THE KAILKOAD TAPER THE THEORY AND APPLICATION OF A COMPOUND TRANSITION CURVE BASED UPON THIRTY- FOOT CHORDS BY LEE PERKINS FORMERLY ASSISTANT ENGINEER, SOUTHERN PACIFIC LINES IN ARIZONA AND MEXICO MEMBER AMERICAN RAILWAY ENGINEERING ASSOCIATION FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITE] 1915 SV COPYRIGHT, 1915 BY LEE PERKINS Stanbope ipress F. H.GILSON COMPANY BOSTON, U.S.A. PREFACE A VARIETY of transition curves are used on the different railroads, most of them fully explained by published hand- books. A number of years ago Mr. Wm. Hood, Chief Engineer of the Southern Pacific, devised a compound transition curve of unusual simplicity, and developed several tables for this curve. For a number of years this taper has been used on many roads with somewhat in- complete data, usually in the form of blue prints. This handbook has been prepared as an extension and elaboration of Mr. Hood's earlier work, in an effort to simplify the use of the taper, and to lessen the work of the man in the field. With this in view, to a brief explanation o^ the taper has been added the solution of all the usual curve problems as applied to tapered curves. Since the various transition curves differ only in their method of development and their numerical values, differing not at all in the method of inserting them between tangent and curve, or between curves, these curve problems for tapered curves are available for use with the tables of any transition curve. The transition curve herein developed is, theoretically, a series of compound curves, each branch of the curve having a thirty-foot chord (in the metric system a ten-meter chord), the last branch compounding with the central circular curve, the degree of curvature of the various branches and ending with the main curve increasing in direct arithmetical progression. The various tables are to enable one to run in this series of compound curves with iii 616268 IV PREFACE the ease of running in a simple curve, and from a single setting of the transit. The primary taper tables give the various taper func- tions for every thirty feet of taper. That all the tapers may be available for use with all degrees and fractional degrees of main curve, avoiding the use of inappropriate taper lengths, additional tables have been furnished giving the taper functions for main curves of intermediate degree, the values having been trigonometrically interpolated. If further interpolation be necessary it may be linear between the values given in these tables of intermediate values. In addition, fairly simple equations are given for all the angular functions, frequently useful in setting intermediate points on the taper. The radius of main curve is assumed based on a fifty- foot chord throughout, mainly because the roads that have always used this taper have always used that radius. However, this usage is in accordance with the present tendency, obviating the necessity of using long chain lengths in setting fifty-foot points, which are set much more frequently than not. All the functions of the taper for both field and drawing- room work are given for eleven different tapers of increas- ing taper increment, which with their forty to eight}^ lengths comprise some seven hundred tapers. Three of these eleven, however, will be found to answer most needs except in mountainous country. The taper numbers used are those which have long been employed for these curves, and are retained for the benefit of present records, though a more logical system of number- ing, such as the degree of curvature increment per thirty feet of taper, might seem desirable, and is suggested for use where this taper is hereafter adopted. The problem of tapering old circular curves is essentially a different one from locating new lines, and is handled separately. The insertion of tapers between the branches PREFACE * V of compound curves should receive more attention than it has generally had. A fairly simple solution of this problem is given. The tapers have been calculated for metric curves for the use of the constantly increasing number of American engineers in Mexico and South America, where this taper is already extensively used. Acknowledgments are due to the following, for the use of tables reprinted from their respective books: Professor Daniel Carhart, Logarithms of Numbers and Trigonometric and Miscellaneous Formulas; Professor J. C. Nagle, Logarithmic Sines and Cosines, Logarithmic Tangents and Cotangents, Functions of a One Degree Curve, Natural Secants and Cosecants, and Metric Con- version Table; Mr. Walter Loring Webb, Logarithmic Versed Sines and External Secants, Natural Sines, Cosines, Tangents and Cotangents, and Natural Versed Sines and External Secants; and the American Railway En- gineering Association, for the Table of Theoretical Super- elevation of Rails, prepared by its Committee on Track and published in the Manual of that Association. LEE PERKINS. SILVER CITY, NEW MEXICO, March, 1915. CONTENTS CHAPTER I INTRODUCTION PAGE 1. Purpose of the Taper 1 DEVELOPMENT OF THE TAPER 2. Series of Compound Curves 1 3. Central Angle, o 3 4. Co-ordinates x and y 3 5. Deflection Angles 3 6. Long Chords 3 7. t and d, giving location of B.C 6 CHAPTER II LOCATION PROBLEMS SIMPLE CURVES 8. Semi-Tangent, T 7 9. External, E 11 10. Tangent Offset and Distance along the Tangent to any Point on a Curve 11 11. To Pass a Curve through a Given Point, uniting two Given Tangents 11 12. To Pass a Curve of Given Radius through a Given Point, and uniting with a Given Tangent 15 13. To Pass a Curve through two Given Points and uniting with a Given Tangent 16 14. To fix upon a Tangent at the End of a Curve. 18 15. To Locate a Tangent to two Located Curves 20 16. To Relocate a Curve making the E.G. fall at a Given Point in a Given Parallel Tangent f . 24 17. The Intersection of a Line with a Curve 26 COMPOUND CURVES 18. Partial Taper between the Branches of the Compound Curve 27 vii Vlll CONTENTS PAGE 19. Taper Functions of the Partial Taper 28 20. To find the Semi-Tangents, both Radii and both Central Angles given 31 21. To Compound a Simple Curve so the E.G. shall fall in a Given Parallel Tangent, both Radii given 33 22. To Compound a Simple Curve so the E.G. shall fall in a new Located Tangent, both Radii, or both Central Angles given 34 23. To find the Second Semi-Tangent and the Central Angles, one Semi-Tangent, the Intersection Angle and both Radii given 36 CURVES BEYOND THE FROG 24. Taper beginning at the Heel of Frog 37 CHAPTER III FIELD WORK 25. Selection of the Taper 40 26. Laying out a Tapered Curve 41 OBSTRUCTIONS ON THE TAPER 27. Curve run from Main Curve Produced 42 28. Curve run from a Point beyond the C.C 44 29. Taper Points set by Offset. ..', 45 30. Taper Points set from Point on Semi-Tangent 45 31. Curve Notes for the Field Book 46 CHAPTER IV OFFICE WORK 32. Plotting a Tapered Curve 47 33. Paper Location with Tapers 47 34. Curve Compensation with Tapers 47 35. Right of Way Descriptions with Tapers 48 CHAPTER V TAPERING OLD TRACK 36. Condition of Track 49 37. Operating Conditions 50 CONTENTS IX METHODS OF INSERTING TAPERS SIMPLE CURVES PAGE 38. Shifting the Curve 50 39. Shifting the Tangents 52 40. Shifting Curve and Tangents 53 41. Increasing Degree of Curve 53 42. Increasing Degree and Shifting Curve 54 43. Maintaining Length of Line Unchanged 55 44. Increasing Degree and Shifting Curve and Tangents 57 45. Combination of Methods 57 46. Increasing Degree of Portion of Curve 59 COMPOUND CURVES 47. Similarity of Procedure 59 48. Shifting the Curves 59 49. Shifting Curves and Tangents 60 50. Shifting Lighter Branch and Increasing Degree of Sharper Branch 60 51. Field Work with Compound Curves 61 TABLES Radii of Curves and their Logarithms 63 Notation used in the Taper Tables 69 Important Equations abstracted from the Text 69 Taper , A = 1 71 1 ,A = 'J 74 U, A = f 78 2 , A = 1 82 21, A = 11 86 2i A = U - 90 2f, A = 2 94 3 , A = 2 97 3i, A = 3 100 3, A = 4 103 3,f A =5 105 Comparative Values of I, d, and o V 108 Maintaining Length of Line Unchanged 115 Logarithms of Numbers 117 Logarithmic Sines and Cosines 135 Logarithmic Tangents and Cotangents 150 Logarithmic Versed Sines and External Secants 165 X CONTENTS PAGE Functions of a One Degree Curve 210 Natural Sines, Cosines, Tangents and Cotangents 240 Natural Secants and Cosecants 263 Natural Versed Sines and External Secants 276 METRIC TABLES Radii of Curves and their Logarithms 299 Notation used in the Taper Tables 305 Important Equations abstracted from the Text 305 Taper , A = 307 1 , A = i 310 U, A = f 313 2 , A = 1 317 2i A = H. 321 2, A = H 325 2f, A = 2 329 3 , A = 2^ 333 3J, A = 3 337 3, A = 4 340 3f , A = 5 342 Comparative Values of I, d, and o 344 Mutual Conversion of Feet and Inches into Meters and Centi- meters 351 Trigonometric and Miscellaneous Formulas 352 Super-elevation of Rails on Curves , 355 THE RAILROAD TAPER CHAPTER I INTRODUCTION PURPOSE OF THE TAPER 1. Tapered curves have gained such wide us^age that it is needless to dwell at length on their advantages. Where circular curves alone are used, the transition from tangent to curve is sudden, giving a decided lurch to the cars, unpleasant to passengers and damaging alike to rolling stock and track. Running the super-elevation back on the tangent does not obviate this, and it can be remedied only by increasing the curvature gradually, in- creasing the super-elevation of rails at the same time. This the tapered curve accomplishes in a satisfactory and practical manner, much as it has always been [done by track foremen, who lined in an approximate taper by eye. The proper use of the taper, however, insures a , uniform curve beyond the taper, a result not easily obtained by the trackman unaided. DEVELOPMENT OF THE TAPER 2. The railroad taper consists, theoretically, of a series of short curves, each of thirty-foot chord lengttj, and of uniformly increasing curvature, compounding at:; the end with the central main curve. The various tapirs differ merely in the curvature increment, and, therefore, in the length needed to lead up to the curvature of the main curve. Fig. 1 illustrates the first one hundred and twenty 1 THK RAILROAD TAPER DEVELOPMENT OF THE TAPER 3 feet of Taper 2, compounding with the main five-degree curve. 3. The central angle (o) of the taper is, of course, the summation of the central angles of its parts. 4. In Fig. 2 is shown the same curve represented by its thirty-foot chords, showing the method of obtaining the x and y co-ordinates of the chord-points referred to the initial tangent and the B.C. (beginning of curve) of the taper. The angle k is the azimuth of the various chords, referred to the initial tangent, and the x and y co-ordinates are the departures and latitudes of the chord-points. These co-ordinates are given in the tables. 5. From x and y were calculated for the tables the deflec- tion angles (/) from the initial tangent at the B.C. to the various chord-points, by the equation tan/ = I- (1) y The backsight deflection (6) from tangent at any chord- point to the B.C. equals the central angle of the taper up to that point minus the deflection at the B.C. to that chord-point, and is given in the tables. As shown in Fig. 3, the deflection from tangent at any chord-point to any other chord-point equals the azimuth of the long chord between the chord-points, minus the central angle of the taper up to the first chord-point. x" x' This azimuth is the angle whose tangent is -77 -,- y y As shown in Fig. 4, the backsight deflection from tangent at any chord-point to any other chord-point is and is given in the tables. 6. The long chord (Ic) from the B.C. to any chord- point, given in the tables, is Ic = -^-,- (2) cos/ THE RAILROAD TAPER FlQ. 3. FIG. 4. FIG. 6. (5) THE RAILROAD TAPER Similarly, as in Fig. 5, the long chord between any other chord-points is IP'-" - y y o\ cos (e' +/'-") 7. In order to make room for the taper, the simple curve may be considered as offset from its natural position a distance from the tangent that will be called d, the taper beginning at a point back of this offset position of the B.C. of the simple curve a distance along the tangent that will be called t. This tangent distance, t, is very nearly half the length of the taper, its exact value, from Fig. 6, being t = BD - IE = y R sin o, (4) where R is the radius of the main curve, and o the central angle of the taper. The value of the circular curve offset, d, from Fig. 6, is d = DE - HI = x R vers o. (5) The values of t and d are given in the tables. These two equations, 4 and 5, are the fundamental equations of the taper, locating the center of curvature of the main curve with reference to the tangents and the beginning and end of curve. The equations are identical in form for all transition curves, and similar values under different letter designations are invariably given in their tables. On these two distances, t and d y are based all the solutions of curve problems in the succeeding chapters. CHAPTER II LOCATION PROBLEMS 8. To Find the Semi-tangents of a Tapered Curve. Given R and / to find T. FIG. 7. CASE I. Tapered at both ends with the same taper. From Fig. 7, T = AD + AB = t + OA tan J 7 = + (# + d) tan 1 7. (6) 7 8 THE RAILROAD TAPER In the tables is given a value D = R + d, so the above may be written T = t + D tan } 7. (7) Example. Find the semi-tangent of a 5 00' curve, tapered with Tap. 2, total angle of curve 35 04'. By Eqn. 7 log D 5 Tap 2 3 . 05949 log tan 17 32' 9.49960 2.55909 362.32 t = 59.99 T = 422.31 CASE II. Tapered at one end only. It may occasionally be necessary to taper one end only of a curve, or, considering the parallel offset tangent, to treat the curve as tapered at one end only. FIG. 8. In this case the two semi-tangents will be of different lengths, the one for the tapered end being the longer, when I is less than 90, by an amount less than t. From Fig. 8. MN = MK = dcscI, T t = t + OAtauI - MN = t + D tan J / - d esc I, (8) T n = R tan J I + d esc 7. (9) LOCATION PROBLEMS 9 These equations do not change for angles greater than 90, the algebraic sign of the last term remaining un- changed. Example. Find the semi-tangents of a 7 00' curve, tapered at the B.C. with Tap. 2J, no taper at the E.C. t total angle of curve 54 22'. By Eqn. 8 esc 54 22' 1 . 23037 d 0.95 615185 1 007333 1.0688515 2| 2.91360 Iogtan27ll / 9.71059 2.62419 420.91 + t 54.98 475.89 1.07 B.C. to P. I. 474.82 By Eqn. 9 log R r 2.91309 log tan 27 11' 9.71059 2.62368 420.42 + 1.07 P.I. to E.G. 421.49 CASE III. Tapered at both ends, with different tapers. From Fig. 9, MN = MK = d l cseI, BM = dz esc I, BJ = BN = MN - BM = (di d%) esc /, Ti = ^ + A tan J / - (di - 4)' esc 7. (10) T 2 = t* + I> 2 tan J7 - (d z - di) esc/. (11) These equations do not change for angles greater than 90, the algebraic sign of the last term remaining un- changed. 10 THE RAILROAD TAPER Example. Find the semi-tangents of a 10 00' curve, tapered at the B.C. with Tap. 2i, and at the E.G. with Tap. 3, total angle of curve 113 42'. -t r - By Eqn. 10 FIG. 9. esc 113 42' (esc 66 18') 1.09211 4 - d, = 2.84 - 0.98 = 1.86 655266 873688 1 09211 log DIO Tap. 2i log tan 56 51 By Eqn. 11 B.C. to P.I. log DIO Tap. 3 log tan 56 51 2.03 2.76041 0.18500 2.94541 881.88 84.88 966.76 2.03 964.73 2.75900 0.18500 2.94400 LOCATION PROBLEMS 11 879.02 +fc 44.97 923.99 + 2.03 P. I. to E.G. 926.02 9. Given M and J, to find the External, E. CASE I. Tapered at both ends with the same taper. From Fig. 7, E = OB - R = D sec j 7 - #, (12) or ' E = (72 + d) sec J 7 - # - d + d = (Jf2 + d) (sec J 7 - 1) + d = D exsec J 7 + d. (13) Example. Find the external of a 5 00' curve, Tap. 2, total angle 27 12'. By Eqn. 12 or 13, E = 33.88. CASE II. Tapered at one end only, or at both ends, with different tapers. From Fig. 10, E = VOA 2 + AB 2 -R = VD* + (T - o 2 - R, (14) or E = D sec tan- 1 ^^ - R, (15) T t T t where tan" 1 ^ is the angle whose tangent is j? Example. Find the external to the curve given in the example under equations 8 and 9. By Eqn. 14 or 15, E = 102.24. 10. To Find the Tangent Offset and Tangent Distance to any Point on the Curve. From Fig. 11, PD = AC + d = RversI + d, (16) ED = BD + t = R sin / + t. (17) 12 THE RAILROAD TAPER FIG. 10. FIG. 11. LOCATION PROBLEMS 13 Or from the Table of Functions of a 1 curve, AC is the mid-ordinate for a central angle equal to 2 /, and BD is half the long chord for a similar central angle. 11. Given a Point, P, referred to the JP.J., to find the Radius of a Curve passing through F uniting the Given Tangents. Since the solution of this problem depends upon the taper chosen and its value of d, and the choice of taper depends largely upon the degree of curve, a trial solution for an untapered curve is first made. From Fig. 12, FIG. 12. OB . sin y = sin a R sec \ I . -TT- sina sin a cosf R = PO sin a sinx PB. (18) (19) 14 THE RAILROAD TAPER Since this problem occurs mainly in location, and it is necessary merely to have the curve pass close to the determining point without using a curve of fractional degree, to allow room for the taper a curve of next higher even degree is chosen, and when desirable it may be determined as follows how close the chosen curve will pass to the given point. In Fig. 13, P is the point through which the curve should pass, P r a point on the chosen curve radially opposite. PP'=QP-R = (OA - AD) 2 + (AB - CB) 2 - R = V(D-PB sin 6) 2 + (D tan J / - PB cos 6) 2 - R. (20) FIG. 13. Example. In Fig. 13, let PB = 194.3, b = 42 24', ^4 22' r I = 54 22'. Then o = 90 - 2 - 42 24' = 20 25'. By Eqn. 18, y = 23 05' x = y -a= 2 40', By Eqn. 19, R = 1456.8. 3 56' curve. LOCATION PROBLEMS 15 Deciding now upon a 4 00' curve with Tap. 1, to determine how close this curve will pass to point P: By Eqn. 20, (D - PB sin 6) 2 =? 1698100 (D tan % I-PB cos 6) 2 = 351675 2049775 V2049775 = 1431.7 - R 1432.5 PP' = ~ - 0.8 The negative result shows that P is inside the curve 0.8 ft., which on most location is sufficiently close. 12. Find the B.C. of a Curve of Given Radius that will pass through a Given Point referred to a Given Tangent. \ FIG. 14. From Fig. 14, AB = R sin vers" 1 BP-d D (21) f R times the sine of the angle whose versine is - ~ J or R 2 = (D- BP)* AB = - (D- BP) 2 . (22) 16 THE RAILROAD TAPER Example. Given BP = 82 ft., M = 6 00', Tap. 2, then D = 956.41, R = 955.04. r>p 7 By Eqn. 21, vers- 1 -~ - = 22 45J', 3G9.45. 13. Given Two Points referred to a Tangent, to find the Radius and B.C. of a Curve passing through them and joining the Given Tangent with a Given Taper. Since the solution of this problem depends upon the taper chosen and its value of d, the choice of which depends largely upon the degree of curve, a trial solution for an untapered curve is first made. FIG. 15. In Fig. 15, given points P and S, referred to the tangent by lengths DP and DS, and the angle a, to find the radius OA and distance AD. By geometry AD 2 = DP X DS, AD = VDP X DS. (23) DB and BP may be obtained from a solution of the tri- angle BDP. LOCATION PROBLEMS 17 D 1 = = AB 2 + (R - BP) 2 = AB 2 + R 2 AB 2 + BP 2 BP 2 , or 2BP AB sin 2 tan" 1 -r (24) (25) A curve of next higher even degree is now chosen with an appropriate taper, passed through S as in paragraph 12, and the distance of P from the curve determined as follows. FIG. 16. From Fig. 16, A'C = R sin vers" 1 or - (D - CS) 2 ', PP' = OP-R = VOE 2 + EP 2 -R ' or PP' = = V(D - BP) 2 + A'B 2 - R, D-BP cos tan~ A'B D-BP -R. (26) (27) (28) (29) 18 THE RAILROAD TAPER Example. DP = 94 ft., PS = 127 ft., angle a = 16 12'. By Eqn. 23, AD = 144.1. FromABDP, BP = 26.2 and DB = 90.3; then AB = AD + DB = 234 . 4. 7?P By Eqn. 25, tan- 1 =% = 6 23', and # = 1060.7 5 24' curve. Deciding now upon a 5 30' curve, Tap. 2, through S: From A DSC, CS = 61.66 - d 1.05 CS-d= 60.61 and DC = 212.2 -DB 90.3 #C = 121.9. By Eqn. 26, vei&- lCS ~ d = 19 38' tt and A'C = 350.06 -BC 121.9 A'B = 228.16. By Eqn. 29, tan' 1 = 12 39', and PP' = 0.14. From the positive value of PP', P lies outside the curve assumed. If the curve had not passed sufficiently close to P, a slightly sharper curve would pass further out at P', or if the curve had passed outside of P, a slightly lighter curve would pass closer to P. 14. To Fix upon a Tangent at the End of a Curve. In the previous problem of passing the curve through two determining points and tangent to the back-located tangent, it is not practical to first locate the tangent LOCATION PROBLEMS 19 ahead and make an intersection. The curve must be run in beyond the determining points, then a meander line run to the next controlling point on the tangent ahead and the tangent located as follows: When the curve has reached a point A, Fig. 17, from which a view may be had of the tangent desired, or of some controlling point, B, on it, through which it is desired to run the tangent because of properly supporting ground, a flag may be sent to C, opposite B, (the uncer- FIG. 17. tainty of right angle direction will affect this very little), and a distance therefrom equal to d of the taper. With improvised targets of some nature, or double points and angle between, a fairly accurate stadia shot may be obtained, and with the angle from tangent at A to C, the distance BL and the central angle of the rest of the curve may readily be calculated. From Fig. 17, AG = 2R sin a, OK = R cos a, CF = VCG X AC, OK (30) Rcosa AC Rsina 20 THE RAILROAD TAPER R tan m = R (3D V(AC -2Rsma)AC b = a + m n. (32) Given b, AD and Z)L may be calculated by equations 9 and 8. Then LB = CF -t = V(AC - 2 # sin a) AC - t. (33) Example. Assume a 5 30' curve, Tap. 2, to have been run to point A. Distance AC by stadia is 610 ft., and angle a is 6 10'. By Eqn. 30, n = 64 19'. By Eqn. 31, m = 66 01'. By Eqn. 32, a 6 10' + ra 66 01' - n 64 19' 6=7 52'. By Eqn. 33, LB = 417.9. 15. To Connect two Located Curves by a Tangent. In the preceding problem, if the next controlling point had been on a curve, instead of a tangent, it would often be necessary to locate this next curve first and then con- nect the two curves by a tangent. CASE I. Curves of contrary flexure. Let AB and CD, Fig. 18, be the two located curves, to locate tangent LM fitting the two curves with given tapers, the curves being offset from the tangent di and d 2 , respectively, to provide room for the tapers. From some point E on tangent at B, run one or more courses, EF, to some point F on tangent at D. Calculate the latitude and departures of N and with reference to tangent at B as base and radius BN as meridian. The difference of the latitudes equals OG, the difference of the departures equals NG. LOCATION PROBLEMS 21 Then, NO = VOG 2 + NG 2 , OG + A + #2 + cos c a = 90 - e - c, / _ g, = a _ 6, tt- f n/\*A LM = ST - (t, + = NO sine- (34) (35) (36) (37) (38) (39) 22 THE RAILROAD TAPER Example. Given a 5 30' curve AB and a 4 00' curve CD, to locate tangent connecting them, using Taper 2 on the 5 30' curve and Taper l on the 4 00' curve. At B turn tangent to curve and set point E, and at D turn tangent and set F. Assume BE = 100 ft., angle / = 9 11', EF = 384.0 ft., angle g = 6 38', FD = 100 ft. Then a traverse from N to is as follows: Course. Length. Azimuth. Lat. Dep. NB . 1041.8 180 00' -1041.8 BE.. EF.. 100.0 384.0 90 00' 80 49' + 61.3 + 100.0 4-379.1 - FD DO 100.0 1432 5 87 27' 177 27' + 4.4 -1431.1 + 99.9 + 63.7 NO 2491.5 165 03' -2407.2 +642.7 From this traverse NO = 2491.5, . NG = 642.7, GO = 2407.2, e = 75 03'. By Eqn. 36, c = 6 23'. By Eqn. 37, a = 8 34'. By Eqn. 38, 6 = 6 01'. By Eqn. 39, LM = 144.5. By Eqn. 9, BH = 85.08. By Eqn. 8, HL = 138.54. By Eqn. 8, MK = 133.54. By Eqn. 9, KD = 82.05. If in this solution LM had been too short, or even negative, one of the curves would have to be shifted or sharpened, or the tapers shortened. CASE II. Curves of similar flexure. Determine NG and GO as in Case I. LOCATION PROBLEMS A FIG. 19. Then from Fig. 19, NO = VOG 2 + NG 2 , OG i^e = m , cos c = a = e + c - 90, 6=/ + <7-a, LM = NOsmc- (ti 24 THE RAILROAD TAPER Example. Assume the same conditions as in the example under Case I, angle g merely being in the opposite direction. Then a traverse from N to is as follows: Course. Length. Azimuth. Lat. Dep. NB 1041 8 180 00' -1041.8 BE 100 90 00' + 100 EF.. 384.0 80 49' + 61.3 +379.1 FD 100 74 ir + 27 3 + 96 2 DO 1432 5 344 IT + 1378 3 -390 4 NO 485.4 23 30' + 425.1 +184.9 From this traverse By Eqn. 36a, By Eqn. 37a, By Eqn. 38a, By Eqn. 39, NO = 485. 4, e = 90 - 23 30' = 66 30'. c = 3629'. a = 12 59'. 6=2 50'. LM = 156.1. 16. To Relocate a Curve so the E.C. shall fall at a Given Point in a Given Parallel Tangent. In Fig. 20 let ABC be the located curve, tapers omitted for clearness, and DEF the required curve ending at F. CG and GF are known. By a solution of triangle KM I, KF becomes known. The value of R 2 now depends solely upon the taper chosen and its value of t% and d 2 , and may best be determined by trial, using the tables of functions of a one-degree curve, assuming at first a value of fe somewhat larger, or smaller as the case may be, than t\. It will usually be sufficient for the semi-tangent of the new curve approximately to equal KF. LOCATION PROBLEMS 25 Example. Assume curve ABC to be a 6 00' curve, Taper 2, total angle 36 14'. CG = 7.1, GF = 42.4. K I FIG. 20. From A KM I, KM = 9.7 and KI = 12.0. By Eqn. 7, 1C = 387.88 + KM 9.7 + GF 42.4 KF = 440.0. From Tables, 1 curve TWW = 1874.7. If the new curve were untapered, 1874.7 M 4.26. 440.0 If Taper 2 be used on the new curve, then M 1847.7 M \l Apprbx. T. 4 45' 394.7 56.2 450.9 50' 387.9 57.5 445.4 55' 381.3 58.7 440.0 5 00' 374.9 60.0 434.9 26 THE RAILROAD TAPER Assuming a 4 55' curve, Taper 2, then by Eqn. 7, T 2 = 440.29. In this particular case it is unnecessary to interpolate to obtain a more exact value for the semi-tangents, 0.29 ft. being sufficiently close. Linear interpolation between the intermediate values given in the tables would be quite accurate when necessary. 17. Intersection of a Line with a Curve. In Fig. 21, given R, d, AB and angle 6, to find BP and angle a. tan n = FIG. 21. D AB' c = 180 - n -6, ' = n + 6' - 180, sin n OB . sin e = ^ sin c 1 LOCATION PROBLEMS 27 (42) n = (e-c) (43) sine =/+i tan J /i - * esc /i. (62) By Eqn. 9, DA = Ri tan \ /i + di esc I lf (63) By Eqn. 10 AG = t e + *D C tan \ 7 2 - (d c - eft,) esc /,. (64) By Eqn. 11, GH = t 2 + D 2 tan \ I 2 - (d 2 - d e ) esc 7 2 , (65) DG = DA + AG. (66) Solving triangle DNG gives values for DN and AfG. Then SN = SD + DN, (67) NH = NG + GH. (68) Where it is desirable to obtain the semi-tangents of the compound curve without first finding the taper functions of the partial taper, DG may be drawn tangent to MI at F and equations 62 to 68 become as follows: SD = ti + Di tan i c - di esc c, (62a) DF = #1 tan i c + di esc c, (63a) FG = fA- tan } - (d r - A) esc g, (64a) GH = k + D 2 tan J g - (4 - *) esc g, (65a) DG = DF + TO, (66a) >S^V = ^D + D^V, (67) NH = NG + GH. (68) Example. Given 7 = 114 24', c = 34 40', g = 7944', Mi = 3 00', M-2. = 8 00 r . Use Taper 1 between tangent * D c = B, + d c . t D r = ft + dr. 32 THE RAILROAD TAPER and 3 00' curve, Taper 2 between 3 00' and 8 00' curves, and Taper 2J between the 8 00' curve and tangent. Find the semi-tangents. 30 By Eqn. 48, l c = y (8 - 3 - 1) = 120. By Eqn. 49, and the tables, d r = 0.79. By Eqn. 50, o c = 9 (8 + 3) (8 - 3 - 1) = 396', = 6 36'. 9 X 3 (8 - 3 - 1) By Eqn. 51a, a = - ^ = 108' = 1 48'. By Eqn. 52, t c = 37.47. By Eqn. 53, d c = 1.38. By Eqn. 54, y c = 119.80. By Eqn. 55, x c = 6.13. By Eqn. 57, f e = 175J' = 2 55|'. By Eqn. 58, 6s = 3 40'. By Eqn. 59, lc c = 119.96. By Eqn. 60, /i = 32 52 ; . By Eqn. 61, / 2 = 81 32'. By Eqn. 62, SD = 637.26. By Eqn. 63, DA = 564.59. By Eqn. 64, AG = 656.32. By Eqn. 65, GH = 683.74. By Eqn. 66, DG = 1220.91. FromADGW, NG = 727.57, and DN =1326. 06. By Eqn. 67, SN = 1963.32. By Eqn. 68, NH = 1411 .31. LOCATION PROBLEMS 33 21. Given a Located Curve, to Compound with Curve of Given Radius so as to End on a Given Parallel Tangent, using Three Tapers. In Fig. 23, given located curve SAFV of radius Ri, parallel tangent NH distant p from the original tangent WV, Rz, di, d 2) d r , ti and fe, to find I lt 7 2 and NH. FIG. 23. OE = A - p - D 2 , OP = Ri-dr- R 2 , OE COS = OP = Rj - R* + dj - d* - p RI R% d r c = I - g. *(69) (70) * Where the new tangent is beyond the original tangent, requir- ing Rz to be greater than K b p and d r in equations 69, 71 and 72 change sign. For d r see Eqn. 49. 34 THE RAILROAD TAPER The new semi-tangents may now be determined by the methods of the preceding problem. If the semi-tangents of the two parts of the curve, very useful in running in the curve, are not needed, the semi-tangents of the entire curve may be obtained more easily as follows: SN = (R! + di) tan J / - p esc / + *,, *(71) NH = (Ri + di) tan J / - (Ri - Ri - d r ) sin g 'fe. *f(72) Example. Given a located 4 30' curve, Taper 1J, in- tersection angle 82 14', to compound onto a parallel tangent 71 ft. beyond the original tangent, using a 2 45' curve, Taper 1, with Taper 1 between the two curves. ByEqn.69, C osg = g = 24 14'. By Eqn. 70, c = 58 00'. By Eqn. 71, SN = 1258.99. By Eqn. 72, NH = 1502.68. 22. Given a Located Curve, to Compound onto a new Located Tangent, Point of Compound Approximately Fixed, full Tapered. In Fig. 24, AN' is the located curve, to be compounded at about N' so as to unite with located tangent UF. Run tangent to located curve at N', intersecting located tangent. From Functions of a One-degree Curve deter- mine degree of curve, M', with which to compound if tapers were neglected. Then use such a degree of curve that fl 2 . ff _ * + *, (73) and the resulting g will be very nearly g'. * See footnote, page 33. t Care should be taken to use a negative value for cot / when / is greater than 90. LOCATION PROBLEMS 35 OE = OB + BF - FE = R l cos]g f + N'C sin g' - (R 2 + A), OP = Ri -R 2 -d r , OE cos g = Qp _ Ri cos g' + N'C sin g' - (ft + (fe) *,-,, Rt-Rt- d r Example. Given located 4 00' curve AN X , to com- pound at about N f (N f to be about the center of the taper between parts of the compound), so as to end on located tangent CF. N'C = 340.5 ft., g' = 42 12', both meas- ured on the ground. i c u FIG. 24. From Functions of a 1 Curve: T4212' = 2211.0, R' = 5730 X == 340.5 2211 = 882.4, ' = 629'+. * For d r see Eqn. 49. 36 THE RAILROAD TAPER By Eqn. 73, d 2 = 1 . 75 for 6 30' Tap. 2, d r = 0.09. A D 1.75 + 0.09 Approx. R 2 = 882 . 4 EB = 881.52, = 881.59. 406.53 Use 6 30' curve, Tap. 2. By Eqn. 74, COS0 = 550.79 = 42 26'. 23. Given one of the Semi-tangents of a Compound Curve, the Intersection Angle, and both Radii, to find the two Central Angles and the other Semi-tangent. In Fig. 25, given SN, I, RI and R z , to find c, g, and NH. FIG. 25. OE = OTF + WU - UE = (Ri + di) cos / + TN sin / - (R 2 + 4), f)~D D _ E> J 0# LOCATION PROBLEMS 37 (Ri + di) cos / + TN sin / - (fa + d) Ri-R 2 - *d r NH = LU - LN - UV + Z 2 = (#1 H- di) sin I - TN cos 7 - (fii - # 2 - *d r ) sin + fe. f(76) Example. Given semi-tangent /V = 1963.32, MI = 3 00' Tap. 1, M 2 = 8 00' Tap. 2}, and Tap. 2 between Mi and M 2 , / = 114 24', to find c, and N#. 212.62 By Eqn. 75, cos g = UQ2 7g > g = 79 44'. By Eqn. 76, NH = 1411.32. TAPERS BEYOND THE FROG 24. When the Track Beyond the Turnout curves back parallel to the main line, there is seldom room and little need for a taper. When the track beyond the turn- out curves almost immediately in the same direction as the turnout, the taper is of much use, both as to looks and from practical considerations. It has been the practice on many roads to introduce a short tangent, of twenty to thirty feet, beyond the frog, before beginning the curve. This answers the double purpose of giving the trackman room for an approximate taper, and by removing the B.C. from the heel of the frog, prevents the kink that otherwise is very likely to occur there due to the rail joint on the outside rail. However, this presents the very bad appearance and gives the bad riding of the carefully avoided " broken-backed curve." If the curve begin at the heel of the frog with a taper of moderate length, both appearance and service are much improved. * For d r see Eqn. 49. f When #2 is greater than Ri, d r becomes negative. When 7 is greater than 90, cos / becomes negative. 38 THE RAILROAD TAPER Considerable ease in location will often be obtained if the curve beyond the taper may be handled as a simple curve tangent to a parallel with the main line, instead of with the tangent through the frog. In Fig. 26, / is the distance from frog angle point to actual point of frog, usually given by standard drawings. It is / = \ (g + v) cot J F, (77) where g = gauge, v = thickness of frog point, usually one-half inch, and F = frog angle, m = length of frog from actual point to heel. L _ f _ *u.~ w~J \ [p FIG. 26. Then p = HE + KO - OD = (f-+m + t)wnF+ (R + d)cosF-R, (78) w = CH - SH - CP = (f + m + cosF - (R + d) sinF -/. (79) LOCATION PROBLEMS 39 From Eqn. 10, AP = w + (R+p) tan J (F+I). - (p-d) esc (F+I). (80) From Eqn. 11, AN = t + (R+d) tan } (^+7) - (d-p) esc (F+7), (81) sin/ The resulting calculations for the entire problem may be longer than if the semi-tangent BT be calculated by Eqn. 7, / + ra added, and triangle ACB solved. However, p and w are very useful in a preliminary consideration of the location of the turnout. Example. Given a No. 9 turnout from a side track to a branch line, with a 6 00' curve, Tap. 2J, intersection angle 72 48', beginning at the heel of the frog, g = 4'-8i", v = J", m = 8'-10", F = 6 22': By Eqn. 77, / = 42.70. By Eqn. 78, p = 10.70 + 949.72 - 955.04 = 5.38. By Eqn. 79, w = 95.92 - 105.97 - 42.70 = -52.75. The negative value of w means that S lies between P andC. By Eqn. 80, AP = - 52.75 + 794.08 - 4.88 = 736.45. By Eqn. 81, AN = 44.99 + 790. 10 + 4.88 = 839.97. By Eqn. 82, AB = 90.45. CHAPTER III FIELD WORK SELECTION OF THE TAPER 25. The Selection of the Proper Taper for use on each curve is an important function of the engineer. As suggested elsewhere, the proper length of taper to use is one on which the super-elevation of rails given the main curve may be run off on the taper at from one-half to one inch per rail length. This will suggest the use of Taper 1, 1J or 2, for curves that do not limit the speed of trains, with a maximum length of about 360 ft., and thence gradually decreasing in length as velocity and super- elevation decrease, to a minimum of 90 ft. Although some very short tapers are given in the tables, they are for exceptional conditions. The Table of Comparative Values will aid in the selection of the taper. A table of the theoretical elevation needed to counter- act the centrifugal forces is included with the other tables, copied from the Proceedings of the American Railway Engineering Association, but as the super-elevation de- pends more upon the velocity of trains than upon the degree of curve, and as this velocity varies with trains, seasons and traffic, an arbitrary rule, usually a compromise, is in force on most roads. The beginning and end of every taper should be clearly marked for the track foreman, so that he may know between what points he is to give the rails a gradually increasing super-elevation. The proper maintenance of this gradual increase, however, is no easy task, even when the points are well marked, and if the distance over which 40 FIELD WORK 41 it occurs be too great, and the increase in super-elevation be too gradual, it is very likely to become and remain very irregular, totally destroying the value of the taper. Therefore, though theoretically a curve tapered through- out might seem perfection, as maintained practically it would be a very poorly riding curve. Too long a taper is as bad as none. The somewhat prevalent practice of fitting a circular curve to the ground, and tangents at some undetermined offset therefrom, then connecting the two by the neces- sary taper to suit . the offset, has the support of a few eminent engineers, but should be condemned as slipshod work. It is rarely the case that more careful thought and investigation will fail to place some other curve or tangents in a different location, with a suitable taper which should be chosen for totally different reasons. The weight and velocity of trains, the weight and char- acter of the engines and cars whose direction is to be changed, and the amount and quickness of the change, should alone govern the choice of taper. If any additional cost of construction be involved in the proper choice of taper, which is doubtful with good choice of location, it would be justified by the resulting lessened operating costs. Most certainly the higher quality of locating ability required to properly place a line to suit economic conditions is well able to solve this lesser problem. LAYING OUT A TAPERED CURVE 26. After the location of tangents and main curve has been decided upon and choice of taper made, the B.C. and E.C. should be set by measurement from the P.I. The taper at each end should then be run from the B.C. and E.G., the C.C. to the main curve being set preferably by long chord from the B.C. and E.G., and then the main curve should be run from C.C. to C.C. of tapers, or from 42 THE RAILROAD TAPER each C.C. to the middle of the curve if it is unusually long. It is usual, especially for final track centers, to stake the taper every thirty feet from B.C. and E.G., closing to the C.C. with a short chord where the length of taper, I, is not a multiple of 30 ft. ; but on location, and especially for slope staking on construction, it is often desirable to stake the station and plus fifty points, which may be done by obtaining the deflections from Eqn. 107. If the main curve is of moderate length, it is usual, with the transit at the C.C., to turn the transit at the B.C. with the backsight deflection set on the reverse side of the vernier, thus bringing the instrument into tangent at the C.C. with the vernier at zero, and thence run in the main curve with foresight on the other C.C. If the main curve is unusually long, and the other C.C. not visible, this involves running a comparatively long curve from a short backsight, frequently resulting in poor closing. In this case a flag may be set at an offset from the initial tangent equal to x of the taper, preferably opposite the P.I., but in any event as far from the C.C. as the curve to be run from it. With the vernier set at the total central angle of the taper, o, and the instrument directed at this flag, bringing it into a direction parallel to the initial tangent, the instrument will then be tangent to the taper at the C.C. when the vernier is brought to zero. PASSING OBSTRUCTIONS ON THE TAPER 27. Curve run from Main Curve Produced. It frequently happens that either the B.C. or C.C. of the taper is inaccessible, when it is often necessary to run the main curve from the offset B.C. of a simple curve, or from a point on the simple curve produced backward as far as necessary beyond the B.C. of the simple curve, such point being set by offset from the initial tangent. FIELD WORK 43 In Fig. 27, point C is the B.C. of the simple curve, D the B.C. and B the C.C. of the taper. The central angle of the portion of the simple curve from C to B is the central angle of the taper, o. Point A, a distance back on the simple curve equal to CB, and about opposite the B.C. of the taper, is a point frequently chosen from which FIG. 27. to run the main curve, because of the ease with which its position may be determined. Its offset from tangent is x of the taper, and its distance along the tangent from the B.C. of the taper is n = y 2 R sin o. From Eqn. 4, R sin o = y t, then n = 2t-y. (83) Any other point, E, may be chosen, its offset from tangent being, r=d + R vers I', (84) where I' is the central angle of the curve back of the B.C. of the simple curve. The distance along the tangent from the B.C. of the taper is s = Z- fivers/'. (85) This is merely equivalent to suggesting that the main curve may be run in beyond the C.C. from any point back of the C.C. by considering it a simple curve from a parallel tangent. 44 THE RAILROAD TAPER 28. Curve run from a Point Beyond the C.C. When on location the C.C. is in a bad position to occupy with the transit, a point may be set on the main curve beyond the C.C. from a point on the semi-tangent, by working the problem under paragraph 17 backwards. The plus of the point to be set is assumed, and from that angle a, Fig. 28, is known. D and A B are known. Then tann=^, (86) OB = , (87) cosn g = a - (90 - n) = a + n - 90. (88) Solving triangle OBP, two sides and the included angle given, tan (c + }(/)= |^| tan \ g, (89) b = 180 - n - c, (90) sine For point P', equations 88 and 90 become g' = 90 - (a' + n), (88a) &' = 180 - n + c', (90a) the other equations remaining the same. Example. Given a 4 00' curve, Taper 1|, C.C. in a poor location for a transit set up, the second station beyond, 137.9 ft. from the C.C., being a particularly good transit point. On the same ridge with this second station is a point on the semi-tangent, PoST, 260.89 ft. from the B.C. Determine the angle and distance from PoST to PoC. a = 4 X 0.6' X 137.9 + 2 36' = 8 07'. By Eqn. 86, n = 82 13'. By Eqn. 87, OB = 1446.50. By Eqn. 88, By Eqn. 89, By Eqn. 90, By Eqn. 91, FIELD WOKK = 20'. \- 1 g = 30 50', c = 30 40'. 6 = 67 07'. BP = 16.34. 45 FIG. 28. FIG. 29. 29. Taper Points set by Offset. When both the B.C. and the C.C. are inconvenient to occupy with the transit, the taper points may be set by offset from the tangent, using x and y, or they may be set by offset from the tangent on the first half and from the curve on the second half. In the latter case the offset from the curve is the same as from tangent at equal distances from B.C. and C.C. 30. Taper Points set from Point on Semi-tangent. A few or all of the taper points are occasionally set by measurement and angle from a point on the semi-tangent, PoST, by a solution of the triangle in Fig. 29. tan b = AB = CB-y 1 CB-y cos b (92) (93) 46 . THE RAILROAD TAPER 31. Curve Notes for the Field Book. Assuming the B.C. at Sta. 489 + 15.2, the transit notes for a tapered curve would be as follows: 497 O +62.2 441i' 2 37*' E.G. +32.2 4 14i' 498 +02.2 341i' + 72.2 3 02|' +42.2 2 17J' 495 +12.2 l26i' +82.2 029i' I = 195 +67.2 1323J' 13 23*' C.C. Tap. 2 o = 7 18|' 494 10 52' +50 8 59*' TA = 41 24' 493 7 07' T = 387.16 +50 . 5 14*' Ext. = 55.64 492 3\22' R = 764.08 +50 129*' L m = 357.0 O 491 +10.2 2 37*' 4 e 41i' C.C. 7 30' R A = 26 46*' + 95.2 2 16*' +65.2 139' t = 97.41 + 35.2 107*' d = 2.71 490 +05.2 042' x = 8.93 + 75.2 022*' lc = 194.86 +45.2 009' I = 195 +15.2 00' B.C. Tap. 2 o = 7 18|' 489 In the above, the deflections for the taper at the E.G. are given from tangent at the C.C., calculated from the equation ,, _ J 300 A an empirical equation derived from the tables by the method of second differences, in which I is the distance from B.C. or E.G. to the point for which deflection is wanted, and s is the distance of the transit from B.C. or E.G. The equation gives the deflection in minutes, and is from tangent at the transit. The deflections from tangent at the C.C. given in the notes above are merely to make the notes complete. Usually both tapers are run from B.C. and E.G. towards the main curve, using the same set of deflections. CHAPTER IV OFFICE WORK 32. Plotting a Tapered Curve. In the plotting of tapered curves, it is customary, after plotting the tangents and locating the B.C. and E.G., to draw the simple curve in its offset position, then draw a curve of radius twice that of the main curve extending from B.C. and E.G. a distance equal to I, the length of the taper, coinciding with the main curve at the C.C. The difference between such a curve and the exact taper is noticeable only if the drawing be on a very large scale. 33. Paper Location with Tapers. The usual method of paper location with tapers, as indeed frequently in location on the ground, is to offset the simple curve from its tangents an amount equal to d for the taper chosen. If curves predominate, they will be located first, and with parallel curves near their prob- able ends of radius greater by d, the tangents may be drawn in their offset position. If tangents predominate, they will be drawn first, with short parallel tangents near their probable ends, and the curves drawn tangent to these offset tangents. After the location of the simple offset curves, and the tangents, the tapers may be drawn or merely indicated as desired, depending upon the scale of map used. 34. Curve Compensation with Tapers. It is a practice on some roads to use a compensation on the taper of half that on the main curve. An exactly equivalent result, with a simpler and more satisfactory grade line, is obtained if the change of grade from tangent 47 48 THE RAILROAD TAPER to curve be made at the point opposite the offset position of the B.C. of the simple curve, which is at the middle of the taper, and a suitable vertical curve inserted. This brings correct theoretical and practical results, increasing the compensation in exact ratio with the curva- ture, and is well suited to location in rough country, where final consideration of the taper must be postponed until the location of curves and tangents. 35. Right of Way Descriptions with Tapered Curves. A suitable description of right of way when the taper is used would be: ". . . , the center line of said rail- road extending thence along a tangent N 1 14' W to station 4937 -f- 48.2, a distance of 465 ft., thence along a taper -to the right of curvature increasing one degree every thirty feet to station 4938 + 98.2, a distance of 150 ft., thence along a 6 00' curve to the right (radius 955.04) to station 4945 + 42.8, a distance of 644.6 ft., thence along a taper to the right of curvature decreasing one degree every thirty feet to station 4946 + 92.8, a distance of 150 ft., thence along a tangent N 46 26|' E to station 5052 + 47.1, a distance of 554.3 ft., . . . ." CHAPTER V TAPERING OLD TRACK 36. The Tapering of Old Track, located and con- structed with circular curves only, calls for most careful analysis of the conditions encountered, and generally repays in results obtained a great deal of otherwise need- less transit work. Although after long experience in this particular work one may decide almost offhand upon the taper to use and the changes to make in tangents and curves, tjie beginner must usually make more than one trial before arriving at the best results obtainable, which usually combines with the most satisfactory curve for operation the lowest cost in making the change in align- ment. Such work is frequently complicated by bridges on the curves or tangents near the curves, limiting the amount track may be thrown, or by the slow, though frequently very decided,' sliding of embankment on side hill work, distorting the original circular curve. It will also frequently be found that the trackman has thrown the B.C. in towards the center of curvature, lining in an approximate taper, and if the curve be short, may have taken care of the additional curvature by throw- ing the middle of the curve out. If the curve be long, it will probably be found that a short portion beyond this approximate taper is of unduly heavy curvature. On the other hand, the passage of heavy engines of rigid wheel base has frequently thrown the curve out at the B.C. and E.C., with which is coupled unduly heavy curvature just beyond. When allowed sufficient time, the engineer will rerun the original simple curve to determine just where the track is, before attempting to insert tapers. The 49 50 THE RAILROAD TAPER resulting transit notes, with measured offsets to the track as it is found, together with notes showing all bridges and openings, narrow embankments and cuts, and points where embankments and cuts may be cheaply widened and where not, will give the engineer the data for a care- ful office study of the conditions to be encountered. With an understanding of the various methods suggested below of inserting tapers, a tentative plan may be decided upon for each curve, subject to revision when running in the curves. Since the method involving the lowest maxi- mum shift of track is the shifting of both curve and tan- gents, which will then influence the next curve in each direction, the entire series of curves and tangents must be studied as a whole. The insertion of tapers in old circular track will gener- ally be accompanied by a resurfacing out of face of the entire portion of the line involved, so that usually there is no need to confine the shifting of track to the curves, if the shifting of tangents also will improve results. If the track may be ballasted at the same time the cost of inserting tapers is practically eliminated. 37. Although, unfortunately, in the insertion of tapers in old circular curve alignment, the physical conditions encountered must very largely influence the choice of taper, frequently necessitating the use oT short tapers, the operating conditions due to amount and direction of grade, velocity, weight and character of trains, amount of traffic, direction of heaviest tonnage, etc., must not be lost sight of, but must, as largely as possible, influence the choice of taper. METHODS OF INSERTING TAPERS IN CIRCULAR TRACK. SIMPLE CURVES 38. Shifting the Curve. The simplest, though usually the least desirable, method of making room for the taper is by shifting the curve radially inward, the degree of TAPERING OLD TRACK 51 curve remaining unchanged. The amount track will be shifted will vary from zero at the new B.C. and E.G., to MN, Fig. 30, at the middle of the curve (with equal tapers at the two ends) , varying approximately with the distance from B.C. or E.G. MN = 00' = AB = d sec } L (95) As this method will shorten the line somewhat, by an amount equal to S = 2 necessitating the cutting of rails under usual conditions, it is to be avoided if possible except at summits, where the rail joints are likely to be quite open, or at sags, where the cutting of rails would be desirable to relieve joints already too tight. This method is limited to small values of d and 7, to avoid throwing the inner rail onto or close to fresh em- bankment, accentuating the tendency of the inner rail to settle, especially on single and double track roads, where the super-elevation must fit the speed of the fastest trains, throwing a large part of the weight of slower trains upon the inner rail. 52 THE RAILROAD TAPER 39. Shifting the Tangents. Where consecutive curves are of contrary flexure, and in a lesser degree where of similar flexure, shifting a short tangent, Fig. 31, forms a desirable method of making room for the taper. The tangent is shifted at its ends amounts approximately equal to d of the tapers adopted. b FIG. 31. With this method it is usual first to fit the main portion of each curve as closely as desired by trial, not over- looking provision for inserting a proper taper at the other end of each curve, then connecting the two curves by a meander line, preferably the old tangent, and proceeding as in paragraph 15. Due to irregularities found in the track it is not likely that these curves, fitted to the track as found, will be exactly tangent to the old tangent, but this is immaterial. With this method the length of track is slightly in- creased, but seldom enough to be objectionable. TAPERING OLD TRACK 53 40. Shifting Curve and Tangents. This method, Fig. 32, usually gives the most desirable rearrangement of alignment, involving the 'lowest maximum throw of track, and maintaining the length of line practically unchanged. To obtain this latter result the tangents should be thrown out about two-thirds of d of the taper opposite the old B.C. and E.G., then run to an intersection and the curve run in. The new curve will cross the old one on the FIG. 32. second half of the taper, the entire central portion of the new curve lying inside the old curve, about one-third d from it at the middle of the curve. These amounts will vary with the length of the tangents and their resulting angular shift. 41. Increasing the Degree of Curve. Where the pre- ceding methods involve too great a throw of track with the taper desired, the curve may be sharpened, as in Fig. 33, enough to make room for the taper. d = (R - R') vers \ I. (97) As the value of d depends upon both R' and the taper chosen, which latter itself is influenced by R f , trial solu- tions of the second half of this equation must be made 54 THE RAILROAD TAPER until a desirable value is obtained for d. For a desirable simplification of this point see the solution in the next problem. With this method of inserting tapers, track will be thrown in toward the center of curvature throughout the curve, the throw being a maximum about at the C.C. of the tapers. This will be h = (R- R') vers (} 7 - o). (98) 42. Increasing Degree and Shifting Curve. Curves of odd minutes curvature are to be avoided as much as possible, as, the trackman being unfamiliar with them, they are difficult to maintain. If the curve of the pre- ceding problem be sharpened so that (R R') vers \ I will give a value greater than d, and the curve shifted away from the center of curvature to bring it tangent to the original tangents, Fig. 34, a desirable realignment is obtained. At the middle of the curve track will now be thrown out an amount or h = [(R- R') vers } I - d] sec J 7, (99) h= (R-D f ) exsec } 7 - d. (99a) TAPERING OLD TRACK 55 For preliminary study, where I is not large, the factor sec \ I may be disregarded as insignificant. The new curve will now lie within the old throughout the tapers and a portion of the main curve, and outside the old curve on the middle portion of the main curve, a very desirable rearrangement. The best balance is obtained by making h equal about one-third d, when the maximum throw of track at the three points of greatest divergence will be about equal. 43. Maintaining Length of Line Unchanged. With the proper value of h, Eqn. 99, R' may be assumed such as to leave the length of line unchanged. The length of line from A to B, Fig. 34, by the new alignment is (100) The length of line from A to B by the old alignment is 1007 L = 2AC M 1007 M (101) 56 THE RAILROAD TAPER The table on page 115 gives the values of M', I and h for a large range of M and / with the more important tapers. Where curves are sharper than ten-degree, tangents will be short, and paragraphs 39 and 40 will give better results. Where I is greater than given in the tables, paragraph 46 will usually give better results unless curves of fractional minutes curvature are used. When the value of M or 7 lies between those given in the table, M' may be interpolated and then checked by equations 100 and 101, recalculating when necessary. If on the first trial L' L is too great, decrease M 1 '. The irregularity of h in the table is due to using the nearest minute for M'. Equations 100 and 101 are based upon fifty-foot chord measurement of both old and new curves, but any dis- crepancy over arc measurement introduced is small and may usually be neglected. With both curves based upon a fifty-foot chord, increasing a 5 00' curve to a 5 15' curve introduces a discrepancy due to arc excess of only 0.0002 ft. per hundred feet. With the old curve based upon a hundred-foot chord and the new curve based upon a fifty-foot chord the difference in arc excess is quite large, especially with the higher degrees of curve, but this is almost exactly balanced by the decreased value of AC, equation 101, over that assumed for the table, due to the increased value of R. For arc measurement equations 100 and 101 become L'(arc)=Z + ^, (lOOa) L(arc) = 2*-2(fl-')tani/ Using these equations, if a 10 00' curve, I = 50 00', be sharpened to a 10 17' curve, Tap. 3, as given in the table, the new alignment would be 0.04 ft. longer than the old. The curve would have to be sharpened to a 10 16' curve, Tap. 3, to maintain the length unchanged. TAPERING OLD TRACK 57 The length of line may be maintained unchanged with many of the other problems in this chapter by arranging by the manner of shifting curves and tangents that approximately half the new alignment lie within and half without the old. 44. Increasing Degree and Shifting Curve and Tan- gents* Where / is large the smallest increase of curvature Avoiding odd minutes will frequently give too large a value for h of equation 99, when to avoid throwing the curve out too much at the middle the tangents may be thrown in at the B.C. and C.C. of the old curve, as in Fig. 35. c + h = (R-R') vers \l - d (approx.). (102) For the best balance of track h should about equal c. The new tangents may be run to an intersection, new external calculated and h checked on the ground, and if satisfactory the curve run in. 45. Any of the preceding methods may be combined in any one curve, by shifting the tangent at one end only 58 THE RAILROAD TAPER FlG. 38. TAPERING OLD TRACK 59 either the full or partial amount of d, and the curve at the other end, or the tangents may be shifted different amounts at the two ends of the curve, thus taking advantage of different widths of roadbed available at different points. On heavy grades a shorter taper may be used at the lower end of the curve when room is limited, due to the lower speed of ascending trains. 46. Increasing Degree of Portion of Curve. Where I is too great to sharpen the entire curve, and the tangents too long to be shifted advantageously, a portion only of the curve may be sharpened to make room for the taper, as in Fig. 36. The difference between the degree of the old curve and the portion sharpened should preferably not be greater than the curvature increment of the taper to be used, and may often be less. -r (103) /' must of course be larger than the central angle of the taper, or the increase in curvature to provide the offset has been too great. I' should not be less than twice 0, and may usually be as much as three times o. COMPOUND CURVES 47. Any combination of the preceding problems of this chapter may be used in inserting tapers at the ends or between the branches of old compound curves. A few of the possible combinations follow. 48. Shifting the Curves. In Fig. 37 the relative values of the various d's are exaggerated, increasing the angular change in the position of the point of compounding. How- ever, this arrangement disturbs the branch of sharper curvature disproportionately, which may be modified as in Fig. 38 by shifting the branch of lighter curvature for the entire amount of d r . 60 THE RAILROAD TAPER 49. Shifting Curves and Tangents. Where both the curves and tangents may be shifted, as in Fig. 39, the FIG. 39. maximum throw of track may be much reduced. This aids particularly with the branch of sharper curvature. 50. Shifting the Lighter Branch, and Increasing the Curvature of the Sharper Branch. This change of align- FIG. 40. ment is shown in Fig. 40. With this may of course be combined the shifting of the end of the sharper branch to avoid curvature of odd minutes, as in paragraph 42, or also the shifting of one or both tangents as in paragraph 44. TAPERING OLD TRACK FIELD WORK 61 51. The principal objection to compound curve align- ment is the difficulty trackmen find in maintaining it to proper curvature. It follows that in inserting tapers in compound curves it is seldom profitable to make any lengthy calculations based upon the theoretical location of the curves. They will be somewhere else. The best, method of procedure is to pick out portions of the curves of apparently fairly uniform curvature, which can be tested with the mid-ordinate to a hundred-foot tape held against the rail at the ends of the tape, set up the transit on these portions, extend them as far in each direc- tion as they fit the track fairly well, then turn tangents and intersect as in Fig. 41. It will usually be found that these FIG. 41. curves, fitted to the track, are not tangent to each other nor to the original tangents, the offsets from which may or may not be in a direction to facilitate the insertion of tapers. 62 THE RAILROAD TAPER At the tangent ends, the existing offset from tangent is, from Fig. 41, c = AB sin a R vers a. (104) Between the two branches of the compound curve, the existing offset along the common radial line may be found from Fig. 41. A solution of right triangles OPK and O'PS will give the sides O'P and OP and the included angle / of triangle O'PO, from the solution of which 00' and angle g may be obtained, from which angle n may be obtained. Then m = b n, (105) e = R' - R - O'O. (106) With these data the shifting of curves and tangents most desirable may be decided upon. 'Then to insure perfect fitting of the curves to each other and to their tangents, the tangents may be located in their new posi- tions, the longer of the two branches located in its new position by the problem in paragraph 12, and then the shorter of the two branches located by the problem in paragraph 22. TABLES 63 RADII 25 , sin , Deg. Radius. Log rad. Deg. Radius. Log. rad. 1 Deg. Radius. Log. rad. 000' Infinite 100' 5729.60 3.75812 2 00' 2864.82 3.45710 1 343775.76 5.53627 1 5635.67 3.75095 1 2841.15 3.45349 2 171887.41 5.23524 2 5544.77 3.74388 2 2817.86 3.44992 3 114591.60 5.05915 3 5456.76 3.73693 3 2794.95 3.44637 4 85943.67 4.93421 4 5371.50 3.73010 4 2772.41 3.44286 5 68754.94 4.83730 5 5288.86 3.72336 5 2750.23 3.43937 6 57295.78 4.75812 6 5208.73 3.71673 6 2728.41 3.43591 7 49110.67 4.69118 7 5130.99 3.71020 7 2706.93 3.43248 8 42971.83 4.63318 8 5055.53 3.70377 8 2685.78 3.42907 9 37197.19 4.58203 9 4982.26 3.69743 9 2664.96 3.42569 10 34377.47 4.53627 10 4911.09 3.69118 10 2644.46 3.42234 11 31252.24 4.49488 11 4841.92 3.68502 11 2624.27 3.41901 12 28647.89 4.45709 12 4774.67 3.67895 12 2604.39 3.41571 13 26444.80 4.42234 13 4709.26 3.67295 13 2584.81 3.41243 14 24555.34 4.39016 14 4645.63 3.66704 14 2565.52 3.40918 15 22918.30 4.36018 15 4583.69 3.66121 15 2546.52 3.40595 16 21485.92 4.33215 16 4523.37 3.65546 16 2527.79 3.40274 17 20222.04 4.30583 17 4464.63 3.64979 17 2509.35 3.39956 18 19098.60 4.28100 18 4407.39 3.64418 18 2491.16 3.39640 19 18093.80 4.25753 19 4351.60 3.63865 19 2473.24 3.39327 20 17188.74 4.23524 20 4297.21 3.63319 20 2455.57 3.39015 21 16370.20 4.21405 21 4244.16 3.62779 21 2438.16 3.38706 22 15626.13 4.19385 22 4192.40 3.62246 22 2420.99 3.38399 23 14946.70 4.17555 23 4141.87 3.61720 23 2404.06 3.38095 24 14323.89 4.15606 24 4092.58 3.61200 24 2387.36 3.37792 25 13750.99 4.13833 25 4044.43 3.60686 25 2370.90 3.37491 26 13222.11 4.12130 26 3997.39 3.60177 26 2354.67 3.37193 27 12732.40 4.10491 27 3951.46 3.59676 27 2338.65 3. ,36896 28 12277.67 4.08912 28 3906.56 3.59179 28 2322.85 3.36602 29 11854.34 4.07388 29 3862.66 3.58689 29 2307.26 3.36310 30 11459.17 4.05915 30 3819.75 3.58203 30 2291.88 3.36019 31 11089.52 4.04491 31 3777.77 3.57724 31 2276.70 3.35731 32 10742.97 4.03112 32 3736.71 3.57249 32 2261 . 72 3.35444 33 10417.66 4.01777 33 3696.53 3.56779 33 2246.94 3.35159 34 10111.03 4.00480 34 3657.21 3.56315 34 2232.35 3.34876 35 9822.15 3.99221 35 3618.71 3.55855 35 2217.95 3.34595 36 9549.31 3.97997 36 3581.02 3.55401 36 2203.73 3.34316 37 9291.65 3.96809 37 3544.10 3.54951 37 2189.70 3.34038 38 9046.71 3.95649 38 3507.93 3.54505 38 2175.84 3.33763 39 8814.75 3.94521 39 3472.50 3.54064 39 2162.15 3.33489 40 8594.38 3.93421 40 3437.78 3.53628 40 2148.64 3.33216 41 8384.76 3.92341 41 3403.74 3.53196 41 2135.29 3.32946 42 8185.12 3.91303 42 3370.37 3.52768 42 2122.12 3.32677 43 7994.77 3.90281 43 3337.65 3.52344 43 2109.09 3.32410 44 7813.13 3.89282 44 3305.56 3.51924 44 2096.24 3.32144 45 7639.45 3.88306 45 3274.08 3.51509 45 2083.53 3.31880 46 7473.38 3.87352 46 3243.19 3.51097 46 2070.98 3.31618 47 7314.37 3.86418 47 3212.88 3.50689 47 2058.56 3.31357 48 7161.99 3.85503 48 3183.13 3,50285 48 2046.33 3.31098 49 7015.82 3.84608 49 3153.93 3.49885 40 2034.22 3.30840 50 6875.51 3.83730 50 3125.26 3.49489 50 2022.26 3.30584 51 6740.70 3.82870 51 3097.10 3.49096 51 2010.43 3.30329 52 6611.09 3.82027 52 3069.45 3.48706 52 1998.74 3.30076 53 6486.33 3.81200 53 3042.29 3.48320 53 1987.19 3.29824 54 6366.22 3.80388 54 3015.60 3.47937 54 1975.77 3.29574 55 6250.46 3.79591 55 2989.38 3.47558 55 1964.48 3.29325 56 6138.85 3.78809 56 2963.61 3.47182 56 1953.32 3.29077 57 6031 . 15 3.78640 57 2938.28 3.46809 57 1942.28 3.28831 58 5927.17 3.77285' 58 2913.38 3.46440 58 1931.37 3.28587 59 5826.71 3.76542 59 2888.89 3.46073 59 1920.58 3.28343 64 THE RAILROAD TAPER RADII (Continued) Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 3 00' 1909.91 3.28101 4 00' 1432.47 3.15608 5 00' 1146.01 3.05919 1 1899.36 3.27861 1 1426.52 3.15430 1 1142.20 3.05774 2 1888.93 3.27622 2 1420.63 3.15248 2 1138.42 3.05630 3 1878.61 3.27384 3 1414.78 3.15069 3 1134.66 3.05487 4 1868.40 3.27147 4 1408.99 3.14891 4 1130.93 3.05344 5 1858.30 3.26912 5 1403.24 3.14713 5 1127.20 3.05201 6 1848.31 3.26677 6 1397.53 3.14536 6 1123.54 3.05059 7 1838.42 3.26445 7 1391.88 3.14360 7 1119.88 3.04917 8 1828.65 3.26213 8 1386.26 3.14185 8 1116.24 3.04776 9 1818.97 3.25983 9 1380.70 3.14010 9 1112.63 3.04635 10 1809.40 3.25753 10 1375.17 3.13836 10 1109.04 3.04495 11 1799.93 3.25525 11 1369.70 3.13662 11 1105.48 3.04355 12 1790.56 3.25299 12 1364.26 3.13490 12 1101.94 3.04216 13 1781.27 3.25073 13 1358.87 3.13318 13 1098.42 3.04077 14 1772.09 3.24849 14 1353.53 3.13146 14 1094.92 3.03938 15 1763.01 3.24625 15 1348.21 3.12976 15 1091.44 3.03800 16 1754.01 3.24403 16 1342.95 3.12806 16 1087.99 3.03662 17 1745.11 3.24182 17 1337.72 3.12637 17 1084.56 3.03525 18 1736.30 3.23962 18 1332.54 3.12468 18 1081.15 3.03389 19 1727.57 3.23744 19 1327.39 3.12300 19 1077.76 3.03252 20 1718.93 3.23526 20 1322.29 3.12133 20 1074.39 3.03116 21 1710.38 3.23309 21 1317.22 3.11966 21 1071.05 3.02981 22 1701.92 3.23094 22 1312.19 3.11800 22 1067.72 3.02846 23 1693.53 3.22879 23 1307.21 3.11634 23 1064.42 3.02711 24 1685.23 3.22666 24 1302.26 3.11470 24 1061.13 3.02577 25 1677.01 3.22454 25 1297.34 3.11305 25 1057.87 3.02443 26 1668.87 3.22242 26 1292.47 3.11142 26 1054.62 3.02310 27 1660.81 3.22032 27 1287.63 3.10979 27 1051.40 3.02177 28 1652.83 3.21823 28 1282.82 3.10817 28 1048.19 3.02044 23 1644.92 3 21614 29 1278.05 3.10655 29 1045.01 3.01912 30 1637.09 3.21407 30 1273.32 3.10494 30 1041.84 3.01780 31 1629.33 3.21201 31 1268.62 3.10333 31 1038.69 3.01649 32 1621.64 3.20996 32 1263.96 3.10173 32 1035.57 3.01518 33 1614.03 3.20791 33 1259.33 3.10014 33 1032.46 3.01387 34 1606.49 3.20588 34 1254.73 3.09855 34 1029.37 3.01257 35 1599.02 3.20385 35 1250.17 3.09697 35 1026.29 3.01127 36 1591.62 3.20184 36 1245.64 3.09539 36 1023.24 3.00998 37 1584.28 3.19983 37 1241.15 3.09382 37 1020.20 3.00869 38 1577.01 3.19784 38 1236.68 3.09226 38 1017.19 3.00740 39 1569.81 3.19585 39 1232.25 3.09070 39 1014.19 3.00612 40 1562.68 3.19387 40 1227.85 3.08915 40 1011.20 3.00484 41 1555.61 3.19190 41 1223.48 3.08760 41 1008.24 3.00356 42 1548.60 3.18994 42 1219.14 3.08606 42 1005.29 3.00229 43 1541.66 3.18799 43 1214.84 3.08452 43 1002.36 3.00102 44 1534.78 3.18605 1 44 1210.56 3.08299 44 999.45 2.99976 45 1527.96 3.18411 45 1206.31 3.08146 45 996.55 2.99850 46 1521.19 3.18218 46 1202.10 3.07994 46 993.67 2.99724 47 1514.49 3.18027 47 1197.91 3.07842 47 990.81 2.99599 48 1507.85 3.17736 48 1193.75 3.07691 48 987.96 2.99474 49 1501.27 3.17646 49 1189.62 3.07541 49 985.13 2.99350 50 1494.74 3.17457 50 1185.51 3.07391 50 982.32 2.99225 51 1488.27 3.17268 51 1181.44 3.07241 51 979.52 2.99101 52 1481.86 3.17081 52 1177.40 3.07092 52 976.74 2.98978 53 1475.50 3.16894 53 1173.38 3.06944 53 973.97 2.98855 54 1469.19 3.16708 54 1169.39 3.06796 54 971.22 2.98732 55 1462.94 3.16523 55 1165.43 3.06649 55 968.48 2.98609 56 1456.75 3.16338 56 1161.49 3.06502 56 965.76 2.98487 57 1450.60 3.16155 57 1157.58 3.06355 57 963.06 2.98365 58 1444.50 3.15972 58 1153.70 3.06209 58 960.37 2.98244 59 1438.46 3.15790 59 1149.84 3.06064 59 957.70 2.98123 TABLES 65 RADII (Continued) Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 6 00' 955.04 2.98002 7 00' 818.64 2.91309 8 00' 716.34 2.85512 952.39 2.97882 1 816.69 2.91206 1 714.85 2.85422 2 949.76 2.97762 2 814.76 2.91103 2 713.37 2.85332 3 947.15 ' 2.97642 3 812.83 2.91000 3 711.90 2.85242 4 944.55 2.97522 4 810.92 2.90898 4 710.43 2.85152 5 941.96 2.97403 5 809.01 2.90795 5 708.96 2.85062 6 939.39 2.97284 6 807.11 2.90693 6 707.50 2.84973 7 936.83 2.97166 7 805.22 2.90592 7 706.05 2.84884 8 934.28 2.97048 8 803.34 2.90490 8 704.60 2.84795 9 931.75 2.96930 9 801.47 2.90389 9 703.16 2.84706 10 929.23 2.96812 10 799.61 2.90288 10 701.73 2.84617 11 926.73 2.96695 11 797.75 2.90187 11 700.30 2.84528 12 924.24 2.96578 12 795.91 2.90086 12 698.88 2.84440 13 921.76 2.96462 13 794.07 2.89986 13 697.46 2.84352 14 919.29 2.96346 14 792.24 2.89886 14 696.05 2.84264 15 916.84 2.96230 15 790.42 2.89786 15 694.64 2.84176 16 914.41 2.96114 16 788.61 2.89686 16 693.24 2.84089 17 911.98 .2.95999 17 786.80 2.89587 17 691.85 2.84t)01 18 909.57 2.95884 18 785.01 2.89487 18 690.46 2.83914 19 907.17 2.95769 19 783.22 2.89388 19 689.08 2.83827 20 904.78 2.95655 20 781.44 2.89290 20 687.70 2.83740 21 902.41 2.95540 21 779.67 2.89191 21 686.33 2.83653 22 900.05 2.95427 22 777.91 2.89093 22 684.96 2.83567 23 897.70 2.95313 23 776.15 2.88995 23 683.60 2.83480 24 895.36 2.95200 24 774.40 2.88897 24 682.25 2.83394 25 893.04 2.95087 25 772.66 2.88799 25 680.90 2.83308 26 890.72 2.94974 26 770.93 2.88702 26 679.55 2.83222 27 888.42 2.94862 27 769.21 2.88604 27 678.21 2.83136 28 886.13 2.94750 28 767.49 2.88507 28 676.88 2.83051 29 883.86 2.94638 29 765.78 2.88410 29 675.55 2.82966 30 881.59 2.94527 30 764.08 2.88314 30 674.22 2.82880 31 879.34 2.94416 31 762.39 2.88218 31 672.90 2.82795 32 877.09 2.94305 32 760.70 2.88121 32 671.59 2.82710 33 874.86 2.94194 33 759.02 2.88025 33 670.28 2.82626 34 872.64 2.94084 34 757.35 2.87930 34 668.98 2.82541 35 870.44 2.93974 35 755.69 2.87834 35 667.68 2.82457 36 868.24 2.93864 36 754.03 2.87739 36 666.39 2.88373 37 866.05 2.93754 37 752.38 2.87644 37 665.10 2 82289 38 863.88 2.93645 38 750.74 2.87549 38 663.81 2^82205 39 861.71 2.93536 39 749.10 2.87454 39 662.54 2.82121 40 859.56 2.93428 40 747.48 2.87360 40 661.26 2.82037 41 857.42 2.93319 41 745.85 2.87265 41 659.99 2.81954 42 855.28 2.93211 42 744.24 2.87171 42 658.73 2.81871 43 853.16 2.93103 43 742.63 2.87077 43 657.47 2.81788 44 851.05 2.92996 44 741.03 2.86984 44 656.22 2.81705 45 848.95 2.92888 45 739.44 2.86890 45 654.97 2.81622 46 846.86 2.92781 46 737.86 2.86797 46 653.72 2.81539 47 844.78 2.92674 47 736.28 2.86704 47 652.48 2.81457 48 842.71 2.92568 48 734.70 2.86611 48 651.25 2.81375 49 840.65 2.92461 49 733.14 2.86519 49 650.02 2.81293 50 838.60 2.92355 50 731.58 2.86426 50 648.79 2.81211 51 836.56 2.92250 51 730.03 2.86334 31 647.57 2.81129 52 834.53 2.92144 52 728.48 2.86242 52 646.35 2.81047 53 832.51 2.92039 53 726.94 2.86150 53 645.14 2.80966 54 830.50 2.91934 54 725.41 2.86058 54 643.93 2.80884 55 828.50 2.91829 55 723.88 2.85967 55 642.73 2.80803 56 826.51 2.91725 56 722.36 2.85875 56 641.53 2.80722 57 824.53 2.91620 57 720.85 2.85784 57 640.34 2.80641 58 822.55 2:91516 58 719.34 2.85693 58 639.15 2.80560 59 820.59 2.91413 59 717.84 2.85603 59 637.96 2.80480 66 THE RAILROAD TAPER RADII (Continued) Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 9 00' 636.78 2.80399 10 00' 573.14 2.75826 11 00' 521.07 2.71690 1 635.60 2.80319 1 572.19 2.75754 1 520.28 2.71624 2 634.43 2.80239 2 571.24 2.75682 2 519.50 2.71558 3 633.27 2.80159 3 570.29 2.75610 3 518.71 2.71493 4 632.10 2.80080 4 569.35 2.75538 4 517.93 2.71427 5 630.94 2.79999 5 568.41 2.75466 5 517.16 2.71362 6 629.79 2.79920 6 567.47 2.75394 6 516.38 2.71297 7 628.64 2.79840 7 566.53 2.75323 7 515.61 2.71232 8 627.49 2.79761 8 565.60 2.75251 8 514.84 2.71167 9 626.35 2.79682 9 564.67 2.75180 9 514.07 2.71102 10 625.21 2.79603 10 563.75 2.75109 10 513.30 2.71037 11 624.08 2.79524 11 562.83 2.75038 11 512.54 2.70972 12 622.95 2.79445 12 561 91 2.74967 12 511.77 2.70908 13 621.82 2.79367 13 560.99 2.74896 13 511.02 2.70843 14 620.70 2.79288 14 560.08 2.74825 14 510.26 2 70779 15 619.58 2.79210 15 559.17 2.74754 15 509.50 2.70714 16 618.47 2.79132 16 558.26 2.74684 16 508.75 2.70650 17 617.36 2.79054 17 557.36 2.74613 17 508.00 2.70586 18 616.25 2.78976 18 556.46 2.74543 18 507.25 2.70522 19 615.15 2.78898 19 555.56 2.74473 19 506.50 2.70458 20 614.05 2.78821 20 554.66 2.74403 20 505.76 2.70394 21 612.96 2.78743 21 553.77 2.74333 21 505.02 2.70330 22 611.87 2.78666 22 552.88 2.74263 22 504.28 2.70267 23 610.78 2.78589 23 551.99 2.74193 23 503.54 2.70203 24 609.70 2.78512 24 551.11 2.74124 24 502.80 2.70140 25 608.62 2.78435 25 550.23 2.74054 25 502.06 2.70076 26 607.55 2.78358 26 549.35 2.73985 26 501.33 2.70013 27 606.48 2.78281 27 548.47 2.73916 27 500.60 2.69950 28 605.41 2.78205 28 547.60 2.73847 28 499.88 2.69887 29 604.35 2.78129 29 546.73 2.73777 29 499.16 2.69824 30 603.29 2.78052 30 545.87 2.73709 30 498.43 2.69761 31 602.23 2.77976 31 545.00 2.73640 31 497.72 2.69698 32 601 . 18 2.77900 32 544.14 2.73571 32 497.00 2.69635 33 600.13 2.77824 33 543.28 2.73502 33 496.28 2.69572 34 599.08 2.77749 34 542.42 2.73434 34 495.56 2.69510 35 598.04 2.77673 35 541.57 2.73365 35 494.85 2.69447 36 597.00 2.77598 36 540.72 2.73297 36 494.14 2.69385 37 595.97 2.77523 37 539.87 2.73229 37 493.43 2.69323 38* 594.93 2.77447 38 539.03 2.73161 38 492.73 2.69261 39 593.91 2.77372 39 538.18 2.73093 39 492.02 2.69198 40 592.89 2.77297 40 537.34 2.73025 40 491.32 2.69136 41 591.87 2.77223 41 536.51 2.72957 41 490.62 2.69074 42 590.85 2.77148 42 535.67 2.72890 42 489.92 2.69013 43 589.84 2.77074 43 534.84 2.72822 43 489.22 2.68951 44 588.83 2.76997 44 534.01 2.72755 44 488.53 j 2.68889 45 587.83 2.76925 45 533.18 2.72687 45 487.84 2.68828 46 586.82 2.76851 46 532.35 2.72620 46 487.15 2.68766 47 585.82 2.76777 47 531.53 2.72553 47 486.46 2.68705 48 584.83 ' 2.76703 48 530.71 2.72486 48 485.77 2.68643 49 583.84 2.76629 49 529.90 2.72419 49 485.09 2.68582 50 582.85 2.76556 50 529.08 2.72352 50 484.40 2.68521 51 581.86 2.76482 51 528.27 2.72286 51 483.72 2.68460 52 580.88 2.76409 52 527.46 2.72219 52 483.04 2.68400 53 579.90 2.76335 53 526.65 2.72152 53 482.37 2.68338 54 578.93 2.76262 54 525.85 2.72086 54 481 . 69 2.68277 55 577.95 2.76189 55 525.05 2.72020 55 481.02 2.68216 56 576.98 2.76116 56 524.25 2.71953 56 480.35 2.68156 57 576.02 2.76044 57 523.45 2.71887 57 479.68 2.68095 58 575.05 2.75971 58 522.65 2.71821 58 479.01 2.68035 59 574.10 2.75899 59 521.86 2.71755 59 478.35 2.67974 TABLES 67 RADII (Continued) Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 12 00' 477.68 2.67914 13 00' 440.97 2.64441 14 00' 409.51 2.61226 1 477.02 2.67854 1 440.41 2.64386 1 409.02 2.61175 2 476.36 2.67794 2 439.85 2.64330 2 408.54 2.61123 3 475.70 2.67734 3 439.29 2.64275 3 408.05 2.61072 4 475.05 2.67674 - 4 438.73 2.64219 4 407.57 2.61020 5 474.39 2.67B14 5 438.17 2.64164 5 407.09 2.60969 6 473.74 2.67554 6 437.61 2.64109 6 406.61 2.60918 7 473.09 2.67494 7 437.05 2.64054 7 406.13 2.60867 8 472.44 2.67435 8 436.50 2.63999 8 405.66 2.60816 9 471.79 2.67375 9 435.95 2.63944 9 405.18 2.60764 10 471.15 2. .67316 10 435.40 2.63889 10 404.70 2.60713 11 470.50 2.67256 11 434.85 2.63834 11 404.22 2.60662 12 469.86 2.67197 12 434.30 2.63779 12 403.75 2.60611 13 469.22 2.67138 13 433.75 2.63724 13 403.28 2.60560 14 468.58 2.67078 14 433.21 2.63669 14 402.81 2.60510 15 467.94 2.67019 15 432.66 2.63615 15 402.33 2.60459 16 467.31 2.66960 16 432.12 2.63560 16 401.87 2.60408 17 466.68 2.66901 17 431.58 2.63506 17 i 401.40 2.60357 18 466.04 2.66843 18 431.04 2.63451 18 400.93 2.60307 19 465.41 2.66784 19 430.50 2.63397 19 400.46 2.60256 20 464.79 2.66725 20 429.96 2.63343 20 400.00 2.60206 21 464.16 2.66667 21 429.42 2.63289 21 399.53 2.60155 22 463.53 2.66608 22 428.89 2.63235 22 399.07 2.60105 23 462.91 2.66550 23 428.36 2.63181 23 398.61 2.60055 24 462.29 2.66491 24 427.82 2.63127 24 398.15 2.60005 25 461.67 2.66433 25 427.29 2.63073 25 397.69 i 2.59954 26 461.05 2.66375 26 426.76 2.63019 26 397.23 2.59904 27 460.43 2.66317 27 426.23 2.62965 27 396.77 2.59854 28 459.82 2.66259 28 425.71 2.62911 28 396.32 2.59804 29 459.20 2.66201 29 425.18 2.62858 29 395.86 2.59754 30 458.59 2.66143 30 424.66 2.62804 30 395.41 2.59704 31 457 98 2.66085 31 424.14 2.62750 31 394 95 2.5C655 32 457.37 2.66027 32 423.62 2.62697 32 394.50 2.59605 33 456.77 2.65970 33 423.09 2.62644 33 394.05 2 59555 34 456.16 2.65912 34 422.57 2.62590 34 393 . 60 2.59505 35 455.56 2.65855 35 422.06 2.62537 35 393.15 2 59456 36 454.96 | 2.65797 36 421.54 2.62484 36 3S2.70 2.59406 37 454.36 2.65740 37 421.02 2.62431 37 392.56 2.59357 38 453 . 76 2.65682 38 420.51 2.62378 38 391.81 2.59307 39 453.16 2.65625 39 420.00 2.62325 39 391.36 2.59258 40 452.57 2.65568 40 419.49 2.62272 40 390.91 2.59209 41 451.97 2.65511 41 418.98 2.62219 41 390.48 2.59160 42 451.38 2.65454 42 418.47 2.62166 42 390.04 2.59110 43 450.79 2.65397 43 417.96 2.62133 43 389.60 2.59061 44 450.20 2.65340 44 417.45 2.62061 44 389.16 2.59012 45 449.61 2.65284 45 416.95 2.62008 45 388.72 2.58963 46 449.02 2.65227 46 416.44 2.61955 46 388.28 2.58914 47 448.44 2.65170 47 415.94 2.61903 47 387.84 2.58865 48 447.86 2.65114 48 415.44 2.61851 48 387.40 2.58816 49 447.28 2.65057 49 414.94 2.61798 49 386.97 2.58767 50 446.70 2.65001 50 414.44 2.61746 50 386.53 2.58719 51 446.12 2.64945 51 413.94 2.61694 5*1 386 . 10 2.58670 52 445.54 2.64889 52 413.44 2.61642 52 385.67 2.58621 53 444.96 2.64832 53 412.95 2.61589 53 385.24 2.58573 54 444.39 2.64776 54 412.45 2.61537 54 384.81 2.58524 55 443.82 2.64720 55 411.96 2.61485 55 384 '.38 2 58476 56 443.25 2.64664 56 411.47 2.61434 56 383.95 2.58423 57 442.67 2.64608 57 410.98 2.61382 57 383.52 2 58379 58 442.11 2.64553 58 410.49 2.61330 58 383.09 2.58331 59 441.54 2.64497 59 410.00 2.61278 59 382.67 2.58282 68 THE RAILROAD TAPER RADII (Continued) Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 15 00' 382.25 2.58234 19 00' 301.91 2.47988 23 00' 249.53 2.39712 05 380.14 2.57994 05 300.59 2.47797 15 246.86 2.39245 10 378.05 2.57755 10 299.28 2.47608 30 244.24 2.38782 15 375.99 2.57518 15 297.99 2.47420 45 241.68 2.38324 20 373.95 2.57281 20 296.71 2.47233 25 371.93 2.57046 25 295.44 2.47047 24 00' 239.17 2.37871 30 369.93 2.56812 30 294.18 2.46861 15 236.71 2.37422 35 367.96 2.56580 35 292.94 2.46678 30 234.30 2.36978 40 366.00 2.56348 40 291.69 2.46492 45 231.95 2.36539 45 364.07 2.56118 45 290.46 2.46309 50 362.16 2.55890 50 289.25 2.46127 25 00' 229.64 2.36104 55 360.26 2.55662 55 288.04 2.45945 15 227.37 2.35673 30 225.15 2.35248 16 00' 358.39 2.55436 20 00' 286.84 2.45764 45 222.98 2.34827 05 356.54 2.55211 05 285.66 2.45585 10 354.70 2.54986 10 284.48 2.45405 26 00' 220.84 2.34408 15 352.88 2.54763 15 283.31 2.45226 15 218.75 2.33995 20 351.09 2.54542 20 282.15 2.45048 30 216.69 2.33584 25 349.31 2.54321 25 281.00 2.44871 45 214.68 2.33179 30 347.55 2.54101 30 279.86 2.44695 35 345.80 2.53883 35 278.73 2.44518 27 00' 212.70 2.32776 40 344.08 2.53666 40 277.61 2.44344 15 210.76 2.32379 45 342.37 2.53450 45 276.50 2.44170 30 208.85 2.31983 50 340.68 2.53234. 50 275.40 2.43996 45 206.98 2.31593 55 339.00 2.53020 55 274.30 2.43823 28 00' 205.14 2.31205 17 00' 337.34 2.52807 21 (xy 273.22 2.43651 15 203.33 2.30820 05 335.70 2.52595 05 272.14 2.43479 30 201.56 2.30440 10 334.07 2.52384 10 271.07 2.43309 45 199.81 2.30062 15 332.46 2.52174 15 270.01 2.43138 20 330.87 2.51965 20 268.96 2.42969 29 00' 198.10 2.29688 25 329.29 2.51758 25 267.91 2.42799 15 196.42 2.29318 30 327.72 2.51551 30 266.88 2.42632 30 194.76 2.28950 35 326.17 2.51344 35 265.86 2.42465 45 193.13 2.28585 40 324.64 2.51140 40 264.83 2.42298 30 00' 191.53 2.28224 45 323.12 2.50936 45 263.82 2.42131 50 321.61 2.50733 50 262.83 2.41966 55 320.12 2.50531 55 261.82 2.41800 18 OO 7 318.64 2.50330 22 00' 260.83 2.41637 05 317.17 2.50129 05 259.85 2.41472 10 315.72 2.49930 10 258.88 2.41310 15 314.28 2.49732 15 257.91 2.41147 20 312.86 2.49534 20 256.95 2.40986 25 311.44 2.49337 25 256.00 2.40824 30 310.04 2.49142 30 255.05 2.40664 35 308.66 2.48948 35 254.12 2.40504 40 307.28 2.48754 40 253.19 2.40344 45 305.92 2.48561 45 252 . 26 2.40185 50 304.57 2.48368 50 251.35 2.40027 55 303.23 2.48177 55 250.44 2.39870 TABLES 69 NOTATION A = Curvature increment of taper per 30 ft. of taper. M = Degree of main circular curve. I Length of taper from the B.C. of the taper. / = Deflection from tangent at the B.C. of the taper. ft /i> /2> etc. = Deflection from tangent at some point on the taper. 6 = Deflection from tangent at the end of the taper, C.C. of main curve, to the B.C. of the taper. o = Central angle of the taper at each end of the curve. d = Circular curve offset from tangent to provide room for the taper. t = Distance along the tangent, from opposite the offset position of the B.C. of the simple curve to the B.C. of the taper. D = R + d, where R is the radius of the main circular curve. x and y are co-ordinates of points on the taper, referred to the tangent at the B.C. of the taper as meridian, and to a radial line at the B.C. of the taper as base. Ic = Long chord of the taper, from the B.C. of the Taper to points on the taper. 5 = Distance along the taper, from the B.C. of the taper to the transit, when an intermediate set-up is necessary. T = Semi-tangent of the entire tapered curve, from P.I. to B.C. or E.G. E = External, the distance from the P.I. to a point on the curve radially opposite. EQUATIONS Eqn. No. T = t + D tan | 7. (7) Both ends, the same taper. T t = t + D tan 7 - d esc 7. (8) Tapered end other end not tapered. T n = R tan \ I + d esc 7. (9) Untapered end other end Xapered. T 1 = ti + Di tan \ I - (di - dj esc 7. (10) Both ends tapered with different tapers. T-2 = t-i + D 2 tan / - ( - dj esc 7. (11) Both ends tapered, with different tapers. E = D sec \ I - R. (12) Both ends the same taper. 70 THE EAILROAD TAPER EMPIRICAL EQUATIONS DERIVED FROM THE TABLES I = ~ (M - A). (45) (1 + 15)0 + 30) / 300 A (l + 28 + 45)0-iQ+450 I (I + 30) = "loo (10S) TABLES 71 TAPER INTERMEDIATE VALUES A-J 11 I / b o D logZ) d t z y Ic 030 30.00 0021 0021 04* 11459.18 4.05915 0.01 15.00 0.02 30.00 30.00 35 40.00 0031 003| 007 9822.17 3.99221 0.02 20.00 0.04 40.00 40.00 40 50.00 0041 005f 010 8594 41 3.93422 0.03 25.00 0.06 50.00 50.00 45 60.00 005* 008 13* 7639.49 3.88306 0.04 30.00 0.10 60.00 60.00 50 70.00 007 010* 017* 6875.56 3.83731 0.05 35.00 0.14 70.00 70.00 55 80.00 008f 0131 022 6250.53 3.79592 0.07 40.00 0.20 80.00 80.00 100 90.00 010* 016* 027 5729.70 3.75813 0.10 45.00 0.27 90.00 90.00 05 100.00 012* 020 032* 5288.99 3.72337 0.13 50.00 0.36 100 00 100.00 10 110.00 014* 024 038* 4911 25 3.69119 0.16 55.00 0.47 110.00 110.00 15 120.00 016f 0281 045 4583.89 3.66123 0.20 60.00 0.59 120.00 120.00 20 130.00 0191 032| 052 4297.45 3.63321 0.24 65.00 0.73 130.00 130.00 25 140.00 022 D37* 059* 4044.72 3.60689 0.29 70.00 0.89 140.00 140.00 30 150.00 024f 42f 107* 3820.09 3.58207 0.34 75.00 1.08 149.99 150.00 35 160.00 027f 0481 1 16 3619.11 3.55860 0.40 80.00 1.29 159.99 160.00 40 170.00 030f 0541 125 3438.25 3.53634 0.47 85.00 1.52 169.99 170.00 45 180.00 034 1 00* 134* 3274.63 3.51516 0.55 90.00 1.79 179.98 179.99 50 190.00 037* 107 144* 3125.89 3.49497 0.63 94.99 2.08 189.98 189.99 55 200.00 0411 1 13| 1 55 2990.10 3.47569 0.72 99.99 2.40 199.98 199.99 200 210.00 045 121 206 2865.64 3.45722 0.82 104.99 2.75 209.97 209.99 05 220.00 049 128* 217* 2751.16 3.43952 0.93 109.99 3.13 219.96 219.98 10 230.00 053 136* 229* 2645.51 3.42251 .05 114.99 3.55 229.95 229.98 15 240.00 0571 144| 242 2547.70 3.40615 .18 119.99 4.00 239.94 239.97 20 250.00 lOlf 1531 255 2456.89 3.39039 .32 124.98 4.49 249.93 249.97 25 260.00 106* 202 308* 2372.36 3.37518 .46 129.98 5.02 259.92 259.97 30 270.00 nil 2111 322* 2293.50 3.36050 .62 134.98 5.59 269.90 269.98 35 280.00 1161 220f 337 2219.74 3.34630 1.79 139.97 6.20 279.88 279.95 40 290.00 1211 2 301 352 2150.61 3.33256 1.97 144.97 6.86 289.86 289.94 45 300.00 1 26* 241 407* .2085.69 3.31925 2.16 149.96 7.56 299.84 299.94 50 310.00 1 32 251* 423* 2024.62 3.30634 2.36 154.96 8.30 309.81 309.93 55 320.00 1 37| 3021 440 1967.06 3.29382 2.58 159.95 9.09 319.78 319.91 300 330.00 143* 313* 457 1912.72 3.28165 2.81 164.94 9.93 329.74 329.89 05 340.00 149* 325 5 14* 1861.35 3.26983 3.05 169.93 10.82 339.70 339.87 10 350.00 155* 337 5 32* 1812.70 3.25833 3.30 174.93 11.76 349.66 349.85 15 360.00 201| 3491 551 1766.58 3.24713 3.57 179.92 12.75 359.61 359.83 20 370.00 2081 401f 610 1722.78 3.23623 3.85 184.90 13.80 369.55 369.81 25 380.00 2 15 414* 629* 1681.16 3.22561 4.15 189.89 14.90 379.49 379.78 30 390.00 221| 427f 649* 1641.56 3.21526 4.47 194.88 16.07 389.42 389.75 35 400.00 228f 4411 710 1603.82 3.20516 4.80 199.86 17.29 399.35 399.72 40 410.00 235f 4551 731 1567.82 3.19530 5.14 204.85 18.57 409.27 409.69 45 420.00 243 509* 752* 1533.46 3.18567 5.50 209.83 19.91 419.18 419.65 50 430.00 250* 524 814* 1500.61 3.17627 5.87 214.81 21.31 429.08 429 61 55 440.00 2581 538| 837 1469.20 3.16708 6.26 219.79 22.78 438.97 439.56 400 450.00 306 554 900 1439.14 3.15810 6.67 224.77 24.31 448.85 449.51 72 THE RAILROAD TAPER u cS 08 US 08 02 08 dt, OS i rrtT FHH " - CO "*< O 0 "^ CO - ioC^ iC^OCO O O r TABLES 73 12 8 OiOiOOOOt- ^cocccocococoeococoeoeococcco TficOt^-iOOOC^OOiOO^OOit^-CO i-H^iO"tfC'OeOC OS i-H^fb-OCOCOOiC^iO o oOr-ii-Hi ii i 74 THE RAILROAD TAPER dS 08 0?, 08 OS, 68 o 08 OS 08 08 08 OS o't, 08 -*f ^ -^ Tf * "^ CO CO i O ' O i i CO i-- . - - - - (N CO CO CO - N OQ.Ci ' CO CO CO O Oi Oi itOrfiCOfN . (M O Tti (N i-H ^-i T-I 0 ooo ; o o i i - i -coccc5^i>-i>- C^ ^D ^^ Oi O^ Oi Oi Oi Oti Oi QO 00 t*^* ^O '->Oo0( CO "^ CO t^ !> CO O5 "^ T-H ^^ t^ t> ^^ l> *-O "^ CO s -! C^l ^H T j CD ^^ OOi(M^OiO O" iOi ilMT^ii 8888888^888888 THE RAILROAD TAPER TAPER 1. INTERMEDIATE VALUES M I / b o D logZ) d t z. V Ic 1 100 30.00 004$ 004! 009 5729.62 3.75813 0.02 15.00 0.04 30.00 30.00 05 35.00 005* 006 Oil! 5288.89 3.72336 0.03 17.50 0.06 35.00 35.00 10 40.00 006! 007! 14 4911.12 3.69118 0.03 20.00 0.08 40.00 40.00 15 45.00 007! 0091 16f 4583.73 3.66122 0.04 22.50 0.10 45.00 45.00 20 50.00 008| 0111 020 4297.26 3.63319 0.05 25.00 13 50.00 50.00 25 55.00 010 13! 023! 4044.49 3.60686 0.06 27.50 0.16 55.00 55.0'J 30 60.00 OllJ 015! 027 3819.83 3.58204 0.08 30.00 0.20 60.00 60.00 35 65.00 012f 18 030! 3618.80 3.55856 0.09 32.50 0.24 65.00 65.00 40 70.00 014J 020! 035 3437.89 3.53629 0.11 35.00 0.29 70.00 70.00 45 75.00 015f 023f 039! 3274.21 3.51511 0.13 37.50 0.35 75.00 75.00 50 80.00 om 026! 044 3125.41 3.49491 0.15 40.00 0.41 80.00 80.00 55 85.00 0191 029! 048! 2989.55 3.47561 0.17 42.50 0.47 85.00 85.00 200 90.00 021 033 054 2865.02 3.45713 0.20 45.00 0.55 90.00 90.00 05 95.00 023 036! 059! 2750.45 3.43940 0.22 47.50 0.63 95.00 95.00 10 100.00 025 040 105 2644.71 3.42238 0.25 50.00 0.72 100.00 100.00 15 105.00 027 043! 10! 2546.80 3.40599 0.28 52.50 0.82 105.00 105.00 20 110.00 0291 047| 17 2455.89 3.39021 0.32 55.00 0.93 109.99 110.00 25 115.00 031! 052 23! 2371.25 3.37498 0.35 57.50 .05 114.99 115.00 30 120.00 033! 0561 30 2292.27 3.36027 0.39 60.00 .18 119 99 120.00 35 125.00 0361 100! 36! 2218.39 3.34604 0.44 62 50 .31 124.99 125.00 40 130.00 038| 1051 44 2149.12 3.33226 0.48 65.00 .46 129.99 130.00 45 135.00 0411 1101 151! 2084.06 3.31891 0.53 67.49 .62 134.98 134.99 50 140.00 044 1 15 159 2022.84 3.30596 0.58 69.99 .79 139.98 139.99 55 145.00 046! 120 206! 1965.11 3 . 29339 0.63 72.49 .97 144.98 144.99 300 150.00 049| 125! 215 1910.60 3.28117 0.69 74.99 2.16 149.97 149.99 05 155.00 052^ 131 223! 1859.05 3.26929 0.75 77.49 2.36 154.97 154.99 10 160.00 055! 136! 232 1810.21 3.25773 0.81 79.99 2.57 159.97 159.99 15 165.00 058| 1421 240! 1763.89 3.24647 0.88 82.49 2.80 164.96 164.98 20 170.00 101! 1481 250 1719.88 3.23550 0.95 84.99 3.05 169.96 169.98 25 175.00 105 153! 259! 1678.03 3.22480 1.02 87.49 3.30 174.95 174.98 30 180.00 1081 200| 309 1638.19 3.21436 1.10 89.98 3.57 179.94 179.98 35 185.00 llf 207 318| 1600.20 3.20417 1.18 92.48 3.86 184.93 184.97 40 190.00 151 2 13! 329 1563.95 3.19422 1.27 94.98 4.16 189.92 189.97 45 195.00 18f 220! 339! 1529.32 3.18450 1.36 97.48 4.47 194. U 194.96 50 200.00 22! 227! 350 1496.19 3.17499 1.45 99.97 4.80 199.90 199.96 55 205.00 261 234! 400f 1464.49 3.16569 1.55 102.47 5.14 204.89 204.95 TABLES 77 TAPER 1. INTERMEDIATE VALUES (Continued) A = \ M I / b D logZ) d t z y Ic 400 210.00 130 242 412 1434.12 3.15659 1.65 104.97 5.50 209.88 209.95 05 215.00 134 249$ 423$ 1405.00 3.14768 1.76 107.47 5.87 214.87 214.95 10 220.00 138 257 435 1377.04 3.13895 1.87 109.96 6.26 219.85 219.94 15 225.00 142 3041 4461 1350.19 3.13039 1.98 112.46 6.67 224.83 224.93 20 230.00 1461 3 121 459 1324.39 3.12201 2.10 114.95 7.10 229.82 229.93 25 235.00 150* 321 511$ 1299.57 3.11380 2.23 117.45 7.54 234.80 234.92 30 240.00 1 541 3291 524 1275.68 3.10574 2.36 119.94 8.01 239.77 239.91 35 245.00 1591 337* 5361 1252.66 3.09783 2.49 122.44 8.49 244.75 244.90 40 250.00 2031 3461 550 1230.48 3.09007 2.63 124.93 8.99 249.73 249.89 45 255.00 2081 3551 603$ 1209.08 3.08246 2.77 127.43 9.51 254.70 254.88 50 260.00 213 404 617 1188.43 3.07497 2.92 129.92 10.05 259.67 259.87 55 265.00 2171 413 6301 1168.51 3.06763 3.08 132.41 10.60 264.64 264.85 500 270.00 222$ 422* 645 1149.25 3.06041 3.24 134.91 11.18 269.61 269.84 05 275.00 227$ 432 659$ 1130.60 3.05331 3.40 137.40 11.78 274.57 274.82 10 280.00 232$ 441$ 714 1112.61 3.04634 3.57 139.89 12.40 279.53 279.81 15 285.00 237| 4501 7281 1095.19 3.03949 3.75 142.38 13.04 284.49 284.80 20 290.00 242f 5011 744 1078.33 3.03275 3.94 144.87 13.71 289.45 289.78 25 295.00 248 5 1U 759$ 1062.00 3.02612 4.13 147.36 14.39 294.40 294.76 30 300.00 2531 5211 815 1046.16 3.01960 4.32 149.85 15.10 299.35 299.73 35 305.00 258f 532 8301 1030.81 3.01318 4.52 152.34 15.83 304.30 304.71 40 310.00 3041 5421 847 1015.93 3.00686 4.73 154.83 16.58 309.24 309.69 45 315.00 309f "5531 903$ 1001.49 3.00065 4.94 157.32 17.36 314.18 314.66 50 320.00 315| 604$ 920 987.48 2.99453 5.16 159.81 18.16 319.12 319.63 55 325.00 3211 615$ 9361 973.86 2.98850 5.38 162.30 18.98 324.06 324.61 600 330.00 327 627 954 960.65 2.98257 5.61 164.77 19.83 328.97 329.57 15 345.00 345 7011 10461 923.18 2.96529 6.34 172.22 22.53 343.73 344.47 - 30 360.00 4031 7381 1142 888.73 2.94877 7.14 179.67 25.46 358.44 359.34 45 375.00 4231 8161 1239$ 856.95 2.93296 8.00 187.10 28.63 373.10 374.20 700 390.00 443$ 855| 1339 827.56 2.91780 8.92 194.52 32.05 387.71 389.03 15 405.00 504$ 9361 14401 800.33 2.90327 9.91 201.93 35.73 402.25 403.84 30 420.00 5261 10181 1545 775.05 2.88933 10.97 209.33 39.67 416.73 418 62 78 THE RAILROAD TAPER 08 do 0*8 dS 03 08 02 d^ C5S o'S 08 OS 08 o't, OS H|N|^ Hh#H|Ni-t|NH|c>*rt|-* mfa lO Tt* iO CO OO I-H O O CO CO OQTt< TjH T-l I -^ t -^H TJH i/5 O -Oi-HCM^uOt^CX) .O'-HO i-i^Hr-HtOCO . Tt< CO CO CO 1C < T-IT-II- . CO t^ O C oo o ooo ; o r-i i i c<< co o co i>. oi I-H - ' . - -- M tX)>O(Mt^(NiOt^O5 I-H I-H (M CO O CO T-I O OO 'OOrH (M CO -^ iOt^OOO(M W . HIM WHMW H|MHN H|M CO . CO t^ O i i (N i i O OO Tf O O .I-H CO O >OTjn ^ iCOCO r^ O O O i-H 1-1 1CN TABLES 79 . *-H'^t | l>OCO "e co ooo^H^*^' T [jj o O O * i Oi ^D C^ C^l CO i-~i CO GO i>- CO CD CO ^ ^ i^ T-H TT *O OO ^H ^^ tO *-O Oi t^ !> O I^^C^T irHOOiOt)OOQO001>*i>* tf q o CD CO THE RAILROAD TAPER TAPER 1|. INTERMEDIATE VALUES A = | M I / b D log> d t X y Ic 130 30.00 006| 006! 013* 3819.78 3.58204 0.03 15.00 0.06 30.00 30.00 35 33.33 007! 008 015f 3618.74 3.55856 0.03 16.67 0.07 33.33 33.33 40 36.67 008| 009* 0181 3437.82 3.53628 0.04 18.33 0.09 36.67 36.67 45 40.00 009f 0111 021 3274.13 3.51509 0.05 20.00 0.11 40.00 40.00 50 43.33 OlOf 013 0231 3125.32 3.49489 0.06 21.67 0.13 43.33 43.33 55 46.67 Ollf 015 0261 2989.45 3.47559 0.07 23.33 0.16 46.67 46.67 200 50.00 013 017 030 2864.90 3.45726 0.08 25.00 0.19 50.00 50.00 05 53.33 0141 019 0331 2750.32 3.43938 0.09 26.67 0.22 53.33 53.33 10 56.67 015* 0211 0361 2644.56 3.42235 0.10 28.33 0.26 56.67 56.67 15 60.00 016! 023! 040* 2546.64 3.40597 0.12 30.00 0.30 60.00 60.00 20 63.33 0181 026 0441 2455.70 3.39018 0.13 31.67 0.34 63.33 63.33 25 66.67 019| 028* 0481 2371.05 3.37494 0.15 33.33 0.38 66.67 66.67 30 70.00 0211 0311 052* 2292.04 3.36022 0.16 35.00 0.43 70.00 70.00 33 73.33 022! 034 056| 2218.13 3.34599 0.18 36.67 0.49 73.33 73.33 40 76.67 024* 036! 1011 2148.84 3.33220 0.20 38.33 0.55 76.66 76.67 45 80.00 0261 39! 06 2083.75 3.31885 0.22 40.00 0.61 80.00 80.00 50 83.33 028 42! 1101 2022.50 3.30589 0.24 41.66 0.68 83.33 83.33 55 86.67 029f 046 115! 1964.75 3.29331 0.27 43.33 0.75 86.66 86.66 300 90.00 031* 049* 121 1910.20 3.28108 0.29 45.00 0.82 89.99 89.99 05 93.33 033* 0521 1261 1858.62 3.26919 0.32 46.66 0.90 93.33 93.33 10 96.67 035* 0561 131! 1809.75 3.25762 0.35 48.33 0.99 96.66 96.66 15 100.00 037* 100 137* 1763.39 3.24635 0.38 50.00 .09 99.99 99.99 20 103.33 039* 1 031 1431 1719.34 3.23536 0.41 51.66 .19 103.32 103.33 25 106.67 041* 1071 1491 1677.45 3.22465 0.44 53.33 .29 106.65 106.66 30 110.00 043f llf 155* 1637.56 3.21420 0.47 54.99 .40 109.99 109.99 35 113.33 046 151 201! 1599.53 3.20399 0.51 56.66 .52 113.32 113.33 40 116.67 0481 20 2081 1563.23 3.19402 0.55 58.33 1.64 116.65 116.66 45 120.00 050* 24* 215 1528.55 3.18428 0.59 59.99 1.77 119.98 119.99 50 123.33 053 281 221! 1495.37 3.17475 0.63 61.66 1.90 123.31 123.32 55 126.67 055* 1.331 228! 1463.61 3.16543 0.67 63.33 2.04 126.64 126.65 400 130.00 058 1 38 236 1433.18 3.15630 0.71 64.99 2.19 129.97 129.99 05 133.33 100* 1 421 2431 1404.00 3.14737 0.76 66.66 2.35 133.30 133.32 10 136.67 1031 1 47* 250! 1375.98 3.13861 0.81 68.32 2.51 136.63 136.65 15 140.00 105| 1 521 258* 1349.07 3.13003 0.86 69.99 2.68 139.96 139.99 20 143.33 108* 157f 3061 1323.20 3.12163 0.91 71.65 2.86 143.29 143.32 25 146.67 111* 2021 3 141 1298.31 3.11338 0.97 73.32 3.05 146.62 146.65 TABLES 81 TAPER 1J. INTERMEDIATE VALUES (Continued) A = f M I / 6 D logD d t X y Ic 430 150.00 114} 208} 322* 1274.35 3.10529 1.03 74.98 3.24 149.94 149.98 35 153.33 1 17} 212* 330! 1251.26 3.09735 .09 76.65 3.44 153.27 153.31 40 156.67 120J 219 339} 1229.00 3.08955 1.15 78.31 3.65 156.60 156.64 45 160.00 123| 224! 348 1207.52 3.08189 .21 79.98 3.87 159.92 159.97 50 163.33 126} 230| 356! 1186.79 3.07437 .28 81.64 4.10 163.25 163.30 55 166.67 129i 236* 405f 1166.78 3.06699 .35 83.31 4.33 166.58 166.63 500 170.00 132* 242* 415 1147.43 3.05973 .42 84.97 4.57 169.90 169.96 05 173.33 135| 248* 424} 1128.69 3.05257 .49 86.64 4.83 173.22 173.29 10 176.67 139 254| 433! 1110.61 3.04556 .57 88.30 5.09 176.55 176.62 15 180.00 142} 301} 443* 1093.09 3.03865 .65 89.96 5.36 179.87 179.95 20 183.33 145| 307| 453} 1076.12 3.03186 .73 91.63 5.64 183.19 183.28 25 186.67 149} 314 503} 1059.68 3.02517 1.81 93.29 5.93 186.51 186.60 30 190.00 152| 319! 513* 1043.74 3.01859 1.90 94.95 6.23 189.83 189.93 35 193.33 156} 3 27J 523! 1028.28 3.01211 1.99 96.62 6.54 193.15 193.25 40 196.67 200 334} 534} 1013.28 3.00573 2.08 98.28 6.86 196.47 196.59 45 200.00 203| 341} 545 998.72 2.99944 2.17 99.94 7.19 199.79 199.71 50 203.33 207* 348} 555! 984.59 2.99326 2.27 101.61 7.53 203.10 203.24 55 206.67 2 11} 355* 606! 970.85 2.98715 2.37 103.27 7.88 206.42 206.57 600 210.00 215 403 618 957.51 2.98114 2.47 104.93 8.24 209.73 209.89 15 220.00 226! 425f 652* 919.64 2.96362 2.80 109.92 9.39 219.66 219.86 30 230.00 239} 449} 728* 884.74 2.94682 3.15 114.90 10.64 229.58 229.83 45 240.00 252i 513! 806 852.48 2.93068 3.53 119.88 12.00 239.49 239.79 700 250.00 305* 539* 845 822.58 2.91518 3.94 124.85 13.47 249.38 249.75 15 260.00 319* 606 925* 794.80 2.90026 4.38 129.82 15.05 259.26 259.70 30 270.00 333f 633! 1007* 768.93 2.88589 4.85 134.79 16.76 269.12 269.64 45 280.00 348| 702* 1051 744.80 2.87204 5.36 139.76 18.58 278.95 279.57 800 290.00 404 732 1136 722.24 2.85868 5.90 144.72 20.53 288.76 289.49 15 300.00 420 802* 12 22f 701.11 2.84579 6.47 149.67 22.61 298.54 299.40 30 310.00 436} 834} 1310* 681.30 2.83334 7.08 154.62 24.83 308.29 309.29 45 320.00 453 907 1400 662.70 2.82132 7.73 159.56 27.18 318.02 319.17 900 330.00 510* 940* 1451 645.20 2.80969 8.42 164.49 29.68 327.70 329.04 15 340.00 528* 1015 15 43* 628.72 2.79846 9.14 169.42 32.32 337.34 338.88 30 350.00 546| 1050! 1637* 613.19 2.78760 9.90 174.34 35.11 346.94 348.72 45 360.00 605* 1127* 1733 598.53 2.77709 10.70 179.25 38.06 356.50 358.54 1000 370.00 625 1205 1830 584.69 2.76693 11.55 184.15 41.16 366.01 368.33 15 380.00 645| 1242! 1928* 571.61 2.75710 12.44 189.04 44.43 375.47 378.09 30 390.00 705} 13 23} 2028* 559.24 2.74760 13.37 193.92 47.85 384.87 387.82 82 THE RAILROAD TAPER 8 O Wg 08 uS u'S US US CO l^ !>. t>- t> O O O rJH CO r- 1 i ( o -^ *o to 00 0000 00 00 OOt^ CO to -^ CO rt|N rt|M H|N rH|M HM H CD CO CO OS C^ Tt< CO i 1 CD O ' O5 i-i -^ OrHOo w^ to^co ;coco iO CO O O 10 "3 rt< CO .OOOSO' icNCO 84 THE RAILROAD TAPER TAPER 2. INTERMEDIATE VALUES A = l Af I / b D log/) d t X y k 200 30.00 009 009 018 2864.86 3.45710 0.04 15.00 0.08 30.00 30.00 05 32.50 010 010* 020* 2750.28 3.43938 0.05 16.25 0.09 32.50 32.50 10 35.00 OlOf 012 022! 2644.51 3.42234 0.05 17.50 0.11 35.00 35.00 15 37.50 Oil! 013* 025* 2546.58 3.40596 0.06 18.75 0.13 37.50 37.50 20 40.00 0121 015* 028 2455.64 3.39016 0.07 20.00 0.15 40.00 40.00 25 42.50 014 016| 030! 2370.98 3.37493 0.08 21.25 0.17 42.50 42.50 30 45.00 015 018! 033! 2291.97 3.36021 0.09 22.50 0.20 45.00 45.00 35 47.50 016* 020* 036! 2218.05 3.34597 0.10 23.75 0.22 47.50 47.50 40 50.00 017* 022! i040 2148.75 3.33219 0.11 25.00 0.25 50.00 50.00 45 52.50 018* 024| 043* 2083.65 3.31882 0.12 26.25 0.28 52.50 52.50 50 55.00 019f 027 046! 2022.39 3.30586 0.13 27.50 0.32 55.00 55.00 55 57.50 021* 029 050* 1964.52 3.29326 0.14 28.75 0.35 57.50 57.50 300 60.00 022* 031* 054 1910.07 3.28105 0.16 30.00 0.39 60.00 60.00 05 62.50 024 033! 057f 1858.47 3.26916 0.17 31.25 0.43 62.50 62.50 10 65.00 025* 036* lOlf 1809.59 3.25758 0.19 32.50 0.48 65.00 65.00 15 67.50 026f 039 105! 1763.21 3.24630 0.20 33.75 0.53 67.50 67.50 20 70.00 028J 041! 110 1719.15 3.23531 0.22 35.00 0.58 70.00 70.00 25 72.50 030 044* 114* 1677.25 3.22460 0.24 36.25 0.63 72.50 72.50 30 75.00 031* 047* 118! 1637.35 3.21414 0.26 37.50 0.69 75.00 75.00 35 77.50 033* 050 123* 1599.30 3.20393 0.28 38.75 0.75 77.50 77.50 40 80.00 034| 053* 128 1562.98 3.19395 0.30 40.00 0.81 79.99 80.00 45 82.50 036* 056* 132! 1528.28 3.18420 0.32 41.25 0.88 82.49 82.50 50 85.00 038* 059* 137! 1495.08 3.17437 0.34 42.50 0.95 84.99 85.00 55 87.50 040* 102* 142! 1463.31 3.16534 0.37 43.75 .02 87.49 87.50 400 90.00 042 104 148 1432 86 3.15620 0.39 45.00 .10 89.99 90.00 05 92.50 044 109* 153* 1403.66 3.14726 0.42 46.24 .18 92.49 92.50 10 95.00 045f 1 13 158! 1375.62 3.13850 0.45 47.49 .27 95.99 95.00 15 97.50 047f 116* 204* 1348.69 3.12991 0.48 48.74 .36 97.49 97.50 20 100.00 049f 20* 210 1322.80 3.12149 0.51 49.99 1.45 99.98 99.99 25 102.50 052 23! 215! 1297.87 3.11323 0.54 51.24 1.55 102.48 102.49 30 105.00 054 27! 221! 1273.89 3.10513 0.57 52.49 1.65 104.98 104.99 35 107.50 056* 31* 227! 1250.77 3.09718 0.60 53.74 1.76 107.48 107.49 40 110.00 058* 135! 234 1228.48 3.08937 0.63 54.99 1.87 109.98 109.99 45 112.50 100* 139! 240* 1206.98 3.08170 0.67 56.24 1.98 112.47 112.49 50 115.00 102f 144 246! 1186.22 3.07416 0.71 57.49 2.10 114.97 114.99 55 117.50 105* 148 253* 1166.18 3.06677 0.75 58.74 2.23 117.47 117.49 500 120.00 107* 152* 300 1146.80 3.05949 0.79 59.99 2.36 119.96 119.99 05 122.50 1 10 156! 306! 1128.03 3.05232 0.83 61.24 2.49 122.46 122.49 10 125.00 112* 201* 3 13! 1109.91 3.04529 0.87 62.48 2.63 124.96 124.99 TABLES 85 TAPER 2. INTERMEDIATE VALUES (Continued) A = l u I / b D logD d t X V Ic 5 15 127.50 14f 206 3201 1092.35 3.03836 0.91 63.73 2.77 127.45 127.49 20 130.00 m 2101 328 1075.35 3.03155 0.96 64.98 2.92 129.95 129.98 25 132.50 20 2151 3351 1058.87 3.02484 .00 66.23 3.08 132.44 132.48 30 135.00 22* 2201 3421 1042.89 3.01823 .05 67.48 3.24 134.94 134.98 35 137.50 251 225 3501 1027.39 3.01174 .10 68.73 3.41 137.44 137.48 40 140.00 27f 2301 358 1012.35 3.00490 .15 69.98 3.58 139.93 139.98 45 142.50 30* 2351 4051 997.76 2.99903 .21 71.22 3.75 142.42 142.48 50 145.00 331 240* 4 131 983.58 2.99281 .26 72.47 3.93 144.92 144.97 55 147.50 361 245* 4211 969.80 2.98668 .32 73.72 4.12 147.41 147.47 600 150.00 39 251 430 ' 956.41 2.98064 .37 74.97 4.32 149.90 149.97 15 157.50 147f 307* 4551 918.39 2.96303 .55 78.71 4.94 157.37 157.47 30 165.00 57 324f 5211 883.34 2.94613 .75 82.45 5.61 164.84 164.96 45 172.50 206* 342| 5491 850.92 2.92989 .97 86.19 6.34 172.32 172.44 700 180.00 2 16* 401* 618 820.84 2.91426 2.20 89.93 7.14 179.77 179.92 15 187.50 226| 421 6471 792.86 2.89920 2.44 93.67 8.00 187.22 187.39 30 195.00 237| 4411 7 181 766.79 2.88468 2.71 97.41 8.93 194.66 194.86 45 202.50 248* 5021 7501 742.43 2.87066 2.99 101 . 14 9.92 202.10 202.34 800 210.00 300 524 824 719.63 2.85711 3.29 104.87 10.98 209.52 209.81 15 217.50 3 m 546* 8581 698.26 2.84402 3.62 108.60 12.12 216.93 217.26 30 225.00 324 609f 9331 678.18 2.83134 3.96 112.33 13.33 224.33 224.72 45 232.50 336| 633f 10101 659.29 2.81908 4.32 116.05 14.62 231.72 232.18 900 240.00 349* 658* 1048 641.49 2.80719 4.71 119.78 15.99 239.10 239.63 15 247.50 402f 724 11261 624.69 2.79566 5.11 123.49 17.44 246.46 247.07 30 255.00 4 16* 7501 12061 608.83 2.78450 5.54 127.21 18.97 253.80 254.51 45 262.50 430* 8 17} 12471 593.82 2.77365 5.99 130.92 20.59 261.12 261.92 1000 270.00 445 845 1330 579.61 2.76314 6.47 134.63 22.30 268.42 269.36 15 277.50 459| 9 13* 14131 566.14 2.75292 6.97 138.33 24.11 275.71 276.76 30 285.00 5 15 942f 14 571 553.36 2.74301 7.49 142.03 26.00 282.96 284.16 45 292.50 530* 10 121 15 431 541.22 2.73337 8.04 145.72 27.99 290.20 291.54 1100 300.00 546* 1043* 1630 529.69 2.72402 8.62 149.41 30.08 297.40 298.92 15 307.50 602| 1115 17171 518.73 2.71494 9.23 153.09 32.26 304.58 306.28 30 315.00 619* 1147J 18061 508.29 2.70611 9.86 156.77 34.55 311.72 313.64 45 322.50 636* 12201 18561 498.36 2.69754 10.52 160.44 36.94 318.84 319.50 1200 330.00 654 1254 1948 488.88 2.68920 11.20 164.10 39.44 325.91 328.29 15 337.50 7111 1328* 20401 479.86 2.68111 11.92 167.76 42.04 332.95 335.59 30 345.00 730 14 031 21 331 471.25 2.67325 12.66 171.41 44.76 339.95 342.88 45 352.50 748* 1439| 22281 463.05 2.66563 13 44 175.05 47.59 346.90 350.15 1300 360.00 807* 1516* 2324 455.22 2.65822 14.25 178.68 . 50.51 353.81 357.40 86 THE RAILROAD TAPER o 02 do OS 08 8 08 ds Win t>- 1O 00 b- 00 rH OS O U5 T-H ' (M O "*! CO i i T-I TP i-* i i CO 1C CO CO CO CO CO O (M O: (N OOO O rH CO 'f CD OO O CO GOC4 (M O5 O -O rH rHCO ^ CO GO O -l > -OCO COCOO(NO.i (CO OirHOOOOOI^.tOI>l>OcDl^O O5 C^ ^* i^ CO *O t^- T-H CO C^l OS o O --I O CO JOO TfH CO CO O5 (N iO 00 i i THE RAILROAD TAPER TAPER 2J. INTERMEDIATE VALUES M I / b o D logD d t X y Ic 230 30.00 oil} 111 022* 2291.93 3.36020 0.05 15.00 0.10 30.00 30.00 35 32.00 012 012! 024f 2218.01 3.34596 0.06 16.00 0.11 32.00 32.00 40 34.00 013 0141 0271 2148.70 3.33218 0.06 17.00 0.13 34.00 34.00 45 36.00 014 15J 029! 2083.60 3.31881 0.07 18.00 0.15 36.00 36.00 50 38.00 015 0171 0321 2022.34 3.30585 0.08 19.00 0.17 38.00 38.00 55 40.00 016 019 035 1964.57 3.29327 0.09 20.00 0.19 40.00 40.00 300 42.00 017 020! 0371 1910.00 3.28103 0.09 21.00 0.21 42.00 42.00 05 44.00 018} 022* 0401 1858.40 3.26914 0.10 22.00 0.23 44.00 44.00 10 46.00 019} 024* 043f 1809.51 3.25756 0.11 23.00 0.26 46.00 46.00 15 48.00 020| 0261 0461 1763.13 3.24628 0.12 24.00 0.29 48.01) 48.00 20 50.00 021| 0281 050 1719.06 3.23529 0.13 25.00 0.32 50.00 50.00 25 52.00 023 0301 0531 1677.16 3.22457 0.15 26.00 0.35 52.00 52.00 30 54.00 024} 032* 0561 1637.25 3.21411 0.16 27.00 0.38 54.00 54.00 35 56.00 025* 034f 1001 1599.19 3.20390 0.17 28.00 0.41 56.00 56.00 40 58.00 026f 037 1031 1562.86 3.19392 0.18 29.00 0.45 58.00 58.00 45 60.00 028 039* 107* 1528.16 3.18417 0.20 30.00 0.49 60.00 60.00 50 62.00 029* 041| HI 1494.95 3.17463 0.21 31.00 0.53 62.00 62.00 55 64.00 031 0441 151 1463.17 3.16529 0.23 32.00 0.57 64.00 64.00 400 66.00 032* 046! 191 1432.71 3.15616 0.24 33.00 0.62 66.00 66.00 05 68.00 034 0491 231 1403.50 3.14721 0.26 34.00 0.67 68.00 68.00 10 70.00 035* 052 27* 1375.45 3.13844 0.28 35.00 0.72 69.99 70.00 15 72.00 037 054f 31! 1348.50 3.12985 0.29 36.00 0.77 71.99 72.00 20 74.00 038* 057! 361 1322.60 3.12143 0.31 37.00 0.83 73.99 74.00 25 76.00 040} 100* 40! 1297.67 3.11316 0.33 38.00 0.89 75.99 76.00 30 78.00 041! 103* 451 1273.67 3.10506 0.35 39.00 0.95 77.99 78.00 35 80.00 043* 106* 150 1250.54 3.09710 0.37 40.00 .01 79.99 80.00 40 82.00 045J 109* 154! 1228.25 3.08929 0.40 40.99 .08 81.99 82.00 45 84.00 047 12! 159! 1206.73 3.08161 0.42 41.99 .15 83.99 84.00 ,50 86.00 048f 16 204! 1185.95 3.07407 0.44 42.99 .22 85.99 86.00 55 88.00 50* 191 209! 1165.90 3.06666 0.47 43.99 .30 87.99 88.00 500 90.00 052* 22* 2 15 1146.50 3.05937 0.49 44.99 .37 89.98 90.00 05 92.00 054* 25! 220} 1127.72 3.05220 0.52 45.99 .45 91.98 91.99 10 94.00 0561 29* 225! 1109.58 3.04516 0.54 46.99 .54 93.98 93.99 15 96.00 0581 133 231} 1092.01 3.03823 0.57 47.99 1.63 95.98 95.99 . 20 98.00 1001 136* 236! 1074.99 3.03140 0.60 48.99 1.72 97.98 97.99 25 100.00 1021 1401 2 42* 1058.50 3.02469 0.63 49.99 1.81 99.98 99.99 30 102.00 1041 144 248} 1042.50 3.01808 0.66 50.99 1.91 101.97 101.99 35 104.00 106* 147f 254} 1026.98 3.01156 0.69 51.99 2.01 103.97 103.99 40 106.00 108* 151f 300} 1011.92 3.00515 0.72 52.99 2.11 105.97 105.99 TABLES 89 TAPER 2i INTERMEDIATE VALVES (Continued) A = U M \ I / b D log-Z) d t X V fc 545 108.00 1 10| 1 55* 306* 997.31 2.99883 0.76 53.99 2.22 107.97 107.99 '50 110.00 1 13 159* 3 12* 983.11 2.99260 0.79 54.98 2.33 109.96 109.99 55 112.00 1 15i 203* 3 18! 969.31 2.98646 0.83 55.98 2.45 111.96 111.99 600 114.00 ii7* 207! 325* 955.90 2.98041 0.86 56.98 2.57 113.95 113.98 15 120.00 124* 220* 345 917.82 2.96276 0.98 59.98 2.95 119.94 119.98 30 126.00 1 31| 234 405! 882.70 2.94581 .11 62.97 3.36 125.93 125.97 45 132.00 139* 248 427* 850.19 2.92952 .24 65.97 3.81 131.91 131.97 700 138.00 147 302! 449!' 820.03 2.91383 .39 68.96 4.30 137.89 137.96 15 144.00 155i 318 513* 791.97 2.89871 .55 71.96 4.83 143.87 143.95 30 150.00 203| 333! 537* 765.80 2.88412 .72 74.95 5.40 149.84 149.94 45 156.00 212| 350* 602! 741.34 2.87002 .90 77.94 6.01 155.81 155.93 800 162.00 221* 407* 628! 718.43 2.85638 2.09 80.93 6.67 161.78 161.91 15 168.00 231 424! 655! 696.93 2.84319 2.29 8,3.92 7.37 167.73 167.89 30 174.00 240| 443 723! 676.73 2.83042 2.51 86.91 8.12 173.69 173.88 45 180.00 250! 501! 752* 657.71 2.81803 2.74 89.90 8.92 179.63 179.86 900 186.00 301 521* 822* 639.77 2.80602 2.99 92.88 9.77 185.57 185.84 15 192.00 3 11* 541* 852! 622.83 2.79437 3.25 95.86 10.68 191.51 191.82 30 198.00 322* 602 924* 606.81 2.78305 3.52 98.84 11.64 197.43 197.79 45 204.00 333* 623* 956! 591.64 2.77206 3.81 101.82 12.65 203.34 203.75 1000 210.00 345 645 1030 577.26 2.76137 4.12 104.80 13.71 209.25 209.70 15 216.00 356f 707* 1104* 563.61 2.75098 4.44 107.78 14.84 215.14 215.65 30 222.00 408f 730* 1139* 550.64 2.74087 4.77 110.75 16.03 221.02 221.60 45 228.00 42H 754 1215* 538.30 2.73102 5.12 113.72 17.27 226.89 227.55 1100 234.00 434 818* 1252* 526.56 2.72145 5.49 116.69 18.58 232.74 233.49 15 240.00 447 843 1330 515.38 2.71213 5.88 119.65 19.96 238.59 239.42 30 246.00 500* 908* 1408! 504.71 2.70304 6.28 122.61 21.40 244.42 245.36 45 252.00 513! 934* 1448* 494.54 2.69420 6.70 125.57 22.90 250.23 251.29 1200 258.00 527| 1001* 1528! 484.82 2.68558 7.14 128.52 24.47 256.02 257.20 15 264.00 541! 1028* 16 10* 475.54 2.67719 7.60 131.47 26.11 261.79 263.09 30 270.00 556* 1056* 1652* 466.67 2.66901 8.08 134.42 27.82 267.54 268.98 45 276.00 611 1124! 1735! 458.19 2.66105 8.58 137.36 29.61 273.27 274.87 1300 282.00 626 1153! 18 19! 450.06 2.65327 9.09 140.30 31.47 278.98 280.75 15 288.00 641* 1223* 1904! 442.29 2.64571 9.63 143.23 33.40 284.66 286.61 30 294.00 657* 1253* 1950! 434.85 2.63834 10.19 146.16 35.40 290.32 292.47 45 300.00 713* 1324^ 2037* 427.71 2.63115 10.76 149.08 37.49 295.95 298.32 1400 306.00 729* 1355! 2125* 420.87 2.62415 11.36 152.00 39.65 301.55 304.15 15 312.00 746 14 27| 2213! 414.31 2.61733 11.98 154.91" 41.89 307.12 309.96 30 318.00 802! 1500* 2303* 408.04 2.61070 12.63 157.81 44.21 312.66 315.76 45 324.00 820 1523! 2353! 402.01 2.60424 13.29 160.71 46.61 318.16 321.55 1500 330.00 837* 1607* 2445 396.23 2.59795 13.98 163.60 49.09 323.63 327.33 90 THE RAILROAD TAPER d8 d8 qg OS. 08 08 OiOOCOOOCOiOCOOltO CO O OO T ( CO t^ ^ t^ 10 i i -^ CO CO rH T-t t I O ! T I O(M COC5 -^ ^OCOCO ^ TH O OO .OrH O T- ( -iO' iiOO ^D CD OS OS OS OS GO t^ COCOGO' i -^ l> O CO OOOSOSOSGOt^iOT-HCO CO I"* Oi ^D ^H O5 i"H ^ QC ^^ 00 00 ^D CO OO CO Oi *^O 00 *O *O GO OS C^l TH r-H t^-iOCOCO^fGOCOOS iH I-H i i (M (M d t 2 y Ic / 400 30 036 1432.55 3.15611 0.08 15.00 0.16 30.00 30.00 600 60 148 955.35 2.98016 0.31 29.99 79 59.99 60.00 800 90 336 717.12 2.85559 0.78 44.98 2.20 89.96 89.98 1000 120 600 574.71 2.75945 1.57 59.95 4.70 119.85 119.95 1200 150 900 480.42 2.68162 2.74 74.87 8.63 149.60 149.85 1400 180 1236 413.90 2.61690 4.39 89.73 14.25 179.07 179.64 1600 210 1648 364.96 2.56225 6.57 104.50 21.87 208.08 209.24 1800 240 21 36 328.02 2.51590 9.38 119.12 31.77 236.44 238.54 2000 270 2700 299.71 2.47670 12.87 133.54 44.13 263.76 267.43 TABLES 95 TAPER 2|. INTERMEDIATE VALUES A =2 M I / b o D logD d t X V Ic 1 400 30.00 018 018 036 1432.55 3.15611 0.08 15.00 0.16 30.00 30.00 05 31.25 019 019J 0381 1403.33 3.14716 0.09- 15.62 0.17 31.25 31.25 10 32.50 019| 021 040f 1375.26 3.13838 0.09 16.25 0.19 32.50 32.50 ' 15 33.75 020J 022J 043 1348.31 3.12979 0.10 16.87 0.20 33.75 33.75 20 35.00 021f 023! 045} 1322.40 3.12136 0.11 17.50 0.22 35.00 35.00 25 36.25 022! 0251 048 1297.45 3.11309 0.11 18.12 0.24 36.25 36.25 30 37.50 023f 027 050! 1273.44 3.10498 0.12 18.75 0.26 37.50 37.50 35 38.75 024! 028} 0531 1250.30 3.09701 0.13 19.37 0.28 38.75 38.75 40 40.00 025f 0301 056 1227.99 3.08919 0.14 20.00 0.30 40.00 40.00 45 41.25 026f 032 058! 1206.46 3.08151 0.15 20.62 0.32 41.25 41.25 50 42.50 027f 033f OH 1185.66 3.07396 0.15 21.25 0.34 42.50 42.50 55 43.75 029 035| 104} 1165.59 3.06655 0.16 21.87 0.37 43.75 43.75 500 45.00 030 037^- 107} 1146.18 3.05925 0.17 22.50 0.39 45.00 45.00 05 46.25 03H 0391 10* 1127.38 3.05207 0.18 23.12 0.42 46.25 46.25 10 47.50 0321 0411 Wi 1109.23 3.04502 0.19 23.75 0.44 47.50 47.50 15 48.75 033} 0431 16! 1091.64 3.03808 0.20 24.37 0.47 48.75 48.75 20 50.00 034! 0451 20 1074.60 3.03125 0.21 25.00 0.50 50.00 50.00 25 51.25 036 0471 231 1058.09 3.02452 0.22 25.62 0.54 51.25 51.25 30 52.50 037 049} 1 26} 1042.08 3.01790 0.24 26.25 0.57 52.50 52.50 35 53.75 038} 051| 130 1026.54 3.01138 0.25 26.87 0.60 53.75 53.75 40 55.00 039! 053! 1 33| 1011.46 3.00495 0.26 27.50 0.63 55.00 55.00 45 56.25 041 056 137 996.83 2.99862 0.28 28.12 0.67 56.25 56.25 50 57.50 0421 0581 140} 982.61 2.99238 0.29 28.75 0.71 57.50 57.50 55 58.75 043! 100} 1441 968.78 2.98623 0.30 29.37 0.75 58.75 58.75 600 60.00 045 103 148 955.35 2.98016 0.31 29.99 0.79 59.99 60.00 15 63.75 0491 1101 159} 917.20 2.96246 0.36 31.87 0.91 63.74 63.75 30 67.50 053! 1 18 211! 882.00 2.94547 0.41 33.74 1.05 67.49 67.50 45 71.25 0581 126 2241 849.41 2.92912 0.46 35.62 1.21 71.24 71.25 700 75.00 103 134} 237} 819.15 2.91336 0.51 37.49 1.38 74.98 75.00 15 78.75 108 1431 2511 790.99 2.89817 0.57 39.36 1.56 78.73 78.74 30 82.50 1 13 152} 305} 764.72 2.88350 0.64 41.24 1.76 82.47 82.49 45 86.25 1 18} 202 320} 740.15 2.86932 0.71 43.11 1.97 86.22 86.24 800 90.00 124 212 336 717.12 2.85559 0.78 44.98 2.20 89.96 89.98 15 93.75 129! 2221 352 695.50 2.84230 0.86 46.85 2.45 93.70 93.73 30 97.50 135f 233 408f 675.17 2.82941 0.95 48.73 2.71 97.44 97.48 45 101.25 141! 244 425! 656.01 2.81691 1.04 50.60 2.99 101.18 101.23 96 THE RAILROAD TAPER TAPER 2|. INTERMEDIATE VALUES (Continued) A = 2 M I / b o D log/) d t X V Ic / 900 105.00 148 255} 4 43} 637.91 2.80476 1.13 52.47 3.30 104.92 104.97 15 108.75 154J 307} '501| 620.81 2.79296 1.23 54.34 3.62 108.66 108.71 30 112.50 2.01 3 19$ 520} 604.63 2.78149 .34 56.21 3.96 112.39 112.46 45 116.25 208 332 540 589.28 2.77032 .45 58.08 4.32 116.12 116.21 1000 120.00 2 15 345 600 574.71 2.75945 .57 59.95 4.70 119.85 119.9* 15 123.75 2221 3581 620} 560.86 2.74885 .69 61.82 5.11 123.58 123.69 30 127.50 229$ 4 12 641} 547.69 2.73853 .82 63 68 5.54 127.31 127.43 45 131.25 2371 426 7031 535.14 2.72847 .96 65.55 6.00 131.03 131.17 1100 135.00 245 440} 725} 523.17 2.71864 2.10 67.42 6.48 134.75 134.91 15 138.75 253 4551 7481 511.75 2.70906 2.25 69.28 6.98 138.47 138.65 30 142 50 301 510} 811} 500.84 2.69970 2.41 71.15 7.51 142.18 142.39 45 146.25 309} 526 835} 490.41 2.69056 2.57 73.01 8.06 145.89 146.12 1200 150.00 3 18 542 900 480.42 2.68162 2.74 74.87 8.63 149.60 149.85 15 153.75 326| 558} 9251 470.86 2.67289 2.92 76.74 9.23 153.30 153.58 30 157.50 335} 615 950} 461.69 2.66435 3.10 78.60 9.86 157.00 157.31 45 161.25 344f 632 1016| 452.91 2.65601 3.30 80.46 10.52 160.69 161.04 1300 165.00 354 649} 1043} 444.47 2.64784 3.50 82.32 11.20 164.37 164.76 15 168.75 403} 7071 11 10| 436.37 2.63985 3.71 84.17 11.92 168.06 168.48 30 172.50 413 725} 1138} 428.59 2.63204 3.93 86.03 12.67 171.74 172.20 45 176.25 423 744 1207 421.10 2.62439 4.15 87.89 13.45 175.41 175.92 1400 180.00 433 803 1236 413.90 2.61690 4.39 89.73 14.25 179.07 179.64 15 183.75 4431 8221 1305} 406.96 2.60955 4.63 91.59 15.09 182.73 183.35 30 187.50 453} 842 1335} 400.29 2.60237 4.88 93.44 15.96 186.38 187.06 45 191.25 5041 902 14061 393.86 2.59534 5.14 95.29 16.86 190.02 190 76 1500 195.00 515 932} 1437} 387.66 2.58845 5.41 97.14 17.79 193.65 194.46 30 202.50 5371 1004} 1541! 375.90 2.57507 5.97 100.82 19.76 200.89 201.86 1600 210.00 600 1048 1648 364.96 2.56225 6.57 104.50 21.87 208.08 209.24 30 217.50 623} 1133 1756} 354.76 2.54993 7.21 108.18 24.12 215.25 216.60 1700 225.00 6 48 12 19} 1907} 345.23 2.53811 7.89 111.85 26.52 222.37 223.94 30 232.50 7 13 13 07} 2020} 336.33 2.52677 8.61 115.49 29.07 229.43 231.26 1800 240.00 7 39 1 13 57 2136 328.02 2.51590 9.38 119.12 31.77 236.44 238.54 30 247.50 805} 1448 2253} 320.22 2.50545 10.18 122.76 34.63 243.38 245.83 1900 255.00 833 1540} 24 13} 312.94 2.49546 11.03 126.38 37.64 250.26 253.07 30 262.50 901 1634} 25 35} 306 10 2.48586 11.92 129.98 40.81 257.06 260.27 2000 270.00 930 1730 2700 299.71 2.47670 12.87 133.54 44.13 263.76 267.43 TABLES 97 TAPER 3. CURVATURE INCREASING 2 30' EVERY 30 FT. A Deflections to rr 1 rjinsit at B.C. c.c. C.C. C.C. C.C. C.C. C.C. C.C. C.C. 2 30' 5 00' 7 30' 10 00' 12 30' 15 00' 17 30' 20 00 ' 22 30' I 30 60 90 120 150 180 210 240 , 022* 056i 1 45 248f 407* 54H 730 933| 30 022* 045 141i 252* 4 18f 600 756* 1007* 60 1 18f 045 1 07* 226i 400 548f 752* 10 in 90 245 203f 1 07* 130 3 Hi 507* 718f 945 120 44l| 352* 248| 1 30 152* 356i 615 848| 150 707* 6 11J 500 333| 152* 215 4411 722* 180 1003f 900 741i 607* 4 18| 215 237* 526i 210 1330 1218f 10 52* 91U 715 503f 237* 300 240 1726i 1607* 1433f 1245 10 4H 822* 5481 300 M I o D logZ) d t X V U 1 500 30 045 1146.11 3.05923 0.10 15.00 0.20 30.00 30.00 730 60 215 764.47 2.88336 0.39 29.99 0.98 59.99 60.00 1000 90 430. 574.12 2.75900 0.98 44.97 2.75 89.94 89.98 1230 120 730 460.55 2.66328 1.96 59.91 5.88 119.77 119.91 1500 150 11 15 385.68 2.58623 3.43 74.80 10.77 149.37 149.76 1730 180 1545 333.21 2.52270 5.48 89.59 17.78 178.54 179.41 2000 210 2100 295.04 2.46988 8.20 104.22 27.25 207.01 208.78 2230 240 2700 266.74 2.42609 11.69 118.63 39.49 234.42 237.72 98 THE RAILROAD TAPER TAPER 3. INTERMEDIATE VALUES M I / 6 o D log> d t X y Ic 500 30.00 022* 022* 045 1146.11 3.05923 0.10 15.00 0.20 30.00 30.00 05 '31.00 023* 023! 047} 1127.30 3.05204 0.10 15.50 0.21 31.00 31.00 10 32.00 024} 025} 049* 1109.15 3.04499 0.11 16.00 0.22 32.00 32.00 15 33.00 025} 026f 052 1091.56 3.03805 0.12 16.50 0.24 33.00 33.00 20 34.00 026} 028} 054* 1074.51 3.03121 0.12 17.00 0.26 34.00 34.00 25 35.00 027 030 057 1058.00 3.02449 0.13 17.50 0.28 35.00 35.00 30 36.00 028 031* 059* 1041.98 3.01786 0.14 18.00 0.29 36.00 36.00 35 37.00 029 033 1 02 1026.44 3.01133 0.15 18.50 0.31 37.00 37.00 40 38.00 030 034* 104* 1011.35 3.00490 0.15 19.00 0.33 38.00 38.00 45 39.00 031 036} 107} 996.71 2.99857 0.16 19.50 0.35 39.00 39.00 50 40.00 032 038 1 10 982.49 2.99233 0.17 20.00 0.37 40.00 40.00 55 41.00 033i 039* 1 12! 968.66 2.98617 0.18 20.50 0.40 41.00 41.00 600 42.00 034} 041} 1 15* 955.23 2.98011 0.19 21.00 0.42 42.00 42.00 15 45.00 037* 046f 124} 917.05 2.96239 0.21 22.50 0.49 45.00 45.00 30 48.00 041 052* 133* 881.83 2.94538 0.24 24.00 0.57 48.00 48.00 45 51.00 044* 058| 143} 849.23 2.92903 0.28 25.50 0.66 51.00 51.00 700 54.00 048} 1 05} 153* 818.95 2.91326 0.31 26.99 0.76 53.99 54.00 15 57.00 052} nil 204 790.77 2.89805 0.35 28.49 0.87 56.99 57.00 30 60.00 056} 1 18f 2 15 764.47 2.88336 0.39 29.99 0.98 59.99 60.00 45 63.00 100* 1 26 226* 739.88 2.86916 0.44 31.49 1.11 62.99 63.00 800 66.00 1 04f 133! 238* 716.82 2.85541 0.48 32.99 1.24 65.98 66.00 15 69.00 109} 141* 250| 695.17 2.84209 0.53 34.49 1.39 68.98 69.00 30 72.00 1 14 149* 303* 674.81 2.82918 0.59 35.99 1.55 71.98 71.99 45 75.00 1 18! 158 316! 655.61 2.81665 0.64 37.48 1.72 74.97 74.99 900 78.00 123f 206f 330* 637.48 2.80447 0.70 38.98 1.90 77.97 77.99 15 81.00 128f 216 344! 620.35 2.79264 0.77 40.48 2.09 80.96 80.99 30 84.00 1 34 2 25* 359* 604.13 2.78113 0.84 41.98 2.30 83.95 83.99 45 87.00 1 39* i 2 35 414* 588.74 2.76992 0.91 43.47 2.52 86.95 86.98 1000 90.00 1 45 2 45 430 574.12 2.75900 0.98 44.97 2.75 89.94 89.98 15 ! 93.00 150| 255} 446 560.23 2.74837 1.06 46.47 2.99 92.93 92.98 30 ! 96.00 156* 306 502* 547.01 2.73800 1.14 47.96 3.25 95.92 95.97 45 ', 99.00 202* 316! 519} 534.41 2.72787 1.23 49.46 3.53 98.90 98.96 1100 102.00 208f 327f 536* 522.39 2.71799 1.32 50.95 3.82 101.89 101.96 15 105.00 215 339f 554} 510.92 2.70835 1.42 52.45 4.12 104.87 104.95 30 108.00 221* 351 612* 499.95 2.69893 1.52 53.94 4.44 107.86 107.95 45 111.00 228 403} 631} 489.46 2.68972 1.62 55.44 4.78 110.84 110.94 TABLES 99 TAPERS. INTERMEDIATE VALUES (Continued} A = 2| M I / b D log> d t X V Ic 1200 114.00 234! 415* 650i 479.41 2.68071 1.73 56.93 5.13 113.82 113.93 15 117.00 2 41f 4 28i 710 469.78 2.67189 1.84 58.42 5.50 116.80 116.92 30 120.00 248! 44U 730 460.55 2.66328 1.96 59.91 5.88 119.77 119.91 45 123.00 256 454* 750* 451.69 2.65484 2.08 61.41 6.29 122.75 122.90 1300 126.00 303J 508i 811* 443.18 2.64658 2.21 62.90 6.71 125.72 125.89 15 129.00 310| 522 832! 435.00 2.63849 2.34 64.39 7.15 128.69 128.88 30 132. OOj 3 18* 536 754* 427.14 2.63057 2.48 65.88 7.61 131.65 131.87 45 135.00 326| 550| 916! 419.58 2.62281 2.63 67.37 8.09 134.61 134.86 1400 138.00 334i 605i 939* 412.29 2.61520 2.78 68.86 8.58 137.57 137.84 15 141.00 342i 620* 1002! 405.26 2.60773 2.93 70.34 9.10 140.53 140.82 30 144.00 350* 636 1026* 398.50 2.60043 3.09 71.83 9.64 143.48 143.80 45 147.00 359 651* 1050* 391.98 2.59326 3.26 73.32 10.19 146.43 146.78 1500 150.00 407| 707* 11 15 385.68 2.58623 3.43 74.80 10.77 149.37 149.76 30 156. OOJ 425 740* 1205* 373.72 2.57255 3.79 77.77 11.99 155.25 155.71 1600 162.00 443i 8 14i 1257* 362.56 2.55938 4.17 80.73 13.30 161.11 161.65 30 168.00 502 849* 1351* 352.13 2.54670 4.58 83.69 14.70 166.94 167.58 1700 174.00 52H 926i 1447* 342.36 2.53448 5.02 86.64 16.19 172.75 173.50 30 180.00 54U 1003J 1545 333.20 .52270 5.48 89.58 17.78 178.54 179.41 1800 186.00 601! 1042f 1644* 324.61 .51136 5.97 92.52 19.47 184.30 185.31 30 192.00 623 11 22* 1745* 316.53 .50041 6.49 95.46 21.26 190.03 191.20 1900 198.00 644f 1203! 1848* 308.94 .48987 7.03 98.39 23.15 195.73 197.07 30 204.00 707 1246* 1953* 301.78 .47969 7.60 101.31 25.14 201.39 202.93 2000 210.00 730 1330 2100 295.04 .46988 8.20 104.22 27.25 207.01 208.78 30 216.00 753* 1415 22 08* 288.69 .46043 8.83 107.12 29.47 212.59 214.62 2100 222.00 817! 1500! 2318* 282.72 .45136 9.50 110.02 31.80 218.13 220.44 30 228.00 842* 1548 2430* 277.07 2.44259 10.19 112.90 34.25 223.62 226.23 2200 234.00 907! 1636! 2544* 271.75 2.43417 10.92 115.77 36.81 229.05 231.99 30 240.00 933! 1726J 2700 266.74 2.42609 11.69 118.63 39.49 234.42 237.72 100 THE RAILROAD TAPER TAPER 3|. CURVATURE INCREASING 3 00' EVERY 30 FT. Deflections to at B.C. C.C. C.C. C.C. C.C. C.C. C.C. C.C. 3 00' 6 00' 9 00' 12 00' 15 00' 18 00' 21 00' 24 00' I 30 60 90 120 150 180 210 o / 027 1 07! 206 O / 322| 457 649! 900 30 027 054 2 01! 327 5 10| 712 93H 60 1 34! 054 1 21 2 55^ 448 658| 927 90 3 18 228! 121 148 349| 609 846! 120 5 37| 439 3 22! 1 48 2 15 443! 730 150 833 725! 600 4164 215 242 537! 180 1204| 1048 913| 721 5 10| 242 309 210 16 12 1446| 1303 11 OH 842 604! 309 M l D log/) d t X V lc 600 30 054 955.16 2.98008 0.12 15. OC ) 0.24 30.00 30.00 900 60 242 637.25 2.80431 0.47 29. 9^ 1 1.18 59.98 60.00 1200 90 524 478.84 2.68021 1.18 44. 9f 3.30 89.91 89.97 1500 120 900 384.60 2.58501 2.35 59. 8 > 7.06 119.67 119.88 1800 150 1330 322.75 2.50887 4.11 74.71 12.91 149.10 149.65 2100 180 1854 279.78 2.44682 6.56 89.4021.29 177.91 179.17 2400 210 25 12 248.99 2.39618 9.82 103.8* 32.58 205.71 208.28 TABLES 101 TAPER 3i INTERMEDIATE VALUES' A- 3 M I / b D logZ) d t X V Ic 600 30.00 027 027 054 955.16 2.98008 0.12 15.00 0.24 30.00 30.00 15 32.50 029| 0311 101 916.98 2.96236 0.14 16.26 0.28 32.50 32.50 30 35.00 032| 035! 108* 881.75 2.94535 0.16 17.50 0.33 35.00 35.00 45 37.50 035* 040* 1 16 849.13 2.92897 0.18 18.75 0.39 37.50 37.50 700 40.00 038* 045* 124 818.84 2.91320 0.20 20.00 0.45 40.00 40.00 15 42.50 41| 50! 1 32* 790.65 2.89798 0.23 21.25 0.52 42.50 42.50 30 45.00 045 056* 141* 764.34 2.88329 0.26 22.49 0.59 45.00 45.00 45 47.50 048* 102 150* 739.73 2.86907 0.29 23.74 0.67 47.49 47.50 800 50.00 052 108 200 716.66 2.85531 0.32 24.99 0.76 49.99 50.00 15 52.50 055f 114| 210 694.99 2.84198 0.35 26.24 0.85 52.49 52.50 30 55.00 059| 120f 220* 674.61 2.82905 0.39 27.49 0.95 54.99 55.00 45 57.50 103| 127* 231 655.40 2.81651 0.43 28.74 1.06 57.49 57.50 900 60.00 107* 134* 242 637.25 2.80431 0.47 29.99 1.18 59.98 60.00 15 62.50 1 11| 1 411 253* 620.09 2.79245 0.51 31.24 1.31 62.48 62.50 30 65.00 1 16 149| 305* 603.85 2.78093 0.56 32.48 1.44 64.98 .64.99 45 67.50 2()| 157 317* 588.44 2.76970 0.61 33.73 1.58 67.48 67.49 1000 70.00 25 205 330 573.80 2.75876 0.66 34.98 1.73 69.97 69.99 15 72.50 29| 213| 343 559.88 2.74810 0.71 36.23 1.89 72.47 72.49 30 75.00 34* 221! 356* 546.64 2.73770 0.77 37.48 2.06 74.96 74.99 45 77.50 39* 230* 4 10 534.01 2.72755 0.83 38.73 2.24 77.45 77.49 1 1 00 80.00 44* 239* 424 521.96 2.71764 0.89 39.97 2.43 79.95 79.98 15 82.50 149f 248! 438* 510.46 2.70796 0.96 41.22 2.63 82.44 82.48 30 85.00 155 258* 453* 499.46 2.69850 1.03 42.46 2.84 84.93 84.98 45 87.50 200* 308 508* 488.94 2.68926 1.10 43.71 3.06 87.42 87.47 1200 90.00 206 318 524 478.86 2.68021 1.18 44.96 3.29 89.91 89.97 15 92.50 211! 3281 540 469.20 2.67136 1.26 46.20 3.54 92.40 92.47 30 95.00 217* 338! 556* 459.93 2.66269 1.34 47.45 3.80 94.89 94.96 45 97.50 223| 349* 613 451.03 2.65421 1.42 48.69 4.07 97.37 97.46 102 THE RAILROAD TAPER TAPfiR 3|. INTERMEDIATE VALUES (Continued) A = M I / ft D log-D d t X V Ic 1300 100.00 229* 400* 630 442.48 2.64589 1.51 49.94 4.35 99.86 99.95 15 102.50 235f 411f 647J 434.26 2.63775 1.60 51.18 4.64 102.35 102.45 30 105.00 242 423| 705* 426.36 2.62978 1.70 52.42 4.94 104.82 104.94 45 107.50 248i 435 723| 418.752.62195 1.80 53.67 5.26 107.30 107.43 1400 110.00 255 447 742 411.41 2.61427 1.90 54.91 5.59 109.78 109.92 15 112.50 301f 459* 801 404.34 2.60675 2.01 56.15 5.94 112.26 112.41 30 115.00 308| 511! 820* 397.53 2.59937 2.12 57.39 6.30 114.73 114.90 45 117.50 315| 524| 840 390.95 2.59212 2.23 58.64 6.67 117.20 117.39 1500 120.00 322| 537| 900 384.60 2.58501 2.35 59.88 7.06 119.67 119.88 30 125.00 337 604* 941* 372.53 2.57116 2.60 62.36 7.88 124.61 124.85 1600 130.00 352 632 1024 361.26 2.55782 2.87 64.84 8.76 129.54 129.82 30 135.00 407 700! 1108* 350.70 2.54494 3.15 67.31 9.70 134.45 134.79 1700 140.00 423 730| 1154 340.79 2.53249 3.45 69.78 10.70 139.34 139.75 30 145.00 440 801J 1241* 331.49 2.52047 3.77 72.25 11.77 144.22 144.70 180Q 150.00 457 833 1330 322.75 2.50887 4.11 74.71 12.91 149.10 149.65 30 155.00 514| 905! 1420* 314.51 2.49763 4.47 77.17 14.12 153.96 154.59 1900 160.00 532J 939| 1512 306.75 2.48678 4.84 79.63 15.40 158.78 159.52 30 165.00 551 1014* 1605* 299.42 2.47628 5.24 82.08 16.76 163.59 164.44 2000 170.00 610 1050 1700 292.50 2.46613 5.66 84.52 18.19 168.39 169.36 30 175.00 629J 1126! 17 56* 285.96 2.45631 6.10 86.96 19.70 173.16 174.27 2100 180.00 649J 12041 1854 279.78 2.44682 6.56 89.40 21.29 177.91 179.17 30 185.00 710 12 43 4 l 1953* 273.92 2.43762 7.04 91.83 22.96 182.62 184.06 2200 190.00 731 1323 2054 268.382.42875 7.55 94.25 24.71 187.30 188.93 30 195.00 752* 1403f 21 56* 263.132.42017 8.08 96.67 26.55 191.95 193.79 2300 200.00 814| 1445 2300 258.172.41191 8.64 99.08 28.47 196.59 198.63 2400 210.00 900 1612 25 12 248.99 2-.39618 9.82 103.88 32.58 205.71 208.28 1 TABLES 103 TAPER 3*. CURVATURE INCREASING 4 00' EVERY 30 FT. Deflections to at B.C. C.C. C.C. C.C. C.C. C.C. C.C. 4 00' 8 00' 12 00' 16 00' 20 00' 24 00' 28 00' I 30 60 90 120 150 180 / 036 1 30 2 48 430 636 906 30 036 1 12 242 436 6 54 936 60 206 1 12 1 48 3 54 624 9 18 90 424 3 18 1 48 224 506 8 12 120 730 612 430 224 300 618 150 1124 954 800 542 300 336 180 1606 1424 1218 948 654 336 M l D logD d t X I lc 800 30 1 12 716.50 2.85522 0.16 15.00 0.31 30. OC 30.00 1200 60 336 478.31 2.67971 0.63 29.98 1.57 59.97 59.99 1600 90 712 359.96 2.55625 1.57 44.92 4.39 89.84 89.95 2000 120 1200 289.97 2.46235 3.13 59.78 9.40 119.42 119.79 2400 150 1800 244.63 2.38851 5.46 74.49 17.17 148. 4C 149.39 2800 180 25 12 213.85 2.33011 8.71 88.95 28.24 176.28 178.52 104 THE KAILROAD TAPER TAPER 3. INTERMEDIATE ^VALUES A = 4 M I / b o D *D d t X V Ic 800 30.00 036 036 12 716.50 2.85522 0.16 15.00 0.31 30.00 30.00 15 31.88 038! 040 181 694.82 2.84187 0.18 15.93 0.36 31.87 31.87 30 33.75 041$ 044$ 26 674.42 2.82893 0.20 16.87 0.41 33.74 33.75 45 35.62 0441 0491 33$ 655.19 2.81637 0.22 17.81 0.46 35.62 35.62 900 37.50 0471 054 411 637.02 2.80415 0.24 18.74 0.52 37.50 37.50 15 39.38 0501 059 491 619.84 2.79228 0.26 19.68 0.58 39.37 39.37 30 41.25 053$ 104 1 57$ 603.58 2.78073 0.29 20.62 0.64 41.24 41.25 45 43.12 056! 1 09$ 2061 588.15 2.76949 0.32 21.55 0.71 43.12 43.12 1000 45.00 100 1 15 2 15 573.49 2.75853 0.35 22.49 0.79 44.99 45.00 15 46.88 103$ 1 20! 2241 559.55 2.74784 0.38 23.43 0.87 46.87 46.87 30 48.75 107 1 26| 233$ 546.28 2.73742 0.41 24.36 0.95 48.74 48.75 45 50.62 110$ 132f 2431 533.62 2.72723 0.44 25.30 .04 50.61 50.62 1100 52.50 141 139 2531 521.54 2.71729 0.47 26.24 .13 52.48 52.50 15 54.38 18 145$ 303$ 510.01 2.70758 0.51 27.17 .23 54.35 54.37 30 56.25 22 1 52 3 14 498.98 2.69808 0.55 28.11 .34 56.22 56.24 45 58.12 26 1 58! 324! 488.43 2.68880 0.59 29.04 .45 58.10 58.12 1200 60.00 130 206 336 478.31 2.67971 0.63 29.98 .57 59.97 59.99 15 61.88 1341 2131 347$ 468.61 2.67081 0.67 30.91 .69 61.84 61.87 30 63.75 138| 2 20$ 359 459.31 2.66211 0.72 31.85 .82 63.71 63.74 45 65.62 142! 2281 411 450.38 2.65358 0.77 32.79 .96 65.59 65.62 1300 67.50 1471 236 4231 441.79 2.64522 0.82 33.72 2.11 67.46 67.49 15 69.38 151! 244 435! 433.53 2.63702 0.87 34.65 2.26 69.33 69.36 30 71.25 1 56f 252 448$ 425.58 2.62898 0.92 35.59 2.42 71.19 71.23 45 73.12 2011 300$ 501! 417.92 2.62109 0.97 36.52 2.58 73.06 73.11 1400 75.00 206 309 515 410.54 2.61336 1.03 37.46 2.75 74.93 74.98 15 76.88 2 11 317! 528! 403.42 2.60576 .09 38.39 2.93 76.79 76.85 30 78.75 216 326$ 542$ 396.56 2.59831 .15 39.33 3.12 78.66 78.72 45 80.62 221 335! 556! 889.94 2.59100 .22 40.26 3.31 ' 80.53 80.59 1500 82.50 2 261 345 6111 383.53 2.58380 .28 41.19 3.51 82.39 82.48 30 86.25 237 404 6*41 371.35 2.56978 .42 43.06 3 93 86.12 86.21 1600 90.00 248 424 712 359.96 2.55625 .57 44.92 4.39 89.84 89.95 30 93.75 259$ 444$ 744 349.28 2.54317 .73 46.79 4.89 93.56 93.69 1700 97.50 3 111 506 8171 339.23 2.53049 .89 48.65 5.42 97.27 97.43 30 101.25 323 i 528 851$ 329.79 2.51824 2.07 50.51 5.98 100.98 101.16 1800 105.00 336 551 927 320.90 2.50637 2.26 52.37 6.59 ;104.68 104.89 30 108.75 349 614$ 1003$ 312.50 2.49485 2.46 54.22 7.23 108.38 108.62 1900 112.50 4021 639 10411 304.58 2.48370 2.67 56.08 7.91 112.07 112.34 30 116.25 416 704 1120 297.07 2.47286 2.89 57.93 8.63 '115.75 116.07 2000 120.00 430 730 1200 289.97 2.46235 3.13 59.78 9.40 1119.42 119.79 30 123.75 444$ 756$ 1241 283.24 2.45215 3.38 61.63 10.21 123.08 123.51 2100 127.50 4591 824 13231 276.86 2.44226 3.64 63.48 11.06 126.74 127.22 30 131.25 514$ 852 1406$ 270.79 2.43263 3.91 65.32 11.96 130.38130.93 TABLES 105 TAPER 3|. INTERMEDIATE VALUES (Continued) A = 4 M I / b D logZ) d t X V Ic 2200 135.00 530 921 1451 265.02 2.42328 4.19 67.16 12.90 134.01 134.63 30 138.75 546 950 1536 259.53 2.41419 4.48 69.00 13.90 137.63 138.33 2300 142.50 602i 1021 1623i 254.33 2.40540 4.80 70.83 14.94 141.23 142.02 2400 150.00 636 1124 1800 244.63 2.38851 5.46 74.49 17.17 148.40 149.39 2500 157.50 7 llj 1230 19 4U 235.82 2.37258 6.18 78.14 19.61 155.50 156.73 2600 165.00 748 1339 2127 227.81 2.35757 6.97 81.76 22.26 162.52 164.04 2700 172.50 826^ 1451 23 m 220.51 2.34343 7.81 85.37 25.14 169.45 171.31 2800 180.00 906 1606 2512 213.85 2.33011 8.71 88.95 28.24 176.29 178.52 TAPER 3f. CURVATURE INCREASING 5 00' EVERY 30 FT. A = 5 Transit at Deflections to B.C. 5 00' C.C. 10 00' C.C. 15 00' C.C. 20 00' C.C. 25 00' C.C. 30 00' I 30 60 90 120 150 30 60 90 120 150 045 237 530 922^ 1415 045 "l30' 407 745 12 22| / 1521 130 / 330 322| 215 "300" 707 537 545 452i 300 '345" 8 15 837 800 622i 345 215 537^ 1000 M I D log> d t X y lc / 1000 1500 2000 2500 3000 30 130 60 430 90 900 120 15 00 150 22 30 573.34 383.13 288.80 233.54 198.34 2.75841 2.58323 2.45939 2.36836 2.29741 0.20 0.78 1.96 3.90 6.81 14.99 29.97 44.88 59.66 74.21 0.39 1.96 5.49 11.73 21.89 30.00 59.96 89.75 119.09 147.50 30.00 59.99 89.92 119.67 149.04 106 THE RAILROAD TAPER TAPER 3|. INTERMEDIATE VALUES M I / b D logD d t X f Ic 1000 30.00 045 / 045 1 30 573.34 2.75841 0.20 14.99 0.39 30.00 30.00 15 31.50 0471 049 136| 559.38 2.74771 0.21 15.74 0.43 31.50 31.50 30 33.00 050} 053! 144 546.10 2.73727 0.23 16.49 0.48 33.00 33.00 45 34.50 053} 058 151} 533.43 2.72708 0.25 17.24 0.53 34.50 34.50 1100 36.00 056 102| 158f 521.34 2.71712 0.27 17.99 0.59 36.00 36.00 15 37.50 059} 107} 206* 509.79 2.70739 0.29 18.74 0.65 37.49 37.50 30 39.00 102 1 12* 214* 498.74 2.69787 0.31 19.49 0.71 38.99 39.00 45 40.50 105} 117* 222! 488.18 2.68858 0.34 20.24 0.77 40.49 40.50 1200 42.00 108* 1 221 231} 478.05 2.67947 0.37 20.99 0.84 41.99 42.00 15 43.50 1 111 1 28 2391 468.34 2.67056 0.40 21.74 0.91 43.49 43.50 30 45.00 1 15 133| 2481 459.02 2.66183 0.43 22.49 0.98 44.99 45.00 45 46.50 1 18} 139} 2571 450.07 2.65328 0.46 23.23 .06 46.49 46.50 1300 48.00 122 145} 307} 441.46 2.64489 0.49 23.98 .14 47.98 48.00 15 49.50 125* 151} 3161 433.18 2.63667 0.52 24.73 .23 49.48 49.50 30 51.00 129 157* 326* 425.21 2.62860 0.55 25.48 .32 50.98 51.00 45 52.50 132f 2 03 f 336* 417.53 2.62069 0.58 26.23 .42 52.48 52.50 1400 54.00 136| 210} 3461 410.13 2.61292 0.62 26.98 1.52 53.97 53.99 15 55.50 1 40* 2161 357} 402.99 2.60529 0.66 27.72 62 55.47 55.49 30 57.00 144} 223f 408 396.11 2.59782 0.70 28.47 .73 56.97 56.99 45 58.50 148} 230* 4181 389.46 2.59046 0.74 29.22 1.84 58.46 58.49 1500 60.00 152* 237* 430 383.03 2.58323 0.78 29.97 1.96 59.96 59.99 30 63.00 201 252 453 370.80 2.56914 0.87 31.46 2.21 62.95 62.99 TABLES 107 TAPER 3|. INTERMEDIATE VALUES - (Continued) A = 5 M I / b D logD d t X y Ic 1600 66.00 209| 307 1 5161 359.35 2.55552 0.96 32.96 2.48 64.44 64.99 30 69.00 218* 323 541* 348.61 2.54234 1.06 34.45 2.77 68.92 68.98 1700 72.00 227f 339* 6071 338.51 2.52957 1.17 35.94 3.09 71.90 71.97 30 75.00 237* 356i 6331 329.00 2.51720 .28 37.44 3.43 74.88 74.97 1800 78.00 247* 4 131 701i 320.04 2.50520 .40 38.92 3.79 77.86 77.96 30 81.00 257* 432 729* 311.57 2.49356 .53 40.41 4.18 80.84 80.95 1900 84.00 308 4-501 7581 303.58 2.48227 .67 41.90 4.59 83.81 83.94 30 87.00 319 510 829 295.99 2.47128 .81 43.39 5.03 86.78 86.93 2000 90.00 330 530 900 288.80 2.46060 .96 44.88 5.49 89.75 89.92 30 93.00 341* 550| 932 281.98 2.45022 2.12 46.36 5.98 92.71 92.91 2100 96.00 353 611f 10 04J 275.50 2.44012 2.28 47.85 6.50 95.67 95.89 30 99.00 405 633* 1038* 269.33 2.43028 2.45 49.33 7.05 98.62 98.87 2200 102.00 417* 6551 11 13J 263.46 2.42071 2.63 50.81 7.63 101.56 101.85 30 105.00 430 7 181 11481 257.872.41140 2.82 52.29 8.23 104.50 104.82 2300 108.00 443 742J 12251 252.55 2.40235 3.02 53.76 8.86 107.44 107.78 2400 114.00 509* 83U 13401 242.61 2.38491 3.44 56.72 10.23 113.28 113.73 2500 120.00 537* 922* 1500 233.54 2.36836 3.90 59.66 11.73 119.09 119.67 2600 126.00 606* 1016J 16221 225.24 2.35265 4.40 62.59 13.37 124.87 125.59 2700 132.00 637 11 121 17491 217.64 2.33774 4.94 65.51 15.15 130.61 131.49 2800 138.00 708* 12 10f 19 191 210.66 2.32358 5.52 68.42 17.07 136.29 137.36 2900 144.00 741 13111 20521 204.24 2.31014 6.14 71.32 19.15 141.93 143.21 3000 150.00 815 14 15 2230 198.34 2.29741 6.81 74.21 21.39 147.50 149.04 108 THE RAILROAD TAPER * 9 .:::::::; ;;;;;: <* CO* . . . . . . . . . . . . . . CO Hw CO : : : : : : : : : : : : : : : ,^: cq w 2r c? i-i o ^^ 10 ^^ ^O CO ^ O T-I CO -^ o o o o o o O O I-H ^ 1-1 i-< TABLES 109 Oo o lO o O O Oo 880 8 SH s safe > Oo OO T ~ t QC'o I O O o C^ r_| CO O * 883 888s OOo & SSff C^ O (M O ~>'^O ^-~>^O f-o"QO i^i^O O "-^ 'tS O 8 8 (N 110 THE RAILROAD TAPER $ ft ^ Oo 52* CO o 2 3 82~ 82g S8g 82 r Hr :* 893- ! ^r I ISs 15? X. > JS? o co ^ ~^"t3O ~~s^Ci ^"eo o -<.-e TABLES 111 O CO O ^ O C3 ^ O^t 1 ^ Oo COOo cO O *^ O OS *" O ^f ' -H -* o O 8 I, 8 00 - Oo c ^ 10 rH' CO o "- 1 CO 83 88;? ' CO ScO^ ^^So* P^c Oo CO' (M O 22 -S O nnoco C o OO o (M CO 8^8 O T i o | T i o Or- < OI-HO O 1 S 838 O-^o og 8Ssg, 88g| 0< OCOo OOo 8^^ 8^ : OOo OI-HO m co co 888a 81 O OOo OOo S co 22 ^ ^ CQ ,-^ *^ ^- 8 8 8 oo TABLES St">- *" co^ 11 113 33? SBfe 889 -o OiojOQ C^^^fN i* ?? |t PI P| p| i ^? T * ^. 4O00 10 ^ ^ ^ s ST CO 2 ^ J ?Q ^^C^ 1 "s "^3 O "-o ^ O ~o ^J O 4 8 ' 8 g 8 8 o o o o o 22 S 8^ 8 S TABLES ALIGNMENT REVISION MAINTAINING LENGTH UNCHANGED (See page 55, par. 43) 115 M r 00' 130' 2 00' A 1 1 1 i Taper. i * * 1 I I h M' I h M' I h M' I h M' 2 30' 5 10 15 20 25 110.00 96.00 0.04 0.14 103' m. 60 158.00 154.00 0.01 0.23 0.35 148' 134' 132' 266!6o 224.00 218.00 214.00 QA2 0.37 0.50 0.26 2 25' 2 07' 2 04' 2 02' 110.00 94.00 92.00 0.07 0.22 0.19 2 20' 2 04' 2 02' M 2 30' 3 00' A 1 i i ! Taper. i i 1 H ' I k M' i h M' I h M' I h 0.03 0.12 0.17 M' 10 15 20 25 30 40 50 350.00 296.00 286.00 280.00 276.00 274.00 0.84 0.71 0.87 0.75 1.05 1.25 3 10' 2 43' 2 38' 2 35' 2 33' 2 32' 132.00 125.00 123.00 0.15 0.19 0.27 2 42' 2 35' 2 33' 165.00 157.00 154.00 153.00 0.31 0.32 0.25 0.35 3 14' 3 07' 3 04' 3 03' 96.67 93.33 92.00 3 10' 3 05' 3 03' M 4 00' A f i > Taper. 1 H 2 I I h M' I h M' I h M' 10 15 20 25 30 40 50 60 i56!66 '6.34 4 30' 139.33 0.35 4 14' 135.33 0.31 4 08' 133.33 0.24 4 05' 132.00 0.35 4 03' 110.00 0.15 4 40' 96.00 0.12 4 12' 93.00 0.10 4 06' 92.00 0.15 4 04' 91.50 0.20 4 03' 259! 66 233.00 224.00 217.00 214.00 213.00 0.81 0.76 0.69 0.51 0.75 4 49' 4 23' 4 14' 4 07' 4 04' 4 03' 116 THE RAILROAD TAPER MAINTAINING LENGTH UNCHANGED (Continued} M 5 00' A 1 1 1 If Taper. 1 | 2 21 I I h M' I h AT I h . M' I i M' 5 30' 5 13' 5 08' 5 05' 5 03' 15 20 25 30 40 50 60 70 80 90 154.00 132.50 127.00 124.50 122.50 121.50 0.35 0.34 0.32 0.29 0.32 0.28 6 08' 5 25' 5 14' 5 09' 5 05' 5 03' 102.00 95.20 93.20 92.00 91.20 0.25 0.17 0.18 0.13 0.19 6 19' 5 26' 5 15' 5 10' 5 07' 5 05' 5 04' 223.33 190.00 182.67 176.67 174.00 172.67 172.00 0.76 0.58 0.64 0.70 0.66 0.62 0.72 6 20' 5 30' 5 19' 5 10' 5 06' 5 04' 5 03' 349.00 296.00 285.00 280.00 277.00 275.00 274.00 1.64 1.43 1.50 1.60 1.53 1.31 1.48 M 6 00' A f 1 li 1^ Taper. 5 2 2i 2* I z h M' I h M' I h M' I h M' 15 20 25 30 40 50 60 70 80 90 0^63 0.58 0.66 0.52 0.76 0.58 111.00 98.33 95.00 93.33 92.00 91.00 0.25 0.22 0.24 0.25 0.34 0.14 7 03' 6 25' 6 15' 6 10' 6 06' 6 03' 6^47' 6 27' 6 14' 6 08' 6 06' 6 04' 131.60 123.60 120.00 117.20 116.00 115.20 0.35 0.36 0.33 0.35 0.34 0.20 6 44' 6 24' 6 15' 6 08' 6 05' 6 03' 258.67 228.67 220.67 217.33 215.33 214.00 212.67 1.21 1.12 1.09 1.25 1.35 1.39 0.72 7 13' 6 28' 6 16' 6 11' 6 08' 6 06' 6 04' 173.50 163.50 157.00 154.00 153.00 152.00 :::: ::::: 5 8 00' A 1 H H - 2 Taper. 2 2J 2* 2! / I h M' I h M' I h M' I h 0.43 0.40 0.43 0.31 0.31 0.43 0.33 M' 20 25 30 40 50 60 70 80 90 112.25 101.00 97.00 93.25 92.00 91.50 91.00 9 29' 8 44' 8 28' 8 13' 8 08' 8 06' 8 04' 176.67 150.00 139.00 135.33 133.67 132.67 132.00 131.67 0.71 0.67 0.64 0.63 0.70 0.75 0.74 0.95 10 20' 9 00' 8 27' 8 16' 8 11' 8 08' 8 06' 8 05' 258 '50 232.50 224.00 219.50 217.00 215.50 !ei .58 .57 .48 .50 .65 9 d 37' 8 45' 8 28' 8 19' 8 14' 8 11' 220.40 179.60 172.00 168.40 166.40 165.60 164.80 1.04 0.96 1.08 0.92 0.81 1.14 1.18 10 26' 8 44' 8 25' 8 16' 8 11' 8 09' 8 07' M 10 00' A U ij 2 2^ Taper. 2i 2* 2| 3 I I h M' J h A/' I h M' I h M' 25 30 40 50 60 70 80 90 258 '.00 234.80 226.40 221.60 218.80 '6!76 0.69 0.65 0.73 0.64 0.71 0.82 12 22' 10 50' 10 28' 10 19' 10 13' 10 10' 1008' 112.00 101.40 95.40 93.40 92.20 91.60 91.20 91.00 0.45 0.41 0.39 0.45 0.39 0.42 0.41 0.53 1150' 1057' 1027' 10 17' ioir 1008' 1006' 10 05' L99 1.90 1.95 1.86 1.90 12 bb' iro2 10 41 10 29 10 22 2i9!6o 190.67 182.67 178.67 176.33 175.00 ^36 .32 .33 .31 .30 1.46 12 27' 11 02' 10 38' 10 26' 10 19' 10 15 155.50 132.50 127.00 124.75 123.25 122.50 122. OC LOGARITHMS OF NUMBERS 117 100 1 2 3 4 5 6 7 8 9 110 1 4 5 6 7 8 9 120 1 2 3 4 5 6 7 8 9 130 1 2 3 4 5 6 7 8 9 140 1 2 3 4 5 6 7 8 9 150 3456789 00000 00043 00087 00130 00173 00217 00260 00303 00346 00389 0432 0475 0518 0561 0604 0647 0689 0732 0775 0817 0860 0903 0945 0988 1030 1072 1115 1157 1199 1242 1284 1326 1368 1410 1452 1494 1536 1578 1620 1662 1703 1745 1787 1828 1870 1912 1953 1995 2036 2078 2119 2160 2202 2243 2284 2325 2366 2407 2449 2490 2531 2572 2612 2653 2694 2735 2776 2816 2857 2898 2938 2979 3019 3060 3100 3141 3181 3222 3262 3302 3342 3383 3423 3463 3503 3543 3583 3623 3663 3703 3743 3782 3822 3862 3902 3941 3981 4021 4060 4100 04139 04179 04218 04258 04297 04336 04376 04415, 04454 04493 4532 4571 4610 4650 4689 4727 4766 4805 4844 4883 4922 4961 4999 5038 5077 5115 5154 5192 5231 5269 5346 5385 5423 5461 5500 5538 5576 5614 5652 5690 5729 5767 5805 5843 5881 5918 5956 5994 6032 6070 6108 6145 6183 6221 6258 6296 6333 6371 6408 6446 6483 6521 6558 6595 6633 6670 6707 6744 6781 6819 6856 6893 6930 6967 7004 7041 7078 7115 7151 7188 7225 7262 7298 7335 7372 7408 7445 7482 7518 7555 7591 7628 7664 7700 7737 7773 7809 7846 7882 07918 07954 07990 08027 08063 08099 08135 08171 08207 08243 8279 8314 8350 8386 8422 8458 8493 8529 8565 8600 8636 8672 8707 8743 8778 8814 8849 8884 8920 8955 8991 9026 9061 9096 9132 9167 9202 9237 9272 9307 9342 9377 9412 9447 9482 9517 9552 9587 9621 9656 9691 9726 9760 9795 9830 9864 9899 9934 9968 10003 10037 10072 10106 10140 10175 10209 10243 10278 10312 0346 0380 0415 0449 0483 0517 0551 0585 0619 0653 0687 0721 0755 0789 0823 0857 0890 0924 0958 0992 1025 1059 1093 1126 1160 1193 1227 1261 1294 1327 1361 11394 11428 11461 11494 11528 11561 11594 11628 11661 11694 1727 1760 1793 1826 1860 1893 1926 1959 1992 2024 2057 2090 2123 2156 2189 2222 2254 2287 2320 2352 2385 2418 2450 2483 2516 2548 2581 2613 2646 2678 2710 2743 2775 2808 2840 2872 2905 2937 2969 3001 3033 3066 3098 3130 3162 3194 3226 3258 3290 3322 3354 3386 3418 3450 3481 3513 3545 3577 3609 3640 S672 3704 3735 3767 3799 3830 3862 3893 3925 3956 3988 4019 4051 4082 4114 4145 4176 4208 4239 4270 4301 4333 4364 4395 4426 4457 4489 4520 4551 4582 14613 14644 14675 14706 14737 14768 14799 14829 14860 14891 4922 4953 4983 5014 5045 5076 5106 5137 5168 5198 5229 5259 5290 5320 5351 5381 5412 5442 5473 5503 5534 5564 5594 5625 5655 5685 5715 5746 5776 5806 5836 5866 5897 5927 5957 5987 6017 6047 6077 6107 6137 6167 6197 6227 6256 6286 6316 6346 6376 6406 6435 6465 6495 6524 6554 6584 6613 6643 6673 6702 6732 6761 6791 6820 6850 6879 6909 6938 6967 6997 7026 7056 7085 7114 7143 7173 7202 7231 7260 7289 7319 7348 7377 7406 7435 7464 7493 7522 7551 7580 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 118 LOGARITHMS OF NUMBERS N 150 1 2 3 4 5 6 7 8 9 160 1 2 3 4 5 6 7 170 1 2 3 4 5 6 7 8 9 180 1 2 3 4 5 6 7 8 190 1 2 3 4 5 6 7 8 9 200 23456789 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 7898 7926 7955 7984 8013 8041 8070 8099 8127 8156 8184 8213 8241 8270 8298 8327 8355 8384 8412 8441 8469 8498 8526 8554 8583 8611 8639 8667 8696 8724 8752 8780 8808 8837 8865 8893 8921 8949 8977 9005 9033 9061 9089 9117 9145 9173 9201 9229 9257 9285 9312 9340 9368 9396 9424 9451 9479 9507 9535 9562 9590 9618 9645 9673 9700 9728 9756 9783 9811 9838 9866 9893 9921 9948 9976 20003 20030 20058 20085 20112 20140 20167 20194 20222 20249 0276 0303 0330 0358 0385 20412 20439 20466 20493 20520 20548 20575 20602 20629 20656 0683 0710 0737 0763 0790 0817 0844 0871 0898 0925 0952 0978 1005 1032 1059 1085 1112 1139 1165 1192 1219 1245 1272 1299 1325 1352 1378 1405 1431 1458 1484 1511 1537 1564 1590 1617 1643 1669 1696 1722 1748 1775 1801 1827 1854 1880 1906 1932 1958 1985 2011 2037 2063 2089 2115 2141 2167 2194 2220 2246 2272 2298 2324 2350 2376 2401 2427 2453 2479 2505 2531 2557 2583 2608 2634 2660 2686 2712 2737 2763 2789 2814 2840 2866 2891 2917 2943 2968 2994 3019 23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 3300 3325 3350 3376 3401 3426 3452 3477 3502 3528 3553 3578 3603 3629 3654 3679 3704 3729 3754 3779 3805 3830 3855 3880 3905 3930 3955 3980 4005 4030 4055 4080 4105 4130 4155 4180 4204 4229 4254 4279 4304 4329 4353 4378 4403 4428 4452 4477 4502 4527 4551 4576 4601 4625 4650 4674 4699 4724 4748 4773 4797 4822 4846 4871 4895 4920 4944 4969 4993 5018 5042 5066 5091 5115 5139 5164 5188 5212 5237 5261 5285 5310 5334 5358 5382 5406 6431 5455 5479 5503 25527 25551 25575 25600 25624 25648 25672 25696 25720 25744 5768 5792 5816 5840 5864 5888 5912 5935 5959 5983 6007 6031 6055. 6079 6102 6126 6150 6174 6198 6221 6245 6269 6293 6316 6340 6364 6387 6411 6435 6458 6482 6505 6529 6553 6576 6600 6623 6647 6670 6694 6717 6741 6764 6788 6811 6834 6858 6881 6905 6928 6951 6975 6998 7021 7045 7068 7091 7114 7138 7161 7184 7207 7231 7254 7277 7300 7323 7346 7370 7393 7416 7439 7462 7485 7508 7531 7554 7577 7600 7623 7646 7669 7692 7715 7738 7761 7784 7807 7830 7852 27875 27898 27921 27944 27967 27989 28012 28035 28058 28081 8103 8126 8149 8171 8194 8217 8240 8262 8285 8307 8330 8353 8375 8398 8421 8443 8466 8488 8511 8533 8556 8578 8601 8623 8646 8668 8691 8713 8735 8758 8780 8803 8825 8847 8870 8892 8914 8937 8959 8981 9003 9026 9048 9070 9092 9115 9137 9159 9181 9203 9226 9248 9270 9292 9314 9336 9358 9380 9403 9425 9447 9469 9491 9513 9535 9557 9579 9601 9623 9645 9667 9688 9710 9732 9754 9776 9798 9820 9842 9863 9885 9907 9929 9951 9973 999430016300383006030081 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 LOGARITHMS OF NUMBERS 119 N|0123456789 200 1 2 3 4 5 6 7 210 1 2 3 4 5 6 7 8 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 032& 0341 0363 0384 0406 0428 0449 0471 0492 0514 0535 0557 0578 0600 0621 0643 0664 068*5 0707 0728 0750 0771 0792 0814 0835 0856 0878 0899 0920 0942 0963 0984 1006 1027 1048 1069 1091 1112 1133 1154 1175 1197 1218 1239 1260 1281 1302 1323 1345 1366 1387 1408 1429 1450 1471 1492 1513 1634 1555 1576 1597 1618 1639 1660 1681 1702 1723 1744 1765 1785 1806 1827 1848 1869 1890 1911 1931 1952 1973 1994 2015 2035 2056 2077 2098 2118 2139 2160 2181 2201 32222 32243 32263 32284 32305 32325 32346 32366 32387 32408 2428 2449 2469 2490 2510 2531 2552 2572 2593 2613 2634 2654 2675 2695 2715 2736 2756 2777 2797 2818 2838 2858 2879 2899 2919 2940 2960 2980 3001 3021 3041 3062 3082 3102 3122 3143 3163 3183 3203 3224 3244 3264 3284 3304 3325 3345 3365 3385 3405 3425 3445 3465 3486 3506 3526 3546 3566 3586 3606 3626 3646 3666 3686 3706 3726 3746 3766 3786 3806 3826 3846 3866 3885 3905 3925 3945 3965 3985 4005 4025. 4044 4064 4084 4104 4124 4143 4163 4183 4203 4223 34242 34262 34282 34301 34321 34341 34361 34380 34400 34420 4439 4459 4479 4498 4518 4537 4557 4577 4596 4616 4635 4655 4674 4694 4713 4733 4753 4772 4792 4811 4830 4850 4869 4889 4908 4928 4947 4967 4986 5005 5025 5044 5064 5083 5102 5122 5141 5160 5180 5199 5218 5238 5257 5276 5295 5315 5334 5353 5372 5392 5411 5430 5449 5468 5488 5507 5526 5545 5564 5583 5603 5622 5641 5660 5679 5698 5717 5736 5755 5774 5793 5813 5832 5851 5870 5889 5908 5927 5946 5965 5984 6003 6021 6040 6059 6078 6097 6116 6135 6154 36173 36192 36211 36229 36248 36267 36286 36305 36324 36342 6361 6380 6399 6418 6436 6455 6474 6493 6511 6530 6549 6568 6586 6605 6624 6642 6661 6680 6698 6717 6736 6754 6773 6791 6810 6829 6847 6866 6884 6903 6922 6940 6959 6977 6996 7014 7033 7051 7070 7088 7107 7125 7144 7162 7181 7199 7218 7236 7254 7273 7291 7310 7328 7346 7365 7383 7401 7420 7438 7457 7475 7493 7511 7530 7548 7566 7585 7603 7621 7639 7658 7676 7694 7712 7731 7749 7767 7785 7803 7822 7840 7858 7876 7894 7912 7931 7949 7967 7985 8003 38021 38039 38057 38075 38093 38112 38130 38148 38166 38184 8202 8220 8238 8256 8274 8292 8310 8328 8346 8364 8382 8399 8417 8435 8453 8471 8489 8507 8525 8543 8561 8578 8596 8614 8632 8650 8668 8686 8703 8721 8739 8757 8775 8792 8810 8828 8846 8863 8881 8899 8917 8934 8952 8970 8987 9005 9023 9041 9058 9076 9094 9111 9129 9146 9164 9182 9199 9217 9235 9252 9270 9287 9305 9322 9340 9358 9375 9393 9410 9428 9445 9463 9480 9498 9515 9533 9550 9568 9585 9602 9620 9637 9655 9672 9690 9707 9724 9742 9759 9777 120 LOGARITHMS OF NUMBERS N 250 1 2 3 4 5 6 7 260 1 2 3 4 5 6 7 8 9 270 1 2 3 4 5 6 7 8 9 280 1 2 3 4 5 6 7 290 1 2 3 4 6 6 7 8 300 678 39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 9967 9985 40002 40019 40037 40054 40071 40088 40106 40123 4014040157 0175 0192 0209 0226 0243 0261 0278 0295 0312 0329 0346 0364 0381 0398 0415 0432 0449 0466 0483 0500 0518 0535 0552 0569 0586 0603 0620 0637 0654 0671 0688 0705 0722 0739 0756 0773 0790 0807 0824 0841 0858 0875 0892 0909 0926 0943 0960 0976 0993 1010 1027 1044 1061 1078 1095 1111 1128 1145 1162 1179 1196 1212 1229 1246 1263 1280 1296 1313 1330 1347 1363 1380 1397 1414 1430 1447 1464 1481 41497 41514 41531 41547 41564 41581 41597 41614 41631 41647 1664 1681 1697 1714 1731 1747 1764 1780 1797 1814 1830 1847 1863 1880 1896 1913 1929 1946 1963 1979 1996 2012 2029 2045 2062 2078 2095 2111 2127 2144 2160 2177 2193 2210 2226 2243 2259 2275 2292 2308 2325 2341 2357 2374 2390 2406 2423 2439 2455 2472 2488 2504 2521 2537 2553 2570 2586 2602 2619 2635 2651 2667 2684 2700 2716 2732 2749 2765 2781 2797 2813 2830 2846 2862 2878 2894 2911 2927 2943 2959 2975 2991 3008 3024 3040 3056 3072 3088 3104 3120 43136 43152 43169 43185 43201 43217 43233 43249 43265 43281 3297 3313 3329 3345 3361 3377 3393 3409 3425 3441 3457 3473 3489 3505 3521 3537 3553 3569 3584 3600 3616 3632 3648 3664 3680 3696 3712 3727 3743 3759 3775 3791 3807 3823 3838 3854 3870 3886 3902 3917 3933 3949 3965 3981 3996 4012 4028 4044 4059 4075 4091 4107 4122 4138 4154 4170 4185 4201 4217 4232 4248 4264 4279 4295 4311 4326 4342 4358 4373 4389 4404 4420 4436 4451 4467 4483 4498 4514 4529 4545 4560 4576 4592 4607 4623 4638 4654 4669 4685 4700 44716 44731 44747 44762 44778 44793 44809 44824 44840 44855 4871 4886 4902 4917 4932 4948 4963 4979 4994 5010 5025 5040 5056 5071 5086 5102 5117 5133 5148 5163 5179 5194 5209 5225 5240 5255 5271 5286 5301 5317 5332 5347 5362 5378 5393 5408 5423 5439 5454 5469 5484 5500 5515 5530 5545 5561 5576 5591 5606 5621 5637 5652 5667 5682 5697 5712 5728 5743 5758 5773 5788 5803 5818 5834 5849 5864 5879 5894 5909 5924 5939 5954 5969 5984 6000 6015 6030 6045 6060 6075 6090 6105 6120 6135 6150 6165 6180 6195 6210 6225 46240 46255 46270 46285 46300 46315 46330 46345 46359 46374 6389 6404 6419 6434 6449 6464 6479 6494 6509 6523 6538 6553 6568 6583 6598 6613 . 6627 6642 6657 6672 6687 6702 6716 6731 6746 6761 6776 6790 6805 6820 6835 6850 6864 6879 6894 6909 6923 6938 6953 6967 6982 6997 7012 7026 7041 7056 7070 7085 7100 7114 7129 7144 7159 7173 7188 7202 7217 7232 7246 7261 7276 7290 7305 7319 7334 7349 7363 7378 7392 7407 7422 7436 7451 7465 7480 7494 7509 7524 7538 7553 7567 7582 7596 7611 7625 7640 7654 7669 7683 7698 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 LOGARITHMS OF NUMBERS 121 6789 300 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 1 310 1 2 3 4 5 6 7 8 320 1 2 3 4 5 6 7 8 9 330 1 2 3 4 5 6 7 8 9 340 1 2 3 4 5 6 7 350 7857 7871 7885 7900 7914 7929 7943 7958 7972 7986 2 8001 8015 8029 8044 8058 8073 8087 8101 8116 8130 3 I 8144 8159 8173 8187 8202 8216 8230 8244 8259 8273 8287 8302 8316 8330 8344 8359 8373 8387 8401 8416 8430 8444 8458 8473 8487 8501 8515 8530 8544 8558 8572 8586 8601 8615 8629 8643 8657 8671 8686 8700 8714 8728 8742 8756 8770 8785 8799 8813 8827 8841 8855 8869 8883 8897 8911 8926 8940 8954 8968 8982 8996 9010 9024 9038 9052 9066 9080 9094 9108 9122 49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 9276 9290 9304 9318 9332 9346 9360 9374 9388 9402 9415 9429 9443 9457 9471 9485 9499 9513 9527 9541 9554 9568 9582 9596 9610 9624 9638 9651 9665 9679 9693 9707 9721 9734 9748 9762 9776 9790 9803 9817 9831 9845 9859 9872 9886 9900 9914 9927 9941 9955 9969 9982 9996 50010 50024 50037 50051 50065 50079 50092 50106 50120 50133 0147 0161 0174 0188 0202 0215 0229 0243 0256 0270 0284 0297 0311 0325 0338 0352 0365 0379 0393 0406 0420 0433 0447 0461 0474 0488 0501 50515 50529 50542 50556 60569 50583 50596 50610 50623 50637 0651 0664 0678 0691 0705 0718 0732 0745 0759 0772 0786 0799 0813 0826 0840 0853 0866 0880 0893 0907 0920 0934 0947 0961 0974 0987 1001 1014 1028 1041 1055 1068 1081 1095 1108 1121 1135 1148 1162 1175 1188 1202 1215 1228 1242 1255 1268 1282 1295 1308 1322 1335 1348 1362 1375 1388 1402 1415 1428 1441 1455 1468 1481 1495 1508 1521 1534 1548 1561 1574 1587 1601 1614 1627 1640 1654 1667 1680 1693 1706 1720 1733 1746 1759 1772 1786 1799 1812 1825 1838 51851 51865 51878 51891 51904 51917 51930 51943 51957 51970 1983 1996 2009 2022 2035 2048 2061 2075 2088 2101 2114 2127 2140 2153 2166 2179 2192 2205 2218 2231 2244 2257 2270 2284 2297 2310 2323 2336 2349 2362 2375 2388 2401 2414 2427 2440 2453 2466 2479 2492 2504 2517 2530 2543 2556 2569 2582 2595 2608 2621 2634 2647 2660 2673 2686 2699 2711 2724 2737 2750 2763 2776 2789 2802 2815 2827 2840 2853 2866 2879 2892 2905 2917 2930 2943 2956 2969 2982 2994 3007 3020 3033 3046 3058 3071 3084 3097 3110 3122 3135 53148 53161 53173 53186 53199 53212 53224 53237 53250 53263 3275 3288 3301 3314 3326 3339 3352 3364 3377 3390 3403 3415 3428 3441 3453 3466 3479 3491 3504 3517 3529 3542 3555 3567 3580 3593 3605 3618 3631 3643 3656 3668 3681 3694 3706 3719 3732 3744 3757 3769 3782 3794 3807 3820 3832 3845 3857 3870 3882 3895 3908 3920 3933 3945 3958 3970 3983 3995 4008 4020 4033 4045 4058 4070 4083 4095 4108 4120 4133 4145 4158 4170 4183 4195 4208 4220 4233 4245 4258 4270 4283 4295 4307 4320 4332 4345 4357 4370 4382 4394 54407 54419 54432 54444 54456 54469 54481 54494 54506 54518 122 LOGARITHMS OF NUMBERS. NO12345678 350 1 2 3 4 5 6 7 8 9 360 1 2 3 4 5 6 7 8 370 1 2 3 4 5 6 7 8 9 380 1 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 400 54407 64419 54432 54444 54456 54469 54481 54494 54506 54518 4531 4543 4555 4568 4580 4593 4605 4617 4630 4642 4654 4667 4679 4691 4704 4716 4728 4741 4753 4765 4777 4790 4802 4814 4827 4839 4851 4864 4876 4888 4900 4913 4925 4937 4949 4962 4974 4986 4998 5011 5023 5035 5047 5060 5072 5084 5096 5108 5121 5133 5145 5157 5169 5182 5194 5206 5218 5230 5242 5255 5267 5279 5291 5303 5315 5328 5340 5352 5364 5376 5388 6400 5413 5425 5437 5449 5461 5473 5485 6497 5509 5522 5534 5546 5558 6570 5582 5594 5606 5618 55630 55642 55654 55666 55678 55691 55703 55715 55727 55739 5751 5763 5775 6787 5799 6811 6823 5835 5847 5859 5871 5883 5895 5907 5919 5931 6943 5955 5967 5979 5991 6003 6015 6027 6038 6050 6062 6074 6086 6098 6110 6122 6134 6146 6158 6170 6182 6194 6205 6217 6229 6241 6253 6265 6277 6289 6301 6312 6324 6336 6348 6360 6372 6384 6396 6407 6419 6431 6443 6455 6467 6478 6490 6502 6514 6526 6538 6549 6561 6573 6585 6597 6608 6620 6632 6644 6656 6667 6679 6691 6703 6714 6726 6738 6750 6761 6773 6785 6797 6808 56820 56832 56844 56855 56867 56879 56891 56902 56914 56926 6937 6949 6961 6972 6984 6996 7008 7019 7031 7043 7054 7066 7078 7089 7101 7113 7124 7136 7148 7159 7171 7183 7194 7206 7217 7229 7241 7252 7264 7276 7287 7299 7310 7322 7334 7345 7357 7368 7380 7392 7403 7415 7426 7438 7449 7461 7473 7484 7496 7507 7519 7530 7542 7553 7565 7576 7588 7600 7611 7623 7634 7646 7657 7669 7680 7692 7703 7715 7726 7738 7749 7761 7772 7784 7795 7807 7818 7830 7841 7852 7864 7875 7887 7898 7910 7921 7933 7944 7955 7967 57978 57990 58001 58013 58024 58035 58047 58058 58070 58081 8092 8104 8115 8127 8138 8149 8161 8172 8184 8195 8206 8218 8229 8240 8252 8263 8274 8286 8297 8309 8320 8331 8343 8354 8365 8377 8388 8399 8410 8422 8433 8444 8456 8467 8478 8490 8501 8512 8524 8535 8546 8557 8569 8580 8591 8602 8614 8625 8636 8647 8659 8670 8681 8692 8704 8715 8726 8737 8749 8760 8771 8782 8794 8805 8816 8827 8838 8850 8861 8872 8883 8894 8906 8917 8928 8939 8950 8961 8973 8984 8995 9006 9017 9028 9040 9051 9062 9073 9084 9095 59106 59118 59129 59140 59151 59162 59173 59184 59195 59207 9218 9229 9240 9251 9262 9273 9284 9295 9306 9318 9329 9340 9351 9362 9373 9384 9395 9406 9417 9428 9439 9450 9461 9472 9483 9494 9506 9517 9528 9539 9550 9561 9572 9583 9594 9605 9616 9627 9638 9649 9660 9671 9682 9693 9704 9715 9726 9737 9748 9759 9770 9780 9791 9802 9813 9824 9835 9846 9857 9868 9879 9890 9901 9912 9923 9934 9945 9956 9966 9977 9988 9999 60010 60021 60032 60043 60054 60065 60076 60086 60097 60108 0119 0130 0141 0152 0163 0173 0184 0195 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 LOGARITHMS OF NUMBERS 123 NO12345678 400 1 2 3 4 5 9 410 1 2 3 4 5 6 7 8 9 420 1 2 3 4 5 6 7 8 9 430 1 2 3 4 5 6 7 8 9 440 1 2 3 4 5 6 7 8 9 450 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 0314 0325 0336 0347 0358 0369 0379 0390 0401 0412 0423 0433 0444 0455 0466 0477 0487 0498 0509 0520 0531 0541 0552 0563 0574 0584 0595 0606 0617 0627 0638 0649 0660 0670 0681 0692 0703 0713 0724 0735 0746 0756 0767 0778 0788 0799 0810 0821 0831 0842 0853 0863 0874 0885 0895 0906 0917 0927 0938 0949 0969 0970 0981 0991 1002 1013 1023 1034 1045 1055 1066 1077 1087 1098 1109 1119 1130 1140 1151 1162 1172 1183 1194 1204 121^ 1225 1236 1247 1257 1268 61278 61289 61300 61310 61321 61331 61342 61352 61363 61374 1384 1395 1405 1416 1426 1437 1448 1458 1469 1479 1490 1500 1511 1521 1532 1542 1553 1563 1574 1584 1595 1606 1616 1627 1637 1648 1658 1669 1679 1690 1700 1711 1721 1731 1742 1752 1763 1773 1784 1794 1805 1815 1826 1836 1847 1857 1868 1878 1888 1899 1909 1920 1930 1941 1951 1962 1972 1982 1993 2003 2014 2024 2034 2045 ' 2055 2066 2076 2086 2097 2107 2118 2128 2138 2149 2159 2170 2180 2190 2201 2211 2221 2232 2242 2252 2263 2273 2284 2294 2304 2315 62325 62335 62346 62356 62366 62377 62387 62397 62408 62418 2428 2439 2449 2459 2469 2480 2490 2500 2511 2521 2531 2542 2552 2562 2572 2583 2593 2603 2613 2624 2634 2644 2655 2665 2675 2685 2696 2706 2716 2726 2737 2747 2757 2767 2778 2788 2798 2808 2818 2829 2839 2849 2859 2870 2880 2890 2900 2910 2921 2931 2941 2951 2961 2972 2982 2992 3002 3012 3022 3033 3043 3053 3063 3073 3083 3094 3104 3114 3124 3134 3144 3155 3165 3175 3185 3195 3205 3215 3225 3236 3246 3256 3266 3276 3286 3296 3306 3317 3327 3337 63347 63357 63367 63377 63387 63397 63407 63417 63428 63438 3448 3458 3468 3478 3488 3498 3508 3518 3528 3538 3548 3558 3568 3579 3589 3599 3609 3619 3629 3639 3649 3659 3669 3679 3689 3699 3709 3719 3729 3739 3749 3759 3769 3779 3789 3799 3809 3819 3829 3839 3849 3859 3869 3879 3889 3899 3909 3919 3929 3939 3949 3959 3969 3979 3988 3998 4008 4018 4028 4038 4048 4058 4068 4078 4088 4098 4108 4118 4128 4137 4147 4157 4167 4177 4187 4197 4207 4217 4227 4237 4246 4256 4266 4276 4286 4296 4306 4316 4326 4335 64345 64355 64365 64375 64385 64395 64404 64414 64424 64434 4444 4454 4464 4473 4483 4493 4503 4513 4523 4532 4542 4552 4562 4572 4582 4591 4601 4611 4621 4631 4640 4650 4660 4670 4680 4689 4699 4709 4719 4729 4738 4748 4758 4768 4777 4787 4797 4807 4816 4826 4836 4846 4856 4865 4875 4885 4895 4904 4914 4924 4933 4943 4953 4963 4972 4982 4902 5002 5011 6021 5031 5040 5050 5060 5070 6079 5089 5099 5108 5118 5128 6137 5147 6157 5167 6176 5186 5196 5205 6215 5225 6234 5244 6254 6263 6273 5283 5292 5302 6312 65321 65331 65341 65350 65360 65369 65379 65389 65398 65408 124 LOGARITHMS OF NUMBERS 1 450 1 2 3 4 5 6 7 3 a 460 1 2 3 4 5 6 7 9 470 1 2 3 4 5 6 7 8 9 480 1 2 3 4 5 6 7 8 9 490 1 2 3 4 5 6 7 8 500 3 4: 5 6 7 8 9 65321 65331 65341 65350 65360 65369 65379 65389 65398 65408 5418 5427 5437 5447 5456 5466 5475 5485 5495 5504 5514 5523 5533 5543 5552 5562 5571 5581 5591 5600 5610 5619 5629 5639 5648 5658 5667 5677 5686 5696 5706 5715 5725 5734 5744 5753 5763 6772 5782 5792 5801 5811 5820 5830 5839 5849 5858 5868 6877 5887 5896 5906 5916 5925 5935 5944 5954 5963 5973 5982 5992 6001 6011 6020 6030 6039 6049 6058 6068 6077 6087 6096 6106 6115 6124 6134 6143 6153 6162 6172 6181 6191 6200 6210 6219 6229 6238 6247 6257 6266 66276 66285 66295 66304 66314 66323 66332 66342 66351 66361 6370 6380 6389 6398 6408 6417 6427 6436 6445 6455. 6464 6474 6483 6492 6502 6511 6521 6530 6539 6549 6558 6567 6577 6586 6596 6605 6614 6624 6633 6642 6652 6661 6671 6680 6689 6699 6708 6717 6727 6736 6745 6755 6764 6773 6783 6792 6801 6811 6820 6829 6839 6848 6857 6867 6876 6885 6894 6904 6913 6922 6932 6941 6950 6960 6969 6978 6987 6997 7006 7015 7025 7034 7043 7052 7062 7071 7080 7089 7099 7108 7117 7127 7136 7145 7154 7164 7173 7182 7191 7201 67210 67219 67228 67237 67247 67256 67265 67274 67284 67293 7302 7311 7321 733"0 7339 7348 7357 7367 7376 7385 7394 7403 7413 7422 7431 7440 7449 7459 7468 7477 7486 7495 7504 7514 7523 7532 7541 7550 7560 7569 7578 7587 7596 7605 7614 7624 7633 7642 7651 7660 7669 7679 7688 7697 7706 7715 7724 7733 7742 7752 7761 7770 7779 7788 7797 7806 7815 7825 7834 7843 7852 7861 7870 7879 7888 7897 7906 7916 7925 7934 7943 7952 7961 7970 7979 7988 7997 8006 8015 8024 8034 8043 8052 8061 8070 8079 8088 8097 8106 8115 68124 68133 68142 68151 68160 68169 68178 68187 68196 68205 8215 8224 8233 8242 8251 8260 8269 8278 8287 8296 8305 8314 8323 8332 8341 8350 8359 8368 8377 8386 8395 8404 8413 8422 8431 8440 8449 8458 8467 8476 8485 8494 8502 851.1 8520 8529 8538 8547 8556 8565 8574 8583 8592 8601 8610 8619 8628 8637 8646 8655 8664 8673 8681 8690 8699 8708 8717 8726 8735 8744 8753 8762 8771 8780 8789 8797 8806 8815 8824 8833 8842 8851 8860 8869 8878 8886 8895 8904 8913 8922 8931 8940 8949 8958 8966 8975 8984 8993 9002 9011 69020 69028 69037 69046 69055 69064 69073 69082 69090 69099 9108 9117 9126 9135 9144 9152 9161 9170 9179 9188 9197 9205 9214 9223 9232 9241 9249 9258 9267 9276 9285 9294 9302 9311 9320 9329 9338 9346 9355 9364 9373 9381 9390 9399 9408 9417 9425 9434 9443 9452 9461 9469 9478 9487 9496 9504 9513 9522 9531 9539 9548 9557 9566 9574 9583 9592 9601 9609 9618 9627 9636 9644 9653 9662 9671 9679 9688 9697 9705 9714 9723 9732 9740 9749 9758 9767 9775 9784 9793 9801 9810 9819 9827 9836 9845 9854 9862 9871 9880 9888 69897 69906 69914 69923 69932 69940 69949 69958 69966 69975 LOGARITHMS OF NUMBERS 125 3456789 500 1 2 3 4 5 6 7 8 9 510 1 2 3 4. 5 6 7 8 9 520 1 2 3 4 5 6 7 8 9 530 1 2 3 4 5 6 7 540 1 2 3 4 5 6 7 550 69897 69906 69914 69923 69932 69940 69949 69958 69966 69973 9984 9992 70001 70010 70018 70027 70036 70044 70053 70062 70070 70079 0088 0096 0105 0114 0122 0131 0140 0148 0157 0165 0174 0183 0191 0200 0209 0217 0226 0234 0243 0252 0260 0269 0278 0286 0295 0303 0312 0321 0329 0338 0346 0355 0364 0372 0381 0389 0398 0406 0415 0424 0432 0441 0449 0458 0467 0475 0484 0492 0501 0509 0518 0526 0535 0544 0552 0561 0569 0578 0586 0595 0603 0612 0621 0629 0638 0646 0655 0663 0672 0680 0689 0697 0706 0714 .0723 0731 0740 0749 70757 70766 70774 70783 70791 70800 70808 70817 70825 70834 0842 0851 0859 0868 0876 0885 0893 0902 0910 0919 0927 0935 0944 0952 0961 0969 0978 0986 0995 1003 1012 1020 1029 1037 1046 1054 1063 1071 1079 1088 1096 1105 1113 1122 1130 1139 1147 1155 1164 1172 1181 1189 1198 1206 1214 1223 1231 1240 1248 1257 1265 1273 1282 1290 1299 1307 1315 1324 1332 1341 1349 1357 1366 1374 1383 1391 1399 1408 1416 1425 1433 1441 1450 1458 1466 1475 1483 1492 1500 1508 1517 1525 1533 1542 1550 1559 1567 1575 1584 1592 71600 71609 71617 71625 71634 71642 71650 71659 71667 71675 1684 1692 1700 1709 1717 1725 1734 1742 1750 1759 1767 1775 1784 1792 1800 1809 1817 1825 1834 1842 1850 1858 1867 1875 1883 1892 1900 1908 1917 1925 1933 1941 1950 1958 1966 1975 1983 1991 1999 2008 2016 2024 2032 2041 2049 2057 2066 2074 2082 2090 2099 2107 2115 2123 2132 2140 2148 2156 2165 2173 2181 2189 2198 2206 2214 2222 2230 2239 2247 2255 22(53 2272 2280 2288 2296 2304 2313 2321 2329 2337 2346 2354 2362 2370 2378 2387 2395 2403 2411 2419 72428 72436 72444 72452 72460 72469 72477 72485 72493 72501 2609 2518 2526 2534 2542 2550 2558 2567 2575 2583 2591 2599 2607 2616 2624 2632 2640 2648 2656 2665. 2673 2681 2689 2697 2705 2713 2722 2730 2738 2746 2754 2762 2770 2779 '2787 2795 2803 2811 2819 2827 2835 2843 2852 2860 2868 2876 2884 2892 2900 2908 2916 2925 2933 2941 2949 2957 2965 2973 2981 2989 2997 3006 3014 3022 3030 3038 3046 3054 3062 3070 3078 3086 3094 3102 3111 3119 3127 3135 3143 3151 3159 3167 3175 3183 3191 3199 3207 3215 3223 3231 73239 73247 73255 73263 73272 73280 73288 732*96 73304 73312 3320 3328 3336 3344 3352 3360 3368 3376 3384 3392 3400 3408 3416 3424 3432 3440 3448 3456 3464 3472 3480 3488 3496 3504 3512 3520 3528 3536 3544 3552 3560 3568 3576 3584 3592 3600 3608 3616 3624 3632 3640 3648 3(556 3664 3672 3679 3687 3695 3703 3711 3719 3727 3735 3743 3751 3759 3767 3775 3783 3791 3799 3807 3815 3823 3830 3838 3846 3854 3862 3870 3878 3886 3894 3902 3910 3918 3926 3933 3941 3949 3957 3965 3973 3981 3989 3997 4005. 4013 4020 4028 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 126 LOGARITHMS OF NUMBERS N O 1 2 3 4 5 6 7 8 9 ! 550 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 ; 1 4115 4123 4131 4139 4147 4155 4162 4170 4178 4186 2 4194 4202 4210 4218 4225 4233 4241 4249 4257 4265 3 4273 4280 4288 4296 4304 4312 4320 4327 4335 4343 4 4351 4359 4367 4374 4382 4390 4398 4406 4414 4421 5 4429 4437 4445 4453 4461 4468 4476 4484 4492 4500 6 4507 4515 4523 4531 4539 4547 4554 4562 4570 4578 7 4586 4593 4601 4609 4617 4624 4632 4640 4648 4656 8 4663 4671 4679 4687 4095 4702 4710 4718 4726 4733 9 4741 4749 4757 4764 4772 4780 4788 4796 4803 4811 560 74819 74827 74834 74842 74850 74858 74865 74873 74881 74889 1 4896 4904 4912 4920 4927 4935 4943 4950 4958 4966 2 4974 4981 4989 4997 5005 5012 5020 5028 5035 5043 3 5051 5059 5066 5074 5082 5089 5097 5105 5113 5120 4 5128 5136 5143 5151 5159 5166 5174 5182 5189 5197 5 5205 5213 5220 5228 5236 5243 5251 5259 5266 5274 6 5282 5289 5297 5305 5312 5320 5328 5335 5343 5351 7 5358 5366 5374 5381 5389 5397 5404 5412 5420 5427 8 5435 5442 5450 5458 5465 5473 5481 5488 5496 5504 9 5511 5519 5526 5534 5542 5549 5557 5565 5572 5580 670 75587 75595 75603 75610 75618 75626 75633 75641 75648 75656 1 5664 5671 5679 5686 5694 5702 5709 5717 5724 5732 2 5740 5747 5755 5762 5770 5778 5785 5793 5800 5808 3 5815 5823 5831 5838 5846 5853 5861 5868 5876 5884 4 5891 5899 5906 5914 5921 5929 5937 5944 5952 5959 5 5967 5974 6982 5989 5997 6005 6012 6020 6027 6035 6 6042 6050 6057 6065 6072 6080 6087 6095 6103 6110 7 6118 6125 6133 6140 6148 6155 6163 6170 6178 6185 8 6193 6200 6208 6215 6223 6230 6238 6245 6253 6260 9 6268 6275 6283 6290 6298 6305 6313 6320 6328 6335 580 76343 76350 76358 76365 76373 76380 76388 76395 76403 76410 1 6418 6425 6433 6440 6448 6455 6462 6470 6477 6485 2 6492 6500 6507 6515 6522 6530 6537 6545 6552 6559 3 6567 6574 6582 6589 6597 6604 6612 6619 6626 6634 ' 4 6641 6649 6656 6664 6671 6678 6686 6693 6701 6708 5 6716 6723 6730 6738 6745 6753 6760 6768 6775 6782 6 6790 6797 6805 6812 6819 6827 6884 6842 6849 6856 7 6864 6871 6879 6886 6893 6901 6908 6916 6923 6930 8 6938 6945 6953 6960 6967 6975 6982 6989 6997 7004 9 7012 7019 7026 7034 7041 7048 7056 7063 7070 7078 590 77085 77093 77100 77107 77115 77122 77129 77137 77144 77151 1 7159 7166 7173 7181 7188 7195 7203 7210 7217 7225 2 7232 7240 7247 7254 7262 7269 7276 7283 7291 7298 3 7305 7313 7320 7327 7335 7342 7349 7357 7364 7371 4 7379 7386 7393 7401 7408 7415 7422 7430 7437 7444 5 7452 7459 7466 7474 7481 7488 7495 7503 7510 7517 6 7525 7532 7539 7546 7554 7561 7568 7576 7583 7590 7 7597 7605 7612 7619 7627 7634 7641 7648 7656 7663 8 7670 7677 7685 7692 7699 7706 7714 7721 7728 7735 9 7743 7750 7757 7764 7772 7779 7786 7793 7801 7808 600 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 LOGARITHMS OF NUMBERS 127 N 600 1 2 3 4 5 6 7 8 9 610 1 3 6 7 8 9 620 1 2 3 4 5 6 7 8 9 680 1 2 3 4 5 6 7 8 9 640 1 2 3 4 5 . 7 650 6789 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 7887 7895 7902 7909 7916 7924 7931 7938 7945 7952 7960 7967 7974 7981 7988 7996 8003 8010 8017 8025 8032 8039 8046 8053 8061 8068 8075 8082 8089 8097 8104 8111 8118 8125 8132 8140 8147 8154 8161 8168 8176 8183 8190 8197 8204 8211 8219 8226 8233 8240 8247 8254 8262 8269 8276 8283 8290 8297 8305. 8312 8319 8326 8333 8340 8347 8355 8362 8369 8376 8383 8390 8398 8405 8412 8419 8426 8433 8440 8447 8455. 8462 8469 8476 8483 8490 8497 8504 8512 8519 8526 78533 78540 78547 78554 78561 78569 78576 78583 78590 78597 8604 8611 8618 8625 8633 8640 8647 8654 8661 8668 8675 8682 8689 8696 8704 8711 8718 8725 8732 8739 8746 8753 8760 8767 8774 8781 8789 8796 8803 8810 4 I 8817 8824 8831 8838 8845 8852 8859 8866 8873 8880 5 8888 8895 8902 8909 8916 8923 8930 8937 8944 8951 8958 8965 8972 8979 8986 8993 9000 9007 9014 9021 9029 9036 9043 9050 9057 9064 9071 9078 9085 9092 9099 9106 9113 9120 9127 9134 9141 9148 9155 9162 9169 9176 9183 9190 9197 9204 9211 9218 9225 9232 79239 79246 79253 79260 79267 79274 79281 79288 79295 79302 9309 9316 9323 9330 9337 9344 9351 9358 9365 9372 9379 9386 9393 9400 9407 9414 9421 9428 9435 9442 9449 9456 9463 9470 9477 9484 9491 9498 9505 9511 9518 9525 9532 9539 9546 9553 9560 9567 9574 9581 9588 9595 9602 9609 9616 9623 9630 9637 9644 9650 9657 9664 9671 9678 9685 9692 9699 9706 9713 9720 9727 9734 9741 9748 9754 9761 9768 9775 9782 9789 9796 9803 9810 9817 9824 9831 9837 9844 9851 9858 9865 9872 9879 9886 9893 9900 9906 9913 9920 9927 79934 79941 79948 79955 79962 79969 79975 79982 79989 79996 80003 80010 80017 80024 80030 80037 80044 80051 80058 80065 0072 0079 0085 0092 0099 0106 0113 0120 0127 0134 0140 0147 0154 0161 0168 0175 0182 0188 0195 0202 0209 0216 0223 0229 0236 0243 0250 0257 0264 0271 0277 0284 0291 0298 0305 0312 0318 0325 0332 0339 0346 0353 0359 0366 0373 0380 0387 0393 0400 0407 0414 0421 0428 0434 0441 0448 0455 0462 0468 0475 0482 0489 0496 0502 0509 0516 0523 0530 0536 0543 0550 0557 0564 0570 0577 0584 0591 0598 0604 0611 80618 80625 80632 8063P 80645 80652 80659 80665 80672 80679 0686 0693 0699 0706 0713 0720 0726 0733 0740 0747 0754 0760 0767 0774 0781 0787 0794 0801 0808 0814 0821 0828 0835 0841 0848 0855 0862 0868 0875 0882 0889 0895 0902 0909 0916 0922 0929 0936 0943 0949 0956 0963 0969 0976 0983 0990 0996 1003 1010 1017 1023 1030 1037 1043 1050 1057 1064 1070 1077 1084 1090 1097 1104 1111 1117 1124 1131 1137 1144 1151 1158 1164 1171 1178 1184 1191 1198 1204 1211 1218 1224 1231 1238 1245 1251 1258 1265 1271 1278 1285 81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 128 LOGARITHMS OF NUMBERS 0123456789 650 1 2 3 4 5 680 1 2 3 4 5 6 7 8 9 670 1 2 3 4 5 6 7 8 9 680 1 2 3 4 5 6 7 8 9 690 81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 1358 1365 1371 1378 1385 1391 1398 1405 1411 1418 1425 1431 1438 1445 1451 1458 1465 1471 1478 1485 1491 1498 1505 1511 1518 1525 1531 1538 1544 1551 1558 1564 1571 1578 1584 1591 1598 1604 1611 1617 1624 1631 1637 1644 1651 1657 1664 1671 1677 1684 1690 1697 1704 1710 1717 1723 1730 1737 1743 1750 1757 1763 1770 1776 1783 1790 1796 1803 1809 1816 1823 1829 1836 1842 1849 1856 1862 1869 1875 1882 1889 1895 1902 1908 1915 1921 1928 1935 1941 1948 81954 81961 81968 81974 81981 81987 81994 82000 82007 82014 2020 2027 2033 2040 2046 2053 2060 2066 2073 2079 2086 2092 2099 2105 2112 2119 2125 2132 2138 2145 2151 2158 2164 2171 2178 2184 2191 2197 2204 2210 2217 2223 2230 2236 2243 2249 2256 2263 2269 2276 2282 2289 2295 2302 2308 2315 2321 2328 2334 2341 2347 2354 2360 2367 2373 2380 2387 2393 2400 2406 2413 2419 2426 2432 2439 2445 2452 2458 2465 2471 2478 2484 2491 2497 2504 2510 2517 2523 2530 2536 2543 2549 2556 2562 2569 2575 2582 2588 2595 2601 82607 82614 82620 82627 82633 82640 82646 82653 82659 82666 2672 2679 2685 2692 2698 2705 2711 2718 2724 2730 2737 2743 2750 2756 2763 2769 2776 2782 2789 2795 2802 2808 2814 2821 2827 2834 2840 2847 2853 2860 2866 2872 2879 2885 2892 2898 2905 2911 2918 2924 2930 2937 2943 2950 2956 2963 2969 2975 2982 2988 2995 3001 3008 3014 3020 3027 3033 3040 3046 3052 3059 3065 3072 3078 3085 3091 3097 3104 3110 3117 3123 3129 3136 3142 3149 3155 3161 3168 3174 3181 3187 3193 3200 3206 3213 3219 3225 3232 3238 3245 83251 83257 83264 83270 83276 83283 83289 83296 83302 83308 3315 3321 3327 3334 3340 3347 3353 3359 3366 3372 3378 3385 3391 3398 3404 3410 3417 3423 3429 3436 3442 3448 3455 3461 3467 3474 3480 3487 3493 3499 3506 3512 3518 3525 3531 3537 3544 3550 3556 3563 3569 3575 3582 3588 3594 3601 3607 3613 3620 3626 3632 3639 3645 3651 3658 3664 3670 3677 3683 3696 3702 3708 3715 3721 3727 3734 3740 3746 3753 3759 3765 3771 3778 3784 3790 3797 3803 3809 3816 3822 3828 3835 3841 3847 3853 3860 3866 3872 3879 83885 83891 83897 83904 83910 83916 83923 83929 83935 83942 3948 3954 3960 3967 3973 3979 3985 3992 3998 4004 4011 4017 4023 4029 4036 4042 4048 4055 4061 4067 4073 4080 4086 4092 4098 4105 4111 4117 4123 4130 4136 4142 4148 4155 4161 4167 4173 4180 4186 4192 4198 4205 4211 4217 4223 4230 4236 4242 4248 4255 4261 4267 4273 4280 4286 4292 4298 4305 4311 4317 4323 4330 4336 4342 4348 4354 4361 4367 4373 4379 4386 4392 4398 4404 4410 4417 4423 4429 4435 4442 4448 4454 4460 4466 4473 4479 4485 4491 4497 4504 700 84510 84516 84522 84528 84535 84541 84547 84553 84559 8456(3 LOGARITHMS OF NUMBERS 129 N 3456789 700 1 2 3 4 5 6 7 8 9 710 1 2 3 4 5 6 7 8 9 720 1 2 3 4 5 6 7 730 1 2 3 4 5 6 7 8 9 740 1 2 3 4 5 6 7 8 9 750 84510 84516 84522 84528 84535 84541 84547 84553 84559 84566 4572 4578 4584 4590 4597 4603 4609 4615 4621 4628 4634 -4640 4646 4652 4658 4665 4671 4677 4683 4689 4696 4702 4708 4714 4720 4726 4733 4739 4745 4751 4757 4763 4770 4776 4782 4788 4794 4800 4807 4813 4819 4825 4831 4837 4844 4850 4856 4862 4868 4874 4880 4887 4893 4899 4905 4911 4917 4924 4930 4936 4942 4948 4954 4960 4967 4973 4979 4985 4991 4997 5003 5009 5016 5022 5028 5034 5040 5046 5052 5058 5065 5071 5077 5083 5089 5095 5101 5107 5114 5120 85126 85132 85138 85144 85150 85156 85163 85169 85175 85181 5187 5193 5199 5205 5211 5217 5224 5230 5236 5242 5248 5254 5260 5266 5272 5278 5285 5291 5297 5303 5309 5315 5321 5327 5333 5339 5345 5352 5358 5364 5370 5376 5382 5388 5394 5400 5406 5412 5418 5425. 5431 5437 5443 5449 5455 5461 *467 5473 5479 5485 5491 5497 5503 5509 '5516 5522 6528 5534 5540 5546 5552 5558 5564 5570 5576 5582 5588 5594 5600 5606 5612 5618 5625 5631 5637 5643 5649 5655 5661 5667 5673 5679 5685 5691 5697 5703 5709 5715 5721 5727 85733 85739 85745 85751 85757 85763 85769 85775 85781 85788 5794 5800 5806 5812 5818 5824 5830 5836 5842 5848 5854 5860 5866 5872 5878 5884 5890 5896 5902 6908 5914 5920 5926 5932 5938 5944 5950 5956 5962 5968 5974 5980 5986 5992 5998 6004 6010 6016 6022 6028 6034 6040 6046 6052 6058 6064 6070 6076 6082 6088 6094 6100 6106 6112 6118 6124 6130 6136 6141 6147 6153 6159 6165 6171 6177 6183 6189 6195 6201 6207 6213 6219 6225 6231 6237 6243 6249 6255 6261 6267 6273 6279 6285 6291 6297 6303 6308 6314 6320 6326 86332 86338 86344 86350 86356 86362 86368 86374 86380 86386 6392 6398 6404 6410 6415 6421 6427 6433 6439 6445 6451 6457 6463 6469 6475 6481 6487 6493 6499 6504 6510 6516 6522 6528 6534 6540 6546 6552 6558 6564 6570 6576 6581 6587 6593 6599 6605 6611 6617 6623 6629 6635 6641 6646 6652 6658 6664 6670 6676 6682 6688 6694 6700 6705 6711 6717 6723 6729 6735 6741 6747 6753 6759 6764 6770 6776 6782 6788 6794 6800 6806 6812 6817 6823 6829 6835 6841 6847 6853 6859 6864 6870 6876 6882 6888 6894 6900 6906 6911 6917 86923 86929 86935 86941 86947 86953 86958 86964 86970 86976 6982 6988 6994 6999 7005 7011 7017 7023 7029 7035 7040 7046 7052 7058 7064 7070 7075 7081 7087 7093 7099 7105 7111 7116 7122 7128 7134 7140 7146 7151 7157 7163 7169 7175 7181 7186 7192 7198 7204 7210 7216 7221 -7227 7233 7239 7245 7251 7256 7262 7268 7274 7280 7286 7291 7297 7303 7309 7315 7320 7326 7332 7338 7344 7349 7355 7361 7367 7373 7379 7384 7390 7396 7402 7408 7413 7419 7425 7431 7437 7442 7448 7454 7460 7466 7471 7477 7483 7489 7495 7500 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 130 LOGARITHMS OF NUMBERS NO 1234567 8 9 750 1 2 3 4 5 6 7 8 9 760 1 2 3 4 5 6 7 8 9 770 1 2 3 4 5 6 7 8 9 780 1 2 3 4 5 6 7 8 9 790 1 2 3 4 5 6 7 800 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 7564 7570 7576 7581 7587 7593 7599 7604 7610 7610 7622 7628 7633 7639 7645 7651 7656 7662 7668 7674 7679 7685 7691 7697 7703 7708 7714 7720 7726 7731 7737 7743 7749 7754 7760 7766 7772 7777 7783 7789 7795 7800 7806 7812 7818 7823 7829 7835 7841 7846 7852 7858 7864 7869 7875 7881 7887 7892 7898 7904 7910 7915 7921 7927 7933 7938 7944 7950 7955 7961 7967 7973 7978 7984 7990 7996 8001 8007 8013 8018 8024 8030 8036 8041 8047 8053 8058 8064 8070 8076 88081 88087 88093 88098 88104 88110 88116 88121 88127 88133 8138 8144 8150 8156 8161 8167 8173 8178 8184 8190 8195 8201 8207 8213 8218 8224 8230 8235 8241 8247 8252 8258 8264 8270 8275 8281 8287 8292 8298 8304 8309 8315 8321 8326 8332 8338 8343 8349 8355 8360 8366 8372 8377 8383 8389 8395 8400 8406 8412 8417 8423 8429 8434 8440 8446 8451 8457 8463 8468 8474 8480 8485 8491 8497 8502 8508 8513 8519 8525 8530 8536 8542 8547 8553 8559 8564 8570 8576 8581 8587 8593 8598 8604 8610 8615 8621 8627 8632 8638 8643 88649 88655 88660 88666 88672 88677 88683 88689 88694 88700 8705 8711 8717 8722 8728 8734 8739 8745 8750 8756 8762 8767 8773 8779 8784 8790 8795 8801 8807 8812 8818 8824 8829 8835 8840 8846 8852 8857 8863 8868 8874 8880 8885 8891 8897 8902 8908 8913 8919 8925 8930 8936 8941 8947 8953 8958 8964 8969 8975 8981 8986 8992 8997 9003 9009 9014 9020 9025 9031 9037 9042 9048 9053 9059 9064 9070 9076 9081 9087 9092 9098 9104 9109 9115 9120 9126 9131 9137 9143 9148 9154 9159 9165 9170 9176 9182 9187 9193 9198 9204 89209 89215 89221 89226 89232 89237 89243 89248 89254 89260 9265 9271 9276 9282 9287 9293 9298 9304 9310 9315 9321 9326 9332 9337 9343 9348 9354 9360 9365 9371 9376 9382 9387 9393 9398 9404 9409 9415 9421 9426 9432 9437 9443 9448 9454 9459 9465 9470 9476 9481 9487 9492 9498 9504 9509 9515 9520 9526 9531 9537 9542 9548 9553 9559 9564 9570 9575 9581 9586 9592 9597 9603 9609 9614 9620 9625 9631 9636 9642 9647 9653 9658 9664 9669 9675 9680 9686 9691 9697 9702 9708 9713 9719 9724 9730 9735 9741 9746 9752 9757 89763 89768 89774 89779 89785 89790 89796 89801 89807 89812 9818 9823 9829 9834 9840 9845 9851 9856 9862 9867 9873 9878 9883 9889 9894 9900 9905 9911 9916 9922 9927 9933 9938 9944 9949 9955 9960 9966 9971 9977 9982 9988 9993 9998 90004 90009 90015 90020 90026 90031 90037 90042 90048 90053 0059 0064 0069 0075 0080 0086 0091 0097 0102 0108 0113 0119 0124 0129 0135 0140 0146 0151 0157 0162 0168 0173 0179 0184 0189 0195 0200 0206 0211 0217 0222 0227 0233 0238 0244 0249 0255 0260 0266 0271 0276 0282 0287 0293 0298 0304 90309 90314 90320 90325 90331 90336 90342 90347 90352 90358 LOGARITHMS OF NUMBERS 131 NO 12 34 567 89 800 1 2 3 4 5 9 810 1 2 3 4 5 6 7 8 820 1 2 3 4 5 6 7 8 830 1 2 3 4 5 6 7 340 1 2 3 4 5 6 7 90309 90314 90320 90325 90331 90336 90342 90347 90352 90358 0363 0369 0374 0380 0385 0390 0396 0401 0407 0412 0417 0423 0428 0434 0439 0445 0450 0455 0461 0466 0472 0477 0482 0488 0493 0499 0504 0509 0515 0520 0526 0531 0536 0542 0547 0553 0558 0563 0569 0574 0580 0585 0590 0596 0601 0607 0612 0617 0623 0628 0634 0639 0644 0650 0655 0660 0666 0671 0677 0682 0687 0693 0698 0703 0709 0714 0720 0725 0730 0736 0741 0747 0752 0757 0763 0768 0773 0779 0784 0789 0795 0800 0806 0811 0816 0822 0827 0832 0838 0843 90849 90854 90859 90865 90870 90875 90881 90886 90891 90897 0902 0907 0913 0918 0924 0929 0934 0940 0945 0950 0956 0961 0966 0972 0977 0982 0988 0993 0998 1004 1009 1014 1020 1025 1030 1036 1041 1046 1052 1057 1062 1068 1073 1078 1084 1089 1094 1100 1105 1110 1116 1121 1126 1132 1137 1142 1148 1153 1158 1164 1169 1174 1180 1185- 1190 1196 1201 1206 1212 1217 1222 1228 1233 1238 1243 1249 1254 1259 1265 1270 1275 1281 1286 1291 1297 1302 1307 1312 1318 1323 1328 1334 1339 1344 1350 1355 1360 1365 1371 1376 91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 1434 1440 1445 1450 1455 1461 1466 1471 1477 1482 1487 1492 1498 1503 1508 1514 1519 1524 1529 1535 1540 1545 1551 1556 1561 1566 1572 1577 1582 1587 1593 1598 1603 1609 1614 1619 1624 1630 1635 1640 1645 1651 1656 1661 1666 1672 1677 1682 1687 1693 1698 1703 1709 1714 1719 1724 1730 1735 1740 1745 1751 1756 1761 1766 1772 1777 1782 1787 1793 1798 1803 1808 1814 1819 1824 1829 1834 1840 1845 1850 1855 1861 1866 1871 1876 1882 1887 1892 1897 1903 91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 1960 1965 1971 1976 1981 1986 1991 1997 2002 2007 2012 2018 2023 2028 2033 2038 2044 2049 2054 2059 2065 2070 2075 2080 2085 2091 2096 2101 2106 2111 2117 2122 2127 2132 2137 2143 2148 2153 2158 2163 2169 2174 2179 2184 2189 2195 2200 2205 2210 2215 2221 2226 2231 2236 2241 2247 2252 2257 2262 2267 2273 2278 2283 2288 2293 2298 2304 2309 2314 2319 2324 2330 2335 2340 2345 2350 2355 2361 2366 2371 2376 2381 2387 2392 2397 2402 2407 2412 2418 2423 92428 92433 92438 92443 92449 92454 92459 92464 92469 92474 2480 2485 2490 2495 2500 2505 2511 2516 2521 2526 2531 2536 2542 2547 2552 2557 2562 2567 2572 2578 2583 2588 2593 2598 2603 2609 2614 2619 2624 2629 2634 2639 2645 2650 2655 2660 2665 2670 2675 2681 2686 2691 2696 2701 2706 2711 2716 2722 2727 2732 2737 2742 2747 2752 2758 2763 2768 2773 2778 2783 2788 2793 2799 2804 2809 2814 2819 2824 2829 2834 2840 2845 2850 2855 2860 2865 2870 2875 2881 2886 2891 2896 2901 2906 2911 2916 2921 2927 2932 2937 850 92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 LOGARITHMS OF NUMBERS 850 1 2 3 4 5 6 7 8 9 860 1 2 3 4 5 6 7 8 9 870 1 2 3 4 5 6 7 880 1 2 3 4 5 6 7 8 890 1 2 3 4 5 900 5678 92942 92947 02952 92957 92962 92967 92973 92978 92983 92988 2993 2998 3003 3008 3013 3018 3024 3029 3034 3039 3044 3049 3054 3059 3064 3069 3075 3080 3085 3090 3095 3100 3105 3110 3115 3120 3125 3131 3136 3141 3146 3151 3156 3161 3166 3171 3176 3181 3186 3192 3197 3202 3207 3212 3217 3222 3227 3232 3237 3242 3247 3252 3258 3263 3268 3273 3278 3283 3288 3293 3298 3303 3308 3313 3318 3323 3328 3334 3339 3344 3349 3354 3359 3364 3369 3374 3379 3384 3389 3394 3399 3404 3409 3414 3420 3425 3430 3435 3440 3445 93450 93455 93460 93465 93470 93475 93480 93485 93490 93495 3500 3505 3510 3515 3520 3526 3531 3536 3541 3546 3551 3556 3561 3566 3571 3576 3581 3586 3591 3596 3601 3606 3611 3616 3621 3626 . 3631 3636 3641 3646 3651 3656 3661 3666 3671 3676 3682 3687 3692 3697 3702 3707 3712 3717 3722 3727 3732 3737 3742 3747 3752 3757 3762 3767 3772 3777 3782 3787 3792 3797 3802 3807 3812 3817 3822 3827 3832 3837 3842 3847 3852 3857 3862 3867 3872 3877 3882 3887 3892 3897 3902 3907 3912 3917 3922 3927 3932 3937 3942 3947 93952 93957 93962 93967 93972 93977 93982 93987 93992 93997 4002 4007 4012 4017 4022 4027 4032 4037 4042 4047 4052 4057 4062 4067 4072 4077 4082 4086 4091 4096 4101 4106 4111 4116 4121 4126 4131 4136 4141 4146 4151 4156 4161 4166 4171 4176 4181 4186 4191 4196 4201 4206 4211 4216 4221 4226 4231 4236 4240 4245 4250 4255 4260 4265 4270 4275 4280 4285 4290 4295 4300 4305 4310 4315 4320 4325 4330 4335 4340 4345 4349 4354 4359 4364 4369 4374 4379 4384 4389 4394 4399 4404 4409 4414 4419 4424 4429 4433 4438 4443 94448 94453 94458 94463 94468 94473 94478 94483 94488 94493 4498 4503 4507 4512 4517 4522 4527 4532 4537 4542 4547 4552 4557 4562 4567 4571 4576 4581 4586 4591 4596 4601 4606 4611 4616 4621 4626 4630 4635 4640 4645 4650 4655 4660 4665 4670 4675 4680 4685 4689 4694 4699 4704 4709 4714 4719 4724 4729 4734 4738 4743 4748 4753 4758 4763 4768 4773 4778 4783 4787 4792 4797 4802 4807 4812 4817 4822 4827 4832 4836 4841 4846 4851 4856 4861 4866 4871 4876 4880 .4885 4890 4895 4900 4905 4910 4915 4919 4924 4929 4934 94939 94944 94949 94954 94959 94963 94968 94973 94978 94983 4988 4993 4998 5002 5007 5012 5017 5022 5027 5032 5036 5041 5046 5051 5056 5061 5066 5071 5075 5080 5085 5090 5095 5100 5105 5109 5114 5119 5124 5129 5134 5139 5143 5148 5153 5158 5163 5168 5173 5177 5182 5187 5192 5197 5202 5207 5211 5216 5221 5226 5231 5236 5240 5245 5250 5255 5260 5265 5270 5274 5279 5284 5289 5294 5299 5303 5308 5313 5318 5323 5328 5332 5337 5342 5347 5352 5357 5361 5366 5371 5376 5381 5386 5390 5395 5400 5405 5410 5415 5419 95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 LOGARITHMS OF NUMBERS 133 0123456789 900 95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 1 5472 5477 5482 5487 5492 5497 5501 5506 5511 5516 2 5521 5525 5530 5535 5540 5545 5550 5554 5559 5564 3 5569 5574 5578 5583 5588 5593 5598 5602 5607 5612 4 5617 5622 5626 5631 5636 5641 5646 5650 5655 5660 5 5665 5670 5674 5679 5684 5689 5694 5698 5703 5708 6 5713 5718 5722 5727 5732 5737 5742 5746 5751 5756 7 5761 5766 5770 5775 5780 5785 5789 5794 5799 5804 5809 5813 5818 5823 5828 5832 5837 5842 5847 5852 5856 5861 5866 5871 5875 5880 5885 5890 5895 5899 910 95904 95909 95914 95918 95923 95928 95933 95988*95942 95947 1 5952 5957 5961 5966 5971 5976 5980 5985 5990 5995 2 I 5999 6004 6009 6014 6019 6023 6028 6033 6038 6042 3 6047 6052 6057 6061 6066 6071 6076 6080 6085 6090 9 920 1 2 3 4 5 6 7 8 9 980 1 2 3 4 5 6 7 8 9 940 1 2 3 4 5 6 7 8 9 950 6095 6099 6104 6109 6114 6118 6123 6128 6133 6137 6142 6147 6152 6156 6161 6166 6171 6175 6180 6185 6190 6194 6199 6204 6209 6213 6218 6223 6227 6232 6237 6242 6246 6251 6256 6261 6265 6270 6275 6280 6284 6289 6294 6298 6303 6308 6313 6317 6322 6327 6332 6336 6341 6346 6350 6355 6360 6365 6369 6374 96379 96384 96388 96393 96398 96402 96407 96412 96417 96421 6426 6431 6435 6440 6445 6450 6454 6459 6464 6468 6473 6478 6483 6487 6492 6497 6501 6506 6511 6515 6520 6525 6530 6534 6539 6544 6548 6553 6558 6562 6567 6572 6577 6581 6586 6591 6595 6600 6605,6609 6614 6619 6624 6628 6633 6638 6642 6647 6652 6656 6661 6666 6670 6675 6680 6685 6689 6694 6699 6703 6708 6713 6717 6722 6727 6731 6736 6741 6745 6750 6755 6759 6764 6769 6774 6778 6783 6788 6792 6797 6802 6806 6811 6816 6820 6825 6830 6834 6839 6844 96848 96853 96858 96862 96867 96872 96876 96881 96886 96890 6895 6900 6904 6909 6914 6918 6923 6928 6932 6937 6942 6946 6951 6956 6960 6965 6970 6974 6979 6984 6988 6993 6997 7002 7007 7011 7016 7021 7025 7030 7035 7039 7044 7049 7053 7058 7063 7067 7072 7077 7081 7086 7090 7095 7100 7104 7109 7114 7118 7123 7128 7132 7137 7142 7146 7151 7155 7160 7165 7169 7174 7179 7183 7188 7192 7197 7202 7206 7211 7216 7220 7225 7230 7234 7239 7243 7248 7253 7257 7262 7267 7271 7276 7280 7285 7290 7294 7299 7304 7308 97313 97317 97322 97327 97331 97336 97340 97345 97350 97354 I 7359 7364 7368 7373 7377 7382 7387 7391 7396 7400 ' 7405 7410 7414 7419 7424 7428 7433 7437 7442 7447 7451 7456 7460 7465 7470 7474 7479 7483 7488 7493 7497 7502 7506 7511 7516 7520 7525 7529 7534 7539 7543 7548 7552 7557 7562 7566 7571 7575 7580 7585 7589 7594 7598 7603 7607 7612 7617 7621 7626 7630 7635 7640 7644 7649 7653 7658 7663 7667 7672 7676 7681 7685 7690 7695 7699 7704 7708 7713 7717 7722 7727 7731 7736 7740 7745. 7749 7754 7759 7763 7768 97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 134 LOGARITHMS OF NUMBERS O12345678 950 1 2 3 4 5 6 7 960 970 1 2 3 4 5 6 7 980 1 2 3 4 5 6 7 8 9 990 1 2 3 4 5 f? 7 8 9 97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 7818 7823 7827 7832 7836 7841 7845 7850 7855 7859 7864 7868 7873 7877 7882 7886 7891 7896 7900 7905 7909 7914 7918 7923 7928 7932 7937 7941 7946 7950 7955 7959 7964 7968 7973 7978 7982 7987 7991 7996 8000 8005 8009 8014 8019 8023 8028 8032 8037 8041 8046 8050 8055 8059 8064 8068 8073 8078 8082 8087 8091 8096 8100 8105 8109 8114 8118 8123 8127 8132 8137 8141 8146 8150 8155 8159 8164 8168 8173 8177 8182 8186-8191 8195 8200 8204 8209 8214 8218 8223 .98227 98232 98236 98241 98245 98250 98254 98259 98263 98268 8272 8277 8281 8286 8290 8295 8299 8304 8308 8313 8318 8322 8327 8331 8336 8340 8345 8349 8354 8358 8363 8367 8372 8376 8381 8385 8390 8394 8399 8403 8408 8412 8417 8421 8426 8430 8435 8439 8444 8448 8453 8457 8462 8466 8471 8475 8480 8484 8489 8493 8498 8502 8507 8511 8516 8520 8525 8529 8534 8538 8543 8547 8552 8556 8561 8565 8570 8574 8579 8583 8588 8592 8597 8601 8605 8610 8614 8619 8623 8628 8632 8637 8641 8646 8650 8655 8659 8664 8668 8673 98677 98682 98686 98691 98695 98700 98704 98709 98713 98717 8722 8726 8731 8735 8740 8744 8749 8753 8758 8762' 8767 8771 8776 8780 8784 8789 8793 8798 8802 8807 8811 8816 8820 8825 8829 8834 8838 8843 8847 8851 8856 8860 8865 8869 8874 8878 8883 8887 8892 8896 8900 8905 8909 8914 8918 8923 8927 8932 8936 8941 8945 8949 8954 8958 8963 8967 8972 8976 8981 8985 8989 8994 8998 9003 9007 9012 9016 9021 9025 9029 9034 9038 9043 9047 9052 9056 9061 9065 9069 9074 9078 9083 9087 9092 9096 9100 9105 9109 9114 9118 99123 99127 99131 99136 99140 99145 99149 99154 99158 99162 9167 9171 9176 9180 9185 9189 9193 9198 9202 9207 9211 9216 9220 9224 9229 9233 9238 9242 9247 9251 9255 9260 9264 9269 9273 9277 9282 9286 9291 9295 9300 9304 9308 9313 9317 9322 9326 9330 9335 9339 9344 9348 9352 9357 9361 9366 9370 9374 9379 9383 9388 9392 9396 9401 9405 9410 9414 9419 9423 9427 9432 9436 9441 9445 9449 9454 9458 9463 9467 9471 9476 9480 9484 9489 9493 9498 9502 9506 9511 9515 9520 9524 9528 9533 9537 9542 9546 9550 9555 . 9559 99564 99568 99572 99577 99581 99585 99590 99594 99599 99603 9607 9612 9616 9621 9625 9629 9634 9638 9642 9647 9651 9656 9660 9664 9669 9673 9677 9682 9686 9691 9699 9704 9708 9712 9717 9721 9726 9730 9734 9739 9743 9747 9752 9756 9760 9765 9769 9774 9778 9782 9787 9791 9795 9800 9804 9808 9813 9817 9822 9826 9830 9835 9839 9843 9848 9852 9856 9861 9865 9870 9874 9878 9883 9887 9891 9896 9900 9904 9909 9913 9917 9922 9926 9930 9935 9939 9944 9948 9952 9957 9961 9965 9970 9974 9978 9983 9987 9991 9996 000 00000 00004 00009 00013 00017 00022 00026 00030 00035 00039 LOGARITHMIC SINES AND COSINES o 1 Sine Cosine Sine Cosine Sine Cosine oo 6.46373 76476 94085 7.06579 16270 24188 41797 7.46373 50512 54291 57767 60985 66784 69417 71'JOO 74248 7.76475 78594 80615 82545 84393 86166 87870 89509 91088 92612 7.94084 95508 96887 99520 8.00779 02002 03192 04350 05478 8.06578 07650 08696 09718 10717 11693 12647 13581 14495 15391 8.16268 17128 17971 18798 19610 20407 21189 21958 22713 23456 24186 10.00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 10.00000 00000 00000 00000 00000 00000 00000 9.99999 99999 99999 99999 . 99998 99998 99998 99998 99997 99997 99997 .99997 99997 99997 99997 99996 99996 99996 99996 .99995 99995 99995 99995 99995 99994 99994 27661 28324 29621 5. 30879 31495 32103 32702 33875 34450 35018 35578 36131 .36678 37217 37750 38276 39310 40320 40816 41307 .41792 42272 42746 43216 43680 44139 44594 45044 45489 45930 48485 48896 49304 49708 50108 .50504 50897 51287 51673 52055 52434 52810 53183 53552 53919 54282 .24186 24903 25609 26304 99993 99992 99992 99992 99990 99990 99987 99987 99987 9.99985 9998E 99984 99984 99983 99983 99982 8.46366 9.99982 46799 47226 47650 99981 99979 99979 9.99978 99977 99977 99977 99975 99974 99974 99974 8.54282 54642 54999 55354 55705 56054 56400 56743 57084 57421 58419 58747 59072 59395 59715 60662 8.60973 61282 61589 61894 62196 62497 62795 63385 63678 8.63968 64256 64543 64827 65110 65391 65670 65947 66497 8.60769 67575 67841 68104 68367 68627 69144 8.69400 69654 69907 70159 70409 70658 70905 71151 71395 71638 71880 9.99974 99973 99973 99972 99972 69971 99971 99970 9.99991 8.57757 9 99967 99966 99966 99964 9.99964 99963 99962 99960 99959 .99959 99958 99957 99956 99956 99955 99955 99954 9!)954 .99953 99952 99952 99951 99951 99950 99949 9.99947 99945 99942 99942 Cosine Sine 89 Cosine Sine Cosine Sine 88 130 LOGARITHMIC SINES AND COSINES t 3 4 6 Sine Cosine Sine Cosine Sine Cosine 8.71880 9.99940 8.84358 9.99894 8.94030 9.99834 60 1 72120 99940 84539 99893 94174 99833 59 2 72359 99939 84718 99892 94317 99832 58 3 72597 !>9938 84897 99891 944(51 99831 57 4 72834 99938 85075 99891 94603 99830 56 5 . 73069 99937 85252 99890 94746 99829 55 6 73303 99936 85429 99889 94887 99828 54 7 73535 99936 85605 99888 95029 99827 53 8 73767 99935 85780 99887 95170 99825 52 9 73997 99934 85955 99886 95310 99824 51 10 8.74226 9.99934 8.86128 9.99885 8.95450 9.99823 50 11 74454 99933 86301 99884 95589 99822 49 12 74680 99932 86474 99883 95728 99821 48 13 74906 99932 86645 99882 95867 99820 47 14 75130 99931 86816 99881 9(5005 99819 46 15 75353 99930 86987 99880 96143 99817 45 16 75575 99929 87156 99879 96280 99816 44 17 75795 99929 87325 99879 96417 99815 43 18 76015 99928 87494 99878 96553 99814 42 19 76234 99927 8766 1 99877 96689 99813 41 20 8.76451 9.99926 8.87829 9.99876 8.96825 9.99812 40 21 76667 99926 87995 99875 96960 99810 39 22 76883 99925 88161 99874 97095 99809 38 23 77097 99924 88326 99873 97229 93808 37 24 77310 99923 88490 99872 97363 99807 36 25 77522 99923 88654 99871 97496 99806 35 26 77733 99922 88817 99870 97629 99804 34 27 77943 99921 88980 99869 97762 90803 33 28 78152 99920 89142 99868 97894 99802 33 29 78360 99920 89304 99867 98026 99801 31 30 8.78568 9.99919 8.89464 9.99866 8.98157 9.99800 30 31 78774 99918 89625 9981)5 98288 99798 29 32 78979 99917 89784 99864 98419 99797 28 33 79183 99917 89943 99863 98549 99796 27 34 79386 99916 P0102 99862 98679 99795 26 35 79588 99915 90260 99861 98808 99793 25 36 79789 99914 90417 99860 98937 99792 24 87 79990 99913 ' 90574 99859 99066 99791 23 38 80189 99913 90730 99858 99194 99790 22 39 80388 99912 90885 99857 99322 99788 21 40 8.80585 9.99911 8.91040 9.99856 8.99450 9.99787 20 41 8078-2 99910 91195 99855 99577 99786 19 42 80978 99909 91349 99854 99704 99785 18 43 81173 99909 91502 99853 99830 99783 17 44 81367 99908 91655 99852 99956 99782 16 45 81560 99907 91807 99851 9.00082 99781 15 46 81752 99906 91959 99850 00207 99780 14 47 81944 99905 92110 99848 00332 99778 13 48 82134 99904 92261 99847 00456 99777 12 49 82324 99904 92411 99846 00581 99776 11 50 8.82513 9.99903 8.92561 9.99845 9.00704 9.99775 10 51 82701 99902 92710 99844 00828 99773 9 52 82888 99901 92859 99843 00951 99772 8 53 83075 99900 93007 99842 01074 99771 7 54 83261 9'.899 93154 99S41 01196 99769 6 55 83446 99898 93301 99840 01318 99768 5 56 83630 99898 93448 99839 01440 99767 4 57 83813 99897 93594 99838 01561 99705 3 58 83996 99896 93740 99837 01682 99764 2 59 84177 99895 93885 99836 01803 99763 1 60 84358 99894 94030 99834 01923 99761 / Cosine Sine Cosine Sine Cosine Sine / 86 - 85 84 LOGARITHMIC SINES AND COSINES Sine Cosine Sine Cosine Sine Cosine 9.01923 02043 02163 02283 . 02402 02520 02639 02757 02874 02992 9.03109 03226 03342 03 158 03574 !690 03805 03920 04034 04149 9.04262 04376 04490 04603 04715 04828 04940 05052 05164 05275 9.05386 05497 05607 05717 05827 05937 06046 06155 06264 06372 9.06481 06589 06804 07018 07124 07231 07337 07442 .07548 07653 07758 07863 07968 08072 08176 08280 08383 08589 9.99761 99760 99759 99757 99756 99755 99753 99752 99751 99749 9.99748 99747 99745 99744 99742 99741 99740 99737 99736 9.99734 99733 99731 99730 99728 99727 99726 99724 99723 99721 9.99720 99718 99717 99716 99714 9CC13 99711 99710 99707 9.99705 99704 99701 99699 99692 99687 99684 99678 99677 99675 08795 09101 09202 09506 09707 09807 09907 10006 10106 10205 i0304 10402 10501 9.10599 10697 10795 10990 11087 11184 11281 11377 11474 9.11570 11666 11761 11857 11952 12047 12142 12236 12331 12425 9.12519 12612 12706 12799 13078 13171 13263 13355 9.13447 13539 13630 13722 13813 18904 14085 14175 9.99675 99674 99666 99663 99661 99656 99653 99651 99650 99648 99647 99645 .99643 99637 99635 99633 99625 99615 99612 &. 99610 99595 9.99593 99589 99586 99584 14356 99581 99579 99577 99575 9.14356 14445 14535 14624 14714 14803 15069 15157 9.15245 15333 15421 15508 15683 15770 15857 15944 9.16116 16203 16289 16374 16460 16545 16631 16716 16801 17055 17139 17223 ]7807 17391 17474 17558 17641 17724 .17807 17890 17973 18055 18137 18220 18302 18465 18547 9.18628 18709 18790 18871 18952 19033 19113 19193 19273 19358 Cosine Sine Cosine ..: 88 Sine Cosine 9.99575 99574 99572 99570 99568 99565 99561 9.99557 99556 99554 99552 99550 99548 99546 99545 99543 99541 9.99539 99537 99535 99533 99532 99530 89528 99526 99524 99522 9.99520 99518 99517 995 J 5 99513 99511 99509 99507 99505 9.99501 99492 99490 99486 99484 9.99482 99478 99476 99474 99472 99470 99468 99464 99462 Sine 81' 138 LOGARITHMIC SINES AND COSINES / 9 10 11 Sine Cosine Sine Cosine Sine Cosine 9.19433 9.99462 9.23967 9.99335 9.28060 9.99195 60 1 19513 99460 24089 99333 28125 99192 59 2 19592 99458 24110 99331 28190 99190 58 3 19672 99456 24181 99328 28254 99187 57 4 19751 99454 24253 99326 28319 99185 56 5 19830 99452 24324 99324 28384 99183 55 6 19909 99450 24395 99322 28448 99180 54 7 19988 99448 24466 99319 28512 99177 53 8 20067 99446 24536 99317 28577 99175 52 9 20145 99444 24607 99315 28641 99172 51 10 9.20223 9.99442 9.24677 9.99313 9.28705 9.99170 50 11 20302 99440 24748 99310 28769 99167 49 12 20380 99438 24818 99308 28833 99165 48 13 20458 99436 24888 99306 28896 99162 47 14 20535 99434 24958 99304 28960 99160 46 15 20613 99432 25028 99301 29024 99157 45 16 20691 99429 25098 99299 29087 99155 ' 44 17 20768 99427 25168 99297 29150 99152 43 18 20845 99425 25237 99294 29214 99150 42 19 20922 99423 25307 99-292 29277 99147 41 20 9.20999 9.99421 9.25376 9.99290 9.29340 9.99145 40 21. 21076 99419 25445 99288 29403 99142 39 22 21153 99417 25514 99285 29466 99140 38 23 21229 99415 25583 99283 29529 99137 37 24 21306 99413 25652 99281 29591 99135 36 25 21382 99411 25721 99278 29654 99132 35 23 21458 99409 25790 99276 29716 99130 34 27 21534 99407 25858 99274 29779 99127 33 28 21610 99404 25927 99271 29841 99124 32 29 21685 99402 25995 99269 29903 99122 31 30 9.21761 9.99400 9.260C3 9.99267 9.29966 9.99119 30 31 21836 99398 26131 99264 30028 99117 29 32 21912 99396 26199 99262 30090 99114 28 33 21987 99394 26267 99260 30151 99112 27 34 22062 99392 26335 99257 30213 99109 26 as 22137 99390 26403 99255 30275 99106 25 36 22211 99388 26470 99252 30336 99104 24 37 22286 99385 26538 99250 30398 99001 23 38 22361 99383 26605 99248 30459 99099 22 39 22435 99381 26672 99245 30521 99096 21 40 9.22509 9.99379 9.26739 9.99243 9.30582 9.99093 20 41 22588 99377 26806 99241 30643 99091 19 42 22657 99375 26873 99238 30704 99088 18 43 22731 99372 26940 99236 30765 99086 17 44 22805 99370 27007 99233 30826 99083 16 45 22878 99368 27073 99231 30887 99080 15 46 22952 99366 27140 99229 30947 99078 14 47 23025 99364 27206 99226 31008 99075 13 48 23098 99362 27273 99224 31068 99072 12 49 23171 99359 27339 99221 31129 99070 11 50 9.23244 9.99357 9.27405 9.99^19 9.31189 9/J9067 10 51 23317 99355 27471 99217 81250 99064 9 52 23390 99353 27537 99214 31310 99062 8 53 23462 99351 27602 99212 31370 99059 7 54 23535 99348 27668 99209 31430 99056 6 55 23607 99346 27734 99207 31490 99054 5 56 23679 99344 27799 99204 31549 99051 4 57 23752 9934? 27864 99202 31609 99048 3 58 23823 99340 27930 99200 31669 99046 2 59 23895 99337 27995 99197 31728 99043 1 60 23967 99335 28060 99195 31788 99040 1 Cosine Sine Cosine Sine Cosine Sine 4 80 79 78 LOGARITHMIC SINES AND COSINES 12 13= Sine Cosine 9.31788 31847 31907 31966 320:25 32084 32143 32202 32261 32319 9.32378 38437 32495 32553 32612 32786 32844 32902 9.32960 33018 ' 33075 33133 33190 33248 33305 33362 33420 33477 9.33534 3359 J 33647 33704 33761 33818 33874 33931 33987 34043 9.34100 34156 34212 34324 35154 35209 Sine Cosine Sine Cosine 99030 99027 99024 99022 99019 9a016 9.99013 99011 99008 9900:5 99002 99000 98991 98975 98972 98967 98964 98953 98947 98944 9.98930 . 98927 98910 34436 34491 34547 34602 98904 9.34658 9, 34713 ' 34769 34824 34879 34934 34989 35014 98898 98875 98872 .35209 35263 35318 35373 35427 35481 35536 35590 35644 35698 .35752 35806 35860 35914 36022 36075 36129 36182 36449 36502 36555 36660 36713 36766 36871 36924 37028 37081 37133 37185 37237 37289 9.37341 37445 37497 37549 37600 37652 37703 37755 37806 .37858 37909 37960 38011 38062 38113 38164 38215 38266 38317 9.98872 1)8867 98864 98858 98855 98849 98346 9.98843 98840 98831 98828 988. '5 98822 98819 98816 9.98813 98810 98804 98798 98795 98789 98786 98777 98774 98771 98765 98762 98759 98756 9.98753 98750 98743 98740 98737 98734 98731 98725 9.98722 98719 98715 98712 98709 98706 98703 98700 9.38368 38418 38519 38570 38620 38721 38771 38821 39021 39071 39121 39170 39220 39270 39319 39418 39467 39517 39566 39615 39713 39762 39811 39958 40006 40055 40103 40152 40200 40249 40297 9.40346 40442 40490 40538 40586 40634 40730 40778 40873 40921 40968 41016 41063 41111 41158 41205 41252 41300 9.98690 98675 98671 98668 98665 98662 >. 38871 9.98659 98652 98649 98643 98633 98030 9.98627 98623 98614 98607 98604 98597 9.98594 98588 98584 98581 9S578 98574 98571 98565 9.98561 98558 98555 98551 98548 98545 98541 98535 98531 9.-J8528 98525 98521 98518 98515 98511 98505 98501 98494 Cosine Sine Cosine Sine Cosine 76 Sine 140 LOGARITHMIC SINES AND COSINES / 15 16 17 Sine Cosine Sine Cosine Sine Cosine | 9.41300 9.98494 9.44034 9.98284 9.46594 9.98060 60 1 41347 98491 44078 98281 46635 98050 59 2 41394 98488 44122 98277 46676 98052 58 3 41441 98484 44166 98273 46717 98048 57 4 41488 98481 44210 98270 46758 9t?044 56 5 41535 98477 44253 98266 46800 98040 55 6 41 582 98474 44297 98262 46841 98036 54 7 41028 98471 44341 98259 46882 98032 53 8 41675 98467 44385 98255 46923 98029 52 9 41722 98464 41428 98251 46964 9b025 51 10 9.41768 9.98460 9.44472 9.98248 9.47005 9.98021 50 11 41815 98457 44516 9S244 47045 98017 49 12 41861 98453 44559 96240 47086 98013 48 13 41908 98450 44602 98237 47127 98009 47 14 41954 98447 44646 98233 47168 98005 4(i 15 42001 98 143 44689 98229 47209 98001 45 16 42047 98440 44733 98226 47249 97997 44 17 42093 98136 44776 98222 47290 97993 43 18 42140 ib'433 44819 98218 47330 97989 42 19 42186 98429 44862 98215 47371 97986 41 20 9.-J2232 9.98426 9.44905 9.98211 9.47411 9.97982 40 21 42278 98422 44948 98207 47452 97978 39 22 42324 98419 44992 98204 47492 97974 38 23 42370 98415 45035 98200 47533 97970 37 24 42416 98412 45077 98196 47573 97906 36 25 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43502 98327 46095 98106 48529 97870 12 49 43546 98324 46136 98102 48568 97866 11 50 9.43591 9.98320 9.46178 9.98098 9.48607 9.97861 10 51 43635 98317 46220 98094 48647 97857 9 52 43680 98313 46262 98090 48686 97853 8 53 43724 98309 46303 98087 48725 97849 7 54 43769 98306 46345 98083 48764 97845 6 55 43813 98302 46386 98079 48803 97841 5 56 43857 98299 46428 98075 48842 97837 4 57 43901 98295 46469 98071 48881 97833 3 58 43946 98291 46511 98067 48920 97829 2 59 43990 98288 46552 98063 48959 97825 1 60 44034 98284 46594 98060 48998 97821 / Cosine Sine Cosine Sine Cosine Sine / 74 78 72 LOGARITHMIC SINES AND COSINES 141 18 19 20 Sine Cosine Sine Cosine Sine Cosine 9.48998 9.97821 9.51264 9.97567 9.53405 9.97299 60 1 49037 97817 51301 97563 53440 97294 59 2 490T6 97812 51338 97558 53475 97289 58 3 49115 97808 51374 97554 53509 97285 57 4 49153 97804 51411 97550 53544 97280 56 5 49192 97800 51447 97545 53578 97276 55 6 49231 97796 51484 97541 5S613 97271 54 7 49269 97792 51520 97536 53647 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9.54433 9.97159 30 31 50le>5 97691 52385 97430 54466 97154 29 32 50223 97687 52421 97426 54500 97149 28 33 50261 97683 52456 97421 54534 97145 27 34 50298 97679 52492 97417 54567 97140 26 35 50336 97674 52527 97412 54601 97135 25 36 50374 97670 52563 97408 54635 97130 24 37 50111 97666 52598 97403 54668 97126 23 38 50449 97662 52634 97399 54702 97121 22 39 50486 97657 52669 97394 64735 97116 21 40 9.50523 9.97653 9.52705 9.97390 9.54769 9.97111 20 41 50561 97649 52740 97385 54802 97107 19 42 50598 97645 52775 97381 54836 97102 18 43 50635 97640 52811 97376 54869 97097 17 44 50673 97636 52846 97372 54903 97092 16 45 50710 97632 52881 97367 54936 97087 15 46 50747 97628 52916 97363 54969 97083 14 47 50784 97623 52951 97858 55003 97078 13 48 50821 97619 52986 97353 55036 97073 12 49 50858 97615 53021 97349 55069 97068 11 50 9.50896 9.97610 9.53056 9.97344 9 55102 9.97063 10 51 509:53 97606 53092 97340 55136 97059 9 52 50970 97602 53126 97335 55169 97054 8 53 51007 97597 53161 97331 55202 97049 7 54 51043 97593 53196 97326 55235 97044 6 55 51080 97589 53231 97322 55268 97039 5 56 51117 97584 53266 97317 55301 97035 4 57 51154 97580 53301 97312 55334 97030 3 58 51 191 97576 53336 97308 55367 97025 2 59 51227 97571 53370 97303 55400 97020 1 60 51264 97567 53405 97299 55433 97015 / Cosine Sine Cosine Sine Cosine Sine / 71 70 69 142 LOGARITHMIC SINES AND COSINES / 21 22 23 / Sine Cosine Sine Cosine Sine Cosine 9.55433 9.97015 9.57358 9.96717 9.59188 9.96403 60 1 55466 97010 5739 96711 59218 96397 59 2 55499 97005 57420 96706 59247 96392 58 3 55532 97001 57451 96701 59277 96387 57 4 55564 96996 57482 96696 59307 96381 56 5 55597 96991 57514 96691 59336 96376 55 6 55630 96986 57545 96686 59366 96370 54 7 556(33 96981 57576 96681 59396 96365 53 8 55695 96976 57607 96676 59425 96360 52 9 55728 96971 57638 96670 59455 96354 51 10 9.55761 9.96966 9.57669 9.96665 9.59484 9.963-19 50 11 55793 96962 57700 96C60 59514 96343 49 12 55826 96957 57731 96655 59543 96338 48 13 55858 96952 57762 96650 59573 96333 47 14 55891 96947 57793 96645 59602 96327 46 15 55923 96942 57824 96640 59632 96322 45 16 55956 96937 57855 96634 59661 96316 44 17 55988 96932 57885 96629 59690 96311 43 18 56021 96927 57916 96624 59720 96305 42 19 56053 9G922 57947 96619 59749 96300 41 20 I'. 56085 9.96917 9.57978 9.96614 9.59778 9.96294 40 ' 21 561 18 96912 5800 ! 96608 59808 96289 39 22 56150 90907 58039 96003 59837 96284 38 23 56182 96903 58070 96598 59866 96278 37 24 56215 96898 58101 96593 59895 96273 36 25 56*247 96893 58131 96588 59924 96267 35 26 56279 96888 58162 96582 59954 96262 34 27 56311 96883 58192 96577 59963 96256 33 28 56343 96878 58223 96572 60012 96251 32 29 56375 96873 58253 96567 60041 96245 31 30 9.56108 9.96868 9.58284 9.9G562 9.60070 9.96240 30 31 56440 96863 58314 96556 60099 96234 29 32 56472 96858 58345 96551 60128 96229 28 33 56504 96853 58375 96546 60157 962',3 27 34 56536 96848 58406 96541 60186 96218 26 a5 56568 96843 58436 06535 60215 96212 25 ae 56599 96838 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9.93077 9.73022 9 92603 30 31 70568 93525 71829 93069 73041 92595 29 32 70590 93517 71850 93061 73061 92587 28 33 70611 93510 71870 93053 73081 92579 27 34 70633 93502 71891 93046 73101 92571 26 35 70654 93495 71911 93038 73121 9-2563 25 36 70675 93487 71932 93030 73140 92555 24 37 70697 93480 71952 93022 73160 9-2546 23 38 70718 93472 71973 9: JO 14 73180 92538 22 39 70739 93465 71994 93007 73200 92530 21 40 9.70761 9.93457 9.72014 9.92999 9.73219 9.92522 20 41 70782 93450 72034 9-2991 73239 92514 19 42 70803 93442 72055 92983 73259 92506 18 43 70824 93435 72075 92976 73-278 92498 17 44 70846 93427 72096 92968 73298 92490 16 45 70867 93420 72116 92960 73318 92482 15 46 70888 93412 72137 92952 73337 92473 14 47 70909 93405 72157 92944 73357 92465 13 48 70931 93397 72177 92936 73377 92457 12 49 70952 93390 72198 92929 73396 92449 11 50 9.70973 9.93382 9.72218 9.92921 9.73416 9 92441 10 51 70994 93375 72238 92913 73435 92433 9 52 71015 93367 72259 92905 73455 92425 8 53 71036 93360 72279 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87255 18 48 80625 88552 81519 87909 82382 87243 12 49 80641 88542 81534 87898 82396 87232 11 50 9.80656 9.88531 9.81549 9.87887 9.82410 9.87S21 10 51 80671 88521 81563 87877 82424 87209 9 52 80686 88510 81578 87866 82439 87198 8 53 80701 88499 81592 87855 824f3 87187 7 54 80716 88489 81607 87844 82467 87175 6 55 80731 88478 81622 87833 82481 87164 5 56 80746 88468 81686 87822 82495 87153 4 57 80762 88457 81651 87811 82509 87141 3 58 80777 88447 81665 87800 82523 87130 2 59 80792 88436 81680 87789 82537 87119 1 60 80807 88425 81694 87778 82551 87107 ~ Cosine Sine Cosine Sine Cosine Sine , 50 49 48 LOGARITHMIC SINES AND COSINES 149 / 42 48 44 ] Sine Cosine Sine Cosine Sine Cosine o 9.82551 9.87107 9.88378 9.86413 9.84177 9.85693 60 82565 87096 83392 86401 84190 85681 59 2 82579 87085 83405 8G889 84203 85669 58 3 82593 87073 83419 86377 84216 85657 57 4 82607 87062 83432 86366 84229 85645 56 5 82621 87050 83446 86354 84242 85632 55 6 82635 87039 83459 86342 84255 85620 54 7 82649 87028 83473 86330 84269 85608 53 8 82663 87016 83486 86318 84282 85596 52 9 82677 87005 83500 86306 84295 85583 51 10 9.82691 9.86993 9.83513 9.86295 9.84308 9.85571 50 11 82705 86982 83527 86283 84321 85559 49 12 82719 86970 83540 86271 84334 85547 48 13 82733 86959 83554 86259 84347 85534 47 14 82747 86947 83567 86247 84360 85522 46 15 82rei 86936 83581 86235 84373 85510 45 16 82775 86924 83594 80888 84385 85497 44 17 82788 86913 83608 86211 84398 85485 43 18 82802 86902 83621 86200 84411 85473 42 19 82816 86890 83634 86188 84424 85460 41 20 9.82830 9.86879 9.83648 9.86176 9.84437 9.85448 40 21 82844 86867 83661 86164 84450 85436 39 22 82858 86855 83674 86152 84463 85423 38 23 82872 86844 83688 86140 84476 85411 37 24 82885 86832 83701 86128 84489 85399 36 25 82899 86821 83715 86116 84502 85386 35 26 82913 86809 83728 86104 84515 85874 34 27 82927 86798 83741 86092 84528 65361 33 28 82-941 86786 83755 86080 84540 85349 32 29 82955 86775 83768 86068 84553 85337 31 30 9.82968 9.86763 9.83781 9.86056 9.84566 9.85324 30 31 82982 86752 83795 86044 84579 85312 29 32 82996 86740 83808 86032 84592 85299 28 33 83010 86728 83821 86020 84605 85287 27 34 83023 86717 83834 86008 84618 85274 26 35 88037 86705 83848 85996 84630 85262 25 36 83051 86694 83861 85984 84643 85250 24 37 83065 86682 83874 85972 84656 85237 23 38 83078 86670 83887 859CO 84669 85225 22 39 83092 86659 83901 85948 84682 85212 21 40 9.83106 9.86647 9.83914 9.85936 9.84694 9.85200 20 41 83120 86635 83927 85924 84707 85187 19 42 83133 86624 83940 85912 84720 85175 18 43 83147 86612 83954 85900 84733 85162 17 44 83161 86600 83967 85888 84745 85150 16 45 83174 86589 83980 85876 84758 85137 15 46 83 188 86577 83993 85864 84771 85125 14 47 83202 86565 84006 85851 81784 85112 13 48 83215 86554 84020 85839 84796 85100 12 49 83229 86542 840.33 85827 84809 85087 11 50 9.83242 9.86530 9.84046 9.85815 9.84822 9.85074 10 51 83256 86518 84059 85803 84835 85062 9 52 83270 86507 84072 85791 84847 85049 8 53 83283 86495 84085 85779 84860 85037 7 54 83297 86483 84098 85766 84873 86024 6 55 83310 86472 84112 85754 84885 85012 5 56 83324 86460 84125 85742 84898 84999 4 57 83338 86448 84138 85730 84911 84986 3 58 83351 86436 84151 85718 84923 84974 2 59 83365 86425 84164 85706 84936 84961 1 60 83378 86413 84177 85693 84949 84949 Cosine Sine Cosine Sine Cosine Sine | 470 46 46 150 LOG. TANGENTS AND COTANGENTS / 1 20 Tan Cotan Tan Cotan Tan Cotan oo OD 8.21192 11.75808 8.54308 11.45692 60 1 6.46373 13.53627 24910 75090 54669 45331 59 2 76476 23524 25616 74384 55027 44973 58 3 94085 05915 26312 73688 55382 44618 57 4 7.06579 12.93421 26996 73004 55734 44266 56 5 16270 83730 27669 72331 56083 43917 55 6 24188 75812 28332 71668 56429 43571 54 7 30882 69118 28986 71014 56773 43227 53 8 36682 63318 29629 70371 57114 42S86 52 9 41797 58203 30263 69737 57452 42548 51 10 7.46373 12.53627 8.30888 11.69112 8.57788 11.42212 50 11 50512 49488 31505 68495 58121 41879 49 12 54291 45709 32112 67888 58451 41549 48 13 57767 42233 32711 67289 58779 41221 47 14 60986 39014 33302 66698 59105 40895 46 15 6398-2 36018 33886 66114 59428 40572 45 16 66785 33215 34461 65539 59749 40251 44 17 (19418 30582 35029 64971 60068 39932 43 18 71900 28100 35590 64410 60384 39616 42 19 74248 25752 36143 63857 60698 89302 41 20 7 76476 12.23524 8.36689 11.63311 8.61009 11.38991 40 21 78595 21405 37229 62771 61319 38681 39 22 80615 19385 37762 62238 61626 38374 38 23 82546 17454 38289 61711 61081 38069 37 24 S4394 15606 38809 61191 62234 37766 36 25 86167 13833 39323 . 60677 62535 37465 i 35 26 87871 12129 39832 60168 62834 371 C6 34 27 89510 10490 40334 59666 63131 36869 33 28 91089 08911 40880 59170 63426 36574 32 29 92613 07387 41321 58679 63718 36282 31 30 7.94086 12.05914 8.41807 11.58193 8.64009 11.35991 30 31 95510 04490 42287 57713 64298 35702 29 32 96889 03111 42762 57238 64585 35415 28 33 98225 01775 43232 56768 64870 35130 27 34 99522 00478 43696 56304 05154 34846 26 35 8.00781 11.99219 44156 55844 65435 34565 25 36 02004 97996 44611 55389 65715 34285 24 37 03194 96S06 45061 54939 65993 34007 23 38 04353 95647 45507 54493 66269 33731 22 39 05481 94519 45948 54052 66543 33457 21 40 8.06581 11.93419 8.46885 11.53615 8.66816 11.23184 20 41 07653 92347 46817 53183 67087 32913 19 42 08700 91300 47245 52755 67356 32644 18 43 09722 90278 47669 52331 67624 32376 17 44 10720 89280 48089 51911 67890 32110 16 45 11696 88304 48505 51495 68154 31846 15 46 12651 87349 48917 51083 68417 31583 14 47 13585 86415 493:25 50675 68678 31322 13 48 14500 85500 49729 50271 68938 31062 12 49 15395 84605 50130 49870 69196 30804 11 50 8.16273 11.83727 8.50527 11 .49473 8.69453 11.30547 10 51 17133 82867 50920 49080 69708 30292 9 52 17976 82024 51310 48690 69962 30038 8 53 18804 81196 51696 48304 70214 29786 7 54 19616 80384 52079 47921 70465 29535 6 55 20413 79587 52459 47541 70714 29286 5 56 21195 78805 52835 47165 70962 29038 4 57 21964 78036 53208 46792 71208 28792 3 58 22720 77280 53578 46422 71453 28547 2 59 23462 76538 53945 46055 71697 28303 1 60 24192 75808 54308 45692 71940 28060 / Cotau Tan Cotan Tan Cotan Tan / 89 88 87 LOG. TANGENTS AND COTANGENTS 151 / 8 4 5 / Tan Cotan Tan Cotan Tan Cotan o 8.71940 11.28060 8.84464 11.15536 8.94195 11.05805 60 I 72181 27819 84646 15354 94340 05660 59 2 72420 27580 84826 15174 94485 05515 58 3 72659 27341 85006 14994 94630 05370 57 4 72896 27104 85185 14815 94773 05227 56 5 73132 26868 85363 14637 94917 05083 55 6 73366 26634 85540 14460 95060 04940 54 7 73600 26400 85717 14283 95202 04798 53 8 73333 2616S 85893 14107 95344 04656 52 9 74063 25937 86069 13931 95486 04514 51 10 8.74292 11.25708 8.86243 11.13757 8.95627 11.04373 50 11 74531 25479 86417 13583 95767 04238 49 12 747-18 25252 86591 13409 95908 04092 48 18 74974 25026 86763 13237 96047 03953 47 14 75199 24801 86935 13065 96187 03813 46 15 75423 24577 8710(5 12894 96325 03675 45 16 75645 24355 87277 12723 96464 03536 44 17 75867 24133 87447 12553 96602 03398 43 18 76087 23913 87616 12384 96739 03261 42 19 76306 23694 87785 12215 96877 02133 41 20 8 76535 11.23475 8.87953 11.12047 8.97013 11.02987 40 21 76743 23258 88120 11880 97150 02850 39 22 76958 23042 88287 11713 97285 02715 38 n 77173 22827 88453 11547 97421 02579 37 24 77387 22613 88618 11382 97556 02444 36 25 77600 22400 88783 11217 97691 02309 35 26' 77811 22189 88948 11053 97885 02175 34 27 78022 21978 89111 10889 97959 02041 33 28 78232 21768 89274 10726 98092 01908 32 29 78441 21559 89437 10563 98225 01775 31 30 8.78649 11.21351 8.89598 11.10402 8.98358 11.01642 30 31 78855 21145 89760 10240 98490 01510 29 32 79061 20939 89920 10080 98622 01378 28 83 79266 20734 90080 09920 98753 01247 27 34 79470 20530 90240 09760 98884 01116 26 35 79673 20327 90399 09601 99015 00985 25 36 79875 20125 90557 09443 99145 00855 24 37 80076 19924 90715 09285 99375 00735 23 38 80277 19723 90872 09128 99405 00595 22 39 80476 19524 91029 08971 99534 00466 21 40 8.80674 11.19326 8.91185 11.08815 8.99662 11.00338 20 41 80872 19128 91340 08660 99791 00209 19 42 81068 18932 - 91495 08505 99919 00081 18 43 81264 18736 91650 08350 9.00046 10.99954 17 44 81459 18541 91803 08197 00174 99826 16 45 81653 18347 91957 08043 00301 99699 15 46 81840 18154 92110 07890 00427 99573 14 47 82038 17962 92262 07738 00553 99447 18 48 82230 17770 92414 07586 00679 99331 12 49 82420 17580 92565 07435 00805 99195 11 50 8.82610 11.17390 8.92716 11.07284 9.00930 10.99070 10 51 82799 17201 92866 07134 01055 98045 9 52 82987 17013 93016 06984 01179 98821 8 53 83175 16825 93165 06835 01303 98697 7 54 83361 16639 93313 06687 01427 98573 6 55 83547 16453 93462 06538 01550 98450 5 56 83T32 16268 93609 06391 01673 98327 4 57 83916 16084 93756 06244 01796 96204 3 58 84100 15900 93903 06097 01918 96082 2 59 84282 15718 94049 05951 02040 97960 1 60 81464 15536 94195 05805 02162 97838 C / Cotan Tan Cotan Tan Cotan Tan t 86 86 84 152 LOG. TANGENTS AND COTANGENTS / ( 7 8 Tan Cotan Tan Cotan Tan Cotan 9.02162 10.97838 9.08914 10.91086 9.14780 10.86220 60 02283 97717 09019 90981 14872 85128 59 2 02404 9T596 09123 90877 14963 85037 58 3 025-^5 97475 09227 90773 15054 84946 57 4 02645 97355 09330 90670 15145 84855 56 5 02766 97234 09434 90566 15236 84764 55 6 02885 97115 09537 90463 15327 84673 54 7 03005 96995 09640 90360 15417 84583 53 8 03124 96876 09742 90258 15508 84492 52 9 03242 96758 09845 90155 15598 84402 51 10 9.03361 10.96639 9.09947 10.90053 9.15688 10.84312 50 11 03479 96521 10049 89951 15777 84228 49 12 03597 96403 10150 89850 15867 84133 48 13 03714 96286 10252 89748 15956 84044 47 14 03832 96168 10353 89647 16046 83954 46 15 03948 96052 10454 89546 16)35 83865 45 16 04065 95935 10555 89445 16224 83776 44 17 04181 95819 10656 89344 16812 83688 43 18 04297 95703 10756 89244 16401 83599 42 19 04413 95587 10856 89144 16489 83511 41 20 9.04528 10.95472 9.10956 10.89044 9.16577 10.83423 40 21 04643 95357 11056 88944 16665 83335 39 22 04758 95242 11155 H8845 16753 83247 38 23 04873 95127 11254 88746 16841 83159 87 24 04987 95013 11358 88647 16928 83072 86 25 05101 94899 11452 88548 17016 82984 35 26 05214 94786 11551 88449 17103 82897 34 27 05328 94672 11649 88351 17190 82810 33 28 05441 94559 11747 88258 17277 82723 82 29 05553 94447 11845 88155 17363 82637 31 30 9.05666 10.94334 9.11943 10.88057 9.17450 10.82550 30 31 05778 94222 12040 87960 17536 82464 29 32 05890 94110 12138 87862 17622 82378 28 33 06002 93998 12235 87765 17708 82292 27 34 06118 93887 12332 8T668 17794 82206 26 35 06224 93776 12428 87572 17880 82120 25 86 06335 93665 12525 87475 17965 82035 24 37 06445 93555 12621 87379 . 18051 81949 23 38 06556 93444 12717 87283 18136 81864 22 39 06666 93334 12813 87187 18221 81779 21 40 9.06775 10.93225 9.12909 10.87091 9.18306 10.81694 20 41 06885 93115 13004 86996 18391 81609 19 42 06994 93006 13099 86901 18475 81525 18 43 07103 92897 13194 86806 18560 81440 17 44 07811 92789 13289 86711 18644 81856 16 45 07820 92680 13384 8G616 18728 81272 15 46 07428 92572 13478 86522 18812 81188 14 47 \536 92464 13573 86427 18896 81104 13 48 07643 92857 13667 86338 18979 81021 12 49 07751 92249 13761 86239 19063 80937 11 50 9.07858 10.92142 9.13854 10.86146 9.19146 10.80854 10 51 07964 92036 13948 86052 19229 80771 9 52 08071 91929 14041 85959 19312 80688 8 53 08177 91828 14134 85866 19395 80605 7 54 08288 91717 14227 85773 19478 80522 6 55 08389 91611 14320 85680 19561 80439 5 56 08495 91505 14412 85588 19643 80357 4 57 08600 91400 14504 85496 19725 80275 3 58 08705 91295 14597 85403 19807 80193 2 59 08810 91190 14688 85312 19889 80111 1 60 08914 91086 14780 85220 19971 80029 f Cotan Tan Cotan Tan Cotan Tan { 83 82' 81 LOG. TANGENTS AND COTANGENTS 153 9 10 11 t Tan Cotan Tan Cotan Tan Cotan 9.19971 10.80029 9.24632 10.75368 9.28865 10.71135 60 1 20053 79947 24706 75294 28933 71067 59 2 20134 79S66 24779 75221 29000 71000 58 3 20216 79784 24853 75147 29067 70933 57 4 20297 79703 24926 75074 29134 70866 56 5 20378 79622 25000 75000 29201 70799 55 6 20459 79541 25073 74927 29268 70732 54 7 20540 79460 25146 74854 29335 70665 53 8 20621 79379 25219 74781 29402 70598 52 9 20701 79299 25292 74708 29468 70532 51 10 9.20782 10.79218 9.25365 10.74685 9.29535 10.70465 50 11 20862 79138 25437 74563 29601 70399 49 12 20942 79058 25510 74490 29668 70332 48 13 21022 78978 25582 74418 29734 70266 47 14 21102 78898 25655 74345 29800 70200 46 15 21182 78818 25727 74273 29866 70134 45 16 21261 78739 25799 74201 29932 70068 44 17 21341 78659 25871 74129 29998 . 70002 43 18 21420 78580 25943 74057 30064 69936 42 19 21499 78501 26015 73985 30130 69870 41 80 9.21578 10.78422 9.26086 10.73914 9.30195 10.69805 40 21 21657 78343 26158 73842 30261 69739 39 22 21736 78264 26229 73771 30326 69674 38 23 21814 78186 26301 73699 30391 69609 37 24 21893 78107 26372 73628 30457 69543 36 25 21971 78029 26443 73557 30522 69478 35 26 22049 77951 2(5514 73486 30587 69413 34 27 22127 77873 26585 73415 30652 69348 83 28 22205 77795 26655 73345 30717 69283 32 29 22283 77717 26726 73274 30782 69218 31 30 9.22361 10.77639 9.26797 10.73203 9.30846 10.69154 30 31 22438 77562 26S67 73133 30911 69089 29 32 22516 77484 26937 73063 30975 69025 28 33 22598 77407 27008 72992 31040 68960 27 34 22670 77330 27078 72922 31104 68896 26 91 22747 77253 27148 72852 31168 68832 25 36 22821 77176 27218 72782 31233 68767 '24 37 22901 77099 27288 72712 31297 68703 23 38 22977 77023 27357 72648 31361 68639 22 39 23054 76946 27427 72573 31425 68575 21 40 9.23130 10.76870 9.27496 10.72504 9.31489 10.68511 20 41 28206 76794 27566 72434 31552 68448 19 42 23283 76717 27635 72365 31616 68384 18 43 23359 76641 27704 72296 31679 68321 17 44 23435 76565 27773 72227 31743 68257 16 45 23510 76490 27842 72158 81806 68194 15 46 23586 76414 27911 72089 31870 68130 14 47 23661 76339 27980 72020 31933 68067 13 48 23737 76263 28049 71951 31996 68004 12 49 23312 76188 28117 71883 32059 67941 11 50 9.23887 10.76113 9.28186 10.71814 9.32122 10.67878 10 51 23962 76038 28254 71746 32185 67815 9 52 24037 759(53 28323 71677 32248 67752 8 53 24112 75888 28391 71609 3231 1 67689 7 54 21186 75814 28459 71541 32873 67627 6 55 24261 75739 28527 71473 32436 67564 5 56 24335 75665 28595 71405 32498 67502 4 57 24410 75590 28662 71338 32561 67439 3 58 24481 75516 28730 71270 32623 67377 2 59 24558 75442 28798 71202 32685 67315 1 60 24632 75368 28865 71135 32747 67253 ' t f Cotan Tan Cotan Tan Cotan Tan 80* 79 78 154 LOG. TANGENTS AND COTANGENTS 12 18 14 / Tan Cotan Tan Cotan Tan Cotan o 9.32747 10.67253 9.36336 10.63664 9.39677 10.60323 60 1 32810 67190 36394 63606 39731 602C9 59 2 32872 67128 36452 63548 39785 60215 58 3 32933 67067 36509 63491 39838 60162 57 4 32995 67005 36566 63434 89892 60108 56 5 1 33057 66943 36624 63376 39945 60055 55 6 33119 66881 36681 63319 39999 60001 54 7 83180 66820 36738 68263 40052 59948 53 8 83>42 66758 36795 63205 40106 59894 52 9 83303 66(597 36852 63148 40159 59841 51 10 9.33365 10.66635 9.36909 10.63091 9.40212 10.59788 50 11 38426 66574 36966 63034 40266 59734 49 12 38487 66513 37023 62977 40319 59681 48 13 33548 66452 37080 62920 40372 59628 47 14 33609 66391 37137 62863 40425 59575 46 15 33670 66330 37193 62807 40478 59522 45 16 33731 66269 37250 62750 40531 59469 44 17 33792 66208 37306 62G94 40584 59416 43 18 33853 66147 37363 62637 40636 593G4 42 19 33913 66087 37419 62581 40689 59311 41 20 9.33974 10.66026 9.37476 10.02524 9.40742 10.59258 40 21 34034 65966 37532 62468 40795 59205 89 22 34095 65905 37588 62412 40847 59153 i 38 23 84155 65845 37644 62356 40900 59100 ! 37 24 34215 65785 37700 62300 40952 59048 ! 36 25 34276 65724 37756 62244 41005 58995 35 26 34336 65664 87812 62188 41057 58943 34 27 34396 65604 37868 62132 41109 8891 33 28 34456 65544 37924 62076 41161 58889 32 29 34516 65484 37980 62020 41214 58786 31 30 9.34576 10.65424 9.38035 10.61965 9.41266 10.58734 30 31 34635 65365 38091 61909 41318 58682 29 32 34695 65305 38147 61853 41370 58630 1 28 33 34755 65245 38202 61798 41422 58578 I 27 34 34814 65186 38257 61743 41474 58526 26 35 34874 65126 38313 61687 41526 58474 25 36' 34933 65067 38368 61632 41578 58422 24 37 34992 65008 38423 61577 41629 58371 23 38 35051 64949 38479 61521 41681 58319 22 39 35111 64889 38534 61466 41733 58267 21 40 9.35170 10.64830 9.38589 10.61411 9.41784 10.58216 20 41 35229 64771 38644 61356 41836 58164 19 42 35288 64712 38699 61301 41887 58113 18 43 35347 64653 38754 61246 41939 58061 17 44 35405 64595 38808 61192 41990 58010 16 45 35464 64536 38863 61137 42041 57959 15 46 35523 64477 38918 61082 42093 57907 14 47 35581 64419 38972 61028 42144 57856 13 48 35640 64360 39027 60973 42195 57805 12 49 35698 64302 39082 60918 42246 57754 11 50 9.35757 10.64243 9.39136 10.60864 9.42297 10.57703 10 51 35815 64185 39190 60810 42348 57652 9 52 35873 64127 39245 60755 42390 57601 8 53 35931 64069 39299 60701 42450 57550 7 54 35989 64011 39353 60647 42501 57499 6 55 36047 63953 39407 60593 42552 57448 5 56 36105 63895 89461 60539 42603 57397 4 57 36163 63837 39515 60485 42653 57347 8 58 36221 63779 39569 60431 42704 57296 2 59 36279 63721 39623 60377 42755 57245 1 60 36336 63664 39677 60328 42805 57195 / Cotan Tan Cotan Tan Cotan Tan 77 76 76 LOG. TANGENTS AND COTANGENTS 155 f 15 16 17 / Tan Cotan Tan Cotan Tan Cotan 9.42805 10.57195 9.45750 10.54250 9.48534 10.51466 60 1 42856 57144 45797 54203 48579 51421 59 2 42906 57094 45845 54155 48624 51376 58 3 42957 57043 45892 54108 48669 51331 57 4 43007 56993 45940 51060 48714 51286 56 5 43057 56943 45987 54013 48759 51241 55 6 43108 56892 46035 53965 48804 51196 54 7 43158 56842 46082 53918 48849 51151 53 8 43208 56792 46130 53870 48894 51106 52 9 43258 56742 46177 53823 48939 51061 51 10 9.43308 10.56692 9.4G224 10.53776 9.48984 10.51016 50 11 43358 56642 46271 53729 49029 50971 49 12 43408 56592 46319 53681 49073 50927 48 18 43458 56542 46366 53634 49118 50882 47 14 43508 56492 46413 53587 49163 50837 46 15 43558 56442 46460 53540 49207 50793 45 16 43607 56393 46507 53493 49252 50748 44 17 43657 56343 46554 5344G 49-^96 50704 43 18 43707 56293 46601 53399 49341 50659 42 19 43756 56244 46648 53352 49385 50615 41 20 9 43806 10.56194 9.46694 10.53306 9.49430 10.50570 40 21 43855 56145 46741 53259 49474 50526 39 22 43905 56095 46788 53212 49519 50481 38 23 43954 56046 46835 53165 49563 50437 37 24 44004 55996 46881 53119 49607 50393 S6 25 44053 55947 46928 53072 49652 50348 35 26 44102 55898 46975 53025 49696 50304 34 27 44151 55849 47021 52979 49740 50260 33 28 44201 55799 47068 52932 49784 50216 3.' 29 44250 55750 47114 52886 49828 50172 31 30 9.44299 10.55701 9.47160 10.52840 9.49872 10.50128 30 31 41348 55652 47207 52793 49916 50084 29 32 44397 55603 47253 52747 49960 50010 28 83 44446 55554 47299 52701 50004 49996 27 34 44495 55505 47346 52654 50048 49952 26 35 44544 55456 47392 52608 50092 49908 25 36 44592 55108 47438 52562 50136 49864 24 37 44641 55359 47484 52516 50180 49820 23 38 44690 55310 47E30 52470 50223 49777 22 39 44738 55262 47576 52424 50267 49733 21 40 9.44787 10.55213 9.47622 10.52378 9.50311 10.49689 20 41 44836 55164 47668 52332 50355 49645 19 42 44884 55116 47714 52286 50398 4960-3 18 43 44933 55067 47760 52240 50442 49558 17 44 44981 5.5019 47806 52194 50485 49515 16 45 45029 54971 4785:? 52148 50529 49471 15 46 45078 54922 47897 52103 50572 49428 14 47 45126 54874 47943 52057 50616 493 C 4 13 48 45174 5482(5 47989 52011 50659 49341 12 49 45222 54778 48035 51965 50703 49297 11 50 9.45271 10.54729 9.48080 10.51920 9.50746 10.49254 JO 51 45319 54681 48126 51874 50789 49211 9 52 45367 54633 48171 51829 50833 491G7 8 53 45415 54585 48217 51783 50876 49124 7 54 45463 54537 48262 51738 50919 49081 6 55 45511 54489 48307 51693 50962 49088 5 56 45559 54441 48353 51647 51005 48995 4 57 45606 54394 48398 51602 51048 48952 3 58 45654 54346 48443 51557 51092 48908 2 59 45702 54298 48489 51511 51135 48H65 60 45750 54250 48534 51466 51178 48822 / Co tan Tan Cotan Tan Cotan Tan / 74 73 72 156 LOG. TANGENTS AND COTANGENTS / 18 19 20 / Tan Cotan Tan Cotan Tan Cotan 9.51178 10.48822 9.53697 10.46303 9.56107 10.43893 60 1 51221 48779 53738 46262 56146 43854 59 2 51264 48736 53779 46221 56185 438.5 58 3 51306 48694 53820 46180 56-J24 43776 57 4 51349 48651 53861 46139 56264 43730 56 5 51392 48608 53902 46098 56303 . 43697 55 6 51435 48565 53943 46057 56342 43658 54 7 51478 48522 53984 46016 56381 43619 53 8 51520 48480 54025 45975 56420 43580 52 9 51563 4S437 54065 45935 56459 43541 51 10 9.51606 10.48394 9.54106 10.45894 9.56498 10.43502 50 11 51648 48352 54147 4S&53 56537 43463 49 12 51691 48309 54187 45813 56576 43424 48 13 51734 48266 54228 45772 56815 43385 47 14 51770 48224 54269 45731 56654 *B46 46 15 51819 48181 54309 45691 56693 43307 45 16 51861 48139 54350 45650 56732 43268 44 17 51903 48097 54390 45610 56771 4:'2J9 43 18 51946 48054 54431 45569 56810 43190 42 19 51988 48012 54471 45529 56849 43151 41 20 9.52031 10.47969 9.54512 10.45488 9.56887 10.43118 40 21 52073 47927 54552 45448 56026 43074 39 22 52115 47885 54593 45407 5(5965 43035 38 23 52157 47843 54633 45367 57004 42996 37 24 52200 47800 54673 45327 57042 42958 30 25 52244 47758 54714 45286 57081 42919 35 26 52284 47716 54754 45246 57120 42880 34 27 52326 47674 54794 45206 57158 42842 33 28 52368 47632 54835 45165 57197 42803 82 29 52410 47590 54875 45125 57235 42765 31 30 9.52452 10.47548 9.54915 10.45085 9.57274 10.42726 30 31 52494 47506 54955 45045 57312 42688 29 32 52536 47464 54995 45005 57851 42649 28 33 52578 47422 55035 44965 57389 42611 27 34 52620 47380 55075 44925 57428 42572 26 ;-;.-. 52661 47339 55115 44885 57466 42534 25 36 52703 47'.'97 55155 44845 57504" 42496 24 37 52745 47255 55195 44805 57543 42457 23 38 52787 47213 55235 44765 57581 42419 22 39 52829 47171 55275 44725 576)9 42381 21 40 9.52870 10.47130 9.55315 10.44685 9.57658 10.42342 20 41 52912 47088 65855 44645 57696 42304 19 42 52953 47047 55395 44605 57734 42266 18 43 52995 47005 55434 44566 57772 42-^28 17 44 53037 46963 55474 445-26 57810 42190 16 45 53078 4G922 55514 44486 57849 42151 15 46 53120 46880 55554 44446 57887 42113 14 47 53161 46839 55593 44407 57925 42075 13 48 53202 46798 55633 44367 579C3 42037 12 49 53244 46756 55673 44327 58001 41999 11 50 9.53285 10.46715 9.55712 10.44288 9.58039 10.41961 10 51 53327 46673 55752 44248 58077 41923 9 52 53368 46632 55791 44209 58115 41885 8 53 53409 46591 55831 44169 58153 41847 7 54 53450 46550 55870 44130 58191 41809 6 55 53492 46508 55910 44090 58229 41771 5 56 53533 46467 55949 44051 58267 41733 4 57 53574 46426 55989 44011 58304 41696 3 58 53615 46385 56028 43972 58342 41658 2 59 53656 46344 56067 43933 58380 41620 1 60 53697 46303 56107 43893 58418 41582 i Cotan Tan Cotan Tan Cotan Tan / 71 70 69 LOG. TANGENTS AND COTANGENTS 157 21 22 23 / Tan Cotan Tan Cotau Tan Cotan o 9.58418 10.41582 9.60641 10.39359 9.62785 10.37215 60 1 58455 41545 60677 39323 62820 37180 59 2 58493 41507 60714 39286 62855 37145 58 3 58531 41469 60750 39250 62890 37110 57 4 58569 41431 60786 39214 62926 37074 56 5 58606 41394 60823 39177 62961 37039 55 6 58644 41356 60859 39141 62996 37004 54 58681 41319 60895 39105 63031 36969 53 8 58719 41281 60931 39069 63066 36934 52 9 58757 41243 60967 39033 63101 36899 51 10 9.58794 10.41206 9.61004 10.38996 9.63135 10.36865 50 11 58832 41168 61040 38960 63170 36830 49 12 58869 41131 61076 38924 63205 36795 48 13 58907 41093 61112 38888 63240 36760 47 14 58944 41056 61148 38852 63275 36725 46 15 58981 41019 61184 38816 63310 3U690 45 16 59019 40981 61220 38780 68:345 36655 44 17 59056 40J4J 61256 38744 63379 36621 43 18 59094 40906 61292 38708 63414 36586 42 19 59131 40869 61328 38672 63449 36551 41 20 9.59168 10.40882 9.61364 10.38636 9.63484 10.36516 40 21 59205 40795 61400 38600 63519 36481 39 22 59243 40757 61430 8856* 63553 36447 38 *3 59280 40720 61472 38528 63588 36412 37 24 59317 40683 61508 3^492 63623 36377 36 25 59854 40646 61544 38456 63657 36343 35 26 59391 40609 61579 38421 03692 36308 34 27 59429 40571 61615 38385 03726 36274 33 28 59466 4(1534 61651 38349 63761 36219 32 29 59503 40497 61687 38313 63796 36204 31 30 9.59540 10.40460 9.61722 10.38278 9.63830 10.36170 30 31 59577 40423 C.1758 3S242 63865 36135 29 32 59614 40386 61794 38206 63899 36101 28 33 59651 40349 61830 38170 63934 86066 27 34 59688 40312 61865 38135 63968 36032 26 35 59725 40275 61901 38099 64003 35997 25 36 59762 40238 61936 38064 64037 35963 24 37 59799 40201 61972 :-!8028 64072 35928 23 38 59835 40165 62008 37992 64106 35894 22 39 59872 40128 62043 37-957 64140 35860 21 40 9.59909 10.40091 9.62079 10.37921 9.64175 10.35825 20 41 59946 40054 62114 37886 64209 35791 19 42 59983 40017 62150 37850 64243 35757 18 43 60019 39981 62185 37815 64278 35722 17 44 60056 39944 62221 37779 6*312 35688 16 45 60093 39907 62256 37744 64346 35654 15 46 (50130 39870 62292 37708 64381 35619 14 47 60166 39834 62327 37673 64415 35585 13 48 60203 39797 62362 37638 64449 35551 12 49 60240 39760 62398 37602 64483 35517 11 50 9.60276 10.39724 9.62433 10.37567 9.64517 10.35483 10 51 60313 39687 62468 37532 64552 35448 9 52 60349 39651 62504 37496 64586 35414 8 53 60386 39614 62539 37461 64620 35380 7 54 60422 39578 62574 37426 64654 35346 6 55 60459 39541 62609 37391 64688 35312 5 56 60495 39505 62645 37355 64722 35278 4 57 60532 39468 62680 37320 64756 35244 8 58 60568 39432 62715 37285 64790 35210 2 59 60605 39395 62750 37250 64824 35176 1 60 60641 39359 62785 87215 64858 85142 / Cotan Tan Cotan Tan Cotan Tan / 68* 67 66 158 LOG. TANGENTS AND COTANGENTS / 84 25 26 / Tan Cotan Tan Cotan Tan Cotan 9.64858 10.35142 9.66867 10.33133 9.68818 10.31182 60 1 64892 35108 66900 33100 68850 31150 59 2 64926 35074 60933 33067 68882 31118 58 3 649GO 35040 66966 33034 68914 31086 57 4 64994 35006 66999 33001 68946 31054 56 5 65028 34972 67032 32968 68978 31022 55 6 65062 34938 67065 32935 69010 80990 54 7 65096 34904 67098 32902 69042 30958 53 8 65130 34870 67131 32869 69074 30926 52 9 65164 34836 67163 32837 69106 30894 51 10 9.65197 10.34803 9.67196 10.32804 9.69138 10.30862 50 11 65231 34769 67229 32771 69170 30830 49 12 65265 34735 67262 32738 69202 80798 48 13 65299 34701 67295 32705 69234 30766 47 14 65333 34667 67327 32673 69266 30734 46 15 65366 34634 67360 32640 69298 30702 45 16 65400 34600 67393 32607 69329 30671 44 17 65434 34566 67426 32574 69361 30639 43 18 654G7 34533 67458 32542 69393 30607 42 19 65501 34499 67491 325C9 69425 30575 41 20 9.65535 10.34465 9.67524 10.32476 9.69457 10.30543 40 21 65568 34432 67556 32444 69488 30512 39 22 65602 34398 67589 30411 69520 30480 38 23 65636 34364 67622 32378 69552 30448 37 24 656C9 34331 67654 32346 69584 30416 36 25 65703 34297 67687 32313 69615 30385 35 26 65736 34264 67719 32281 69647 30353 34 27 65770 34230 67752 32248 69679 80321 33 28 65S03 34197 67785 32215 69710 30290 32 29 65837 34163 67817 32183 69742 30258 31 30 9.65870 10.34130 9.67850 10.32150 9.69774 10.30226 30 31 65904 34096 67882 32118 t>9805 30195 29 32 65937 34063 67915 32085 69837 30163 28 33 65971 34029 67947 32053 69868 30132 27 ' 34 66004 33996 67980 32020 C9900 30100 26 35 66038 33962 68012 31988 69932 30068 25 6 66071 83929 68044 31956 69963 30037 24 37 66104 a3896 68077 31923 69995 30005 23 38 66138 33862 68109 31891 70026 29974 22 39 66171 33829 68142 31858 70058 29942 21 40 9.66204 10.33796 9.68174 10.31826 9.70089 10.29911 20 41 66238 33762 68206 31794 70121 29879 19 42 66271 33729 68239 31761 70152 29848 18 43 66304 33696 68271 31729 70184 29816 17 44 66337 33663 68303 31697 70215 29785 16 45 66371 33629 68336 31664 70247 29753 15 46 66404 33596 68368 31632 70278 29722 14 47 66437 33563 68400 31600 70309 29691 13 48 66470 33530 68432 31568 70341 29659 if 49 66503 33497 68465 31535 70372 29628 11 50 9.66537 10.33463 9.68497 10.31503 9.70404 10.29596 10 51 66570 33430 68529 31471 70435 29565 9 52 66603 33397 68561 31439 70466 29534 8 53 66636 33364 68593 31407 70498 29502 7 54 66669 33331 68626 31374 70529 29471 6 55 66702 33298 68658 31342 70560 29440 5 56 66735 33265 68690 31310 70592 29408 4 57 66768 33232 68722 31278 70623 29377 3 58 66801 33199 68754 31246 70654 29346 2 59 66834 33168 68786 31214 70685 29315 1 60 66867 33133 68818 31182 70717 29283 / Cotau Tan Cotan Tan Cotan Tan / 66 64 68 LOG. TANGENTS AND COTANGENTS 159 27 28 29 / Tan Cotan Tan Cotan Tan Cotan o 9.70717 10.29283 9.72567 10.27433 9.74375 10.25625 60 1 70748 29252 72598 27402 74405 25595 59 2 70779 29221 72628 27372 74435 25565 58 3 70810 29190 72659 27341 74465 25535 57 4 70841 29159 72689 27311 74494 25506 56 5 70873 29127 72720 27280 74524 25476 55 6 70904 29096 72750 27250 74554 25446 54 7 70935 29065 72780 27220 74583 25417 53 8 70906 29034 72811 27189 7461 3 25387 52 9 70997 29003 72841 27159 74643 25357 51 10 9.71028 10.28972 9.72872 10.27128 9.74673 10.25327 50 11 71059 28941 72902 27098 74702 25298 49 12 71090 28910 72932 27'068 74732 25268 48 13 71121 28879 72963 27037 74762 25238 47 ! 14 71153 28847 72993 27007 74791 25209 46 I 15 71184 28816 73023 26977 7'4821 25179 45 i 16 71215 28785 73054 26946 74851 25149 44 1 17 71246 28754 73084 26916 74880 25120 43 18 71277 28723 78114 26886 74910 25090 42 19 71308 28692 73144 26856 74939 25061 41 20 9.71339 10.28661 9.73175 10.26825 9.74969 10.25031 40 21 71370 28630 73205 26795 74998 25002 39 22 71401 28599 73235 26765 75028 24972 38 -,3 71431 28569 73265 26735 75058 24942 37 24 71462 28538 73295 26705 75087 24913 36 25 71493 28507 73326 26674 75117 24883 35 26 71524 28476 7:3356 26644 75146 24854 34 27 71555 28445 73386 26614 75176 24824 33 28 71536 28.414 73416 26584 75205 24795 32 29 71617 28383 73446 26554 75235 24765 31 30 9.71648 10.28352 9.73476 10.26524 9.75264 10.24736 30 .'31 71679 28321 73507 26493 75294 24706 29 32 71709 28291 73537 26463 75323 24677 28 33 71740 28260 73567 26433 75353 24647 27 31 71771 28229 73597 26403 75382 24618 26 35 71802 28198 73627 26373 75411 21589 25 36 71833 28167 73657 26343 75441 24559 24 37 71863 28137 73687 26313 75470 24530 23 38 71894 28103 73717 26283 75500 24500 22 39 71925 28075 73747 26253 75529 24471 21 40 9.71955 10.28045 9.73777 10.26223 9.75558 10.24442 20 41 71986 28014 73807 26193 75588 24412 19 42 72017 27983 73837 26163 75617 24383 18 43 72048 279.V2 73867 26133 75647 24353 17 44 72078 27922 73897 26103 75676 24324 16 45 72109 27891 73927 26073 75705 24295 15 46 72140 27860 73957 26043 75735 242C5 14 47 72170 27830 73987 26013 75764 24236 13 48 72201 27799 74017 25983 75793 24207 12 49 72231 27769 74047 25953 75822 24178 11 i 50 9.72262 10.27738 9.74077 10.25923 9.75852 10.24148 10 51 72293 27707 74107 25893 75881 24119 9 52 72323 27677 74137 25863 75910 24090 8 53 72354 27646 74166 25834 75939 24061 7 54 72384 27616 74196 25804 75969 24031 6 55 72415 27585 74226 25774 75998 24002 5 56 72445 27555 74256 25744 76027 23973 4 57 72476 27524 74286 25714 76056 23944 3 58 72506 27494 74316 25684 76086 28914 2 59 72537 27463 74345 25655 76115 23885 1 60 72567 27433 74375 25625 76144 23856 / Cotan Tan Cotan Tan Cotan Tan , 62* 61 60 100 LOG. TANGENTS AND COTANGENTS 30 31 32 Tan Cotan Tan Cotan Tan Cotan 9.76144 10.23856 9.77877 10.22123 9.79579 10.20421 60 76173 23827 77906 22094 79607 20393 59 2 76202 23798 77935 220G5 79G35 20365 58 3 76231 23769 77963 22037 79663 20337 57 4 76261 23739 77992 22008 79691 20309 56 5 76290 23710 78020 21980 79719 20281 55 6 76319 23681 78049 21951 79747 20253 54 7 76348 23652 78077 21923 79776 20224 53 8 76377 23623 78106 21894 79804 20196 52 9 76406 23594 78135 21865 79882 20168 51 10 9.76435 10.23565 9.78163 10.21837 9.79860 10.20140 50 11 76464 23536 78192 21808 79888 20112 49 12 76493 23507 78220 21780 79916 20084 48 13 76522 23478 78249 21751 79944 20056 47 14 76551 23449 78277 21723 79972 20028 46 15 76580 23420 78306 21694 80000 20000 45 16 76609 23391 78334 21666 80028 19972 44 17 76639 23361 78363 21637 80056 19944 43 18 76668 23332 78391 21609 80084 19916 42 19 76697 23303 78419 21581 80112 19888 41 20 9.76725 10.23275 9.78448 10.21552 9.80140 10.19860 40 21 76754 23246 78476 21524 80168 19832 39 22 7678S 23217 78505 21495 80195 19805 38 23 76812 23188 78533 21467 80223 19777 37 24 76841 23159 78562 21438 80251 19749 36 25 76870 23180 78590 21410 80279 19721 35 26 76899 23101 78618 21382 80307 19693 34 27 76928 23072 78647 21353 80335 19665 33 28 76957 23043 78675 21325 8G363 19637 32 29 76986 23014 78704 21296 80391 19609 31 30 9.77015 10.22985 9.78732 10.21268 9.80419 10.19581 30 31 77044 22956 78760 21240 80447 19553 29 32 77073 22927 78789 21211 80474 19526 28 33 77101 22899 78817 21183 80502 19498 27 34 77130 22870 78845 21155 80530 19470 26 35 77159 22841 78874 21126 80558 19442 25 36 77188 22812 78902 21098 80586 19414 24 37 77217 22783 78930 21070 80614 19386 23 38 77246 22754 78959 21041 80642 19358 22 39 77274 22726 78987 21013 80669 19331 21 40 9.77303 10.22697 9.79015 10.20985 9.80697 10.19303 20 41 77332 22668 79043 20957 80725 19275 19 42 77361 22639 79072 20928 80753 19247 18 43 77390 22610 79100 20900 80781 19219 17 44 77418 22582 79128 20872 80808 19192 16 45 77447 22553 79156 20844 80836 19164 15 46 77476 22524 79185 20815 80864 19136 14 47 77505 22495 79213 20787 80892 19108 13 48 77533 22467 79241 20759 80919 19081 12 49 77562 22438 79269 20731 80947 19053 11 50 9.77591 10.22409 9.79297 10.20703 9.80975 10.19025 10 51 77619 22381 79326 20674 81003 18997 9 52 77648 22352 79354 20646 81030 18970 8 53 77677 22323 79382 20618 81058 18942 7 54 77706 22294 79410 20590 81086 18914 6 55 77734 22266 79438 20562 81113 18887 5 56 77763 22237 79466 20534 81141 18859 4 57 77791 22209 79495 20505 81169 18831 3 58 77820 22180 79523 20477 81196 18804 2 59 77849 22151 79551 20449 81224 18776 1 60 77877 22123 79579 20421 81252 18748 / Cotan Tan Cotan Tan Cotan Tan , 69 68 67 LOG. TANGENTS AND COTANGENTS 161 88 84 36 / Tan Cotan Tan Cotan Tan Cotan o 9.81252 10.18748 9.82899 10.17101 9.84523 10.15477 60 1 81-279 18721 82926 17074 84550 15450 59 2 81307 18693 82953 17047 84576 15424 58 3 81335 18665 82980 17020 84603 15397 57 4 81362 18638 83008 16992 84630 15370 56 5 81390 18610 83035 16965 84657 15343 55 g 81418 18582 83062 16938 84684 15316 54 7 81445 18555 83089 16911 84711 15289 53 81473 18527 83117 16883 84738 15262 52 9 81500 18500 83144 16856 84764 15236 51 10 9.81528 10.18472 9.83171 10.16829 9.84791 10.15209 50 81556 18444 83.198 16802 84818 15182 49 12 81583 18417 83225 16775 84845 15155 48 12 81611 18389 83252 16748 84872 15128 47 14 81638 18362 83280 16720 84899 15101 46 15 81666 18334 83307 16693 84925 15075 45 16 81693 18307 83334 16666 84952 15048 44 17 81721 18279 83361 16639 84979 15021 43 18 81748 18252 83388 16612 85006 14994 42 19 81776 18224 83415 16585 85033 14967 41 20 9.81803 10.18197 9.88442 10.16558 9.85059 10.14941 40 21 81831 18169 83470 16530 85086 14914 39 22 81858 18142 83497 165015 85113 14887 38 23 81886 18114 83524 16476 85140 14860 37 \ 24 81913 18087 83551 16449 85166 14834 36 25 81941 18059 83578 16422 85193 14807 35 26 81968 18032 83605 16395 85220 14780 34 27 81996 18004 83632 16368 85247 14753 33 28 82023 17977 83659 16341 85273 14727 82 29 82051 17949 83686 16314 85300 14700 31 30 9.82078 10.17922 9.83713 10.16287 9.85327 10.14673 80 31 82106 17894 83740 16260 85354 14646 29 32 82133 1787 83768 16232 85380 14620 28 83 82161 17839 83795 16205 85407 14593 27 34 82188 17812 83822 16178 85434 14566 26 35 82215 17785 83849 16151 85460 14540 25 36 82243 17757 83876 16124 85487 14513 24 37 82270 17730 83903 16097 85514 14486 23 38 82298 17702 83930 16070 85540 14460 22 39 82325 17675 83957 16043 85567 14433 21 40 9.82352 10.17648 9.83984 10.16016 9.85594 10.14406 20 41 82880 17620 84011 15989 85620 14380 19 42 82407 17593 84038 15962 85647 14353 18 43 82435 17565 84065 15935 85674 14326 17 44 82462 17538 84092 15908 85700 14300 16 45 82489 17511 84119 15881 85727 14273 15 46 82517 17483 84146 15854 85754 14246 14 47 82544 17456 84173 15827 85780 14220 18 48 82571 17429 84200 15800 85807 14193 12 49 82599 17401 84227 15773 85834 14166 11 50 9.82626 10.17374 9.84254 10.15746 9.85860 10.14140 10 51 82653 17347 84280 15720 85887 14113 9 52 82681 17319 84307 15693 85913 14087 8 53 82708 17292 84334 15666 85940 14060 7 54 55 82735 82762 17265 17238 84361 84388 15639 15612 85967 85993 14033 14007 6 5 56 82790 17210 84415 15585 86020 13980 4 57 82817 17183 84442 15558 86046 13954 3 58 82844 17156 84469 15531 86073 13927 2 59 82871 17129 84496 15504 86100 13900 1 60 82899 17101 84523 15477 86126 13874 ~ f Cotan Tan Cotan Tan Cotan Tan / 56 66 64 162 LOG. TANGENTS AND COTANGENTS / 86 87 88 / Tan Cotan Tan Cotan Tan Cotau 9.86126 10.13874 9.87711 10.12289 9.89281 10.10-719 60 1 86153 13847 87738 12262 89307 10693 59 2 86179 13821 87764 12236 89333 10667 58 3 86206 13794 87790 12210 89359 10641 57 4 86232 13768 87817 12183 89385 10615 56 5 86259 13741 87843 12157 89411 10589 55 6 86285 13715 87869 12131 89437 10563 54 7 86312 13668 87895 12105 89463 10537 53 8 86338 13662 87922 12078 89489 10511 52 9 863C5 13635 87948 12052 89515 10485 51 10 9.86392 10.13608 9.87974 10.12026 9.89541 10.10459 50 11 86418 13582 88000 12000 89567 10433 49 12 86445 13555 88027 11973 89593 10407 48 13 86471 13529 88053 11947 89619 10381 47 14 86498 13502 88079 11921 89645 10355 46 15 86524 13476 88105 11895 89671 10329 45 16 86551 13449 88131 11869 89697 10303 44 17 86577 13423 88158 11842 89723 10277 43 18 86603 13397 88184 11816 89749 10251 42 19 86630 13370 88210 11790 89775 10225 41 20 9.86656 10.13344 9.88236 10.11764 9.89801 10.10199 40 21 86683 13317 88262 11738 89827 10173 39 22 86709 13291 88289 11711 89853 10147 38 23 86736 13264 88315 11685 89879 10121 37 24 86762 13238 88341 11659 89905 10095 36 25 . 86789 13211 88367 11633 89931 10069 35 26 86815 13185 88893 11607 89957 10043 34 27 86842 13158 88420 11580 89983 10017 33 28 86868 13132 88446 11554 90009 09991 32 29 86894 13106 88472 11528 90035 09965 31 30 9.86921 10.13079 9.88498 10.11502 9.90061 10.09939 30 31 86947 13053 88524 11476 90086 09914 29 32 86974 13026 88550 11450 90112 09888 28 33 87000 13000 88577 11423 90138 09862 27 34 87027 12973 88603 11397 90164 09836 26 35 87053 12947 88629 11371 90190 09810 25 36 87079 12921 8H655 11345 90216 09784 24 37 87106 12894 88681 11319 90242 09758 23 38 87132 12868 88707 11293 902(58 09732 22 39 87158 12842 88733 11267 90294 09706 21 40 9.87185 10.12815 9.88759 10.11241 9.90320 10.09080 20 41 87211 12789 88786 11214 90346 09654 19 42 87238 12762 88812 11188 90371 09629 18 43 87264 12736 88838 11162 90397 09603 17 44 87290 12710 88884 11136 90423 09577 16 45 87317 12683 88890 11110 90449 09551 15 46 87343 12657 88916 11084 90475 09525 14 47 87369 12631 88942 11058 90501 09499 13 48 87396 12604 88968 11032 90527 09473 12 49 87422 12578 88994 11006 90553 09447 11 50 9.87448 10.12552 9.89020 10.10980 9.90578 10.09422 10 51 87475 13525 89046 10954 90604 OOH96 9 52 87501 12499 89073 10927 90630 09370 8 53 87527 12473 89099 10901 90656 09344 7 54 87554 12446 89125 10875 90682 09818 6 55 87580 12420 89151 10849 90708 09292 5 56 87606 12394 89177 10823 90734 09266 4 57 87633 12367 89203 10797 90759 09241 8 58 87659 12341 89229 10771 90785 09215 2 59 87685 12315 89255 10745 90811 09189 1 60 87711 12289 89281 10719 90837 09163 i Cotan Tan Cotan Tan Cotan Tan / 63 52 61 LOG. TANGENTS AND COTANGENTS 163 89 40 41 / Tan Cotan Tan Cotau Tan Cotan o 9.90837 10.09163 9.92381 10.07619 9.93916 10.06084 60 1 90863 09137 92407 07593 93942 06058 59 2 90889 09111 92433 07567 93967 06033 58 3 90914 09086 92458 07542 93993 06007 57 4 90940 09060 92484 07516 91018 05982 56 5 90966 09034 92510 07490 94044 05956 55 6 90992 09008 92535 07465 94069 05931 54 7 91018 08982 92561 07439 94095 05905 53 8 91043 08957 92587 07413 94120 05880 52 9 910J9 08931 92612 07388 94146 05854 51 10 9 91095 10.08905 9.92638 10.07362 9.91171 10. 05829 50 11 91121 08879 92663 07337 94197 05803 49 12 91147 08853 92689 07311 94222 05778 48 13 91172 08828 92715 07285 91248 05752 47 14 91198 08802 92740 07260 94273 05727 46 15 9J224 08776 92766 07234 94299 05701 45 16 91250 08750 92792 07208 94324 05676 44 17 91276 08724 92817 07183 94350 05650 43 18 91301 08699 92843 07157 94375 05625 42 19 91327 08673 92868 07132 94401 05599 41 20 9.91858 10.08647 9.92894 10.07106 9.94426 10.05574 40 21 91379 08621 92920 07080 94452 05548 39 22 91404 08596 92945 07055 94477 05523 38 23 91430 08570 ' 92971 07029 94503 05497 37 24 91456 08544 92996 07004 94528 05472 36 25 9148.! 08518 93022 06978 94554 05446, 35 26 91507 08493 93048 06952 94579 05421 34 27 91533 08467 93073 06927 94604 05396 33 28 91559 08441 93091) 06901 94630 05370 32 29 91583 08415 93124 06876 94655 05345 31 30 9.91610 10.08390 9.93150 10.06850 9.94681 10.05319 30 31 91630 08364 93175 06825 94706 05294 29 32 91662 08338 93201 06799 94732 05268 28 33 9168S 08312 93227 06773 94757 05243 27 34 91713 08287 93252 06748 94783 05217 26 35 91739 08261 93278 06722 94808 05192 25 36 91765 08235 93303 06697 94834 05166 24 37 91791 08209 03329 06671 94859 05141 23 38 91816 08184 93354 06646 94884 05116 22 39 91842 08158 93380 06620 'J4910 05090 21 40 9.91868 10.08132 9.93406 10.06594 9.94935 10.05065 20 41 91893 08107 93431 06569 94961 05039 19 42 91919 08061 93457 06543 94986 05014 18 43 91945 08055 93482 06518 95012 04988 17 44 91971 OS029 93508 06492 95037 04963 16 45 91996 08004 93533 06467 'J5002 04938 15 46 92022 07978 93559 06441 95088 04912 14 47 92048 07952 93584 06416 95113 04887 13 48 92073 07927 93610 06390 95139 04861 12 49 92099 07901 93636 06364 95164 04836 11 50 9.92125 10.07875 9.93661 10.06339 9.95190 10.04810 10 51 92150 07850 93687 06313 95215 04785 9 52 92176 07824 93712 06288 95240' 04760 8 .',3 92202 07798 93738 06262 95266 04734 7 5"i 92227 07773 93763 06237 95291 04709 6 55 92253 07747 93789 06211 95317 04683 5 r.6 92279 07721 93814 06186 95342 04658 4 57 92304 07696 93840 06160 95368 04632 3 58 92330 07670 93865 06135 95393 04607 2 59 92356 07644 93891 06109 95418 04582 1 00 92381 07619 93916 06084 95444 04556 / Cotan Tan Cotau Tan Cotan Tan , 50 49 48 164 LOG. TANGENTS AND COTANGENTS / 42 43 44 / Tan Cotan Tan Cotan Tan Cotan 9.95444 10.04556 9.96966 10.03034 9.98484 10.01516 60 1 95469 04531 96991 03009 98509 01491 59 2 95495 04505 97016 02984 98534 01466 58 3 95520 04480 97042 02958 98560 01440 57 4 95545 04455 97067 02933 98585 01415 56 5 95571 04429 97092 02908 98610 01390 55 6 95596 04404 97118 02882 98635 01365 54 7 95622 04378 97143 02857 98661 01339 53 8 95647 04353 97168 02832 98686 01314 52 9 95672 04328 97193 02807 98711 01289 51 10 9.95698 10.04302 9.97219 10.02781 9.98737 10.01263 50 11 95723 04277 97244 02756 98762 01238 49 18 95748 04252 97269 02731 98787 01213 48 13 95774 04226 97295 02705 98812 01188 47 14 95799 04201 97320 02680 98838 01162 46 15 95825 04175 97345 02655 98863 01187 45 16 95850 04150 97371 02629 98888 01112 44 17 95875 04125 97396 02604 98913 01087 43 18 95901 04099 97421 02579 98939 01061 42 19 959*6 04074 97447 02553 98964 01086 41 20 9.95952 10.04048 9.97472 10.02528 9.98989 10.01011 40 21 95977 04023 97497 02503 99015 00985 39 22 96002 03998 97523 02477 99040 00960 38 23 96028 03972 97548 02452 '99065 00935 37 24 96053 03947 97573 02427 99090 00910 86 25 96078 0392-2 97598 02402 99116 00884 35 26 96104 03896 97624 02376 99141 00859 34 27 96129 03871 97649 02351 99166 00834 33 28 96155 03845 97674 02326 99191 00809 82 29 96180 03820 97700 02300 99217 00783 31 30 9.96205 10.03795 9.97725 10.02^75 9.99242 10.00758 30 31 96231 03769 97750 02250 99267 00733 29 32 96256 03744 97776 02224 99293 00707 28 33 96281 08719 97801 02199 99318 00682 27 34 96307 03693 97826 02174 99343 00657 26 35 96332 03668 97851 02149 ' 99368 00632 25 36 96357 03643 97877 02123 99394 00606 24 37 96383 03617 97902 02098 99419 OC581 23 38 96408 03592 97927 02073 99444 00556 22 39 96433 03567 97953 02047 99469 00531 21 40 9.96459 10.03541 9.97978 10.02022 9.99495 10.00505 20 41 96484 03516 98003 01997 99520 00480 19 42 96510 03490 98029 01971 99545 00455 18 43 96535 03465 98054 01946 99570 00430 17 44 96560 03440 98079 01921 99596 00404 16 45 96586 03414 98104 01896 99621 00379 15 46 96611 03389 98130 01870 99646 00354 14 47 96636 03364 98155 01H45 99672 00328 13 48 96662 03338 98180 01820 99697 00303 12 49 96687 03313 98206 01794 99722 00278 11 50 9.96712 10.03288 9.98231 10.01769 9.99747 10.00253 10 51 96738 03262 98256 01744 99773 00227 9 96763 03237 98281 S 01719 99798 00202 8 58 96788 03212 98307 01693 99823 00177 B 64 96814 03186 98332 01668 99848 00152 6 65 96839 03161 98357 01643 99874 00126 5 56 96864 03136 98383 01617 99899 00101 4 57 96890 03110 98408 01592 99924 00076 3 58 96915 03085 98433 01567 99949 00051 2 59 96940 03060 98458 01542 99975 00025 1 60 96966 08034 98484 01516 10.00000 00000 f Cotan Tan Cotan Tan Cotan Tan , 47 48* 46 LOG. VERSED SINES AND EXTERNAL SECANTS 165 o 1 ' Log. Vers. D .og. Exsec. D Log. Vers. I> Log, Exsec. D f 1 2 3 4 oo 2-62642 3-22848 3-58066 3-83054 60206 35218 24987 CO 2.62642 3-22848 3-58066 3-83054 60206 35218 24987 6-18271 -19707 .21119 22509 23877 1435 1412 1389 1368 6.18278 -19714 -21126 . -22516 23884 1436 1412 1390 1368 O 1 2 3 4 5 6 7 8 9 4-02436 -18272 31662 -43280 -53490 19382 15836 13389 11598 10230 4-02436" -18272 31662 .43260 53491 19382 15836 13389 11598 10230 6-25223 -26549 .27856 29142 -30410 1346 1326 1306 1286 1268 6-25231 26557 -27864 29151 30419 1347 1326 1306 1287 1268 5 6 7 8 9 10 11 12 13 14 4-62642 -70920 78478 .85431 .91868 9151 8278 7558 6953 6437 4-62642 70921 .78478 85431 .91868 9151 8279 7557 6952 6437 6-31660 -32892 34107 35305 -36487 1250 1232 1214 1198 1182 6-31669 -32901 -34116 -35315 -36497 1250 1232 1215 1198 1182 1O 11 12 13 14 15 16 17 18 19 4-97880 5-03466 -08732 -13696 -18393 5992 5605 5266 4964 4696 4-97861 5-03466 .08732 .13697 .18393 5993 5605 5266 4964 4696 6-37653 .38803 .39938 .41059 -42165 1166 1150 1135 1121 1106 6-37663 -38814 -39949 41070 -42177 1166 1151 1135 1121 1106 15 16 17 18 19 20 21 22 23 24 5-22348 .27086 .31126 34987 -38684 4455 4238 4040 3861 3697 5-22849 .27087 -31127 .34988 .38685 4456 4238 4040 3861 3697 6-43258 -44337 .45403 46455 .47496 1093 1078 1066 1052 1040 6-43270 44349 45415 46468 47509 1093 1079 1066 1053 1040 30 21 22 23 24 25 26 27 28 29 5-42230 -45636 48915 .52073 .55121 3545 3406 3278 3158 3048 5-42231 .45638 .48916 .52075 .55123 3545 3407 3278 3159 3048 6-48524 .49539 .50544 .51536 .52518 1028 1016 1004 992 981 6-48537 -49553 50557 .51550 -52532 1028 1015 1004 993 982 25 26 27 28 29 .30 31 32 33 34 5-58068 60914 -63372 -66344 -68937 2941 2848 2757 2672 2593 5-58068 .60916 .63674 .66346 68940 2945 2848 2758 2672 2593 6-53488 . 54448 .55397 .56336 .57265 970 960 949 939 929 6-53503 . 54463 55413 .56352 57281 970 960 950 939 929 30 31 32 33 34 35 36 37 38 39 5-71455 .73902 76282 .78598 .80854 2518 2447 2379 2316 2256 5-71457 .73904 .76284 .78601 -80857 2517 2447 2380 2316 2256 6-58184 .59093 .59993 .60884 .61766 919 909 900 891 882 6-58201 .59110 .60011 .60902 .61784 919 909 900 891 882 35 36 37 38 39 40 41 42 43 44 5-83053 85198 .87291 .89335 .91332 2199 2145 2093 2044 1996 5-83056 85201 .87295 .89338 .91335 2199 2145 2093 2043 1997 6-62639 .63503 .64359 65206 .66045 872 864 855 847 839 6-62657 .63522 64378 .65226 -66065 873 864 856 848 839 40 41 42 43 44 45 46 47 48 49 5-93284 95193 .97061 5-98890 6-00680 1909 1868 1829 1790 5-93288 95197 97065 5-98894 6-00685 1952 1909 1868 1829 1791 6-66876 67700 .68515. .69323 70124 831 823 815 808 800 6-66897 67720 .68536 .69345 -70145 831 823 816 808 800 45 46 47 48 49 50 51 52 53 54 6-02435 -04155 .05842 07496 -09120 1720 1686 1654 1623 6-02440 04160 05847 07501 .09125 1755 1720 1687 1654 1623 6-70917 71703 72482 73254 74019 793 786 779 772 765 8-70939 -71725 -72505 73277 74043 794 786 779 772 765 SO 51 52 53 54 55 56 57 58 59 6-10714 12279 .13816 15327 .16811 1565 1537 1511 1484 6-10719 12284 13822 15333 16818 1594 1565 1537 1511 1485 6- 74777 -75529 76275 77014 -77747 758 752 745 739 733 6-74802 75554 76300 77040 77773 759 752 746 739 733 55 56 57 58 5fl 60 6-18271 6.18278 1460 6 - 78474 726 6-78500 727 fifi D Log. Exsec JO Log. Vers. JO Log. Exsec. D 166 LOG. VERSED SINES AND EXTERNAL SECANTS 2 3 / Log, Vers, r> ,og. Exsec, J> Log, Vers, J> .og, Exsec. 1> ' o 1 2 3 4 6-78474 .79195 79909 .80618 .81322 721 714 709 703 6-78500 79221 .79937 80646 .81350 721 715 709 703 7-13687 .14168 14646 .15122 .15595 481 478 475 473 7-13746 14228 .14707 .15183 .15657 481 479 476 474 1 2 3 4 5 6 7 8 9 6-82019 82711 .83398 84079 84755 697 692 686 681 676 6-82048 .82740 .83427 84109 .84785 698 692 687 682 676 7-16066 16534 .17000 .17463 17923 475 468 466 463 460 7-16129 .16598 17064 17528 -17989 471 469 466 464 461 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 6-85425 -86091 86751 -87407 .88057 6-88703 .89344 .89980 .90612 -91239 670 665 655 650 646 641 636 631 627 6-85457 .86123 .86783 87439 .88090 6-88737 89378 90015 90647 -91275 67l 666 660 656 651 646 641 636 632 628 7-18382 .18837 .19291 .19742 .20191 7-20637 .21081 .21523 .21963 -22400 458 455 453 451 448 446 444 442 440 437 7-18448 .18905 19359 .19811 -20260 7-20707 -21152 -21595 -22035 -22473 459 456 454 452 449 447 445 442 440 438 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 6-91862 .92480 -93093 93703 .94308 622 618 613 609 605 6-91898 92516 93131 .93741 -94346 623 618 614 610 605 7-22836 .23269 .23700 .24129 -24555 435 433 431 429 426 7-22909 -23343 -23775 -24204 -24632 436 434 431 429 427 20 21 22 23 24 25 26 27 28 29 6-94909 9550G .96099 .96688 -9727? 601 597 592 589 584 6-94948 95545 .96139 .96728 -97313 601 597 593 589 585 7-24980 .25402 .25823 26241 26658 424 422 420 418 416 7-25057 -25485 -25902 26321 -26738 425 423 421 419 417 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 6-97853 .98430 .99004 6-99573 7-00139 7-00701 .01259 .01814 -02366 .cmi4 581 577 573 569 535 562 558 555 551 548 6-97895 .98472 .99046 6-99616 7-00182 7 - 00745 01304 -01860 -02412 .02960 581 577 574 570 566 563 559 555 552 548 7-27072 27485 .27895 28304 -28711 7-29116 .29518 .29919 30319 -30716 414 412 410 409 406 405 402 40 i 399 397 7-27153 .27567 .27978 .28387 -28795 7-29205 -29604 -30006 30406 3P804 415 413 411 409 407 405 404 402 400 398 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 7-03458 .03999 -04537 -05071 -05R03 542 541 537 534 531 7-03505 .04047 .04585 .05125 -05652 545 541 538 535 531 7-31112 .31505 .31897 .32288 -3267G 395 393 392 390 388 386 7-31201 31595 .31988 32379 -32768 dUb 394 393 391 389 388 4O 41 42 43 44 45 46 47 48 49 7-06135 06655 .07177 .07695 .08211 527 52R 521 518 515 7-06185 .06706 .07228 .07747 .08263 528 525 522 519 516 7-33063 -33448 -33831 -34213 -34593 385 383 382 380 7-33156 -33542 -33926 -34309 34689 385 384 382 385 Q7Q 46 46 47 48 49 50 51 52 53 54 7-08723 .09232 .09739 .10242 10743 512 509 50R 503 500 7-08776 .09236 .09793 .10297 -107HR 513 509 507 503 501 7-34971 -35348 -35723 -36097 -36468 378 377 375 373 371 7-35069 -35446 -35822 -36196 -36569 377 376 374 373 07-1 50 51 52 53 54 55 56 57 58 59 7-11240 .11735 .12227 -12716 .13203 497 495 492 489 486 7-11297 .11792 .12285 .12775 -13262 498 495 493 490 487 7-36839 .37207 .37574 37945 -38304 370 368 367 366 364 ncn 7-36945 -37310 -37678 -38044 -38409 369 368 366 365 363 55 56 57 58 59 60 7-13687 484 7 13746 484 7.38667 362 7-38773 tJO Log, Vers", D Log. Exsec. 1) ' Log. Vers, D Log. Exsec D LOG. VERSED SINES AND EXTERNAL SECANTS 167 4 5 ' Lg.Vers D Log.'Exs. D Lg.Vers. D Log.Exs D ' P.P. o 1 2 1 3 T 6 7 9 7-38667 -39028 -39387 39745 40102 361 359 358 356 355 353 352 350 349 348 346 345 343 342 341 339 338 337 335 333 332 330 329 328 327 325 324 323 322 321 III 317 316 315 314 313 311 7-38773 -39134 -39495 -39854 -40211 7-40567 -40922 -41275 -41627 -41977 361 360 359 357 356 354 353 352 350 349 347 346 345 343 342 340 339 338 337 335 ili it* 329 328 327 325 324 323 322 321 319 318 317 316 315 313 313 311 310 309 308 307 306 305 304 303 302 301 299 299 298 296 296 295 294 292 292 291 7-58039 -58328 -58615 -58902 59188 289 287 287 286 285 284 283 282 281 280 279 279 277 277 276 275 274 273 272 272 271 27C 269 268 267 266 266 265 264 263 2G3 261 261 260 259 258 258 257 256 255 255 254 258 252 252 251 25C 25C 24g 248 247 247 246 245 245 in 242 242 241 7-58204 -58494 -58783 59071 -59358 290 289 288 28V7 286 285 284 283 282 281 28C 28C 27S 278 277 276 275 274 274 273 272 271 270 269 269 268 267 266 265 264 264 263 262 261 261 260 259 258 257 257 256 255 254 254 253 252 252 251 250 250 248 248 247 246 246 245 245 244 243 242 O 1 2 3 4 5 6 7 8 9 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 O 9 Id 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 360 36-0 42-0 48-0 54-0 60-0 120-0 180-0 240-0 300-0 330 33-0 38-5 44-0 49.5 55-0 110-0 165-0 220.0 275-0 300 30-0 35-0 40-0 45-0 50-0 100-0 150-0 200-0 250-0 270 27-0 31-5 36-0 40-5 45-0 90-0 135-0 180-0 225-0 240 24-0 28-0 32-0 36-0 40-0 80-0 120-0 160-0 200-0 210 21-0 24-5 28-0 31-5 35-0 70-0 105-0 140-0 175-0 35O 35-0 40.8 46-6 51-5 58-3 116-6 175-0 233-3 291-6 320 32-0 37-3 42-6 48-0 53-3 106-6 160-0 213-3 266-6 290 29-0 33-8 38-6 43-5 48-3 96. G 145-0 193-3 241-6 260 26-0 30-3 34-6 39-0 43-3 86-6 130-0 173-3 216-6 230 23-0 26-8 30-6 34-5 38-3 78-6 115-0 153-3 191-6 200 20-0 23-3 26-6 30-0 33-3 66-6 100-0 133-3 166-6 34O 34-0 39-6 45-3 51-0 56-6 113-3 170-0 226-6 283-3 310 31-0 36-T 41-3 46-5 51- S 103-3 155-0 206-6 258-3 28O 28-0 32 6 37-3 42-0 46-6 93-3 140-0 186-B 233-3 250 25-0 29- T 33-3 37-5 41-6 83-3 125-0 166-6 208-3 220 22- Q 25-6 29-3 33-0 36-6 73-3 110 146-6 183-3 190 19-0 22-T 25-3 28-5 31-6 63-3 95-0 126-6 158-3 7-40457 40810 41163 41513 41863 7-59473 -59758 -60041 - 60323 -60604 7-59645 -59930 -60214 -60498 -60780 TO 11 12 13 14 7-42211 42557 -42903 -43246 -43589 7-43930 -44270 -44608 -44946 -45281 7-42326 -42673 -43019 -43364 -43708 7-60885 -61164 -61443 -61721 -61998 7-61062 61342 61622 61901 62179 10 11 12 13 14 15 16 17 18 -11 21 22 23 24 25 26 27 28 29 15 16 17 18 19 7-44050 -44390 -44730 -45068 -45405 7-62274 -62549 -62823 63096 .63369 7-62456 -62733 -63008 -63282 63556 20 21 22 23 24 25 26 27 28 29 30 31 32 33 31 35 36 37 38 39 7-45616 -45949 -46281 46612 -46941 7-45740 -46075 -46407 -46739 -47070 7-63641 -63911 -64181 64451 64719 7-63829 -64101 -64372 - 64643 -64912 7-47270 -47597 -47922 -48247 -48570 7-47399 -47727 48054 48379 .48703 7-64986 65253 65519 65784 66048 7-65181 -65449 -65716 -65982 -66247 7-48892 -49213 -49533 -49852 -50169 7-49026 -49348 -49669 49989 50307 7-66311 -66574 66836 67097 67357 7-66512 -66776 -67039 -67301 -67562 30 31 32 33 34 7-50485 -50800 -51114 -51427 .51739 7-50624 -50941 -51256 51569 -51882 7-67617 67875 68133 68390 -68647 7-67823 -68083 -68342 -68601 -68858 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49_ 50 51 52 53 54 7-52050 -52359 -52667 -52975 53281 309 308 307 306 305 304 303 300 299 297 297 295 295 293 293 III 290 7-52194 -52504 -52814 53122 -53429 7-68902 69157 .69411 69665 69917 7-69115 69371 -69627 -69881 -70135 40 41 42 43 44 7-53586 -53890 -54193 -54495 -54796 7-53735 -54041 -54345 -54648 -54950 7-70169 70421 70671 70921 71170 7-70388 -70641 -70893 -71144 -71394 45 46 47 48 49 7-55096 55395 -55692 -55989 -56285 7-55251 -55550 -55849 -56147 - 56444 7-71418 71666 71913 72159 72404 7-71644 -71892 72141 72388 72635 50 51 52 53 54 55 56 57 58 59 7-56580 -56873 -57166 -57458 -57749 7-56740 -57035 -57329 57621 57913 7-58204 7-72649 -72893 73137 73379 73621 7-72881 73126 73371 .73615 -73859 55 56 57 58 59 BO 7-58039 7-73863 7-74101 BO ' Lg.Vers. JJ Log.Exs. J > Lg. Vers.| D Log.Exs. JJ 1 P. P. 168 LOG. VERSED SINES AND EXTERNAL SECANTS 6 7 ' Lg.Vers, I> Log.Exs, X> 7-8723 .8744 .87650 87855 .88060 7-87563 87771 .87978 .88185 -88391 -88672 -88875 JJ9077 7 - 89279 -89481 -76492 76728 :?} SSI -77431 7-8962C -8982 -90025 -9022 -9042E 7 - 90630 .90831 .9103 -91231 -91431 -90082 7-90282 .90481 7-9127 .9147 .91667 -91863 -92058 7-92255 -924 .9264 .9283 .9302 7-9163 .9182 -92027 -92224 -92421 7-92618 -92815 .93010 .93206 -93401 -93222 -93415 .93607 .93799 -93990 7-93596 .93790 -93984 .94177 .94370 82201 8242 8264 82862 8308 83300 7-94181 .94371 .94561 .94751 .94940 9513 .95328 7-95519 .95709 95898 7.95129 .95317 .95505 .95693 -95880 7-96066 -96253 -96439 -9662" -96465 -96653 .96841 -97028 -97215 7-97401 .97587 .9777 .9795 -98143 7-96994 97178 -97362 -97546 -97729 7-98327 -98512 .98695 .98879 -99062 7-97912 -98094 -98276 -98458 -98639 1> P, P, ISO 9 9 18-0:0.9 0-J 21-0 24-0 27-0 30-0 60-0 90-0 120-0 150- C 8_ 8 6 0-8 7 8 10 202-i 304- i 40 50 7-1 6-6 6 1.0 1.2 1-3 1-5 3-0 4-5 6-0 7.5 0-9 1-0 5- 7 6_ 6 60.70.60-8 70. 80- _ 9 1-0 10 1-1 20 2-3 303-5 404-6 50 5-8 0-7 0.8 00-9 1.0 2-0 3-0 4-0 5-4 5-0 5544 ).40-4 BIO.5iO.5lO 70- 80- 90-80- 100.! 201.1 30 2- 1 40 3- i 504-64. 3 3 60-30 70- 0- ;0-5 0-80. 0-Z 0-7 1-5 2.2 3-02 3-7 0-8 0-6 1-3 2-0 3-; 8 90-50 502-9 0-i 10 0-60-5 0-4.0 2 2 0-20-2 0.3JO. 0-5 0-2 0.4'0-S 1.00.80 2.52-l'l.6 e'o-i 7|0.20 8,0-: 9;o-i 100-i 200-1 300-' 401-1 501-! 0-1 6 o o" 0-0 0-1 0-1 0-1 8:1 0-4 P.P. LOG. VERSED SINES AND EXTERNAL SECANTS 169 8 9 1 g, Vers, n og.Exs, D g.Vers. D og.Exs, J> ' P. P. 1 2 3 4 5 6 7 8 JL 10 11 12 13 14 -98820 -99000 99180 .99360 -99539 80 79 79 .n n 76 76 176 175 175 175 174 174 1 172 17: 171 171 170 170 170 169 169 169 168 168 168 167 167 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 99244 .99427 .99609 .99790 .99971 82 82 181 181 180 180 180 180 179 179 178 178 178 177 i77 -09031 .09192 .09352 .09512 09671 60 60 60 59 59 159 158 158 158 157 157 157 III 156 III III 155 154 154 154 153 153 153 153 152 152 152 152 15' 15' 151 151 150 150 15C 149 15C 14 149 149 148 14 14 14 14 14 14 14 14 14" 14 14" 14" 14 14 144 F .09569 .09732 .09894 .10056 .10217 62 62 62 61 61 61 60 60 160 16C 159 159 159 158 158 158 158 157 157 15 r 157 156 15 15 15 15 15 15 15 15 15^ 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 14 14 14 14 14 14 14 14 14 14 14 14 14 1 2 3 4 1 6 1 7 2 8 2 9 2 1C 3 20 6 30 9 4C 12 50 15 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 SO 8-0 1-0 4-C 7-C 0-C 0-C 0-C 0-C 0-C If 15 17 20 22 25 50 70 ICO 125 I \. 1- 1- 3- 1 8 o. 0- 1- 1- 1- 2- 4- 5- 1 1 1 2 2 2 5 8 11 14 .0 -5 .0 5 .0 .0 .0 .0 -0 li \\ 1 1 61 1 3 7 4 3 6 97 80 9 6 1 g 2 03 3 5 6 6 ) (> 6 )-? )-8 L.O L-l !i 1.3 3.4 3*5 I.B 3-7 3-8 3-9 !:! J.F 70 160 7-0 16-0 9-8 18-6 2-6 21-3 5-5 24-Q 8-3 26-6 6-6 53-3 5-0 80 3-3 106-6 1.6 133-3 14O 14-0 16-3 18-6 21-0 23-3 46-6 70-0 93-3 116-6 > 8 90-8 01-0 21-1 5 l-'i 2-8 54-2 5-6 57-1 ? 70 7 7 .90-; .00- ; '.2l.\ 52-i 73- -04- .25-8 6 0-6 0-7 0-8 0-9 1-0 2-0 3-0 40 5-0 5 0-5 0-6 0-6 0-7 0-8 2^5 3-3 4-1 -99718 -99897 8-00075 .00253 .00431 .00152 .00332 .00512 .00692 .0087! .09830 .09989 .10148 .10306 .10464 .10378 .10539 .10700 .10860 .11020 8-11180 .11340 .11499 .11658 -11816 5' 6 7 8 9 8-00608 .00784 .00961 .01137 .01313 8-01050 .01229 .01407 .01585 .01763 -10622 .10779 .10936 .11093 .11250 10 11 12 13 14 15 16 17 18 19 8-01488 .01663 .01838 .02012 .02186 8-01940 .02117 .02293 02469 .02645 .11406 .11562 .11718 .11873 -12029 8-11975 12133 12291 12448 -12605 15 16 17 18 li 20 21 22 23 24 20 21 22 23 24 8-02359 .02533 .02706 .02878 .03050 8-02820 .02995 .03170 .03345 .03519 175 175 174 17< 17! 173 173 172 172 17: 17: 170 170 170 170 169 169 169 168 168 168 167 167 167 166 166 166 165 165 165 16 16" 164 16 16 16 .12184 .12335 .12492 .12647 .12800 8-12762 .12919 .13075 .13232 .1338" 8-1354" .1369 .1385 1400" -1416 25 26 27 28 29 8-03222 .03394 .03565 .03736 .03906 8-03692 .03866 .04039 .04212 04384 .12954 .13107 .13260 .13413 .13565 25 26 27 28 29 30 31 32 33 34 8-04076 04246 .04416 .04585 .04754 8-04556 .04728 .04899 05070 -05241 8-13717 .13869 .14021 .14172 .14323 8.1431 1447 .1462 .1477 1493 30 31 32 33 34 35 36 37 38 39_ 40 41 42 43 44 8-04922 .05090 .05258 .05426 -05593 8-05411 .0558" .05751 .05921 .06090 8-14474 .1462 .1477 .1492" 1507 8-1508 1523 .1539 .1554 .1569 35 36 37 38 39 8-05760 -05926 -06093 -06259 -0642^ 8-06589 .0675' .06915 .0708; .07247 8-06259 .06427 .06595 .06763 06931 8 1522 .15374 -15523 1567 -15820 8-1584 .1599 .1614 1629 .1645 40 41 42 43 44 45 46 47 48 49 50 D- 6( 7( 8( 9 10 20 30 40' 50 61 7( 8 91 101 20 30 40 50 45 46 47 48 49 8-07098 -07265 .0743' .07598 .07784 8-1596 .1611 1626 1641 -1655 8-1660 -1675 -1690 -1705 -1719 50 51 52 53 54 8-07411 .07575 .07738 .07900 .08063 8-07929 .08095 .08260 .0842! .08589 8-1670 1685 1699 1714 1729 8-1734 .1749 1764 1779 .1794 50 51 52 53 54 55 56 57 58 59 55 56 57 58 59 8-08225 08387 .08549 08710 08871 8-08753 .08917 .09081 09244 -09407 8-1743 17582 1772 1787 .1801 8-1809 1823 .1838 1853; -18680 8-0903 8-09569 8-1816 8-18827 60 Lg. Vers ^ lo,Exs JJ Lg.Vers Log.Exs, *> * P.P. 170 LOG. VERSED SINES AND EXTERNAL SECANTS 10 11 / Lg.Vers J> Log, Exs /> Lg.Vers JD Log, Exs D / P, P, 1 2 3 4 8-18162 18306 18450 18594 18738 144 144 144 143 143 14; 142 14! 142 142 142 142 14! 14] 14] l^C 14( 140 139 139 139 139 138 138 138 138 137 138 137 137 136 137 136 136 136 135 136 III 135 135 134 134 134 134 133 133 133 133 133 132 133 132 132 132 ii? 131 8-18827 18973 .19120 19266 19411 14 14 14 14 14 14 14 145 14- 144 144 144 144 143 L4c || 142 11 L41 ^4C 140 140 139 39 39 39 38 38 38 38 8-2641 i / 2654E 2667 2681C 26941 13 13 13 13 13 13 13 13 129 129 129 128 129 128 128 128 128 128 127 127 127 127 127 126 8-27223 27356 27490 27623 27756 133 133 133 133 133 132 132 132 132 132 13! 13: 131 131 131 130 130 130 130 1 2 3 4 6 7 8 9 10 20 30 40 50] 6 7 8 9 10 20 30 40 50 1 j 1( 2( 3( 4( 5( f r J 1C 2C 3C 4C 5C i / 1C 2C 3C 4C 5C 130 13- C 15-1 17-3 19-5 21-6 43-3 65-0 86-6 08-3 0-40 0-50 0-60 0.70 D.7iO 1.51 2-2:2 3-02 3-7l3 3 30-3 70-5 50-4 )0-4 )0-5 )1-0 ) 1-5 )2-0 12-5 2 JO- 2 ' 0-2 0-2 0-3 0-3 0-6 1-0 1-5 1-6 1 0-1 0-1 0-1 0-1 0-1 0-3 0-5 0-6 0-8 ] ( i 1( 4 4 6 2 0- 0- 0- 0- 0- 1. 1. 2- 0- 0- o. o. o. o. o. 1. 1. o 0- o. o. o. o. 0- o. 120 L2-0 L4-0 L6-0 L8-0 !0-0 10- 50-0 JO-0 )0-0 o S S 81 0-5 2^9 2 I 4 1 2 j 1 5 1 4 5 6 7 8 9 8-18881 -19024 -19167 19309 -19452 8.1S357 19702 19847 19992 20137 8-2707] 27201 27331 27461 -27590 8-27889 28021 28153 28286 28418 8-28550 28681 28813 28944 .29075 g 6 7 8 9 10 11 12 13 14 10 11 12 13 14 15 16 17 18 19 8-19594 19736 19878 20019 20180 8.20281 20425 .2056J 2071J .20857 8-27719 .27849 27977 28108 .28235 8-28363 28491 28619 -28747 28875 8-20301 20442 -20582 20723 20883 8-21000 21143 21286 21428 21571 8-29206 .29336 29467 -29597 29727 15 16 17 J8 19 20 21 22 23 24 25 26 27 28 29 8-21003 21142 21282 21421 21560 8-21713 21855 21996 22138 .22279 8-29002 29129 29256 29383 29510 8-29857 29987 30117 30246 -30375 130 130 III 129 129 129 128 128 128 128 128 127 127 127 127 127 127 126 126 20 21 22 23 24 8-21698 21837 21975 22113 22251 8-22420 22561 22701 22842 22982 8-29636 29763 29889 30015 30140 126 126 126 126 125 125 125 125 III HI 124 124 124 123 124 123 123 123 123 122 122 122 122 122 III 121 121 121 121 120 120 120 120 ~D 8-30504 30633 30762 -30890 -31019 25 26 27 28 29 30 31 32 33 34 30 31 32 33 34 35" 36 37 38 39 22389 22526 22663 22800 22937 8-23122 23262 23401 23540 23679 30266 30391 30516 30642 30766 8-31147 31275 .31402 31530 31657 -31785 .31912 32039 32165 32292 23073 23209 23346 23481 23617 8-23818 23957 24095 -24234 24372 30891 31015 31140 31264 31388 35 36 37 38 39 4 a 41 42 43 44 23752 23888 24023 24158 24292 24509 24647 24784 24922 25059 38 37 37 37 36 36 36 36 36 36 35 35 35 35 34 11 34 34 134 31511 31635 31758 31882 32005 3241S 32544 .32670 32796 32922 126 126 126 125 125 125 125 125 124 125 124 124 124 123 124 124 123 123 123 123 ^ 40 41 42 43 44 45 46 47 48 49 24426 24561 24695 24828 24962 25195 25332 25468 25604 25740 32128 32250 32373 -32495 -32617 33047 33173 .33298 .33423 33547 33672 33797 33921 34045 34169 45 46 47 48 49 50 51 52 53 54 25095 25228 25361 25494 25627 25876 26012 26147 26282 26417 32739 32861 32983 33104 33225 "50 51 52 53 54 55 56 57 58 59 6rt 25759 25891 26023 26155 26286 26552 26686 26821 26955 27089 -33347 .33468 -33588 -33709 -33829 34293 34417 34540 34663 34786 55 56 57 58 59 26417 8-27223 33950 34909 HO g.Vers, 1) Log. Exs. Z> g.Vers. Log. Exs. ' P. P, LOG. VERSED SINES AND EXTERNAL SECANTS 13 13 171 1 Lg.Vers J> Log, Exs D Lg.Vers I> Log, Exs, I> ' P.P. 1 2 3 ^4 5 6 7 8 9 8-33950 34070 34190 34309 -34429 120 120 119 120 119 119 119 119 119 118 118 118 118 118 117 118 117 117 117 117 116 III 116 116 Lie 115 115 115 115 115 114 III 114 114 8-34909 -35032 .35155 35277 35399 123 122 122 122 122 122 121 122 121 121 121 121 120 121 L20 120 120 L20 il9 119 .18 '.il 118 118 118 118 117 117 117 117 117 117 116 III 116 116 .16 115 115 115 ss ii! 114 114 114 8-40875 40985 41096 41206 41317 110 110 110 110 110 110 110 109 110 109 109 109 109 109 109 108 109 108 108 108 108 108 108 107 107 107 107 107 107 106 107 iS! 106 ioe 106 105 106 105 105 105 105 105 105 105 104 105 104 104 10? 104 104 103 104 103 103 103 103 103 103 8-42002 42116 42229 42343 -42456 113 111 113 113 113 113 113 113 112 112 112 112 112 112 112 111 111 111 111 111 111 111 lie 110 110 no lie nc 109 nc nc 109 109 109 109 109 109 10F 109 ioe 108 108 108 108 108 107 18? 107 107 107 107 107 106 107 10P 106 10P 106 106 T> 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 3C 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 130 12- 14- 16- 18- 20- 40- 60- 80. 100- 117 11-7 13-6 15-6 17-5 19-5 39-0 58-5 78-0 97-5 114 11-4 13-3 15-2 17-1 19-C 38-0 57-0 76-0 95-0 111 11-1 12-9 14-8 16-6 18-5 37-0 55-5 74-0 92-5 108 10-8 12-6 14-4 16-2 18-0 36-0 54-0 72-0 90-0 105 10-5 12-2 14-Q 15-7 17-5 35-0 52-5 70-0 R7.R Hi Oil- 13- 015- D 17- 019- 039- D59- J 79- D99- 116 11-6 13-5 15-4 17-4 19-3 38-6 58-0 77-3 96-6 113 11-3 13-2 15-0 16-0 18-8 37-6 56-5 75-3 94-1 110 11-0 12-8 14-6 16-5 18-3 36-6 55-0 73-3 91. S 107 10-7 12-5 14-2 16-0 17-8 35-6 53-5 71-3 89-1 104 10-4 12-1 13-8 15-6 17-3 34-6 52-0 69-3 86. R > 118 911-8 a 13-7 ) 15-7 i 17-7 S19-6 B 39-3 5 59-0 3 78-6 [98-3 115 11-5 13-4 15-3 17-2 19-1 38-3 57-5 76-6 95-8 113 11-2 13-5 14-9 16-8 18-6 37-3 56-0 74-6 93-3 109 10-9 12-7 14-5 16- 1 18-1 36-3 54-5 72-6 90-8 106 10-6 12-3 14-1 15-9 17- 35-3 53-0 70-6 88-3 -5 0-6 o -8 0-1 0-1 0-1 0.2 0-3 0-4 8-34549 34668 34787 34906 35025 8-35522 -35644 -35765 -35887 36009 8-41427 41537 41647 41757 -41867 8-42569 42682 42795 -42908 .43021 5 6 7 8 9 10 11 12 13 14 10 11 12 13 14 8-35143 -35262 -35380 35498 35616 8-36130 36251 36372 -36493 36614 8-41976 42086 42195 42304 42413 8-43133 43246 -43358 -43470 -43582 15 16 17 18 19 8-35734 35852 -35969 -36086 -36204 8-36734 -36855 36975 37095 37215 8-42522 42630 42739 42847 -42956 8-43694 -43805 43917 44028 -44139 15 16 17 18 -i! 20 21 22 23 24 20 21 22 2? 24 8-36321 36437 36554 -36671 36787 8-37335 37454 37574 37693 37812 8-43064 .43172 43280 43388 .43495 8-44251 -44362 -44473 44583 -44694 25 26 27 28 29 8-36903 37019 37135 37251 -37366 8-37931 38050 38169 38287 -38406 8-38524 38642 38760 -38878 38995 8-43603 .43710 43817 43924 44031 8-44804 -44915 45025 45135 -45245 25 26 27 28 29 30 81 82 33 84 8-37482 37597 -37712 -37827 37942 8-44138 .44245 44351 44458 44564 8-45355 45465 45574 45684 -45793 30 31 32 33 34 85 86 87 38 82. 40 41 42 43 44 8-38057 38171 38286 38400 -38514 -39113 39230 39347 39464 39531 8-44670 44776 44882 44.988 8-45902 -46011 46120 46229 -46338 8-46446" -46555 46663 46771 46879 35 36 37 38 39 8-38628 38741 38855 38969 .39082 113 III 113 113 113 113 113 112 III 112 112 112 iii si m 39698 39814 39931 40047 40163 8-45199 45304 45409 45514 45619 40 41 42 43 44 35 46 47 48 49_ 50 51 52 53 54 55 56 57 58 59 no 39195 39308 -39421 39534 .39646 40279 40395 40511 40626 .40742 40857 40972 .41087 41202 41317 8-45724 45829 45934 4603P -46142 8-46987 -47095 -4720? -473K .4.7417 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 RO 39758 39871 39983 40095 40207 8.46247 -46351 4645R -46558 46662 8-47525 -47632 -4773P -47846 -47953 8-40318 40430 -40541 40652 40764 8-41431 -41546 -41665 41775 -41888 8-4676R -46869 -46972 47076 47179 o 47900 8-48060 -48166 .48273 -48379 -48485 w Lg, Vers. Log. Exs, Lg.Vers. -Og.Exs. P. P: 172 LOG. VERSED SINES AND EXTERNAL SECANTS 14 15 1 I Lg.Vers. 8-47282 .47384 .47487 .47590 .47692 1> Log.Exs. J> Lg.Vers. 1> Log.Exs. & ' P.P. 102 103 102 102 102 102 102 102 102 101 101 101 101 101 101 101 101 100 101 100 100 100 100 100 100 100 99 100 99 9? 99 99 99 98 99 98 98 98 98 98 98 98 97 98 97 97 97 97 97 97 11 97 9 96 96 96 -& 8-48591 48697 .48803 -48909 .49014 106 106 105 105 105 105 105 105 105 105 104 105 104 104 104 104 104 104 104 103 104 103 103 103 ioi '.03 102 103 102 102 102 102 102 102 102 102 101 101 101 101 101 101 101 101 101 101 100 100 100 100 100 100 100 100 100 99 100 99 99 8-53242 .53338 .53434 -53530 53625 96 95 96 95 95 95 95 95 95 95 94 95 94 94 94 94 94 94 94 11 94 93 93 93 93 93 93 93 93 92 92 93 92 92 92 92 92 92 92 92 91 91 91 91 91 91 91 91 91 90 91 90 90 90 90 90 90 90 90 8 54748 54847 .54946 -55045 -55144 99 99 99 99 99 99 98 C8 98 98 98 98 98 98 97 98 97 97 97 97 97 97 97 97 97 96 97 96 96 96 96 96 96 96 95 96 95 96 95 95 95 95 95 95 95 94 94 11 94 94 94 94 94 94 93 94 93 2 3 4 6 7 8 9 10 20 30 40 50 6 7i 8, i?; IS 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 ( ' ! 1! 2( 3( 4( 5( 103 10-3 12-0 13-7 15-4 III 51-5 68-6 85-8 100 10-0 11-6 13-5 15-0 16-6 33-3 50- G 66-6 83-3 97 9.7 11.3 12-9 14-5 16-1 32-3 48-5 64-6 80-8 94 9-4 10-9 12.5 14-1 15-6 31-3 47-0 62-6 78-3 91 * 9. :& !S:- )30- )45. )60- )75-i 103 10-2 11.9 13-6 15-3 17-0 34-0 51-0 68-0 85-0 99 9-9 11-5 13-2 14-8 16-5 33-0 49-5 66.0 82-5 96 9-6 11-2 12-8 14-4 16-0 32-0 48-0 64-0 80-0 93 9.3 10. 8 12-4 13-9 15-5 31-0 46-5 62-0 77-5 90 L 9-( 5 10-. : 12. ( nil 330-( 545-( 360-( j 75- < 101 10-1 11-8 13-3 15-1 16-8 33-6 50-5 67-3 84-1 98 9-8 11-4 13-0 14-7 16-3 32-6 49-0 65-3 82-6 95 9-5 12'e 14-2 31^6 47-5 63- 79-1 92 10.7 12.2 13-8 15-3 30.6 46-0 61-3 76-6 o o-5 0-0 0-0 0.1 o-l 0-1 0-2 0.3 0-4 8 _9_ 13 14 16 16 17 13 11 20 21 22 23 24 8.47795 .47897 .47999 .48101 .48203 8-48304 .48406 .48507 .48609 .48710 8-49120 .49225 .49331 .49436 .49541 8-53721 53816 .53911 54007 54102 8-55^3 55342 55141 55539 55638 5 6 7 8 9 8-49646 .49750 .49855 .49960 .50064 8-54197 .54291 .54386 .54481 54575 8-54670 .54764 .54858 .54952 55046 8-55736 55834 .55933 -5603: -56129 8-56226 56324 -56422 -56519 56617 10 11 12 13 14 15 16 17 18 19 8-48811 .48912 .49013 .49114 .49215 8-50168 .50273 .50377 .50481 -50585 8-49315 .49415 .49516 .49616 -49716 8-50688 .50792 .50896 .50999 -51102 8-55140 55234 .55328 .55421 .55515 8-56714 -56812 -56909 .57006 57103 20 21 22 23 24 25 27 28 29 30 31 32 33 81 35 36 37 38 39 8-49816 -49916 -50015 .50115 -50215 8-50314 50413 .50512 50611 .50710 8-51205 .51309 '51412 .51514 51617 8.55608 .55701 55795 .55888 55981 8-57200 .57296 .57393 .57490 57586 25 26 27 28 29 30 31 32 33 34 8-51720 .51822 .51925 .52027 .52129 8-56074 -56166 .56259 .56352 56444 8-57682 57779 57875 .57971 -58067 8-50809 .50908 .51006 51105 .51903 8-52231 .52333 .52435 .52537 526?,^ 8-565C6 .56629 .56721 .56813 .56905 8-58163 58259 .58354 -58450 -58546 35 36 37 38 39 40 41 42 43 44 8-51301 .51399 .51497 .51595 .51693 8-5274C .52841 52943 .53044 .53145 8-56997 .57089 .57180 .57272 .57363 8-58641 -58736 -58832 .58927 -59022 40 41 42 43 44 45 46 47 48 49 8.51791 51888 .51986 .52083 -52180 8-53246 .53347 53448 .53548 53649 8-57455 .57546 .57637 .57728 .57819 8-59117 -59211 -59306 -59401 .59495 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 8-52277 52374 -52471 .52568 52665 8.53749 .53850 53950 .54050 .54150 8.57910 .58001 .58092 .58182 .58273 8-59590 59684 -59779 -59873 -59967 50 51 52 53 54 8-52761 52858 52954 53050 .53146 8-54250 .54350 . 54449 .54549 .54649 8-58363 58453 .58544 .58634 58724 8-60061 -60155 .60249 60342 .60436 55 56 57 58 59 6018-53242 ' [Lg.Vers, 8-54748 8-58814 8-60530 60 Log.Exs, *> Lg. Vers, D Log.Exs, ' P.P. LOG. VERSED SINES AND EXTERNAL SECANTS 16 17 173 ' Lg.Vers, J> Log.Exs, Lg.Vers, Log.Exs P.P. 8-58814 .58904 .58993 .59083 .59173 -59262 .59351 59441 59530 -59619 -59708 59797 59886 .59974 60063 8-60152 -60240 60328 .60417 .60505 8-60593 60681 60769 60857 -60944 8-61032 61119 61207 .61294 -6138! 8-61469 61556 61643 .61730 61816 8-61903 61990 .62076 .62163 .62249 8-62336 62422 62508 .62594 8-62766 .62852 .62937 6302 6310 8-63194 63279 63364 63449 -63534 8-636li .6370; .63874 .63959 8 - 64043 Lg.Vers 86 85 8-60530 .6062 .6071 .60810 60903 .61089 61182 .61275 61368 8-61460 .61553 .61645 61738 .61830 8.61922 62014 .62106 .62198 -62290 8-62382 .62474 .62565 .62657 -62748 8.62840 .62931 63022 .63113 .63204 8-63295 63477 .63567 .63658 8-63748 .63839 .63929 .64019 -64109 8-64199 .64289 64379 .64469 .64559 8 - 64649 64738 .64828 .64917 -65006 8-65096 .65185 65274 65363 .65452 8-65541 65629 .65718 .65807 .65895 8-65984 Log.Exs 8-64465 .64549 64633 64717 .64801 - 64884 64968 .65052 .65135 -65218 8-65302 65385 65468 .6555] 8-65717 65800 .65883 .65965 -66048 8-66131 .66213 66295 .66378 8.67359 .67440 .67521 67602 -67683 8-67764 67845 .67926 .68007 -68087 8-65984 l> Lg.Vers. 8-6642! 6651S -66776 8-66863 .66951 .67039 67126 -67213 8-67301 67388 .67475 6756: .67649 -67736 67822 .67909 67996 -68082 8-6816Q 68255 .68341 .68428 .68514 8-6860C .68686 .6877 .68858 .68944 8-6902 .6911 .69201 69286 69372 1-69457 -69542 .6962 .69712 .69798 .7005 .7013 -70222 8-70306 .70391 .70475 .7056 .7064 88 88 87 87 85 85 8-70728 10 11 12 IS li 15 16 17 18 Jl 20 21 22 23 24 25 26 27 28 29 30 31 32 33 li 35 36 37 38 -Si 40 41 42 43 ii 45 46 47 48 ii 51 52 53 5i 55 56 57 58 59 Log.Exs 60. 91 9-1 10-6 12- 1 13-6 15-1 30-5 3046-5 46- 045- 5 40 62-0 61- i 50 93 C 710- 812. 13 10 15-5 2031-C 92 9-2 10-7 12-2 13-8 15. r 30- i 77-5 76- 90 89 10. 11 13. 14- i 29-6 10-5 812-0 13-5 10 15-0 30- C 75-8 88 8-8 10-2 11.7 13-2 14-6 29-3 45-0 44-5144 4060-0 5075-0 87 8-7 10.1 11, 13- 59.3J58- 74-1 10 J.W Xt U J.:t O 2029.028-6 30 43-5 43- C 40 50 58-0 72-5 71- 86 8-6 10- C 11-4 12-1 14-5 14- S 57- 73-3 85 85 9-9 11-3 12-7 14- 1 28-3 42-5 56-6 70-8 84 83 82 10 40 11.2 12-6 14-0 2028-0 30 42-041- 56-055 11-0 12-4 13-8 27-6 5070-1 81 80 10-8 12-1 10-6 12-0 13-5 13-3 20 27-0 26-6 3040-540-0 54-0 53-3 67-5166-6 10-8 12- S 13- 6 27- S 41. C 54- 6 68- 1 79 7-t 9. 10. 11. 13- 26. 39. 52. 65- O 60- S 70-0 80-5 90-1 100-1 200-1 30 0-2 40 0-3 Ff 0-4 P. P. 174 LOG. VERSED SINES AND EXTERNAL SECANTS 18 19 ' Lg.Vers. D Log.Exs. D Lg.Vers. D Log. Exs / P.P. o 1 2 3-68969 .69049 69129 79 80 8-71149 .71232 .71316 83 84 8.73625 .73700 .73775 75 75 8-76058 .76137 76217 79 II C 1 2 3 .69208 -69288 79 .71400 .71484 84 73851 .73926 75 .76297 76376 79 3 4 84 83 82 5 8-69367 7 A 8-71567 83 8.74001 Vb 8-76456 bu 7Q 5 6 0.4 90 9 6 7 - 69446 -69526 79 7Q .71651 71734 83 .74076 74151 75 76536 76615 79 6 7 8 11-2 1 9 fi 192 10 : 1 9 1 8 9 -69605 69684 79 71817 71901 83 74226 74301 75 76694 76774 79 8 9 10 20 14-0 28-0 13-8 27-6 13- 27. ! 10 11 12 8-69763 69842 69921 V9 79 79 75 8-71984 72067 72150 83 83 83 00 8-74376 .74451 .74526 Vb 74 75 8-76853 76932 77011 V9 79 79 70 10 11 12 30 40 50 42-0 56-0 70-0 41-5 55-3 69-1 41. 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Q i 9-12937 12984 13031 13078 13125 9-15710 15755 15801 15846 15891 9 22441 -22494 22547 22600 -22653 10 11 12 13 14 15 16 17 18 19 9-13172 13219 .13266 13313 13359 9-19492 .19546 .19601 .19655 .19709 9-15937 -15982 -16027 16073 -16118 9-22706 -22759 22812 -22865 -22918 10 11 12 13 14 9-13406 13453 13500 13546 13593 9-19763 .19817 .19871 .19925 -19979 9-16163 16208 .16253 16298 -16343 9-22971 .23024 .23076 23129 -23182 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 9-13639 13686 13733 13779 13826 46 47 46 46 46 46 46 46 46 46 46 46 46 46 46 if 46 9-20033 .20087 .2014! 20195 .20249 9-16388 -16434 16479 16523 16568 HI 45 44 45 45 45 45 45 44 if 45 44 45 i! 45 44 44 44 it ii 44 it if 44 if 44 44 44 it 44 44 44 9-23235 -23287 23340 -23393 - 23446 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 9.13872 13919 13965 14011 14058 9.20303 .20357 .20411 20465 20518 9-16613 16658 16703 16748 16793 9-23498 -23551 -23603 -23656 -23709 30 31 32 33 34 35 36 37 38 39 9-14104 14151 14197 14243 14289 9-20572 -20626 -20680 -20733 -20787 9-16838 -16882 16997 16972 17017 9-23761 23814 -23866 -23919 -23971 9-14336 14382 14428 14471 11520 9-20841 20894 20948 -21002 -21055 9-17061 17106 17151 17195 -17240 9 24024 -24076 -24128 -24181 24233 35 36 37 38 39 40 41 42 43 44 9-14566 14612 14658 14704 14750 46 46 46 46 46 46 46 46 46 45 46 45 46 n 45 46 45 45 45 46 9-21109 21162 21216 21269 21323 9-17284 -17329 17373 17418 -17462 9.24285 -24138 -24390 - 24442 -24495 40 41 42 43 44 45 46 47 48 49 9-14796 14842 14888 14934 14980 9-21376 21430 21483 21537 -21590 9-17507 17551 17596 -17640 .17684 9-24547 -24599 24651 24704 -24756 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 9.15026 15071 15117 15163 15209 9-21643 21697 -21750 -21803 -21857 9-17729 -17773 -17817 17861 17906 9-24808 -24860 -24912 -24964 -25016 50 51 52 53 54 9-15254 15300 15346 15391 15437 9-21910 21963 22016 .22070 22123 9-17950 17994 -18038 .18082 18126 9-25068 -25120 -25172 -25224 -25276 55 56 57 58 59 SO 9-15483 9-22176 9.18170 9-25328 BO ' Lg.Vers, JL> Log.Exs, 1> Lg.Vers, D Log.Exs. ' P, P, LOG. VERSED SINES AND EXTERNAL SECANTS 32 33 181 1 ~f) i 2 3 4 Lg.Vers. 9-18170 .18214 18258 18302 18346 n Log.Exs. D Lg.Vers. D Log.Exs. D ' P. P, 44 44 44 44 44 44 44 44 43 44 44 44 43 44 43 43 43 44 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 42 43 43 43 42 43 42 43 43 42 43 9-25328 -25380 -25432 -25484 -25536 52 52 11 52 52 52 51 1 51 52 a 51 51 51 51 51 51 51 51 51 11 51 51 51 51 1 51 51 51 51 51 51 51 51 51 51 51 55 51 51 51 51 50 9-20771 -20814 -20856 20899 -20942 42 42 42 43 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 a 42 a 42 41 41 42 41 4l 41 41 41 41 41 41 41 41 41 41 41 41 41 D 9-28412 -28463 -28514 -28564 -28615 51 11 51 1 51 88 8} 51 50 50 5Q 5Q 50 50 50 50 50 50 50 88 88 50 50 50 50 50 50 50 50 50 50 50 50 49 50 50 50 50 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 52 5-2 6.0 6-9 s'e 17-3 26-0 34-6 43-3 50 5-0 5-9 6-7 7-6 8-4 16-8 25-2 33-6 42-1 44 4-4 5-1 5-8 6-6 7-3 14-6 22-0 29-3 36-6 43 4-2 4-9 5.6 6-4 7-1 14-1 21-2 28-3 35-4 6 7 8 1C 20 30 40 50 61 5-1 6-0 6-8 7-7 8-6 17-1 25-7 34-3 42-9 50 5-0 5-1 7^5 8-3 16-6 25-0 33-3 41-6 43. 4-3 5-1 5-8 l'.% 14-5 21-7 29-0 36-2 4.2 4-9 5-6 6-3 7-0 14-0 21-0 28-0 35-0 41 4-] 4-1 5:! i81 20. e 27.3 34.] 51 U ?:l 8-i 17. 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( S ! > ) > ) > i 5 i i I l i i 5 6 7 8 J 10 11 12 13 14 9-18390 18434 18478 18522 18566 9-25588 -25640 25692 -25743 -25795 9 - 20984 21027 21069 21112 21154 9-28666 -28717 -28768 -28818 -28869 9-18610 18654 18697 18741 -18785 9-25847 -25899 -25950 -26002 -26054 9-21196 21239 21281 21324 21366 9-28920 28970 29021 29072 29122 15 16 J7 18 19 9-18829 18872 18916 .18959 -19003 9-26105 26157 26209 .26260 26312 9-21408 21451 .21493 21535 -21577 9-29173 29223 29274 -29324 29375 15 16 17 18 Ji 20 21 22 23 24 20 22 23 24 9-19047 19090 19134 19177 -19221 9-26364 26415 .26467 26518 26570 9-21620 .21662 21704 .21746 -21788 9-29426 29476 29527 29577 29627 25 !Z7 28 29 9-19264 - 19308 19351 19395 19438 9-26621 .26673 26724 .26776 26827 9-21830 21872 2i9i< 21956 21998 9-29678 29728 2977J 2982S 29879 25 26 27 28 .29 30 31 32 33 34 30 31 32 33 34 9-19481 19525 19568 19611 19654 9-26878 26930 26981 27032 27084 9 - 22040 .22082 .22124 .22166 .22208 9-29930 2998C 30030 -30081 30131 35 36 37 38 21 40 41 42 43 44 9-19698 19741 19784 19827 -19870 9-27135 27186 27238 27289 27340 9-22250 22292 22334 22376 -22417 9-30181 -30231 -30282 -30332 30382 35 36 37 38 11 40 41 42 43 44 9-19914 19957 20000 20043 20086 9-27391 27443 27494 27545 27596 9-22459 22501 .22543 22584 -22626 9-30432 -30482 -30533 -30583 -30633 45 46 47 48 49 50 51 52 53 54 9-20129 20172 20215 20258 -20301 9-27647 27698 27749 .27800 27852 9-22668 22709 22751 -22792 22834 9-30683 -30733 30782 3083 .30883 45 46 47 48 49 9-20343 20386 20429 20472 -20515 9-27903 27954 .28005 .28056 28107 9-22876 .22917 .22959 23000 -23042 9.30933" -30983 -31033 -31083 -31133 50 51 52 53 54 55 56 57 58 59 9-20558 20600 . 20643 20686 20728 9-28157 28208 .28259 .28310 28361 9-23083 -23124 .23166 23207 -23248 9-31183 31233 31283 .31333 .31383 9-31432 55 56 57 38 59 60 9-20771 9.28412 9 - 23290 60 Lg.Vers. JL> Log.Exs. D Lg.Vers. Log.Exs. /> ' P. P. 182 LOG. VERSED SINES AND EXTERNAL SECANTS 34 35 / Lg.Vers, D Log.Exs, D Lg.Vers. D Log.Exs, D ' P. P. o 1 2 3 4 5 6 7 8 9 9-23290 23331 23372 23414 23455 41 a 41 41 a 41 41 a 41 41 41 41 41 41 41 41 40 41 41 41 40 a a 41 40 I 40 40 i a a 40 40 40 40 40 40 40 9-31432 .31482 31532 .31582 31632 50 50 49 50 49 50 49 50 49 49 50 49 49 49 49 50 t! il 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49, 49 "W 9-25731 -2577' -25811 .25851 25891 40 40 40 40 40 40 II 40 40 40 39 40 40 39 40 39 40 39 39 40 39 39 39 39 40 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 38 39 39 39 9-34395 - 34444 -34492 34541 34590 49 48 49 49 49 48 49 48 49 49 48 48 49 48 49 48 49 48 48 48 49 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 O 1 2 3 4 5( 6 5- 7 5- 8 6. 9 7 10 8 20 16 3025 4033 5041 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 ) 41 4 8 5 6 6 5 7 3 8 6 16 24 3 33 641 48 4-8 5 6 6-4 7-3 8-1 16- T 24-2 32-3 40-4 41 !:i 6-2 6-9 13-8 20-7 27-6 34-6 40 4-0 4-7 5-4 6-1 6-7 13-5 20-2 27-0 33-7 39 3-9 4-6 5-2 5-9 6-6 13-1 19.7 28-3 32-9 3 6 3 7 4 8 5 9 10 e 20K 30 19 40 25 50 35 > 49 9 4-9 8 5-7 6 6-5 5 16-3 7 24-5 32-6 240-8 48 4-8 5-6 6-4 7-2 8-0 16-0 24-0 32-0 40-0 41 4-1 4 8 !; 13-6 20-5 27-3 34-1 40 "-0 & 6-0 1,8 13-3 26^6 33-3 39 3-9 4-5 5-2 5-8 6-5 13-0 19-5 26-0 32-5 % :l 8 i .6 .1 9-23496 23537 23579 .23620 .23661 9-31681 31731 31781 31831 .31880 9-25931 -2597" 2601" 26051 -26091 9-34639 34688 34737 34785 34834 5 6 7 8 9 10 11 12 13 14 15 16 17 18 12. 20 21 22 23 24 9.23702 23743 . 23784 .23825 .23866 9.31935 .31980 .32029 32079 .32129 9-26131 26171 26210 -26250 -26290 9-34883 .34932 34980 34029 35078 10 11 12 13 14 9 2390'7 23948 -23989 . 24030 .24071 9-32178 .3222* .32277 .32327 32377 9-26330 26370 26409 26449 .26489 9-35127 35175 35224 -35273 .35321 15 16 17 18 -li 20 21 22 23 24 9-24112 24153 .24194 .24235 .24275 9-32426 .32476 .32525 .32575 .32624 9-26528 2656! 26608 26647 -26687 9-35370 35419 35467 -35516 35564 25 26 27 28 29 9-24316 .24357 .24398 .24438 . 24479 9-32673 3272S .32772 .32822 32871 9-26726 26766 26806 26845 -26885 9-35613 .35661 35710 35758 .35807 25 26 27 28 29 30 31 32 33 34 9-24520 24561 . 24601 .24642 - 24682 9-32920 .32970 33019 .33069 -33118 9-26924 26964 2700! . 27042 27082 9-35855 35904 35952 36001 .36049 30 31 32 33 34 35 86 37 38 89. 40 41 42 43 44 45 46 47 48 49 9-24723 .24764 .24804 .24845 .24885 9-33167 -33216 -33266 33315 .33364 9-27121 27161 27200 2723J 27278 9.36098 36146 -36194 36243 .36291 35 36 37 38 .39. 40 41 42 43 44 9-24926 .24966 .25007 .25047 -25087 9-33413 33463 33512 3356] 33610 9-27318 27357 27396 27435 27475 9-36340 -36388 .36436 36484 36533 9-25128 .25168 .25209 .25249 .25289 9-33659 33708 .33758 .33807 33856 9-27514 .27553 .27592 -27631 -27670 9-36581 36629 .36678 36726 -36774 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 9.25329 .25370 .25410 2545C .25490 9-33905 -33954 . 34003 .34052 .34101 9-27709 -27749 -27788 -27827 -27866 9-36822 36870 36919 -36967 37015 50 51 52 53 54 55 56 57 58 59 9-25531 .25571 25611 .25651 -25691 9-34150 -34199 -34248 -34297 .34346 9-27905 .27944 -27982 -28021 .28060 9-37063 37111 -37159 37207 37255 9-25731 9 34395 9-28099 9-37303 60 Lg. Vers. D Log.Exs Lg.Vers. 1) Log.Exs. JJ P. P, LOG. VERSED SINES AND EXTERNAL SECANTS 36 37 T 2 I Lg.Vers. 9-28099 28138 .28177 .28816 -28255 1> Log.Exs, i> Lg.Vers. j> og.Exs. D ' P.P. 39 38 39 39 9-37303 -37352 37400 37448 -37496 48 48 48 48 48 48 8 48 48 48 48 48 47 48 48 48 47 48 47 8 48 47 48 47 48 i 48 47 47 8 48 47 47 48 47 8 I 8 47 47 47 3 47 47 47 47 47 47 D 9.30398 -30436 -30474 .30511 30549 37 i! 37 38 37 37 38 37 37 37 37 37 37 37 37 37 37 37 37 37 3Z 37 37 40163 40210 40258 .40305 .40352 % 8 47 47 47 8 47 8 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 47 46 8 47 46 47 46 47 46 47 II II 46 46 i? 46 46 46 TT 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 80 40 50 6 7 8 9 10 20 80 J40 50 6 7 8 10 20 80 40 50 6 7 8 9 10 20 80 40 50 48 -, 4-8 5-6 6-4 7-3 8-1 16-1 24-2 32-3 40.4 47 4-7 5-5 6-3 7-1 7-9 15-8 23-7 31-6 39-6 4 6 4 7 E 8 6 9 7 10 7 20 IE 8023 4031 50 Si 39 U 5-2 5-8 6-5 13-0 10-5 26-0 82-5 38 3-8 4-4 5-C 5-7 6-3 12-6 19-0 25-3 31-6 37 8-7 4-3 4-9 5-5 6-1 12-3 18-5 24-6 30. f 48 4-8 5-6 6-4 7-2 8-0 16 24-0 32-0 40.0 47 4-7 5.! 6-: 7-i 7. 15- ii 23- 31.; 39- 5 5 4 2 : :! :? 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Vers. Log.Exs. P, P. 184 LOG. VERSED SINES AND EXTERNAL SECANTS 38 rfkCOtOH-'Ol " 1 9-32631 .32668 32704 32741 .32778 9-32814 .32851 32888 32924 1 -32961 D Log.Exs, J> Lg.Vers, D Log.Exs. JD > P.P. 36 36 37 36 36 36 37 36 36 36 36 36 36 36 36 38 36 36 36 36 36 36 36 36 36 36 38 36 36 36 36 36 36 36 36 36 36 33 36 38 38 38 36 38 36 36 36 36 35 36 36 35 i* i 35 36 35 9-42978 -43024 -43071 .43118 -43164 46 47 46 46 n 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 9-34802 34837 34873 34909 34944 35 36 35 35 35 36 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 34 35 35 35 35 34 35 34 35 34 35 9-45752 45797 45843 45889 -45935 45 46 46 46 45 46 46 45 46 il il 46 45 46 45 46 45 il 45 46 45 45 46 45 45 45 46 45 45 45 45 46 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 1 45 45 45 45 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 140 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 47 4-7 5-5 6-2 7-0 7-8 15-6 23-5 31-5 39-1 46 4-6 5-3 6-1 M 15-5 23-0 30-6 38-3 4 6 4 7 5 8 6 9 6 10 7 2015 3022 4030 5037 37 3-7 4-3 11 6-1 12-3 18-5 24-6 30-8 36 3-6 4-2 4-8 5-4 6-0 12-0 18-0 24-0 30.0 35 3-5 1:1 8:1 11-6 17-5 23- 5 29- T 46_ 4-6 5-4 6-2 15*5 23-2 31- Q 38-7 68 7-8 15.J 22-7 30-3 37.9 5 if 5 5 5 5^5 6-1 12-1 18-2 24-3 30-4 35 3-5 4-1 4-7 5-3 5-9 11.8 17-7 23-6 29-6 3_ 3-4 4-0 4-6 ills 17-2 23-0 28.7 ! ! 9 9-43211 -43257 -43304 .43350 -43396 9 34980 35016 35051 35087 35122 9-45981 -46027 -46073 -46118 .46164 5 6 7 8 9 10 11 12 13 14 15 16 \l il 20 21 22 23 24 25 26 27 28 30 31 32 33 34 9-32997 33034 33070 33107 ^33143 9-33180 33216 33252 33289 9-33381 33398 33434 33470 33507 9 - 43443 .43489 -43536 -43582 -43629 9-43675 43721 43768 -43814 -43861 9-35158 .35193 35229 .35264 -35300 9-35335 -35370 -35406 35441 -35477 9-46210 46256 .46302 46347 46393 9 46439 46485 46530 46576 .46622 10 11 12 13 14 15 16 17 18 19 9-43907 43953 43999 44046 44092 9-35512 35547 35583 35618 35653 9-46668 46713 46759 46805 46850 2O 21 22 23 24 9-33543 -33579 33615 33652 33688 9 - 44138 -44185 .44231 .44277 - 44323 9-35689 -35724 35759 35794 -35829 9-46896 . 46942 46987 .47033 _.47078 9-47124 47170 47215 -47261 -47306 25 26 27 28 29 9-33724 33760 33796 33833 33869 9-44370 -44416 - 44462 -44508 44554 9-44601 - 44647 .44693 44739 -44785 9.35865 -35900 35935 -35970 36005 30 31 32 33 34 35 36 37 38 39. 40 41 42 43 44 9-33905 33941 33977 34013 . 31049 9.34085 34121 -34157 34193 .34229 9-36040 .36076 .36111 .36146 36181 9-47352 -47398 47443 -47489 47534 35 36 37 38 -31 40 41 42 43 44 9-44831 -44877 -44924 44970 45016 9-36216 36251 36286 36321 36356 9-47580 47625 47671 47716 47762 45 46 47 48 49 9-34265 -34301 34337 34373 _ 34408 9 34444 . -34480 34516 34552 34587 9-45062 45108 45154 .45200 45246 9-36391 -36426 -36461 36495 36530 9-36565 -36600 36635 .36670 .36705 9-47807 -47852 -47898 -47943 47989 45 46 47 48 49 50 51 52 53 54_ 55 56 57 58 59 9-45292 -45338 -45384 45430 45476 9-48034 -48080 -48125 -48170 .48216 50 51 52 53 54 9-34623 34659 34695 -34730 -347R6 9-34802 Lg, Vers 9-45522 -45568 -45614 -45660 45706 9 36739 36774 36809 36844 36878 9-4826T 48306 .4835? 48397 -48442 55 56 57 58 59 V 9-45752 9.36913 9.48488 60 Log.Exs, JJ Lg. Vers, ^> Log.Exs. JL> P, P. LOG. VERSED SINES AND EXTERNAL SECANTS 40 41 ' Lg.Vers. j> Log.Exs. 1> Lg.Vers, D Log.Exs. 7> / P, P, 9-36913 7 9-48488 9.38968 9-51190 1 36948 oT 48533 .39002 ^ -51235 45 1 2 3 4 36982 37017 37052 35 34 48578 48624 48669 45 45 .39035 .39069 39103 II 51279 -51324 51369 a 2 3 4 43. 45 5 9-37086 OR 9-48714 4b 9-39137 33 9-51414 a 5 6 7 5 3 4-5 5 2 6 37121 aJ 48759 45 39170 qq 51458 fi 6 6 ft 7 37156 31 .48805 39204 -51503 7 9 6 8 67 8 9 37190 .37225 34 34 .48850 .48895 it 45 -39238 -39271 33 33 -51548 .51592 44 tl 8 9 10 10 20 30 7-6 15-1 22-7 7-5 15-0 22-5 10 9-37259 9 48940 9-39305 9-51637 11 37294 -48986 .39339 .51682 11 11 40 30-3 30-0 .37328 3 A -49031 -39372 51726 12 50 37-9 37-5 13 37363 37 .49076 -39406 -51771 / 7 13 14 -37397 .49121 -39439 -51816 -4 14 15 9-37432 34 9-49166 45 9 . 39473 33 9-51860 44 15 44 44 16 37466 3 A -49211 39507 34 -51905 16 6 4-4 17 18 19 -37501 37535 37570 II -49257 -49302 .49347 45 45 .39540 -39574 -39607 33 33 -51950 -51994 52039 44 17 18 19 7 8 19 5-2 5-9 6-7 5 I 6-6 20 21 22 9-37604 -37639 37673 i 9-49392 -49437 .49482 45 45 45 9-39641 -39674 39708 30 9-52084 -52128 -52173 45 20 21 22 20 30 40 14-8 22-2 29 g 14-6 22-0 29 3 23 24 37707 37742 34 -49527 .49572 45 39741 -39774 33 52217 52262 44 23 24 50 37.1 36.6 25 26 9-37776 -37810 it o7 9-49618 -49663 4b 45 9-39808 .39841 33 9 52306 52351 44 25 26 35 3 27 -37845 -49708 39875 52396 Jl 27 6 3-5 3-4 28 29 -37879 -37913 34 -49753 -49798 45 39908 39941 33 52440 -52485 44 28 29 7 8 4-1 4-6 J:S 30 31 32 33 34 9-37947 37982 .38016 38050 38084 It 11 34 if 9-49843 .49888 -49933 -49978 50023 4b 45 45 45 45 45 45 9-39975 . 40008 40041 40075 40108 33 33 II 9-52529 -52574 52618 .52662 52707 44 4f 4^ 4^ 44 44 44 30 31 32 33 -34 35 36 9 10 20 30 40 50 5-2 5-f 11-6 17-5 23-j 29-1 1:1 11. 17-2 23-0 28-7 35 86 9-38118 38153 9-50068 50113 9 40141 .40175 9-52752 52796 37 38 39 -38187 38221 -38255 34 34 50158 50203 -50248 45 45 40208 .4024 40274 33 33 52841 52885 52930 44 44 37 38 39 6 7 34 3-4 3-q 33 3-3 3-9 40 41 9-38289 -38323 34 34 9-50293 50338 4b 45 9-40307 40341 33 33 9-52974 53018 44 44 40 41 8 9 4.5 5-1 4-3 5-0 42 43 44 38357 38391 38425 34 34 34 50383 5042^ 50472 44 45 40374 -40407 - 40440 33 33 53063 53107 53152 44 44 42 43 44 10 20 30 5-6 11-3 17-C 5.6 11.1 16-2 45 46 9-38459 38493 34 34 9-50517 -50562 4b 45 9 40473 -40506 33 33 9-53196 - 53240 44 44 45 46 40 50 22-6 28-3 22-3 27.9 47 48 49 -38527 .38561 .38595 34 34 34 50607 50652 .50697 41 45 .40540 40573 -40606 33 33 53285 -53329 -53374 44 47 48 49 33 50 51 52 53 54 9.38629 .38663 .38697 -38731 -38765 34 34 34 33 34 9.50742 50787 .50831 .50876 4b 11 45 45 9-40639 .40672 .40705 .40738 40771 38 33 33 33 33 9 - 53418 53462 .53507 .53551 -53595 44 44 44 44 44 50 51 52 53 54 7 8 10 201 4^4 4.9 5-5 1.0 55 56 57 58 9-38799 -38833 38866 -38900 34 34 33 34 9-50966 .51011 .51055 51100 43 45 44 45 9-40804 -40837 40870 .40903 66 33 33 33 9-53640 .53684 . 53728 -53773 44 44 44 44 55 56 57 58 801 402 502 6-5 2.0 7-5 59 -38934 66 .51145 .40936 -53817 AA b9 go 9-38968 34 9-51190 44 Q- 40969 33 Q- 53861 60 T Lg.Vers, 1> Log.Exs, *> Lg.Vers, D Log.Exs, U . P. P, 186 LOG. VERSED SINES AND EXTERNAL SECANTS 42 43 / Lg.Vers. D Log.Exs. J) Lg.Vers. 7> Log.Exs. D ' P.P. 1 3 4 5 6 7 8 9 10 11 12 13 14 9-40969 .41001 .41034 .41067 .41109 32 33 33 33 32 33 33 32 33 33 32 33 32 33 32 33 32 32 32 33 32 11 32 32 32 33 32 32 32 32 32 32 32 32 32 32 32 ! II 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 9-53861 -53906 53950 53994 . 54038 44 a 44 44 44 44 44 44 44 a '44 44 44 44 a 44 a 44 44 44 44 44 44 44 44 44 a 44 44 44 44 44 44 44 44 44 44 44 44 a 44 44 44 44 44 a 44 44 44 44 43 9-42918 -42950 42982 43014 43046 32 32 32 32 32 32 31 32 32 32 32 32 31 32 32 II 32 32 31 32 31 32 32 31 31 32 31 32 31 31 31 32 3 3' 31 31 s! 5 31 3' 3^ 8 31 31 31 S:: 31 31 31 31 IT 31 9-56505 -56549 -56593 -56637 -56680 43 44 44 43 44 44 44 43 44 a a a 44 43 43 44 43 44 1! a 43 a n 44 43 43 43 43 44 11 43 43 43 43 43 43 43 43 44 43 $ a 43 43 1 2 3 4 5 6 7 8 9 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 "e 7 8 9 10 20 30 40 50 I \ 5.2 5-9 6-7 7-4 14-8 22-2 29-6 37-1 43 4-3 5-1 5-8 6-5 7-2 14-5 21-7 29-0 36-2 33 i:l 11 5-5 11-0 16-5 22-0 27-5 32 3-2 3-7 4-2 4-8 5-3 10. B 16-0 21-5 26-6 3 6 3 7 3 8 4 9 4 LO 5 20 10 JO 15 1020 5025 44 4-4 5-1 5-8 6-6 7-3 14-6 22-0 29-3 36-6 43 4-3 5-0 5-7 6-4 7-1 14-3 21-5 28-6 35-8 32 3-2 3-8 4-3 4-9 5-4 10-8 16-2 21-6 27-1 3-7 4-2 4-7 5-2 10-5 15-7 21-0 26-2 1 -1 J 9-41133 .41166 -41199 41231 .41264 9 54083 .54127 54171 54215 .54259 9-43078 -43110 -43142 -43174 -43206 9-56724 - 56?o8 -56812 -56856 -56899 9-41297 -41330 -41362 -41395 -41428 9 54304 54348 .54392 54436 . 54480 9-43238 43270 43302 43334 43365 9 - 56943 -56987 -57031 -57075 -57118 10 11 12 13 14 15 16 17 18 11 20 21 22 23 24 9-41461 -41493 -41526 -41559 41591 9-41624 -41657 -41689 -41722 -41754 9-54525 54569 54613 .54657 54701 9-43397 43429 43461 43493 .43525 9-57162 -57206 -57250 -57293 -57337 15 16 17 18 19 9-54745 .54790 54834 .54878 .54922 9-43557 43588 43620 43652 43684 9-57381 - 57424 -57468 -57512 -57556 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 9-41787 -41819 -41852 41885 .41917 9-54966 .55010 55054 55098 55142 9-43715 43747 .43779 43810 43842 9-57599 -57643 -57687 -57730 -57774 25 26 27 28 29 9-41950 41982 42014 42047 42079 9-55186 .55230 .55275 .55319 55363 9-43874 43906 43937 43969 .44000 9-57818 -57861 -57905 -57949 - 57992 30 31 32 33 34 35 36 37 38 39 35 36 37 38 39 9-42112 -42144 42177 42209 -42241 9-55407 55451 55495 .55539 -55583 9-44032 44064 .44095 44127 44158 9-58036 -58079 -58123 -58167 -58210 40 41 42 43 44 9-42274 42306 -4233J 42371 -42403 9-55627 55671 55715 55759 .55803 9-44190 44221 44253 44284 -44316 9-58254 -58297 . 58341 -58385 -58428 40 41 42 43 44 45 46 47 48 41 50 51 52 53 54 9-42435 -42467 -42500 -42532 -42564 9-55847 -55890 55934 55978 56022 9 - 44347 -44379 - 44410 -44442 -44473 9 - 58472 -58515 -58559 -58602 .58646 45 46 47 48 49 9-42596 -42629 42661 42693 42725 9-56066 56110 56154 56198 56242 9 - 44504 -44536 -44567 -44599 -44630 9-58689 .58733 -58776 -58820 -58864 50 51 52 53 54 55 56 57 58 59 55 56 57 58 59 9-42757 42789 42822 42854 42886 9-56286 -56330 -56374 -56417 -56461 9-44661 .44693 44724 44755 -44787 9-58907 -58951 -58994 -59037 -59081 og.Exs. D g.Vers, D Log.Exs, D ' P. P. o 1 2 3 4 .44818 44849 .44880 .44912 .44943 31 11 31 31 31 11 31 31 SI 31 31 31 Si 31 31 31 31 31 31 31 30 31 31 31 31 Si 31 31 30 31 Si Si Si Si 31 31 30 30 31 30 30 31 Si i 30 -5 59124 -59168 59211 59255 .59298 43 43 43 43 43 43 43 43 43 43 43 43 43 43 -46671 46701 46732 46762 46793 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 .61722 -61765 -61808 61852 61895 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 1 2 3 4 6 7 8 9 10 20 30 40 5G ] i 3 ' , 6 7 8 9 10 20 30 40 50 6 7 4 4 5 5 6 7 14 21 29 36 6 7 8 9 LO 20 JO 10 iC 3 3 3 4 4 5 10 15 21 26 3 3 3 4 4 5 10 15 20 25 6 7 8 9 10 20 30 40 50 ? 3 8 ] 4 4 5 6 7 14 21 28 35 '? 2 '\ 0_ .6 .1 -4 | I c i t. i ( i< i< 2' 43 4-3 i:l i: 14-3 21-5 28-6 35-8 3_ 4 4 31 3-1 3-6 4-1 4-6 5-1 10-3 15-5 20-6 25-8 30 3 t) 3-5 4-0 4-5 50 10-0 15-0 20-0 25-0 >Q 'I 1-9 1-4 L-9 )-8 1-7 ).6 1.6 5 6 7 8 9 44974 45005 .45036 45068 .45099 59342 59385 59429 59472 59515 46823 46853 46884 46914 46945 61938 61981 62024 62067 62110 5 6 7 8 9 10 11 12 13 14 .45130 45161 .45192 45223 -45254 59559 59602 59646 59689 59732 46975 47005 47036 47066 47096 9-62153 -62196 62239 62282 62326 10 11 12 13 14 15 16 17 18 19 15 16 17 18 19 45285 45316 45348 45379 45410 59776 59819 59863 -59906 -59949 43 43 43 43 43 43 43 43 43 43 n 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 "D 9-47127 47157 47187 47218 .47248 9-62369 -62412 -62455 -62498 -62541 20 21 22 23 24 9-45441 45472 .45503 45534 .45565 59993 60036 .60079 60123 .60166 9-47278 -47308 .47339 47369 47399 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 30 29 30 30 29 30 30 29 30 29 9-62584 -62627 6267C 62713 62756 20 21 22 23 24 25 26 27 28 29 9-45595 45626 45657 45688 .45719 9 60209 60253 .60296 60339 60383 9-47429 47459 .47490 47520 47550 9-62799 -62842 -62885 -62928 -62971 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 9-45750 45781 45812 45843 45873 9-45904 -45935 -45966 45997 46027 9 60426 60469 G0512 60556 60599 9-47580 .47610 47640 -4767C 47700 9-63014 -63057 -63100 -63143 -63186 30 31 32 33 34 9 . 60642 60685 .60729 60772 .60815 T. 80851 60902 60945 .60988 61031 9-47731 .47761 .47791 -47821 .47851 9-63229 -63272 -63315 -63358 63401 35 36 37 38 39 40 41 42 43 44 9-46058 46089 46120 46150 46181 9.47881 47911 47941 47971 48001 9 - 63443 -63486 -6352? -63572 -63615 42 43 43 43 43 43 42 43 43 43 43 42 43 43 43 42 43 43 43 42 43 40 41 42 43 44 8 9 10 20 30 40 50 45 46 47 48 49 9-46212 46242 46273 46304 46334 9-61075 61118 61161 61204 61247 9-48031 48061 48096 48120 48150 9-63651 -63701 -63744 -63787 -6383f 45 46 47 48 49 50 51 52 53 54 9-46365 -46396 -46426 -46457 46487 9-61291 61334 61377 61420 61463 9-48180 48210 48240 48270 -48300 9- 63878 -63915 63958 . 6400' - 64044 50 51 52 53 54 55 56 57 58 59 9-46518 46549 46579 46610 .46640 9-61506 -61550 -61593 -61636 .61679 9-48329 48359 48389 48419 -48449 9 - 64087 64130 64173 .64216 64258 55 56 57 58 59 60 9-46671 9-6172? 9.4847P 9 - 64301 60 ' Lg, Vers Log.Exs Lg.Vers 1} Log.Exs JJ ' P. P, 188 LOG. VERSED SINES AND EXTERNAL SECANTS 46 47 [Lg.Vers 1> "Isl 48508 .48538 4856* .48597 9-48627 Log.Exs 643U1 64344 . 64430 64473 64515 6455~ 6460 . 64644 6468 .64729 64772 64815 64858 .64901 64943 .64986 .65029 6507 .6511 ,65157 65200 .6524 6528 6532 65371 .65414 .65456 65499 .65542 .6558 .6562 .656 .6571 .6575 85 27 70 .65798 .65841 -65884 65926 .65969 .66012 .66054 .66097 66140 66182 66225 66268 66310 66353 .66396 -66438 66481 66523 66566 66609 9-66651 66737 66779 L 66822 9-66864 .Og.Exs P. P. Lg.Vers. - Log.Exs 9-5024 50272 50301 50330 50359 9-67075 -6712C 67162 67205 -67248 9-67290 6733 67375 67418 -6746C 9-67503 6754 6758 6763 -6767 9-5082 5085 50879 -50908 50937 .50965 50994 .51023 51052 .51080 9-6771 6775 6780 .6784 JI7888 9-67928 67971 68013 6805 9-51109 51138 5116 5119 .51224 9.51253 51281 5131 5133 .51367 9-68141 .68183 6822 6826 .68311 .51396 .51424 51453 51481 .51510 3-68566 -68608 .6865 .6869 .6873 10 20 3014-5 4019. 50;24- 9-5153 5156 5159 .5162 -51653 9-51681 -51710 51738 51767 -51795 69033 .69075 -69117 -69160 9-69202 69245 -69287 51823 51852 51880 51909 51937 9-51965 LOG. VERSED SINES 48 AND EXTERNAL 49 SECANTS f Lg.Vers, JD Log.Exs, J> Lg.Vers. D Log.Exs. D > P, P, 1 2 3 4 9-51965 51994 52022 52050 .52079 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 28 28 28 27 28 28 28 27 28 i; 28 27 28 27 28 27 28 ~D 9-69414 -69457 .69499 -69542 -69584 42 42 42 42 42 ti n ii 42 42 42 42 a ti 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 9-53648 53676 53704 53731 53759 27 28 27 27 28 27 27 27 28 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 2Z 27 27 27 27 27 27 27 27 27 27 27 % 11 27 27 27 27 27 27 27 -&" 9-71954 .71996 72038 -72081 -72123 42 42 42 42 42 42 42 42 42 42 42 42 11 il 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 42 4-2 4-9 5-6 6-4 ill 21-2 28-3 35-4 28 2-8 3-3 3-8 4-3 4-7 9-5 14-2 19-0 23-7 27 2-7 3-2 3-6 4-1 4-6 9-1 13-7 18-3 22.9 43 4.2 4-9 5-6 6-3 7-0 14-0 21-0 28-0 35-0 28 2-8 3-2 3-7 4-2 4-6 9-3 14.0 18-6 23-3 27 2-7 3-1 3-6 4-0 4.5 9-0 13-5 18-0 22-5 5 6 7 8 9 9-52107 52135 .52164 52192 .52220 9-69626 .69669 69711 69753 69796 9-53787 .53814 53842 53870 .53897 9-72165 72207 72250 72292 -72334 5 6 7 8 _9 10 11 12 13 14 10 11 12 13 14 15 16 17 18 19 9 52249 52277 .52305 52333 52362 9-69838 69881 69923 69965 - 70008 9-53925 53952 53980 54008 54035 9-72376 72419 72461 -72503 -72545 9-52390 52418 .52446 . 52474 .52503 9-70050 -70092 -70135 70177 .70220 9 54063 . 54096 54118 54145 -54173 9 - 54200 .54228 5425b 54283 .54310 9.54338 54365 .54393 . 54420 . 54448 9-72587 -72630 -72672 -72714 -72756 15 16 17 18 11 20 21 22 23 24 190 LOG. VERSED SINES AND EXTERNAL SECANTS 6O 51 t Lg.Vers D Log.Exs J> Lg.Vers J> Log.Exs 1> i P.P. 2 4 9-55292 -5531! . 55847 55374 55401 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 26 27 27 27 27 26 27 27 9 7448 74528 . 74570 .74612 74654 42 11 42 s 42 42 42 42 42 42 11 42 42 42 42 42 9-56900 .5692 .56953 5697 5700 26 26 26 26 26 26 21 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 II II 26 Ii 26 26 26 26 26 II 26 26 2R 26 26 26 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 9-77012 77055 .77097 77139 77181 42 42 42 42 42 11 42 .42 42 42 3 42 42 42 42 42 42 42 42 42 42 42 11 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 ( \ i 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 4 !4.2 4-8 5-6 6.4 7-1 14-1 21-2 28-3 35-4 27 2.7 3.2 3-6 4-1 4.6 9-1 13-7 18-3 22-9 23 2-6 3-1 3-5 4-0 4-4 8-8 13-2 17-6 22-1 , 2 6 2 7 3 8 3 9 3 10 4 20 8 3012 4017 5021 42 4.2 *.9 5.3 6.3 7-0 14.0 21.0 28.0 35.0 R 3-1 3-8 4-0 4-5 9-0 13.5 18-0 22.5 26 3-'0 3-4 ii 8.6 13-0 L7-.I 21.6 8 5 .0 '1 :l 5 6 8 9 9-55428 -55455 55482 55509 .55536 9 74696 74739 . 74781 . 74823 .74865 9.57032 .5705i .57085 5711' .57138 9 77223 77265 77307 .77349 77391 { r t I 10 11 12 13 ii 15 16 17 18 ii 20 21 22 23 2' 9-55563 .55590 .55617 55644 .55671 9-74907 7494J . 74991 7503J .75076 9.57164 57190 .57217 .57243 57269 9-77433 7747! .77517 .77560 .77602 1O 11 12 13 14 9.55698 .55725 -55751 55778 .55805 9-75118 .75160 .75202 .75243 .75286 9-57296 .57322 .57348 .57375 57401 9 77644 .77686 -77728 .77770 77812 15 16 17 18 19 9.55832 .55859 .55886 .55913 .55940 9-75328 .75370 75413 .75455 .75497 42 42 42 42 57427 57454 .57480 .57506 .57532 9.77854 .77896 .77938 .77980 .78022 20 21 22 23 24 25 26 27 28 29 9-55906 55993 56020 . 56047 . 56074 26 27 II 27 27 26 27 26 27 26 11 11 9-75539 .75581 .75623 .75665 .75707 42 42 42 42 42 57559 .57585 .57611 .57637 .57664 .78064 78107 .78149 .78191 .78233 25 26 27 28 29 30 31 32 33 34 9-56101 56127 56154 .56181 .56208 9-56234 .56261 56288 56315 56311 9-75750 .75792 .75834 .75876 75918 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 .57690 57716 .57742 57768 .57794 .78275 .78317 .78359 .78401 .78443 30 31 32 33 34 35 36 37 38 39 .75960 .76002 . 76044 .76086 .76128 .57821 .57847 .57873 .57899 m .58082 .58108 .58134 .58160 58186 .78485 78527 .78569 .78611 .78653 35 36 37 38 39. 40 41 42 43 44 45 46 47 48 49 40 41 42 43 44 45 46 47 48 60 51 52 53 54 9.56368 56395 .56421 . 56448 .56475 27 26 26 27 26 26 26 11 26 26 26 26 27 26 26 26 26 26 26 ^ 76171 .76213 .76255 .76297 .76339 .78696 .78738 .78780 .78822 .78864 9-56501 56528 56554 56581 58608 76381 .76423 .76465 .76507 7854S 78906 .78948 78990 .79032 79074 9-56634 56661 56687 56714 56741 76592 .76634 .76676 .76718 76760 58212 .58238 58264 58290 58316 7&116 .79158 79205 . 79242 .79285 42 42 42 42 42 42 42 42 42 42 42 50 51 52 53 54 55 56 57 58 59 9-56767 58794 56820 56847 56873 76802 .7684? .768<6 .76928 .76970 58342 58367 58393 584i9 58445 .79327 79369 79411 .79453 79495 55 56 57 58 59 60 9-56900 9-77012 . 58471 79537 60 Lg. Vers, Log.Exs. JJ g.Vers, JJ og.Exs, JJ 9 P.P. LOG. VERSED SINES AND EXTERNAL SECANTS 53 53 191 _u 1 2 3 4 Lg.Vers, 9-58471 58497 58523 .58549 58575 D Log.Exs, D Lg.Vers, J> Log.Exs, D f P.P. 26 25 26 26 26 25 26 26 25 9-79537 .79579 79621 79663 79705 42 42 42 42 42 42 n 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 II 42 42 42 42 42 42 42 42 42 42 42 42 42 42 9-60008 60034 .60059 . 60084 60110 25 25 25 25 25 25 25 25 25 9-82062 82104 .82146 82188 82230 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 il 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 il 42 42 42 42 42 42 42 42 42 42 il 42 42 42 42 1> 0, 1 2 3 _4 5 6 7 8 9 10 11 12 13 14 j 9 10 20 30 40 50 6 7 8 9 10 20 30 40 60 6 7 9 10 20 30 40 50 42 4-2" 4-9 5-6 6-4 7-1 14-1 21-2 28-3 35-4 26 2-6 3-0 3-4 3-9 4-3 8-6 13-0 17-3 21-6 ! 5 5 2-9 3-5 3-7 4-1 8-3 12-5 16-6 20-8 43 4 2 4-9 5-6 6-3 7-0 14-0 21-0 28-0 35-0 L 2-5 3-0 3.4 3 1 4-2 8-5 12-7 17-0 21.2 3~7 8J 12-2 16-3 20-4 5 6 7 8 9 9-58601 .58626 .58652 58678 .58704 9-79747 79789 79831 .79874 79916 9-60135 .60160 60185 60211 60236 9 . 60261 60286 60312 .60337 60362 9-82272 .82315 .82357 .82399 82441 10 11 12 13 14 15 16 17 18 il 20 21 22 23 24 g-ss-'so 58755 58781 .58807 58833 26 25 26 26 25 26 25 26 25 26 25 25 26 25 26 25 25 26 25 25 25 26 25 25 25 25 25 26 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 5 9-79958 80000 . 80042 80084 80126 2b 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 2fc 25 25 25 25 25 25 25 25 25 25 25 25 25 24 25 25 11 25 25 25 24 25 24 25 25 D 9-82483 82525 -82567 -82609 -82651 9-58859 58884 .58910 .58936 58962 9-80168 80210 80252 80294 .80S?6 9-80378 - 80420 -80463 -80505 -80547 9 60387 60412 60488 60463 - 60488 9-82694 .82736 .82778 82820 82862 15 16 17 18 .19. 20 21 22 23 24 9-58987 59013 59039 .59064 59090 9-60513 60538 -60563 -60589 -60614 9-82904 .82946 82988 .83031 .83073 25 26 27 28 29 9-59116 -59141 59167 .59193 -59218 9-80589 80631 .80673 80715 80757 9 - 60639 .6066- 6068 1 607i; - 60739 9-83115 83157 -83199 -8324] -83283 25 26 27 28 29 30 31 32 33 34 9-59244 -59270 -59295 -59321 .59346 9-80799 .80841 .80883 80925 .80968 9 -81010 .81052 .81094 81136 -81178 9-60764 6078J 6081; 6083J -60864 9-83325 -83368 83410 83452 -83494 30 31 32 33 34 35 36 37 38 39 9-59372 -59397 -59423 - 59449 -594V4 9 - 60889 -6091? 6093J 6096^ 60989 9-83536 -83571 83620 83663 -83705 35 36 137 38 J9 40 41 42 43 44 40 41 42 43 44 9-59500 .59525 59551 sgs^e 59602 9 -8122(5 .81262 .81304 .81346 -81388 9-61014 61035 610& 6108J 61114 9-83747 -83789 83831 83873 -83916 45 46 47 48 49 9-59627 59653 59678 59704 59729 9-81430 81473 -81515 81557 81599 9-61139 61164 6118S 61214 61239 9-83958 84000 8404| 8408-' -84.26 45 46 47 48 49 50 51 52 53 54 9-59754 5978C 59805 .59831 59856 9-81641 81683 8172E 81767 81809 9-61264 6128! 6131! 6133: 61363 9-84168 -84211 8425J 84295 84337 50 51 52 53 54 55 56 57 58 59 9-59881 59907 .59932 59958 59983 9-81851 81894 81936 .81978 82020 9-61388 61413 61438 .61462 61487 9-84379 84422 .84464 .81506 - 84548 55 56 57 58 59 9-60008 9-82062 9-81512 9-84f)90 60 Lg.Vers Log.Exs, JJ Lg.Vers. Log.Exs P. P. 192 LOG. VERSED SINES AND EXTERNAL SECANTS 54 55 1 "o" 1 2 3 4 Lg.Vers, 7> Log.Exs, J> Lg.Vers, D Log.Exs D / P,P, 9-61512 .61537 61562 .61586 .61611 24 25 24 25 24 25 24 25 24 24 24 25 24 25 M 24 25 24 it 11 24 24 24 24 24 24 2j 2< : 24 24 24 2< 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 ^ 9-84590 84632 84675 84717 84759 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 21 tl 42 42 42 42 42 42 42 42 42 42 42 42 42 42 i 42 42 42 42 tl tl 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 9-62984 63008 63032 63057 .63081 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 9-87125 87167 87209 87252 87294 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 4g 42 42 42 42 42 42 42 42 42 42 42 tl 42 - 1 2 3 _4 5 6 7 8 9 6 8 9 10 20 30 40 50 } 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 fl 4.9 5-6 6-4 7-1 14-1 21-2 28.3 35-4 25 2-5 2-9 3-3 3-7 4-1 8-3 12-5 16-6 20-8 24 2-4 2-8 3-2 3-6 4-0 8-0 12-0 16-0 20-0 a 4. 5.1 6., 7-( 14. 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( 2? 2.^ 2.i 3.; 3. 4. 8-; 12.: 16-! 20-4 2-1 s 2 :i 3- 3.) il 15.1 19. e i I t ) i p tj ! ! y, ; \ 5 6 7 8 9 10 11 12 13 11 15 16 17 18 19 9-61636 .61661 .61685 .61710 61735 9-8480! -84843 84886 84928 84970 9-63105 .63129 63154 .63178 63202 9-87336 87379 -87421 87463 87506 9-61760 61784 .61809 61834 61858 9-85012 85054 85097 .85139 .85181 9-63226 63250 63274 63209 63323 9-87548 -87590 .87633 87675 -87717 10 11 12 13 14 15 16 17 18 19 9-61883 .61908 .61932 .61957 -61982 9-85223 85265 85308 85350 9-85434 85476 85519 85561 85603 9-63347 63371 63395 63419 -63443 9-87760 87802 87844 87887 -87929 20 21 22 23 24 9-62006 -62031 -62055 -62080 -62105 9-63468 63492 63516 63540 63564 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 9-87971 88014 88056 88099 88141 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 9.62129 62154 62178 .62203 -62227 9.85645 85688 85730 85772 85814 9-85857 85899 .85941 85983 86026 9.63588 63612 63636 63660 63684 9-88183 88226 88268 -88310 88353 25 26 27 28 29 30 31 32 33 34 9-62252 -62276 -62301 -62325 -62350 9-63708 .63732 .63756 63780 63804 9-88395 88438 88480 88522 88565 35 36 37 38 39 9-62374 -62399 -62423 - 62448 -62472 9-86068 86110 86152 .86195 88237 9-86279 -86321 .86364 86406 9-86490 .86533 86575 86617 86659 9-63828 63852 63876 63900 63924 9-88607 88650 88692 88734 -88777 35 36 37 38 -39. 40 41 42 43 44 40 41 42 43 44 9-82497 62521 62546 62570 62594 9-63948 63972 63996 .64019 64043 24 24 23 24 24 24 24 23 24 24 23 24 24 23 24 24 23 23 24 9-88819 88862 88904 88947 -88989 45 46 47 48 49 9-62619 62643 .62668 62692 -62716 9-64067 64091 -64115 64139 -64163 9-89031 -89074 89116 -89159 -89201 45 46 47 48 49 50 51 52 53 54 55 56 57 58 5JL 60 9-62741 -62765 62789 62814 62838 9.86702 86744 86786 .86829 86871 9.86913 86956 .86998 .8704(5 87082 9-64187 64210 .64234 64258 .64282 9-89244 89286 -89329 89371 89414 50 51 52 53 54 55 56 57 58 59 60 9-62862 62887 -62911 -62935 -62960 9-64306 . 64330 643u3 64377 . 64401 9-89456 .89499 .89541 89583 -89626 9-62984 9-87125 [Log. Exs. 9.64425 9.89668 ' Lg.Vers. Lg.Vers, D Log.Exs, P, P, LOG. VERSED SINES AND EXTERNAL SECANTS 193 56 57 t k.Vers D Log.Exs D Lg.Vers Z> Log.Exs Z> ' P.P. 2 3 9-64425 . 64441 .64472 64496 64520 23 24 23 24 23 24 23 23 24 23 23 24 23 23 24 23 23 23 24 9-89668 .89711 89753 89796 .89838 1 42 42 42 42 42 42 43 42 42 42 42 42 42 42 42 42 43 42 43 42 42 42 42 43 42 9-65835 65859 .6588? .65905 .65928 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 22 23 23 23 22 23 23 22 23 23 23 9-92224 92267 .92310 .92353 92395 43 42 43 42 a ti 43 42 43 42 43 42 43 43 42 43 42 43 43 42 43 43 42 43 43 42 43 43 42 43 43 43 42 43 43 43 42 43 43 43 43 43 42 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 O ] r 4 3 4 43 4-3 5-0 5-7 6-4 7-1 14-3 21-5 28-6 35-8 24 2-4 2-8 3-2 3-6 4-0 8-0 12-0 16-0 20-0 23 2-3 2-7 li 3-8 7-6 11-5 15-3 19-1 42 4-2 4-9 5-6 6-4 7-1 14-1 21-2 28-3 35-4 3- '5 3-9 7-8 11-7 15-6 19.6 s 22 2-2 2-8 3-0 3-4 3-7 nil 15-0 18-7 5 6 7 8 9 9-64543 .64567 64591 64614 64638 9.89881 .89923 .89966 .90008 90051 9.65952 65975 65998 66021 . 66044 9-92438 92481 9252' .92561 92609 5 6 7 8 9 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 10 11 12 13 14 9-64662 64685 .64709 64733 .64756 9-90094 90136 90179 90221 .90264 9-66068 66091 66114 66137 66160 9-92652 92695 92737 92780 92823 10 11 12 13 14 15 16 17 18 19 9-64780 64804 64827 64851 -64875 9-90306 . 90349 90391 90434 90476 9-66183 .66207 .66230 .66253 66276 9.92866 .92909 9295! .92994 93037 15 16 17 18 -19 20 21 22 23 24 25 26 27 28 29 20 21 22 23 24 9-64898 64922 64945 -64969 -64992 23 23 23 23 24 23 23 23 23 23 23 23 23 23 23 23 23 23 9-90519 90561 90604 90647 -90689 9-66299 66322 .66345 .66368 66391 9-93080 93123 93165 93208 .93251 25 26 27 28 29 9-65016 65040 65063 65087 -65115 9-90732 90774 90817 90880 90902 9-66415 66438 66461 66484 66507 9-93294 93337 93380 93422 .93465 30 31 32 33 34 -65134 -65157 65181 .65204 65228 9-90945 90987 91030 91073 91115 9-66530 66553 66576 66599 66622 9-93508 93551 93594 93637 93680 30 31 32 33 34 35 36 37 38 39 65251 65275 65298 65321 65345 91158 91200 91243 91286 .91328 66645 66668 66691 66714 -66737 9-93722 93765 93808 93851 93894 35 36 37 38 -39. 40 41 42 43 44 40 41 42 43 44 9-65368 -65392 -65415 -65439 -65462 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 91371 91414 .91455 .91499 .91541 43 42 42 42 43 42 42 43 42 42 43 42 a 42 43 42 43 42 42 66760 66783 66805 66828 66851 9-93937 93980 94023 94066 94109 45 46 47 48 49 -65485 65509 65532 65556 -65579 -65602 -65626 -65649 65672 65696 9-91584 91627 .91669 91712 .91755 66874 66897 .6692(5 66943 .66966 9-94151 94194 94237 94286 94323 45 46 47 48 49 50 51 52 53 54 91797 .91840 .91883 9192B 91968 66989 67012 .67034 .67057 6708?) 67103 67126 67149 67171 67194 94366 .94400 94452 94495 94538 50 51 52 53 54 55 56 57 58 59 9-65719 65742 65765 65789 -65812 9-92011 92054 .92096 .92139 .92182 94581 94624 94667 94710 -94753 55 56 57 58 59 2 9.65835 P. 9222* -67217 Jl 94796 BO Lg, Vers. Log.Exs. D Ig.Vers. 1> og.Exs. JJ ' P. P. 194 LOG. VERSED SINES AND EXTERNAL SECANTS 58 59 ' [ Lg, Vers. D Log, Exs D Lg, Vers D Log. Exs, D ' P.P. 1 2 3 4 5 6 7 8 9 9-67217 .67240 .67263 .67285 .67308 9- 67331 67354 .67376 .67399 67422 i 23 i 23 oH 23 22 23 92 23 22 23 22 22 23 22" 22 23 22 22 23 22 22 23 22 22 22 22 il 22 22 22 22 22 22 22 22 22 22 22 9-94796 .94835 94882 94925 .94968 9-95011 .95054 .95097 .95140 .95183 n 43 43 43 43 43 43 43 43 43 43 43 43 43 43 1! 43 9.68571 68593 .68615 .68637 .68660 22 22" 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 9-97387 .97430 .97473 .97517 .97560 43 43 43 43 43 43 43 43 43 43 43 43 1! 43 43 43 II 43 43 43 43 43 43 43 43 1! 43 43 43 43 43 43 44 43 43 43 43 44 43 43 43 i 43 It If 44 43 43 It 44 1 2 3 4 e 7 8 S 10 2C 30 40 50 6 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 44 4-' 5." 5-i 6- 7. 14.1 22. ( 29.: 36- ( 6 7 8 9 10 201 302 402 503 23 2.3 2-7 3-0 3-4 3-8 7-6 11.5 15-3 19-1 22 2-2 2-5 2-9 3-3 3-6 7-3 11.0 14-6 18-3 18 i 6-5 i 7-2 i 14-5 )21-7 ;29.Q .36-2 43 4-3 5.0 5-7 6 i 44 1-5 8-6 2^6 3-0 34 3.7 7-5 11.2 15-Q 18-7 3t 5S. 2.5 2-8 3-2 3-B 7-1 10. z 14-3 i7a 9-68682 .68704 68727 .68749 .68771 9.97603 97647 97690 97734 97777 5 6 7 8 9 10 11 12 13 14 9.67445 67467 67490 .67513 67535 9-95226 .95269 .95313 .95356 .95399 9-68793 .68816 .68838 .68860 .68882 9-97820 .97864 .97907 .97951 97994 10 11 12 13 14 15 16 17 18 JL 20 21 22 23 24 25 26 27 28 _29_ 3 31 32 33 34 9-67558 67581 67603 .67626 .67649 J. 67671 67694 67717 67739 67762 9.67784 67807 67830 67852 .67875 9-67897 67920 67942 67965 67987 95442 .95485 .95528 .95571 .95614 95657 .95703 .95744 95787 .95830 95873 .95916 95959 .96002 96046 9-68905 .68927 .68949 .68971 .68993 9 .98038 .98081 .98125 .98168 -9821T 15 16 17 18 .12. 20 21 22 23 24 43 1! 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 8 It 9-69016 .69038 .69060 .69082 .69104 9 .98255 .98298 .98342 .98385 98429 9-69126 .69149 .69171 .69193 69215 9.98472 98516 .98559 .98603 .98647 25 26 27 28 29 96089 .96132 .96175 .96211 9626T 9.69237 .69259 6928T .69303 .69325 9.98690 .98734 .98777 .98821 98864 30 31 32 33 34 35 36 37 38 -39. 40 41 42 43 44 -68010 68032 68055 68077 -68100 -68122 68145 68167 68190 -68212 96305 .96348 .96391 .96434 .96478 9.69347 .69369 69392 .69414 69436 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 21 22 22 9.98908 .98952 98995 .99039 99082 35 36 37 38 _39_ 40 41 42 43 44 96521 .96564 .96607 .96650 .96694 9-69458 69480 .69502 69524 .69546 9-99126 .99170 .99213 .99257 -99300 45 46 47 48 49 9-68235 .68257 .68280 .68302 .68324 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 96737 .96780 96824 96867 .96910 11 43 43 43 43 43 43~ 1 43 9-69568 69590 .69612 .69634 69656 9-99344 -99388 9943T 99475 .99519 45 46 47 48 49 50 51 52 53 54 9-68347 .68369 .68392 68414 -68436 96953 .96997 97040 .97083 .97127 9-69673 .69700 .69721 -69743 -69765 9-99562 99606 99650 .99694 .99737 50 51 52 53 54 55 56 57 58 59 9-68459 -68481 .68503 .68526 68548 97170 97213 .97257 .97300 .97343 9-69787 -69809 .69831 .69853 .69875 22 22 I! 22 22 9-99781 .99825 -99868 .99912 9-99956 55 56 57 58 59 ftO 60 9 68571 97387 .69897 o.ooooo Lg, Vers, D og, Exs, D g. Vers, D Log, Exs. D P.P. LOG. VERSED SINES AND EXTERNAL SECANTS 60 61 195 ' Lg, Vers, D Log. Exs. D Lg, Vers. D log, Exs, D ' P.P. 1 2 3 4 9-69897 .69919 - 69940 69962 .69984 If 22 22 22 21 22 22 21 22 21 22 22 21 i 21 22 2T 22 21 22 21 2T 22 21 22 2T 21 22 21 21 21 22 21 21 21 22 2T 21 21 21 21 22 2 2'\: 2 2 21 2".'. 2.: 2:'. 2 2~ 2". 2". 2 2". 21 10-00000 . 00044 00087 00131 -00175 a 44 43 44 44 43 44 44 43 44 44 44 44 44 43 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 a 44 44 a 44 44 a 44 44 44 44 44 9-71197 .71218 .71239 .71261 .71282 21 21 21 21 21 21 l\ 21 21 21 21 21 21 21 21 21 21 21 21 21 2T 21 2T 21 21 21 21 a a 21 21 21 a 21 a 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 10-02639 .02684 .02728 .02772 .02816 44 It 44 44 44 44 44 44 11 3 44 44 44 44 44 44 44 44 I i 44 44 45 44 44 44 44 44 44 45 44 44 44 45 44 44 44 45 44 45 44 44 45 44 a a 45 O 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 45 4-5 5-2 6-0 6-7 7-5 15-0 22-5 30-0 37-5 44 5^8 6-6 7-3 14-6 22-0 29- 36-6 22 2-2 2-5 2-9 3-3 3-6 7-3 11-0 14-6 18-3 ! 6 ! 7 ! 8 : 9 ; 10 5 20 ' 30 1( 40 1< 50 r 5-2 5-9 6-7 7-4 14-8 22-2 29-6 37-1 a 5-1 5-8 6-5 7-2 14-5 21-7 29- Q 36.2 a 1:1 3-2 7-1 10-7 14-3 17. b 1:1 i.5 K5 L.O M 5 6 7 8 9 9-70006 70028 .70050 70072 .70093 9.70115 .70137 70159 70181 70202 10-00219 00262 .00306 -00350 -00394 9-71304 .71325 .71346 71368 -71389 10-02861 .02905 .02949 .02994 -03038 5 6 7 8 9 10 11 12 13 14 10 11 12 13 14 10-00438 00482 .00525 .00569 .00613 9-71411 -71432 -71453 -71475 - 71495 10-03082 03127 -.03171 .03215 .03260 15 16 17 18 -19- 20 21 22 23 24 9.70224 70246 70268 -70289 9.70333 .70355 .70376 .70398 70420 10-00657 .00701 00745 00789 -00833 9-71517 -71539 -71560 -71581 71603 10-03304 .03348 .03393 .03437 .03481 15 16 17 18 .12. 20 21 22 23 24 10-00876 .00920 00964 .01008 .01052 9-71624 71645 -71667 -71688 -71709 10-03526 03570 .03615 .03659 03704 25 26 27 28 29 9 . 7044T .70463 70485 70507 .70528 10-01096 01140 .01184 .01228 -01272 9.71730" 71752 71773 .71794 71815 10-03748 03793 03837 .03881 .03926 25 26 27 28 29 30 31 32 33 34 9-70550 70572 70593 70615 70636 10-01316 -01360 01404 01448 01492 9.71837 71858 71879 71900 71922 10-03970 .04015 .04059 .04104 .04149 30 31 32 33 34 35 36 37 38 "40" 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 9-70658 70680 70701 70723 -70745 9-70766 -70788 .70809 70831 -70852 10-01536 -01580 .01624 01668 10-01756 .01800 .01844 01889 .01933 9.71943 71964 71985 72006 -72028 10-04193 .04238 04282 .04327 04371 35 36 37 38 19. 40 41 42 43 44 9-72049 72070 72091 .72112 .72133 10-04416 .04461 -04505 -04550 -04594 9-70874 70896 70917 .70939 70960 10-01977 .02021 -02065 -02109 -02153 9-72154 72176 72197 72218 .72239 10-04639 -04684 .04728 04773 .04813 45 46 47 48 49 9-70982 .71003 .71025 71045 71068 10-02197 .02242 .02286 .02330 -02374 9.72260 72281 .72302 72323 .72344 10.04862 04907 .04952 .04996 .05041 50 51 52 53 54 55 56 57 58 59 9-71089 71111 .71132 .71154 .71175 10-02418 .02463 .02507 .02551 02595 9.72365 .72386 .72408 .72429 72450 9-72471 10-05086 .05131 .05175 .05220 .05265 60 9.71197 10-02639 10.05310 Log, Exs, 60 ' Lg, Vers, D Log, Exs. D Lg, Vers. D D ' P. P. 196 LOG. VERSED SINES AND EXTERNAL SECANTS 62 63 t Lg.Vers, J> Log, Exs. D Lg.Vers. & Log, Exs. JD < P . P. 1 9.72471 .72492 21 10.05310 .05354 44 9-73720 -73740 20 9n 10-08015 .08061 45 1 2 .72513 91 .05399 73761 .08106 2 3 4 .72534 .72555 21 .05444 05489 44 .73782 .73802 20 08151 .08197 45 3 4 4 6_ 46 5 9-72576 21 91 10-05534 9-73823 20 10.08242 45 5 6 7 t g /t 4-0 K 6 .72597 .72618 21 05579 05623 44 .73843 .73864 20 .08288 .08333 45 6 7 8 g 6 7 -2 6-1 6 9 8 9 .72639 -72660 21 .05668 -05713 45 .73884 -73905 21 08379 -08424 45 8 9 10 20 7 1 ^ 7 7-6 15-3 10 9-72681 *4 10-05758 9-73926 2(J 10-08470 4b 10 30 23 2 23-0 11 .72701 o-j -05803 73946 2Q -08515 11 40 31 -0 30-6 12 .72722 -05848 7396-7 oH 08561 1?, 50 38 . 7 38-3 13 .72743 n-i .05893 73987 20 -08606 13 14 -72764 05938 74008 _ -08652 14 15 16 9-72785 72806 21 21 10-05983 .06028 45 4! 9 - 74028 . 74049 20 10-08697 -08743 4b a 15 16 6 4 4 % 45 4-5 17 18 72827 72848 20 06072 .06117 45 74069 74090 20 08789 08834 45 17 18 7 8 5 6 '$ 5-2 6-Q 19 -72869 -06162 -74110 -08880 19 9 b 8 6-7 20 21 22 23 9-72890 72911 .72931 72952 72973 21 21 20 21 21 10-06207 .06252 .06297 .06342 .06387 4b 45 45 45 45 9.74131 .74151 .74172 .74192 .74213 '20 20 20 20 20 10-08926 0897! 09017 .09062 -09108 46 45 45 45 46 20 21 22 23 24 20 30 40 50 M 22-7 30-3 37-9 15-0 22-5 30-0 37-5 25 9-72994 $ 10-06432 45 AK 9-74233 20 nf) 10-09154 45 40 25 26 97 28 29 -73015 -73036 -73057 -73077 21 2 J .06477 .06522 .06568 -06613 45 45 45 .74254 74274 . 74294 -74315 20 20 20 -09200 -09245 -09291 -09337 45 45 46 26 27 28 29 6 7 8 4-4 5-2 5-9 30 31 32 9-73098 -73119 -73140 21 i 10-06658 .06703 .06748 45 45 45 9-74335 -74356 - 74376 20 20 10-09382 -09428 -09474 4b 46 30 31 3?, 9 10 20 6-7 7-4 14-8 33 34 -73161 -73181 20 21 91 .06793 .06838 45 45 45 -74396 -74417 20 on -09520 -09566 46 45 4fi 33 34 30 40 50 22-2 29-6 37-1 35 9-73202 10-06883 9 - 74437 10-09011 35 36 73223 on -06928 AC 74458 .09657 36 37 38 39 -73244 -73265 -73285 21 20 21 nS -06974 .07019 -07064 45 45 45 . 74478 .74498 -74519 y 20 20 20 09703 .09749 -09795 45 46 46 4c 37 38 39 6 7 8 21 2-1 2-4 2-8 20 2-0 2-4 2-7 40 9-73306 10-07109 9-74539 10-09841 40 41 .73327 .07154 -74559 4(J -09886 41 9 3 3-1 42 - 73348 21 .07200 A K -74580 (j 27) -09932 4?, 10 3 .5 3-4 43 44 -73368 -73389 21 .07245 .07290 45 A 5 -74600 -74620 20 .09978 .10024 *0 - HI " 20 30 7 10 5 6-8 10-2 45 46 47 9-73410 7343Q -73451 Q 20 21 90 10-07335 -07380 .07426 45 45 45 4C 9-74641 -7466! -7468! 1 10-10070 -10116 -10162 46 46 d! 45 46 47 40 50 14 17 .0 5 13-6 17-1 48 49 -73472 -73493 21 .07471 .07516 45 -74702 -74722 20 -10208 -10254 46 48 49 20 50 9-73513 ^U 10-07562 4b 4C 9 - 74742 20 10-10300 46 50 6 7 2-0 2-3 bl 52 -73534 -73555 21 07607 .07652 45 -74762 -74783 20 -10346 -10392 46 51 5?, 8 g 2-6 3-0 b3 54 .73575 73596 20 .07697 07743 45 -74803 - 74823 20 .10438 -10484 46 53 54 10 20 3-3 6-6 55 56 57 58 59 9-73617 .73637 .73658 73679 -73699 1 10-07788 07834 07879 .07924 07970 4b 45 45 45 45 9 - 74844 -74864 .7488^ . 7490^ - 7492^ 20 % 20 20 10-10530 -10576 .10622 .10668 10714 46 46 46 46 46 55 56 57 58 59 30 40 50 10-0 13-3 16-6 9-73720 20 10-08015 4b -D 9.74945 UU ~ 10-10760 46 60. Lg.Vers, JJ Log. Exs. Lg.Vers, Log. Exs, P , P. LOG. VERSED SINES AND EXTERNAL SECANTS 197 64 65 / Lg.Vers, it Log.Exs, J> Lg.Vers, J> Log.Exs. D P.P. o 1 2 3 4 9 74945 74965 74985 -75005 75026 20 20 20 20 20 20 20 20 20 20 20 20 11 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 19 20 20 20 20 20 19 20 20 20 19 20 20 20 19 20 20 10-10760 -10807 10853 .10899 .10945 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 9-76146 76166 76186 76206 76225 19 20 20 19 20 19 20 20 a ?i 20 19 20 19 19 20 19 20 19 19 20 19 19 20 19 19 19 20 19 19 20 19 19 19 19 19 19 20 19 19 19 19 19 19 19 19 19 19 10-13551 13598 13645 .13692 13739 47 47 47 47 47 47 47 47 47 47 47 47 8 47 47 8 47 47 47 8 47 8 47 47 47 47 47 47 47 47 1 II 11 47 47 47 I 48 1 2 3 _i 5 6 7 8 9 6 7 8 9 10 2C 3C 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 48 4.8 5-6 6-4 7-2 8-0 16-0 24- C 32-0 40-0 47 4-7 5-5 6-2 7-0 7-8 15-6 23-5 31-3 39-1 4 6 t 7i e 8 f 9 ( 10 ' 20 1J 30 2: 40 3( 50 3i 2-0 2.4 2.7 3-1 3-4 6-8 10-2 13-6 17-1 ] 6 ] 7 I 8 : 9 S 10 J 20 e 30 < 40 11 50 1( 47 4-7 5-5 63 7-1 79 15 23 7 31.8 39-6 6*4 6-2 ?: 15 * 23 2 31-0 38.7 16 t-8 3 i-1 9 r.g > 3 10 ) 6 5-3 20 2-0 i'i 3-0 3-3 66 10-0 13-3 16. G L9. L-9 5-3 5-6 !: J 1:8 5 6 7 8 9 10 11 12 13 14 9-75046 .75066 .75086 -75106 .75126 10-10991 11037 .11084 11130 -11176 9 - 76245 76265 76285 76304 -76324 10-13786 13833 13880 13927 13974 9-75147 75167 .75187 75207 75227 10-11222 11269 11315 11361 -11407 9 - 76344 76364 76384 76403 -76423 10-14021 14068 -14115 14162 -14210 10 11 12 13 14 15 16 17 18 19 9 75247 .75267 75287 75308 -75328 10-11454 11500 11546 11593 -11639 9 - 76443 76463 76482 .76502 -76522 10-14257 -14304 14351 14398 - 14445 15 16 17 18 19 20 21 22 23 24 9-75348 75368 .75388 .75408 -75428 10-11685 -11732 .11778 11825 -11871 46 46 46 46 9-76541 76561 76581 -76600 -76620 10-14493 14540 14587 .14634 .14682 20 21 22 23 24 25 26 27 28 29 9-75448 .75468 -75488 -75508 -75528 10-11917 11964 .12010 .12057 .12103 4b 46 46 46 46 46 46 46 46 46 9-76640 .76659 76679 -76699 -76718 10-14729 14776 14823 14871 14918 25 26 27 28 .21 30 31 32 33 34 35 36 37 38 39 30 31 32 33 34 9-75548 -75568 -75588 -75808 -75628 10-12150 12196 -12243 .12289 -12336 9-76738 76758 76777 .76797 -76817 10.14965 .15013 .15060 .15108 .15155 35 36 37 38 39 40 41 42 43 44 9-75648 -75668 -75688 -75708 -75728 10-12383 .12429 .12476 -12522 -12569 L 46 46 46 46 47 46 46 47 46 46 H II 47 47 46 47 46 47 47 47 46 47 47 9-76836 .76856 .76875 .76895 -76915 10.15202 .15250 15297 15345 -15392 9-75748 -75768 .75788 -75808 -75828 10-12616 .12662 12709 12756 -12802 9-76934 -76954 -76973 76993 -77012 10-15440 15487 .15535 15582 .15630 40 41 42 43 44 45 46 47 48 49 9-75848 -75868 .75888 .75908 -75928 10-12849 -12896 -12942 -12989 -13036 9-77032 77052 77071 77091 77115 9-77130 .77149 .77169 .77188 .77208 10-15678 15725 15773 .15820 -15868 45 46 47 48 49 50 51 52 53 54 50 51 52 53 54 55 56 57 58 59 j< 9-75947 75967 .75987 -76007 -76027 10-13083 .13130 .13176 .13223 .13270 10.15916 15963 .16011 .16059 16106 8 8 47 8 48 8 48 9 76047 -76067 -76087 -76106 -76126 10-13317 .13364 13411 -13457 -13504 9-77227 .77247 .77266 77286 77305 10-16154 16202 16250 .16298 .16345 55 56 57 58 59 9-76146 10.13551 9-77325 10-16393 60 Lg. Vers /> Log.Exs. J> Lg, Vers, 1> Log, Exs, X> ' P. P, 198 LOG. VERSED SINES AND EXTERNAL SECANTS 66 67 t Lg.Vers. J> Log.Exs, 1> Lg.Vers. J> Log.Exs, & i P.P. o 1 9-77325 77344 19 10. 16393 .16441 if 9-78481 78500 19 10-19293 19342 49 1 a -77363 .16489 78519 .19391 Aft 2 3 4 .77383 .77402 19 19 .16537 .16585 10-16633 .16680 48 AQ .78538 78557 19 19 19 19439 19488 49 49 49 AQ 3 ~5 6 6 8 53 5-0 5-8 6-6 49 4.9 5.8 6-6 ti 6 9-77422 . 77441 9-78576 78595 10-19537 .19586 7 .77461 4 o .16728 78614 .19635 7 9 7-5 7-4 8 .77480 t .16776 .78633 .19684 8 10 8.3 8-1 9 .77499 19 . 16824 -78652 19733 9 20 lfi.fi 16-5 10 11 9.77519 .77538 .77557 19 19 19 10.16872 16920 .16968 48 48 48 9-78671 .78690 .78709 19 19 19 10-19782 .19831 .19880 49 49 49 10 11 12 30 40 50 25.0 33-3 41-6 24-7 33-0 41-2 13 14 77577 .77596 19 .17016 .17064 48 .78728 78747 19 19 .19929 .19979 49 13 14 15 16 17 18 19 9-77616 .77635 .77654 77674 .77693 19 19 19 19 19 10-17112 .17160 .17209 .17257 .17305 48 48 48 48 48 9-78766 .78785 78804 .78823 .78842 19 19 19 19 19 10-20028 20077 .20126 20175 .20224 49 49 49 49 49 15 16 17 18 19 6 7 8 9 10 49 4-9 y 8 T 7^3 8 i 20 21 9-77712 77732 19 19 10-17353 .17401 48 48 AQ 9-78861 78880 19 19 10-20273 .20323 49 49 20 21 20 30 16-i 24.5 16-1 24-2 22 7775] .17449 AQ 78899 .20372 AQ 22 40 32 6 32-3 23 24 77770 .77790 19 .17498 .17546 48 .78918 78937 18 .2042! .20470 49 23 24 5040.8 40.4 25 26 9.77809 77828 19 19 10-17594 .17642 48 48 9-78956 78975 19 19 10-20520 .20569 4y 49 25 26 48 47 27 77847 .17690 .78994 .20618 27 6 4-8 ! 28 29 .77867 .77886 19 17739 .17787 48 .79013 79032 19 .20668 .20717 49 28 29 7 8 5-6 6-4 5.5 6-3 30 9-77905 19 1 Q 10-17835 48 9-79051 19 1 5 10-20767 30 9 7-2 7-1 31. 32 33 34 .77925 77944 77963 77982 19 19 19 19 .17884 .17932 .17980 .18029 48 48 48 .79069 .79088 .79107 .79126 19 19 19 .20816 .20865 .20915 .20964 49 49 49 49 40 31 32 33 34 20 30 40 50 16-0 24-0 32.0 40-0 15.8 23-7 31-6 39-6 35 9-78002 10.18077 9.79145 10-21014 85 36 .78021 1 Q .18126 A! .79164 .21063 36 37 . 78040 .18174 Aft .79183 .21113 37 19 19 38 78059 I Q .18222 Q .79202 19 .21162 en 38 6 1.5 1.9 5 78078 -18271 79220 .21212 39 7 2-3 2-2 40 9-78098 19 10-18319 48 40 9-79239 19 10-21262 49 40 8 2-6 2-5 41 78117 1 Q .18368 40 .79258 19 1 .21311 41 9 2.9 2-8 42 43 44 .78136 .78155 .78174 19 19 .18416 .18465 .18514 49 48 .79277 79296 79315 19 19 .21361 .21410 .21460 49 50 42 43 44 10 20 30 3-2 6-5 9.7 3.1 6-3 9.5 45 46 9-78194 .78213 19 19 10.18562 .18611 48 48 AR 9.79333 .79352 18 19 10-21510 .21560 49 50 AQ 45 46 40 50 13-0 16-2 12.fi 15.8 47 .78232 .18659 A8 .79371 .21609 47 48 49 78251 78270 19 .18708 .18757 49 .79390 .79409 18 19 .21659 21709 49 48 49 IS ft l 5 50 51 52 9-78289 78309 78328 19 19 19 1 Q 10-18805 .18854 .18903 48 48 49 AQ 9.79427 .79446 .79465 18 19 19 10-21759 .21808 .21858 bU 49 50 50 51 52 '/ 2.1 8 2-4 9 2-8 53 78347 in .18951 .79484 .21908 oy 58 10 3-1 54 .78366 .19000 79503 .21958 54 20 64 55 56 57 9-78385 78404 78423 19 19 19 1 Q 10.19049 .19098 .19146 48 49 48 AQ 9.79521 .79540 .79559 18 18 10-22008 22058 .22108 bU 50 50 55 56 57 30 9-2 40 12-3 5015.4 58 . 78442 \l .19195 .79578 .22158 58 59 78462 .19244 .79596 22208 50 59 60 9-78481 19 D 10-19293 48 9-79R15 19 10.^22258 bO 60 Lg, Vers, Log.Exs. Lg.Vers. Log.Exs. ** * P. P. LOG. VERSED SINES AND EXTERNAL SECANTS 68 69 199 ' Lg.Vers. l> 18 18 18 Log.Exs, - Lg.Vers. 10-22258 .22308 .22358 .22408 22458 10-22508 .2255 .2260 .2265 .2270 10-22759 .22809 .22859 22909 -22960 LO-2301 .230 .2311 .2316 .2321 10.2326: .2331! .2336! .2341! -2346; 0-2351' .2356; .2361! .23661 237li 10-2376^ .2381: .2386! .2391! -2396! 10 - 24020 .24071 .2412 .2417 .2422 10-2427 .24325 .24376 .24427 .24478 10-24529 .24580 .24631 .24682 .24733 10 - 24784 .24835 .24886 .24988 10- ^5039 .25090 .25142 .25193 .25244 10.25295 ~ 35295 .Exs, i. 80728 80747 .80765 80783 -80802 -80820 80839 80857 -80875 9-80912 .80930 9-81003 .81022 .81040 .81058 81077 9-81095 .81113 .81131 .81150 -81168 9-81186 . 81204 .81223 .81241 81259 t.8127 .81295 .81314 9.81361 81381 .81405 .81423 9 -8145 8147 8149 8151 81532 81550 .81568 .81586 .81640 .81658 .81676 .81695 .81713 9-81731 .81749 .81767 .81785 -81803 r.Vers. Log, Exs. ,1> 10-25295 -25347 -25398 .25449 .25501 10".25552 -25604 -25655 25707 25758 10-2581 2586 25913 -25964 2S016 10-2606 -2611 2617 2622 -2627 10-22627 .27680 .27732 -27785 .27837 10-2789 .2794 .27995 .28047 .28100 10-28152 28205 .28258 -28310 .28363 0-28416 Log, Exs, I> P.P. 53 5-3 17.1 10 20 3026-5 40 35-3 60)44. 1 52 3C26. 4034. 5043- 4C 51 5-1 5-9 17-0 8025-525 34 033 52 5-2 6-1 7-0 7-9 8-7 17-5 26-2 7-7 86 17-1 25-7 34-3 42-9 50 5-B 5-9 6-7 7-6 5042-542-1 50 5-0 16- 3025-0 4033-3 5041-6 19 1! 1-9 2-5 2-5 3-1 6-3 4012-i 5015-i 12-5 15-4 18 18 2.1 2-4 27 3-0 6-0 40 12-0 50 IB P, P, 200 LOG. VERSED SINES AND EXTERNAL SECANTS 70 71' ' Lg.Vers. D Log. Exs. n Lg.Vers. B Log, Exs, D r P , P, 2 3 4 9-81821 .81839 81857 .81875 -81893 18 18 18 18 18 18 18 18 18 18 18 18 18 18 if 18 18 18 18 18 18 11 18 18 18 if 18 18 18 17 18 18 if 10-28416 .28469 28521 .28574 .28627 53 53 52 53 53 53 53 53 53 53 53 53 53 53 II 53 53 53 53 53 53 53 53 53 53 53 11 it II 54 54 54 54 54 54 54 54 54 i! 54 54 55 1 54 54 9-82894 .82911 '.829^9 -82947 -82964 il if 18 i! 17 18 17 17 17 18 17 18 17 17 17 17 18 17 17 17 17 17 18 17 17 17 17 17 17 17 17 17 17 17 17 i 17 17 17 17 17 17 17 17 17 D 10-31629 .31684 .31738 31793 -31847 5< 5] 5^ 5' 5; il 54 il il 55 55 55 55 55 55 55 55 55 11 55 | 55 55 55 56 55 55 56 55 56 56 55 56 56 56 56 56 56 56 56 56 56 56 56 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 1 6 1 7 2 8 2 9 2 10 3 20 6 30 9 4012 5015 fl 5 6 7 8 9 18 28 37 47 r> 6 7 8 9 18 27 37 46 5 5 6 7 8 9 18 27 36 45 5 5 6 7 8 8 17 26 35 44 7 8 9 10 20 30 40 50 8 .8 1 4 -7 -0 -0 -0 -0 1 ^5 4 'l ^4 '1 2 4 3_ 'o 6 1 F 1 \ \ r 21 81 4c 1 1 2 2 2 2 5 8 11 14 56 5-6 6-5 7-4 8-4 9-3 18- 6 28-0 37-3 46-6 55 5-5 6-- 7-i 8-i 9. is.; 27-5 36- 45-8 54 5-4 6-3 7-2 8-1 9-0 18-0 27-0 36-0 45-0 53 5-3 6-2 7-5 8.8 17-6 26-5 35-3 44-1 "i -0 ':? :! -0 -7 7 17 -7 1-7 .0 2-0 000 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ii 20 21 22 23 24 25 26 27 28 29 30 31 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 9-81911 8192! .81947 .81965 -81983 9-82001 -82019 82037 82055 82073 10-28680 .28733 .28786 .28839 . 28892 10-28945 .28998 .29051 .29104 29157 9-82982 8300C .83017 83035 _. 83053 9-83070 .83088 .83106 .83123 .-83141 9-83159 .83176 83194 83211 -83229 10-31902 31956 .32011 .32066 10-32175 32230 .32284 .32339 32394 5 6 7 8 9 10 11 13 14 15 16 17 18 19 9-82091 82109 .82127 82145 82163 10-29215 29263 29316 29370 -29423 10 - 32449 .32504 32558 32613 -32668 9-82181 -82199 82217 82235 -82252 10-29476 .29529 .29583 .29636 29689 9-83247 83264 83282 .83299 .83317 10-32723 .32778 .32833 .32888 -32944 30 21 22 23 24 9-82270 82288 82306 -82324 -82342 10-29743 29796 .29850 .29903 29957 9.83335 83352 .83370 .83387 83405 10-32999 33054 .33109 .33164 -33220 25 26 27 28 29 9-82360 -82378 -82391) 8241J! -82431 10-30010 .30064 .30117 30171 .30225 9 83422 .83440 .83458 .83475 .83493 10-33275 .33330 .33385 .33441 .33496 30 31 32 33 34 9 - 82449 -82467 -82485 82503 -82520 10-30278 30332 .30386 .30440 .30493 9-83510 .83528 .83545 83563 -83580 10.33552 33607 33663 .33718 -33774 35 36 37 38 -39. 40 41 42 43 .44 45 46 47 48 49 9-82538 82556 82574 82592 .82609 18 if 18 18 17 18 17 18 17 18 17 18 17 18 17 18 ^ 10-30547 .30601 .30655 .30709 .30763 9-83598 83615 83633 83650 .83668 10-33829 .33885 33941 -33996 -34052 9-82627 -82645 82663 -82681 -82698 10-30817 .30871 30925 30979 31033 9-83685 83703 .83720 83737 83755 10-34108 34164 34220 34275 3433! 50 51 52 53 54 55 56 57 58 59 9-82716 -82734 -82752 -82769 -82787 10-31087 31141 .31195 .3124? .31303 9-83772 83790 83807 83825 83842 10-34387 - 34443 -34495 .34555 .34611 50 51 52 53 54 55 56 57 58 59 9-82805 -82823 -82849 -82858 -82876 10-31358 31412 .31466 31521 .31575 9-83859 .83877 83894 83912 83929 10-34667 34723 .34780 .34836 .34892 -6 2-5 ! 5J -7 8-5 .611-J .6 14-1 go 9-82894 Lg.Vers. 10-31629 9-83946 10-34948 60 Log. Exs, D Lg.Vers Log, Exs, ' P, P, LOG. VERSED SINES AND EXTERNAL SECANTS 73 73 201 1 Lg.Vers It Log.Exs. D Lg.Vers D Log. Exs. D ' P. P. 1 9-83946 .83964 10-34948 .35005 56 9-84980 .84997 17 10-38387 38445 58 1 6 61 6-1 60 6-0 2 8 .83981 .83999 17 17 35061 .35117 56 .85014 85031 17 1 7 .38504 .38562 58 2 3 7 8 7. 8- B'.Q 4 840ie .8504$ -38621 4 9 9.' 9-1 B 9-84033 12 10-35230 bb 9-85066 IV 10-38679 b8 5 10 10- 10-1 6 .84051 35286 -85083 3873J 8 6 20 20. 20-1 7 .84088 17 .35343 56 .85100 17 .38796 58 7 30 30- b 30-2 8 9 .84085 .84103 17 .35399 35456 57 -85117 -85134 17 .38855 .38914 59 58 8 9 40 50 40- b 50-8 40 3 50-4 K 11 9-84120 84137 IV iz 10-35513 .35569 bb 56 9-85151 .85168 IV 17 10-38973 39031 fa9 58 10 11 60 59 12 13 14 .84155 .84172 .84189 17 17 .35626 .35685 .35739 57 '56 85185 -85202 -85219 17 17 .39090 .39149 39208 59 59 58 12 13 14 7 8 p 7-0 8-0 90 6-9 7-9 15 16 17 18 19 9-84207 . 84224 . 84241 .84259 .84276 17 10-35796 .35853 .35910 .35967 .36023 bV 56 57 il 9-85236 -85253 -85270 -85287 .85304 IV 17 17 17 17 10-39267 39326 39385 39444 -3&5G3 59 59 59 59 59 15 16 17 18 19 10 20 30 40 50 10-0 20-0 30-0 40-0 50-0 9-9 19-8 29-7 39-6 49-6 20 21 9-84293 .84310 IV 17 10-36080 36137 bV 57 9-85321 85338 IV 17 10.3S562 -89621 59 59 20 21 59 58 22 . 84328 |Z .36194 57 85355 1 7 39681 b M 6 b-9 5-8 23 .84345 36253 85372 39740 28 7 6.9 6-8 24 .84362 3630J .85389 .39799 24 8 V-8 7-8 25 26 27 28 29 9-84380 .84397 .84414 . 84431 . 84449 IV 17 17 10-36366 36423 36480 36537 36594 bV 57 57 57 57 9 - 85405 8542J -8543< -85456 -85473 16 17 17 17 17 10-39859 39918 39977 4CG37 -40C96 59 59 59 59 59 25 26 27 28 9 10 20 30 40 en 8-8 9-8 19-6 29-5 39-3 8-8 9-7 19-5 29-2 39-0 30 9-84466 17 10-36652 bV 9-85490 IV 10-40156 59 80 81 32 33 34 84483 84500 -84517 -84535 II II 36709 36766 36824 -36881 57 57 57 85507 .85524 85541 .85558 II 17 17 40216 -4C275 -4C335 -40395 60 59 59 60 31 32 33 34 6 7 8 58 5.8 6-7 7-7 57 5-7 6-7 7-6 35 9-84552 IV 1 7 10-36938 5Z 9-85575 IV 10-40454 59 35 9 8-7 8-6 36 .84569 36996 57 85592 i 40514 ?c 36 10 9-b 9-6 37 38 39 .84586 .84603 .84620 17 17 17 37054 37111 .37169 57 57 57 K7 .85608 8562J -85642 17 17 17 40574 40634 40694 b9 60 60 60 37 38 40~ 20 30 40 50 19-3 29-0 38-6 48-3 19-1 28-7 38-3 47-9 40 9-84838 10-37226 9-85659 10-40754 41 42 43 44 84655 84672 -84689 -84706 17 17 17 37284 37342 -37399 -37457 58 57 58 85676 .85693 85710 -85726 17 11 .40814 40874 40934 40994 60 60 60 60 41 42 43 44 6 7 57 5-7 6-6 7 B 56 5-6 66 7 5 45 46 9-84724 84741 IV 17 10-37515 37573 b8 57 9-85743 85760 IV il 10-41054 41114 60 60 45 46 9 10 Li 9.5 8-5 9-4 47 84758 1 7 37631 CO 85777 1 7 41174 60 47 20 19-0 18-8 48 84775 37689 85794 .41235 c 48 30 28-5 28-2 49 -84792 17 1 7 37747 58 CO .85811 16 1 7 41295 10-41355 60 fin 49 40 38 - C 50147.5 37-6 47-1 50 9-84809 10-37805 9-85827 50 bi 52 53 54 -84828 84844 84861 84878 1-7 17 17 37863 37921 37979 38037 58 58 58 85844 .85861 85878 .85895 il 17 41416 41476 41537 .41597 60 60 60 51 52 53 54 17 17 16 6 1-7 1-7 1-6 7 2-0 2-0 1-9 55 9-84895 17 10-38095 58 KO 9-85911 Ib ', 7 10-41658 60 55 9 2-6 2-5 25 bb 84912 17^ 38153 rP .85928 i 7 41719 1 56 10 29 28 27 b 1 / 58 59 60 84929 -84946 -84963 17 17 17 3821P 38270 -38328 58 58 58 85945 .85962 -85979 s 17 16 41779 41840 41901 60 60 61 61 57 58 59 20 5-8 5-6 5-5 30 8-7 85 8-2 4011-6 11 3 11-0 FP 14.6 14- T '3-7 9-84980 10-3S387 85995 10-41962 60 Lg. Vers. ^ Log.Exs. 5 Lg.Vers. D Log. Exs. Z> ' P. P. 202 LOG. VERSED SINES AND EXTERNAL SECANTS 74 75 / m 2 3 4 g.Vers, .85995 .86012 .86029 .86046 .86062 D Log, Exs, J> Lg.Vers. D Log. Exs, n < P, P, 17 16 17 16 17 16 17 16 17 16 16 17 16 17 16 16 17 16 16 17 16 18 17 16 16 17 16 16 16 17 16 16 16 16 a 16 16 16 16 16 17 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 D 10-41962 42022 .42083 .42144 .42205 60 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 6l 61 62 9-86992 87009 .87025 .87042 87058 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 10-45693 45756 45820 45884 45947 63 63 64 63 64 64 64 64 64 64 It !i 64 64 64 64 65 64 65 64 65 65 65 65 65 65 65 65 65 65 65 65 65 65 66 65 65 66 66 66 65 66 66 66 66 66 66 66 66 66 66 II 11 67 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 140 50 6 7 8 9 10 20 30 40 50 67 6-7 7-8 8.9 10-6 11-1 22-3 33-5 44-6 55-8 65 6-5 7-6 8-7 9-8 10-9 21-8 32-7 43-6 54-6 64 6-4 l\ 9-6 10-6 21- S 32.0 42-6 53-3 62 6-2 7-3 8-3 9-4 10-4 20-8 31-2 41. B 52-1 f 6 ( 7 ' 8 i 9 { 10 1( 20 2( 30 3( 40 4( 50 5( 17 1-7 2-0 2-2 2-5 2-8 5-6 8-5 11-5 14-1 66_ 6-6 7-7 8-8 10-0 11.1 22-1 33-2 44.3 55-4 65 6-5 7-6 8-6 9-7 10-8 21- B 32-5 43.3 54-1 63_ 6-3 7-4 8-4 9-5 10-6 21-1 31-7 42-3 52-9 62 6-2 7-2 8-2 9-3 10- S 31-0 41-3 51-6 51 ( -l ( M ' I- 1 J I.I ( ).I 1( )-3 2( )-5 3( >-64( ).8I5( 16 1-6 1.9 2-2 2-5 2-7 5-5 8-2 11-0 13-7 66 6-6 7-7 8-8 9.9 11 22.0 33-0 44-0 55-0 61 6-4 7-5 8-6 9-7 10-7 21-5 32-2 43 -Q 53-7 63 6-3 7-3 8-4 9-4 10-5 21-0 31-5 42 52-5 61 6-1 7-2 8-2 9-2 10-2 20-5 30-7 41-0 51-2 JO i -0 t-5 M )-l )! ). )-3 ).4 16 1-6 1.1 2-' 2. 2. 5-1 8- 10- 13- 5 6 7 8 9 86079 .86096 .86113 86129 .86146 10-42266 42327 .42388 .42450 .42511 9-87074 .87091 .87107 .87124 .87140 10-46011 .46075 46139 46203 46267 5 6 7 8 9 10 11 12 13 14 10 11 12 13 14 86163 .86179 .86196 .86213 .86230 10-42572 .42633 .42695 42756 .42817 9.87157 .87173 .87189 .87206 9-87239 87255 .87271 .87288 .87304 10-46331 -46395 .46460 46524 -46588 15 16 17 18 11 20 21 22 23 24 .86246 .86263 .86280 .86296 .86313 86330 .86346 .86363 .86380 .86396 10-42879 .42940 .43002 .43063 -43125 10-46652 .46717 .46781 46846 -46910 15 16 17 18 J2. 20 21 22 23 24 10-43187 .43249 .43310 .43372 43434 62 61 62 61 62 62 62 62 62 62 62 62 62 62 62 62 62 62 63 62 62 II 63 62 63 63 63 63 63 63 63 63 63 63 63 63 63 63 64 D 9-87320 -87337 .87353 87370 -87386 10-46975 47040 47104 47169 47234 25 26 27 28 29 .86413 .86430 .86446 .86463 .86479 10-43496 .43558 43620 .43882 .43744 9-87402 .87419 .87435 .87451 -87468 10-47299 47364 47429 47494 47559 25 26 27 28 29 30 31 32 33 34 9-86496 86513 -86529 .86546 .86562 10-43806 .43868 43931 .43993 .44055 9-87484 87500 .87516 -87533 .87549 10-47624 -47689 47754 47820 -47885 30 31 32 33 34 35 36 37 38 39 9-86575 86596 .86612 .86629 .86645 10.44118 .44180 .44242 .44305 .44368 9-87565 .87582 .87598 .87614 -87631 10-47950 48016 48081 .48147 -48213 35 36 37 38 _39. 40 41 42 43 44 40 41 42 43 44 9-86662 .86678 86695 -86712 .86728 10-44430 .44493 .44556 .44618 .44681 9-87647 .87653 87679 87696 .87712 10-48278 48344 48410 .48476 48542 45 46 47 48 49 9-86745 .86761 86778 -86792 -86811 10-44744 .44807 .44870 .44933 .44996 9.87728 .87744 .87761 87777 .87793 10-48607 48674 .48740 48806 48872 45 46 47 48 49 50 51 52 53 54 9.86827 . 8684< .86860 .86877 86893 10-45059 45122 4518! 45248 45312 9-87809 .87825 87842 .87858 .87874 10-48938 .49004 -43071 49137 -49204 50 51 52 53 54 55 56 57 58 59 9-86910 .86926 86943 86959 86976 10-45375 .45439 .45502 .45565 .45629 9-87890 .87906 .87923 .87939 .87955 9-8797T 10-49270 .49337 49403 49470 49537 55 56 57 58 59 60 9-86992 10-45693 10-49604 60 Lg.Vers Log, Exs. Lg.Vers, Log, Exs, JJ ' P, P, LOG. VERSED SINES AND EXTERNAL SECANTS 76 77 203 ' Lg.Vers. D Log.Exs. I> Lg.Vers, 7> Log.Exs. 7) 70 a 71 71 71 71 ?i ?i 71 71 72 71 71 72 71 72 72 72 72 72 72 72 72 72 73 72 72 ?1 73 73 73 73 73 73 73 75 73 73 74 73 74 74 73 74 74 74 74 74 ?! ?! 74 75 75 75 ' P.P. 1 2 3 4 9-87971 .87987 88003 .88020 88036 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 10 16 10-49604 -49670 49737 49804 49871 66 67 67 67 67 67 67 67 67 67 67 67 87 68 6? II 68 68 68 68 68 68 68 68 68 68 68 69 68 68 69 69 69 69 69 69 69 69 69 69 69 69 69 70 69 6S 70 70 70 70 70 70 70 70 70 75 70 9-88933 1 -88949 -88964 -88980 -88996 ii 16 16 ii 16 ii 16 16 16 15 16 15 16 16 15 16 8 ii 16 16 15 15 16 15 16 15 ii 16 15 15 16 15 16 15 15 ii 15 16 15 15 16 15 15 ii 15 16 15 10-53724 53794 -53865 53936 54007 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 75 7-5 8-7 10-0 11.2 12.5 25.0 37-5 50-0 62-5 72 7-2 8-4 9-6 10-8 12-0 24-0 36-0 48-0 60-0 69 6-9 8-0 9-2 10-3 11-5 23-0 34-5 46-0 57-5 6 7 8 9 10 20 : 30 ; 40 ^ 50! 1 1^9 2-2 2-5 2-7 5-5 8-2 11-0 13-7 74 7-4 8-6 9/8 11-1 12-3 24-6 37-0 49-3 61-6 71 7-1 8-3 9-4 10-6 11.5 23-6 35-5 47-3 59-1 68 6-8 7-9 9-0 10-2 11-3 22-6 34-0 45-3 56-6 66 6.6 7-7 8-8 ( 9-9( Ll-0( J2-0( J3-0( 14-0( 55. 0( 16 1-6 1-8 2-1 2-4 2-6 5-3 8-0 10-6 13-3 73 7-3 8-5 9-7 10-9 12-1 24-3 36-5 48-6 60.8 70 7-0 8-2 9-4 10-6 11-7 23-3 35.2 47-0 58-7 67 6-7 8^9 10-0 11- 1 22-3 33-5 44-6 55-5 & !:i )-l )-l )-! )-2 )-3 )-4 15 1:8 2-3 2-8 i 10-3 12-9 5 6 7 8 9 9-88052 .88068 .88084 .88100 .88116 10-49939 50006 50073 50140 50208 9-89012 89028 89044 89060 89075 10-54078 54149 54220 54291 -54362 5 6 7 8 9 10 11 12 13 14 9-88133 .88149 .88165 .88181 -88197 10-50275 .50342 50410 50477 .50545 9-89091 89107 .89123 89139 89155 10 - 54433 54505 54576 54647 54719 10 11 12 13 14 15 16 17 18 ii 30 21 22 23 24 9-88213 .88229 88245 .88261 .88277 10-50613 .50681 50743 50816 .50884 9-89170 89186 89202 89218 -89234 10-54791 .54862 .54934 .55006 -55078 15 16 17 18 01 20 21 22 23 24 9-88294 .88310 88326 .88342 -88358 9-88374 -88390 88406 .88422 .88438 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 10-50952 51020 51088 51157 51225 9-89249 -89265 89281 89297 -89312 10-55150 55222 .55294 .55366 -55438 25 26 27 28 29 10-51293 51361 51430 51498 -51567 9-89328 89344 .89360 .89376 89391 10-55511 55583 55655 55728 55801 25 26 27 28 29 30 31 32 33 34 9.88454 .88470 .88486 .88502 -88518 10-51636 51704 .51773 51842 -51911 9.89407 .89423 .89438 .89454 89470 10-55873 -55946 .56019 56092 -56165 30 31 32 33 34 35 36 37 38 39 35 36 37 38 39_ 40 41 42 43 44 9-88534 -88555 .88560 .88582 88598 9-88614 .88630 .88646 .88662 -88678 10-51980 52049 52118 52187 52256 9-89486 89501 89517 89533 89548 10-56238 -56311 .56384 .56457 56531 16 16 16 15 16 16 16 16 16 16 16 16 15 16 16 16 16 15 16 16 16 10-52325 52394 52464 52533 -52603 9-89564 .89580 89596 89611 89627 10-56604 .56678 .56751 56825 -56899 40 41 42 43 44 45 46 47 48 49 9-88694 -88710 .88726 -88742 .88758 10-52672 52742 52812 52881 52951 9-89643 .89658 89674 89690 89705 10-56973 .57047 57120 -57195 .57269 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 9.88774 -88790 .88805 -88821 -88837 10-53021 53091 53161 53231 53301 9-89721 .89737 .89752 89768 89783 10-57343 -57417 .57491 57566 57640 50 51 52 53 34 9-88853 -88869 -88885 -88901 .88917 10.53372 53442 53512 53583 -53653 9-89799 .89815 .89830 89846 -89862 10-57715 -57790 .57864 .57939 -58014 55 56 57 58 59 BO 9 88933 10 53724 9-89877 10.58089 60 ' Lg. Vers 1) Log.Exs, L> Lg.Vers. 1> Log.Exs. 1> ' P. P. 204 LOG. VERSED SINES AND EXTERNAL SECANTS 78 79 ' Lg.Vers. 1) Log.Exs, D Lg.Vers. D Log. Exs. J> ' P, P. 1 2 3 4 9-89877 .89893 .89908 .89924 .89939 15 15 15 15 16 15 15 15 15 15 15 15 16 15 1 1 1 1 1 1 1 1 1 1 15 15 jS 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 11 15 15 15 15 10-58089 .58164 58239 58315 .58390 75 75 75 75 i 76 76 75 75 76 76 76 76 76 76 76 77 76 77 77 77 i 78 78 78 78 78 78 78 78 78 79 78 79 79 79 79 79 ?! 8 8 ie 80 80 9-90805 90820 90835 90851 90866 15 15 15 15 15 15 15 15 15 15 15 11 15 15 11 15 15 15 15 a 15 15 15 11 15 15 15 15 15 15 11 15 15 15 15 15 15 15 15 iS 15 15 15 a 15 15 15 15 15 15 15 15 15 -D 10-62745 62825 .62906 62986 .63067 80 80 80 81 86 81 81 81 81 81 81 81 81 81 82 82 82 82 82 82 82 82 82 83 82 83 83 83 83 83 83 83 83 84 83 84 it 84 84 84 84 84 85 85 85 85 85 85 85 85 85 86 86 86 86 86 11 86 1 2 3 4 5 6 7 8 9 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 86 8-6 10.6 11-4 12.9 14-3 28-6 43-0 57-3 71-6 83 8.3 9-7 11-0 12-4 13-8 27-6 41-5 55-3 69-1 80 8.0, 9-3 10-6 12-0 13-3 26-6 40-0 53-3 66-6 77 7-7 9-0 10-2! 11-51 12-8"! 25-6 38-5 51-3 64-1! e ! * i K 2( 3( 4( 5( 16 1-6 1.8 2-1 2-4 2-6 5-3 8-0 10-6 13-3 85 8-5 9-9 11.3 12-7 14-1 28-3 42-5 56-6 70-8 82 8-2 9-5 10-9 12-3 13-6 27-3 41-0 54-6 68-3 79 7-9 9-2 10-5 11-8 13-1 26-3 39-5 52-6 65- 8 76 7-6 8-8 10-1 11-4 12-6 25-3 38-0 50-6 B3-3 O.Q 0-6 0-5 0-1 lo'.j )0-2 )0-3 )0-4 15 1.5 1-8 2-0 2-3 2.6 5-1 7-7 10-3 12-9 84 8-4 9.8 11-2 12-6 14-0 28-0 42-0 56-0 70-0 81 13-5 27-0 40-5 54-0 67-5 78 7-8 9-1 10-4 11-7 13-0 26-0 39-0 52-0 65-0 75 7-5 8-7 10-0 11-2 12-5 25-0 37 5 50-0 62-5 15 11 1:1 2-5 5-0 7-5 10-0 12-5 5 6 7 8 9 10 11 12 13 14 9-89955 -89971 -89986 .90002 -90017 10-58465 58541 58616 58692 .58768 9-90881 -90897 .90912 .90927 -90943 9-90958 .90973 90988 .91004 .91019 9-91034 .91049 .91065 91080 .91095 10-63148 63229 .63310 -63391 .63472 9-90033 .90048 90064 .90080 .90095 10-58844 -58920 58995 -59072 .59148 10-63553 .63634 63716 63797 63879 10 11 12 13 14 15 16 17 18 19 15 16 17 18 19 9-90111 .90126 .90142 90157 90173 10-59224 .59300 .59377 59453 .59530 10-63961 64043 .64125 64207 -64289 20 21 22 23 24 25 26 27 28 29 9-90188 90204 90219 90235 90250 10-59606 59683 59760 .59837 .59914 9-91110 .91126 -91141 .91156 .91171 9-91187 .91202 .91217 91232 91247 10-64371 .64453 64536 64618 -64701 20 21 22 23 24 25 26 27 28 29 9-90266 .90281 90297 90312 90328 10-59991 -60068 -60145 -60223 - 60300 10-64784 64867 64950 65033 65116 30 31 32 33 34 9 90343 90359 90374 -90389 90405 10-60378 .60455 -60533 -60611 .60688 9-91263 -91278 -91293 .91308 91323 10.65199 .65283 .65366 .65450 .65534 30 31 32 33 34 35 36 37 38 39. 40 41 42 43 44 45 46 47 48 49 9 90420 90436 -90451 .90467 -90482 10-60766 . 60844 .60923 .61001 .61079 9-91338 -91354 -91369 91384 -91399 10-65617 .65701 .65785 .65870 .65954 35 36 37 38 -39. 40 41 42 43 44 ^45 46 47 48 49 50 51 52 53 54 9-90497 90513 .90528 .90544 .90559 10.61158 61236 61315 .61393 .61472 9-91414 91429 91445 91460 91475 10-66038 .66123 -66207 .66292 .66377 9-90574 90590 .90605 90621 .90636 10-61551 61630 61709 .61788 .61867 9-91490 91505 91520 91535 91550 10-66462 .66547 .66632 66717 66803 50 51 52 53 54 9-90651 90667 90682 90697 .90713 10-61947 -62026 62105 -62185 .62265 9-91565 91581 91596 91611 -91626 10-66888 .66974 .67059 .67145 -67231 55 56 57 58 59 9-90728 - 90744 90759 90774 90790 10-62345 . 62424 .62504 62585 .62665 9-91641 -91656 91671 91686 91701 10-67317 .67403 67490 -67576 67663 55 56 57 58 59 eg 9-90805 10-62745 9-91716 10-67749 Lg. Vers Log.Exs 1> Lg.Vers Log.Exs, P, P, LOG. VERSED SINES AND EXTERNAL SECANTS 205 80 81 1 *o" i 2 3 4 Lg.Vers. 7> Log.Exs. D Lg.Vers JD Log, Exs, D t P, P, 9-91716 91731 .91746 91761 91776 15 15 15 15 5 lo 15 15 15 15 15 15 15 15 15 15 15 15 15 14 15 15 15 15 15 15 15 ii 15 15 15 15 15 14 15 15 15 15 13 15 15 i! 15 15 14 15 15 14 15 ll 15 ll 15 \l 15 10-67749 .67836 67923 68010 68097 86 87 87 87 87 87 87 87 87 88 88 88 88 88 89 89 89 89 CO 89 89 90 90 90 90 90 91 90 91 91 91 91 91 91 91 92 92 92 92 93 92 93 93 93 93 93 93 94 94 94 94 S4 9-92612 .92626 92641 92656 .92671 14 i! 15 Ji- ll 15 14 a ii 15 14 14 ii 15 14 ii 15 14 14 15 14 14 !! 15 14 14 14 15 14 14 15 1 4 ii 14 14 15 14 14 14 i! 15 14 14 14 14 14 14 ~w 10-73178 .73273 .73368 .73463 73558 95 94 95 95 95 95 95 96 95 96 96 Si 96 96 97 97 97 97 97 98 97 98 98 98 98 98 99 99 99 99 99 100 99 LOO 100 101 101 101 101 101 102 102 102 102 102 102 103 103 103 103 104 10E 104 104 104 1 2 3 4 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 - 30 40 50 90 9-0 10-5 12-0 13-5 15-0 30-0 15-0 80.0 75-0 9 0.9 1.0 1-2 1-3 1.5 3-0 4-5 6.0 7-5 7 0-7 : i:? 2-3 3-5 4-6 5-8 5 0-5 0-6 0-6 0-7 0-8 2~5 3-3 4-1 & *:l 2-3 2-6 5-1 7-7 10-3 12-9 ] 6 7 8 9 10 20 30 40 50 1 80 8-0 9.3 10.6 12-0 13-3 26-6 40.0 53-3 66-6 8 0-8 0-9 i's 2-6 4.0 5-3 6.6 6 0-6 0-7 0.8 0-9 1.0 2-0 3.0 4.0 5-0 4 81 0.5 0.6 0-6 2-0 2-6 3-3 15 i:? 2.Q 2.2 2-5 5.0 7-5 10.0 12.5 ft ii 2-2 2.4 4-8 j'e 2.1 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ii 20 21 22 23 24 9-91791 91807 .91822 91837 91852 10-68184 68272 .68359 68447 68534 9-92686 -9270C 82715 92730 92745 10-73653 73748 . 73844 73940 74035 5 6 7 8 9 10 11 12 13 14 9-91867 91882 91897 91912 -91927 10-68622 68710 68798 68886 68975 9-92759 92774 92789 92804 92818 10-74131 74227 . 74324 . 74420 .74517 9-91942 -91957 91972 91987 -92002 10-69063 .69152 69240 69329 69418 9-92833 .92848 .92862 .92877 -92892 10-74613 74710 .74807 74905 .75002 15 16 17 18 19 9-92016 -92031 -92046 92061 -92076 10-69507 69596 69686 69775 69865 9-92907 92921 92936 92951 92965 10-75099 75197 .75295 .75393 75491 20 21 22 23 24 25 26 27 28 29 9-92091 92106 .92121 .92136 .92151 10-69955 . 70044 70134 .70224 70315 9-92980 92995 .93009 .93024 93039 10-75589 .75688 .75786 .75885 10 - 76083 76182 .76282 .76382 76481 25 26 27 28 _?i 30 31 32 33 34 30 31 32 33 34 9-92166 .92181 .92196 92211 .92226 9 - 92240 92255 92270 92285 -92300 10-70405 .70495 70586 70677 70768 9-93053 93068 .93083 93097 93112 35 36 37 38 39 10-70859 70950 71041 .71133 71224 9-93127 93141 .93156 93171 -93185 10-76581 .76681 .76782 76882 76983 35 36 37 38 39 40 41 42 43 44 9-92315 -92330 .92345 92360 -92374 10-71316 .71408 .71500 .71592 .71684 9-93200 .93214 93229 .93244 93258 10-77083 77184 .77286 .77387 .77488 40 41 42 43 44 45 46 47 48 49. 50 51 52 53 54 9-92389 -92404 92419 92434 92449 10-71776 .71869 .71961 .72054 .72147 9-93273 .93287 93302 .93317 93331 10-77590 .77692 .77794 .77896 .77998 45 46 47 48 49 9-92463 92478 92493 -92508 92523 10-72240 72333 .72427 72520 .72614 9-93346 .93360 .93375 93389 . 93404 10.78101 .78203 .78306 . 78409 .78513 50 51 52 53 54 55 56 57 58 59 9-92538 -92552 92567 92582 92597 10-72707 72801 .72895 .72990 . 73084 9-93419 .93433 . 93448 93462 .93477 10-78616 .78720 7882E .78927 .79031 55 56 57 58 59 BO ?- 92612 10-73178 9-93491 10-79136 60 ' g.Vers 1> Log.Exs, /> Lg.Vers Log, Exs, /> P P, 206 LOG. VERSED SINES AND EXTERNAL SECANTS 82 83 p,p, Lg.Vers. -D Log, Exs, 1> Lg.Vers 10-7913 7924 .7934 79450 79555 10-85766 .85884 .86001 86119 86237 9350 9352 .93535 .93549 120 12-0 14-0 16-0 18.0 20-0 40.0 60.0 80-0 100-0 10-86355 86474 .86592 86711 86831 93578 .93593 93607 .93622 10-8695C 87070 87190 87310 87431 0-80723 .80831 .8093 .8104 .811 J-9370 9372 9373 9375 .93767 10-87552 8767 .8779 87916 .88038 94584 .9459 .9461 .94626 0-81262 81371 .8147 .8158 .8169 0-81806 .8191 .8202 .8213 .8224 94711 .94726 . 94740 94754 .94768 10-8877. -88898 .89022 8914 .89271 3 6u-; 70-i so-- 90.: 100-50. 201-00 82356 .82466 .82577 .82688 .82799 0-82910 .83 .8313 .8324 -83358 } 93997 -9401 9402 . 94041 94055 10-90025 90152 90279 . 90406 90 70 80 90 100 200 300 400 500 LO. 83470 .8358 .8369 .83809 83922 90789 90917 .91046 91175 94127 9414 9415 9417 9418 .9419 LO. 84035 8414 .8426 8437 -84492 LO- 84607 .84721 .84837 .84952 10-91956 .9208 9221 923 .924 LO-9261 .9274 9288 93014 -93147 LO 93281 -94227 -94241 -94256 -94270 - 94284 .94299 .94313 .9432 LO-8518 .8529 .85418 .8553 .8564 P.P. LOG. VERSED SINES AND EXTERNAL SECANTS 207 84 C 85 g.Vers, /> Log.Exs, J> g.Vers, Z> Log.Exs, Z> ' P.P. 1 3 _4 5 6 7 8 9 -95205 -95219 -95233 -95247 -95261 14 it 14 14 14 14 14 14 14 14 14 14 14 14 14 14 0-93281 -93416 -93551 -93686 -93821 134 135 135 135 135 136 136 137 137 137 137 138 138 139 139 139 140 141 142 142 142 143 143 .44 1 118 i 148 149 149 149 150 150 151 ill III 153 -96039 -96053 -96067 -96081 96095 14 13 14 14 13 i! 11 i! 14 13 13 i! 14 11-02010 02168 .02327 .02487 .02646 158 160 .61 161 161 162 163 163 164 164 165 165 166 167 167 167 168 169 169 169 170 171 171 172 173 173 174 174 175 176 176 177 177 178 179 179 180 III III 183 184 185 185 186 186 187 188 189 189 190 191 19; 192 193 195 O 1 2 3 4 5 6 7 8 9 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 140 50 f ! e s 1C 2C . 3C 4( 5C 6 7 8 9 10 20 30 40 50 19O 19- C 22-1 25-3 28- 31- e 63-1 95- C 126- 6 158-3 170 17-C 19 -I 22- f 25- 28- J 56- f 85- ( 113. i 141- ( 150 15- ( 17-5 20- ( 22-! 25- ( 50- ( 75- ( 100-1 125-( 130 13-0 15-1 17-3 19-5 21- B 43-3 65- C 86- 6 108-3 7 0-7C 0-8C 0-9C 1-OC 1-11 2-3 2 '3-53 ,ft:!i 1:1 2-2 2-4 4-8 7-2 9-6 12-1 ] ISO 18-0 21-0 24-0 27-0 30.0 60-0 90-0 120-0 150-0 160 16-0 18-6 21-5 24-0 26-6 53-3 80-0 106-6 133-3 140 ) 14-0. > 16-3 ) 18-6 5 21-0 ) 23-3 ) 46-6 ) 70-0 ) 93-5 j 116-6 9 8 0-90-8 1-00-9 1-2 1-5 -95275 -95289 -95303 -95317 -95331 0-93957 -94093 .94229 .94366 -94503 96108 96122 96136 96150 .96163 11.02807 -02968 .03129 -03291 -03453 10 11 12 13 14 -95345 -95359 95373 95387 -95401 0-94641 -94778 -94917 -95055 -95194 9-96177 -96191 -96205 -96218 -96232 11-03616 -03780 -03944 -04108 -04273 10 11 12 13 11 15 16 17 18 Oi 20 23 24 15 16 17 18 11 20 23 24 9-95415 95429 -95443 -95457 95471 0-95333 -95473 -95613 -95753 -95894 9-96246 -96259 -96273 -96287 -96301 11-04438 -04604 -04771 -04938 -05106 9.95485 95499 95513 -95527 -95540 14 14 14 it 14 14 14 14 14 14 13 14 14 14 13 14 14 14 14 0-96035 -96176 -96318 -96461 -96603 -96314 -96328 -96342 -96355 -96369 14 13 13 14 13 14 13 14 15 15 14 13 13 15 14 13 13 14 13 13 13 13 15 13 13 if 13 15 13 13 14 13 13 13 13 15 11-05274 -05443 -05612 .05782 -05952 25 26 27 28 29 9-95554 95568 -95582 95596 .95615 0-96746 96889 97033 97177 97322 9-96383 -96397 -96410 -96424 -96438 11-06125 -06295 -06467 -06640 -06813 25 26 27 28 29 30 31 32 33 34 35 36 37 33 39 9-95624 -95638 -95652 -95666 -95680 0-97467 97612 97758 -97904 -98050 9-96451 -96465 -96479 96492 -96506 11-06987 .07161 .07336 -07512 -07688 30 31 32 33 34- 35 36 37 38 J! 40 41 42 43 44 9-95693 -95707 95721 9573B -95749 10-98197 -98345 -98492 93640 -98789 9-96519 -96533 -96547 -96560 -96574 11-07865 -08043 -08221 -08400 -08579 1-3 1-5 3-0 4-5 B.O 7-5 6 ( 7 ( 8< 9< .0 .0 .0 .0 .0' 14 i -4 1:1 2-1 2-3 4.f 7-0 9-3 1.1 1-3 2-6 4-0 5-3 6-6- 5 ) 5 )-6 )-8 2-5 }-3 l-I 13 1-3 1-6 1-8 2-0 2-5 4-5 6-7 9-0 11-2 40 41 42 43 44 9-95763 -95777 -95791 -95804 -95818 14 14 13 14 14 if 14 14 11 14 14 13 14 11 V. 10-98938 -99087 -99237 -99387 -99538 9-96588 -96601 -96615 -9662S -96642 11-08759 -08940 -09121 -09303 -09486 45 46 47 48 49 9-95835 -95846 -95860 -95874 -95888 10-99689 -99841 10-99993 11-00145 -00298 9-96656 -96669 -96683 -96697 -96715 11-09669 -09853 -10038 .10223 . 10409 45 46 47 48 49 50 51 52 53 54 9-95901 -95915 -95929 .95943 -95957 11-00451 -00605 -0075E -00914 -01069 Ib3 154 154 155 155 155 156 156 157 157 158 D 9-96724 -96737 -96751 .9676< -96778 11-10595 -10783 -10971 .11160 -11349 50 51 52 53 54 55 56 57 58 59 9-95976 -9598; -95998 -96012 -96026 11-01225 -01381 -01537 01692 -01852 9-96792 -96805 -96815 -96832 -96848 11-11539 -11730 .1192? 12114 -12307 55 56 57 58 59 Q 9-96039 11-0201C 9-96859 11-12501 60 Lg, Vers *> Log.Exs. Lg.Vers D Log.Exs, ^ ' P. P, 208 LOG. VERSED SINES AND EXTERNAL SECANTS 86 87 Lg. Vers D Log. Exs. D Lg. Vers. D Log. Exs. D > P. P. 1 2 3 4 9-96859 96837 96887 96900 96914 13 13 13 11-12501 12696 -12891 .13087 13284 19 19 19 19 9-97665 .97679 97692 .97705 97718 13 13 13 13 11.25785 26040 26297 26554 26814 255 256 257 259 I? 2 u 6 7 8 9 25O 25-0 29-1 33-3 37-5 24O 24-0 28.0 32-0 36-0 5 6 7 8 9 9-96927 96941 96954 96968 9698T if 11 11 13 11.13482 .13680 .13879 . 14079 14280 19 19 19< 20 20 9.97732 .97745 97758 97772 97785 lei 13 13 13 13 11.27074 .27336 .27599 27864 28131 261 262 263 265 266 b~ \ [j 10 20 30 40 50 41 -b 83-3 125-0 166-6 208-3 40-0 80.0 120-0 160-0 200-0 10 96995 11-14482 9-97798 11-28398 267 ho 230 220 11 12 13 14 97008 97022 97035 97049 13 I 14684 14887 .15092 -15297 203 20^ 205 97811 .97825 .97838 .97851 13 13 13 -28668 -28938 -29211 -29485 26S 27C 272 274 11 12 13 14 6 7 8 9 23-0 26-8 30-6 34-5 22-0 25-6 29-3 33.0 15 16 17 18 19 97062 97076 97089 97103 97116 13 13 11.15502 .15709 -15917 .16125 16334 208 208 209 9-97864 -97978 97891 .97904 97917 13 13 13 13 11-29760 30037 30316 30596 30878 277 278 28C 282 15 16 17 18 19 10 20 30 40 50 38-3 76-6 115-0 153-3 191-6 36-6 73-3 110-0 146 - 6 183-3 20 21 22 23 24 97130 97143 97157 -97170 97183 13 11 1-16544 .16755 16967 .17180 17394 210 211 212 213 214 9.97931 .97944 .97957 .97970 .97984 13 13 13 11 11.31162 .31447 31734 32023 32313 2 si 287 288 29C 21 22 23 24 6 7 8 9 210 21-0 24-5 28-0 31-5 200 20-0 23-3 26-6 30-0 25 26 97197 97210 13 1.17609 .17824 2 it 9.97997 .98010 13 13 1 3 11-32606 32900 29^ 25 26 10 20 35-0 70-0 33-3 68-6 27 -97224 .18041 9-1 n .98023 13 33196 27 30 105-0 100-0 28 29 97237 97251 13 18259 18477 211 98036 .98050 13 33494 33793 29< 28 29 40 50 140-0 175-0 133-3 166-6 30 31 32 33 34 97264 97277 97291 97304 97318 13 13 13 13 13 1-18697 .18917 19138 .19361 .19581 220 i 9-98063 .98076 .98089 .98102 .98116 13 13 13 13 13 11-34095 -34398 -34704 35011 35321 301 303 305 307 309 30 31 32 33 34 6 7 8 190 19-0 22-1 25 3 4 3 0-410.3 0-40-3 O-JO-4 35 36 37 38 -32. 40 97331 97345 97358 9737T 973RR 97398 13 13 13 13 13 13 1-19809 .20034 20261 20489 -207V7 1-20947 224 225 227 227 228 230 9.n 9.98129 98142 98155 .98168 -98181 13 13 13 13 13 13 13 11-35632 35946 36261 36579" 36899 6\.i 313 315 318 320 322 35 36 37 38 39 40 Ld 20 30 40 50 31-6 63-3 95-0 126-6 158-3 0-605 1-3 1.0 2.01-5 2-6 2-0 3-32-5 9-98195 11-37221 41 97412 .21178 98208 37546 41 210 42 97425 .21410 000 .98221 13 37872 000 42 ( JO- 20 .10-0" 43 44 97438 97452 13 .21643 .21877 234 98234 .98247 13 38201 38532 331 43 44 t 1 fO-20 50-20 . 0-0 . 0-0 45 46 47 48 49 97465 97478 97492 97505 97519 13 13 13 13 13 1-22112 .22349 -22586 .22825 -23065 23f 237 239 239 9.98260 .98273 .98287 .98300 -98313 13 13 13 13 13 11-38866 .39201 39540 39880 40224 666 335 338 340 342 45 46 47 48 49 ( 1( 2( 3( 4( )0-30 )0-30 )0-60 ) 1-00 ) 1-50 - 0-1 0-1 - 0-1 - 5 - 2 .60.3 50 97532i 13 1.23306 241 n j o 9.98326 13 11-40569 50 50 1- D U-O U- 51 52 53 54 97545 97559 97572 97585 13 13 13 .23548 .23792 .24037 24283 243 245 246 -98339 .98352 -98365 -98378 13 13 13 40918 -41269 41622 41979 351 353 356 51 52 53 54 6 7 g 14 1 1-4 1 1-6 1 1-8 1 3 13 -3 1.3 -6 1.5 8 1-7 55 97599 13 1-24530 247 9-98392 13 11.42338 55 9 2-1 2 56 97612 24778 248 .98405 42699 3R/T 56 10 2-3 2 2 v 1 57 58 59 97625 .97639. 97652 13 13 .25028 25279 25531 251 252 98418 .98431 98444 13 13 .43064 -43431 .43802 367 370 57 58 59 20 30 40 4-6 4 7-0 6 9.3 9 5 4- 5 7 6-5 60 -97665 13 11-25785 264 9-98457 13 11.44175 616 (50 501 1-6 11 .2 10.8 ' g. Vers. D Log. Exs. D Lg, Vers. D Log. Exs, D f P.P. LOG. VERSED SINES AND EXTERNAL SECANTS 88 89 209 Lg.Vers It Log.Exs. D Lg.Vers. D Log. Exs z> ' P. P. 1 2 3 4 9-98457 .98470 98483 98496 98509 13 13 13 13 11-44175 .44551 44931 45313 45699 37 379 382 386 9.99235 .99248 .99261 .99274 .99287 12 13 13 11-7505 7579 .7654 .7731 .7809 74 75 76 78~ ] s 4 5 6 7 8 9 9-98522 98535 .98548 .98562 -98575 13 13 13 13 13 11.46088 .46480 46876 47275 47677 389 392 395 399 402 9-99299 9931J -99325 .99338 .99351 12 13 11 13 11-7889 7970 -8052 8136 82223 79 80" 82 840 856 E e 7 8 9 to 11 1'2 18 14 9-98588 .98601 -98614 .98627 -98640 13 13 13 13 13 11.48083 48493 48906 49323 .49743 4Ub 409 413 417 420 9-99363 .99376 .99389 .99402 99415 12 13 12 13 11.83095 .83986 84894 85821 .86768 872 89i 908 927 947 10 11 12 13 14 15 J6 17 18 19 9-98693 .98666 .98679 -98692 .98705 13 13 13 13 13 11-50168 .50597 51029 -51466 51906 42b III ft 9-99428 99440 .99453 .99466 99479 13 12 13 12 13 , s 11-87735 -88724 89735 -90769 .91829 967 989 1009 103; 1059 15 16 17 18 -li 20 21 22 23 24 9-98718 98731 .98744 .98757 .98770 13 13 13 13 13 11-52351 .52801 53255 .53713 54176 445 449 454 458 463 9-99491 99504 99517 .99530 99543 12 13 13 12 13 11-92914 94026 95167 .96338 -97541 1085 1112 1140 1171 1203 20 21 22 23 24 25 26 27 28 29 9-98783 98796 98809 -98822 .98835 13 13 13 13 13 11-54643 55116 55593 .56076 -56563 4bV 472 477 482 487 9-99555 .99568 99581 .99594 99606 12 13 12 13 12 11.98777 12.00048 01358 .02707 .04098 1236 1271 1309 1349 1391 25 26 27 28 29 30 31 32 33 34 9-98848 -98861 -98874 -98887 -98900 13 13 13 13 13 11-57056 57554 58058 58567 .59082 492 498 504 509 515 9-99619 99632 99645 99657 99670 ii 13 12 13 12-05535 .07020 .08557 .10149 11801 1436 1485 1537 1592 1652 30 31 32 33 34 3o 3fi 37 38 39 -98913 98925 .98938 .98951 -98964 13 12 13 13 13 11-59602 -60129 60662 .61202 -61747 520 527 533 i39 9-99683 99695 .99708 .99721 -99734 y 13 12 13 12-13517 .15302 1716J 19106 21139 1716 1785 1861 1943 2033 35 36 37 38 39 40 41 42 43 44 -98977 98990 99003 -99016 99C29 13 13 13 12 1.62300 .62859 .63425 .63998 .64579 5b2 559 566 573 581 9-99740 99759 99772 99784 -99797 13 12 12 13 i"o 12-23271 2551] .27872 30367 -33013 2131 2240 2361 2495 2645 40 41 42 43 44 13 13 6 1-3 1.3 7 1.6 1. 8 1-8 1.' 9 2-0 1.9 45 46 47 48 49 99042 99055 99068 99081 -99093 13 13 13 13 12 1-65167 65762 66366 66978 -67598 ill ft 620 9-99810 99823 .99835 99848 99861 12 13 12 13 12-35828 .38837 42068 45557 .49349 2815 3009 3231 3489 3791 45 46 47 48 49 10 2-2 2. 20 4-5 4.3 30 6-7 6-i 40 9-0 8. 50 11.2 10-8 50 51 52 53 54 -99106 -99119 -99132 99145 -99158 13 13 13 13 12 1.68227 .68865 69511 70168 70834 628 638 646 656 666 9-99873 .99886 .99899 99911 S9924 12 12 13 12 12 2-53501 58089 .63217 .69029 75736 4152 4588 5127 5812 6707 50 51 52 53 54 12 6 1-2 7 I.I 8 1.6 55 56 57 58 59 -99171 -99184 -99197 99209 99222 13 13 12 13 1-71509 .72196 72892 73600 74319 '96 9-99937 99949 .99962 99974 .99987 18 12 12 12 13 2-83667 .93371 3-05877 -23499 .53615 7931 9704 2506 7621 0116 55 56 57 58 59 91-9 10 2-1 20 4-1 30 6.2 40 8-3 50 10-4 60 99235 13 1-75050 30 o.ooooo 12 Infinity 60 ' Lg.Vers. J> Log.Exs, D Lg.Vers. | X> Log.Exs, 1> 7 P, P, 210 THE RAILROAD TAPER FUNCTIONS OF A ONE-DEGREE CURVE The Long Chords, Mid-Ordinates, Externals, and Tangent Distances of this table are for a curve of 5730 feet radius. To find the corresponding functions of any other curve divide the tabular values by the degree of curve. For metric curves having a 20-meter chord, this is a table of the functions of a two-tenth degree curve. Multiply the values in the table by 0.2 to obtain the functions, in meters, of a one-degree metric curve. 1 / L. 0. M. E. T. L. C. M. E. T. 0.00 0.000 0.000 0.00 100.00 0.',M8 0.218 50 00 2 3.33 0.000 0.000 1.67 103.33 0.233 0.233 51.67 2 4 6.6? 001 0.001 3.33 106.66 0.2J8 0.248 53.33 4 6 10.00 0.002 0.002 5.00 310.00 0.264 0.264 55.00 6 8 13.33 0.004 0.004 6.67 113.33 0.280 280 56.67 8 10 16.67 0.006 0.006 8.33 116.66 0.297 0.297 58.33 10 12 20.00 0.009 0.009 10.00 120 00 0.314 0.314 60.00 12 14 23.33 0.012 0.012 11.67 123.33 0.332 0.332 61.67 14 16 26.67 0.015 0.015 13.33 126.66 0.350 0.350 63 33 16 18 30.00 0.019 0.019 15.00 130.00 0.3G8 0.368 65.00 18 20 33.33 0.024 0.024 16.67 133.33 0.388 0.388 66 67 20 22 36.67 0.029 0.029. 18.33 136.66 0.407 0.407 68.33 22 24 40.00 0.035 0.035 20.00 140.00 0.427 0.427 70 00 24 26 43.33 0.041 0.041 21.67 143.33 0.448 0.448 71.67 26 28 4667 0.048 0.048 23.33 146.66 0.469 0.469 73.33 28 30 50.00 0.054 0.054 25.00 15000 0.491 0.491 75.00 30 32 53 33 062 O.OC2 26.67 153.33 0.513 0.513 76.67 32 34 5667 0070 0070 28.33 156.66 0.536 0.536 78.33 34 36 6000 0.079 0.079 30.00 160.00 0.559 0.559 80.00 36 38 63.33 0.088 0.088 31.67 163.33 0.582 0.582 81.07 38 40 66.67 0.097 0.097 3333 166 66 0.606 6.606 83.33 40 42 70.00 0.107 0.107 35.00 J70.00 0.630 0.630 85.00 42 44 73.33 0.117 0.117 36. G7 173 33 0.655 0.655 86.67 44 46 76.6? 0128 0.128 38.33 176.66 0.681 0.681 88 33 46 48 80.00 0.140 0.140 40.00 180.00 0.706 0.706 90.00 48 50 83.33 0.151 0.151 41.67 18333 0.733 0.733 91.07 50 52 86 67 0.164 0.164 43.33 186.66 0.760 0.760 93.33 52 54 90 00 176 0.176 45.00 190.00 0.788 0788 95.00 54 56 9333 0.190 0.190 46.67 193.33 0.815 0.815 96.67 56 58 96.67 0.204 0.204 4833 196.66 0.844 0.844 98.33 58 60 100.00 0.218 0.218 5000 199.98 0.873 0873 100.00 60 FUNCTIONS OF A ONE-DEGREE CURVE 211 Jjo 3 9 L. C. M. E. T. L. C. M. E. T. / o 199.98 0.873 0.873 100.00 299.96 1.964 a. 964 150.07 2 203.31 0.902 0.902 101.67 303.29 2.008 2.009 151 74 2 4 206.64 0.932 0.932 103.34 306.62 2.053 2.054 153.41 4 6 209.97 0.9C2 0.962 105.01 309.95 2.098 2.099 155 08 6 8 213.31 0.993 0.993 106.68 313.29 2.143 2.144 156.75 8 10 216.64 .024 1.024 108.35 316.62 2.188 2.189 158.42 10 12 219.97 .056 1.056 110.02 319.95 2.235 2.236 160.09 12 14 223.30 .088 1.088 111.69 323.28 2.282 2.283 161.76 14 16 226.64 .121 1.121 113.36 326.62 2.329 2.330 163.43 16 18 229.97 .154 1.154 115.02 329.95 2.376 2.377 165.09 18 20 233.30 .188 1.188 116.69 333.28 ?.424 2.425 166.76 20 22 236.63 .222 1.222 118.36 336.61 2.473 2.474 168.43 22 24 239.97 .256 1.256 120.03 339.95 2.523 2.523 170.10 24 26 243.30 .292 1.292 121.70 343.28 2.572 2.573 171.77 26 28 246.63 .328 1.328 123.37 346.61 2.622 2.623 17344 28 30 249.96 .364 1.364 125.03 349.94 2.672 2.G73 175.10 30 32 253.29 .399 1.399 126.70 353.27 2.724 2.725 176.72 32 34 256.62 .437 1.437 128.37 356.60 2.776 2 777 178.39 34 36 259.96 .475 1.475 130.04 359.94 2.828 2.829 180.06 36 38 263.29 .513 1.513 131.71 363.27 2.880 2.881 181.73 38 40 266.62 .552 1.552 133.38 366.60 2.933 2.934 183.40 40 42 269.96 .592 1.592 135.05 369.94 2.987 2.988 185.07 42 44 273.29 .632 1.632 136.72 373.27 3.042 3 043 186.74 44 46 276.62 .672 1.672 138.38 376.60 3.096 3 097 188.40 46 43 279.96 .712 1.712 140.05 379.94 3.151 3.152 190.07 48 50 283.29 .752 1.752 141.72 383.27 3.206 3.207 191.74 50 52 286.62 .794 1.794 143.39 386.60 3.263 3.264 193.41 52 54 289.96 .836 1.836 145.06 389.94 3.320 3.321 195.08 54 56 293.29 .878 1.878 146.73 393.27 3.377 3 378 196.75 56 58 296.62 .921 1.921 148.40 396.60 3.434 3.435 198.42 58 60 299.96 .964 1.964 150.07 399.94 3.491 3.492 2uO 09 60 4 6 L. C. M. E. T. L. C M. E. T. 8 399.94 3.491 3.492 200.09 499.88 5.454 5.459 250.17 2 403.27 3.550 3.551 201.76 503.21 5.&27 5.o33 251.84 2 4 406.00 3.609 3.610 203.43 506.54 5.601 5.607 253.51 4 6 409.93 3.668 3.670 205.10 509.87 5.675 5681 255.18 6 8 413.26 3.727 3.730 206.77 513.20 5.749 5.755 256.85 8 10 416.59 3.787 3.790 208.44 51653 5.823 5.829 258.52 10 12 419.92 3.848 3.851 210.11 519.86 5.899 5.905 260.20 12 14 423.26 3.910 3.913 211.77 523.19 5.975 5 981 261.86 14 16 426.59 3.972 3.975 213.45 526.52 6.052 6.058 263.54 16 18 429.92 4.034 4.037 215.11 529.85 6.129 6.135 265.20 18 20 433.25 4.096 4.099 216.78 533.18 6.206 6.212 266.87 20 22 436.58 4.160 4.163 218.45 536.51 6.284 6.290 268.54 22 24 439.91 4.224 4.227 220.12 539.84 6.362 6.369 270.21 24 26 443.24 4.288 4.291 221.79 543.17 6.441 6.448 971.88 26 28 446.58 4.353 4.356 223.46 546.50 6.520 6.527 273.54 28 30 449.91 4.418 4.421 225.13 549.83 6.599 6.606 275.21 30 32 453.24 4.484 4.487 226.80 553 17 6.680 6.687 276.88 32 34 456.57 4.550 4.554 228.47 556.50 6.761 6.768 278.55 34 .36 45990 4.617 4.621 230.14 559.83 6.842 6.849 280.23 36 38 463.23 4.684 4.G88 231.81 563.16 6.923 6.931 281.90 38 40 466.56 4.751 4.755 23348 566.49 7.005 7013 283.57 40 42 469.89 4.820 4.824 235.15 569.82 7.088 7.096 285.24 42 44 473.23 4.889 4.6Q3 236.82 573.15 7 171 7.180 28691 44 46 476 56 4.958 4 962 238.48 576.48 7 255 7.264 288.59 46 48 479.89 5.027 5.031 240.15 579.81 7339 7.348 290.26 48 50 483.22 5.096 5 100 241.82 583.14 7423 7.432 291.93 50 52 486.55 5.167 5.171 243.49 586.47 7.508 7.517 293.60 52 54 489 88 5.238 5.243 245.16 589.80 7.593 7.603 295.27 54 56 493.21 5.310 5.315 24683 593.13 7.678 7 689 296 95 56 58 496 54 5.382 5.387 248.50 596.46 7.764 7 775 298.62 58 60 499.88 5 454 5.459 250.17 599.80 7.850 7 861 300.30 60 212 FUNCTIONS OF A ONE-DEGREE CURVE 6 7 / L. C. M. E. T. L. C. M. E. T. / 599.80 7.850 7.861 300.30 699.60 10.69 10.71 350.44 2 603.13 7.940 7.951 301.97 702.93 10.79 10.81 352.11 2 4 606 46 8.030 8.041 303.64 706.26 10.90 10.92 353.79 4 6 609.78 8.120 8.131 305.31 709.58 11.00 11.02 355.46 6 8 613.11 8.210 8.221 306.98 712.91 11.11 11.13 357.13 8 10 616.44 8.300 8.311 308.65 716.24 11.21 11.23 35881 10 12 619.76 8.390 8.401 310.32 719.56 11.31 11.33 360.48 12 14 623.09 8.480 8.491 311.99 722.89 11.42 11.44 362.15 14 16 626.42 8.570 8.581 313.66 726.21 11.52 11.54 36383 16 13 629.74 8.660 8.671 315.33 729.53 11.63 11.65 365.50 18 20 633.07 8.750 8.761 317.00 732.86 11.73 11.75 367.17 20 22 636.40 8.844 8.856 318.67 736.19 11.84 11.86 368.85 22 24 639.72 8.939 8.951 320.34 739.51 11.95 11.97 370.52 24 26 643.05 9.033 9.046 322.01 742.84 12.06 1208 37219 26 28 646.38 9.128 9.141 323.68 746.17 12.17 12.19 373.80 28 30 649.70 9.222 9.236 325 35 749.49 12.27 12.30 375.54 30 32 653.03 9.317 9.331 327.02 752.82 12.38 12.41 377.22 32 34 656.36 9.411 9.426 328.69 756.15 12.49 12 52 378. H9 34 36 659.69 9.506 9.521 330.37 759.47 12.60 12.63 380 57 36 38 663.02 9.600 9.616 332 04 762.80 12.71 12.74 382.24 38 40 666.34 9.695 9.712 333.71 766.13 12.82 12.85 383.92 40 42 669.67 9 794 9.812 335.38 769.45 12.93 12.96 385.60 42 44 673.00 9.894 9.912 337.05 772.78 13.04 13.08 387.27 44 46 676.32 9.993 10.01 338.78 776.11 13.15 13.19 388.95 46 48 679.65 10.09 10.11 340.40 779.43 13.26 13.31 390.62 48 50 68298 10.19 10.21 342.07 782.76 13.37 13.42 392.30 50 52 686.30 10.29 10.31 343.74 786.09 13.48 13.53 393.98 52 54 68963 10.39 10.41 345.41 789.41 13.59 13.65 395.65 54 56 692.96 10.49 10.51 347.08 792.74 13-. 70 13.76 397.33 56 58 696.28 10.59 10.61 348.76 796.07 13.81 13.88 399.01 58 60 699.60 10.69 10.71 350.44 799.40 13.96 13.99 400.70 60 8 9 L. C. M. E. T. L. C. M. E. T. 799.40 13.96 13.99 400.70 899.10 r.C6 17.71 450.95 2 802.72 14.07 14.10 402.37 902.42 17.79 17.84 452.63 2 4 806.04 14.19 14.22 40405 905.74 17.92 17.98 454.31 4 6 809.37 14.31 14.34 405.72 909.07 18.06 18.11 455.98 6 8 812.69 1443 14.46 407.39 912.39 18.19 18.25 457.66 8 10 816.01 14.55 14.58 409.06 915.71 18.32 18.38 459.34 10 12 819.34 14.66 14.70 410.74 919.04 18.46 18.52 461.02 12 14 822.66 14.78 14.82 412.41 922.36 18.59 18.65 462.70 14 16 825.98 14.90 14.94 414.08 925.68 18.72 18.79 464.37 16 18 829.31 15.02 15.06 415.75 929.01 18.86 18.92 466.05 18 20 832.63 15.14 15.18 417.43 932.33 18.99 19.06 467.73 20 22 835.95 15.26 15.30 419.10 935.65 19.12 19.19 469.41 22 24 839.28 15.38 15.43 420.77 938.98 19.26 19.33 471.08 24 26 842.60 15.51 15.55 422.45 942.30 19.40 19.47 472.76 26 28 845.92 15.63 15.68 424.12 945.62 19.54 19.61 474.43 28 30 849.25 15.75 15.80 425.79 948.95 19.68 19.75 476.10 30 32 852.57 15.88 15.93 427.47 952.27 19.82 19.89 477.78 32 34 855.89 16.00 16.05 429.15 955.59 19.96 20.03 479.46 34 36 859.22 16.12 16.18 430.82 958.92 20.10 20.17 481.14 36 38 862.54 16.25 16.30 432.50 962.24 20.24 20.31 482.83 38 ' 40 865.86 16.38 16.43 434.18 965.56 20.38 20.45 484.51 40 42 869.19 16.50 16.55 435.86 968.89 20.52 20.59 486.19 42 44 872.51 16.63 16.68 437.54 972.21 20.66 20.74 487.87 44 46 875.83 16.76 16.81 439.21 975.53 20.80 20.88 489.56 46 48 879.16 16.89 16.94 440.89 978.86 20.94 21.03 491.24 48 50 882.48 17.02 17.07 442.57 982.18 21.09 21.17 492.92 50 52 885.80 17.14 17.19 444.25 985.50 21.23 21.31 494.60 52 54 889.13 17.27 17.32 445.93 988.83 21.37 21.46 496.28 54 56 892.45 17.40 17.45 447.60 992.15 21.51 21.60 497.96 56 58 895.77 17.53 17.58 449.28 995.47 21.65 21.75 499.65 58 60 899.10 17.66 17.71 450.95 998.80 21.80 21.89 501.32 60 FUNCTIONS OF A ONE-DEGREE CURVE 213 10 11 ' L C. M. E. T. L. C. M. E. T. / 998.8 21.80 21.89 501.32 1098.4 26.38 26.50 551.74 2 1002.1 21.94 22.03 503.00 1101.7 26.54 26.86 553.42 2 4 1005.4 22.09 82.18 504.68 1105.0 26.70 26.83 555.10 4 6 1008.8 22.24 22.33 506.36 1108 3 26.86 26.99 556.78 6 8 1012.1 22.39 22.48 508.04 1111.7 27.02 27.16 558.46 8 10 1015.4 22.54 22.63 509.72 1115.0 27.19 27.32 560.14 10 12 1018.7 22.68 22.78 511.40 1118.3 27.35 27.48 561.82 12 14 1022 32.83 22.93 513.08 1121.6 27.51 27.65 563.50 14 16 1025.4 22.98 23.08 514.76 1124.9 27.67 27.81 565.18 18 18 1028.7 23.13 23.23 516.44 1128.2 27.83 27.98 566.86 18 20 1032.0 23.28 23.38 518 12 1131 6 28.00 28.14 568.54 20 22 1035.3 23.43 23.53 519.80 11349 26.17 28.30 570.22 22 24 1038.6 23.58 23 68 521.48 1138.2 28.34 28.47 571.90 24 26 1042.0 23.73 23.84 523.16 1141.5 28.50 2864 573.58 26 28 1045.3 23.88 23.99 524.85 1144.8 28.67 28.81 575 27 28 30 1048.6 24.04 24.14 526.53 1148.1 2884 28.98 576.95 30 32 1051.9 24.19 24.30 528.21 1151.5 29.00 29.14 578.63 32 34 1055.2 24.34 24.45 529.89 1154.8 29.17 29.31 580.32 34 36 1058.6 24.49. 24.60 531.57 1158.1 29.34 29.48 582.00 36 .38 1061.9 24.64 24.76 53325 1161.4 29.50 29.05 583.69 38 40 1065.2 24.80 24.91 534.93 1164.7 29.67 29.82 585.37 40 42 1068.5 24.95 25.06 536.65 1168.0 29.84 29.99 587.05 42 44 1071.8 25.11 25.22 53829 1171 4 30.01 30.17 588.74 44 46 1075.2 25.27 25.38 539.97 1174.7 30.18 30.34 590.42 46 48 1078.5 25.43 25.54 541.65 1178.0 30.35 30.52 592.11 48 50 1081.8 25.59 25.70 54333 1181.3 30.53 3069 593.79 50 52 1085.1 25.74 25.86 545.01 1184.6 30.70 30.86 595 47 52 54 1088.4 25.90 26.02 546.69 1187.9 30.87 31.04 597.16 54 56 10U1.8 26.06 26.18 548.37 1191.3 31.04 31.21 598.84 56 58 1095.1 26.22 26.34 550 06 1104.6 31.21 31.39 600.53 58 60 1098.4 26.38 26.50 551.74 1197.9 31.39 31.56 602.22 60 12 13 L. C. M. E. T. L. C. M. E. T. 1197.9 31.39 31.56 602.22 1297.3 36.83 37.07 652.87 I 1201.2 31.57 31.73 603.91 1300.6 3". 02 37.26 654.56 2 4 1204.5 31.74 31.91 605.60 1303.9 3". 21 37.46 656.25 4 6 1207.8 31.93 32.09 607.28 1307.2 3". 40 37.65 657.93 6 8 1211 1 32.09 32.27 608.97 1310.5 3 .59 37.85 659.62 8 10 1214.5 32.27 32.45 610.66 1313.8 3-79 38.04 661.31 10 12 1217.8 32.45 32.63 612.35 1317.2 3". 98 38.23 663. CO 12 14 1221.1 32.62 32.81 614.04 1*20.5 38.17 38.43 664.69 14 16 1224.4 32.80 32.99 615.72 1323.8 38.86 38. C2 666.37 16 18 1227.7 32.97 33.17 617.41 1327.1 38.55 38.82 668.06 18 20 1231.0 33.15 33.35 619.10 1330.4 38.75 39.01 669.75 20 ss 1231.3 33.33 33.53 6'.!0.79 1333.7 38.95 39. 2C 671.44 22 24 1237.7 33.51 33.72 622 48 1337.0 39.15 39.40 673.13 24 26 1241.0 33.69 33.90 624.16 1340.3 39.35 39.60 674.81 26 28 1244.3 33.87 34.09 6. '5. 85 1343.6 39.54 39.80 676.51 28 30 1247.6 34.06 34.27 627.55 1346.9 39.74 40.00 678.20 30 32 1250.9 34.24 34 45 629.24 1350.3 39.94 40.19 679.89 32 34 1254.2 34.42 34 64 630.93 1353.6 40.13 40.39 681.58 34 36 1257.5 34,60 34 82 632.61 1356.9 40.33 40 59 683.26 36 38 1260.8 34.78 35.01 634.30 1360.2 40.52 40.79 684.95 38 40 1264.2 34 97 35.19 635.99 1363,5 40.71 40.99 686.64 40 42 1267.5 35.16 35.37 637.68 1366.8 40.91 41.19 68833 42 44 1270.8 36.34 35.56 639.37 1370.1 41.11 41.40 690.02 44 I 46 1274.1 35.53 35.75 641.05 1373.4 41.31 41.60 691.70 46 48 1277.4 35.71 85.94 642.74 1376.7 41.51 41.81 693.39 48 50 1280.7 35.90 36.13 644.43 1380.0 41.71 42.01 695.08 50 52 1284.0 36.09 36.31 646.12 1383.4 41.91 4221 696.77 52 54 1287.4 36.27 36.50 647.81 1386.7 42.11 42.42 698.46 54 56 1290.7 36.46 30.09 619.49 1390.0 4231 42.62 700.14 56 58 1294.0 36.64 36.88 651.18 1393.3 42.51 42.83 701.83 58 60 1297.3 36.83 3r.07 652.87 1396.6 42.71 43.03 703.53 60 214 FUNCTIONS OF A ONE-DEGREE CURVE 14 16 / L. C. M. E. T. L. C. M. E. T. / 1396.6 42.71 43.03 ?03.53 1495.9 49.02 49.44 754.85 2 1399.9 42.92 43.23 705.23 1499.2 49.24 49.66 756.05 2 4 1403.2 43.12 43.44 706.92 1502.5 49.46 49.89 757.74 4 6 1406.5 43.33 43.65 708.62 1505.8 49.68 50.11 759.44 6 8 1409.8 43.53 43.86 710.31 1509.1 4990 50.34 761.13 8 10 1413.1 43.74 44.07 712.01 1512.4 50.12 50.56 762.83 10 12 1416.5 43.94 44.28 713.71 1515.7 50.34 50.78 764.53 12 14 1419.8 44.15 44.49 715.40 1519.0 50.56 51.01 766.22 14 16 1423.1 44.35 44.70 717.10 1522.3 5078 51.23 767.92 16 18 1426.4 44.56 44.91 718.79 1525.6 51.00 51.46 769.61 18 20 1429.7 44.77 45.12 720.49 1528.9 51.22 51.68 771.31 20 22 1433.0 44 98 45.33 722.20 1532.2 51.44 51.90 773.01 22 24 1436.3 45.19 45.54 723.89 1535.5 51.67 52.13 774.70 24 26 1439.6 45.40 45.76 725.59 1538.8 51.89 52.36 776.40 26 28 1442.9 45.61 45.97 727.28 1542.1 52.12 52.59 778.09 28 30 1446 2 45.82 46.18 728.97 1545.4 52.34 52.82 779.79 30 32 1449.6 46.03 46.40 730.66 1548.7 52.57 53.05 781.49 32 34 1452.9 4^.24 46.61 732.35 1552.0 52.79 53.28 783.19 34 36 1456.2 46.45 46.82 734.05 1555.3 53.02 53.51 784.89 36 38 14M).5 46.66 47.04 785.74 1558.6 53.24 53.74 786.59 38 40 1462.8 46.87 47.25 737.43 1561.9 53.47 53.97 788.29 40 42 1460.1 47.08 47.46 739.12 1585.2 53.69 54.20 789.99 42 44 1469.4 47.30 47.68 740.81 1568.5 53.92 54.44 791.69 44 46 1472.7 47.51 47.90 742.51 1571.8 54.15 54.67 793.39 46 48 1476.0 47.73 48.12 744.20 1575.1 54.38 54.91 795.09 48 50 1479.3 47.94 48.34 745.89 1578.4 54.61 55.14 796.79 50 52 1482.7 48.16 48.56 747.58 1581.7 54.84 55.37 798.49 52 54 1486.0 48.37 48.78 749.27 1585.0 55.07 55.61 800.19 54 56 1489.3 48.59 49.00 750.97 1588.3 55.30 55.84 801.89 56 58 1492.6 48.80 49.22 752.66 1591.6 5553 56.08 803.59 58 60 1495.9 49.02 49.44 754.35 1594.9 55.76 56.31 805.29 60 16 17 L. C. M. E. T. L. C. M. E. T. / 1594.9 55.76 56.31 805.29 1693.9 62.94 63.64 856.35 2 1598.2 55.99 56.54 806.99 1697.2 63.18 63.89 858.05 2 4 1601.5 56. S3 56.78 808.64 1700.5 63.43 64.15 859.76 4 6 1604.8 56.46 57.02 810.39 1703.8 63.68 64.40 861.46 6 8 1608.1 56.70 57.26 812.09 1707.1 63.93 64.66 863.16 8 10 1611.4 56.93 57.50 813.79 1710.4 64.18 64.91 864.87 10 12 1614.7 57.17 57.74 815.49 1713.7 64.42 65.16 866.57 12 14 1618.0 57.40 57.98 817.19 1716.9 64.67 65.42 868.27 14 16 1621.3 57.64 58.22 818.89 1720.2 64.92 65.67 869.98 16 18 1624.6 57.87 58.46 820.59 1723.5 65.17 65.93 871.68 18 20 1627.9 58.11 58.70 822.29 1726.8 65.42 66.18 873.38 20 22 1631.2 58.34 58.94 823.99 1730.1 65.67 66.43 875.09 22 24 1634.5 58.58 59 19 825.69 1733.4 65.93 66.69 876.79 24 26 1637.8 58.82 59.43 82739 1736.7 66.18 66.95 878.49 26 28 1641.1 59.06 59.68 829.09 1740.0 66.44 67.21 880.20 28 30 1644.4 59.30 59 92 83079 1743.3 66.69 67.47 881.90 30 32 1647.7 59.54 60.16 832.49 1746.6 66.94 67.72 883.61 32 34 1651.0 59.78 6041 834.20 1749.9 67.20 67.98 885.32 34 36 1654.3 60.02 60.65 835.90 1753.2 67.45 68.24 887.02 36 38 1657.6 60.26 60.90 837.61 1756.5 67.71 68.50 888.73 38 40 1660.9 60.50 61.14 839.31 1759.8 67.96 68.76 890.44 40 42 1664.2 60.74 61.39 841.01 1763.1 68.21 69.03 892.15 42 44 1667.5 60.99 61.64 842.72 1766.3 68.47 69.29 893.86 44 46 1670.8 61.23 61.89 844.42 1769.6 68.73 69.56 895.56 46 48 1674.1 61.48 62.14 846.13 1772.9 68.99 69.82 897.27 48 50 1677.4 61.72 62.39 847.83 1776 2 69.25 70.09 898.98 50 52 1680.7 61.96 6-^.64 849.53 1779.5 69.50 70.36 900.69 52 54 1684.0 62.21 62.89 851 24 1782.8 69.76 70.62 902.40 54 56 1687.3 62.45 63.14 852.94 1786.1 70.02 70.89 904.10 56 58 1690.6 62.70 63 39 854.65 1789.4 70.28 71.17 905.81 58 60 1693.9 62.94 63 64 856 35 1792.7 70.54 71.42 907.52 60 FUNCTIONS OF A ONE-DEGREE CURVE 215 18 19 / L. C. M. E. T. L. C. M. E. T. / 1792.7 70.54 71.42 907.52 1891.5 78.58 79.65 958.86 2 1796.0 70.80 71.69 909.23 1894.8 78.86 79.94 960.57 2 4 1799.3 71.06 71.96 910.94 1898.1 79.13 80.22 962.30 4 6 1802.6 71.33 72.23 912.65 1901.3 79.41 80.51 964.00 6 8 1805.9 71.59 72.50 914.36 1904.6 79.68 80.79 965.72 8 10 1809.2 71.85 72.77 916.07 1907.9 79.96 81.08 967.43 10 12 1812.5 72.12 73.04 917.78 1911.2 80.24 81.37 969.15 12 14 1815.7 72.38 73.31 919.49 1914.5 80.51 81.65 970.86 14 16 1819.0 72.64 73.58 921.20 1917.8 80.79 81.94 972.58 16 18 1822.3 72 91 73.85 922.91 1921.0 81.07 82.22 974.29 18 20 1825.6 73.17 74.12 924.63 1924.3 81.35 82.51 976.01 20 22 1828.9 73.43 74.39 926.34 1927.6 81.63 82.80 977.72 22 24 1832.2 73.70 74.67 928.05 1930.9 81.91 83.09 979.44 24 26 1835.5 73.97 74.94 929.76 1934.2 82.20 83.38 981.15 26 28 1838.8 74.24 75.22 931.47 1937.5 82.48 83.67 982.86 28 30 1842.1 74.51 75.49 933.18 1940.7 82.76 83.97 984.58 30 32 1845.4 74.77 75.77 934.89 1944.0 83.05 84.26 986.30 32 34 1848.7 75.04 76.04 936.60 1947.3 83.33 84.55 988.02 34 36 1852.0 75.31 76.32 938.32 1950.6 83.61 84.84 989.74 36 38 1855.3 75.58 76.59 940.03 1953.9 83.90 85.13 991.46 38 40 1858.6 75.85 76.87 941.74 1957.2 84.18 85.43 993.18 40 42 1861.9 76.12 77.14 943.45 1960.4 84.47 85.73 994.90 42 44 1865.1 76.39 77.42 945.16 1963.7 84.75 86.02 996.62 44 46 1868.4 76.67 77.70 946.88 1967.0 85.04 86.32 998.34 46 48 1871.7 76.94 77.98 948.59 1970.3 85.32 86.61 1000.0 48 50 1875.0 77.21 78.26 950.30 1973.6 85.61 86.91 1001.8 50 52 1878.3 77.49 78.53 952.01 1976.9 85.90 87.21 1003.5 52 54 1881.6 77.76 78.81 953.72 1980.1 86.19 87.50 1005.2 54 5(5 1884.9 78.03 79.09 955.44 1983.4 86.47 87.80 1006.9 56 58 1888.2 78.31 79.37 957.15 19S6.7 86.76 88.09 1008.6 58 60 1891.5 78.58 79.65 958.86 1990.0 87.05 88.39 1010.4 60 20 21 / L. C. M. E. T. L. C. M. E. T. / 1990.0 87.05 88.39 1010.4 2088.5 95.95 97.58 1062.0 2 1993.3 87.34 88.69 1012.1 2091.8 96.26 97.90 1063.7 2 4 1996.6 87.63 88.99 1013.8 2095.0 96.56 98.21 1065.4 4 6 1999.8 87.92 89.29 1015.5 2098.3 96.87 98.53 1067.2 6 8 2003.1 88.21 89.59 1017.2 2101.6 97.17 98.84 1068.9 8 10 2006.4 88.50 89.89 1019.0 2104.9 97.48 99.16 1070.6 10 18 2009.7 88.79 90.19 1020.7 2108.1 97.79 99.48 1072.4 12 14 2013.0 89.08 90.49 1022.4 2111.4 98.09 99.79 1074.1 14 16 20163 89.37 90.79 1024.1 2114.7 98.40 100.1 1075.8 16 18 2019.5 89.66 91.09 1025.8 2118.0 98.70 100.4 1077.5 18 20 2022.8 89.96 91.40 1027.6 2121.2 99.00 JOO. 7 1079.3 20 22 2026.1 90.25 91.71 1029.3 2124.5 99.30 J01.1 1081.0 22 24 2029-4 90.55 92.01 1031.0 2127.8 99.60 101.4 1082.7 24 26 2032.7 90.85 92.32 1032.7 2131.0 99.90 101.7 1084.4 26 28 2036.0 91.15 92.62 1034.4 2134.3 100.2 102.0 1086.2 28 30 2039.2 91.45 92.93 1036.1 2137.6 100.5 102.3 1087.9 30 32 2042.5 91.74 93.24 1087.9 2140.9 100.8 102.7 1089.6 32 34 2045.8 92.04 93.54 1039.6 2144.1 101.1 103.0 1091.3 34 36 2049.1 92.34 93.85 1041.3 2147.4 101.4 103.3 1093.1 36 38 2052.4 92.64 94.15 1043.0 2150.7 101.7 103.6 1094.8 38 40 2055.7 92.94 94.46 1044.8 2154.0 102.1 104.0 1096.5 40 42 2058.9 98.24 94.78 1046.5 2157.2 102.4 104.3 1098.3 42 44 2062.2 93.54 95.09 1048.2 2160.5 102.7 104.6 1100.0 44 46 2065.5 93.84 95.40 1049.9 2163.8 103.0 104.9 1101.7 46 48 2068.8 94.14 85.71 1051.7 2167.1 103.3 105.3 1103.4 48 50 2072.1 94.44 96.03 1053.4 2170.3 103.6 105.6 1105.2 50 52 2075.4 94.74 96.34 1055.1 2173.6 103.9 105.9 1106.9 52 54 2078.6 95.04 96.65 1056.8 2176.9 104.2 106.3 1108.6 54 56 2081.9 95.34 96.96 1058.6 2180.1 104.5 106.6 1110.3 56 58 2085.2 95.64 97.27 1060.3 2183.4 104.8 106.9 1112.1 58 60 2088.5 95.95 97.58 1062.0 2186.7 105.2 107.2 1118.8 60 216 FUNCTIONS OF A ONE-DEGREE CURVE 22 23 L. C. M. E. T. L. G. M. E. T. 2186.7 105.2 107.2 1113.8 2284.8 115.0 117.4 1165.8 2 2190.0 105.6 107.6 1115.5 2288.1 115.3 117.7 1167.5 2 4 2193.2 105.9 107.9 1117.3 2291.3 115.7 118.1 1169.2 4 6 2196.5 106.2 108.2 1119.0 2294.6 116.0 118.4 1171.0 6 8 2199.8 106.5 108.6 1120.7 2297.8 116.4 118.8 1172:7 8 10 2203.0 106.8 108.9 1122.4 2301.1 116.7 119.1 1174.4 10 12 2206.3 107.1 109.2 1124.2 2304.4 117.0 119.5 1176.2 12 14 2209.6 107.4 109.6 1125.9 2307.6 117.4 119.8 1177.9 14 16 22129 107.7 109.9 1127.6 2310.9 117.7 120.2 1179.7 16 18 2216.1 108.0 110.2 1129.4 2314.1 118.1 120.5- 1181.4 18 20 2219.4 108.4 110.6 1131.1 2317.4 118.4 120.9 1183.1 20 22 2222.7 108.7 110.9 1132.8 2320.7 118.7 121.2 1184.9 22 24 2225.9 109.0 111.2 1134.6 2323.9 119.1 121 6 1186.6 24 26 2229.2 109.4 111.6 1136.3 2327.2 119.4 121.9 1188.4 26 28 2232.5 109.7 111.9 1138.0 2330.4 119.8 122.3 1190.1 28 30 2235.7 110.0 112.3 1139.7 2333.7 120.1 122.6 1191.8 30 32 2239.0 110.4 112.6 1141.5 2337.0 120.4 123.0 1193.6 82 34 22423 110.7 112.9 1143.2 2340.2 120.8 123.3 1195.3 34 36 2245.6 111.0 113.3 1144.9 2343.5 121.1 123.7 1197.1 36 38 2248.8 111.4 113.6 1146.7 2346.7 121.5 124.1 1198.8 38 40 2252.1 111.7 113.9 11J8.4 2350.0 121.8 124.4 1200.5 40 42 2255.4 112.0 114.3 1150.1 2353.3 122.1 124.8 1202.3 42 44 2258.6 112.3 114.6 1151.9 2356.5 122.5 125.1 1204.0 44 46 2261.9 112.7 115.0 1153.6 2359.8 122.8 125.5 1205.8 46 48 2265.2 113.0 115.3 1155.4 2303.0 123.2 125.8 1207.5 48 50 2268.4 113.3 115.7 1157.1 2366.3 123.5 126.2 1209.2 50 52 2271.7 113.7 116.0 1158.8 2369.6 123.8 126.6 1211.0 52 54 2275.0 114.0 116.3 1160.6 2372.8 124.2 126.9 1212.7 54 56 2278.3 114.3 116.7 116-2.3 2376.1 124.5 127.3 1214.5 56 58 2281.5 114.7 117.0 1164.0 2379.3 124.9 127.6 1216.2 58 60 2284.8 115.0 117.4 11G5.8 2382.6 125.2 128.0 1218.0 60 24 25 L. 0. M. E. T. L C. M. E. T. 23S2.6 125.2 128.0 1218.0 2480.4 135.8 139.1 1270.3 2 2385.9 125.5 128.4 1219.7 2483.6 JJ6.2 139.5 1272.0 2 4 2389.1 125.9 128.7 1221.4 2486.9 136.5 139.9 1273.8 4 6 2392.4 126.2 129.1 1223.2 2490.1 136 9 140.3 1275.5 6 8 2395.6 126.6 129.5 1224.9 2493.4 137.2 140.6 1277.3 8 10 2398.9 126.9 129.8 1226.7 2496.6 137.6 141.0 1279.0 10 12 2402.2 127.3 130.2 1228.4 2499.9 138.0 141.4 1280.8 12 14 2405.4 127.6 130.6 1230.2 2503.1 1383 141 8 1282.5 14 1 2408.7 128.0 130.9 1231.9 2506.4 138.7 142.2 1284.3 16 18 2411.9 128.3 131.3 1233.6 2509.6 139.0 142.5 1286.1 18 20 2415.2 128.7 131.7 1235.4 2512.9 139.4 142.9 1287.8 20 22 2418.5 129.0 132.0 1237.1 2516.1 139.8 143.3 1289.6 22 24 2421.7 129.4 132.4 1233.9 2519.4 140.1 143.7 1291.3 24 26 2425.0 129.7 132.8 1240.6 2522.6 140.5 144.1 1293.1 26 28 2428.2 130.1 133 1 1242.4 2525.9 140.8 144.5 1294.8 28 30 2431.5 130.4 133.5 1244.1 2529 1 141.2 144.9 1296.6 30 32 2434.8 130.8 133.9 1215.8 2532.4 141.6 145.3 1298.3 32 34 2433.0 131.1 134.2 1247.6 2535.6 142.0 145.6 1300.1 34 36 2441.3 131.5 134.6 1249.3 2538.9 142.3 146.0 1301.8 36 38 2444.5 131.8 135.0 1251.1 2542.1 142.7 146.4 1303.6 38 40 2447.8 132.2 135.4 1252.8 2545.4 143.1 146.8 1305.3 40 42 2451.1 132.6 135.7 1254.6 2548.6 143.5 147.2 1307.1 42 44 2454.3 132.9 136.1 1256.3 2551.9 143.8 147.6 1308.8 44 46 2457.6 133.3 136.5 1258.1 2555.1 144.2 148.0 1310.6 46 48 2460.S 133.6 136.9 1259.8 2558.4 144.5 148 4 1312.4 48 50 2464.1 134.0 137.2 1261.5 2561.6 144.9 148.8 1314.1 50 52 2467.4 134.4 137.6 1263.3 2564.9 145.3 149.2 1315.9 52 54 2470.6 134.7 138.0 1265.0 2568.1 145.7 149.5 1317.6 54 56 2473.9 135.1 138.4 1266.8 2571.4 146.0 149.9 1319.4 56 58 2477.1 135.4 138.7 1268.5 2574.6 146.4 150.3 1321.1 58 60 2480.4 135.8 139.1 1270.3 2577.9 146.8 150.7 1322.9 60 FUNCTIONS OF A ONE-DEGREE CURVE 217 26 27* / L. C. M. . E. T. L. C. M. E. T. / 2577.9 146.8 150.7 1322.9 2675.3 158.3 162.8 1375.6 2 2581.1 147.1 151.1 1324.6 2678.5 158.6 163.2 1377.4 2 4 2584.4 147.5 151.5 1326.4 2681.8 159.0 163.7 1379.2 4 6 2587.6 147.9 151.9 1328.1 2685.0 159.4 164.1 1380.9 6 8 2590.9 148.3 152.3 1329.9 2688.2 159.8 164.5 1882 7 8 10 2594.1 148.7 152.7 1331.6 2691.5 160.2 164.9 1384.5 10 12 2597.4 149.1 153.1 1333.4 2694.7 160.6 165.3 1386.2 12 14 2600.6 149.4 153.5 1335.2 2698.0 181.0 165.7 1388.0 14 16 2603.9 149.8 153.9 1336.9 2701.2 161.4 1661 1389.8 16 18 2607.1 150.2 154.3 1338.7 2704.4 161.8 166.5 1391.5 18 20 2610.4 150.6 154.7 1340.4 2707.7 162.2 167.0 1393.3 20 22 2613.6 151.0 155.1 1342.2 2710.9 162.6 167.4 1395.0 22 24 2616.9 151.4 155.5 1343.9 2714.1 163.0 167.8 1396.8 24 26 2620.1 151.7 155.9 1345.7 2717.4 163.4 168.2 1398.6 26 28 2623.4 152.1 156.3 1347.4 2720.6 163.8 168.6 1400.3 28 30 2626.6 152.5 156.7 1349.2 2723.8 164.2 169.1 1402.1 30 32 2629.8 152.9 157.1 1351.0 2727.1 164.6 169.5 1403.9 32 34 2633.1 153.3 157.5 1352.7 2730.3 165.0 169.9 1405.6 34 36 26363 153.7 157.9 1354.5 2733.6 165.4 170.3 1407.4 36 38 2639.6 154.0 158.3 1356.2 2736.8 165.8 170.8 1409.2 38 40 2642.8 154.4 158.7 1358.0 2740.0 166.2 171.2 1410.9 40 42 2646.1 154.8 159.1 1359.8 2743.3 166.6 171.6 1412.7 42 44 2649.3 155.2 159.5 1361.5 2746.5 167.0 172.0 1414.5 44 46 2652.6 155.6 160.0 1363.3 2749.7 1674 172.5 1416.3 46 48 2655.8 156.0 160.4 1365.1 2753.0 167.8 172.9 1418.0 48 50 2659.1 156.3 160.8 1366.8 2756.2 168.2 173.3 1419.8 50 52 2662.3 156.7 161.2 1368.6 2759.5 168.6 173.7 1421.6 52 54 2665.6 157.1 161.6 13~0.4 2762.7 169.0 174.1 1423.3 54 56 2668.8 157 5 162.0 13?2.1 2765.9 169.4 174.6 1425.1 56 58 2672.1 157.9 162.4 1373.9 2769.2 169.8 175.0 1426.9 58 60 2675.3 158.3 162.8 1375.6 2772.4 170.2 175.4 1428.6 60 28 29 L. C. M. E. T. L. C. M. E. T. 2772.4 170.2 175.4 1428.6 2869.4 182.5 188.5 1481.9 2 2775.6 170.6 175.8 1430.4 2872.6 182.9 189.0 1483.7 2 4 2778.9 171.0 176.3 1432.2 2875.8 183 3 189.4 1485.4 4 6 2782.1 171.4 176.7 1434.0 2879.1 183.7 189.9 1487.2 6 8 2785.3 171.8 177.1 1435.7 2882.3 184.2 190.3 1489.0 8 10 2788.6 172.2 177.6 1437.5 2885.5 184.6 190.8 1490.8 10 12 2791.8 178.6 178.0 1439.3 2888.7 185.0 191.2 1492.6 12 14 2795.0 173.0 178.4 1441.1 2892.0 185.4 191.7 1494.3 14 16 2798.3 173.4 178.9 1442.8 2895.2 185.8 192.1 1496.1 16 18 2801.5 173.8 179.3 1444.6 2898.4 186.3 192.5 1497.9 18 20 2804.7 174.3 179.7 1446.4 2901.6 186.7 193.0 1499.7 20 22 2808.0 174.7 180.2 1448.2 2904.8 187.1 193.5 1501.5 22 24 2811.2 175.1 180.6 1449.9 2908.1 187.5 193.9 1503.2 24 26 2814.4 175.5 181.0 1451.7 2911.3 188.0 194.4 1505.0 26 28 2817.7 175.9 181 5 1453.5 2914.5 188.4 194.8 1506.8 28 30 2820.9 176.3 181.9 1455.2 2917.7 188.8 195.3 1S08.6 30 32 2824.1 176.7 182.3 1457.0 2921.0 189.2 195.7 1510.4 32 34 2827.4 177.1 182.8 1458.8 2924.2 189.7 196.2 1512.1 34 36 2830.6 177.5 183.2 1460.6 2927.4 190.1 196.7 1513.9 36 38 2833.8 177.9 183.6 1462.3 2930.6 190.5 197.1 1515.7 38 40 2837.1 178.4 184.1 1464.1 2933.9 190.9 197.6 1517.5 40 4-2 2840.3 178.8 184.5 1465.9 2937.1 191.4 198.0 1519.3 42 44 2843 5 179.2 185.0 1467.7 2940.3 191.9 198.5 1521.0 44 46 2846.8 179.6 185.4 1469.5 2943.5 192.4 198.9 1522.8 46 48 2850.0 1800 185.9 1471.2 2946.8 192.8 199.4 1524.6 48 50 2853.2 180.4 186.3 1473.0 2950 193.2 199.8 1526.4 50 52 2856.5 180.8 186.8 1474.8 2953.2 193.6 200.3 1528.2 52 54 2859.7 181.2 187.2 1476.6 2956.4 194.0 200.8 1530.0 54 56 2862.9 181.6 187.6 1478.3 2959.6 194.4 201.2 1531.7 56 58 2866.2 1820 188.1 14801 2962.9 194.8 201.7 1533.5 58 60 2869.4 1825 188.5 1481.9 2966.1 195.2 202.1 15353 60 218 FUNCTIONS OF A ONE-DEGREE CURVE 80 31 / L. C. M. E. T. L. C. M. E. T. 2966.1 195.2 202.1 1535.3 3062.6 208.4 2163 1589.0 2 2969.3 195.6 202.6 1537.1 3065.8 208.8 216.8 1590.8 2 4 2972.5 190.1 203.1 1538.9 3069.0 209.3 217.2 1592.6 4 6 2975.7 196.5 -203.5 1540.7 3072.2 209.7 217.7 1594.4 6 8 2^79 197.0 X'04.0 1542.5 3075.4 210.2 218.2 1596.2 8 10 2982.2 197.4 204.5 1544 3 3078.6 210.6 218.7 1598.0 10 12 2985.4 197.8 204.9 154G.O 3081.8 211.1 219.2 1599.8 12 14 2988.6 198.2 205.4 1547.8 3085.0 211.5 219.6 1601.6 14 16 2991.8 198.6 2059 1549.6 3088.3 212.0 220.1 1603.4 16 18 2995.0 199.1 2063 1551.4 3091.5 212.4 220.6 1605.2 18 20 2998.3 199.5 206.8 1553.2 3094.7 212.9 221.1 1607.0 20 22 3001.5 199.9 207.3 1555.0 3097.9 213.3 221 6 1608.8 22 24 3004.7 200.4 207.7 1556.8 3101.1 213.8 222.1 1610.6 24 26 3007.9 200.8 208.2 1558.6 3104.3 214.2 222.6 1612.4 26 28 3011.1 201.3 208.7 1560.4 3107.5 214.7 223.0 1614.2 28 30 3014.3 201.7 209.1 1562.2 3110.7 215.1 223.5 1616.0 30 32 3017.6 202.1 209.6 1564.0 3113.9 215.6 224.0 1617.8 32 34 3020.8 202.6 210.1 1565.7 3117.1 216.0 224.5 1619.6 34 36 3024.0 203.0 210.5 1567.5 3120.3 216.5 225.0 1621.4 36 38 3027.2 203.5 211.0 1569.3 3123.5 216.9 225.5 1623.2 38 40 3030.4 203.9 211 5 1571.1 3126.7 217.4 226.0 1625.0 40 42 3033.6 204.3 212 1572.9 3129.9 217.8 226.5 1626.8 42 44 3036.9 204.8 212.4 1574.7 3133.1 218.3 227.0 1628.6 44 46 3040.1 205.2 212.9 1576.5 3136.4 218.7 227.5 1630.5 46 48 3043.3 205.7 213.4 1578.3 3139.6 219.2 228.0 1632.3 48 50 3046.5 206.1 213.9 1580.1 3142.8 219.6 228.4 1634.1 50 52 3049.7 206.5 214.4 1581.9 3146.0 220.1 228.9 1635.9 52 54 3052.9 207 214.8 1583.7 3149.2 220.5 229.4 1637.7 54 56 3056.2 207.4 215.3 1585.5 3152.4 221.0 2- ) 9.9 1639.5 56 58 3059.4 207.9 215.8 1587.2 3155.6 221.5 230.4 1641.3 58 60 3062.6 208.4 216 3 1589.0 3158.8 2-J2.0 230.9 1643.1 60 32 33 L. C. M. E. T. L. C. M. E. T. o 3158.8 222.0 230.9 1G43.1 3254.9 236.0 246.1 1697 3 2 3162.0 2225 231.4 1644.9 3258 1 236.4 246.6 1699.1 2 4 3165.2 222.9 231.9 1646.7 3261.3 236.9 247.1 17009 4 6 3168.4 223.4 232.4 1648.5 3264.5 237.4 247.7 1702.7 6 8 3171.6 223.8 232.9 1650.3 3267 7 237.9 248.2 1704.5 8 10 3174.8 224.3 233.4 1652.1 3270.8 238.4 248.7 1706.4 10 12 3178.0 224 8 233.9 1653.9 3274 238 9 249.2 1708.2 12 14 3181.2 225.2 234.4 16557 3277 2 239 3 249.7 1710.0 14 16 3184.4 225.7 234.9 1657.5 32804 239.8 250.2 1711.8 16 18 3187.6 226.1 235.4 1659.3 3-J83.6 240.3 250.8 1713.6 18 20 3190.8 226.6 235.9 1661.1 3286.8 240.8 251.3 1715.5 20 22 3194.0 227.1 236.4 1662.9 3290 241.2 251.8 1717.3 22 24 3197 2 227.5 236 9 1664.7 3293.2 241 7 252.3 1719.1 24 26 3200.4 228.0 237.4 1666 5 3296.4 242.2 252.9 1720.9 26 28 3203.6 228.4 237.9 1668.3 3299.6 242.7 253.4 1722.7 28 30 3206.8 228 9 238.4 1670.1 3302.7 2432 253.9 1724.6 30 32 3210.0 229.4 239.0 1671.9 3305 9 243.6 254.4 1726.4 32 34 3213.2 229 8 239.5 1673.7 3309 1 244.1 255.0 1728.2 34 36 3216.5 230.3 240.0 1675.5 3312.3 244.6 255.5 1730.0 36 88 8219.7 230.7 240.5 1677.4 33155 245.1 256.0 1731.8 38 40 3222.9 231 2 241.0 1679.2 3318.7 245.6 256 5 1733.6 40 42 32261 231.7 241.5 1681.0 3321.9 2460 2571 1735.5 42 44 3629.3 232.2 242.0 1682.8 3325.1 246.5 257.6 1737.3 44 46 3232.5 232.6 242.5 1684.6 3328.3 247.0 258.1 1739.1 46 48 3235.7 233.1 243.0 16864 3331.5 247 5 258 6 1740.9 48 50 3238.9 233.5 243.5 1688.2 3334.6 248.0 259.2 1742.7 50 52 3242.1 234.0 244 1 1690.0 3337 8 248.4 259.7 1744.6 52 54 3245.3 234.5 244.6 1691.8 3341.0 248.9 260 2 1746.4 54 56 3248.5 2350 245.1 1693.7 3344-2 249.4 260,8 1748.2 56 58 3251.7 235.5 245.6 1695 5 3347.4 249.9 261.3 1750.0 58 60 3254.9 236.0 246.1 1697.3 3350-6 250.4 261 .B 1751.8 60 FUNCTIONS OF A ONE-DEGREE CURVE 219 34 35 / L. O. M. E. T. L. C. M. E. T. 3350.6 250.4 261.8 1751.8 3446.1 265.2 278.1 1806.7 2 3353.8 250.8 262.3 1753.7 3449.3 265.7 278.6 1808.5 2 4 3357.0 251.2 262.9 1755.5 3452.5 266.2 279.2 1810.3 4 6 3360.1 251.7 263.4 1757.3 3455.6 266.7 279.7 1812.2 6 8 3363.3 252.2 204.0 1759.1 3458.8 267.2 280.3 1814.0 8 10 3366.5 252.7 264.5 1761.0 3462.0 267*7 280.8 1815.8 10 12 3369.7 253.2 265.0 1762.8 3465.2 268.2 281.4 1817.7 12 14 3372.9 253.7 265.6 1764.6 3468.3 268.7 281.9 1819.5 14 16 3376.1 254.2 266.1 766.4 3471.5 269.2 282.5 1821.3 16 18 3379.2 254.7 266.7 768.3 3474.7 269.7 283.0 1823.2 18 20 3382.4 255.2 267.2 1770.1 34V7.9 270.2 283.6 1825.0 20 22 3385.6 255.7 267.7 771.9 3481.0 270.7 284.2 1826 8 22 24 3388.8 256.2 268.3 773.7 3484.2 271.2 284.7 18'28.7 24 26 339-2.0 256.7 268.8 775.6 3487.4 271.7 285.3 1830.5 26 28 3395.2 257.2 269.3 777.4 3490.6 272.2 285.9 1832.3 28 30 3398.3 257.7 269.9 779.2 3493.7 272.7 286.4 1834.2 30 32 3401.5 258.2 270.4 781.0 3496.9 273.2 287.0 1836.0 32 34 3404.7 258.7 271.0 782.9 3500.1 273. Y 287.5 1837.8 34 36 3407.9 259 2 271.5 784.7 3503.3 274.2 288.1 1839 7 36 38 3411.1 259.7 2720 1786.5 3506.5 274.7 288.7 1841.5 38 40 3414.3 260.2 272.6 1788.4 3509.6 275.2 289,2 1843.4 40 42 3417.4 260.7 273.1 1790.2 3512.8 275.7 289.8 1845.2 42 44 3420.6 261.2 273.7 1792.0 3516.0 2762 290.4 1847.1 44 46 3423.8 261.7 274.2 1793.9 3519.2 276.7 290.9 1848.9 46 48 3427.0 262.2 274.8 1795.7 35-22.3 277.2 291.5 1850.7 48 50 3430.2 262.7 275.3 1797.5 3525.5 277.7 292.0 185'2.6 50 52 3433.4 263.2 275.9 1799.3 3528.7 278.2 292.6 1854.4 52 54 3436.5 263.7 276.4 1801.2 3531.9 278.7 293.2 1856.3 54 56 3439.7 264.2 277.0 1803.0 3535.0 279.2 293.7 1858.1 56 58 3442.9 264.7 277.5 1804.8 3538.2 279.8 294.3 1859.9 58 60 3446.1 265.2 278.1 1806.7 3541.4 280.4 294.9 1861.8 60 36 37 / L. C. M. E. T. L. C. M. E. T. / 3541.4 280.4 294.9 1861.8 3636.3 296.1 312.3 1917.3 2 3544.6 280 9 295.4 1863.6 3639.5 296.6 312.8 1919.1 2 4 3547.7 281.4 296.0 1865.5 3642 6 297 1 313.4 1921.0 4 6 3550.9 281.9 296.6 1867.3 3645.8 297.7 314.0 1922.8 6 8 3554.0 282.5 297.2 1869.2 3648.9 298.2 314.6 1924.7 8 10 3557.2 283.0 297.7 1871.0 3652.1 298.7 315.2 1926.5 10 12 3560.4 283.5 298.3 1872.9 3655.2 299.3 315.8 1928.4 12 14 3563.5 284.0 298.9 1874.7 3658.4 299.8 316.4 1930 2 14 16 3566.7 284.6 299.5 1876.5 3661.6 300.3 317.0 1932.1 16 18 3569.9 285.1 300.0 187'8.4 3061.7 300.9 317.5 1933.9 18 20 3573.0 285.6 300.6 1880.2 3667.9 301.4 318.1 1935.8 20 22 3576.2 286.1 301.2 1882.1 3671.0 301.9 318.7 1937.6 22 24 3579.4 286.7 301.8 1883.9 3674.2 302.5 319.3 1939.5 24 26 3582.5 2S7.2 302.3 1885.8 3677.3 303.0 319.9 1941.3 26 28 3585.7 287.7 302.9 1887.6 3680.5 303.5 320.5 1943.2 28 30 3588.8 288.2 303.5 1889.5 3683.6 304.1 321.1 1945.0 30 32 3592.0 288.8 304.1 1891.3 3686.8 304.6 321.7 1946.9 32 34 3595.2 289.3 304.6 1893.2 3690.0 305.1 322.3 1948.8 34 36 3598.3 289 8 305.2 1895.0 3693.1 305.7 322.9 1950.6 36 38 3601.5 290.3 305.8 1896.9 3696.3 306.2 323.5 1952.5 38 40 3604.7 290.9 306.4 1898.7 3699.4 306.7 324.2 1954.4 40 42 3607.8 291.4 307.0 1900.6 3702.6 307.3 324.8 1956.2 42 44 3611.0 291.9 307.5 1902.4 3705.7 307.8 325.4 1958.1 44 46 3614.1 292.4 308.1 1904.3 3708.9 308.3 326.0 1960.0 46 48 3617.3 293.0 308.7 1906.1 3712.1 308.9 3266 1961.8 48 50 3620.5 293.5 309.3 1908.0 3715 2 309.4 327 2 1963. V 50 52 3623.6 294.0 309.9 1909.8 3718.4 309.9 327.8 1965.5 52 54 3626.8 294.5 310.5 1911.7 3721 5 310.5 3-28.4 1967.4 54 56 3630.0 295.1 311.1 1913.5 3724.7 311.0 329.0 1969.3 56 58 3633.1 295.6 311.7 1915.4 3727.8 311.6 329.6 1971.1 58 60 3636.3 296.1 312.3 1917.3 3731.0 312.2 330.2 1973.0 60 220 FUNCTIONS OF A ONE-DEGREE CURVE 38 39 1 / L. C. M. E. T. L. C. M E. T. 3731.0 312.2 330.2 1973.0 3825.5 328-7 348.7 2029.1 2 3734.1 312.7 330.8 1974.9 3828.6 329.2 349 3 2031.0 2 4 3737.3 313.3 331.4 1976-7 3831.8 329.8 349.9 2032 9 4 6 3740.4 313.8 332.0 1978.6 3834.9 330-3 350.6 2034.7 6 8 3743.6 314.4 332.6 1980.5 3838.0 330.9 351.2 2036.6 8 10 3746.7 314.9 333.2 1982.3 3841.2 331 5 351.8 2038.5 10 12 3749.9 315.5 333.8 1984.2 3844 3 332.0 352-4 8040.4 12 14 37530 3160 334-5 1986.1 3847.4 332.6 358.1 2042 3 14 16 37562 316.6 335.1 1987.9 3850.6 333.2 353.7 2044.1 16 18 3759.3 317.1 335 7 1989.8 3853 7 333 7 354.3 2046.0 18 20 3762 5 317.7 336 3 1991.7 3856.8 334.3 354.9 2047.9 20 22 3765.6 318 2 336 9 1993-6 3860 334.9 355.6 2049.8 22 24 3768.8 318.8 337.5 1995.4 3863.1 335.4 356.2 2051.7 24 26 3771 9 319.3 338.1 1997.3 3866.2 336-0 356.9 2053.5 26 28 3775.1 319.9 338.7 1999.2 3869.4 336.6 357 5 2055.4 28 30 3778.2 3204 339.4 2001.0 3872.5 337.1 358.1 2057-3 30 32 3781.4 321.0 340.0 2002.9 3875 6 337 7 358.8 2059.2 32 34 3784 5 321.5 340-6 2004.8 3878.8 338.3 359 4 2061.1 34 36 3787.7 322.1 341.2 2006.6 3881.9 338.8 360.1 2063.0 36 38 37908 322.6 341.8 2008.5 3885.0 339 4 360.7 2064-8 38 40 3794.0 338.2 342.4 2010 4 3888.2 340.0 361.3 2066.7 40 4-2 37971 323.7 343.1 2012.3 3891.3 340.5 362.0 2068 .6 42 44 3800.3 324.3 343.7 2014.1 3894 4 341 1 362.6 2070.5 44 46 3803.4 324.8 344.3 2016 3897.6 341.7 363.3 2072.4 46 48 3806.6 325.4 344.9 2017.9 3900 7 342 2 363.9 2074 2 48 50 3809.7 325 9 345.6 2019.7 3903.8 342.8 364.5 2076 1 50 52 3812.9 3265 346.2 2021.6 3907.0 343 4 365.2 2078.0 52 54 3816.0 3270 346.8 2023.5 3910.1 343.9 365.8 2079.9 54 56 3819.2 327.6 347.4 2025 4 3913.2 344.5 3665 2081.8 56 68 3822.3 323.1 3481 2027.2 3916.4 345 1 367.1 2083.7 58 60 3825.5 328.7 348-7 2029.1 3919 5 345.6 367.7 2085-5 60 40 41 L. C. M. E. T. L. C. M. E. T. 3919.5 345.6 367.7 2085.5 4013.4 362.9 387.4 2142.3 2 3922.6 346.1 368.4 2087 4 4016.5 G63.4 888.1 2144.2 2 4 3925.8 346 7 309 2089.3 4019.6 364.0 388.8 2146.1 4 6 3928.9 347.2 369.7 2091 2 4022.7 364.5 389.4 2148.0 6 8 3932 347.8 370.3 20H3.1 4025.9 365.1 390 1 2149.9 8 10 3935.1 348.4 371.0 2095 4029.0 365.6 390.7 2151.9 10 12 3938.3 348.9 371 6 2096 9 4032.1 366.2 391.4 2158.8 12 14 3941.4 349.5 372.3 2098.8 4035.2 366.8 392.1 2)55.7 14 16 3944 5 350.1 372.9 2100.7 4038.3 367.4 392.7 2157.6 16 18 8947.7 350.7 373.6 2102 6 4041.4 368.0 393.4 2159.5 18 20 3950.8 351.3 374.3 2104.5 4044.6 368.6 394.1 2161.4 20 2:2 3953.9 351 8 374.9 2106.3 4047.7 369.2 394.7 2163.3 22 24 3957.1 352.4 375.6 2108.2 4050.8 369.8 395.4 2165.2 24 26 3960.2 353.0 376.2 2110.1 4053.9 370.4 396.1 2167.1 26 28 3963.3 353.6 376.9 21120 4057.0 371.0 396.8 2169.0 28 30 3966.4 354.2 377.5 2113 9 4060.1 371.6 397.5 2170.9 30 32 3969.6 354.7 378.2 2115 8 4063.3 372.2 398.1 2172.8 32 34 3972.7 355.3 378.8 2117 7 4066.4 372.8 398.8 2174.7 34 36 3975.8 355.9 379.5 2119 6 4069.5 373.4 399.5 2176.6 36 38 3979.0 356.5 380 1 2121.5 4072.6 374.0 400.2 2178.5 38 40 3982.1 357.1 380.8 2123.4 4075.7 374.6 400.9 2180.4 40 4-> 3985.2 357 6 381.4 2125 3 4078.8 375.2 401.5 2182.4 42 44 3988.4 358.2 382.1 2127.2 4082.0 375.8 402.2 2184.8 44 46 3991 5 358 8 382 8 2129.1 4085.1 376.4 402.9 2186.2 46 48 3994.6 359.4 383.4 2131.0 4088.2 377.0 403.6 2188.1 48 50 3997 7 360 384.1 2132.9 4091.3 377.6 404.3 2190.0 50 52 4000 9 360 5 384.8 2134.7 4094.4 378.2 404.9 2191.9 52 54 4004.0 361 1 385.4 2136.0 4097.5 378.8 405.6 2198.8 54 56 4007 1 361.7 386.1 2138.5 4100.7 379.4 406.3 2195.7 56 58 40103 362.3 3868 21404 4103.8 380.0 407.0 2197.6 58 60 4013.4 362.9 387.4 2142.3 4106.9 380.6 407.7 2199.5 60 FUNCTIONS OF A ONE-DEGREE CURVE 221 42 43 & L. C. M. E. T. L. C. M. E. T. 4106.9 380.6 407.7 2199.5 4200.1 398.7 428.6 2257.1 2 4110.0 381.2 408.3 2:201.4 4203.2 399.3 429.3 2259.0 2 4 4113.1 381.8 409.0 2-J03.3 4206.3 399.9 430.0 2261.0 4 6 4116.2 382.4 409.7 2205.3 4209.4 400.5 430.7 2262.9 6 8 4119.3 383.0 410.4 2207.2 4212.5 401.1 431.4 2264.8 8 10 4122.4 383.6 411.1 2209.1 4215.6 401.7 432.1 2266.7 10 12 4125.5 384.2 411.8 2211.0 4218.7 402.4 432.8 2268.7 12 14 4128.6 384.8 412.5 2212.9 4221.8 403.0 433.5 2270.6 14 16 4131.8 38r 4 413.2 2214.9 4224.9 403.6 434.2 2272.5 16 18 4134.9 38b 413.9 2216.8 4228.0 404.2 434.9 2274.5 18 20 4138.0 386 6 414.6 2218.7 4231.1 404.8 435.6 2276.4 20 22 4141.1 387.2 415.3 2220.6 4234.2 405.4 436.3 S278.3 22 24 4144.2 387.8 416.0 2222.5 4237.3 406.1 437.0 2280.2 24 26 4147.3 388.4 416.6 2224.4 4240.4 406.7 437.8 2282.2 26 28 4150.4 389.0 417.3 2226.4 4243.5 407.3 438.5 2284.1 28 30 4153.5 389.6 418.0 2228.3 4246.5 407.9 439.2 2286.0 30 32 4156.6 390,2 418.7 2230.2 4249.6 408.5 439.9 2288.0 32 34 4159.7 390.8 419.4 223'?. 1 4252.7 409.1 440.6 2289.9 34 36 4162.8 391.4 420.1 2234.0 4255.8 409.8 441.4 2291.8 36 38 4165.9 392.0 420.8 2236.0 4258.9 410.4 442.1 2293.8 38 40 4169.0 392.6 421.5 2237.9 4262.0 411.0 442.8 2295.7 40 42 4172.1 393.2 422.2 2239.8 4265.1 411.6 443.5 2297.7 42 44 4175.2 393.8 422.9 2241.7 4268.2 412.2 444.2 2299.6 44 46 4178.4 394.4 423.6 2243.6 4271.3 412.8 445.0 2301.5 46 48 4181.5 395.0 424.3 2245.6 4274.4 413.5 445.7 2303.5 48 50 4184.6 395.6 425.0 2247.5 4277.5 414.1 446.4 2305.4 50 52 4187.7 396.2 425.7 2249.4 4280.6 414.7 447.1 2307.3 52 54 4190.8 396.8 426.4 2251.3 4283.7 415.3 447.8 2309.3 54 56 4193.9 397.4 427.1 2253.3 4286.8 415.9 448.6 2311.2 56 58 4197.0 398.0 427.8 2255.2 4289.9 416.5 449.3 2313.1 58 60 4-,'00.1 398.7 428.6 2257.1 4293.0 417.2 450.0 2315.1 60 440 45 L. C. M. E. T. L. C. M. E. T. f 4293.0 417.2 450.0 2315.1 438"). 5 436.2 472.1 2373.4 2 4296 1 417.8 450.7 2317.0 4388.6 436.8 472.9 2375.4 2 4 4-299.2 418.4 451.5 2319.0 4391.7 437.5 473.6 2377.3 4 6 4302.2 419.1 452.2 2320.9 4394.7 438.1 474.4 2379.3 6 - 8 4305.3 419.7 452 9 2322.8 4397.8 438.8 475.1 2381.2 8 10 4308.4 420.3 453.7 2324.8 4400.9 439.4 475.9 2383.2 10 VI 4311.5 421.0 454.4 2326.7 4404.0 440.0 476.6 2385.2 12 14 4314.6 421.6 4f,5.1 2328.7 4407.0 440.7 477.4 2387.1 14 16 4317.7 422.2 455.9 2330.6 4410.1 441.3 478.1 2389.1 16 18 4320.7 422.9 456.6 2332.6 4413.2 442.0 478.9 2891.0 18 20 4323.8 423 5 457.3 2334.5 4416.3 442.6 479.6 2393.0 20 22 4326.9 424.1 458.1 2336.4 4419.3 443.2 480.4 2394.9 22 24 4330.0 424.8 458.8 2338.4 4422.4 443.9 481.1 2396.9 24 26 4333.1 425 4 459.5 2340 3 4425.5 444.5 481.9 2398.8 26 28 4336 2 426.0 460.3 2342.3 4428.6 445.2 4S2.6 2400.8 28 30 4339.2 426.7 461 2344.2 4431.6 445.8 483.4 2402.8 30 3-4 4342.3 427.3 461.7 2346.1 4434.7 446.4 484.2 2404.7 32 34 4345.4 427 9 462.5 2348.1 4437.8 447.1 484.9 2406.7 34 36 4348.5 4286 463.2 2350.0 4440.9 447.7 485.7 2408.6 36 38 4351.6 429.2 463.9 2352.0 4444.0 448.3 486.5 24106 sL 40 4354.7 429.8 464.7 2353.9 4447.0 448.9 487.2 2412.6 40 42 4357.7 430.5 485.4 2355.9 4450.1 449,5 488.0 2414.5 42 44 4360.8 431.1 466.2 2357.8 4453 2 450.2 488.7 2416.5 44 46 4363.9 431.7 466.9 2359.8 4456.3 450.8 489.5 2418.5 46 48 4367.0 432.4 467.7 2361.7 4459.3 451.5 490.3 2420.4 48 50 4370.1 433.0 468 4 2363.7 4462.4 452.1 491.0 2422.4 50 52 4373.2 433.6 469.1 2365.6 4465.5 452.7 491 8 2424.4 52 54 43762 4343 469.9 2367.6 4468.6 453 4 492.5 2426.3 54 56 43^9.3 434.9 470.6 2369.5 4471.6 454.1 493.3 2428.3 56 58 4382.4 435 b 471.4 28*1.5 4474.7 454.8 494.1 2430.2 58 60 4385.5 436.2 472.1 2373.4 4477.8 455.5 494.8 2432.2 60 222 FUNCTIONS OF A ONE-DEGREE CURVE 46 47 L. C. M. E. T. L. C. M. E. T. 4477.8 455.5 494.8 2432.2 4569.7 475.2 518.3 2491.5 2 4480.9 456.1 495.6 2434.2 4572.7 475.9 519.0 2493.4 2 4 4483.9 456.8 496.5 2436.1 4575.8 476.5 519.8 2495.4 4 6 4487.0 457.4 497.2 2438.1 4578.8 477.2 520.6 2497.4 6 8 4490.0 458.1 497.9 2440.1 4581.9 477.8 521.4 2499.4 8 10 4493.1 458.7 498.7 2442.1 4584.9 478.5 522.2 2501.4 10 12 4496.2 459.4 499.5 2444.0 4588.0 479.2 523.0 2503.4 12 14 4499.2 460.0 500.3 2446.0 4591.0 479.8 523.8 2505.4 14 16 4502.3 460.7 501.0 2448.0 4594.1 480.5 524.6 2507.3 16 18 4505.4 461.3 501.8 2449.9 4597.1 481.1 525.4 2509.3 18 20 4508.4 462.0 502.6 2451.9 4600.2 481.7 526.2 2511.3 20 22 4511.5 462.7 503.4 2453.9 4603.2 482.3 527.0 2513.3 22 24 4514.6 463.3 504.1 24559 4606.3 483.0 527.8 2515.3 24 26 4517.6 464.0 504.9 2457.8 4609.3 483.7 528.6 2517.3 26 28 4520.7 464.6 505.7 2459.8 4612.4 484.3 529.4 2519.3 28 30 4523.7 465.3 506.5 2461.8 4615.4 485.0 530.2 2521.2 30 32 4526.8 466.0 507.3 2463. S 4618.5 485.7 531.0 2523.2 32 34 4529.9 466.6 508.0 2465.7 4621.5 486.3 531.8 2525.2 34 36 4532.9 467.3 508.8 2467.7 4624.6 487.0 532.6 2527.2 36 88 4536.0 467.9 509.6 2469.7 4627.6 487.7 533.4 2529.2 38 40 4539.1 468.6 510.4 2471.7 4630.7 488.4 534.2 2531.2 40 42 4542.1 469.3 511.1 2473.6 4633.7 489.1 535.0 2533.2 42 44 4545.2 469.9 511.9 2475.6 4636.8 489.8 535.8 2535.2 44 46 4548.2 470.6 512.7 2477.6 4639.8 490.5 536.6 2537.2 46 48 4551.3 471.2 513.5 2479.6 4642.9 491.2 537.4 2539.2 48 50 4554.4 471.9 514.3 2481.6 4645.9 491.9 53S.2 2541.2 50 52 4557.4 472.6 515.1 2483.5 4649.0 492.6 539.0 2543.1 52 54 4560.5 473.2 515.9 2485.5 4652.0 493.3 539.8 2545.1 54 56 4563.6 473.9 516.7 2487.5 4655.1 494.0 540.6 2547.1 56 58 4566.6 474.5 517.5 2489.5 4658.1 494.7 541.4 2549.1 58 60 4569.7 475.2 518.3 2491.5 4661.2 495.4 542.3 2551.1 60 48 4U / L. C. M. E. T. L. C. M. E. T. 4661.2 495.4 542.3 2551. 4752.3 515.9 567.0 2611.3 2 4664.2 496.0 543.1 2553. 4755.3 516.5 567.8 2613.3 2 4 4667.3 496.7 543.9 2555. 4758.4 517.2 568.7 2615.3 4 6 4670.3 497.4 544.7 2557. 4761.4 517.9 569.5 2617.3 6 8 4673.3 498.1 545.5 2559. 4764.4 518.6 570.3 2619.3 8 10 4676.4 498.8 546.4 2561. 4767.4 519.3 571.2 2621.4 10 12 4679.4 499.4 547.2 2563. 4770.5 520.0 572.0 2623.4 12 14 4682.5 500.1 548.0 2565. 4773.5 520.7 572.8 2625.4 14 16 4685.5 500.8 548.8 2567. 4776.5 521.4 573.7 2627.4 16 18 4688.5 501.5 549.6 2569. 4779.6 522.1 574.5 2629.4 18 20 4691.6 502.2 550.5 2571. 4782.6 522.8 575.3 2631.4 20 22 4694.6 502.8 551.3 2573. 4785.6 523.5 576.2 2633.5 22 24 4697.6 503.5 552.1 2575. 4788.7 524.2 577.0 2635.5 24 26 4700.7 504.2 552.9 2577.1 4791.7 524.9 577.9 2637.5 26 28 4703.7 504.9 553.7 2579. 4794.7 525.6 578.7 2639.5 28 30 4706.7 505.6 554.6 2581. 4797.7 5','6.3 579.6 2641.5 30 32 4709.8 506.2 555.4 2583.1 4800.8 527.0 580.4 2643.5 32 34 4712.8 506.9 556.2 2585.1 4803.8 527.7 581.3 2645.6 34 36 4715.9 507.6 557.0 2587.2 4806.8 528.4 582.1 2647.6 36 38 4718.9 508.3 557.8 2589.2 4809.9 529.1 583.0 2649.6 38 40 4721.9 509.0 558.7 2591.2 4812.9 529.8 583.8 2651.6 40 42 4725.0 509.6 559.5 2593.2 4815.9 530.5 584.7 2653.7 42 44 4728.0 510.3 560.3 2595.2 4819.0 531.2 585.5 2655.7 44 46 4731.0 511.0 561.2 2597.2 4822.0 531.9 586.4 2657.7 46 48 4734.1 511.7 562.0 2599.2 4825.0 532.6 587.2 2659.7 48 50 4737.1 512.4 562.8 2601.2 4828.0 533.3 588.1 2661.8 50 52 4740.2 513.1 563.7 2603.2 4831.1 534.0 5S8.9 2663.8 52 54 4743.2 513.8 564.5 2605.2 4834.1 534.7 589.8 2665.8 54 56 4746.2 514.5 565.3 2607.2 4837.1 535.4 590.6 2667.8 56 58 4749.3 515.2 566.2 2609.3 4840.2 536.1 591.5 2669 9 58 60 4752.3 515.9 567.0 2611.3 4843.2 536.8 592.4 2671.9 60 FUNCTIONS OF A ONE-DEGREE CURVE 223 60 61 L. C. M. E. T. L. C. M. E. T. 4843.2 536.8 592.4 2671.9 4933 6 558.2 618.5 2733.0 2 4846.2 537.5 593.2 2673 9 4936.6 558.9 619.3 2735.1 2 4 4849.2 538.2 594.1 2676.0 4939.6 559.7 620.2 2737.1 4 6 4852.2 538.9 594.9 2678.0 4942.6 560.4 621.1 2739.2 6 8 4855.2 539.6 595.8 2680.0 4945.6 561.1 622.0 2741.2 8 10 4858.3 540.3 596.7 2682.1 4948.6 561.8 622.9 2743.3 10 re 4861.3 541.0 597.5 2684.1 4951.6 562.5 623.7 2745.3 12 14 4864.3 541.7 598.4 2686.1 4954.6 563.3 624.6 2747.4 14 16 4867.3 542.4 599.3 2688.2 4957.6 584.0 625.5 2749.4 16 18 4870.3 543.1 600.1 2690.2 4960.6 564.7 626.4 2751.5 18 80 4873.3 543.9 601.0 2692.3 4963.6 565.4 627.3 2753.5 20 22 4876.3 544.6 601.9 2694.3 4966.6 566.2 628.2 2755.6 22 24 4879.4 545.3 602.7 2696.3 4969.6 566.9 629.9 2757.7 24 26 4882.4 546.0 603.6 2698.4 4972.6 567.6 630.0 2759.7 26 28 4885.4 546.7 604.5 2700.4 4975.6 568.3 630.9 2761.8 28 30 4888.4 547.4 605.3 2702.4 4978.6 569.1 631.8 2763.8 30 32 4891.4 548.1 606.2 2704.5 4981.6 569.8 632.7 2765.9 32 34 4894.4 548.8 607.0 2706.5 4984.6 570.5 633.6 2767.9 34 33 4897.4 549.5 607.9 2708.6 4987. 571.2 G34.5 2770.0 36 38 4900.4 550.2 608.8 2710.6 4990. 572.0 635.3 2772.0 38 40 4903.5 551.0 609.7 2712.6 4993. 572.7 636.2 2774.1 40 42 4906.5 551.7 610.5 2714.7 4996.7 573.4 637.1 2776.2 42 44 4909.5 552.4 611.4 2716.7 4999. 574.1 638.0 2778.2 44 46 4912.5 553.1 612.3 2718.8 5002. 574.9 638.9 2780.3 46 48 4915.5 553.8 613.2 2720.8 5005. 575.6 639.8 2782.3 48 50 4918.5 554.5 614.1 2722.8 5008. 576.3 640.7 2784.4 50 52 4921.5 555.2 614.9 2724.9 5011.7 577.0 641.6 2786.4 52 54 4924.6 555.9 615.8 2726.9 5014.7 577.8 642.5 2788.5 54 56 4927.6 556.6 616.7 2729.0 5017.7 578.5 643.4 2790.6 56 58 4930.6 557.4 617.6 2731.0 5020.7 579.2 644.3 2792.6 58 60 4933.6 558.2 618.5 2733.0 5023.7 579.9 645.2 2794.7 60 52 53 L. C. M. E. T. L. C. M. E. T. 5023.7 579.9 645.2 2794.7 5113.5 602.0 672.7 2856.9 5026.7 580.6 646.1 2796.8 5116.5 602.8 673.7 2858.9 2 4 5029.7 581.3 647.0 2798.8 5119.4 603.5 674.6 2861.0 4 6 5032.7 582.1 647.9 2800.9 5122.4 604.3 675.5 2863.1 6 8 5035.7 582.8 648.9 2803.0 5125.4 605.0 676.4 2865.2 8 10 5038.7 583.5 649.8 2805.0 5128.4 605.8 677.4 2867.3 10 12 5041.7 584.3 650.7 2807.1 5131.3 606.5 678.3 2869.4 12 14 5044.7 585.0 651.6 2809.2 5134.3 607.3 6T9.2 2871.5 14 16 5047.7 585.7 652.5 2811.2 5137.3 608.0 680.2 2873.5 16 18 5050.7 586.5 653.4 2813.3 5140.3 608.8 681.1 2875.6 18 20 5053.6 587.2 654.3 2815.4 5143.2 609.5 682.0 2877.7 20 22 5056.6 587.9 655.2 2817.4 5146.2 610.3 683.0 2879.8 22 24 5059.6 588.7 656.2 2819.5 5149.2 611.0 683.9 2881.9 24 26 5062.6 589.4 657.1 28-'1.6 5152.1 611.8 684.9 2884.0 26 28 5065.6 590.1 658.0 2823.6 5155.1 612.5 685.8 2886.1 28 30 5068.6 590.9 658.9 2825.7 5158.1 613.3 686.7 2888.1 30 32 5071.6 591.6 659.8 2827.8 5161.1 614.0 687.7 2890.2 32 34 5074.6 592.3 660.7 2829.8 5164.0 614.8 688.6 2892.3 34 36 5077.6 593.1 661.6 2831.9 5167.0 615.5 689.6 2894.4 36 38 5080.6 593.8 662.5 2834.0 5170.0 616.3 690.5 2896.5 38 40 5083.6 594.5 663.5 2836.1 5173.0 617.0 691.5 2898.6 40 42 5086.6 595.3 664.4 2838.2 5175.9 617.8 692.4 2900.7 42 44 5089.6 596.0 665.3 2840.2 5178.9 61S.5 693.4 2902.8 44 46 5092.6 596.7 666.2 2842.3 5181.9 619.3 694.3 2904.9 46 48 5095.6 597.5 667.2 2844.4 5184.9 620.1 695.3 2907.0 48 50 5098.6 598.2 668.1 2846.5 51b7.8 620.8 696.2 2909.1 50 52 5101.6 598.9 669.0 '2848. 5 5190.8 621.5 697.1 2911.2 52 54 5104.6 599.7 669.9 2850.6 5193.8 622.3 698.1 2913.3 54 56 5107.6 600.4 670.9 2852.7 5196.7 623.0 699.0 2915.4 56 58 5110.6 601.2 671.8 2854.8 5199.7 6',>3.8 700.0 2917.5 58 60 5113.5 602.0 672.7 2856.9 S20-J.7 624.6 700.9 2919.5 60 224 FUNCTIONS OF A ONE-DEGREE CURVE 54 65 / L.C. M. E. T. L. C. M. E. T. 5202.7 624.6 700.9 2919.5 5291.7 647.4 7'29.9 2982.8 2 5205.7 625.4 701.9 2921.6 5294.6 648.1 730.9 2984.9 2 4 5208.6 626.1 702.8 2923.8 5297.6 648.9 731.9 2987.1 4 6 5211.6 626.9 703.8 2925.9 5300.5 649.6 732.9 2989.2 6 8 5214.6 627.6 704.8 2928.0 5303.5 650.4 733.8 2991.3 8 10 5217.5 628.4 705.7 2930.1 5306.4 651.2 734.8 2993.4 10 12 5220.5 629.2 706.7 2932.2 5309.4 652.0 735.8 2995 5 12 14 5223.5 629.9 707.7 2934.3 5312.3 652.7 736.8 2997.7 14 16 5226.4 630.7 708-6 2936.4 5315.3 653.5 737.8 2999.8 16 18 5229.4 631.4 109.6 2938.5 5318.2 654.3 733.7 3001.9 18 20 5232.4 632.2 710.5 2940.6 53-21.2 655.1 739.7 3004.0 20 22 5235.3 633.0 711.5 2942.7 5324.1 655.8 740.7 3006.2 22 24 5238.3 633.7 712.5 2944.8 5327.1 656.6 741.7 3008.3 24 26 5241.3 634.5 713.4 2946.9 53.30.0 057. 4 742.7 3010.4 26 28 5244.2 635.2 714.4 2949.0 5333.0 658.2 7'43.7 3012.5 28 30 5247.2 636.0 715.3 2951.1 5335.9 658.9 744.7 3014.7 30 32 5250.2 636.8 716.3 2953.2 5338.8 659.7 745 7 3016.8 32 34 5253.1 637.5 717.3 2955.3 5341.8 660.5 746.7 3018.9 34 36 5256.1 638.3 718.2 2957.5 5344.7 661.3 747.7 3021.1 36 38 5259.1 639.0 719.2 2959.6 5347.7 662.0 748.7 3023.2 38 40 5262.0 639.8 720.2 2961.7 5350.6 66-2.8 749.7 3025.3 40 42 5265.0 640.6 721.1 2963.8 5353.6 6636 750.7 3027.5 42 44 5268.0 641.3 722.1 2965.9 5356.5 664.4 751.7 3029.6 44 46 5270.9 642.1 723.1 2968.0 5359.5 665.1 752.6 3031.7 46 48 5-273.9 642.8 724.1 2970.1 5362.4 665.9 753.6 3033.8 48 50 5276.9 643.6 725.0 2972.2 5365.4 666.7 754.6 3036.0 50 52 5279.8 644.4 726.0 2974.4 5368.3 667.5 755.6 3038.1 52 54 5282.8 645.1 727.0 2976.5 5371.3 668.3 756.6 3040.2 54 56 5285.8 645.9 728.0 2978.6 5374.2 669.1 757.6 3042.4 56 58 5288.7 646.6 729.0 2980.7 5377.2 669.9 758.6 3044.5 58 60 5291.7 647.4 729.9 2382.8 5380.1 670.7 759.6 3046.6 60 66 57 ' L. C. M. E. T. L.C. M. E. T. 5380.1 670.7 759.6 3046.6 5468.2 694.4 790.2 3111.1 2 5383.0 671.4 760.6 3048.8 5471.1 695.2 791.2 3113.3 2 4 5386.0 672.2 761.6 3050.9 5474.0 696.0 792.2 3115.4 4 6 5388.9 672.9 762.7 3053.1 5477.0 696.8 793.3 3117.6 6 8 5391.8 673.7 763.7 3055.2 5479.9 697.6 794.3 3119.7 8 10 5394.8 674.4 764.7 3057.4 5482.8 698.4 795.3 3121.9 10 12 5:597.7 675.2 705.7 3059.5 5485.7 699.2 796.3 3124.1 12 14 5400.7 676.0 7G6.7 3061.6 5488. 7 700.0 797.4 3126.2 14 16 5403.6 676.8 767.7 3063.8 5491.6 700.8 798.4 3128.4 16 18 5406.5 677.6 768.7 3065.9 5494.5 701.6 799.4 3130.6 18 20 5409.5 678.4 769.7 3068.1 5497.4 702.4 800.5 3132.7 20 22 5412.4 679.2 770.8 3070.2 5500.3 703.2 801.5 3134.9 2'2 24 5415.3 680.0 771.8 3072.4 5503.3 704.0 802.6 3137.0 24 26 5418.3 680.8 772.8 307'4.5 5506.2 704.8 803.6 3139.2 26 28 5421.2 681.6 773.8 3076.6 5509.1 705.6 804.7 3141.4 28 30 5424.1 682.4 774.8 3078.8 5512.0 706.4 805.7 3143.5 30 3-2 5427.1 683.2 775.8 3080.9 5515.0 707.2 806.8 3145.7 32 34 5430.0 684.0 776.8 3083.1 5517.9 708.0 807.8 3147.9 34 36 5433.0 684.8 777.8 3085.2 5520.8 708.8 808.8 3150.0 36 38 5435.9 685.6 778.9 3087.4 55:23.7 709.6 809.9 3152.2 38 40 5438.8 686.4 779.9 3089.6 55-26.7 710.4 810.9 3154.4 40 42 5441.8 687.2 780.9 3091.7 5529.6 711.2 812.0 3156.6 42 44 5444.7 688.0 781.9 3093.9 5532.5 712.0 813.0 3158.7 44 46 5447.6 688.8 783.0 3096.0 5535.4 712.8 814.1 3160.9 46 48 5450.6 689.6 784.0 3098.2 5538.4 713.6 815.1 3163.1 48 50 5453.5 690.4 785.0 3100.3 5541.3 714.4 816.2 3165.3 50 52 5456.5 691.2 786.0 3102.5 5544.2 715.2 817.2 3167.4 52 54 5459.4 692.0 787.1 3104.6 5547.1 716.0 818.3 3169.6 54 56 5462.3 69-2.8 788.1 3106.8 5550.0 716.8 819.3 3171.8 56 58 5465.3 693.6 789.1 3108.9 5553.0 717.6 820.4 3174.0 58 60 5468.2 694.4 790.2 3111.1 5555.9 718.4 821.4 3170. 1 60 FUNCTIONS OF A ONE-DEGREE CURVE 225 1 68 69 1 L. C. M. E. T. L. C. M. E. T. 5555.9 718.4 821.4 3176.1 5643.1 742.8 853.5 3241.9 2 5558.8 719.2 822.5 3178.3 5646.0 743.6 854.6 3244.1 2 4 5561.7 720.0 823.5 3180.5 5648.9 744 4 855.7 3246.3 4 c 5564.6 720.8 824.6 3182.7 5651.8 745.3 856.8 3248.5 6 8 5567.5 721.6 825.7 3184.9 5654.7 746.1 857.9 3250.7 8 10 5570.4 722.4 826.7 3187.1 5657.6 746.9 859.0 8252.9 10 12 5573.3 723.2 827.8 3189.2 5660.5 747.7 860.0 3255.1 12 14 5576.2 724.0 828.9 3191.4 5663.4 748.6 861.1 3257.3 14 16 5579.2 724.8 829.9 3193.6 5666.3 749.4 862.2 3259.5 16 ' 18 5582.1 725.6 831.0 3195.8 5669.2 750.2 863.3 3261.7 18 20 5585.0 726.5 832.1 3198.0 5672.1 751.1 864.4 3263.9 20 22 5587.9 727.3 833.1 3200.2 5675.0 751.9 865-5 3266.1 22 24 5590.8 728.1 834.2 3202.4 5677.9 752.7 866.6 3268 3 24 26 5593.7 728.9 835.3 3204.5 5680.8 753.5 867.7 3270.5 26 28 5596.6 729.7 836.3 3206.7 5683.7 754.4 868.8 3272.7 28 30 5599.5 730 5 837.4 3208.9 5686 5 755.2 869.9 3274.9 30 32 5602.4 731.3 838.4 3211.1 5689.4 756.0 871.0 3277.1 82 34 5605.3 732.1 839.5 3213.3 5692.3 756.9 872.1 3279.4 34 36 5608.2 732.9 840.6 3215.5 5695.2 757.7 873.2 3281 6 36 38 5611.1 733.7 841.6 3217.7 5698 1 758.5 874.3 3283.8 38 40 5614.0 734.6 842.7 3219.9 5701.0 7594 875.4 3286.0 40 42 5616.9 735.4 843.8 3222.1 5703.9 760.2 876.5 3288.2 42 44 5619.8 730.2 844.9 3224.3 5706.8 761 877.6 3290.5 44 4G 5622.8 737.0 846.0 3226.5 5709 7 761.9 878.7 3292.7 46 4S 5625.7 7378 847.0 3228 7 5712.6 762.7 879.8 3294.9 48 50 5628.6 738.6 848.1 32309 5715.5 763.5 880.9 3297.1 50 58 5631.5 739.4 849.2 3233.1 5718.4 764.4 882.0 3299.3 52 54 5634.4 740.2 850.3 3235.3 5721.3 765.2 883.1 3301.5 54 50 5637.3 741 851.4 3237.5 5724.2 766.0 884.2 3303.8 56 58 5040.3 741.9 85-.>.5 3239.7 5727.1 766.8 885.3 3306.0 58 60 5643.1 742.8 853.5 3241.9 5730.0 767.7 886.4 3308.2 60 60 61 L. C. M, E. T. L. C. M. E. T. 5730.0 767.7 886.4 3308.2 5816.4 792.9 920.2 3375.2 2 5732.9 768.5 887.5 3310.4 5819.3 793.7 921.4 3377.4 2 4 5735.8 769.4 888.7 3312.7 5822.1 794.6 922.5 3379.7 . 4 6 5738.6 770.2 889.8 3314.9 5825.0 795.4 923 6 3381.9 6 8 5741.5 771.1 890.9 3317.1 5827.9 796.3 924.8 3384.2 8 10 5744 4 771.9 892.0 3319.3 5830.7 797.1 935.0 3386.4 10 12 5747.3 772.7 893.1 3321.6 5833.6 798.0 927.1 3388.7 12 14 5750.2 773.6 894.3 3323.8 5836.5 798.8 928.2 3390.9 14 16 5753.0 774.4 895.4 3326.0 5839.3 799.7 929.3 3393.2 16 18 5755.9 775.3 896.5 3328.3 5842.2 800.5 930.5 3395.4 18 20 5758.8 776.1 897.6 3330.5 5845.1 801.4 931.6 33S7.7 20 22 5761.7 776.9 898.8 3332.7 5847.9 802.2 932.8 8399.9 22 24 5764.6 777.8 899.9 3334.9 5850.8 803.1 933.9 3402.2 24 26 5767.4 778.6 901.0 3337.2 5853.7 803.9 935.1 3404.4 26 28 5770.3 779.5 902.1 3339.4 5856.5 804.8 936.3 3406.7 28 30 5773.2 780.3 903.2 3341.6 5859.4 805.6 937.. 4 3408.9 30 32 5776.1 781.1 904.4 3343.9 5862.3 806.5 938.6 3411.2 32 34 5779.0 782.0 905.5 3346.1 5865.1 807.3 939.7 3413.5 34 36 5781.8 782.8 906.6 3348.3 5868.0 808.2 940.9 3415.7 36 38 5784.7 783.7 907.7 3350.6 5870.9 809.0 942.1 3418.0 38 40 5787.6 784.5 908.8 3352.8 5873.7 809.9 943.2 3420.3 40 42 5790.5 785.3 910.0 3355.0 5876.6 810.7 944.4 3422.5 42 44 5793.4 786.2 911.1 3357.3 5879.5 811.6 945.5 3424.8 44 46 5796.2 787.0 912.3 3359.5 5882.3 812.4 946.7 3427.1 46 48 5799.1 787.9 913.4 3361.8 5885.2 813.3 947.8 3429.3 48 50 5802.0 788.7 914.5 3364.0 5888.1 814.1 949.0 3431.6 50 52 5804.9 789.5 915.7 3366.2 5890.9 815.0 950.2 3433.9 52 54 5807.8 790.4 916.8 3368.5 5893.8 815.8 951.3 3436.1 54 56 5810.6 791.2 918.0 3370.7 5896.7 816.7 952.5 3438.4 56 58 5813.5 792.1 919.1 3373.0 5899.5 817.5 953.6 3440.7 58 60 5816.4 792.9 920.2 3375.2 5902.4 818.4 954.8 3442.9 60 j 226 FUNCTIONS OF A ONE-DEGREE CURVE 62 63 / L. C. M. E. T. L. C. M. E. T. / 5902.4 818.4 954.8 3442.9 5987.8 844.4 990.3 3511.3 2 5905.2 819.3 956.0 3445.2 5990.6 845.3 991.5 3513.6 2 4 5908.1 820.1 957.2 3447.5 5993.5 846.2 992.7 3515.9 4 6 5910.9 821.0 958 3 3449.7 5096.3 847.1 993.9 3518.2 6 8 5913.8 821.8 959.5 3452.0 5999.1 847.9 995.1 3520.5 8 10 5916.6 822.7 960.7 3454.3 6002 848.8 996.3 3522.8 10 12 5919.5 823.6 961.9 3456.6 6004.8 849.7 997.5 3525.1 12 14 5922.3 824.4 963.0 3458.8 6007.7 850.6 998.7 3527.4 14 16 5925.2 825.3 9G4.2 3461.1 6010.5 851.4 999.9 3529.7 16 18 5928.0 826.1 965.4 3463.4 6013.3 852.3 1001.1 3532.0 18 20 5930.9 827.0 966.6 3465.7 6016.2 853.2 1002.3 3534.3 20 22 5933.7 827.9 967.8 3107.9 6019.0 854.1 1003.5 3536.6 22 24 5936.6 828.7 968.9 3470.2 6021.8 854.9 1004.7 3538.9 24 26 5939.4 829.6 970.1 3472.5 6024.7 855.8 1005.9 3541.2 26 28 5942.3 830.4 971.3 3474.7 6027.5 856.7 1007.1 3543.5 28 30 5945.1 831.3 972.5 3477.0 6030.3 857.6 1008.4 3545.8 30 82 5947.9 832.2 973.6 3479.3 6033.2 858.4 1009.6 3548.1 32 34 5950.8 833.0 974.8 3481.6 6036.0 859.3 1010.8 3550.4 34 36 5953.6 833.9 976.0 3483.9 6038.9 860.2 1012.0 3552.7 36 U8 5956.5 834.7 977.2 3486.2 6041.7 861.1 1013.2 3555.0 38 40 5959.3 835.6 978.4 3488.5 6044.5 861.9 1014.5 3557.3 40 42 5962 2 836.5 979.6 3490.7 6047.4 862-8 1015.7 3550.6 42 44 5965.0 837.4 980.8 3493.0 6050.2 863.7 1016.9 356-J.O 44 46 5967.9 838.3 982.0 3495.3 6053.0 864.6 1018.1 3564.3 46 48 5970.7 839.1 983.2 3497.6 C055.9 865.4 1019.3 3566.6 48 50 5973.6 840.0 984.4 3499.9 6058.7 866.3 1020.6 3568.9 50 52 5976.4 840.9 985.5 3502.2 6061.6 867.2 1021.8 3571.2 52 54 5979.3 841.7 986.7 3504.5 6064.4 868.1 1023.0 3573.5 54 56 5982.1 842.6 987.9 3506.8 6067.2 868.9 1024.2 3575.8 56 58 5985.0 843.5 989.1 3509.0 6070.1 869.8 1025.4 3578.1 58 60 5987.8 844.4 990.3 3511.3 6072.9 870.7 1026.7 3580.4 60 64: 66 L. C. M. E. T. L. C. M. E. T. 6072.9 870.7 1026.7 3580.4 6157.5 897.3 1064.0 3650.4 2 6075.7 871.5 1027.9 3582.8 6160.3 898.2 1065.2 3652.8 2 4 6078.5 872.4 1029.2 3585.1 6163.1 899.1 1066.5 3655.1 4 6 6081.4 873.3 1030.4 3587.4 6165.9 900.0 1067.7 3657.5 6 8 6084.2 874.2 1031.7 3589.7 6168.7 900.9 1069.0 3659.8 8 10 6087.0 875.1 1032.9 3592.1 6171.5 901.8 1070.2 3662.2 10 12 6089.8 875.9 1034.1 3594.4 6174.3 902.7 1071.5 3664.5 12 14 609:2.6 876.8 1035.4 3596.7 6177.1 903.6 1072.7 3666.9 14 16 6095.5 877.7 1036.6 3599.1 6179.9 904.5 1074.0 3669.2 16 18 0098.3 878.6 1037.9 3601.4 6182.7 905.4 1075.2 3671.6 18 20 6101.1 879.5 1039.1 3603.7 6185.5 906.3 1076.6 3673.9 20 22 6103.9 880.3 1040.3 3606.0 6188.3 907.2 1077.8 3676.2 22 24 6106.7 881.2 1041. G 3608.4 6191.1 908.1 1079.1 3678.6 24 26 6109.6 882.1 1042.8 3610.7 6193.9 909.0 1080.4 3680.9 26 28 6112.4 883.0 1044.1 3613.0 6196.7 909.9 1081.7 3683.3 28 30 6115.2 883.9 1045.3 3615.3 6199.5 910.8 1083.0 3685.6 30 32 6118.0 884.7 1046.5 3617.7 6202.3 911.7 1084.2 3688.0 32 34 6120.8 885.6 1047.8 3620.0 6205.1 912.6 1085.5 3690.4 34 36 6123.7 886.5 1049.0 3622.3 6208.0 913.5 1086.8 3692.7 36 38 6126.5 887.4 1050.3 3624.7 6210.8 914.4 1088.1 3695.1 38 40 6129.3 888.3 1051.5 3627.0 6213.6 915.3 1089.4 3697.4 40 42 6132.1 889.2 1052.7 3629.4 6216.4 916.2 1090.6 3699.8 42 44 6134.9 890.1 1054.0 3631.7 6219.2 917.1 1091.9 3702.2 44 46 6137.8 891.0 1055.2 3634.0 6222.0 918.0 1093.2 3704.5 46 48 6140.6 891.9 1056.5 3636.4 6224.8 918.9 1094.5 3706.9 48 50 6143.4 892.8 1057.7 3638.7 6227.6 919.8 1095.8 3709-3 50 52 6146.2 893.7 1059.0 3641.1 6230.4 920.7 1097.0 3711.6 52 54 6149.0 894.6 1060.2 3643.4 6233.2 921.6 1098.3 3714.0 54 56 6151.9 895.5 1061.5 3645.7 6236.0 922.5 1099.6 3716.3 56 58 6154.7 896.4 10(52.7 3648.1 6238.8 923.4 1100.9 3718.7 58 60 6157.5 897.3 1064.0 3650.4 6241.6 924.3 1102.2 3721.1 60 FUNCTIONS OF A ONE-DEGREE CURVE 227 66 67 L. C. M. E. T. L. C. M. E. T. 6241.6 924.3 1102.2 3721.1 6325.2 951.8 1141.5 3792.6 2 6244.4 925.2 1103.5 3723.4 6328.0 952.7 1142.8 3795.0 2 4 6247.2 926.1 1104.8 3725.8 6330.7 953.6 1144.1 3797.4 4 6 6250.0 927.0 1106.1 3728.2 C333.5 954.5 1145.4 3799.8 6 8 6252.7 927.9 1107.4 3730.6 6336.3 955.5 1146.7 3802.2 8 10 6255 5 928.8 1108.7 3732.9 6339.0 956.4 1148.1 3804.6 10 12 6258.3 929.8 1110.0 3735.3 6341.8 957.3 1149.4 3807.0 12 14 6201.1 930.7 1111.8 3737.7 6344.6 958.2 1150.7 3809.4 14 16 6263.9 931 6 1112.6 3740.1 6347.4 959.2 1152.0 3811.8 16 18 6266.7 ( J32.5 1113.9 3742.4 6350.1 960.1 1153.3 3814.2 18 20 6269.5 933.4 1115.2 3744.8 6352.9 961.0 1154.7 3816.6 20 22 6272.3 934.3 1116.5 3747.2 6355.7 961.9 1156.0 3819.0 22 24 6275 935.3 1117.8 8749.6 6358.4 962.9 1157.4 3821.4 24 26 6277 8 936.2 1119.1 3751.9 6361.2 963.8 1158.7 3823.8 26 28 6280.6 937.1 1120.4 3754.3 6364.0 964.7 1160.1 3826.2 28 30 6283.4 938.0 1121.7 3756.7 6366.7 965.6 1161.4 3828.6 30 32 6286.2 938.9 1123.0 3759.1 6369.5 966.6 1162.8 3831.0 32 34 6289.0 939.8 1124.3 3761.5 6372.3 907.5 1164.1 3833.4 34 36 6291.8 940.8 1125.6 3763.9 6375.1 968.4 1165.5 3835.9 36 38 6294.5 941.7 1126.9 3766.3 6377.8 969.3 1166.8 3838.3 38 40 6297.3 942 6 1128.3 3768.7 6380.6 970.3 1168.2 3840.7 40 42 6300.1 943.5 1129.6 3771.0 6383.4 971.2 1169.5 3843.1 42 44 6302.9 944.4 1130.9 3773.4 6386.1 972.1 1170.9 3845.5 44 46 6305.7 945.3 1132.2 3775.8 6388.9 973.0 1172.2 3847.9 46 48 6308.5 946.3 1133.5 3778.8 6391.7 974.0 1173.6 3850.4 48 50 6311.3 947.2 1134.9 3780.6 6394.4 974.9 1174.9 3852.8 50 52 6314.1 948.1 1136.2 3783.0 6397.2 975.8 1176.3 3855.2 52 51 6316.8 949.0 1137.5 3785.4 6400.0 976.8 1177.6 3857.6 54 56 6319.6 949.9 1138.8 3787.8 6402.8 977.7 1179.0 3860.0 56 58 6322.4 950.8 1140.1 3790.2 6405.5 978.6 1180.3 3862.5 58 60 6325.2 951.8 1141.5 3792.6 6408.3 979.6 1181.6 3864.9 60 68 69 L C. M. E. T. L. C. M. E. T. 6408.3 979.6 1181.6 3864.9 6491.1 1007.7 1222.9 3938.1 2 6411.1 980.5 1183.0 3867.3 6493.8 1008.7 1224.3 3940.6 2 4 6413.8 981.4 1184.4 3869.7 6496.6 1009.6 1225.7 3943.0 4 6 6416.6 982.4 1185.7 3872.2 6499.3 1010.6 1227.1 3945.5 6 8 6419.3 983.3 1187.1 3874.6 6502.1 1011.5 1228.5 3947.9 8 10 6422.1 984.2 1188.5 3877.0 6504.8 1012.5 1229.9 3950.4 10 12 6421.9 985.2 1189.8 3879.5 6507.5 1013.4 1231.3 3952.9 12 14 6427.6 986.1 1191.2 3881.9 6510.3 1014.4 1232.7 3955.3 14 10 6430.4 987.0 1192.6 3884.3 6513.0 1015.3 1234.1 3957.8 18 18 6433.1 988.0 1193.9 3886.8 6515.8 1016.3 1235.5 3960.2 18 20 6435.9 988.9 1195.3 3889.2 6518.5 1017.2 1236.9 3962.7 20 22 6438.7 989.8 1196.7 3891.6 6521.2 1018.2 1238.3 3965.2 22 24 6141.4 990.8 1198.0 3894.1 6524.0 1019.1 1239.7 3967.6 24 26 6444.2 991.7 1199.4 3896.5 6526.7 1020.1 1241.1 3970.1 26 28 64415. 9 992.6 1200.8 3898.9 6529.5 1021.0 1242.5 3972.5 28 30 6149.7 991.6 1202.1 3901.4 6532.2 1022.0 1243.9 3975.0 30 32 6452.5 994.5 1203.5 3903.8 6534.9 1022.9 1245.3 3977.5 32 34 6455.2 995.4 1204.9 3906.3 6537.7 1023.9 1246.7 3980.0 34 36 6458.0 996.4 1206.2 3908.7 6640.4 1024.8 1248.1 3982.4 36 38 6460.7 997.3 1207.6 3911.2 6543.2 1025.8 1249.5 3984.9 38 40 64G3.5 998.2 1209.0 3913.6 6545.9 1026.7 1250.9 3987.4 40 42 6466.3 999.2 1210.3 3916.1 6548.6 1027.7 1252.3 3989.9 42 44 6469.0 1000.1 1211.7 3918.5 6551.4 1028.6 1253.7 3992.3 44 46 6471.8 1001.0 1213.1 3921.0 6554.1 1029.6 1255.1 3994.8 46 48 6474.5 1002.0 1214.5 3923.4 6556.9 1030.5 1256.5 3997.3 48 50 6477.3 1002.9 1215.9 3925.9 6559.6 1031.5 1257.9 3999.8 50 52 6480.1 1003.8 217.3 3928.3 6562.3 1032.4 1259.3 4002.2 52 51 6482.8 1001.8 218.7 3930.8 6565.1 1033.4 1260.7 4004.7 54 56 6485.6 1005.7 220.1 3933.2 6567.8 1034.3 1262.1 4007.2 56 58 6488.3 1006.7 221.5 3935.7 6570.6 1035.3 1263.5 4009.7 58 60 6491.1 1007.7 222.9 3938.1 6573.3 1036.3 1265.0 4012.1 60 228 FUNCTIONS OF A ONE-DEGREE CURVE 70 71 L. C. M. E. T. L. C M. E. T. f 6573.3 1036.3 1265.0 4012.1 6654.9 1065.1 1308.4 4087.1 2 6576.0 1037.3 1266.4 4014.6 6657 .6 1066.1 1309.9 4089.7 2 4 6578.7 1038.2 1267.9 4017.1 6660.3 1067.0 1311.3 4092.2 4 6 6581.5 1039.2 1269.3 4019.6 6663.0 1068.0 1312.8 4094!7 6 8 6584.2 1040.1 1270.8 4022.1 6665.7 1068.9 1314.2 4097.2 8 10 65 C 6.9 1041.1 1272.2 4024.6 6668.4 1069.9 1315.7 4099.8 10 18 6589.6 104v!.l 1273.6 4027.1 6671.1 1070.9 1317.2 4102.3 12 14 6592.3 1043.0 1275.1 4029.6 6673.8 1071.9 1318.6 4104.8 14 16 6595.1 1044.0 1276.5 4032.1 6676.6 1072.9 1320.1 4107.3 16 18 6597.8 1044.9 1278.0 4034.6 6679.3 1073.8 1321.5 4109.8 18 20 6600.5 1045.9 1279.4 4037.1 6682.0 1074.8 1323.0 4112.4 20 22 6603.2 1046.9 1280.8 4039.6 6684,7 1075.8 13'.>4.4 4114.9 22 24 6605.9 1047.8 1282.3 4042.1 6687 4 1076.8 1325.9 4117.4 24 26 6608.7 1048.8 1283.7 404-1.6 6690.1 1077.7 1327.4 4119.9 26 28 6611.4 1049.7 1285.2 4047.1 6692.8 1078.7 1328.9 4122.4 28 30 6614.1 1050.7 1286.6 4049.6 6695.5 1079.7 1330.4 4125.0 30 32 6616.8 1051.7 1288.0 4052.1 6693.2 1080.7 1331.8 4127.5 32 34 6619.5 1052.6 1289.5 4054.6 6700.9 1081.6 1333.3 4130.1 34 36 6622.3 1053.6 1290.9 4057.1 6703.6 1082.6 1334.8 4132.6 36 38 6625.0 1054.5 1292.4 4059.6 670(3.3 1083.6 1336.3 4135.1 38 40 6627.7 1055.5 1293.8 4062.1 6709.0 1084.5 1337.8 4137.7 40 42 6630.4 1056.5 1295.3 4064.6 6711.7 1085.5 1339.2 4140.2 42 44 6633.1 1057.4 1296.7 4067.1 6714.4 1086.5 1340.7 4142.7 44 46 6635.9 1058 4 1298.2 4069.6 6717.2 1087.5 1342.2 4145. 3 46 48 6638.6 1059.3 1299.6 4072.1 6719.9 1088.4 1343.7 4147.8 48 50 6641.3 1060.3 1301.1 4074.6 6722.6 1089.4 1345.2 4150.4 50 52 6644. C 1061.3 1302.6 4077.1 6725.3 1090.4 1346.7 4152.9 52 4 6646.7 1002.2 1304.0 4079.6 6728.0 1091.3 1348.2 4155.4 51 56 6649.5 1063.2 1305.5 4082.1 6730.7 1092.3 1349.7 4158.0 56 58 6652.2 1064.1 1306.9 4084.6 6733.4 1093.3 1351.2 4160.5 58 60 6654.9 1065.1 1308.4 4087.1 6736.1 1094.3 1352.7 4163.1 60 72 73 L. C. M. E. T. L. C. M. E. T. ' 6736.1 1094.3 1352.7 4163.1 6816.6 1123.9 1398.1 4240.0 2 6738.8 1095 2 1354.2 4165.6 6819.3 1124.8 1399.6 4242.6 2 4 6741.5 1096.2 1355.7 4168.2 6821.9 1125.8 1401.2 4245.1 4 6 6744.1 1097.2 1357.2 4170.7 68-J4.6 1126.8 1402.7 4247.7 6 8 6746.8 1098.2 1358.7 4173.3 6827.3 1127.8 1404.2 4250.3 8 10 6749.5 1099.2 1360.2 4175.8 6830.0 1128.8 1405.8 4252.9 10 12 6752 2 1100.1 1361.7 4178.4 6832.6 1129.8 1407.3 4255.5 12 14 6754.9 1101.1 1363.2 4181.0 6835.3 1130.8 1408.8 4258.1 14 16 6757.6 1102.1 1364.7 4183.5 6838.0 1131.8 1410.4 4260.7 16 18 6760.2 1103.1 1366.2 4186.1 6840.7 1132.8 1411.9 4263.2 18 20 6762.9 1104.1 1367.7 4188.6 6843.3 1133.8 1413.5 4265.8 20 22 6765.6 1105.1 1369.2 4191.2 684G.O 1134.8 1415.1 4268.4 22 24 6768.3 1106.0 1370.7 4193.7 6848.7 1135.8 1416.6 4271.0 24 26 6771.0 1107.0 1372.2 4196.3 6851.3 1136.8 1418.2 4273.6 26 28 6773.7 1108.0 1373.7 4198.8 6854.0 1137.8 1419.7 4276.2 28 30 6776.3 1109.0 1375.2 4201.4 6856.7 1138.8 1421.3 4278.8 30 32 6779.0 1109.9 1376.7 4204.0 6859.4 1139.8 1422.9 4281.4 32 34 6781.7 1110.9 1378.2 4206.5 6862.0 1140.8 1424.4 4284.0 34 36 6784.4 1111.9 1379.7 4209.1 6864.7 1141.8 1426.0 4286.6 36 38 6787.1 1112.9 1381.2 4211.7 6867.4 1142.8 1427.5 4289.2 38 40 6789.8 1113.9 1382.8 4214.3 6870.1 1143.8 1429.1 4291.8 40 42 6792.4 1114.9 1384.3 4216.8 6872.7 1144.8 1430.7 4294.4 42 44 6795.1 1115.9 1385.8 4219.4 6875.4 1145.8 1432.2 4297.0 44 46 6797.8 1116.9 1387.4 4222.0 6878.1 1146.8 1433.8 4299.6 46 48 6800.5 1117.9 1388.9 4224.5 6880.8 1147.8 1435.3 4302.2 48 50 6803.2 1118.9 1390.4 4227.1 6883.4 1148 8 1436.9 4304.8 50 52 6805.9 1119.9 1392.0 4229.7 6886.1 1149.8 1438.5 4307.4 52 54 6808.5 1120.9 1393.5 4232.3 6888.8 1150.8 1440.0 4310.0 54 56 6811.2 1121.9 1395.0 4234.8 6891.4 1151.8 1441.6 4312.6 56 68 6813.9 1122.9 1396.6 4237.4 6894.1 1152.8 1443.1 4315.2 58 60 6816.6 1123.9 1398.1 4240.0 6896.8 1153.8 1444.7 4317.8 60 FUNCTIONS OF A ONE-DEGREE CURVE 229 740 76 L. C. M. E. T. L. C. M. E. T. 6890.8 1153.8 1444.7 4317.8 6976.4 1184.1 1492.5 4396.7 2 6899.4 1154.8 1446.2 4320.5 6979.0 1185.1 1494.1 4399.4 2 4 6902.1 1155.8 1447.8 4323.1 6981.7 1186.1 1495.7 4402.1 4 6 6904.8 1156.8 1449.4 4325.7 6984.3 1187.1 1497.3 4404.7 6 8 0907.4 1157.8 1451.0 4328.3 6986.9 1188.1 1499.0 4407.4 8 10 6910.1 1158.8 1452.6 4330.9 6989.6 1189.2 1500.6 4410.0 10 12 6912.7 1159.8 1454.1 4333.6 699-,'. 2 1190.2 1502.2 4412.7 12 14 6915.4 1160.8 1455.7 4336.2 6994.9 1191.2 1503.8 4415.3 14 16 6918.0 1161.8 1457.3 4338.8 6997.5 1192.2 1505.4 4418.0 16 18 6920.7 1162.8 1458.9 4341.4 7000.1 1193.2 1507.0 4420.7 18 20 6923.3 1163.9 1460.5 4344.0 7002.8 1194.3 1508.7 4423.3 20 22 C926.0 1164.9 1462.0 4346.7 7005.4 1195.3 1510.3 4426.0 22 24 6928.6 1165.9 1463.6 4349.3 7008.0 1196.3 1512.0 4428.6 24 26 6931.3 1166.9 1465.2 4351.9 7010.7 1197.3 1513.6 4431.3 26 28 6933.9 1167.9 1466.8 4354.5 7013.3 1198.3 1515.3 4434.0 28 30 6936.6 1168.9 1468.4 4357.1 7015.9 1199.4 1516.9 4436.6 30 32 6939.2 1169.9 1469.9 4359.8 7018.6 1200.4 1518.5 4439.3 32 34 6941.9 1170.9 1471.5 4362.4 7021.2 1201.4 1520.2 4442.0 34 3 6944.6 1171.9 1473.1 4365.1 7023.9 1-202.4 1521.8 4444.6 36 38 6947.2 1172.9 1474.7 4367.7 7026.5 1203.4 1523.5 4447.3 33 40 6949.9 1174.0 1476.4 4370.3 7029.1 3204.5 1525.1 4450.0 40 42 6952.5 1175.0 1478.0 4373.0 7031.8 1205.5 1526.7 4452. 7 42 44 6955.2 1176.0 1479.6 4375.6 7034.4 1206.5 1528.4 4455.3 44 46 6957.8 1177.0 1481.2 4378.3 7037.0 1207.5 1530.0 4458.0 46 48 0960.5 1178.0 1-182.8 4380.9 7039.7 1208.5 1531.7 4460.7 48 50 6963.1 1179.0 1484.4 4383.5 7042.3 1209.6 1533.3 4463.4 50 5:2 6965.8 1180.0 1486.0 4386.2 7045.0 1210.6 1534.9 4466.0 52 54 6968.4 1181.0 1487.7 4388.8 7047.6 1211.6 1536.6 4468.7 54 56 6971.1 1182.0 1489.3 4391.5 7050.2 1212.6 1538.2 4471.4 56 58 6973.7 1183.0 1490.9 4394.1 705-2.9 1213.6 1539.9 4474.1 58 60 6976.4 118-1.1 149:2.5 439(5.7 7055.5 1211.7 1541.5 4476.7 60 ! , 76 JJO L. C. M. E. T. L. C. M E. T. 7055.5 1214.7 1541.5 4476.7 7134.0 1245.6 1591.7 4557.8 2 7058.1 1215.7 1543.2 4479.4 7136.6 1246.6 1593.4 4560.5 2 4 7060.7 1216.7 1544.9 4482.1 7139.2 1247.7 1595.1 4563.3 4 6 7063.3 1217.8 1546.5 4484.8 7141.8 124S.7 1596.8 4566.0 6 8 7066.0 1218.8 1548.2 4487.5 7144.4 1249.8 1598.5 4568.7 8 10 7068.6 1219.8 1549.9 4490.2 7147.0 1250.8 1600.2 4571.5 10 12 7071.2 1220.9 1551.5 4492.9 7149.6 1251.8 1601.9 4574.2 12 14 7073.8 1221.9 1553.2 4495.6 7152.2 1252.9 1603.6 4576.9 14 16 7076.4 1222.9 1554.9 4498.3 7154.8 1253 9 1605 3 4579.7 16 18 7079.0 1224.0 1556.5 4501.0 7157.4 12550 1607.0 4582.4 18 20 7081.7 1225.0 1558.2 4503.7 7160.0 1256.0 1608.7 4585.1 20 22 7084.3 1226.0 1559.9 4506.8 7162.6 1257.0 1610.4 4587.9 22 24 7086.9 1227.1 1561.5 4509.0 7165.2 1258.1 1612.1 4590.6 24 26 7089.5 1228.1 1563.2 4511.7 7167.8 1259.1 1613.8 4593.3 26 28 7092.1 1229.1 1564 9 4514.4 7170.4 1260.2 1615.5 4596.0 28 30 7094.7 1230.2 1566.5 4517.1 7173.0 1261 2 16'17.3 4598 8 30 32 7097.4 1231.2 1568.2 4519.8 7175.6 12622 1619.0 4601.5 32 31 7100.0 1232.2 1569.9 4522.5 71782 1263.3 1620.7 4604.3 34 36 7102.6 1233.3 1571.5 4525.3 7180 8 12S4.3 1622.4 4607 36 38 7105.2 1234.3 1573.2 4528.0 7183.4 1265.4 1624.1 46098 38 40 7107.8 1235.3 1574.8 4530.7 7186.0 1266.4 1625.9 4612.5 40 42 7110.4 1236.4 1576.4 4533.4 7188 6 1267.4 1627.6 4615.3 42 44 7113.1 1237.4 1578.1 4536.1 7191.2 1268.5 1629.3 4618.0 44 46 7115.7 1238.4 1579.8 4538.8 7193.8 1269.5 1631.0 4620.8 46 48 7118.3 12395 1581.5 4541.5 7196.4 1270 6 1632.7 4623.5 48 50 7120.9 1240.5 1583.2 4544.2 7199.0 1271.6 1634 5 4626.3 50 52 7123.5 1241.5 1584.9 4547.0 7201.6 1272.7 16362 4629.0 52 54 7126.1 1242.6 1586.6 4549.7 72042 1273.7 1687.9 4631.8 54 56 7128.8 1243.6 1588.3 4552.4 7206.8 1274.8 1639.6 4634.5 56 58 7131.4 1244.6 15900 4555.1 7209.4 1275.8 1641.3 4637.3 58 6C 7134.0 1245.6 1591.7 4557.8 7212.0 1276.9 1643.1 4640.0 60 230 FUNCTIONS OF A ONE-DEGREE CURVE 78 79 L. C. M. E. T. L. C. M. E. T. 7212.0 1276.9 1643.1 4640.0 7289.5 1308.5 1696.0 4723.4 2 7214.6 1278.0 1644.8 4642.8 7292.1 1309.5 1697.7 4726.2 2 4 7217.2 1279.0 1646.6 4645.6 7294.6 1310.6 1699.5 4729.0 4 6 7219.7 1280.1 1648.3 4648.3 7297.2 1311.7 1701.3 4731.8 6 8 7222.3 1281.1 1650.1 4651.1 7299.7 1312.7 1703.1 4734.7 8 10 7224.9 1282.2 1651.8 4653.9 7302.3 1313.8 1704.9 4737.5 10 12 7227.5 1283.2 1653.6 4656.7 7304.9 1314.9 1706.6 4740.3 12 14 7230.1 1284.3 1655.3 4659.4 7307.4 1315.9 1708.4 4743.1 14 16 7232.7 1285.3 1657.1 4662.2 7310.0 1317.0 1710.2 4745.9 16 18 7235.2 1286.4 1658.8 4665.0 7312.6 1318.1 1712.0 4748.7 18 20 7237.8 1287.4 1660.6 4667.7 7315.1 1319.1 1713.8 4751.5 20 22 7240.4 1288.5 5662.3 4670.5 7317.7 1320.2 1715.6 4751.3 22 24 7243.0 1289.5 1664.1 4673.3 7320.3 1321.3 1717.4 4757.1 24 26 7245.6 1290.6 1665.8 4676.0 7322.8 1322.3 1719.2 4760.0 26 28 7248.2 1291.6 1667.6 4678.8 7325.4 1323.4 1721.0 4762.8 28 30 7250.7 1292.7 1G69.3 4681.6 7327.9 1324.5 1722.8 4765.6 30 82 7253.3 1293.7 1671.1 4684.4 7330.5 1325.5 1724.6 4768.4 32 34 7255.9 1294.8 1672.8 4687.2 7333.1 13-J6.6 1726.4 4771.2 34 36 7258.5 1295.8 1674.6 4689.9 7335.6 1327.7 1728.2 4774.1 36 38 7261.1 1296.9 1676.3 4692.7 7338.2 1328. 7 17*0.0 4776.9 38 40 7263.7 1297.9 1678.2 4695.5 7340.8 1329.8 1731.9 4779.7 40 42 7266.2 1299.0 1679.9 4698.3 7343.3 1330.8 173S.7 4782.6 4'^ 44 7268.8 1300.0 1681.7 4701.1 7345.9 1331.9 1735.5 4785.4 44 46 7271.4 1301.1 1683.5 4703.9 7348.4 1333.0 1737.3 4788.2 46 48 7274.0 1302.1 1685.3 4706.7 7351.0 1334.1 1739.1 4791.0 48 50 7276.6 1303.2 1687.1 4709.5 7353.6 1335.2 1740.9 4793.9 50 52 7279.2 1304.2 1688.8 4712.2 7356.1 1336.2 1742.7 4796.7 52 54 7281.7 1305.3 1690.6 4715.0 7358.7 1337.3 1744.5 4799.5 54 56 7284.3 1306.3 1692.4 4717.8 7361.3 1338.4 1746.3 4802.4 56 58 7286.9 1307.4 1694.2 4720.6 7363.8 1339.5 1748.1 4805. -2 58 60 7289.5 1308.5 1696.0 4723.4 7366.4 1340.6 1750.0 4808.0 60 80 81 L. C. M. E. T. L. C. M. E. T 7366.4 1340.6 1750 4808.0 7442.7 1372.8 3805.5 4893.9 2 7368 9 1341.7 1751.8 4810.9 7445.2 1373.9 1807.3 4896.8 2 4 7371.5 1342.7 1753.7 4813.7 7447.7 1375.0 1809.2 4899.7 4 6 7374.0 1343.8 1755.5 4816.6 7450.3 1376.1 1811.1 4902.6 6 8 7376.6 1344.9 1757.4 4819.4 74M.8 1377.1 1813.0 4905.4 8 10 7379.1 1346.0 1759.2 4822.3 7455.3 1378.2 1814.9 4908.3 10 12 7381.7 1347.0 1761.0 4825.1 7457.8 1379.3 1816 8 4911.2 12 14 7384.2 1348.1 1762.9 4828.0 7460.4 1380.4 1818.6 4914.1 14 16 7386.7 1349.2 1764.7 4830. H 7462.9 1381.4 1820.5 4917.0 16 18 7389.3 1350.3 1766.6 4833.7 7465.4 1382.5 1822.4 4919.9 18 20 7391.8 1351.3 1768.4 4836.5 7467.9 1383.6 1824.2 4922.8 20 22 7394.4 1352.4 1770.2 4839.4 7470.4 1384.7 1826.1 4925.7 22 24 7396.9 1353.5 1772.1 4842.2 7473.0 1385.7 1828.0 4928.6 24 26 7399.5 1354.6 1773.9 4845.1 7475.5 1386.8 1829.9 4931.5 26 28 7402.0 1355.6 1775.8 4847.9 7478.0 1387.9 1831.8 4934.4 28 30 7404.5 1356.7 1777.6 4850.8 7480.5 1389.0 1833.7 4937.2 30 32 7407.1 1357.8 1779.4 4853.7 7483.1 1390.1 1835.6 4940.2 82 34 7409.6 1358.9 1781.3 4856.5 7485.6 1391.2 1837.5 4943.1 34 36 7412.2 1359.9 1783.1 4859.4 7488. J 1392.3 1839.4 4946.0 36 38 7414.7 1361.0 1785.0 4862.3 7490.6 1393.4 1841.3 4948.9 38 40 7417.3 1362.1 1786.8 4865.1 7493.2 1394.5 1843.2 4951.8 40 42 7419 8 1363 2 1788.6 4868.0 7495.7 1395.6 1845.1 4954.7 42 44 7422.3 1364.2 1790.5 4870.9 7498.2 1396.7 1847.0 4957.6 44 46 7424.9 1365.3 1792.4 4873.8 7500.7 1397.8 1848.9 4960.6 46 48 7427.4 1366.4 1794 3 4876.6 7503.3 1398.9 1850.8 4963.5 48 50 7430.0 1367.5 1796.2 4879.5 7505.8 1400.0 1852 7 4966.4 50 52 7432.5 1368.5 1798.0 4882.4 7508.3 1401.1 1854 6 4969.3 52 54 7435.1 1369.6 1799.9 4885 3 7510.8 1402.2 1856 5 4972.2 54 56 7437.6 1370 7 1801.8 4888.1 7513.3 1403.3 185S.4 4975.1 56 58 7440.1 1371.8 1803.7 4891.0 7515.9 1404 4 1860.3 4978.0 58 60 7442.7 1372.8 1805.5 4893.9 7518.4 1405.5 1862 3 4981. C SO FUNCTIONS OF A ONE-DEGREE CURVE 231 82 83 L. C. M. E. T L. C. M. E. T. 7513.4 1405.5 1862.3 4981.0 7593.6 1438.5 1920.6 5069.4 2 7520.9 1406.6 1864.2 4983.9 7596.1 1439.6 1922.6 5072.4 2 4 7523.4 1407.7 1866.1 4986.8 7598.6 1440.7 1924.6 5075.4 4 6 7525.9 1408 8 1868.1 4989.8 7601.1 1441.8 1926.5 5078.4 6 8 7528.4 1409.9 1870.0 4992.7 7603.6 1442.9 1928.5 5081.4 8 10 7530.9 1411.0 1871 9 4995.7 7606.0 1444.0 1930.5 5084.4 10 12 7533.4 1412.1 1873 9 4998.6 7608.5 1445.1 1932.4 5087.3 12 14 7535.9 1413.2 1875.8 5001.5 7611.0 1446.2 1934.4 5090.3 14 16 7538.5 1414.3 1877.7 5004.5 7613.5 1447.3 1936.4 5093.3 16 18 7541.0 1415.4 1879.7 5007.4 7616.0 1448.4 1938.4 5096. 3 18 20 7543.5 1416.5 1881.6 5010.3 7618.5 1449.6 1940.4 5099.3 20 22 7546.0 1417.6 1883.5 5013.3 7621.0 1450.7 1942.4 5102.3 22 24 7548.5 1418.7 1885.5 5016.2 7623.5 1451.8 1944.4 5105.2 24 26 7551.0 1419.8 1887.4 5019.2 7626.0 1452.9 1946.4 5108.2 26 28 7553.5 1420.9 1889.3 5022.1 7628.5 1454.0 1948.4 5111.2 28 30 7556.0 1422.0 1891.3 5025.0 7630.9 1455.1 1950.4 5114 2 30 32 7558.5 1423.1 1893.2 5028.0 7633.4 1456.2 1952.4 5117.2 32 34 7501.0 1424.2 1895.1 5031.0 7G35.9 1457.3 1954.4 5120.2 34 36 7563.5 14-25.3 1897.1 5033.9 7638.4 1458.4 1956.4 5123.2 36 38 7566.0 1426.4 1899.0 5036.9 7640.9 1459.5 1958.4 5126.2 38 40 7568.5 1427.5 1901.0 5039.8 7643.4 1460.7 1960.4 5129.2 40 42 7571.0 1428.6 1902.9 5042.8 7645.9 1461.8 1962.4 5132.2 42 44 7573.5 1429.7 1904.9 5045.8 7648.4 1462.9 1964.4 5135.2 44 46 7576.1 1430.8 1906.9 5048.7 7650.9 1464.0 1966.4 5138.2 46 48 7578.6 H31.9 1908.8 5051.7 7653.4 1465.1 1968.4 5141.2 48 50 7581.1 1433.0 1910.8 5054.6 7655.8 1466.2 1970.4 5144.3 50 52 7583.6 1434.1 1912.8 5057.6 7658.3 1467.3 1972.4 5147.3 52 54 7586.1 1435.2 1914.7 5060.6 7660.8 1468.4 1974.4 5150.3 54 56 7588.6 1436.3 1916.7 5063.5 7663.3 14C9.5 1976.4 5153.3 56 58 7591.1 1437.4 1918.7 5066.5 7665.8 1470.6 1978.4 5156.3 58 60 7593.6 1438.5 1920.6 5069.4 7668.3 1471.8 1980.5 5159.3 60 84- 85 / L. C. M. E. T. L. C. M. E. 1 7668.3 1471.8 1980.5 5159.3 7742.4 1505.4 2041.8 5250.6 2 7670.8 1472.9 1982.5 5162.3 7744.8 1506.5 2043.9 5253.6 2 4 7673.2 1474.0 1984.5 5165.3 7747.3 1507.6 2046.0 5256.7 4 6 7675.7 1475.1 1986.6 5168.4 7749.7 1508.8 2048.0 5259.8 6 8 7(378.2 1476.2 1988.6 5171.4 7752.2 1509.9 2050.1 5262.9 8 10 7680.6 1477.4 1990.6 5174.4 7754.6 1511.0 2052.2 5266 10 12 7683.1 1478.5 1992.7 5177.5 7757.1 1512.2 2054.2 5269.0 12 14 7685 6 1479.6 1994.7 5180.5 77595 1513.3 2056.3 5272.1 14 16 7688.1 1480.7 1990.7 5183.5 7762.0 1514.4 2058.4 5275.2 16 18 7690.5 1481.8 1998.8 5186.6 7764 4 1515.6 2060.5 5278.3 18 20 7693.0 14&3.0 2000.8 5189.6 7766.9 1516.7 2062.6 5281.4 20 22 7695.5 1484 1 2002.8 5192.6 7769.3 1517.8 2064.7 5284.4 22 24 769r.9 1485.2 2004.9 5195.6 7771.8 1519.0 2066.8 5287.5 24 26 7700.4 1486.3 2006.9 5198.7 7774.2 1520.1 2068.9 5290.6 26 ass 770'2.9 1487.4 2008.9 5201.7 7776.7 1521.2 2071.0 5293.7 28 30 7705.3 1488 6 2011.0 5204.7 7779.1 1522.4 2073. f 5296.7 30 32 7707.8 1489.7 2013.0 5207.8 7781.5 1523.5 2075 2 5299 8 32 34 7710.3 1490.8 2015.0 5210 8 7784.0 1524.6 2077 3 5302.9 34 36 7712.8 1491.9 2017.0 5213.9 7786.4 1525.8 2079.4 5306.1 86 38 7715.2 1493.0 2019.1 5216.9 7788.9 1526.9 2081.5 5309.2 38 40 7717.7 1494.2 2021.2 5220.0 7791.3 1528.0 2083.7 5312.3 40 42 77202 1495.3 2023.2 5223.1 7793.8 1529.2 2085.8 5315.4 42 44 7722 6 1496.4 2025.3 5226.1 7796 2 1530.3 2087.9 5318.5 44 46 7725.1 1497-5 2027.4 5229.2 77987 1531.4 2090.0 5321.6 46 48 7727.6 1498.6 2029.4 5232.2 7801.1 1532.6 2092.1 5324.7 48 50 7730.0 14998 2031.5 5235.3 7803.6 1533.7 2094.2 5327.8 50 52 7732 5 1500.9 2033.6 5238.3 7806.0 1534 8 2096.3 5330 9 52 54 7735.0 1502.0 2035.6 5241 4 7808.5 1536.0 2098.4 5334.0 54 56 7737 5 1503.1 2037.7 5244 5 7810.9 1537.1 2100.6 5337.1 56 58 7739 9 1504.2 2039.8 5247.5 7813 4 1538.2 2102.7 5340.2 58 60 7742.4 15054 2041.8 5250.6 7815.8 1539.3 2104.8 5343.3 60 232 FUNCTIONS OF A ONE-DEGREE CURVE 86 87 L. C. M. E. ,T. L. C. M. E. T. / 7815.8 1639.3 2104.8 5343.3 7888.5 1573.6 2169.5 5437.5 2 7818.2 1540.4 2106.9 5346.4 7890.9 1574.8 2171.6 5440.7 2 4 7820.6 1541.6 2109.1 5349.5 7893.3 1575.9 2173.8 5443.9 4 6 7823.1 1542.7 2111.2 5352.7 7895.7 1577.1 2176.0 5447.1 6 8 7825.5 1543.9 2113.4 5355.8 7898.1 1578.2 2178.2 5450.3 8 10 7827.9 1545.0 2115.5 5358.9 7900.5 1579.4 2180.4 5453.4 10 12 7830.3 1546.1 2117.6 5362.0 7903.0 1580.5 2182.5 5456.6 12 14 7832.8 1547.3 2119.8 5365.2 7905.4 1581.7 2184.7 5459.8 14 16 7835.2 1548.4 2121.9 5368.3 7907.8 1582.9 2186.9 5463.0 16 18 7837.6 1549.6 2124.1 5371.4 7910.2 1584.0 2189.1 5466.2 18 20 7840.0 1550.7 2126.2 5374.6 7912.6 1585.1 2191.3 5469.4 20 22 7812.4 1551.8 2128.3 5377.7 7915.0 1586.3 2193.5 5472.5 22 24 7844.9 1553.0 2130.5 5 9 80.8 7917.4 1587.4 2195.7 5475.7 24 26 7847.3 1554.1 2132.6 5383.9 7919.8 1588.6 2197.9 5478.9 26 28 7849.7 1555.3 2134.8 5387.1 7922.2 1589.7 2200.1 5482.1 28 30 7852.1 1556.4 2136.9 5390.2 7924.6 1590.9 2202.3 5485.3 30 32 7854.6 1557.5 2139.0 5393.4 7927.1 1592.0 2204.5 5488.5 S2 34 7857.0 1558.7 2141.2 5396.5 7929.5 1593.2 2206.8 5491.7 34 36 7859.4 1559.8 2143.3 5399.7 7931.9 1594.3 2209.0 5494.9 36 38 7861.8 1561.0 2145.5 5402.8 7934.3 1595.5 2211.2 5498.1 38 40 7864.3 1562.1 2147.7 5406.0 7936.7 1596.6 2213.4 5501.3 40 42 7866.7 1563.2 2149.8 5409.1 7939.1 1597.8 2215.6 5504.5 42 44 7869.1 1564.4 2152.0 5412.3 7941.5 1598.9 2217.8 5507.7 44 46 7871.5 1565.5 2154.2 5415.4 7943.9 1600.1 2220.0 5510.9 46 48 7874.0 1566.7 2156.4 5418.6 7946.3 1601.2 2222.3 5514.1 48 50 7876.4 1567. 2158.6 5421.8 7948.7 1602.4 2224.5 5517.3 50 52 7878.8 1568.9 2160.7 5424.9 7951.2 1603.5 2226.7 5520.5 52 54 7881.2 1570.1 2162.9 5428.1 7953.6 1604.7 2228.9 5523.7 54 56 7883.6 1571.2 2165.1 5431.2 7956.0 1605.8 2231.1 5526.9 56 58 7886.1 1572.4 2167.3 5434.4 7958.4 1607.0 2233.3 5580.1 58 60 7888.5 1573.6 2169.5 5437.5 7960.8 1608.2 2235.6 5533.3 60 88 89 L. C. M. E. T. L. C. M. E. T. 7960.8 1608.2 2235.6 5533.3 8032.4 1643.0 2303.6 5630.8 2 7963.2 1609.4 2237.8 5536.6 8034.8 1644.1 2305.9 5634.1 2 4 7965.6 1610.5 2240.1 5539.8 8037.1 1645.3 2308.2 5637.4 4 6 7968.0 1611.7 2242.3 5543.1 8039.5 1646.5 2310.5 5640.7 6 8 7970.3 1612.8 2344.6 5546.3 8041.9 1647.7 2312.8 5644.0 8 10 7972.7 1614.0 2246.8 5549.5 8044.2 1648.9 2315.1 5647.3 10 1? 7975.1 1615.2 2249.1 5552.8 8046.6 1650.0 2317.4 5650.6 12 14 7977.5 1616.3 2251.3 5556.0 8049.0 1651.2 2319.7 5653.9 14 16 7979.9 1617.5 2253.6 5559.2 8051.4 1652.4 2322.0 5657.1 16 18 7982.3 1618.6 2255.8 5562.5 8053.7 1653.6 2324.3 5660.4 18 20 7984.7 1619.8 2258.1 5565.7 8056.1 1654.8 2326.7 5663.7 20 22 7987.1 1621.0 2260.4 5568.9 8058.5 1655.9 2329.0 5667.0 22 24 7989.4 1622.1 2262.7 5572.2 8060.8 1657.1 2331.3 5670.3 24 26 7991.8 1623.3 2264.9 5575.4 8063.2 1658.3 2333.7 5673.6 26 28 7994.2 1624.4 2267.2 5578.6 8065.6 1659.5 2336.0 5676.9 28 30 7996.6 1625.6 2269.5 5581.9 8067.9 1660.7 2338.3 5680.2 30 32 7999.0 1626.8 2271.7 5585.1 8070.3 1661.8 2340.7 5683.5 32 34 8001.4 1627.9 2273.9 5588.4 8073.7 1663.0 2343.0 5686.8 34 36 8003.8 1629.1 2276.2 5591.7 8075.1 1664.2 2345.3 5690.2 36 38 8006.1 1630.2 2278.5 5594.9 8077.4 1665.4 2347.7 5693.5 38 40 8008.5 1631.4 2080.8 5598.2 8079.8 1666.6 2350.0 5696.8 40 42 8010.9 1632.6 2283.0 5601.4 8082.2 1667.7 2352.3 5700.1 42 44 8013.3 1633.7 2285.3 5604.7 8084.5 1668.8 2354.7 5703.4 44 46 8015.7 1634.9 2^87.6 5608.0 8086.9 1670.0 2357.0 5706.8 46 48 8018.1 1636.0 2289 9 5611.2 8089.3 1671.2 2359.3 5710.1 48 50 8020.5 1637.2 2292.2 5614.5 8091.6 1672.4 2361.7 5713.4 50 52 8022.9 1638.4 2294.4 5617.8 8094.0 1673.5 2364.0 5716.7 52 54 8025.2 1639.5 2296.7 5621.0 8096.4 1674.7 2366.3 5720.0 54 56 8027.6 1640.7 2299.0 5624.3 8098.8 1675.9 2368.7 5723.4 56 58 8030.0 1641.8 2301.3 5627.5 8101.1 1677.1 2371.0 5726.7 58 60 8032.4 1643.0 2303.6 5630.8 8103.5 1678.3 2373.4 5730.0 60 FUNCTIONS OF A ONE-DEGREE CURVE 233 90 91 L. C. M. E. T. L. C. M. E. T. / 8103.5 1678.3 2373.4 5730.0 8173.9 1713.8 2445.1 5830.9 2 8105.8 1679.5 2375.8 5733.3 8176.2 1715.0 2447.5 5834.3 2 4 8108.2 1680.6 2378.2 5736.7 8178.5 1716.2 2450.0 5837.7 4 6 8110.5 1681.8 2380.5 5740.0 8180.9 1717.4 2452.4 5841.1 6 8 8112.9 1683.0 2382.9 5743.4 8183.2 1718.6 2454.8 5844.5 8 10 8115.2 1684.2 2385.3 5746.7 8185.5 1719.7 2457.2 5847.9 10 12 8117.6 1685.4 2387.6 5750.0 8187.9 1720.9 2459.7 5851.3 12 14 8119.9 1C86.5 2390.0 5753.4 8190.2 1722.1 2462.1 5854.7 14 16 8122.3 1687.7 2392.4 5756.7 8192.5 1723.3 2464.5 5858.1 16 18 8124.6 1688.9 2394.7 5760.1 8194.8 1724.5 2467.0 5861.5 18 20 8127.0 1690.1 2397.1 5763.4 8197.2 1725.7 2469.4 5864.9 20 2-2 81 -,'9. 3 1691.3 2399.5 5766.8 8199.5 1726.9 2471.9 5868.3 22 24 8131.7 1692.5 2401.9 5770.1 8201.8 1728.1 2474.3 5871.8 24 20 8134.0 1693.6 2404.3 5773.5 8204 .2 1 729 . 3 2476 . 7 5875 . 2 26 28 8136.4 1694.8 2406.6 5776.9 8206.5 730.5 2479.2 5878.6 28 30 8138.7 1696.0 2409.0 5780.2 8208.8 731.7 2481.6 5882.0 30 32 8141.1 1697.2 2411.4 5783.6 8211.1 732.9 2484.1 5885.4 32 34 8143.4 1698. 4 2413.8 5787.0 8213.5 734.1 2486.5 5888.9 34 36 8145.8 1699.6 2416.2 5790.3 8215.8 735.3 2489.0 5892.3 36 38 8148.1 1700.7 2418.6 5793.7 8218.1 736.4 2491.5 5895.7 38 40 8150.4 1701.9 2421.0 5797.1 8220.4 737.6 2493.9 5899.2 40 4-2 8152.8 1703.1 2423.4 5800.4 8222.8 738.8 2496.4 5902.6 42 44 8155.1 1704.3 2425.8 5803.8 8225.1 740.0 2498.9 5906.0 44 46 8157.5 1705.5 2428.2 5807.2 8227.4 741.8 2501.3 5909.4 46 48 8159.8 1706.7 2430.6 5810.6 8229.7 742.4 2503.8 5912.9 48 50 8162.2 1707.9 2433.0 5814.0 8232.0 743.6 2506.3 5916.3 50 52 8164.5 1709.0 2435.4 5817.3 8234.3 744.8 2508.7 5919.8 52 54 8166.8 1710.2 2437.9 5820.7 8236.7 746.0 2511.2 5923.2 54 56 8169.2 1711.4 2440.3 5824.1 8239.0 747.2 2513.7 5926.7 56 58 8171.5 1712.6 2442.7 5827.5 8241.3 1748.4 2516.2 5930.1 58 60 8173.9 1713.8 2445.1 5830.9 8243.6 1749.6 2518.7 5933.6 60 92 93 L. C M. E. T. L.C. M. E. T. 8243.6- 1749.6 2518.7 5933.6 8312.8 1785.7 2594.2 6088.2 0~ 2 8245.9 1750.8 2521.2 5937.0 8315.1 1786.9 2596.8 6041 7 2 4 8248.2 1752.0 2523.6 5940.5 8317.4 1788.2 2599.3 6045.2 4 6 8250.6 1753.2 2526.1 5944.0 8319.7 1789.4 2601.9 6048.7 6 8 8252.9 1754.4 2528.6 5947.4 8322.0 1790,6 2604.4 6052.2 8 10 8255.2 1755.6 2531.1 5950.9 8324.3 1791.8 2(507.0 6055.8 10 12 8257.5 1756.8 2533.6 5954.4 8306.6 1793.0 2609.6 6059.3 12 14 8259.8 1758.0 2536.1 5957.8 8308.8 1794.2 2612.1 6062.8 14 16 8262.2 1759.2 2538.6 5961.3 8331.1 1795.4 2614.7 6066.4 16 18 8264.5 1760.4 2541.1 5964.8 8333.3 1796.6 2617.3 6069.9 18 20 8266.8 1761.6 2543.6 5968.2 8335.6 1797.8 2619.8 6073.4 20 82 8269.1 1762.8 2546.1 5971.7 8337.9 1799.1 2622.4 6077.0 22 24 8271.4 1764.0 2548.6 5975.2 8340.2 1800.3 2625.0 6080.5 24 26 8273.7 1765.2 2551.2 5978.7 8342.5 1801.5 2627.6 6084.1 26 28 8276.0 1766.4 2553.7 5982.2 8344.8 1802.7 2630.2 6087.6 28 30 8278.3 1767.6 2556.2 5985.6 8347.1 1803.9 2632.7 6091.2 30 32 8280.6 1768.8 2558.7 5989.1 8349.4 1805.1 2635.3 6094.7 32 34 8282.9 1770.0 2561.2 5992.6 8351.7 1806.3 2637.9 6098.3 34 36 8285.2 1771.2 2563.8 5996.1 8354.0 1807.6 2640.5 6101.8 36 38 8287.5 1772.5 2566.3 5999.6 8356.3 1808.8 2643.1 6105.4 38 40 8289.8 1773.7 2568.8 6003.1 8358.5 1810.0 2645.7 6109.0 40 42 8292.1 1774.9 2571.3 6006.6 8360.8 1811.2 2648.3 6112.5 42 44 8294.4 1776.1 2573.9 6010.1 8363.1 1812.4 2650.9 6116 1 44 46 8296.7 1777.3 2576.4 6013.6 8365.4 1813.6 2653.5 6119.7 46 48 8299.0 1778.5 2578.9 6017.1 8367.7 1814.9 2656.1 6123.2 48 50 8301.3 1779.7 2581.5 6020.6 8369.9 1816.1 2658.7 6106 8 50 52 8303.6 1780.9 2584.0 C024.1 8372.2 1817.3 2661.3 6130.4 52 51 8805.9 1782.1 2586.6 6027.6 8374.5 1818.5 2663.9 6133.9 54 56 8308.2 1783.3 2589.1 6031.1 8376.8 1819.7 2666.6 6137.5 56 58 8310.5 1784.5 2591.7 6034.6 8379.1 1820.9 2669.2 6141.1 58 60 8312.8 1785.7 2594.2 6038.2 8381.3 1822.2 2671.8 6144.7 60 234 FUNCTIONS OF A ONE-DEGREE CURVE 94 95 L. C. M. E. T. L. C. M. E. T. 8381.3 1822.2 2671.8 6144.7 8449.2 1858.9 2751.5 6253.2 2 8383.6 1823.4 2674.4 6148.3 8451.5 1860.1 2754.2 6256.9 2 4 8385.9 1824.6 2677.0 6151.9 8453.7 1861.3 2756.9 6260.5 4 6 8388.1 1825.8 2679.7 6155.4 8456.0 1862.6 2759.6 6264.2 6 8 8390.4 1827.0 2G82.3 6159.0 8458.2 1863.8 2762.3 6267.8 8 10 8392.7 1828.3 2684.9 6162.6 8460.4 1865.0 2765.0 6271.5 10 12 8395.0 1829.5 2687.6 6166.2 8462.7 1866.3 2767.7 6275.2 12 14 8397.2 1830.7 2690.2 6169.8 8464.9 1867.5 2770.4 6278.8 14 16 8399.5 1831.3 2692.8 0173.4 8467.2 1868.7 2773.1 6282.5 16 18 8401.7 1833.1 2695.6 6177.0 8469.4 1869.9 2775.8 6286.2 18 20 8404. C 1834.4 2698.1 6180.6 8471.7 1871.2 2778.5 6289.8 20 22 8406.3 1835.6 2700.8 6184.2 8473.9 1872.4 2781.2 6293.5 22 24 8408.5 1836.8 2703.4 6187.8 8476.2 1873.6 2784.0 6297.2 24 26 8410.8 1838.0 2706.1 6191.5 8478.4 1874.9 2786.7 6300.9 26 28 8413.1 1839.3 2708.7 6195.1 8480.7 1876.1 2789.4 6304.6 28 30 8415.3 1840.5 2711.4 6198.7 8482.9 1877.3 2792.1 6308.2 30 32 8417.6 1841.7 2714.0 6202.3 8485.1 1878.6 2794.9 6311.9 32 31 8419.9 1842.9 2716.7 6205.9 8487.4 1879.8 2797.6 6315.6 34 36 8422.1 1844.2 2719.3 6209.5 8489.6 1881.0 2800.3 6319.3 36 38 8424.4 1845.4 2722.0 6213.2 8491.9 1882.3 2803.1 6323.0 38 40 8426.6 1846.6 2724.7 6216.8 8494.1 1883.5 2805.8 6326.7 40 42 8428.9 1847.8 5J727.3 6220.4 8496.3 1884.8 2808.6 6330.4 42 44 8431.2 1849.1 2730.0 6224.1 8498.6 1886.0 2811.3 6334.1 44 46 8433.4 1850.3 2732.7 6227.7 8500.8 1887.2 2814.1 6337.8 46 48 8435.7 1851.5 2735.4 6231.3 8503.0 1888.5 2816.8 6341.5 48 50 8437.9 1852.7 2738.0 6235.0 8505.3 1889.7 2819.6 6345.2 50 52 8440.8 1854.0 2740.7 6238.6 8507.5 1890.9 2822.3 6349.0 52 54 8442.4 1855.2 2743.4 6242.3 8509.8 1892.2 2825.1 6352.7 54 56 8444.7 1856.4 2746.1 6245.9 85120 1893.4 2827.8 6356.4 56 58 8447.0 1857.6 2748.8 6249.6 8514.2 1894.6 2830.6 6360.1 58 60 8449.2 1858.9 2751.5 6253.2 8516.4 1895.9 2833.4 6363.8 60 96 97 j / L. C. M. E. T. L. C. M. E. T. 8516 4 1895.9 2833.4 6363.8 8583.0 1933.2 2917.5 6476.6 2 8518.7 1897.1 2836.1 6367.5 8585.2 1934.4 2920.3 6480.4 2 4 85209 1898.4 2838.9 6371.3 8587.5 1935.7 2923.2 6484.2 4 6 8523.1 1899.6 2841.7 6375.0 8589.7 1936.9 2926.0 6488.0 6 8 8525 4 1900. S 2844.5 6378.7 8591.9 1938.2 2928.9 6491.8 8 10 8527.6 1902.1 2847.2 6382.5 8594 1 1939.4 2931.7 6495.6 10 12 8529 8 1903.3 2850.0 6386.2 8596.3 1940.7 2934.6 6499.4 12 14 8532.0 1904.6 2852.8 6389.9 859S.5 1941.9 2937.5 6503.2 14 16 8534.3 1905.8 2855.6 6393.7 8000.7 1943.2 2940.3 6507.1 16 18 8536.5 1907.0 2858.4 6397.4 8602.9 1944.4 2943.2 6510.9 18 20 8538.7 1908.3 2861.2 6401.2 8605.1 1945.7 2946.1 6514.7 20 22 8540 9 1909.5 2864.0 6404.9 8607.3 1946.9 2948.9 6518.5 22 24 8543.2 1910.8 2866.7 6408.7 8609.5 1948.2 2951.8 6522.3 24 26 8545.4 1912.0 2869.5 6412.4 8611.7 1949.4 2954.7 6526.2 26 28 8547 6 1913 3 2872.3 6416.2 8613.9 1950.7 2957.6 6530.0 28 30 8549.8 1914.5 2875.1 6419.9 8616.1 1952.0 2960.4 6533.8 30 32 8552 1915.7 2877.9 6423.7 8618.3 1953.2 2963.3 6537.7 32 34 8554.3 1917.0 2880.8 6427 5 8620.5 1954.5 2966.2 6541.5 34 36 8556.5 1918.2 2883.6 6431.2 8622.7 1955.7 2969.1 6545.3 36 38 8558.7 1919.5 2886.4 64350 8624.9 1957.0 2972.0 6549.2 38 40 8560.9 1920.7 2889.2 6438.8 8627.1 1958.2 2974.9 6553.0 40 42 8563.1 1922.0 2892.0 6442 5 8629.3 1959 5 2977.8 6556.9 42 44 8565.3 1923.2 2894.8 6446.3 8631.5 1960.7 2980.7 6560.7 44 46 8567 6 1924 5 2897.7 6450.1 8633.7 1962.0 2983.6 6564.6 46 48 8569.8 1925.7 2900.5 6453.9 8635.8 1963.2 2986.5 6568.4 48 50 8572.0 1927.0 2903.3 6457.6 8638.0 1964.5 2989.4 6572.3 50 52 8574.2 1928.2 2906.! 6461.4 8040.2 1965.8 2992.3 6576.2 52 54 8576.4 1929 4 2909.0 6465.2 8642.4 1967.0 2995.2 6580.0 54 56 8578.6 1930.7 2911.8 6469.0 8044.6 1968.3 2998.1 6583.9 56 58 8580.8 1931.9 2914.7 6472.8 8646 $ 1969.5 3001.1 6587.7 58 60 8583.0 1933.2 2917.5 6476.6 86490 1970.8 3004 6591.6 60 FUNCTIONS OF A ONE-DEGREE CURVE 235 98 99 L. C. M. . T. L. C. M. E. T. 8649.0 1970.8 3004.0 6591.6 8714.3 2008.7 3092.9 6709.0 2 8651.2 1972.0 3006.9 6595.5 8716.4 2009.9 3095.9 6712.9 2 4 8653.3 1973.3 3009.8 6599.4 8718.6 2011.2 3098.9 6716.9 4 6 8655.5 1974.6 3012.8 6603.2 8720.7 2012.5 3101.9 6720.8 6 8 8657.7 1975.8 3015.7 6607.1 8722.9 2013.7 3104.9 6724.8 8 10 8659.9 1977.1 3018.6 6611.0 8725.1 2015.0 3107.9 6728.8 10 12 8662.1 1978.3 3021.6 6614.9 8727.2 2016.3 3111.0 6732.7 12 14 8664.3 1979.6 3024.5 6618.8 8729.4 2017.5 3114.0 6736.7 14 16 8666.4 1980.9 3027.5 6(122.7 8731.5 2018.8 3117.0 6740.7 16 18 8668.6 1982.1 3030.4 6C26.6 8733.7 2020.1 3120.0 6744.6 18 20 8670.8 1983.4 3033.3 6630.5 8735.9 2021.4 3123.1 6748.6 20 22 8673.0 1984.6 3036.3 6634.4 8738.0 2022.6 3126.1 6752.6 22 24 8675.2 1985.9 3039.3 6638.3 87.7 9'245.8 2344.5 3968.1 7824.3 34 36 9188.4 2305.6 3858.0 7687.4 9247.8 2345.8 3971.9 7829.1 86 38 9190.4 2306.9 3861.7 7692.1 9249.7 2347.2 3975.8 7833.8 38 40 9192.4 2308.3 3865.4 7696.7 9251.7 2348.5 3979.6 7838.6 40 42 9194.4 2309.6 3869.2 7701.4 9-253.7 2349.9 3983.5 7843.4 42 44 9196.3 2311.0 3873.0 7706.1 9255.6 2351.2 3987.4 7848.2 44 46 9198.3 2312.3 3876.7 7710.8 9257.6 2352.6 3991.3 7853.0 46 48 9200.3 2313.6 3880.5 7715.5 9259.6 2353.9 3995.1 7857.8 48 50 9202.3 2315.0 3884.2 7720.1 9261.5 2355.3 3999.0 7862.6 50 52 9204.3 2316.3 3888.0 7724.8 9263.5 2356.6 4002.9 7867.4 52 54 9206.3 2317.7 3891.8 7729.5 9265.4 2358.0 4006.8 7872.2 54 56 9208.2 2319.0 3895.6 7734.2 9267.4 2359.3 4010 7 7877.0 56 58 9210.2 2320.3 3899.3 7739.0 9269.4 2360.7 4014.6 7881.9 58 60 9212.2 2321.7 3903.1 7743.7 9-271.8 2362.0 4018.5 7886.7 60 108 109 / L. C. M. E. T. L. C. M. E. T. 9271.3 2362.0 4018.5 7886.7 9329.8 2402.6 4137.4 8033.2 g 9273.3 2363.3 4022.4 7891.5 9331.7 2403.9 4141.4 8038.1 2 4 9275.3 2364.7 40:26.3 7896.3 9333.6 2405.3 4145.4 8043.1 4 6 9277.2 2366.0 4030.2 7901.2 9335.6 2406.6 4149.5 8048.0 6 8 9279.2 2367.4 4034.1 7906.0 9337.5 2408.0 4153.5 8053.0 8 10 9281.1 2368.7 4038.0 7910.8 9339.4 240S.4 4157.5 8057.9 10 12 9283.1 2370.1 4042.0 7915.7 9341.4 2410.7 4161.6 8062.9 12 14 9285.0 2371.4 4045.9 7920.5 9343.3 2412.1 4165.6 8067.9 14 16 9287.0 2372.8 4049.8 7925.4 9345.2 2413.4 4169.7 8072.8 16 18 9288.9 2374.1 4053.8 7930.3 9347.2 2414.8 4173.8 8077.8 18 20 9290.9 2375.5 4057.7 7935.1 9349.1 2416.2 4177.8 8082.8 20 22 9292.8 2376.8 4061.6 7940.0 9351.0 2417.5 4181.9 8087.8 22 24 9294.8 2378.2 4065.6 7944.8 9352.9 2418.9 4186.0 8092.8 24 26 9296.7 2379.5 4069.5 7949.7 9354.9 2420.2 4190.0 8097.8 26 28 9298.7 2380.9 4073.5 7954.6 9356.8 2421.6 4193.1 > 8102.8 28 30 9300.6 2382.3 4077.5 7959.5 9358.7 2423.0 4198.2 8107.8 30 82 9302.6 2383.6 4081.4 7964.4 9360.6 2424.3 4202.3 8112.8 32 34 9304.5 2385.0 4085.4 7969.3 9362.6 2425.7 4206.4 8117.8 34 36 9306.5 2386.3 4089.4 7974.1 9364.5 2427.0 4210.5 8122.8 36 38 9308.4 2387.7 4093.4 7979.0 9366.4 2428.4 4214.6 8127.8 38 40 9310.4 2389.0 4097.3 7983.9 9368.3 2430.0 4218.7 8132.8 40 42 9312.3 2390.4 4101.3 7988.8 9370.2 2431.1 4222.8 8137.9 42 44 9314.2 2391.7 4105.3 7993.8 9372.2 2432.5 4'226.9 8142.9 44 46 9316.2 2393.1 4109.3 7998.7 9374.1 2433.9 4231.0 8147.9 46 48 9318.1 2394.4 4113.3 8003.6 9376.0 2435.2 4235.1 8153.0 48 50 9320.1 2395.8 4117.3 8008.5 9377.9 2436.6 4239.3 8158.0 50 52 9322.0 2397.2 4121.3 8013.4 9379.8 2438.0 4'243.4 8163.1 52 54 9323.9 2398.5 4125.3 8018.4 9381.7 2439.3 4247.5 8168.1 54 56 9325.9 2399.9 4129.3 8023 3 9383.7 2440.7 4251.7 8173.2 56 58 9327.8 2401.2 4133.4 8028.2 9385.6 2442.1 4255.8 8178.2 58 60 9320.8 240-2.6 4137.4 8033.2 9387.5 24434 4260.0 8183.3 60 238 FUNCTIONS OF A ONE-DEGREE CURVE 110 111 L. C. M. E. T. L. C. M. E. T. 9387.5 2443.4 4260.0 8183.3 9444.5 2484.5 4386.4 8337.2 2 9389.4 2444.8 4264.1 8188.4 9446.4 2485.9 4390.7 8342.4 2 4 ' 9391.3 2446.1 4268.3 8193.4 9448.3 2487.2 4395 8347.6 4 6 9393 2 2447.5 4272.4 8198.5 9450.1 2488.6 4399.3 8352.8 6 8 9395.1 2448.9 4276.6 8203.6 9452.0 2490.0 4403.6 8358.0 8 10 9397.0 2450.2 4280.8 8208.7 9453.9 2491.4 4407.9 8363.2 10 12 9398.9 2451.6 4284.9 8213.8 9455.8 2492.7 4412. "2 8368.5 12 14 9400.8 2453.0 4289.1 8218.9 9457.7 2494.1 4416.5 8373.7 14 16 9402.7 2454.3 4293.3 8224.0 9459.6 2495.5 4420.8 8378 9 16 18 9404.7 2455.7 4297.5 8229.1 9461.4 2496.9 4425.1 8384 1 18 20 9406.6 2457.1 4301.7 8234.2 9463.3 2498.2 4429.5 8389.4 20 22 9408.5 2458.4 4305.9 8239.3 9465.2 2499.6 4433 8 8394 6 22 24 9410.4 24598 4310.1 8244.4 9467.1 2501.0 4438.1 839.9 24 26 9412.3 2461.2 4314.3 8249.5 9469.0 2502.4 4442.5 8405.1 26 28 9414.2 2462.6 4318.5 8254.6 9470.8 2503.8 4446.8 8410.4 28 30 9416.1 2463.9 4322.7 8259.8 9472.7 2505.1 4451.2 8415.6 30 '6-2 9418.0 2465.3 4326.9 8:264.9 9474.6 2506.5 4455.5 84^0 9 32 34 9419.9 2466.7 4331.1 8270.0 9476.5 2507.9 4459.9 8426.2 34 36 9421.8 2468.0 4335.4 8275.2 9478.3 2509.3 4464.2 8431.4 36 38 9423.7 2469.4 4339.6 8280.3 9480.2 2510.6 4468.6 8436.7 38 40 9425.6 2470.8 4343.8 8285.5 9482.1 2512.0 4473.0 8442.0 40 42 9427.5 2472.1 4348.1 8290.6 9484.0 2513.4 4477.3 8447.3 42 44 9429.3 2473.5 4352.3 8295.8 9485.8 2514.8 4481.7 8452.6 44 46 9431.2 2474.9 4356.6 8300.9 9487.7 2516.2 4486.1 8457.9 46 48 9433.1 2476.3 4360.8 8306.1 9489.6 2517.5 4490.5 8463.2 48 50 9435.0 2477.6 4365.1 8311.3 9491.4 2518.9 4494.9 8468.5 50 52 9436.9 2479.0 4369.3 8316.5 9493.3 2520.3 4499.3 8473.8 52 54 9438.8 2480.4 4373.6 8321.6 9495.2 2521.7 4503.7 8479.1 54 56 9440.7 2481.7 4377.9 8326.8 9497.0 2523.1 4508.1 8484.4 56 58 9442.6 2483.1 4382.2 8332.0 9498.9 2524.5 4512.5 8489.7 58 60 9444.5 2484 5 4386,4 8337.2 9500.8 2525.8 4516.9 8495.1 60 112 113 L. C. M. E. T. L. C. M. E T. 9500.8 2525.8 4516.9 8495.1 9556.3 2567.4 4651.6 8657.1 2 9502.6 2527.2 4521.4 8500.4 9558.2 2568.8 4656.2 8662.6 2 4 9504.5 2528.6 4525.8 8505.8 9560.0 2570.2 4660.8 8668.0 4 6 9506.4 2530.0 4530.2 8511.1 9561.8 2571.6 4665.4 8673.5 6 8 9508.2 2531.4 4534.6 8516.4 9563.7 2573.0 4669.9 8679.0 8 10 9510.1 2532.7 4539.1 8521.8 9565.5 2574.4 4674.5 8684.5 10 12 9511.9 2534.1 4543.5 8527.1 9567.4 2575.8 4679.1 8690.0 12 14 9513.8 2535.5 4548.0 8532.5 9569.2 2577.1 4683.7 8695.5 14 16 9515.7 2536.9 4552.4 8537.9 9571.0 2578.5 46883 8701.0 16 18 9517.5 2538.3 4556.9 8543.2 9572.9 2579.9 4692.9 8706.5 18 20 9519.4 2539.7 4561.3 8548.6 9574.7 2581.3 4697.5 8712.0 20 22 9521.2 2541.0 4565.8 8554.0 9576.5 2582.7 4702.1 8717.6 22 24 9523.1 2542.4 4570.3 8559.4 9578.4 2584.1 4706.8 8723.1 24 26 9524.9 2543.8 4574.8 8564.8 95802 2585.5 4711.4 8728.6 26 28 9526.8 2545.2 4579.3 8570.2 9582.0 2586.9 4716.0 8734.2 28 30 9528.6 2546.6 4583.7 8575.6 9583.8 2588.3 4720.6 8739.7 30 32 9530.5 2548.0 4588.2 8581.0 9585.7 2589.7 4725.3 8745.3 32 34 9532.3 2549.4 4592.7 8586.4 9587.5 2591.1 4729.9 8750.8 34 36 9534.2 2550.7 4597.2 8591.8 9589.3 259-2.5 4734.6 8756.4 36 38 9536.0 2552.1 4601.7 8597.2 9591.1 2593.9 4739.2 8761.9 38 40 9537.9 2553.5 4606 2 8602.6 9593.0 2595.3 4743.9 8767.5 40 42 9539.7 2554.9 4610.8 8608.0 9594.8 2596.7 4748.5 8773.1 42 44 9541.6 2556.3 4615.3 8613.5 9596.6 2598.1 4753.2 8778.6 44 46 9543.4 2557.7 4619.8 8618.9 9598.4 2599.4 4757.9 8784.2 46 48 9545.3 2559.1 4624.3 8624.3 9600.3 2600.8 4762.6 8789.8 48 50 9547.1 2560.5 4628.9 8629.8 9602.1 2602.2 4767.2 8795.4 50 52 9549.0 2561.8 4633.4 8035.2 9603.9 2603.6 4771.9 8801.0 52 54 9550.8 2563 2 4638.0 8640.7 9605.7 2605 4776.6 8806.6 54 56 9552 6 2564.6 4642.5 8646.2 9607.5 2606.4 4781.3 8812.2 56 58 <>554.5 2566.0 4647.1 8651.6 9609.4 2607.8 4786.0 8817.8 58 60 95563 2567.4 4651.6 8657.1 9611.2 2609.2 4790.7 8823.4 60 FUNCTIONS OF A ONE-DEGREE CURVE 239 i . 114 116 / L. C. M. E. T. L. 0. M. E. T. 9611 2 2609.2 4790.7 8823.4 9665.3 2651.3 4934.4 8994.3 2 9613.0 2610.6 4795.5 8829.1 9667.1 2652.7 4939.3 9000.1 2 4 9614.8 2612.0 4800.2 8834.7 9668.8 2654.1 4944.2 9005.9 4 6 9616.6 2613.4 4804.9 8840.3 9670.6 2655.5 4949.1 9011.6 6 8 9618.4 2614.8 4809.6 8846.0 9672.4 2656.9 4954.0 9017.4 8 10 9620.2 2616.2 4814.4 8851.6 9674.2 2658.3 4958.9 9023.2 10 12 9622.0 2617.6 4819.1 8857.2 9676.0 2659.7 4963.8 9029.0 12 14 9623.8 2619.0 4823.9 8862.9 9677.8 2661.1 4968.7 9034.8 14 16 9625.7 2620.4 4828.6 8868.5 9679.6 2662.5 4973.6 9040.7 16 18 9627.5 2621.8 4833.4 8874.2 9681.4 2663.9 4978.5 9046.5 18 20 9629.3 2623.2 4838.1 8879.9 9683.1 2665.4 4983.4 9052.3 20 22 9631.1 2624.6 4842.9 8885.5 9684.9 2666.8 4988.3 9058.1 22 24 9632.9 2626.0 4847.7 8891.2 9686.7 2668.2 4993.3 9064.0 24 26 9634,7 2627.4 4852.4 8896.9 9688.5 2669.6 4998.2 9069.8 26 28 9636.5 2628.8 4857.2 8902.6 9690.3 2671.0 5003.2 9075.7 28 30 9638.3 2630.2 4862.0 8908.3 9692.0 2672.4 5008.1 9081.5 30 32 9640.1 2631.6 4866.8 8914.0 9693.8 2673.8 5013.1 9087.4 32 34 9641.9 2633.0 4871.6 8919.7 9695.6 2675.2 5018.0 9093.2 84 36 9643.7 2634.4 4876.4 8925.4 9697.4 2676.6 5023.0 9099.1 36 38 9645.5 2635.8 4881.2 8931.1 9699.1 2678.0 5028.0 9105.0 38 40 9647.3 2637.2 4885.0 8936.8 9700.9 2679.5 5032.9 9110.8 40 42 9649.1 2638.6 4890.9 8942.6 9702.7 2680.9 5037.9 9116.7 42 44 9650.9 2640.0 4895.7 8948.3 9704.5 2682.3 5042.9 9122.6 44 46 9652.7 2641.4 4900.5 89540 9706.2 2683.7 5047 9 9128.5 46 48 9654.5 2642.9 4905.3 8959.8 9708.0 2685.1 5052.9 9134.4 48 50 9656.3 2644.3 4910.2 8965.5 9709.8 2686.5 5057.9 9140.3 50 52 9658.1 2645.7 4915.0 8971.3 9711.6 2687.9 5062.9 9146.2 5',? 54 9659.9 2647.1 4919.9 8977.0 9713.3 2689.3 5067.9 9152.1 54 56 9061.7 2648.5 4984.7 8982.8 9715.1 2690.7 5072.9 9158.1 56 58 9663.5 2649.9 4929.6 8988.5 9716.9 2692.2 5078.0 9164.0 58 60 9665.3 2651.3 4934.4 8994.3 9718.6 2693.6 5083.0 9169.9 60 116 117 L/C. M. E. T. L. C. M. E. T. 9718.6 2693.6 5083.0 9169.9 9771.3 2736.1 5236.6 9350.5 9720.4 2695.0 5088.0 9175.9 9773.0 2737.5 5241.8 9356.6 2 4 9722.2 2696.4 5093.1 9181.8 9774.7 2788.9 5247.0 9362.7 4 6 9723.9 2697.8 5098.1 9188.8 9776.5 2740.4 5252.2 9368.9 6 8 9725.7 2699.2 5103.2 9193.7 9778.2 2741.8 5257.4 9375.0 8 10 9727.4 2700.6 5108.2 9199.7 9779.9 2743.2 5262.6 9381.1 10 12 9729.2 2702.1 5113.3 9205.6 9781.7 2744.6 5267.9 9387.3 12 14 9731.0 2703.5 5118.4 9211.6 9783.4 2746.0 5273.1 9393.4 14 16 9732.7 2704.9 5123.4 9217.6 9785.2 2747.5 5278.4 9399.5- 16 18 9734.5 2706.3 5128.5 9223.6 9;86.9 2748.9 5283.6 9405.7 18 20 9736.3 2707.7 5133.6 92296 9788.6 2750.3 5288.9 9411.9 20 22 9738.0 2709.1 5138.7 9235 5 9790.4 2751.7 5294.2 9418.0 22 24 9739.8 2710.6 5143.8 9241.5 9792.1 2753.2 5299.5 9424.2 24 26 9741.5 2712.0 5148.9 9247.6 9793.8 2754.6 5304.7 9430.4 26 28 9743.3 2713.4 5154 9">53.6 9795.6 2756.0 5310.0 9436.6 28 30 9745.0 2714.8 5159.1 9259.6 9797.3 2757.4 5315:3 9442.8 30 32 9746.8 2716.2 5164.2 9265 6 9799.0 2758.9 5320.6 9449.0 32 34 9748.5 2717.6 5169.4 9271.6 9800.7 2760.3 5325.9 9455.2 34 36 9750.3 2719.1 5174.5 9277.7 9802.5 2761.7 5331.2 9461.4 36 38 9752.0 2720.5 5179.7 9283.7 9804.2 2763.1 5336.5 9467.6 38 40 9753.8 2721.9 5184.8 9289.8 9805.9 2764.6 5341.8 9473.8 40 42 9755.6 2723.3 5190.0 9295.8 9807.7 2766.0 5347.2 9480.0 42 44 9757.3 2724.7 5195.1 9301.9 9809.4 8767. 4 5352.5 9486.3 44 46 9759.0 2726.2 5200.3 9307.9 9811.1 2768.8 5357.9 9492.5 46 48 9760.8 2727.6 5205.4 9314.0 9812.8 2770.3 5363.2 9498.7 48 60 9762.5 2729.0 5210.6 9320.1 9814.5 2771.7 5368.5 9505.0 50 52 9764.3 2730.4 5215.8 9326.1 9816.3 2773.1 5373.9 9511.2 52 54 9766.0 2731.8 5221.0 9332.2 9818.0 2774.6 5379.3 9517.5 54 56 9767.8 2733.3 5226.2 9338.3 9819.7 2776.0 53S4.7 9523.8 56 58 9769.5 2734.7 5231.4 9344.4 9821.4 2777.4 5390.0 9530.0 58 60 9771.3 2736.1 5236.6 93505 9823.1 2778.8 5395.4 9536.3 60 240 NATURAL SINES, COSINES, TANGENTS, COTAN. 1 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. r o 1 2 3 JL_ 5 6 8 g .00000 .00029 .00058 .00087 .00116 One One One One One 00000 .00029 .00058 .00087 .00116 Infinite 3437-75 1718-87 1145-92 839-436 01745 .01774 .01803 .01832 .01862 .01891 -01920 .01949 .01978 .02007 .99985 -99984 .99984 .99983 .99983 .01746 .01775 -01804 .01833 .01862 57.2900 56-3506 55-4415 54-5613 53-7086 60 59 58 57 58 .00145 .00175 .00204 .00233 .00262 One One One One One .00145 .00175 00204 .00233 .00262 687-549 572-957 491-106 429-718 381.971 -99982 -99982 -99981 -99980 ^99980 ~99979 -99979 .99978 -99977 .99977 01891 -01920 .01949 .01978 02007. .02036 -02066 -02095 .02124 .02153 52-8821 52-0807 51-3032 50-5485 49-8157 55 54 53 52 51 10 11 12 13 !i_ 15 16 17 18 19 20 21 22 23 24 .00291 .00320 .00349 .00378 .00407 One .99999 .99999 .99999 .99999 .00291 .00320 .00349 .00378 .00407 343 . 774 312-521 286.478 264-441 245-552 .02036 .02065 -02094 .02123 .02152 49-1039 48-4121 47-7395 47-0853 46-4489 50 49 48 47 46 .00436 .00465 .00495 .00524 .00553 .99999 .99999 .99999 .99999 .99998 .00436 .00465 .00495 -00524 .00553 229.182 214.858 202.219 190-984 180-932 02181 .02211 .02240 -02269 .02298 ,99976 .99976 -99975 .99974 .99974 .02182 .02211 .02240 .02269 .02298 45-8294 45-2261 44-6386 44-0661 43-5081 45 44 43 42 -i! 40 39 38 37 _36 35 34 33 32 31 .00582 .00611 .00640 .00689 .00698 .99998 .99998 .99998 .99998 .99998 00582 .00611 .00640 .00869 .00698 171-885 163-700 156-259 149-465 143-237 -02327 .02356 -02385 .02414 . 02443__ ^02472 .02501 -02530 .02560 .02589 .99973 .99972 .99972 .99971 .99970 .99969 .99969 -99968 .99967 .99966 02328 .02357 .02386 .02415 .02444 42-9641 42-4335 41-9158 41-4106 40-9174 25 26 27 28 2P 00727 .00756 00785 .00814 .00844 .99997 .99997 .99997 .99997 .99996 .00727 .00756 .00785 .00815 .OC844 137-507 132.219 127-321 122.774 118-540 .02473 .02502 .02531 .02560 .02589 40-4358 39-9655 39.5059 39-0568 38-6177 30 31 32 33 34_, 35 36 37 38 39 40 41 42 43 44 .00873 i. 99996 .00902 -99993 .00931 .99998 .00960 .99995 .00989 .99995 .00873 .00902 .00931 .00960 .00989 114.589 110-892 107-426 104-171 101-107 .02618 -02647 -02676 .02705 .02734 .99966 -99965 -99964 -99963 .99963 .02619 .02648 .02677 .02706 .02735 38-1885 37.7686 37-3579 36-9560 36-5627 36-1776 35-8006 35-4313 35-0695 34-7151 30 29 28 27 26 .01018 .01047 .01076 .01105 .01134 .99995 .99995 .99994 .99994 .99994 .01018 .01047 .01076 .01105 .01135 98-2179 95-4895 92.9085 90.4633 88-1436 -02763 -02792 -02821 -02850 .02879 .99962 .99961 .99960 .99959 -99959 .02764 .02793 .02822 .02851 .02881 25 24 23 22 21 .01164 .01193 .01222- .01251 .01280 .99993 .99993 .99993 .99992 .99992 .01164 .01193 .01222 .01251 .01280 85-9398 83-8435 81-8470 79-9434 78.1263 02908 -02938 .02967 02996 -03025 99958 -99957 .99956 .99955 .99954 .02910 .02939 .02968 .02997 .03026 34-3678 34-0273 33-6935 33-3662 33-0452 20 19 18 17 -IB 15 14 13 12 _!! 10 9 8 7 6 45 46 47 48 49 .01309 .01338 .01367 .01396 .01425 .99991 .99991 .99991 .99990 .99990 01309 .01338 .01367 .01396 01425 .01455 .01484 .01513 .01542 .01571 76-3900 74-7292 73-1390 71.6151 70.1533 03054 -03083 03112 03141 .03170 .99953 .99952 -99952 .99951 .99950 .99949 .99948 .99947 .99946 .99945 .99944 .99943 .99942 .99941 .99940 03055 .03084 -03114 -03143 .03172 32-7303 32-4213 32-1181 31-8205 31-5234 50 51 52 53 54 .01454 .01483 01513 01542 .01571 .99939 .99989 .99989 .99988 .99988 68-7501 67-4019 66.1055 64.8580 63.6567 03199 .03228 .03257 .03236 .03316 .03201 .03230 -03259 -03288 03317 31-2416 30-9599 30.6833 30-4116 30.1446 55 56 57 58 59 00 .01600 01629 01658 01687 01716 .99987 .99987 .99988 .99986 .99985 .01600 .01629 .01658 .01687 .01716 62.4992 61-3829 60.3058 59.2659 58-2612 03345 .03374 -03403 -03432 .03461 -03346 03376 03405 03434 .03463 29.8823 29-6245 ,?9.3711 29.1220 28-8771 5 4 3 2 1 01745 .99985 .01746 57-2900 03490 .99939 03492 28-6363 O Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 89 88 NATURAL SINES, COSINES, TANGENTS, COTAN. 241 2 3 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 2 3 4 .03490 .03519 .03548 .03577 .03606 -99939 99938 .99937 .99936 .99935 -03492 .03521 .03550 .03579 .03609 28-6363 28.3994 28-1664 27-9372 27-7117 -05234 .05263 -05292 -05321 .05350 .99863 .99861 .99860 .99858 99857 .05241 .05270 .05299 .05328 .05357 19-0811 18-9755 18-8711 18-7678 18.6656 6O 59 58 57 56 5 6 7 8 9 .03635 .03664 .03693 .03723 .03752 .99934 .99933 .99932 .99931 .99930 .03638 .03667 .03696 .03725 .03754 27-4899 27.2715 27-0566 26.8450 26-6367 .05379 .05408 .05437 -05466 .05495 .99855 .99854 .99852 .99851 .99849 .05387 .05416 .05445 .05474 -05503 18-5645 18-4645 18-3655 18-2677 18-1708 55 54 53 52 51 10 11 12 13 14 .03781 .03810 .03839 .03868 .03897 .99929 .99927 .99926 .99925 .99924 .03783 .03812 .03842 .03871 .03900 26-4316 26-2296 26-0307 25.8348 25-6418 -05524 .05553 .05582 .05611 .05640 .99847 .99846 .99844 .99842 .99841 .05533 .05562 .05591 .05620 .05649 18.0750 17-9802 17-8863 17-7934 17-7015 5O 49 48 47 46 15 16 17 18 19 .03926 .03955 .03984 .04013 .04042 .99923 .99922 .99921 .99919 .99918 .03929 -03958 .03987 .04016 .04046 25-4517 25-2644 25-0798 24-8978 24-7185 .05669 .05698 -05727 -05756 .05785 .99839 .99838 .99836 .99834 -99833 .05678 .05708 .05737 .05766 .05795 17-6106 17-5205 17-4314 17-3432 17-2558 45 44 43 42 41 20 21 22 23 ?A .04071 .04100 .04129 .04159 .04188 -99917 -99916 -99915 .99913 .99912 -04075 -04104 -04133 -04162 -04191 24-5418 24-3675 24-1957 24-0263 23.8593 -05814 .05844 -05873 -05902 .05931. -99831 -99829 -99827 -99826 -99824 .05824 -05854 -05883 .05912 -05941 17-1693 17-0837 16-9990 16-9150 16-8319 4O 39 38 37 36 25 26 27 28 29 .04217 .04246 .04275 .04304 .04333 .99911 .99910 -99909 .99907 .99906 .04220 .04250 .04279 -04308 -04337 23-6945 23-5321 23-3718 23-2137 23-0577 -05960 -05989 -06018 .06047 .06076 -99822 .99821 .99819 .99817 .99815 .05970 .05999 .06029 .06058 .06087 16.7496 16-6681 16-5874 16-5075 16-4283 35 34 33 32 31 30 31 32 33 84 04362 -04391 .04420 - 04449 .04478 -99905 -99904 -99902 -99901 -99900 .04366 .04395 .04424 .04454 -04483 22-9038 22-7519 22-6020 22-4541 22-3081 .06105 .06134 .06163 .06192 -06221 .99813 -99812 -99810 -99808 -99806 .06116 -06145 -06175 .06204 -06233 16-3499 16-2722 16.1952 16.1190 16-0435 30 29 28 27 26 85 36 37 38 39 .04507 .04536 .04565 -04594 .04623 -99898 .99897 .99896 -99894 -99893 -04512 -04541 -04570 -04599 .04628 22J 640 22-0217 21-8813 21-7426 21-6056 -06250 .06279 -06308 .06337 .06366 -99804 -99803 .99801 .99799 .99797 .06262 -06291 .06321 -06350 -06379 15-9687 15-8945 15-8211 15-7483 15-6762 25 24 23 22 21 40 41 42 43 44 04653 .04682 .04711 . 04740 .04769 -99892 -99890 -99889 99888 .99886 .04658 .04687 -04716 - 04745 -04774 21-4704 21-3369 21-2049 21-0747 20-9460 -06395 06424 06453 06482 -06511 .99795 -99793 -99792 -99790 -99788 -06408 -06437 -06467 06496 -06525 15-6048 15-5340 15-4638 15-3943 15-3254 20 19 18 17 16 45 46 47 48 49 -04798 -04827 .04856 -04885 -04914 -99885 .99883 .99882 -99881 99879 -04803 .04833 04862 -04891 .04920 20-8188 20-6932 20-5691 20-4465 20-3253 -06540 .06569 06598 .06627 .06656 -99786 -99784 -99782 -99780 -9.9778 -06554 .06584 -06613 -06642 .06671 15-2571 15-1893 15-1222 15-0557 14-9898 15 14 13 12 11 50 51 52 53 54 -04943 -04972 -05001 -05030 .05059 -99878 -99876 -99875 -99873 -99872 .04949 -04978 -05007 -05037 .05066 20-2056 20-0872 19-9702 19-8546 19-7403 -06685 -06714 -06743 -06773 -06802. .99776 -99774 .99772 .99770 .99768 -06700 -06730 .06759 -06788 .06817 14.9244 14-8596 14-7954 14.7317 14-6685 10 9 7 3 55 56 U 99 -05088 -05117 -05146 -05175 -05205 -99870 -99869 .99867 -99886 .99864 -05095 -05124 -05153 -05182 -05212 19-6273 19-5156 19-4051 19-2959 19-1879 -06831 .06860 .06889 .06918 -06947 .99766 -99764 -99762 .99760 .99758 .06847 .06876 .06905 .06934 -06963 14-6059 14-5438 14.4823 14.4212 14.3607 5 3 1 60 .05234 ,99863 -05241 19-0811 .06976 -99756 -OR993 14-3007 -7~ Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. * 87 86 242 NATURAL SINES, COSINES, TANGENTS, COTAN. 4 5 t Sin. Cos. Tan. Cot. Sin. Cos. Tari. Cot. / ? 2 JL .06976 .07005 .07034 .07063 .07092 99756 99754 .99752 .99750 .99748 .06993 07022 .07051 .07080 .07110 14-3007 14.2411 14.1821 14-1235 14.0655 08716 .08745 .08774 .08803 -08831 .99619 .99617 .99614 -99612 .99609 -08749 -08778 -08807 .08837 .08866 11.4301 11.3919 11.3540 11.3163 11.2789 60 59 58 57 56 6 J J .07121 .07150 .07179 .07208 .07237 .99746 .99744 .99742 .99740 .99738 .07139 .07168 .07197 .07227 .07256 14-0079 13-9507 13.8940 13-8378 13-7821 08860 .08889 .08918 .08947 .08976 .99607 .99604 .99602 .99599 .99596 08895 08925 -08954 -08983 -09013 11-2417 11-2048 11-1681 11-1316 11.0954 55 54 53 52 51 10 11 12 13 14 .07266 .07295 .07324 .07353 .07382 .99736 .99734 .99731 .99729 .99727 .07285 .07314 .07344 .07373 .07402 13-7267 13.6719 13-6174 13-5634 13-5098 -09005 .09034 .09063 .09092 .09121 .-99594 99591 .99588 .99586 .99583 09042 09071 -09101 .09130 09159 11.0594 11-0237 10-9882 10-9529 10-9178 50 49 48 47 4ft 15 16 17 18 19 .07411 .07440 -07469 .07498 .07527 .99725 .99723 .99721 .99719 .99716 .07431 .07461 .07480 .07519 .07548 13 4566 13-4039 13.3515 13-2996 13-2480 .09150 -09179 .09208 .09237 .09266 -99580 .99578 .99575 .99572 .99570 .09189 .09218 .09247 .09277 .09306 10-8829 10-8483 10-8139 10-7797 10.7457 45 44 43 42 41 20 21 22 23 24 .07556 .07585 .07614 .07643 .07672 .99714 .99712 .99710 .99708 .99705 .07578 .07607 .07636 .07665 .07695 13-1969 13.1461 13-0958 13-0458 12-9962 .09295 .09324 -09353 .09382 .09411 .99567 .99564 .99562 .99559 -99556 -09335 -09365 .09394 .09423 .09453 10.7119 10.6783 10-6450 10-6118 10.5789 40 39 38 37 36 25 26 27 28 29 .07701 .07730 .07759 .07788 .07817 .99703 .99701 .99699 .99696 .99694 .07724 .07753 .07782 .07812 .07841 12-9469 12-8981 12-8496 12-8014 12.7536 . 09440 .09469 .09493 .09527 -09556 .99553 .99551 .99548 .99545 .99542 .09482 .09511 .09541 .09570 .09600 10 5462 10-5136 10.4813 10-4491 10-4172 35 34 33 32 31 30 31 32 33 34 .07846 .07875 .07904 .07933 .07962 .99692 .99689 .99687 .99685 .99683 .07870 .07899 .07929 .07958 .07987 12-7062 12.6591 12-6124 12.5660 12.5199 .09585 .09614 .09642 .09671 .09700 .99540 .99537 .99534 .99531 .99528 .09629 .09658 .09688 -09717 -09746 10-3854 10-3538 10-3224 10.2913 10-2602 30 29 28 27 26 35 36 37 38 39 .07991 .08020 .08049 .08078 .08107 .99680 .99678 .99676 .99673 -99671 .08017 .08046 .08075 .08104 -08134 12-4742 12-4288 12-3838 12-3590 12-2946 .09729 .09758 -09787 .09816 -09845 .99526 .99523 -99520 .99517 .99514 .09776 -09805 -09834 .09864 .09893 10.2294 10-1988 10-1683 xO.1381 10-1080 25 24 23 22 21 40 41 42 43 44 .08136 08165 -08194 .08223 .08252 .99668 .99666 .99664 .99661 .99659 .08163 .08192 .08221 .08251 .08280 12-2505 12.2067 12-1632 12.1201 12-0772 -09874 09903 .09932 09961 .09990 -99511 -99508 -99506 .99503 -99500 .09923 .09952 -09981 .10011 .10040 10-0780 10-0483 10 0187 9-98931 9.96007 20 19 18 17 16 45 46 47 48 49 08281 .08310 .08339 .08368 .08397 .99657 .99654 .99652 .99649 .99647 .08309 .08339 .08368 .08397 .08427 12-0346 11-9923 11-9504 11-9087 11-8673 10019 10048 .10077 .10106 .10135 .99497 .99494 .99491 .99488 -99485 .10069 .10099 .10128 .10158 -10187 9-93101 9-90211 9.87338 9 84482 _O184i_ 15 14 13 12 _U 50 51 52 53 54 .08428 .08455 08484 08513 08542 99644 .99642 .99639 .99637 .99635 .08456 .08485 .08514 .08544 .08573 11.8262 11.7853 11.7448 11.7045 11.6645 .10164 .10192 .10221 .10250 .10279 -99482 .99479 .99476 .99473 .99470 .10216 .10246 .10275 .10305 .10334 9-78817 9-76009 9-73217 9 70441 9.67680 10 9 8 7 6 55 56 57 58 59 08571 08600 08829 08658 08687 .99632 .99630 .99627 .99625 .99622 .08602 .08632 .08661 .08690 .08720 11-6248 11-5853 11.5461 11.5072 11.4685 10308 .10337 .10366 -10395 .10424 .99467 99464 99461 99458 99455 .10363 10393 10422 10452 10481 9-64935 9-62205 9-59490 9-56791 9-54106 5 4 3 2 1 go 08716 .99619 .08749 11-4301 .10453 99452 10510 9-51436 Cos. Sin. Cot. Tan. Cos. Sin. Cot. | Tan. 85 84 NATURAL SINES, COSINES, TANGENTS, COTAN. 243 6 7 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 1 2 3 4 10453 10482 10511 10540 10569 .99452 99449 -99446 .99443 .99440 10510 10540 10569 10599 10628 9-51436 9-48781 9-46141 9-43515 9-40904 12187 12216 12245 12274 12302 99255 99251 99248 99244 99240 .12278 12308 12338 12367 12397 8-14435 8-12481 8-10536 8-08600 8-06674 60 59 58 57 _5J 55 54 53 52 51 5O 49 48 47 46 45 44 43 42 U 5 6 7 8 9 10 11 12 13 14 10597 10626 10655 .10684 .10713 .99437 .99434 .99431 99428 .99424 10657 10687 10716 10746 10775 9-38307 9-35724 9-33155 9-30599 9-28058 12331 12360 12389 12418 12447 99237 99233 99230 99226 99222 99219 99215 99211 .99208 .99204 12426 12456 12485 12515 .12544 8-04756 8-02848 8-00948 7-99058 7.97176 7-95302 7-93438 7-91582 7-89734 7-87895 7-86064" 7-84242 7-82428 7-80622 7-78825 .10742 .10771 .10800 .10829 .10858 .99421 .99418 .99415 .99412 .99409_ .99406 .99402 .99399 .9L396 .99393 10805 10834 10883 10393 _109_22_ 10952 10981 11011 11040 .11070 9-25530 9 23016 9-20516 9.18028 9.15554 9-13093 9-10646 9.08211 9-05789 9-03379 12476 12504 .12533 .12562 12591 .12574 .12603 .12633 .12662 . .12692^ 15 16 17 18 19 20 21 22 23 24 ,10887 .10916 .10945 .10973 .11002 .12620 .12649 .12678 .12706 .12735 99200 .99197 .99193 -99189 -99186 .12722 .12751 .12781 .12810 .12840 .11031 .11060 .11089 11118 .11147 .99390 -99386 .99383 -99380 .99377 11099 11128 11158 11187 11217 9-00983 8-98598 8-96227 8-93867 8.91520 8-89185 8-86862 8-84551 8-82252 8-79964 -12764 -12793 12822 .12851 ^12880 .99182 .99178 -99175 -99171 .99167 .12869 .12899 12929 .12958 .12988 7-77035 7-7525-4 7-73480 7-71715 7.69957 40 39 38 37 36 25 26 27 28 29 11176 .11205 11234 .11263 .11291 .99374 -93370 .99367 -99364 .99360 11246 11276 11305 11335 .11364 .12908 12937 .12968 -12995 .13024 .99163 99160 -99156 .99152 .99148 .13017 .13047 .13076 .13106 .13136 7.68208 7-66466 7-64732 7.63005 7-61287 35 34 33 32 31 30 31 32 33 34 .11320 .11349 .11378 11407 -11436 .99357 .99354 .99351 .99347 -99344_ .99341 .99337 99334 .99331 .99327 .11394 11423 .11452 .11482 .11511 8.77689 8-75425 8-73172 8-7093-1 8-68701 .13053 .13081 .13110 .13139 .13168 .99144 .99141 .99137 99133 .99129 .13165 .13195 .13224 .13254 .13284 7-59575 7-57872 7-56176 7-54487 7-52806 30 29 28 27 28 25 24 23 22 21 35 36 37 38 39 .11465 .11494 .11523 -11552 11580 .11541 .11570 .11600 .11629 .11659 8-66482 8-64275 8-62078 8-59893 8-57718 .13197 -13226 .13254 .13283 .13312 .99125 .99122 -993.18 99114 -99110 .13313 -13343 .13372 .13402 .13432 7-51132 7-49465 7-47806 7-46154 7-44509 40 41 42 43 44 45 46 47 48 49 11609 .11638 11667 -11696 .11725 .99324 -99320 .99317 .99314 .99310 .11688 .11718 .11747 -11777 -11806 8.55555 8-53402 8-51259 8.49128 8-47007 .13341 .13370 .13399 -13427 -13456 -99106 -90102 -99098 .99094 .99091 .13461 .13491 .13521 .13550 13580_ .13609 .13639 .13669 .13698 .13728 7-42871 7-41240 7.39616 7-37999 7.36389 2O 19 18 17 16 15 14 13 12 11 -11754 .11783 .11812 -11840 -11869 .99307 99303 .99300 .99297 99293 .11836 .11865 .11895 11924 .11954 '.1 1983" ,12013 .12042 12072 .12101 8-44896 8.42795 8-40705 8-38625 8-36555 -13485 .13514 .13543 .13572 .13600 99087 .99083 -99079 -99075 -99071 7-34786 7-33190 7-31600 7-30018 7-28442 50 51 52 53 54_ 55 56 57 58 59 .11898 .11927 .11956 .11985 -12014 99290 .99286 99283 .99279 .99276 8.34496 8-32446 8-30406 8.28376 8-26355 .13629 .13658 .13687 13716 .13744 .99067 -99063 .99059 .99055 .99051 13758 .13787 .13817 .13846 .13876 7-26873 7-25310 7-23754 7-22204 7-20661 10 9 8 7 6 5 3 2 1 o 12043 -12071 .12100 12129 .12158 .99272 .99269 .99265 .99262 .99258 .12131 .12160 12190 12219 .12249 8 . 24345 8-22344 8-20352 8-18370 8.16398_ .13773 .13802 13831 -13860 -13889 .99047 99043 99039 99035 99031 .13906 .13935 .13965 .13995 .14024 7-19125 7-17594 7.16071 7-14553 7-13042 S2. .12187 .99255 12278 8 14435 .13917 99027 14054 7-11537 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 8b NATURAL SINES, COSINES, TANGENTS, COTAN. 8 9 1 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. o 2 13917 13946 13975 14004 14033 .99027 .99023 .99019 .99015 .99011 14054 14084 14113 14143 14173 7.11537 7-10038 7-08546 7-07059 7-05579 15643 15672 15701 15730 15758 98769 .98764 -98760 -98755 -98751 15838 15868 .15898 15928 .15958 6-31375 6-30189 6-29007 6-27829 6.26655 60 59 58 57 56 5 6 7 8 9 14061 14090 14119 14148 14177 .99006 .99002 .98998 .98994 .98990 14202 14232 14262 14291 14321 7-04105 7-02637 7-01174 6-99718 6-98268 15787 15816 15845 15873 15902 -98746 -98741 .98737 98732 -98728 .15988 .16017 .16047 .16077 .16107 6-25486 6-24321 6-23160 6-22003 6-20851 55 54 53 52 51 10 11 12 13 14 14205 14234 14263 14292 14320 .98986 .98982 98978 98973 98969 14351 14381 14410 14440 .14470 6-96823 6-95385 6-93952 6-92525 6-91104 15931 15959 15988 16017 16046 -98723 -98718 -98714 -98709 -98704 .16137 .16167 .16196 .16226 .16256 6-19703 6-18559 6-17419 6-16283 6-15151 5O 49 48 47 46 15 16 17 18 19 14349 .14378 .14407 .14436 -14464 98965 98961 98957 .98953 .98948 14499 .14529 .14559 .14588 .14618 6-89688 6-88278 6-86874 6-85475 6-84082 16074 16103 16132 16160 16189 -98700 -98695 -98690 .98686 .98681 .16286 .16316 .16346 -16376 .16405 6-14023 6-12899 6-11779 6-10664 6-09552 45 44 43 42 41 20 21 22 23 24 . 14493 .14522 .14551 .14580 .14608 .98944 .98940 .98936 .98931 .98927 .14648 .14678 .14707 .14737 .14767 6-82694 6-81312 6-79936 6-78564 6-77199 -16218 .16246 .16275 .16304 .16333 98676 98671 .98667 .98662 .98657 .16435 .16465 -16495 16525 .16555 6-08444 6-07340 6-06240 6-05143 6-04051 40 39 38 37 36 25 26 27 28 29 .14637 .14666 .14695 .14723 .14752 .98923 .98919 98914 .98910 .98906 .14796 .14826 .14856 .14886 .14915 6-75838 6-74483 6-73133 6-71789 6-70450 .16361 .16390 .16419 .16447 ,16476 .98652 .98648 .98643 -98638 .98633 .16585 .16615 -16645 .16674 .16704 6-02962 6-01878 6-00797 5-99720 5-98646 35 34 33 32 31 30 31 32 33 34 .14781 .14810 .14838 .14867 .14896 .98902 .98897 .98893 .98889 .98884 .14945 .14975 .15005 .15034 .15064 6-69116 6-67787 6-66463 6-65144 6-63831 .16505 .16533 .16562 .16591 .16620 .98629 .98624 .98619 -98614 -98609 .16734 .16764 .16794 .16824 .16854 5-97576 5-96510 5-95448 5-94390 5-93335 30 29 28 27 26 35 36 37 38 39 .14925 .14954 .14982 .15011 .15040 98880 .98876 .98871 .98867 .98863 .15094 .15124 -15153 .15183 .15213 6-62523 6-61219 6-59921 6-58627 6-57339 -16648 .16677 .16706 16734 -16763 .98604 .98600 -98595 .98590 .98585 16884 .16914 .16944 .16974 .17004 5-92283 5-91236 5-90191 5-89151 5- ,88114 25 24 23 22 21 40 41 42 43 44 .15069 .15097 .15126 .15155 .15184 .98858 .98854 .98849 .98845 -98841 15243 .15272 15302 .15332 .15362 6.56055 6.54777 6.53503 6-52234 6-50970 .16792 .16820 ,16849 .16878 .16906 .98580 .98575 .98570 .98565 .98561 -17033 17063 .17093 .17123 .17153 5-87080 5-86051 5-85024 5.84001 5.82982 20 19 18 17 16 45 46 47 48 49 .15212 .15241 .15270 .15299 .15327 98836 98832 98827 98823 .98818 .15391 .15421 .15451 .15481 .15511 6-49710 6-48456 6-47206 6-45961 6-44720 .16935 .16964 -16992 17021 17050 .98556 .98551 .98548 -98541 -98536 .17183 .17213 .17243 -17273 -17303 5.81966 5-80953 5-79944 5-78938 5-77936 15 14 13 12 11 50 51 52 53 54 .15356 .15385 .15414 15442 .15471 98814 .98809 .98805 .98800 .98796 .15540 .15570 .15600 .15630 .15660 6-43484 6-42253 6.41026 6-39804 6-38587 -17078 -17107 -17136 17164 17193 .98531 -98526 98521 98516 .98511 17333 17363 .17393 .17423 .17453 5-76937 5-75941 5-74949 5-73960 5-72974 10 9 8 7 6 55 56 57 58 59 .15500 .15529 .15557 15586 .15615 .98791 .98787 -98782 -98778 .98773 .15689 .15719 .15749 15779 .15809 6-37374 6-36165 6-34961 6-33761 6-32566 -17222 -17250 .17279 .17308 .17336 .98506 .98501 .98496 -98491 .98486 17483 .17513 .17543 17573 .17603 5-71992 5-71013 5-70037 5-69064 5-68094 5 4 3 2 _J 60 .15643 98769 .15838 6-31375 .17365 -98481 -17633 5-67128 ' Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ' 81 e NATURAL SINES, COSINES, TANGENTS, COTAN. 245 10 11 1 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. ' o 2 3 4 .17365 17393 -17422 17451 .17479 .98481 .98476 .98471 .98466 .98461 .17633 .17663 .17693 .17723 .17753 5-67128 5-66165 5-65205 5 64248 5-63295 -19081 -19109 -19138 -19167 -19195 -98163 98157 .98152 .98146 .98140 19438 -19468 -19498 -19529 19559 5-14455 5-13658 5-12862 5-12069 5-11279 60 59 58 57 56 5 6 7 8 9 .17508 .17537 .17565 .17594 17623 .98455 .98450 .98445 .98440 .98435 .17783 .17813 .17843 .17873 .17903 5-62344 5-61397 5-60452 5-59511 5-58573 -19224 -19252 -19281 -19309 -19338 .98135 -98129 98124 98118 98112 19589 19619 19649 .19680 .19710 5-10490 5-09704 5-08921 5-08139 5-07360 55 54 53 52 51 10 11 12 13 14 .17651 .17680 .17708 -17737 .17766 .98430 98425 .98420 98414 .98409 17933 .17963 .17993 .18023 .18053 5-57638 5-56706 5.55777 5-54851 5-53927 -19366 19395 19423 -19452 -19481 98107 98101 -98096 .98090 .98084 -19740 19770 .19801 19831 19861 5-06584 5-05809 5-05037 5-04267 5-03499 50 49 48 47 46 15 16 17 18 19., .17794 .17823 .17852 17880 .17909 .98404 .98399 .98394 .98389 .98383 .18083 .18113 .18143 .18173 18203 5-53007 5-52090 5-51176 5-50264 5-49356 19509 19538 -19566 -19595 -19623 -98079 .98073 98067 -98061 98056 .19891 .19921 .19952 -19982 -20012 5-02734 5-01971 5-01210 5-00451 4-99695 45 44 43 42 41 30 21 22 23 24 17937 .17966 .17995 .18023 .18052 .98378 .98373 .98368 .98362 .98357 .18233 18263 .18293 .18323 .18353 5-48451 5-47548 5-46648 5-45751 5-44857 -19652 .19680 .19709 -19737 -19766 .98050 .98044 .98039 .98033 .98027 -20042 -20073 ,20103 -20133 20164 4.98940 4-98188 4-97438 4-96690 4.95945 40 39 38 37 36 25 26 27 28 29 .18081 .18109 .18138 .18166 18195 .98352 98347 .98341 .98336 .98331 .18384 .18414 . 18444 .18474 .18504 5-43966 5-43077 5-42192 5-41309 5 40429 .19794 -19823 -19851 .19880 .19908 .98021 .98016 -98010 -98004 -97998 .20194 .20224 20254 -20285 -20315 4-95201 4-94460 4-93721 4-92984 4-92249 35 34 33 32 31 30 31 32 33 34 .18224 .18252 .18281 .18309 .18338 98325 98320 .98315 98310 .98304 .18534 .18564 .18594 -18624 .18654 5-39552 5-38677 5-37805 5-36936 5-36070 19937 .19965 19994 .20022 -20051 -97992 -97987 -97981 -97975 -97969 -20345 -20376 - 20406 -20436 -20466 4-91516 4-90785 4-90056 4-89330 4-88605 30 29 28 27 26 35 36 37 38 9_ .18367 .18395 .18424 .18452 .18481 .98299 .98294 .98288 982a3 .98277 18634 .18714 .18745 .18775 .18805 5-35206 5-34345 5-33487 5-32631 5-31778 -20079 -20108 -20136 -20165 -20193 -97963 -97958 -97952 -97946 -97940 -20497 -20527 -20557 -20588 .20618 4-87882 4-87162 4-86444 4-85727 4 85013 25 24 23 22 21 40 41 42 43 44 .18509 .18538 18567 .18595 .18624 .98272 .98267 .98261 .98256 .98250 18835 18865 .18895 .18925 .18955 5-30928 5-30080 5-29235 5-28393 5-27553 20222 20250 -20279 20307 -20336 -97934 -97928 -97922 -97916 -97910 -20648 20679 -20709 -20739 -20770 4-84300 4-83590 4-82882 4-82175 4-81471 20 19 18 17 16 45 46 47 48 49 18652 .18681 .18710 18738 .18767 .98245 98240 .98234 98229 .98223 .18986 .19016 .19046 19076 .19106 5-26715 5-25880 5-25048 5-24218 5.23391 -20364 -20393 -20421 -20450 -20478 -97905 -97899 -97893 -97887 -97881 -20800 -20830 -20861 .20391 .20921 4-80769 4-80068 4-79370 4-78673 4-77978 15 14 13 12 _ii 50 51 52 53 54 -18795 .18824 .18852 .18881 .18910 98218 98212 98207 98201 .98196 .19136 19166 .19197 .19227 .19257 5-22566 5-21744 5-20925 5-20107 5-19293 -20507 20535 .20563 -20592 -20620 -97875 -97869 -97863 -97857 -97851 -20952 20982 .21013 .21043 .21073 4-77286 4-76595 4-75908 4-75219 4-74534 10 9 7 6 55 56 57 58 59 18938 .18967 .18995 19024 .19052 .98190 .98185 .98179 98174 98168 .19287 .19317 19347 19378 19408 5-18480 5-17671 5-16863 5-16058 5-15256 - 20649 20677 .20706 -20734 -20763 -97845 -97839 -97833 -97827 -97821 -21104 -21134 -21164 -21195 -21225 4-73851 4-73170 4-72490 4-71813 4-71137 5 4 3 ! 60 .19081 .98163 .19438 5-14455 -20791 -97815 -21256 4-70463 o T- Cos. Sin. Cot. Tan. Co's. Sin. Cot. Tan. , 79 78 246 NATURAL SINES, COSINES, TANGENTS, COTAN. 12 13 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. ' 1 2 3 4 20791 20820 20848 .20877 .20905 .97815 .97809 .97803 .97797 .97791 .21256 21286 21316 .21347 .21377 i 4-70463 4-69791 4-69121 4-68452 4-67786 22495 22523 22552 22580 22608 97437 97430 97424 97417 97411 23087 23117 23148 23179 23209 4-33148 4-32573 4-32001 4-31430 4-30860 60 59 58 57 56 5 6 7 8 9 .20933 .20962 .20990 .21019 21047 97784 .97778 97772 .97766 .97760 .21408 21438 .21469 .21499 21529 4-67121 4-66458 4-65797 4-65138 4 64480 22637 22665 22693 22722 22750 97404 -97398 .97391 97384 .97378 .23240 23271 23301 -23332 -23363 4-30291 4-29724 4-29159 4-28595 4-28032 55 54 53 52 51 10 11 12 13 14 .21076 .21104 .21132 .21161 21189 .97754 .97748 .97742 .97735 97729 21560 .21590 -21621 .21651 -21682 4-63825 4-63171 4-62518 4-61868 4-61219 22778 .22807 -22835 22863 -22892 97371 .97365 -97358 -97351 97345 23393 -23424 -23455 -23485 -23516 4-27471 4-26911 4-26352 4-25795 4-25239 50 49 48 47 46 15 16 17 18 19 .21218 .21246 .21275 .21303 -21331 .97723 .97717 97711 97705 97698 21712 .21743 .21773 .21804 .21834 4-60572 4-59927 4-59283 4-58641 4-58001 .22920 .22948 .22977 -23005 .23033 -97338 97331 .97325 -97318 .97311 -23547 -23578 -23608 -23639 .23670 4-24685 4-24132 4-23580 4-23030 4-22481 45 44 43 42 41 20 21 22 23 24 -21360 .21388 21417 21445 .21474 .97692 .97686 .97680 .97673 .97667 .21864 .21895 .21925 .21956 .21986 4-57363 4-56726 4-56091 4-55458 4-54826 .23062 .23090 .23118 .23146 .23175 -97304 -97298 -97291 -97284 -97278 -23700 -23731 -23762 -23793 -23823 4-21933 4-21387 4-20842 4-20298 4-L9756 40 39 38 37 36 25 26 27 28 29 .21502 .21530 .21559 .21587 .21616 .97661 .97655 .97648 .97642 97636 .22017 22047 .22078 .22108 .22139 4-54196 4-53568 4-52941 4-52316 4-51693 .23203 .23231 .23260 23288 -23316 -97271 -97264 .97257 -97251 .97244 -23854 -23885 -23916 .23946 23977 4-19215 4-18675 4-18137 4-17600 4-17064 35 34 33 32 31 30 31 32 33 34 .21644 .21672 21701 21729 .21758 97630 .97623 .97617 .97611 .97604 .22169 .22200 .22231 .22261 .22292 4-51071 4-50451 4-49832 4-49215 4-48600 23345 -23373 23401 -23429 -23458 -97237 -97230 .97223 97217 .97210 24008 .24039 24069 24100 24131 4-16530 4-15997 4-15465 4-14934 4-14405 30 29 28 27 26 35 38 37 38 39 .21786 21814 .21843 .21871 21899 .97598 .97592 .97585 .97579 -97573 22322 .22353 22383 .22414 .22444 4-47986 4-47374 4-46764 4-46155 4-45548 23486 -23514 .23542 23571 .23599 -97203 -97196 .97189 -97182 -97176 24162 .24193 24223 24254 -24285 4-13877 4-13350 4-12825 4-12301 4-11778 25 24 23 22 -21 40 41 42 43 44 .21928 .21956 .21985 22013 22041 .97566 -97560 .97553 .97547 .97541 .22475 .22505 22536 .22567 .22597 4-44942 4-44338 4-43735 4-43134 4-42534 23627 .23656 .23684 23712 23740 -97169 .97162 .97155 .97148 .97141 -24316 -24347 -24377 - 24408 -24439 4-11256 4-10736 4-10216 4-09699 4-09182 20 19 18 17 16 45 46 47 48 49 22070 .22098 22126 .22155 22183 .97534 .97528 .97521 .97515 .97508 .22628 .22658 .22689 22719 22750 4-41936 4-41340 4-40745 4-40152 4-39560 23769 23797 -23825 -23853 23882 .97134 .97127 97120 .97113 .97106 .24470 .24501 -24532 -24562 -24593 4-08666 4-08152 4-07639 4-07127 4-06616 15 14 13 12 11 50 51 52 53 54 22212 22240 22268 .22297 .22325 .97502 .97496 .97489 .97483 .97476 22781 22811 .22842 .22872 2290S 4-38969 4-38381 4-37793 4-37207 4-36623 -23910 23938 23966 -23995 .24023 .97100 .97093 .97086 .97079 .97072 24624 -24655 24686 24717 24747 4-06107 4-05599 4-05092 4-04586 4-04081 10 9 8 7 _6 55 56 57 58 59 .22353 .22382 22410 22438 .22467 .97470 97463 .97457 97450 .97444 22934 22964 22995 .23026 23056 4-36040 4-35459 4-34879 4-34300 4-33723 .24051 24079 24108 .24136 .24164 .97065 -97058 .97051 .97044 .97037 24778 24809 24840 24871 - 24902 4-03578 4-03076 4-02574 4-02074 4-01576 5 4 3 2 1 60 22495 97437 . 23087 4-33148 24192 97030 - 24933 4-01078 " 7 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. . 77* 76 NATURAL SINES, COSINES, TANGENTS, COTAN. 247 14 15 < Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. 1 2 3 4 .24192 . 24220 . 24249 .24277 .24305 .97030 97023 .97015 -97008 .97001 24933 24964 .24995 25026 25056 4-01078 4-00582 4-00086 3-99592 3-99099 -25882 -25910 -25938 -25966 -25994 96593 -96585 96578 96570 96562 -26795 -26826 26857 -26888 26920 3-73205 3-72771 3-72338 3-71907 3-71476 o;: 5S 58 57 56 5 6 7 8 9 .24333 -24362 -24390 -24418 - 24446 96994 .96987 .96980 .96973 -96966 25087 25118 25149 25180 25211 3-98607 3-98117 3-97627 3-97139 3-96651 26022 -26050 -26079 26107 26135 96555 96547 96540 -96532 96524 26951 26982 27013 27044 27076 3-71046 3-70616 3-70188 3-69761 3-69335 55 54 53 52 51 10 11 12 13 14 . 24474 .24503 -24531 .24559 .24587 -96959 -96952 .96945 .96937 .96930 25242 25273 25304 25335 25366 3-96165 3-95680 3-95196 3-94713 3-94232 26163 .26191 26219 .26247 26275 96517 96509 96502 96494 96486 27107 27138 27169 27201 27232 3-68909 3-68485 3-68061 3-67638 3-67217 50 49 48 47 46 15 16 17 18 19 .24615 24644 -24672 .24700 .24728 .96923 .96916 -96909 .96902 .96894 .25397 25428 .25459 .25490 -25521 3-93751 3-93271 3-92793 3-92316 3-91839 -26303 26331 .26359 -26387 .26415 96479 96471 -96463 96456 -96448 27263 27294 27326 27357 27388 3-66796 3-66376 3-65957 3-65538 3-65121 45 44 43 42 41 20 21 22 23 24 24756 24784 .24813 .24841 .24869 -96887 -96880 -96873 -96866 -96858 -25552 25583 25614 25645 25676 3-91364 3-90890 3-90417 3 89945 3-89474 .28443 .26471 26500 -26528 26556 -96440 -96433 96425 -96417 96410 -27419 27451 -27482 27513 27545 3-64705 3-64289 3-63874 3-63461 3-63048 4O 39 38 37 36 25 26 27 28 29 24897 24925 24954 24982 25010 96851 -96844 -96837 -96829 -96822 25707 25738 25769 -25800 25831 3-89004 3-88536 3-88068 3-87601 3-87136 26584 26612 26640 26668 26696 96402 -96394 -96386 96379 96371 27576 27607 27638 27670 27701 3-62636 3-62224 3-61814 3-61405 3-60996 35 34 33 32 31 30 31 32 33 34 25038 25066 25094 25122 25151 96815 96807 96800 96793 96786 .25862 25893 .25924 .25955 -2598R 3-86871 3-86208 3-85745 3-85284 3-84824 26724 26752 26780 26808 26836 -96363 -96355 -96347 -96340 96332 27732 27764 27795 27826 27858 3-60588 3-60181 3-59775 3-59370 3-58966 30 29 28 27 26 35 38 37 38 39_ 25179 25207 25235 25263 25291 96778 96771 96764 96756 96749 26017 26048 26079 26110 -26141 3-84364 3-83906 3-83449 3-82992 3-82537 26864 26892 26920 26948 26976 96324 96316 96308 96301 96293 27889 .27921 -27952 27983 28015 3-58562 3-58160 3-57758 3-57357 3-56957 25 24 23 22 21 40 41 42 43 44 25320 25348 25376 25404 25432 96742 96734 96727 96719 96712 26172 26203 28235 26266 26297 3-82083 3-81630 3-81177 3-80726 3-80276 27004 27032 27060 27088 27116 96285 96277 96269 96261 96253 28046 28077 28109 28140 28172 3-56557 3-56159 3-55761 3-55364 3-54968 20 19 18 17 16 45 46 47 48 49_ 25460 25488 25518 25545 25573 96705 96697 96690 96682 96675 28328 26359 26390 26421 26452 3-79827 3-79378 3-78931 3-78485 3 78040 27144 27172 27200 27228 27256 96246 96238 96230 96222 S6214 28203 28234 28266 28297 28329 3-54573 3-54179 3-53785 3-53393 3-53001 15 14 13 12 11 50 51 52 53 54 25601 25629 25857 25885 25713 96667 96660 96653 96645 96638 .26483 26515 26546 26577 26608 3-77595 3-77152 3-76709 3-76268 3-75828 27284 27312 27340 27368 27396 96206 96198 96190 96182 96174 28360 28391 28423 28454 28486 3-52609 3-52219 3-51829 3-51441 3-51053 1O 9 8 7 6 55 56 57 56 59 25741 25769 25798 25826 25854 96630 96623 96615 96608 96600 .26639 .26670 -26701 26733 26764 3-75388 3-74950 3-74512 3-74075 3-73640 27424 27452 27480 27508 27536 96166 96158 96150 96142 96134 28517 28549 28580 28612 28643 3-50666 3-50279 3-49894 3-49509 3-49125 5 4 3 2 1 60 25882 -96593 26795 3-73205 27564 96126 28675 3-48741 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ' 74 74 248 NATURAL SINES, COSINES, TANGENTS, COTAN. 16 17 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / o 1 2 3 4 27564 .27592 .27620 .27648 .27676 96126 .96118 .96110 .96102 .96094 28675 28706 28738 28769 .28800 3-48741 3-48359 3-47977 3-47596 3-47216 .29237 29265 29293 .29321 -29348 95630 95622 .95613 .95605 -95596 30573 30605 30637 30669 .30700 3-27085 3-26745 3-26406 3-26067 3-25729 60 59 58 57 56 5 6 7 8 9 .27704 .27731 .27759 .27787 .27815 .96086 .96078 .96070 .96062 .96054 .28832 28864 .28895 .28927 .28958 3-46837 3-46458 3-46080 3-45703 3-45327 .29376 .29404 .29432 .29460 -29487 -95588 .95579 .95571 .95562 .95554 30732 30764 .30796 .30828 .30860 3-25392 3-25055 3-24719 3-24383 3 24049 55 54 53 52 51 10 11 12 13 14 .27843 .27871 .27899 .27927 .27955 .96046 .96037 .96029 .96021 .96013 .28990 .29021 .29053 .29084 .29116 3-44951 3-44576 3-44202 3-43829 3-43456 -29515 -29543 .29571 .29599 .29626 .95545 .95536 .95528 .95519 .95511 -30891 30923 -30955 -30987 -31019 3-23714 3-23381 3-23048 3-22715 3-22384 50 49 48 47 46 15 16 17 18 19 .27983 .28011 .28039 .28067 -2809iL .96005 .95997 .95989 .95981 95972 .29147 .29179 .29210 .29242 .29274 3-43084 3-42713 3-42343 3-41973 3-41604 .29654 .29682 .29710 29737 -29765 .95502 .95493 .95485 95476 -95467 .31051 .31083 .31115 .31147 .31178 3-22053 3 21722 3-21392 3-21063 3-20734 45 44 43 42 41 20 21 22 23 24 .28123 .28150 .28178 .28206 .28234 .95964 .95956 .95948 .95940 .95931 .29305 .29337 .29368 .29400 .29432 3-41236 3-40869 3-40502 3-40136 3.39771 .29793 .29821 -29C49 -29876 -29904 .95459 954eO 9544i .95433 .95424 -31210 .31243 .31274 31306 .31338 3-20406 3-20079 3-19 r /52 3-19426 3-19100 4O 39 38 37 36 25 26 27 28 29 .28262 .28290 -28318 .28346 .28374 .95923 .95915 .95907 .95898 .95890 29463 .29495 .29526 .29558 .29590 3-39406 3-39042 3-38679 3-38317 3-37955 .29932 .29960 .29987 -30015 .30043 .95415 .95407 .95398 .95389 .95380 31370 .31402 31434 .31466 .31498 3-18775 3-18451 3-18127 3-17804 3.17481 35 34 33 32 31 30 31 32 33 34 .28402 .28429 .28457 .28485 .28513 .95882 .95874 .95865 .95857 .95849 .29621 .29653 .29685 .29716 .29748 3-37594 3-37234 3-36875 3-36516 3-36158 -30071 30098 .30126 .30154 .30182 .95372 .95363 .95354 .95345 .95337 .31530 -31562 -31594 -31626 .31658 3-17159 3-16838 3-16517 3-16197 3-15877 30 29 28 27 28 35 36 37 38 39 .28541 28569 .28597 .28625 .28652 .95841 .95832 .95824 .95816 .95807 29780 .29811 .29843 .29875 .2990fi_ 3-35800 3-35443 3-35087 3-34732 3-34377 .30209 .30237 .30265 .30292 .30320 .95328 .95319 -95310 -95301 .95293 .31690 .31722 .31754 .31786 .31818 3-15558 3-15240 3-14922 3-14605 3-14288 25 24 23 22 -21 40 41 42 43 44 .28680 .28708 .28736 .28764 .28792 .95799 .95791 .95782 .95774 .95766 .29938 .29970 .30001 .30033 .30065 3-34023 3-33670 3-33317 3-32965 3-32614 30348 30376 30403 .30431 .30459 -95284 -95275 -95266 -95257 -95248 .31850 .31882 .31914 .31946 .31978 3-13972 3-13656 3-13341 3-13027 3-12713 20 19 18 17 16 45 46 47 48 49 .28820 28847 28875 28903 28931 .95757 95749 .95740 .95732 .95724 .30097 .30128 .30160 .30192 .30224 3-32264 3-31914 3-31565 3-31216 3-30868 30486 .30514 .30542 .30570 .30597 -95240 -95231 -95222 -95213 -95204 32010 .32042 -32074 .32106 .32139 3-12400 3-12087 3-11775 3-11464 3-11153 15 14 13 12 11 50 51 52 53 54 28959 28987 29015 29042 29070 .95715 95707 .95698 95690 .95681 30255 30287 30319 30351 30382 3-30521 3-30174 3-29829 3-29483 3-29139 30625 30653 .30680 .30708 .30736 -95195 .95186 .95177 .95168 .95159 32171 -32203 32235 32267 32299 3-10842 3-10532 3-10223 3-09914 3-09606 1O 9 8 7 6 55 56 57 58 59 29098 29126 29154 29182 29209 .95673 95664 95656 95647 95639 30414 30446 30478 30509 30541 3-28795 3-28452 3.28109 3-27767 3-27426 -30763 -30791 30819 30846 30874 .95150 95142 .95133 .95124 .95115 32331 32363 32396 32428 32460 3-09298 3-08991 3-08685 3-08379 3-08073 5 3 2 1 go. 29237 95630 30573 3-27085 30902 95106 32492 3-07768 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ' 73 78 NATURAL SINES, COSINES, TANGENTS, COTAN. 249 18 19 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 1 2 3 4 .30902 .30929 .30957 .30985 .31012 .95106 .95097 .95088 .95079 .95070 32492 32524 32556 32588 .32621 3-07768 3-07464 3-07160 3-06857 3-06554 32557 32584 32612 32639 32667 94552 94542 94533 -94523 -94514 . 34433 34465 .34498 -34530 .34563 2-90421 2-90147 2-89873 2-89600 2-89327 6O 59 58 57 56 5 6 7 8 9 .31040 .31068 .31095 .31123 .31151 .95061 .95052 .95043 .95033 .95024 32653 .32685 .32717 .32749 .32782 3-06252 3-05950 3-05649 3-05349 3-05049 .32694 .32722 32749 .32777 .32804 -94504 .94495 94485 .94476 .94466 .34596 -34628 .34661 -34693 -34726 2-89055 2-88783 2-88511 2-88240 2-87970 55 54 53 52 51 10 11 12 13 14 .31178 .31206 .31233 .31261 .31289 .95015 .95006 .94997 .94988 .94979 .32814 .32846 .32878 .32911 .32943 3 04749 3 04450 3-04152 3-03854 3-03556 .32832 32859 32887 32914 .32942 -94457 . 94447 -94438 .94428 -94418 -34758 .34791 .34824 .34856 .34889 2-87700 2 87430 2-87161 2-86892 2-86624 5O 49 48 47 46 15 16 17 18 19 -31316 .31344 31372 .31399 .31427 .94970 .94961 .94952 .94943 .94933 .32975 .33007 .33040 .33072 .33104 3-03260 3-02963 3-02667 3-02372 3-02077 .32969 .32997 .33024 .33051 .33079 - 94409 .94399 -943SO .94380 .94370 -34922 -34954 -34987 .35020 .35052 2-86356 2-86089 2-85822 2-85555 2-85289 45 44 43 42 41 20 21 22 23 24 .31454 .31482 .31510 .31537 -31565 .94924 .94915 .94906 .94897 .94888 -33136 -33169 -33201 .33233 .33266 3-01783 3-01489 3-01196 3-00903 3-00611 -33106 .33134 .33161 .33189 .33216 -94361 -94351 -94342 .94332 .94322 .35085 .35118 35150 .35183 -35216 2-85023 2-84758 2-84494 2-84229 2-83965 40 39 38 37 36 25 26 27 28 29 31593 31620 -31648 31675 .31703 .94878 .94869 94860 94851 .94842 -33298 -33330 .33363 .33395 .33427 3-00319 3-00028 2-99738 2-99447 2-99158 33244 -33271 33298 .33326 -33353 -94313 .94303 .94293 .94284 .94274 .35248 .35281 .35314 35346 .35379 2-83702 2-83439 2-83176 2-82914 2-82653 35 34 33 32 31 30 31 32 33 34 31730 .31758 .31786 .31813 31841 .94832 .94823 .94814 94805 .94795 33460 33492 33524 33557 33589 2-98868 2-98580 2-98292 2-98004 2-97717 .33381 .33408 33436 33463 33490 .94264 .94254 .94245 .94235 .94225 .35412 .35445 .35477 .35510 .35543 2-82391 2-82130 2-81870 2-81610 2-81350 30 29 28 27 38 35 36 37 38 39 31868 .31896 31923 .31951 .31979 .94786 .94777 .94768 .94758 .94749 33621 .33654 -33686 -33718 -33751 2-97430 2-97144 2-96858 2-96573 2.96288 -33518 33545 .33573 33600 .33627 -94215 94206 .94196 94186 .94176 -35576 -35608 35641 .35674 .35707 2-81091 2-80833 2-80574 2-80316 2 80059 25 24 23 22 21 40 41 42 43 44 .32006 32034 .32061 32089 321i6 .94740 .94730 94721 .94712 94702 .33783 33816 33848 33381 .33913 2-96004 2-95721 2-95437 2-95155 2-94872 .33655 33682 33710 33737 33764 -94167 -94157 -94147 -94137 -94127 -35740 -35772 .35805 .35838 35871 2-79802 2-79545 2-79289 2-79033 2-78778 20 19 18 17 16 45 46 47 48 49 .32144 .32171 .32199 .32227 .32254 .94693 .94684 .94674 94865 -94856 .33945 33978 .34010 34043 .34075 2-94591 2-94309 2-94028 2-93748 2-93468 33792 33819 33846 33874 33901 -94118 94108 .94098 -94088 -94078 .35904 .35937 .35969 .36002 .36035 2-78523 2-78269 2-78014 2-77761 2-77507 15 14 13 12 11 50 51 52 53 54 .32282 .32309 .32337 .32364 .32392 .94646 94637 .94627 .94618 .94609 34108 .34140 34173 .34205 .34238 2-93189 2-92910 2-92632 2-92354 2-92076 -33929 33956 33983 -34011 34038 -94068 94058 . 94049 -94039 94029 .36068 .36101 .36134 36167 .36199 2-77254 2-77002 2-76750 2-76498 2-76247 10 9 7 6 55 56 57 58 59 32419 .32447 32474 32502 -32529 94599 .94590 .94580 94571 .94561 .34270 .34303 .34335 34368 . 34400 2-91799 2-91523 2-91246 2-90971 2-90696 34065 34093 34120 34147 34175 -94019 .94009 -93999 93989 -93979 36232 .36265 .36298 .36331 .36364 2-75996 2-75746 2-75496 2-75246 2-74997 5 4 3 2 60 .32557 94552 .34433 2.90421 -34202 .93969 .38397 2-74748 Q ' Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 71 70 250 NATURAL SINES, COSINES, TANGENTS, COTAN. 20 21 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 1 2 3 4 .34202 .34229 34257 34284 .34311 93969 93959 93949 .93939 -93929 36397 .36430 36463 36496 36529 2-74748 2-74499 2-74251 2-74004 2-73756 -35837 -35864 -35891 .35918 -35945 .93358 -93348 -93337 -93327 -93316 38386 38420 -38453 .38487 .38520 2-60509 2-60283 2-60057 2-59831 2-59606 6O 59 58 57 56 5 6 7 8 9 .34339 .34366 34393 . 34421 .34448 93919 .93909 .93899 .93889 .93879 36562 36595 36628 36661 .36694 2-73509 2-73263 2-73017 2-72771 2-72526 -35973 36000 -36027 .36054 .36081 -93306 -93295 .93285 93274 93264 .38553 .38587 38620 .38654 .38687 2-59381 2-59156 2-58932 2-58708 2-58484 55 54 53 52 51 10 11 12 13 14 34475 34503 .34530 .34557 .34584 .93869 .93859 .93849 93839 .93829 -36727 -36760 .36793 .36826 .36859 2-72281 2-72036 2-71792 2-71548 2-71305 36108 .36135 36162 .36190 .36217 .93253 93243 .93232 93222 -93211 -38721 38754 .38787 38821 .38854 2-58261 2-58038 2-57815 2-57593 2-57371 50 49 48 47 4B 15 16 17 18 i?_ .34612 34639 .34666 .34694 34721 .93819 .93809 93799 .93789 .93779 36892 .36925 .36958 .36991 -37024 2-71062 2-70819 2-70577 2-70335 2-70094 .36244 -36271 -36298 .36325 .36352 -93201 -93190 -93180 -93169 -93159 .38888 .38921 .38955 .38988 39022 2-57150 2-56928 2-56707 2-56487 2-56266 45 44 43 42 41 20 21 22 23 24 -34748 .34775 .34803 .34830 .34857 -93769 93759 93748 93738 .93728 -37057 .37090 -37123 .37157 -37190 2-69853 2-69612 2-69371 2-69131 2-68892 36379 .36406 -36434 -36461 -36488 -93148 -93137 -93127 -93116 -93106 39055 39089 .39122 39156 39190 2-56046 2-55827 2-55608 2-55389 2-55170 40 39 38 37 SB 25 26 27 28 29 34884 -34912 34939 .34966 .34993 93718 93708 .93698 93688 .93677 -37223 37256 .37289 .37322 .37355 2-68653 2-68414 2.68175 2-67937 2.67700 -36515 36542 .36569 .38596 .36623 93095 -93084 .93074 93063 .93052 -39223 39257 39290 .39324 39357 2-54952 2-54734 2.54516 2.54299 2-54082 35 34 33 32 31 30 31 32 33 34 .35021 .35048 -35075 -35102 -35130 .93667 .93657 93647 .93637 93626 37388 .37422 37455 .37488 .37521 2.67462 2.67225 2-66989 2-66752 2-66516 36650 .36677 .36704 .36731 .36758 93042 93031 93020 93010 .92999 39391 -39425 -39458 -39492 -39526 2-53865 2.53648 2-53432 2-53217 2-53001 3O 29 28 27 26 35 36 37 38 39 -35157 .35184 .35211 35239 -35266 93616 93606 93596 93585 93575 37554 -37588 37621 .37654 .37687 2-66281 2-66046 2-65811 2-65576 2-65342 36785 .36812 .36839 .36867 .36894 92988 92978 92967 92956 .92945 39559 .39593 39626 39660 .39694 2-52786 2-52571 2-52357 2-52142 2-51929 25 24 23 22 21 4O 41 42 43 44 -35293 -35320 -35347 -35375 -35402 .93565 93555 93544 .93534 .93524 -37720 37754 37787 .37820 .37853 2-65109 2-64875 2-64642 2-64410 2-64177 36S21 36948 -36975 -37002 -37029 .92935 92924 -92913 -92902 -92892 -39727 .39761 39795 .39829 -39862 2-51715 2-51502 2-51289 2-51076 2-50864 20 19 18 17 16 45 46 47 48 49 -35429 35456 .35484 -35511 -35538 93514 93503 93493 .93483 93472 -37887 .37920 37953 37986 .38020 2-63945 2-63714 2-63483 2-63252 2-63021 .37056 -37083 -37110 .37137 -37164 .92881 92870 92859 92849 .92838 39896 .39930 -39963 39997 .40031 2-50652 2-50440 2-50229 2-50018 2-49807 15 14 13 12 11 50 51 52 53 54 35565 .35592 35619 .35647 35674 93462 93452 93441 -93431 93420 -38053 -38086 .38120 38153 .38186 2-62791 2-62561 2.62332 2-62103 2-61874 -37191 -37218 - 37245 37272 .37299 92827 -92816 -92805 -92794 -92784 .40065 40098 .40132 .40166 - 40200 2-49597 2-49386 2-49177 2-48967 2-48758 10 9 8 7 6 55 56 57 58 59 35701 35728 35755 35782 35810 .93410 .93400 93389 .93379 93368 -38220 .38253 38286 .38320 .38353 2.61646 2-61418 2-61190 2.60963 2-60736 .37326 .37353 37380 37407 -37434 -92773 -92762 .92751 92740 92729 40234 40267 40301 40335 40369 2-48549 2-48340 2-48132 2-47924 2.47716 5 4 3 2 1 60 35837 93358 38386 2-60509 -37461 92718 40403 2-47509 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ' 69 68 NATURAL SINES, COSINES, TANGENTS, COTAN. 251 33 33 1 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. f 1 2 3 37461 .37488 .37515 .37542 .37569 92718 .92707 92697 92686 92675 . 40403 .40436 . 40470 .40504 .40538 2-47509 2-47302 2.47095 2.46888 2-46682 39073 39100 39127 39153 .39180 92050 92039 .92028 92016 .92005 42447 42482 42516 42551 42585 2-35585 2-35395 2-35205 2-35015 2-34825 60 59 58 57 5fi 5 6 7 8 9 .37595 .37622 .37649 .37676 .37703 92664 .92653 .92642 .92631 92620 .40572 40606 .40640 .40674 .40707 2-46476 2-46270 2-46065 2-45860 2-45655 39207 39234 39280 .39287 -39314 .91994 91982 .91971 .91959 -91948 42619 42654 42688 .42722 .42757 2-34636 2-34447 2-34258 2-34069 2-33881 55 54 53 52 51 10 11 12 13 14 .37730 .37757 37784 .37811 .37838 .92609 .92598 .92587 .92576 .92565 40741 .40775 40809 40843 .40877 2-45451 2-45246 2-45043 2-44839 2.44636 -39341 -39367 -39394 -39421 - 39448 -91936 .91925 .91914 .91902 .91891 42791 .42826 42860 .42894 .42929 2-33693 2-33505 2-33317 2-33130 2-32943 50 49 48 47 46 15 16 17 18 19 , 37865 .37892 .37919 .37946 .37973 .92554 .92543 92532 .92521 .92510 .40911 40945 .40979 .41013 .41047 2-44433 2-44230 2-44027 2-43825 2-43623 -39474 -39501 -39528 -39555 .39581 .91879 91868 91856 91845 .91833 42963 .42998 43032 43067 43101 2-32756 2-32570 2-32383 2-32197 2-32012 45 44 43 42 41 20 21 22 23 24 .37999 .38026 38053 .38080 .38107 92499 92488 .92477 92466 92455 .41081 .41115 .41149 .41183 .41217 2-43422 2-43220 2-43019 2-42819 2-42618 -39608 -39635 -39661 .39688 -39715 91822 .91810 .91799 91787 .91775 43136 43170 .43205 .43239 .43274 2-31826 2-31641 2-31456 2-31271 2-31086 40 39 38 37 3 25 26 27 28 29 .38134 .38161 .38188 -38215 .38241 .92444 .92432 .92421 .92410 92399 .41251 .41285 41319 41353 .41387 2-42418 2-42218 2-42019 2-41819 2-41620 39741 .39768 .39795 .39822 .39848 91764 91752 91741 .91729 -91718 43308 .43343 .43378 43412 .43447 2-30902 2-30718 2-30534 2-30351 2-30167 35 34 33 32 31 30 31 32 33 34 .38268 .38295 .38322 .33349 .38376 92388 .92377 92366 92355 92343 .41421 41455 41490 .41524 .41558 2 41421 2-41223 2-41025 2-40827 2-40629 39875 .39902 .39928 -39955 39982 .91706 -91694 -91683 91671 91660 .43481 .43516 .43550 43585 43620 2-29984 2-29801 2-29619 2-29437 2-29254 30 29 28 27 26 35 36 37 38 39 38403 .38430 .38456 .38483 .38510 .92332 .92321 92310 92299 .92287 .41592 41626 41660 41694 .41728 2-40432 2-40235 2-40038 2-39841 2-39645 - 40008 .40035 -40062 -40088 -40115 91648 .91636 .91625 91613 -91601 43654 .43689 .43724 .43758 .43793 2-29073 2-28891 2-28710 2-28528 2-28348 25 24 23 22 21 40 41 42 43 44 38537 38564 .38591 .38617 -38644 .92276 .92265 .92254 .92243 92231 .41763 41797 41831 .41865 .41899 2-39449 2-39253 2-39058 2-38863 2-38668 -40141 -40168 -40195 40221 40248 .91590 91578 .91566 .91555 .91543 43828 .43862 43897 43932 .43966 2-28167 2-27987 2-27806 2-27626 2 - 27447 20 19 18 17 16 45 46 47 48 49 -38671 .38698 -38725 38752 .33778 92220 92209 .92198 .92186 92175 .41933 .41968 .42002 .42036 .42070 2-38473 2-38279 2-38084 2-37891 2-37697 40275 40301 .40328 .40355 .40381 91531 91519 .91508 .91496 91484 44001 .44036 .44071 .44105 44140 2-27267 2-27088 2-26909 2-26730 2-26552 15 14 13 12 11 50 51 52 53 54 38805 .38832 .38859 38886 .38912 .92164 92152 92141 92130 .92119 .42105 .42139 .42173 .42207 42242 2-37504 2-37311 2-37118 2-36925 2-36733 . 40408 -40434 -40461 -40488 -40514 91472 91461 91449 91437 91425 44175 44210 . 44244 .44279 44314 2-26374 2-26196 2-26018 2-25840 1O 9 8 7 6 55 56 57 58 59 38939 38966 38993 .39020 .39046 .92107 .92096 92085 92073 .92062 42276 .42310 42345 42379 42413 2-36541 2-36349 2-36158 2-35967 2-35776 .40541 40567 40594 40621 40847 91414 .91402 .91390 .91378 91366 .44349 44384 44418 44453 . 44488 2-25486 2-25309 2-25132 2-24956 2-24780 5 3 2 1 60 -39073 .92050 . 42447 2-35585 40674 91355 -44523 2 24604 _o Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. f 67 66 252 NATURAL SINES, COSINES, TANGENTS, COTAN. 24* 25 t Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / o 1 2 3 40674 .40700 40727 .40753 .40780 91355 91343 .91331 .91319 9i307 44523 44558 -44593 .44627 .44662 2-24604 2-24428 2-24252 2-24077 2.23902 42262 .42288 .42315 -42341 -42367 -90631 -90618 -90606 90594 90o82 -46631 46666 46702 46737 46772 2-14451 2-14288 2-14125 2-13963 2.13801 60 59 58 57 56 5 6 7 8 9 .40806 .40833 .40860 .40886 .40913 91295 .91283 .91272 91260 .91248 44697 44732 44767 44802 44837 2-23727 ?. 23553 2.23378 2-23204 2-23030 42394 -42420 -42446 42473 -42499 -90569 -90557 90545 90532 90520 .46808 .46843 46879 46914 46950 2-13639 2-13477 2-13316 2-13154 2-12993 55 54 53 52 51 10 11 12 13 14 40939 40966 40992 41019 .41045 .91236 .91224 .91212 .91200 .91188 44872 44907 44942 44977 45012 2-22857 2-22683 2.22510 2-22337 2-22164 -42525 -42552 -42578 42604 .42631 90507 90495 .90483 - 90470 -90458 46985 .47021 .47056 47092 47128 2-12832 2-12671 2-12511 2-12350 2-12190 50 49 48 47 46 15 16 17 18 19 .41072 .41098 .41125 .41151 .41178 91176 91164 91152 .91140 .91128 45047 45082 45117 45152 .45187 2.21992 2-21819 2-21647 2-21475 2-21304 .42657 .42683 .42709 42736 42762 -90446 -90433 90421 -90408 90396 .47163 .47199 .47234 .47270 -47305 2-12030 2-11871 2.11711 2-11552 2-11392 45 44 43 42 41 20 21 22 23 24 .41204 41231 41257 41284 41310 .91116 91104 .91092 .91080 91068 -45222 45257 -45292 -45327 -45362 2-21132 2-20961 2-20790 2-20619 2-20449 42788 .42815 .42841 42867 -42894 .90383 90371 -90358 -90346 -90334 -47341 .47377 .47412 .47448 -47483 2-11233 2.11075 2-10916 2-10758 2-10600 40 39 38 37 36 25 26 27 28 29 41337 41363 41390 41416 41443 -91056 -91044 -91032 .91020 91008 45397 45432 45467 45502 .45538 2-20278 2-20108 2.19938 2-19769 2.19599 .42920 42946 42972 -42999 -43025 90321 90309 90296 90284 90271 .47519 -47555 .47590 .47626 .47662 2 10442 2-10284 2-10126 2-09969 2-09811 35 34 33 32 31 30 31 32 33 34 41469 41496 41522 41549 41575 .90996 90984 90972 .90960 .90948 45573 -45608 -45643 -45678 45713 2-19430 2-19261 2-19092 2-18923 2-18755 43051 43077 -43104 -43130 -43156 90259 90246 90233 90221 .90208 .47698 .7733 .47769 .47805 47840 2-09354 2-09498 2-09341 2-09184 2-09028 30 29 28 27 26 35 36 37 38 39 41602 41628 41655 .41681 .41707 90936 -90924 -90911 90899 .90887 45748 45784 46819 45854 45889 2-18587 2-18419 2-18251 2-18084 2-17916 -43182 -43209 43235 -43261 -43287 90196 90183 90171 90158 .90146 47876 47912 47948 47984 -48019 2-08872 2-08716 2-08560 2-08405 2-08250 25 24 23 22 21 40 41 42 43 44 41734 .41760 41787 41813 41840 .90875 90863 90851 90839 -90826 45924 45960 45995 46030 46065 2-17749 2-17582 2-17416 2-17249 2-17083 -43313 -43340 43366 -43392 .43418 .90133 90120 .90108 .90095 .90082 48055 48091 48127 48163 48198 2-08094 2-07939 2-07785 2-07630 2-07476 20 19 18 17 16 45 46 47 48 49 41866 41892 41919 41945 41972 90814 90802 90790 90778 90766 46101 46136 46171 46206 46242 2-16917 2-16751 2.16585 2-16420 2-16255 - 43445 43471 43497 43523 -43549 90070 90057 90045 90032 -90019 -48234 -48270 -48306 -48342 -48378 2.07321 2-07167 2-07014 2-06860 2-06706 15 14 13 12 11 50 51 52 53 54 41998 42024 42051 42077 42104 90753 -90741 -90729 .90717 90704 46277 46312 -46348 46383 46418 2-16090 2-15925 2-15760 2-15596 2-15432 -43575 -43602 43628 43654 .43680 -90007 89994 89981 .89968 .89956 .48414 .48450 -48486 -48521 48557 2-06553 2 - 06400 2-06247 2-06094 2-05942 10 9 8 6 55 56 57 58 59 42130 42156 42183 42209 42235 .90692 .90680 .90668 .90655 .90643 46454 .46489 .46525 46560 .46595 2-15268 2-15104 2-14940 2 14777 2-14614 43706 43733 -43759 -43785 -43811 89943 -89930 -89918 89905 -89892 48593 .48629 -48665 48701 -48737 2-05790 2-05637 2-05485 2-05333 2-05182 5 3 2 1 60 .42262 90631 46631 2-14451 -43837 -89879 -48773 2-05030 _o Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 65 64 NATURAL SINES, COSINES, TANGENTS, COTAN. 253 86 37 t Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 1 2 3 4 43837 43863 .43889 .43916 .43942 89879 89867 .89854 .89841 .89828 48773 .48809 .48845 .48881 .48917 2-05030 2-04879 2-04723 2.04577 2-04426 45399 -45425 .45451 .45477 .45503 .89101 .89087 .89074 .89061 .89048 -50953 .50989 -51026 -51063 -51099 1-96261 1.96120 1-95979 1.95838 1.95698 60 59 58 57 56 5 6 7 8 9 43968 .43994 .44020 . 44046 .44072 .89816 .89803 .89790 .89777 .89764 .48953 .48989 .49026 .49062 .49098 2-04276 2-04125 2-03975 2-03825 2-03675 .45529 .45554 .45580 .45606 -45632 89035 -89021 -89008 -88995 .88981 51136 .51173 .51209 .51246 -51283 1-95557 1-95417 1-95277 1-95137 1-94997 55 54 53 52 51 10 11 12 13 14 .44098 .44124 .44151 -44177 .44203 .89752 .89739 .89726 .89713 .89700 .49134 .49170 49206 49242 .49278 2-03526 2-03376 2-03227 2-03078 2-02929 45658 .45684 .45710 .45736 .45762 -88968 .88955 .88942 -88928 .88915 -51319 -51356 -51393 .51430 51467 1-94858 1-94718 1.94579 1 . 94440 1-94301 50 49 48 47 46 15 16 17 18 19 44229 44255 .44281 .44307 .44333 .89687 .89674 .89662 .89649 -89636 .49315 .49351 .49387 .49423 .49459 2-02780 2-02631 2-02483 2-02335 2-02187 45787 .45813 .45839 .45865 .45891 -88902 -88888 -88875 .88862 .88848 .51503 -51540 -51577 .51614 -51651 1.94162 1-94023 1-93885 1.93746 1-93608 45 44 43 42 41 30 21 22 23 24 .44359 .44385 .44411 44437 .44464 .89623 .89610 .89597 -89584 .89571 .49495 .49532 .49568 .49604 .49640 2-02039 2-01891 2-01743 2-01596 2-01449 .45917 .45942 .45968 45994 .46020 .88835 -88822 88808 -88795 -88782 -51688 51724 51761 .51798 .51835 1-93470 1-93332 1.93195 1-93057 1-9?920 40 39 38 37 36 25 26 27 28 29 .44490 .44516 .44542 .44568 .44594 .89558 .89545 .89532 .89519 .89506 -49677 .49713 .49749 .49786 -49822 2-01302 2-01155 2-01008 2-00862 2-00715 .46046 .46072 46097 .46123 -46149 -88768 -88755 -88741 .88728 .88715 .51872 51909 51946 .51983 -52020 1-92782 1-92645 1-92508 1-92371 1-92235 35 34 33 32 31 30 81 32 33 34 . 44620 .44646 -44672 .44698 .44724 89493 .89480- .89467 .89454 .89441 .49858 .49894 .49931 .49967 .50004 2-00569 2-00423 2-00277 2-00131 1.99986 .46175 .46201 .46226 .46252 46278 .88701 -88688 -88674 -88661 .88647 .52057 52094 .52131 .52168 .52205 1-92098 1-91962 1-91826 1-91690 1.91554 30 29 28 27 28 35 36 37 38 39 44750 .44776 .44802 .44828 .44854 .89428 .89415 .89402 .89389 .89376 . 50040 .50076 .50113 .50149 .50185 1-99841 1-99695 1-99550 1-99406 1-99261 .46304 .46330 46355 46381 -46407 88634 -88620 -88607 -88593 -88580 -52242 .52279 -52316 .52353 -52390 1-91418 1-91282 1.91147 1.91012 1.90876 25 24 23 22 21 40 41 42 43 44 .44880 .44906 .44932 44958 .44984 .89363 .89350 .89337 .89324 .89311 .50222 .50258 .50295 .50331 .50368 1-99116 1-98972 1-98828 1-98684 1.98540 46433 -46458 -46484 -46510 -46536 -88566 -88553 -88539 -88526 -88512 -52427 .52464 .52501 -52538 .52575 1-90741 1-90607 1-90472 1-90337 1-90203 3O 19 18 17 16 45 46 47 48 49 .45010 45036 .45062 .45088 -45114 .89298 .89285 .89272 89259 .89245 . 50404 . 50441 .50477 .50514 .50550 1-98396 1-98253 1.98110 1.97966 1-97823 -46561 -46587 .46613 -46639 .46664 -88499 -88485 -88472 -88458 . 88445 .52613 52650 -52687 .52724 .92761 1-90069 1-89935 1-89801 1-89667 1-89533 15 14 13 12 11 50 51 52 53 54 -45140 .45166 45192 .45218 .45243 .89232 .89219 -89206 .89193 -89180 .50587 -50623 50660 -50696 -50733 1-97681 1-97538 1-97395 1-97253 1-97111 .46690 .46716 .46742 .46767 .46793 .88431 88417 88404 88390 88377 52798 .52836 .52873 .52910 .52947 1-89400 1-89266 1.89133 1.89000 1-88867 10 9 8 7 6 55 56 57 58 59 .45269 .45295 .45321 -45347 45373 -89167 -89153 -89140 .89127 -89114 -50769 -50806 -50843 -50879 50916 1-96969 1-96827 1.96685 1-96544 1-96402 -46819 - 46844 -46870 -46896 -46921 88363 88349 88336 88322 88308 52985 53022 53059 53096 53134 1-88734 1-88602 1-88469 1-88337 1-88205 5 4 3 2 1 60 45399 .89101 -50953 1.96261 -46947 88295 53171 1-88073 / Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 63 63 254 NATURAL SINES, COSINES, TANGENTS, COTAN. 28 29 I Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. \ y 2 3 .46947 .46973 .46999 .47024 .47050 .88295 .88281 .88267 .88254 .88240 -53171 .53208 53246 .53283 .53320 1.88073 1-87941 1-87809 1.87677 1.87546 48481 .48506 -48532 -48557 -48583 87462 .87448 87434 87420 .87406 -55431 -55469 .55507 55545 .55583 i 80405 1.80281 1.80158 1-80034 1.79911 60 59 58 57 56 5 6 7 9 .47076 47101 .47127 .47153 .47178 .88226 88213 .88199 .88185 .88172 .53358 .53395 .53432 53470 .53507 1.87415 .1.87283 1-87152 1.87021 1-86891 -48608 48634 .48659 48684 48710 .87391 .87377 .87363 87349 .87335 .55621 -55659 .55697 -55736 -55774 1 79788 1 79665 1.79542 1.79419 1-79296 55 54 53 52 51 10 11 12 13 14 .47204 .47229 .47255 .47281 .47306 .88158 .88144 .88130 .88117 .88103 .53545 .53582 .53620 .53657 .53694 1.86760 1.86630 1.86499 1.86369 1.86239 48735 .48761 48786 .48811 48837 87321 .87306 .87292 87278 .87264 -55812 -55850 -55888 .55926 -55964 1-79174 1 7905] 1-78929 1-78807 1-78685 50 49 48 47 46 15 16 17 18 *9 47332 .47358 .47383 .47409 .47434 .88089 .88075 .88062 88048 .88034 53732 .53769 -53807 -53844 .53882 1-86109 1.85979 .85850 : -85720 -85591 .48862 .48888 .48913 .48938 -48964, 87250 .87235 87221 .87207 -87193 -56003 -56041 -56079 -56117 .56156 1-78563 1-78441 1-78319 1-78198 1-78077 45 44 43 42 ,,.41 20 21 22 23 24 47460 .47486 .47511 -47537 .47562 .88020 .88006 .87993 .87979 .87965 .53920 .53957 .53995 .54032 .54070 .85462 .85333 83204 1-85075 1.84946 .48989 .49014 .49040 -49065 -49090 .87178 .87164 .87150 .87136 .87121 .56194 -56232 -56270 -56309 56347 1.77955 1-77834 1-77713 1-77592 1-77471 40 39 38 37 36 25 26 27 28 29 47588 47614 47639 .47665 .47690 .87951 .87937 87923 .87909 .87896 .54107 .54145 54183 . 54220 . 54258 1.84818 1-84689 1-84561 1.84433 1-84305 -49116 -49141 -49166 -49192 -49217 87107 87093 .87079 -87064 -87050 -56385 -56424 -56462 .56501 .56539 1-77351 1-77230 1 77110 1-76990 1-76869 35 34 33 32 31 30 31 32 33 34 .47716 .47741 .47767 -47793 .47818 .87882 .87868 .87854 .87840 .87826 54296 54333 .54371 . 54409 - 54446 1.84177 1 - 84049 1.83922 1.83794 1.83667 -49242 -49268 -49293 -49318 .49344 -87036 -87021 -87007 -86993 -86978 -56577 -56616 5'6654 .56693 .56731 1.76749 1-76629 1.76510 1-76390 1.76271 30 29 28 27 26 35 36 37 38 39 .47844 47869 .47895 .47920 .47946 .87812 .87798 .87784 .87770 -87756 - 54484 -54522 -54560 -54597 -54635 .83540 -83413 -83286 .83159 -83033 .49369 -49394 -49419 49445 .49470 -86964 -86949 -86935 -86921 .86906 56769 .56808 .56846 -56885 -56923 1-76151 1-76032 1-75913 1-75794 1-75675 25 24 23 22 -21 40 41 42 43 44 .47971 -47997 .48022 .48048 -48073 87743 .87729 87715 87701 .87687 .54673 54711 - 54748 -54786 -54824 .82906 .82780 1-82654 1-82528 1-82402 49495 49521 49546 49571 .49596 -86892 -86878 86863 86849 .86834 -56962 57000 -57039 57078 .57116 1-75556 1 75437 1-75319 1-75200 1.75082 20 19 18 17 16 45 46 47 48 49 .48099 48124 .48150 48175 .48201 .87673 .87659 .87645 .87631 .87617 .54862 . 54900 .54938 .54975 .55013 1.82276 1-82150 1-82025 1-81899 1-81774 .49622 -49647 49672 -49697 -49723 -86820 .86805 .86791 -86777 -86762 .57155 .57193 .57232 .57271 -57309 1-74964 1 . 74846 1-74728 1-74610 1 . 74492 15 14 13 12 11 50 51 52 53 54 .48226 .48252 .48277 .48303 .48328 87603 .87589 -87575 .87561 87546 .55051 .55089 .55127 .55165 . 55203 1-81649 1.81524 1.81399 1-81274 1-81150 49748 .49773 -49798 -49824 -49849 .86748 -86733 -86719 .86704 -86690 .57348 -57386 -57425 .57464 -57503 1-74375 1 . . 4257 1 - 74140 1 - 74022 1-73905 10 9 8 7 6 55 56 57 58 59 .48354 -48379 48405 .48430 .48456 87532 .87518 .87504 87490 .87476 55241 .55279 .55317 55355 .55393 1.81025 1-80901 1.80777 1.80653 1-80529 -49874 49899 -49924 49950 49975 -86675 .86661 .86646 .86632 .86617 57541 .57580 .57619 .57657 -57696 1-73788 1-73671 1.73555 1-73438 1.73321 5 4 3 2 1 90 -48481 .87462 55431 1.80405 50000 .86603 57735 1-73205 Cos. Sin. Cot. Tan. Cos, Sin. Cot. Tan. ' 61 60 NATURAL SINES, COSINES, TANGENTS, COTAN. 255 30 31 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. .Cot,. ' o 1 2 3 4 .50000 .50025 .50050 .50076 .50101 .86603 .86588 .86573 .88559 .86544 -57735 .57774 .57813 .57851 .57890 1.73205 1-73089 1.72973 1.72857 1.72741 .51504 51529 .51554 .51579 .51604 .85717 85702 85687 .85672 .85657 60086 60126 60165 60205 60245 1.66428 1-66318 1-66209 1-66099 i- 65990 60 59 58 57 56 5 6 7 8 9 .50126 .50151 .50176 .50201 .50227 .86530 .86515 .86501 .86486 .86471 .57929 .57968 .58007 .58046 .58085 1-72625 1.72509 1.72393 1-72278 1.72163 .51628 51653 51678 .51703 .51728 85642 85627 85612 .85597 -85582 60284 .60324 60364 60403 . 60443 1-65881 1-65772 1-65663 1 65554 1-65445 55 54 53 52 51 10 11 12 13 14 .50252 .50277 .50302 .50327 -50352 86457 .86442 .86427 .86413 .86398 .58124 .58162 .58201 .58240 -58279 1-72047 1.71932 1.71817 1.71702 1.71588 -51753 51778 -51803 51828 51852 85567 85551 .85536 -85521 .85506 60483 60522 .60562 60602 60642 1-65337 1-65228 1-65120 1.65011 1 . 64903 5O 49 48 47 48 15 16 17 18 19 .50377 50403 50428 .50453 .50478 86384 .86369 86354 .86340 .86325 .58318 .58357 .58396 -58435 .58474 1-71473 1.71358 1.71244 1-71129 1-71015 .51877 .51902 -51927 51952 .51977 .85491 .85476 -85461 -85446 85431 60681 .60721 .60761 .60801 .60841 1-64795 1-64687 1-64579 1-64471 1-64363 45 44 43 42 41 20 21 22 23 24 .50503 .50528 -50553 .50578 .50603 .86310 86295 .86281 .86266 .86251 58513 -58552 -58591 -58631 .58670 1-70901 1-70787 1-70673 1-70560 1 . 70446 52002 .52026 52051 .52076 -52101 -.85416 .85401 -85385 -85370 -85355 60881 .60921 60960 61000 61040 1-64256 1-64148 1 -64041 1-63934 1-63826 40 39 38 37 36 25 28 27 28 89 .50628 .50654 .50679 -50704 50729 .86237 86222 ,86207 .86192 .86178 58709 .58748 .58787 .58826 .58865 1-70332 1-70219 1-70106 1-69992 1.69879 .52126 .52151 .52175 .52200 -52225 85340 .85325 85310 .85294 .85279 .61080 .61120 .61160 .61200 -61240 1-63719 1-63612 1-63505 1-63398 1-63292 35 34 33 32 31 30 31 32 33 34 50754 .50779 . 50804 .50829 .50854 .86163 86148 .86133 86119 .86104 .58905 .58944 .58983 .59022 .59061 1-69766 1-69653 1-69541 1-69428 1-69316 52250 52275 .52299 .52324 52349 85264 .85249 85234 85218 .85203 -61280 -61320 .61360 .61400 .61440 1-63185 1-63079 1-62972 1-62866 1-62760 30 29 28 27 26 35 36 37 38 39 50879 .50904 50929 50954 50979 .86089 -86074 .86059 86045 86030 .59101 59140 .59179 59218 .59258 1-69203 1.69091 1-68979 1.68866 1-68754 .52374 .52399 52423 52448 52473 .85188 85173 85157 .85142 -85127 61480 61520 61561 61601 .61641 1-62654 1-62548 1-62442 1-62336 1-62230 25 24 23 22 21 40 41 42 43 44 51004 51029 51054 .51079 51104 .86015 .86000 85985 -85970 .85956 .59297 .59336 .59376 .59415 .59454 1-68643 1.68531 1.68419 1-68308 1.68196 .52498 52522 52547 52572 .52597 .85112 .85096 85081 85066 -85051 61681 61721 61761 .61801 .61842 1-62125 ]- 62019 1-61914 1-61808 1-61703 20 19 18 17 18 45 46 47 48 49 51129 51154 51179 51204 .51229 85941 .85926 .85911 .85896 .85881 .59494 59533 59573 .59612 .59651 1-68085 1-67974 1-67863 1.67752 1-67641 -52621 -52646 -52671 -52696 .52720 85035 85020 85005 -84989 -84974 61882 .61922 61962 .62003 62043 1-61598 1.61493 1.61388 1-61283 1-61179 15 14 13 12 11 50 51 52 53 54 .51254 51279 51304 51329 .51354 85866 .85851 85836 85821 .85806 59691 59730 59770 59809 59849 1-67530 1-67419 1-67309 1 67198 1-67088 .52745 -52770 -52794 -52319 .52844 84959 . 84943 .84928 84913 .84897 62083 62124 62164 62204 .62245 1-61074 1-60970 1 60865 1-60761 1.60657 10 9 8 7 6 55 56 57 58 59 51379 51404 .51429 51454 51479 85792 85777 .85762 .85747 85732 .59888 59928 .59967 60007 60046 1-66978 1-66867 1-66757 1-66647 1-66538 .52869 .52893 -52918 .52943 .52967 84882 84866 .84851 84836 .84820 62285 62325 62366 62406 . 62446 1.60553 1 . 60449 1-60345 1 60241 1-60137 5 4 3 2 1 (60 .51504 85717 60086 1-66428 52992 .84805 62487 1 - 60033 O Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 69 58 256 NATURAL SINES, COSINES, TANGENTS, COTAN. 33 33 t Sin. Cos Tan. Cot Sin. Cos. Tan. Cot. * o. 2 3 .52992 .53017 .53041 .53066 .53091 .84805 84789 .84774 .84759 . 84743 .62487 62527 .62568 .62608 62649 -60033 .59930 .59826 59723 .59620 54464 -54488 54513 54537 -54561 .83867 .83851 83835 83819 -83804 64941 64982 65024 65065 65106 1-53986 1-53888 1-53791 1-53693 1-53595 60 59 58 57 56 6 j 9 .53115 .53140 .53164 .53189 .53214 .84728 84712 84697 84681 .84666 -62689 .62730 .62770 .62811 -62852 59517 .59414 .59311 .59208 .59105 - 54583 .54610 54635 54659 . 54683 -83788 83772 83756 83740 .83724 65148 65189 -65231 65272 65314 1-53497 1-53400 1-53302 1-53205 1.53107 55 54 53 5L 51 10 i* 13 14 .53238 .53263 .53288 .53312 .53337 84650 .84635 84619 .84604 .84588 .62892 .62933 -62973 -63014 .63055 .59002 .58900 .58797 .58695 .58593 -54708 -54732 -54756 -54781 -54805 83708 83692 83676 83660 83645 65355 -65397 -65438 -65480 -65521 1-53010 1-52913 1-52816 1-52719 1-52622 50 49 48 47 48 15 16 17 18 19.. .53361 .53386 .53411 .53435 .53460 .84573 .84557 .84542 .84526 .84511 .63095 -63136 .63177 .63217 -63258 .58490 .58388 -68286 .58184 -58083 -54829 .54854 .54878 54902 -54927 83629 83613 83597 83581 .83565 65563 -65604 65646 65688 -65729 1-52525 1-52429 1-52332 1-52235 1-52139 45 44 43 42 41 2O 21 22 23 24 -53484 .53509 .53534 .53558 .53583 .84495 84480 .84464 . 84448 .84433 -63299 -63340 - 63380 .63421 . 63462 .57981 ' .57879 .57778 .57676 .57575 54951 -54975 54999 55024 .55048 .83549 .83533 83517 83501 83485 .65771 .65813 65854 .65896 .65938 1.52043 1.51946 1-51850 1-51754 1.51658 4O 39 38 37 36 25 26 27 28 29 53607 .53632 .53656 -53681 .53705 84417 . 84402 84386 .84370 .84355 63503 .63544 .63584 .63625 .63666 .57474 .57372 .57271 .57170 .57069 55072 -55097 55121 -55145 .55169 83469 83453 83437 83421 83405 .65980 .66021 .66063 .66105 -66147 1-51562 1-51466 1-51370 1-51275 1.51179 35 34 33 32 31 30 81 32 33 34 .53730 .53754 .53779 .53804 .53828 .84339 .84324 .84308 .84292 .84277 .63707 -63748 -63789 .63830 .63871 .56969 .56868 .56767 .56667 .56566 .55194 .55218 .55242 .55266 -55291 83389 83373 -83356 -83340 -83324 .66189 .66230 .66272 .66314 .66356 1-51084 1.50988 1-50893 1.50797 1.50702 30 29 28 27 26 35 36 37 38 39 .53853 .53877 .53902 .53926 .53951 84261 .84245 -84230 .84214 .84198 .63912 .63953 -63994 .64035 -64076 .56466 .56366 56265 .56165 1-56065 .55315 -55339 .55363 .55388 -55412 83308 83292 83276 83260 83244 .66398 .66440 66482 .66524 .66566 1-50607 1-50512 1.50417 1.50322 1.50228 25 24 23 22 21 40 41 42 43 44 .53975 . 54000 .54024 .54049 .54073 -84182 .84167 .84151 .84135 .84120 .64117 .64158 .64199 . 64240 -64281 1.55966 1.55866 1.55766 1.55666 1-55567 -55436 .55460 .55484 -55509 -55533 83228 .83212 33195 .83179 83163 .66608 .66650 .66692 .66734 .66776 1.50133 1.50038 1-49944 1.49849 1-49755 30 19 18 17 16 45 46 47 48 49 .54097 .54122 .54146 .54171 .54195 84104 .84038 -84072 -84057 . 84041 .64322 .64363 . 64404 . 64446 - 64487 1-55467 1.55368 1.55269 1.55170 1.55071 .55557 -55581 -55605 .55630 55654 .83147 .83131 .83115 83098 -83082 .66818 .66860 .66902 .66944 .66986 1.49661 1-49566 1.49472 1.49378 1.49284 15 14 13 12 11 50 51 59, 53 54 .54220 . 54244 .54269 .54293 .54317 84025 -84009 .83994 -83978 83962 -64528 64569 -64610 64652 -64693 1.54972 1.54873 1 . 54774 1.54675 1.54576 .55678 -55702 55726 -55750 .55775 83066 -83050 .83034 .83017 83001 .67028 .67071 .67113 .67155 .67197 1.49190 1-49097 1.49003 1.48909 1.48816 1O 9 8 7 6 55 56 57 58 59 54342 54366 54391 .54415 . 54440 83946 83930 .83915 -83899 .83883 . 64734 -64775 .64817 -64858 64899 1 . 54478 1-54379 1.54281 1.54183 1-54085 55799 -55823 -55847 .55871 -55895 82985 82969 -82953 82936 82920 .67239 .67282 .67324 .67366 .67409 1.48722 1.48629 1.48536 1-48442 1-48349 5 3 2 60 . 54464 .83867 -64941 1-53986 .55919 82904 .67451 1-48256 .O ' Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 66 NATURAL SINES, COSINES, TANGENTS, COTAN. 257 34 35 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / o 2 4 .55919 55943 .55968 .55992 .56016 -82904 82887 82871 82855 82839 67451 67493 .67536 67578 67620 1-48256 1.48163 1.48070 1-47977 1.47885 57358 57381 57405 57429 57453 -81915 81899 81882 81865 -81848 -70021 -70064 -70107 .70151 .70194 1.42815 1-42726 1-42638 1-42550 1-42462 60 59 58 57 56 5 ? 1 -56040 .56064 .56088 .56112 .56136 .82822 82806 82790 .82773 .82757 67663 67705 67748 67790 67832 1-47792 1-47699 1-47607 1-47514 1-47422 -57477 -57501 -57524 -57548 -57572 -81832 -81815 81798 -81782 -81765 70238 70281 .70325 70368 70412 1-42374 1-42286 1-42198 1-42110 1-42022 55 54 53 52 51 1O 11 12 13 14 56160 .56184 .56208 .56232 56256 .82741 .82724 .82708 .82692 .82675 67875 67917 .67960 .68002 .68045 1-47330 1-47238 -47146 47053 46962 -57596 57619 57643 57667 57691 -81748 81731 81714 -81698 -81681 . 70455 .70499 70542 70586 70629 1-41934 1-41847 1-41759 1-41672 1-41584 5O 49 48 47 46 15 16 17 18 19 .56280 .56305 .56329 56353 56377 82659 .82643 .82626 .82610 -82593 .68088 .68130 68173 -68215 68258 .46870 .46778 .46686 .46595 .46503 .57715 .57738 .57762 57786 57810 81664 81647 81631 81614 81597 70673 70717 70760 70804 70848 1-41497 1-41409 1-41322 1-41235 1-41148 45 44 43 42 41 20 21 22 23 24 .56401 .56425 . 56449 .56473 .56497 82577 82561 82544 82528 82511 .68301 .68343 68386 .68429 68471 46411 -46320 -46229 .46137 1-46046 .57833 57857 57881 57904 57928 -81580 81563 81546 81530 81513 70891 70935 70979 .71023 -71066 1-41061 1-40974 1-40887 1-40800 1-40714 40 39 38 37 36 25 26 27 28 29 56521 -56545 .56569 -56593 .56617 82495 82478 82462 .82446 .82429 68514 .68557 68600 68642 68685 1-45955 1.45864 1-45773 1-45682 1-45592 -57952 57976 57999 58023 58047 -81496 81479 81462 -81445 -81428 -71110 -71154 -71198 -71242 -71285 1.40627 1 40540 1.40454 1.40367 1-40281 35 34 33 32 31 30 31 32 33 34 .56641 .56665 .56689 .56713 -56736 .82413 .82396 .82380 82363 .82347 68728 68771 68814 .68857 .68900 1-45501 1-45410 1-45320 1-45229 1-45139 -58070 58094 58118 58141 -58165 -81412 -81395 -81378 -81361 -81344 .71329 -71373 -71417 -71461 -71505 1-40195 1-40109 1-40022 1-39936 1.39850 3O 29 28 27 26 35 36 37 38 39 -56760 . 5678-ft . 56803 56832 .568?i6_ -82330- .82314 .82297 82281 -82264 .68942 68985 .69028 .69071 .69114 1.45049 1.44958 1-44868 1-44778 1-44688 -58189 -58212 -58236 -58260 -58283 -81327 -81310 81293 81276 81259 -71549 71593 .71637 .71681 .71725 1-39764 1-39679 1-395'S 1-39507 1-39421 25 24 23 22 21 40 41 42 43 44 56880 .56904 .56928 56952 .56976 .82248 .82231 .82214 82198 -82181 .69157 -69200 .69243 -69286 -69329 1-44598 1-44508 1-44418 1-44329 1-44239 -58307 -58330 -58354 -58378 -58401 81242 .81225 81208 81191 81174 .71769 .71813 .71857 .71901 71946 1-39336 1-39250 1.39165 1-39079 1-38994 20 19 18 17 16 45 46 47 48 49 . 57000 .57024 .57047 .57071 .57095 .82165 82148 .82132 82115 .82098 -69372 .69416 69459 .69502 69545 1.44149 1 44060 1-43970 1-43881 1-43792 58425 . 58449 58472 58496 .58519 -81157 81140 81123 81106 -81089 -71990 .72034 .72078 .72122 72167 1.38909 1.38824 1.38738 1-38653 1-38568 15 14 13 12 11 50 51 52 53 54 .57119 57143 .57167 .57191 57215 82082 82065 82048 82032 82015 -69588 69631 69675 69718 69761 1-43703 1-43614 1-43525 1-43436 1-43347 -58543 -58567 -58590 -58614 -58637 81072 81055 .81038 -81021 81004 .72211 72255 72299 72344 .72388 1-38484 1-38399 1-38314 1-38229 1-38145 10 9 8 7 6 55 56 57 58 59 57238 57262 .57286 57310 57334 81999 -81982 -81965 81949 81932 69804 69847 69891 -69934 69977 1-43258 1-43169 1-43080 1-42992 1-42903 58661 -58684 -58708 .58731 58755 80987 .80970 80953 80936 80919 . 72432 .72477 72521 .72565 72610 1-38060 1-37976 1-37891 1-37807 1-37722 5 4 3 2 60 57358 81915 .70021 1-42815 -58779 -80902 .72654 L37638 o * Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 65 64 258 NATURAL SINES, COSINES, TANGENTS, COTAN. 36 37 Sin. Cos. Tan. Cot. Sin. 1 Cos. Tan. Cot. , 2 3 58779 58802 -58826 58849 .58873 .80902 80885 80867 80850 80833 .72654 72699 .72743 .72788 72832 1.37638 1-37554 1-37470 1-37386 1-37302 60182 60205 60228 60251 60274 -79864 - 79846 -79829 .79811 79793 75355 75401 75447 -75492 -75538 1-32704 1-32624 1-32544 1-32464 1-32384 60 59 58 57 56 5 6 7 8 9 58896 58920 58943 58967 58990 80816 80799 80782 80765 80748 72877 72921 72966 73010 73055 1-37218 1.37134 1.37050 1-36967 1.36883 60298 60321 60344 -60367 -60390 79776 -79758 .79741 .79723 -79706 75584 75629 75675 -75721 -75767 1-32304 1-32224 1-32144 1-32064 1-31984 55 54 53 52 51 10 11 12 13 14 59014 59037 59061 59084 59108 80730 80713 80696 80679 80662 73100 73144 73189 73234 73278 1-36800 1.36716 1.36633 1.36549 1.36466 60414 60437 60460 60483 .60506 -79688 79671 -79653 -79635 -79618 75812 -75858 -75904 75950 -75996 1-31904 1-31825 1-31745 1-31666 1-31586 50 49 48 47 46 15 16 17 18 19 59131 59154 59178 59201 59225 80644 80627 80610 80593 80576 73323 73368 73413 73457 73502 1-36383 1.36300 1.36217 1.36134 1-36051 60529 60553 60576 60599 60622 79600 79583 -79565 -79547 -79530 76042 76088 76134 76180 76226 1-31507 1-31427 1-31348 1-31269 1-31190 45 44 43 42 41 21 22 23 24 59248 59272 59295 59318 59342 80558 80541 80524 80507 80489 73547 73592 73637 73681 73726 1.35968 1-35885 1-35802 1-35719 1-35837 60645 60668 60691 '$0714 60738 79512 .79494 .79477 79459 - 79441 76272 76318 76364 76410 76456 1-31110 1.31031 1.30952 1-30873 1 30795 40 39 38 37 36 25 26 27 28 29 .59385 59389 59412 59436 59459 80472 80455 80438 80420 80403 73771 73816 73861 73906 73951 1-35554 1-35472 1-35389 1.35307 1-35224 60761 6C784 60807 60830 60853 - 79424 79406 .79388 79371 79353 -76502 -76548 -76594 -76640 .76686 1-30716 1-30637 1-30558 1 - 30480 1 30401 35 34 33 32 31 30 31 32 33 34 59482 59506 59529 59552 59576 80386 80368 80351 80334 80318 73996 74041 74086 74131 74176 1.35142 1-35060 1.34978 1.34896 1-34814 60876 60899 60922 60945 -60968 .79335 -79318 -79300 -79282 -79264 .76733 .78779 76825 76871 76918 1-30323 1-30244 1-30166 1-30087 1-30009 30 29 28 27 26 35 36 37 38 39. 59599 59622 59646 59669 59693. 80299 -80282 -80264 80247 80230 74221 74267 74312 74357 74402 1-34732 1.34650 1-34568 1.34487 1.34405 -60991 -61015 -61038 61061 61084 79247 .79229 .79211 -79193 -79176 76964 .77010 -77057 .77103 .77149 1-29931 1.29853 1-29775 1-29696 1-29B18 25 24 23 22 21 40 41 42 43 44 .59716 .59739 .59763 .59786 .59809 .80212 80195 80178 80160 80143 74447 74492 74538 74583 .74628 1.34323 1-34242 1-34160 1-34079 1-33998 61107 61130 61153 61176 61199 -79158 -79140 -79122 -79105 -79087 .77196 77242 -77289 -77335 -77382 1-29541 1-29463 1-29385 1-29307 1-29229 20 19 18 17 Ifi 45 46 47 48 49_ 59832 .59856 .59879 59902 .59926 .80125 .80108 .80091 .80073 .80056 74674 .74719 74764 74810 .74855 1-33916 1-33835 1.33754 1.33673 1-33592 -61222 61245 -61268 -61291 -61314 -79069 -79051 -79033 -79016 -78998 -77428 77475 .77521 .77568 .77615 1-29152 1-29074 1-28997 1-28919 1-28842 15 14 13 12 11 50 51 52 53 5i_ 59949 59972 .59995 60019 .60042 .80038 .80021 .80003 .79986 .79968 74900 .74946 74991 75037 .75082 1-33511 1.33430 1.33349 1.33268 1-33187 -61337 -61360 .61383 6H06 61429 -78980 -78962 -78944 -78926 -78908 .77661 .77708 77754 -77801 -77848 1-28764 1-28687 1-28610 1-28533 1-28456 10 9 8 7 6 55 56 57 58 59 60065 60089 60112 60135 60158 .79951 .79934 .79916 .79899 .79881 .75128 75173 .75219 75264 .75310 1.33107 1.33026 1.32946 1-32865 1-32785 .61451 61474 61497 61520 -61543 -78891 -78873 -78855 78837 78819 .77895 -77941 77988 78035 .78082 1-28379 1-28302 1-28225 1-28148 1-28071 5 4 3 2 1 60 .60182 .79864 ,75355 1-32704 -61566 78801 .78129 1-27994 . Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 63* NATURAL SINES, COSINES, TANGENTS, COTAN. 259 38 39 1 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. ' o 1 2 3 4 61566 -61589 .61612 -61635 -61658 -78801 78783 -78765 78747 -78729 78129 78175 78222 78269 -78316 1-27994 1-27917 1-27841 1-27764 1-27688 62932 .62955 -62977 - 63000 -63022 77715 77696 77678 77660 77641 80978 81027 81075 81123 -81171 1-23490 1-23416 1-23343 1-23270 1-23196 60 59 58 57 56 5 6 7 8 9 -61681 -61704 -61726 -61749 .61772 78711 -78694 -78676 -78658 - 78640 78363 78410 78457 -78504 -78551 27611 .27535 27458 27382 27306 63045 63068 -63090 .63113 -63135 77623 77605 77586 77568 .77550 81220 81268 81316 81364 81413 1-23123 1-23050 1-22977 1-22904 1-22831 55 54 53 52 51 10 11 12 13 14 -61795 -61818 -61841 -61864 -61887 -78622 78604 78586 78568 -78550 -78598 -78645 -78692 78739 78786 27230 27153 .27077 27001 26925 -63158 -63180 -63203 -63225 .63248 .77531 .77513 .77494 .77476 .77458 .81461 .81510 -81558 -81606 .81655 1-22758 1-22685 1.22612 1-22539 1-22467 50 49 48 47 46 15 16 17 18 19 -61909 -61932 -61955 -61978 .62001 .78532 78514 -78496 - 78478 .78460 -78834 -78881 78928 -78975 .79022 .26849 26774 .26698 .26622 -26546 63271 63293 63316 -63338 .63361 . 77439 77421 . 77402 77384 .77366 .81703 .81752 -81800 -81849 .81898 1-22394 1-22321 1-22249 1.22-176 1-22104 45 44 43 42 41 20 21 22 23 24 62024 62046 .62069 -62092 62115 -78442 -78424 .78405 -78387 78369 79070 79117 -79164 79212 -79259 .26471 1-26395 1-26319 1-26244 1-26169 -63383 -63406 .63428 .63451 .63473 77347 77329 77310 .77292 . 77273 .81946 81995 -82044 -82092 82141 1.22031 1-21959 1-21886 1-21814 1-21742 40 39 38 37 36 25 26 27 28 29 -62138 -62160 .82183 . 62206 62229 78351 .78333 -78315 -78297 78279 79306 79354 79401 79449 79496 1-26093 1.26018 1-25943 1.25867 1-25792 -63496 .63518 -63540 -63563 -63585 77255 77236 77218 77199 77181 82190 82238 82287 -82336 -82385 1.21670 1.21598 1.21526 1-21454 1-21382 35 34 33 32 31 30 31 32 33 34 -62251 62274 62297 62320 -62342 78261 - 78243 -78225 -78206 -78188 79544 79591 79639 79686 79734 1-25717 1-25642 1-25567 1-25492 1.25417 -63608 -63630 -63653 -63675 63698 77162 .77144 .77125 .77107 .77088 82434 82483 -82531 -82580 -82629 1-21310 1.21238 1-21166 1-21094 1-21023 30 29 28 27 26 35 36 37 38 39 -62365 -62388 -62411 .62433 62456 78170 78152 -78134 78116 .78098 -79781 .79829 .79877 79924 -79972 1-25343 1-25268 1-25193 1-25118 1.25044 -63720 -63742 -63765 -63787 -63810 .77070 77051 .77033 77014 -76996 82678 -82727 -82776 82825 82874 1-20951 1-20879 1-20808 1-20736 1-20665 25 24 23 22 21 40 41 42 43 44 62479 .62502 -62524 -62547 .62570 .78079 78061 . 78043 78025 78007 80020 -80067 .80115 .80163 80211 1-24969 1-24895 1.24820 1.24746 1.24672 -63832 .63854 63877 .63899 -63922 -76977 -76959 -76940 -76921 76903 82923 -82972 83022 83071 -83120 1-20593 1-20522 1-20451 1-20379 1-20308 20 19 18 17 16 45 46 47 48 49 62592 62615 -62638 -62860 -62683 -77988 77970 77952 77934 77916 -80258 -80306 -80354 -80402 . 80450 1-24597 1-24523 1 24449 1.24375 1-24301 63944 63966 .63989 -64011 - 64033 76884 -76866 -76847 -76828 76810 -83169 -83218 -83268 -83317 -83366 1-20237 1-20166 1-20095 1 - 20024 1-19953 15 14 13 12 11 50 51 52 53 54 -62706 -62728 -62751 -62774 -62796 77897 -77879 77861 77843 -77824 -80498 -80546 -80594 -80642 -80690 1-24227 1-24153 1 - 24079 1 - 24005 1.23931 -64056 -64078 -64100 -64123 64145 76791 76772 76754 76735 76717 83415 -83465 -83514 -83564 -83613 1-19882 1-19811 1-19740 1-19669 1-19599 10 9 8 7 6 55 56 57 58 59 62819 -62842 .62864 .62887 -62909 -77806 -77788 -77769 77751 -77733 .80738 80786 .80834 .80882 80930 1-23858 1-23784 1-23710 1-23637 1-23563 -64167 .64190 .64212 . 64234 -64256 76698 76679 76661 76642 76623 -83662 -83712 -83761 -83811 -83860 1-19528 1-19457 1-19387 1-19316 1-19246 5 4 3 2 1 60 -62932 -77715 80978 1-23490 64279 -76604 .83910 1-19175 ' Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ' 61 50 260 NATURAL SINES, COSINES, TANGENTS, COTAN. 40 41 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t o 1 2 3 4 64279 64301 64323 64346 64368 76604 76586 76567 76548 76530 83910 83960 84009 84059 84108 1-19175 1-19105 1-19035 1-18964 1-18894 65606 65628 65650 65672 65694 75471 75452 75433 75414 75395 86929 86980 87031 87082 87133 1-15037 1-14969 1.14902 1.14834 1-14767 60 59 58 57 56 5 6 8 9 64390 64412 64435 64457 64479 76511 76492 76473 76455 76436 84158 84208 84258 84307 .84357 1-18824 1-18754 1-18684 1-18614 1-18544 65716 65738 65759 65781 65803 75375 75356 75337 75318 75299 87184 87236 87287 -87338 -87389 1-14699 1.14632 1-14565 1-14498 1.14430 55 54 53 52 51 10 11 12 13 14 64501 64524 64546 64568 64590 .76417 76398 .76380 76361 76342 . 84407 84457 -84507 -84556 -84606 1-18474 1-18404 1-18334 1-18264 1-18194 65825 65847 65869 65891 65913 75280 75261 -75241 -75222 75203 87441 87492 -87543 87595 87646 1-14363 1-14296 1-14229 1-14162 1-14095 50 49 48 47 46 15 16 17 18 19 64612 64635 64657 64679 64701 .76323 .76304 .76286 .76267 76248 84656 .84706 84756 .84806 84856 1.18125 1-18055 1.17986 1-17916 1-17846 -65935 - 65956 -65978 - 66000 .66022 75184 75185 75146 -75126 -75107 87698 87749 87801 -87852 .87904 1-14028 1-13961 1-13894 1-13828 1-13761 45 44 43 42 41 20 21 22 23 24 64723 64746 64768 64790 64812 76229 76210 76192 76173 76154 .84906 i/4956 85006 85057 .85107 1.17777 1-17708 1-17638 1-17569 1-17500 - 66044 -66066 -66088 -66109 -66131 -75088 -75069 -75050 -75030 75011 87955 .88007 .88059 88110 88162 1-13694 1-13627 1-13561 1-13494 1-13428 40 39 38 37 36 25 26 27 28 29 .64834 64856 .64878 64901 .64923 76135 76116 76097 76078 76059 85157 85207 85257 -85308 -85358 1-17430 1-17361 1.17292 1-17223 1.17154 -66053 66175 66197 66218 . 66240 74992 74973 74953 74934 74915 88204 88265 88317 88369 .88421 1-13361 1-13295 1-13228 1-13162 1.13096 35 34 33 32 31 30 31 32 33 34 .64945 64967 64989 65011 -65033 76041 .76022 76003 75984 75965 85408 85458 85509 .85559 85609 1-17085 1-17016 1-16947 1-16878 1-16809 -66262 66284 66306 -66327 66349 74896 74876 74857 74838 74818 .88473 .88524 .88576 .88628 -88680 1-13029 1-12963 1-12897 1-12831 1-12765 30 29 28 27 26 35 36 37 38 39 65055 65077 .65100 .65122 -65144 75946 75927 .75908 -75889 -75870 .85660 85710 85761 -85811 -85862 1-16741 1-16672 1-16603 1-16535 1-16466 -66371 -66393 -66414 66436 -66458 74799 . 74780 . 74760 . 74741 -74722 -88732 -88784 -88836 -88888 -88940 1-12699 1-12633 1-12567 1.12501 1-12435 25 24 23 22 21 40 41 42 43 44 -65166 .65188 65210 65232 -65254 -75851 -75832 -75813 -75794 -75775 -85912 -85963 86014 86064 .86115 1-16398 1.16329 1-16261 1-16192 1-16124 66480 66501 66523 66545 -66566 . 74703 . 74683 74664 - 74644 -74625 88992 89045 89097 89149 89201 1-12369 1.12303 1.12238 1.12172 1-12106 2 18 18 17 16 45 46 47 48 49 65276 -65298 -65320 -65342 -65364 .75756 75738 .75719 75700 .75680 86166 86216 .86267 86318 -86368 1-16056 1-15987 1-15919 1.15851 1-15783 -66588 -66610 -66632 -66653 -66675 - 74606 -74586 -74567 74548 74528 89253 -89306 -89358 .89410 89463 1.12041 1-11975 1.11909 1.11844 1-11778 15 14 13 12 11 50 51 52 53 54 -65386 -65408 .65430 .65452 65474 .75661 .75642 75623 -75604 75585 -86419 86470 86521 86572 -.86623 1-15715 1-15647 1-15579 1-15511 1-15443 -66697 -66718 -66740 -66762 -66783 74509 - 74489 74470 74451 -74431 89515 89567 -89620 -89672 -89725 1-11713 1-11648 1-11582 1-11517 1-11452 10 g 8 7 6 55 56 57 58 59 65496 .65518 .65540 65562 .65584 75566 75547 .75528 75509 75490 .86674 .86725 .86776 86827 86878 1-15375 1-15308 1.15240 1-15172 1-15104 -66805 -66827 -66848 .66870 66891 74412 74392 74373 74353 74334 89777 .89830 -89883 -89935 -89988 1.11387 1-11321 1-11256 1-11191 1-11126 5 4 3 2 1 6O .65606 75471 -86929 1-15037 -66913 74314 - 90040 1-11061 1 Cos. Sin. j Cot. Tan. Cos. Sin. Cot. Tan. 49 48 NATURAL SINES, COSINES, TANGENTS, COTAN. 261 42 43 * Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 2 3 4 .66913 66935 .66956 .66978 .66999 74314 74295 74276 74256 . 74237 .90040 90093 .90146 .90199 .90251 1.11061 1 10996 1.10931 1-10867 1-10802 68200 -68221 -68242 -68264 -68285 73135 -73116 -73096 -73076 -73056 93252 93306 93360 93415 93469 1-07237 1.07174 1-07112 1 07049 1-06987 6O 59 58 57 56 5 6 7 8 9 .67021 67043. .67064 .67086 .67107 .74217 74198 74178 74159 74139 ! 90304 .90357 .90410 90463 .90516 1.10737 1.10672 1.10607 1-10543 1 . 10478 68306 -68327 .68349 .68370 .68391 73036 73016 72996 72976 -72957 93524 93578 93633 -93688 93742 1.06925 1.06862 1-06800 1-06738 1 06676 55 54 53 52 51 10 11 12 13 14 .67129 .67151 .67172 .67194 67215 .74120 .74100 . 74080 .74061 . 74041 .90569 90621 .90674 90727 90781 1-10414 1.10349 1-10285 1-10220 1.10156 .68412 -68434 68455 68476 68497 -72937 -72917 -72897 -72877 -72857 -93797 -93852 93906 -93961 .94016 1-06813 1-06551 1.06489 1-06427 1 06365 5O 49 48 47 46 15 16 17 18 19 67237 .67258 67280 67301 67323 . 74022 74002 73983 .73963 73944 90834 90887 90940 90993 .91046 1.10091 1.10027 1-09963 1.09899 1.09834 68518 .68539 68561 -68582 . 68603 .72837 .72817 -72797 -72777 72757 94071 -94125 -94180 94235 .94290 1-06303 1-06241 1.06179 1-06117 1.06056 45 44 43 42 41 20 21 22 23 24 67344 67366 67387 67409 67430 .73924 . 73904 73885 73865 73846 .91099 91153 91206 91259 91313 1-09770 1-09706 1.09642 1-09578 1.09514. 68624 68645 68666 68688 68709 72737 72717 -72697 -72677 .72657 .94345 - 94400 -94455 .94510 .94565 1-05994 1-05932 1.05870 1-05809 1-05747 40 39 38 37 36 25 26 27 28 29 67452 67473 .67495 67516 .67538 .73826 .73806 .73787 .73767 73747 91366 91419 91473 91526 91580 1-09450 1.09386 1-09322 1-09258 1.09195 68730 68751 68772 68793 68814 72637 72617 72597 72577 .72557 .94620 .94676 -94731 -94786 - 94841 1-05685 1-05624 1-05562 1-05501 1-05439 35 34 33 32 31 30 31 32 33 34 .67559 67580 67602 67623 67645 .73728 -73708 .73688 .73669 .74549 91633 91687 91740 91794 91847 1-09131 1-09067 1-09003 1-08940 1-08876 68835 68857 68878 68899 68920 72537 72517 -72497 72477 .72457 -94896 -94952 -95007 -95062 -95118 1-05378 1-05317 1.05255 1-05194 1-05133 30 29 28 2? 26 35 36 37 38 39 67666 67688 67709 67730 67752 .73629 .73810 73590 73570 .73551 91901 91955 92008 92062 92116 1-08813 1-08749 1-08686 1-08622 1 -08559 68941 68962 68983 69004 69025 .72437 .72417 .72397 72377 -72357 -95173 95229 95284 -95340 -95395 1-05072 1.05010 1 - 04949 1-04888 1-04827 25 24 23 22 21 40 41 42 43 44 -67773 67795 67816 67837 -67859 .73531 .73511 .73491 .73472 .73452 92170 92224 92277 92331 92385 1-08498 1-08432 1-08369 1-08306 1 - 08243 69046 69067 -69088 -69109 .69130 72337 72317 72297 -72277 .72257 -95451 95506 .95562 .95618 -95673 1-04766 1-04705 1 04644 1-04583 1.04522 20 19 18 17 16 45 16 47 48 49 -67880 -67901 -67923 - 67944 -67965 -73432 73413 73393 73373 73353 92439 92493 92547 92601 92655 1.08179 1-08116 1-08053 1-07990 1-07927 69151 69172 69193 69214 69235 -72236 -72216 72196 72176 72156 95729 .95785 .95841 .95897 .95952 1-04461 1-04401 1 - 04340 1-04279 1.04218 15 14 13 12 11 50 51 52 53 54_ 67987 -68008 -68029 68051 -68072 73333 73314 73294 -73274 73254 92709 92763 92817 .92872 .92926 1-07864 1-07801 1-07738 1-07678 1.07613 -69258 -69277 69298 .69319 -69340 72138 72116 72095 .72075 .72055 .96008 -96064 -96120 96176 -96232 1-04158 1-04097 1.04036 1-03978 1-03915 10 9 8 _J 55 56 57 58 59 68093 68115 68136 68157 .68179 .73234 73215 73195 .73175 .73155 .92980 93034 .93088 93143 93197 1-07550 1.07487 1-07425 1-07362 1-07299 -69361 69382 -69403 69424 -69445 .72035 .72015 .71995 71974 71954 -96288 96344 96400 96457 .96513 1.03855 1-03794 1-03734 1-03874 1-03613 5 1 60 . 68200 .73135 .93252 1-07237 69466 71934 .96569 1 03553 Q 1 Cos. Sin. Cot, Tan. Cos. Sin. Cot. Tan. 47 46 262 NATURAL SINES, COSINES, TANGENTS, COTAN. 44 44 ' Sin. Cos. Tan. Cot, , i / Sin. Cos. Tan. Cot. / 1 2 J_ 5 6 7 8 9 10 11 12 13 y_ 15 16 17 18 12_ 20 21 22 23 24_ 25 26 27 28 29_ 3 o 2 3 .94766 94795 .94825 .94854 .94883 18-10732 18-21397 18-32182 18-43088 18-54119 .96510 .96539 .96568 96597 -96626 27-65371 27-89440 28-13917 28-38812 28-64137 98255 98284 98313 -98342 -98371 53-29869 57.26976 58-27431 59-31411 60-39105 2 3 4 5 6 7 8 9 94912 94941 .94970 .94999 .95028 18-65275 18-76560 18-87976 18-99524 19-11208 .96655 .96684 .96714 .96743 .96772 28-89903 29-16120 29-42802 29-69960 29.97607 98400 98429 .98458 .98487 .98517 61.50715 62.66460 63-86572 65-11304 66-40927 5 6 8 9 10 11 12 13 14 .95057 .95086 95115 .95144 .95173 19.23028 19-34989 19-47093 19-59341 19-71737 .96801 .96830 -96859 -96888 -96917 30-25758 30-54425 30-83623 31-13366 31-43671 98546 98575 -98604 .98633 .98662 67-75736 69-16047 70-62207 72-14583 73-73586 10 11 12 13 14 15 16 17 18 19_ -95202 95231 .95260 .95289 .95318 19-84283 19-96982 20-09838 20-22852 20-36027 -96946 -96975 .97004 .97033 .97062 31-74554 32.06030 32-38118 32-70835 33-04199 98691 98720 98749 .98778 .98807 75-39655 77-13274 78-94968 80-85315 82-84947 15 16 17 18 19 20 21 22 23 24 .95347 .95377 .95406 .95435 .95464 20-49368 20-62876 20-76555 20-90409 21 . 04440 .97092 .97121 .97150 .97179 .97208 33-38232 33-72952 34-08380 34-44539 34-81452 .98836 98866 .98895 .98924 98953 84-94561 87-14924 89-46886 91-91387 94-49471 20 21 22 23 24 25 26 27 28 29 .95493 95522 95551 .95580 .95609 21-18653 21.33050 21-47635 21-62413 21-77386 97237 .97266 .97295 .97324 97353 35-19141 35-57633 35-96953 36-37127 36-78185 .98982 .99011 .99040 -99069 -99098 97-22303 100-1119 103-1757 106-4311 109-8966 25 26 27 28 29 30 31 32 33 34 .95638 .95667 95696 .95725 95754 21-92559 22-07935 22-23520 22-39316 22-55328 .97382 .97411 .97440 .97470 .97499 37-20155 37-63068 38-06957 38-51855 38-97797 .99127 .99156 -99186 .99215 .99244 113-5930 117-5444 121-7780 126-3253 131-2223 30 31 32 33 34 35 36 37 38 39_ 95783 .95812 95842 95871 95900 22.71563 22-88022 23-04712 23-21637 23-38802 .97528 .97557 .97586 -97615 .97644 39-44820 39-92963 40-42266 40-92772 41-44525 .99278 .99302 .99331 .99360 99389 136-5111 142-2406 148-4684 155-2623 162-7033 35 36 37 38 39 40 41 42 43 44 95929 95958 95987 .96016 96045 23-56212 23-73873 23-91790 24-09969 24-28414 97673 .97702 .97731 .97760 .97789 41-97571 42-51961 43-07746 43-64980 44-23720 -99418 -99447 .99476 .99505 .99535 170-8883 179-9350 189-9868 201-2212 213-8600 40 41 42 43 44 45 46 47 48 49 96074 .96103 .96132 .96161 .96190 24-47134 24-66132 24-85417 25-04994 25-24869 97819 97848 97877 .97906 .97935 44-84026 45-45963 46-09596 46-74997 47-42241 99564 .99593 .99622 .99651 .99680 228-1839 244.5540 263-4427 285-4795 311.5230 45 46 47 48 49 50 51 52 53 54 .96219 .96248 .96277 .96307 .96336 25-45051 25-65546 25-86360 26.07503 26-28981 97964 -97993 -98022 -98051 .98080 48-11406 48-82576 49-55840 50.31290 51.09027 .99709 .99738 .99767 .99796 .99825 342.7752 380-9723 428-7187 490-1070 571-9581 50 51 52 53 54 55 56 57 58 59_ .96365 .96394 .96423 .96452 .96481 26-50804 26-72978 26-95513 27-18417 27-41700 .98109 -98138 -98168 .98197 98226 51-89156 52-71790 53-57048 54-45053 55-35946 .99855 .99884 .99913 99942 99971 686-5496 858-4369 1144-916 1717-874 3436-747 55 58 57 58 59 60 .96510 27-65371 .98255 56-29869 1-00000 Infinite 60 TABLES 299 RADII METRIC R " Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 000' 00 oo 100' 1145.930 3.05916 2 00' 572.987 2.75814 01 68754.931 4.83730 01 1127.145 3.05198 01 568.252 2.75454 02 34377.468 4.53627 02 1108.966 3.04492 02 563.595 2.75097 03 22918.310 4.36018 03 1091.364 3.03797 03 559.013 2.74742 04 17188.735 4.23524 04 1074.311 3.03113 04 554.505 2.74391 05 13750.987 4.13833 05 1057.785 3.02440 05 550.070 2.74042 06 11459.157 4.05912 06 1041.757 3.01777 06 545.705 2.73696 07 9822.134 3.99221 07 1026.209 3.01124 07 541.408 2.73352 08 8594.369 3.93421 08 1011.118 3.00480 08 537.179 2.73012 09 7639.438 3.88306 09 996.465 2.99846 09 533.015 2.72674 10 6875.496 3.83730 10 982.230 2.99221 10 528.916 2.72339 11 6250.451 3.79591 11 968.396 2.98605 11 524.879 2.72006 12 5729.582 3.75812 12 954.947 2.97998 12 520.903 2.71676 13 5288.845 3.72336 13 941.866 2.97399 13 516.985 2.71348 14 4911.070 3.69118 14 929.139 2.96808 14 513.129 2.71023 15 4583.666 3.66121 15 916.751 2.96225 15 509.328 2.70700 16 4297.187 3.63318 16 904.689 2.95650 16 505.584 2.70379 17 4044.412 3.60686 17 892.940 2.95082 17 501.894 2.70061 18 3819.723 3.58203 18 881.492 2.94522 18 498.258 2.69745 19 3618.685 3.55855 19 870.335 2.93969 19 494.674 2.69432 20 3437.752 3.53627 20 859.456 2.93422 20 491.141 2.69121 21 3274.050 3.51509 21 848.846 2.92883 21 487.658 2.68812 22 3125.230 3.49488 22 838.495 2.92350 22 484.224 2.68505 23 2989.351 3.47558 23 828.393 2.91824 23 480.838 2.68200 24 2864.795 3.45709 24 818.532 2.91304 24 477.500 2.67897 25 2750.203 3.43936 25 808.902 2.90790 25 474.207 2.67597 26 2644.427 3.42233 26 799.497 2.90282 26 470.960 2.67298 27 2546.485 3.40594 27 790.308 2.89780 27 467.756 2.67002 28 2455.540 3.39015 28 781.327 2.89283 28 464.596 2.66708 29 2370.867 3.37491 29 772.549 2.88793 29 461.479 2.66415 30 2291.838 3.36018 30 763.966 2.88307 30 458.403 2.66125 31 2217.909 3.34594 31 755.571 2.87828 31 455.367 2.65836 32 ' 2148.600 3.33216 32 747.359 2.87353 32 452.372 2.65550 33 2083.491 3.31879 33 739.323 2.86883 33 449.416 2.65265 34 ! 2022.212 3.30583 34 731.458 2.86419 34 446.498 2.64982 35 1 1964.435 3.29324 35 723.759 2.85959 35 443.618 2.64701 36 | 1909.868 3.28100 36 716.221 2.85505 36 440.775 2.64422 37 ; 1858.251 3.26910 37 708.837 2.85055 37 437.968 2.64144 38 ! 1809.350 3.25752 38 701.605 2.84609 38 435.196 2.63868 39 1762.957 3.24624 39 694.518 2.84168 39 432.460 2.63595 40 1718.883 3.23525 40 687.574 2.83732 40 429.757 2.63322 41 1676.959 3.22452 41 680.766 2.83300 41 427.088 2.63052 42 1637.032 3.21406 42 674.093 2.82872 42 424.452 2.62783 43 1598.962 3.20384 43 667.549 2.82448 43 421.849 2.62516 44 1562.623 3.19385 44 661 . 130 2.82029 44 419.277 2.62250 45 1527.899 3.18409 45 654.834 2.81613 45 416.737 2.61986 46 1494.684 3.17455 46 648.657 2.81202 46 414.227 2.61724 47 1462.883 3.16521 47 642.595 2.80794 47 411.747 2.61463 48 1432.406 3.15607 48 636.646 2.80390 48 409.296 2.61204 49 1403.174 3.14711 49 630.806 2.79990 49 406.875 2.60946 50 1375.111 3.13834 50 625.072 2.79593 50 404.482 2.60690 51 1348.148 3.12974 51 619.441 2.79200 51 402.117 2.60435 52 1322.223 3.12130 52 613.910 2.78810 52 399.780 2.60182 53 1297.276 3.11303 53 608.478 2.78424 53 397.469 2.59930 54 1273.253 3.10491 54 603.141 2.78042 54 395.186 2.59680 55 1250.103 3.09695 55 597.897 2.77663 55 392.928 2.59431 56 1227.780 3.08912 56 592.743 2.77287 56 390.696 2.59184 57 1206.241 3.08143 57 587.677 2.76914 57 388.489 2.58938 58 1185.444 3.07388 58 582.698 2.76544 58 386.307 2.58693 59 1165.352 3.06646 59 577.802 2.76178 59 384.149 2.58450 300 THE RAILROAD TAPER RAJDII METRIC (Continued) Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 3 00' 382.016 2.58208 4 00' 286.537 2.45718 i 5 00' 229.256 2.36032 [01 379.906 2.57968 01 285.348 2.45537 01 228.495 2.35888 02 377.818 2.57728 02 284.170 2.45358 02 227.739 2.35744 03 375.754 2.57490 03 283.001 2.45179 03 226.987 2.35600 04 373.713 2.57254 04 281.842 2.45001 04 226.241 2.35457 05 371.693 2.57018 05 280.692 2.44823 05 225.500 2.35315 06 369.695 2.56784 06 279.551 2.44646 06 224.764 2.35173 07 367.719 2.56552 07 278.420 3.44470 07 224.032 2.35031 08 365.763 2.56320 08 277.298 2.44295 08 223.305 2.34890 09 363.828 2.56090 09 276.185 2.44120 09 222.583 2.34749 10 361.914 2.55861 10 275.080 2.43946 10 221.865 2.34609 11 360.020 2.55633 11 273.985 2.43773 11 221.152 2.34469 12 358.145 2.55406 12 272.898 2.43600 12 220.444 2.34330 13 356.290 2.55180 13 271.820 2.43428 13 219.740 2.34191 14 354.454 2.54956 14 270.750 2.43257 1 14 219.041 2.34053 15 352.637 2.54733 15 269.689 2.43086 15 218.346 2.33915 16 350.838 2.54511 16 268.636 2.42916 j .16 217.656 2.33777 17 349.058 2.54290 17 267.591 2.42747 1 17 216.969 2.33640 18 347.295 2.54070 18 266.555 2.42579 18 216.288 2.33503 19 345.550 2.53851 19 265.526 2.42411 19 215.610 2.33367 20 343.823 2.53633 20 264.505 2.42243 20 214.937 2.33231 21 342.113 2.53417 21 263.492 2.42077 21 214.268 2.33096 22 340.420 2.53202 22 262.487 2.41911 22 213.603 2.32961 23 338.743 2.52987 23 261.489 2.41745 23 212.942 2.32826 24 337.084 2.52774 24 260.499 2.41581 24 212.285 2.32692 25 335.440 2.52561 25 259.517 2.41417 25 211.632 2.32558 26 333.812 2.52350 26 258.542 2.41253 26 210.984 2.32425 27 332.200 2.52140 27 257.574 2.41090 27 210.339 2.32292 28 330.603 2.51931 28 256.613 2.40928 28 209.698 2.32159 29 329.022 2.51722 29 255.660 2.40766 29 209.061 2.32027 30 327.455 2.51515 30 254.713 2.40605 30 208.428 2.31896 31 325.904 2.51309 31 253.774 2.40445 31 207.799 2.31764 32 324.367 2.51104 32 252.841 2.40285 32 207.174 2.31634 33 322.845 2.50899 33 251.916 2.40126 33 206.552 2.31503 34 321.337 2.50696 34 250.997 2.39967 34 205.934 2.31373 35 319.842 2.50494 35 250.085 2.39809 35 205.320 2.31243 36 318.362 2.50292 36 249.179 2.39651 36 204.709 2.31114 37 316.896 2.50092 37 248.280 2.39494 37 204.102 2.30985 38 315.442 2.49892 38 247.387 2.39338 38 203.499 2.30856 39 314.003 2.49693 39 246.501 2.39182 39 202.899 2.30728 40 312.576 2.49496 40 245.621 2.39027 40 202.303 2.30600 41 311.162 2.49299 41 244.748 2.38872 41 201.710 2.30473 42 309.761 2.49103 42 243.880 2.38718 42 201.121 2.30346 43 308.372 2.48907 43 243.019 2.38564 43 200.535 2.30219 44 306.996 2.48713 44 242.164 2.38411 44 199.952 2.30093 45 305.632 2.48520 45 241.314 2.38258 45 199.373 2.29967 46 304.280 2.48327 46 240.471 2.38106 46 198.798 2.29841 47 302.940 2.48136 47 239.634 2.37955 47 198.225 2.29716 48 301.612 2.47945 48 238.802 2.37804 48 197.656 2.29591 49 300.295 2.47755 49 237.976 2.37653 49 197.090 2.29466 50 298.990 2.47566 50 237.156 2.37503 50 196.528 2.29342 51 297.696 2.47377 51 236.342 2.37354 51 195.968 2.29219 52 296.414 2.47190 52 235.533 2.37205 52 195.412 2.29095 53 295.142 2.47003 53 234.730 2.37057 53 194.859 2.28972 54 293.881 2.46817 54 233.932 2.36909 54 194.309 2.28849 55 292.631 2.46632 55 233.139 2.36761 55 193.762 2.28727 56 291.392 2.46448 56 232.352 2.36615 56 193.218 2.28605 57 290.163 2.46264 57 231.570 2.36468 57 192.677 2.28483 58 288.944 2.46081 58 230.794 2.36322 58 192.140 2.28362 59 287.735 2.45899 59 230.022 2.36177 59 191.605 2.28241 TABLES RADII METRIC (Continued) 301 Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 6 00' 191.073 2.28120 7 00' 163.804 2.21432 8 00' 143.356 2.15642 01 190.544 2.28000 01 163.415 2.21329 01 143.058 2.15551 02 190.018 2.27879 02 163.029 2.21226 02 142.762 2.15461 03 189.496 2.27760 03 162.644 2.21124 03 142.467 2.15371 04 188.976 2.27641 04 162.261 2.21021 04 142.173 2.15282 05 188.458 2.27521 05 161.880 2.20919 05 141.880 2.15192 06 187.944 2.27403 06 161.500 2.20817 06 141.589 2.15103 07 187.432 2.27284 07 161.122 2.20715 07 141.299 2.15014 08 186.923 2.27166 08 160.746 2.20614 08 141.010 2.14925 09 186.417 2.27049 09 160.372 2.20513 09 140.722 2.14836 10 185.914 2.26931 10 160.000 2.20412 10 140.435 2.14748 11 185.413 2.26814 11 159.629 2.20311 11 140.150 2.14659 12 184.915 2.26697 12 159.260 2.20211 12 139.865 2.14571 13 184.420 2.26581 13 158.892 2.20110 13 139.582 2.14483 14 183.927 2.26465 14 158.527 2.20010 14 139.300 2.14395 15 183.437 2.26349 15 158.163 2.19910 15 139.019 2.14307 16 182.950 2.26233 16 157.800 2.19811 16 138.739 2.14220 17 182.465 2.26118 17 157.440 2.19712 17 138.460 2.14132 18 181.983 2.26003 18 157.081 2.19612 18 138.183 2.14045 19 181.503 2.25888 19 156.724 2.19514 19 137.906 2.13958 20 181.026 2.25774 20 156.368 2.19415 20 137.631 2.13872 21 180.552 2.25660 21 156.014 2.19316 21 137.357 2.13785 22 180.079 2.25546 22 155.661 2.19218 22 137.084 2.13699 23 179.610 2.25433 23 155.310 2.19120 23 136.812 2.13612 24 179.142 2.25320 24 154.961 2.19022 24 136.541 2.13526 25 178.678 2.25207 25 154.613 2.18925 25 136.271 2.13440 26 178.215 2.25094 26 154.267 2.18827 26 136.002 2.13355 27 177.755 2.24982 27 153.923 2.18730 27 135.734 2.13269 28 177.298 2.24870 28 153.580 2.18633 28 135.468 2.13184 29 176 842 2.24759 29 153.238 2.18537 29 135.202 2.13098 30 176.389 ' 2.24647 30 152.898 2.18440 30 134.937 2.13013 31 175.939 2.24536 31 152.559 2.18344 31 134.674 2.12928. 32 175.490 2.24425 32 152.222 2.18248 32 134.411 2.12843 33 175.044 i 2.24315 33 151.887 2.18152 33 134.150 2.12759 34 174.600 ; 2.24204 34 151.553 2.18056 34 133.889 2.12674 35 174.159 2.24095 35 151.220 2.17961 35 133.630 2.12590 36 173.720 2.23985 36 150.889 2.17866 36 133.371 2.12506 37 173.283 2.23876 37 150.559 2.17771 37 133.114 2.12422 38 172.848 2.23766 38 150.231 2.17676 38 132.857 2.12338 39 172.415 2.23658 39 149.904 2.17581 39 132.602 2.12255 40 171.984 2.23549 40 149.579 2.17487 40 132.347 2.12171 41 171.556 2.23441 41 149.255 2.17393 41 132.094 2.12088 42 171.130 2.23333 42 148.932 2.17299 42 131.841 2.12005 43 170.705 2.23225 43 148.611 2.17205 43 131.589 2.11922 44 170.284 2.23117 44 148.291 2.17111 44 131.339 2.11839 45 169.863 2.23010 45 147.973 2.17018 45 131.089 2.11757 46 169.446 2.22903 46 147.656 2.16925 46 130.840 2.11674 47 169.030 2.22796 47 147.340 2.16832 47 130.593 2.11592 48 168.616 2.22690 48 147.026 2.16739 48 130.346 2.11510 49 168.204 2.22584 49 146.713 2.16647 49 130.100 2.11428 50 167.794 2.22478 50 146.401 2.16554 50 129.855 2.11346 51 167.387 2.22372 51 146.091 2.16462 51 129.611 2.11264 52 166.981 2.22267 52 145.782 2.16370 52 129.368 2.11183 53 166.577 2.22162 53 145.474 2.16278 53 129.125 2.11101 54 166.175 2.22057 54 145.168 2.16187 54 128.884 2.11020 55 165.775 2.21952 55 144.862 2.16095 55 128.644 2.10939 56 165.377 2.21848 56 144.559 2.16005 56 128.404 2.10858 57 164.981 2.21743 57 144.256 2.15913 57 128.166 2.10777 58 164.587 2.21640 58 143.955 2.15823 58 127.928 2.10697 59 164.195 2.21536 59 143.655 2.15732 59 127.691 2.10616 302 THE RAILROAD TAPER RADII METRIC (Continued) Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. jog. rad. 9 00' 127.455 2.10536 10 00' 114.737 2.05970 11 00' 104.334 2.01843 01 127.220 2.10456 01 114.547 2.05898 01 104.177 2.01777 02 126.986 2.10376 02 114.357 2.05826 02 104.020 2.01712 03 126.752 2.10295 03 114.168 2.05754 03 103.864 2.01646 04 126.520 2.10216 04 113.979 2.05682 04 103.708 2.01581 05 126.288 2.10136 05 113.791 2.05611 05 103.552 2.01516 06 126.057 2.10057 06 113.604 2.05539 06 103.397 2.01451 07 125.827 2.09977 07 113.417 2.05468 07 103.243 2.01386 08 125.598 2.09898 08 113.231 2.05397 08 103.089 2.01321 09 125.370 2.09819 09 113.046 2.05326 09 102.935 2.01256 10 125.142 2.09740 10 112.861 2.05254 10 102.782 2.01192 11 124.916 2.09662 11 112.677 2.05184 11 102.629 2.01127 12 124.690 2.09583 12 112.493 2.05113 12 102.477 2.01063 13 124.465 2.09505 13 112.310 2.05042 13 102.325 2.00998 14 124.241 2.09426 14 112.128 2.04971 14 102.174 2.00934 15 124.017 2.09348 15 111.946 2.04901 15 102.023 2.00870 16 123.795 2.09270 16 111.765 2.04831 16 101.873 2.00806 17 123.573 2.09192 17 111.584 2.04760 17 101.723 2.00742 18 123.352 2.09115 18 111.404 2.04690 18 101.573 2.00678 19 123.132 2.09037 19 111.224 2.04620 19 101.424 2.00614 20 122.912 2.08959 20 111.046 2.04550 20 101.275 2.00550 21 122.694 2.08882 21 110.867 2.04480 21 101.127 2.00487 22 122.476 2.08805 22 110.689 2.04410 22 100.979 2.00423 23 122.259 2.08728 23 110.512 2.04341 23 100.832 2.00360 24 122.043 2.08651 24 110.336 2.04272 24 100.685 2.00296 25 121.827 2.08574 25 110.160 2.04202 25 100.538 2.00233 26 121.612 2.08498 26 109.984 2.04133 26 100.392 2.00170 27 121.398 2.08421 27 109.809 2.04064 27 . 100.247 2.00107 28 121.185 2.08345 28 109.635 2.03995 28 100.102 2.00044 29 120.973 2.08269 29 109.461 2.03926 29 99.957 1.99981 30 120.761 2.08193 30 109.288 2.03857 30 99.812 1.99918 31 120.550 2.08117 31 109.115 2.03788 31 99.668 1.99856 32 120.340 2.08041 32 108.943 2.03720 32 99.525 1.99793 33 120.130 2.07965 33 108.771 2.03651 33 99.382 1.99731 34 119.921 2.07890 34 108.600 2.03583 34 99.239 1.99668 35 119.713 2.07814 35 108.430 2.03515 35 99.097 1.99606 36 119.506 2.07739 36 108.260 2.03447 36 98.955 1.99544 37 119.299 2.07664 37 108.090 2.03379 37 98.813 1.99481 38 119.093 2.07589 38 107.921 2.03311 38 98.672 1.99419 39 118.888 2.07514 39 107.753 2.03243 39 98.532 1.99358 40 118.684 2.07439 40 107.585 2.03175 40 98.391 1.99296 41 118.480 2.07364 41 107.418 2.03108 41 98.251 .99234 42 118.277 2.07290 42 107.251 2.03040 42 98.112 .99172 43 118.074 2.07215 43 107.084 2.02972 43 97.973 .99111 44 117.873 2.07141 44 106.919 2.02905 44 97.834 .99049 45 117.672 2.07067 45 106.753 2.02838 45 97.696 .98988 46 117.471 2.06993 46 106.588 2.02771 46 97.558 .98926 47 117.272 2.06919 47 106.424 2.02704 47 97.420 .98865 48 117.073 2.06846 48 106.261 2.02637 48 97.283 .98804 49 116.875 2.06772 49 106.097 2.02570 49 97.147 .98743 50 116.677 2.06699 50 105.935 2.02504 50 97.010 .98682 51 116.480 2.06625 51 105.772 2.02437 51 96.874 .98621 52 116.284 2.06552 52 105.611 2.02371 52 96.739 .98560 53 116.088 2.06479 53 105.449 2.02304 53 96.604 .98500 54 115.893 2.06406 54 105.289 2.02238 54 96.469 .98439 55 115.699 2.06333 55 105.128 2.02172 55 96.334 .98378 56 115.505 2.06260 56 104.968 2.02106 56 96.200 .98318 57 115.312 2.06187 57 104.809 2.02040 57 96.067 .98257 58 115.120 2.06115 58 104.650 2.01974 58 95.933 .98197 59 114.928 2.06043 59 104.492 2.01908 59 95.800 .98137 TABLES RADII METRIC (Continued) 303 Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 12 00' 95.668 1.98077 13 00' 88.337 1.94614 14 00' 82.055 .91410 01 95.536 1.98017 01 88.224 1.94559 01 81.958 .91359 02 95.404 1.97957 02 88.112 1.94504 02 81.861 .91308 03 95.272 1.97897 03 88.000 1.94448 03 81.765 .91257 04 95.141 1.97837 04 87.888 1.94393 04 81.668 .91205 05 95.010 1.97777 05 87.776 1.94338 05 81.572 .91154 06 94.880 1.977f7 06 87.665 1.94283 06 81.476 .91103 07 94.750 1.97658 07 87.554 1.94228 07 81.380 .91052 08 94.620 1.97598 08 87.444 1.94173 08 81.285 .91001 09 94.491 .97539 09 87.333 1.94118 09 81.190 .90950 10 94.362 .97480 10 87.223 1.94063 10 81.095 .90899 11 94.233 .97420 11 87.114 1.94009 11 81.000 .90848 12 94.105 .97361 12 87.004 1.93954 12 80.905 .90798 13 93.977 .97302 13 86.895 1.93899 13 80.811 .90747 14 93.850 .97243 14 86.786 1.93845 14 80.717 .90696 15 93.723 .97185 15 86.677 1.93790 15 80.623 .90646 16 93.596 .97126 16 86.569 1.93736 16 80.529 ..90595 17 93.469 .97067 17 86.461 1.93682 17 80.436 .90545 18 93.343 .97008 18 86.353 1.93628 18 80.342 .90494 19 93.217 .96950 19 86.245 1.93573 19 80.249 .90444 20 93.092 .96891 20 86.138 1.93519 20 80.156 1.90394 21 92.967 .96833 21 86.031 1.93465 21 80.064 1.90344 22 92.842 .96774 22 85.924 1.93411 22 79.971 1.90293 23 92.717 .96716 23 85.818 1.93358 23 79.879 1.90243 24 92.593 .96658 24 85.711 1.93304 24 79.787 1.90193 25 92.469 .96600 25 85.605 1.93250 25 79.696 1.90144 26 92.346 .96542 26 85.500 1.93197 26 79.604 .90093 27 92.223 .96484 27 85.394 1.93143 27 79.513 .90044 28 92.100 .96426 28 85.289 1.93089 28 79.422 .89994 29 91.977 .96368 29 85.184 1.93036 29 79.331 .89944 30 91.855 .96310 30 85.079 1.92982 30 79.240 .89894 ' 31 91.733 .96253 31 84.975 1.92929 31 79.149 .89845 32 91.612 .96195 32 84.871 1.92876 32 79.059 .89795 33 91.491 .96138 33 84.767 1.92823 33 78.969 .89746 34 91.370 .96080 34 84.663 1.92769 34 78.879 .89696 35 91.249 .96023 35 84.560 1.92716 35 78.790 .89647 36 91.129 .95966 36 84.457 1.92664 36 78.700 .89597 37 91.009 .95908 37 84.354 1.92611 37 78.611 .89548 38 90.890 .95852 38 84.251 1.92558 38 78.522 .89499 39 90.770 .95794 39 84.149 1.92505 39 78.433 .89450 40 90.652 1.95738 40 84.047 1.92452 40 78.344 .89401 41 90.533 1.95681 41 83.945 1.92399 41 78.256 .89352 42 90.415 1.95624 42 83.843 1.92347 42 78.168 .89303 43 90.297 1.95567 43 83.742 .92294 43 78.080 .89254 44 90.179 1.95511 44 83.640 .92241 44 77.992 .89205 45 90.061 1.95454 45 83.540 .92189 45, 77.904 .89156 46 89.944 1.95397 46 83.439 .92137 46 77.817 .89107 47 89.828 1.95341 47 83.339 .92085 47 77.729 .89058 48 89.711 1.95285 48 83.238 .92032 48 77.642 .89010 49 89.595 1.95228 49 83 . 138 .91980 49 77.556 .88962 50 89.479 1.95172 50 83.039 .91928 50 77.469 .88913 51 89.363 1.95116 51 82.939 .91876 51 77.382 .88864 52 89.248 1.95060 52 82.840 .91824 52 77.296 .88816 53 89.133 1.95004 53 82.741 .91772 53 77.210 .88767 54 89.019 1.94948 54 82.642 .91720 54 77.124 .88719 55 88.904 1.94892 55 82.544 .91669 55 77.039 .88671 56 88.790 1.94836 56 82.446 .91617 56 76.953 .88623 57 88.676 1.94781 57 82.348 .91565 57 76.868 .88575 58 88.563 1.94725 58 82.250 1.91514 58 76.783 1.88527 59 88.450 1.94670 59 82.152 1.91462 59 76.698 1.88478 304 THE RAILROAD TAPER RADII METRIC (Continued) Deg. Radius. Log. rad. Deg. Radius. Log. rad. Deg. Radius. Log. rad. 15 00' 76.613 1.88430 19 00' 60.589 1.78239 23 00' 50.159 1.70035 05 76.192 1.88191 05 60.326 1.78050 15 49.626 1.69571 10 75.776 1.87953 10 60.067 .77864 30 49.106 .69113 15 75.364 1.87716 15 59.809 .77677 45 48.596 .68660 20 74.957 1.87481 20 59.554 .77491 24 00' 48.097 .68212 25 74.554 1.87247 25 59.300 .77305 15 47.609 .67769 30 74.156 1.87015 30 59.050 .77122 30 47.130 .67330 35 73.762 1.86783 35 58.801 .76938 45 46.662 .66896 40 73.372 1.86553 40 58.554 .76756 25 00' 46.202 .66466 45 72.986 1.86324 45 58.309 .76574 15 45.752 .66041 50 72.604 1.86096 50 58.067 .76393 30 45.311 .65620 55 72.227 1.85870 55 57.826 .76212 45 44.878 .65203 16 00' 71.853 1.85644 20 00' 57.588 .76033 26 00' 44.454 .64791 05 71.483 1.85420 05 57.351 .75854 15 44.038 .64383 10 71.117 1.85197 10 57.117 .75677 30 43.630 .63979 15 70.755 1.84976 15 56.884 .75499 i 45 43.230 .63579 20 70.396 1.84755 20 56.653 .75322 27 00' 42.837 .63182 25 70.041 1.84535 25 56.425 .75147 15 42.451 .62789 30 69.690 1.84317 30 56.198 .74972 30 42.072 .62399 35 69.342 1.84100 35 55.973 .74798 45 41.701 .62015 40 68.998 1.83884 40 55.749 .74624 28 00' 41.336 .61633 45 68.657 1.83668 45 55.528 .74451 15 40.977 .61254 50 68.320 1.83455 50 55.308 .74279 30 40.625 .60879 55 67.986 1.83242 55 55.090 .74107 45 40.279 .60508 17 00' 67.655 1.83030 21 00' 54.874 .73937 29 00' 39.939 .60140 05 67.327 1.82819 05 54.660 .73767 15 39.605 .59775 10 67.003 1.82609 10 54.447 .73597 30 39.277 .59414 15 66.682 1.82401 15 54.236 .73429 45 38.954 .59055 20 66.363 1.82193 20 54.026 .73260 30 00' 38.637 .58700 25 66.058 1.81986 25 53.819 .73094 30 65.736 1.81780 30 53.612 .72926 35 65.427 .81576 35 53.408 .72761 40 65.121 .81372 40 53.205 .72595 45 64.818 .81170 45 53.003 .72430 50 64.517 .80967 50 52.804 .72267 55 64.219 .80766 55 52.605 .72103 18 00' 63.924 .80566 22 00' 52.408 .71940 05 63.632 .80368 05 52.213 .71778 10 63.343 .80170 10 52.019 .71616 15 63.056 .79973 15 51.827 .71456 20 62.772 .79777 20 51.636 .71295 25 62.490 .79581 25 51.446 .71135 30 62.211 .79387 30 51.258 .70976 35 61.935 .79194 35 51.072 .70818 40 61.661 .79001 40 50.886 .70660 45 61.389 .78809 45 50.702 .70503 50 61.120 .78618 50 50.520 .70346 55 60.853 .78428 55 50.338 .70190 TABLES 305 NOTATION A = Curvature increment of taper per 10 meters of taper. M = Degree of main circular curve. I = Length of taper from the B.C. of the taper. / = Deflection from tangent at the B.C. of the taper. f'j /i> /2j etc. = Deflection from tangent at some point on the taper. b = Deflection from tangent at the end of the taper, C.C. of main curve, to the B.C. of the taper. o = Central angle of the taper at each end of the curve. d = Circular curve offset from tangent to provide room for the taper. t = Distance along the tangent, from opposite the offset posi- tion of the B.C. of the simple curve to the B.C. of the taper. D = R + d, where R is the radius of the main circular curve. x and y are co-ordinates of points on the taper, referred to the tangent at the B.C. of the taper as meridian, and to a radial line at the B.C. of the taper as base. Ic = Long chord of the taper, from the B.C. of the taper to points on the taper, s = Distance along the taper, from the B.C. of the taper to the transit, when an intermediate set-up is necessary. T = Semi-tangent of the entire tapered curve, from P.I. to B.C. or E.G. E = External, the distance from the P.I. to a point on the curve radially opposite. EQUATIONS Eqn. No. T = t + D tan \I. (7) Both ends the same taper. T t = t + D tan \ I - d esc 7. (8) Tapered end other end not tapered. T n = Rtan \ I + tan | 7 - (d* - di) esc 7. (11) Both ends tapered, with different ta- pers. E = D sec % I - R. (12) Both ends the same taper. 306 THE RAILROAD TAPER EMPIRICAL EQUATIONS DERIVED FROM THE METRIC TABLES A (45m) (1 + 5) (Z + 10) (107m) (94m) 20 A 20 A 3 1 (I + 10) (108m) 20 A The equations under paragraph 18 as derived from the metric taper tables become : I = (M A). (45m) A In = (M 2 A). (46m) 10 M, Zi = - -v (47m) lc = ^ (M, - M! - A). (48m) = 15(M 2 + Mi)(M 2 -M 1 -A) t A a = 15Mi(M 2r Mi-A). (51am) 20 A 2M 2 + A)(M 2 -M 1 -A) ' (56m) + A!- (57m) TABLES 307 08 - O * CO 00 t"- i i O * Ot^COOO CO ^ Tt< T^l rj< Tfl CO CO CO - . ' (M O CO O CO . TjH (M O qS o?,, . IO rH C- t^ rH O O CO "*! rf CO 05 rH IO CO . CO rH IO -^ ^ * "^ ^H rH rH rH rH rH O O ' O rH rH (M CO ^ >O 08 ^i00500CO 1C "* CO i 1 ' o !>. OO GO 00 OO QO l>. !> CO O ^t 1 CO rH ' i-H . HHMnn|- HNrnlNWl-* H-di^lNnW . ^|N^|-* * CO b 00 O T-I (M CO 1C CO t^ OO O . t^ CO Tf O i 1 (M i 1 *ft C* t& 1C >O -* 1 CO . CO C^ o CO t t^. b !> CO CO ift TJH CO CO O CO C^ 00 iO . Oi Ofl - Oi TH TH T-H T-H O ' O TH O3 ^ 1C t>- 00 O C CO IOCO C^ TH Q OO t^ CO CO .CO(N(NCO>OOt>OOO(N^ OO O O TH C1 CO -^ COJ> O5 TH CO iO TH TH TH o O ' O O TH f*" 1 '^ t^- CD O5 C^ CD OS C^ > t>- ^ 00 O CO Ci >O i Ci Oi 00 t>- CD CO O o T 8^SS88S*SSS J o K, OOOO'-lT-HCO'*CDo6O'*t^i-H rj i-ii-H i-KN PH w ooo.*-** 1 !^ 881 o o . ^^ , ,^ , , ^^ ,.. v >^, ^^^ ...^ 4. CO T-H O5 Oi '"^ o o o o o o o T-H T-H OO lO t>- TH O CO O O * O "tf r-t CO CO C CO lO CO QO iO t^ -^ ^ iQ iO CO O C O - OS rH rH TjH t^ O rHrHrH(N i OS? O 'OrH rHCOTfH CO 00 O ia cOOO5 lOiOiOiOOt^iOi-HCOOOt^-^ COCO O i l i OO i i Oi i COiO. rH|MH|M HNrtlN rtlMrtlN H|N oooooooooooo cqco^ocoj>ooo5Or-Nco (M CO CO ^ O CO CO t^ 00 O5 O5 O TABLES 315 --< oso^i-KMeacocOTtt^i^ d d d o" d o" d o" o ooooooooo OO CO OO CO !> < oooooooooo oooooooooooo ooooooooooo CO CJiCOCC on ea c< CCOf aoit^'O-^oq (O I 7 - !li S! SCOCOCO fOt~--fcC'M !5H^50^CSCOTt<50C^eO i ^ TJ< in D c s 10 5 co o co o t~ oo oo a o o ~ eo OJ TABLES 317 p8 T-H O5 1>- >o co ot^ioc^iot^-^oc^oi^-io ! o * CO O - ^0 - CO "* . o co CO rfrl CO CO C 00 t>* I-HI iiO' iO5>OO5T lOOOOi' i OOOiOOQOi iCOi i COi it^COO T-(T-iOOO CO OO COCOOiT-iOiOOOOO5T-iCO OOGOcooo>c^iooiOJioco*-H^i^ oeo3coH^< JCOCOClCOCCO^COCOdCOSOOi CO CO Oi q c-H eot^ ^H CO CO -* OOOOOOOOOi 320 THE RAILROAD TAPER t-H t- oo o ) C5 CO CO CO < I T-H CM COCO < jScOcOeoS ntttS^OOOOOOOO SSSc5So< oooooa eco ^ '^ C^l CO t'- r-l 1C O5 Tj< O5 ci CN csi CN ci co co co *' TH' 2 ^ H c< " c w ** *^ "* os cc oo Tf ^< o t^os t^ CO O OO CO ^ CO CO CM CO -^f >O t^- OS C^l CO O ^ OS kO to t>^ oo oo oi o -j co^ jjfjjo^s '' H ' e< T < "' t ^ o ^ OOOOOO odi-H-li-lT-I^HT-l CMCMCMCMCMCOCOCO CO TiT*Tj^ O* CO 00 id -i clT^'ti'i-i o * O I-H << O CM T< O CO >O i t^"s; CM r^ o o^o -i CM" l CO O CO O CO IO CO O CO S O3 O O ~H ^H -rt CM CO CO !**g8 i co t^ 06 o> o 00 0> O TABLES 321 g H 8 dt, ds S o (N . 00 00 00 i> CO rP O OO i I tO COCOCO(NT-1 : TH Tj< CO O CO l> 85 :?^^^S8 - C (N TH THiMOOCOCO tO OO :oOOiCO O '. O i-( CO iO t^ O CO t^ T- i CO t^ T- i to i I T-H 00 O T-H CO CO OS CO CO c CO C5 CO CO O5 CO CO < JCOOCOCOCTJCOCoaxOCOOi O5O3O5 CO CO O iO C$ O O5 O CO t>- CO O OO OO O CO OO iO ^* *** CO O CO -"cH *^ C t-- 5 53 ** l>- O ?>) CO O) 03 CO O CO t^ - t- OO OO O5 C5 O O *-( ~H (M 0000 000000000000 OOOOi o CO CO Oi CO CO O CO CO C5 CO CD O Oi O> O5 *O 1C *O CO CO CD t>- t^- t^ OO OO GO OS O * ' ^>O^I CO CO CO CO * 5 t^ Oi CNHC OO . CO O5 CO -^ - OO O5 O *^ CO -^ *O t^- OO O *-H CO ^ CD OO O CN ^* O OO -^ -H OO i I rH i I i-H i i rH i ( i 1 i ( -H C^I C^ CM o'o'o'o'ooo'o'oo'o'o' odd iO*O$Ot>- OO O5 *~ i C^l T* < ^^^! 5J^^^5' t^os oo i I t^- CO i-H IO CO Cv| OO i-H OO CNI O CD r-H I Tf O O t^- CO CO O t>- O O O5 ^ -^ O C t^ -H t-. co 06tta5FCb CM* 00 C i-l 00 5 * i-icoo>co>OT-Hco--ct-. eoo^ot^cooscocq os co co -* CD o CO CO TH <*< "5 >0 O O I rH . O'CM c"oO i-i TJ< t-T O CO -*iOlO to 1C O O O O i-H ,-H rH IM CN1 IM CO CO CO -* * T* U5 5 >0 O OOO O O (M C^t ?5 * co co ?5 I-H c c it^t^ co -* ^ OO O3 O i (C^^HiOCO rtrt(MC^C^^ci co rfJ ic c> t^ od o O --i osi cc *' i COCOCOCOCOCOCOCO CO*^ 1 ^^^^' ^S! Hi ^1 Pr- us i Tt< r-i O5 1^ o co ^ o O5C30OCOOOOO o O^h ?3 O-^tl o2 O 1O S 00 OO O5 O O i-H .C^t>CO>IO5 O5OiOCOCOOCOC-1 OCOi-HO CO T-I O -<*M <-H O Tt< COCM 03 rt OO O O t>OOOOO5OiCiOO T-MI iC^C^C^JCOCO-^t ' O o co !coi> - O CO - 1^ l> t> CO r^ (M :cocOO IH|NW H-<*H|N nl^HNrHl-* rH ! CO OO (N CO * 00 (N CO O Tj C^l O^ '""i ^^ CC ^O C^ C5OiO5t>-OO' it^-l^ OOiOiOiOlOiCsOOCO-^ T-HT-iC^cO^iOCOt^OOO - .- -oocoo ' ICOO5O5I>. " ScoSc^Scoocooco TABLES 327 i O O 5 1C i i t>- C^ t- < O O -< 1 o o o o o o' o o o o o o o o o o o o' T-I T-! i-I I-H i-I -H oi c^i -- -^ *O O *O CC CO * S^ OCoS O iQ^t- * co i- o -< CM * us t^ os o o co us cooooi-* ^ US 1C o US US US CO CO CO CO CO CO t~ t- t t t- t~ t^- 00 r>^ t od oi oi o r-< ,-! CM co co * us co i O OS CM CO CO -^ CO -^ CO t-- O> O 1-1 t^- 1C 5 t-* 1C CO CN O OO CO -*t C^l -t O5 t^- WS CO i I OS !> O CO O OO CO - CM C^l CO l^ CO CO ^ OO O5 ** CM CO H CO OO IN- ^ O5 i t ^ t OS 1C O ^^.^^t^g^u^Sj -*OCOCOOOOI>.I^ t^-OS'-ICOt^'-U i-l CN CO * IO CO 00 O O . O5 t^ O5 O O >C CO CO CO cTo o?O5 o O ? 00 1> O lO CO '. CO t^ (N 10 >o 10 -^ .OO'-iOi i OO t^* t^* 888888888 ^888888888 Tt< CO OO O W Tfi CO OO O TABLES 331 i < CO lO CO *H ^D Oi " ^f C5 *-O CO *O Os C^l ^^ *O *O ^^ - . I O rf( O ^ O ^ OO CO OO CO t C^ CO O ^O O^ CO l>- i-H id OO ' 88 '^8r? g| CO 00 rH -^ t^ O CO O OS rH (N CO ^f ^ * CO C3 O 1>- Tfl O irH-H^CpOQrHCO 1 ^?^ OC^'^COOOOC^'^t 1 COt^-OSrH ^ o COCOCO^^^>OOO CO CO 1> !> 00 00 OS O OrH rH rH C<1 l> t>- 00 00 OS rH rH rH rH rH rH C<1 C^ C Tt< 1C tO "O lO O tO lO iO CO CO CO CO CO CO t^ t^ 1> t^ 00 00 OO 00 OSCDoico^to'cdl^OsCDrHeqco COOOCDOO>OiO>OCOCOCOCO COCOt^-lS-t-t-t^OOOOOO t>-rHTfiOOCOOCO>O>O'^ti I rHiOi it^-C^iOt^t^iOC^ CO 00 OS O rH rH (N CO CO CO CO CO CO C^l O OC - OS O O rH ^ rH O 00 OS rH -* OS l> OS CO OS rHrHrH(MCOTtlC005(M0 -a ._ .__ rHrHrHC CO C^l rH C^ ^O CD 1^* CO !> CD ^ CD OO *O t^* C^J CO CD CO C^ t^ OO ^t 1 OSOSOSOSOSOSO5O5O5OSOSOSO5 OOOOOOOOOOOOOOOOOOOO t^-O^COC^COt^CQlOCOrHOO^f COrHOSCOOCOOOOOCOI>- rH CO lO 00 C^ 1>OCOrHCOOOI>-lO'^t O5OSO5O5O500OOOOOC HMH!N H|MiH|M OS OS O O rH rH (N C^ CO CO T^ tO IO CO OO OS O rH CO T^ CO t^ OS IO "5 iO CO CO CO l> !> 1> i> OO 00 00 OS O O rH - OO O T-H C 1 s " . CO CO i ( OO rt< O lO ,3 ^^^^^ Si O* >O'^t | CO'-HOOiOOt > -CO iO^COC^r-iOOiOOt~>- ,_, ,_, ^H ^H rH r-t T-l O O O O O O O O O O Oi OS Oi CO Oi (N "* ^ i i i i Ir^ 00 i-< i 1 1 i I-H C^ C^ C^ (M C* C^l '^ CO 05 C^ ^^ CO *O "^ CO 05 05 CO CO 00 ^"i CO C^l t^" ^O *O COOt^-'^OCO'-Hf^C^ Ot^-C^iOOCOOi-HOCO 00 CO 05 00 C^ |> J> CO CO CO *O *O "^ '^ CO *"^ ^2 OO CO "^ ^H 0i CO C^ 00 ^f 05 "^ 05 3^3h$$W3 ss?ig^8SS^o ^^^^^ OJC^^OOiCOOO^OOS Oit v -COOO^fi-H' i^fiHC^ OOOOOOCO T (COi (t^-iOCOPOT^iO C s ^ T t 1 ' idO5r- lOOOOOi i Oi^COOOd CO ^H O5 t^ t^* CO t^ t^* O5 ^^ CO CO O^ CO - t>- 00 O CO iOt>- O < ^T-J^^T-HT-JT-H C^fMC^C^eO GO OO *-O CO T 4 1>- CO C^ CO CO OS 00 CO CO o' OS O5 O^ OS O5 OS OS OS GO OO OO OO OO 00 00 00 00 OO t^ I s * t> i I 1C CO CO O CO OO CO CO t^ rt< lO OS i I lO CO I> O ^O OO -^ OS OS O COI> O OS OS rf * O O <* iO I-H T i iO O t^ TJH OO !> -^ to OS -^ OS O C^J OS l> >O C<| i i CN -^ 1> C^ (N ^ CN O O O T i CO CO oJ t>^ co Tt< co CN o os 06 co ^ IN o 06 i>i to -* co c l>- OO 00 OS O O i I i-< C^ CO CO T^ to O CO CO CO OS i* ^ CO OS T ( CO CO OO CO *""< CO ^f t>. iO CO CN i i l>- , o irj i i t^ rJH j-* OS O i I W CO ^ to l>- 00 O5 Or- ICO^O1>. OO O i-H C^ CO tO CO !>. OO OS O ^-H Ofl THOOC^cOO^t 1 OOCOI>'-HOO5 COt--' itOOTt^OO ^H to iO OOC5OOOO OOOOOO OOOOOOO ^ (M O OS t^ co TJH os os os oo oo oo oo "* Tt< iO iO O CO CO OS OS O O T-H 1-1 (M t^O5(NiOOO(McO O CO^OT-KNr^to 1-1 (M t i O i i C^ co cOTfitOO 1-1 T-I iM (N * CO CO I>* CO CO !> CO CO COOCOCOO COCOOCOCOO COCOOCOCOOCO COtOCOi-nO OO CO to CO T i O OOCOtOCO'-HOOO TABLES 339 OS CO t>. i r-t ^ CO I ^ r-n CO CO CO t^ OO OO OS i i CN Th CO b- OS O CM CO to CO OO OS O CM CO CO CO CO CO CO CO TjH -^ Tti "* Tt< T^ to iO to iO to to to CO CD CO CO 8ii o CD o OS OS OS r1 COCOCOCOCO T}HTtiiOOCOI>. 1^0005001-1 (MCO^Ot^ i TfH CO ^H ^^ C*^ Ot> O^ ' i 00 CO O^ *O T-H ii I OO i I CM OS Tf to CO i co co oo os os co co co t^ CO OO CO ' T i os CO -^ OO 00 OS OS OS O T-H CM CO CO O (M ( ' OO OS O O i i CO CO !> CD !> OS CO OS 00 "53 CO *~^ '"H CM CM ^f* *O b* OS CO CM "^ CO 00 '"^ CO to GO TI "^ t^* I i-H rH i I r-t i I T-I T-| rH CM CM CM CM CM CO CO CO ~^> "^ CO OS CO ^t^ OS OO CO CO t^* CO CO OS T-H 00 OS "^f CO ( T ( r- 1 O OS OS t^-COtO^fCOCM i ( O OS OO l^- CO tOtO^COCM OS OS OS 00 OO OO GO 00 00 00 OO OOOOI>i>t^l> l>t^l>I SCO ^H OO CM CM to 00 t^. CO ' _ CO rH CO rH OS CO CM IO rH OO CO tO CO OS ^ l>- rHl>.TtlCMOS tO^f^ftOOOrH COON 1 Q . . CO rH CO OS !> tO CO rH OS t N * CO Tf CO rH CO OS 00 sssfs cf^Il8 s^fsss fsiss iO t^ OO 00 OS OS O O rH CM CO Tt| tpcOi>.oqOSO rHCMcOT^t^ i CM CM CM CM CM CM Tt< to iO to to CO CO CO l> !> 00 OO OS OS O O i ( i ( C^ CO CO iO CO > t^ CO i ' CO CO ' ' CO CO ' CO CO CO CO ^ -t 1 "* -^ ^ rt< to to iO to iO to CO COCOcOCO 88888 340 THE RAILROAD TAPER TAPER 3 METRIC. CURVATURE INCREASING 4 00' EVERY 10 M. A = 4 Transit at Deflections to B.C. 4 00' C.C. 8 00' C.C. 12 00' C.C. 16 00' C.C. 20 00' C.C. 24 00' C.C. 28 00' I 10 20 30 40 50 60 10 20 30 40 50 60 , 100 230 200 / 440 430 300 '400 930 1620 730 740 630 400 'sod 11 30 1100 1130 1040 830 500 600 / 1510 1600 1530 1340 1030 600 100 330 720 1230 1900 2650 200 530 1020 1630 2400 300 730 1320 2030 TAPER 3 METRIC. CURVATURE INCREASING 4 00' EVERY 10 M. (Continued) A = 4 M l o D log> d i X y lc 800 10 200 143.443 2.15668 0.087 4.996 0.175 9.999 10.000 1200 20 600 96.016 1.98234 0.348 9.974 0.872 19.974 19.993 1600 30 1200 72.719 1.86165 0.866 14.912 2.437 29.851 29.951 2000 40 2000 59.309 1.77312 1.721 19.767 5.195 39.463 39.804 2400 50 3000 51.083 1.70828 2.986 24.477 9.432 48.526 49.434 2800 60 4200 46.067 1.66339 4.731 28.957 15.347 56.616 58.66^) TABLES 341 TAPER 3| METRIC. INTERMEDIATE VALUES A=4 M I / 6 o D log D d t X V Ic 1200 20.000 230 330 600 96.016 1.98234 0.348 9.974 0.872 19.974 19.993 15 20.625 237 342 619 94.095 1.97357 0.372 10.284 0.941 20.596 20.617 30 21.250 244 353* 638* 92.252 1.96498 0.397 10.594 1.013 21.217 21.241 45 21.875 251J 407 658i 90.484 1.95657 0.423 10.904 1.089 21.838 21.865 1300 22.500 258! 420 718! 88.787 1.94835 0.450 11.214 1.169 22.458 22.488 15 23.125 306i 433J 739* 87.155 1.94029 0.478 11.523 1.252 23.077 23.111 30 23.750 314 447 801 85.587 1.93241 0.508 11.833 1.339 23.696 23.734 45 24.375 322 500! 822! 84.078 1.92468 0.538 12.142 .430 24.315 24.357 1400 25.000 330 515 845 82.625 1.91711 0.570 12.450 .525 24.933 24.979 15 25.625 338i 529* 907! 81.225 1.90969 0.602 12.759 .624 25.550 25.602 30 26.250 346* 544* 931 79.876 1.90242 0.636 13.067 .727 26.167 26.224 45 26.875 355 559* 954* 78.575 1.89528 0.671 13.375 .834 26.783 26.846 1500 27.500 403! 615 1018! 77.321 1.88830 0.708 13.683 .945 27.398 27.467 30 28.750 421* 647 1108* 74.940 1.87471 0.784 14.298 2.182 28.626 28.710 1600 30.000 - 440 720 1200 72.719 1.86165 0.866 14.912 2.437 29.851 29.951 30 31.250 459 754* 1253* 70.643 1.84906 0.953 15.524 2.710 31.072 31.190 1700 32.500 518! 830 1348! 68.700 1.83696 1.045 16.135 3.002 32.288 32.427 30 33.750 539 907 1446 66.879 1.82529 1.143 16.745 3.315 33.499 33.662 1800 35.000 600 945 1545 65.170 .81405 1.246 17.353 3.648 34.704 34.895 30 36.250 621* 1024* 1646 63.567 .80323 1.356 17.959 4.002 35.904 36.126 1900 37.500 643! 1105 1748! 62.060 .79281 1.471 18.563 4.377 37.097 37.355 30 38.750 706* 1147 1853* 60.643 .78278 1.593 19.166 4.775 38.284 38.581 2000 40.000 730 1230 2000 59.309 .77312 1.721 19.767 5.195 39.463 39.804 30 41.250 754 13 14* 2108* 58.053 .76382 1.855 20.365 5.639 40.633 41.023 2100 42.500 8 18! 1400 2218! 56.870 .75488 1.996 20.961 6.106 41 . 794 42.238 30 43.750 844 1447 2331 55.755 .74628 2.143 21.554 6.598 42.945 43.449 2200 45.000 910 1535 2445 54.706 .73804 2.298 22.145 7.114 44.086 44.656 30 46.250 936* 1624* 2601 53.717 .73011 2.459 22.733 7.655 45.215 45.858 2300 47.500 1003! 1715 2718! 52.787 .72253 2.628 23.317 8.222 46.332 47.056 2400 50.000 1100 1900 3000 61.083 .70828 2.986 24.477 9.432 48.526 49.434 2500 52.500 1158! 2050 3248! 49.577 .69528 3.375 25.623 10.749 50.660 51.787 2600 55.000 1300 2245 3545 48.248 .68348 3.794 26.753 12,173 52.725 64.111 2700 57.500 1403! 2445 3848! 47.083 .67286 4.246 27.865 13.705 54.714 56.403 2800 60.000 1510 2650 4200 46.067 .66339 4.731 28.957 15.347 56.616 58.660 342 THE RAILROAD TAPER TAPER 3f METRIC. CURVATURE INCREASING 5 00' EVERY 10 M. A = 5 Transit at 10 20 30 40 50 Deflections to B.C. 5 00' C.C. 10 00' C.C. 15 00' C.C. 20 00' C.C. 25 00' C.C. 30 00' 10 20 30 40 50 1 15 1 15 307^ 230 / 550 537| 1 922| 935 1345 1422| 422^ 910 15371 2345 230 652 1255 20371 '345" 922^ 1640 345 "SOO" 11521 807 500 6 15 1320 1037| 615 TAPER 3| METRIC. CURVATURE INCREASING 5 00' EVERY 10 M. (Continued] A = 5 M I o D log D d t X y lc 1000 10 230 114.847 2.06012 0.110 4.993 0.218 9.998 10.000 1500 20 730 77.048 1.88676 0.435 9.960 1.090 19.960 19.990 2000 30 1500 58.667 1.76839 1.079 14.863 3.041 29.767 29.922 2500 40 2500 48.338 1.68429 2.136 19.638 6.465 39.164 39.694 3000 50 3730 42.331 1.62666 3.694 24.192 11.675 47.713 49.120 TABLES 343 ' 1^ CO OS Tt< OO i ICO"*' ^t^ C^ ^^ "^ ^2 *O OQ ^^ i O t^ tM to CO Tf OS O 1-- T-H T-H to CO O 00 CO -* OO "* 00 CO CO O "f C^ r-t OS b- to CS O CO CO OS Tt^ OS -^ t^. CO CO O T-I (M OS T-H os os os os oo oo oo oo oo b- t^ - 00 >O iO 00 O CO i I T it>-tX)T^ i I O O i I Tt< OO Tf< i I O5 O5 i I t^ CO CO O >O iO * CO ^ CO rH OS t^ TfH OS CO !> OS C^ C^ t"^ CO CO to Tt^ "^ CO T-H i >O O O5 00 00 OS OS O O rH (M (M ^f 1C l> 00 O rH CO O C^l ' ^ t^^O i o'cq'os CO oTrn'o OSOo'oo'osO'^rcOCq' T-H T^T^ oo"tO O 1> t^ 00 OS OT-HT-M tO CO i>. 8 88888888 88888 8 3 8 838 344 THE RAILROAD TAPER 3" > r 2> 00 'Oo o ^ o O TABLES 345 ;o 82 00 5'-' i O o CD O o 1 "S O -~i^3 O P-0 CO 8 TABLES 347 :s . CO co gbj, IS co S i2 i CO fA T* ;H t^ (M CO 0 "^ ' o o (5 CO t ^ CO 00 *^** CD ^^ *^ ^^ C^ ^^ CO L O **^ ^^ C^ ^^ rt "*O *'O *'O **O **O 'S co^S ^"^in S^; S^Si _ n '0 -J' L^/v' o o ^ o . . -. O O >_ > 00 Q Q >O X So oooo O r "Q O 88888 350 THE RAILROAD TAPER CO* S O < 8 SB* 8 is 8 S -^,^3 r-o'XS r-0^ ^^ 8 8 TABLES 351 CONVERSION OF ENGLISH INCHES INTO CENTIMETERS. Ins. 1 2 3 4 5 6 7 8 9 Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. 0.000 2.540 5.080 7.620 10.16 12.70 15.24 17.78 20.32 22.86 10 25.40 27.94 30.48 33.02 35.56 38.10 40.64 43.18 45.72 48.26 20 50.80 53.34 55.88 58.42 60.96 63.50 66.04 68.58 71.12 73.66 30 76.20 78.74 81.26 83.82 86.36 88.90 91.44 93.98 96.52 99.06 40 101.60 104.14 106.68 109.22 111.76 114.30 116.84 119.38 121.92 124.46 50 127.00 129.54 132.08 134.62 137.16 139.70 142.24 144.78 147.32 149.86 60 152.40 154.94 157.48 160.02 162.56 165.10 167.64 170.18 172.72 175. 96 70 177.80 180.34 182.88 185.42 187.96 190.50 193.04 195.58 198.12 200. &6 80 203.20 205.74 208.28 210.82 213.36 215.90 218.44 220.98 223.52 226.06 90 228.60 231.14 233.68 236.22 238.76 241.30 243.84 246.38 248.92 251.46 100 254.00 256.54 259.08 261 .62 264.16 266.70 269. 24! 271. 78 274.32 276. W CONVERSION OF CENTIMETERS INTO ENGLISH INCHES. Cm. 1 2 3 4 5 6 .7 8 9 Ins. Ins. Ins Ins. Ins. Ins. Ins. Ins. Ins. Ins. 0.000 0.394 0.787 1.181 1.575 1.969 2.362 2.756 3.150 3.543 10 3.937 4.331 4.742J 5.118 5.512 5.906 6.299 6.693 7.087 7.480 20 7.874 8.268 8.662 9.055 9.449 9.843 10.236 10.630 11.024 11.418 30 11.811 12.205 12.599 12.992 13.386 13.780 14.173;14.567 14.961 15.356 40 15.748 16.142 16.530 16.929 17.323 17.717 18.11i;i8.504 18.898 19.292 50 19.685 20.079 20.473 20.867 21.260 21.654 22.04822.441 22.835128.229 * 60 23.622 24.016 24.410 24.804 25.197 25.591 25.98526.378 26.772 27.166 70 27.560 27.953 28.347 28.741 29.134 29.528 29.92230.316 30.709 31.103 80 31.497 31.890| 82.884' 32.678 33.071 33.46533.85934.253 34.646 35.040 90 35.434 35.8271 36.221 36.615 37.009 37.40237.79638.190 38.58338.977 100 39.370| 39.764 1 40.158 40.552 40.945 41.33941.73342.126 42.520l42.914 CONVERSION OF ENGLISH FEET INTO METERS. Feet. 1 2 3 4 5 6 7 8 9 Met. Met. Met. Met. Met. Met. Met. Met. Met. Met. 0.000 0.3048 0.6096 0.9144 1.2192 1.5239 1.8287 2.1335 2.4383 2.7431 10 3.0479 3.3527 8.6575 3.9623 4.2671 4.5719 4.87675.1815 5.4863 5.7911 20 6.0959 6.4006 6.7055 7.0102 7.3150 7.6198 7.92468.2294 8.5342 8.8390 30 9.1438 9.4486 9.7534 10.058 10.363 10.668 10.97211.277 11.582 11.887 40 12.192 12.496 12.801 13.106 13.411 13.716 14.02014.325 14.630 14.935 50 15.239 15.544 15.849 16.154 16.459 16.763 17.06817.373 17.678 17.983 60 18.287 18.592 18.897 19.202 19.507 19.811 20.11620.421 20.726 21.031 70 21.335 21.640 21.945 22.250 22.555 22.859 23.16423.469 23.774 24.079 80 24.383 24.688 24.993 25.298 25.602 25.907 26.21226.517 26.822 27.126 90 27.431 27.736 28.041 28.346 28.651 28.955 29.26029.565 29.870 30.174 100 30.479 30.784 31.089 31.394 31.698 32.003 32.30832.613 32.918 33.222 CONVERSION OF METERS INTO ENGLISH FEET. Met. 1 2 3 4 5 6 7 8 9 Feet. Feet. Feet, i Feet. Feet. Feet. Feet. Feet. Feet. Feet. 0.000 3.2809 6.5618 9.8427 13.123 16.404 19.685 22.966 26.247 29.528 10 82.809 36.090 39.371 42.651 45.932 49.213 52.494 55.775 59.056 62.337 20 65.618 68.899 72.179 75.461 78.741 82.022 85.303 88.584 91.865 95.146 30 98.427 101.71 104.99 108.27 111 55 114.83 118.11 121.39 124.67 127.96 40 131.24 134.52 137.80 141.08 144.36 147.64 150.92 154.20 157.48 160.76 50 164.04 167.33 170.61 173.89 177.17 180.45 183.73 187.01 190.29 193.57 60 196.85 200.13 203.42 206.70 209.98 213.26216.54 219.82 223.10 226.38 70 229.66 232.94 236.22 239.51 242.79 246.07,249.35 252.63 255.91 259.19 80 262.47 265.75 269.03: 272.31 275.60 278.88282.16 285.44 288.72 292.00 90 295.28 298.56 301. 84 1 305.12 308.40 311.69314.97 318.25 321.53 324.81 100 328.09 331.37 334.65' 337.93 341.21 344.49347.78 351.06 354.34 .357.02 352 THE RAILROAD TAPER TRIGONOMETRIC AND MISCELLANEOUS FORMULAS. SOLUTION OF RIGHT TRIANGLES. Required. A, (7, C A, C, b C, a, c C, a, 6 Given, a, 6 a, c , a A,c Formulas. = cot 5 = - ; 6 = a ) (6 - a), + c a . SOLUTION OF OBLIQUE TRIANGLES. Required. 5 Given. y4 7? /r Formulas. A CL b sin A ij fee f a tir\lf4 7?\ ^~ ton if A 4- 7?\ ^4 P . a + o , , f ., ,, , , p . n j ^_ /(S ft)(.S C) a, &, c \ &c ^ i j /(s 6) (s c) V 8(8 -a) cm 1 2^8(-a)(-6)(8-c) 6c Area Area Arpa A, 6, c >4 7? r Area = */8 (s a) (s 6) (s c) Area = 6c sin A Arca _ c 2 sin ^ sin B -A, />, C TABLES 353 TRIGONOMETRIC AND MISCELLANEOUS FORMULAS. GENERAL FORMULAS. sin A = /l cos 2 .4 sin.4 cosec A cos A = - = vsin 2 ^. = cot A sin A, sec A cosA =1 2sin i 4^1 = l ver&A. cosJ. _vl cos 2 A ___ sin2A cosA ~1 + ~~cotA~~ s'm2A cot A = -1- = = tan A sin yl cot^l = sin 2 ^4 _ 1 + "~ sec^l = =the reciprocal of any expression for cos A cos A cosec A = = the reciprocal of any expression for sin .4. sin-4 vers^. = 1 cos.4 = 2 sin 2 1 A. versA 354 THE BAILROAD TAPER TRIGONOMETRIC AND MISCELLANEOUS FORMULAS. i j _ inn A 1 cos^i sin A 1 + secu4 sinJ. 1 + sin .4 I cos A sin 2 .4 = 2 sin .4 cos A, cos 2 .4 = cos 2 ^. sin 2 .4 = 2 cos 2 J. 1. _ll5a4- 1 tanM 2 cot A sin (^4 7^) = sin A cosB cos A sin B. cos (A B) = cos^l cos!? q= sin^l sinB. 1 =p tan ^1 tan B Bin A +sinB = 2 sin (A + B) cos| (^L B>- sin A sin B = 2 cos| (A + B) sin $ (A B). cos A +-cosB=2coal(A + B)cos$(A B). costf cosA = 2sm$(A + B)sin^(^4 7^). sin 2 ^4 sin 2 B = cos 2 B cos' 2 A = sin (A -f 73) GIP (^ cos 2 -4 sin 2 7? = cos(J. + 73) cos (A B). cos A cos 73 TABLES TRIGONOMETRIC AND MISCELLANEOUS FORMULAS. MISCELLANEOUS FORMULAS. 355 Required. Given. F Area of Parallel sides = in and n p Trapezoid Perp. dist. bet. them =p 2 (m Regular Polygon Length of side = I Number of sides = n T c ' Circle Radius = r itr' 1 [: Ellipse Semi-axes = a and b Ttab Parabola Base = b, height = h Ibh Surface of Radius of base = r Cone Slant height = s Ttrs Cylinder Radius = r, height = h 2rtrh Sphere Radius = r litr* Zone Height = h 2itrh Radius of its sphere = r Volume of Prism or cylinder Area of base = b Height == h bh Pyramid or cone Area of base = b Height = h bh 3 Frustum of Pyramid or cone ! Area of bases = 6 and &' Height = h 3 Sphere \ Radius = r farr 8 Formulas. i. 1414J 356 MISCELLANEOUS FORMULAS THEORETICAL SUPER-ELEVATION IN INCHES Degree of Curve. Velocity in Miles per Hour. . $ bfl 3 Q 10 15 20 25 30 35 40 45 50 55 60 65 70 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 j ' . . ' " 1 1 1 1 1 1 1 . I I 1 4 H H if H if U 2 21 2l 2! 21 3 1 3 4 it if 21 2f 2f 21 si 31 31 41 a 4| 5 51 il 2 21 21 31 3| 41 41 5f 51 61 6f 7| 81 11 If 2f 3 4| 51 51 71 7f 8| 81 If 2| 31 4 41 5f 71 j i 21 31 41 51 61 il If 2f 4 5f 61 8 If 41 6f 81 2 4 6 8 2| 71 2| si 31 9| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ... i j THIS BOOK IS DUE ON THE LAST DA: STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. /V-Y* < : n 2 LJjU JLAjLLASn - LD 21-95m-7,'37 616268 UNIVERSITY OF CALIFORNIA LIBRARY