"9-NRLF 572 273 WALLS 7 LIBRARY OF THE UNIVERSITY OF CALIFORNIA. l^ceived j^) 12 1893 .189 Accessions No.l\C(C(C\L^ . Class No, H * - RETAINING- WALLS FOR EARTH. THE THEORY OF EARTH-PRESSURE AS DEVELOPED FROM THE ELLIPSE OF STRESS. AN APPENDIX PRESENTING THE THEORY OF PROF. WEYRAUCH. BY MALVERD A. HOWE, O.E., Professor of Civil Engineering, Rose Polytechnic Institute. Seconlr Btiition, 3acbtscti aitlr NEW YORK: JOHN WILEY & SONS, 53 EAST TENTH STREET. 1891, *Wx^ THE 'UHIVERSITTl COPYRIGHT, 1891, BY JOHN WILEY & SONS. FERRIS BROS., ROBERT DRCMMOND, printers, Electrotype,; pearl street| 414 & 446 Pearl Street, N Mew York. CONTENTS. PART I. PAGE NOMENCLATURE, vii FORMULAS FOR THE THRUST OF EARTH, 1 FORMULAS FOR THE BREADTH OF BASE OF A WALL, .... 6 FORMULAS FOR THE DEPTH OF FOUNDATIONS 9 EXAMPLES, 11 PART II. DEMONSTRATION OF THE FORMULAS FOR THE THRUST OF EARTH, . . 27 DEMONSTRATION OF THE FORMULAS FOR THE BREADTH OF THE BASE OF A WALL, 48 DEMONSTRATION OF THE Fo MULAS FOR THE DEPTH OF FOUNDATIONS, 54 APPENDIX. WEYRAUCH'S THEORY OF EARTH-PRESSURE, ...... 59 REFERENCES, 103 DIAGRAM I, 10? TABLES, ....,,.,...,,. 109, PREFACE. THE first edition of this work was based upon the theory advanced by Prof. Weyrauch in 1878, but owing to the length of the demonstrations used by him, it was thought advisable to present different and shorter demonstrations in this edition. To show that the new demonstrations give identical results with those obtained by Prof. Weyrauch, his demonstrations have been given in an appendix as they appeared in the first edition. The new demonstrations are based upon the theory first advanced by Prof. Rankine in 1858. Those readers who are familiar with Rankine's Ellipse of Stress can omit pages 27 to 35, inclusive, in following the demonstrations. An attempt has been made to present the theory in a shape easily followed by those who have only a knowledge of algebra, geometry, and trigonometry; whenever cal- culus has been resorted to, the work has been simplified as much as possible. For convenience in practice, the formu- las have been arranged in a condensed shape in Part I, and are followed by numerous examples illustrating their application. The values of various coefficients have been computed and tabulated and will be found to very materially decrease the labor of substitution in the formulas, v vi PREFACE. It is hoped that the introduction of a brief treatment of the supporting power of earth in the case of foundations, as well as the formula for determining the breadth of the base of a retaining- wall, will prove acceptable. For valuable help in the verification of proofs of formu- las, and the critical reading of the whole text, I acknowl edge the kind assistance of Prof. Thos. Gray. M. A. H. TEBRE HAUTE, IND. , March, 1891. NOMENCLATURE. = the angle of repose, or the maximum angle which any force acting upon any plane within the mass of earth can make with the normal to the plane. e = the angle made by the surface of the earth with the horizontal; e is positive when measured above and negative when measured below the horizontal. a = the angle which the back of the wall makes with the vertical passing through the heel of the wall; a is positive when measured on the left and nega- tive when measured on the right of the vertical. 6 = the angle which the direction of the resultant earth- pressure makes with the horizontal. 0' = the angle of friction between the wall and its foun- dation. 0" = the angle of friction between the back of the wall and the earth. If = the vertical height of the wall in feet. h '= the depth of earth in feet which is equivalent to a given load placed upon the surface of the earth. B' = the width in feet of the top of the wall. B = the width in feet of the base of the wall. Q = the distance in feet from the toe of the wall to the point where R cuts the base. vii Vlll NOMENCLATURE. P the resultant earth-pressure in pounds against a ver- tical wall. E = the resultant earth-pressure in pounds against any wall. R = the resultant pressure in pounds on the base of the wall. G = the total weight in pounds of material in the wall. y = the weight in pounds of a cubic foot of earth. W = the weight in pounds of a cubic foot of wall. p = the intensity of the pressure in pounds on the base of the wall at the toe. p' the intensity of the pressure in pounds on the base of the wall at the heel. p 9 = the average intensity of the pressure in pounds on the base of the wall, x H tan a. OF THE UNIVERSITY RETAINING-WALLS FOR EARTH FORMULAS FOR EARTH-PRESSURE. IN the following formulas a and e are considered as positive, and the wall is assumed-to be one foot long. CASE I. General case of inclined earth-surface and in- clined back of wall. = % cos 2 a cos e t / \/ V . ( cos e i/cos' 2 e cos' 2 ) a in 2 a -{- cos 9 (e a) 1 ( cos e -f- I'cos* e cos 2

\ -f 2 sin esiu acos (e - a) 4 - r - _= - ( cos e + I/cos 2 e cos^ ) ' ^ ' or V(C) tan * : sn tf cos e -f sin e cos (e oi]A , A - or tan d = - - cos (e o-) 2 RETAIN1NG-WALLS FOR EARTH. where cos e l/cos 2 e cos" , 7 . A = cos e = == . . . (a) cos e -f r cos 2 e cos 2

from to 90 and of e from to 90, vary- ing by 5. or for all vertical walls tlie direction of the earth-pressure is parallel to the surface of the earth. CASE III. The surface of the earth parallel to the surface of repose. E _ H^r cos (0 ^) |/sin j a -f- cos a (0 - a) ,^\ 2 cos 2 a cos ^ -j- 2 sin <^ sin cos (0 fl')' tan sin o = -yy cos 0, and (10) becomes 8 RETAINING -WALLS FOR EARTH. Formulas for the width of the base of triangular walls under the condition that the resultant R cuts the base at a point distant from the toe of the wall equal to one third the width of the base, or Q = ^B. CASE I. The general case in which the hack of the wall is inclined) and E makes an angle with the horizontal. B*+B psin tf - * = (tfcos tf + a sin #). (11) CASE II. The hack of the wall vertical. CASE III. The hack of the wall vertical, and the thrust normal to the wall. x = and d = 0. *=* / W- ...... (13) TJie above formulas do not contain the condition that R shall not make an angle greater than ' with the normal to the base of the wall. From Fig. 3, which expresses the condition under which the wall will not slidc t FORMULAS FOR EARTH-PRESSURE. 9 DEPTH OF FOUNDATIONS. CASE I. When the intensity of the pressure on the earth is uniform. Letting x' equal the depth of the foundation below the surface, - JF(L-sin 0) 2 ' when the weight of the foundation is included; and sm y when the weight of the foundation is not included. x' is the minimum depth to which the foundation must be extended for equilibrium. The actual depth should be based upon the minimum value which is likely to have, under any condition of the earth. CASE II. When the intensity of the pressure on the earth is uniformly varying. x , _ P. ~ sin 0) 8 ~ y 1 + sin'~0 ' ' where x' is the minimum depth to which the foundation must be extended for equilibrium; _ 1 sin ^ ~ 3 1 -f sin 8 0' ..... where x is the maximum distance from the centre of the base of the foundation to the point where the resultant pressure cuts the base of the foundation. 10 RETAINING-WALLS FOR EARTH. ABUTTING POWER OF EAKTH. _ (x'Y Y 1 + sin /91 x 2 1 - sin 0' where P represents the maximum resultant pressure which horizontal earth can resist, when P is applied against \\ vertical plane of the depth x'. APPLICATIONS. The determination of the earth-pressure by the pre- ceding formulas and graphical constructions is a very simple operation when the angle has been determined or assumed. That care and judgment be used in assuming the value of is very important, since a change of a few degrees in the value of sometimes causes a large change in the value of E. An inspection of Diagram I shows that the value of the coefficient A increases very rapidly as decreases. When the earth to be retained contains springs, the bank must be thoroughly drained if it is to be retained by an economical tight wall; if it is not drained, the angle will be likely to become very small as the earth becomes wet. When the location of the earth to be retained is sub- jected to jars, the value of will be decreased. Hence, in assuming the value of 0, the engineer must be sure that the value assumed will be the least value which, in his judgment, it is likely to have. In constructing the Avail the judgment and authority of the engineer must again be exercised in order that the wall be constructed as designed. In all cases, to insure perfect drainage between the back FORMULAS FOR EARTH-PRESSURE. 11 of the wall and the earth, numerous " weep-holes" should be provided in the body of the wall, or proper arrange- ments made to carry away the water at the base of the wall. To facilitate drainage, the backing resting against the wall sfionld be sand or gravel. In no case should water be permitted to get under the foundation of the wall, neither should the earth in front of the wall be allowed to become wet. In cold localities the back of the wall near the top should have a large batter to prevent the frost from moving the top courses of stone. As a guard against sliding, the courses of the wall should have very rough beds. The strength of a wall is increased the nearer it approaches a monolith. Care should be taken to have the foundation broad and deep enough to prevent sliding and upheaving of the earth in front. In clay the foundation should be deep, while in sand or gravel it may be broad and shallow. The following examples illustrate the application of the formulas : Ex. 1. Design a trapezoidal wall of sandstone, weighing 150 Ibs. per cubic foot, having a width of 3 ft. on top, a height of 30 ft., and the back inclining forward 5, to re- tain a bank of sand sloping upward at an angle of 20. Data. y - 100 Ibs., W= 150 Ibs.; e = 20, = 39, a = 5; H = 30 ft., B' = 3 ft., x - 2.63 ft. 1. Graphical determination of the values of E and 6\ The graphical solution of the problem is shown in Fig. 4, where E is found to equal 15,000 pounds. 6 lies between 35 and 3G. 12 RETA1NING-WALLS FOR EARTH. 2. Algebraic determination of E and d. =- (B) V(0) * + (E)A. FIG. 4. Substituting the values of B, C, D, and E as given in the tables, and that of A as given by Diagram I, this becomes l / (0.008)+(1.057)(0264) a +(0.061)0.264, ^-45,000 (1.036) 1/0098 = 14,500 Ibs. tan d = sin a cos (e a)A + tan e, . . (I' a) FORMULAS FOR EARTH- PRESSURE. 13 tan 6 = 0.705 = tan 35 11', about. 3. Algebraic determination of the value of B under the assumption that Q = %B. ". . (8) B* -j- 7.797? = 172.53, B- - 3.89 A/172.53 -f 3^9 a ; .-. B = 13.69 3.89 = 9.80 ft; or, practically, 10 feet is the required width of the base. 4. To determine if the watt will slide on a foundation of sandstone. From (14), , E cos d Taking B = 10 ft., (? = -- 30x150 = 29250 Ibs. fl 14 RETAINING-WALL8 FOR EARTH. d = 35 11', cos d = 0.817, and sin 6 = 0.576, then Ecos $ 14500 X 0.817 G -f E sin d ~ 29250 + 14500"x"oT57G ~ From Table II, the value of tan 0' for masonry is 0.6 to 0.7; hence there is no danger of the wall sliding on the foundation. 5. To determine the minimum depth to which the foun- dation must extend consistent with the stability of the earth. First determine the maximum value of x . From (20), 1 sin 3 1 + sin 3 0' where must be assumed at its minimum value. Assume that the minimum value of in this case is 30; then showing that the resultant must cut the base of the foun- dation within 0.133 feet of its centre. The resultant cuts the base of the wall 1.67 feet from the centre- of its base; hence the width of the foundation must be increased. Assuming that the depth to which the foundation ex- tends is 4 feet, and that it is vertical in the rear; then the direction of the resultant pressure (not including the addi- tional weight of the foundation) will cut the base of the foundation 7.93 feet from the ret;r or heel. The required width of the base of the foundation is (7.93 0.13)2 = 15.6; say, 16 feet. The value of ;? can now be found, which corresponds to the assumed value of x 9 4 feet. FORMULAS FOR EARTH-PRESSURE. 15 From (19), , 1 + sin* Po ~ X r (l - sin 0) 2 > p 9 = 400 '-^ = 2960 Ibs. The average intensity of the pressure on the base of the foundation due to the resultant R is 29250 + 14500 sin d 16 The foundation adds an intensity equal to 4 X 150 = 600 pounds approximately; hence the actual value of /? = 2350 4. 600 = 2950 pounds ; therefore, if tlie foundation has a depth of 4 feet and a base of 16 feet, the wall will not sink nor the earth in front of the wall heave, until becomes less than 30. 6. To determine if the wall and foundation will slide on the earth. This is resisted in two ways by the friction between the masonry und the earth, and by a prism of earth in front of the wall. The horizontal force tending to make the wall slide equals E sin 6, or 14500-0.576 = 8352 pounds. The hori- zontal force tending to make the foundation slide equals the resultant earth-pressure on the rear face of the founds* tion, which is vertical and 4 feet in height. From (6), E = j (30+ J>! _ ?C ( * * or E = 12800 X 0.226 = 2893. 16 RETAINING- WALLS FOR EARTH. Then the total horizontal force tending to make the wall slide is 8352 + 2893 = 11245 Ibs. From Table II the tangent of the angle of friction be- tween masonry and moist clay is 0.33, which evidently is much smaller than the tangent of the actual angle of fric- tion between masonry and dry earth. Assume this tangent to be 0.500. The total vertical pressure upon the base of the foun- dation is 37600 pounds, hence the ability to resist sliding is 37600 (0.5) = 18800 pounds, which is much Lirger than 11245; hence there is no danger of the wall slipping, even if the earth in front of the wall does not act. Ex. 2. Design a trapezoidal wall of sandstone weighing 150 Ibs. per cubic foot, having a width of 3 ft. on top, a height of 30 ft., and the back inclining backward 15, to retain a bank of sand sloping upward at an angle of 30. Data. y = 100 Ibs., W = 1501bs.;e = 30, = 33, a =- 15; H = 30 ft., B' = 3 ft., x = 8 ft. 1. Graphical determination of the values of E and d. In Fig. 5, let EG represent the surface of the earth, and AB the back of the wall. Draw AF parallel to BG, and from any point D' in A F lay off D' F equal to the vertical D'G, and draw FL horizontal; lay off the angle 1FD'

the intensity q. The following identities can be written: P = UP + 9) + l(p - q), and 30 RETAINING-WALLS FOR EARTH. or the resultant intensity on the plane CB may be con- sidered as being the resultant of two intensities, one being the intensity of the resultant stress caused by two like prin- cipal stresses having the same intensity |(;j -[- q), and the other the intensity of the resultant stress caused by two unlike principal stresses having the same intensity %(p ({) FIG. 3. The intensity of the resultant stress caused by the first two principal stresses will be, by Case I, \(p + > y, ~ $) s i Q LX y, + %) cos e DX* y. Now DY=DG' = DG-2GX, or DG' - y = DG - y (p q) cos G? = UP + 0) cos e - (p - q) cos GO, \(P + 4) sil1 <*> :: i(/> - y) : sin e, and p -\- q sin 09 = ^- ! * sin e, P-flf or At . A cos GL> = i/ 1 K i-i sin 2 e = ^ / x \P - fiv (^ - fl') and since J(p + g') sin = J(^ g'), cos oo = - - f/cos 2 e cos a 0. gin 38 RETAINING-WALLS FOR EARTH. Substituting this value for cos GO in the equation for DG' . y, it becomes 1 DG' -y = %(p + q) cos e l(p q) . -. 1/cos 2 e cos 2 0, u sin 1 p-\-q or since - * -r - . sin p q DG' . Y l(p + q) ! cos 6 ^cos 3 e cos 2 0J. In a similar manner, e - cos 2 and DG' _ cos e 4/cos 2 e cos 2 DG cos e -j- 4/cos 2 e cos 2 hence T^nr r> ^COS e 4/COS 2 COS 2 JJ(JT = cos e -{- I/cos 2 e cos 2 Let x = the vertical distance between the two planes and AD t then cos e Vcos 2 e cos 2 .-.DG'-y = (x) y cose cos e v cos e cos which is the expression for the intensity of the resultant earth-pressure on a vertical plane at any depth x below the surface. Let cos e Vcos 2 e cos * A = cos e - =. . . (d) cos e -f- T cos 2 e cos 2 * See Rankine's Applied Mechanics ; Alexander's Applied Me- chanics ; Theories of Winkler and Mohr. THEORY OF EARTH-PRESSURE. 39 The average intensity of the resultant earth-pressure on a vertical plane of the length x will be and hence the total pressure will be Since the intensities of the pressures are uniformly varying from the surface, and increasing as x increases, the appli- cation of the resultant thrust will be at a depth of \x be- low the surface. Considering the earth as an unconfined mass, the above formula is perfectly general and can be applied under all conditions, including the case when e is negative. The resultant stress on any plane as AB, Fig. 6, can be found by applying the principles of Case IV. Draw PA parallel to RL, make AN= ZZ)and NO = LG\ then AO represents the direction of the resultant pressure on AB. Make AC = AO', then the area of the triangle ABC mul- tiplied by y is the total pressure on the plane AB, and this pressure is applied at \AE below B. In unconfined earth this construction is perfectly gen- eral and applies to any plane. It also applies equally well to curved profiles. An example illustrating the applica- tion of the method will be given in the applications. See pages 22 and 23. The following graphical construction, Fig. 7, is more con- venient than that of Fig. 6. As before, let BE represent the surface of the earth, and 40 RETAINING-WALLS FOR EARTH. AD a, plane parallel to the surface. At any point D in this plane, draw DE vertical and make DF DE \ draw FG horizontal and make the angle HFD = 0. With L as a centre, describe an arc passing through G and tangent to MF', then with L as a centre and LF as B| FIG. 7. radius, describe the circumference FON, cutting AD at N; through N draw NO parallel to AB, then draw AC nor- mal to AB and equal to OG. The area of the triangle AB C multiplied by y will be the total earth-pressure on AB. To determine the direction of the thrust prolong OG to ft then QN is the direction of the thrust. That this construction is equivalent to that of Fig. 6 is THEORY OF EARTH-PRESSURE. 41 proved as follows. The triangle GLF of Fig. 7 equals the triangle OLD of Fig. 6. . GL'Y \(p q] and LF-y = LO-y = {(p + q). In Fig. 6, the angle NAP = NPA = 90 $(GO e) a. . . ONA = GO e + 2a. In Fig. 7, the angle OLN=2e-2a. But GLN= c+e. and GO of Fig. 7 equals A of Fig. 6. In Fig. 7, the angle QNO = 90 - /?'. In Fig. 6, the angle GAB = 90 - ft'. Therefore the direction of the thrust is the same in both constructions. The two constructions given above are all that is re- quired to determine the thrust of earth upon any plane within the mass of earth, as one can be used as a check upon the other; but as a formula is often very convenient, a general formula will now be deduced which will enable one to determine the values of E and d for any plane with- in a mass of earth. GENERAL FORMULA FOR THE THRUST OF EARTH. In Fig. 8, let BQ represent the surface of the earth and AB any plane upon which the earth-pressure is desired. Draw AD parallel to BQ and let the vertical distance 42 RETAINING-WALLS FOR EARTH. From (e) the earth-pressure upon FA is parallel to the surface and equal to FIG. 8. But AF= x = //(I + tan a tan e) = cos a cos e 2 cos COS 8 (6 - Of) 2 ' ' ' Now the thrust P combined with the weight of the prism ABFmn&i produce the resultant pressure upon AB. THEORY OF EARTH-PRESSURE. 43 Then from Fig. 8, V ~ tan a (1 -f tan a tan e) 6 H*y sin a cos (e a) E = V( V+P sin e) a +(P cos e) a = I/ F 2 +P a + 2 FP sin e. Substituting (/) and (g) in this it becomes jBV cos (e - a) -tv = - - ^ -- X 2 cos* a cos e / ~A A* 4/sin 2 a 4- 2 sin a sin e cos (e a) -- 1- cos 2 (e a) , cos e ' x cos- e which becomes, by replacing A by its value from (d ), 2 cos 2 a cos e -}- sin 2 a cos e i/cos 8 e cos 2 4- 2 sin nr sin e cos (e a) cos e-j- y cos 2 e cos 2 <> /j\ , cos e I/cos 2 e cos 2 (p ) 2 cos 2 (e a) ] ; ( cos e -f- \ cos 2 e cos- ) which is the general equation for the thrust of earth upon any plane within the mass. To determine the direction of the thrust of the earth, let 8 be the angle which the direction of the thrust makes with the horizontal; then, from Fig. 8, y i&n 6 = -=- - -f tan e. P cos e ' 44 RETAINING-WALLS FOR EARTH. Substituting the values of V and P given above, this becomes . sin a cos e -f sm e cos (e a] A tan d = - '- . . (la) cos e cos (e a) A where cos e Vcos 2 e cos a A = cos e =. . . (r/) cos e -j- fcos a e cos 2 Equations (1) and (la) are readily reduced to more sim- ple forms for special cases. These forms will be found in Part I. The Plane of Rupture. Although it is not necessary to know the position of the plane of rupture in order to deter- mine the thrust of the earth, yet it may be of interest to know its position, which can be easily determined as fol- lows : The plane of rupture will be back of the wall and pass through the heel of the wall. The resultant earth-pressure will make the angle with the normal to this plane. Now the tangent of the angle which the direction of the result- ant earth -pressure on any plane makes with the horizontal is determined from the formula sin a tan o = -. r r ~\- tan e. cos (e oi)A If GO represents the angle which the plane of rupture makes with the vertical passing through the heel of the wall, a = GO and d = -J- GO. tan (0 + GO) = -. r-7 + tan e, cos (e - GO) A from which the value of GO can be determined for any case. THEORY OF EARTH-PRESSURE. 46 For the case where e = 0, e being positive with respect to the wall and negative with respect to the plane of rupture, the above equation becomes sin a> tan (0 4- G?) = --- T -- 7 tan 0, cos (0 + a?) cos which is satisfied when GO = 90 0. For the case where e 0, sn tan (0 + GO) = - cos GO tan 2 which is satisfied when G? = 45 . Reliability of the Preceding Theory. The preceding theory is based upon the assumptions that the earth is a homogeneous mass and without cohesion, and the formulas are deduced under the assumption that the surface of the earth is a plane. All writers on the subject have considered the earth as a homogeneous mass and, with a few exceptions, without cohesion. Old and recent experiments indicate that cohesion has very little effect upon the pressure of the earth, which ex- plains why it has not been considered by most writers. The assumption of a plane earth-surface is necessary whenever practical formulas and direct graphical construc- tions for obtaining the thrust of the earth are obtained. General formulas can be deduced for any character of sur- face, but they are too complex for practical use. Tliose graphical constructions which do not require a plane earth- 46 RETAINING- WALLS FO& EARTH. surface are not direct in tlieir solution of the problem, but require a series of trials to obtain the maximum thrust. If the earth-surface is not a plane, one can be assumed which .will give the thrust of the earth sufficiently exact for all practical purposes. For uncon fined earth no exceptions can be taken to the preceding theory, the assumptions upon which it is based being accepted, and for confined earth the theory must be true when the direction of the principal stress passing through the heel of the wall lies entirely within the earth. For all cases in which a and e are positive the theories of Rankine, Winkler, Weyraucli, and Mohr agree and give identical results with the preceding theory, as they should, being founded upon the same assumptions. When a is negative Weyraiich does not consider his theory reliable, and his equations lead to indeterminate re- sults. WinTcler and Mohr consider their theories reliable when- ever the direction of the principal stress passing through the heel of the wall lies entirely within the earth. Rankings method of considering the case where a is negative is equivalent to assuming that the introduction of a wall does not affect the stresses within the mass. It may be concluded that the preceding theory is per- fectly exact when a and e are positive; and when a or e is negative that the stresses obtained will be the maximum which under any circumstances can exist. For the case where e is negative the stress obtained (which represents the maximum thrust the wall can have against the earth and have equilibrium) will be considerably larger than the actual stress (when a wall is introduced), depending upon the magnitude of e. For small values of e the results will be practically correct. For large values of e THEORY OF EARTH-PRESSURE. 47 the following method can be employed in determining the thrust of the earth. The method depends upon the assump- tion that the pressure of the earth is normal to the back of the wall. This may or may not be the case, but it appears to be the most consistent assumption to make for this rare and not important case. Fio. 8a. * In Fig. 80, let AB be the back of the wall and Bfi\\Q surface of the earth. Make Ba = ab = be = cd = etc. Some prism BAa or BAb or BAc, etc., will produce the maximum thrust on the wall; and when this maximum thrust is produced, the resultant pressure on the plane Aa * See Van Nostrand's Magazine, xvn, 1877, p. 5. "New Con- structions in Graphical Statics," by H. T. Eddy, C.E., Ph.D. 48 HETAININO-WALLS FOR EAHT8. or A b or A c, etc., will make the angle with the normal to the plane. On the vertical line Ad' la,yoB.Aa'=a'b' = b'c', etc., and draw Aa" making the angle with the normal to Aa, Ab" making the angle with the normal to Ab, etc.; then draw a' a", b'b", etc., perpendicular to AB, and draw a curve through Aa", b", c" , etc. Then there will be a maximum distance parallel to a'a" between Ad' and this curve which will be proportional to the thrust of the earth against AB. This maximum distance multiplied by the altitude Ac -f- 2 and the product by y, the weight of a cubic foot of earth, will be the pressure of the earth. This method is perfectly general and can be applied in any case. If the earth-pressure is assumed to have the direction given by the formulas of the preceding theory, the con- struction will give the same value of E, the pressure of the earth. Some writers assume that the direction of E makes the angle 0" = with the normal to the back of the wall in all cases. This assumption cannot be correct until the wall commences to tip forward, and then it is doubtful that such is the case unless the earth and wall are perfectly dry. To be on the side of safety in every case, it is better to take the direction of E as given by the above theory. The construction of Fig. 8a will give the maximum thrust for any assumed direction for any case. TRAPEZOIDAL WALLS. It will be assumed that the direction and magnitude of the earth-pressure is known, that the position and extent of the back of the wall and the width of the top are given, T8EORY OF EARTH-PRESSURE. 49 to determine the width of the base for stability against over- turning, sliding, and crushing of the material. FIG. 9. Stability against Overturning. Let AB CD, Fig. 9, rep- resent a section of a trapezoidal wall, TR the direction of the earth-thrust, JG the vertical passing through the cen- tre of gravity of the wall, and JO the direction of the re- sultant pressure on the base AD caused by ^and G. As long as R cuts the base AD, the wall will be stable against overturning. When R takes the direction JQ, the wall may be said to be on the point of overturning; then ON the factor of safety against overturning is ~~., where ON is the actual value of E, and QNthe value of E required to make the resultant R pass through D. Stability against Sliding. Since the wall will not slide 50 RET AININO-W ALLS FOR EARTH. along the surface DA until the resultant R makes an angle with the normal to DA greater than the angle of friction 0', the factor of safety against sliding can be obtained as follows: Draw JP making the angle JMU '= 0'; then PN the factor of safety against sliding is -^, where PN is the force required in the direction of E to make R make the angle 0' with the normal to AD, and ON the actual value of K Stability against the Crushing of the Material. In ordi- nary practice walls for retaining earth are not of sufficient height to cause very large pressures at their bases, but it is necessary to consider the subject on account of the ten- dency of the bed-joints to open under certain conditions. Let AB, Fig. 10, represent any bed- joint in the wall, P the vertical resultant pressure upon the joint, and x the distance of the point of application from the centre of the joint. The intensity of P can be considered as composed of a p uniform intensity p = j, and a uniformly varying inten- sity p 9 ', so ih&t p x = p -\- p. Let a equal the tangent of the angle CDE, then ;;/ ax and p x = p -j- ax. THEORY OF EARTH-PRESSURE. 51 The pressure upon a surface (dx) the joint heing con- sidered unity in the dimension normal to the page is p x dx = p dx -f- axdx, and the moment of this about DB is (p dx -f- axdx)x. The algebraic sum of these moments for values of x be- tween the limits must equal Px , or Px = (p xdx + ax*dx). Integrating, _ B 3 and or - I2xx and if x be replaced by ^B Q, where Q is the distance from A to the point where P cuts the base, (Fig. 11,) and if e=i5, p' = and 52 RETA1NING-WALLS FOR EARTH. from which it is seen that when R cuts the base outside the middle third, the joint will have a tendency to open at points which are at a maximum distance from R where it cuts the base. Therefore in no case should the resultant pressure be permitted to cut the base outside the middle third. This makes it unnecessary to consider the stability against over- turning. C B B Then in designing a wall the following conditions must exist for stability : I. The resultant R must cut the base for stability against overturning. II. The resultant R must not make an angle with the normal to the base of the wall greater than the angle of fric- tion 0'. THEORY OF EARTH-PRESSURE. 53 III. The resultant R must not cut the base outside of the middle third, in order that there may be no tendency for the bed-joints to open. The above three conditions apply to any bed-joint of the wall; but if they are satisfied at the base and the wall has the section shown in Fig. 11, it will not be necessary to consider any joints above the base unless the character of the stone or the bonding is different. Determination of the width of the base of a retaining- w all under the condition that R cuts the base at a point rom the toe of the wall. Let H, B', x, d, and E be given to determine B. From Fig. 11, KF - sin -{- cos 8 -- sin tf, DO O _ -Bx- 2B'x - B" nv-nn B _B* + BB>- Bx- "~ B') For equilibrium E(KF) = G(HF) = E \ B ' HW(HF). Substituting the values of A^and HFin the above and reducing, it becomes , . (8) 54 RETAINING -WALLS FOR EARTH. which is the general equation for the width of the base of a trapezoidal wall. For a rectangular wall B' = B. For a triangular wall B' 0. For a wall with a vertical front B' -\-x-B or B' = B - x. For a wall with a vertical back x = 0. Equation (8) is easily transformed to satisfy the require- ments of special cases. The width of the base can be found graphically by as- suming a value for B and finding the value of Q-, if it is less than %B another value of B must be assumed, and so on until Q is equal to or greater than ^B. Depth of Foundations. Given the angle of repose of any earth, to determine the depth to which it is necessary to sink a foundation to support a given load. The surface of the earth is assumed to be horizontal. CASE I. When the intensity of the pressure on the base of the foundation is uniform. In Fig. 12, let p represent the intensity of the pressure on the base of the foundation. THEORY OF EARTH-PRESSURE. 55 Now when the masonry is about to sink (see Eq. (c)), p. 14- sin 1 sin * - . ' _ ' Q| Q - ftj _ '_ q ' 1 sin 1 + sin 0' If x' represents the depth to which the foundation extends below the surface of the earth and y the weight of a cubic foot of earth, then yx' equals the vertical intensity of the earth-pressure on a plane at the depth of the lowest point of the foundation. When the wall is on the point of sinking, the earth must be on the point of rising, or q _ 1 -|- sin yx' 1 sin 0' or \L + r 1 sin ) In any case p 9 must not have- a greater value than that ob- tained from (15) = p. y -- sin y _ 4>\ 2J The value of x' as obtained from (16) is the least allow- able value consistent with equilibrium. Since x' is a func- tion of tan 4 U5 |-j, care must be taken that is assumed at its least value. As becomes smaller the value of x' increases rapidly. CASE II. When the intensity of the pressure on the base is uniformly varying. Let p represent the maximum intensity of the pressure on the earth and p' the minimum intensity; then for 56 RETAINING -WALLS FOR EARTH. equilibrium p must not exceed the value obtained from the following equation : Also, p r must never be less than x'y; then 7; -I-?/ X r "V ( TJ = ' ' ' \ l-|-( ~ ' ^ I \. %'y *- I "*" V" /I Q\ 2 2 ( U sin0/ [ r (1 sin0) a> which expresses the maximum value which p can have for the equilibrium of the earth. Solving (18) for x', I I o^2~ ^/> > .... (^/ which is the minimum value x' can have for the equilibrium of the earth. In order that p 9 may never be less than x'y the result- ant pressure on the base of the foundation must cut the base within a certain distance of the centre of the base. If x equal this distance, then (see page 51) Substituting the value oip a from (18) and solving for x 9 , 1 sin < 2 > which is the maximum value x a can have, consistent with the stability of the earth. Abutting Power of Earth. Let the surface of the earth be horizontal and the body pushing the earth have a verti- THEORY OF EARTH-PRESSURE. 57 cal face; then at the depth x' the maximum horizontal pressure per unit of area is (see Case I above) , 1 -f- sin and since q varies directly as x', the total thrust P which the earth is capable of resisting is _ (*')> 1 + Bin 4> m . 5} 1 - sin 0' ' W APPENDIX. WEYRAUCH'S THEORY OF THE RETAINING-WALL* Itf the following the earth is supposed without cohesion, and its pressure is determined independently of any arbi- trary assumptions as to direction of the earth-pressure, and with sole reference to the three necessary conditions of equilibrium. The single and only supposition, then, is as follows: That the forces upon any imaginary plane-section through the mass of earth have the same direction. This assumption lies at the foundation of all theories of earth-pressure against retaining-walls. For those cases, therefore, to which the following discussion does not apply no complete or satisfactory theory is yet possible. In what follows, the ordinary assumption as to the direction of the earth-pressure will be proved to be incorrect, except for special cases. * Zeitsclirift fur Baukunde, Band I. Ilcft 2, 1878. 60 THEORY OF THE RETAILING -WALL. GENERAL RELATIONS. Let the surface of the earth have any form, and the wall AB, Fig. 1, have any inclination. The earth-pres- sure makes any angle, d, with the normal to the wall. Suppose through the point A the plane AC. Then the weight G of the prism ABC is held in equilibrium by the reaction of the wall, E, arid by the resultant, R, of all the forces acting upon A C. Now decompose E, G, and R into components parallel and normal to AC; then for every unit in length of the wall, denoting by e, g, and r the lever-arms of E, G, and R respectively with reference to A, the sum of the forces parallel to A C = 0, or P-P,_P =0; (1) GENERAL RELATIONS. 61 the sum of the forces perpendicular to A = 0, or Q + C. - ; 62 THEORY OF THE RETAINING -WALL. and when GO refers to the surface of rupture, the earth - pressure upon AB becomes -p cos GO sin GO tan cp ~ sin (GO -\-a-\- d) -f- cos (GO-\- a -\- 6) tan cp Substituting the value of tan cp or -. this becomes cos cp cos cp cos GO sin GO sin cp sin (GO -j- a -f- 6) cos cp -f- cos (GO -j- a -}- d) sin <> which becomes cos (p + GO) In order to refer to the surface of rupture, the following relation must exist : , ,'G cos GO E sin (GO -\- a -}- fG c \6r si sina J + J &oo B (< B +4 1 tf) i = ^ca Performing the differentiation indicated in the equation to), considering 6r and <# as the variables, it becomes + [dG COSO) - Sin o>rfo> - J?7COS (w + a+ 8)da>l [G Sin w 4 ^COS (+ a+ )] [d(r sin a> + cos a> G E sin (a> 4- a + 8)da)] [G cos a> - -g sin (a> -\- a 4- 5)] _ [G sin a> + COS (+ ^ CO S (a> + a + 5)] r dg sin a> cos w -i7 sin (+ a-f 5)]] [G cos co - #sin ( + a + 5)] ~~ L aw [& sin w + E cos (w + a 4- 6)J 2 ...... (re) GENERAL RELATIONS. 63 or dcosco r ^, ..... .___, , M _[ + cos 2 GL) 1, by clearing of fractions this becomes og(+j) + + 6) _ 2 cos (.ft + to) ^Y^^_+I), (8a) cos (^ + w) " 1 ~" 64 THEORY OF THE RETAINING -WALL. whence which reduces to _ __ cos ( -r to) sin (0 -f -|-+w) sin ( -f w+a+S>fcos 2 (+)] ' Since sin ((p + c 4~ + tf) si n (^ + ^ cos ( a + ^) + cos (9? + ct>) sin (a + d), the parenthetical portion of the denominator becomes -f 2 sin (<*+#) cos ((p+Go) sin (^4-^) cos + cos 2 (^?+() sin 2 (a+6) 2sin(a4-(J) cos (^+c^) sin 2 sin ^(5' cos -^ cos or cos 2 (or+tf) 2 sin 2 (tf+tf) cos 2 or sin' () cos 2 (+^) cos 3 (q>-{-Go) sin 2 ( + COS 2 (+<) [1-sin 2 (a+3)], ?) cos 3 (-f or cos* (+o v ) [sin 2 GENERAL RELATIONS. 65 which equals cos 2 ( ,-,,rw, / x,, I *\ ^o / ^/ I x*\ O * * V / cos FIG. 2. And, since the sum of the horizontal components of E, G, and R must be equal to 0, or Fig. 2, R cos (GO-\- cp\ and ~ Jit 66 THEORY OF THE RETAINING-WALL. which becomes, after substituting the value of E from equation (1.0), . . (11) Let AD, Fig. 2, be the natural slope of the ground. From C let fall the perpendicular CH, and draw CJ, mak- ing the angle (<*-}-#) with CH ; then cos FIG. 2. The expression for AJ is obtained in the following man ner (Fig. 2): CH k cos (q> +(*)), AH & sin (\ -- K I cos and hence, according to equation (9), Also, if ^4^Tis perpendicular to CJ, CH cos ) [cos ( + w ) cos 6 sin ($ +t>> + a + S) sin a] = cos (a e) [cos (a -f 6) cos < - sin ( -f- w + a -j- 5) sin w]. . . (226) By means of the two equations (16#) and (22b) the two unknown quantities d and &) are completely determined. As soon as these are known, ,Z?can be found from equation (19) or (20). Also by the relations in equations (16) and (22), or (\6a) and (22Z>), the surface of rupture and direc- tion of the earth -pressure may be determined, and can therefore be found by a graphical construction. THEORY OF THE RETAINING- WALL. III. HORIZONTAL EARTH-SURFACE. FOR this most important practical case it is simply nec- essary to make = in equation (19). The proper values of d and GO in this case are found from (16) and (221). Making = in equation (22b), it becomes cos GO [cos (cp -{-GO) cos d sin (cp -(- GO ~\- a -f #) sin a~\ - cos a [cos (a -\- d) cos cp sin (cp-}- GO-\- a-\- d)&\\\Go] Q. Since sin (cp-}-Go-{- a-\- d) = sin (

+ GL>) cos GO sin a cos acos d sin (9? -f- GO) -]- cos G? sin 2 a sin # sin (cp -f- 6?) cos GO sin or cos <* sin 6 cos (

cos (or + 6) -}- cos 2 sin 6? cos #sin (cp -{ GO) cos a sin GO sin # sin d sin (<^ -j- GO) + cos 2 sin a? sin d cos (^? -f- GO) -|-cos ^ sin GO sin # cos d cos (cp -j- G?) HORIZONTAL EARTH-SURFACE. 75 fr which reduces to COS ft? COS (<>+ GO) COS S ~] sin a cos a [sin () cos # sin a cos cos $ sin cp sin a cos a sin # cos cp -j- sin (J sin 2 ^ sin (cp-\-oo) cos c^-f-sin tf sin c cos (cp-\-co) sin 2 ** sin GO sin d cos (cp -j- &?) -j- cos d cos 2 ar sin (<>-]- GS?) sin GO cos (Jcosc^cos (^-j-^) + cos 2 a cos (^ cos GO cos (<7? -j- GO) - cosV cos 9? cos d" -f- sin a cos a cos

) | sin ^>, this becomes _ 2sin ) express sin (cp -j- ca -f- a -j- o") in terms of sin and cos (cp -f GL>) and (or +o'), and then the sin and cos of (a -f- 6) in terms of the sin and cos^of a and #. Mak- ing s = in equation (16#), it becomes sin (a -f- G?) cos (a -f tf) cos G? sin (cp + GL> -j- a + 6") [cos (9* + a?) cos a]. (24) Since sin (cp -\- GO -\-a -f- 3) sin (^? -f- c?) cos (a' -|- 6) 4- cos (^? -j- a?) sin (a -|- ") sin (a -+- d) = sin a cos d -f~ cos a' sin # cos (or + d) = cos a cos d sin a sin #; hence sin (cp -\- GO + a + #) = sin (cp -\- GO) cos a cos # sin (^ -j- GO) sin <* sin d 4- cos (^ + GO) sin <* cos d -j- cos (^> 4~ ^ cos 78 THEORY OF THE RETAINING-WALL. and equation (24) reduces to cos oo sin (a -\-GO) cos a cos 6 ^ cos oo sin (a -j- GO) sin a sin d cos 2 a cos (cp -4- GO) sin ( -4- GO] cos # V _ f\ / f\ t -j \ -f cos -f- sin GO cos G> cos 2 a, and likewise cos GO sin <* sin (<* -f- GO) tan # = cos 2 GO sin 2 <* tan d - cos or sin a cos ca sin GO tan #, equation (24c) becomes sin a cos a [cos 2 (

] cos 2 a [sin (cp-\-co) cos (^> + <*>) sin G!?COS c^ [cos 2 a cos 2 (^ -f GO) -f- sin 2 ^ cos 2 &?] tan d -)- sin a cos [sin (cp -j- GJ) cos (<^ -f- GJ) r- sin cy cos GO] tan ^ , ^ = 0. HORIZONTAL EARTH-SURFACE. 79 Now COS 2 (q) -\- GO) COS 2 GO = cos2(4- GO) cos 2 GO * /Vl V ' ' ' which equals 2 sin \ [Zoo sn _ 2 sin ( -f- G^) -j- cos 2 sin 2 Y sin 2(

-\- GO) cos (cp -J- o?) tan sin %~| j = cos 2 -\- GO) sin " ~T~" ~2~ _ 2 sin j (2

and tan d cos 2a cos 2 (^? + GL>) -f- 2 tan # cos 2 a cos 8 /. . . cos 2a \ = (by making cos a = - --- 1- ^ 1 - tan 6 cos 2a [cos 2 ((p + GL>) cos 2 &?] + tan ^ cos 2 pr tan 6 1 cos 2a sin (2&7 4- q>) sin ^ -j- tan # cos 2 G? ; HORIZONTAL EARTH-SURFACE. 81 tan d sin [ Also, - cos 2 (cp -f- GO) tan d -f- tan $ cos 2 GO = tan d [cos'" (cp -}- GO) cos 2 GO] = sin

) tan #. Also, tan d sin 2 sin (cp + G?) cos (cp + GL>) sin 2<* sin GI> cos GJ tan # = tan d sin 2a [sin (9? -f" **) cos (^ ~i~ ^ ~ g i n ^ cos sin 2(^> + GO) sin 2&9~| - a ~J = tan # sin 2(Y sin

+ <^>); and hence equation (240) becomes + sin [sin (Su + 4>) sin 2a - cos (2a> + ft) cos 2a] - sin cos (2a> + - sin [sin (2w+<) cos Sa+cos (2w+ sin (2w+) 2 sin a cos a cos (2to4) 2cos 2 w-sin^[sin(2w-H>)2cos 2 a-sin(2w+/.)+cos(2w+ sin a [sin (2to+ft) cos a+cos (2) 82 THEORY OF THE RETAIN ING-WALL. which reduces to tan d = sin cp sin a sin (%GJ -f- cp -\- oi) sin cp cos (2&) -f- cp) cos 2 oo sin cp cos a sin (2&) -\- cp -f a) Equating this value of tan d with that in equation ('23), sin cp sin a sin (2a) -\- cp -\- a) sin cp cos (2oo -f- 9^) cos" &? sn ^ cos a sn sin <> sin 2a -j- <^> -}- ~ sin (2ct? -j- cp) sin 90 cos 2a ' Dividing by sin cp, clearing of fractions and dividing by sin a, also transposing, this becomes sin (2cj -[- cp -f a) sin (2&J -f- ) sin cp cos 2or] sin a J Since 2 cos 2 f cos 2] 2 cos a cos a c0 /)= 0, or cos a cos cy - -- = The formulae for &?, d, and E can now be found in the simplest manner. Equation (25) is satisfied for2co-\-(p= 90. Hence, Substituting this value in equation (23), it becomes sin cp sin 2a tan # = sin (90 ] sin cp cos 2 (27) 1 sin ^? cos 2' or the equivalent, but more convenient expression for cal- culation, . . . . (28) 84 THEORY OF THE RETAINING-WALL. If, finally, GO = 45 ^ in equation (10), it becomes, re 72 membering that k* = , cos'cy E = 3 008' (45- |) cos ' (45 - 1 hence ^ tan' or, from equation (28), . tan a This last expression, however, when a = takes the in- determinate form . The earth-pressure upon a portion of the wall reaching from the depth A to the depth H = A -{- A, may be found HORIZONTAL EARTH-SURFACE. 85 from equation (29) by substituting H* h * in place of If, as is evident from the following: 7T2 Suppose the wall to have a height H, then E = U*^rY> and likewise for a height h E l = C y . . E = E Q E\ = C Q - y, . . (29&) C representing the constant quantity. From equation (29) E = C(H* 7* 2 ); hence dE = 2 CHdH 2 Ch dk . Now let x equal the distance of the centre of pressure below the top of the wall, then Ex=2C f H H*dH- 2C Phfdh, V Q C or C(H> - h*)x = %CH* - f 6V, 2 tr - n: or x ~* ~fn y^j and if y = the distance from bottom, ^ -,,f^- (3) Equation (30) holds good when the earth-surface is loaded and the loading is equal to a distributed load of the height h . Still, even then, h is often so small that can o be substituted for it just as for unloaded earth-surface.. In all cases # is determined by equation (28). 86 THEORY OF THE RETAINING- WALL. Instead of using equations (28) and (29), the following simple construction can be used : FIG. 4. Draw (Fig. 4) AC and A D vertically and horizontally, each equal to h, also DP making the angle FDG = 45 - - *| with the horizontal. Through the points D and ^describe a circle whose centre lies in AD. Then draw GH parallel to AB, and through A the straight line HJ. Then JG is the direction of the earth-pressure upon the wall AB. If AK'is made perpendicular to AB, and equal to AH, then the AABK gives the intensity and distribution of the earth-pressure, or E=yAABK. The proof of this construction is as follows : Conceive, in Fig. 4, JD and FG drawn, then A G cos a A nr ~ _ _ ~ PH~ HG-[AGsma=PGY in which AP represents the perpendicular let fall from A upon GH. HORIZONTAL EARTH SURFACE. 87 but AG : AF :: AF : AD=h, therefore AG = -^- =h tan Now HG = GD sin a (A G + AD) sin a h sin a -f- li tan 2 (45 - - ~) sin or; tan AHG = h tan 2 (45 - 2\ cos f\ / fj \ 45 -^1 sin a h tan 2 (45 - ^j sin a 2J \ / therefore tan AHG = ^-^ tan 2 ^45 -} = cot a tan 2 (45 *?}. sin a \ 2J \ 2] From Fig. 4, -\- GO -\- a) cos # + cos (^ + GO -f a') sin tf, and making s cp it becomes -f- cos (cp -{- GO) cos (

-\- GO -f- f) sin tf sin a? ; AKGLE e ANGLE cp. 91 dividing by cos d and transposing, cos (ex cp) cos (a -\- 6) cos cp cos d -j- cos (a cp) sin (cp -}- GO -j- a) sin GO { > = + cos (cp -\- GO) cos (

) COS (a -(- 8) COS _ COS (a ) COS (COS a COS 5 sin a sin 5) cos 8 cos 8 = COS (a ) COS COS a -(- COS (a <) sin a COS 0, the above expression reduces to tan 8 = COS a COS(a-<^ COS COS a COS (<^>+a>) COS ( + w + a ) ~ COS(a ) sin a> sill((f-f w+a) sin a cos (a ^)"cos"^"sin a cos(+ a>) cos( +w+a) -(- cos(a ) sin to cos(+w -(-a) and this equation fulfils the condition that the sum of the moments of G, E, and R sliall be zero. If equation (1G&) is treated in a like manner, the result- ing equation will fulfil the condition that the sum of the forces parallel to the surface of rupture shall equal zero. Making f = cp in equation (16#), it reduces to sin (a -j- GO) cos (cp -f- G?) cos (a -\- d) sin (cp + OL -\- GO -j- $) cos (cp -j- GO) cos (a cp) = ? 3 92 THEORY OF THE RETAININQ-WALL. or sin (a -|- GO) cos (a ~\- d) sin (cp -\- GO -\- a) cos ( ) cos or cos ^ sin (a -f- co) sin # sin ^ cos 6" cos d \ / \ cos(-|-Gi>-[-tf)cos(a' <)sin# + a )ooB(a-p) -^^ v -=0; therefore _ cos a sin ( a + ^ sin (9? + &9 + a) cos (a cp) ~ sin (a -f- G;?) sin a -{- cos (f/> + GJ -j-or) cos (: cp) Setting both values of tan 6" equal to each other and clear- ing of fractions^ the following expression is obtained: -j- cos a cos cp sin a sin (&) -f- a) cos (a cp) - cos a sin a sin (co -\- a] cos (&? + -f- GO -f- tf) -)- cos a cos 9? cos (a cp) cos (

) cos a cos (<> -j- 62?) cos 2 (cp -{- GJ -J- a) cos ( cos 2 (a cp) cos (

-f for the second member. ANGLE e = ANGLE (p. 93 The first terms, second terms, and sixth terms cancel. Divide the equation by cos (a cp). Terms number 3 combined give sinw sin(w + a) [sin a sin ( -f a> -f- a) -f COS a COS ( + + a)], which becomes sin GO sin (GO -{- a) cos (cp -f- GO). Terms number 5 combined give COS ( + w) COS ( 4- w -|- a) [COS a COS (< + a> -j-a) -fsin a sin (< + <> + a)], which becomes cos (cp + GO -\- a) cos (cp -\- GO) cos (cp + GO). Terms number 4 combined give -j- cos ^ cos (cr ^>) [cos a cos (p -\- GO -\- a) -\- sin a sin (

) sin GO cos (<> -f- &? -j- a) j Substituting for GO, 90 cp, this becomes f -j- cos nr cos (

cos a] 6? or [sin 2 a -\- cos ( cp) (sin 9? sin a -j- cos

sin (cp 2a)]doj- therefore tan cos (Var (p)' or sin c> cos (

= . tan $. 1 -j- sin cp sin ( COS (^ a) tan * - sin

, and 2 / tan (45 -f sin [45 K+f) If the surface is loaded, substitute IT + 7/ 2 for A 2 , or con- sider li to be the height of the earth increased by the height of an amount of earth weighing as much as the applied load. RECAPITULA T10N OF FORM UL^fl. 101 NOMENCLATURE. Height of wall H Thickness at base b Thickness at top b r Batter in inches per foot of Hon front face. . . d Weight per cubic foot W Total weight of wall G Angle of repose of earth cp Angle made by surface of rupture with vertical GO Weight of cubic foot of earth y Total thrust of earth against wall E Angle made with the horizontal by the surface of the earth s Angle made by rear face of wall with the ver- tical a Angle made with normal by E. d Dist. of point where the resultant pressure cuts the base from the front edge of the wall . . q The resultant pressure due to E and G R NOTE. FOR the translation of Prof. Wey ranch's paper the writer is indebted to the labor of Prof. A. J. Du Bois, of the Sheffield Scientific School, Yale College, who had copies printed by the electric-pen process. However, only the leading equations of Prof. Weyrauch were given ; hence a great deal of labor has been devoted to expanding, verifying, and filling in the intermediate steps of the work, and this nucleus of the mathematical part alone has grown to about double the original quantity. M. A. H. REFERENCES. A brief outline of the theories advanced by the follow- ing writers can be found in " Neue Theorie des Erd- druckes," Dr. E. Winkler, Wien, 1872: D' Antony, Hoffmann, Poncelet, Ande, Holzhey, Prony, Andoy, de Lafont, Rankine, Belidor, Levi, Rebhann, Blaveau, deKoszegh Martony, Rondelet, Bullet, Maschek, Saint-Guilhem, Considere, Mayniel, Saint- Venant, Coulomb, Mohr, Sallonnier, Couplet, Montlong, Scheffler, Culmann, Moseley, Trincaux, Frangais, Navier, Vauban, Gadroy, Ortmann, Winkler, Gauthey, v. Ott, Woltmann. Hagen, Persy, AUDE. Poussee des Terres. Nouvelles experiences sur la poussee des terres. Paris, 1849. BAKER-CURIE. Note sur la brochure de M. B. Baker theorie. Annales des Pouts et Chaussees, pp. 558-592, 1882. The actual lateral pressure of earthwork. Van Nos- trand's Magazine, xxv, 1881; also Van Nostrand's Science Series, No, 5G. 103 104 REFERENCES. BOUSSINESQ. Complement a de precedentes notes sur la poussee des terres. *Annales P. et C., 1884. BOUSIN. Equilibrium of pulverulent bodies. The equilib- rium of earth when confined by a wall. fVan N., 1881. CAIN. Modification of Weyrauch's Theory. Van N., 1880. - Earth-pressure. Modification of Weyrauch/s Theory. Criticism of Baker's articles. Van N., 1882. - Uniform cross-section, and T abutments: their proper proportions and sizes, deduced from Rankine's general formulas. Van N., 1872. - Practical designing of retaining-walls. Van N. Science Series, No. 3, 1888. CHAPERON. Observations sur le memoire de M. de Sazilly (1851). Stabilite et consolidation des talus. Annales P. et C., 1853. CONSIDERS. Note sur la poussee des terres. Annales P. et C., 1870. COUSINERY. Determination graphique de 1'epaisseur des inurs de soutenement. Annales P. et C., 1841. DE LAFONT. Sur la poussee des terres et sur les dimensions a donner, suivant leurs profils, aux murs de soutene- ment et de reservoirs d'eau. Annales P. et C., 1866. DE SAZILLY. Sur les conditions d'equilibre des massifs de terre, et sur les revetements des talus. Annales P. et C., 1851. EDDY. Retaining-walls treated graphically. Van N., 1877. FLAMANT. Note sur la poussee des terres. Annales P. et C., 1882. Resume d'articles publies par la Societe des Inge- nieures Civils de Londres sur la poussee des terres. An- nales P. et C., 1883. * Annales des Fonts et Chaussees. f Van Nostrand's Magazine. REFERENCES. 105 FLAMANT. Note sur la poussee des terres. Annales P. et 0., 1872. - Memoire sur la stabilite de la terre sans cohesion par W. J. Macquorm Eankine (Extrait 1856-57). An- nales P. et C., 1874. GOBIN. Determination precis de la stabilite des murs de soutenement et de la poussee des terres. Annales P. et C., 1883. GOULD. Theory of J. Dubosque. Van N., 1883. - Designing. Van N., 1877. JACOB. Practical designing of retaining-walls. Van N., 1873; also Van N. Science Series, No. 3. JACQUIEB. Note sur la determination graphique de la poussee des terres. Annales P. et 0., 1882. KLEITZ. Determination de la poussee des terres et eta- blissement des murs de soutenement. Annales P. et C., 1844. LAGREUE. Note sur la poussee des terres avec ou sans sur- charges. Annales P. et C.. 1881. L'EvEiLLE. De 1'emploi des contre-forts. Annales P. et C. 1844. LEYGUE. Sur les grands murs de soutenement de la ligne de Mezamet a Bedarieux. Annales P. et C., 1887. - Nouvelle recherche sur la poussee des terres et le profil de revetement le plus economique. Annales P. et C., 1885. MERRIMAN. On the theories of the lateral pressure of sand against retaining walls. (School of Mines Quarterly.) Engineering News, 1888. The theory and calculation of earthwork. Engineer- ing News, 1885. Theorie des Erddruckes und der Futtermauern. Wien, 1870 and 1871, 106 REFERENCES. SAINT-GUILHEM. Sur la poussee des terres avec ou sans surcharge. Annales P. et C., 1858. ScHEFFLER-FouRNiE. Traite de la stabilitc des construc- tions. Paris, 1864. TATE. Surcharged and different forms of retaining- walls. Van N., 1873; also Van N. Science Series, No. V Also published by E. & F. N. Spon. THORNTON. Theory. Van N., 1879. DIAGRAM I. 107 TABLES. Table I contains the crushing-strengths and the average weights of stone likely to be used in the construction of retaining-walls and foundations; also the average weights of different earths. Table II contains the coefficients of friction, limiting angles of friction, and the reciprocals of the coefficients of friction for various substances. Tables III, IV, and V contain the values of the coeffi- cients [see equation (I')] (B), (C), (D) and (E), where x cos (e a) /ri . . 2 ( cos (e a] (B) ~ --, (C) = sin 2 a, (D) \ cos a cos e ( cos e , _,. . . . cos (e a) and (E) = 2 sm a sin e - -. cos e The tables were computed with a Thacher calculating in- strument and checked by means of diagrams. It is believed that they are correct to the second place of decimals; an error in the third place of decimals does not affect the re- sults for practical purposes. Table VI contains the natural sines, cosines and tan- gents. 109 110 TABLES. TABLE I. VALUES OF W. Name of Substance. Crushing Lds. in tons per sq. ft. Average weight in Ibs. per cu. ft. Alab.ister 144 Urick best pressed. 40 to 300 150 " common hard 125 " soft inferior . 100 Chalk 20 to 30 150 ( 'ement loose 49 6 to 102 Flint 162 Feldspar .... .... . 160 Granite 300 to 1200 170 Gneiss 168 Greenstone, trap 187 Hornblende black 203 Limestones and Marbles ordinary 250 to 1000 j 164.4 Mortftr hardened I 108 103 Quartz common 165 Sandstone 1 50 to 550 151 Shales . .... .... 162 Slate 400 to 800 175 Soapstone 170 VALUES OF Name of Substance. Average weight in Ibs. per cu. ft. Earth, common loam, Gravel loose 72 to 80 82 92 90 100 90 106 90 106 104 120 118 129 shaken rammed moderately Sand Sand nerfectlv wet . ill TABLE II. * ANGLES AND COEFFICIENTS OF FRICTION. tan <. * tan Dry masonry and brickwork Masonry and brickwork with damp mortar .... O.Gto 0.7 74 31 to 35 364 1.67 to 1.43 1 35 Timber on stone about 0.4 0.7 toO.3 22 35 tolfif 2.5 1.43 to 3 33 Timber on timber 05 "02 26 10 " lli 2 " 5 Timber on metals 06 "02 31 " 11J 1 67 " 5 Metals on metals 25 " 0.15 14 " 8-T 4 " 6 67 Masonry on dry clay " " moist clay 0.51 33 . 27.- 18J 1.96 3 Earth on earth Earth on earth, dry sand, clay, and mixed earth. . . . Earth on earth, damp clay . Earth on earth, wet clay. . Earth on earth, shingle and 0.25 to 1.0 0.38 "0.75 1.0 0.31 81 14 to 45 21 " 37 45 17 39 to 48 4 to 1 2.63 " 1.33 1 3.23 1 23 to 9 From Rankine's Applied Mechanics. 112 TABLES. TABLE III. e a = 5 a = 6 a = 7 a = 6 a = 9 (B) (B) (B) (#) (B) 1.004 1.0U5 1.007 1.010 1.012 5 1.012 1.015 1.018 1.022 1.026 10 1.019 1.024 1.029 1.035 1.040 15 1.027 1.034 1.041 .048 1.055 20 1.036 1.044 1.052 .062 1.071 25 1.045 1.055 1.065 .076 1.088 30 1.055 1.006 1.079 .092 1.105 35 1.065 1.079 1.094 1.109 1.124 40 1.078 1.094 1.111 1 . 129 1.147 45 1.093 1 111 1.131 1.152 1.173 (C) (O) (C) (C) () 0.008 0.011 0.015 0.019 0.0-^4 TABLE IV. a =5 a = 6 a = ? a = 8 a = 9 (D) (D) (D) (D) (D) 0.992 0.989 0.985 0.981 0.976 5 1.008 1.008 1.006 1.005 1.003 10 1.023 1.026 1.028 1.030 1.031 15 1.040 1.046 1.051 1.056 1.060 20 1.057 1.066 1.075 1.084 1.092 25 1.075 1.089 1.102 1.114 1.125 30 1.096 1.113 1.130 1.147 1.163 35 1.118 1.140 1.164 1.183 1.204 40 1.144 1.172 1.199 1.226 1.253 45 1.174 1.208 1.242 1.276 1.309 TABLE V. 6 a = 5 a = 6 a = 7 a = 8- a = 9 (E) (K) (E) (E) (E) 5 0.015 0.018 0.021 0.024 0.027 10 0.031 0.037 0.043 0.049 0.055 15 0.046 0.055 0.065 0.074 0.083 20 0.061 0.074 0.086 0.099 0.112 25 0.076 0.092 0.108 0.124 0.140 30 0.091 0.110 0.130 0.149 0.169 35 0.106 0.128 0.151 0.174 0.197 40 0.120 0.145 0.172 0.198 0.225 45 0.134 0.162 0.192 0.222 0.253 TABLES. 113 TABLE HI Continued. e a= 10 a =llo a= 12 a= 13 a= 14 () (#) CB) (fi) (#) 1.015 1.019 1.022 1.026 1.031 5 1.031 1.037 1.041 1.047 1.053 10 1.046 1.055 1.061 1.068 1.076 15 1.063 1.073 1.081 : .090 1.100 20 1.081 1.092 1.103 .112 1.120 25 1.099 1.112 1.124 .136 1.150 30 1.119 1.135 1.151 .163 1.179 35 1.141 1.159 1.175 .195 1.211 40 1.166 1.186 1.205 .225 1.245 45 1.195 1.218 1.240 1.263 1.288 (O) CO (C) (C) (C) 0.030 0.036 0.043 0.051 0.029 TABLE IV Continued. a= 10 a= 11 a= l:> a = 13 a= 14 () (C>) CD) (*>) (>) 0.970 0.964 0.957 0.950 0.943 5 .000 0.997 0.993 0.988 0.983 10 .031 1.031 .030 1.028 1.026 15 .064 1.067 .069 1.061 1.072 20 .099 1.105 .110 1.116 1.121 25 .136 1.147 .156 1.165 1.173 30 .178 1.194 .204 1.220 1.232 35 .224 1.244 .262 1.281 1.300 40 .291 1.304 1.328 1.353 .1.377 45 .342 1.375 1.407 1.438 1.469 TABLE V Continued. e a= 10 a= 11 a= !> a = 13 a= 14 (#) (&') (E) CS) (#) 5 0.030 0.032 0.036 0.039 0.042 10 0061 0.067 0.073 0.079 0.085 15 0.093 0.102 0.111 0.119 0.130 20 0.124 0.137 0.150 0.163 0.175 25 0.156 0.173 0.189 0.205 0.221 30 0.188 0.208 0.216 0.248 0.269 35 0.220 0.244 268 0.292 0.316 40 0.252 0.280 0.308 0.336 0.365 45 0.284 0.316 0.349 0.382 0.415 UHI7BRSIT7 114 TABLES. TABLE III Continued. a = 15 a= 16 a = 17 a= 18 a =20 (#) (fi) f/0 (JB) OS) 1.035 .040 1.048 1.051 1.062 5 1.059 .060 1.076 1.081 1.098 10 1.084 .093 l.lt)4 1.112 1.132 15 1.110 .120 1.134 1.138 1.1(58 20 1.135 .149 1.165 1.177 1.218 25 1.165 .179 1.197 1.211 1.245 30 1.195 .212 1.233 1.248 1.288 35 1.229 .249 1.272 1.291 1.339 40 1.268 .21)1 1.317 1.340 1.389 45 1.313 .338 1.369 1.393 1.451 (O) CO) (C') CO) (CO 0.067 0.076 086 0.095 11? TABL E IV Continued. e a=: 15 a= 16 a= 17 a = 18 a = 20 CM) CD) (D) ID) (>) 933 0.924 0.915 0.905 0.883 5 0.977 0.971 0.964 957 0.940 10 1.023 1.018 1.016 1.011 1 . 000 15 1.072 1.073 1.071 1 069 1.068 20 1.124 1.127 1.129 1.181 1.132 25 1.181 1.188 1.194 1.200 1.208 30 1.244 1.256 1.266 1.276 1.293 35 1.316 1.332 1.348 1.363 1.390 40 1.400 1.422 1.444 1.465 1.505 45 1.500 1.530 1.559 1.588 1.643 TABLE V- Continued. e a = 15 a= 16 a= 17 a = 18 a= 20 CE) (0) () (K) (#) 5 0.045 0.047 0.050 0.053 0.058 10 0.091 097 0.102 0.108 0.119 15 0.139 0.148 0.157 0.165 0.183 20 0.188 0.200 0.213 0.225 0.249 25 0.238 0.254 0.270 0.177 0.318 30 0.289 . 309 0.3>9 0.349 0.389 35 341 0.365 0.390 0.414 0.463 40 0.394 0.42, 0.452 0.481 0.539 45 0.448 0.482 0.516 0.551 0.620 TABLE VI. NATURAL SINES, COSINES, TANGENTS AND COTANGENTS. NATURAL SINES AND COSINES. Sine Tooooo .00029 .00058 .00087 .00116 .00145 .00175 One. One. One. One. One. One. One. One. One. One. One. .01745 .01774 .01803 .01832 .01862 .99983 .99983 .99982 .99982 .99981 .99980 .99980 .99935 .99934 .99933 .99932 .99931 .99930 .01920 .01949 .01978 .02007 99846' 99844 99842 99841 00320 00349 00378 00407 00436 00465 00495 00524 00553 00582 00611 00640 02065 02094 02123 02152 02181 02211 02240 02269 02298 02327 99979 99978 99977 99977 99976 99976 99975 99974 99974 .99888 .99830 .99834 .99833 .99831 02414 02443 02472 02501 02530 99824 99822 99821 99819 99817 99815 99813 00727 00756 00785 00814 00844 00873 0-.W.I 02618 02647 02676 02705 02734 02763 02792 02821 02850 06134 06163 06192 06221 06250 06279 06308 06337 06366 06395 01018 01047 01076 01105 01134 01164 01193 01222 01251 01280 01309 01338 01367 01396 01425 01454 99957 99956 99955 99954 99953 99952 90962 06424 06453 06482 06511 06540 06569 .02967 .02996 .03025 .03054 99991 99991 99991 99990 .03112 .03141 .03170 .03199 06627 06656 06685 06714 06743 06773 .01483 .01513 .01542! .01571 .01600 1 .01629 .01658 .01687 ,01716 01745 99876 99875 99873 99872 99870 .99774 .99772 .99770 .99768 .99766 .99764 .99762 .99760 .99758 .99756 03257 03286 03316 03345 03374 03403 03432 03461 .99947 .99946 .99945 .99944 .99943 .99942 89' 85 NATURAL SINES AND COSINES. 117 Sine 08716 08745 08774 08831 .08976 .09005 .09034 .09063 09092 .09121 .09150 .09179 Cosin .99617 .99614 .99612 .99607 .99604 .09237 .09353 .09382 .09411 .09440 .09498 .09527 .09556 .09585 .09614 .09642 .09671 .09700 .09729 .09758 .09787 .09816 .09845 .09874 .09903 .09932 .09961 .09990 .10019 .10048 .10077 .10106 .10135 .10164 .10192 .10221 .10250 .10279 .10308 .10337 10366 .10395 .10424 .10453 Cosin 99580 99578 99575 99572 99570 99567 99564 99562 99559 99556 99553 99551 99548 99545 99542 99540 99537 99534 99531 99528 99526 99523 99520 ,99517 ,99514 ,99511 ,99506 ,99503 .99500 ,99497 ,99494 ,99491 ,99488 ,99485 ,99482 ,99479 .99476 .99473 .99470 .99467 .99464 .99461 .99458 .99455 .99452 Sine Sine 10453 .10511 .10540 .10597 .10626 .10655 .10684 .10713 .10742 .10771 .10800 .10858 .10887 .10916 .10945 .10973 .11002 .11031 .11060 .11089 .11118 .11147 .11176 .11205 .11234 .11263 .11291 .11320 .11349 .11378 .11407 .11436 .11465 .11494 .11523 .11552 .11580 .11609 .11638 .11667 .11696 .11725 .11754 .11783 .11812 .11840 .11869 .11898 .11927 .11956 .11985 .12014 .12043 .12071 .12100 .12129 .12158 .12187 Cosin Cosin 99443 99440 99437 99434 99431 99428 99424 99421 99418 99415 99412 99409 99406 99402 99393 99377 99374 99370 99367 99364 99357 99354 99351 99347 99344, 99341 99337 99334 99331 99327 99324 99317 99314 99310 99307 .99297 .99293 .99283 .99279 .99276 .99272 .99265 .99262 ^99255 Sine 83' Sine .12187 .12216 .12245 .12274 .12302 .12331 .12360 .12389 .12418 .12447 .13476 .12504 .12533 .12562 .12591 .12620 .12649 .12678 .12706 .12735 .12764 .12793 .12822 .12851 .12880 ,12937 ,12966 ,12995 ,13024 ,13053 ,13081 ,13110 ,13139 ,13168 ,13197 ,13226 ,13254 ,13283 ,13312 13341 13370 ,13399 ,13427 ,13450 ,13485 ,13514 ,13543 ,13572 ,13600 ,13629 ,13658 ,13687 .13716 .13744 .13773 .13802 .13831 .13917 Cosin Cosin .99248 .99244 .99240 .99237 .99233 .99226 .99222 .99219 99215 99211 99208 99204 99197 99193 ,99182 ,99178 ,99175 ,99171 ,99167 ,99163 ,99156 ,99152 ,99144 ,99141 ,99137 ,99133 ,99129 .99125 ,99122 .99118 ,99114 .99110 .99106 .99102 99094 ,99091 ,99087 ,99083 ,99079 ,99075 ,99071 ,99067 .99059 ,99055 ,99051 ,99047 ,99043 ,99039 .99031 .99027 Sine 82- ^ 13917 13946 13975 14004 14033 14061 14090 14119 14148 14177 14205 .14234 .14263 .14292 .14320 .14349 .14378 .14407 .14436 .14464 .14493 .14522 .14551 .14580 .14608 .1463' .14666 .14695 .14723 .1475 .14781 .14810 .14838 .14867 .14925 .14954 .14982 .15011 .15040 .15069 .15097 .15126 .15155 .15184 .15212 .15241 .15270 .1529 .15327 .15356 .15385 .15414 .15442 .15471 .15500 .15529 .15557 .15586 .15615 .15643 Cosin Cosin .99019 .99015 .99011 .99006 .99002 .98998 .98990 ,98978 ,98973 ,98969 ,98965 ,98961 ,98957 .98944 98927 .98914 ,98910 ,98876 ,98871 ,98867 ,98863 ,98854 ,98849 ,98845 .98832 ,98818 .98814 ,98796 ,98791 ,98787 ,98782 98778 98773 Sine .15643 .15672 .15701 .15730 .15758 .15787 .15816 .15845 .15873 .15902 .15931 .15959 .15988 .16017 .16046 .16074 .16103 .16132 .16160 .16189 .16218 .16246 .16275 .16304 .16333 .16361 .16390 .16419 .16447 .16476 .16505 .16533 .16562 .16591 .16620 .16648 .16677 .16706 .16734 .16763 .16792 .16820 .16849 .16878 .16906 .16935 .16964 .16992 .17021 .17050 .17078 .17107 .17136 .17164 .17193 .17222 .17250 .17279 .17308 .17336 .17365 Cosin !osin .98764 .98760 .98755 .98751 .98746 .98741 .98737 .98732 .98728 .98723 .98718 .98714 .98709 .98704 .98700 .98695 .98676 .98671 .98667 .98662 .98657 .98652 98629 98624 98619 98614 .98585 .98575 .98570 .98565 .98561 .98556 .98551 .98546 .98541 .98536 .98531 .98526 .98521 .98516 .98511 .98506 .98501 .98496 .98491 .98486 .98481 Sine 80' 118 NATtRAL SINES AND COSINES. 34 37 10 ^ .17365 .17393 .17422 .17451 .17479 .17508 .17587 .17565 .17594 .17623 .17651 17708 17737 17766 17794 17823 17852 17880 17909 17937 17966 17995 18023 18052 18081 18109 18138 18166 18195 18224 18367 18395 18452 18481 18509 18538 18567 18595 18624 18652 18681 18710 18738 18767 18795 18910 18967 19024 19052 19081 Cosin Cosin .98481 .98476 .98471 .98466 .98461 .98455 .98450 .98445 .98435 .98430 98425 98420 98414 98404 98399 98394 98378 98373 98368 98362 98352 98347 98341 98336 98320 98315 98304 98277 98272 98256 98250 98245 98240 98234 98229 98223 98218 98190 98185 98179 .98174 .98168 .98163 ^790 11 Sine .19081 .1910S .19188 -19167 19195 19224 19252 19: 19; 19423 19452 19481 19509 19566 19595 19652 19709 19737 19766 19794 19823 19851 19880 19908 19937 19965 19994 20022 20051 20079 20108 20136 20165 20279 20307 20421 .20450 '.20478 .20507 .20563 .20592 .20620 : 20849 '.20677 ".20706 f. 20734 >20763 '.20791 Cosin .98157 .98153 .98146 .98140 .98135 .98129 .98124 .98118 .98112 .98107 98101 ,98080 98084 98079 98073 98050 98044 93033 98027 98021 98016 98010 98004 97998 97992 97987 97981 97975 97969 97963 97958 97952 97940 97934 97922 97916 97910 97905 97887 97881 97875 97863 97857 97851 '.97845 : 97839 '.97833 '.97827 .97821 ,97815 k>sin Sinej * 178-r Sine .20791 .20848 .20877 .20905 .20933 .20962 .20990 .2101 .21047 .21076 .21104 .21132 .21161 .21189 .21218 .21246 .21275 .21303 .21331 .21388 .21417 .21445 .21474 .21502 .21530 .21559 .21587 .21616 .21644 .21672 .21701 21729 .21753 21786 .21814 .21843 .21871 .21899 121956 '.21985 '.22013 -.22041 .22070 .22098 .22126 .22155 ; 22183 '.22212 '.22297 f. 22325 * 22353 *22410 '.22438 22467 f. 22495 Cosin .97815 .97809 .97803 .97797 .97791 .97784 .97778 .97772 .97766 .97760 .97754 .97748 .97742 .97735 .97729 .97723 .97717 .97711 .97705 97673 97667 97661 97655 97648 97642 97G36 97630 97623 .97017 97611 97604 9^598 97592 .97585 97579 97573 .97566 .'97560 .97553 .97547 .97541 .97534 .97528 97521 .97515 .97508 97502 ?97496 '.97489 .97483 '.97476 .97470 '.97463 197457 .97450 ; 97444 197437 Cosin ! ' '77 13 C Sine .22495 .22523 .22552 .22665 .22693 .22722 .22750 .82778 .22807 ,22977 ,23005 .23118 .23146 .23175 .23260 .23283 .23316 .23345 .23373 .23401 .23423 .23458 .23486 .23514 .23542 .23571 .23599 .23627 $23656 .23712 .23740 .23769 '.23797 '.23853 .24023 .24051 .24079 .24108 .24164 ".24192 Cosin Cosin .97437 .97430 .97424 .97417 .97411 .97404 ,.07398 .97391 .97384 .97378 .97371 .97365 .97358 .97351 .97345 .97338 .97331 .97325 .97318 .97311 .97304 .97298 .97291 .97284 .97278 .97271 .97264 .97257 .97251 .97244 .97237 97230 97223 97217 97210 97203 97196 ,97189 97182 97176 97169 ,97162 ,97155 ,97148 .97141 ,97134 .97127 .97120 .97113 .97106 .97100 .97093 ,97086 ,97079 ,97072 ,97065 .97058 .97051 ,97044 97037 Sine 176- 14- Sine .24192 .24220 .24249 .24277 .84305 .24333 .84362 .84390 .84418 .84446 .24474 .84503 .24531 .24559 .84587 .24615 .84644 .84672 .24700 .24728 .84756 .24784 .24813 .24841 .24869 .24897 .24925 .24954 .24982 .25010 .25038 .85066 .25094 .25122 .25151 .25179 .25207 .25235 .25263 .25291 ! .25320J .25348! .25376 .25404 .25432, .25460 .25488 .25516 .25545 .25573 .25601 1 .25629 .25657 .256851 .25713 .25741 .25769 .25798 .25826 .25854 Cosin .97030 .97023 .97015 .97008 .97001 96945 96894 .96873 .96851 .96844 .96837 .96815 .96807 .96778 96771 96764 .96756 96749 .96742 96734 .96727 .96719 .96712 .96705 .96675 .96667 .96660 .96653 .96645 .96638 96630 .96623 96615 96608 96600 Cosin | Sine NATURAL SINES AND COSINES. 119 21 15 Sine 25883 25910 25966 Coein .96578 .96570 .96562 .96555 ,260501.96547 .96540 16- Sine 26107 26135 .96524 .96517 .26191 1.96509 .26219 96502 96494 .26359 .262751.96486 96479i 96471 96463 .96456 .26415 .96448 .26443 .96440 .26471 .26500 .96425 .96417 ,96410 .26556 .26584 .26612 .26724 .26752 .26780 .87004 .27032 .27060 .27068 .27116 .27144 .27172 .27200 .27256 .27284 .27812 .27340 .27396 .27424 .27452 .27480 .27508 .27586 .27564 %379 96371 96347 96340 .96301 .96277 .96269 .96261 ,96214 Cosin I .96198 .96190 .96182 .96174 .96166 .96158 .96150 .96142 .96184 .98188 Sine 74 .27564 .27592 ,27620 ,27648 ,27676 ,27704 ,27731 ,27759 ,27787 ,27815 ,27843 ,27871 ,27899 ,27927 ,27955 ,27983 ,28011 .28039 ,28067 Cosin .96126 .96118 .96110 .96102: .96094 .96078 96070 .96062 .96054 .28123 .28150 .28178 28234 ,28200 ,28318 28346 ,28374 ,28429 ,28457 ,28485 ,28513 .28541 ,28569 ,28597 ,28625 ,28652 ,28736 ,28764 17 ,28875 ,29015 ,29042 ,29070 ,29098 .29126 ,29154 .29182 9G021 96013 96005 95997 95981 95972 95956 95948 95940 ,95931 ,95923 ,95915 ,95907 ,95898 ,95890 .95874 ,95863 ,95857 ,95849 ,95841 .95832 .95824 .95816 .95807 .95799 .95791 ,95782 .95774 .95766 ,95757 .95749 .95740 .95732 .95724 ,95715 ,95707 ,95681 ,95673 Cosin Sine .95656 .95647 .82887 .82914 .32942 .32969 .82997 .33024 .33031 .83079 .83106 .83134 .33161 .83189 .83216 .83244 .83271 .33298 .83326 .83353 .83381 .94979 .94970 .94961 .94952 .94943 95450 95441 95433 93424 95415 95407 .94888 .94878 .94869 .94860 .94851 .94842 .94832 .94303 .94293 .94284 .94274 .94264 .94254 .94245 .83408 .83436 .3346? .83490 .83518 .83545 .3357? .83600 .33627 .33655 .94814 .94805 .94793 .94786 .94777 .94768 .94758 .94740 .94740 .95319 .95310 .95301 .952C3 .94730 .94721 .94712 .94702 .94693 .94684 .94674 .94665 .94656 .9404o .95275 .93260 .95257 .95248 .95240 .95231 .95222 .95213 .95204 .95195 .95186 .95177 .95168 .95159 .95150 .95142 .951&3 .95124 .95115 .95106 Sine" .84120 .84147 .84175 34202 120 NATURAL SINES AND COSINES. 20* 34202 34229 34257 34284 34311 34339 34366 34393 34421 34448 84475 .93337 .359181.93327 .93316 .35973 .36000 .36027 .37757 .37784 .37811 .36135 .36162 .36190 .36217 .36244 .36271 .36298 .36325 .36352 .36379 .34530 .34557 .34584 .34612 .34639 .93232 .93222 .93211 .93201 .93190 .93180 .93169 .93159 .93148 .37865 .37892 .37919 .37946 .37973 .37999 .39474 .39501 .39528 .39555 .39581 93799 93789 93779 93769 93759 93748 .34830 .34857 .34884 .34912 .34939 .34966 .34993 93728 93718 93708 93698 .39715 .39741 .39768 .39795 .39822 .36515 .36542 .36569 .36590 .36623 .86650 93677 93667 93657 93647 93637 36677 36704 36731 36758 36785 .93031 .93020 .93010 .92999 350T5 35102 35130 35157 35184 35211 .38349 .38376 .384C3 .38430 .38450 .40008 .40035 .40062 .40088 .40115 .40141 .92978 .92967 .92956 .92945 .92935 93606 93596 93585 93575 93565 .38564 .38591 .38617 .38644 .38071 .38G03 93555 93544 93534 93524 93514 93503 93493 93483 93472 93402 .40195 .40221 .40248 .40275 .40301 .40320 .40355 .40381 .40408 35347 35375 35402 35429 35456 35484 35511 35538 35565 36975 37002, 87029! 37056 37083 37110 37137 37164 37191 92881 92870 92859 92849 92838 92837 .92816 .92805 .92794 .92784 .92773 .92762 .92751 .92740 .92729 .92718 35619 35647 35674 35701 35728 35755 35782 35810 .93441 .93431 .93420 .93410 .93400 .38912 .38939 .38966 .38993 .39020 .39046 .39073 Cosin 02060 .92016 .92005 .91994 .91982 .91971 .91959 .91948 91925 91914 91902 91891 91879 91868, 918561 91845 | 91833 91822 1 91810 ! 91799 917871 91775 91764 91752 91741 91729 91718 91706 .91683 .91671 .91660 .91648 .91636 .91625 .91613 .91601 .91590 .91578 .91566 .91555 .91543 .91531 .91519 .91508 .91496 .91484 .91472 .40434 '.91461 .91449 .91437 .91425 .91414 .91402 .91378 .40647 .91366 . 40674 1. 91355 Cosin | Sine _ .40674 .40700 .40727 .40753 .40780 .40806 Cosin .91355 60 .91343 ! 59 .91331 ! 58 .91319 .91307 .91295 .91283 .40860 .91272 .40886 .91260 .409131.91248 .40939 .91236 91224 .40992 .41019 .41045 .41072 .41098 .41125 .41151 .41178 .41204 .41231 .41257 .41284 .41310 .41337 .41363 .41390 .41416 .41443 .41469 .41496 .41522 .41549 .41575 .41602 .41628 .41655 .41681 .41707 .41734 .41760 .41787 .41813 .41840 .41866 .41892 .41919 .41945 .41972 .41998 .42024 .42051 .42077 .42104 .42130 .42156 .42183 .42209 .42235 .42262 .91212 .91200 .91188 .91176 .91164 .91152 .91140 .91128 .91116 .91104 .91092 .91080 .91068 .91056 .91044 .91032 .91020 .91008 ,90972 ,90960 ,90911 ,90875 .90851 .90814 .90802- .90790 .90778 .90766 .90753 .90741 .90729 .90717 .90704 .90692 .90680 .90668 .90655 .90643 .90631 69 Cosin Sine 65 NATURAL SINES AND COSINES. 25 26" 27 28 29 9 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin $ "o .42262 .90631 .43837 .89879 .45399 .89101 .46947 .88295 .48481 .87462 60 i .42288 .90618 .43863 .89867 .45425 .89087 .46973 .88281 .48506 .87448 59 2 .42315 .90606 .43889 .89854 .45451 .89074 .46999 .88267 .48532 .87434 58 3 .42341 .90594 .43916 .89841 .45477 .89061 .47024 .88254 .48557 .87420 57 4 .42367 .90582 .43942 .89828 .45503 .89048 .47050 .88240 .48583 .87406 56 5 .42394 .90569 .43968 .89816 .45529 .89035 .47076 .88226 .48608 .87391 55 6 .42420 .90557 .43994 .89803 .45554 .89021 .47101 .88213 .48634 .87377 54 7 .42446 .90545 .44020 .89790 .45580 .89008 .47127 .88199 .48659 .87363 53 8 .42473 .90532 .44046 .89777 .45606 .88995 .47153 .88185 .48684 .87349 52 9 .42499 .90520 .44072 .89764 .45632 .88981 '.47178 .88172 .48710 .87335 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .88158 .48735 .87321 50 11 .42552 .90495 .'44124 .89739 ;'45684 .88955 T47229 T88144 .48761 .87306 49 12 .42578 .90483 .44151 .89726 .45710 .88942 '.47255 .88130 .48786 .87292 48 13 .42604 .90470 .44177 .89713 .45736 .88928 -.47281 .88117 .48811 .87278 47 14 .42631 .90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 15 .42657 .90446 .44229 :89687 .45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48888 .87235 44 17 .42709 .90421 .44281 .89662 .45839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .90403 .44307 .89649 .45865 .88862 .47409 .88048 .48938 .87207 42 19 .42762 .90396 .44333 .89636 .45891 .88848 .47434 .88034 .48964 .87193 41 20 .42788 .90383 .44359 .89623 .45917 - -.* .88835 .47460 .88020 .48989 .87178 40 21 .42815 .90371 ;44385 .'89610 \45942 .88822 ?47486 .88006 .49014 .87164 39 22 .42841 .90358 .44411 .89597 .45968 .88808 *.47511 .87993 .49040 .87150 38 23 .42867 .90346 .44437 .89584 .45994 .83795 .47537 .87979 .49065 .87136 37 24 .42894 .90334 .44464 .89571 .46020 .88782 '.47562 .87965 .49090 .87121 36 25 .42920 .90321 .44490 .89558 .40046 .88708 147588 .87951 .49116 .87107 35 26 .42946 .90309 .44516 .89545 .46072 .88755 .47614 .87937 .49141 .87093 34 27 .42972 .90296 .44542 .89532 .4G097 .88741 .47639 .87923 .49166 .87079 33 23 .42999 .90284 .44568 .89519 .40123 .88728 .47665 .87909 .49192 .87064 32 29 .43025 .90271 .44594 .89506 .46149 .88715 .47690 .87890 .49217 .87050 31 30 .43051 .90259 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .87036 30 31 .43077 .90246 f44646 .89480 146201 .'88688 f47741 .87868 .49268 .87021 29 32 .43104 .90233 .44672 .89467 .40226 .88674 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .4-1698 .89454 .40252 .88061 .47793 .87840 .49318 .86993 27 34 .43156 .90203 .44724 .89441 .46278 .88647 .47818 .87826 .49344 .86978 26 35 .43182 .90196 .44750 .89428 .40304 .88634 .47844 .87812 .49369 .86964 25 38 .43209 .90183 .44776 .89415 .40330 .88620 .47869 .87703 .49394 .86949 24 37 .43235 .90171 .44802 .89402 .46355 .88007 .47895 .87784 .49419 .86935 23 38 .43261 .90158 .44828 .89389 .40381 .88593 .47020 .87770 .49445 .86921 22 39 .43287 .90146 .44854 .89376 .40407 .88580 .47940 .87756 .49470 .86906 21 40 .43313 .90133 .44880 .89363 .46433 .88566 .47971 .87743 .49495 .86892 20 41 .43340 .90120 .44906 .89350 .46458 .88553 .47997' .87729 .49521 .86878 19 42 .43366 .90108 .44932 .89337 .40484 .8G539 .48022 .87715 .49546 .86863 18 43 .43392 .90095 .44958 .89324 .40510 .88526 .48048 .87701 .49571 .86849 17 44 .43418 .90082 .44984 .89311 .40536 .88512 .48073 .87687 .49596 .86834 16 45 .43445 .90070 .45010 .89298 .40501 .88409 .48099 .87673 .49622 .86820 15 46 .43471 .90057 .45036 .89285 .46587 .88485 .48124 .87659 .49647 .86805 14 47 .43497 .90045 .45062 .89272 .46613 .88472 .48150 .87645 .49672 .80791 13 43 .43523 .90032 .45088 .89259 .46639 .88458 -.48175 .87631 .49697 .86777 12 49 .43549 .90019 .45114 .89245 .46064 .88445 .48201 .87617 .49723 .86762 11 50 .43575 .90007 .45140 .89232 .46690 .88431 .48226 .87603 .49748 .86748 10 51 .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 .49773 .86733 9 52 .43628 .89981 .45192 .89206 .40742 .88404 .48277 .87575 .49798 .86719 8 53 .43654 .89968 .45218 .89193 .46767 .88390 .48303 .87561 .49824 .86704 7 54 .43680 .89956 .45243 .89180 .46793 .88377 .48328 .87546 .49849 .86090 6 55 .43706 .89943 .45269 .89167 .46819 .88363 .48354 .87532 .49874 .86675 5 56 .43733 .89930 .45295 .89153 .46844 .88349 .48379 .87518 .49899 .86661 4 57 .43759 .89918 .45321 .89140 .46870 .88336 .48405 .87504 .49924 .86646 3 58 .43785 .89905 .45347 .89127 .46896 .88322 .48430 .87490 .49950 .86632 2 59 .43811 .89892 .45373 .89114 .46921 .88308 .48456 .87476 .49975 .86617 1 60 .43837 .89879 .45399 .89101 .46947 .88295 .48481 .87462 .50000 86603 _0 I Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine i 64 63 62 1 61 1 60 NATURAL SINES AND COSINES. 30* 31 32 33 34* / Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin / "o .50000 .86603 .51504 85717 .52992 84805 .54464 .83867 .55919 .82904 60 1 .50025 .86588 .51529 85702 .53017 84789 .54488 .83851 .55943 .82887! 59 2 .50050 .86573 .51554 85687 .53041 84774 .54513 .83835 .55968 .82871 58 3 .50076 .86559 .51579 85672 .53066 84759 .54537 .83819 .55992 .82855 57 4 .50101 .86544 .51604 85657 .53091 .84743 .54561 .8C804 .56016 .82839 56 5 .50126 .86530 .51628 85642 .53115 .84728 .54586 .83788 .56040 .828221 55 6 .50151 .86515 .51653 85627 .53140 .84712 .54610 .83772 .56064 .82806 ! 54 7 .50176 .86501 .51678 85612 .53164 .84697 ; .54635 .83756 .56088 .82790 53 8 .50201 .86486 .51703 85597 .53189 .84681 .54659 .83740 .56112 .82773 52 9 .50227 .86471 .51728 85582 .53214 .84666 .54683 .83724 .56136 .82757 51 10 .50252 .86457 .51753 .85567 .53238 .84650, .54708 .83708 .56160 .82741 50 11 .50277 .86442 .51778 .85551 .53263 .84635 .54732 .83692 .56184 .82724 49 12 .50302 .86427 .51803 .85536 .53283 .84619 .54750 .83676 .56208 .82708 48 13 .50327 .86413 .51828 85521 .53312 .84604 1 .54781 .83660 .50232 .82692 47 14 .50352 .86398 .51852 .85506 .53337 .84588! .54805 .83045 .56256 .82675 1 46 15 .50377 .86384 .51877 .85491 .53361 .84573 .54829 .83029 .56280 .82659 1 45 16 .50403 .86369 .51902 .85476 .53386 .84557 .54854 .83013 .56305 .82643 44 17 .50428 .86354 .51927 .85461 .53411 .84542 .54878 .83597 .56329 .82626 43 18 .50453 .86340 .51952 .85446 .53435 .84526 .54902 .83581 .50353 .82610 42 19 .50478 .86325 .51977 .85431 .534GO .84511 .64927 .83505 .50377 .82593 41 20 .60503 .86310 .52002 .85416 .53484 .84495 .54951 .83549 .56401 .82577 40 21 .50528 86295 .52026 .85401 .53509 .84480 .54975 .83533 .56425 .82561 89 22 .50553 .86281 .52051 .85383 .53534 .84464 .54999 .83517 .56449 .82544! 38 23 .50578 .86266 .52076 .85370 .53558 .84448 .5502-1 .83501 .56473 .82528 37 24 .50603 .86251 .52101 .85355 .53583 .84433 .55043 .83485 .56497 .82511 36 25 .50628 .86237 .52126 .85340 .53007 .84417 .55072 .83409 .56521 .82495 35 26 .50654 .86222 .52151 .85325 .53032 .84402 .55097 .83453 .56545 .82478 34 27 .50679 .86207 .52175 .85310 .53056 .84386 .55121 .83437 .56509 .82462 33 28 .50704 .86192 .52200 .85294 .53681 .84370 .5514-5 .83421 .56593 .82446 32 29 .50729 .86178 .52225 .85279 .53705 .84355 .55109 .83405 .50017 .82429 31 80 .50754 .86163 .52250 .85264 .53730 .84339 .55194 .83389 .56641 .82413 36 31 .50779 .86148 .52275 .85249 .53754 .84324 .55218 .83373 .56665 .82396 29 82 .50804 .86133 .52293 .85234 .53779 .84308 .55242 .83356 .56689 .82380 28 33 .50829 .86119 .52324 .85218 .53804 .84292 .55206 .83340 .56713 .82363 27 34 .50854 .86104 .52349 .85203 .53828 .84277 .55291 .83324 .56730 .82347 26 35 .50879 .86089 .52374 .85188 .53853 .84261 .55315 .83308 .56760 .82330 25 36 .50904 .86074 .52399 .85173 .53877 .84245 .55339 .83292 .56784 .82314 24 37 .50929 .86059 .52423 .85157 .53902 .84230 .55303 .83276 .56808 .82297 23 38 .50954 .86045 .52448 .85142 .63926 .84214 .55388 .83200 .56832 .82281 22 39 .50979 .86030 .52473 .85127 .03951 .84198 .65412 .83244 .56856 .82264 21 40 .51004 .86015 .52498 .85113 .53975 .84182 .65436 .83228 .56880 .82248 20 41 .51029 .86000 .53522 .85096 .54000 .84167 .55460 .83212 .56904 .82231 19 42 .51054 .85985 .52547 .85081 .54024 .84151 .55484 .83195 .56928 .82214 i 18 43 .51079 .85970 .52572 .85066 .54049 .84135 .55509 .83179 .56952 .82198 17 44 .51104 .85956 .52597 .85051 .54073 .84120 .55533 .83103 .66976 .82181 16 45 .51129 .85941 .52621 .85035 .54097 .84104 .65557 .83147 .57000 .82165] 15 46 .51154 .85926 .52646 .85020 .54122 .84088 .55581 .83131 .57024 . 82148 j 14 47 .51179 .85911 .52671 .85005 .54146 .84072 .55605 .83115! .67047 .821321 13 48 .51204 .85896 .52698 .84989 .54171 .84057 .55630 .83098 : ! .57071 .82115 12 49 .51229 .85881 .52720 .84974 .64195 .84041 .55654 .83082J .67095 .82098 11 50 .51254 .85866 .52745 .84959 .54220 .84025 .55678 .83Q66 .57119 .82082 10 51 .51279 .85851 .52770 .84943 .54244 .84009 .55702 .83050 .57143 .82065 9 52 .51304 .85836 .52794 .84928 .54209 .83994 .55726 .83034 .67107 .82048 8 53 .51329 .85821 .52819 .84913 .54293 .83978 .65750 .83017 .57191 .82032 7 54 .51854 .85806 .52844 .84897 .54317 .83902 .55775 .83001 .57215 .82015 6 55 .51379 .85792 .52869 .84882 .54342 .83946 .55799 .82985 .57238 .81999 5 56 .51404 .85777 .52893 .84866 .54306 .83930 .55823 .82909 .67262 .81982 4 57 .51429 .85762 .52918 .84851 .54391 .83915 .55847 .829.53 .57286 .81905 3 58 .51454 .85747 .52943 .84836 .54415 .83899 .55871 .82936 .67310 .81949 2 59 .51479 .85732 .52967 .84820 .54440 .83883 .65895 .82920 .57334 .81932 1 60 .51504 .85717 .52992 .84805 .54464 .83867 .55919 .82904 .57358 .81915 / Sine Cosin Sine Cosin Sine I Cosin Sine Cosin Sine I 59 58 57 56 55 NATURAL SINES AND COSINES. 35 | SB* 37* SB 39* 9 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin ~o .57358 .81915' .58779 .80902 .60182 .79864 .61566 .78801 .62932 .77715 60 i .57381 .81899 .58802 .80885 .60205 .79846 .61589 .78783 .62955 .77696 69 2 .57405 .81882 .58826 .80867 .60228 .79829 .61612 .78765 .62977 .77678 58 3 .57429 .81865! .58849 .80850 .60251 .79811 .61635 .78747 .63000 .77660 67 4 .57453 .81848 .58873 .80833 .60274 .79793 .61658 .78729 .63022 .77641 56 5 .57477 .818321 .58896 .80816 .60298 .79776 .61681 .78711 .63045 .77623 55 6 .57501 .81815 .58920 .80799 .60321 .79758 .61704 .78694 .63068 .77605 54 7 .57524 .81798 .58943 .80782 .60344 .79741 .61726 .78676 .63090 .77586 53 8 .57548 .81782 .58967 .80765 .60367 .79723 .61749 .78658 .63113 .77568 52 9 .57572 .81765 .58990 .80748 .60390 .79706 .61772 .78640 .63135 .77550 61 10 .57596 .81748 .59014 .80730 .60414 .79688 .61795 .78622 .63158 .77531 60 11 .57619 .81731 .59037 .80713 .60437 .79671 '61818 .78604 .63180 .77513 40 12 .57043 .81714 .59061 .80096 .60460 .79653 .61841 .78586 .63203 .77494 48 13 .57667 .81698 .59084 .80679 .60483 .79635 .61864 .78508 .63225 .77476 47 14 .57691 .81681 .59108 .80662 .60506 .79618 .61887 .78550 .63248 .77458 46 15 .57715 .81664 .59131 .80644 .60529 .79600 .61909 .78532 .63271 .77439 45 16 .57738 .81647 .59154 .80627 .60553 .79583 .61932 .78514 .63293 .77421 44 17 .57762 .81631 .59178 .80610 .G0576 .79565 .61955 .78496 .63316 .77402 43 18 .57786 .81614 .59201 .80593 .60599 .79547 .61978 .78478 .63338 .77384 42 19 .57810 .81597 .59225 .80576 .60622 .79530 .62001 .78460 .63361 .77366 41 .57833 .81580 .59248 .80558 .60645 .79512 .62024 .78442 .63383 .77347 40 21 .57857 .81563 .59272 .80541 .60668 .79494 162046 .78424 .63406 .77329 39 23 .57881 .81546 .59295 .80524 .60691 .79477 .62069 .78405 .63428 .77310 33 23 .57904 .81530 .59318 .80507 .60714 .79459 .62092 .78387 .63451 .77292 87 24 .57928 .81513 .59342 .80489 .60738 .79441 .62115 .78369 .63473 .77273 86 25 .57952 .81406 .59365 .80472 .60761 .79424 .62138 .78351 .63496 .77255 35 26 .57976 .81479 .59389 .80455 .60784 .79406 .62160 .78333 .63518 .77236 34 27 .57999 .81462 .59412 .80438 .60807 .79388 .62183 .78315 .63540 .77218 33 28 .58023 .81445 .59436 .80420 .60830 .79371 .62206 .78297 .63563 .77199 32 23 .58047 .81423 .59459 .80403 C0853 .79353 .62229 .78279 .63585 .77181 31 SO .58070 .81412 .59483 .80386 .60876 .79335 .62251 .78261 .63608 .77162 30 31 .58094 .81395 .59506 .80368 .60899 .79318 .62274 .78243 .63630 .77144 20 82 .58118 .81378 .59529 .80351 .60922 .79300 .62297 .78225 .63653 .77125 23 33 .58141 .81301 .59552 .80334 .60945 .79282 .62320 .78206 .63675 .77107 27 34 .58165 .81344 .59576 .80316 .60968 .79264 .62342 .78188 .63698 .77088 2G 85 .58189 .81327 .59599 .80299 .60991 .79247 .62365 .78170 .63720 .77070 25 36 .58212 .81310 .59622 .80282 .61015 .79229 .62388 .78152 .63742 .77051 24 37 .58236 .81293 .59646 .80264 .61038 .79211 1 .62411 .78134 .63765 .77033 23 38 .58260 .81276 .59669 .80247 .61061 .79193 .62433 .78116 .63787 .77014 23 39 58283 .81259 .59C93 .80230 .61084 .79176 .62456 .78098 .63810 .76996 21 40 .58307 .81242 .59716 .80212 .61107 .79158 .62479 .78079 .63832 .76977 go 41 .58330 .81225 .59739 .80195 .61130 .79140 .62502 .78061 .63854 .76959 19 42 .58354 .81208 .59763 .80178 .61153 .79122 .62524 .78043 .63877 .76940 18 43 .58378 .81191 .59786 .80160 .61176 .79105 .62547 .78025 .63899 .76921 17 44 .58401 .81174 .59809 .80143 .61199 .79C87 .62570 .78007 .63922 .76903 16 45 .58425 .81157 .59832 .80125 .61222 .79069 .62592 .77988 .63944 .76884 15 46 .58449 .81140 .59856 .80108 .61245 .79051 .62615 .77970 .63966 .76868 14 47 .58472 .81123 .59879 .80091 .61268 .79033 .62638 .77952 .63989 .76847 13 48 .58496 .81106 .59902 .80073 .61291 .79016 .62660 .77934 .64011 .76828 12 49 .58519 .81089 .59926 .80056 .61314 .78998 .62683 .77916 .64033 .76810 11 50 .58543 .81072 .59949 .80038 .61337 .78980 .62706 .77897 .64056 .76791 10 51 .58567 .81055 .59972 .80021 .61360 .78962 .62728 .77879 .64078 .76772 9 52 .58590 .81038 .59995 .80003 .61383 .78944 .62751 .778G1 .64100 .76754 8 53 .58614 .81021 .60019 .79986 .61406 .78926 .62774 .77843 .64123 .76735 7 54 .58637 .81004 .60042 .79968 .61429 .78908 .62796 .77824 .64145 .76717 6 55 .58661 .80987 .60065 .79951 .61451 .78891 .62819 .77806 .64167 .76698 6 56 .58684 .80970 .60089 .79934 .61474 .78873 .62842 .77788 .64190 .76679 4 57 .58708 .80953 .60112 .79916 .61497 .78855 .62864 .77769 .64212 .76661 X 58 .58731 .80936 .60135 .79899 .61520 .78837 .62887 .77751 .64234 .76642 2 59 .58755 .80919 .60158 .79881 .61543 .78819 .62909 .77733 .64256 .76623 1 60 .58779 .80902 .60182 .79864 .61566 .78801 .62932 .77715 .64279 .76604 Cosin Sine Cosin Sine Cosin Sine Cosin, Sine Cosin Sine i i 54 53 52 51- 50 124 NATURAL SINES AND COSINES. 40 41 42 43" 44 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin i "o .64279 .76604 .65606 .75471 .66913 .74314 .68200 .73135 .69466 .71934 60 1 .64301 .76586 .65628 .75452 .66935 .74295 .68221 .73116 .69487 .71914 59 2 .64323 .76567 .65650 .75433 .66956 .74276 .68242 .73096 .69508 .71894 58 3 .64346 .76548 .65672 .75414 .66978 .74256 .68264 .73076 .69529 .71873 57 4 .64368 .76530 .65694 .75395 .66999 .74237 .68285 .73056 .69549 .71853 56 5 .64399 .76511 .65716 .75375 .67021 .74217 .68306 .73036 .69570 .T1833 55 6 .64412 .76492 .65738 .75356 .6704S .74198 .68327 .73016 .69591 .71813 54 7 .64435 .76473 .65759 .75337 .67064 .74178 .68349 .72996 .69612 .71792 53 8 .64457 .76455 .65781 .75318 .67086 .74159 .68370 .72976 .69633 .71772 52 9 .64479 .76433 .65803 .75299 .67107 .74139 .68391 .72957 .69654 .71752 51 10 .64501 .76417 .65825 .75280 .67129 .74120 .68412 .72937 .69675 .71732 50 11 .64524 .76398 .65847 .75261 .67151 .74100 .68434 .72917 .69696 .71711 49 13 .64548 .7G330 .65339 .75241 .67172 .74080 .68455 .72897 .69717 .71091 48 13 .64533 .76331 .65891 .75222 .67194 .74061 .68476 .72877 .69737 .71671 47 14 .64590 .76342 .65913 .75203 .67215 .74041 .68497 .72857 .69758 .71650 46 15 .64612 .76323 .65935 .75184 .67237 .74022 .68518 .72837 .69779 .71630 45 16 .64635 .76304 .65953 .75165 .67258 .74002 .68539 .72817 .69800 .71610 44 17 .64857 .76238 .65078 75146 .67280 .73983 .68561 .72797 .69821 .71590 43 18 .64679 .76287 .63000 75123 .67301 .73963 .68582 .72777 .69842 .71569 42 19 .64701 .76248 .63022 75107 .67323 .73944 .68603 .72757 .69862 .71549 41 20 .64723 .70229 .66044 75088 .67344 .73924 .68624 .72737 .69883 .71529 40 21 .64746 .76210 .66066 75069 .67366 .73904 .68645 .72717 .69904 .71508 39 22 .64768 .76192 .63033 75050 . 67307 .73885 .68GG3 .72697 .69925 .71488 33 23 .64790 76173 .63103 75030 .67409 .73805 .68683 .72677 .69946 .71468 37 24 .64812 76154 .63131 75011 .67430 .73846 .68709 .72657 .69966 .71447 30 25 .64834 76135 .68153 74992 .67452 .73823 .68730 .72637 .69987 .71427 35 26 .64856 76116 .68175 74973 .67473 .73803 .68751 .72617 .70008 .71407 34 27 .64878 76097 .66197 74953 .67495 .73787 .68772 .72597 .70029 .71386 33 28 .64901 76078 .66218 74934 .67516 .73767 .68793 .72577 .70049 .71366 32 29 .64923 76059 .68240 74915 .67533 .73747 .68814 .72557 .70070 .71345 31 30 .64945 76041 .66262 74896 .67559 .73728 .68835 .72537 .70091 .71325 30 31 .64967 76022 .66284 74876 67580 .73708 .68857 .72517 .70112 .71305 29 32 .64989 76003 .63303 74857 67602 .73683 .68878 .72497 70132 .71284 23 33 .65011 75984 .68327 74833 67C23 .73609 .68899 .72477 .70153 .71264 27 34 .65033 75965 .68349 74818 67645 .73643 .68920 .72457 .70174 .71243 26 .65055 7594S .68371 74799 67006 .73623 .68941 .72437 .70195 .71223 35 38 .65077 75927 .63393 74780 67G33 .73010 .68902 .72417 .7C215 .71203 24 37 .65100 75903 .66414 74760 67709 73590 .68983 .72397 .70236 .71182 23 S3 .65122 75889 .66433 74741 67730 73570 .69004 .72377 .70257 .71162 22 39 .65144 75870 .63453 74722 67752 73551 .69025 .72357 .70277 .71141 21 40 .65166 75851 .66480 74703 67773 73531 .69046 .72337 .70298 .71121 20 41 .65188 75832 .66501 74683 67795 .73511 .69067 .72317 .70319 .71100 19 42 .65210 75813 .63523 74664 678fG .73491 .69038 .72297 .70339 .71080 18 43 .65232 75794 .68545 74644 67837 .73472 .69109 .72277 .70360 .71059 17 44 .65254 75775 .66563 74625 67859 .73452 .69130 .72257 .70381 .71039 16 45 .65276 75756 .66588 74608 .67880 .73432 .69151 .72236 .70401 .71019 15 46 .65298 75738 .66610 74588 .67901 .73413 .69172 .72216 .70422 .70998 14 47 .65320 75719 .66632 74567 .67923 .73393 .69193 .72196 .70443 .70978 13 48 .65342 75700 .66653 74548 .67944 .73373 .69214 .72176 .70463 .70957 12 49 .65364 75680 .66675 74523 .67965 .73353 .69235 .72156 .70484 .70937 11 50 .65386 75661 .66697 74509 .67987 .73333 .69256 .72136 .70505 .70916 10 51 .65408 .75642 .66718 74489 .68008 .73314 .69277 .72116 .70525 .70896 9 52 .65430 .75623 .66740 74470 .68029 .73294 .69298 .72095 .70546 .70875 8 53 .65452 .75604 .66762 74451 .68051 .73274 .69319 .72075 .70567 .70855 7 54 .65474 .75585 .66783 .74431 .68072 .73254 .69340 .72055 .70587 .70834 6 55 .65496 .75566 .66805 .74412 .68093 .73234 .69361 .72035 .70608 70813 5 56 .65518 .75547 .66827 .74392 .68115 .73215 .69382 .72015 .70628 70793 4 57 .65540 .75528 .66848 .74373 .68136 .73195 .69403 .71995 .70649 70772 3 58 .65562 .75509 .66870 .74353 .68157 .73175 .69424 .71974 .70670 .70752 2 59 .65584 .75490 .66891 .74334 .68179 .73155 .69445 .71954 .70690 .70731 1 60 .65606 .75471 .66913 .74314 .68200 .73135 .69466 .71934 .70711 .70711 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine 49 48 47 40 i 45 | NATURAL TANGENTS AND COTANGENTS. 125 1 2 8 Tang Cotang Tang Cotang Tang Cotang Tang Cotang / .00000 Infinite. .01746 57.2900 .03492 28.6363 .05241 19.0811 60 1 .00029 3437.75 .01775 56.3506 .03521 28.3994 .05270 18.9755 59 2 .00058 1718.87 .01804 55.4415 .03550 28.1664 .05299 18.8711 58 3 .00087 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7678 57 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.6656 56 5 .00145 687.549 .01891 52.8821 .03638 27.4899 .05387 18.5645 55 .00175 572.957 .01920 52.0807 .03667 27.2715 .05416 18.4645 54 7 .00204 491.106 .01949 51.3032 .03096 27.0566 .05445 18.3G55 53 8 .00233 429.718 .01978 50.5485 .03725 26.8450 .05474 18.2G77 52 .00262 381.971 .02007 49.8157 .03754 26.63G7 .05503 18.1708 51 10 .00291 &43.T74 .02036 49.1039 .03783 26.4316 .05533 18.0750 50 11 .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.9802 49 12 .00349 236.478 .02095 47.7395 .03342 26.0307 .05591 17.88G3 48 13 .00378 234.441 .02124 47.0853 .03871 25.8348 .05620 17.7934 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 .05649 17.7015 46 15 .00436 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 45 1(5 .00465 214.858 .02211 45.2261 .03958 25.2644 .05708 17.5205 44 17 .00495 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 .00524 100.984 .02269 44.0661 .04016 24.8978 .05766 17.3432 42 19 .00553 180.932 .02298 43.5081 .04046 24.7185 .05795 17.2558 41 80 .00583 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 40 21 .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 39 82 .00040 156.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 38 23 ,OOGG9 149.465 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 21 . 00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 x).> .00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 35 ';; .00756 132.219 .02502 39.9G55 .04250 23.5321 .05999 16.6681 34 27 .00785 127.321 .02531 39.5059 .04279 23.3718 .06029 16.5874 33 2s .00315 122.774 .02560 39.05G8 .04308 23.2137 .06058 16.5075 32 20 .00844 118.540 .02589 38.6177 .04337 23.0577 .06087 16.4283 31 CO .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 30 81 .00002 110.892 .02648 37.7683 '.04395 22.7519 .06145 16.2722 29 32 .00931 107.426 .02677 37.3579 .04424 22.6020 .06175 16.1952 28 88 .009GO 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 27 84 .00989 101.107 .02735 36.5G27 .04483 22.3081 .06233 16.0435 26 85 .01018 98.2179 .02764 36.1776 .04512 22.1640 .06262 15.9687 25 36 .01047 95.4895 .02793 35.8006 .04541 22.0217 .06291 15.8945 24 87 .01076 93.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 23 ::s .01105 90.4633 .02851 35.0005 .04599 21.7426 .06350 15.7483 22 .7;l .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 20 11 .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 19 12 .01222 81.8470 .029G8 33.6935 .04716 21.2049 .06467 15.4638 18 ta .01251 79.9434 .02997 33.3662 .04745 21.0747 .06496 15.3943 17 41 .01280 78.1263 .03026 33.0452 .04774 20.9460 .06525 15.3254 16 45 .01309 76.3900 .03055 32.7303 .04803 20.8188 .06554 15.2571 15 16 .C1338 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.1893 14 17 .01367 73.1390 .03114 32.1181 .048G2 20.5691 .06613 15.1222 13 48 .01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.0557 12 ID .01425 70.1533 .03172 31.5284 .04920 20.32.53 .06671 14.9898 11 50 .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 9 52 .01513 66.1055 .03259 30.6833 .05007 19.9702 .06759 14.7954 8 53 .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 7 54 .01571 63.6567 .03317 30.1446 .05066 19.7403 .06817 14.6685 6 55 .01600 62.4992 .03346 29.8823 .05095 19.6273 .06847 14.6059 5 56 .01629 61.3829 .03376 29.6245 .05124 19.5156 .06876 14.5438 4 57 .01658 60.3058 .03405 29.3711 .05153 19.4051 .06905 14.4823 3 58 .01687 59.2659 .03434 29.1220 .05182 19.2959 .06934 14.4212 2 59 .01716 58.2612 .03463 28.8771 .05212 19.1879 .06963 14.3607 1 GO .01746 57.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang t 89 88 87* - 86 126 NATURAL TANGENTS AND COTANGENTS. 4 5 6 70 Tang Cotang Tang Cotang Tang Cotang Tang Cotang / .06993 14.3007 .08749 11.4301 .10510 9.51436 .12278 8.14435 GO 1 .07022 14.2411 .08778 11.3919 .10540 9.48781 .12308 8.12481 58 2 .07051 14.1821 .08807 11.3540 .10569 9.46141 .12338 8.10536 58 3 .07080 14.1235 .08837 11.3163 .10599 9.43515 .12307 8.08600 57 4 .07110 14.0655 .08866 11.2789 .10628 9.40904 .12397 8.06674 56 5 .07139 14.0079 .08895 11.2417 .10657 9.38307 .19426 8.04756 K 6 .07168 13.9507 .08925 11.2048 .10687 9.35724 .12456 8.02848 54 7 .07197 13.8940 .08954 11.1681 .10716 9.33155 .12485 8.00948 53 8 .07227 13.8378 .08983 11.1316 .10746 9.30599 .12515 7.99058 52 9 .07256 13.7821 .09013 11.0954 .10775 9.28058 .12544 7.97176 51 10 .07285 13.7267 .09042 11.0594 .10805 9.25530 .12574 7.95302 60 11 .07314 13.6719 .09071 11.0237 .10834 9.23016 .12603 7.93438 40 12 .07344 13.6174 .09101 10.9882 .1CSG3 9.20516 .12CS3 7.91582 48 18 .07373 13.5634 .09130 10.9529 .10893 9.18028 .12602 7.89734 47 14 .07402 13.5098 .09159 10.9178 .1CC22 9.15554 .12692 7.87895 46 15 .07431 13.4566 .09189 10.8829 .10952 9.13093 .12722 7.86064 45 1G .07461 13.4039 .09218 10.8483 .10901 9.10(k6 .12751 7.84242 44 IT .07490 13.3515 .09247 10.8139 .11011 9.08211 .12781 7.82428 48 IS .07519 13.2996 .09277 10.7797 .11040 9.057C9 .12810 7.80022 42 19 .07548 13.2480 .09306 10.7457 .11C70 9.03379 .12840 7.78825 41 20 .07578 13.1969 .09335 10.7119 .11099 9.00983 .12869 7.77035 40 21 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 7.75254 30 22 .07636 13.0958 .00394 10.6450 .11158 8.90227 .12929 7.73480 3!< 23 .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 37 21 .07695 12.9962 .09453 10.5789 .11217 8.91520 .12988 7.C9957 86 25 .07724 -12.9469 .09482 10.54G2 .11246 8.89185 .13017 7.C8208 85 20 .07753 12.8981 .09511 10.5136 .11276 8.86862 .13047 7.CG4G6 34 27 .07782 12.8496 .C9541 10.4813 .11305 8.84551 .13076 7.64732 88 28 .07812 12.8014 .09570 10.4491 .11335 8.82252 .13100 7.63005 82 29 .07841 12.7536 .oceoo 10.4172 .11364 8.79904 .13136 7.01287 81 oO .07870 12.7062 .09629 10.3854 .11394 8.77689 .13165 7.59575 30 81 .07899 12.6591 .09658 10.3538 .11423 8.75425 .13195 7.57872 20 :J2 .07929 12.6124 .09688 10.3224 .11452 8.73172 .13224 7.56176 28 33 .07958 12.EGGO .00717 10.2913 .11482 8.70931 .13254 7.54487 27 34 .07987 12.5199 .09746 10.2602 .11511 8.68701 .13284 7.52806 26 35 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 7.51132 25 30 .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 7.49465 24 37 .08075 12.3838 .09834 10.1683 .11600 8.02078 .13372 7.47806 23 38 .08104 12.3390 .09864 10.1381 .11629 8.59893 .13402 7.46154 22 39 .08134 12.2946 .00893 10.1080 .11659 8.57718 .13432 7.44509 31 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 7.42871 20 41 .08192 12.2067 .09952 10.0483 .11718 8.53402 .13491 7.41240 10 42 .08221 12.1632 .09981 10.0187 .11747 8.51259 .13521 7.39616 18 43 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 .11806 8.47007 .13580 7.36389 16 45 .08309 12.0346 .10069 9.93101 .11836 8.44896 .13609 7.34786 15 40 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13639 7.33190 14 47 .08368 11.9504 .10128 9.87338 .11895 8.40705 .13669 7.31600 13 48 .08397 11.9087 .10158 9.84482 .11924 8.38G25 .13698 7.30018 12 41) .08427 11.8673 .10187 9.81641 .11954 8.36555 .13728 7.28442 11 50 .08456 11.8262 .10216 9.78817 .11983 8.34496 .13758 7.26873 10 51 .08485 11.7853 .10246 8.76009 .12013 8.32446 .13787 7.25310 9 52 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 7.23754 8 53 .08544 11.7045 .10305 9.70441 .12072 8.28376 .13846 7.22204 7 54 .08573 11.6645 .10334 9.67680 .12i01 8.26355 .13876 7.206G1 G 55 .08002 11.6248 .10363 9.64935 .12131 8.24345 .13906 7.19125 6 50 .08032 11.5853 .10393 9.62205 .12100 8.22344 .13935 7.17594 4 57 .08661 11.5461 .10422 9.59490 .12190 8.20352 .13965 7.16071 3 58 .08690 11.5072 .10452 9.56791 .12219 8.18370 .13995 7.14553 2 5!) .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 7.13042 1 GO .08749 11.4301 .10510 9.51436 .12278 8.14435 .14054 7.11537 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 85 84 83 82 NATURAL TANGENTS AND COTANGENTS. 127 8 < 9 1 1 1- / Tang Cotang Tang Cotang Tang Cotang Tang Cotang ~0 .14054 7.11537 .15838 6.31375 .17633 5.67128 .19438 5.14455 60 1 .14084 7.10038 .15868 6.30189 .17663 5.66165 .19468 5.13658 59 c .14113 7.08546 .15898 6.29007 .17693 5.65205 .19498 5.12862 58 s .14143 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 57 4 .14173 7.05579 .15958 6.26655 .17753 5.63295 .19559 5.11279 56 5 .1420S 7.04105 .15988 6.25486 .17783 5.62344 .19589 5.10490 55 G .14232 7.02637 .16017 6.24321 .17813 5.61397 .19619 5.09704 54 7 .14262 6.91174 .16047 6.23160 .17843 5.60452 .19640 5.08921 53 8 .14291 6.99718 .16077 6.22003 .17873 5.59511 .19680 5.08139 52 9 .14321 6.982G8 .16107 6.20851 .17903 5.58573 .19710 5.07360 51 10 .14351 6.96823 .16137 6.19703 .17933 5.57638 .19740 5.06584 50 11 .14381 6.95385 '.16167 6.18559 .17963 5.56706 .19770 5.05809 49 12 .14410 6.93952 .10190 6.17419 .17993 5.55777 .19801 5.05037 48 13 .14440 6.92525 .16226 6.10283 .18023 5.54851 .19831 5.042G7 47 14 .14470 6.91104 .16256 6.15151 .18053 5.53927 .19861 5.03499 46 15 .14499 6.89388 .16286 6.14023 .18033 5.53007 .19891 5.02734 45 1C .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17 .14559 6.8G874 '.16346 6.11779 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 ; 16376 6.10604 .18173 5.50264 .19982 5.00451 42 19 .14618 6.84082 .16405 6.09552 .18203 5.49356 .20012 4.99695 41 20 .14648 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 21 .14678 6.81312 116465 6.07340 .18263 5.47548 .20073 4.98188 39 22 .14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4.97438 38 23 .14737 6.73564 .16525 6.05143 .18323 5.45751 .20133 4.96690 37 24 .14767 6.77199 .16555 6.04051 .18353 5.44857 .20164 4.95945 36 23 .14796 6.75838 .16585 6.02902 .18384 5.43966 .20194 4.95201 35 23 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .14356 6.73133 .16645 6.00797 .18444 5.42192 .20254 4.93721 as 28 .14386 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 32 29 .14915 6.70450 .16704 5.98646 .18504 5.40429 .20315 4.92249 31 SO .14945 6.69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 30 81 .14975 6.67787 .16764 5.96510 .18564 5.38677 '.20376 .90785 29 32 .15005 6.66463 .16794 5.95448 .18594 5.37805 .20406 .90056 28 33 .15034 6.65144 .16824 5.94390 .18624 5.36936 .20436 .89330 27 34 .15064 6.63831 .16854 5.93365 .18654 5.36070 .20466 .88605 26 35 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 .87882 25 30 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 .87162 24 37 .15153 6.50021 .16944 5.90191 .18745 5.33487 .20557 .86444 23 30 .15183 6.53627 .16974 5.89151 .18775 5.32631 .20588 .85727 22 CO .15213 6.57339 .17094 5.88114 .18805 5.31778 .20618 .85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 .84300 20 41 .15272 6.54777 .17063 5.86051 .18865 5.30080 .20679 .83590 19 42 .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 .82882 18 43 .15332 6.52234 .17123 5.84001 .18925 5.28393 .20739 .82175 17 44 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 .81471 16 45 .15391 6.49710 .17183 5.81966 .18986 5.26715 .20800 .80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.258SO .20830 .80068 14 47 .15451 6.47206 .17243 5.79944 .19046 5.25048 .20861 .79370 13 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 .78673 12 49 .15511 6.44720 .17303 5.77936 .19106 6.23391 .20921 .77978 11 50 .15540 6.43484 .17333 5.76937 .19136 5.22566 .20952 .77286 10 51 .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 .76595 9 52 .15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 .75906 8 53 .15630 6.39804 .17423 5.73960 .19227 5.20107 .21043 .75219 7 54 .15660 6.38587 .17453 5.72974 .19257 5.19293 .21073 .74534 6 55 .15689 6.37374 .17483 5.71992 .19287 5.18480 .21104 .73851 5 56 .15719 6.30165 .17513 5.71013 .19317 5.17671 .21134 .73170 4 57 .15749 6.31961 .17543 5.70037 .19347 5.16863 .21164 .72490 3 53 .15779 6.33761 .17573 5.69064 .19378 5.16058 .21195 .71813 2 59 .15809 6.32566 .17603 5.68094 .19408 5.15256 .21225 .71137 1 60 .15838 6.31375 .17633 5.67128 .19438 5.14455 .21256 .70463 f Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 8 ! 1 8 7 9 7 B . 128 NATURAL TANGENTS AND COTANGENTS. 12 13 14 ] 15 Tang Cotang Tang Cotang Tang Cotang Tang Cotang f .21256 4.70463 .23087 4.33148 .24933 4.01078 .26795 3.73205 60 1 .21286 4.69791 .23117 4.32573 .24904 4.00582 .26826 3.72771 59 2 .21316 4.G9121 .23148 4.32001 .24995 4.00086 .26857 3.72338 58 3 .21347 4.C8452 .23179 4.31430 .25026 3.99592 .26888 3.71907 57 4 .21377 4.67786 .23209 4.30860 .25056 3.99099 .26920 3.71476 56 5 .21408 4.67121 .23240 4.30291 .25087 3.98607 .26951 3.71046 55 G .21438 4.6G458 .23271 4.29724 .25118 3.98117 .2G982 3.70616 54 7 .21469 4.65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 53 8 .21499 4.65138 .23332 4.28595 .25180 3.97139 .27044 3.69761 52 9 .21529 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.69335 51 10 .21560 4.63825 .23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 .21590 4.63171 .23424 4.26911 .25273 3.95680 .27138 3.68485 49 13 .21621 4.62518 .23455 4.26352 .25304 3.95196 .27109 3.680G1 48 13 .21G51 4.61868 .23485 4.25795 .25335 3.94713 .27201 3.67638 47 14 .21G82 4.61219 .23516 4.25239 .253G6 3.94232 .27232 3.67217 46 15 .21712 4.60572 .23547 4.24685 .25397 3.93751 .272G3 3.66796 45 1C .21743 4.59927 .23578 4.24132 .25428 3.93271 .27294 3.66376 44 17 .21773 4.59283 .23608 4.23580 .25459 3.92793 .27326 3.65957 43 10 .21004 4.58641 .23639 4.23030 .25490 3.92316 .27357 3.65538 42 19 .21834 4.58001 .23G70 4.22481 .25521 3.91639 .27388 3.65121 41 20 .21864 4.57363 .23700 4.21933 .25552 3.91364 .27419 3.64705 40 21 .21895 4.56726 .23731 4.21387 .25583 S.90890 .27451 3.64289 39 23 .21925 4.5G091 .23762 4.20842 .25014 3.90417 .27482 3.63874 38 23 .21956 4.55458 .23793 4.20298 .25045 3.89945 .27513 3.63461 37 21 .21986 4.54826 .23823 4.19756 .25076 3.89474 .27545 3.63048 36 23 .22017 4.54196 .23854 4.19215 .25707 3.89004 .27576 3.62636 35 23 .22047 4.53568 .23885 4.18675 .25738 3.88536 .27007 3 62224 34 27 .22078 4.52941 .23916 4.18137 .25769 3.88068 .27038 3.61814 33 23 .22108 4.52316 .23946 4.17600 .25800 3.87601 .27670 3.61405 32 29 .22139 4.51693 .23977 4.17064 .25831 3.87136 .27701 3.60996 31 30 .22169 4.51071 .24008 4.16530 .25862 3.86671 .27733 3 60588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 3.60181 29 S3 .22231 4 49832 .24069 4.15465 .25924 3.85745 .27795 3.59775 28 83 .22261 4.49215 .24100 4.14934 .25955 3.85284 .27826 3.59370 27 24 .22292 4.48600 .24131 4.14405 .25986 3.84824 .27858 3.58966 26 33 .22322 4.47986 .24163 4.13877 .26017 3.84364 .27889 3.58562 25 30 .22353 4.47374 .24193 4.13350 .26048 3.83906 .27921 3.58100 24 37 .22383 4.46764 .24223 4.12825 .26079 3.83449 .27952 3.57758 23 38 .22414 4.46155 .24254 4.12301 .26110 3.82992 .27983 3.57357 22 29 .22444 4.45548 .24285 4.11778 .26141 3.82537 .28015 3.56957 21 40 .22475 4.44942 .24316 4.11256 .26172 3.82083 .28046 3.56557 20 41 .22505 4.44338 .24347 4.10736 .26203 3.81630 .28077 3.56159 19 42 .22536 4.43735 .24377 4.10216 .26235 3.81177 .28109 3.55761 18 13 .22567 4.43134 .24408 4.09699 .26266 3.80726 .28140 3.55364 17 44 .22597 4.42534 .24439 4.09182 .26297 3.80276 .28172 3.54968 16 43 .22028 4.41936 .24470 4.08666 .26328 3.79827 .28203 3.54573 15 40 .22058 4.41340 .24501 4.08152 .26359 3.79378 .28234 3.54179 14 47 .22089 4.40745 .24532 4.07639 .26390 3.78931 .282G6 3.53785 13 <18 .22719 4.40152 .24562 4.07127 .26421 3.78485 .28297 3.53393 12 49 .22750 4.39560 .24593 4.06616 .26452 8.78040 .28329 3.53001 11 50 .22781 4.38969 .24624 4.06107 .26483 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 .28391 3.52219 9 .22842 4.37793 .24686 4.05092 .26546 3.76709 .28423 3.51829 8 Ha .22872 4 37207 .24717 4.04586 .26577 3.76268 .28454 3.51441 7 51 .22903 4.3G623 .24747 4.04081 .26608 3.75828 .28486 3.51053 6 "> .22934 4 30040 .24778 4.03578 .26639 3.75388 .28517 3.50666 5 DC .22904 4.35459 .24809 4.03076 .26670 3.74950 .28549 3.50279 4 57 .22995 4 34879 .24840 4.02574 .26701 3.74512 .28580 3.49894 3 58 .23026 4.34300 ..24871 4.02074 .26733 3.74075 .28612 3.49509 2 59 .23056 4.33723 .24902 4.01576 .26764 3.73640 .28643 3.49125 1 CO .23087 4.33148 .24933 4.01078 .2G795 3.73205 .28675 3.48741 Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 77 76 75 74" NATURAL TANGENTS AND COTANGENT?. 129 16 17 18 19 Tang Cotang Tang Cotang Tang Cotang Tang Cotang .28675 3.48741 .30573 3.27085 .32492 3.07768 .34433 2.90421 60 1 .28706 3.48359 .30605 3.26745 .32524 3.07464 .34465 2.90147 59 2 .28738 3.47977 .30637 3.26406 .32556 3.07160 .34498 2.89873 58 3 .28769 3.47596 .30669 3.26067 .32588 3.06857 .34530 2.89600 57 4 .28800 3.47216 .30700 3.25729 .32621 3.06554 .34563 2.89327 56 5 .28832 3.46837 .30732 3.25392 .32653 3.06252 .34596 2.89055 55 6 .28864 3.46458 .30764 3.25055 .32685 3.05950 .34628 2.88783 54 7 .28895 3.46080 .30796 3.24719 .32717 3.05649 .34661 2.88511 53 8 .28927 3 45703 .30828 3.24383 .32749 3.05349 .34693 2.88240 52 9 .28958 3.45327 .30800 3.24049 .32782 3.05049 .34726 2.87970 51 10 .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2.87700 50 11 .29021 3.44576 .30923 3.23381 .32846 3.04450 .34791 2.87430 49 12 .29053 3.44202 .30955 3.23048 .32878 3.04152 .34824 2.87161 48 1-3 .29084 3.43829 .30987 3.22715 .32911 3.03854 .34856 2.86892 47 14 .29116 3.43456 .31019 3.22384 .32943 3.03556 .34889 2.86624 46 15 .29147 3.43084 .31051 3.22053 .32975 3.032GO .34922 2.86356 45 16 .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.86089 44 17 .29210 3.42343 .31115 3.21392 .33040 3.02GG7 .34987 2.85822 43 18 .29242 3.41973 .31147 3.21063 .33072 3.02372 .35020 2.85555 42 19 .29274 3.41604 .31178 3.20734 .33104 3.02077 .35C52 2.85289 41 20 .29305 3.41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 40 21 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 8.84758 29 22 .29368 3.40502 .31274 3.19752 .33201 3.011C6 .351CO 2.84494 38 23 .29400 3.40136 .31208 3.19426 .33233 3.00C03 .35183 2.84229 37 24 .29432 3.39771 .31338 3.10100 .33266 3.00611 .35216 2.83965 36 25 .29463 3.39406 .31370 3.18775 .33298 3.00319 .35248 2.83702 35 2G .29495 3.39042 .314C2 3.10451 .33330 3.00028 .35231 2.83439 34 27 .29526 3.38679 .31434 3.10127 .33363 2.99738 .35314 2.83176 33 28 .29558 3.38317 .31466 3.17804. .33305 2.99447 .35346 2.82914 32 29 .29590 3.37955 .31498 8.17401 .33427 2.99158 .35379 2.82653 31 30 .29621 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 30 31 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 29 33 .29685 3.36375 .31594 3.16517 .33524 2.98292 .35477 2.81070 28 33 .29716 3.3G516 .31626 3.16197 .33557 2.98004 .35510 2.81610 27 34 .29748 3.30158 .31658 3.15877 .33589 2.97717 .35543 2.81350 26 85 .29780 3.35800 .31690 3.15558 .33621 2.97430 .35576 2.81091 25 36 .29811 3.35443 .31722 3.15240 .33654 2.97144 .35603 2.80833 24 37 .29843 3.35087 .31754 3.14922 .33686 2.96850 .35641 2.80574 23 38 .29875 3.34732 .31786 3.14605 .33718 2.96573 .35674 2.80316 22 39 .29906 3.34377 .31018 3.14288 .33751 2.96288 .35707 2.80059 21 40 .29938 3.34023 .31850 3.13972 .33783 2.96004 .35740 2.79802 20 41 .29970 3.33670 .31882 3.13656 .33816 2.95721 .35772 2.79545 19 42 .30001 3.33317 .31914 3.13341 .33848 2.95437 .35805 2.79289 18 43 .30033 3.32965 .31946 3.13027 .33881 2.95155 .35838 2.79033 17 44 .30065 3.32614 .31978 3.12713 .33913 2.94872 .35871 2.78778 16 45 .30097 3.32264 .32010 3.12400 .33945 2.94591 .35904 2.78523 15 ^6 .30128 3.31914 .32043 3.12087 .33978 2.94309 .35937 fc. 78269 14 4, .30160 3.31565 .32074 3.11775 .34010 2.94028 .85969 2.78014 13 48 .30192 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 49 .30224 3.30868 .32139 3.11153 .34075 2.934C8 .36035 2.77507 11 50 .30255 3.30521 .32171 3.10843 .34108 2.93189 .36068 2.77254 10 51 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 9 52 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2.76750 8 53 .30351 3.29483 .32267 3.09914 .34205 2.92354 .36167 2.76498 7 54 .30382 3.: 29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 6 55 .30414 3.28795 .32331 3.09298 .34270 2.91799 .36232 2.75996 5 56 .30446 3.28452 .32363 3.08991 .34303 2.91523 .36265 2.75746 4 57 .30478 3.28109 .32396 3.08685 .34335 2.91246 .36298 2.75496 3 58 .30509 3.27767 .32428 3.08379 .34368 2.90971 .36331 2.75246 2 59 .30541 3.27426 .32400 3.08073 .34400 2.90696 .36364 2.74997 1 GO .30573 3.27085 .32492 3.07768 .34433 2.90421 .36397 2.74748 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 73 72 71 70 130 NATURAL TANGENTS AND COTANGENTS. 80- 21* 22 23 Tang Cotang Tang Cotang Tang Cotang Tang Cotang / .36397 2.74748 .38386 2.60509 .40403 2.47509 .42447 2.35585 60 1 .36430 2.74499 .38420 2.60283 .40436 2.47302 .42482 2.35395 59 2 .36463 2.74251 .38453 2.60057 .40470 2.47095 .42516 2.35205 58 3 .36496 2.74004 .38487 2.59831 .40504 2.4G888 .42551 2.35015 57 4 .36529 2.73756 .38520 2.59606 .40538 2.46082 .42585 2.34825 56 5 .36562 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.34636 55 6 .36595 2.73263 .38587 2.59156 .40606 2.46270 .42654 2.34447 54 7" .36628 2.73017 .38620 2.58932 .40643 2.46065 .42688 2.34258 53 8 .36661 2.72771 .38654 2.58708 .40074 2.45860 .42722 2.34069 52 9 .30694 2.72526 .38687 2.58484 .40707 2.45655 .42757 2.83881 51 10 .36727 2.72281 .38721 2.58261 .40741 2.45451 .42791 2.33693 50 11 .36760 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.33505 49 12 .36793 2.71793 .38787 2.57815 .40809 2.45043 .42800 2.33317 48 13 .30826 2.71548 .38821 2.57593 .40843 2.44839 .42894 2.33130 47 14 .30859 2.71305 .38854 2.57371 .40877 2.44G36 .42929 2.32943 46 15 .30892 2.71062 .38888 2.57150 .40911 2.44433 .42963 2.32756 45 16 .33925 2.70819 .38921 2.56928 .40945 2.44230 .42998 2.32570 44 17 .30958 2.70577 .38955 2.56707 .40979 2.44027 .43032 2.32383 43 10 .30991 2.70335 .88988 2.56487 .41013 2.43825 .43067 2.32197 42 19 .37024 2.70094 .39023 2.56266 .41047 2.43623 .43101 2.32012 41 20 .37057 2.69853 .89055 2.56046 .41081 2.43422 .43136 2.31826 40 21 .87090 2.69612 .39089 2.55827 .41115 2.43220 .43170 2.31641 39 22 .37123 2.CD371 .39123 2.55608 .41149 2.43019 .43205 2.31456 38 23 .37157 2.C3131 .39156 2.55389 .41183 2.42819 .43239 2.31271 |37 24 .37190 2.63392 .89190 2.55170 .41217 2.42018 .43274 2.31086 36 25 .3?223 2.63G53 .39223 2.54952 .41251 2.42418 .43308 2.30902 3T> 20 .37256 2.C3414 .39257 2.54734 .412C5 2.42218 .43343 2.30718 34 27 .37289 2.C3175 .39290 2.54516 .41319 2.42019 .43378 2.30534 33 23 .37323 2.67037 .39324 2.54299 .41353 2.41819 .43412 2.30351 32 29 .37355 2.67700 .39357 2.54082 .41307 2.41620 .43447 2.30167 31 30 .37388 2.67462 .89391 2.53865 .41421 2.41421 .43481 2.29984 30 31 .37422 2.67225 .39425 2.53648 .41455 2.41223 .43516 2.29801 29 32 .37455 2.63989 .33458 2.53432 .41490 2.41025 .43550 2.29619 28 33 .37488 2.GG752 .33492 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.63516 .39526 2.53001 .41553 2.40029 .43620 2 29254 26 35 .37554 2.GG281 .39559 2.53708 .41592 2.40432 .43654 2.. 29073 25 36 .37588 2.60046 .39593 2.52571 .416C6 2.40235 .43689 2.28891 24 37 .37621 2.65811 .39626 2.52357 .41600 2.40038 .43724 2.28710 23 33 .37654 2.65576 .39060 2.52142 .41694 2.89841 .43758 2.28528 22 39 .37687 2.C5342 .39094 2.51929 .41728 2.39045 .43793 2.28348 21 40 .37720 2.65109 .39727 2.51715 .41763 2.39449 .43828 2.28167 20 41 .37754 2.64875 .89761 2.51502 .41797 2.39253 .43862 2.27987 19 42 .37787 2.61G42 .39795 2.51289 .41831 2.39058 .43897 2 27806 18 43 .37820 2.64410 .39829 2.51076 .41805 2.38803 .43932 2.27626 17 44 .37853 2.64177 .39802 2.50804 41899 2.38008 .43966 2.27447 16 45 .37887 2.63045 .39898 2.50652 .41933 2.38473 .44001 2.27267 15 46 .37920 2.63714 .39930 2.50440 .41908 2.38279 .44036 2.27088 14 47 .37953 2.63483 .33963 2.50229 .42003 2.38084 .44071 2.20909 13 48 .37986 2.63252 .39997 2.50018 .42036 2.37891 .44105 2.26730 12 49 .33020 2.63021 .40031 2.49807 .42070 2.37097 .44140 2.26552 11 50 .38053 2.62791 .40065 2.49597 .42105 2.87504 .44175 2.26374 10 51 .38086 2.62561 .40098 2.49386 .42139 2.87311 .44210 2.26196 9 52 33120 2.62332 .40132 2.49177 .42173 2.37118 .44244 2.26018 8 53 .38153 2.62103 .40166 2.48967 .42207 2.36925 .44279 2.25840 7 54 .33186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25603 6 55 .38220 2.61646 .40234 2.48549 .42276 2.36541 .44349 2.25486 5 56 .38253 2.61418 .40267 248340 .42310 2.36349 .44384 2.25309 4 57 .38286 2.61190 .40301 2.48132 .42345 2.36158 .44418 2.25132 3 58 38320 2 60963 40335 2.47924 .42379 2.35967 .44453 2.24956 2 59 .38353 2.60736 .40369 2.47716 .42413 2.35776 .44488 2 24780 1 GO .38386 2.60509 .40403 2.47509 .42447 2.35585 .44523 2.24604 j '/ Cotang Tang, Cotang Tang Cotang Tang Cotang Tang / 69 > 68 67' II 66 NATURAL TANGENTS AND COTANGENTS. 131 2 40 2 5 2 6' 2 7 o Tang Cotang Tang Cotang Tang Cotang Tang Cotang .44523 2.24604 .46631 2.14451 .48773 2.05030 .50953 1.96261 60 1 .44558 2.24428 .46666 2.14288 .48809 2.04879 .50989 1.96120 59 2 .44593 2.24252 .46702 2.14125 .48845 2.04728 .51026 1.95979 58 3 .44627 2.24077 .46737 2.13963 .48881 2.04577 .51063 1.95838 57 4 .44662 2.23902 .46772 2.13801 .48917 2.04426 .51099 1.95698 56 .44697 2.23727 .46808 2.13639 .48953 2.04276 .51136 1.95557 55 G .44732 2.23553 .46843 2.13477 .48989 2.04125 .51173 1.95417 54 7 .44767 2.23378 .46879 2.13316 .49026 2.03975 .51209 1.95277 53 8 .44802 2.23204 .46914 2.13154 .49062 2.03825 .51246 1.95137 52 9 .44837 2.23030 .46950 2.12993 .49098 2.03675 .51283 1.94997 51 10 .44872 2.22857 .46985 2.12833 .49134 2.03526 .51319 1.94858 50 11 144907 2.22683 .47021 2.12671 .49170 2.03376 .51356 1.94718 49 12 .44942 2.22510 .47056 2.12511 .49206 2.03227 .51393 1.94579 48 13 .44977 2.22337 .47092 2.12350 .49242 2.03078 .51430 1.94440 47 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 .51467 1.94301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 .51503 1.94162 45 18 .45082 2.21819 .47199 2.11871 .49351 2.02631 .51540 1.94023 44 ir .45117 2.21647 .47234 2.11711 .49337 2.02403 .51577 1.93885 43 18 .45152 2.21475 .47270 2.11552 .49423 2.02335 .51614 1.93746 42 19 .45187 2.21304 .47C05 2.11392 .49459 2.02187 .51651 1.93608 41 90 .45222 2.21133 .47341 2.11233 .49495 2.02039 .51688 1.93470 40 21 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 1.93332 39 29 .45292 2.20790 .47412 2.10916 .495C8 2.01743 .51761 1.93195 38 28 .45327 2.20619 .47448 2.10758 .49604 2.01596 .51798 1.93057 37 J4 .45362 2.20449 .47483 *. 10600 .49640 2.01449 .51835 1.92920 36 JJ5 .45397 2.20278 .47519 2.10442 .49677 2.01302 .51872 1.92782 35 26 .45432 2.20108 .47555 2.10284 .49713 2.01155 .51909 1.92645 34 2T .45407 2.19938 .47590 2.10126 .49749 2.01008 .51946 1.92508 33 28 .45502 2.19709 .47626 2.09909 .49786 2.00862 .51983 1.92371 32 29 .45538 2.19599 .47062 2.09811 .49822 2.00715 .52020 1.92235 31 30 .45573 2.19430 .47698 2.09854 .49858 2.00569 .52057 1.92093 30 31 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52094 1.91962 29 3-2 .45643 2.19092 .47769 2.09341 .49931 2.00277 .52131 1.91826 28 83 .45678 2.18923 .47005 2.09184 .49967 2.00131 .52168 1.91690 27 84 .45713 2.18755 .47840 2.09028 .50004 .99986 .52205 1.91554 26 35 .45748 2.18587 .47876 2.08872 .50040 .99841 .52242 1.91418 25 3G .45784 2.18419 .47912 2.08716 .50076 .99695 .52279 1.91282 24 37 .45819 2.10251 .47948 2.08560 .50113 .9550 .52316 1.91147 23 88 .45854 2.18084 .47984 2.08405 .50149 .C3406 .52353 1.91012 22 39 .45889 2.17916 .43019 2.03250 .50185 .C9261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .50222 .93116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 .98972 .52464 1.90607 19 42 .45995 2.17416 .43127 2.07785 .50295 .93828 .52501 1.90472 18 43 .40030 2.17249 .48163 2.07630 .50331 .93684 .52538 1.90337 17 41 .4G065 2.17083 .48198 2.07476 .50368 .98540 .52575 1.90203 16 45 .40101 2.10917 .48234 2.07321 .50404 .08396 .52613 1.90069 15 46 .40136 .2.10751 .48270 2.07167 .50441 .88253 .52650 1.89935 14 47 .40171 2.10585 .48306 2.07014 .50477 .93110 .52687 1.8S801 13 48 .46206 2.10420 .48342 2.00860 .50514 .97966 .52724 1.89667 12 49 .46242 2.10255 .43378 2.06706 .50550 .97823 .52761 1.89533 11 50 .46277 2.16090 .48414 2.06553 .50587 .97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 .97538 .52836 1.8926B 9 52 .40348 2.157GO .43486 2.00247 .50660 .97395 .52873 1.89133 8 53 .46383 2.15596 .43521 2.00094 .50696 .97253 .52910 1.89000 7 54 .46418 2.15432 .43557 2.05942 .50733 .97111 .52947 1.88807 6 68 .40454 2.15268 .48593 2.05790 .50769 .96969 .52985 1.88734 5 B6 .40489 2.15104 .48629 2.05637 .50806 .96827 .53022 1.88602 4 57 .46525 2.14940 .48665 2.05485 .50843 .96685 .53059 1.88469 3 58 .40560 2.14777 .48701 2.05333 .50879 .96544 .53096 1.88337 2 59 .46595 2.14614 .48737 2.05182 .50916 .96402 .53134 1.88205 1 60 .46631 2.14451 .48773 2.05030 .50953 .96261 .53171 1.88073 J) Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 6 5 6 4 6 3 6 z- 1 132 NATURAL TANGENTS AND COTANGENTS. f 28 29 30 31 Tang Cotang Tang Cotang Tang Cotang Tang Cotang j "o .53171 1.88073 .55431 1.80405 .57735 1.73205 .60086 1.66428 60 1 .53208 1.87941 .55469 1.80281 .57774 1.73089 .60126 1.66318 59 2 .53246 1.87809 .55507 1.80158 .57813 1.72973 .60165 1.66209 58 3 .53283 1.87677 .55545 1.80034 .57851 1.72857 .60205 1.66099 57 4 .53320 1.87546 .55583 1.79911 .57890 1 .72741 .60245 1.65990 56 i .53358 1.87415 .55621 1.79788 .57929 1.72625 .60284 1.65881 55 6 .53395 1.87283 .55659 1.79665 .57968 1.72509 .60324 1.65772 54 7 .53432 1.87152 .55697 1.79542 .58007 1.72393 .60364 1.65663 53 6 .53470 1.87021 .55736 1.79419 .58046 1.72278 .60403 1.65554 52 9 .53507 1.86891 .55774 1.79296 .58085 1.72163 .60443 1.65445 51 10 .53545 1.86760 .55813 1.79174 .58124 1.72047 .60483 1.65337 50 11 .53582 1.86630 .55850 1.79051 .58162 1.71932 .60522 1.65228 49 12 .53620 1.86499 .55888 1.78929 .58201 1.71817 .60562 1.65120 48 13 .53657 1.86369 .55926 1.78807 .58240 1.71702 .60602 1.65011 47 14 .53694 1.86239 .55964 1,. 78685 .58279 1.71588 .60642 1.64903 46 15 .53732 1.86109 .56003 1.78563 .58318 1.71473 .60681 1.64795 45 16 .53769 1.85979 .56041 1.78441 .58357 1.71358 .60721 1.64687 44 17 .53807 1.85850 .56079 1.78319 .58396 1.71244 .60761 1.64579 43 18 .53844 1.85720 .56117 1.78198 .58435 1.71129 .60801 1.64471 42 19 .53882 1.85591 .56156 1.78077 .58474 1.71015 .60841 1.64363 41 20 .53920 1.85462 .56194 1.77955 .58513 1.70901 .60881 1.64256 40 21 .53957 1.85333 .56232 1.77834 .58552 1.70787 .60921 1.64148 39 22 .53995 1.85204 .56270 1.77713 .58591 1.70673 .60960 1.64041 38 23 .54032 1.85075 .56309 1.77592 .58631 1.70560 .61000 1.63934 37 24 .54070 1.84946 .56347 1.77471 .58670 1.70446 .61040 1.63826 36 25 .54107 1.84818 .56385 1.77351 .58709 1.70332 .61080 1.63719 35 26 .54145 1.84689 .56424 1.77230 .58748 1.70219 .61120 1.63612 34 27 .54183 1.84561 .56462 1.77110 .58787 1.70106 .61160 1.63505 33 28 .54220 1.84433 .56501 1.76990 .58826 1.69992 .61200 1.63398 32 29 .54258 1.84305 .5C539 1.76869 .588C5 1.69879 .61240 1.63292 31 30 .54296 1.84177 .56577 1.76749 .58905 1.69766 .61280 1.63185 30 31 .54333 1.84049 .56616 1.76629 .58944 1.69653 .61320 1.63079 29 32 .54371 1.83922 .50654 1.76510 .58983 1.69541 .61360 1.62972 28 33 .54409 1.83794 .56693 1.76390 .59022 1.69428 .61400 1.62866 27 34 .54446 1.83667 .56731 1.76271 .59061 1.69316 .61440 1.62760 26 35 .54484 1.83540 .56769 1.76151 .59101 1.69203 .61480 1.62654 25 36 .54522 1.83413 .56808 1.76032 .59140 1.69091 .61520 1.62548 24 37 .54560 1.83286 .56846 1.75913 .59179 1.68979 .61561 1.62442 23 38 .54597 1.83159 .56885 1.75794 .59218 1.68866 .61601 1.62336 22 39 .54635 1.83033 .56923 1.75675 .59258 1.68754 .61641 1.62230 21 40 .54673 1.82906 .56962 1.75556 .59297 1.68643 .61681 1.62125 20 41 .54711 1.82780 .57000 1.75437 .59336 1.68531 .61721 1.62019 19 42 .51748 1.82654 .57039 1.75319 .59376 1.68419 .61761 1.61914 18 43 .54786 1.82528 .57078 1.75200 .59415 1.68308 .61801 1.61808 17 44 .5^964 1.82402 .57116 1.75082 .59454 1.68196 .61842 1.61703 16 45 .54862 1.82276 .57155 1.74964 .59494 1.68085 .61882 1.61598 15 46 .54900 1.82150 .57193 1.74846 .59533 1.67974 .61922. 1.61493 14 47 .54938 1.82025 .57232 1.74728 .59573 1.67863 .61962 1.61388 13 48 .54975 1.81899 .57271 1.74610 .59612 1.67752 .62003 1.61283 12 49 .55013 1.81774 .57309 1.74492 .59651 1.67641 .62043 1.61179 11 50 .55051 1.81649 .57348 1.74375 .59691 1.67530 .62083 1.61074 10 51 .55089 1.81524 .57386 1.74257 .59730 1.67419 .62124 1.60970 9 52 .55127 1.81399 .57425 1.74140 .59770 1.67309 .62164 1.60865 8 53 .55165 1.81274 .57464 1.74022 .59809 1.67198 .62204 1.60761 7 54 .55203 1.81150 .57503 1.73905 .59849 1.67088 .62245 1.60657 6 55 .55241 1.81025 .57541 1.73788 .59888 1.66978 .62285 1.60553 5 56 .55279 1.80901 .57580 1.73671 .59928 1.66867 .62325 1.60449 4 57 .55317 1.80777 .57619 1.73555 .59967 1.66757 .62366 1.60345 3 58 .55355 1.80653 .57657 1.73438 .60007 1.66647 .62406 1.60241 2 59 .55393 1.80529 .57696 1.73321 .60046 1.66538 .62446 1.60137 60 .55431 1.80405 .57735 1.73205 .60086 1.66428 .62487 1.60033 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang i 61- 60 59 58 i NATURAL TANGENTS AND COTANGENTS. 133 32" 83 34 | 35' Tang Cotang Tang Cotang Tang Cotang Tang Cotang ~o 62487 1. 600*3 .64941 1.53986 .67451 1.48256 i .70021 1.42815 60 i 62527 1.59930 .64983 1.53888 .67493 1.48163 .70064 1.42726 59 2 62568 1.59826 .65024 1.53791 .67536 1.48070 .70107 1.42638 58 3 62608 1.59723 .65065 1.53693 .67578 1.47977 .70151 1.42550 57 4 62649 1.59620 .65106 1.53595 .67620 1.47885 .70194 1.42462 56 5 62689 1.59517 .65148 1.53497 .67663 1.47792 .70238 1.42374 55 6 62730 1.59414 .65189 1.53400 .67705 1.47699 .70281 1.42286 54 7 62770 1.59311 .65231 1.53302 .67748 1.47607 .70325 1.42198 53 8 62811 1.59208 .65272 1.53205 .67790 1.47514 .70368 1.42110 52 g 62852 1.59105 .65314 1.53107 .67832 1.47422 .70412 1.42022 51 10 62893 1.59003 .65355 1.53010 .67875 1.47330 .70455 1.41934 50 11 62933 1.58900 .65397 1.52913 .67917 1.47238 .70499 1.41847 49 12 62973 1.58797 .65438 1.52816 .67900 1.47146 .70542 1.41759 48 13 63014 1.58695 .65480 1.52719 .68002 1.47053 .70586 1.41672 47 14 63055 1.58593 65521 1.52623 .68045 1.46962 .70629 1.41584 46 15 63095 1.58490 .65563 1.52525 .68088 1.4G870 .70673 1.41497 45 16 .63136 1.58388 ! 65604 1.52429 .68130 1.46778 .70717 1.41409 44 17 .63177 1.58286 .65646 1.52333 .68173 1.46G86 .707GO 1.41322 43 18 .63217 1.581&4 .65G88 1.52235 .68215 1.46595 .70804 1.41235 42 19 .63258 1.58083 .65729 1.E2139 .68258 1.4G503 .70S48 1.41148 41 20 .63299 1.57981 .65771 1.52043 .68301 1.46411 .70891 1.41061 40 21 .63340 1.57879 .65813 1.51946 .68343 1.46320 .70935 1.40974 39 22 .63380 1.577T3 .65854 1.51850 .68386 1.4G229 .70979 1.40887 38 23 .63421 1.57076 .65893 1.51754 .68429 1.46137 .71023 1.40800 37 24 .63462 1.57575 .65938 1.51058 .68471 1.4G046 .71066 1.40714 3G 25 .63503 1.57474 .65980 1.61563 .68514 1.45955 .71110 1.40G27 35 26 .63544 1.57372 .66021 1.51406 .68557 1.458G4 .71154 1.40540 S4 27 .63584 1.57271 .66003 1.51370 .68600 1.45773 .71108 1.40454 33 28 .63625 1.57170 .66105 1.51275 .68642 1.45683 .71242 1.40367 32 29 .63666 1.57009 .66147 1.51179 .68685 1.45593 .71285 1.40281 31 30 .63707 1.56969 .66189 1.51084 .68728 1.45501 .71329 1.40195 30 31 .63748 1.56868 .66230 1.50988 .68771 1.45410 .71373 1.40109 29 12 .63789 1.56767 .66272 1.50893 .68814 1.45320 .71417 1.40022 28 33 .63830 1.56667 .66314 1.50797 .68857 1.45229 .71461 1.39936 27 34 .63871 1.56566 .66356 1.50702 .G8900 1.45139 .71505 1.39850 2G 35 .63912 1.56466 .66398 1.60607 .68943 1.45049 .71549 1.39764 25 36 .63953 1.56366 .66440 1.50513 .68985 1.44958 .71593 1.39679 24 37 .63994 1.56265 .66482 1.50417 .69028 1.44868 .71637 1.39593 23 38 .64035 1.56165 .66524 1.50323 .C9071 1.44778 .71681 1.39507 22 39 .64076 1.56065 .66566 1.50228 .C9114 1.44688 .71725 1.39421 21 40 .64117 1.55966 .66608 1.50133 .69157 1.44598 .71769 1.39336 20 41 .W158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.39250 19 42 .64199 1.55766 .66692 1.49944 .C9243 1.44418 .71857 1.89165 18 43 .64240 1.55666 .66734 1.49849 .69286 1.44329 .71901 1.39079 17 44 .64281 1.55567 .66776 1.49755 .69329 1.44239 .71946 1.38994 1G 45 .64322 1.55467 .66818 1.49661 .69372 1.14149 .71990 1.38909 15 46 .64363 1.55368 .66860 1.49566 .69416 1.44060 .72034 1.38824 14 47 .64404 1.55263 .66902 1.49473 .69459 1.43970 .72078 1.38738 13 48 .64446 1.55170 .66944 1.49378 .69503 1.43881 .72122 1.38G53 12 4( .64487 1.55071 .66CCS 1.49284 .69545 1.43793 .72167 1.385C8 11 & .64528 1.54972 .67023 1.49190 .69588 1.43703 .72211 1.38484 10 5 .64560 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.38399 9 5 .64610 1.54774 .67113 1.49003 .69675 1.43525 .72299 1.38314 8 53 .64652 1 .54675 .67115 1.48909 .69718 1.43436 .72344 1.38229 54 .64693 1.54576 .67197 1.48816 .69761 1.43347 .72388 1.38145 6 5o .64734 1.54478 .67239 1.48722 .69804 1.43358 .72432 1.88060 5 5 .64775 1.54379 .67282 1.48629 .69847 1.43169 .72477 1.37976 4 ti .64817 1.54281 .67324 1.48536 .69891 1.43080 .72521 1.37801 3 5* .64858 1.54183 .67CGe 1.48442 .69934 1.42992 .72565 1.37807 2 5 .64899 1.54085 .67409 1.48349 .69977 1.42903 .72610 1.37722 1 (X .64941 1.53986 .67451 1.48256 .70021 1.42815 .72654 1.37638 ( Cotang Tang i Cotang Tang Cotang Tang , Cotang Tang 57' 56 55 II 54 134 NATURAL TANGENTS AND COTANGENTS. 31 5 3 7 3 B" 3 90 1 Tangf Cotang Tang Cotang Tang Cotang Tang Cotang .72654 1.37638 .75355 1.32704 .78129 1.27994 .80978 .23490 00 1 .72699 1.37554 .75401 1.32624 .78175 1.27917 .81027 .23416 59 2 .72743 1.37470 .75447 1.32544 .78222 1.27841 .81075 .23343 58 3 .72788 1.37386 .75492 1.32464 .78269 1.27764 .81123 .23270 57 4 .72832 1.37302 .75538 1.32384 .78316 1.27688 .81171 .23196 56 5 .72877 1.37218 .75584 1.32304 .78363 1.27611 .81220 .23123 55 6 .72921 1.37134 .75629 1.32224 .78410 1.27535 .81268 .23050 54 7 .72966 1.37050 .75675 1.32144 .78457 1.27458 .81316 .22977 53 8 .73010 1.36967 .75721 1.32064 .78504 1.27382 .81364 .22904 52 9 .73055 1.36883 .75767 1.31984 .78551 1.27306 .81413 .22*31 51 10 .73100 1.36800 .75812 1.31904 .78598 1.27230 .81461 .22758 50 11 .73144 1.36716 .75858 1.31825 .78645 1.27153 .81510 .22685 49 12 .73189 1.36633 .75904 1.31745 .78692 1.27077 .81558 .22612 48 13 .73234 1.36549 .75950 1.31666 .78739 1.27001 .81606 .22539 47 14 .73278 1.36466 .75996 1.31586 .78786 1.26925 .81655 .22467 46 15 .73323 1.36383 .76042 1.31507 .78834 1.26849 .81703 .22394 45 16 .73368 1.36300 .76088 1.31427 .78881 1.26774 .81752 .22321 44 17 .73413 1.36217 .76134 1.31348 .78928 1.26698 .81800 .22249 43 18 .73457 1.36134 .76180 1.31269 .78975 1.26622 .81849 .22176 42 19 .73502 1.36051 .76226 1.31190 .79022 1.26546 .81898 .22104 41 20 .73547 1.35968 .76272 1.31110 .79070 1.26471 .81946 .22031 40 21 .73592 1.35885 .76318 1.31031 .79117 1.26395 .81995 .21959 39 22 .73637 1.35802 .76364 1.30952 .79164 1.26319 .82044 .21886 38 23 .73681 1.35719 .76410 1.30873 .79212 1.26244 .82092 .21814 37 24 .73726 1.35637 .76456 1.30795 .79259 1.26169 .82141 .21742 36 25 .73771 1.35554 .76502 1.30716 .79306 1.26093 .82190 .21670 35 26 .73816 1.35472 .76548 1.30637 .79354 1.26018 .82238 .21598 34 27 .73861 1.35389 .76594 1.30558 .79401 1.25943 .82287 .21526 33 28 .73906 1.35307 .76640 1.30480 .79449 1.25867 .82336 .21454 32 29 .73951 1.35224 .70686 1.30401 .79496 1.25792 .82385 .21382 31 30 .73996 1.35142 .76733 1.30323 .79544 1.25717 .82434 .21310 30 31 .74041 1.35060 .76779 1.30244 .79591 1.25642 .82483 .21238 29 32 .74086. 1.34978 .76825 1.30166 .79639 1.25567 .82531 .21166 28 33 .74131 1.34896 .76871 1.30037 .79686 1.25492 .82580 .21094 27 34 .74176 1.34814 .76918 1.30009 .79734 1.25417 .82629 .21023 26 35 .74221 1.34732 .76964 1.29931 .79781 1.25343 .82678 .20951 25 36 .74267 1.34650 .77010 1.29853 .79829 1.25268 .82727 .20879 24 37 .74312 1.34568 .77057 1.29775 .79877 1.25193 .82776 .20808 23 38 .74357 1.34487 .77103 1.29696 .79924 1.25118 .82825 .20736 22 39 .74402 1.34405 .77149 1.29618 .79972 1.25044 .82874 .20665 21 40 .74447 1.34323 .77196 1.29541 .80020 1.24969 .82923 .20593 20 41 .74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 .20522 19 42 .74538 1.34160 .77289 1.29385 .80115 1.24820 .83022 .20451 18 43 .74583 1.34079 .77335 1.29307 .80163 1.24746 .83071 .20379 17 44 .74628 1.33998 .77382 1.29229 .80211 1.24672 .83120 .20308 16 45 .74674 1.33916 .77428 1.29152 .80258 1.24597 .83169 .20237 15 46 .74719 1.33835 .77475 1.29074 .80306 1.24523 .83218 .20166 14 47 .74764 1.33754 .77521 1.28997 .80354 1.24449 .83268 .20095 13 48 .74810 1.33C73 .77568 1.28919 .80402 1.24375 .83317 .20024 12 49 .74855 1.33592 .77615 1.28842 .80450 1.24301 .83366 .19953 11 50 .74900 1.33511 .77661 1.28764 .80498 1.24227 .83415 .19882 10 51 .74946 1.33430 .77708 1.28687 .80546 1.24153 .83465 .19811 9 52 .74991 1.33349 .77754 1.28610 .80594 1.24079 .83514 .19WO 8 53 .75037 1.33268 .77801 1.28533 .80642 1.24005 .82564 .19669 7 54 .75082 1.33187 .77848 1.28456 .80690 1.23931 .83613 .19599 6 55 .75128 1.33107 .77895 1.28379 .80738 1.23858 .83662 .19528 5 56 .75173 1.33026 .77941 1.28302 .80786 1.23784 .83712 .19457 4 57 .75219 1.32946 .77988 1.28225 .80834 1.23710 .83761 .19387 8 58 .75264 1.32865 .78035 1.28148 .80882 1.23637 .83811 .19316 2 59 .75310 1.32785 .78082 1.28071 ,80930 1.23563 .83860 .19246 1 60 .75355 1.32704 .78129 1.27994 .80978 1.23490 .83910 .19175 t Cotang Tang Cotang Tang Cotang Tang Cotang Tang t I 3 C 2" 1 ! j 0* NATURAL TANGENTS AND COTANGENTS. 135 40 Tang .83910 .84009 .84108 .84158 .84258 .84307 .84357 .84407 .84457 .84507 .84556 .84606 .84656 .84706 .84756 .84806 .84056 .85057 .85107 .85157 .85207 .85257 .85358 .85408 .85458 .85660 .85761 .85811 .85862 .85963 .86014 .86115 .86216 .86419 .86470 .86572 .86674 .86725 .86776 .86878 Cotang Cotang .19175 .19105 .19035 .18964 .18824 .18754 .18614 .18544 .18474 .18404 .18264 .18194 .18125 .18055 .17986 .17916 .17846 .17777 .17708 .17638 .17500 .17430 .17361 .17292 .17154 .17085 .17016 .16947 .16878 .16809 .16741 .16672 .16535 .16466 .16329 .16192 .16124 .16056 .15987 .15919 .15851 .15783 .15715 .15647 .15579 .15511 .15443 .15375 .15308 .15240 .15172 .15104 .15037 Tang 49 41 Tang .87031 .87082 .87133 .87184 .87287 .87338 .87389 .87441 .87492 .87543 .87595 .87646 .87749 .87801 .87904 .87955 .88007 .88214 .88265 .88421 .88524 .88576 .88784 .89410 .89463 .89515 .89777 .89935 .89988 .90040 Cotang Cotang 1.15037 1.14969 1485* 14767 .14565 .14498 .14430 .14363 .14296 .14229 .14162 .14095 .13761 .13627 .13561 .13494 .13428 .13361 .13096 .13029 .12963 .12897 .12831 .12765 .12633 .12567 .12501 .12435 .12303 .12172 .12106 .12041 .11975 .11909 .11844 11778 .11713 .11648 .11582 .11517 .11452 .11387 .11321 .11256 .11191 .11126 .11061 Tang Tang .90040 .90146 .90199 .90251 .90304 .90357 .90410 .90516 .90674 .90727 .90781 .90940 .91046 .91099 .91153 .91206 .91259 .91313 .91419 .91473 .91580 .91687 .91740 .91794 .91847 .91901 .91955 .92062 .92116 .92170 .92224 .92277 .92331 .92493 .92547 .92601 .92655 .92709 .92817 .92872 .93143 .93197 .93252 Cotang Cotang 1.11061 1 .10802 .10737 .10672 .10607 .10543 .10478 .10414 .10349 .10285 .10156 .10091 .10027 .09770 .09706 .09578 .09514 .09450 .09195 .09131 .09067 .08876 .C3813 .C8749 .03559 .084% .CS369 .08306 .08243 .C3179 .08116 .08053 .07090 .07927 .07864 .07801 .07738 .07676 .07613 .07550 .07487 .07425 .07209 .07237 Tang 48 47 43 Tang .93415 .93742 .93797 .94016 .94071 .94125 .94180 .94345 .94400 .94455 .94510 .94565 .94620 .94676 .94731 .94786 .94841 .94952 .95007 .95118 .95173 .95229 .95451 .95506 .95562 .95618 .95673 .95729 .95785 .95841 .95897 .95952 .96008 .96064 .96120 .96176 .96400 .96457 .96513 .96569 Cotang Cotang .07237 .07174 .07112 .07049 .06738 .06676 .06613 .06551 .06489 .06427 .06179 .06117 .06056 .05994 .05870 .05809 .05747 .05685 .05624 .05562 .05501 .05439 .05378 .05317 .05255 .05194 .05133 .05072 .05010 .04949 .04766 .04705 .04644 .04583 .04522 .04461 .04401 .04310 .04279 .04218 .04158 .04097 .04036 .03976 .03915 .03855 .03794 .03734 03553 Tang 46 136 NATURAL TANGENTS AND COTANGENTS. 440 440 44. Tang Cotang Tang Cotang Tang Cotang .96569 .03553 60 20 .97700 .02355 40 40 .98843 .01170 20 1 .96625 .03493 59 21 ,97756 .02295 '39 41 .98901 .01112 19 2 .96681 .03433 58 22 .97813 .02236 38 42 .98958 .01053 18 8 .96738 .03372 57 23 .97870 .02176 87 43 .99016 .00994 17 4 .9679-1 .03312 56 24 .97927 .02117 36 44 .99073 .00935 16 5 .96850 .03252 55 25 .97984 .02057 35 45 .99131 .00876 15 6 .96907 .03192 54 26 .98041 .01998 34 46 .99189 .00818 14 .96963 .0313* 53 27 .98098 .01939 33 47 .99247 .00759 13 8 .97020 .03072 52 28 .98155 .01879 32 48 .99304 .00701 12 9 .97076 .03012 51 29 .98213 .01820 81 49 .99362 .00642 11 10 .97133 .02952 50 30 .98270 .01761 30 50 .99420 .00583 10 11 .97189 .02892 49 31 .98327 .01702 20 51 .99478 .00525 9 12 .97246 .02832 48 32 .98384 .01642 28 52 .99536 .00467 8 13 .97302 .02772 47 .98441 .01583 27 53 .99594 .00408 7 14 .97359 .02713 46 34 .98499 .01524 26 54 .99652 .00350 6 15 .97416 .02653 45 .35 .98556 .01465 25 55 .99710 .00291 5 16 .97472 1.02593 44 36 .98613 .01406 24 56 .99768 .00233 4 17 .97529 1.02533 43 37 .98671 .01347 23 57 .99826 .00175 3 18 .97586 1.02474 42 38 .98728 .01288 22 58 .99884 .00116 2 19 90 .97643 .97700 1.02414 1.02355 41 40 39 40 .98786 .98843 .01229 .01170 21 20 59 60 .99942 1.00000 .00058 .00000 1 Cotang Tang ' Cotang Tang t Cotang Tang / 45 45- 45- /&' OF THE^^^. {TJ'HIVEKSITTJ X^JFogl^: UNIVERSITY OF CALIFORNIA LIBRARY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW 7 11 APR 16 1946 30m 6/14 YB 11067