 393 269 334 400 276 341 406 7 8 9 4-9 5.6 6.3 667 668 669 1 670 671 672 673 413 478 543 419 484 549 426 491 556 432 497 562 439 504 569 445 510 575 452 517 582 458 523 588 465 530 595 471 536 601 607 614 620 627 633 640 646 653 659 666 672 737 802 679 743 808 685 750 814 692 756 821 698 763 827 705 769 834 711 776 840 718 782 847 724 789 853 730 795 860 674 675 676 866 82 \ 930 83/995 872 937 *001 879 943 *008 885 950 *014 892 956 *020 898 963 *027 905 969 *033 911 975 *040 918 982 *046 924 988 *052 677 678 679 059 123 187 065 129 193 072 136 200 078 142 206 085 149 213 091 155 219 097 161 225 104 168 232 110 174 238 117 181 245 680 251 257 264 270 276 283 289 296 302 308 681 682 683 315 378 442 321 385 448 327 391 455 334 398 461 340 404 467 347 410 474 353 417 480 359 423 487 366 429 493 372 436 499 684 685 686 506 569 632 512 575 639 518 582 645 525 588 651 531 594 658 537 601 664 544 607 670 550 613 677 556 620 683 563 626 689 I 6 0.6 687 688 689 696 759 822 702 765 828 708 771 835 715 778 841 721 784 847 727 790 853 734 797 860 740 803 866 746 809 872 753 816 879 3 4 5 6 7 8 9 1.8 2.4 ti 5-4 690 33.885 84/948 'on 073 891 954 017 080 897 904 910 916 923 929 992 055 117 935 942 691 692 693 960 023 086 967 029 092 973 036 098 979 042 105 985 048 111 998 061 123 *004 067 130 694 695 696 136 198 261 142 205 267 148 211 273 155 217 280 161 223 286 167 230 292 173 236 298 180 242 305 186 248 311 192 255 317 697 698 699 700 323 386 448 330 392 454 336 398 460 522 342 404 466 528 348 410 473 354 417 479 361 423 485 547 367 429 491 373 435 497 379 442 504 566 510 516 535 541 553 559 N. L. 1 2 3 4 5 6 7 8 9 P.P. 116 N. L. 1 2 3 4 5 6 7 8 9 P.P. 700 84 510 516 522 528 535 541 547 553 559 566 701 572 578 584 590 597 603 609 615 621 628 702 634 640 646 652 658 665 671 677 683 689 703 696 702 708 714 720 726 733 739 745 751 704 757 763 770 776 782 788 794 800 807 813 705 819 825 831 837 844 850 856 862 868 874 7 0.7 1.4 706 880 887 893 899 905 911 917 924 930 936 707 942 948 954 960 967 973 979 985 991 997 2 708 85 003 009 016 022 028 034 040 046 052 058 3 2.1 709 710 065 126 187 071 132 077 083 089 150 095 101 107 169 114 175 120 181 5 6 7 8 3-5 4.2 4-9 5.6 138 144 156 217 163 224 711 193 199 205 211 230 236 242 712 248 254 260 266 272 278 285 291 297 303 9 6.3 713 309 315 321 327 333 339 345 352 358 364 714 370 376 382 388 394 400 406 412 418 425 715 431 437 443 449 455 461 467 473 479 485 716 491 497 503 509 516 522 528 534 540 546 717 552 558 564 570 576 582 588 594 600 606 718 612 618 625 631 637 643 649 655 661 667 719 673 733 679 739 685 691 697 757 703 709 715 775 721 781 727 788 720 745 751 763 769 721 794 800 806 812 818 824 830 836 842 848 722 854 860 866 872 878 884 890 896 902 908 t> 723 86/(334 920 926 932 938 944 950 956 962 968 I 2 0.6 1.2 724 980 986 992 998 *004 *010 *016 *022 *028 3 1.8 725 040 046 052 058 064 070 076 082 088 4 5 6 2.4 3.0 3.6 726 094 100 106 112 118 124 130 136 141 147 727 153 159 165 171 177 183 189 195 201 207 7 g 4.2 4.8 S.4 728 213 219 225 231 237 243 249 255 261 267 9 729 273 332 279 338 285 344 291 350 297 356 303 308 314 374 320 326 730 362 368 380 386 731 392 398 404 410 415 421 427 433 439 445 732 451 457 463 469 475 481 487 493 499 504 733 510 516 522 528 534 540 546 552 558 564 734 570 576 581 587 593 599 605 611 617 623 735 629 635 641 646 652 658 664 670 676 682 736 688 694- 700 705 711 717 723 729 735 741 737 747 753 759 764 770 776 782 788 794 800 738 806 812 817 823 829 835 841 847 853 859 5 739 864 870 876 882 888 894 900 906 911 917 I 2 3 4 o.S I.O 1.5 2.0 ' 740 923 ^^ J 040 929 935 941 947 953 958 964 970 976 741 988 994 999 *005 *011 *017 *023 *029 *035 742 046 052 058 064 070 075 081 087 093 1 2.5 3-0 3-5 743 099 105 111 116 122 128 134 140 146 151 7 744 157 163 169 175 181 186 192 198 204 210 8 4.0 745 216 221 227 233 239 245 251 256 262 268 746 274 280 286 291 297 303 309 315 ,320 326 747 332 338 344 349 355 361 367 373 379 384 748 390 396 402 408 413 419 425 431 437 442 749 448 454 460 466 471 529 477 483 489 495 500 558 750 506 512 518 523 535 541 547 552 N. L. 1 2 3 4 5 6 7 8 9 P.P. 117 N. L. 1 2 3 4 5 6 7 8 1 9 P.P. 750 87 506 564 622 679 512 570 628 685 518 576 633 691 523 581 639 697 529 587 645 703 535 593 651 708 541 599 656 714 547 604 662 720 552 610 668 726 558 616 674 731 751 752 753 754 755 756 737 795 852 743 800 858 749 806 864 754 812 869 760 818 875 766 823 881 772 829 887 777 835 892 783 841 898 789 846 904 757 758 759 88 1 967 88 i Q24 081 915 973 030 087 921 978 036 093 927 984 041 098 933 990 047 938 996 053 944 *001 058 116 950 *007 064 121 178 235 292 955 *013 070 127 961 *018 076 133 I 2 3 4 1 6 0.6 1.2 1.8 2.4 3.0 3.6 760 761 762 763 104 110 138 195 252 144 201 258 150 207 264 156 213 270 161 218 275 167 224 281 173 230 287 184 241 298 190 247 304 764 765 766 309 366 423 315 372 429 321 377 434 326 383 440 332 389 446 338 395 451 343 400 457 349 406 463 355 412 468 360 417 474 7 8 9 4.2 4.8 5.4 767 768 769 480 536 593 485 542 598 491 547 604 497 553 610 502 559 615 508 564 621 513 570 627 519 576 632 525 581 638 530 587 643 770 649 655 660 666 672 677 734 790 846 683 739 795 852 689 694 700 756 812 868 771 772 773 705 762 818 711 767 824 717 773 829 722 779 835 728 784 840 745 801 857 750 807 863 774 775 776 874 II }S ^ 042 098 154 209 880 936 992 885 941 997 891 947 *003 897 953 *009 902 958 *014 908 964 *020 913 969 *025 919 975 *031 925 981 *037 777 778 779 048 104 159 053 109 165 059 115 170 064 120 176 070 126 182 076 131 187 081 137 193 087 143 198 092 148 204 780 215 221 226 232 287 343 398 237 293 348 404 243 298 354 409 248 304 360 415 254 260 781 782 783 265 321 376 271 326 382 276 332 387 282 337 393 310 365 421 315 371 426 784 785 786 432 487 542 437 492 548 443 498 553 448 504 559 454 509 564 459 515 570 465 520 575 476 526 581 476 531 586 481 537 592 I 2 5 0.5 I.O 787 788 789 597 653 708 603 658 713 609 664 719 614 669 724 620 675 730 625 680 735 631 686 741 636 691 746 642 697 752 647 702 757 3 4 5 6 7 8 9 1. 5 2.0 2.5 3.0 3-5 4.0 4-5 790 763 768 774 779 785 790 796 801 807 812 791 792 793 818 873 091 823 878 933 829 883 938 834 889 944 840 894 949 845 900 955 851 905 960 856 911 966 862 916 971 867 922 977 794 795 796 988 042 097 993 048 102 998 053 108 *004 059 113 *009 064 119 *015 069 124 *020 075 129 *026 080 135 *031 086 140 1 797 798 799 146 200 255 151 206 260 157 211 266 162 217 271 168 222 276 173 227 282 179 233 287 184 238 293 189 244 298 195 249 304 358 800 309 314 320 325 331 336 342 347 352 N. L. 1 2 3 4 5 6 7 8 9 P.P. 118 N. L. 1 2 3 4 5 6 7 8 9 P.P. 800 90 309 314 320 325 331 336 342 347 352 358 412 466 520 801 802 803 363 417 472 369 423 477 374 428 482 380 434 488 385 439 493 390 445 499 396 450 504 401 455 509 407 461 515 804 805 806 526 580 634 531 585 639 536 590 644 542 596 650 547 601 655 553 607 660 558 612 666 563 617 671 569 623 677 574 628 682 807 808 809 810 687 741 795 693 747 800 698 752 806 703 757 811 709 763 816 714 768 822 720 773 827 725 779 832 730 784 838 891 736 789 843 897 950 *004 057 849 QQ.902 9?} 956 9^ ^ 009 854 907 961 014 859 865 870 875 881 934 988 041 886 I 2 3 4 5 to 0.6 1.2 1.8 2.4 3.0 811 812 813 913 966 020 918 972 025 924 977 030 929 982 036 940 993 046 945 998 052 814 815 816 062 116 169 068 121 174 073 126 180 078 132 185 084 137 190 089 142 196 094 148 201 100 153 206 105 158 212 110 164 217 6 7 8 9 3.6 4.2 4.8 S.4 817 818 819 222 275 328 228 281 334 233 286 339 238 291 344 243 297 350 249 302 355 408 461 514 566 254 307 360 259 312 365 418 265 318 371 424 477 529 582 270 323 376 429 482 535 587 820 381 434 487 540 387 392 397 403 413 821 822 823 440 492 545 445 498 551 450 503 556 455 508 561 466 519 572 471 524 577 824 825 826 593 645 698 598 651 703 603 656 709 609 661 714 614 666 719 619 672 724 624 677 730 630 682 735 635 687 740 640 693 745 827 828 829 751 803 855 91)^^^ 92 ^ 012 065 756 808 861 913 965 018 070 761 814 866 918 971 023 075 766 819 871 772 824 876 777 829 882 782 834 887 939 787 840 892 944 793 845 897 950 *002 054 106 798 850 903 955 *007 059 111 830 924 976 028 080 929 981 033 085 934 986 038 091 831 832 833 991 044 096 997 049 101 834 835 836 117 169 221 122 174 226 127 179 231 132 184 236 137 189 241 143 195 247 148 200 252 153 205 257 158 210 262 163 215 267 I 2 3 5 o.S I.O 1.5 837 838 839 273 324 376 278 330 381 283 335 387 288 340 392 293 345 397 298 350 402 304 355 407 309 361 412 314 366 418 319 371 423 4 S 6 7 8 9 2.0 2.5 3.0 3.5 4.0 4.5 840 428 480 531 583 433 438 443 495 547 598 449 500 552 603 454 459 464 516 567 619 469 521 572 624 474 526 578 629 841 842 843 485 536 588 490 542 593 505 557 609 511 562 614 844 845 846 634 686 737 639 691 742 645 696 747 650 701 752 655 706 758 660 711 763 665 716 768 670 722 773 675 727 778 681 732 783 847 848 849 788 840 891 793 845 896 799 850 901 804 855 906 809 860 911 814 865 916 819 870 921 824 875 927 829 881 932 834 886 937 850 942 947 952 957 962 967 973 978 983 988 N. L. 1 2 3 4 5 6 7 8 9 P.P. 119 N. L. 1 2 3 ! 4 5 6 7 8 9 P.P. 850 92 942 921993 93/044 095 947 952 957 1 962 967 *018 069 120 973 *024 075 125 978 983 988 *039 090 141 851 852 853 998 049 100 *003 054 105 *008 059 110 *013 064 115 *029 080 131 *034 085 136 854 855 856 146 197 247 151 202 252 156 207 258 161 212 263 166 217 268 171 222 273 176 227 278 181 232 283 186 237 288 192 242 293 I 2 6 0.6 I 2 857 858 859 298 349 399 450 303 354 404 455 308 359 409 313 364 414 318 369 420 323 374 425 328 379 430 334 384 435 339 389 440 490 541 591 641 344 394 445 495 546 596 646 3 4 1 7 8 9 1.8 2.4 3.6 4.2 4.8 5-4 860 460 510 561 611 465 515 566 616 470 475 526 576 626 480 485 536 586 636 861 862 863 500 551 601 505 556 606 520 571 621 531 581 631 864 865 866 651 702 752 656 707 757 661 712 762 666 717 767 671 722 772 676 727 777 682 732 782 687 737 787 692 742 792 697 747 797 867 868 869 802 852 902 952 807 857 907 812 862 912 817 867 917 822 872 922 827 877 927 832 882 932 982 837 887 937 842 892 942 847 897 947 870 957 962 967 972 977 987 992 997 871 872 873 94 002 052 101 007 057 106 012 062 111 017 067 116 022 072 121 027 077 126 032 082 131 037 086 136 042 091 141 047 096 146 I 2 5 0.5 I.O 874 875 876 151 201 250 156 206 255 161 211 260 166 216 265 171 221 270 176 226 275 181 231 280 186 236 285 191 240 290 196 245 295 3 4 5 6 7 8 9 i.S 2.0 2.5 3.0 35 4.0 4.5 1 877 878 879 880 300 349 399 305 354 404 310 359 409 458 315 364 414 320 369 419 325 374 424 473 330 379 429 335 384 433 483 340 389 438 488 537 586 635 345 394 443 493 542 591 640 448 453 463 512 562 611 468 478 881 882 883 498 547 596 503 552 601 507 557 606 517 567 616 522 571 621 527 576 626 532 581 630 884 1 885 i 886 645 694 743 650 699 748 655 704 753 660 709 758 665 714 763 670 719 768 675 724 773 680 729 778 685 734 783 689 738 787 1 887 888 889 792 841 890 797 846 895 802 851 900 807 856 905 812 861 910 817 866 915 822 871 919 827 876 924 832 880 929 836 885 934 I 2 3 4 1 4 li 1.2 1.6 2.0 2.4 890 939 ^|}988 ^^ ^ 036 085 944 949 954 959 963 968 973 978 983 *032 080 129 891 892 893 993 041 090 998 046 095 *002 051 100 *007 056 105 *012 061 109 *017 066 114 *022 071 119 *027 075 124 894 895 896 134 182 231 139 187 236 143 192 240 148 197 245 153 202 250 158 207 255 163 211 260 168 216 265 173 221 270 177 226 274 7 8 9 2.8 3.2 3.6 897 898 899 279 328 376 284 332 381 289 337 386 294 342 390 439 299 347 395 444 303 352 400 448 308 357 405 313 361 410 318 366 415 463 323 371 419 468 900 424 429 434 453 458 N. L. 1 2 3 4 5 6 7 8 9 P.P. 120 ^H N. L. 1 2 3 4 5 6 7 8 9 P.P. 900 901 902 903 95 424 429 434 439 444 448 453 458 463 468 472 521 569 477 525 574 482 530 578 487 535 583 492 540 588 497 545 593 501 550 598 506 554 602 511 559 607 516 564 612 904 905 906 617 665 713 622 670 718 626 674 722 631 679 727 636 684 732 641 689 737 646 694 742 650 698 746 655 703 751 660 708 756 907 908 909 910 911 912 913 761 809 856 904 951^^2 766 813 861 909 770 818 866 775 823 871 780 828 875 923 785 832 880 928 976 *023 071 789 837 885 794 842 890 799 847 895 804 852 899 914 918 933 938 942 947 I 2 3 4 5 0.5 I.O i-S 2.0 957 *004 052 961 *009 057 966 *014 061 971 *019 066 980 *028 076 985 *033 080 990 *038 085 995 *042 090 914 915 916 095 142 190 099 147 194 104 152 199 109 156 204 114 161 209 118 166 213 123 171 218 128 175 223 133 180 227 137 185 232 S 6 7 8 o 2.5 3.0 3.5 4.0 ,1 c 917 918 919 920 921 922 923 237 284 332 379 426 473 520 242 289 336 246 294 341 251 298 346 256 303 350 261 308 355 265 313 360 270 317 365 275 322 369 280 327 374 384 388 393 398 445 492 539 402 407 412 417 464 511 558 421 468 515 562 431 478 525 435 483 530 440 487 534 450 497 544 454 501 548 459 506 553 924 925 926 567 614 661 572 619 666 577 624 670 581 628 675 586 633 680 591 638 685 595 642 689 600 647 694 605 652 699 609 656 703 927 928 929 930 931 932 933 934 935 936 708 755 802 713 759 806 717 764 811 722 769 816 727 774 820 731 778 825 736 783 830 741 788 834 745 792 839 750 797 844 848 853 858 862 867 872 876 881 886 890 895 96X988 97/988 ^ 035 081 128 900 946 993 039 086 132 904 951 997 044 090 137 909 956 *002 049 095 142 914 960 *007 053 100 146 918 965 *011 058 104 151 923 970 *016 063 109 155 928 974 *021 067 114 160 932 979 *025 072 118 165 937 984 *030 077 123 169 I 2 3 1.2 T A 937 938 939 174 220 267 313 179 225 271 183 230 276 188 234 280 192 239 285 197 243 290 202 248 294 206 253 299 211 257 304 216 262 308 1 7 8 9 2.0 li 940 317 322 327 331 336 340 345 391 437 483 350 354 941 942 943 359 405 451 364 410 456 368 414 460 373 419 465 377 424 470 382 428 474 387 433 479 396 442 488 400 447 493 944 945 946 497 543 589 502 548 594 506 552 598 511 557 603 516 562 607 520 566 612 525 571 617 529 575 621 534 580 626 539 585 630 947 948 949 635 681 727 640 685 731 644 690 736 649 695 740 653 699 745 658 704 749 663 708 754 667 713 759 672 717 763 676 722 768 950 772 777 782 786 791 795 800 804 809 813 N. L. 1 2 3 4 5 6 7 8 9 P.P. 121 N. L. 1 2 3 4 5 6 7 8 9 P.P. ; 950 97 772 777 782 786 791 795 800 804 809 813 951 952 953 818 864 909 823 868 914 827 873 918 832 877 923 836 882 928 841 886 932 845 891 937 850 896 941 855 900 946 859 905 950 954 955 956 955 98 000 046 959 005 050 964 009 055 968 014 059 973 019 064 978 023 068 982 028 073 987 032 078 991 037 082 996 041 087 957 958 959 091 137 182 096 141 186 100 146 191 105 150 195 109 155 200 114 159 204 118 164 209 123 168 214 259 127 173 218 263 132 177 223 268 313 358 403 I 2 3 4 5 6 5 0.5 I.O i-S 2.0 2.5 3.0 960 227 1 232 236 241 245 290 336 381 250 295 340 385 254 299 345 390 961 962 963 272 318 363 277 322 367 281 327 372 286 331 376 304 349 394 308 354 399 964 965 966 408 453 498 412 457 502 417 462 507 421 466 511 426 471 516 430 475 520 435 480 525 439 484 529 444 489 534 448 493 538 7 8 9 3.5 4.0 4-5 967 968 969 543 588 632 547 592 637 552 597 641 556 601 646 561 605 650 565 610 655 570 614 659 574 619 664 579 623 668 583 628 673 970 677 682 686 691 695 700 704 749 793 838 709 753 798 843 713 717 971 972 973 722 767 811 726 771 816 731 776 820 735 780 825 740 784 829 744 789 834 758 802 847 762 807 851 974 975 976 856 900 .. 945 860 905 949 865 909 954 869 914 958 874 918 963 878 923 967 883 927 972 887 932 976 892 936 981 896 941 985 977 978 979 '^ ^ 034 078 994 038 083 998 043 087 *003 047 092 *007 052 096 *012 056 100 *016 061 105 149 193 238 282 *021 065 109 154 198 242 286 *025 069 114 158 202 247 291 *029 074 118 162 207 251 295 980 123 127 131 136 140 145 981 982 983 167 211 255 171 216 260 176 220 264 180 224 269 185 229 273 189 233 277 984 985 986 300 344 388 304 348 392 308 352 396 313 357 401 317 361 405 322 366 410 326 370 414 330 374 419 335 379 423 339 3S3 427 I 2 4 987 988 989 432 476 520 436 480 524 441 484 528 572 445 489 533 449 493 537 454 498 542 585 458 502 546 590 463 506 550 594 467 511 555 471 515 559 3 4 1 7 8 9 1.2 1.6 2.0 2.4 2.8 3.2 3.6 990 564 I 568 577 581 599 603 991 992 993 607 651 695 612 656 699 616 660 704 621 664 708 625 669 712 629 673 717 634 677 721 638 682 726 642 686 730 647 691 734 994 995 996 739 782 826 743 787 830 747 791 835 752 795 839 756 800 843 760 804 848 765 808 852 769 813 856 774 817 861 778 822 865 997 998 999 870 913 957 874 917 961 878 922 965 883 926 970 887 930 974 891 935 978 896 939 983 900 944 987 904 948 991 035 909 952 996 039 1000 00 000 004 009 013 017 022 026 030 N. L. 1 1 2 3 4 5 1 6 7 8 9 P.P. TABLE II, LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS. 123 124 0° S. — 4. 557 T. It / L. Sin. L. Tan. L. Cot. L. Cos. 60 59 58 57 557 0.00000 557 557 557 557 557 557 60 120 180 1 2 3 6.46373 6.76476 6.94085 6.46373 6.76476 6.94085 3.53627 3.23524 3.05915 0.00000 0.00000 0.00000 557 557 557 558 558 558 240 300 360 4 5 6 7.06579 7.16270 7.24188 7.06579 7.16270 7.24188 2.93421 2.83730 2.75812 0.00000 0.00000 0.00000 56 55 54 557 557 557 558 5S8 558 420 480 540 7 8 9 7.30882 7.36682 7.41797 7.30882 7.36682 7.41797 2.69118 2.63318 2.58203 0.00000 0.00000 0.00000 53 52 51 50 49 48 47 557 558 600 10 7.46373 7.46373 2.53627 0.00000 557 557 557 558 558 558 660 720 780 11 12 13 7.50512 7.54291 7.57767 7.50512 7.54291 7.57767 2.49488 2.45709 2.42233 0.00000 0.00000 0.00000 557 557 557 558 358 558 840 900 960 14 15 16 7.60985 7.63982 7.66784 7.60986 7.63982 7.66785 2.39014 2.36018 2.33215 0.00000 0.00000 0.00000 46 45 44 557 557 557 558 558 558 1020 1080 1 140 17 18 19 7.69417 7.71900 7.74248 7.69418 7.71900 7.74248 2.30582 2.28100 2.25752 9.99999 9.99999 9.99999 43 42 41 40 39 38 37 557 558 I2CX) 20 7.76475 7.76476 2.23524 9.99999 557 557 557 558 558 558 1260 1320 1380 21 22 23 7.78594 7.80615 7.82545 7.78595 7.80615 7.82546 2.21405 2.19385 2.17454 9.99999 9.99999 9.99999 557 557 557 558 558 558 1440 1500 1560 24 25 26 7.84393 7.86166 7.87870 7.84394 7.86167 7.87871 2.15606 2.13833 2.12129 9.99999 9.99999 9.99999 36 35 34 557 557 557 558 558 559 1620 1680 1740 27 28 29 7.89509 7.91088 7.92612 7.89510 7.91089 7.92613 2.10490 2.08911 2.07387 9.99999 9.99999 9.99998 2>^ 32 31 30 557 559 1800 30 7.94084 7.94086 2.05914 9.99998 557 557 557 559 559 559 i860 1920 1980 31 32 33 7.95508 7.96887 7.98223 7.95510 7.96889 7.98225 2.04490 2.03111 2.01775 9.99998 9.99998 9.99998 29 28 27 557 557 557 559 559 559 2040 2100 2160 34 35 36 7.99520 8.00779 8.02002 7.99522 8.00781 8.02004 2.00478 1.99219 1.97996 9.99998 9.99998 9.99998 26 25 24 557 557 557 559 559 559 2220 2280 2340 3^ 38 39 8.03192 8.04350 8.05478 8.03194 8.04353 8.05481 1.96806 1.95647 1.94519 9.99997 9.99997 9.99997 23 22 21 20 557 559 2400 40 8.06578 8.06581 1.93419 9.99997 556 556 556 560 560 560 2460 2520 2580 41 42 43 8.07650 8.08696 8.09718 8.07653 8.08700 8.09722 1.92347 1.91300 1.90278 9.99997 9.99997 9.99997 19 18 17 556 556 556 560 560 560 2640 2700 2760 44 45 46 8.10717 8.11693 8.12647 8.10720 8.11696 8.f2651 1.89280 1.88304 1.87349 9.99996 9.99996 9.99996 16 15 14 556 556 556 560 560 560 2820 2880 2940 47 48 49 8.13581 8.14495 8.15391 8.13585 8.14500 8.15395 1.86415 1.85500 1.84605 9.99996 9.99996 9.99996 13 12 11 556 S6i 3000 50 8.16268 8.16273 1.83727 9.99995 10 9 8 7 556 556 556 561 561 561 3060 3120 3180 51 52 53 8.17128 8.17971 8.18798 8.17133 8.17976 8.18804 1.82867 1.82024 1.81196 9.99995 9.99995 9.99995 556 556 556 561 561 3240 3300 3360 54 55 56 8.19610 8.20407 8.21189 8.19616 8.20413 8.21195 1.80384 1.79587 1.78805 9.99995 9.99994 9.99994 6 5 4 555 555 555 561 562 562 3420 3480 3540 57 58 59 8.21958 8.22713 8.23456 8.21964 8.22720 8.23462 1.78036 1.77280 1.76538 9.99994 9.99994 9.99994 3 2 1 555 562 3600 60 8.24186 8.24192 1.75808 9.99993 1 L. Cos. L. Cot. L. Tan. L. Sin. V 125 s. 4. 555 T. n t L. Sin. L. Tan. L. Cot. L. Cos. 562 3600 8.24186 8.24192 1.75808 9.99993 60 59 58 57 555 555 555 562 562 562 3660 3720 3780 1 2 3 8.24903 8.25609 8.26304 8.24910 8.25616 8.26312 1.75090 1.74384 1.73688 9.99993 9.99993 9.99993 555 555 555 563 563 563 3840 3900 3960 4 5 6 8.26988 8.27661 8.28324 8.26996 8.27669 8.28332 1.73004 1.72331 1.71668 9.99992 9.99992 9.99992 56 55 54 555 555 555 563 563 563 4020 4080 4140 7 8 9 8.28977 8.29621 8.30255 8.28986 8.29629 8.30263 1.71014 1.70371 1.69737 9.99992 9.99992 9.99991 53 52 51 50 554 563 4200 10 8.30879 8.30888 1.69112 9.99991 554 554 554 564 564 564 4260 4320 4380 11 12 13 8.31495 8.32103 8.32702 8.31505 8.32112 8.32711 1.68495 1.67888 1.67289 9.99991 9.99990 9.99990 49 48 47 554 554 554 564 564 565 4440 4500 4560 14 15 16 8.33292 8.33875 8.34450 8.33302 8.33886 8.34461 1.66698 1.66114 1.65539 9.99990 9.99990 9.99989 46 45 44 554 554 554 565 565 56s 4620 4680 4740 17 18 19 8.35018 8.35578 8.36131 8.35029 8.35590 8.36143 1.64971 1.64410 1.63857 9.99989 9.99989 9.99989 43 42 41 554 56s 4800 20 8.36678 8.36689 1.63311 9.99988 40 553 553 553 566 566 566 4860 4920 4980 21 22 23 8.37217 8.37750 8.38276 8.37229 8.37762 8.38289 1.62771 1.62238 1.61711 9.99988 9.99988 9.99987 39 38 37 553 553 553 566 566 567 5040 5100 5160 24 25 26 8.38796 8.39310 8.39818 8.38809 8.39323 8.39832 1.61191 1.60677 1.60168 9.99987 9.99987 9.99986 36 35 34 553 553 553 567 567 567 5220 5280 5340 27 28 29 8.40320 8.40816 8.41307 8.40334 8.40830 8.41321 1.59666 1.59170 1.58679 9.99986 9.99986 9.99985 33 32 31 30 553 567 5400 30 8.41792 8.41807 1.58193 9.99985 552 552 552 568 568 568 5460 5520 5580 31 32 8.42272 8.42746 8.43216 8.42287 8.42762 8.43232 1.57713 1.57238 1.56768 9.99985 9.99984 9.99984 29 28 27 552 552 552 568 569 569 5640 5700 5760 34 35 36 8.43680 8.44139 8.44594 8.43696 8.44156 8.44611 1.56304 1.55844 1.55389 9.99984 9.99983 9.99983 26 25 24 552 552 551 569 569 569 5820 5880 5940 37 38 39 8.45044 8.45489 8.45930 8.45061 8.45507 8.45948 1.54939 1.54493 1.54052 9.99983 9.99982 9.99982 23 22 21 551 570 6000 40 8.46366 8.46385 1.53615 9.99982 20 19 18 17 551 551 551 570 570 570 6060 6120 6180 41 42 43 8.46799 8.47226 8.47650 8.46817 8.47245 8.47669 1.53183 1.52755 1.52331 9.99981 9.99981 9.99981 551 551 551 571 571 571 6240 6300 6360 44 45 46 8.48069 8.48485 8.48896 8.48089 8.48505 8.48917 1.51911 1.51495 1.51083 9.99980 9.99980 9.99979 16 15 14 550 550 550 572 572 572 6420 6480 6540 47 48 49 8.49304 8.49708 8.50108 8.49325 8.49729 8.50130 1.50675 1.50271 1.49870 9.99979 9.99979 9.99978 13 12 11 10 9 8 7 ' 550 572 6600 50 8.50504 8.50527 1.49473 9.99978 550 550 550 573 573 573 6660 6720 6780 51 52 53 8.50897 8.51287 8.51673 8.50920 8.51310 8.51696 1.49080 1.48690 1.48304 9.99977 9.99977 9.99977 550 549 549 573 574 574 6840 6900 6960 ^4 55 56 8.52055 8.52434 8.52810 8.52079 8.52459 8.52835 1.47921 1.47541 1.47165 9.99976 9.99976 9.99975 6 5 4 549 549 549 574 575 575 7020 7080 7140 57 58 59 8.53183 8.53552 8.53919 8.53208 8.53578 8.53945 1.46792 1.46422 1.46055 9.99975 9.99974 9.99974 3 2 1 549 575 7200 60 8.54282 8.54308 1.45692 9.99974 L. Cos. L. Cot. L. Tan. L. Sin. / 126 2° s. 549 T. 68— >.• n t L. Sin. L. Tan. L. Cot. L. Cos. 575 7200 8.54282 8.54308 1.45692 9.99974 60 549 548 548 575 576 576 7260 7320 7380 1 2 3 8.54642 8.54999 8.55354 8.54669 8.55027 8.55382 1.45331 1.44973 1.44618 9.99973 9.99973 9.99972 59 58 57 548 548 548 576 577 577 7440 7500 7560 4 5 6 8.55705 8.56054 8.56400 8.55734 8.56083 8.56429 1.44266 1.43917 1.43571 9.99972 9.99971 9.99971 56 55 54 548 547 547 577 578 578 7620 7680 7740 7 8 9 8.56743 8.57084 8.57421 8.56773 8.57114 8.57452 1.43227 1.42886 1.42548 9.99970 9.99970 9.99969 53 52 51 547 578 579 579 579 7800 10 8.57757 8.57788 1.42212 9.99969 50 547 547 547 7860 7920 7980 11 12 13 8.58089 8.58419 8.58747 8.58121 8.58451 8.58779 1.41879 1.41549 1.41221 9.99968 9.99968 9.99967 49 48 47 546 546 546 579 580 580 8040 8100 8160 14 15 16 8.59072 8.59395 8.59715 8.59105 8.59428 8.59749 1.40895 1.40572 1.40251 9.99967 9.99967 9.99966 46 45 44 546 546 546 580 581 581 8220 8280 8340 17 18 19 8.60033 8.60349 8.60662 8.60068 8.60384 8.60698 1.39932 1.39616 1.39302 9.99966 9.99965 9.99964 43 42 41 545 582 8400 20 8.60973 8.61009 1.38991 9.99964 40 39 38 37 545 545 545 582 582 583 8460 8520 8580 21 22 23 8.61282 8.61589 8.61894 8.61319 8.61626 8.61931 1.38681 1.38374 1.38069 9.99963 9.99963 9.99962 545 545 544 583 583 584 8640 8700 8760 24 25 26 8.62196 8.62497 8.62795 8.62234 8.62535 8.62834 1.37766 1.37465 1.37166 9.99962 9.99961 9.99961 36 35 34 544 544 544 584 584 585 8820 8880 8940 27 28 29 8.63091 8.63385 8.63678 8.63131 8.63426 8.63718 1.36869 1.36574 1.36282 9.99960 9.99960 9.99959 32 31 544 544 543 543 585 9000 30 8.63968 8.64009 1.35991 9.99959 30 29 28 27 585 586 586 9060 9120 9180 31 32 33 8.64256 8.64543 8.64827 8.64298 8.64585 8.64870 1.35702 1.35415 1.35130 9.99958 9.99958 9.99957 543 543 543 587 587 587 9240 9300 9360 34 35 36 8.65110 8.65391 8.65670 8.65154 8.65435 8.65715 1.34846 1.34565 1.34285 9.99956 9.99956 9.99955 26 25 24 542 542 542 588 588 588 9420 9480 9540 37 38 39 8.65947 8.66223 8.66497 8.65993 8.66269 8.66543 1.34007 1.33731 1.33457 9.99955 9.99954 9.99954 23 22 21 542 589 9600 40 8.66769 8.66816 1.33184 9.99953 20 542 541 541 589 590 590 9660 9720 9780 41 42 43 8.67039 8.67308 8.67575 8.67087 8.67356 8.67624 1.32913 1.32644 1.32376 9.99952 9.99952 9.99951 19 18 17 541 541 541 590 591 591 9840 9900 9960 44 45 46 8.67841 8.68104 8.68367 8.67890 8.68154 8.68417 1.32110 1.31846 1.31583 9.99951 9.99950 9.99949 16 15 14 540 540 540 540 540 539 539 592 592 592 593 10020 10080 10140 47 48 49 8.68627 8.68886 8.69144 8.68678 8.68938 8.69196 1.31322 1.31062 1.30804 9.99949 9.99948 9.99948 13 12 11 10200 50 8.69400 8.69453 1.30547 9.99947 10 593 594 594 10260 10320 10380 51 52 53 8.69654 8.69907 8.70159 8.69708 8.69962 8.70214 1.30292 1.30038 1.29786 9.99946 9.99946 9.99945 9 8 7 539 539 539 595 595 595 10440 10500 10560 54 55 56 8.70409 8.70658 8.70905 8.70465 8.70714 8.70962 1.29535 1.29286 1.29038 9.99944 9.99944 9.99943 6 5 4 538 538 538 596 596 597 10620 10680 10740 57 58 59 8.71151 8.71395 8.71638 8.71208 8.71453 8.71697 1.28792 1.28547 1.28303 9.99942 9.99942 9.99941 3 2 1 538 597 loSoo 60 8.71880 8.71940 1.28060 9.99940 L. Cos. L. Cot. L. Tan. L. Sin. 1 3° 127 s. 538 T. II f L. Sin. L. Tan. L. Cot. L. Cos. 597 10800 8.71880 8.71940 1.28060 9.99940 60 537 537 537 598 598 599 10860 ioy20 10980 1 2 3 8.72120 8.72359 8.72597 8.72181 8.72420 8.72659 1.27819 1.27580 1.27341 9.99940 9.99939 9.99938 59 58 57 537 537 536 599 599 600 1 1040 IIIOO 11160 4 5 6 8.72834 8.73069 8.73303 8.72896 8.73132 8.73366 1.27104 1.26868 1.26634 9.99938 9.99937 9.99936 56 55 54 536 536 536 600 601 6oi 11220 11280 11340 7 8 9 8.73535 8.73767 8.73997 8.73600 8.73832 8.74063 1.26400 1.26168 1.25937 9.99936 9.99935 9.99934 53 52 51 535 602 11400 10 8.74226 8.74292 1.25708 9.99934 50 535 535 535 602 603 603 1 1460 1 1 520 11580 11 12 13 8.74454 8.74680 8.74906 8.74521 8.74748 8.74974 1.25479 1.25252 1.25026 9.99933 9.99932 9.99932 49 48 47 534 534 534 604 604 605 1 1 640 II 700 II 760 14 15 16 8.75130 8.75353 8.75575 8.75199 8.75423 8.75645 1.24801 1.24577 1.24355 9.99931 9.99930 9.99929 46 45 44 534 533 533 605 606 606 11820 11880 11940 17 18 19 8.75795 8.76015 8.76234 8.75867 8.76087 8.76306 1.24133 1.23913 1.23694 9.99929 9.99928 9.99927 43 42 41 533 607 12000 20 8.76451 8.76525 1.23475 9.99926 40 39 38 37 533 532 532 607 608 608 12060 12120 12180 21 22 23 8.76667 8.76883 8.77097 8.76742 8.76958 8.77173 1.23258 1.23042 1.22827 9.99926 9.99925 9.99924 532 532 531 609 609 610 12240 12300 12360 24 25 . 26 8.77310 8.77522 8.77733 8.77387 8.77600 8.77811 1.22613 1.22400 1.22189 9.99923 9.99923 9.99922 36 35 34 531 531 531 610 6ii 611 12420 12480 12540 27 28 29 8.77943 8.78152 8.78360 8.78022 8.78232 8.78441 1.21978 1.21768 1.21559 9.99921 9.99920 9.99920 33 32 31 530 612 12600 30 8.78568 8.78649 1.21351 9.99919 30 530 530 530 612 613 613 12660 12720 12780 31 32 8.78774 8.78979 8.79183 8.78855 8.79061 8.79266 1.21145 1.20939 1.20734 9.99918 9.99917 9.99917 29 28 27 529 529 529 614 614 61S 12840 12900 12960 34 35 36 8.79386 8.79588 8.79789 8.79470 8.79673 8.79875 1.20530 1.20327 1.20125 9.99916 9.99915 9.99914 26 25 24 529 528 528 615 616 616 13020 13080 13140 37 38 39 8.79990 8.80189 8.80388 8.80076 8.80277 8.80476 1.19924 1.19723 1.19524 9.99913 9.99913 9.99912 23 22 21 528 617 13200 40 8.80585 8.80674 1.19326 9.99911 20 528 527 527 617 618 618 13260 13320 13380 41 42 43 8.80782 8.80978 8.81173 8.80872 8.81068 8.81264 1.19128 1.18932 1.18736 9.99910 9.99909 9.99909 19 18 17 527 526 526 619 620 620 13440 13500 13560 44 45 46 8.81367 8.81560 8.81752 8.81459 8.81653 8.81846 1.18541 1.18347 1.18154 9.99908 9.99907 9.99906 16 15 14 526 526 525 621 621 622 13620 13680 13740 47 48 49 8.81944 8.82134 8.82324 8.82038 8.82230 8.82420 1.17962 1.17770 1.17580 9.99905 9.99904 9.99904 13 12 11 525 622 13800 50 8.82513 8.82610 1.17390 9.99903 10 525 525 524 623 623 624 13860 13920 13980 51 52 53 8.82701 8.82888 8.83075 8.82799 8.82987 8.83175 1.17201 1.17013 1.16825 9.99902 9.99901 9.99900 9 8 7 524 524 523 62s 625 626 14040 14100 14160 54 55 56 8.83261 8.83446 8.83630 8.83361 8.83547 8.83732 1.16639 1.16453 1.16268 9.99899 9.99898 9.99898 6 5 4 523 523 522 626 627 628 14220 14280 14340 57 58 59 8.83813 8.83996 8.84177 8.83916 8.84100 8.84282 1.16084 1.15900 1.15718 9.99897 9.99896 9.99895 3 2 1 522 628 14400 60 8.84358 8.84464 1.15536 9.99894 L. Cos. L. Cot. L.Tan. L. Sin. / l^O * S. 4. 522 T. It / L. Sin. L. Tan. L. Cot. L. Cos. 628 14400 8.84358 8.84464 1.15536 9.99894 60 522 522 521 629 629 630 14460 14520 14580 1 2 3 8.84539 8.84718 8.84897 8.84646 8.84826 8.85006 1.15354 1.15174 1.14994 9.99893 9.99892 9.99891 - 59 58 57 521 521 520 631 631 632 14640 14700 14760 4 5 6 8.85075 8.85252 8.85429 8.85185 8.85363 8.85540 1.14815 1.14637 1.14460 9.99891 9.99890 9.99889 56 55 54 520 520 520 632 633 634 14820 14880 14940 7 8 9 8.85605 8.85780 8.85955 8.85717 8.85893 8.86069 1.14283 1.14107 1.13931 9.99888 9.99887 9.99886 53 52 51 519 634 15000 10 8.86128 8.86243 1.13757 9.99885 50 519 519 518 635 635 636 15060 15120 15180 11 12 13 8.86301 8.86474 8.86645 8.86417 8.86591 8.86763 1.13583 1.13409 1.13237 9.99884 9.99883 9.99882 49 48 47 518 518 517 637 637 638 15240 15300 15360 14 15 16 8.86816 8.86987 8.87156 8.86935 8.87106 8.87277 1.13065 1.12894 1.12723 9.99881 9.99880 9.99879 46 45 44 517 517 516 638 639 640 15420 15480 15540 17 18 19 8.87325 8.87494 8.87661 8.87447 8.87616 8.87785 1.12553 1.12384 1.12215 9.99879 9.99878 9.99877 43 42 41 516 640 15600 20 8.87829 8.87953 1.12047 9.99876 40 5x6 515 515 641 642 642 15660 15720 15780 21 22 23 8.87995 8.88161 8.88326 8.88120 8.88287 8.88453 1.11880 1.11713 1.11547 9.99875 9.99874 9.99873 39 38 37 515 514 514 643 644 644 15840 15900 15960 24 25 26 8.88490 8.88654 8.88817 8.88618 8.88783 8.88948 1.11382 1.11217 1.11052 9.99872 9.99871 9.99870 36 35 34 514 513 513 645 646 646 16020 16080 16140 27 28 29 8.88980 8.89142 8.89304 8.89111 8.89274 8.89437 1.10889 1.10726 1.10563 9.99869 9.99868 9.99867 32 31 513 647 16200 30 8.89464 8.89598 1.10402 9.99866 30 512 512 512 648 648 649 16260 16320 16380 31 32 Z2> 8.89625 8.89784 8.89943 8.89760 8.89920 8.90080 1.10240 1.10080 1.09920 9.99865 9.99864 9.99863 29 28 27 5" 5" 650 650 651 16440 16500 16560 34 35 36 8.90102 8.90260 8.90417 8.90240 8.90399 8.90557 1.09760 1.09601 1.09443 9.99862 9.99861 9.99860 26 25 24 510 510 510 652 652 653 16620 16680 16740 37 38 39 8.90574 8.90730 8.90885 8.90715 8.90872 8.91029 1.09285 1.09128 1.08971 9.99859 9.99858 9.99857 23 22 21 509 654 16800 40 8.91040 8.91185 1.08815 9.99856 20 509 509 508 654 655 656 16860 16920 16980 41 42 43 8.91195 8.91349 8.91502 8.91340 8.91495 8.91650 1.08660 1.08505 1.08350 9.99855 9.99854 9.99853 19 18 17 508 508 507 656 657 658 17040 17100 17160 44 45 46 8.91655 8.91807 8.91959 8.91803 8.91957 8.92110 1.08197 1.08043 1.07890 9.99852 9.99851 9.99850 16 15 14 507 507 506 659 659 660 17220 17280 17340 47 48 49 8.92110 8.92261 8.92411 8.92262 8.92414 8.92565 1.07738 1.07586 1.07435 9.99848 9.99847 9.99846 13 12 11 506 661 17400 50 8.92561 8.92716 1.07284 9.99845 10 S06 505 505 661 662 663 17460 17520 17580 51 52 53 8.92710 8.92859 8.93007 8.92866 8.93016 8.93165 1.07134 1.06984 1.06835 9.99844 9.99843 9.99842 9 8 7 505 504 504 664 664 665 17640 17700 17760 54 55 56 8.93154 8.93301 8.93448 8.93313 8.93462 8.93609 1.06687 1.06538 1.06391 9.99841 9.99840 9.99839 6 5 4 503 503 503 666 666 667 17820 17880 17940 57 58 59 8.93594 8.93740 8.93885 8.93756 8.93903 8.94049 1.06244 1.06097 1.05951 9.99838 9.99837 9.99836 3 2 1 502 668 18000 60 8.94030 8.94195 1.05805 9.99834 L. Cos. L. Cot. L. Tan. L. Sin. f s.f;° 129 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 L. Sin. 8.94 030 8.94 174 8.94 317 8.94 461 8.94 603 8.94 746 8.94 887 8.95 029 8.95 170 8.95 310 .95 450 .95 589 .95 728 .95 867 .96 005 .96 143 .96 280 .96 417 .96 553 .96 689 8.96 825 8.96 960 8.97 095 8.97 229 8.97 363 8.97 496 8.97 629 8.97 762 8.97 894 8.98 026 8.98 157 8.98 288 8.98 419 8.98 549 8.98 679 8.98 808 8.98 937 8.99 066 8.99 194 8.99 322 8.99 450 8.99 577 8.99 704 8.99 830 8.99 956 9.00 082 9.00 207 9.00 332 9.00 456 9.00 581 9.00 704 9.00 828 9.00 951 9.01 074 9.01 196 9.01 318 9.01 440 9.01 561 9.01 682 9.01 803 9.01 923 L. Cos. L. Tan. c.d. L. Cot 8.94 195 8.94 8.94 8.94 8.94 8.94 8.95 8.95 8.95 8.95 340 485 630 773 917 060 202 344 486 .95 627 8.95 8.95 8.96 8.96 8.96 8.96 8.96 8.96 8.96 767 908 047 187 325 464 602 739 877 8.97 013 8.97 8.97 8.97 8.97 8.97 8.97 8.97 8.98 8.98 150 285 421 556 691 825 959 092 225 8.98 358 8.98 8.98 8.98 8.98 8.99 8.99 8.99 8.99 8.99 490 622 753 884 015 145 275 405 534 8.99 662 8.99 8.99 9.00 9.00 9.00 9.00 9.00 9.00 9.00 791 919 046 174 301 427 553 679 805 9.00 930 9.01 9.01 9.01 9.01 9.01 9.01 9.01 9.01 9.02 055 179 303 427 550 673 796 918 040 9.02 162 L. Cot. 145 145 145 143 144 143 142 142 142 141 140 141 139 140 138 139 138 137 138 136 137 135 136 135 135 134 134 133 133 133 132 132 131 131 131 130 130 130 129 128 129 128 127 128 127 126 126 126 126 125 125 124 124 124 123 123 123 122 122 122 1.05 805 .05 660 .05 515 .05 370 .05 227 .05 083 .04 940 .04 798 .04 656 ,04 514 ,04 373 .04 233 ,04 092 ,03 953 ,03 813 ,03 675 ,03 536 ,03 398 ,03 261 ,03 123 ,02 987 ,02 850 ,02 715 ,02 579 ,02 444 ,02 309 ,02 175 ,02 041 ,01 908 ,01 775 .01 642 .01 510 .01 378 .01 247 .01 116 .00 985 .00 855 .00 725 .00 595 .00 466 .00 338 .00 209 .00 081 0.99 954 0.99 826 0.99 699 0.99 573 0.99 447 0.99 321 0.99 195 0.99 070 0.98 945 0.98 821 0.98 697 0.98 573 0.98 450 0.98 327 0.98 204 0.98 082 0.97 960 0.97 838 L. Tan. L. Cos. 9.99 834 9.99 833 9.99.832 9.99 831 9.99 830 9.99 829 9.99 828 9.99 827 9.99 825 9.99 824 9.99 823 9.99 822 9.99 821 9.99 820 9.99 819 9.99 817 9.99 816 9.99 815 9.99 814 9.99 813 9.99 812 9.99 810 9.99 809 9.99 808 9.99 807 9.99 806 9.99 804 9.99 803 9.99 802 9.99 801 9.99 800 9.99 798 9.99 797 9.99 796 9.99 795 9.99 793 9.99 792 9.99 791 9.99 790 9.99 788 9.99 787 9.99 786 9.99 785 9.99 783 9.99 782 9.99 781 9.99 780 9.99 778 9.99 777 9.99 776 9.99 775 9.99 773 9.99 772 9.99 771 9.99 769 9.99 768 9.99 767 9.99 765 9.99 764 9.99 763 9.99 761 L. Sin. I 20 19 18 17 16 15 14 13 12 1]_ 9 8 7 6 5 4 3 2 _1^ P.P. 145 144 143 12. 1 12.0 1 1.9 24.2 36.2 48.3 60.4 24.0 36.0 48.0 60.0 23.8 35.8 47.7 59.6 72.S 84.6 96.7 108.8 72.0 84.0 96.0 108.0 71.S 83.4 95.3 107.2 120.8 120.0 119. 2 132.9 132.0 131. 1 141 11.8 23.5 35.2 47.0 58.8 70.5 82.2 94.0 105.8 117.S 129.2 140 11.7 23-3 35.0 46.7 58.3 70.0 81.7 93.3 1 05.0 1 16.7 128.3 139 11.6 23.2 34.8 46.3 57-9 69.5 81.1 92.7 104.2 11S.8 127.4 137 136 135 1 1.4 1 1.3 11.2 22.8 22.7 22.5 34.2 34.0 33.8 45.7 45.3 45.0 57.1 .56.7 .■>6.2 68.5 68.0 67.5 79.9 79.3 78.8 91.3 90.7 90.0 102.8 102.0 101.2 114.2 1 13.3 112.5 125.6 124.7 123.8 II.8 23.7 35'S 47.3 59.2 71.0 82.8 94-7 106.5 118.3 130.2 138 ii.S 23.0 34-5 46.0 57.S 69.0 80.S 92.0 103.5 iiS.o 126.S 11.2 22.3 33.5 44-7 SS.8 67.0 78.2 89.3 100.5 111.7 122.8 "I 133 132 131 130 II. I 22.2 33.2 44.3 55.4 66.5 77-6 88.7 99.8 110.8 121.9 II.O 22.0 33.0 44.0 55.0 66.0 77.0 88.0 99.0 IIO.O 10.9 21.8 32.8 43.7 54.6 65.5 76.4 87.3 98.2 109.2 10.8 21.7 32.5 43.3 54.2 65.0 75.8 86.7 97.5 108.3 119.2 129 128 127 126 10.8 10.7 10.6 lo.s 21.5 21.3 21.2 21.0 32.2 32.0 31.8 31.5 43.0 42.7 42.3 42.0 53.8 53.3 52.9 52.5 64.5 64.0 63.5 63.0 75.2 74-7 74.1 73.S 86.0 85.3 84.7 84.0 96.8 96.0 95.2 94.5 107.S 106.7 105.8 105.0 1X8.2 II7.3 1 16.4 115.S " 125 124 5 10.4 10.3 10 20.8 20.7 15 31.2 31.0 20 41.7 41.3 25 52.1 51.7 30 62.5 62.0 35 72.9 72.3 40 83.3 82.7 45 93.8 93-0 50 104.2 103-3 55 1 14.6 1 13.7 10.2 10.2 20.5 20.3 30.8 30.5 41.0 40.7 51.2 50.8 61.5 61.0 71.8 71.2 82.0 81.3 92.2 9I.S 102. 5 101.7 II2.8 III.8 P.P. 130 6° L. Sin. d. L. Tan. c.d. L. Cot, L. Cos. P.P. 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 3& 39_ 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.01 923 9.02 043 9.02 163 9.02 283 9.02 402 9.02 520 9.02 639 9.02 757 9.02 874 9.02 992 9.03 109 9.03 226 9.03 342 9.03 458 9.03 574 9.03 690 9.03 805 9.03 920 9.04 034 9.04 149 9.04 262 9.04 376 9.04 490 9.04 603 9.04 715 9.04 828 9.04 940 9.05 052 9.05 164 9.05 275 9.05 386 9.05 497 9.05 607 9.05 717 9.05 827 9.05 937 9.06 046 9.06 155 9.06 264 9.06 372 9.06 481 9.06 589 9.06 696 9.06 804 9.06 911 9.07 018 9.07 124 9.07 231 9.07 337 9.07 442 9.07 548 9.07 653 9.07 758 9.07 863 9.07 968 9.08 072 9.08 176 9.08 280 9.08 383 9.08 486 9.08 589 120 120 120 119 118 119 118 117 118 117 117 116 116 116 116 115 115 114 115 113 114 114 113 112 113 112 112 112 111 111 111 110 110 110 110 109 109 109 108 109 108 107 108 107 107 106 107 106 105 106 105 105 105 105 104 104 104 103 103 103 .02 162 .02 283 02 404 .02 525 .02 645 02 766 02 885 03 005 03 124 03 242 03 361 03 479 03 597 03 714 03 832 03 948 04 065 04 181 04 297 04 413 04 528 04 643 04 758 04 873 04 987 05 101 05 214 05 328 05 441 05 553 05 666 05 778 05 890 06 002 06 113 06 224 06 335 06 445 06 556 06 666 06 775 06 885 06 994 07 103 07 211 07 320 07 428 07 536 07 643 07 751 07 858 07 964 08 071 08 177 08 283 08 389 08 495 08 600 08 705 08 810 08 914 121 121 121 120 121 119 120 119 118 119 118 118 117 118 116 117 116 116 116 115 115 115 115 114 114 113 114 113 112 113 112 112 112 111 111 111 110 111 110 109 110 109 109 108 109 108 108 107 108 107 106 107 106 106 106 106 105 105 105 104 0.97 838 9.99 761 0.97 717 0.97 596 0.97 475 0.97 355 0.97 234 0.97 115 0.96 995 0.96 876 0.96 758 0.96 639 0.96 521 0.96 403 0.96 286 0.96 168 0.96 052 0.95 935 0.95 819 0.95 703 0.95 587 0.95 472 0.95 357 0.95 242 0.95 127 0.95 013 0.94 899 0.94 786 0.94 672 0.94 559 0.94 447 0.94 334 0.94 222 0.94 110 0.93 998 0.93 887 0.93 776 0.93 665 0.93 555 0.93 444 0.93 334 0.93 225 0.93 115 0.93 006 0.92 897 0.92 789 0.92 680 0.92 572 0.92 464 0.92 357 0.92 249 0.92 142 0.92 036 0.91 929 0.91 823 0.91 717 0.91611 0.91 505 0.91 400 0.91 295 0.91 190 0.91 086 9.99 760 9.99 759 9.99 757 9.99 756 9.99 755 9.99 753 9.99 752 9.99 751 9.99 749 9.99 748 9.99 747 9.99 745 9.99 744 9.99 742 9.99 741 9.99 740 9.99 738 9.99 737 9.99 736 9.99 734 9.99 733 9.99 731 9.99 730 9.99 728 9.99 727 9.99 726 9.99 724 9.99 723 9.99 721 9.99 720 9.99 718 9.99 717 9.99 716 9.99 714 9.99 713 9.99 711 9.99 710 9.99 708 9.99 707 9.99 705 9.99 704 9.99 702 9.99 701 9.99 699 9.99 698 9.99 696 9.99 695 9.99 693 9.99 692 9.99 690 9.99 689 9.99 687 9.99 686 9.99 684 9.99 683 9.99 681 9.99 680 9.99 678 9.99 677 50 20 9.99 675 119 9.9 19.8 29.8 39.7 49.6 59.5 69.4 79.3 89.2 99.2 1 09. 1 118 117 116 121 120 10. 1 lO.O 20.2 20.0 30.2 30.0 40.3 40.0 50.4 So.o 00.5 60.0 70.6 70.0 80.7 80.0 90.8 90.0 100.8 lOO.O 1 10.9 IIO.O 9.8 19.7 29-5 39-3 49.2 59-0 68.8 78.7 88.5 98.3 108.2 9.8 19-5 29.2 39-0 48.8 S8.5 68.2 78.0 87.8 97.5 107.2 115 114 9.6 9.5 19.2 19.0 28.8 28.S 38.3 38.0 47.9 47.5 57.5 S7.0 67.1 66.5 76.7 76.0 86.2 85.5 95.8 95-0 10S.4 104.5 112 111 9.3 9.2 18.7 la..-; 28.0 27.8 37.3 37-0 46.7 46.2 56.0 55.5 65.3 64.8 74-7 74-0 84.0 83.2 93.3 92.S 102.7 101.8 9.7 19-3 29.0 38.7 48.3 58.0 67.7 77.3 87.0 96.7 106.3 9.4 18.8 28.2 37.7 47.1 56.5 65-9 75-3 84.8 94-2 103.6 110 9.2 18.3 27.5 36.7 45.8 S5.0 64.2 73-3 82.S 91.7 100.8 1091 108 5 9-1 9 10 18.2J18 15 27.2,27 20I36.3 36 25|4S.4'45 30 54-5 54 35,63.663 40 72.7I72 45,8i.8|8i 5090.8 90 55 99.9'99 106 105 8.8 8.8 17.7 17-5 26.5 26.2 35.3 35-0 44.2 43.8 53.0 52.5 61.8 61.2 70.7 70.0 79.5 78.8 88.3 87.5 97.2 96.2 107 o 8.9 17.8 o 26.8 035.7 044.6 053.5 062.4 oi7i.3 0)80.2 o 89.2 o'98.i 104 8.7 17.3 26.0 34-7 43.3 52.0 60.7 09-3 78.0 86.7 95.3 L. Cos. d. L. Cot. c.d. L. Tan L. Sin. P.P. 7° 131 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. P.P. 10 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 60 08 589 08 692 08 795 08 897 08 999 09 101 09 202 09 304 09 405 09 506 9.09 606 09 707 09 807 09 907 10 006 10 106 10 205 10 304 10 402 10 501 9.10 599 9 9 9 10 697 10 795 10 893 10 990 11087 11 184 11281 11377 11474 11 666 11 761 11857 11952 12 047 12 142 12 236 12 331 12 425 12 519 12 612 12 706 12 799 12 892 12 985 13 078 13 171 13 263 13 355 9.13 447 13 539 13 630 13 722 13 813 13 904 13 994 14 085 14 175 14 266 9.11 570 14 356 103 103 102 102 102 101 102 101 101 100 101 100 100 99 100 99 99 98 99 98 98 98 98 97 97 97 97 96 97 96 96 95 96 95 95 95 94 95 94 94 93 94 93 93 93 93 93 92 92 92 92 91 92 91 91 90 91 90 91 90 9.08 914 9.09 019 9.09 123 9.09 227 9.09 330 9.09 434 9.09 537 9.09 640 9.09 742 9.09 845 9.09 947 9.10 049 9.10 150 9.10 252 9.10 353 9.10 454 9.10 555 9.10 656 9.10 756 9.10 856 9.10 956 9.11 056 9.11 155 9.11 254 9.11353 9.11452 9.11551 9.11649 9.11 747 9.11845 9.11943 12 040 12 138 12 235 12 332 12 428 12 525 12 621 12 717 9.12 813 9.12 909 9.13 004 9.13 099 9.13 194 9.13 289 13 384 13 478 13 573 13 667 13 761 9.13 854 9.13 948 9.14 041 9.14 134 9.14 227 9.14 320 9.14 412 9.14 504 9.14 597 9.14 688 9.14 780 105 104 104 103 104 103 103 102 103 ,102 102 101 102 101 101 101 101 100 100 100 100 99 99 99 99 99 98 98 98 98 97 98 97 97 96 97 96 96 96 96 95 95 95 95 95 94 95 94 94 93 94 93 93 93 93 92 92 93 91 92 0.91 086 9.99 675 60 0.90 981 0.90 877 0.90 773 0.90 670 0.90 566 0.90 463 0.90 360 0.90 258 0.90 155 9.99 674 9.99 672 9.99 670 9.99 669 9.99 667 9.99 666 56 55 54 9.99 664 9.99 663 9.99 661 53 52 51 0.90 05 3 0.89 951 0.89 850 0.89 748 0.89 647 0.89 546 0.89 445 0.89 344 0.89 244 0.89 144 9.99 65 9 9.99 658 9.99 656 9.99 655 9.99 653 9.99 651 9.99 650 9.99 648 9.99 647 9.99 645 50 0.89 044 9.99 643 0.88 944 0.88 845 0.88 746 0.88 647 0.88 548 0.88 449 0.88 351 0.88 253 0.88 155 9.99 642 9.99 640 9.99 638 9.99 637 9.99 635 9.99 633 9.99 632 9.99 630 9.99 629 0.88 057 9.99 627 0.87 960 0.87 862 0.87 765 0.87 668 0.87 572 0.87 475 0.87 379 0.87 283 0.87 187 9.99 625 9.99 624 9.99 622 9.99 620 9.99 618 9.99 617 9.99 615 9.99 613 9.99 612 0.87 091 9.99 610 20 0.86 996 0.86 901 0.86 806 0.86 711 0.86 616 0.86 522 0.86 427 0.86 333 0.86 239 9.99 608 9.99 607 9.99 605 9.99 603 9.99 601 9.99 600 9.99 598 9.99 596 9.99 595 0.86 146 9.99 593 0.86 052 0.85 959 0.85 866 0.85 773 0.85 680 0.85 588 0.85 496 0.85 403 0.85 312 9.99 591 9.99 589 9.99 588 9.99 586 9.99 584 9.99 582 9.99 581 9.99 579 9.99 577 0.85 220 9.99 575 104! 103 102 5\ 8 io|i7 15 26 2034 25 43 3o|S2 35i6o 40 1 69 4578 5086 SSI95 .7 8.6 8.5 .3;i7-2 17-0 .0 25.8 25.S •7 34-3 34-0 .342.942.5 •0 51.S Si.o .7!6o.i 59.5 .3 68.7 68.0 .0 77.2 76.S .7 85.8 85.0 •394.493.5 1011 lOOi 99 5 8.4 8 io!i6.8 16 is'25.2125 20 33.7[33 2542.1 41 30 50.5 50 3558.958 40 67.3 66 45 75.8 75 SO 84.2 83 55 92.6 91 .3! 8.2 •7 16.5 .0 24.8 ■333.0 .7:41.2 .049.5 .3 57.8 .7 66.0 .0 74.2 .382.5 .790.8 S 8.2 io|i6.3 I5;24.5 20 32.7 25 40.8 3049.0 35 57.2 40 65.3 45 73.5 5081.7 5589.8 97 I 96 8.1 i 8.0 16.2 16.0 24.2 24.0 32.3I32.0 40. 4 1 40.0 48.5148.0 56.6 56.0 64.7164.0 72.8 72.0 180.880.0 l88.9'88.o 95 I 94 93 7.8 15-5 23.2 20 31 25 39 3047 3S!55 4063 .9 7.8 •8 15.7 .823.5 .7 31.3 31.0 .6 39.2 38.8 .5 47-0 46.5 .4 54.8 54-2 62.7 62.0 70.5 69.8 78.3 77.5 I 86.2,85.2 92 I 91 I 90 5 7.7 10,15.3 15 23.0 20130.7 25 38.3 30 46.0 35 53-7 40 61.3 45 69.0 50 76.7 55 84.3 7.6 75 15.2:15.0 22.8 22.5 30.3:30.0 37-9 37.5 45-5 45.0 53-1 52.5 60.7 60.0 68.2 67.5 75.8 75.0 83.4 82.5 L. Cos. d. L. Cot. c.d. L. Tan. 82° L. Sin. P.P. 132 8° ' L. Sin. d. L. Tan. | c.d. L. Cot. L. Cos. P.P. 9.14 356 89 90 89 90 89 88 89 89 88 88 88 88 87 88 87 87 87 87 86 86 87 86 85 9.14 780 92 91 91 91 91 91 90 91 90 90 89 90 89 90 89 89 88 89 88 88 88 88 88 0.85 220 9.99 575 60 1 9.14 445 9.14 872 0.85 128 9.99 574 59 2 9.14 535 9.14 963 0.85 037 9.99 572 58 3 9.14 624 9.15 054 0.84 946 9.99 570 57 " j 91 90 89 4 9.14 714 9.15 145 0.84 855 9.99 568 56 5 7.6 7-5 7-4 5 9.14 803 9.15 236 0.84 764 9.99 566 55 lojis.a I5.0|i4.8 ISj22.8|22.Sj22.2 20|30.3[30.0 29.7 6 9.14 891 9.15 327 0.84 673 9.99 565 54 7 9.14 980 9.15 417 0.84 583 9.99 563 53 25!37.9'37.Sl37.l ■30 45-5 45.0 44-5 35 53-1 52. 5 51-9 8 9.15 069 9.15 508 0.84 492 9.99 561 52 9 9.15 157 9.15 598 9.15 688 9.15 777 0.84 402 9.99 559 51 40 60.7 60.0 59-3 45 68.2 67. 5 66.8 50:75.8,75.0 74-2 SS83.482.S 81.6 10 11 9.15 245 0.84 312 9.99 557 50 9.15 333 0.84 223 9.99 556 49 12 9.15 421 9.15 867 0.84 133 9.99 554 48 13 9.15 508 9.15 956 0.84 044 9.99 552 47 14 9.15 596 9.16 046 0.83 954 9.99 550 46 15 9.15 683 9.16 135 0.83 865 9.99 548 45 16 9.15 770 9.16 224 0.83 776 9."^9 546 44 17 9.15 857 9.16 312 0.83 688 9.99 545 43 18 9.15 944 9.16 401 0.83 599 9.99 543 42 19 9.16 030 9.16 489 0.83 511 9.99 541 41 5 10 »» 87 Ht> 7.3 7.2 7.2 14.7 14-5 14.3 20 9.16 116 9.16 577 0.83 423 0.83 335 9.99 539 40 21 9.16 203 9.16 665 9.99 537 39 I5|22.0;2I.8|2I.S 2029.329.028.7 2S'36.7 36.235.8 22 9.16 289 9.16 753 0.83 247 9.99 535 38 23 9.16 374 9.16 841 0.83 159 9.99 533 37 30I44.0143.5 43.0 24 9.16 460 86 85 86 85 85 85 84 85 84 84 84 84 83 9.16 928 8'/ 88 87 87 87 86 87 86 86 86 86 86 85 0.83 072 9.99 532 36 35 51.3 50.8 50.2 40 58.7 58.0 57.3 75 9.16 545 9.17 016 0.82 984 9.99 530 35 45 66.0 65.2 64.S 26 9.16 631 9.17 103 0.82 897 9.99 528 34 50 73.3 72.5 71-7 55 80.7I79.8 78.8 27 9.16 716 9.17 190 0.82 810 9.99 526 33 28 9.16 801 9.17 277 0.82 723 9.99 524 31 29 9.16 886 9.17 363 0.82 637 9.99 522 9.99 520 31 30 30 9.16 970 9.17 450 0.82 550 31 9.17 055 9.17 536 0.82 464 9.99 518 29 32 9.17 139 9.17 622 0.82 378 9.99 517 28 33 9.17 223 9.17 708 0.82 292 9.99 515 27 34 9.17 307 9.17 794 0.82 206 9.99 513 26 // 85 84 1 83 35 9.17 391 9.17 880 0.82 120 9.99 511 25 5 7.1 7.0 6.9 36 9.17 474 9.17 965 0.82 035 9.99 509 24 10 14.2 14.0 13.8 37 9.17 558 84 83 83 83 83 83 82 82 83 82 81 82 82 81 81 81 81 81 81 80 80 80 80 80 9.18 051 86 85 85 85 85 84 85 84 84 84 84 83 84 83 83 83 S3 83 S3 s? 0.81 949 9.99 507 23 15 21.2 21.0 20.8 20 28.3 28.o!27.7 38 9.17 641 9.18 136 0.81 864 9.99 505 22 2S'35.4|35.0 34-6 39 40 9.17 724 9.18 221 0.81 779 0.81 694 9.99 503 21 30 35 40 45 50 42.5 49.6 56.7 63.8 70.8 42.041.5 49.o'48.4 56.055.3 63.0 62.2 70.0 69.2 9.17 807 9.18 306 9.99 501 20 41 9.17 890 9.18 391 0.81 609 9.99 499 19 42 9.17 973 9.18 475 0.81 525 9.99 497 18 S5l77.9'77.0 76.1 43 9.18 055 9.18 560 0.81 440 9.99 495 17 44 9.18 137 9.18 644 0.81 356 9.99 494 16 45 9.18 220 9.18 728 0.81 272 9.99 492 15 46 9.18 302 9.18 812 0.81 188 9.99 490 14 47 9.18 383 9.18 896 0.81 104 9.99 488 13 48 9.18 465 9.18 979 0.81 021 9.99 486 12 49 50 9.18 547 9.19 063 0.80 937 9.99 484 11 5 82 6.8 81,80 6.8 6.7 I3.5ii3.3 20.2 20.0 9.18 628 9.19 146 0.80 854 9.99 482 10 51 9.18 709 9.19 229 0.80 771 9.99 480 9 15 20.5 52 9.18 790 9.19 312 0.80 688 9.99 478 8 20 27.3 27.026.7 53 9.18 871 9.19 395 0.80 605 9.99 476 7 25,34-2 33.8 33.3 30 41.0 40.5 40.0 54 9.18 952 9.19 478 0.80 522 9.99 474 6 35 47.8 47.2'46.7 55 9.19 033 9.19 561 0.80 439 9.99 472 5 40 54-7 54o'S3.3 45 61. 5 60.8 60.0 56 9.19 113 9.19 643 82 82 82 0.80 357 9.99 470 4 50 68.3 67.S 66.7 57 9.19 193 9.19 725 0.80 275 9.99 468 3 55 75.2 74-2 73.3 58 9.19 273 9.19 807 0.80 193 9.99 466 2 59 60 9.19 353 9.19 889 82 0.80 111 9.99 464 1 9.19 433 9.19 971 0.80 029 9.99 462 L. Cos. 1 d. L. Cot. c.d. L. Tan. L. Sin. / P.P. i 81 9° 133 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. P.P. 10 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26 27 28 _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 9.19 433 9.19 513 9.19 592 9.19 672 9.19 751 9.19 830 9.19 909 9.19 988 9.20 067 9.20 145 9.20 223 9.20 302 9.20 380 9.20 458 9.20 535 9.20 613 9.20 691 9.20 768 9.20 845 9.20 922 9.20 999 9.21076 9.21 153 9.21 229 9.21 306 9.21 382 9.21 458 9.21 534 9.21 610 9.21 685 9.21 761 9.21 836 9.21912 9.21 987 9.22 062 9.22 137 9.22 211 9.22 286 9.22 361 9.22 435 9.22 509 9.22 583 9.22 657 9.22 731 9.22 805 9.22 878 9.22 952 9.23 025 9.23 098 9.23 171 9.23 244 9.23 317 9.23 390 9.23 462 9.23 535 9.23 607 9.23 679 9.23 752 9.23 823 9.23 895 9.23 967 80 79 80 79 79 79 79 79 78 78 79 78 78 77 78 78 77 77 77 77 77 77 76 77 76 76 76 76 75 76 75 76 75 75 75 74 75 75 74 74 74 74 74 74 73 74 73 73 73 73 73 73 72 73 72 72 73 71 72 72 9.19 971 9.20 053 9.20 134 9.20 216 9.20 297 9.20 378 9.20 459 9.20 540 9.20 621 9.20 701 9.20 782 9.20 862 9.20 942 9.21 022 9.21 102 9.21 182 9.21 261 9.21 341 9.21 420 9.21 499 9.21 578 9.21 657 9.21 736 9.21 814 9.21 893 9.21971 9.22 049 9.22 127 9.22 205 9.22 283 9.22 361 9.22 438 9.22 516 9.22 593 9.22 670 9.22 747 9.22 824 9.22 901 9.22 977 9.23 054 9.23 130 9.23 206 9.23 283 9.23 359 9.23 435 9.23 510 9.23 586 9.23 661 9.23 737 9.23 812 9.23 887 9.23 962 9.24 037 9.24 112 9.24 186 9.24 261 9.24 335 9.24 410 9.24 484 9.24 558 9.24 632 82 81 82 81 81 81 81 81 80 81 80 80 80 80 80 79 80 79 79 79 79 79 78 79 78 78 78 78 78 78 77 78 77 77 77 77 77 76 77 76 76 77 76 76 75 76 75 76 75 75 75 75 75 74 75 74 75 74 74 74 0.80 029 9.99 462 60 0.79 947 0.79 866 0.79 784 0.79 703 0.79 622 0.79 541 0.79 460 0.79 379 0.79 299 9.99 460 9.99 458 9.99 456 9.99 454 9.99 452 9.99 450 9.99 448 9.99 446 9.99 444 0.79 218 9.99 442 0.79 138 0.79 058 0.78 978 0.78 898 0.78 818 0.78 739 0.78 659 0.78 580 0.78 501 0.78 422 9.99 440 9.99 438 9.99 436 9.99 434 9.99 432 9.99 429 9.99 427 9.99 425 9.99 423 9.99 421 40 0.78 343 0.78 264 0.78 186 0.78 107 0.78 029 0.77 951 0.77 873 0.77 795 0.77 717 9.99 419 9.99 417 9.99 415 9.99 413 9.99 411 9.99 409 9.99 407 9.99 404 9.99 402 0.77 639 0.77 562 0.77 484 0.77 407 0.77 330 0.77 253 0.77 176 0.77 099 0.77 023 0.76 946 9.99 398 9.99 396 9.99 394 0.76 870 9.99 379 0.76 794 0.76 717 0.76 641 0.76 565 0.76 490 0.76 414 0.76 339 0.76 263 0.76 188 9.99 377 9.99 375 9.99 372 9.99 370 9.99 368 9.99 366 9.99 364 9.99 362 9.99 359 0.76 113 9.99 357 0.76 038 0.75 963 0.75 888 0.75 814 0.75 739 0.75 665 0.75 590 0.75 516 0.75 442 9.99 355 9.99 353 9.99 351 9.99 348 9.99 346 9.99 344 9.99 342 9.99 340 9.99 337 0.75 368 9.99 335 9.99 400 30 9.99 392 9.99 390 9.99 388 26 25 24 9.99 385 9.99 383 9.99 381 23 22 21 L. Cos. d. L. Cot. c.d L. Tan. L. Sin. 39 38 37 36 35 34 33 31 31 " i 82 s! 6.8! 10|I3.7 15^20.5 20 27.3 25 34-2 30 41-0 35 47-8 40 54-7 4561.5 50;68.3, 55I75.2I 81 I 80 6.81 6.7 i3S|i3-3 20.2 20.0 27.0J26.7 33.8j33.3 40.5 40.0 47.2 46.7 54-0 53-3 60.8 60.0 67-5 66.7 74.2173.3 79 78 I 77 5! 6.6 1013.2 15 19.8 20;26.3 2S'32.9 30J39.5 35146.1 40,52.7 45 59-2 5065.8 5572.4 6.5 6.4 13.0 12.8 I9.5'i9.2 26.0 25.7 32.S 32.1 39.0 38.5 45-5 44-9 52.0 51-3 58.5 57.8 65.0164.2 7 1. 5 1 70.6 ' 76 I 75 I 74 3 6.2 6.2 7112.5 12.3 o 18.8I18.S 25.0 24.7 31.2 30.8 37.537.0 43.843.2 50.0 49.3 56.255.5 62.5161.7 68.8167.8 S 6.1' 10 12.2 15I18.2 20 24.3 25 30.4 3036. 5 35-42.6 40148.7 4554.8 50 60.8 55166.9 72 71 6.0 5.9 12.0 11.8 18.0 17.8 24.0 23.7 30.o|29.6 36.0)35.5 42.0 41.4 48.047.3 54.0 53-2 60.0 59.2 66.oi65.i P.P. kO° 134 10° 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. 9.23 967 9.24 039 9.24 110 9.24 181 9.24 253 9.24 324 9.24 395 9.24 466 9.24 536 9.24 607 9.24 677 9.24 748 9.24 818 9.24 888 9.24 958 9.25 028 9.25 098 9.25 168 9.25 237 9.25 307 9.25 376 9.25 445 9.25 514 9.25 583 9.25 652 9.25 721 9.25 790 9.25 858 9.25 927 9.25 995 9.26 063 9.26 131 9.26 199 9.26 267 9.26 335 9.26 403 9.26 470 9.26 538 9.26 605 9.26 672 9.26 739 9.26 806 9.26 873 9.26 940 9.27 007 9.27 073 9.27 140 9.27 206 9.27 273 9.27 339 9.27 405 9.27 471 9.27 537 9.27 602 9.27 668 9.27 734 9.27 799 9.27 864 9.27 930 9.27 995 9.28 060 L. Cos. d. 72 71 71 72 71 71 71 70 71 70 71 70 70 70 70 70 70 69 70 69 69 69 69 69 69 69 68 69 68 68 68 68 68 68 68 67 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 65 66 66 65 65 66 65 65 9.24 632 9.24 706 9.24 779 9.24 853 9.24 926 9.25 000 9.25 073 9.25 146 9.25 219 9.25 292 9.25 365 9.25 437 9.25 510 9.25 582 9.25 655 9.25 727 9.25 799 9.25 871 9.25 943 9.26 015 9.26 086 9.26 158 9.26 229 9.26 301 9.26 372 9.26 443 9.26 514 9.26 585 9.26 655 9.26 726 9.26 797 9.26 867 9.26 937 9.27 008 9.27 078 9.27 148 9.27 218 9.27 288 9.27 357 9.27 427 9.27 496 9.27 566 9.27 635 9.27 704 9.27 773 9.27 842 9.27 911 9.27 980 9.28 049 9.28 117 9.28 186 9.28 254 9.28 323 9.28 391 9.28 459 9.28 527 9.28 595 9.28 662 9.28 730 9.28 798 9.28 865 74 73 74 73 74 73 73 73 73 73 72 73 72 73 72 72 72 72 72 71 72 71 72 71 71 71 71 70 71 71 70 70 71 70 70 70 70 69 70 69 70 69 69 69 69 69 69 69 68 69 68 69 68 68 68 68 67 68 68 67 0.75 368 0.75 294 0.75 221 0.75 147 0.75 074 0.75 000 0.74 927 0.74 854 0.74 781 0.74 708 0.74 635 0.74 563 0.74 490 0.74 418 0.74 345 0.74 273 0.74 201 0.74 129 0.74 057 0.73 985 0.73 914 0.73 842 0.73 771 0.73 699 0.73 628 0.73 557 0.73 486 0.73 415 0.73 345 0.73 274 0.73 203 0.73 133 0.73 063 0.72 992 0.72 922 0.72 852 0.72 782 0.72 712 0.72 643 0.72 573 0.72 504 0.72 434 0.72 365 0.72 296 0.72 227 0.72 158 0.72 089 0.72 020 0.71 951 0.71 883 0.71 814 0.71 746 0.71 677 0.71 609 0.71 541 0.71 473 0.71 405 0.71 338 0.71 270 0.71 202 0.71 135 L. Cot. c.d. L. Tan. L. Sin. d. 9.99 335 9.99 333 9.99 331 9.99 328 9.99 326 9.99 324 9.99 322 9.99 319 9.99 317 9.99 315 9.99 313 9.99 310 9.99 308 9.99 306 9.99 304 9.99 301 9.99 299 9.99 297 9.99 294 9.99 292 9.99 290 9.99 288 9.99 285 9.99 283 9.99 281 9.99 278 9.99 276 9.99 274 9.99 271 9.99 269 9.99 267 9.99 264 9.99 262 9.99 260 9.99 257 9.99 255 9.99 252 9.99 250 9.99 248 9.99 245 9.99 243 9.99 241 9.99 238 9.99 236 9.99 233 9.99 231 9.99 229 9.99 226 9.99 224 9.99 221 9.99 219 9.99 217 9.99 214 9.99 212 9.99 209 9.99 207 9.99 204 9.99 202 9.99 200 9.99 197 9.99 195 P.P. " 74 73 1 Sj 6.2 6.1 lO 12.3 12.2 IS i8.s i8.2 20 24.7,24.3 25 30.8 30.4 30 37.036.5 35 43.2^42.6 40:49.3 48.7 45 55-5 54-8 SO 61.7 60.8 55 67.8 66.9 72 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 71 70 5 5 10 II 15I17. 20I23. 25 29. 3035. 35 41. 5.8 ii.S 17.2 23.0 28.8 5-8 11.7 17-5 23.3 29.2 35.034.5 40.8 40.2 3'46.7 46.0 2 52.5 51.8 2 58.3 57-5 I '64.2 63.2 30,34 35 39 404s 45 51 SO 56 SSI62 67 I 66 5.6 5.5 11.2 II.O 16.8J16.5 22.3 22.0 27.9 27.S 33.5 33.0 39-1 38.5 44.7 44-0 50.2149.5 55.8 55-0 61.4 60.5 65 I 3 I 2 5 5-4 10 10.8 I5ii6.2 20 21.7, 25|27.i 3o;32.5' 35 37-9| 4043.3' 4548.81 SO 54.2 55 59.61 lAI 0.2J0.2 0.5 0.3 0.8,0.5 i.o 0.7 2.5)1.7 2.8 1.8 P.P. n/Cko 11 135 10 11 12 13 14 15 16 17 18 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 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. 9.28 060 9.28 125 9.28 190 9.28 254 9.28 319 9.28 384 9.28 448 9.28 512 9.28 577 9.28 641 9.28 705 9.28 769 9.28 833 9.28 896 9.28 960 9.29 024 9.29 087 9.29 150 9.29 214 9.29 277 9.29 340 9.29 403 9.29 466 9.29 529 9.29 591 9.29 654 9.29 716 9.29 779 9.29 841 9.29 903 9.29 966 9.30 028 9.30 090 9.30 151 9.30 213 9.30 275 9.30 336 9.30 398 9.30 459 9.30 521 9.30 582 9.30 643 9.30 704 9.30 765 9.30 826 9.30 887 9.30 947 9.31 008 9.31 068 9.31 129 9.31 189 9.31 250 9.31310 9.31 370 9.31 430 9.31 490 9.31 549 9.31 609 9.31 669 9.31 728 9.31 788 L. Cos. 65 65 64 65 65 64 64 65 64 64 64 64 63 64 64 63 63 64 63 63 63 63 63 62 63 62 63 62 62 63 62 62 61 62 62 61 62 61 62 61 61 61 61 61 61 60 61 60 61 60 61 60 60 60 60 59 60 60 59 60 9.28 865 9.28 9.29 9.29 9.29 9.29 9.29 9.29 9.29 9.29 933 000 067 134 201 268 335 402 468 9.29 535 9.29 9.29 9.29 9.29 9.29 9.29 9.29 9.30 9.30 601 668 734 800 866 932 998 064 130 9.30 195 9.30 9.30 9.30 9.30 9,30 9.30 9.30 9.30 9.30 261 326 391 457 522 587 652 717 782 9.30 846 9.30 9.30 9.31 9.31 9.31 9.31 9.31 9.31 9.31 911 975 040 104 168 233 297 361 425 9.31 489 9.31 9.31 9.31 9.31 9.31 9.31 9.31 9.31 9.32 552 616 679 743 806 870 933 996 059 9.32 122 9.32 9.32 9.32 9.32 9.32 9.32 9.32 9.32 9.32 185 248 311 373 436 498 561 623 685 9.32 747 d. L. Cot. c.d 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 66 66 66 65 66 65 65 66 65 65 65 65 65 64 65 64 65 64 64 65 64 64 64 64 63 64 63 64 63 64 63 63 63 63 63 63 63 62 63 62 63 62 62 62 0.71 135 0.71 067 0.71 000 0.70 933 0.70 866 0.70 799 0.70 732 0.70 665 0.70 598 0.70 532 0.70 465 0.70 399 0.70 332 0.70 266 0.70 200 0.70 134 0.70 068 0.70 002 0.69 936 0.69 870 0.69 805 0.69 739 0.69 674 0.69 609 0.69 543 0.69 478 0.69 413 0.69 348 0.69 283 0.69 218 0.69 154 0.69 089 0.69 025 0.68 960 0.68 896 0.68 832 0.68 767 0.68 703 0.68 639 0.68 575 0.68 511 0.68 448 0.68 384 0.68 321 0.68 257 0.68 194 0.68 130 0.68 067 0.68 004 0.67 941 0.67 878 0.67 815 0.67 752 0.67 689 0.67 627 0.67 564 0.67 502 0.67 439 0.67 377 0.67 315 0.67 253 9.99 195 9.99 192 9.99 190 9.99 187 9.99 185 9.99 182 9.99 180 9.99 177 9.99 175 9.99 172 9.99 170 9.99 167 9.99 165 9.99 162 9.99 160 9.99 157 9.99 155 9.99 152 9.99 150 9.99 147 9.99 145 9.99 142 9.99 140 9.99 137 9.99 135 9.99 132 9.99 130 9.99 127 9.99 124 9.99 122 9.99 119 9.99 117 9.99 114 9.99 112 9.99 109 9.99 106 9.99 104 9.99 101 9.99 099 9.99 096 9.99 093 9.99 091 9.99 088 9.99 086 9.99 083 9.99 080 9.99 078 9.99 075 9.99 072 9.99 070 9.99 067 9.99 064 9.99 062 9.99 059 9.99 056 9.99 054 9.99 051 9.99 048 9.99 046 9.99 043 9.99 040 L. Tan. L. Sin. 78° d. ^0 59" 58 57 56 55 54 53 52 AL 50 49 48 47 46 45 44 43 42 il JO 39 38 37 36 35 34 33 31 IL 30 P.P. 5 5-7 10 II.3 IS 17-0 67 I 5.6 II.2 II.O i6.8!i6.s 20 22.7:22.3'22.0 25 28.3 27.9 27.5 30 34-0 33-5 33-0 35,39-7 39-1 38.5 40 45.3 44.7J44.0 45|5i.o 50.2 49.5 50 56.7 55-8 55.0 55162.361.4160.5 65 64 I 63 5| 54 10 10.8 15 16.2 20 21.7 25 27.1 3032.5 35 37.9' 40'43.3i 45|48.8 5054.2I 55 59.6' 5-3 5-2 10.7,10.5 i6.o;i5.8 21.3 21.0 26.7 26.2 32.0 31.5 37.3,36.8 42.7I42.0 48.0J47.2 53.3 52.5 58.7IS7.8 62 61 60 5 5-2 10 10.3 I5;i5.5 20 20.7 2525.8 3031.0 35 36.2 4041.3 4546.5 50 lO.O 15.0 S.I 10.2 15-2 20.3 20.0 25.4 25.0 30.5 30.0 35.635.0 40.7 40.0 45.845.0 50 5 1. 7 50.8 50.0 5556.8155.955.0 "I 59 5 4.9 10 9.8 15 14-8 20 19.7 24.6 29.5 34-4 39.3 4S[44.2 50 49.2 55-54.1 3j2 0.2 0.2 0.50.3 0.8 0.5 i.o 0.7 1.2 0.8 1.5 I.O l.8;l.2 2.0 1.3 2.2 I.S 2.5 1.7 2.8 1.8 P.P. 136 12 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 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d 9.31 788 9.31 847 9.31 907 9.31 966 9.32 025 9.32 084 9.32 143 9.32 202 9.32 261 9.32 319 9.32 378 9.32 437 9.32 495 9.32 553 9.32 612 9.32 670 9.32 728 9.32 786 9.32 844 9.32 902 9.32 960 9.33 018 9.33 075 9.33 133 9.33 190 9.33 248 9.33 305 9.33 362 9.33 420 9.33 477 9.33 534 9.33 591 9.33 647 9.33 704 9.33 761 9.33 818 9.33 874 9.33 931 9.33 987 9.34 043 9.34 100 9,34 156 9.34 212 9.34 268 9.34 324 9.34 380 9.34 436 9.34 491 9.34 547 9.34 602 9.34 658 9.34 713 9.34 769 9.34 824 9.34 879 9.34 934 9.34 989 9.35 044 9.35 099 9.35 154 9.35 209 59 60 59 59 59 59 59 59 58 59 59 58 58 59 58 58 58 58 58 58 58 57 58 57 58 57 57 58 57 57 57 56 57 57 57 56 57 56 56 57 56 56 56 56 56 56 55 56 55 56 55 56 55 55 55 55 55 55 55 55 9.32 747 9.32 810 9.32 872 9.32 933 9.32 995 9.33 057 9.33 119 9.33 180 9.33 242 9.33 303 9.33 365 9.33 426 9.33 487 9.33 548 9.33 609 9.33 670 9.33 731 9.33 792 9.33 853 9.33 913 9.33 974 9.34 034 9.34 095 9.34 155 9.34 215 9.34 276 9.34 336 9.34 396 9.34 456 9.34 516 9.34 576 9.34 635 9.34 695 9.34 755 9.34 814 9.34 874 9.34 933 9.34 992 9.35 051 9.35 111 9.35 170 9.35 229 9.35 288 9.35 347 9.35 405 9.35 464 9.35 523 9.35 581 9.35 640 9.35 698 9.35 757 9.35 815 9.35 873 9.35 931 9.35 989 9.36 047 9.36 105 9.36 163 9.36 221 9.36 279 9.36 336 L. Cos. d. L. Cot. c.d 63 62 61 62 62 62 61 62 61 62 61 61 61 61 61 61 61 61 60 61 60 61 60 60 61 60 60 60 60 60 59 60 60 59 60 59 59 59 60 59 59 59 59 58 59 59 58 59 58 59 58 58 58 58 58 58 58 58 58 57 0.67 253 0.67 190 0.67 128 0.67 067 0.67 005 0.66 943 0.66 881 0.66 820 0.66 758 0.66 697 0.66 635 0.66 574 0.66 513 0.66 452 0.66 391 0.66 330 0.66 269 0.66 208 0.66 147 0.66 087 0.66 026 0.65 966 0.65 905 0.65 845 0.65 785 0.65 724 0.65 664 0.65 604 0.65 544 0.65 484 0.65 424 0.65 365 0.65 305 0.65 245 0.65 186 0.65 126 0.65 067 0.65 008 0.64 949 0.64 889 0.64 830 0.64 771 0.64 712 0.64 653 0.64 595 0.64 536 0.64 477 0.64 419 0.64 360 0.64 302 0.64 243 0.64 185 0.64 127 0.64 069 0.64 011 0.63 953 0.63 895 0.63 837 0.63 779 0.63 721 0.63 664 L. Tan. L. Sin 9.99 040 9.99 038 9.99 035 9.99 032 9.99 030 9.99 027 9.99 024 9.99 022 9.99 019 9.99 016 9.99 013 9.99 011 9.99 008 9.99 005 9.99 002 9.99 000 9.98 997 9.98 994 9.98 991 9.98 989 9.98 986 9.98 983 9.98 980 9.98 978 9.98 975 9.98 972 9.98 969 9.98 967 9.98 964 9.98 961 9.98 958 9.98 955 9.98 953 9.98 950 9.98 947 9.98 944 9.98 941 9.98 938 9.98 936 9.98 933 9.98 930 9.98 927 9.98 924 9.98 921 9.98 919 9.98 916 9.98 913 9.98 910 9.98 907 9.98 904 9.98 901 9.98 898 9.98 896 9.98 893 9.98 890 9.98 887 9.98 884 9.98 881 9.98 878 9.98 875 9.98 872 30 P.P. 63 I 62 I 61 5 5-2 10 lo.s 15,15.8 20 21.0 2526. 2 3o!3i-5 35!36.8 40 42.0 4S'47-2 SO 52.S ss 57.8 1 5-2 5-1 10.3 10.2 jiS-5 15-2 20.7 20.3 25.8 25.4 I31.030.5 36.2 35.6 41-340.7 46.545.8 51.7 50.8 '56.8I55.9 // 60 69 1 58 5 5.0! 4.9! 4.8 10 10.0: 9.8 9.7 15 15-0 I4.8|I4.S 20 20.0 19.7 19.3 25 25.o'24.6 24.2 30 30.0 29. s 29.0 35,35.034-433.8 40|40.o 39.338.7 4545.0 44.2 43.5 50*50.0 49.2 48.3 55 S5.0 54.1 53.2 57 I 4.8 9.5 15 14.2 20 19.0 2523.8: 3028.5 35 33.2 4038.0 4542.8 5047.5 5552.2 56 4.7 9-3 14.0 55 4.6 9.2 13.8 18.7 18.3 23.3,22.9 28.0 27.5 32.7132.1 37.3{36.7 42.0'41.2 46.7145.8 5l.3'S0.4 "13 12 5 0.2 0.2 100.5 0.3 i5lo.8'o.s 20:1.0 0.7 25J1.2 0.8 30 1.51.0 3S'l.8;i.2 40 2.0 1.3 45,2.2 1.5 502.5 1.7 552.8 1.8 P.P. 77° 13° 137 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 J9 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 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. 9.35 209 9.35 263 9.35 318 9.35 373 9.35 427 9.35 481 9.35 536 9.35 590 9.35 644 9.35 698 9.35 752 9.35 806 9.35 860 9.35 914 9.35 968 9.36 022 9.36 075 9.36 129 9.36 182 9.36 236 9.36 289 9.36 342 9.36 395 9.36 449 9.36 502 9.36 555 9.36 608 9.36 660 9.36 713 9.36 766 9.36 819 9.36 871 9.36 924 9.36 976 9.37 028 9.37 081 9.37 133 9.37 185 9.37 237 9.37 289 9.37 341 9.37 393 9.37 445 9.37 497 9.37 549 9.37 600 9.37 652 9.37 703 9.37 755 9.37 806 37 858 37 909 37 960 38 011 38 062 38 113 38 164 38 215 38 266 38 317 38 368 L. Cos. 54 55 55 54 54 55 54 54 54 54 54 54 54 54 54 53 54 53 54 53 53 53 54 53 53 53 52 53 53 53 52 53 52 52 53 52 52 52 52 52 52 52 52 52 51 52 51 52 51 52 51 51 51 51 51 51 51 51 51 51 9.36 336 9.36 9.36 9.36 9.36 9.36 9.36 9.36 9.36 9.36 394 452 509 566 624 681 738 795 852 9.36 909 9.36 9.37 9.37 9.37 9.37 9.37 9.37 9.37 9.37 966 023 080 137 193 250 306 363 419 9.37 476 9.37 9.37 9.37 9.37 9.37 9.37 9.37 9.37 9.37 532 588 644 700 756 812 868 924 980 9.38 035 9.38 9.38 9.38 9.38 9.38 9.38 9.38 9.38 9.38 091 147 202 257 313 368 423 479 534 9.38 589 9.38 9.38 9.38 9.38 9.38 9.38 9.38 9.39 9.39 9.39 644 699 754 808 863 918 972 027 082 136 9.39 9.39 9.39 9.39 9.39 9.39 9.39 9.39 9.39 190 245 299 353 407 461 515 569 623 9.39 677 d. L. Cot. c.d. 58 58 57 57 58 57 57 57 57 57 57 57 57 57 56 57 56 57 56 57 56 56 56 56 56 56 56 56 56 55 56 56 55 55 56 55 55 56 55 55 55 55 55 54 55 55 54 55 55 54 54 55 54 54 54 54 54 54 54 54 0.63 664 0.63 606 0.63 548 0.63 491 0.63 434 0.63 376 0.63 319 0.63 262 0.63 205 0.63 148 0.63 091 0.63 034 0.62 977 0.62 920 0.62 863 0.62 807 0.62 750 0.62 694 0.62 637 0.62 581 0.62 524 0.62 468 0.62 412 0.62 356 0.62 300 0.62 244 0.62 188 0.62 132 0.62 076 0.62 020 0.61 965 0.61 909 0.61 853 0.61 798 0.61 743 0.61 687 0.61 632 0.61 577 0.61 521 0.61 466 0.61411 0.61 356 0.61 301 0.61 246 0.61 192 0.61 137 0.61 082 0.61 028 0.60 973 0.60 918 0.60 864 0.60 810 0.60 755 0.60 701 0.60 647 0.60 593 0.60 539 0.60 485 0.60 431 0.60 377 0.60 323 L. Tan. L. Sin 9.98 872 9.98 869 9.98 867 9.98 864 9.98 861 9.98 858 9.98 855 9.98 852 9.98 849 9.98 846 9.98 843 9.98 840 9.98 837 9.98 834 9.98 831 9.98 828 9.98 825 9.98 822 9.98 819 9.98 816 9.98 813 9.98 810 9.98 807 9.98 804 9.98 801 9.98 798 9.98 795 9.98 792 9.98 789 9.98 786 9.98 783 9.98 780 9.98 777 9.98 774 9.98 771 9.98 768 9.98 765 9.98 762 9.98 759 9.98 756 9.98 753 9.98 750 9.98 746 9.98 743 9.98 740 9.98 737 9.98 734 9.98 731 9.98 728 9.98 725 9.98 722 9.98 719 9.98 715 9.98 712 9.98 709 9.98 706 9.98 703 9.98 700 9.98 697 9.98 694 9.98 690 P.P. n 51 4-8; lo; 9-7| IS 14-5 20 19.3 25i24.2j 30 29.0' 35133-8 4038.71 45 43-5; 5048-3 55 S3-2I 58 I 57 I 56 4.8 4-7 9-5 9-3 14.2 14.0 19.0 18.7 23-8 23.3 28. S 28.0 33-2 32.7 38.037.3 42.8 42.0 47-5 46.7 52.2 S1.3 55 I 54 53 4-6 9-2 13-8 18.3 25122.9 30 27.5 35 32.1 4036.7 45 41-2 50I45.8 55 150.4 4-5 4-4 9.0 13-5 13-2 18.0 17-7 22.5 22.1 27.0 26.5 31-5 30.9 36.0 35-3 40.5 39-8 45-0,44-2 49.5'48.6 52 51 4-3 8.7 13.0 17-3 21.7 30 26.0 35*30.3 40134.7 45j39-0 5043-3 SS!47.7 4.2 8-5 12.8 17.0 21.2 25-5 29.8 340 38.2 42-5 46.8 4 3 2 50.3 100.7 15 i-o 20 1.3 25'i-7 i.o 0.7 1.20.8 302.0 1.5 1.0 35 2.3 i.8|i.2 40 2.712.0 1.3 45i3-0,2.2 i.S 50 3-3 2.5 1.7 55 3-7 2.8 1.8 P.P. 7AO 138 14^ 10 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 60 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d 9.38 368 9.38 418 9.38 469 9.38 519 9.38 570 9.38 620 9.38 670 9.38 721 9.38 771 9.38 821 9.38 871 9.38 921 9.38 971 9.39 021 9.39 071 9.39 121 9.39 170 9.39 220 9.39 270 9.39 319 9.39 369 9.39 418 9.39 467 9.39 517 9.39 566 9.39 615 9.39 664 9.39 713 9.39 762 9.39 811 9.39 860 9.39 909 9.39 958 9.40 006 9.40 055 9.40 103 9.40 152 9.40 200 9.40 249 9.40 297 9.40 346 9.40 394 9.40 442 9.40 490 9.40 538 9.40 586 9.40 634 9.40 682 9.40 730 9.40 778 9.40 825 9.40 873 9.40 921 9.40 968 9.41 016 9.41 063 9.41 111 9.41 158 9.41 205 9.41 252 9.41 300 50 51 50 51 50 50 51 50 50 50 50 50 50 50 50 49 50 50 49 50 49 49 50 49 49 49 49 49 49 49 49 49 48 49 48 49 48 49 48 49 48 48 48 48 48 48 48 48 48 47 48 48 47 48 47 48 47 47 47 48 9.39 677 9.39 731 9.39 785 9.39 838 9.39 892 9.39 945 9.39 999 9.40 052 9.40 106 9.40 159 9.40 212 9.40 266 9.40 319 9.40 372 9.40 425 9.40 478 9.40 531 9.40 584 9.40 636 9.40 689 9.40 742 9.40 795 9.40 847 9.40 900 9.40 952 9.41 005 9.41 057 9.41 109 9.41 161 9.41 214 9.41 266 9.41 318 9.41 370 9.41 422 9.41 474 9.41 526 9.41 578 9.41 629 9.41 681 9.41 733 9.41 784 9.41 836 9.41 887 9.41 939 9.41 990 9.42 041 9.42 093 9.42 144 9.42 195 9.42 246 9.42 297 L. Cos. d. L. Cot. 9.42 348 9.42 399 9.42 450 9.42 501 9.42 552 9.42 603 9.42 653 9.42 704 9.42 755 9.42 805 54 54 53 54 53 54 53 54 53 53 54 53 53 53 53 53 53 52 53 53 53 52 53 52 53 52 52 52 53 52 52 52 52 52 52 52 51 52 52 51 52 51 52 51 51 52 51 51 51 51 51 51 51 51 51 51 50 51 51 50 c.d. 0.60 323 0^60 269 0.60 215 0.60 162 0.60 108 0.60 055 0.60 001 0.59 948 0.59 894 0.59841 0.59 788 0.59 734 0.59 681 0.59 628 0.59 575 0.59 522 0.59 469 0.59 416 0.59 364 0.59 311 0.59 258 0.59 205 0.59 153 0.59 100 0.59 048 0.58 995 0.58 943 0.58 891 0.58 839 0.58 786 0.5 8 734 0.58 682" 0.58 630 0.58 578 0.58 526 0.58 474 0.58 422 0.58 371 0.58 319 0.58 267 0.58 216 0.58 164 0.58 113 0.58 061 0.58 010 0.57 959 0.57 907 0.57 856 0.57 805 0.57 754 0.57 703 0.57 652 0.57 601 0.57 550 0.57 499 0.57 448 0.57 397 0.57 347 0.57 296 0.57 245 0.57 195 9.98 690 9.98 687 9.98 684 9.98 681 9.98 678 9.98 675 9.98 671 9.98 668 9.98 665 9.98 662 9.98 659 9.98 656 9.98 652 9.98 649 9.98 646 9.98 643 9.98 640 9.98 636 9.98 633 9.98 630 9.98 627 9.98 623 9.98 620 9.98 617 9.98 614 9.98 610 9.98 607 9.98 604 9.98 601 9.98 597 9.98 594 9.98 591 9.98 588 9.98 584 9.98 581 9.98 578 9.98 574 9.98 571 9.98 568 9.98 565 9.98 561 9.98 558 9.98 555 9.98 551 9.98 548 9.98 545 9.98 541 9.98 538 9.98 535 9.98 531 9.98 528 9.98 525 9.98 521 9.98 518 9.98 515 9.98 511 9.98 508 9.98 505 9.98 501 9.98 498 9.98 494 L. Tan. 75° L. Sin. d. 60 59 58 57 56 55 54 53 52 50^ 49 48 47 46 45 44 43 42 40 39 38 37 36 35 34 33 31 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1 P.P. 54 52 5 4-5 4- 10 j Q.o! 8. I5|l3.5ll3. 20|i8.oli7, 25J22.S!22, 3o'27.o 26, 3S'3i.5 30. 40 36.0 35. 4540. S39> 50'4S.o:44. SS;49.5i48. 4.3 8.7 2 j 13.0 7 17.3 1 21.7 S'26.0 930.3 3 34-7 839.0 2 43.3 647.7 "I 51 5 4-2 io| 8.5 IS 12.8 20 17.0 2S[2I.2 30:25.5 35,29.8 4034-0 4538.2 5042.5 5546.8 50 I 49 4.21 4.1 8.3 8.2 12.5 12.2 16.7 16.3 20.8 20.4 25.0 24.S 29.2 33.3 37.5 41.7 45.8 28.6 32.7 36.8 40.8 44.9 48 I 47 4-0 3-9 8.0 7.8 i2.o'ii.8 16.0,15.7 20.0|l9.6 24.0 23.5 35]28.o 27.4 4032.031.3 4536.035.2 5o'40.o!39.2 55 44.o'43.i " 4 I 3 5 o.3'o.2 10 0.7 0.5 15 1.0^0.8 20 1.311.0 25 I.7|I.2 30 2.0jl.5 35 2.3I1.8 40 2.7|2.0 453.0 2.2 50 3.3 2.5 55 3.7 2.8 P.P. 15° 139 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 31 32 33 34 35 36 37 38 39 41 42 43 44 45 46 47 48 49 50 60 L. Sin. 9.41 300 9.41 347 9.41 394 9.41 441 9.41 488 9.41 535 9.41 582 9.41 628 9.41 675 9.41 722 9.41 768 9.41815 9.41 861 9.41 908 9.41 954 9.42 001 9.42 047 9.42 093 9.42 140 9.42 186 9.42 232 9.42 278 9.42 324 9.42 370 9.42 416 9.42 461 9.42 507 9.42 553 9.42 599 9.42 644 9.42 690 9.42 735 9.42 781 9.42 826 9.42 872 9.42 917 9.42 962 9.43 008 9.43 053 9.43 098 9.43 143 9.43 188 9.43 233 9.43 278 9.43 323 9.43 367 9.43 412 9.43 457 9.43 502 9.43 546 9.43 591 9.43 635 9.43 680 9.43 724 9.43 769 9.43 813 9.43 857 9.43 901 9.43 946 9.43 990 9.44 034 L. Cos. d. L. Tan. c.d. L. Cot. L. Cos. d. 47 47 47 47 47 47 46 47 47 46 47 46 47 46 47 46 46 47 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 46 45 45 46 45 45 45 45 45 45 45 44 45 45 45 44 45 44 45 44 45 44 44 44 45 44 44 42 805 ,42 856 ,42 906 ,42 957 43 007 43 057 43 108 43 158 43 208 43 258 43 308 43 358 43 408 43 458 43 508 43 558 43 607 43 657 43 707 43 756 43 806 43 855 43 905 43 954 44 004 44 053 44 102 44 151 44 201 44 250 44 299 44 348 44 397 44 446 44 495 44 544 44 592 44 641 44 690 44 738 44 787 44 836 44 884 44 933 44 981 45 029 45 078 45 126 45 174 45 222 45 271 45 319 45 367 45 415 45 463 45 511 45 559 45 606 45 654 45 702 45 750 d. L. Cot. c.d 51 50 51 50 50 51 50 50 50 50 50 50 50 50 50 49 50 50 49 50 49 50 49 50 49 49 49 50 49 49 49 49 49 49 49 48 49 49 48 49 59 48 49 48 48 49 48 48 48 49 48 48 48 48 48 48 47 48 48 48 0.57 195 0.57 144 0.57 094 0.57 043 0.56 993 0.56 943 0.56 892 0.56 842 0.56 792 0.56 742 0.56 692 0.56 642 0.56 592 0.56 542 0.56 492 0.56 442 0.56 393 0.56 343 0.56 293 0.56 244 0.56 194 0.56 145 0.56 095 0.56 046 0.55 996 0.55 947 0.55 898 0.55 849 0.55 799 0.55 750 0.55 701 0.55 652 0.55 603 0.55 554 0.55 505 0.55 456 0.55 408 0.55 359 0.55 310 0.55 262 0.55 213 0.55 164 0.55 116 0.55 067 0.55 019 0.54 971 0.54 922 0.54 874 0.54 826 0.54 778 0.54 729 0.54 681 0.54 633 0.54 585 0.54 537 0.54 489 0.54 441 0.54 394 0.54 346 0.54 298 0.54 250 L. Tan. L. Sin. 9.98 494 9.98 491 9.98 488 9.98 484 9.98 481 9.98 477 9.98 474 9.98 471 9.98 467 9.98 464 9.98 460 9.98 457 9.98 453 9.98 450 9.98 447 9.98 443 9.98 440 9.98 436 9.98 433 9.98 429 9.98 426 9.98 422 9.98 419 9.98 415 9.98 412 9.98 409 9.98 405 9.98 402 9.98 398 9.98 395 9.98 391 9.98 388 9.98 384 9.98 381 9.98 377 9.98 373 9.98 370 9.98 366 9.98 363 9.98 359 9.98 356 9.98 352 9.98 349 9.98 345 9.98 342 9.98 338 9.98 334 9.98 331 9.98 327 9.98 324 9.98 320 9.98 317 9.98 313 9.98 309 9.98 306 9.98 302 9.98 299 9.98 295 9.98 291 9.98 288 9.98 284 d. 40 P.P. I S 4-2 lo! 8.5 I5;i2.8 20|I7.0 25 21.2 3025.5 35 29.8 4034-0 45:38.2 SOj42.5 5SI46.8, 51 ' 50 I 49 4.2 4.1 8.3 8.2 12.5 12.2 16.7 16.3 20.8 20.4 25.0I24.S 29.2 28.6 33.332.7 37.536.8 41.7 40.8 45.844.9 48 I 47 5\ 4-0 10 8.0 IS1I2.0 20J16.0 25 j 20.0 30124.0 35 28.0 40 32.0 45 36.0 SO 40.0 55 44-0 3.9 3.8 7.8; 7.7 11.8 11.5 15.715.3 19.6 19.2 23.5I23.0 27.426.8 31-3 35.2 39-2 43.1 30.7 34-5 38.3 42.2 45 44 3.8 7.5 II. 2 3.7 7.3 II.O 15.0 14.7 18.8 18.3 22.5 22.0 26.2 25.7 30.0;29.3 33.8|33.0 37.5 36.7 55141.2140.3 50.3 10 0.7 15 i.o 20J1.3 25 1.7 30i2.0 35|2.3 40 2.7 2.0 45 3.02.2 50 3.3 2.5 S5I3.7 2.8 P.P. 140 16< 10 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 60 L. Sin. 9.44 034 9.44 078 9.44 122 9.44 166 9.44 210 9.44 253 9.44 297 9.44 341 9.44 385 9.44 428 9.44 472 9.44 516 9.44 559 9.44 602 9.44 646 9.44 689 9.44 733 9.44 776 9.44 819 9.44 862 9.44 905 9.44 948 8.44 992 9.45 035 9.45 077 9.45 120 9.45 163 9.45 206 9.45 249 9.45 292 9.45 334 9.45 377 9.45 419 9.45 462 9.45 504 9.45 547 9.45 589 9.45 632 9.45 674 9.45 716 9.45 758 9.45 801 9.45 843 9.45 885 9.45 927 9.45 969 9.46 011 9.46 053 9.46 095 9.46 136 9.46 178 9.46 220 9.46 262 9.46 303 9.46 345 9.46 386 9.46 428 9.46 469 9.46 511 9.46 552 9.46 594 L. Cos. 44 44 44 44 43 44 44 44 43 44 44 43 43 44 43 44 43 43 43 43 43 44 43 42 43 43 43 43 43 42 43 42 43 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 41 42 42 42 41 42 41 42 41 42 41 42 L. Tan. c.d. L. Cot. L. Cos. d. 9.45 750 9.45 797 9.45 845 9,45 892 9.45 940 9.45 987 9.46 035 9.46 082 9.46 130 9.46 177 9.46 224 9.46 271 9.46 319 9.46 366 9.46 413 9.46 460 9.46 507 9.46 554 9.46 601 9.46 648 9.46 694 9.46 741 9.46 788 9.46 835 9.46 881 9.46 928 9.46 975 9.47 021 9.47 068 9.47 114 9.47 160 47 48 47 48 47 48 47 48 47 47 47 48 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 46 46 47 46 46 47 46 46 46 46 46 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 45 46 45 45 46 4S 9.48 534 L. Cot. c.d 9.47 207 9.47 253 9.47 299 9.47 346 9.47 392 9.47 438 9.47 484 9.47 530 9.47 576 9.47 622 9.47 668 9.47 714 9.47 760 9.47 806 9.47 852 9.47 897 9.47 943 9.47 989 9.48 035 9.48 080 9.48 126 9.48 171 9.48 217 9.48 262 9.48 307 9.48 353 9.48 398 9.48 443 9.48 489 0.54 250 0.54 203 0.54 155 0.54 108 0.54 060 0.54 013 0.53 965 0.53 918 0.53 870 0.53 823 0.53 776 0.53 729 0.53 681 0.53 634 0.53 587 0.53 540 0.53 493 0.53 446 0.53 399 0.53 352 0.53 306 0.53 259 0.53 212 0.53 165 0.53 119 0.53 072 0.53 025 0.52 979 0.52 932 0.52 886 0.52 840 0.52 793 0.52 747 0.52 701 0.52 654 0.52 608 0.52 562 0.52 516 0.52 470 0.52 424 0.52 378 0.52 332 0.52 286 0.52 240 0.52 194 0.52 148 0.52 103 0.52 057 0.52 011 0.51 965 0.51 920 0.51 874 0.51 829 0.51 783 0.51 738 0.51 693 0.51 647 0.51 602 0.51 557 0.51 511 0.51 466 9.98 284 9.98 281 9.98 277 9.98 273 9.98 270 9.98 266 9.98 262 9.98 259 9.98 255 9.98 251 9.98 248 9.98 244 9.98 240 9.98 237 9.98 233 9.98 229 9.98 226 9.98 222 9.98 218 9.98 215 9.98 211 9.98 207 9.98 204 9.98 200 9.98 196 9.98 192 9.98 189 9.98 185 9.98 181 9.98 177 9.98 174 9.98 170 9.98 166 9.98 162 9.98 159 9.98 155 9.98 151 9.98 147 9.98 144 9.98 140 9.98 136 9.98 132 9.98 129 9.98 125 9.98 121 9.98 117 9.98 113 9.98 110 9.98 106 9.98 102 9.98 098 9.98 094 9.98 090 9.98 087 9.98 083 9.98 079 9.98 075 9.98 071 9.98 067 9.98 063 9.98 060 L. Tan. L. Sin. d. P.P. " 48 47 46 5 4.0 3.9 3.8 10 8.0 7.8 7.7 15 12.0 11.8 1 1.5 20 i6.o!i5.7 15.3 25 20.0 19.6 19.2 30 24.0 23.5(23.0 35 28.0 27.4 26.8 40 32.0 31.330.7 45 36.0 35.234.5 SO 40.0 39.238.3 55 44.0 43.1 42.2 45 44 43 3.8 7.5 II. 2 15-0 18.8 3022. 5 35 26.2 40! 30.0 45 33.8 5037.5 S5I41.2 3-7 7.3 II.O 14.7 18.3 22.0 25-7 29-3 33-0 136.7 I40.3 3.6 7.2 10.8 14.3 17.9 2I.S 25.1 28.7 32.2 35.8 39.4 "I 42 5 3.5 10 7.0 15 10.5 20 14.0 17.5 21.0 41 3.4 6.8 10.2 13.7 17. 1 20.5 24.5123.9 28.0|27.3 31.530.8 35.034-2 38.537.6 " ; 4 I 3 5 o.3'o.2 10 0.7 0.5 IS i.ojo.S 20 1.3 i.o 25 1.7 1.2 30 2.0I1.S 35 2.3 1.8 40 2.7 2.0 45 3.0 2.2 50 3-3 2.5 S53.7I2.8 P.P. 17* 141 L. Sin. L. Tan. c.d. L. Cot. L. Cos. d. P.P. 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 9.46 594 9.46 635 9.46 676 9.46 717 9.46 758 9.46 800 9.46 841 9.46 882 9.46 923 9.46 964 9.47 005 9.47 045 9.47 086 9.47 127 9.47 168 9.47 209 9.47 249 9.47 290 9.47 330 9.47 371 9.47 411 9.47 452 9.47 492 9.47 533 9.47 573 9.47 613 9.47 654 9.47 694 9.47 734 9.47 774 9.47 814 9.47 854 9.47 894 9.47 934 9.47 974 9.48 014 9.48 054 9.48 094 9.48 133 9.48 173 9.48 213 9.48 252 9.48 292 9.48 332 9.48 371 9.48 411 9.48 450 9.48 490 9.48 529 9.48 568 9.48 607 9.48 647 9.48 686 9.48 725 9.48 764 9.48 803 9.48 842 9.48 881 9.48 920 9.48 959 9.48 998 41 41 41 41 42 41 41 41 41 41 40 41 41 41 41 40 41 40 41 40 41 40 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 39 40 40 39 40 40 39 40 39 40 39 39 39 40 39 39 39 39 39 39 39 39 39 48 534 48 579 48 624 48 669 48 714 48 759 48 804 48 849 48 894 48 939 48 984 49 029 49 073 49 118 49 163 49 207 49 252 49 296 49 341 49 385 49 430 49 474 49 519 49 563 49 607 49 652 49 696 49 740 49 784 49 828 49 872 49 916 49 960 50 004 50 048 50 092 50 136 50 180 50 223 50 267 50 311 50 355 50 398 50 442 50 485 50 529 50 572 50 616 50 659 50 703 50 746 50 789 50 833 50 876 50 919 50 962 51005 51048 51092 51 135 51 178 45 45 45 45 45 45 45 45 45 45 45 44 45 45 44 45 44 45 44 45 44 45 44 44 45 44 44 44 44 44 44 44 44 44 44 44 44 43 44 44 44 43 44 43 44 43 44 43 44 43 43 44 43 43 43 43 43 44 43 43 0.51 466 9.98 060 0.51421 0.51 376 0.51 331 0.51 286 0.51 241 0.51 196 0.51 151 0.51 106 0.51 061 0.51 016 0.50 971 0.50 927 0.50 882 0.50 837 0.50 793 0.50 748 0.50 704 0.50 659 0.50 615 9.98 056 9.98 052 9.98 048 9.98 044 9.98 040 9.98 036 9.98 032 9.98 029 9.98 025 9.98 021 9.98 017 9.98 013 9.98 009 9.98 005 9.98 001 9.97 997 9.97 993 9.97 989 9.97 986 0.50 570 9.97 982 0.50 526 0.50 481 0.50 437 0.50 393 0.50 348 0.50 304 0.50 260 0.50 216 0.50 172 9.97 978 9.97 974 9.97 970 9.97 966 9.97 962 9.97 958 9.97 954 9.97 950 9.97 946 0.50 128 9.97 942 0.50 084 0.50 040 0.49 996 0.49 952 0.49 908 0.49 864 0.49 820 0.49 777 0.49 733 9.97 938 9.97 934 9.97 930 9.97 926 9.97 922 9.97 918 9.97 914 9.97 910 9.97 906 049^89 0.49 645 0.49 602 0.49 558 0.49 515 0.49 471 0.49 428 0.49 384 0.49 341 0.49 297 9.97 902 9.97 898 9.97 894 9.97 890 9.97 886 9.97 882 9.97 878 9.97 874 9.97 870 9.97 866 0.49 254 9.97 861 0.49 211 0.49 167 0.49 124 0.49 081 0.49 038 0.48 995 0.48 952 0.48 908 0.48 865 9.97 857 9.97 853 9.97 849 9.97 845 9.97 841 9.97 837 9.97 833 9.97 829 9.97 825 0.48 822 9.97 821 10 45 44 43 3.8 7-5 II. 2 15.0 i8.8 22.5 26.2 40 30.0 4533-8 5037.S 5541.2 3.7 3.6 7-3 7.2 ii.o 10.8 I4.7ji4.3 18.3 17-9 22.0 21.5 25.7 25. 1 29.3'28.7 33.032.2 36.7'35.8 40.3:39.4 42 41 51 3.5 10 7.0 15 10.5 20' 14.0 25;I7.S 3021.0 3.4 6.8 10.2 13.7 17.1 20.5 35 24.S 23.9 40 28.0 45 31.5 SO 35.0 5538.5 27.3 30.8 34-2 37.6 " I 40 I 39 5 3.3 3.2 io[ 6.7 6.5 15 lO.O 9.8 20 13.3; 13.0 25:16.7 16.2 30 20.0 35 23.3 40; 26.7 45 30.0 50 33-3 55 36.7 19.S 22.8 26.0 29.2 32.5 35.8 "1 5 1 50.4! 10 0.8 15 1.2 20 1.7 252. 1 30J2.5 35 2.9 4 3 2.7 2.0 3.0:2.2 3.3 2.S 3.7i2.8 L. Cos. d. L. Cot. c.d L. Tan. L. Sin. d. P.P. 142 20 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 L. Sin. 9.48 998 9.49 037 9.49 076 9.49 115 9.49 153 9.49 192 9.49 231 9.49 269 9.49 308 9.49 347 9.49 385 9.49 424 9.49 462 9.49 500 9.49 539 9.49 577 9.49 615 9.49 654 9.49 692 9.49 730 9.49 768 9.49 806 9.49 844 9.49 882 9.49 920 9.49 958 9.49 996 9.50 034 9.50 072 9.50 110 9.50 148 9.50 185 9.50 223 9.50 261 9.50 298 9.50 336 9.50 374 9.50 411 9.50 449 9.50 486 9.50 523 9.50 561 9.50 598 9.50 635 9.50 673 9.50 710 9.50 747 9.50 784 9.50 821 9.50 858 9.50 896 50 933 50 970 51007 51043 51080 51 117 51 154 51 191 51 227 51 264 39 39 39 38 39 39 38 39 39 38 39 38 38 39 38 38 39 38 38 38 38 38 38 38 38 38 38 38 38 38 37 38 38 37 38 38 37 38 37 37 38 37 37 38 37 37 37 37 37 38 37 37 37 36 37 37 37 37 36 37 18^ L. Tan. c.d. L. Cot. L. Cos. d. 9.51 178 9.51 221 9.51 264 9.51 306 9.51 349 9.51 392 9.51 435 9.51 478 9.51 520 9.51 563 9.51 606 9.51 648 9.51 691 9.51 734 9.51 776 9.51 819 9.51 861 9.51 903 9.51 946 9.51 988 9.52 031 9.52 073 9.52 115 9.52 157 9.52 200 9.52 242 9.52 284 9.52 326 9.52 368 9.52 410 9.52 452 9.52 494 9.52 536 9.52 578 9.52 620 9.52 661 9.52 703 9.52 745 9.52 787 9.52 829 9.52 870 9.52 912 9.52 953 9.52 995 9.53 037 9.53 078 9.53 120 9.53 161 9.53 202 9.53 244 9.53 285 0.53 327 9.53 368 9.53 409 9.53 450 9.53 492 9.53 533 9.53 574 9.53 615 9.53 656 9.53 697 43 43 42 43 43 43 43 42 43 43 42 43 43 42 43 42 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 42 42 42 41 42 42 42 42 41 42 41 42 42 41 42 41 41 42 41 42 41 41 41 42 41 41 41 41 41 0.48 822 0.48 779 0.48 736 0.48 694 0.48 651 0.48 608 0.48 565 0.48 522 0.48 480 0.48 437 0.48 394 0.48 352 0.48 309 0.48 266 0.48 224 0.48 181 0.48 139 0.48 097 0.48 054 0.48 012 0.47 969 0.47 927 0.47 885 0.47 843 0.47 800 0.47 758 0.47 716 0.47 674 0.47 632 0.47 590 9.97 821 9.97 817 9.97 812 9.97 808 9.97 804 9.97 800 9.97 796 9.97 792 9.97 788 9.97 784 9.97 779 9.97 775 9.97 771 9.97 767 9.97 763 9.97 759 9.97 754 9.97 750 9.97 746 9.97 742 9.97 738 0.47 548 0.47 506 0.47 464 0.47 422 0.47 380 0.47 339 0.47 297 0.47 255 0.47 213 0.47 171 0.47 130 0.47 088 0.47 047 0.47 005 0.46 963 0.46 922 0.46 880 0.46 839 0.46 798 0.46 756 0.46 715 0.46 673 0.46 632 0.46 591 0.46 550 0.46 508 0.46 467 0.46 426 0.46 385 0.46 344 0.46 303 9.97 734 9.97 729 9.97 725 9.97 721 9.97 717 9.97 713 9.97 708 9.97 704 9.97 700 9.97 696 9.97 691 9.97 687 9.97 683 9.97 679 9.97 674 9.97 670 9.97 666 9.97 662 9.97 657 9.97 653 9.97 649 9.97 645 9.97 640 9.97 636 9.97 632 9.97 628 9.97 623 9.97 619 9.97 615 9.97 610 9.97 606 9.97 602 9.97 597 9.97 593 9.97 589 9.97 584 9.97 580 9.97 576 9.97 571 9.97 567 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin 71° P.P. 43 3.6 7.2 I0.8 14.3 17.9 3.4 6.8 10.2 30121.5 25.1 28.7 32.2 35.8 39-4 42 3.5 7.0 10.5 14.0I13.7 17.5117.1 21.0 20.5 24.5 23.9 28.0:27.3 3I.S30.8 35.0 34.2 38.5'37.6 39 38 37 3.2 6.5 9.8 13.0 16.2 19-5 22.8 4O126.O 45 39.2I28.5 5032.5131.7 55'35.8'34.8 3.2 6.3 9.5 12.7 15.8 19.0 22.2 25.3 3.1 6.2 9.2 12.3 15.4 18.5 21.6 24.7 27.8 30.8 33.9 " I 36 I 5 I 4 51 3.00.4,0.3 10 6.0 0.8 0.7 15' 9-0 1.2 I.O 20 12.0 1.7 1.3 25 15.0 2.1 1.7 30 18.0 2.5 2.0 35 21.02.9 2.3 40 24.0 3.3 2.7 45,27.03.8 3.0 so'30.0 4.2 3.3 5533.04.6 3.7 P.P. 19° 143 10 11 12 13 14 15 16 17 18 19 20 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 L. Sin. d. ,51 264 51 301 .51 338 51374 51411 51447 51 484 51 520 51 557 51 593 9.51629 .51 666 .51 702 .51 738 .51 774 .51811 ,51 847 ,51 883 51919 51955 9.51991 .52 027 .52 063 .52 099 .52 135 .52 171 .52 207 .52 242 .52 278 .52 314 I 9.52 350 .52 385 .52 421 .52 456 .52 492 .52 527 .52 563 .52 598 .52 634 .52 669 ! 9.52 705 .52 740 .52 775 .52 811 .52 846 .52 881 .52 916 .52 951 .52 986 .53 021 9.53 056 I 53 092 53 126 53 161 53 196 53 231 53 266 53 301 53 336 ,53 370 53 405 L. Cos. 37 37 36 37 36 37 36 37 36 36 37 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 35 36 35 36 35 36 35 36 35 35 36 35 35 35 35 35 35 35 36 34 35 35 35 35 35 35 34 35 L. Tan. c.d. L. Cot. 9.53 697 9.53 738 9.53 779 9.53 820 9.53 861 9.53 902 9.53943 9.53 984 9.54 025 9.54 065 9.54 106 9,54 147 9.54 187 9.54 228 9.54 269 9.54 309 9.54 350 9.54 390 9.54 431 9.54 471 9.54 512 9.54 552 9.54 593 9.54 633 9.54 673 9.54 714 9.54 754 9.54 794 9.54 835 9.54 875 9.54 915 9.54 955 9.54 995 55 035 55 075 55 115 55 155 55 195 55 235 55 275 9.55 315 9.55 355 9.55 395 9.55 434 9.55 474 9.55 514 9.55 554 9.55 593 9.55 633 9.55 673 9.55 712 9.55 752 9.55 791 9.55 831 9.55 870 9.55 910 9.55 949 9.55 989 9.56 028 9.56 067 9.56 107 41 41 41 41 41 41 41 41 40 41 41 40 41 41 40 41 40 41 40 41 40 41 40 40 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 40 40 40 39 40 40 40 39 40 40 39 40 39 40 39 40 39 40 39 39 40 L. Cot. I c.d. 0.46 303 0.46 262 0.46 221 0.46 180 9.46 139 0.46 098 0.46 057 0.46 016 0.45 975 0.45 935 0.45 894 0.45 853 0.45 813 0.45 772 0.45 731 0.45 691 0.45 650 0.45 610 0.45 569 0.45 529 0.45 488 0.45 448 0.45 407 0.45 367 0.45 327 0.45 286 0.45 246 0.45 206 0.45 165 0.45 125 0.45 085 0.45 045 0.45 005 0.44 965 0.44 925 0.44 885 0.44 845 0.44 805 0.44 765 0.44 725 0.44 685 0.44 645 0.44 605 0.44 566 0.44 526 0.44 486 0.44 446 0.44 407 0.44 367 0.44 327 0.44 288 0.44 248 0.44 209 0.44 169 0.44 130 0.44 090 0.44 051 0.44 011 0.43 972 0.43 933 0.43 893 L. Cos. 9.97 567 9.97 563 9.97 558 9.97 554 9.97 550 9.97 545 9.97 541 9.97 536 9.97 532 9.97 528 9.97 523 9.97 519 9.97 515 9.97 510 9.97 506 9.97 501 9.97 497 9.97 492 9.97 488 9.97 484 9.97 479 9.97 475 9.97 470 9.97 466 9.97 461 9.97 457 9.97 453 9.97 448 9.97 444 9.97 439 9.97 435 9.97 430 9.97 426 9.97 421 9.97 417 9.97 412 9.97 408 9.97 403 9.97 399 9.97 394 9.97 390 9.97 385 9.97 381 9.97 376 9.97 372 9.97 367 9.97 363 9.97 358 9.97 353 9.97 349 9.97 344 9.97 340 9.97 335 9.97 331 9.97 326 9.97 322 9.97 317 9.97 312 9.97 308 9.97 303 9.97 299 L. Tan. L. Sin. 7(V d. 40 39 38 37 36 35 34 33 32 il 30 29 28 27 26 25 24 23 22 11 20 P.P. 41 I 40 I 39 34 6.8 I0.2 13-7 I7.I 20.5 23-9 27.3 30.8 34.2 37.6 I 3.3; 3-2 I 6.7| 6.5 lO.O 9.8 ,13-3 13-0 ,16.7 16.2 20.0 19.5 23.3'22.8 26.7 j 26.0 30.0 29.2 33.3'32.S 36.735.8 37 36 , 35 3-1 6.2 9.2 12.3 15.4 30|i8.5 3521.6 40,24.7 45:27.8 50I30.8 5533-9 3.0, 2.9 6.0 5-8 9.0 8.8 12.0 11.7 15.01 14.6 18.0 17.5 21.0 20.4 24.0I23.3 27.0 26.2 30.0 29.2 33-ol32.i IM 0.40.3 0.8:0.7 34 I 5 I 4 2.8 5-7 8.5 11.3 14.2 17.0 19.8 22.7 25.5 28.3 15 20 25 30 35 40 45 50 5531.2 P.P. 144 20° 1 2 3 4 5 6 7 8 _9 11 12 13 14 15 16 17 18 20 21 22 23 24 25 26 27 28 _29. 30 31 32 33 34 35 36 37 38 41 42 43 44 45 46 47 48 49^ 50 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. 9.53 405 9.53 440 9.53 475 9.53 509 9.53 544 9.53 578 9.53 613 9.53 647 9.53 682 9.53 716 9.53 751 9.53 785 9.53 819 9.53 854 9.53 888. 9.53 922 9.53 957 9.53 991 9.54 025 9.54 059 9.54 093 9.54 127 9.54 161 9.54 195 9.54 229 9.54 263 9.54 297 9.54 331 9.54 365 9.54 399 9.54 433 9.54 466 9.54 500 9.54 534 9.54 567 9.54 601 9.54 635 9.54 668 9.54 702 9.54 735 9.54 769 9.54 802 9.54 836 9.54 869 9.54 903 9.54 936 9.54 969 9.55 003 9.55 036 9.55 069 9.55 102 55 136 55 169 55 202 55 235 55 268 55 301 55 334 55 367 55 400 55 433 L. Cos. 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 34 34 34 34 34 34 34 34 34 34 34 34 33 34 34 33 34 34 33 34 33 34 33 34 33 34 33 33 34 33 33 33 34 33 33 33 33 33 33 33 33 33 9.56 107 9.56 9.56 9.56 9.56 9.56 9.56 9.56 9.56 9.56 146 185 224 264 303 342 381 420 459 9.56 887 9.56 9.56 9.57 9.57 9.57 9.57 9.57 9.57 9.57 926 965 004 042 081 120 158 197 235 9.57 274 9.57 9.57 57 57 57 57 57 57 9.57 312 351 389 428 466 504 543 581 619 9.57 658 9.57 9.57 9.57 9.57 9.57 9.57 9.57 9.57 9.58 696 734 772 810 849 887 925 963 001 9.58 039 9.58 9.58 9.58 9.58 9.58 9.58 9.58 9.58 9.58 077 115 153 191 229 267 304 342 380 9.58 418 d. L. Cot. c.d. 39 39 39 40 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 38 39 39 39 38 39 39 38 39 38 39 38 39 38 39 38 38 39 38 38 39 38 38 38 38 39 38 38 38 38 3S 38 38 38 38 38 38 37 38 38 38 0.43 893 0.43 854 0.43 815 0.43 776 0.43 736 0.43 697 0.43 658 0.43 619 0.43 580 0.43 541 0.43 502 0.43 463 0.43 424 0.43 385 0.43 346 0.43 307 0.43 268 0.43 229 0.43 190 0.43 151 0.43 113 0.43 074 0.43 035 0.42 996 0.42 958 0.42 919 0.42 880 0.42 842 0.42 803 0.42 765 0.42 726 0.42 688 0.42 649 0.42 611 0.42 572 0.42 534 0.42 496 0.42 457 0.42 419 0.42 381 0.42 342 0.42 304 0.42 266 0.42 228 0.42 190 0.42 151 0.42 113 0.42 075 0.42 037 0.41 999 0.41 961 0.41 923 0.41 885 0.41 847 0.41 809 0.41 771 0.41 733 0.41 696 0.41 658 0.41 620 0.41 582 9.97 299 9.97 294 9.97 289 9.97 285 9.97 280 9.97 276 9.97 271 9.97 266 9.97 262 9.97 257 9.97 252 9.97 248 9.97 243 9.97 238 9.97 234 9.97 229 9.97 224 9.97 220 9.97 215 9.97 210 9.97 206 9.97 201 9.97 196 9.97 192 9.97 187 9.97 182 9.97 178 9.97 173 9.97 168 9.97 163 9.97 159 9.97 154 9.97 149 9.97 145 9.97 140 9.97 135 9.97 130 9.97 126 9.97 121 9.97 116 9.97 111 9.97 107 9.97 102 9.97 097 9.97 092 9.97 087 9.97 083 9.97 078 9.97 073 9.97 068 9.97 063 9.97 059 9.97 054 9.97 049 9.97 044 9.97 039 9.97 035 9.97 030 9.97 025 9.97 020 9.97 015 L. Tan. L. Sin. d. d. 59 58 57 56 55 54 53 52 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 31 3)^ 30 29 28 27 26 25 24 23 22 IL 20^ 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 P.P. " I 40 39 38 5 3-3 10 6.7 is'io.o 13.3 16.7 20.0 ,23.3 '26.7 30.0 50133.3 55 136.7 3-2 3-2 6.5 6.3 9.8 9-5 13.0 16.2 19.5 22.8 26.0 29.2 32.5 35.8 12.7 IS.8 19.0 22.2 25.3 28.5 31.7 34.8 37 35 3.1 2.0 6.2 5.8 9.2 8.8 12.3 II.7! 15.4 14.6 18.5117.51 21.6l20.4l 24.7 23.3 27.8 26.2 30.8 29.2 33.9 32.1 34 2.8 5.7 8.5 11.3 14.2 17.0 19.8 22.7 25-5 28.3 31.2 \ 5 I 4 .8 0.4 0.3 ,5 0.8 0.7 ,2 1.2 oji.7 .8 2.1 5 2.5 ,2 2.9 03.3 ,83.8 ,54.2 ,2 4.6 P.P. ai\o 21 145 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 60 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. 9.55 433 9.55 466 9.55 499 9.55 532 9.55 564 9.55 597 9.55 630 9.55 663 9.55 695 9.55 728 9.55 761 9.55 793 9.55 826 9.55 858 9.55 891 9.55 923 9.55 956 9.55 988 9.56 021 9.56 053 9.56 085 9.56 118 9.56 150 9.56 182 9.56 215 9.56 247 9.56 279 9.56 311 9.56 343 9.56 375 9.56 408 9.56 440 9.56 472 9.56 504 9.56 536 9.56 568 9.56 599 9.56 631 9.56 663 9.56 695 9.56 727 9.56 759 9.56 790 9.56 822 9.56 854 9.56 886 9.56 917 9.56 949 9.56 980 9.57 012 9.57 044 9.57 075 9.57 107 9.57 138 9.57 169 9.57 201 9.57 232 9.57 264 9.57 295 9.57 326 9.57 358 L. Cos. 33 33 33 32 33 33 33 31 33 33 31 33 31 33 32 33 31 33 31 31 33 31 31 33 31 31 31 31 31 33 31 31 31 31 31 31 32 32 32 32 32 31 32 32 32 31 32 31 32 32 31 32 31 31 32 31 32 31 31 32 58 418 58 455 58 493 58 531 58 569 58 606 58 644 58 681 58 719 58 757 58 794 58 832 58 869 58 907 58 944 58 981 59 019 59 056 59 094 59 131 59 168 59 205 59 243 59 280 59 317 59 354 59 391 59 429 59 466 59 503 59 540 59 577 59 614 59 651 59 688 59 725 59 762 59 799 59 835 59 872 59 909 59 946 59 983 60 019 60 056 60 093 60 130 60 166 60 203 60 240 60 276 60 313 60 349 60 386 60 422 60 459 60 495 60 532 60 568 60 605 60 641 d. L. Cot. 37 38 38 38 37 3^ 37 38 3^ 37 38 37 38 37 37 3Z 31 38 37 37 37 38 37 37 37 37 3'^ 31 31 31 31 31 31 31 31 31 31 36 37 37 37 37 36 37 37 37 36 37 37 36 37 36 37 36 37 36 37 36 37 36 41 582 41 545 41 507 41 469 41 431 41 394 41 356 41 319 41 281 41 243 .41 206 41 168 41 131 41093 41056 41019 40 981 40 944 40 906 40 869 40 832 40 795 40 757 40 720 40 683 40 646 40 609 40 571 40 534 40 497 40 460 40 423 40 386 40 349 40 312 40 275 40 238 40 201 40 165 40 128 40 091 40 054 40 017 39 981 39 944 39 907 39 870 39 834 39 797 39 760 39 724 39 687 39 651 39 614 39 578 39 541 39 505 39 468 39 432 39 395 39 359 9.97 015 9.97 010 9.97 005 9.97 001 9.96 996 9.96 991 9.96 986 9.96 981 9.96 976 9.96 971 9.96 966 9.96 962 9.96 957 9.96 952 9.96 947 9.96 942 9.96 937 9.96 932 9.96 927 9.96 922 9.96 917 9.96 912 9.96 907 9.96 903 9.96 898 9.96 893 9.96 888 9.96 883 9.96 878 9.96 873 9.96 868 9.96 863 9.96 858 9.96 853 9.96 848 9.96 843 9.96 838 9.96 833 9.96 828 9.96 823 9.96 818 9.96 813 9.96 808 9.96 803 9.96 798 9.96 793 9.96 788 9.96 783 9.96 778 9.96 772 9.96 767 9.96 762 9.96 757 9.96 752 9.96 747 9.96 742 9.96 737 9.96 732 9.96 727 9.96 722 9.96 717 10 c.d. L. Tan. L. Sin 68° d. 60 59 58 57 56 55 54 53 52 50 49 48 47 46 45 44 43 42 IL 40 39 38 37 36 35 34 33 31 29 28 27 26 25 24 23 22 11. 20 P.P. 38 I 37 36 St 3-2] 10^ 6.3 IS p-s! 20 12.7 25 15.81 30 IQ.O 35 22.2 3-0 6.0 9.0 25-3 28.5 31.7 34-8 12.3 12.0 15-4 15.0 18.5 18.0 21.6 21.0 24.7 24.0 27.8 27.0 30.8|30.o 33.9i33.0 33 32 31 2.8 5.5 8.2 II.O 13.8 16.5 19.2 22.0 24.8 27.5 30.2 2.7 5.3 8.0 10.7 13.3 16.0 18.7 21.3 24.0 26.7 2.6 5-2 7.8 10.3 12.9 I5.S I8.I 20.7 23.2 25.8 29.3128.4 "I 6 SO.s 10 I. o 15,1.5 20 2.0 25 2.5 3030 35 3-5 40 '4.0 45*4.5 50,5.0 S5iS.5 5 4 P.P. 20 21 22 23 24 25 26 27 28 30 40 41 42 43 44 45 46 47 48 50 51 52 53 54 55 56 57 58 60 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d 9.57 358 9.57 389 9.57 420 9.57 451 9.57 482 9.57 514 9.57 545 9.57 576 9.57 607 9.57 638 9.57 669 9.57 700 9.57 731 9.57 762 9.57 793 9.57 824 9.57 855 9.57 885 9.57 916 9.57 947 9.57 978 9.58 008 9.58 039 9.58 070 9.58 101 9.58 131 9.58 162 9.58 192 9.58 223 9.58 253 9.58 284 9.58 314 9.58 345 9.58 375 9.58 406 9.58 436 9.58 467 9.58 497 9.58 527 9.58 557 9.58 588 9.58 618 9.58 648 9.58 678 9.58 709 9.58 739 9.58 769 9.58 799 9.58 829 9.58 859 9.58 889 58 919 58 949 58 979 59 009 59 039 59 069 59 098 59 128 59 158 59 188 31 31 31 31 32 31 31 31 31 31 31 31 a 31 31 31 30 31 31 31 30 31 31 31 30 31 30 31 30 31 30 31 30 31 30 31 30 30 30 31 30 30 30 31 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 L. Cos. 9.60 641 9.60 9.60 9.60 9.60 9.60 9.60 9.60 9.60 9.60 677 714 750 786 823 859 895 931 967 9.61 004 9.61 9.61 9.61 9.61 9.61 9.61 9.61 9.61 9.61 040 076 112 148 184 220 256 292 328 9.61 364 9.61 9.61 9.61 9.61 9.61 9.61 9.61 9.61 9.61 400 436 472 508 544 579 615 651 687 9.61 722 9.61 9.61 9.61 9.61 9.61 9.61 9.61 9.62 9.62 758 794 830 865 901 936 972 008 043 9.62 079 9.62 9.62 9.62 9.62 9.62 9.62 9.62 9.62 9.62 114 150 185 221 256 292 327 362 398 9.62 433 9.62 9.62 9.62 9.62 9.62 9.62 9.62 9.62 9.62 9.62 468 504 539 574 609 645 680 715 750 '785 d. L. Cot. c.d 36 37 36 36 37 36 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 36 36 35 36 35 36 36 35 36 35 36 35 36 35 36 35 35 36 35 35 36 35 35 35 36 35 35 35 35 0.39 359 0.39 323 0.39 286 0.39 250 0.39 214 0.39 177 0.39 141 0.39 105 0.39 069 0.39 033 0.38 996 0.38 960 0.38 924 0.38 888 0.38 852 0.38 816 0.38 780 0.38 744 0.38 708 0.38 672 0.38 636 0.38 600 0.38 564 0.38 528 0.38 492 0.38 456 0.38 421 0.38 385 0.38 349 0.38 313 0.38 278 0.38 242 0.38 206 0.38 170 0.38 135 0.38 099 0.38 064 0.38 028 0.37 992 0.37 957 0.37 921 0.37 886 0.37 850 0.37 815 0.37 779 0.37 744 0.37 708 0.37 673 0.37 638 0.37 602 0.37 567 0.37 532 0.37 496 0.37 461 0.37 426 0.37 391 0.37 355 0.37 320 0.37 285 0.37 250 0.37 215 9.96 717 9.96 711 9.96 706 9.96 701 9.96 696 9.96 691 9.96 686 9.96 681 9.96 676 9.96 670 9.96 665 9.96 660 9.96 655 9.96 650 9.96 645 9.96 640 9.96 634 9.96 629 9.96 624 9.96 619 9.96 614 9.96 608 9.96 603 9.96 598 9.96 593 9.96 588 9.96 582 9.96 577 9.96 572 9.96 567 9.96 562 9.96 556 9.96 551 9.96 546 9.96 541 9.96 535 9.96 530 9.96 525 9.96 520 9.96 514 9.96 509 9.96 504 9.96 498 9.96 493 9.96 488 9.96 483 9.96 477 9.96 472 9.96 467 9.96 461 9.96 456 9.96 451 9.96 445 9.96 440 9.96 435 9.96 429 9.96 424 9.96 419 9.96 413 9.96 408 9.96 403 L. Tan. L. Sin. 67' d. 60 59 58 57 56 55 54 53 52 ii 50 49 48 47 46 45 44 43 42 40^ 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 IL ^0 19 18 17 16 15 14 13 12 J_l 9 8 7 6 5 4 3 2 1 P.P. " I 37 ! 36 35 S 3.li 10 6.2i IS' 9.2: 20 12.3 25 1S.4 30 18. 5 35 21.6 40,24.71 45 27.8| SO 30.8 55 33.9I 2.9 5.8 8.8 12.0 11.7 15.0 14.6 18.0 17.5 21.0 20.4 24.0 23.3 27.0 26.2 30.0 29.2 33.032.1 "I 32 5 2.7 10! 5.3 15I 8.0 20! 10.7 j 25 13-3 3oii6.o 35 18.71 4021.3! 45 24.0 SO 26.71 55 29.3I 31 30 2.6 5-2 7.8 10.3 12.9 15-5 18. 1 2.5 5.0 7.5 10.0 12.S 150 17-5 20.7120.0 23.2'22.5 25.8 25.0 28.4I27.S II 29 6j5 5 2.4 0.5 0.4 10 4.8 i.o 0.8 IS 7.2 i.5il.2 20 9.7 2.0,1.7 25 12. 1 2.52.1 30 14. 5 2.0 2.5 35 16.9 3.5 2.9 40 19.3 403.3 45 21.8 4-5 3.8 SO 24.2 5-0 4-2 55 26.6 5.5 4.6 P.P. 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 41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 L. Sin. 9.59 188 9.59 218 9.59 247 9.59 277 9.59 307 9.59 336 9.59 366 9.59 484 9.59 514 9.59 543 9.59 573 9.59 602 9.59 632 9.59 661 9.59 690 9.59 720 9.59 749 d. L. Tan. c.d. 9.59 778 9.59 808 9.59 837 9.59 866 9.59 895 9.59 924 9.59 954 9.59 983 9.60 012 9.60 041 9.60 070 9.60 099 9.60 128 9.60 157 9.60 186 9.60 215 9.60 244 9.60 273 9.60 302 9.60 331 9.60 359 9.60 388 9.60 417 9.60 446 9.60 474 9.60 503 9.60 532 9.60 561 9.60 589 9.60 618 9.60 646 9.60 675 9.60 704 9.60 732 9.60 761 9.60 789 9.60 818 9.60 846 9.60 875 9.60 903 9.60 931 30 29 30 30 29 30 30 29 30 29 30 29 30 29 30 29 29 30 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 28 29 29 29 28 29 28 29 29 28 29 28 29 28 29 28 28 9.62 785 9.62 820 9.62 855 9.62 890 9.62 926 9.62 961 9.62 996 9.63 031 9.63 066 9.63 101 9.63 135 9.63 170 9.63 205 9.63 240 9.63 275 9.63 310 9.63 345 9.63 379 9.63 414 9.63 449 9.63 484 9.63 519 9.63 553 9.63 588 9.63 623 9.63 657 9.63 692 9.63 726 9.63 761 9.63 796 9.63 830 9.63 865 9.63 899 9.63 934 9.63 968 9.64 003 9.64 037 9.64 072 9.64 106 9.64 140 9.64 175 9.64 209 9.64 243 9.64 278 9.64 312 9.64 346 9.64 381 9.64 415 9.64 449 9.64 483 9.64 517 9.64 552 9.64 586 9.64 620 9.64 654 9.64 688 9.64 722 9.64 756 9.64 790 9.64 824 9.64 858 35 35 35 36 35 35 35 35 35 34 35 35 35 35 35 35 34 35 35 35 35 34 35 35 34 35 34 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 23° L. Cot. 147 0.37 215 0.37 180 0.37 145 0.37 110 0.37 074 0.37 039 0.37 004 0.36 969 0.36 934 0.36 899 0.36 865 0.36 830 0.36 795 0.36 760 0.36 725 0.36 690 0.36 655 0.36 621 0.36 586 0.36 551 0.36 516 0.36 481 0.36 447 0.36 412 0.36 377 0.36 343 0.36 308 0.36 274 0.36 239 0.36 204 0.36 170 0.36 135 0.36 101 0.36 066 0.36 032 0.35 997 0.35 963 0.35 928 0.35 894 0.35 860 0.35 825 0.35 791 0.35 757 0.35 722 0.35 688 0.35 654 0.35 619 0.35 585 0.35 551 0.35 517 0.35 483 0.35 448 0.35 414 0.35 380 0.35 346 0.35 312 0.35 278 0.35 244 0.35 210 0.35 176 0.35 142 L. Cos. d. 9.96 403 9.96 397 9.96 392 9.96 387 9.96 381 9.96 376 9.96 370 9.96 365 9.96 360 9.96 354 9.96 349 9.96 343 9.96 338 9.96 333 9.96 327 9.96 322 9.96 316 9.96 311 9.96 305 9.96 300 9.96 294 9.96 289 9.96 284 9.96 278 9.96 273 9.96 267 9.96 262 9.96 256 9.96 251 9.96 245 9.96 240 9.96 234 9.96 229 9.96 223 9.96 218 9.96 212 9.96 207 9.96 201 9.96 196 9.96 190 9.96 185 9.96 179 9.96 174 9.96 168 9.96 162 9.96 157 9.96 151 9.96 146 9.96 140 9.96 135 9.96 129 9.96 123 9.96 118 9.96 112 9.96 107 9.96 101 9.96 095 9.96 090 9.96 084 9.960^79 9^96 073 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. 6G^ P.P. " 36 35 34 51 3-0 lOj 6.0 15 9.0 20|l2.0 II.7 25ii5.o 14-6 30 i8.o 17.5 3S'2i.o 20.4 40' 2.8 5-7 8.5 III.3 14.2 17.0 19.8 24.0 23.3 22.7 45 27.0 26.2 25.5 50 30.0 29.2 28.3 5533.032.1 31.2 30 ! 29 28 5 2.5 2.4 10 5.0i 4.8 15 7.5 7-2 20JIO.0 25|I2.5 30,15.0 35 17.5 20.0 22.5 25.0 27.5 9.7 12. 1 14-5 16.9 2.3 4.7 7.0 9.3 11.7 14.0 16.3 19.3 18.7 21.8 21.0 24.2 23.3 26.6I25.7 6 I 5 0.5:0.4 i.0|0.8 I.5|1.2 2.0|1.7 2.5 2.1 30 2.5 3-5 2.9 40,4 03.3 45.4.53.8 50 5.04.2 ss's.s 4.6 P.P. 148 24° 10 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 60 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d 9.60 931 .60 960 ,60 988 .61 016 .61 045 ,61 073 .61 101 .61 129 61 158 •.61 186 9.61 214 .61 242 .61 270 .61 298 ,61 326 ,61 354 .61 382 .61411 61 438 .61 466 9.61 494 ,61 522 ,61 550 ,61 578 ,61 606 ,61 634 ,61 662 ,61 689 ,61 717 ,61 745 9.61 773 .61 800 .61 828 .61 856 .61 883 .61911 .61 939 .61 966 .61 994 .62 021 9.62 049 .62 076 .62 104 .62 131 .62 159 .62 186 .62 214 .62 241 .62 268 ,62 296 ,62 323 ,62 350 ,62 377 .62 405 ,62 432 ,62 459 ,62 486 ,62 513 ,62 541 ,62 568 9.62 595 29 28 28 29 28 28 28 29 28 28 28 28 28 28 28 28 29 27 28 28 28 28 28 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 27 28 27 28 27 28 27 28 27 27 28 27 27 27 28 27 27 27 27 28 27 27 9.64 858 9.64 892 9.64 926 9.64 960 9.64 994 9.65 028 9.65 062 9.65 096 9.65 130 9.65 164 9.65 197 9.65 231 9.65 265 9.65 299 9.65 333 9.65 366 9.65 400 9.65 434 9.65 467 9.65 501 9.65 535 9.65 568 9.65 602 9.65 636 9.65 669 9.65 703 9.65 736 9.65 770 9.65 803 9.65 837 9.65 870 9.65 904 9.65 937 9.65 971 9.66 004 9.66 038 9.66 071 9.66 104 9.66 138 9.66 171 9.66 204 9.66 238 9.66 271 9.66 304 9.66 337 9.66 371 9.66 404 9.66 437 9.66 470 9.66 503 9.66 537 9.66 570 9.66 603 9.66 636 9.66 669 9.66 702 9.66 735 9.66 768 9.66 801 9.66 834 9.66 867 34 34 34 34 34 34 34 34 34 33 34 34 34 34 33 34 34 33 34 34 33 34 34 33 34 33 34 33 34 33 34 33 34 33 34 33 33 34 33 33 34 33 33 33 34 33 33 33 33 34 33 33 33 33 33 33 33 33 33 33 0.35 142 0.35 108 0.35 074 0.35 040 0.35 006 0.34 972 0.34 938 0.34 904 0.34 870 0.34 836 0.34 803 0.34 769 0.34 735 0.34 701 0.34 667 0.34 634 0.34 600 0.34 566 0.34 533 0.34 499 0.34 465 0.34 432 0.34 398 0.34 364 0.34 331 0.34 297 0.34 264 0.34 230 0.34 197 0.34 163 0.34 130 0.34 096 0.34 063 0.34 029 0.33 996 0.33 962 0.33 929 0.33 896 0.33 862 0.33 829 0.33 796 0.33 762 0.33 729 0.33 696 0.33 663 0.33 629 0.33 596 0.33 563 0.33 530 0.33 497 0.33 463 0.33 430 0.33 397 0.33 364 0.33 331 0.33 298 0.33 265 0.33 232 0.33 199 0.33 166 0.33 133 9.96 073 9.96 067 9.96 062 9.96 056 9.96 050 9.96 045 9.96 039 9.96 034 9.96 028 9.96 022 9.96 017 9.96 011 9.96 005 9.96 000 9.95 994 9.95 988 9.95 982 9.95 977 9.95 971 9.95 965 9.95 960 9.95 954 9.95 948 9.95 942 9.95 937 9.95 931 9.95 925 9.95 920 9.95 914 9.95 908 9.95 902 9.95 897 9.95 891 9.95 885 9.95 879 9.95 873 9.95 868 9.95 862 9.95 856 9.95 850 9.95 844 9.95 839 9.95 833 9.95 827 9.95 821 9.95 815 9.95 810 9.95 804 9.95 798 9.95 792 9.95 786 9.95 780 9.95 775 9.95 769 9.95 763 9.95 757 9.95 751 9.95 745 9.95 739 9.95 733 9.95 728 P.P. "34 33 5 2.8 2.8 10 5-7 5-5 IS 8.5 8.2 20 1 1.3 II.O 25 14-2 13.8 30 17.0 16.S 35 19-8 19.2 40 22.7 22.0 45 25.5 24.8 SO 28.3 27.S 55 31.2 30.2 5 10 15 20 25 30 35 40 45 21.8 SO 24.2 55 26.6 29 2.4 4.8 7.2 9-71 12. 1 14-5 16.9 19.3 28 27 2.3 4.7 7.0 9.3' 9.0 11.7I11.2 14-0 13.5 16.3 15-8 18.7 18.0 21.0 20.2 23.3 22.5 25.7 24.8 5 0.5 0.4 10 1.00.8 15 1.5 1.2 20 2.0 1.7 25 2.5 2.1 30 3.0:2.5 35 3-5 2.9 40 4-03.3 45 4.5i3.8 50 5.0 4-2 55 5.5 4-6 L. Cos. L. Cot. c.d. L. Tan. L. Sin. 65^ P.P. 25° 149 10 11 12 13 14 15 16 17 18 19 20 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 60 L. Sin. 9.62 595 9.62 622 9.62 649 9.62 676 9.62 703 9.62 730 9.62 757 9.62 784 9.62 811 9.62 838 9.62 865 9.62 892 9.62 918 9.62 945 9.62 972 9.62 999 9.63 026 9.63 052 9.63 079 9.63 106 9.63 133 9.63 159 9.63 186 9.63 213 9.63 239 9.63 266 9.63 292 9.63 319 9.63 345 9.63 372 9.63 398 9.63 425 9.63 451 9.63 478 9.63 504 9.63 531 9.63 557 9.63 583 9.63 610 9.63 636 9.63 662 9.63 689 9.63 715 9.63 741 9.63 767 9.63 794 9.63 820 9.63 846 9.63 872 9.63 898 9.63 924 9.63 950 9.63 976 9.64 002 9.64 028 9.64 054 9.64 080 9.64 106 9.64 132 9.64 158 9.64 184 d. 27 27 27 27 27 27 27 27 27 27 27 26 27 27 27 27 26 27 27 27 26 27 27 26 27 26 27 26 27 26 27 26 27 26 27 26 26 27 26 26 27 26 26 26 27 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 L. Tan. 9.66 867 9.66 900 9.66 933 9.66 966 9.66 999 9.67 032 9.67 065 9.67 098 9.67 131 9.67 163 9.67 196 9.67 229 9.67 262 9.67 295 9.67 327 9.67 360 9.67 393 9.67 426 9.67 458 9.67 491 9.67 524 9.67 556 9.67 589 9.67 622 9.67 654 9.67 687 9.67 719 9.67 752 9.67 785 9.67 817 9.67 850 9.67 882 9.67 915 9.67 947 9.67 980 9.68 012 9.68 044 9.68 077 9.68 109 9.68 142 9.68 174 9.68 206 9.68 239 9.68 271 9.68 303 9.68 336 9.68 368 9.68 400 9.68 432 9.68 465 9.68 497 I L. Cos. I d. L. Cot. c.d 9.68 529 9.68 561 9.68 593 9.68 626 9.68 658 9.68 690 9.68 722 9.68 754 9.68 786 9.68 818 c.d. L. Cot. 33 33 33 33 33 33 33 33 31 33 33 33 33 31 33 33 33 31 33 33 31 33 33 31 33 31 33 33 31 33 31 33 31 33 31 31 33 31 33 31 31 33 31 31 33 31 31 31 33 31 31 31 31 33 31 31 31 32 32 32 0.33 133 0.33 100 0.33 067 0.33 034 0.33 001 0.32 968 0.32 935 0.32 902 0.32 869 0.32 837 0.32 804 0.32 771 0.32 738 0.32 705 0.32 673 0.32 640 0.32 607 0.32 574 0.32 542 0.32 509 0.32 476 0.32 444 0.32 411 0.32 378 0.32 346 0.32 313 0.32 281 0.32 248 0.32 215 0.32 183 0.32 150 0.32 118 0.32 085 0.32 053 0.32 020 0.31 988 0.31 956 0.31 923 0.31 891 0.31 858 0.31 826 0.31 794 0.31 761 0.31 729 0.31 697 0.31 664 0.31 632 0.31 600 0.31 568 0.31 535 0.31 503 0.31471 0.31 439 0.31 407 0.31 374 0.31 342 0.31 310 0.31 278 0.31 246 0.31 214 0.31 182 L. Cos. 9.95 728 9.95 722 9.95 716 9.95 710 9.95 704 9.95 698 9.95 692 9.95 686 9.95 680 9.95 674 9.95 668 9.95 663 9.95 657 9.95 651 9.95 645 9.95 639 9.95 633 9.95 627 9.95 621 9.95 615 9.95 609 9.95 603 9.95 597 9.95 591 9.95 585 9.95 579 9.95 573 9.95 567 9.95 561 9.95 555 9.95 549 9.95 543 9.95 537 9.95 531 9.95 525 9.95 519 9.95 513 9.95 507 9.95 500 9.95 494 9.95 488 9.95 482 9.95 476 9.95 470 9.95 464 9.95 458 9.95 452 9.95 446 9.95 440 9.95 434 9.95 427 9.95 421 9.95 415 9.95 409 9.95 403 9.95 397 9.95 391 9.95 384 9.95 378 9.95 372 9.95 366 L. Tan. L. Sin. d. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 P.P. 33 I 32 2.8' 2.7 S-S\ 5-3 8.2I 8.0 ii.o 10.7 13-8 13-3 16.S 16.0 35!i9.2 18.7 40 22.0 21.3 45124-8 24.0 50 27.5 26.7 5S'30.2 29.3 " I 27 2.2 4-3 6.5 8.7 5, 2.2 10! 4-5 15 6.8 20| 9.0| 25 II.2 10.8 30 13.5 13-0 35 15-8 15-2 40 18.0 17.3 45j20.2 19.S 50 22.5 21.7 S5I24-8 23.8 "I 7 I s'0.60 IO;I. IS;I. 20:2.3 2 25 2.9 2 30|3S 3 35 4-il3 404.7 4 45 5.2;4 50 5.8,5 55 6.4IS 6 I 5 50.4 o 0.8 51.2 1.7 52.1 o 2.5 5 2.9 03-3 53.8 04.2 5 '4.6 P.P. 150 1 2 3 4 5 6 7 8 _9 JO 11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28 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 L. Sin. 9.64 184 9.64 210 9.64 236 9.64 262 9.64 288 9.64 313 9.64 339 9.64 365 9.64 391 9.64 417 9.64 442 9.64 468 9.64 494 9.64 519 9.64 545 9.64 571 9.64 596 9.64 622 9.64 647 9.64 673 9.64 698 9.64 724 9.64 749 9.64 775 9.64 800 9.64 826 9.64 851 9.64 877 9.64 902 9.64 927 9.64 953 9.64 978 9.65 003 9.65 029 9.65 054 9.65 079 9.65 104 9.65 130 9.65 155 9.65 180 9.65 205 9.65 230 9.65 255 9.65 281 9.65 306 9.65 331 9.65 356 9.65 381 9.65 406 9.65 431 9.65 456 9.65 481 9.65 506 9.65 531 9.65 556 9.65 580 9.65 605 9.65 630 9.65 655 9. 65 680 9.65 705 26 26 26 26 25 26 26 26 26 25 26 26 25 26 26 25 26 25 26 25 26 25 26 25 26 25 26 25 25 26 25 25 26 25 25 25 26 25 25 25 25 25 26 25 25 25 25 25 25 25 25 25 25 25 24 25 25 25 25 25 26^ L. Tan. c.d. L. Cot. L. Cos. d. 9.68 818 9.68 850 9.68 882 9.68 914 9.68 946 9.68 978 9.69 010 9.69 042 9.69 074 9.69 106 9.69 138 9.69 170 9.69 202 9.69 234 9.69 266 9.69 298 9.69 329 9.69 361 9.69 393 9.69 425 9.69 457 9.69 488 9.69 520 9.69 552 9.69 584 9.69 615 9.69 647 9.69 679 9.69 710 9.69 742 9.69 774 9.69 805 9.69 837 9.69 868 9.69 900 9.69 932 9.69 963 9.69 995 9.70 026 9.70 058 9.70 089 9.70 121 9.70 152 9.70 184 9.70 215 9.70 247 9.70 278 9.70 309 9.70 341 9.70 372 9.70 404 9.70 435 9.70 466 9.70 498 9.70 529 9.70 560 9.70 592 9.70 623 9.70 654 9.70 685 9.70 717 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 31 32 32 32 32 31 32 32 32 31 32 32 31 32 32 31 32 31 32 32 31 32 31 32 31 32 31 32 31 32 31 31 32 31 32 31 31 32 31 31 32 31 31 31 32 0.31 182 0.31 150 0.31 118 0.31 086 0.31 054 0.31 022 0.30 990 0.30 958 0.30 926 0.30 894 0.30 862 0.30 830 0.30 798 0.30 766 0.30 734 0.30 702 0.30 671 0.30 639 0.30 607 0.30 575 9.95 366 9.95 360 9.95 354 9.95 348 9.95 341 9.95 335 9.95 329 9.95 323 9.95 317 9.95 310 9.95 304 0.30 543 0.30 512 0.30 480 0.30 448 0.30 416 0.30 385 0.30 353 0.30 321 0.30 290 0.30 258 0.30 226 0.30 195 0.30 163 0.30 132 0.30 100 0.30 068 0.30 037 0.30 005 0.29 974 0.29 942 0.29 911 0.29 879 0.29 848 0.29 816 0.29 785 0.29 753 0.29 722 0.29 691 0.29 659 0.29 628 0.29 596 0.29 565 0.29 534 0.29 502 0.29 471 0.29 440 0.29 408 0.29 377 0.29 346 0.29 315 0.29 283 9.95 298 9.95 292 9.95 286 9.95 279 9.95 273 9.95 267 9.95 261 9.95 254 9.95 248 9.95 242 9.95 236 9.95 229 9.95 223 9.95 217 9.95 211 9.95 204 9.95 198 9.95 192 9.95 185 9.95 179 9.95 173 9.95 167 9.95 160 9.95 154 9.95 148 9.95 141 9.95 135 9.95 129 9.95 122 9.95 116 9.95 110 9.95 103 9.95 097 9.95 090 9.95 084 9.95 078 9.95 071 9.95 065 9.95 059 9.95 052 9.95 046 9.95 039 9.95 033 9.95 027 9.95 020 9.95 014 9.95 007 9.95 001 9.94 995 9.94 988 P.P. " I 32 I 31 s! 2.7! 2.6 10 5-3' S.2 15 8o 7.8 2010.7 10.3 25II3.3 12.9 30 16.0 15.5 35'i8.7 18.1 40 21.3 20.7 45 24.0 23.2 SO 26.7 25.8 55 29.3 28.4 "1 26 25 24 S 2.2I 2.1 2.0 10 4.31 4-2 IS 6.5 6.2 20 8.71 8.3 4.0 6.0 8.0 25 10.8 10.4,10.0 30 13.0 3515.2 40 17.3 45|I9.S 50121.7 12.5112.0 14.6 14.0 16.7I16.0 i8.8|i8.0 20.8I20.0 SS'23.8 22.9 22.0 "I 7 50.6 10:1.2 is;i.8 20 2.3I2.0 2.92.S 353.0 4.ii3.5 4.74.0 5.24.5 5.85.0 6.4I5.5 L. Cos. d. L. Cot. c.d. I L. Tan. | L. Sin. P.P. 27° 151 1 ' L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. P.P. 1 2 3 9.65 705 24 25 25 25 24 25 25 24 25 25 24 25 24 25 25 24 25 24 25 24 24 25 24 25 24 24 25 24 24 25 24 24 24 24 25 24 24 24 24 24 24 25 24 24 24 24 24 24 24 23 24 24 24 24 24 24 24 23 24 24 9.70 717 31 31 31 31 32 31 31 31 31 31 31 31 31 32 31 31 31 31 31 31 31 31 30 31 31 31 31 31 31 31 31 30 31 31 31 31 3,0 31 31 30 31 31 31 30 31 31 30 31 30 31 31 30 31 30 31 30 31 30 31 30 0.29 283 9.94 988 6 7 6 7 6 7 6 7 6 7 6 6 7 6 7 6 7 7 6 7 6 7 6 7 6 7 6 7 7 6 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 7 60 59 58 57 9.65 729 9.65 754 9.65 779 9.70 748 9.70 779 9.70 810 0.29 252 0.29 221 0.29 190 9.94 982 9.94 975 9.94 969 4 5 6 9.65 804 9.65 828 9.65 853 9.70 841 9.70 873 9.70 904 0.29 159 0.29 127 0.29 096 9.94 962 9.94 956 9.94 949 56 55 54 " i 32 1 31 1 30 7 8 9 9.65 878 9.65 902 9.65 927 9.70 935 9.70 966 9.70 997 0.29 065 0.29 034 0.29 003 9.94 943 9.94 936 9.94 930 53 52 51 50 49 48 47 s! 2.7 2.6 10; S.3| 5-2 15 8.01 7-8 20| 10.7 10.3 25,13.3 12.9 3016.015.5 35 i8.7!i8.i 40 21.3 20.7 45 24.0 23.2 50 26.7 25.8 -5 5.0 7.5 lO.O 12.5 15.0 17.S 20.0 22.5 9t; n 10 11 12 13 9.65 952 9.71 028 0.28 972 9.94 923 9.65 976 9.66 001 9.66 025 9.71 059 9.71 090 9.71 121 0.28 941 0.28 910 0.28 879 9.94 917 9.94 911 9.94 904 14 15 16 9.66 050 9.66 075 9.66 099 9.71 153 9.71 184 9.71215 0.28 847 0.28 816 0.28 785 9.94 898 9.94 891 9.94 885 46 45 44 55 29.3 28.4 27.S 17 18 19 20 21 22 23 9.66 124 9.66 148 9.66 173 9.71 246 9.71277 9.71 308 0.28 754 0.28 723 0.28 692 9.94 878 9.94 871 9.94 865 43 42 41 40 39 38 37 9.66 197 9.71339 0.28 661 9.94 858 9.66 221 9.66 246 9.66 270 9.71 370 9.71 401 9.71431 0.28 630 0.28 599 0.28 569 9.94 852 9.94 845 9.94 839 24 25 26 9.66 295 9.66 319 9.66 343 9.71 462 9.71 493 9.71 524 0.28 538 0.28 507 0.28 476 9.94 832 9.94 826 9.94 819 36 35 34 " 1 25 1 24 23 27 28 29 9.66 368 9.66 392 9.66 416 9.71 555 9.71 586 9.71617 0.28 445 0.28 414 0.28 383 9,94 813 9.94 806 9.94 799 33 32 31 30 29 28 27 s 10 IS 20 2.1 4.2 6.2 8.3 2.0 4.0 6.0 8.0 1.9 3f 5.8 7.7 9.6 TT C 30 9.66 441 9.71 648 0.28 352 9.94 793 2S| 10.41 lO.O 30 12.5 12.0 31 32 33 9.66 465 9.66 489 9.66 513 9.71679 9.71 709 9.71 740 0.28 321 0.28 291 0.28 260 9.94 786 9.94 780 9.94 773 35 14-6 40 16.7 4518.8 50 20.8 14.0 13.4 16.0 15.3 18.017.2 20.0 10.2 34 35 36 9.66 537 9.66 562 9.66 586 9.71 771 9.71 802 9.71 833 0.28 229 0.28 198 0.28 167 9.94 767 9.94 760 9.94 753 26 25 24 55 22.9'22.0;2I.I 37 3S 39 9.66 610 9.66 634 9.66 658 9.71 863 9.71 894 9.71 925 0.28 137 0.28 106 0.28 075 9.94 747 9.94 740 9.94 734 23 22 21 20 19 18 17 40 9.66 682 9.71 955 0.28 045 9.94 727 41 42 43 9.66 706 9.66 731 9.66 755 9.71986 9.72 017 9.72 048 0.28 014 0.27 983 0.27 952 9.94 720 9.94 714 9.94 707 44 45 46 9.66 779 9.66 803 9.66 827 9.72 078 9.72 109 9.72 140 0.27 922 0.27 891 0.27 860 9.94 700 9.94 694 9.94 687 16 15 14 "\7 .6 47 48 49 9.66 851 9.66 875 9.66 899 9.72 170 9.72 201 9.72 231 0.27 830 0.27 799 0.27 769 9.94 680 9.94 674 9.94 667 13 12 11 10 9 8 7 5 0.6 0.5 I0'l.2 I.O I5;i.8 i.S 20 2.312.0 25!2.9 2.5 30'3.5 3.0 35:4.1 3.5 404.714.0 45 5.24.5 50:5.8,5.0 50 9.66 922 9.72 262 0.27 738 9.94 660 51 52 53 9.66 946 9.66 970 9.66 994 9.72 293 9.72 323 9.72 354 0.27 707 0.27 677 0.27 646 9.94 654 9.94 647 9.94 640 54 55 56 9.67 018 9.67 042 9.67 066 9.72 384 9.72 415 9.72 445 0.27 616 0.27 585 0.27 555 9.94 634 9.94 627 9.94 620 6 5 4 556.45.5 57 58 59 9.67 090 9.67 113 9.67 137 9.72 476 9.72 506 9.72 537 9.72 567 0.27 524 0.27 494 0.27 463 9.94 614 9.94 607 9.94 600 3 2 1 60 9.67 161 0.27 433 9.94 593 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. P.P. 1 152 28° 30 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. 9.67 161 9.67 185 9.67 208 9.67 232 9.67 256 9.67 280 9.67 303 9.67 327 9.67 350 9.67 374 9.67 398 9.67 421 9.67 445 9.67 468 9.67 492 9.67 515 9.67 539 9.67 562 9.67 586 9.67 609 9.67 633 9.67 656 9.67 680 9.67 703 9.67 726 9.67 750 9.67 773 9.67 796 9.67 820 9.67 843 9.67 866 9.67 890 9.67 913 9.67 936 9.67 959 9.67 982 9.68 006 9.68 029 9.68 052 9.68 075 9.68 098 9.68 121 9.68 144 9.68 167 9.68 190 9.68 213 9.68 237 9.68 260 9.68 283 9.68 305 9.68 328 9.68 351 9.68 374 9.68 397 9.68 420 9.68 443 9.68 466 9.68 489 9.68 512 9.68 534 9.68 557 L. Cos. 24 23 24 24 24 23 24 23 24 24 23 24 23 24 23 24 23 24 23 24 23 24 23 23 24 23 23 24 23 23 24 23 23 23 23 24 23 23 23 23 23 23 23 23 23 24 23 23 22 23 23 23 23 23 23 23 23 23 22 23 d. 9.72 567 9.72 598 9.72 628 9.72 659 9.72 689 9.72 720 9.72 750 9.72 780 9.72 811 9.72 841 9.72 872 9.72 902 9.72 932 9.72 963 9.72 993 9.73 023 9.73 054 9.73 084 9.73 114 9.73 144 9.73 175 9.73 205 9.73 235 9.73 265 9.73 295 9.73 326 9.73 356 9.73 386 9.73 416 9.73 446 9.73 476 9.73 507 9.73 537 9.73 567 9.73 597 9.73 627 9.73 657 9.73 687 9.73 717 9.73 747 9.73 777 9.73 807 9.73 837 9.73 867 9.73 897 9.73 927 9.73 957 9.73 987 9.74 017 9.74 047 9.74 077 9.74 107 9.74 137 9.74 166 9.74 196 9.74 226 9.74 256 9.74 286 9.74 316 9.74 345 9.74 375 L. Cot. 31 30 31 30 31 30 30 31 30 31 30 30 31 30 30 31 30 30 30 31 30 30 30 30 31 30 30 30 30 30 31 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 30 30 29 30 c.d. 0.27 433 0.27 402 0.27 372 0.27 341 0.27 311 0.27 280 0.27 250 0.27 220 0.27 189 0.27 159 0.27 128 9.27 098 0.27 068 0.27 037 0.27 007 0.26 977 0.26 946 0.26 916 0.26 886 0.26 856 0.26 825 0.26 795 0.26 765 0.26 735 0.26 705 0.26 674 0.26 644 0.26 614 0.26 584 0.26 554 0.26 524 0.26 493 0.26 463 0.26 433 0.26 403 0.26 373 0.26 343 0.26 313 0.26 283 0.26 253 0.26 223 0.26 193 0.26 163 0.26 133 0.26 103 0.26 073 0.26 043 0.26 013 0.25 983 0.25 953 0.25 923 0.25 893 0.25 863 0.25 834 0.25 804 0.25 774 0.25 744 0.25 714 0.25 684 0.25 655 0.25 625 L. Tan. 9.94 593 9.94 587 9.94 580 9.94 573 9.94 567 9.94 560 9.94 553 9.94 546 9.94 540 9.94 533 9.94 526 9.94 519 9.94 513 9.94 506 9.94 499 9.94 492 9.94 485 9.94 479 9.94 472 9.94 465 9.94 458 9.94 451 9.94 445 9.94 438 9.94 431 9.94 424 9.94 417 9.94 410 9.94 404 9.94 397 9.94 390 9.94 383 9.94 376 9.94 369 9.94 362 9.94 355 9.94 349 9.94 342 9.94 335 9.94 328 9.94 321 9.94 314 9.94 307 9.94 300 9.94 293 9.94 286 9.94 279 9.94 273 9.94 266 9.94 259 9.94 252 9.94 245 9.94 238 9.94 231 9.94 224 9.94 217 9.94 210 9.94 203 9.94 196 9.94 189 9.94 182 L. Sin. P.P. n 31 30 29 5 2.6 2.5 2.4 10 5-2 5.0 4.8 15 7.8 7-5 7.2 20 I0.3 lO.O 9.7 2SjI2.9 12.5 12. 1 30 IS.5 iS-o 14-5 35 i8.i 17.5 16.9 40 20.7 20.0 19.3 45 23.2I22.S 21.8 50 25.8 25.0 24.2 55 28.4 27.S 26.6 24 I 23 22 2.0 1.9 4-0 3-8 6.0 5.8 8.01 7.7 lo.o] 9.6 3o!i2.o 11.5 35 14.0 13-4 40 16.0 15.3 45 18.0 17.2 SO 20.0 19.2 55 22.0 21. 1 1.8 3-7 5-5 7-3 9.2 II.O 12.8 14.7 16.S 18.3 20.2 "I 7 5i0.6 0.5 lo;i.2 i.o 151I.8 l.S 20|2.3 2.0 25|2.9'2.S 303.53.0 35'4.i!3.5 4.74.0 5.24.5 5.8 5.0 55 6.4 S.S P.P. 29' 153 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. 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 9.68 557 9.68 580 9.68 603 9.68 625 9.68 648 9.68 671 9.68 694 9.68 716 9.68 739 9.68 762 10 9.68 784 60 9.68 807 9.68 829 9.68 852 9.68 875 9.68 897 9.68 920 9.68 942 9.68 965 9.68 987 9.69 010 9.69 032 9.69 055 9.69 077 9.69 100 9.69 122 9.69 144 9.69 167 9.69 189 9.69 212 9.69 234 9.69 256 9.69 279 9.69 301 9.69 323 9.69 345 9.69 368 9.69 390 9.69 412 9.69 434 9.69 456 9.69 479 9.69 501 9.69 523 9.69 545 9.69 567 9.69 589 9.69 611 9.69 633 9.69 655 9.69 677 9.69 699 9.69 721 9.69 743 9.69 765 9.69 787 9.69 809 9.69 831 9.69 853 9.69 875 9.69 897 23 23 22 23 23 23 22 23 23 22 23 22 23 23 22 23 22 23 22 23 22 23 22 23 22 22 23 22 23 22 22 23 22 22 22 23 22 22 22 22 23 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 9.74 375 9.74 405 9.74 435 9.74 465 9.74 494 9.74 524 9.74 554 9.74 583 9.74 613 9.74 643 9.74 673 9.74 702 9.74 732 9.74 762 9.74 791 9.74 821 9.74 851 9.74 880 9.74 910 9.74 939 9.74 969 9.74 998 9.75 028 9.75 058 9.75 087 9.75 117 9.75 146 9.75 176 9.75 205 9.75 235 9.75 264 9.75 294 9.75 323 9.75 353 9.75 382 9.75 411 9.75 441 9.75 470 9.75 500 9.75 529 9.75 558 9.75 588 9.75 617 9.75 647 9.75 676 9.75 705 9.75 735 9.75 764 9.75 793 9.75 822 9.75 852 9.75 881 9.75 910 9.75 939 .75 969 .75 998 .76 027 .76 056 .76 086 9.76 115 9.76 144 30 30 30 29 30 30 29 30 30 30 29 30 30 29 30 30 29 30 29 30 29 30 30 29 30 29 30 29 30 29 30 29 30 29 29 30 29 30 29 29 30 29 30 29 29 30 29 29 29 30 29 29 29 30 29 29 29 30 29 29 0.25 625 0.25 595 0.25 565 0.25 535 0.25 506 0.25 476 0.25 446 0.25 417 0.25 387 0.25 357 0.25 327 0.25 298 0.25 268 0.25 238 0.25 209 0.25 179 0.25 149 0.25 120 0.25 090 0.25 061 9.94 182 9.94 175 9.94 168 9.94 161 9.94 154 9.94 147 9.94 140 9.94 133 9.94 126 9.94 119 9.94 112 0.25 031 0.25 002 0.24 972 0.24 942 0.24 913 0.24 883 0.24 854 0.24 824 0.24 795 0.24 765 0.24 736 0.24 706 0.24 677 0.24 647 0.24 618 0.24 589 0.24 559 0.24 530 0.24 500 0.24 471 0.24 442 0.24 412 0.24 383 0.24 353 0.24 324 0.24 295 0.24 265 0.24 236 0.24 207 0.24 178 0.24 148 0.24 119 0.24 090 0.24 061 0.24 031 0.24 002 0.23 973 0.23 944 0.23 914 0.23 885 0.23 856 9.94 105 9.94 098 9.94 090 9.94 083 9.94 076 9.94 069 9.94 062 9.94 055 9.94 048 9.94 041 9.94 034 9.94 027 9.94 020 9.94 012 9.94 005 9.93 998 9.93 991 9.93 984 9.93 977 9.93 970 9.93 963 9.93 955 9.93 948 9.93 941 9.93 934 9.93 927 9.93 920 9.93 912 9.93 905 9.93 898 9.93 891 9.93 884 9.93 876 9.93 869 9.93 862 9.93 855 9.93 847 9.93 840 9.93 833 9.93 826 9.93 819 9.93 811 9.93 804 9.93 797 9.93 789 9.93 782 9.93 775 9.93 768 9.93 760 9.93 753 60 59 58 57 56 55 54 53 52 11. _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 IL 20^ 19 18 17 16 15 14 13 12 n 10 P.P. 30 29 2.5 2.4 5-0 4.8 7-5 7.2 lO.O 9.7 12.5 12. 1 IS-O 14.5 17.516.9 20.0 19.3 22.5 21.8 25.0(24.2 27.5 26.6I 23 1.9 3.8 5.8 7.7 9.6 11.5 13.4 15-3 17.2 19.2 21.1 II 22 8 1 7 5 1.8 0.7,0.6 10 3-7 1.31.2 IS 5.5 2.0:1.8 20 7.3 2.7|2.3 25 9.2 3.32.9 30 II.O 4.03.5 35 12.8 4.7 4.1 40 14.7 5.3 4.7 45 16.5 6.0I5.2 50 18.3,6.715.8 55 20.2 7.316.4 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. p.p. 154 30* 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 L. Sin. 9.69 897 9.69 919 9.69 941 9.69 963 9.69 984 9.70 006 9.70 028 9.70 050 9.70 072 9.70 093 9.70 115 9.70 137 9.70 159 9.70 180 9.70 202 9.70 224 9.70 245 9.70 267 9.70 288 9.70 310 9.70 332 9.70 353 9.70 375 9.70 396 9.70 418 9.70 439 9.70 461 9.70 482 9.70 504 9.70 525 9.70 547 9.70 568 9.70 590 9.70 611 9.70 633 9.70 654 9.70 675 9.70 697 9.70 718 9.70 739 9.70 761 9.70 782 9.70 803 9.70 824 9.70 846 9.70 867 9.70 888 9.70 909 9.70 931 9.70 952 9.70 973 9.70 994 9.71015 9.71 036 9.71 058 9.71 079 9.71 100 9.71 121 9.71 142 9.71 163 9.71 184 L. Cos. 22 22 22 21 22 22 22 22 21 22 22 22 21 22 22 21 22 21 22 22 21 22 21 22 21 22 21 22 21 22 21 22 21 22 21 21 22 21 21 22 21 21 21 22 21 21 21 22 21 21 21 21 21 22 21 21 21 21 21 21 L. Tan. c.d. L. Cot. L. Cos. d. 76 144 76 173 76 202 76 231 76 261 76 290 76 319 76 348 76 377 76 406 76 435 76 464 76 493 76 522 76 551 76 580 76 609 76 639 76 668 76 697 76 725 76 754 76 783 76 812 76 841 76 870 76 899 76 928 76 957 76 986 77 015 77 044 77 073 77 101 77 130 77 159 77 188 77 217 77 246 77 274 77 303 77 332 77 361 77 390 77 418 77 447 77 476 77 505 77 533 77 562 77 591 77 619 77 648 77 677 77 706 77 734 77 763 77 791 77 820 77 849 77 877 d. L. Cot. c.d 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 30 29 29 28 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 29 29 28 29 29 29 29 28 29 29 29 28 29 29 28 29 29 29 28 29 28 29 29 28 0.23 856 0.23 827 0.23 798 0.23 769 0.23 739 0.23 710 0.23 681 0.23 652 0.23 623 0.23 594 0.23 565 0.23 536 0.23 507 0.23 478 0.23 449 0.23 420 0.23 391 0.23 361 0.23 332 0.23 303 0.23 275 0.23 246 0.23 217 0.23 188 0.23 159 0.23 130 0.23 101 0.23 072 0.23 043 0.23 014 0.22 985 0.22 956 0.22 927 0.22 899 0.22 870 0.22 841 0.22 812 0.22 783 0.22 754 0.22 726 0.22 697 0.22 668 0.22 639 0.22 610 0.22 582 0.22 553 0.22 524 0.22 495 0.22 467 0.22 438 0.2 2 409 0.22 381 0.22 352 0.22 323 0.22 294 0.22 266 0.22 237 0.22 209 0.22 180 0.22 151 0.22 123 9.93 753 9.93 746 9.93 738 9.93 731 9.93 724 9.93 717 9.93 709 9.93 702 9.93 695 9.93 687 9.93 680 9.93 673 9.93 665 9.93 658 9.93 650 9.93 643 9.93 636 9.93 628 9.93 621 9.93 614 9.93 606 9.93 599 9.93 591 9.93 584 9.93 577 9.93 569 9.93 562 9.93 554 9.93 547 9.93 539 9.93 532 9.93 525 9.93 517 9.93 510 9.93 502 9.93 495 9.93 487 9.93 480 9.93 472 9.93 465 9.93 457 9.93 450 9.93 442 9.93 435 9.93 427 9.93 420 9.93 412 9.93 405 9.93 397 9.93 390 9.93 382 9.93 375 9.93 367 9.93 360 9.93 352 9.93 344 9.93 337 9.93 329 9.93 322 9.93 314 9.93 307 L. Tan. L. Sin. d. P.P. 30 I 29 I 28 S 2.5 lo] 5.0 I5j 7-5 20:10.0 2.4 4.8i 7.2, 9.7! 2.3 4.7 7.0 9.3 25:12.5 12. I|II.7 !i4-5 14-0 ■16.9 16.3 19.3,18.7 2I.8'2I.O 24.2 23.3 26.6 25.7 30:15.0: 3S|i7.5: 40 20.0 45,22.5' 50 25.0 5527.5 22 21 1.8 3.5 5-2 7.0 30 ii.o lo.s 35 12.8 12.2 40 14.7 45 16.5 50 18.3 55 20.2 14.0 15.8 17.S 19.2 :i 8,7 50.70.6 10 1.3 1.2 1512.011.8 20J2.7,2.3 25 3-3 2.9 4-03.5 4.7|4.I S.34.7 6.0 5.2 S0i6.7 5.8 557.36.4 P.P. 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 37 38 41 42 43 44 45 46 47 48 50 51 52 53 54 55 56 57 58 60 L. Sin. 9.71 184 9.71 205 9.71 226 9.71 247 9.71 268 9.71 289 9.71310 9.71331 9.71 352 9.71 373 9.71 393 9.71 414 9.71 435 9.71 456 9.71 477 9.71498 9.71519 9.71 539 9.71 560 9.71 581 9.71 602 9.71 622 9.71 643 9.71 664 9.71 685 9.71 705 9.71 726 9.71 747 9.71 767 9.71 788 9.71 809 71829 71 850 71870 71891 71911 71932 71952 71973 71994 72 014 9.72 034 9.72 055 9.72 075 9.72 096 9.72 116 9.72 137 9.72 157 9.72 177 9.72 198 9.72 218 72 238 72 259 72 279 72 299 72 320 72 340 72 360 72 381 72 401 72 421 L. Cos. 21 21 21 21 21 21 21 21 21 20 21 21 21 21 21 21 20 21 21 21 20 21 21 21 20 21 21 20 21 21 20 21 20 21 20 21 20 21 21 20 20 21 20 21 20 21 20 20 21 20 20 21 20 20 21 20 20 21 20 20 31' L. Tan. c.d. L. Cot. L. Cos 155 77 877 77 906 77 935 77 963 77 992 78020 78 049 78 077 78 106 78 135 78 163 78 192 78 220 78 249 78 277 78 306 78 334 78 363 78 391 78 419 78 448 78 476 78 505 78 533 78 562 78 590 78 618 78 647 78 675 78 704 78 732 78 760 78 789 78 817 78 845 78 874 78 902 78 930 78 959 78 987 79 015 79 043 79 072 79 100 79 128 79 156 79 185 79 213 79 241 79 269 79 297 79 326 79 354 79 382 79 410 79 438 79 466 79 495 79 523 79 551 79 579 d. L. Cot. c.d 29 29 28 29 28 29 28 29 29 28 29 28 29 28 29 28 29 28 28 29 28 29 28 29 28 28 29 28 29 28 28 29 28 28 29 28 28 29 28 28 28 29 28 28 28 29 28 28 28 28 29 28 28 28 28 28 29 28 28 28 0.22 123 0.22 094 0.22 065 0.22 037 0.22 008 0.21 980 0.21951 0.21 923 0.21 894 0.21 865 0.21 837 0.21 808 0.21 780 0.21 751 0.21 723 0.21 694 0.21 666 0.21 637 0.21 609 0.21 581 0.21 552 0.21 524 0.21 495 0.21 467 0.21 438 0.21 410 0.21 382 0.21 353 0.21 325 0.21 296 0.21 268 0.21 240 0.21 211 0.21 183 0.21 155 0.21 126 0.21 098 0.21 070 0.21 041 0.21 013 0.20 985 0.20 957 0.20 928 0.20 900 0.20 872 0.20 844 0.20 815 0.20 787 0.20 759 0.20 731 0.20 703 0.20 674 0.20 646 0.20 618 0.20 590 0.20 562 0.20 534 0.20 505 0.20 477 0.20 449 0.20 421 L. Tan. L. Sin 9^3^07 9.93 299 9.93 291 9.93 284 9.93 276 9.93 269 9.93 261 9.93 253 9.93 246 9.93 238 9.93 230 9.93 223 9.93 215 9.93 207 9.93 200 9.93 192 9.93 184 9.93 177 9.93 169 9.93 161 9.93 154 9.93 146 9.93 138 9.93 131 9.93 123 9.93 115 9.93 108 9.93 100 9.93 092 9.93 084 9.93 077 9.93 069 9.93 061 9.93 053 9.93 046 9.93 038 9.93 030 9.93 022 9.93 014 9.93 007 9.92 999 9.92 991 9.92 983 9.92 976 9.92 968 9.92 960 9.92 952 9.92 944 9.92 936 9.92 929 9.92 921 9.92 913 9.92 905 9.92 897 9.92 889 9.92 881 9.92 874 9.92 866 9.92 858 9^2850 9.92 842 d. 60 59 58 57 56 55 54 53 52 11 50^ 49 48 47 46 45 44 43 42 11 40 P.P. " I 29 i 28 I 21 SI 2 101 4 IS, 7 20 9 25 12 30 14 35 i6 40 19 4S 21 5024 SS 26 .4 2.3 1.8 .8{ 4.7I 3-5 ,21 7.0; S.2 7| 9.3] 7.0 ,1 11.71 8.8 ,5 14.010.5 ,9 16.3 12.2 ,3] 18.7 14-0 ,821.0 15.8 2 23.3 17.S 6125.7 19.2 " 20 I 8 1 7 1.7I0.7JO.6 3.31.3 1.2 5.0 2.0 1.8 6.7 '2.7 '2.3 8.3 3.3'2.9 0.0 4.0 3-5 35 ii.7 4.7|4-I 40 13.3 5-3 4-7 45 15.0 6.0 5.2 50 16.7 6.7 5-8 55 18.3 7-3 6.4 P.P. 156 32° 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 L. Sin. 9.72 421 9.72 441 9.72 461 9.72 482 9.72 502 9.72 522 9.72 542 9.72 562 9.72 582 9.72 602 9.72 622 9.72 643 9.72 663 9.72 683 9.72 703 9.72 723 9.72 743 9.72 763 9.72 783 9.72 803 9.72 823 9.72 843 9.72 863 9.72 883 9.72 902 9.72 922 9.72 942 9.72 962 9.72 982 9.73 002 9.73 022 9.73 041 9.73 061 9.73 081 9.73 101 9.73 121 9.73 140 9.73 160 9.73 180 9.73 200 9.73 219 9.73 239 9.73 259 9.73 278 9.73 298 9.73 318 9.73 337 9.73 357 9.73 377 9.73 396 73 416 .73 435 .73 455 .73 474 .73 494 .73 513 .73 533 .73 552 .73 572 .73 591 73 611 20 20 21 20 20 20 20 20 20 20 21 20 20 20 20 20 20 20 20 20 20 20 20 19 20 20 20 20 20 20 19 20 20 20 20 19 20 20 20 19 20 20 19 20 20 19 20 20 19 20 19 20 19 20 19 20 19 20 19 20 L. Tan. c.d. L. Cot. L. Cos. d. 9.79 579 9.79 607 9.79 635 9.79 663 9.79 691 9.79 719 9.79 747 9.79 776 9.79 804 9.79 832 9.79 860 9.79 888 9.79 916 9.79 944 9.79 972 9.80 000 9.80 028 9.80 056 9.80 084 9.80 112 9.80 140 9.80 168 9.80 195 9.80 223 9.80 251 9.80 279 9.80 307 9.80 335 9.80 363 9.80 391 9.80 419 9.80 447 9.80 474 9.80 502 9.80 530 9.80 558 9.80 586 9.80 614 9.80 642 9.80 669 9.80 697 9.80 725 9.80 753 9.80 781 9.80 808 9.80 836 9.80 864 9.80 892 9.80 919 9.80 947 9.80 975 9.81 003 9.81 030 9.81 058 9.81 086 9.81 113 9.81 141 9.81 169 9.81 196 9.81 224 9.81 252 28 28 28 28 28 28 29 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 28 28 28 28 28 28 28 28 28 27 28 28 28 28 28 28 27 28 28 28 28 27 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 28 0.20 421 0.20 393 0.20 365 0.20 337 0.20 309 0.20 281 0.20_253 0.20 224 0.20 196 0.20 168 0.20 140 0.20 112 0.20 084 0.20 056 0.20 028 0.20 000 0.19 972 0.19 944 0.19 916 0.19 888 0.19 860 0.19 832 0.19 805 0.19 777 0.19 749 0.19 721 0.19 693 0.19 665 0.19 637 0.19 609 0.19 581 0.19 553 0.19 526 0.19 498 0.19 470 0.19 442 0.19 414 0.19 386 0.19 358 0.19 331 0.19 303 0.19 275 0.19 247 0.19 219 0.19 192 0.19 164 0.19 136 0.19 108 0.19 081 0.19 053 0.19 025 0.18 997 0.18 970 0.18 942 0.18 914 0.18 887 0.18 859 0.18 831 0.18 804 0.18 776 0.18 748 9.92 842 9.92 834 9.92 826 9.92 818 9.92 810 9.92 803 9.92 795 9.92 787 9.92 779 9.92 771 9.92 763 9.92 755 9.92 747 9.92 739 9.92 731 9.92 723 9.92 715 9.92 707 9.92 699 9.92 691 9.92 683 9.92 675 9.92 667 9.92 659 9.92 651 9.92 643 9.92 635 9.92 627 9.92 619 9.92 611 9.92 603 9.92 595 9.92 587 9.92 579 9.92 571 9.92 563 9.92 555 9.92 546 9.92 538 9.92 530 9.92 522 9.92 514 9.92 506 9.92 498 9.92 490 9.92 482 9.92 473 9.92 465 9.92 457 9.92 449 9.92 441 9.92 433 9.92 425 9.92 416 9.92 408 9.92 400 9.92 392 9.92 384 9.92 376 9.92 367 9.92 359 30 P.P. 29 28 27 2.4 4.8 7.2 9-7 12. 1 14-5 2.2 4-5 6.8 9.0 7 II.2 ojia-s 35 i6.9!i6.3ji5.* .7 i8.0 .o'20.2 .3'22.5 .724.8 40 19.31 45 2i.8;2i. 50 24.2j23. SS'26.6'2S. 21 20 19 1.7 3-3 5.0 7.01 6.7 8.81 8.3 io.5|io.o 12.2 11.7 14.0; 13.3 I5.8|I5.0 I7.5ji6.7 S5'i9.2'i8.3 1.6 3.2 4.8 6.3 7.9 9.5 II. I 12.7 14.2 IS.8 17.4 9 I 8 5 0.8 o 10 1.5 I 2.2 2 3-0 2 3.83 4-5 4 5.2 4 6.0 5 45:6.8 6 50:7.56 55 8.2 7 171 70.6 3 1.2 1.8 7 2.3 3|2.9 03.5 74.1 3|4-7 05.2 75.8 3i6.4 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. P.P. 33^ 157 ' L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. P.P. 9.73 611 19 20 19 20 19 19 20 19 19 20 19 19 20 19 19 20 19 19 19 19 20 19 19 19 19 20 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18 19 19 19 19 19 18 19 19 19 18 19 9.81 252 27 28 28 27 28 28 27 28 27 28 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 27 28 27 28 27 27 28 27 28 27 27 28 27 27 28 27 27 28 27 27 27 28 27 27 27 28 0.18 748 9.92 359 8 8 60 59 1 9.73 630 9.81 279 0.18 721 9.92 351 2 9.73 650 9.81 307 0.18 693 9.92 343 8 58 3 9.73 669 9.81 335 0.18 665 9.92 335 9 8 57 4 9.73 689 9.81 362 0.18 638 9.92 326 56 5 9.73 708 9.81 390 0.18 610 9.92 318 8 55 6 9.73 727 9.81418 0.18 582 9.92 310 8 Q 54 "\ 28 27 7 9.73 747 9.81 445 0.18 555 9.92 302 53 5 2.3 2.2 8 9.73 766 9.81 473 0.18 527 9.92 293 8 52 10 4-7 IS 7.0 20 9-3 25 II-7 30 14-0 35 i6.3 4-5 6 8 9 9.73 785 9.81 500 0.18 500 9.92 285 8 8 9 51 50 49 9.0 II. 2 13.S 15.8 10 9.73 805 9.81 528 0.18 472 9.92 277 11 9.73 824 9.81 556 0.18 444 9.92 269 12 9.73 843 9.81 583 0.18 417 9.92 260 8 48 40 18.7 45 21.0 50 23.3 18.0 13 9.73 863 9.81611 0.18 389 9.92 252 8 9 47 22.5 14 9.73 882 9.81 638 0.18 362 9.92 244 46 55 25.7 24.8 15 9.73 901 9.81 666 0.18 334 9.92 235 8 45 16 9.73 921 9.81 693 0.18 307 9.92 227 8 8 9 8 8 9 8 44 17 9.73 940 9.81 721 0.18 279 9.92 219 43 18 9.73 959 9.81 748 0.18 252 9.92 211 42 19 9.73 978 9.81 776 0.18 224 9.92 202 41 40 39 20 9.73 997 9.81 803 0.18 197 9.92 194 21 9.74 017 9.81831 0.18 169 9.92 186 22 9.74 036 9.81 858 0.18 142 9.92 177 38 23 9.74 055 9.81 886 0.18 114 9.92 169 8 9 37 24 9.74 074 9.81913 0.18 087 9.92 161 36 25 9.74 093 9.81 941 0.18 059 9.92 152 8 8 9 8 8 9 8 8 9 8 9 8 8 9 8 9 8 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 35 26 9.74 113 9.81 968 0.18 032 9.92 144 34 " 20 19 18 27 9.74 132 9.81 996 0.18 004 9.92 136 33 5 I.' 1 1.6 I.S 28 9.74 151 9.82 023 0.17 977 9.92 127 31 10 3-: 5 3.2 ) 4.8 1 6.3 5 7.9 3.0 29 30 9.74 170 9.82 051 0.17 949 9.92 119 31 30 20 25 30 6.' 8.: TO f 6.0 7.5 9.74 189 9.82 078 0.17 922 9.92 111 31 9.74 208 9.82 106 0.17 894 9.92 102 29 35 II.' 1 II. I 10.5 32 9.74 227 9.82 133 0.17 867 9.92 094 28 40 13.: J 12.7 12.0 33 9.74 246 9.82 161 0.17 839 9.92 086 27 50 16.' 15.8 15.0 34 9.74 265 9.82 188 0.17 812 9.92 077 26 55 18.: H7.4 16.5 35 9.74 284 9.82 215 0.17 785 9.92 069 25 36 9.74 303 9.82 243 0.17 757 9.92 060 24 37 9.74 322 9.82 270 0.17 730 9.92 052 23 38 9.74 341 9.82 298 0.17 702 9.92 044 22 39 9.74 360 9.82 325 0.17 675 9.92 035 21 20 40 9.74 379 9.82 352 0.17 648 9.92 027 41 9.74 398 9.82 380 0.17 620 9.92 018 19 42 9.74 417 9.82 407 0.17 593 9.92 010 18 43 9.74 436 9.82 435 0.17 565 9.92 002 17 44 9.74 455 9.82 462 0.17 538 9.91 993 16 45 9.74 474 9.82 489 0.17 511 9.91 985 15 46 9.74 493 9.82 517 0.17 483 9.91 976 14 " 9 8 47 9.74 512 9.82 544 0.17 456 9.91 968 13 5 0.8 0.7 48 9.74 531 9.82 571 0.17 429 9.91 959 12 10 i.S 1.3 49 50 9.74 549 9.82 599 0.17 401 9.91 951 11 10 20 25 30 3-0 3.8 1 5 2.7 3-3 4.0 9.74 568 9.82 626 0.17 374 9.91 942 51 9.74 587 9.82 653 0.17 347 9.91 934 9 35 5.2 4-7 52 9.74 606 9.82 681 0.17 319 9.91 925 8 40 b.o 5.8 7.5 1:5 6.7 53 9.74 625 9.82 708 0.17 292 9.91917 7 50 54 9.74 644 9.82 735 0.17 265 9.91 908 6 55 3.2 7.3 55 9.74 662 9.82 762 0.17 238 9.91 900 5 56 9.74 681 9.82 790 0.17 210 9.91 891 4 57 9.74 700 9.82 817 0.17 183 9.91 883 3 -^ 58 9.74 719 9.82 844 0.17 156 9.91 874 2 f ! 59 9.74 737 9.82 871 0.17 129 9.91 866 1 60 9.74 756 9.82 899 0.17 101 9.91 857 1 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. / P.P. 56^ 158 34^ 10 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 L. Sin. 9.74 756 9.74 775 9.74 794 9.74 812 9.74 831 9.74 850 9.74 868 9.74 887 9.74 906 9.74 924 9.74 943 9.74 961 9.74 980 9.74 999 9.75 017 9.75 036 9.75 054 9.75 073 9.75 091 9.75 110 9.75 128 75 147- 75 165 75 184 75 202 75 221 9.75 239 9.75 258 9.75 276 9.75 294 9.75 313 9.75 331 9.75 350 9.75 368 9.75 386 9.75 405 9.75 423 9.75 441 9.75 459 9.75 478 9.75 496 75 514 75 533 75 551 75 569 75 587 75 605 9.75 624 9.75 642 9.75 660 9.75 678 75 696 75 714 75 733 75 751 75 769 75 787 75 805 75 823 75 841 9.75 859 d. L. Tan. c.d. L. Cot. L. Cos. d 19 19 18 19 19 18 19 19 18 19 18 19 19 18 19 18 19 18 19 18 19 18 19 18 19 18 19 18 18 19 18 19 18 18 19 18 18 18 19 18 18 19 18 18 18 18 19 18 18 18 18 18 19 18 18 18 18 18 18 18 9.82 9.82 9.82 9.82 9.83 9.83 9.83 9.83 9.83 9.83 899 926 953 980 008 035 062 089 117 144 9.83 171 9.83 9.83 9.83 9.83 9.83 9.83 9.83 9.83 9.83 198 225 252 280 307 334 361 388 415 9.83 442 9.83 9.83 9.83 9.83 470 497 524 551 578 605 632 659 686 9.83 713 9.83 9.83 9.83 9.83 9.83 9.83 9.83 9.83 9.83 740 768 795 822 849 876 903 930 957 9.83 984 9.84 9.84 9.84 9.84 9.84 9.84 9.84 9.84 9.84 Oil 038 065 092 119 146 173 200 227 9.84 254 9.84 9.84 9.84 9.84 9.84 9.84 9.84 9.84 9.84 280 307 334 361 388 415 442 469 496 9.84 523 27 27 27 28 27 27 27 28 27 27 27 27 27 28 27 27 27 27 27 27 28 27 27 27 27 27 27 27 27 27 27 28 27 27 27 27 27 27 27 27 27 27 27 2'7 27 27 27 27 27 27 26 27 27 27 27 27 27 27 27 27 0.17 101 0.17 074 0.17 047 0.17 020 0.16 992 0.16 965 0.16 938 0.16 911 0.16 883 0.16 856 0.16 829 0.16 802 0.16 775 0.16 748 0.16 720 0.16 693 0.16 666 0.16 639 0.16 612 0.16 585 0.16 558 0.16 530 0.16 503 0.16 476 0.16 449 0.16 422 0.16 395 0.16 368 0.16 341 0.16 314 0.16 287 0.16 260 0.16 232 0.16 205 0.16 178 0.16 151 0.16 124 0.16 097 0.16 070 0.16 043 0.16 016 0.15 989 0.15 962 0.15 935 0.15 908 0.15 881 0.15 854 0.15 827 0.15 800 0.15 773 0.15 746 0.15 720 0.15 693 0.15 666 0.15 639 0.15 612 0.15 585 0.15 558 0.15 531 0.15 504 0.15 477 9.91 857 9.91 849 9.91 840 9.91 832 9.91 823 9.91 815 9.91 806 9.91 798 9.91 789 9.91 781 9.91 772 9.91 763 9.91 755 9.91 746 9.91 738 9.91 729 9.91 720 9.91 712 9.91 703 9.91 695 9.91 686 9.91 677 9.91 669 9.91 660 9.91 651 9.91 643 9.91 634 9.91 625 9.91 617 9.91 608 9.91 599 9.91 591 9.91 582 9.91 573 9.91 565 9.91 556 9.91 547 9.91 538 9.91 530 9.91 521 9.91 512 9.91 504 9.91 495 9.91 486 9.91 477 9.91 469 9.91 460 9.91 451 9.91 442 9.91 433 9.91 425 9.91416 9.91 407 9.91 398 9.91 389 9.91 381 9.91 372 9.91 363 9.91 354 9.91 345 9.91 336 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. | d 55° P.P. 28 27 26 2.3 2.2 4-7 4-5 7.0 6.8 93 9.0 25 11.7,11.2 3o!i4.o!i3.5 3S'i6.3:iS.8 40 i8.7ji8.o 45 21.0 20.2 S0J23.3:22.S 5Sl2S.7'24.8i 2.2 4.3 6.5 8.7 10.8 13.0 15.2 17.3 19.5 21.7 23.8 19 18 1.6 3.2 4.8 6.3 7.9 9.5 II. I 12.7 45 14.2 50 15-8 55 17-4 1.5 3.0 4.S 6.0 7-5 9.0 10.5 12.0 13.S iS.o 16.S 9 8 50.8 ioii.5 15,2.2 203.0 2S|3.8 30|4.5 35{S.2 40 6.0 .217-3 P.P. 35° 159 1 1 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. P.P. 9.75 859 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 17 9.84 523 27 26 0.15 477 9.91 336 8 9 60 59 1 9.75 877 9.84 550 0.15 450 9.91 328 2 9.75 895 9.84 576 27 27 27 27 0.15 424 9.91 319 Q 58 3 9.75 913 9.84 603 0.15 397 9.91 310 9 q 57 4 9.75 931 9.84 630 0.15 370 9.91 301 56 5 9.75 949 9.84 657 0.15 343 9.91 292 9 55 6 9.75 967 9.84 684 27 97 0.15 316 9.91 283 9 8 9 54 N 27 26 7 9.75 985 9.84 711 0.15 289 9.91 274 53 5 2.2 2.2 8 9.76 003 9.84 738 26 0.15 262 9.91 266 52 lO 4-5 6 8 i-l 9 10 9.76 021 9.84 764 27 27 27 0.15 236 9.91 257 9 9 Q 51 50 49 9.0 8.7 1 9.76 039 9.84 791 0.15 209 9.91 248 25 II. 2 I0.8 30,13-5 13-0 35 IS.8, 15.2 11 9.76 057 9.84 818 0.15 182 9.91 239 12 9.76 075 9.84 845 27 0.15 155 9.91 230 9 9 9 9 9 9 48 40 i8.o 17.3 45 20.2 19.5 50 22.5 21.7 13 9.76 093 9.84 872 27 ?6 0.15 128 9.91 221 47 14 9.76 111 9.84 899 0.15 101 9.91 212 46 Ssi24.8 23.8 15 9.76 129 9.84 925 97 0.15 075 9.91 203 45 16 9.76 146 18 18 9.84 952 27 27 0.15 048 9.91 194 44 17 9.76 164 9.84 979 0.15 021 9.91 185 43 18 9.76 182 18 9.85 006 27 0.14 994 9.91 176 9 42 19 9.76 200 18 18 17 9.85 033 26 27 27 0.14 967 9.91 167 9 9 8 9 9 9 9 41 20 9.76 218 9.85 059 0.14 941 9.91 158 40 39 21 9.76 236 9.85 086 0.14 914 9.91 149 22 9.76 253 18 9.85 113 27 0.14 887 9.91 141 38 23 9.76 271 18 18 17 9.85 140 26 97 0.14 860 9.91 132 37 24 9.76 289 9.85 166 0.14 834 9.91 123 36 25 9.76 307 9.85 193 27 0.14 807 9.91 114 35 26 9.76 324 18 18 18 9.85 220 27 26 27 0.14 780 9.91 105 9 9 9 9 9 9 9 9 10 9 9 9 9 9 9 9 9 9 9 9 9 9 10 34 " 18 17 27 9.76 342 9.85 247 0.14 753 9.91 096 ?>?^ s 1.5 1.4 28 9.76 360 9.85 273 0.14 727 9.91 087 32 I5i 4-51 4-2 20; 6.0 5.7 25 7.5 7.1 30: 9.0 8.5 3510.5 9.9 29 9.76 378 17 18 18 17 9.85 300 27 27 26 27 0.14 700 9.91 078 31 30 29 30 9.76 395 9.85 327 0.14 673 9.91 069 31 9.76 413 9.85 354 0.14 646 9.91 060 32 9.76 431 9.85 380 0.14 620 9.91 051 28 2>Z 9.76 448 18 18 9.85 407 27 26 0.14 593 9.91 042 27 5o]i5.o 14.2 34 9.76 466 9.85 434 0.14 566 9.91 033 26 55 16.5 15.6 35 9.76 484 17 9.85 460 27 0.14 540 9.91 023 25 36 9.76 501 18 18 9.85 487 27 96 0.14 513 9.91 014 24 37 9.76 519 9.85 514 0.14 486 9.91 005 23 38 9.76 537 17 9.85 540 27 0.14 460 9.90 996 22 39 9.76 554 18 18 17 9.85 567 27 26 97 0.14 433 9.90 987 21 20 40 9.76 572 9.85 594 0.14 406 9.90 978 41 9.76 590 9.85 620 0.14 380 9.90 969 19 42 9.76 607 18 9.85 647 97 0.14 353 9.90 960 18 43 9.76 625 17 18 9.85 674 26 97 0.14 326 9.90 951 17 44 9.76 642 9.85 700 0.14 300 9.90 942 16 45 9.76 660 17 9.85 727 97 0.14 273 9.90 933 15 46 9.76 677 18 17 18 9.85 754 26 27 97 0.14 246 9.90 924 14 "|10 9 j 8 47 9.76 695 9.85 780 0.14 220 9.90 915 13 5 0.8 0.8 0.7 48 9.76 712 9.85 807 0.14 193 9.90 906 12 10 1.7 15 2.5 20 3-3 25 4-2 30 5.0 1.5 1.3 49 50 9.76 730 17 18 17 18 9.85 834 26 27 26 27 0.14 166 9.90 896 9 9 9 9 11 10 3-0 2.7 3-83.3 4-5 4-0 9.76 747 9.85 860 0.14 140 9.90 887 51 9.76 765 9.85 887 0.14 113 9.90 878 9 35 5.8 5-2 4.7 52 9.76 782 9.85 913 0.14 087 9.90 869 8 406.7 45:7-5 508.3 6.0 5-3 6860 53 9.76 800 17 18 9.85 940 27 26 27 0.14 060 9.90 860 9 9 10 9 9 9 9 7 7-5 6.7 54 9.76 817 9.85 967 0.14 033 9.90 851 6 55 9.218.2 7.3 55 9.76 835 17 9.85 993 0.14 007 9.90 842 5 56 9.76 852 18 17 9.86 020 26 27 0.13 980 9.90 832 4 57 9.76 870 9.86 046 0.13 954 9.90 823 3 1 58 9.76 887 17 9.86 073 27 26 0.13 927 9.90 814 2 59 9.76 904 18 9.86 100 0.13 900 9.90 805 1 » 60 9.76 922 9.86 126 0.13 874 9.90 796 L. Cos. d. L. Cot. c.d. T. Tan. L. Sin. d. 1 P.P. 54° 160 36° ' L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. P.P. 9.76 922 17 18 17 17 18 17 17 18 17 17 17 18 17 17 17 18 17 17 17 18 17 17 17 17 17 17 17 18 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 16 17 17 17 17 17 17 16 17 17 17 17 16 9.86 126 27 26 27 26 27 26 27 26 27 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 26 27 26 26 27 26 27 26 26 27 26 26 27 26 26 27 26 26 27 26 26 27 26 26 26 0.13 874 9.90 796 9 10 9 9 9 9 10 9 9 9 10 9 9 9 10 9 9 9 10 9 9 10 9 9 9 10 9 9 'I 9 10 9 10 9 9 10 9 9 10 9 10 9 10 9 9 10 9 10 9 10 9 10 9 10 9 10 9 10 9 60 59 58 57 1 2 3 9.76 939 9.76 957 9.76 974 9.86 153 9.86 179 9.86 206 0.13 847 0.13 821 0.13 794 9.90 787 9.90 777 9.90 768 4 5 6 9.76 991 9.77 009 9.77 026 9.86 232 9.86 259 9.86 285 0.13 768 0.13 741 0.13 715 9.90 759 9.90 750 9.90 741 56 55 54 // 27 [ 26 7 8 9 9.77 043 9.77 061 9.77 078 9.86 312 9.86 338 9.86 365 0.13 688 0.13 662 0.13 635 9.90 731 9.90 722 9.90 713 53 52 51 50 49 48 47 5 10 15 20 25 30 35 40 1 451 5oi 2.2J 2.2 4.5j 4.3 6.8! 6.5 9.0! 8.7 11.2 10.8 13.5 13.0 15.8 15.2 18.0 17.3 20.2 19.5 22..=; 21.7 10 9.77 095 9.86 392 0.13 608 9.90 704 11 12 13 9.77 112 9.77 130 9.77 147 9.86 418 9.86 445 9.86 471 0.13 582 0.13 555 0.13 529 9.90 694 9.90 685 9.90 676 14 15 16 9.77 164 9.77 181 9.77 199 9.86 498 9.86 524 9.86 551 0.13 502 0.13 476 0.13 449 9.90 667 9.90 657 9.90 648 46 45 44 5si24.8 23.8 17 18 19 9.77 216 9.77 233 9.77 250 9.86 577 9.86 603 9.86 630 0.13 423 0.13 397 0.13 370 9.90 639 9.90 630 9.90 620 43 42 41 40 39 38 37 20 9.77 268 9.86 656 0.13 344 9.90 611 21 22 23 9.77 285 9.77 302 9.77 319 9.86 683 9.86 709 9.86 736 0.13 317 0.13 291 0.13 264 9.90 602 9.90 592 9.90 583 24 25 26 9.77 336 9.77 353 9.77 370 9.86 762 9.86 789 9.86 815 0.13 238 0.13 211 0.13 185 9.90 574 9.90 565 9.90 555 36 35 34 It 18 1 17 16 27 28 29 9.77 387 9.77 405 9.77 422 9.86 842 9.86 868 9.86 894 0.13 158 0.13 132 0.13 106 9.90 546 9.90 537 9.90 527 33 31 31 30 29 28 27 5 10 15 20 I 3 4 6 5 1.4 2.8 5 4.2 5.7 5 7.1 01 8.5 1.3 2.7 4.0 11 8.0 9.3 107 12.0 13.3 30 31 32 33 9.77 439 9.86 921 0.13 079 9.90 518 25 7 30! 9 9.77 456 9.77 473 9.77 490 9.86 947 9.86 974 9.87 000 0.13 053 0.13 026 0.13 000 9.90 509 9.90 499 9.90 490 35 10.5I 9.9 40 12.0 11.3 45 13.5 12.8 50 15.0 14.2 34 35 36 9.77 507 9.77 524 9.77 541 9.87 027 9.87 053 9.87 079 0.12 973 0.12 947 0.12 921 9.90 480 9.90 471 9.90 462 26 25 24 55 16.S IS.6 14.7 37 38 39 9.77 558 9.77 575 9.77 592 9.87 106 9.87 132 9.87 158 0.12 894 0.12 868 0.12 842 9.90 452 9.90 443 9.90 434 23 22 21 20 19 18 17 i 40 9.77 609 9.87 185 0.12 815 9.90 424 41 42 43 9.77 626 9.77 643 9.77 660 9.87 211 9.87 238 9.87 264 0.12 789 0.12 762 0.12 736 9.90 415 9.90 405 9.90 396 44 45 46 9.77 677 9.77 694 9.77 711 9.87 290 9.87 317 9.87 343 0.12 710 0.12 683 0.12 657 9.90 386 9.90 377 9.90 368 16 15 14 " j 10| 9 1 47 48 49 50 51 52 53 9.77 728 9.77 744 9.77 761 9.77 778 9.87 369 9.87 396 9.87 422 0.12 631 0.12 604 0.12 578 9.90 358 9.90 349 9.90 339 13 12 11 10 9 8 7 5 0.8 10 1.7 152.5 203.3 25 4-2 30 5-0 0.8 1 i.S 2.2 3.0 3-8 4.5 5.2 6.0 6.8 7.5 9.87 448 0.12 552 9.90 330 9.77 795 9.77 812 9.77 829 9.87 475 9.87 501 9.87 527 0.12 525 0.12 499 0.12 473 9.90 320 9.90 311 9.90 301 35 40 45 50 5.8 6.7 7.5 8.3 54 55 56 9.77 846 9.77 862 9.77 879 9.87 554 9.87 580 9.87 606 0.12 446 0.12 420 0.12 394 9.90 292 9.90 282 9.90 273 6 5 4 55 9.218.2 57 58 59 60 9.77 896 9.77 913 9.77 930 9.87 633 9.87 659 9.87 685 0.12 367 0.12 341 0.12 315 9.90 263 9.90 254 9.90 244 3 2 1 9.77 946 9.87 711 0.12 289 9.90 235 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. ' P.P. 53° 37° 161 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 41 42 43 44 45 46 47 48 49 50 60 L. Sin. d. L. Tan. c.d. ! L. Cot. L. Cos. d 9.77 946 9.77 963 9.77 980 9.77 997 .78 013 .78 030 .78 047 .78 063 .78 080 .78 097 9.78 113 9.78 130 9.78 147 9.78 163 9.78 180 9.78 197 9.78 213 9.78 230 9.78 246 9.78 263 9.78 280 9.78 296 9.78 313 9.78 329 9.78 346 9.78 362 9.78 379 9.78 395 9.78 412 9.78 428 9.78 445 9.78 461 9.78 478 9.78 494 9.78 510 9.78 527 9.78 543 9.78 560 9.78 576 9.78 592 9.78 609 9.78 625 9.78 642 9.78 658 9.78 674 9.78 691 9.78 707 9.78 723 9.78 739 9.78 756 9.78 772 9.78 788 9.78 805 9.78 821 9.78 837 9.78 853 9.78 869 9.78 886 9.78 902 9.78 918 9.78 934 9.87 711 9.87 738 9.87 764 9.87 790 9.87 817 9.87 843 9.87 869 9.87 895 9.87 922 9.87 948 9.87 974 9.88 000 9.88 027 9.88 053 9.88 079 9.88 105 9.88 131 9.88 158 9.88 184 9.88 210 9.88 236 9.88 262 9.88 289 9.88 315 9.88 341 9.88 367 9.88 393 9.88 420 9.88 446 9.88 472 9.88 498 9.88 524 9.88 550 9.88 577 9.88 603 9.88 629 9.88 655 9.88 681 9.88 707 9.88 733 9.88 759 9.88 786 9.88 812 9.88 838 9.88 864 9.88 890 9.88 916 9.88 942 9.88 968 9.88 994 9.89 020 9.89 046 9.89 073 9.89 099 9.89 125 9.89 151 9.89 177 9.89 203 9.89 229 9.89 255 9.89 281 27 26 26 27 26 26 26 27 26 26 26 27 26 26 26 26 27 26 26 26 26 27 26 26 26 26 27 26 26 26 26 26 27 26 26 26 26 26 26 26 27 26 26 26 26 26 26 26 26 26 26 27 26 26 26 26 26 26 26 26 0.12 289 0.12 262 0.12 236 0.12 210 0.12 183 0.12 157 0.12 131 0.12 105 0.12 078 0.12 052 0.12 026 0.12 000 0.11973 0.11 947 0.11921 0.11895 0.11 869 0.11 842 0.11816 0.11 790 0.11 764 0.11 738 0.11 711 0.11 685 0.11 659 0.11 633 0.11 607 0.11 580 0.11554 0.11 528 0.11 502 0.11476 0.11 450 0.11423 0.11397 0.11 371 0.11 345 0.11319 0.11 293 0.11 267 0.11 241 0.11 214 0.11 188 0.11 162 0.11 136 0.11 110 0.11 084 0.11058 0.11032 0.11006 0.10 980 0.10 954 0.10 927 0.10 901 0.10 875 0.10 849 0.10 823 0.10 797 0.10 771 0.10 745 0.10 719 9.90 235 9.90 225 9.90 216 9.90 206 9.90 197 9.90 187 9.90 178 9.90 168 9.90 159 9.90 149 9.90 139 9.90 130 9.90 120 9.90 111 9.90 101 9.90 091 9.90 082 9.90 072 9.90 063 9.90 053 9.90 043 9.90 034 9.90 024 9.90 014 9.90 005 9.89 995 9.89 985 9.89 976 9.89 966 9.89 956 9.89 947 9.89 937 9.89 927 9.89 918 9.89 908 9.89 898 9.89 888 9.89 879 9.89 869 9.89 859 9.89 849 9.89 840 9.89 830 9.89 820 9.89 810 9.89 801 9.89 791 9.89 781 9.89 771 9.89 761 9.89 752 9.89 742 9.89 732 9.89 722 9.89 712 9.89 702 9.89 693 9.89 683 9.89 673 9.89 663 9.89 653 10 9 10 9 10 9 10 9 10 10 9 10 9 10 10 9 10 9 10 10 9 10 10 9 10 10 9 10 10 9 10 10 9 10 10 10 9 10 10 10 9 10 10 10 9 10 10 10 10 9 10 10 10 10 10 9 10 10 10 10 60^ 59 58 57 56 55 54 53 52 11 50 49 48 47 46 45 44 43 42 il 40 39 38 37 36 35 34 S3 32 31 _30 29 28 27 26 25 24 23 22 20 19 18 17 16 15 14 13 12 11 JO 9 P.P. 27 26 2.2 2.2 4.5 4..3 6.8 6.5 Q.O 8.7 II.2 10.8 13.5 13.0 i5-8'i5-2' 18.0 17.3 20.2 19.5 22.5 21.7 24.8 23.8I 17 1.4 2.8 4.2 5.7 7.1 8.5 9-9 II.3 12.8 14.2 15.6 16 I 10| 9 10| 1.30.8 0.8 2.7ii-7 1.5 4.0 2.5 2.2 5.3!3.3 3-0 6.7'4.2 3-8 8.0 5.0 4.5 9-3 5.8 5-2 40J10.7 6.7 6.0 45 12.0 7.5 6.8 50; 13.3 8.3 7-5 55:14.79.2 8.2 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin, P.P. 11 52* 162 38° 10 20 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 L. Sin. 9.78 934 9.78 950 9.78 967 9.78 983 9.78 999 9.79 015 9.79 031 9.79 047 9.79 063 9.79 079 9.79 095 9.79 111 9.79 128 9.79 144 9.79 160 9.79 176 9.79 192 9.79 208 9.79 224 9.79 240 9.79 256 9.79 272 9.79 288 9.79 304 9.79 319 9.79 335 9.79 351 9.79 367 9.79 383 9.79 399 9.79 415 9.79 431 9.79 447 9.79 463 9.79 478 9.79 494 9.79 510 9.79 526 9.79 542 9.79 558 9.79 573 9.79 589 9.79 605 9.79 621 9.79 636 9.79 652 9.79 668 9.79 684 9.79 699 9.79 715 79 731 .79 746 .79 762 .79 778 .79 793 .79 809 .79 825 .79 840 .79 856 .79 872 79 887 L. Cos. d. L. Tan. c.d. L. Cot. L. Cos. d. 16 17 16 16 16 16 16 16 16 16 16 17 16 16 16 16 16 16 16 16 16 16 16 15 16 16 16 16 16 16 16 16 16 15 16 16 16 16 16 15 16 16 16 15 16 16 16 15 16 16 15 16 16 15 16 16 15 16 16 15 9.89 281 9.89 9.89 9.89 9.89 9.89 9.89 9.89 9.89 9.89 307 333 359 385 411 437 463 489 515 9.89 541 9.89 9.89 9.89 9.89 9.89 9.89 9.89 9.89 9.89 567 593 619 645 671 697 723 749 775 9.89 801 9.89 9.89 9.89 9.89 9.89 9.89 9.89 9.90 9.90 827 853 879 905 931 957 983 009 035 9.90 061 9.90 9.90 9.90 9.90 9.90 9.90 9.90 9.90 9.90 086 112 138 164 190 216 242 268 294 9.90 320 9.90 9.90 9.90 9.90 9.90 9.90 9.90 9.90 9.90 346 371 397 423 449 475 501 527 553 9.90 578 9.90 9.90 9.90 9.90 9.90 9.90 9.90 9.90 9.90 9.90 604 630 656 682 708 734 759 785 811^ 837 L. Cot. c.d 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 26 26 26 25 26 26 26 26 26 26 25 26 26 26 0.10 719 0.10 693 0.10 667 0.10 641 0.10 615 0.10 589 0.10 563 0.10 537 0.10 511 0.10 485 0.10 459 0.10 433 0.10 407 0.10 381 0.10 355 0.10 329 0.10 303 0.10 277 0.10 251 0.10 225 0.10 199 0.10 173 0.10 147 0.10 121 0.10 095 0.10 069 0.10 043 0.10 017 0.09 991 0.09 965 0.09 939 0.09 914 0.09 888 0.09 862 0.09 836 0.09 810 0.09 784 0.09 758 0.09 732 0.09 706 0.09 680 0.09 654 0.09 629 0.09 603 0.09 577 0.09 551 0.09 525 0.09 499 0.09 473 0.09 447 0.09 422 0.09 396 0.09 370 0.09 344 0.09 318 0.09 292 0.09 266 0.09 241 0.09 215 0.09 189 0.09 163 L. Tan. 9.89 653 9.89 643 9.89 633 9.89 624 9.89 614 9.89 604 9.89 594 9.89 584 9.89 574 9.89 564 9.89 554 9.89 544 9.89 534 9.89 524 9.89 514 9.89 504 9.89 495 9.89 485 9.89 475 9.89 465 9.89 455 9.89 445 9.89 435 9.89 425 9.89 415 9.89 405 9.89 395 9.89 385 9.89 375 9.89 364 9.89 354 9.89 344 9.89 334 9.89 324 9.89 314 9.89 304 9.89 294 9.89 284 9.89 274 9.89 264 9.89 254 9.89 244 9.89 233 9.89 223 9.89 213 9.89 203 9.89 193 9.89 183 9.89 173 9.89 162 9.89 152 9.89 142 9.89 132 9.89 122 9.89 112 9.89 101 9.89 091 9.89 081 9.89 071 9.89 060 9.89 050 L. Sin. d. 10 P.P. 26 25 2.2 4-3 6.5 8.7 10.8 I3-0 15-2 17.3 19-5 21.7 23.8 2.1 4.2 6.2 8.3 10.4 12.5 14.6 16.7 18.8 20.8 22.9 17 16 1.4 1.3 2.8 2.7 4.2 4.0 S.7 ."5.3 7.1 6.7 8.5 8.0 9.9 9.3 11.3 10.7 12.8 12.0 14.2 13-3 1S.6 14.7 1.2 2.5 3.8 50 6.2 7.5 8.8 lO.O II. 2 12.S 13.8 11 10 9 0.9 0.8 1.8 1.7 2.8 2.5 3-7 4.6 5-5 6.4 7-3 8.2 9.2 10. 1 P.P. 51° 39° 163 / L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. P.P. "o" 1 9.79 887 16 15 16 16 15 16 15 16 9.90 837 26 26 25 26 26 ?6 0.09 163 9.89 050 10 10 60 59 9.79 903 9.90 863 0.09 137 9.89 040 2 9.79 918 9.90 889 0.09 111 9.89 030 10 58 3 9.79 934 9.90 914 0.09 086 9.89 020 11 10 57 4 9.79 950 9.90 940 0.09 060 9.89 009 56 5 9.79 965 9.90 966 0.09 034 9.88 999 10 55 6 9.79 981 9.90 992 26 25 0.09 008 9.88 989 11 10 54 7 9.79 996 9.91 018 0.08 982 9.88 978 53 8 9.80 012 15 16 15 16 9.91 043 26 26 26 26 25 26 26 0.08 957 9.88 968 10 52 9 9.80 027 9.91 069 0.08 931 9.88 958 10 11 in 51 50 49 " 5>R 1 5»i; 1 ie 10 9.80 043 9.91 095 0.08 905 9.88 948 11 9.80 058 9.91 121 0.08 879 9.88 937 12 9.80 074 15 16 15 9.91 147 0.08 853 9.88 927 10 48 5 10 2 2 2 I 1.3 2.7 13 9.80 089 9.91 172 0.08 828 9.88 917 11 10 47 4-3 4.2 14 9.80 105 9.91 198 0.08 802 9.88 906 46 15 0.5 8 7 b.2 8.3 10.4 4.0 1:? 15 9.80 120 16 9.91 224 26 0.08 776 9.88 896 10 45 25 10.8 16 9.80 136 15 1 s 9.91 250 26 95 0.08 750 9.88 886 11 10 44 30 13.0 12.S 8.0 17 9.80 151 9.91 276 0.08 724 9.88 875 43 35 40 17.3 16.7 9.3 10.7 18 9.80 166 16 15 16 15 9.91 301 26 26 26 25 26 0.08 699 9.88 865 10 11 10 10 42 45 19.5 18.8 12.0 19 20 9.80 182 9.80 197 9.91 327 0.08 673 9.88 855 41 40 39 50 55' 21.7 2^.8 20.8 22.0 13.3 1A.7 9.91 353 0.08 647 9.88 844 21 9.80 213 9.91 379 0.08 621 9.88 834 22 9.80 228 16 9.91 404 0.08 596 9.88 824 11 38 23 9.80 244 15 15 16 15 1 !? 9.91 430 26 26 25 26 ?6 0.08 570 9.88 813 10 10 37 24 9.80 259 9.91 456 0.08 544 9.88 803 36 25 9.80 274 9.91 482 0.08 518 9.88 793 11 10 1 1 35 26 9.80 290 9.91 507 0.08 493 9.88 782 34 27 9.80 305 9.91 533 0.08 467 9.88 772 33 28 9.80 320 16 9.91 559 96 0.08 441 9.88 761 10 32 29 9.80 336 15 15 16 9.91 585 25 26 26 0.08 415 9.88 751 10 11 10 31 30 9.80 351 9.91 610 0.08 390 9.88 741 30 31 9.80 366 9.91 636 0.08 364 9.88 730 29 32 9.80 382 15 15 16 9.91 662 26 25 96 0.08 338 9.88 720 11 28 33 9.80 397 9.91 688 0.08 312 9.88 709 10 1 1 27 34 9.80 412 9.91 713 0.08 287 9.88 699 26 35 9.80 428 15 15 15 16 9.91 739 26 26 25 76 0.08 261 9.88 688 10 10 11 10 11 10 1 1 25 36 9.80 443 9.91 765 0.08 235 9.88 678 24 37 9.80 458 9.91 791 0.08 209 9.88 668 23 38 9.80 473 9.91 816 0.08 184 9.88 657 22 39 40 9.80 489 15 15 1 1; 9.91 842 26 25 96 0.08 158 9.88 647 21 20 19 9.80 504 9.91 868 0.08 132 9.88 636 41 9.80 519 9.91 893 0.08 107 9.88 626 e I 2 8 42 9.80 534 16 9.91 919 26 26 95 0.08 081 9.88 615 10 11 in 18 10 2.5 1.8 1.7 43 9.80 550 15 1 s 9.91 945 0.08 055 9.88 605 17 15 3.8 2.8 11 44 9.80 565 9.91 971 0.08 029 9.88 594 16 25 i).0 6.2 3-7 4.6 3.3 4.2 45 9.80 580 15 15 15 16 9.91.996 26 26 25 96 0.08 004 9.88 584 11 10 11 10 11 10 11 11 10 11 10 11 10 11 11 15 30 7.5 5.5 5.0 46 9.80 595 9.92 022 0.07 978 9.88 573 14 35' 0.0 40 lo.o 0.4 7.3 5.8 6.7 47 9.80 610 9.92 048 0.07 952 9.88 563 13 45 II.2 8.2 7.S 48 9.80 625 9.92 073 0.07 927 9.88 552 12 SO I2.S y.2 m T 8.3 n 49 9.80 641 15 15 9.92 099 26 25 26 26 25 26 26 0.07 901 9.88 542 11 10 50 9.80 656 9.92 125 0.07 875 9.88 531 51 9.80 671 9.92 150 0.07 850 9.88 521 9 52 9.80 686 9.92 176 0.07 824 9.88 510 8 53 9.80 701 15 15 9.92 202 0.07 798 9.88 499 7 54 9.80 716 9.92 227 0.07 773 9.88 489 6 55 9.80 731 15 9.92 253 0.07 747 9.88 478 5 56 9.80 746 16 15 9.92 279 25 26 0.07 721 9.88 468 4 57 9.80 762 9.92 304 0.07 696 9.88 457 3 58 9.80 777 15 15 9.92 330 26 25 0.07 670 9.88 447 2 59 9.80 792 9.92 356 0.07 644 9.88 436 1 60 9.80 807 9.92 381 0.07 619 9.88 425 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. / P.P. 50^ 164 40° 1 2 3 4 5 6 7 8 9 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 L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. 9.80 807 50 51 52 53 54 55 56 57 58 59^ 60 9.80 822 9.80 837 9.80 852 9.80 867 9.80 882 9.80 897 9.80 912 9.80 927 9.80 942 9.80 957 9.80 972 9.80 987 9.81 002 9.81 017 9.81 032 9.81 047 9.81 061 9.81 076 9.81 091 9.81 106 9.81 121 9.81 136 9.81 151 9.81 166 9.81 180 9.81 195 9.81 210 9.81 225 9.81 240 9.81 254 9.81 269 9.81 284 9.81 299 9.81 314 9.81 328 9.81 343 9.81 358 9.81 372 9.81 387 9.81 402 9.81417 9.81 431 9.81 446 9.81 461 9.81 475 9.81 490 9.81 505 9.81 519 9.81 534 9.81 549 9.81 563 9.81 578 9.81 592 9.81 607 9.81 622 9.81 636 9.81 651 9.81 665 9.81 680 9.81 694 9.92 381 9.92 407 9.92 433 9.92 458 9.92 484 9.92 510 9.92 535 9.92 561 9.92 587 9.92 612 9.92 638 9.92 663 9.92 689 9.92 715 9.92 740 9.92 766 9.92 792 9.92 817 9.92 843 9.92 868 9.92 894 9.92 920 9.92 945 9.92 971 9.92 996 9.93 022 9.93 048 9.93 073 9.93 099 9.93 124 9.93 150 9.93 175 9.93 201 9.93 227 9.93 252 9.93 278 9.93 303 9.93 329 9.93 354 9.93 380 9.93 406 9.93 431 9.93 457 9.93 482 9.93 508 9.93 533 9.93 559 9.93 584 9.93 610 9.93 636 9.93 661 9.93 687 9.93 712 9.93 738 9.93 763 9.93 789 9.93 814 9.93 840 9.93 865 9.93 891 9.93 916 26 26 25 26 26 25 26 26 25 26 25 26 26 25 26 26 25 26 25 26 26 25 26 25 26 26 25 26 25 26 25 26 26 25 26 25 26 25 26 26 25 26 25 26 25 26 25 26 26 25 26 25 26 25 26 25 26 25 26 25 0.07 619 0.07 593 0.07 567 0.07 542 0.07 516 0.07 490 0.07 465 0.07 439 0.07 413 0.07 388 0.07 362 0.07 337 0.07 311 0.07 285 0.07 260 0.07 234 0.07 208 0.07 183 0.07 157 0.07 132 0.07 106 0.07 080 0.07 055 0.07 029 0.07 004 0.06 978 0.06 952 0.06 927 0.06 901 0.06 876 9.88 425 9.88 415 9.88 404 9.88 394 9.88 383 9.88 372 9.88 362 9.88 351 9.88 340 9.88 330 9.88 319 9.88 308 9.88 298 9.88 287 9.88 276 9.88 266 9.88 255 9.88 244 9.88 234 9.88 223 9.88 212 0.06 850 0.06 825 0.06 799 0.06 77:3 0.06 748 0.06 722 0.06 697 0.06 671 0.06 646 0.06 620 0.06 594 0.06 569 0.06 543 0.06 518 0.06 492 0.06 467 0.06 441 0.06 416 0.06 390 0.06 364 0.06 339 0.06 313 0.06 288 0.06 262 0.06 237 0.06 211 0.06 186 0.06 160 0.06 135 0.06 109 0.06 084 9.88 201 9.88 191 9.88 180 9.88 169 9.88 158 9.88 148 9.88 137 9.88 126 9.88 115 9.88 105 9.88 094 9.88 083 9.88 072 9.88 061 9.88 051 9.88 040 9.88 029 9.88 018 9.88 007 9.87 9 96 9.87 985 9.87 975 9.87 964 9.87 953 9.87 942 9.87 931 9.87 920 9.87 909 9.87 898 9.87 887 9.87 877 9.87 866 9.87 855 9.87 844 9.87 833 9.87 822 9.87 811 9.87 800 9.87 789 9.87 778 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. 49° 60 59 58 57 56 55 54 53 52 _M _50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 iL 30^ 29 28 27 26 25 24 23 22 11 20^ 19 18 17 16 15 14 13 12 11 P.P. " j 26 25 15 SI 2.2 2.1 1.2 10 4-3 4.2 2.S is' 6.5 6.2 3.8 20i 8.7 8.3 S.o 25,10.8 10.4 6.2 30 13-0 12.5 7.S 3S IS.2 14.0 8.8 40 17.3 ib.7 lO.O 4S 19.5 18.8 II. 2 S021.7 20.8 12.5 ss:23.8 22.9 13.8 14 1.2 2.3 3-5 4-7 S.8 7.0 8.2 9.3 10.5 5 10 IS 20 25 30 35 40 4S So;ii.7 55 12.8 11 I 10 0.9 0.8 1.8 1.7 2.812.5 3.7 3.3 4.6|4.2 5-55.0 6.415.8 7.3,6.7 8.2 7.5 9.28.3 10.1I9.2 P.P. 41 165 , L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. *d. P.P. 9.81 694 15 14 9.93 916 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 0.06 084 9.87 778 11 11 11 11 11 11 11 1 1 60 59 1 9.81 709 9.93 942 0.06 058 9.87 767 2 9.81 723 15 14 1 1; 9.93 967 0.06 033 9.87 756 58 3 9.81 738 9.93 993 0.06 007 9.87 745 57 4 9.81 752 9.94 018 0.05 982 9.87 734 56 5 9.81 767 14 15 14 9.94 044 0.05 956 9.87 723 55 6 9.81 781 9.94 069 0.05 931 9.87 712 54 7 9.81 796 9.94 095 0.05 905 9.87 701 53 8 9.81 810 15 14 15 14 14 9.94 120 0.05 880 9.87 690 11 11 11 11 11 11 11 12 11 11 11 11 11 11 52 9 9.81 825 9.94 146 0.05 854 9.87 679 51 10 9.81 839 9.94 171 0.05 829 9.87 668 50 49 11 9.81 854 9.94 197 0.05 803 9.87 657 12 9.81 868 9.94 222 0.05 778 9.87 646 48 " 26 25 15 13 9.81 882 15 14 9.94 248 0.05 752 9.87 635 47 5 10 2.2 2.1 4.3 4 2 1.2 2 5 14 9.81 897 9.94 273 0.05 727 9.87 624 46 IS 6.5I 6.2 3.8 15 9.81911 15 9.94 299 0.05 701 9.87 613 45 20: 8.71 8.3] 5.0 2s;io.8 10.4 6.2 30 13.0 i2.s: 7.5 16 9.81 926 14 15 14 9.94 324 0.05 676 9.87 601 44 17 9.81 940 9.94 350 0.05 650 9.87 590 43 35 is.2;i4.61 8.8 40 17.3 16.7 lO.O 45 19.5 18.8 II. 2 18 9.81 955 9.94 375 0.05 625 9.87 579 42 19 9.81 969 14 15 14 9.94 401 0.05 599 9.87 568 41 40 SO 21.7 20.8 12.5 5Si23.8l22.9ll3.8 20 9.81 983 9.94 426 0.05 574 9.87 557 21 9.81 998 9.94 452 0.05 548 9.87 546 39 22 9.82 012 14 9.94 477 26 25 26 25 0.05 523 9.87 535 11 11 12 38 23 9.82 026 15 14 9.94 503 0.05 497 9.87 524 37 24 9.82 041 9.94 528 0.05 472 9.87 513 36 25 9.82 055 14 9.94 554 0.05 446 9.87 501 11 35 26 9.82 069 15 14 9.94 579 25 26 25 26 25 26 25 26 25 26 0.05 421 9.87 490 11 11 11 11 12 11 11 11 11 12 11 11 11 11 12 11 11 12 11 11 11 12 11 11 12 11 11 12 11 11 12 11 11 12 34 27 9.82 084 9.94 604 0.05 396 9.87 479 33 28 9.82 098 14 9.94 630 0.05 370 9.87 468 32 29 9.82 112 14 15 14 14 15 14 9.94 655 0.05 345 9.87 457 31 30 9.82 126 9.94 681 0.05 319 9.87 446 30 29 31 9.82 141 9.94 706 0.05 294 9.87 434 32 9.82 155 9.94 732 0.05 268 9.87 423 28 33 9.82 169 9.94 757 0.05 243 9.87 412 27 34 9.82 184 9.94 783 0.05 217 9.87 401 26 35 9.82 198 14 9.94 808 0.05 192 9.87 390 25 36 9.82 212 14 14 9.94 834 25 9S 0.05 166 9.87 378 24 37 9.82 226 9.94 859 0.05 141 9.87 367 23 38 9.82 240 15 14 14 14 9.94 884 26 25 26 25 0.05 116 9.87 356 22 39 40 9.82 255 9.94 910 0.05 090 9.87 345 21 20 " 14 1 12 ! 11 9.82 269 9 94 935 0.05 065 9.87 334 41 9.82 283 9.94 961 0.05 039 9.87 322 19 5 1.2 I.o' 0.0 1 42 9.82 297 14 9.94 986 26 25 25 26 25 26 25 0.05 014 9.87 311 18 10 2.3 2.0J 1.8 1 43 9.82 311 15 14 9.95 012 0.04 988 9.87 300 17 IS 20 3.5 4.7 3.0 A 2.8 1 1 44 9.82 326 9.95 037 0.04 963 9.87 288 16 25 5.8 5.0 4.6 45 9.82 340 14 9.95 062 0.04 938 9.87 277 15 30 7.0 8.2 Q.3 6.0 11 ■ 7.3 46 9.82 354 14 14 9.95 088 0.04 912 9.87 266 14 40 8.0 47 9.82 368 9.95 113 0.04 887 9.87 255 13 45 10.5 9.0 8.2 48 9.82 382 14 9.95 139 0.04 861 9.87 243 12 55 12.8 II.O 10. 1 49 50 9.82 396 14 14 15 9.95 164 26 25 25 26 25 26 25 0.04 836 9.87 232 11 10 9 9.82 410 9.95 190 0.04 810 9.87 221 51 9.82 424 9.95 215 0.04 785 9.87 209 52 9.82 439 14 9.95 240 0.04 760 9.87 198 8 53 9.82 453 14 14 9.95 266 0.04 734 9.87 187 7 54 9.82 467 9.95 291 0.04 709 9.87 175 6 55 9.82 481 14 9.95 317 0.04 683 9.87 164 5 56 9.82 495 14 14 9.95 342 26 25 7'^ 0.04 658 9.87 153 4 57 9.82 509 9.95 368 0.04 632 9.87 141 3 58 9.82 523 14 9.95 393 0.04 607 9.87 130 2 59 9.82 537 14 9.95 418 26 0.04 582 9.87 119 1 60 9.82 551 9.95 444 0.04 556 9.87 107 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. 1 P.P. 166 42° , L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. d. P.P. 9.82 551 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 13 14 14 14 14 14 14 13 14 14 14 14 14 13 14 14 14 13 14 14 14 13 14 14 14 13 14 14 13 14 14 13 14 13 14 14 13 14 13 14 14 13 14 13 9.95 444 25 26 25 25 26 25 26 25 25 26 25 25 26 25 26 25 25 26 25 26 25 25 26 25 25 26 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 0.04 556 9.87 107 11 11 12 11 12 11 11 12 11 12 11 12 11 12 11 12 11 11 12 11 12 12 11 12 11 12 11 12 11 12 11 12 12 11 12 11 12 12 11 12 12 11 12 12 11 12 12 11 12 12 12 11 12 12 11 12 12 12 11 12 60 59 58 57 1 2 3 9.82 565 9.82 579 9.82 593 9.95 469 9.95 495 9.95 520 0.04 531 0.04 505 0.04 480 9.87 096 9.87 085 9.87 073 4 5 6 9.82 607 9.82 621 9.82 635 9.95 545 9.95 571 9.95 596 0.04 455 0.04 429 0.04 404 9.87 062 9.87 050 9.87 039 56 55 54 7 8 9 10 11 12 13 9.82 649 9.82 663 9.82 677 9.95 622 9.95 647 9.95 672 0.04 378 0.04 353 0.04 328 9.87 028' 9.87 016 9.87 005 53 52 51 50 49 48 47 9.82 691 9.95 698 0.04 302 9.86 993 9.82 705 9.82 719 9.82 733 9.95 723 9.95 748 9.95 774 0.04 277 0.04 252 0.04 226 9.86 982 9.86 970 9.86 959 " 1 26 25 S' 2.2 2.1 10 4-3 4-2 14 1.2 2.3 14 15 16 9.82 747 9.82 761 9.82 775 9.95 799 9.95 825 9.95 850 0.04 201 0.04 175 0.04 150 9.86 947 9.86 936 9.86 924 46 45 44 15 6.5 6.2 3-5 20 8.7 8.3I 4.7 25 10.8 10.4! 5.8 30 13.0 12.51 7-0 17 18 19 9.82 788 9.82 802 9.82 816 9.95 875 9.95 901 9.95 926 0.04 125 0.04 099 0.04 074 9.86 913 9.86 902 9.86 890 43 42 41 40 39 38 37 35 15-2 i4-6i 8.2 40 17.3 16.7! 9.3 45 19-5 18.8 10.5 50 21.7 20.8 II.7 55 23.8 22.9 12.8 20 9.82 830 9.95 952 0.04 048 9.86 879 21 22 23 9.82 844 9.82 858 9.82 872 9.95 977 9.96 002 9.96 028 0.04 023 0.03 998 0.03 972 9.86 867 9.86 855 9.86 844 24 25 26 9.82 885 9.82 899 9.82 913 9.96 053 9.96 078 9.96 104 0.03 947 0.03 922 0.03 896 9.86 832 9.86 821 9.86 809 36 35 34 27 28 29 9.82 927 9.82 941 9.82 955 9.96 129 9.96 155 9.96 180 0.03 871 0.03 845 0.03 820 9.86 798 9.86 786 9.86 775 33 31 31 30 29 28 27 30 9.82 968 9.96 205 0.03 795 9.86 763 31 32 33 9.82 982 9.82 996 9.83 010 9.96 231 9.96 256 9.96 281 0.03 769 0.03 744 0.03 719 9.86 752 9.86 740 9.86 728 34 35 36 9.83 023 9.83 037 9.83 051 9.96 307 9.96 332 9.96 357 0.03 693 0.03 668 0.03 643 9.86 717 9.86 705 9.86 694 26 25 24 37 38 39 9.83 065 9.83 078 9.83 092 9.96 383 9.96 408 9.96 433 0.03 617 0.03 592 0.03 567 9.86 682 9.86 670 9.86 659 23 22 21 20 19 18 17 40 9.83 106 9.96 459 0.03 541 9.86 647 " 13 SI i-i 10 2.2 15 3.2- 12 ^^ II 41 42 43 9.83 120 9.83 133 9.83 147 9.96 484 9.96 510 9.96 535 0.03 516 0.03 490 0.03 465 9.86 635 9.86 624 9.86 612 I.O 2.0 3.0 0.9 1.8 2.8 44 45 46 9.83 161 9.83 174 9.83 188 9.96 560 9.96 586 9.96 611 0.03 440 0.03 414 0.03 389 9.86 600 9.86 589 9.86 577 16 15 14 20 4.3 25 5.4 30 6.5 35 7.6 4.0 5-0 6.0 7.0 80 7-3 47 48 49 9.83 202 9.83 215 9.83 229 9.96 636 9.96 662 9.96 687 0.03 364 0.03 338 0.03 313 9.86 565 9.86 554 9.86 542 13 12 11 10 9 8 7 45 9.8 50 10.8 55 I1.9 9.0 lO.O II.O 8.2 9.2 lO.I 50 9.83 242 9.96 712 0.03 288 9.86 530 51 52 53 9.83 256 9.83 270 9.83 283 9.96 738 9.96 763 9.96 788 0.03 262 0.03 237 0.03 212 9.86 518 9.86 507 9.86 495 54 55 56 9.83 297 9.83 310 9.83 324 9.96 814 9.96 839 9.96 864 0.03 186 0.03 161 0.03 136 9.86 483 9.86 472 9.86 460 6 5 4 57 58 59 9.83 338 9.83 351 9.83 365 9.96 890 9.96 915 9.96 940 0.03 110 0.03 085 0.03 060 9.86 448 9.86 436 9.86 425 3 2 1 60 9.83 378 9.96 966 0.03 034 9.86 413 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. d. ' P.P. 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 L. Sin. 9.83 378 9.83 392 9.83 405 9.83 419 9.83 432 9.83 446 9.83 459 9.83 473 9.83 486 9.83 500 9.83 513 9.83 527 9.83 540 9.83 554 9.83 567 9.83 581 9.83 594 9.83 608 9.83 621 9.83 634 9.83 648 9.83 661 9.83 674 9.83 688 9.83 701 9.83 715 9.83 728 9.83 741 9.83 755 9.83 768 9.83 781 9.83 795 9.83 808 9.83 821 9.83 834 9.83 848 9.83 861 9.83 874 9.83 887 9.83 901 9.83 914 9.83 927 9.83 940 9.83 954 9.83 967 9.83 980 9.83 993 9.84 006 9.84 020 9.84 033 .84 046 84 059 84 072 84 085 84 098 84 112 84 125 84 138 84 151 84 164 84 177 L. Cos. 14 13 14 13 14 13 14 13 14 13 14 13 14 13 14 13 14 13 13 14 13 13 14 13 14 13 13 14 13 13 14 13 13 13 14 13 13 13 14 13 13 13 14 13 13 13 13 14 13 13 13 13 13 13 14 13 13 13 13 13 43' L. Tan. c.d. L. Cot. L. Cos. d. 167 9.96 966 9.96 991 9.97 016 9.97 042 9.97 067 9.97 092 9.97 118 9.97 143 9.97 168 9.97 193 9.97 219 9.97 244 9.97 269 9.97 295 9.97 320 9.97 345 9.97 371 9.97 396 9.97 421 9.97 447 9.97 472 9.97 497 9.97 523 9.97 548 9.97 573 9.97 598 9.97 624 9.97 649 9.97 674 9.97 700 9.97 725 9.97 750 9.97 776 9.97 801 9.97 826 9.97 851 9.97 877 9.97 902 9.97 927 9.97 953 9.97 978 9.98 003 9.98 029 9.98 054 9.98 079 9.98 104 9.98 130 9.98 155 9.98 180 9.98 206 9.98 231 9.98 256 9.98 281 9.98 307 9.98 332 9.98 357 9.98 383 9.98 408 9.98 433 9.98 458 9.98 484 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 25 26 0.03 034 0.03 009 0.02 984 0.02 958 0.02 933 0.02 908 0.02 882 0.02 857 0.02 832 0.02 807 0.02 781 0.02 756 0.02 731 0.02 705 0.02 680 0.02 655 0.02 629 0.02 604 0.02 579 0.02 553 0.02 528 0.02 503 0.02 477 0.02 452 0.02 427 0.02 402 0.02 376 0.02 351 0.02 326 0.02 300 0.02 275 0.02 250 0.02 224 0.02 199 0.02 174 0.02 149 0.02 123 0.02 098 0.02 073 0.02 047 0.02 022 0.01 997 0.01971 0.01 946 0.01921 0.01 896 0.01 870 0.01 845 0.01 820 0.01 794 9.86 413 9.86 401 9.86 389 9.86 377 9.86 366 9.86 354 9.86 342 9.86 330 9.86 318 9.86 306 9.86 295 9.86 283 9.86 271 9.86 259 9.86 247 9.86 235 9.86 223 9.86 211 9.86 200 9.86 188 9.86 176 9.86 164 9.86 152 9.86 140 9.86 128 9.86 116 9.86 104 9.86 092 9.86 080 9.86 068 9.86 056 9.86 044 9.86 032 9.86 020 9.86 008 9.85 996 9.85 984 9.85 972 9.85 960 9.85 948 9.85 936 0.01 769 0.01 744 0.01 719 0.01 693 0.01 668 0.01 643 0.01 617 0.01 592 0.01 567 0.01 542 0.01 516 9.85 924 9.85 912 9.85 900 9.85 888 9.85 876 9.85 864 9.85 851 9.85 839 9.85 827 9.85 815 9.85 803 9.85 791 9.85 779 9.85 766 9.85 754 9.85 742 9.85 730 9.85 718 9.85 706 9.85 693 L. Cot. c.d. L. Tan. L. Sin, ^Ao 30 P.P. ir 26 25 5 2.2 2.1 lO 4-3 4.2 IS 6.5 6.2 20! 8.71 8.3 25 10.8' 10.4 30 13.0; 12. s 35,15.2 14.6 40 17.316.7 45 19.5 18.8 50 21.7 20.8 55 23.8 22.9I 1.2 2.3 3.5 4-7 5.8 7.0 8.2 9-3 lo.s 11.7 12.8 " 13 12 11 5 I.I I.O 0.9 10, 2.2 2.0 1.8 15' 3-2 3.0 2.8 20 4.3 4.0 3.7 25 5.4 5.0 4.6 30 6.5 6.0 5.S 35 7.6 7.0 6.4 40 8.7 8.0 7.3 45 Q.8 9.0 8.2 50 10.8 lO.O 9.2 55 11.9 II.O lO.I P.P. 168 44 1 2 3 4 5 6 7 8 10 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 60 L. Sin. d. 9.84 177 9.84 190 9.84 203 9.84 216 9.84 229 9.84 242 9.84 255 9.84 269 9.84 282 9.84 295 9.84 308 9.84 321 9.84 334 9.84 347 9.84 360 9.84 373 9.84 385 9.84 398 9.84 411 9.84 424 9.84 437 9.84 450 9.84 463 9.84 476 9.84 489 9.84 502 9.84 515 9.84 528 9.84 540 9.84 553 9.84 566 9.84 579 9.84 592 9.84 605 9.84 618 9.84 630 9.84 643 9.84 656 9.84 669 9.84 682 9.84 694 9.84 707 9.84 720 9.84 733 9.84 745 9.84 758 9.84 771 9.84 784 9.84 796 9.84 809 9.84 822 9.84 835 9.84 847 9.84 860 9.84 873 9.84 885 9.84 898 9.84 911 9.84 923 9.84 936 9.84 949 13 13 13 13 13 13 14 13 13 13 13 13 13 13 13 12 13 13 13 13 13 13 13 13 13 13 13 12 13 13 13 13 13 13 12 13 13 13 13 12 13 13 13 12 13 13 13 12 13 13 13 12 13 13 12 13 13 12 13 13 L. Tan. 9.98 484 9.98 509 9.98 534 9.98 560 9.98 585 9.98 610 9.98 635 9.98 661 9.98 686 9.98 711 9.98 737 9.98 762 9.98 787 9.98 812 9.98 838 9.98 863 9.98 888 9.98 913 9.98 939 9.98 964 9.98 989 9.99 015 9.99 040 9.99 065 9.99 090 9.99 116 9.99 141 9.99 166 9.99 191 9.99 217 9.99 242 9.99 267 9.99 293 9.99 318 9.99 343 9.99 368 9.99 394 9.99 419 9.99 444 9.99 469 9.99 495 9.99 520 9.99 545 9.99 570 9.99 596 9.99 621 9.99 646 9.99 672 9.99 697 9.99 722 9.99 747 9.99 773 9.99 798 9.99 823 9.99 848 9.99 874 9.99 899 9.99 924 9.99 949 9.99 975 c.d. 0.00 000 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 25 26 25 L. Cot. 0.01 516 0.01 491 0.01 466 0.01 440 0.01 415 0.01 390 0.01 365 0.01 339 0.01 314 0.01 289 0.01 263 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin 0.01 238 0.01 213 0.01 188 0.01 162 0.01 137 0.01 112 0.01 087 0.01 061 0. 01 036 0.01011 0.00 985 0.00 960 0.00 935 0.00 910 0.00 884 0.00 859 0.00 834 0.00 809 0.00 783 L. Cos. I d. 9.85 693 9.85 681 9.85 669 9.85 657 9.85 645 9.85 632 9.85 620 9.85 608 9.85 596 9.85 583 9.85 571 9.85 559 9.85 547 9.85 534 9.85 522 9.85 510 9.85 497 9.85 485 9.85 473 9.85 460 0.00 758 0.00 733 0.00 707 0.00 682 0.00 657 0.00 632 0.00 606 0.00 581 0.00 556 0.00 531 0.00 505 0.00 480 0.00 455 0.00 430 0.00 404 0.00 379 0.00 354 0.00 328 0.00 303 0.00 278 0.00 253 0.00 227 0.00 202 0.00 177 0.00 152 0.00 126 0.00 101 0.00 076 0.00 051 0.00 025 0.00 000 9.85 448 9.85 436 9.85 423 9.85 411 9.85 399 9.85 386 9.85 374 9.85 361 9.85 349 9.85 337 9.85 324 9.85 312 9.85 299 9.85 287 9.85 274 9.85 262 9.85 250 9.85 237 9.85 225 9.85 212 9.85 200 9.85 187 9.85 175 9.85 162 9.85 150 9.85 137 9.85 125 9.85 112 9.85 100 9.85 087 9.85 074 9.85 062 9.85 049 9.85 037 9.85 024 9.85 012 9.84 999 9.84 986 9.84 974 9.84 961 9.84 949 P.P. 10 26 1 25 2.2j 2.1 4-3 4-2 6.5 6.2 8.7 8.3 io.8jio.4 13-0 12.5 15.2 14.6 17-3 16.7 19.5J18.8 SOJ2I.7;20.8 SS:23.8!22.9 14 , 13 12 1.2 2.3 3-5 4-7 5.8 7.0 8.2 9-3 10.5 11.7 12.8 I.I 2.2 3-2 4-3 5-4 6.5 7.6 8.7 9.8 10.8 11.9 i.o 2.0 30 4.0 5.0 6.0 7.0 8.0 9.0 lO.O II.O P.P. TABLE III. NATURAL TRIGONOMETRIC FUNCTIONS. 169 170 0° N. Sin. N.Tan. N. Cot. N. Cos. JO 11 12 13 14 15 16 17 18 Jl 20 21 22 23 24 25 26 27 28 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 .00000 029 058 087 116 .00145 175 204 233 262 .00291 320 349 378 407 .00436 465 495 524 553 .00582 611 640 669 698 .00727 756 785 814 844 .00873 902 931 960 .00989 .01018 047 076 105 134 .01164 193 222 251 280 .01309 33S 367 396 425 .01454 483 513 542 571 .01600 629 658 687 716 .01745 .00000 029 058 087 116 .00145 175 204 233 262 .00291 320 349 378 407 .00436 465 495 524 553 .00582 611 640 669 698 .00727 756 785 815 844 ,00873 902 931 960 ,00989 .01018 047 076 105 135 .01164 193 222 251 280 .01309 338 367 396 425 .01455 484 513 542 571 .01600 629 658 687 716 .01746 oo 3437.7 1718.9 1145.9 859.44 687.55 572.96 491.11 429.72 381.97 343.77 312.52 286.48 264.44 245.55 229.18 214.86 202.22 190.98 180.93 171.89 163.70 156.26 149.47 143.24 137.51 132.22 127.32 122.77 118.54 114.59 110.89 107.43 104.17 101.11 98.218 95.489 92.908 90.463 88.144 85.940 83.844 81.847 79.943 78.126 76.390 74.729 73.139 71.615 70.153 68.750 67.402 66.105 64.858 63.657 62.499 61.383 60.306 59.266 58.261 57.290 N. Cos. N. Cot. N.Tan. N. Sin. ' l.OCOO 000 000 000 000 1.0000 000 000 000 000 1.0000 .99999 999 999 999 .99999 999 999 999 998 .99998 998 998 998 998 .99997 997 997 997 996 .99996 996 996 995 995 .99995 995 994 994 994 .99993 993 993 992 992 .99991 991 991 990 990 .99989 989 989 988 988 .99987 987 986 986 985 ,99985 60 59 58 i 57 56 55 54 53 52 51 50 49 I 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 31 IL 30 19 j 18 I 17 \ 16 ! 15 i 14 ' 13 ' 12 11 I 10 7 I 6 ! 5 I 4 j 3 ! 2 J_ 1 o 1 ' 1 N. Sin. N.Tan.! N. Cot. N. Cos. [» .01745 .01746 157.290 .99985 60 59 774 775 56.351 984 1 2 803 804 55.442 «>S4 58 1 3 832 833 54.561 933 57 1 4 862 862 53.709 983 56 5 .01891 .01891 52.882 .999«2 55 i ^ 920 920 52.081 982 54 7 949 949 51.303 981 53 8 .01978 .01978 50.549 980 52 : 9 i 10 .02007 .02007 49.816 980 51 .02036 .02036 4t).104 .99979 50 i 11 065 066 48.412 979 49 ' 12 094 095 47.740 978 48 ; 13 123 124 47.085 977 47 Il4 152 153 46.449 977 46 15 .02181 .02182 45.829 .99976 45 16 211 211 45.226 976 44 i 17 240 240 44.639 975 43 i 18 269 269 44.066 974 42 ; 19 20 298 298 43.508 974 41 40 39 .02327 .02328 42.964 .99973 21 356 357 42.433 972 22 385 386 41.916 972 38 : 23 414 415 41.411 971 37 ! 24 443 444 40.917 970 36 1 25 .02472 .02473 40.436 .99969 35 26 501 502 39.965 969 34 27 530 531 39.506 968 33 28 560 560 39.057 967 32 29 30 589 589 38.618 966 31 30 .02618 .02619 38.188 .99966 31 647 648 37.769 965 29 32 676 677 37.358 964 28 33 705 706 36.956 963 27 \ 34 734 735 36.563 963 26 ■ 35 .02763 .02764 36.178 .99962 25 36 792 793 35.801 961 24 37 821 822 35.431 960 23 3% 850 851 35.070 959 22 39 40 879 881 34.715 959 21 20 19 .02908 .02910 34.368 .99958 41 938 939 34.027 957 ■ 42 967 968 33.694 956 18 43 .02996 .02997 33.366 955 17 \ 44 .03025 .03026 33.045 954 16 : 45 .03054 .03055 32.730 .99953 15 1 46 083 084 32.421 952 14 i 47 112 114 32.118 952 13 ' 48 141 143 31-821 951 12 : 49 50 170 172 31.528 950 11 10| 9 .03199 .03201 31.242 .99949 51 228 230 30.960 948 ^ 52 257 259 30.683 947 8 1 53 286 288 30.412 946 7 i 54 316 317 30.145 945 6 ! 55 .03345 .03346 29.882 .99944 5 i 56 374 376 29.624 943 4 57 403 405 29.371 942 3 58 432 434 29.122 941 2 59 60 461 463 28.877 940 1 .03490 .03492 28.636 .99939 N. Cos. N. Cot. N.Tan. N. Sin. ' i 1 2 o 1 N. Sm.|N.Tan.'N. Cot.N.Cos ! [ 3 .03490 .03492 28.636 .99939 60 519 548 577 521 550 579 .399 28.166 27.937 938 937 936 59 58 57 4 5 6 606 .03635 664 609 .03638 667 .712 27.490 .271 935 .99934 933 56 55 54 7 8 9 10 693 723 752 696 725 754 27.057 26.845 .637 932 931 930 53 52 51 50 .03781 .03783 26.432 .99929 11 12 13 810 839 868 812 842 871 .230 26.031 25.835 927 926 925 49 48 47 14 15 1 16 897 .03926 955 900 .03929 958 .642 25.452 .264 924 .99923 922 46 44 I 17 18 19 120 .03984 .04013 042 .03987 .04016 046 25.080 24.898 .719 921 919 918 43 42 41 40 .04071 .04075 24.542 .99917 21 22 23 100 129 159 104 133 162 .368 .196 24.026 916 915 913 39 38 37 24 25 26 188 .04217 246 191 .04220 250 23.859 23.695 .532 912 .99911 910 36 1 35 34 27 28 29 ,30 275 304 333 279 308 337 .372 .214 23.058 909 907 906 33 31 31 1 30 .04362 .04366 22.904 .99905 31 32 33 391 420 449 395 424 454 .752 .602 .454 904 902 901 29 , 28 27 1 34 i 35 i 36 478 .04507 536 483 .04512 541 .308 22.164 22.022 900 99898 897 26 1 37 38 39 565 594 623 570 599 628 21.881 .743 .606 896 894 893 13 11 21 40 .04653 .04658 21.470 .99892 20 41 42 43 682 711 740 687 716 745 .337 .205 21.075 890 889 888 19 18 ; 17 44 45 46 769 .04798 827 774 .04803 833 20.946 20.819 .693 886 .99885 883 16 15 14 47 48 49 50 856 885 914 862 891 920 .569 .446 .325 882 881 879 13 12 11 10 .04943 .04949 20.206 .99878 51 52 53 .04972 .05001 030 .04978 .05007 037 20.087 19.970 .855 876 875 873 9 8 7 54 55 56 059 .05088 117 066 .05095 124 .740 19.627 .516 872 .99870 869 6 5 ! 4 57 58 59 60 146 175 205 153 182 212 .405 .296 .188 867 866 864 3 2 \ 1 i .05234 .05241 19.081 .99863 |n. Cos. N. Cot. N.Tan.|N. Sin.| ' 1 3° 171 1 , N. Sin 'N.Tan. N. Cot ^N.Cos .05234 .05241 19.081 .99863 60 59 58 57 1 2 i 3 263 292 321 270 299 328 18.976 .871 .768 861 860 858 1 4 5 6 350 .05379 408 357 .05387 416 .666 18.564 .464 857 .99855 854 56 55 54 7 i 8 9 437 466 495 445 474 503 .366 .268 .171 852 851 849 53 52 51 10 .05524 .05533 18.075 .99847 50 11 12 13 553 582 611 562 591 620 17.980 .886 .793 846 844 842 49 48 47 14 15 16 640 .05669 698 649 .05678 708 .702 17.611 .521 841 .99839 838 46 45 44 17 18 19 20 727 756 785 737 766 .795 .431 .343 .256 836 834 833 43 42 41 40 .05814 .05824 17.169 .99831 21 22 23 844 873 902 854 883 912 17.084 16.999 .915 829 827 826 39 38 37 1 24 25 26 931 .05960 989 941 .05970 999 .832 16.750 .668 824 .99822 821 36 31 34 27 28 29 30 .06018 047 076 .06029 058 087 .587 507 .428 819 817 815 33 31 31 30 .06105 .06116 16.350 .99813 31 32 33 134 163 192 145 175 204 .272 .195 .119 812 810 808 29 28 27 34 35 ^36 221 .06250 279 233 .06262 291 16.043 15.969 .895 806 .99804 803 26 25 24 1 37 38 ! 39 308 337 366 321 350 379 .821 .748 .676 801 799 797 23 22 21 40 .06395 .06408 15.605 .99795 20 41 42 43 424 453 482 438 467 496 .534 .464 .394 793 792 790 19 18 17 44 45 46 511 .06540 569 525 .06554 584 .325 15.257 .189 788 .99786 784 16 15 14 47 48 49 50 598 627 656 613 642 671 .122 15.056 14.990 782 780 778 13 12 11 .06685 .06700 14.924 .99776 10 51 52 53 714 743 773 730 759 788 .860 .795 .732 774 772 770 9 8 7 54 55 56 802 .06831 860 817 .06847 876 .669 14.606 .544 768 .99766 764 6 5 4 57 58 59 60 889 918 947 905 934 963 .482 .421 .361 762 760 758 0> .06976 .06993 14.301 .99756 N. Cos. N. Cot. N.Tan. N. Sin. / 172 4 o : / N. Sin. N.Tan. N. Cot. N. Cos. .06976 .06993 14.301 .99756 60 1 1 2 3 .07005 034 063 .07022 051 080 .241 .182 .124 754 752 750 59 58 57 4 5 6 092 .07121 150 110 .07139 168 .065 14.008 13.951 748 .99746 744 56 55 54 7 8 9 179 208 237 197 227 256 .894 .838 .782 742 740 738 53 52 51 10 11 12 13 .07266 295 324 353 .07285 314 344 373 13.727 .99736 50 .672 .617 .563 734 731 729 49 48 47 14 15 i 16 382 .07411 440 402 .07431 461 .510 13.457 .404 727 .99725 723 46 45 44 17 18 19 20 21 22 23 469 498 527 490 519 548 .07578 607 636 665 .352 .300 .248 721 719 716 .99714 712 710 708 43 42 41 40 39 38 37 .07556 "585 614 643 13.197 .146 .096 13.046 24 25 26 672 .07701 730 695 .07724 753 12.996 12.947 .898 705 .99703 701 36 35 34 27 28 29 30 759 788 817 .07846 782 812 841 .07870 .850 .801 .754 699 696 694 33 32 31 30 29 28 27 12.706 .659 .612 .566 .99692 31 32 33 875 904 933 899 929 958 689 687 685 34 35 36 962 .07991 .08020 .07987 .08017 046 .520 12.474 .429 683 .99680 678 26 25 24 37 3S 39 40 049 078 107 075 104 134 .08163 .384 .339 .295 676 673 671 23 22 21 .08136 12.251 .99668 20 41 42 43 165 194 223 192 221 251 .207 .163 .120 666 664 661 19 18 17 44 45 46 252 .08281 310 280 .08309 339 .077 12.035 11.992 659 .99657 654 16 47 48 49 50 51 52 53 339 368 397 .08426 368 397 427 .08456 .950 .909 .867 652 649 647 13 i 12 11 10 I 7 11.826 .99644 642 639 637 455 484 513 485 514 544 .785 .745 .705 54 55 56 542 .08571 600 573 .08602 632 .664 11.625 .585 635 .99632 630 6 5 4 57 58 59 60 629 658 687 661 690 720 .546 .507 .468 627 625 622 .99619 3 ! 2 i 1| i .08716 .08749 11.430 N. Cos. N. Cot. N.Tan. N. Sin. j / N. Sin. N.Tan. N. Cot. N.Cos " .08716 .08749 11.430 .99619 60 1 2 i ^ 745 774 803 778 807 837 .392 .354 .316 617 614 612 59 58 57 ! 4 i 5 i 6 831 .08860 889 866 .08895 925 .279 11.242 .205 609 .99607 604 56 55 54 1 7 1 8 1 9 jlO 918 947 .08976 .09005 954 .08983 .09013 .168 .132 .095 602 599 596 53 52 51 50 .09042 11.059 .99594 11 12 13 034 063 092 071 101 130 11.024 10.988 .953 591 588 586 49 48 47 14 15 |16 121 .09150 179 159 .09189 218 .918 10.883 .848 583 .99580 578 46 45 44 |l7 18 19 208 237 266 247 277 306 .814 .780 .746 575 572 570 43 42 41 20 .09295 .09335 10.712 .678 .645 .612 .99567 564 562 559 40 39 38 37 21 22 23 324 353 382 365 394 423 24 ! 25 26 411 .09440 469 453 .09482 511 .579 10.546 .514 556 .99553 551 36 35 34 27 28 29 |30 1 31 32 33 498 527 556 541 570 600 .09629 658 688 717 .481 .449 .417 548 545 542 33 31 31 30 .09585 614 642 671 10.385 .99540 .354 .322 .291 537 534 531 29 28 1 27 34 35 36 700 .09729 758 746 .09776 805 .260 10.229 .199 528 .99526 523 26 25 24 37 38 39 40 41 42 43 787 816 845 .09874 903 932 961 834 864 893 .09923 952 .09981 .10011 .168 .138 .108 520 517 514 23 22 21 20 19 18 17 10.078 .99511 508 506 503 .048 10.019 9.9893 44 45 46 .09990 .10019 048 040 .10069 099 .9601 9.9310 .9021 500 .99497 494 16 15 14 47 48 49 50 077 106 135 128 158 187 .10216 246 275 305 .8734 .8448 .8164 9.7882 .7601 .7322 .7044 491 488 485 13 12 11 10 9 8 7 .10164 .99482 479 476 473 51 52 53 192 221 250 54 55 56 279 .10308 337 334 .10363 393 .6768 9.6493 .6220 470 .99467 464 6 5 4 57 58 59 366 395 424 422 452 481 .5949 .5679 .5411 461 458 455 \ 60 .10453 .10510 9.5144 .99452 1 N. Cos.N. Cot. N.Tan. N. Sin. ' i 6 o I_ 1 2 3 4 5 6 7 8 9 10 N. Sin. 'n. Tan. N. Cot. N. Cos. 1 .10453 .10510 540 569 599 628 .10657 687 716 746 775 9.5144 .99452 449 446 443 440 .99437 434 431 428 424 60 1 59 1 58 57 i 56 ' 55 54 ; 53 52 ! 51 50 49 48 47 . 46 45 44^ 43 1 42 41 40 482 511 5i0 569 .10597 626 655 684 713 .4878 .4614 .4352 .4090 9.3831 .3572 .3315 .3060 .2806 .10742 .10805 9.2553 .99421 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 59 771 800 829 858 .10887 916 945 .10973 .11002 .11031 060 089 118 147 .11176 205 234 263 291 .11320 349 378 407 436 .11465 494 523 552 580 .11609 638 667 696 725 .11754 783 812 840 869 834 863 893 922 .10952 .10981 .11011 040 070 .2302 .2052 .1803 .1555 9.1309 .1065 .0821 .0579 .0338 418 415 412 409 .99406 402 399 396 393 .99390 .11099 128 158 187 217 .11246 276 305 335 364 .11394 423 452 482 511 .11541 570 600 629 659 9.0098 8.9860 .9623 .9387 .9152 8.8919 .8686 .8455 .8225 .7996 8.7769 " .7542 .7317 .7093 .6870 8.6648 .6427 .6208 .5989 .5772 386 383 380 377 .99374 370 367 364 360 39 38 ; 37 36 35 i 34 I 33 i 32 i 31 ' 30 29 28 27 26 25 24 23 22 21 20 .99357 354 351 347 344 .99341 337 334 331 327 .99324 .11688 718 747 777 806 .11836 865 895 924 954 .11983 8.5555 .5340 .5126 .4913 .4701 8.4490 .4280 .4071 .3863 .3656 8.3450 .3245 .3041 .2838 .2636 8.2434 .2234 .2035 .1837 .1640 320 317 314 310 .99307 303 300 297 293 19 18 17 16 15 14 13 12 11 10 .11898 927 956 .11985 .12014 .12043 071 100 129 158 .99290 .12013 042 072 101 .12131 160 190 219 249 286 283 279 276 .99272 269 265 262 258 9 8 7 6 5 4 3 2 1 60 .12187 .12278 8.1443 .99255 1 N.COS.N. Cot.lN.Tan. N. Sin. _l_l 7 o 173 i N. Sin. N.Tan.N. Cot. N. Cos. 60 59 .12187 .12278 308 8.1443 .1248 .99255 251 1 216 2 245 338 .1054 248 58 3 274 367 .0860 244 57 1 4 302 397 .0667 240 56 i 5 .12331 .12426 8.0476 .99237 55 6 360 456 .0285 233 54 7 389 485 8.0095 230 53 8 418 515 7.9906 226 52 9 ! 10 447 544 .12574 .9718 222 .99219 51 50 49 .12476 7.9530 11 504 603 .9344 215 12 533 633 .9158 211 48 13 562 662 .8973 208 47 14 591 692 .8789 204 46 15 .12620 .12722 7.8606 .99200 45 16 649 751 .8424 197 44 17 678 781 .8243 193 43 18 706 810 .8062 189 42 19 20 735 840 .7882 186 41 .12764 .12869 7.7704 .99182 40 21 793 899 .7525 178 39 22 822 929 .7348 175 38 23 851 958 .7171 171 37 24 880 .12988 .6996 167 36 25 .12908 .13017 7.6821 .99163 35 26 937 047 .6647 160 34 27 966 076 .6473 156 33 28 .12995 106 .6301 152 32 ' 29 |30 .13024 136 .6129 148 .99144 141 31 30 29 .13053 .13165 195 7.5958 .5787 31 081 32 110 224 .5618 137 28 33 139 254 .5449 133 27 34 168 284 .5281 129 26 35 .13197 .13313 7.5113 .99125 25 36 226 343 .4947 122 24 37 254 372 .4781 118 23 j 38 283 402 .4615 114 22 ' 39 40 312 432 .4451 7.4287 110 .99106 21 20 .13341 .13461 1 41 370 491 .4124 102 19 ; 42 399 521 .3962 098 18 1 43 427 550 .3800 094 17 i 44 456 580 .3639 091 16 1 45 .13485 .13609 7.3479 .99087 15 !46 514 639 .3319 083 14 47 543 669 .3160 079 13 48 572 698 .3002 075 12 49 50 i 51 600 728 .13758 .2844 7.2687 071 .99067 11 10 .13629 658 787 .2531 063 9 ' 52 687 817 .2375 059 8 53 716 846 .2220 055 7 : 54 744 876 .2066 051 6 55 .13773 .13906 7.1912 .99047 5 56 802 935 .1759 043 4 57 831 965 .1607 039 3 58 860 .13995 .1455 035 2 59 60 889 .14024 .1304 7.1154 031 1 .13917 .14054 .99027 1 N. Cos. N. Cot. N.Tan. N. Sin. / 83< 82° 174 8 ° ' N. Sin. N. Tan. N. Cot. N. Cos. 1 2 3 .13917 .14054 7.1154 .99027 60 946 .13975 .14004 084 113 143 .1004 .0855 .0706 023 019 015 59 1 58 ; 57 4 5 6 033 .14061 090 173 .14202 232 .0558 7.0410 .0264 Oil .99006 .99002 56 55 54 7 8 9 10 11 12 13 119 148 177 262 291 321 7.0117 6.9972 .9827 .98998 994 990 53 52 51 50 .14205 .14351 6.9682 .98986 234 263 292 381 410 440 .9538 .9395 .9252 982 978 973 49 48 47 14 15 16 320 .14349 378 470 .14499 529 .9110 6.8969 .8828 969 .98965 961 46 45 44 17 18 19 20 407 436 464 .14493 559 588 618 .8687 .8548 .8408 957 953 948 43 42 41 .14648 6.8269 .98944 40 39 38 37 21 22 23 522 551 580 678 707 737 .8131 .7994 .7856 940 936 931 24 25 26 608 .14637 666 767 .14796 826 .7720 6.7584 .7448 927 .98923 919 36 35 34 27 28 29 i30_ 31 32 33 695 723 752 856 886 915 .7313 .7179 .7045 6.6912 914 910 906 33 32 31 30 .14781 .14945 .14975 .15005 034 .98902 810 838 867 .6779 .6646 .6514 897 893 889 29 28 27 34 35 26 896 .14925 954 064 .15094 124 .6383 6.6252 .6122 884 .98880 876 26 25 24 37 38 39 40 41 42 43 .14982 .15011 040 153 183 213 .5992 .5863 .5734 871 867 863 23 22 21 .15069 .15243 6.5606 .98858 20 097 126 155 272 302 332 .5478 .5350 .5223 854 849 845 19 18 17 44 45 46 184 .15212 241 362 .15391 421 .5097 6.4971 .4846 841 .98836 832 16 15 14 47 48 49 50 270 299 327 .15356 451 481 511 .4721 .4596 .4472 827 823 818 13 12 11 .15540 6.4348 .98814 10 51 52 53 385 414 442 570 600 630 .4225 .4103 .3980 809 805 800 9 8 7 54 55 56 471 .15500 529 660 .15689 719 .3859 6.3737 .3617 796 .98791 787 6 5 4 57 58 59 60 557 586 615 749 779 809 :r5838 .3496 .3376 .3257 6.3138 782 778 773 3 2 1 .15643 .98769 N. Cos. N. Cot. N.Tan. N. Sin. 9° 1 N. Sin. N.Tan. N. Cot. N. Cos. 60 59 58 57 1 2 3 .15643 .15838 6.3138 .98769 764 760 755 672 701 730 868 898 928 .3019 .2901 .2783 4 5 6 758 .15787 816 958 .15988 .16017 .2666 6.2549 .2432 751 56 .98746/^55 741 54 7 8 9 10 11 12 13 845 873 902 047 077 107 .16137 167 196 226 .2316 .2200 .2085 6.1970 737 732 728 53 52 51 50 49 48 47 .15931 .98723 959 .15988 .16017 .1856 .1742 .1628 718 714 709 14 15 16 046 .16074 103 256 .16286 316 .1515 6.1402 .1290 704 .98700 695 46 45 44 17 18 19 20 21 22 23 132 160 189 346 376 405 .16435 .1178 .1066 .0955 "6:0844 .0734 .0624 .0514 690 686 681 43 42 41 .16218 246 275 304 .98676 40 465 495 525 671 667 662 39 38 37 24 25 26 333 .16361 390 555 .16585 615 .0405 6.0296 .0188 657 .98652 648 36 35 34 27 28 29 30 419 447 476 .16505 645 674 704 .16734 764 794 824 6.0080 5.9972 .9865 643 638 633 .98629 33 31 31 30 5.9758 .9651 .9545 .9439 31 32 33 533 562 591 624 619 614 29 28 27 34 35 36 620 .16648 677 854 .16884 914 .9333 5.9228 .9124 609 .98604 600 26 25 24 37 3^ 39 40 41 42 43 706 734 763 944 .16974 .17004 .9019 .8915 .8811 5.8708 .8605 .8502 .8400 595 590 585 23 22 21 20 19 18 17 .16792 .17033 063 093 123 .98580 820 849 878 575 570 565 44 45 46 906 .16935 964 153 .17183 213 .8298 5.8197 .8095 561 .98556 551 16 15 14 47 48 1 49 50 51 52 53 .16992 .17021 050 243 273 303 .7994 .7894 .7794 546 541 536 13 12 11 10 .17078 .17333 5.7694 .98531 107 136 164 363 393 423 .7594 .7495 .7396 526 521 516 9 8 7 54 55 56 193 .17222 250 453 .17483 513 .7297 5.7199 .7101 511 .98506 501 6 5 4 57 58 59 279 308 336 543 573 603 .7004 .6906 .6809 496 491 486 3 2 1 / 60 .17365 .17633 5.6713 .98481 N. Cos. N. Cot. N.Tan. N. Sin. ^1° 967 .64989 .65011 458 509 559 .1702 .1695 .1688 022 .76003 .75984 34 35 36 033 .65055 077 609 .85660 710 .1681 1.1674 .1667 965 .75946 927 26 25 24 37 38 39 100 122 144 761 811 862 .1660 .1653 .1647 908 889 870 23 22 21 20 19 18 17 40 .65166 .85912 1.1640 .75851 41 42 43 188 210 232 .85963 .86014 064 .1633 .1626 .1619 832 813 794 44 45 46 254 .65276 298 115 .86166 216 .1612 1.1606 .1599 775 .75756 738 16 15 14 47 48 49 320 342 364 267 318 368 .1592 .1585 .1578 719 700 680 13 12 11 10 9 8 7 50 .65386 .86419 1.1571 .75661 51 52 53 408 430 452 470 521 572 .1565 .1558 .1551 642 623 604 54 55 56 474 .65496 518 623 .86674 725 .1544 1.1538 .1531 585 .75566 547 6 5 4 57 58 59 60 540 562 584 776 827 878 .1524 .1517 .1510 528 509 490 3 2 1 / .65606 .86929 1.1504 .75471 N. Cos. N. Cot. N.Tan. N. Sin. 41° '" N. Sin. N.Tan. N. Cot. N. Cos. .65606 .86929 1.1504 .75471 60 1 2 3 628 650 672 .86980 .87031 082 .1497 .1490 .1483 452 433 414 59 58 57 4 5 6 694 .65716 738 133 .87184 236 .1477 1.1470 .1463 395 .75375 356 56 55 54 7 8 9 759 781 803 287 338 389 .1456 .1450 .1443 337 318 299 53 52 51 10 .65825 .87441 1.1436 .75280 50 11 12 13 847 869 891 492 543 595 .1430 .1423 .1416 261 241 222 49 48 47 14 15 16 913 .65935 956 646 .87698 749 .1410 1.1403 .1396 203 .75184 165 46 45 44 17 18 19 20 .65978 .66000 022 801 852 904 .1389 .1383 .1376 146 126 107 43 42 41 .66044 .87955 1.1369 .75088 40 21 22 23 066 088 109 .88007 059 110 .1363 .1356 .1349 069 050 030 39 38 37 24 25 26 131 .66153 175 162 .88214 265 .1343 1.1336 .1329 .75011 .74992 973 36 35 34 27 28 29 30 197 218 240 317 369 421 .1323 .1316 .1310 953 934 915 ZZ 32 31 30 .66262 .88473 1.1303 .74896 31 32 ZZ 284 306 327 524 576 628 .1296 .1290 .1283 876 857 838 29 28 27 34 35 36 349 .66371 393 680 .88732 784 .1276 1.1270 .1263 818 .74799 780 26 25 24 37 38 39 40 414 436 458 836 888 940 .1257 .1250 .1243 760 741 722 23 22 21 20 .66480 .88992 1.1237 .74703 41 42 43 501 523 545 .89045 097 149 .1230 .1224 .1217 683 664 644 19 18 17 44 45 46 566 .66588 610 201 .89253 306 .1211 1.1204 .1197 625 .74606 586 16 15 14 47 48 49 632 653 675 358 410 463 .1191 .1184 .1178 1.1171 567 548 528 13 12 11 10 50 .66697 .89515 .74509 51 52 53 718 740 762 567 620 672 .1165 .1158 .1152 489 470 451 9 8 7 54 55 56 783 .66805 827 725 .89777 830 .1145 1.1139 .1132 431 .74412 392 6 5 4 57 58 59 60 848 870 891 883 935 .89988 .1126 .1119 .1113 373 353 334 3 2 1 .66913 .90040 1.1106 .74314 N. Cos. N. Cot. N.Tan. N. Sin. 49' 48° 42° 48° 191 , N. Sin. N. Tan. N. Cot. N. Cos. 1 / "o" 1 2 3 N. Sin- N.Tan. N. Cot. N. Cos. .66913 .90040 1.1106 .74314 60 ! .68200 .93252 1.0724 .73135 60 1 2 3 935 956 978 093 146 199 .1100 .1093 .1087 295 276 256 59 58 57 221 242 264 306 360 415 .0717 .0711 .0705 116 096 076 59 58 57 4 5 6 .66999 .67021 043 251 .90304 357 .1080 1.1074 .1067 237 .74217 198 56 55 54 4 5 6 285 .68306 327 469 .93524 578 .0699 1.0692 .0686 056 .73036 .73016 56 55 54 7 8 9 064 086 107 410 463 516 .1061 .1054 .1048 178 159 139 53 52 51 7 8 9 349 370 391 633 688 742 .0680 .0674 .0668 .72996 976 957 53 52 51 10 .67129 .90569 1.1041 .74120 50 10 .68412 .93797 1.0661 .72937 50 11 12 13 151 172 194 621 674 727 .1035 .1028 .1022 100 080 061 49 48 47 11 12 13 434 455 476 852 906 .93961 .0655 .0649 .0643 917 897 877 49 48 47 14 15 16 215 .67237 258 781 .90834 887 .1016 1.1009 .1003 041 .74022 .74002 46 45 44 14 15 16 497 .68518 539 .94016 .94071 125 .0637 1.0630 .0624 857 .72837 817 46 45 44 17 18 19 20 280 301 323 940 .90993 .91046 .0996 .0990 .0983 .73983 963 944 43 42 41 17 18 19 561 582 603 180 235 290 .0618 .0612 .0606 797 777 757 43 42 41 .67344 .91099 1.0977 .73924 40 20 .68624 .94345 1.0599 .72737 40 21 22 23 366 387 409 153 206 259 .0971 .0964 .0958 904 885 865 39 38 37 21 22 23 645 666 688 400 455 510 .0593 .0587 .0581. 717 697 677 39 38 37 24 25 26 430 .67452 473 313 .91366 419 .0951 1.0945 .0939 846 .73826 806 36 35 34 24 25 26 709 .68730 751 565 .94620 676 .0575 1.0569 .0562 657 .72637 617 36 35 34 27 28 29 30 495 516 538 473 526 580 .91633 .0932 .0926 .0919 787 767 747 33 32 31 27 28 29 772 793 814 731 786 841 .0556 .0550 .0544 597 577 557 33 32 31 .67559 1.0913 .73728 30 30 .68835 .94896 1.0538 .72537 30 31 32 33 580 602 623 687 740 794 .0907 .0900 .0894 708 688 669 29 28 27 31 32 33 857 878 899 .94952 .95007 062 .0532 .0526 .0519 517 497 477 29 28 27 34 35 36 645 .67666 688 847 .91901 .91955 .0888 1.0881 .0875 649 .73629 610 26 25 24 34 35 36 920 .68941 962 118 .95173 229 .0513 1.0507 .0501 457 .72437 417 26 25 24 37 38 39 40 709 730 752 .92008 062 116 .0869 .0862 .0856 590 570 551 23 22 21 20 37 38 39 .68983 .69004 025 284 340 395 .0495 .0489 .0483 397 377 357 23 22 21 .67773 .92170 1.0850 .73531 40 .69046 .95451 1.0477 .72337 20 41 42 43 795 816 837 224 277 331 .0843 .0837 .0831 511 491 472 19 18 17 41 42 43 067 088 109 506 562 618 .0470 .0464 .0458 317 297 277 19 18 17 44 45 46 859 .67880 901 385 .92439 493 .0824 1.0818 .0812 452 .73432 413 16 15 14 44 45 46 130 .69151 172 673 .95729 785 .0452 1.0446 .0440 257 .72236 216 16 15 14 47 48 49 50 923 944 965 547 601 655 .92709 .0805 .0799 .0793 393 373 353 13 12 11 47 48 49 50 51 52 53 193 214 235 841 897 .95952 .0434 .0428 .0422 196 176 156 13 12 11 10 .67987 1.0786 .73333 10 .69256 .96008 1.0416 .72136 51 52 53 .68008 029 051 763 817 872 .0780 .0774 .0768 314 294 274 9 8 7 277 298 319 064 120 176 .0410 .0404 .0398 116 095 075 9 8 7 54 55 56 072 .68093 115 926 .92980 .93034 .0761 1.0755 .0749 254 .73234 215 6 5 4 54 55 56 340 .69361 382 232 .96288 344 .0392 1.0385 .0379 055 .72035 .72015 6 5 4 57 58 59 136 157 179 088 143 197 .0742 .0736 .0730 195 175 155 3 2 1 57 58 59 60 403 424 445 400 457 513 .0373 .0367 .0361 .71995 974 954 3 2 1 60 .68200 .93252 1.0724 .73135 .69466 .96569 1.0355 .71934 N. Cos. N. Cot. N.Tan. N. Sin. r N. Cos. N. Cot. N.Tan. N. Sin. 1 47' ifV 192 44< / N. Sin. N.Tan. N. Cot. N. Cos. 60 59 1 .69466 .96569 1.0355 .71934 487 625 .0349 914 2 508 681 .0343 894 58 3 529 738 .0337 873 57 4 549 794 .0331 853 56 5 .69570 .96850 1.0325 .71833 55 6 591 907 .0319 813 54 7 612 .96963 .0313 792 53 8 633 .97020 .0307 772 52 9 10 654 076 .0301 752 .71732 51 50 .69675 .97133 1.0295 11 696 189 .0289 711 49 12 717 246 .0283 691 48 13 737 302 .0277 671 47 14 758 359 .0271 650 46 15 .69779 .97416 1.0265 .71630 45 16 800 472 .0259 610 44 17 821 529 .0253 590 43 18 842 586 .0247 569 42 19 862 643 .0241 549 41 40 20 .69883 .97700 1.0235 .71529 21 904 756 .0230 508 39 22 925 813 .0224 488 38 23 946 870 .0218 468 37 24 966 927 .0212 447 36 25 .69987 .97984 1.0206 .71427 35 26 .70008 .98041 .0200 407 34 27 029 098 .0194 386 33 28 049 155 .0188 366 32 29 070 213 .0182 345 31 30 30 .70091 .98270 1.0176 .71325 31 112 327 .0170 305 29 32 132 384 .0164 284 28 33 153 441 .0158 264 27 34 174 499 .0152 243 26 35 .70195 .98556 1.0147 .71223 25 36 215 613 .0141 203 24 37 236 671 .0135 182 23 38 257 728 .0129 162 22 39 277 786 .0123 141 21 40 .70298 .98843 1.0117 .71121 20 41 319 901 .0111 100 19 42 339 .98958 .0105 080 18 43 360 .99016 .0099 059 17 44 381 073 .0094 039 16 45 .70401 .99131 1.0088 .71019 15 46 422 189 .0082 .70998 14 47 443 247 .0076 978 13 48 463 304 .0070 957 12 49 50 484 362 .0064 937 11 10 9 .70505 .99420 1.0058 .70916 51 525 478 .0052 896 52 546 536 .0047 875 8 53 567 594 .0041 855 7 54 587 652 .0035 834 6 55 .70608 .99710 1.0029 .70813 5 56 628 768 .0023 793 4 57 649 826 .0017 772 3 58 670 884 .0012 752 2 59 690 .99942 .0006 731 1 60 .70711 1.0000 1.0000 .70711 N. Cos. N. Cot. N.Tan. N. Sin. r 45° EXPLANATION OF TABLES. 1. Definition of logarithms. The logarithm of a number to the base k is defined as the power to which k must be raised in order to equal the number. That is, if the logarithm of A to the base k is a, then k" = A. We may then write log;^ A = a. This equation is read "The logarithm of A to the base k equals a." Since 2^' = S, we have log2 8 = 3. Similarly since 2^ = 4 and 32 = 9, we have log2 4 = 2 and logs 9 = 2. Example: Find the values of logs 27, logio 1000, log2 32, log^^ k, log, 1. 2. Logarithms of products, quotients, powers and roots. Let A = k'' and B = k* so that log, A = a and log, B = b. Then AB = rk' = k'+K Hence log, (AB) =a + h. Therefore log*(AB) =log,A+log,B. Also A -^ B = k'' -^ k' = k'^. Hence log, (A-i- B) = a-h. Therefore log, {A -^ B) = log, A - log, B. And A" = (kT = k''^ Hence log, (A") = na. Therefore log, (A") = nlog, A. And *^A = (k'')^=k^\ Hence log, ^A = -a. Therefore log, ^A =-log,A. These four results may be stated as follows: 1. The logarithm of the product of two numbers equals the sum of the logarithms of the two numbers. 13 193 194 ELEMENTS OF PLANE TRIGONOMETRY. 2. The logarithm of the quotient of two numbers equals the logarithm of the numerator minus the logarithm of the denominator. 3. The logarithm of the nth. power of a number equals n times the logarithm of the number. 4. The logarithm of the nth root of a number equals 1/n times the logarithm of the number. 3. Common* Logarithms. For numerical calculations the most convenient system of logarithms is that in which the base is 10. Logarithms to the base 10 are called common logarithms. In all that follows common logarithms will be used and logio A will be denoted simply by log A, etc. The Characteristic. Since 10^ = 10 and 10^ = 100, we have log 10 = 1 and log 100 = 2, and hence the logarithm of any number between 10 and 100 equals a number between 1 and 2, and is therefore made up of 1 plus a decimal. The whole number 1 is called the characteristic of the logarithm and the decimal part is called the mantissa. Similarly the logarithm of a number between 100 and 1000 equals 2 plus a decimal, 2 being the characteristic and the decimal the mantissa of this logarithm. Further, since 10'' = 1 and 10^ = 10, the logarithm of a number between 1 and 10 is between and 1 and the characteristic of such a logarithm is then 0. Since W = 1 and 10~^ = 0.1, the logarithm of a number between 0.1 and 1 is between — 1 and and may therefore be written — 1 + a decimal. The characteristic of such a loga- rithm is then — 1. The mantissa is always considered positive whether the logarithm is a positive or negative number. Suppose the characteristic of a logarithm is — 2 and the mantissa .56253. This logarithm is then — 2 + .56253 for which we use the notation 2.56253, the minus sign being written above the characteristic to show that it alone is negative. In almost all numerical calculations it is found to be more con- venient to add and subtract 10, so that the logarithm takes the form 8.56253 — 10. In practice the — 10 is generally left off when there is no danger of confusion, but its existence must be remembered. Example. What are the characteristics of the logarithms of ELEMENTS OF PLANE TRIGONOMETRY. 195 the following numbers: 315.72, 7.6523, 0.354, 0.000673, 27.0054? The Mantissa. Consider two numbers made up of the same sequence of digits but in which the position of the decimal place is different, such as 723.51 and 7.2351. Now 723.51 = 10^ X 7.2351. Hence, by art. 2, rule 1, log 723.51 = log 102 + log 7.2351. But log 10^ = 2, and hence . log 723.51 = 2 + log 7.2351. That is, the logarithms of these two numbers differ only in the characteristics, and the mantissse are the same. Obviously two numbers made up of the same sequence of digits, but differing in the position of the decimal point, are such that either is equal to the product of the other by an integral power of 10, and hence the mantissae of the logarithms of numbers differing only in the position of the decimal point are the same. For this reason in tables of logarithms the mantissse alone are given, the characteristics being determined by the two rules below. From the foregoing we may formulate a rule for finding the characteristic of the logarithm of any number. A number with five digits to the left of the decimal place, such as 13256.7, is obviously between 10,000 and 100,000, that is, between lO'* and 10^ Its logarithm is then between 4 and 5, and the character- istic is therefore 4. Similarly the characteristic of the logarithm of a number having three digits to the left of the decimal place is 2, and finally the logarithm of a number having n digits to the left of the decimal place is n — 1 Hence, the characteristic of the logarithm of a. number is one less than the number of digits to the left of the decimal place. This rule obviously does not apply to the logarithms of numbers less than 1. Consider the logarithm of a number having no integral part and two zeros to the right of the decimal before the first significant digit appears, such as 0.00325. This number is greater than one one-thousandth and less than one one-hundredth and is therefore between 10~^ and 10"^. Its 196 ELEMENTS OF PLANE TRIGONOMETRY. logarithm is then — 3 + a decimal, so that the characteristic is — 3. Similarly the characteristic of log 0.041 is — 2. Hence for the logarithms of numbers less than 1 the character- istic is minus one more than the number of zeros to the right of the decimal place before the first significant digit. Table I. 4. To find the Logarithm of a Number. (a) When the number has four figures. Find the first three figures of the number on pages 104-121 in the column under N and the fourth figure in the top row. The last three figures of the mantissa are found in the column and row so determined, and the first two figures are found in the column under L. The proper characteristic is then prefixed. Thus to find log 62.48 we look for 624 under N. This is found on page 114, and the first two figures of the mantissa are 79. The last three figures of the mantissa are found opposite 624 and under 8. They are 574. Finally, the characteristic is 1, and therefore log 62.48 = 1.79574. Whenever the three last figures of the mantissa are preceded in the tables by an asterisk (*) it will be found that in this row 78 ) under L two numbers are bracketed, as «q V opposite 616 on page 114. In all such cases the upper of the bracketed numbers is to be used for the first two figures of the mantissa when there is not an asterisk, and the lower when there is one. Thus log 616.4 = 2.78986 and log 616.7 = 2.79007. Similarly we find log 0.7562 = 1.87864 or 9.87864-10; log 0.02543 = 2.40535 or 8.40535 - 10. (6) When the number has three or less figures. Add zeros to the right of the number and then find the logarithm as above. Thus log 23 = log 23.00 = 1.36173 (page 106), etc. (c) When the number has five or more figures. Let us be required to find log 277.53. Since the mantissa does not depend upon the position of the decimal point, we may find it by finding the mantissa of log 27753. The number is between 27750 and 27760, and hence its logarithm is between the logarithms of these numbers. It is therefore equal to log 27750 + x, where ELEMENTS OF PLANE TRIGONOMETRY. 197 a; is a correction to be added to log 27750. We assume that the difference between the logarithms is proportional to the differ- ence between the corresponding numbers. This assumption is only approximately true. Now, the mantissa of log 27753 = .44326 + x ] ^'«f ^"^f f .'^"'^^^^^ log 27750 = .44326 tJ' ^""^ °[ ^°^' = ="' ^ c^^^nr. aa^ac, [ difference of numbers log 27760 = .44342 j m ^ a ta *= -' = 10, and of logs = 16. We then assume that 3 : 10 = a: : 16. And hence x = .3 X 16 = 4.8 = 5 - Therefore log 277.53 = 2.44326 + (5* = 2.44331 16 1.6 3-f 4-8 6.4 8.0 9.6 7 II. 2 8 12.8 The multiplication of .3 by 16 is most easily done , by aid of the table of proportional parts found in the ' logarithm tables to the right of the logarithms. Thus J under 16 and opposite 3 we find 4.8 which is the prod- uct of 16 by .3. 11^.1 From the foregoing we see that we may find the logarithm of a number of five figures as follows. Find the mantissa of the logarithm of the first four figures, and add to this a correction found hy multiplying the difference between this mantissa and the one next following in the tables {tabular dif- ference) by the remaining figure of the given number considered as a decimal. Finally to the mantissa so obtained prefix the proper characteristic. If the number whose logarithm is required has more than five figures we find the logarithm of the nearest number of five figures. Thus, for log 380567 we take log 380570. With logarithmic tables giving only five places of decimals there is little or no accuracy gained by making a correction for a sixth figure. In the correction any decimal less than .5 is neglected, and one over .5 is increased to 1. Thus the correction for log 18.202 is found to be (page 105) 4.8, and this is taken as 5 so that log 18.202 = 1.26007 + (5, * The symbol ( is written before the correction 5 to indicate that this cor- rection is to be added to the last figure of the mantissa. 198 ELEMENTS OF PLANE TRIGONOMETRY. the 5, of course, being added to the last figure of the mantissa. Then log 18.202=1.26012. When a correction involves exactly .5 it is customary to either neglect this .5 or to increase it to 1 as may be necessary in order to make the last figure of the mantissa an even number. Thus log 19115 = 4.28126+ (11. 5. We then take the correction as 12 so that log 19115=4.28138. On the other hand, log 19105 = 4.28103+ (11.5, and the correction is here taken as 11, so that log 19105 = 4.28114. This arbitrary rule is followed as such errors are more apt to neutralize one another than would be the case were every .5 to be either neglected or increased to 1. 5. To find the number corresponding to a given logarithm. Let us be required to find the number n when given that log n= 1.08337. The given mantissa .08337 is not to be found in the tables, but it lies between .08314 and .08350. Hence n, apart from the position of the decimal point, lies between 1211 and 1212. Therefore n = 1211+a; where a; is a correction we may find by assuming that the difference between two numbers is proportional to the difference between their loga- rithms. We have, then, i.- r 1 1010 AoorA ) difference of logs = 36, and of mantissa of log 1212 = .08350 i 1 . 1 1211 - 08*^14 -'^ ^^^^^^^' , "" \oooT 1 difference of logs = 23, and of log n = .08337 r 1 ° ) numbers = a;. Therefore a;: 1 = 23: 36. or a; = 23^36 = .6+ Hence the sequence of numbers making up n is 12116. Finally, since the logarithm has the characteristic 1, we must point off two places, and hence w= 12.116. In finding the correction x we take the nearest tenth only. When we find six significant figures of a number by means of its logarithm from tables giving the mantissae to five places only, the sixth figure is very probably incorrect, and any attempt to find a seventh figure would be of absolutely no value. The finding of the correction x may be performed more easily by means of the tables of proportional parts, as follows: ELEMENTS OF PLANE TRIGONOMETRY. 199 36 36 7.2 10.8 14.4 18.0 21.6 25.2 288 9l32-4 As we have seen a; = 23-T-36. That is x is the quo- , tient obtained by dividing the difference between the 3 given mantissa and the next smaller one in the tables \ by the tabular difference (the difference between sue- ^ cessive mantissae in the Tables). In the table of proportional parts under the tabular differ- ence (36 in this case) find the nearest number to 23. This is seen to be 21.6, opposite to which is found 6. We then have x = 6. Examples. 1. Find the logarithms of 25.897, 0.057281, 2537.3, 526.88. 2. Find n when given that log n = . 32380, log n = 8.58720- 10. 3. Verify the equation log 23.6+log 7.0004 = log (23.6X 7.0004). 1. By means of logarithms find to five figures i/227.54. Table II. 6. By log sin d we mean the logarithm of that number which is equal to sin B, and similarly for the logarithms of the other functions of 6. In Table II will be found the logarithms of the sines, cosines, tangents and cotangents of angles from 0° to 90°, computed for each minute. For angles from 0° to and including 44° the number of degrees is printed at the top of the page and the number of minutes in the column under ' (to the left of the column L. Sin.) . The logarithms of the sines are found under the heading L. Sin., those of the tangents under L. Tan., etc. For angles from 45° to 90° the number of degrees is printed at the bottom of the page, the number of minutes is found in the right hand column, and the logarithms of the sines in the column over L. Sin., and similarly for the other func- tions. Thus on page 147, we find log sin 23° 52' = 9.60704, log tan 23° 31' = 9.63865, log sin 66° 18' = 9.96174, etc. Since the sine and cosine of any angle are less than 1, the logarithms of these functions have negative characteristics. Also since the tangent of an angle less than 45° is less than 1, the logarithm of the tangent of such an angle has a negative characteristic. All of the logarithms in the columns under L. Sin., L. Tan., and L. Cos., are too great by 10. Thus, 200 ELEMENTS OF PLANE TRIGONOMETRY. log sin 23° = 9.59188 — 10, etc. In practice it is customary to leave off the — 10 but its existence must be remembered (art. 3). The logarithms in the column under L. Cot. have correct characteristics. 7. To find the logarithm of the sine of an angle. Let us be required to find log sin 42° 24' 35''. The required logarithm will be between log sin 42° 24' and log sin 42° 25'. On page 166 we find log sin 42° 24' = 9.82885, and the difference between this logarithm and log sin 42° 25' is given in the column under d as 14. Then the required logarithm will be 9.82885 + x, where x is a correction found by assuming that the difference between two angles is proportional to the difference between the logarithms of their sines. That is, a: : 14 = 35 : 60, or X = ^^^-^^ = 8.1+. Hence log sin 42° 24' 35" = 9.82885 + (8 = 9.82893. The correction to be added to 9.82885 will be found in the table of proportional parts under the tabular difference (14) and opposite 35". With a little practice this correction may be made mentally and the final logarithm alone written down. 8. To find the logarithm of the tangent of an angle the method employed is exactly similar to that of the last article. Thus log tan 56° 11' 45" = 0.17401 + (21 = 0.17422. 9. To find the logarithms of the cosine and cotangent of an angle. Since the cosine and cotangent of angles less than 90° decrease in value as the angles increase the logarithms of these functions also decrease as the angles increase, and hence the correction for the given number of seconds must be subtracted. Thus to find log cos 57° 32' 25", we have, page 156, log cos 57° 32' = 9.72982 (tabular difference = 20) correction for 25" = 8.3 .-. log cos 57° 32' 25" = 9.72974. Similarly, log cot 33° 25' = 0.18059 (tabular difference = 27) correction for 20" = 9 .-. log cot 33° 25' 20" = 0.18050. ELEMENTS OF PLANE TRIGONOMETRY. 201 10. To find the angle corresponding to the logarithm of one of its functions. Let us be required to find 6 when given log sin = 9.78350. On page 161 we find that the given logarithm is between log sin 37° 24' and log sin 37° 25' (tabular difference = 16). Hence d = 37° 24' + x'\ The difference between the given logarithm (9.78350) and log sin 37° 24' (= 9.78346) is 4. Hence x" : 60" = 4 : 16. Then x" =^-t«^ = 15". And lb therefore 9 = 37° 24' 15". The tables of proportional parts may be used to find the number of seconds as follows: in the column under the tabular difference find the number nearest to the difference between the given logarithm and the logarithm corresponding to the next smaller angle in the tables. Opposite to this will be found the proper number of seconds. Find 6 when given log cos 6 = 9.80541. On page 163 we find that the given logarithm is between log cos 50° 17' = 9.80550 and log cos 50° 18'. The tabular difference is 16 and the dif- ference between the given logarithm and that corresponding to the smaller of these two angles (50° 17') is 9. Hence the number of seconds is found by looking for the number nearest to 9 in the table of proportional parts under 16. This is 9.3 and corre- sponds to 35". Hence 6 = 50° 17' 35". In finding the number of seconds we take the nearest number of seconds given in the tables of proportional parts because we are not sure of obtaining accurate results closer than five seconds with tables giving only five places of decimals. Examples. 1. Find log sin 63° 13' 15", log cos 37° 42' 55", log cot 62° 22' 5", log tan 21° 14' 40", log cot 14° 12' 50". 2. Find d when given log sin d = 9.88645, log cos 6 = 9.32253, log tan e = 0.25436, log cot d = 9.34152, log tan 6 = 9.22335. 3. Given a = 24.762 X tan 36° 42' 45" , find a. 4. Given a = 2.0034 ycos 32° 24' 25", find a. 5. Find the numerical values of cos^ 37° 23' 5" and sin^ 37° 23' 5", and show that their sum is unity. (Owing to the fact that 202 ELEMENTS OF PLANE TRIGONOMETRY. the logarithmic tables give only five places of decimals, there may be an error in the last decimal.) 11. To find the logarithms of the sine and tangent of angles near 0°. The method we have employed for correcting the logarith- mic sines and tangents for seconds does not give reliable resuli% for angles between 0° and 4°. To find such logarithms we make use of two quantities S and T defined by the equations o , sin ^ . . ^ , ^,, >S = log-^ = log sm e - log 61", ^ = log^7^ = logtan0-logC (1) where 0" is the number of seconds in 6. So that log smd = S + log e", and log tan ^ = r + log e'\ (2) On pages 124 to 128 will be found the proper values of S and T, the first three figures being printed at the top of the column and the last three opposite the required angle. The characteristics of S and T in the tables should be decreased by 10. Also in the column headed '' will be found the number of seconds in each angle given in the tables. Let us be required to find log sin 2° 32' 25". On page 126 we find that the number of seconds in 2° 32' = 9120, and hence = 2° 32' 25" = 9120" + 25" = 9145". Hence we have S = 4.68543 - 10 log e" = log 9145 = 3.96118 (page 120) .-. log sin 2° 32' 25" = 8.64661 - 10. Similarly we may find log tan 0° 26' 35" as follows. 26' 35" = 1560" + 35" = 1595". Hence we have T = 4.68558 - 10 and log 1595 = 3.20276 .-. log tan 0° 26' 35" = 7.88834 - 10. 12. To find the logarithm of the cotangent of an angle near 0°. Since cot = -, -, we have log cotO = — log tan 6. We may tan 6 ELEMENTS OF PLANE TRIGONOMETRY. 203 then find log tan d by the method of the last article and, upon changing its sign, obtain the required logarithm. Thus, in article 11 we have found that log tan 0° 26' 35" = 7.88834 - 10. Hence we have log cot 0° 26' 35" = 10 - 7.88834 = 2.11166. 13. To find an angle between 0° and 5° when given the logarithm of its sine, tangent or cotangent. Let us be required to find 6 when given log sin d = 8.34106. On page 125 we find that 6 is between 1° 15' and 1° 16", and that the corresponding value of S is 4.68554 — 10. Then, since *S = log sin 6 — log e, we have log d = log sin 6 — S. But log sin e = 8.34106 - 10 S = 4.68554 - 10 log 0" = 3.65552 and 0" = 4524" = 1° 15' 24". In a similar manner we may find 6 when given log tan 0. Let log tan = 8.44932 - 10. Then d is between 1° 36' and 1° 37' and the corresponding value of T is 4.68569 — 10. Then, log tan d = 8.44932 - 10 T = 4.68569 - 10 therefore log 0" = 3.76363 and 0" = 5803" - = 1° 36' 43" -. As a third example let us find 6 when log cot = 1.63442. Then as in article 12, log tan = — log cot ^ = — 1.63442 = 8.36558 - 10. Hence d is between 1° 19' and 1° 20', and the corresponding value of T is 4,68565 — 10. Then log tan d = 8.36558 - 10 T = 4.68565 - 10 therefore log 0" = 3.67993 and • 0" = 4786 - = 1° 19' 46" -. 13. The ordinary method for finding the correction for seconds also fails when we attempt to find the logarithms of the tan- gent, cotangent and cosine of angles near 90°. Now, tan = cot (90° — 0), and hence we may find log tan by finding log cot (90° - 0). But when is near 90°, 90° - is near 0°, 204 ELEMENTS OF PLANE TRIGONOMETRY. and therefore log cot (90° — 6) may be found by the method of article 12. In like manner we may find log cos 6 by finding log sin (90° - d), and log cot d by log tan (90° - 6>). Thus log cot 89° 44' 45'' = log tan 0° 15' 15" = log tan 915". Then T = 4.68558 - 10 log 915 = 2.96142 log cot 89° 44' 45" = 7.64700 - 10. As a second problem let us find 6 when given log cos 6 = 8.24104 - 10. Then we have log sin (90° - d) = 8.24104 - 10. Hence 90° -6 is between 0° 59' and 1° 0', and the correspond- ing value of S is 4.68555 - 10. Then log sin (90° - d) = 8.24104 - 10 S = 4.68555 - 10 therefore log (90° - 61)" = 3.55549 and (90° - ^)" = 3593" + = 59' 53". Hence = 90° - 59' 53" = 89° 0' 7". 14. To find the logarithms of functions of angles not in the first quadrant. We may find log sin 120° by making use of the relation sin 120° = sin (180° - 120°), and looking up log sin 60°. The case is, however, somewhat different if we need to find log cos 120°, for cos 120° is a negative number and a negative number has no real logarithm. We avoid this dif- ficulty as follows: we have cos 120° = — cos 60°, and we use log cos 60°. Since logarithms are in general only used in problems involving multiplication and division, we may con- sider any product or quotient as made up only of positive quantities, and after all the logarithmic work is completed, give the result the proper sign, positive when an even number of negative quantities are involved and negative when this number is odd. Even though A^ be a negative number it is customary to write log N, this symbol being used instead of the strictly proper log {- N). Thus we write log cos 120° = log cos 60° = 9.69897. Examples. 1. Find log cos 132° 45' 35", log sin 2° 22' 15", log tan 1° 21' 55"; log cot 0° 1' 5", log cos 87° 59' 30", log cot 88° 53' 50". ELEMENTS OF PLANE TRIGONOMETRY. 205 2. Find the numerical value of a = 312.87 X sin 2° 2' 20" X cos 100° 43' 45''. 3. By means of logarithms verify sin 140° 26' 40" = 2 X sin 70° 13' 20" X cos 70° 13' 20". 4. Verify the relation cos 2x = 2 cos (45° — x) sin (45° -- a;), when X = 10° 13' 15" Table III. 15. In this table are found the numerical values of the trigo- nometric functions of angles between 0° and 90°. The method of using this table is exactly similar to that made use of in finding the logarithms of the trigonometric functions in Table II. Tables of proportional parts are, however, not given, and so any necessary corrections must be performed by actual multipli- cation or division. To find the value of sin 36° 24' 25" we proceed as follows: From page 88 we find sin 36° 24' = .59342 and sin 36° 25' = .59365 The difference between these numbers is 23, and hence the correction to be added to sin 36° 24' is Q^ = 9.5 +. The correction is then taken as 10 and we have sin 36° 24' 25" = .59342 + (10 = .59352. Examples. 1. Find the values of tan 65° 51' 50", cos 32° 12' 15", cot 14° 16' 55", cos 154° 24' 25". 2. Find sin 55° 24' 35". Then look up the logarithm of this number in table I, and show that the result so obtained is the same as log sin 55° 24' 35" as found from Table II. nnHE following pages contain advertisements of Mac- millan books on kindred subjects. TRIGONOMETRY By DAVID A. 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The impor- tant engineering curves are thoroughly discussed and the applica. tion of analytic geometry to mathematics and physics is made a special point. Conic Sections* By Charles Smith, Master of Sidney Sus- sex College, Cambridge. Cloth* t2mo* 352 pages » $f*60net* This well-known text, which has been reprinted eighteen times since the second edition was issued in 1833, is still considered a standard. The elementary properties of the straight lines, circle, quadrille, ellipse, and hyperbola are discussed, accompanied by many examples selected and arranged to illustrate principles. CALCULUS AND DIFFERENTIAL EQUATIONS A First Course in the Differential and Integral Calculus^ By William F. Osgood, Professor of Mathematics in Harvard University. Qoth* t2mo. $2M net. Designed as a text for students beginning the study and devoting to it about one year's work. The principal features of the book are the introduction of the integral calculus at an early date ; the introduction of the practical applications of the subject in the first chapters; and the introduction of many practical problems of engineering, physics, and geometry. The problems, over 900 in number, have been chosen with a view to presenting the ap- plications of the subject not only to geometry, but also to the practical problems of physics and engineering. COLLEGE MATHEMATICS— Cbn/mae(/ The Elements of the Differential and Integral Calculus^ By Donald Francis Campbell, Professor of Mathe- matics, Armour Institute of Technology. Cloth, tlmo, 362 pages, $t*90 net Written to meet the need of students desiring a thoroughgoing practical treatise on the subject. Il is primarily designed for use in technical schools, but has given excellent satisfaction in connection with general courses in classical colleges and universities. In order to enable the studeiit to grasp more fully the details of the subject the a*uthor has intro- duced a large number of practical questions which are found in actual experience to produce the desired result better than theoretical proposi- tions. Differential and Integral Calculus for Technical Schools and Colleges* By P. A. Lambert. Qoih, 245 pages, $t^50 net A text for students who have a working knowledge of elementary geom- etry, algebra, trigonometry, and analytical geometry. Its object is three- fold : (1 ) to inspire confidence in the methods of infinitesimal analysis ; (2) to aid in acquiring facility in applying these methods-, and (3) to show the practical value of the calculus by applications to problems in physics and engineering. An Elementary Treatise on the Calculus* By George A. Gibson, Professor of Mathematics in Glasgow and West of Scotland Technical College. Cloth, \2mo, 5t8 pages, $1.90 net A text for college classes which will devote a year's study of approxi- mately three hours per week to the subject. The treatment has been as thorough and as rigid as it is possible to give in an elementary treatise. An Introduction to the Calculus* By George A. Gibson. Cloth, t2mo, 222 pages, $0.90 The elements of calculus have been treated in a way easily understood by immature students. The reason is based essentially on the graphical representation of a function. PUBLISHED BY THE MACMILLAN COMPANY 64-66 FIFTH AVENUE NEV YORK FOURTEEN DAY USE RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 16 Fpn-5fP^ JUNl 1956 LU ' taYall^^Jr,!'^^ Vn^^^r.. YC 22282 THE UNIVERSITY OF CALIFORNIA LIBRARY